estuaries (prandle)

8/21/2019 Estuaries (Prandle)

http://slidepdf.com/reader/full/estuaries-prandle 1/248

http://www.cambridge.org/9780521888868



This page intentionally left blank



ESTUARIES

This volume provides researchers, students, practising engineers and managers

access to state-of-the-art knowledge, practical formulae and new hypotheses for

the dynamics, mixing, sediment regimes and morphological evolution in estuaries.

The objectives are to explain the underlying governing processes and synthesisethese into descriptive formulae which can be used to guide the future development

of any estuary. Each chapter focuses on different physical aspects of the estuarine

system – identifying key research questions, outlining theoretical, modelling and

observational approaches, and highlighting the essential quantitative results. This

allows readers to compare and interpret different estuaries around the world, and

develop monitoring and modelling strategies for short-term management issues and

for longer-term problems, such as global climate change.

The book is written for researchers and students in physical oceanography and

estuarine engineering, and serves as a valuable reference and source of ideas for

professional research, engineering and management communities concerned with

estuaries.

DAVID PRANDLE is currently Honorary Professor at the University of Wales’

School of Ocean Sciences, Bangor. He graduated as a civil engineer from the

University of Liverpool and studied the propagation of a tidal bore in the River Hooghly for his Ph.D. at the University of Manchester. He worked for 5 years as a

consultant to Canada’s National Research Council, modelling the St. Lawrence and

Fraser rivers. He was then recruited to the UK ’s Natural Environment Research

Council’s Bidston Observatory to design the operational software for controlling the

Thames Flood Barrier. He has subsequently carried out observational, modelling

and theoretical studies of tide and storm propagation, tidal energy extraction,

circulation and mixing, temperatures and water quality in shelf seas and their coastal

margins.



ESTUARIES

Dynamics, Mixing, Sedimentation and Morphology

DAVID PRA NDLEUniversity of Wales, UK



CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-88886-8

ISBN-13 978-0-511-48101-7

© Jacqueline Broad and Karen Green 2009

2009

Information on this title: www.cambridge.org/9780521888868

This publication is in copyright. Subject to statutory exception and to the

provision of relevant collective licensing agreements, no reproduction of any part

may take place without the written permission of Cambridge University Press.

Cambridge University Press has no responsibility for the persistence or accuracy

of urls for external or third-party internet websites referred to in this publication,

and does not guarantee that any content on such websites is, or will remain,

accurate or appropriate.

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

eBook (NetLibrary)

hardback


http://www.cambridge.org/


http://www.cambridge.org/



Contents

List of symbols page viii1 Introduction 1

1.1 Objectives and scope 1

1.2 Challenges 3

1.3 Contents 5

1.4 Modelling and observations 13

1.5 Summary of formulae and theoretical frameworks 16

Appendix 1A 17

References 21

2 Tidal dynamics 23

2.1 Introduction 23

2.2 Equations of motion 24

2.3 Tidal response – localised 26

2.4 Tidal response – whole estuary 31

2.5 Linearisation of the quadratic friction term 38

2.6 Higher harmonics and residuals 402.7 Surge – tide interactions 44

2.8 Summary of results and guidelines for application 46

References 48

3 Currents 50

3.1 Introduction 50

3.2 Tidal current structure – 2D ( X-Z ) 53

3.3 Tidal current structure – 3D ( X -Y - Z ) 59

3.4 Residual currents 673.5 Summary of results and guidelines for application 71

Appendix 3A 73

Appendix 3B 75

References 76

v



4 Saline intrusion 78

4.1 Introduction 78

4.2 Current structure for river flow, mixed and stratified saline

intrusion 84

4.3 The length of saline intrusion 904.4 Tidal straining and convective overturning 96

4.5 Stratification 105


Appendix 4A 111

References 120

5 Sediment regimes 123

5.1 Introduction 123

5.2 Erosion 1265.3 Deposition 129

5.4 Suspended concentrations 131

5.5 SPM time series for continuous tidal cycles 135

5.6 Observed and modelled SPM time series 137


Appendix 5A 145

References 149

6 Synchronous estuaries: dynamics, saline intrusion and bathymetry 151


6.2 Tidal dynamics 152

6.3 Saline intrusion 158

6.4 Estuarine bathymetry: theory 161

6.5 Estuarine bathymetry: assessment of theory against

observations 165

6.6 Summary of results and guidelines for application 170References 173

7 Synchronous estuaries: sediment trapping and sorting – stable

morphology 175


7.2 Tidal dynamics, saline intrusion and river flow 179

7.3 Sediment dynamics 182

7.4 Analytical emulator for sediment concentrations and fluxes 184

7.5 Component contributions to net sediment flux 1877.6 Import or export of sediments? 193

7.7 Estuarine typologies 196


References 202

vi Contents



8 Strategies for sustainability 205


8.2 Model study of the Mersey Estuary 206

8.3 Impacts of GCC 218

8.4 Strategies for modelling, observations and monitoring 2238.5 Summary of results and guidelines for application 226

Appendix 8A 228

References 231

Index 234

Contents vii



Symbols

A cross-sectional area B channel breadth

C concentration in suspension

D water depth

E vertical eddy viscosity coefficient

E X tidal excursion length

F linearised bed friction coefficient

dimensionless friction term

H total water depth D + ς

I F sediment in-fill time

J dimensionless bed friction parameter

K z vertical eddy diffusivity coefficient

L estuary length

LI salinity intrusion length

LM resonant estuarine length

M2 principal lunar semi-diurnal tidal constituent M4 M6 over-tides of M2

MS4 MSf over-tides of M2 and S2

P tidal period

Q river flow

RI Richardson number

S R Strouhal number U * P / D

S c Schmid number ( K z/ E )

S t Stratification number S X relative axial salinity gradient 1/ ρ ∂ ρ/ ∂ x

S dimensionless salinity gradient

SL axial bed slope

SP spacing between estuaries

viii



T F flushing time

U axial current

U * tidal current amplitude

Residual current components:U 0 river flow

U s density-induced

U w wind-induced

V lateral current

W vertical current

W s sediment fall velocity

X axial dimension

Y lateral axis Z vertical axis

c wave celerity

d particle diameter

f bed friction coefficient (0.0025)

g gravitational constant

i (−1)½

surface slope

k wave number (2π / λ)

m power of axial depth variations (xm)

n power of axial breadth variation (xn)

s salinity

t time

t 50 half-life of sediment in suspension (α/0.693)

y dimensionless distance from mouth

z = Z / Dα exponential deposition rate

exponential breadth variation (eα x )

tan α side slope gradient ( B/2 D)

β exponential suspended sediment profile

exponential depth variation (e β x )

γ sediment erosion coefficient

ε efficiency of mixing

ς surface elevationς * tidal elevation amplitude

θ phase advance of ζ* relative to U *

λ wavelength

ν funnelling parameter (n + 1)/(2 − m)

Symbols ix



π 3.141592

ρ density

σ frequency

τ stress

φ latitudeφE potential energy anomaly

ψ ellipse direction

ω tidal frequency (P/2π )

Ω Coriolis parameter (2ωs sinφ)

Superscripts

* tidal amplitude

–

depth meanSubscripts

0 residual

1D, 2D, 3D one-, two- and three-dimensional

Note: other notations are occasionally used locally for consistency with referenced

publications. These are defined as they appear.

x Symbols



1

Introduction

1.1 Objectives and scope

This book aims to provide students, researchers, practising engineers and managers

access to state-of-the-art knowledge, practical formulae and new hypotheses cover-

ing dynamics, mixing, sediment regimes and morphological evolution in estuaries.

Many of these new developments assume strong tidal action; hence, the emphasis is on

meso- and macro-tidal estuaries (i.e. tidal amplitudes at the mouth greater than 1 m).

For students and researchers, this book provides deductive descriptions of the-

oretical derivations, starting from basic dynamics through to the latest research publications. For engineers and managers, specific developments are presented in

the form of new formulae encapsulated within generalised Theoretical Frameworks.

Each chapter is presented in a ‘stand-alone’ style and ends with a concise

‘Summary of Results and Guidelines for Application’ outlining the issues involved,

the approach, salient results and how these can be used in practical terms. The goal

throughout is to explain governing processes in a generalised form and synthesise

results into guideline Frameworks. These provide perspectives to interpret and inter-

compare the history and conditions in any specific estuary against comparable

experience elsewhere. Thus, a background can be established for developing mon-

itoring strategies and commissioning of modelling studies to address immediate

issues alongside longer-term concerns about impacts of global climate change.

1.1.1 Processes

Estuaries are where ‘fresh’ river water and saline sea water mix. They act as bothsinks and sources for pollutants depending on (i) the geographical sources of the

contaminants (marine, fluvial, internal and atmospheric), (ii) their biological and

chemical nature and (iii) with temporal variations in tidal amplitude, river flow,

seasons, winds and waves.

1



Tides, surges and waves are generally the major sources of energy input into

estuaries. Pronounced seasonal cycles often occur in temperature, light, waves, river

flows, stratification, nutrients, oxygen and plankton. These seasonal cycles alongside

extreme episodic events may be extremely significant for estuarine ecology. As an

example, adjustments in axial intrusion of sea water and variation in vertical stratifica-tion associated with salinity and temperature may lead to rapid colonisation or,

conversely, extinction of sensitive species. Likewise, changes to the almost impercep-

tible larger-scale background circulations may affect the pathways and hence lead to

accumulation of persistent tracers. Dyer (1997) provides further descriptions of these

processes alongside useful definitions of much of the terminology used in this book.

Vertical and horizontal shear in tidal currents generate fine-scale turbulence,

which determines the overall rate of mixing. However, interacting three-dimensional

(3D) variations in the amplitude and phase of tidal cycles of currents and contaminantsseverely complicate the spatial and temporal patterns of tracer distributions and

thereby the associated mixing. On neap tides, near-bed saline intrusion may enhance

stability, while on springs, enhanced near-surface advection of sea water can lead to

overturning. Temperature gradients may also be important; solar heating stabilises the

vertical density profile, while winds promote surface cooling which can produce

overturning. In highly turbid conditions, density differences associated with suspended

sediment concentrations can also be important in suppressing turbulent mixing.

The spectrum of tidal energy input is effectively constrained within a few tidal

constituents, and, in mid-latitudes, the lunar M2 constituent is generally greater than

the sum of all others – providing a convenient basis for linearisation of the equations

for tidal propagation. However, ‘mixing’ involves a wider spectrum of interacting

non-linear processes and is thus more difficult to simulate. The ‘decay time’ for

tidal, surge, wave and associated turbulent energy in estuaries is usually measured in

hours. By contrast, the flushing time for river inputs generally extends over days.

Hence, simulation of the former is relatively independent of initial conditions, whilesimulation of the latter is complicated by ‘historical’ chronology resulting in

accumulation of errors.

1.1.2 Historical developments

Following the end of the last ice age, retreating ice cover, tectonic rebound and the

related rise in mean sea level (msl) resulted in receding coastlines and consequent

major changes in both the morphology and the dynamics of estuaries. Large post-glacial melt-water flows gouged deep channels with the rate of subsequent in-filling

dependent on localised availability of sediments. Deforestation and subsequent

changes in farming practices substantially changed the patterns of river flows and

both the quantity and the nature of fluvial sediments. Thus, present-day estuarine

2 Introduction



morphologies reflect adjustments to these longer-term, larger-scale effects along-

side more recent, localised impacts from urban development and engineering

‘interventions’.

Ports and cities have developed on almost all major estuaries, exploiting oppor-

tunities for both inland and coastal navigation, alongside supplies of freshwater andfisheries. In more recent times, the scale of inland navigation has generally declined

and the historic benefit of an estuary counterbalanced by growing threats of flood-

ing. Since estuaries often supported major industrial development, the legacy of

contaminants can threaten ecological diversity and recreational use. The spread of

national and international legislation relating to water quality can severely restrict

development, not least because linking discharges with resulting concentrations is

invariably complicated by uncertain contributions from wider-area sources and

historical residues. This combination of legal constraints and uncertainties about impacts from future climate changes threatens planning-blight for estuarine devel-

opment. This highlights the need for clearer understanding of the relative sensitivity

of estuaries to provide realistic perspectives on their vulnerability to change.

1.2 Challenges

Over the next century, rising sea levels at cities bordering estuaries may require

major investment in flood protection or even relocation of strategic facilities. The

immediate questions concern the changing magnitudes of tides, surges and waves.

However, the underlying longer-term (decadal) issue is how estuarine bathymetries

will adjust to consequent impacts on these dynamics (Fig. 1.1; Prandle, 2004). In

addition to the pressing flood risk, there is growing concern about sustainable

exploitation of estuaries. A common issue is how economic and natural environment

interests can be reconciled in the face of increasingly larger-scale developments.

1.2.1 Evolving science and technology agendas

Before computers became available, hydraulic scale models were widely used to

simulate dynamics and mixing in estuaries. The scaling principles were based on

maintaining the ratios of the leading terms in the equation describing tidal propaga-

tion. Ensuing model ‘validation’ was generally limited to reproduction of tidal

heights along the estuary. Subsequent expansion in observational capabilities indi-

cated how difficulties arose when such models were used to study saline intrusion,sediment regimes and morphological adjustments.

Even today, validation of sophisticated 3D numerical models may be restricted to

simulation of an M2 cycle – providing little guarantee of accurate reproduction of

higher harmonics or residual features. Likewise, these fine-resolution 3D models

1.2 Challenges 3



may encounter difficulties in reproducing the complexity and diversity of mixing

and sedimentary processes. Moreover, the paucity of observational data invariably

limits interpretation of sensitivity tests. However, modelling is relatively cheap and

continues to advance rapidly, whilst observations are expensive and technology

developments often take decades. Thus, a major challenge in any estuary study is

how to use theory to bridge the gaps between modelling and available observations.

Both historical and ‘ proxy’ data must be exploited, e.g. wave data constructed from

wind records, flood statistics from adjacent locations, sedimentary records of flora

and fauna as indicators of saline intrusion and anomalous fossilised bed features as

evidence of extreme events.

The evolving foci for estuarine research are summarised in Fig. 1.2. These have

evolved alongside successive advances in theory, modelling and observational

technologies to address changing political agendas.

1.2.2 Key questions

Successive chapters address the following sequence of key questions:(Q1) How can strategies for sustainable exploitation of estuaries be developed?

(Q2) How do tides in estuaries respond to shape, length, friction and river flow? Why are

some tidal constituents amplified yet others reduced and why does this vary from one

estuary to another?

Tides

surges

waves

mean sea levels

Sediment source

sink

Dredging

Bank &marshexchange

U n d e r l y i n g

g e o l o g y

River flow

fluvial load

Coring

Surficial sediments

Reclamation

Fig. 1.1. Schematic of major factors influencing estuarine bathymetry.

4 Introduction



(Q3) How do tidal currents vary with depth, friction, latitude and tidal period?

(Q4) How does salt water intrude and mix and how does this change over the cycles of

Spring – Neap tides and flood-to-drought river flows?

(Q5) How are the spectra of suspended sediments determined by estuarine dynamics?

(Q6) What determines estuarine shape, length and depth?

(Q7) What causes trapping, sorting and high concentrations of suspended sediments?How does the balance of ebb and flood sediment fluxes adjust to maintain bathymetric

stability?

(Q8) How will estuaries adapt to Global Climate Change?

1.3 Contents

1.3.1 Sequence

The chapters follow a deductive sequence describing (2) Tidal Dynamics,

(3) Currents, (4) Saline Intrusion, (5) Sediment Regimes, (6) Synchronous Estuary:

Dynamics, Saline Intrusion and Bathymetry, (7) Synchronous Estuary: Sediment

Trapping and Sorting – Stable Morphology and (8) Strategies for Sustainability.

Analytical solutions for the first-order dynamics of estuaries are derived in Chapter 2

and provide the basic framework of our understanding. Details of associated

currents are described in Chapter 3. Tidal currents and elevations in estuaries are

largely independent of biological, chemical and sedimentary processes – except for their influences on the bed friction coefficient. Conversely, these latter processes are

generally highly dependent on tidal motions. Thus, in Chapters 4 and 5, we consider

how estuarine mixing and sedimentation are influenced by tidal action. Chapters 6

and 7 apply these theories to synchronous estuaries, yielding explicit algorithms

Tidegauges

Tides

Storm

surges Waves

Temperature

salinity

Sediments

algal bloomsprimary productivity

Fish stocks

Ecologicalcommunities

NavigationCoastaldefence

Offshoreindustries

Agriculture(marine &terrestr ial)

Tourism

Sustainableexploitation

Defence

19901980 20102000

MeteorologyIn situ telemetry

SatelliteAircraftradarferries

Fig. 1.2. Historical development in key processes, ‘

end-users’

and observationaltechnologies.

1.3 Contents 5



for tidal currents, estuarine lengths and depths, sediment sorting and trapping and a

bathymetric framework based on tidal amplitude and river flow.

1.3.2 Tidal dynamics

Chapter 2 examines the propagation of tides, generated in ocean basins, into

estuaries, explaining how and why tidal elevations and currents vary within estu-

aries (Fig. 1.3; Prandle, 2004). The mechanisms by which semi-diurnal and diurnal

constituents of ocean tides produce additional higher-harmonic and residual com-

ponents within estuaries are illustrated. Since the expedient of linearising the

relevant equations in terms of a predominant (M2) constituent is extensively used

throughout this book, the details of this process are described. Many earlier texts

and much of the literature focus on large, deep estuaries with relatively low frictioneffects. Here, it is indicated how to differentiate between such deep estuaries and

shallower frictionally dominated systems and the vast differences in their response

characteristics are illustrated.

2

ν 3 4 5

2.5

1.0

0.5

51030

507090

100

B

A

D G

IE

H

F

C

50

25

10

5.0

2.5

–180° –150°

150°

90° 1.0

0.1

0.01

–10°

–1°

–30°

–90°

0.5

30°

050 –30°

–90°

100

100

1

2

3

y 4

5

6

7

Fig. 1.3. Tidal elevation responses for funnel-shaped estuaries. ν represents degreeof bathymetric funnelling and y distance from the mouth, y = 0. Dashed contoursindicate relative amplitudes and continuous contours relative phases. Lengths,

y (for M2), and shapes, ν , for estuaries (A) – (I) shown in Table 2.1.

6 Introduction



1.3.3 CurrentsChapter 3 examines how tidal currents vary along (axially) and across estuaries

and from surface to bed. Changes in current speed, direction and phase (timing

of peak or slack values) are explained by decomposition of the tidal current ellipse

into clockwise and anti-clockwise rotating components. While the main focus is

on explaining the nature and range of tidal currents, the characteristics of wind-

and density-driven currents are also described. A particular emphasis is on deriving

the scaling factors which encapsulate the influence of the ambient environmental

parameters, namely depth, friction factor and Coriolis coefficient, i.e. latitude

(Fig. 1.4; Prandle, 1982).

1.3.4 Saline intrusion

Noting the earlier definition of estuaries as regions where salt and fresh water mix,

Chapter 4 examines the details of this mixing. It is shown how existing theories

derived for saline intrusion in channels of constant cross section can be adaptedfor mixing in funnel-shaped estuaries. Saline intrusion undergoes simultaneous

adjustments in axial location and mixing length – explaining traditional problems

in understanding observed variations over spring – neap and flood-drought condi-

tions (Fig. 1.5; Liu et al ., 2008).

Fig. 1.4. Vertical profiles of tidal current, U *( z )/ U *mean, versus the Strouhalnumber, S R , U * tidal current amplitude, P tidal period, D depth, S R = U*P/D.

1.3 Contents 7



The predominance of mixing by vertical stirring driven by tidally induced tur-

bulence has long been recognised. Here, the importance of incorporating the effects

of tidal straining and resultant convective overturning is described.The ratio of currents, U 0/ U *, associated with river flow and tides, is shown to be

the most direct determinant of stratification in estuaries.

1.3.5 Sediment regimes

Chapter 5 focuses on the character of sediment regimes in strongly tidal estuaries,

adopting a radically different approach to traditional studies of sediment regimes.

Analytical solutions are derived encapsulating and integrating the processes of erosion, suspension and deposition to provide descriptions of the magnitude, time

series and vertical structure of sediment concentrations. These descriptions enable the

complete range of sediment regimes to be characterised in terms of varying sedi-

ment type, tidal current speed and water depth (Fig. 1.6; Prandle, 2004). Theories are

Danshuei river – Tahan stream

–5

–10

–5

–5

0 20 302510 155

–10

–10

D e p

t h ( m )

D e p t h ( m )

D e p t h ( m )

Salinity:ppt

Hsin-Hai Bri.

Hsin-Hai Bri.

Hsin-Hai Bri.

Taipei Bri.

Taipei Bri.

Taipei Bri.

Kuan-Du Bri.

Kuan-Du Bri.

Kuan-Du Bri.

River mouth

River mouth

River mouth

Q75 flo w

Qm flo w

Q10 flo w

Distance from Danshuei River mouth (km)

1 2 3 4

5 6 7 9 1 1

1 3

1 5

1 7

1 9 2

1

2 2 2

3 2 4

2 5

2 7

2 9

3 0 3

1 3 2 3

3

3 2

3 1 2 8

2 4 2 3

2 1 1 9 1 5 1 7

1 3

1 1

9 7 6 5 1 3

3 0

2 8

2 7

2 5

2 2

2 0 1

8

1 5

1 3

1 4

1 2 1 1

1 0

9

7

6 4 3 1 2

Fig. 1.5. Axial variations in salinity, ‰, in the Danshuei River, Taiwan Q75, flowrate exceeded 75% of time, Q10 flow exceeded 10% of time.

8 Introduction



developed by which tidal analyses of suspended sediment time series, obtained from

either model simulations or observations, can be used to explain the underlying

characteristics.

1.3.6 Synchronous estuary: dynamics, saline

intrusion and bathymetry

A ‘synchronous estuary’ is where the sea surface slope due to the axial gradient in

phase of tidal elevation significantly exceeds the gradient from changes in tidal

amplitude. The adoption of this assumption in Chapters 6 and 7 enables the theoreticaldevelopments described in earlier chapters to be integrated into an analytical emu-

lator, incorporating tidal dynamics, saline intrusion and sediment mechanics.

Chapter 6 re-examines the tidal response characteristics for any specific location

within an estuary. The ‘synchronous’ assumption yields explicit expressions for

both the amplitude and phase of tidal currents and the slope of the sea bed.

Integration of the latter expression provides an estimate of the shape and length

of an estuary. By combining these results with existing expressions for the length

of saline intrusion and further assuming that mixing occurs close to the seawardlimit, an expression linking depth at the mouth with river flow is derived. Hence,

a framework for estuarine bathymetry is formulated showing how size and shape

are determined by the ‘ boundary conditions’ of tidal amplitude and river flow

(Fig. 1.7; Prandle et al ., 2005).

Fig. 1.6. Spring – neap patterns of sediment concentrations at fractional heightsabove the bed.

1.3 Contents 9



1.3.7 Synchronous estuary: sediment trapping

and sorting – stable morphology

Chapter 7 indicates how, in ‘synchronous’ estuaries, bathymetric stability is main-

tained via a combination of tidal dynamics and ‘delayed’ settlement of sediments

in suspension. An analytical emulator integrates explicit formulations for tidal and

residual current structures together with sediment erosion, suspension and deposi-

tion. The emulator provides estimates of suspended concentrations and net sediment fluxes and indicates the nature of their functional dependencies. Scaling analyses

reveal the relative impacts of terms related to tidal non-linearities, gravitational

circulation and ‘delayed’ settling.

The emulator is used to derive conditions necessary to maintain zero net flux of

sediments, i.e. bathymetric stability. Thus, it is shown how finer sediments are imported

and coarser ones are exported, with more imports on spring tides than on neaps,

i.e. selective trapping and sorting and consequent formation of a turbidity maximum.

The conditions derived for maintaining stable bathymetry extend earlier concepts of flood- and ebb-dominated regimes. Interestingly, these derived conditions correspond

with maximum sediment suspensions. Moreover, the associated sediment-fall veloci-

ties are in close agreement with settling rates observed in many estuaries. Figure 1.8

(Lane and Prandle, 2006) encapsulates these results, illustrating the dependency on

Fig. 1.7. Zone of estuarine bathymetry. Coordinates (Q, ς ) for Coastal Plain andBar-Built estuaries, Q river Flow and ς elevation amplitude. Bathymetric zone

bounded by E x < L, LI < L and D/ U 3 < 5 0 m2 s−3.

10 Introduction



delayed settlement (characterised by the half-life in suspension t 50) and the phase

difference, θ , between tidal current and elevation. A feedback mechanism between

tidal dynamics and net sedimentation/erosion is identified involving an interaction

between suspended and deposited sediments.

These results from Chapters 6 and 7 are compared with observed bathymetricand sedimentary conditions over a range of estuaries in the USA, UK and Europe.

By encapsulating the results in typological frameworks, the characteristics of

any specific estuary can be immediately compared against these theories and in

a perspective of other estuaries. Identification of ‘anomalous’ estuaries can pro-

vide insight into ‘ peculiar ’ conditions and highlight possible enhanced sensitivity

to change. Discrepancies between observed and theoretical estuarine depths can

be used to estimate the ‘age’ of estuaries based on the intervening rates of sea

level rise.Importantly, the new dynamical theories for estuarine bathymetry take no account

of the sediment regimes in estuaries. Hence, the success of these theories provokes

a reversal of the customary assumption that bathymetries are determined by their

prevailing sediment regimes. Conversely, it is suggested that the prevailing sediment

–90° –45° 0°100

1

0.01

Phase advance,θ , of ζ with respect to Û ˆ

0.1 0.5 0.9 0.99

0

–0.01

–0.1

Import

Exp ort

–0.5

Sand

Silt

10

0.1

4 m

16 m

4 m

16 m

4 m

Tidal amplitude ζ = 4 3 2 1 mˆ

W s = 0.01 0.001 0.0001 m s –1

16 m

H a l f - l i f e t 5 0 ( h )

Fig. 1.8. Net import versus export of sediments as a f (θ , t 50). Theoretical contoursfrom (7.33). Specific examples of spring – neap variability for tidal amplitudes

ς = 1 (open circle), 2, 3 and 4 m; fall velocities, W s = 0.0001, 0.001 and 0.01 m s−1

and depths, D = 4 and 16 m.

1.3 Contents 11



the theories developed in earlier chapters, estimates of likely impacts of global

climate change are quantified across a range of estuaries. It is emphasised how

international co-operation is necessary to access the resources required to ameliorate

the threats to the future viability of estuaries.

1.4 Modelling and observations

Since this book focuses on the development of theories for underpinning modelling

and planning measurements, a background to the capabilities and limitations of

models and observations is presented.

1.4.1 Modelling Models synthesise theory into algorithms and use observations to set-up, initialise,

force, assimilate and evaluate simulations in operational, pre-operational and

‘exploratory’ modes (Appendix 8A). The validity of models is limited by the degree

to which the equations or algorithms synthesise the governing processes and by

numerical and discretisation accuracies. The accuracy of model simulations

depends further on the availability and suitability (accuracy, resolution, representa-

tiveness and duration) of data from observations and linked models (adjacent sea,

meteorological and hydrological).

Parameters of interest include tides, surges, waves, currents, temperature, salinity,

turbidity, ice, sediment transport and an ever-expanding range of biological and

chemical components. The full scope of model simulations spans across atmosphere –

seas – coasts – estuaries, between physics – chemistry – biology – geology – hydrology and

extends over hours to centuries and even millennia. Recent developments expand to

total-system simulators embedding the models described here within socio-economic

planning scenarios.

Resolution

Models can be (i) non-dimensional conceptual modules encapsulated into whole-

system simulations, (ii) one-dimensional (1D), single-point vertical process studies

or cross-sectionally averaged axial representations, (iii) two-dimensional (2D),

vertically averaged representations of horizontal circulation or (iv) fully 3D. Over

the past 40 years, numerical modelling has developed rapidly in scope, from

hydrodynamics to ecology, and in resolution, progressing from the earliest 1D barotropic models to present-day 3D baroclinic – incorporating evolving temperature-

and salinity-induced density variations. Comparable resolutions have expanded from

typically 100 axial sections to millions of elements, exploiting the contemporaneous

development of computing power. Unfortunately, concurrent development in




observational capabilities has not kept pace, despite exciting advances in areas such as

remote sensing and sensor technologies.

Tidal predictions for sea level at the mouth of estuaries have been available for

more than a century. The dynamics of tidal propagation are almost entirely deter-

mined by a combination of tides at the mouth and estuarine bathymetry with somemodulation by bed roughness and river flows. Thus, 1D models, available since

the 1960s, can provide accurate simulation of the propagation of tidal heights and

phases. However, tidal currents vary over much shorter spatial scales reflecting

localised changes in bathymetry, creating small-scale variability in both the vertical

and the horizontal dimensions. Continuous growth in computer power has enabled

these 1D models to be extended to two and three dimensions, providing the resolu-

tion necessary to incorporate such variability. The full influence of turbulence on

the dynamics of currents and waves and their interaction with near-bed processesremains to be clearly understood. Presently, most 3D estuarine models use a 1D

(vertical) turbulence module. Development of turbulence models is supported by

new measuring techniques like microstructure profilers which provide direct com-

parisons with simulated energy dissipation rates.

These latest models can accurately predict the immediate impact on tidal eleva-

tions and currents of changes in bathymetry (following dredging or reclamation),

river flow or bed roughness (linked to surficial sediments or flora and fauna).

Likewise, such models can provide estimates of the variations in salinity distribu-

tions (ebb to flood, spring to neap tides, flood to drought river flows), though with a

reduced level of accuracy. The further step of predicting longer-term sediment

redistributions remains problematic. Against a background of subtly changing

chemical and biological mediation of estuarine environments, specific difficulties

arise in prescribing available sources of sediment, rates of erosion and deposition,

the dynamics of suspension and interactions between mixed sediment types.

Higher resolution can provide immediate improvements in the accuracy of simulations. Similarly, adaptable and flexible grids alongside more sophisticated

numerical methods can reduce problems of ‘numerical dispersion’. In the horizontal,

rectangular grids are widely used, often employing polar coordinates of latitude and

longitude. Irregular grids, generally triangular or curvi-linear, are used for variable

resolution. The vertical resolution may be adjusted for detailed descriptions – near

the bed, near the surface or at the thermocline. The widely used sigma coordinate

system accommodates bottom-following by making the vertical grid size pro-

portional to depth. In computational fluid dynamics, continuously adaptive grids provide a wide spectrum of temporal and spatial resolution especially useful in

multi-phase processes.

Broadly, first-order dynamics are now well understood and can be accurately

modelled. Hence, research focuses on ‘second-order ’ effects, namely higher-order

14 Introduction



(and residual) tides; vertical, lateral and high-frequency variability in currents; and

salinity. For the pressing problems concerning the net exchange of contaminants, non-

linear interactions are important, and accurate time-averaging requires second-order

accuracy in the temporal and spatial distributions of currents, elevations and density.

Numerical simulation of these higher-order effects requires increasingly fine resolu-tion. Thus, ironically, despite the exponential growth in computer power since the

first 1D tidal models, limitations in computing power remain an obstacle to progress.

1.4.2 Observations

Rigorous model evaluation and effective assimilation of observational data into

models require broad compatibility in their respective resolution and accuracy –

temporally and spatially across the complete parameter range. Technologies involvedin providing observational data range from development of sensors and platforms,

design of optimal monitoring strategies to analyses, curation and assimilation of data.

The set-up of estuarine models requires accurate fine-resolution bathymetry, and

ideally, corresponding descriptions of surficial sediments/bed roughness. Subsequent

forcing requires tide, surge and wave data at the open-sea boundary together with river

flows at the head alongside their associated temperature, sediment and ecological

signatures.

Sensors use mechanical, electromagnetic, optical and acoustic media. Platforms

extend from in situ, coastal, vessel-mounted to remote sensing, e.g. satellites, aircraft,

radar, buoys, floats, moorings, gliders, automated underwater vehicles (AUVs),

instrumented ferries and shore-based tide gauges.

Remote-sensing techniques have matured to provide useful descriptions of ocean

wind, waves, temperature, ice conditions, suspended sediments, chlorophyll, eddy

and frontal locations. Unfortunately, these techniques provide only sea-surface

values, and in situ observations are necessary both for vertical profiles and to correct for atmospheric distortion in calibration. The improved spatial resolution provided

from aircraft surveillance is especially valuable in estuaries. High-frequency radars

also provide synoptic surface fields of currents, waves and winds on scales appro-

priate to the validation of estuarine models.

It is convenient to regard observational programmes in three categories: measure-

ments, observations and monitoring. Process measurements aim to understand

specific detailed mechanics, often with a localised focus over a short period, e.g.

derivation of an erosion formula for extreme combinations of tides and waves.Test-bed observations aim to describe a wide range of parameters over a wide area

over a prolonged period (spanning the major cycles of variability). Thus, year-long

measurements of tides, salinity and sediment distributions throughout an estuary

provide an excellent basis for calibrating, assessing and developing a numerical




modelling programme. Monitoring implies permanent recording, such as tide gauges.

Careful site selection, continuous maintenance and sampling frequencies sufficient

to resolve significant cycles of variability are essential. A comprehensive monitoring

strategy is likely to embed all three of the above and include duplication and synergy

to address quality assurance issues. Models can be used to identify spatial andtemporal modes and scales of coherence to establish sampling resolution and to

optimise the selection of sensors, instruments, platforms and locations. Coastal

observatories now extend observational programmes to include physical, chemical

and biological parameters.

Teleconnections

In addition to the immediate, localised requirements, information may be needed

about possible changes in ocean circulation which may influence regional cli-mates and the supplies and sinks for nutrients, contaminants, thermal energy, etc.

Associated data are provided by meteorological, hydrological and shelf-sea models.

Ultimately, fully coupled, real-time (operational) global models will emerge

incorporating the total water cycle (Appendix 8A). The large depths of the oceans

introduce long inertial lags in impacts from Global Climate Change. By contrast, in

shallow estuaries, detection of systematic regional variations may provide early

warning of impending impacts.

1.5 Summary of formulae and theoretical frameworks

The following lists summarise formulae and Theoretical Frameworks presented in

following chapters.

Parameter Dependencies Equation number

(a) Current amplitude U *∝

ς

1/2

D

1/4

f

−1/2

shallow water (6.9)∝ ς D−1/2 deep water (6.9)

(b) Estuarine length L∝ D5/4/ ς 1/2 f 1/2 (6.12)

(c) Depth at the mouth D0∝ (tan α Q)0.4 (6.25)

(d) Depth variation D( x)∝ D0 x0.8 (6.11)

(e) Ratio of friction: inertia F / ω∝ 10 ς / D (6.8)

(f) Stratification limit ς 1 m (6.24)

(g) Salinity intrusion Li,∝ D2/ f U 0 U * (6.16)

(h) Bathymetric zone LI < L, E X < L and D/ U *3

< 5 0 m−2 s3(6.23)

(i) Flushing time T F∝ LI/ U 0 (6.17)

(j) Suspended concentration C ∝ f U * (7.36)

(k) Equilibrium fall velocity W s∝ f U* (7.35)

16 Introduction



ς is tidal elevation amplitude, f bed friction coefficient, Q river flow with current

speed U 0, tan α lateral inter-tidal slope, F linearised friction coefficient, ω tidal

frequency, E x tidal excursion.

Theoretical frameworks have been established to explain both amplitude and

phase variations of elevations and (cross-sectionally averaged) currents for the primary tidal constituents. Qualitative descriptions of vertical current structure

have been derived for (i) oscillatory tidal components and (ii) residual components

associated with river flow, wind forcing and both well-mixed and fully stratified

density gradients. These dynamical results provide the basis for similar frameworks

describing saline intrusion and sedimentation. Further applications of these theories

for synchronous estuaries enable the frameworks to be extended to illustrate con-

ditions corresponding to stable bathymetry and sedimentary regimes.

Theoretical Framework Figure Question

(T1) Tidal response 2.5 Q2(T2) Current structure: (a) tidal 3.3 Q3

(b) riverine, wind and density gradient 4.4(T3) Saline mixing 4.13(T4) Sediment concentrations 5.6 Q5(T5) Bathymetry:

(a) Bathymetric zone 6.12 Q6(b) Stability 7.7 Q7(c) Lengths and depths 8.7(d) W s, C , T F 7.11

W s is the fall velocity for stable bathymetry, C is mean suspended sediment

concentration and T F is flushing time. Q2 to Q8 refer to cardinal questions high-

lighted in the Summary Sections of Chapters 2 to 8. Equation (4.44) addresses

Q4 and Figs 7.9 and 7.10 and Table 8.4 address Q8. Figure 1.2 and Section 1.4

summarise the issues concerned in Q1.

Appendix 1A

1A.1 Tide generation

Much of the theory presented here focuses on strongly tidal estuaries where the M2

constituent amplitudes are used as a basis for parameterising the linearised bed-

friction coefficient, eddy viscosity and diffusivity together with related half-lives of sediments in suspension. Figure 1A.1 shows tidal elevations in the Mersey, illustra-

ting the predominance of the semi-diurnal M2 constituent. Here, we introduce a brief

background to the generation of tides, illustrating their spectral and latitudinal

variations. For a rigorous, historical account of the development of tidal theory see

Appendix 1A 17



Cartwright (1999). For pictorial illustrations and simplified deductive steps for thefollowing theory see Dean (1966).

Newton’s gravitational theory showed that the attractive force between bodies is

proportional to the product of their mass divided by the square of their distance

apart. This means that only the tidal effects of the Sun and the Moon need be

considered. Mathematically, it is convenient to regard the Sun as rotating around a

‘fixed’ Earth – enabling the same theory to be applied to the attraction from both the

Sun and the Moon.

1A.2 Non-rotating Earth

The attractive force on the Earth’s surface due to the Moon’s orbit can be separated

into two components:

tangential 3

2 gM

E

a

d

3

sin2θ (1A:1)

radial gM

E

a

d

3

ð1 3cos2 θ Þ; (1A:2)

where M / E is the ratio of the mass of the Moon to that of the Earth, i.e. 1/81, and a/ d

is the ratio of the radius of the Earth to their distance apart, i.e. 1/60. The longitude,

11.0

Gladstone dock1997

January 1997

10.0

9.0

8.0

7.0

6.0

5.0

M e t r e s a b o v e A C D

4.0

3.0

2.0

1.0

0.0

–1.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 25 26 27 28 29 30 3124

Fig. 1A.1. Month-long recording of tidal heights at the mouth of the Mersey Estuary.

18 Introduction



θ , is measured relative to their alignment along the ecliptic plane of the Moon’s

orbit. The radial force component is negligible compared with gravity, g .

Integrating the tangential force, with the constant of integration determined from

satisfying mass conservation, indicates a surface displacement:

η ¼a

4

M

E

a

d

3

ð3cos2θ þ 1Þ: (1A:3)

This corresponds to bulges on the sides of the Earth nearest and furthest from the

Moon of about 35 cm, with depressions at the ‘ poles’ of about 17 cm.

1A.3 Rotating Earth

Taking account of the Earth’s rotation, cos θ =cos cos λ, where is latitude and λ the angular displacement and hence

η ¼a

2

M

E

a

d

3

ð3 cos2 cos2 λ 1Þ: (1A:4)

Thus, we note the generation of two tides per day (semi-diurnal) with maximum

amplitude at the equator = 0 and zero at the poles = 90°. The period of the

principal solar semi-diurnal constituent, S2, is 12.00 h. The Moon rotates in 27.3

days, extending the period of the principal lunar semi-diurnal constituent to 12.42 h.The ubiquitous spring – neap variations in tides follow from successive intervals of

coincidence and opposition of the phases of M2 and S2. The two constituents are in

phase when the Sun and the Moon are aligned with the Earth, i.e. both at ‘full moon’

and ‘new moon’.

1A.4 Declination

The Moon’s orbit is inclined at about 5° to the equator; this introduces a daily

inequality in (1A.4), producing a principal lunar diurnal constituent, O1. The equiva-

lent solar declination is 27.3°, producing the principal solar diurnal constituent P1

alongside the principal lunar and solar constituent K 1. The lunar declination varies

over a period of 18.6 years changing the magnitude of the lunar constituents by

up to ± 4%.

1A.5 Elliptic orbit

The Moon and the Sun’s orbits show slight ellipticity, changing the distance d in

(1A.4). For the Moon, this introduces a lunar ellipse constituent N2, while for the

Sun constituents at annual, Sa, and semi-annual period, Ssa, are introduced.

Appendix 1A 19



1A.6 Relative magnitude of the Sun’ s attraction

Although the ratio of masses, S / E =3. 3×105, overshadows that of M / E , this is

counterbalanced by the corresponding ratio of distances d s/ d m 390. Thus, the

relative impact of Moon: Sun is given from (1A.4) as (S / M )/(d s/ d m )3 0.46.

1A.7 Equilibrium constituents

In consequence of the above, ‘equilibrium’ magnitudes of the principal constituents

relative to M2 are S2−0.46, N2−0.19, O1−0.42, P1−0.19 and K 1−0.58.

1A.8 Tidal amphidromes

The integration of tidal potential over the spatial extent of the deep oceans means

that ‘direct ’ attraction in adjacent shelf seas can be neglected compared with the

propagation of energy from the oceans. In consequence, tides in enclosed seas and

lakes tend to be minimal. In practice, the world’s oceans respond dynamically to the

above tidal forces. Responses in ocean basins and within shelf seas take the form of

amphidromic systems – as shown in Fig. 1A.2 (Flather, 1976) for the M2 constituent

in the North Sea. The amplitudes of such systems are a maximum along their coastal

boundaries, and the phases rotate (either clockwise or anti-clockwise) such that high

water on one side of the basin is balanced by low water on the other side. While

these surface displacements propagate around the system in a tidal period, the net

ebb or flood excursions of individual particles seldom exceeds 20 km.

These co-oscillating systems can accumulate energy over a number of cycles (see

Section 2.5.4), resulting in spring tides occurring several days after new or full Moon.

Basin morphology can selectively amplify the amphidromes for different constituents.

In general, the observed amplitudes of semi-diurnal constituents relative to diurnal

are significantly larger than indicated from their equilibrium ratios shown above.

1A.9 Monthly, fortnightly and quarter-diurnal constituents

In shallow water and close to abrupt changes in bathymetry, tidal constituents

interact (see Section 2.6). From the trignometric relationship

cosω1 cos ω2 ¼ 0:5 ðcosðω1 þ ω2Þ þ cosðω1 ω2ÞÞ; ð1A:5Þ

a product of two constituents ω1 and ω2 results in constituents at their sum anddifference frequencies. Thus, terms involving products of M2 and S2 generate

constituents at the quarter-diurnal frequency MS4 and the fortnightly frequency

MSf . Similarly, M2 and N2 generate constituents at the quarter-diurnal frequency

MN4 and the monthly Mm.

20 Introduction



References

Cartwright, D.E., 1999. Tides: A Scientific History. Cambridge University Press,Cambridge.

Dean, R.G., 1966. Tides and harmonic analysis. In: Ippen, A.T. (ed.), Estuary and Coastline Hydrodynamics. McGraw-Hill, New York, pp. 197 – 230.

Fig. 1A.2. M2 tidal amphidromes in the north west European continental shelf.

References 21



Dyer, K.R., 1997. Estuaries: A Physical Introduction, 2nd ed. John Wiley, Hoboken, NJ.Flather, R.A., 1976. A tidal model of the north west European Continental Shelf, Memoires

Societe Royale des Sciences de Liege, Ser , 6 (10), 141 – 164.Lane, A. and Prandle, D., 2006. Random-walk particle modelling for estimating bathymetric

evolution of an estuary. Estuarine, Coastal and Shelf Science, 68 (1 – 2), 175 – 187.

Liu, W.C., Chen, W.B., Kuo, J-T, and Wu, C., 2008. Numerical determination of residencetime and age in a partially mixed estuary using a three-dimensional hydrodynamicmodel. Continental Shelf Research, 28 (8), 1068 – 1088.

Prandle, D., 1982. The vertical structure of tidal currents and other oscillatory flows.Continental Shelf Research, 1, 191 – 207.

Prandle, D., 2004. How tides and river flows determine estuarine bathymetries. Progressin Oceanography, 61, 1 – 26.

Prandle, D., Lane, A., and Manning, A.J., 2005. Estuaries are not so unique. Geophysical Research Letters, 32 (23).

22 Introduction



2

Tid al d y n am ics

2.1 Introduction

T id a l p r op a g at i on i n e s tu a ri e s c a n b e a c cu r at e ly s i mu l at e d u s in g e i th e r n u me r ic a l

o r h yd ra ul ic s ca le m od e ls . H ow ev e r, s uc h m od el s d o n o t d ir ec tl y p ro v id e u nd e r-

s t a n d i n g o f t h e b a s i c m e c h a n i s m s o r i n s i g h t i n t o t h e s e n s i t i v i t i e s o f t h e c o n t r o l l i n g

parameters. Thus, while terms representing friction and bathymetry appear expli-

c it ly i n (2.8) a nd ( 2. 11 ), i t i s n ot i mm ed ia te ly e vi de nt w hy t id es a re g re at ly a mp li fi ed

i n c er ta in e st ua rie s y et q ui ck ly d is si pa te d in o th er s. T he a im h er e i s to d eri ve

a n a l y t i c a l s o l u t i o n s , a n d t h e r e b y T h e o r e t i c a l F r a m e w o r k s , t o g u i d e s p e c i f i c m o d e l -

l in g a nd m o ni to ri ng s tu d ie s a nd p ro v id e i ns ig ht i nt o a nd p er sp ec ti ve o n e st ua ri ne

responses generally.

M u ch o f t he t he o ry d ev e lo p ed h e re a ss um es t ha t t id al p ro p ag at io n i n e st ua ri es

c an b e r ep re se nt ed b y t he s ha ll ow -w at er w av e e qu at io ns r ed uc ed t o a 1 D c ro ss -

s e ct i on a ll y a v er a ge d f o rm . S e ct i on 2 . 2 d e sc ri be s t he b as es o f t hi s s im p li fi ca ti on .

B y f u r t h e r r e d u c i n g t h e s e e q u a t i o n s t o a l i n e a r f o r m , l o c a l i s e d s o l u t i o n s a r e r e a d i l y

o b t a i n e d , t h e s e a r e e x a m i n e d i n S e c t i o n 2 . 3.I t i s s h o w n i n S e c ti o n 2 . 4 t h at b y i n tr o du c in g g e om e tr i c e x p re s si o ns t o a p pr o x-

i ma te e st ua ri ne b a th y me tr y, w ho le -e st ua ry r es po n se s c an b e d e te rm in ed . Ti da l

r es po n se s i n e st u ar ie s a re s ho w n f o r g eo me tr ie s a pp ro x im at ed b y ( i) b re ad th a nd

d e p t h v a r i a t i o n s o f t h e f o r m BL( X / λ)n and H L( X / λ)m, w h e r e X i s t h e d i s t a n c e f r o m

t h e h e a d o f t h e e s t u a r y , i . e . t h e l o c a t i o n o f t h e u p s t r e a m b o u n d a r y c o n d i t i o n a t t h e

l im it o f t id al i nf lu en ce ; ( ii ) b re ad th a nd d ep th v ar yi ng e xp on en ti al ly a nd ( ii i) a

‘synchronous’ estuary. C h a p t e r s 6 a n d 7 p r o v i d e d e t a i l s o f ‘s y n c h r o n o u s e s t u a r i e s’,

t he ir g eo me tr y i s s ho wn t o c or re sp on d t o ( i) w it h m = n = 0 .8 . B y e xp re ss in g t her e le v an t e q u at i on s i n d i me n si o nl e ss f o rm , t h es e a n al y ti c al s o lu t io n s a r e t r an s po s ed

i n to T h eo r et i ca l F r am e wo r ks , d e s cr i bi n g t i da l e l ev a ti o ns a n d c u rr e nt s o v e r a w i de

r a ng e o f e s tu a ri n e c o nd i ti o ns . F u rt h er d e ta i ls o f c u rr e nt r e sp o n se s a r e d e s cr i be d i n

C h a p t e r 3.

23



W he re a s in gl e ( M2) c on st it ue n t p re do mi na te s, t hi s p ro v id es a r ob u st b as is f or

l in ea ri sa ti o n o f t he f ri ct io n t er m a s o ut li ne d i n S e ct i on 2 . 5. Ti da l p ro p ag a ti on i n

e s tu a ri e s o f te n i n v ol v es l a rg e e x cu r si o n s o v e r r a pi d ly v a ry i ng s h al l ow t o po g ra p h y.

W hi le f ir st -o rd er t id a l p ro pa ga ti on i s r el at iv el y i ns en si ti ve t o s ma ll t op o gr ap hi c

c h a n g e s ( I a n n i e l l o , 1979), S e c t i o n 2 . 6 i l l u s t r a t e s h o w t h e a s s o c i a t e d n o n - l i n e a r i t i e sr es ul t i n t he g en er at io n o f s ig n if ic an t h i gh er h ar mo ni c a nd r es id u al c om po n en ts w it h

pronounced spatial gradients.

Finally, S e c t i o n 2 . 7 i n d i c a t e s s o m e o f t h e p e c u l i a r i t i e s o f s u r g e – t i d e i n t e r a c t i o n s .

2.2 Equations of motion

T he e qu ati on s o f m ot io n a t a ny h ei gh t Z ( m e as ur ed v er ti ca ll y u p wa rd s a bo v e

t he b e d) a lo ng o rt ho go n al h or iz on ta l a xe s, X and Y , m ay b e w ri tt en i n C ar te si anc o - o r di n a t e s ( n e g l e c t i ng v e r t i ca l a c c e l er a t i o ns ) a s f o l l o ws :

A c c e l e r a t i o n s i n X -direction:

@ U

@ t þ U

@ U

@ X þ V

@ U

@ Y þ g

@&

@ X ΩV ¼

@

@ Z E

@ U

@ Z (2:1)

A c c e l e r a t i o n s i n Y -direction:

@ V

@ t þ U

@ V

@ X þV

@ V

@ Y þ g

@&

@ Y þ ΩU ¼

@

@ Z E

@ V

@ Z (2:2)

Continuity:

@ U

@ X þ

@ V

@ Y þ

@ W

@ Z ¼ 0; (2:3)

where U , V and W a r e v e l o c i t i e s a l o n g X , Y and Z , ς i s s u r f a c e e l e v a t i o n , Ω = 2ω sin φ

i s t he C or io li s p ar am et er r ep re se nt in g t he i nf lu en ce o f t he e ar th’s r ot at io n (ω =

2π / 2 4 h ) , φ i s l a t i t u d e a n d E i s a v e r t i c a l e d d y v i s c o s i t y c o e f f i c i e n t . F o r c i n g d u e t ow i n d a n d v a r i a t i o n s i n d e n s i t y o r a t m o s p h e r i c p r e s s u r e i s o m i t t e d i n ( 2.1 ) a n d ( 2 . 2 ) .

F or m an y a pp li ca ti on s, i t i s c on ve ni en t t o v er ti ca ll y i nt eg ra te b et we en t he b ed a nd

t h e s u r f a c e . T h e d e p t h - a v e r a g e d e q u a t i o n s r e t a i n t h e s a m e f o r m e x c e p t t h a t

( 1 ) t h e n o n - li n ea r c o n v ec t i ve t e rm s U (∂U / ∂ X ) + V (∂U / ∂Y ) i n (2.1 ) a n d U (∂V / ∂ X ) + V (∂V /

∂Y ) i n ( 2.2) a r e m u l ti p li e d b y c o ef f ic i en t s d e pe n de n t o n t h e v e r t ic a l s t r u ct u re o f U and V ;

t h e s e c o e f f i c i e n t s a r e o f t e n a s s u m e d t o e q u a l 1 f o r s i m p l i c i t y ;

( 2) w it h z er o s ur fa ce s tr es s, t he v er ti ca l v is co si ty t er ms a re r ep la ce d b y b e d s tr es s t er ms

τ x/ ρ D and τ y/ ρ D, a ss um ed t o b e p ro po rt io na l t o t he r es pe ct iv e c om po ne nt s o f b ed

v e l o c i t y s q u a r e d , i . e .

τ x ¼ ρ fU ðU 2 þ V 2Þ1=2; τ y ¼ ρ fV ðU 2 þ V 2Þ1=2; (2:4)

where ρ i s w a t e r d e n s i t y a n d f i s t h e b e d s t r e s s c o e f f i c i e n t (≈0.0025)

24 Tidal dynamics



( 3) t he k in em a ti c b ou nd ar y c on di ti on a t t he s ur fa ce a nd b ed i s:

W s ¼@&

@ tþ U

@&

@ X þ V

@&

@ Y

and

W 0 ¼ U @ D

@ X V

@ D

@ Y : (2:5)

T h es e y i el d a d e pt h -i n te g ra t ed c o n ti n ui t y e q u at i on :

D þ & ð Þ@&

@ tþ

@

@ X U D þ & ð Þ þ

@

@ Y V D þ & ð Þ ¼ 0 (2:6)

I a nn i el l o ( 1977) i nd ic at es t ha t t ra ns ve rs e v el oc it ie s c an b e n eg le ct ed i f t he K el vi n

Number Ω B/( gD)1/2≪1 a n d t h e h or iz on ta l a sp ec t r at io B2ω2/( gD)≪1 ( B breadth,

ω = 2π / P , P ti d al p e ri o d) . T h en c e a d op t in g X a x ia l ly, b y i n te g ra t in g a c ro s s

bo th bre adt h an d dep th , ( 2 . 1 ) m a y b e r e wr i tt e n i n c r os s -s e ct i on a ll y a v er a ge d

pa ram ete rs as

@ U

@ t þ U

@ U

@ X þ g

@&

@ X þ f

U jU j

D þ & ð Þ¼ 0 (2:7)

a n d t h e c o n t i n u i t y e q u a t i o n (2.6) as

B D þ & ð Þ@&

@ tþ

@

@ X BUA ¼ 0; (2:8)

where A i s t h e c r o s s - s e c t i o n a l a r e a .

A l t h o u g h l a t e r a l v e l o c i t i e s m a y b e r e s t r i c t e d i n e s t u a r i e s , t h e t r a n s v e r s e C o r i o l i s

term ΩU , in (2.2), m us t b e b al an ce d, g en er al ly b y a l at er al s ur fa ce g ra di en t. T hi s

g ra di en t p ro du ce s a n e le va ti on p ha se a dv an ce o n t he r ig ht -h an d s id e ( lo ok in g

l an dw ard s i n t he n or th er n h em is ph ere ) o f t he o rd er o f BΩ/(2( gD)

1/2

) r ad ia ns(Larouche et al ., 1987).

T h e r e l a t i v e m a g n i t u d e s o f t h e t e r m s i n (2.7) f o r a p r e d o m i n a n t t i d a l f r e q u e n c y ω

a r e a p p r o x i m a t e l y

ωU :

2π U 2

λ :

2π & g

λ :

fU 2

D ; (2:9)

where λ i s t h e w a v e l e n g t h o v e r w h i c h b o t h U and ς vary. Assuming λ = ( gD)½ P , t h e

r e l a t i v e m a g n i t u d e s o f t h e f i r s t t w o t e r m s a r e

ð gDÞ1=2: U : (2:10)

T h u s , t h e r a t i o o f t h e m a g n i t u d e s o f t h e c o n v e c t i v e t e r m a n d t h e t e m p o r a l a c c e l e r a -

t io n t er m i s e qu al t o t he F ro ud e n um be r f or t he f lo w. T hi s i s g en er al ly s ma ll , a nd

2.2 Equations of motion 25



h en ce f or f ir st -o r de r t id al s im ul at io n, t he c on ve ct iv e t er ms c an b e n e gl ec te d. T h e

r el at iv e m ag ni tu de o f t he f ri ct io n t er m t o t he t em po ra l a cc el er at io n t er m f or t he

s e m i - d i u r n a l f r e q u e n c y a n d f = 0 .0 0 25 i s a pp ro x im at el y 2 0 U */ D s−1, i . e . p r e d o m i -

n an t i n f as t- fl ow in g s ha ll ow e st u ar ie s ( se e S e ct i on 2 . 3. 2) . I n s uc h e st ua ri es , t he

f r i c t i o n a l f o r c e g r e a t l y e x c e e d s t h e a c c e l e r a t i o n ( i n e r t i a l ) t e r m o v e r m o s t o f t h e t i d a lc y c l e a n d w a v e p r o p a g a t i o n i s d i f f u s i v e i n c h a r a c t e r ( L e B l o n d , 1978).

2.3 Tidal response – localised

It is shown in Section 2.4.2 t ha t c om bi ni ng t he t wo e qu at io ns , (2.7) a n d ( 2 .8 ) , p ro d uc e s

a n e xp re ss io n f or t id al r es po ns e a lo ng a n e st ua ry s im il ar t o t he s pe ct ra l r es po ns e f or

a l i ne a rl y d a mp e d, s i ng l e- d eg r ee - of -f r ee d om o s ci l la to r y s y st e m e x ec u ti n g ‘simple

harmonic motion’. T h us , w e e x pe c t h a rm o ni c s o lu t io n s w i th a x ia l v a ri a ti o ns i n t i da la m pl i tu d es a n d p h as e s d e sc r ib e d b y B e ss e l f u nc t io n s, a s i ll u st ra t ed i n Section 2.4.1.

2.3.1 Linearised solution

Negle cting the convec tive term and linea rising the frict ion term in ( 2 . 7 ) ( s e e

S e c t i o n 2 . 5 f o r d e t a i l s o f t h i s l i n e a r i s a t i o n ) y i e l d s

@ U

@ t þ g @& @ X þ FU ¼ 0: (2:11)

I t i s r ea di ly s ho wn t ha t i n a p r is m at ic c h an ne l o f i nf in it e l en gt h a nd z er o f ri ct io n,

(2.8) a nd ( 2. 11 ) i nd ic at e a w av e c el er it y, c = ( gD)1/2 and U * = ς ( g / D)1/2 (Lamb,

1932) . M ax im um a mp li fi ca ti on t he n o c cu rs f or q u ar te r- wa ve r es o na nc e a t l en gt h

L = 0 .2 5 λ = 0 .2 5 P ( gD)1/2. It is sh own in S ec t io n 2 . 4. 1 t ha t e ve n i n d am pe d,

f u nn e l- s ha p e d e s tu a r ie s , m a x im u m a m pl i fi c at i o n o f te n o c cu r s f o r v a lu e s o f L close

t o t h i s v a l u e .

I n t r o d u c i n g a s u r f a c e g r a d i e n t f o r a p r e d o m i n a n t c o n s t i t u e n t i n t h e f o r m

@&

@ X ¼ & x cos ωt (2:12)

from (2.11), w e o b t a i n

U ¼ g& x

ðF 2 þ ω2ÞðF cos ωt þ ω sin ωtÞ: (2:13)

T h u s , f o r a f r i c t i o n a l l y d o m i n a t e d s y s t e m F ≫ω,

U ¼ g& x=F cos ωt; (2:14)

w h i l e f o r a f r i c t i o n l e s s s y s t e m F ≪ω,

26 Tidal dynamics



U ¼ g& x =ω sin ωt: (2:15)

F i gu r e 2 . 1 i n d i c a t e s t h e s o l u t i o n o f (2.13) f o r p r e s c r i b e d v a l u e s o f s u r f a c e g r a d i e n t

u p t o 0 .0 00 25 g a nd d ep th s o f 4 , 1 6 a nd 6 4 m . F or t he d ee pe st c as e, t he s ol ut io n

approximates (2.15) w h i l e t h e s h a l l o w e s t c a s e a p p r o x i m a t e s (2.14).

F ig ur e 2 .2 s how s t he m ag ni tu de o f t he t er ms i n (2.7) a t t wo p os it io ns i n t he T ham es

f o r t h e p r e d o m i n a n t M2 c o n s t i t u e n t a n d t h e r e l a t e d h i g h e r h a r m o n i c s M4 a n d M6, as

c a l c u l a t e d i n a n u m e r i c a l m o d e l s i m u l a t i o n ( P r a n d l e , 1980) . F o r M2, t h e i n e r t i a l a n d

f ri ct io na l t er ms a re o rt ho go na l i n p ha se a nd b al an ce t he s ur fa ce g ra di en t t er m.

B y c on tr as t, f or M4 a n d M6, t he s pa ti al g ra di en t t er m i s a c on se qu en ce o f r at he r

t h a n a d r i v i n g f o r c e f o r c u r r e n t s ( s e e S e c t i o n 2 . 6 ) a n d h e n c e d i f f e r e n t r e l a t i o n s h i p s

apply.

2.3.2 Synchronous estuary solution

A ‘s y n ch r on o us e s tu a ry’ i s o ne w he re s ur fa ce g ra di en ts a ss oc ia te d w it h a xi al

a m p li t ud e v a ri a ti o ns i n ς * a re s ig n if ic an tl y l es s t ha n t ho se a ss oc ia te d w it h c or re -

s po nd in g p ha se v ar ia ti on s. I n d er iv in g s ol ut io ns t o (2.8) a nd ( 2. 11 ), a s im il ar

2.0

1.5

1.0

0.5 V e l o c i t y a m p l i t u d e ( m

s – 1 )

0.00 5.0 × 10 –5 1.0 × 10 –4

Surface gradient amplitude ( × g )

1.5 × 10 –4 2.0 × 10 –4 2.5 × 10 –4

No friction

Depth = 4 m

No inertia

D = 64 m

D = 16 m

F i g . 2 . 1 . M2 t i d a l c u r r e n t a m p l i t u d e a s a f u n c t i o n o f s u r f a c e g r a d i e n t . C o n t i n u o u sl i ne s s h ow s o lu t i on (2.13) for D = 4 , 1 6 a nd 6 4 m . Da sh ed l in es s ho w s ol ut io n(2.14) for D = 4 m a n d (2.15) for D = 6 4 m .




a pp ro xi ma ti on i s a ss um ed t o a pp ly t o a xi al v ar ia ti on s i n U *. Fo r the d eriv ed‘synchronous’ b at h ym et ry, t he se a ss um pt io ns f or b ot h U * and ς * h ave b een

s ho wn t o b e v al id , e xc ep t i n t he s ha ll ow es t c o nd it io ns a t t he t id al l im it ( Pr an dl e,

2003) . I nt ro du ct io n o f t he s ol ut io n h er e p er mi ts a r ea dy c om pa ri so n w it h o th er

s ol ut io ns a nd p ro vi de s c on v en ie nt e xp re ss io ns f or U * in terms of ς * and D –

s u b s e q u e n t l y u s e d t h r o u g h o u t t h i s b o o k .

C o n ce n t ra t in g o n t h e p r o pa g at i on o f o n e p r ed o m in a nt t i da l c o n st i tu e n t, M2, t h e

s o l u t i o n s f o r U and ς a t a n y l o c a t i o n c a n b e e x p r e s s e d a s

& ¼ & cosðK 1X ωtÞ and U ¼ U cosðK 2X ωt þ θ Þ; (2:16)

where K 1 and K 2 a r e t h e w a v e n u m b e r s , ω i s t h e t i d a l f r e q u e n c y a n d θ i s t h e p h a s e

l a g o f U r e l a t i v e t o ς .

F ur th er a ss um in g a t ri an gu la r c ro ss -s ec ti on w it h c on st an t s id e s lo pe s, (2.8)

r e d u c e s t o

@&

@ t þ U

@&

@ X þ@ D

@ X

þ1

2

@ U

@ X & þ Dð Þ ¼ 0: (2:17)

F ri ed ri ch s a nd A u br ey (1994) i nd ic at e t ha t U (∂ A/ ∂ X )≫ A(∂U / ∂ X ) i n c on ve rg en t

c h a n n e l s . L i k e w i s e , a s s u m i n g ∂ D/ ∂ X ≫∂ζ*/ ∂ X , w e a d o p t t h e f o l l o w i n g f o r m o f t h e

c o n t i n u i t y e q u a t i o n :

du dt

F

F

F

F F

F

G

G

G

G G

G

Scale

Scale

M2

Section 6

Section 40

1×10 –4

ft s –2

1×10 –3 ft s –2

M4 M6

M2 M4 M6

du dt

du dt

du dt

du dt

du dt

F i g . 2 . 2 . M2, M4 a n d M6 c o n s ti t u e n ts o f (2.7) n ea r t h e m o ut h ( t op ) a n d u p st r ea mi n t h e T h a m e s , b a s e d o n n u m e r i c a l m o d e l s i m u l a t i o n .

28 Tidal dynamics



@&

@ tþ U

@ D

@ X þ

D

2

@ U

@ X ¼ 0: (2:18)

S u b s t i t u t i n g s o l u t i o n (2.16) i n t o E q s (2.11) a n d ( 2 . 1 8 ) , f o u r e q u a t i o n s ( p e r t a i n i n g a t

a ny s pe ci fi c l oc at io n a lo ng a n e st ua ry ) r ep re se nt in g c o mp on en ts o f c os ωt a nd s in ωt a re o bt ai ne d. B y s pe ci fy in g t he s yn ch ro no us e st ua ry c on di ti on t ha t t he s pa ti al

g ra di en t i n t id al e le va ti on a mp li tu de i s z er o, t he c o nd it io n K 1 = K 2 = k i s d e r iv e d ,

i .e . i de n ti ca l w av e n um be rs f or a x ia l p ro pa ga ti on o f ς and U . T h en , t h e f ol lo w in g

s o lu t io n s f o r t h e a m pl i tu d e, U * , a nd p ha se , θ , o f t id al c ur re nt t og et he r w it h b ed

slope, SL = ∂ D/ ∂ X , a r e o b t a i n e d :

tan θ ¼ F

ω

¼ SL

0:5Dk

; U ¼ & g k

ðω2

þ F 2

Þ1=2

; k ¼ ω

Dg=2ð Þ1=2

: (2:19)

Results

T h e a b o v e s o l u t i o n s a r e c o n s i s t e n t w i t h (2.13), t h e c e l e r i t y 0 . 5 ( gD)½ f o l l o w s f r o m

t h e a s s u m p t i o n o f a t r i a n g u l a r c r o s s s e c t i o n . C h a p t e r 6 i l l u s t r a t e s h o w t h e s e e x p l i c i t

s o l u t i o n s f o r U *, θ and SL e n a b l e o t h e r r e l a t e d p a r a m e t e r s , s u c h a s e s t u a r i n e l e n g t h ,

t o b e d et er mi ne d, y ie ld in g a r an ge o f T he or et ic al F ra me wo rk s i n t er ms o f t he

parameters D and ς * . T h e p a r a m e t e r r a n g e s s e l e c t e d a r e ς * (0 – 4 m ) a n d D (0 – 4 0 m ),

r e pr es e nt i ng a l l b u t t h e d e ep e st o f e s tu a ri e s.

Current amplitudes U

F ig ur e 2 .3 s h ow s t he s ol ut io n (2.19) w it h c ur re nt a mp li tu de s e xt en di ng t o 1 .5 m s−1

( Pr an d le , 2 00 4 ). Th e c on to ur s s ho w t ha t m ax im um v al u es o f U * o cc ur a t a pp ro xi ma te ly

D = 5 + 1 0 ς * ( m) ; h ow ev er, t he se a re n ot p ro no un c ed m ax im a. T hi s f ig ur e e xp la in s

w hy o bs er ve d v al ue s o f U * are so often in the range 0.5 – 1 . 0 m s−1 despite large

variations in ς * o v e r t h e s p r i n g – n e a p c y c l e a n d t h e w i d e r a n g e o f e s t u a r i n e d e p t h s .

Role of bed friction

F ri ed ri ch s a nd A ub re y (1994) s ho we d t he p re do mi na nc e o f t he f ri ct io n t er m i n

s t ro n gl y c o nv e rg e n t c h a nn e ls , i r re s p ec t iv e o f d e pt h . F i gu r e 2 . 4 s ho ws t he r at io o f

f r i c t io n : i n e r t ia , F / ω, f ro m (2.19) ( P r an d l e , 2 0 0 4 ) . F / ω i s a pp ro x im at el y e qu al t o

u n it y f or ς * = D/ 10 . F o r ς *≪ D/ 10 , c ur re nt s a re i n se ns it iv e t o f ri ct io n, w hi le f or

ς *≫ D/ 1 0 , t i d a l d y n a m i c s b e c o m e f r i c t i o n a l l y d o m i n a t e d a n d c u r r e n t s d e c r e a s e b ya f a c t o r o f t w o a s t h e f r i c t i o n c o e f f i c i e n t i n c r e a s e s o v e r i t s t y p i c a l r a n g e f r o m 0 . 0 0 1

t o 0 .0 04 . P ra nd le (2003 ) p ro vi de s a d et ai le d a na ly si s o f t he s en si ti vi ti es t o t he

f r i c ti o n p a r a m et e r. F r o m (2.19), f o r F ω; U / & 1=2D1=4 f 1=2while for F ω;

U / & D1=2:




From (2.19), F / ω = 0 . 1 c o rr e sp o nd s t o a p h as e d i ff e re n ce b e tw e en t i da l e l ev a ti o na nd c ur re nt s o f θ = −6 ° . S i m i l ar l y, F / ω = 0 .5 c or re sp on d s t o θ = −2 7 ° , 1 . 0 t o −45°,

2 to −63°, 5 to −77 ° an d 10 to −8 4° . T he se v al ue s o f θ e mp ha si se h ow t he t id al w av e

propagation changes from ‘ progressive’ in d ee pe r w at er c lo se r t o t he m ou th t o

‘standing’ i n s h a l l o w e r w a t e r a t t h e h e a d .

F i g . 2 . 4 . R a t i o F : ω o f t h e f r i c t i o n t e r m F t o t h e i n e r t i a l t e r m a s a f ( D, ς *) (2.19).

F i g. 2 . 3. T i d al C u r r en t a m p l i t ud e , U * ( m s−1) , a s a f ( D, ς *) ( 2 . 1 9 ).

30 Tidal dynamics



Rate of funnelling in a synchronous estuary

B y i n t e g r a t i o n o f t h e s o l u t i o n f o r b e d s l o p e , SL, in (2.19), i t c a n b e s h o w n t h a t t h e

s y n ch r on o us s o lu t io n c o rr e sp o n ds t o d e p th s a n d b r ea d t hs p r op o r ti o na l t o X 0.8, i . e .

m = n = 0 . 8 i n (2.20) a n d ( 2. 21 ). C om pa ri ng t he l oc al is ed s yn ch ro n ou s s o lu ti on s

t o t h e w ho le -e st ua ry r es p on se i n S e c t io n 2 . 4. 1, t h is s y n ch r on o u s g e om e tr y c o rr e -s p o n d s t o ν = 1 .5 . F r om F i g . 2 . 5, t h i s i s c l o s e t o t h e c e n t r e o f t h e r a n g e o f g e o m e t r i e s

e n co u n te r ed ( P ra n dl e , 2 0 0 4) . M o re o ve r, t h e e s tu a ri n e l e ng t hs d e te r mi n ed f o r s y n-

c h r o n o u s e s t u a r i e s , i n c o r p o r a t e d i n F i g . 2 . 5, r a n g e f r o m a s m a l l f r a c t i o n u p t o c l o s e

t o t h a t f o r ‘quarter-wavelength’ ( f i r s t n o d e ) , r e s o n a n c e a t t h e M2 frequency.

2.4 Tidal response – whole estuary

T hi s s ec ti on i s c on c er ne d w it h t he f ir st -o rd er r es po ns e o f w h ol e e st ua ri es t o t id alf o r c i n g . T h e a i m i s t o c o n s t r u c t s i m p l i f i e d a n a l y t i c a l f r a m e w o r k s t h a t a n s w e r s u c h

basic questions as to why (i) tides are large in some estuaries, (ii) semi-diurnal

c on s ti tu en t s a re s o me ti me s a mp li fi ed w hi le d iu rn al s a re o ft en d am pe d a nd ( ii i) s om e

e st ua ri es a re s en si ti ve t o s ma ll c ha ng e s i n b ed f ri ct io n, l en gt h o r d ep th . S im pl if ie d

F i g. 2 . 5. Ti d al e l ev a t io n r e sp o n se s (2.24) for s − 2π . ν r ep re se nt s t he d eg re e o f bathymetric funnelling and y d i st an ce f ro m t he m o ut h , y = 0 . D as he d c on to ur si n di c at e r e la t iv e a m pl i t ud e s, c o nt i nu o us c o nt o ur s r e la t iv e p h as e s. Ve r ti c al l i ne a t ν = 1 .5 s ho ws t yp i ca l l en gt h s o f s yn ch ro no us e st ua ri es ( se e C h ap t er 6) . L e ng t hs ,

y ( f o r M2) a n d s h a p e s , ν , f o r e s t u a r i e s ( A ) – ( I ) s h o w n i n Ta b l e 2 . 1.




a n al y ti c al s o lu t io n s t o (2.8) a nd ( 2. 11 ) h av e b ee n p re se nt ed b y Ta yl or (1921),

D o rr e st e in ( 1961) a n d H u n t ( 1964) . H er e w e d is cu s s m o re g en e ra li se d s ol u ti o ns

f o r ( i ) b r e a d t h a n d d e p t h v a r y i n g w i t h p o w e r s o f d i s t a n c e X ( P r an d le a n d R a hm a n,

1980, s u b s e q u e n t l y P r a n d l e a n d R a h m a n , 1980) a n d ( i i ) b r e a d t h a n d d e p t h v a r y i n g

e x po n e nt i al l y w i th X (Prandle, 1985) . S in ce th es e r es po ns es a re b as ed o n t hel i ne a ri s ed e q ua t io n s, t h ey a r e g e ne r al l y a p p li c ab l e t o e s tu a ri e s w i th a p r ed o m in a nt

t i d a l c o n s t it u e n t .

Taylor ’s f ri ct io nl es s s ol ut io n f or a n e st ua ry w it h l in ea rl y v ar yi ng d ep th a nd b re ad th

r ep re se nt s a s pe ci al c as e o f ( i) . H un t ’s a n al yt i ca l s o lu ti o ns f o r e st u ar ie s w i th e x po -

n e n t i a l l y i n c r e a s i n g b r e a d t h a n d c o n s t a n t d e p t h a r e p r e s e n t e d i n S e c t i o n 2 . 4 . 2.

2.4.1 Breadth and depth varying with powers of distance X(Prandle and Rahman, 1980 )

B r e a d t h a n d d e p t h a r e a s s u m e d t o v a r y b y

BðX Þ ¼ BL

X

λ

n

(2:20)

and

H ðX Þ ¼ H L X λ

m

(2:21)

with X m e as u re d f r om t h e h e a d o f t h e e s t u ar y. To c o n ve r t t o a d i me n si o nl e ss f o rm a t,

w e a d o p t λ a s a u n i t o f h o r i z o n t a l d i m e n s i o n , H L a s a u n i t o f v e r t i c a l d i m e n s i o n a n d

P , t h e t i d a l p e r i o d , a s a u n i t o f t i m e , w i t h

λ ¼ ð gH LÞ1=2P (2:22)

c o rr e sp o nd i n g t o t h e t i da l w a ve l en g th f o r H L c o n s t an t . D i m e n s io n l e s s p a r a m et e r sa r e i n t r o d u c e d a s f o l l o w s :

x ¼ X = λ; t ¼ T =P; h ¼ H =H L; b ¼ B= λ; u ¼ UP= λ; frictional parameter s ¼ F P:

(2:23)

P r a n d l e a n d R a h m a n ( 1980) s ho we d t ha t t he s ub s ti tu ti o n o f (2.20) a n d ( 2. 21 ) i nt o

(2.8) a nd ( 2. 11 ) y ie ld s t he f ol lo wi ng s o lu ti on f or t id a l e le v at io n ς a t a ny l oc at io n x, at

a n y t i m e t , f o r a n y t i d a l p e r i o d P :

& ¼ & ky

kyM

1ν Jν 1ðkyÞ

Jν 1ðkyMÞei2π t; (2:24)

where ς * ei2π t i s t h e t i d a l e l e v a t i o n a t t h e m o u t h xM and

32 Tidal dynamics



ν ¼ n þ 1

2 m; k ¼

1 is

2π

1=2

y ¼

4π

2 m x

2m

2 (2:25)

a n d Jν − 1, i s a B e s s e l f u n c t i o n o f t h e f i r s t k i n d a n d o f o r d e r v − 1.

T h e s o l u t i o n (2.24) i s i l l u s t r a t e d i n d i a g r a m m a t i c f o r m i n F i g . 2 . 5 f o r t h e c a s e o f

s = 2π , i . e . F = ω. P r a n d l e a n d R a h m a n ( 1980 ) s h o w t h e c o r r e s p o n d i n g s o l u t i o n s f o r

s = 0 . 2π . A w a y f r o m t h e r e s o n a n t c o n d i t i o n s i l l u s t r a t e d i n F i g . 2 . 6 , t h e r e s p o n s e s f o r

t h e t w o f r ic t io n al c o ef f ic i en t s a r e e s se n ti a ll y s i mi l ar, w i th r e du c ed a m pl i tu d e s a n d

e n h a n c e d p h a s e d i f f e r e n c e s f o r t h e l a r g e r f r i c t i o n c o e f f i c i e n t . T h i s f i g u r e c o n s t i t u t e s

a g en er al r es po ns e d ia gr am s ho wi ng t he v ar ia ti on i n a mp li tu de a nd p h as e o f t id ale l e v a t i o n s a l o n g t h e l e n g t h o f a n e s t u a r y . F o r t h e M2 s e m i -d i u r n al c o n s t it u e n t , t h e

positions indicated (A) – ( I) d es ig na te t he m ou th s o f t he m aj or e st ua ri es l is te d i n

T a b l e 2 . 1.

C o n f i d e n c e i n t h e v a l i d i t y o f t h i s a p p r o a c h w a s s h o w n b y c o m p a r i n g r e s u l t s f o r

M2 e l e v a t i o n r e s p o n s e f o r t h e t e n m a j o r e s t u a r i e s l i s t e d i n T a b l e 2 . 1. I n a l l o f t h e s e

s = 0.2π

s = 2π

F i g . 2 . 6 . F r e q u e n c y r e s p o n s e f o r t i d a l e l e v a t i o n s i n t h e B a y o f F u n d y (2.24), w i t h s = 0 . 2π a n d 2π . V e r t i c a l s c a l e s h o w s a m p l i f i c a t i o n a t t h e h e a d r e l a t i v e t o v a l u e s a t

t h e s h e l f e d g e e n c i r c l e d d o t s r e p r e s e n t o b s e r v e d d a t a .




e s t u a r i e s , g o o d a g r e e m e n t w a s f o u n d u s i n g s = 2π , e x c e p t f o r t h e B a y o f F u n d y ( G ) ,

w he re w it h d ep th s e xc ee di ng 2 0 0 m b et te r a g re em e nt w a s f ou n d f or s = 0 . 2π .

G i v e n t h e f u n n e l l i n g f a c t o r v a n d l e n g t h yM f o r a p a r t i c u l a r e s t u a r y , t h e v a r i a t i o n

i n a m p l i t u d e a n d p h a s e i n t h e e s t u a r y c a n b e r e a d a l o n g t h e c o r r e s p o n d i n g v e r t i c a l

l i n e . M o r e o v e r , t h e v a l u e o f yM i s i n v er s el y p r o po r ti o na l t o t h e t i da l p e ri o d P , t h u s

doubling P halves yM. U si ng t hi s r el at io ns hi p, F ig . 2 .6 i ll u st r at e s t h e s p ec t ra l

r es po ns e f or t he B a y o f F u nd y. S tr ic tl y, s om e a d ju s tm en t i s n ec es sa ry t o r ef le ct

t he 5 0% i nc re as e i n t he f ri ct io n f ac to r a pp ro pr ia te t o c on st it ue nt s o th er t ha n t he

predominant M2 – a s d es cr ib ed i n S e ct i on 2 . 5. S i m il ar r es po ns e d ia gr am s t o F ig . 2 .5

c a n b e c o n s t r u c t e d f o r a m p l i t u d e s a n d p h a s e s o f t i d a l c u r r e n t s .

I n s u mm a ry, F ig . 2 .5 c o ns ti tu te s a g e ne ra l t id al r es p on se d i ag ra m i nd ic at in g

a mp li tu de s a nd p ha se s ( re la ti ve t o t he m ou th ) a t a ll p o si ti on s, f or a ll t id al p er io ds ,f or a ll e st ua ri es w hi ch r ea so n ab ly c or re sp o nd t o (2.11) a n d ( 2. 12 ). T h is r es po n se

d i a g r a m e x p l a i n s a n u m b e r o f f e a t u r e s c o m m o n l y e n c o u n t e r e d :

( 1 ) T h e q u ar t er -w a ve l en g t h r e so n an c e o r p r im a ry m o de f o un d i n s u ff i ci e nt l y l o ng e s tu a ri e s

i s i n d i c a t e d b y t h e t h i c k l i n e t h r o u g h t h e a m p l i t u d e n o d e s .

( 2 ) F o r a d iu rn al t id al c on st it u en t, t he yM v al u es ( A ) – ( I) a re h al ve d; h en ce , w e e xp ec t r el at iv el y

s mal l a mpl i f ic a ti o n o f s u ch c o ns ti t u en t s . F o r MSf , a 1 4 -d a y c o n s ti t u en t , t h e r e d u ct i on i n

the yM values would indicate little amplification or phase difference along any estuary.

( 3) F or q ua rt er -d iu rn al s o r o th er h ig he r h ar mo ni cs , i n r el at iv e t er ms w e e xp ec t h ig h

a m pl i f ic a ti o n , l a rg e p h as e d i ff e re n ce s a n d o n e o r m o re n o da l p o si t i on s . H o we v er , i t i s

i mp o rt an t t o d i st in gu is h b et we en t he r es p on se t o e xt er na l f or ci ng r ep re se nt ed b y t he

present analysis versus the internal generation of higher harmonics by non-linear pro-

c e s s e s w i t h i n a n e s t u a r y d i s c u s s e d i n Section 2.6 a n d i l l u s t r a t e d i n F i g . 2 . 2.

Ta b l e 2 . 1 Geometrical parameters for ten estuaries shown in Fig. 2.5

H M (m) L (km) n m ν y0 H 0 (m) α β α + 2 β

A Fraser 44 135 −0.7 0.7 0.2 3.0 2.3 −2.8 2.8 2.8

B Rotterdam Waterway 13 99 0 0 0.5 1.2 13.0 0 0 0C Hudson 17 248 0.7 0.4 1 .1 4.2 4.8 2.2 1.3 4 .8D Potomac 13 184 1.0 0.4 1.3 3.7 3.5 3.6 1.4 6.4E Del aware 5 214 2.1 0.3 1 .8 5.3 2.3 5.3 0.8 6 .9F Mirami chi 7.0 55 2.7 0 1.9 0.9 7.0 46.6 0 46.6G Bay Fundy 2000 635 1.5 1.0 2.4 3.8 21.4 3.9 2.6 9.1H Thames 80 95 2.3 0.7 2.5 1.77 2.7 14.1 4.3 22.7I B ristol Channel 5000 623 1.7 1.2 3.4 5.20 12.5 3.4 2.4 8.2J St. Lawre nce 300 418 1.5 1.9 19.5 1 1.3 1.6 4.5

Notes: H M d e p t h a t m o u t h , H 0 d e p t h a t h e a d , L and y0 e s t ua r i n e l e n g t h s ( f r o m ( 2 . 2 5) )n, m, ν , α and β b a t h y me t r i c p a r ame t e r s.Source: P r an d l e a n d R a h man , 1980; P r an d le , 1985

34 Tidal dynamics



Quarter-wavelength resonance

T h e p o s i t i o n o f t h e f i r s t n o d a l l i n e i n F i g. 2 .5, c o rr e sp o n di n g t o m a xi m um a m pl i -

f i c a t i o n i n e s t u a r i e s w i t h v a r y i n g d e g r e e s o f f u n n e l i n g , i s a p p r o x i m a t e d b y

y ¼ 1:25 þ 0:75ν ; i:e: x ¼ 3n 5m þ 13

16π

ð2=2mÞ

: (2:26)

T he r at io o f r es on an t l en gt h s, (2.26), a s a f ra ct io n o f t ha t f or a p ri sm at ic c ha nn el ,

(2.22), is x(1−m/2)/0.25.

Ta b l e 2 . 2 s ho ws t hi s r at io , f or t he t yp ic al r an ge s o f 0 < n < 2 . 5 a n d 0 < m < 1,

i nc lu d in g t he v a lu e m = n = 0 .8 c or re sp on di ng t o t he s ol ut io n f or a s yn ch ro no us

e st ua ry (S e ct i on 2 . 3. 2) . T he se r es u lt s i nd ic at e t ha t u ps tr ea m r ed u ct io ns i n d e pt hsd ec re as e r es on an t l en gt hs w hi le b re ad th c on ve rg en ce h as t he o pp os it e e ff ec t, e mp ha -

s i s i n g t h e c o m p l e x i t y o f t i d a l r e s p o n s e i n f u n n e l - s h a p e d e s t u a r i e s .

2.4.2 Depth and breadth varying exponentially

(Hunt, 1964; Prandle, 1985 )

A s s u m i n g a b r e a d t h a n d d e p t h v a r i a t i o n :

BðX Þ ¼ B0 expðnX Þ

H ðX Þ ¼ H 0 expðmX Þ; (2:27)

where B0 and H 0 a re the respective values at the head X = 0 , we conv ert to

d i m e n s i o n l e s s u n i t s b a s e d o n t h e t i d a l p e r i o d P a s t h e u n i t o f t i m e . H 0 a s t h e v e r t i c a l

d i m e n s i o n a n d λ t h e h o r i z o n t a l d i m e n s i o n g i v e n b y

λ ¼ ð gH 0Þ1=2P: (2:28)

T hu s , w e o b ta in t ra ns fo rm ed d im en si on le ss v ar ia bl es : x = X/ λ , t = T/P, z = Z/H 0,b = B / λ , h = H/H 0, u = U ( P/ λ), s = FP and

bðxÞ ¼ b0 expðαxÞ (2:29)

and

Ta b le 2 . 2 Resonant lengths for funnel-shaped estuaries, B∝Xn

and D∝Xm , (2.26) , as a fraction of the prismatic value (2.22)

m =n 0 0.8 1.0

0 1.04 0.74 0.570.8 1.23 0.91 0.832.5 1.64 1.33 1.26




hðxÞ ¼ expð β xÞ; (2:30)

where α = n λ and β = m λ.

Substituting (2.29) a nd ( 2. 30 ) i nt o (2.8) a nd ( 2. 11 ), t he se e qu at io ns m ay b e

r ea rr an g ed t o f or m s ep ar at e e xp re ss io n s f or e it he r ς or u. T he t im e d er iv at iv es i nt he se e xp re ss io ns m ay b e e li mi na te d b y c on si de ri ng a mp li tu de s p er ta in in g t o a

s i n g l e p e r i o d P , t h u s

& ¼ & expði2πtÞ and u ¼ u expði2π tÞ: (2:31)

T h e e x p r e s s i o n s f o r t h e t i d a l a m p l i t u d e s ς and u a r e t h e n a s f o l l o w s :

@ 2

@ x2 & þ ðα þ β Þ

@&

@ x

þ ð4π 2 2π isÞ &

expð β xÞ

¼ 0 (2:32)

@ 2u

@ x2 þ ðα þ 2 β Þ

@ u

@ x þ β ðα þ β Þ þ

ð4π 2 2π isÞ

expð β xÞ

u ¼ 0: (2:33)

B y i n t r o d u c i n g a p p r o p r i a t e t r a n s f o r m a t i o n s , t h e m i d d l e t e r m s ( i n v o l v i n g t h e s i n g l e

d e ri v at i ve i n x in (2.32) a nd ( 2. 33 )) m ay b e e li mi na te d. T he r es ul ti ng e qu a ti on s

m a y t h e n b e s o l v e d a n a l y ti c a l l y ( G i ll , 1982 , S e ct i on 8 . 12) . S u c h s o l u t io n s h a v e be e n

e x a m i n e d b y X i u (1983 ) ; h o w e v e r , t h e i r c o m p l e x i t y o b s c u r e s d i r e c t u n d e r s t a n d i n g .

I n t h e f o l l o w i n g s e c t i o n , w e c o n s i d e r s i m p l e r a n a l y t i c a l s o l u t i o n s r e l a t i n g t o c e r t a i ns p e c i a l c a s e s a l o n g s i d e a n u m e r i c a l s o l u t i o n t o i l l u s t r a t e t h e n a t u r e o f t h e r e s p o n s e s .

(1) S o l u t i o n s f o r c o n s t a n t d e p t h , β = 0 : H u n t ( 1964) s h o w e d t h a t f o r t h i s c a s e t h e s o l u t i o n s

to (2.32 ) a n d ( 2 . 3 3 ) a r e

& ¼ & 0 exp α x

2

cosh ωx þ

α

2ω sinh ωx

(2:34)

u ¼ & 0 exp α x

2 2π i

α sinh ωx; (2:35)

where ω = ω1 + i ω2, ω12− ω2

2= α2/4 − 4π 2, ω1·ω2 = π s and ς *0 i s t h e e l ev a ti o n a m pl i -

t u d e a t t h e h e a d , x = 0.

(2) S o lu t io n s f o r c o ns t an t d e pt h , β = 0 a nd z er o f ri ct io n: F or t hi s c as e, (2.32) a n d ( 2 .3 3 )

d es cr ib ed t he f re e v ib ra ti on s o f a d am pe d s im pl e h ar mo ni c o sc il la to r. U si ng t hi s

a n al o g y, f o r α < 4π t h e s ys t em i s u nd er -d am pe d, α = 4π r e p re s en t s c r it i ca l d a mp i n g

while α > 4π i s o v e r -d a mp ed .

For α > 4π , t he s ol ut io ns r et ai n t he f or m s ho wn i n (2.34) a nd ( 2. 3 5) w it h ω12 = α2/4 − 4π 2

and ω2 = 0.For α < 4π , t h e s o l u t i o n s s i m p l i f y t o

& ¼ & 0 exp αx

2

cos ω2x þ

α

2ω2

sin ω2x

(2:36)

36 Tidal dynamics



u ¼ & 0 exp αx

2

2π i

ω2

sin ω2x (2:37)

with ω22 = −α2/ 4 + 4π 2.

For α = 4π , t h e f o l l o w i n g s p e c i f i c s o l u t i o n s a p p l y

& ¼ & 0 expð2π xÞð1 þ 2π xÞ (2:38)

u ¼ & 0 expð2π xÞ2π ix: (2:39)

(3) N u m e r i c a l s o l u t i o n f o r b o t h d e p t h a n d b r e a d t h v a r y i n g e x p o n e n t i a l l y a n d f r i c t i o n .

T he g en er al r es po n se d ia gr am i s s ho w n i n F i g. 2 .7 ( P ra n dl e , 1 9 85 ) , f o r s = 2π , t h e

o r th o go n al a x es r e fe r t o t h e p a ra m et e rs α and β . T h e c o nt o u rs s h ow t h e a m pl i fi c at i o n

between the amplitude of the tidal elevation at the head of the estuary relative to the

v al u e a t t he f ir st n od al p os it i on . H ow ev er, f or e st ua ri es w it h v al ue s o f α + 2 β > 1 0 , n o

n o da l p o si t io n o c cu r s a n d i n t h is c a se t h e a m p l if i ca t io n s h ow n i s r e la t iv e t o t h e v a l u e f o r x = 1 w h er e t h e l a tt er v a lu e c l os e ly a p pr o xi m at e s t h e a s ym p to t e a t x =∝. T h i s d e m a r c a -

t i o n i n t h e r e s p o n s e o f e s t u a r i e s a t α + 2 β = 1 0 w as n ot e vi de nt i n t h e s ol ut io n i n Section

2.4.1. T h e s y m b o l s ( A ) – ( J ) a g ai n i n di ca t e t h e a m pl i fi ca t io n f o r a l l t e n m a jo r e s tu ar i es ,

l i s t e d i n Table 2.1, b e t w e e n t h e h e a d a n d t h e f i r s t n o d a l p o s i t i o n o r x = 1 ( no t t h e m o u th )

f o r t h e s e mi - d i u r n a l c o n s t i t u e n t M2.

F i g . 2 . 7 . T i d a l e l e v a t i o n a m p l i f i c a t i o n a s a f u n c t i o n o f α and β , s = 2π (2.32) .




To d e te r mi n e t h e m a xi m u m r e sp o n se f o r o t he r t i da l c o ns t i tu e nt s f r om F i g. 2 .7, we

n ot e t ha t t he v al ue s f or α and β a r e d ir ec tl y p ro po rt io na l t o p er io d; t hu s, f or d iu rn al

constituents α and β a re d o ub l e d w h i le f o r q u ar t er - di u rn a l c o ns t i tu e n ts α and β are

h al ve d. I n c on se qu en ce , w e m ay d ed uc e t h e f ol lo w in g c on cl u si on s f ro m t he g en er al

r e s p o n s e d i a g r a m, F i g . 2 . 7 a n d t h e a n a l y t i c a l s o l u t i o n s i n S e c t i o n 2 . 4 . 2.

( a) T he r es po n se o f a ny e st ua ry m ay b e l ik en ed t o t he f re e v ib ra t io ns o f a d am p ed s im pl e

h a r mo n i c o s c i l l a t o r . E s t u a r i e s o f t y p e I w i t h α + 2 β < 1 0 a r e u n d e r- d am p ed a n d e l e v at i o n

a mp l it ud es v ar y i n a n o sc il l at or y m an ne r a lo ng t he x- ax is . E st ua ri es o f t yp e I I w it h

α + 2 β ~ 1 0 a re c ri t ic al l y d am pe d a nd p ro du ce m ax im um a mp li fi ca ti o n. E st ua ri es o f t yp e

I II w it h α + 2 β ≫ 1 0 a r e o v er - da m pe d a n d e l ev a t io n s i n cr e as e m o no t o ni c al l y t o wa r ds

t h e h e a d w i t h l i t t l e a m p l i f i c a t i o n a n d l i t t l e s e n s i t i v i t y t o f r i c t i o n a l e f f e c t s . T h e v a l u e o f

1 0 i s a p p r o x i m a t e , t h e c o r r e s p o n d i n g v a l u e o f 4π w a s d e te r mi n ed f o r t h e z e ro - fr i ct i o n

c a s e i n S e c t io n 2 . 4 . 2 ( 2).( b) T he B ri st o l C h a nn el ( I) a nd B ay o f F un dy ( G) b ot h l ie c lo se t o t he d em ar ca ti on l in e,

α + 2 β 1 0 , i n F i g . 2 . 7 , e x p l a i n i n g t h e s e n s i t i v e ‘near-resonant ’ n a t u r e o f t h e r e s p o n s e

o f t h e s e t w o s y s t e m s . T h e s e r e s p o n s e s a r e a c o n s e q u e n c e o f t h e i r p a r t i c u l a r v a l u e s o f α

and β ( f o r t h e M2 c o n s t i t u e n t ) a n d n o t s i m p l y d u e t o t h e i r r e s o n a n t l e n g t h s a s g e n e r a l l y

assumed.

2.5 Linearisation of the quadratic friction term

2.5.1 Single constituent

T h e p r ec e di n g a n al y se s r e qu i re d t h e q u ad r at i c f r ic t io n t e rm f 1U |U | t o b e a pp ro xi -

m a t e d b y a l i n e a r t e r m f 2U . F o r a s i n g l e c o n s t i t u e n t U = U 1cos ωt , e q u a t i n g e n e r g y

d i s s i pa t i o n , p r o p o r ti o n a l t o U 3, o v e r a t i d a l c y c l e r e q u i r e s f 2 = ( 8/ 3π ) f 1U 1.

2.5.2 Two constituents

P r ou d m an (1923 ) a n d J e f f r e y s ( 1970) s ho we d t ha t w he re t wo t id al s tr ea m c on st i-tuents U 1 cos ωt and U 1

0 cos ω0t c o i n c i d e , t h e f r i c t i o n t e r m i s g i v e n b y

f 1ðU 1 cos ωt þ U 01 cos ω0 tÞ U 1 cos ωt þ U 01 cos ω0t : (2:40)

T h e n , f o r s m a l l v a l u e s o f U 01, t h e l i n e a r i s e d f r i c t i o n c o m p o n e n t F ω

a t t h e f r e q u e n c y

of ω i s g i v e n , t o a f i r s t a p p r o x i m a t i o n , b y

F ω ¼ 8 f 1

3π U 21 cos ωt ¼ f 2 U 1 cos ωt (2:41)

w h i l e t h e c o m p o n e n t a t t h e f r e q u e n c y ω0 is

F ω0 ¼ 4

π

f 1 U 1 U 01 cos ω0 t ¼

3

2

f 2 U 01 cos ω0t: (2:42)

38 Tidal dynamics



I n t he c as e o f a s ma ll r es id ua l c ur re nt U 0 c oi nc id en t w it h a l ar ge t id al s tr ea m

U 1 cos ωt , Bowden (1953 ) s ho we d t ha t, w it h U 1≫U 0, t he l in ea r f ri ct io n t er m

a s s o c i a t e d w i t h t h e r e s i d u a l c u r r e n t i s g i v e n b y

F 0 ¼ 4

π

f 1 U 1 U 0 ¼ 3

2

f 2 U 0: (2:43)

H en ce , (2.41), (2,42) a nd ( 2. 43 ) s ho w th at t o a cc ou nt f or t he i nt er ac ti on w it h a

predominant M2 c o n s t i t u e n t i n a l i n e a r i s e d m o d e l , w h e n s i m u l a t i n g s e p a r a t e l y a n y

o th er t id a l c on st it ue nt o r a r es id ua l f lo w, t h e l in ea ri se d f ri ct io na l c oe ff ic ie nt s f R must

be computed according to

f R ¼

3

2

f 2: (2:44)

T h e m a g n i t u d e o f t h e t i d a l s t r e a m s d u e t o M2 i n m i d - l a t i t u d e s i s t y p i c a l l y o f o r d e r

t wo o r t hr ee t im es t he m ag ni tu de o f t he n ex t l ar ge st c on st it ue nt , S2. Hen ce, it is

r ea so na b le , i n s im ul at in g M2 a lo ne , t o n eg le ct t he f ri ct io na l i nt er ac ti on d ue t o

o th er c o ns ti tu en ts a nd u s e (2.41). H ow ev er, i n s im ul at in g a ny o th e r t id a l c on st i-

t ue nt , i t i s a pp ro pri at e t o l in ea ri se t he f ri ct io na l t erm b y r ef er en ce t o th e t id al

v e l o c i t i e s a s s o c i a t e d w i t h M2 b y u s i n g (2.44). H e n c e , r e l a t i v e t o o t h e r t e r m s i n t h e

m o m e n t u m e q u a t i o n , t h e l i n e a r i s e d f r i c t i o n c o e f f i c i e n t f o r o t h e r c o n s t i t u e n t s , ω0 , isa factor 1.5 U M2

=U ω0 l arger. In th e north west shelf seas, S2 a m pl i tu d es a r e

t yp ic al ly 0 .3 3 o f M2, w he re as t he r at io o f t he ir ‘equilibrium’ p o te nt ia ls i s 0 .4 6 ,

( A pp e n di x 1 A ). T hu s, i nc re as in g t he l in ea ri se d S2 f ri ct io n f ac to r b y a f ac to r o f

4 .5 ( re la ti ve t o t ha t f or p ro pa ga ti on o f M2) a pp ea rs t o r ed uc e S2 a m p li t ud e s b y

a b o ut 1 / 3.

G a r r e t t (1972 ) , H u n t e r ( 1975) a n d S a u n d e r s (1977 ) d i s c u s s t h e e x t e n s i o n o f t h e

a b o v e r e s u l t s f o r f l o w i n t w o d i m e n s i o n s .

2.5.3 Triangular cross section

F or t he s yn ch ro n ou s s o lu ti on , S e ct i on 2 . 3. 2, t he c om po ne nt o f f (U |U |/ D) a t t h e

predominant tidal frequency M2 w a s a p p ro x im a te d b y

8

3π

25

16 f

jU jU

D ¼ FU ; (2:45)

i.e. F = 1 .3 3 f U */ D, wh ere 8/3π d er iv es f ro m t he l in ea ri sa ti on o f t he q u ad r at ic

v e l o c i t y t e r m d e s c r i b e d a b o v e . T h e f a c t o r 2 5 / 1 6 i s i n c l u d e d t o w e i g h t t h e f r i c t i o n a l

t e r m c l o s e t o t h e l a r g e s t f l o w w h i c h o c c u r s i n t h e d e e p e s t w a t e r w h e r e , f r o m (2.19)

with F ≫ω, t h e m a x i m u m v e l o c i t y i s 5 / 4 o f t h e c r o s s - s e c t i o n a l m e a n .

2.5 Linearisation of the quadratic friction term 39



2.5.4 Q factor (Garrett and Munk, 1971 )

A na lo gi es b et we en t he p ro p ag at io n o f t id es i n a c ha nn el a nd t he t ra ns mi ss io n o f

e le ct ri ca l e ne rg y i n a n A C c ir cu it w er e m ad e b y Va n Ve en (1947) a nd P ra nd le

(1980) . T h e a na lo gy l in k s t id al e le va ti on s t o v o lt ag e, t id al s tr ea ms t o c ur re nt , b ed

f r ic t io n t o r e si s ta n ce a n d, s o me w ha t l e ss d i re c tl y, i n er t ia l e f fe c ts t o i n d uc t an c e a n d

s u r f a c e a r e a t o c a p a c i t a n c e . T h e a n a l o g y i s d e p e n d e n t o n t h e a s s u m p t i o n t h a t t i d a l

propagation is sensibly linear and lightly damped. Evidence that tidal propagation is

e s s e n t i a l l y l i n e a r i n o c e a n s a n d s h e l f s e a s f o l l o w s f r o m t h e a c c u r a c y o f t i d a l p r e d i c -

t io n t ec hn iq u es , i .e . t he h ar mo ni c m et ho d a nd , i n p ar ti cu la r, t he r es po ns e m et ho d

( M u n k a n d C a r t w r i g h t , 1966).

T h e r e l a t i v e i n f l u e n c e o f t h e f r i c t i o n t e r m i n (2.11) w a s d i s c u s s e d i n S e c t i o n 2 . 3 .

U si ng t he a bo ve a na lo g y w it h a n A C e le ct ri ca l c ir cu i t, a s uc ci nc t q ua nt if ic at io n

o f f ri ct io n al i nf lu en ce i s p ro vi de d b y t he q ua nt it y Q, k no wn a s t he q ua li ty f ac to r

(Q f ac to r) . A n o sc il la ti ng s ys te m d is s ip a te s 2π / Q o f i ts e ne rg y e ac h c yc le . F or

s y s t e ms n e a r r e s o n an c e ,

Q ¼ ωo

ω2 ω1

(2:46)

i s a m ea su re o f t he s ha rpn es s o f r es on an ce , b as ed o n t he r ati o o f t he n at ur al f re qu enc y,

ω0, t o t he f re qu en cy d if fe re nc e b et we en p oi nt s o n t he r es po ns e c ur ve , ω2 and ω1,

c o r r e s p o n d i n g t o h a l f t h e p o w e r d i s s i p a t i o n o f t h a t a t t h e n a t u r a l f r e q u e n c y .

G od in (1988 ) s ho we d t ha t f or (2.11), the Q f ac to r o f t id al b as in s i s g iv en b y

Q = ω/ F . Thus, the v alue of s = FP = 2π u se d i n re pro du ci ng t he re sp on se s o f

e s tu a ri e s i n S e c ti o n 2 . 4 i s e q ui va le nt t o Q = 1 a nd e mp h as is e s t ha t m os t e s tu a ri es

a r e h i g h l y d i s s i p a t i v e . C o n v e r s e l y , i n F u n d y , f o r w h i c h s = 0 . 2π , Q = 1 0, i n di ca ti ng

a m or e s ha rp ly r es on an t s ys te m . T he se e st ua ri ne v al ue s f or Q m ay b e c om pa re d w it h

t he v al ue o f Q ≈ 1 7 c om pu te d f or t he N or th A tl an ti c b y G ar re tt a nd G re en be rg

(1977).In ‘spinning-up’ a n um er ic al s im ul at io n o f t id al p ro pa ga ti on s ta rt in g f ro m

s t il l -w a te r c o nd i ti o n s, i t s i mi l ar l y f o ll o ws f r om (2.11) t h a t c y cl i ca l c o n ve r ge n ce o f

t i d a l a m p l i t u d e s i s a p p r o a c h e d a s y m p t o t i c a l l y a t a r a t e ( 1 – exp(− Ft )).

2.6 Higher harmonics and residuals

T h e a b ov e t h eo r ie s p r o v id e r o bu s t d e sc r ip t io n s o f t h e f i rs t -o r de r e s t u ar i ne r e sp o n se s

f or t he p ri ma ry t id al c on st it ue n ts . H o we ve r, s ub s eq ue nt c ha pt er s e mp h as is e t hel on ge r- te rm i mp or ta nc e t o b o th m ix in g p ro ce ss es a nd s ed im en t d y na mi cs o f s ee -

mingly ‘second-order ’ e f fe ct s, n am el y h ig he r- or de r a nd r es id u al t id es a lo n gs id e

v er ti ca l, l at er al a nd h ig h- fr eq ue nc y v ar ia bi li ty i n c ur re nt s a nd s al in it y. W hi le

t he f ir st -o rd e r e ff ec ts c an b e a cc u ra te ly m od el le d, n um er ic al s im ul at io n o f t he se

40 Tidal dynamics



‘second-order ’ e f f e c t s r e q u i r e s i n c r e a s i n g l y f i n e r e s o l u t i o n i n b o t h s p a c e a n d t i m e .

T h u s , i r o n i c a l l y , d e s p i t e t h e r a p i d g r o w t h i n c o m p u t e r p o w e r s i n c e t h e f i r s t s i m p l e ,

but successful, numerical tidal models, limitations in computing power remain an

o bs ta cl e t o p ro gr es s. H er e, w e e xp la in h ow t he se h ig he r h ar mo ni c a nd r es id ua l

t e r m s a r e g e n e r a t e d .

2.6.1 Trigonometry

A lt ho ug h t he t id al f orc in g a t th e m ou th o f a n e st ua ry i s p ri ma ri ly a t th e s em i-

d i u rn a l o r d i u rn a l f r eq u e nc i es , t h e n o n- l in e ar t e rm s i n (2.1) t o ( 2 . 5 ) a l m o s t a l w a y s

produce significant higher harmonics in shallow water (Aubrey and Speer, 1985).

T h i s p r o c e s s c a n b e u n d e r s t o o d f r o m s i m p l e t r i g o n o m e t r y i n w h i c h

U 1 cosðω1tÞ U 2 cosðω2tÞ ¼U 1U 2

2 cosðω1 ω2Þt þ cosðω1 þ ω2Þtð Þ:

(2:47)

T h us , t er ms i nv o lv in g a p ro du ct o f t he p re do mi n an t M2 c o n s ti t ue n t g e ne r at e b o th

M4 a n d Z0 c o n s t i t u e n t s . S i m i l a r l y , w h e n e v e r t w o l a r g e c o n s t i t u e n t s a r e p r e s e n t ( e . g .

M2 an d S2) , t he s am e m ec ha n is ms g en e ra te c o ns ti tu e nt s a t t he ir s um f re q ue nc y

(ω1 + ω2) = ωMS4a n d t h e d i f f e r e n c e f r e q u e n c y (ω2 – ω1) = ωMSf

, i . e . q u a r t e r - d i u r n a l

a nd f or tn ig ht ly p er io ds , o r f or ( M2 + N2) , q u ar te r- di ur na l a nd m on th ly p e ri o ds .

T he se r es id ua ls a nd h ig h er h ar mo n ic s m an if es t t he ms el ve s v ia f ea tu re s s uc h a s

a s y m m e t r y b e t w e e n e b b a n d f l o o d f l o w s ( p a r t i c u l a r l y M4) , o c c a s i o n a l d o u b l e h i g h

w a t e r s a n d t i d a l p u m p i n g ( i n w h i c h a n e s t u a r y e x c h a n g e s w a t e r o v e r a n e a p – spring

c y cl e c o mp l ic a ti n g m a ss - ba l an c e c a lc u la t io n s b a s ed o n o b s er v at i on s m a d e o v e r a

s in g le s em i- di u rn a l p er io d) . I n f or mu la ti ng m od e ls t o e x am in e t he se r es id ua l a nd

h i g h e r h a r m o n i c s , i t s h o u l d b e r e c o g n i s e d t h a t t h e s c a l i n g a r g u m e n t s u s e d t o d e r i v e

(2.11) , b a s e d o n t h e p r e d o m i n a n t c o n s t i t u e n t , m a y b e i n v a l i d a t e d . I n p a r t i c u l a r , f o r

h i gh e r h a rm o n ic s , t h e w a ve l en g th λ a ss oc ia te d w it h t he v ar ia bi li ty o f U and ς m ay b e

d e t e r mi n e d b y b a t h y m et r y ( Z i m m er m a n , 1978).

2.6.2 Non-linear terms in the tidal equations

A c on ve ni en t d ev ic e t o i ll us tr at e t he n at ur e o f t he se n on -l in ea ri ti es i s t o r ew ri te (2.1)

a n d ( 2 . 6 ) i n t e r m s o f m a s s t r a n s p o r t s . T h e s e e q u a t i o n s t h e n t a k e t h e f o l l o w i n g f o r m

(Prandle, 1978):

@

@ tQx þ

@

@ X

Q2x

H þ

@

@ X

QxQy

H þ Hg

@&

@ X þ g

f QxðQ2x þ Q2

yÞ1=2

H 2 ΩQy ¼ 0

(2:48)

2.6 Higher harmonics and residuals 41



@

@ t& þ

@

@ X Qx þ

@

@ Y Qy ¼ 0; (2:49)

where Qx = UH , Qy = VH and H = D + ς .

Since (2.49) i s l in ea r, i t i s s uf fi ci en t t o c on s id er o n ly (2.48). B y r es tr ic ti ng t he

a na ly si s t o t he c on si de ra ti on o f t he p ro p ag at io n o f a s in gl e t id al c on s ti tu e nt , t he

f o ll o wi n g a s su m pt i on s m a y b e i n tr o d uc e d:

U ¼ U 0 þ U 1 cosðωt þ θ Þ; V ¼ V 0 þ V 1 cosðωt þ ψ Þ and

& ¼ & 0 þ & 1 cosðωtÞ ;(2:50)

w h e r e t h e p a r a m e t e r s U 0, V 0 and ς 0 d e n o t e r e s i d u a l s . I n t h e f o l l o w i n g a n a l y s i s , i t i s

a l s o a s s u m e d t h a t

U 055U 1; V 055V 1 and & 055& 1: (2:51)

Inertial term

Incorporating (2.50) a n d i n t e g r a t i n g t h e f i r s t t e r m i n (2.48) o v e r a t i d a l c y c l e g i v e s a

r e s i d u a l f l o w :

Q0 ¼ U 0D þ 0:5 U 1& 1 cos θ : (2:52)

T h i s s h o w s t h a t , e x c e p t i n t h e c a s e o f θ = −π / 2 f o r a s ta nd in g w av e, t he r es id u al f lo wc o mp r is e s b o t h a t e rm a s so c ia t ed w i th n e t r e s i du a l c u rr e nt a n d a s e co n d t e r m k n o w n

a s t h e S t o k e s’ t r a n s p o r t . I n a c l o s e d e s t u a r y w i t h n e g l i g i b l e r i v e r f l o w , a c u r r e n t , U 0,

m us t b al an ce t he S to k es’ d ri ft . In t he c as e o f a p ur el y p ro gr es si ve w av e, θ = 0,

U 0 = −05U 1ς 1/ D, i . e . s e a w a r d s .

Convective term

B y e x p a n d i n g t h e s e c o n d , c o n v e c t i v e , t e r m i n (2.48), i t m a y b e s h o w n t h a t , t o a f i r s t

approximation:

@

@ X

Q2x

H

@

@ X

DU 212

cos 2ωt þ 1ð Þ: (2:53)

T h us , t h is c o nv e ct i ve t e rm a s so c ia t ed w i th t h e p r ed o mi n an t c o n st i tu e nt f r eq u en c y,

ω, g en e ra te s b o th a r es id u al ( st ea dy c o mp on en t) a nd a c on st it ue nt w it h f re qu e nc y

2ω, i . e . M2 g e n e r a t e s Z0 a n d M4.

S i m i l a r l y , e x p a n d i n g t h e t h i r d , c o n v e c t i v e , t e r m i n (2.48),

@

@ Y

QxQy

H

@

@ Y

DU 1V 1

2

cos 2ωt þ 1ð Þ (2:54)

w i t h s i m i l a r r e s u l t s t o (2.53).

42 Tidal dynamics



S i n ce , f r om (2.13), t id al c u rr en t a mp li tu de s a dj us t r ap id ly t o c ha n ge s i n b at hy -

m e tr y, b o th (2.53) a nd ( 2. 54 ) e mp ha si se h ow b at hy me tr ic c ha ng es c an g e ne ra te

pronounced non-linearities in tidal propagation.

Surface gradient E x p a n d i n g t h e f o u r t h t e r m i n (2.48) p r o d u c e s t w o d i s t i n c t , r e s i d u a l t e r m s :

Hg @&

@ X gðD þ & 0Þ

@& 0@ X

þ1

2 g& 1

@& 1@ X

: (2:55)

T he se t er ms a re n o t o bt ai ne d e x pl ic it ly i n t he c or re sp o nd in g a na ly si s o f r es id ua l

t e r m s a s s o c i a t e d w i t h (2.7). I n t h e p r e s e n t c a s e , w h e r e t h e e q u a t i o n i s i n t r a n s p o r t s ,

t he t er ms m a y b e c on s id er ed t o r ep re se nt t he n on -l in ea ri ti es d ue t o s ha ll ow -w at er

e ff ec ts i n t he c on ti nu it y e q u a ti o n ( 2 .3 ). T he f ir st o f t he t wo t erm s r ep re se nt s t hev ar ia ti on i n m sl . N ih ou l a nd R on da y (1975 ) d es cr ib ed t he s ec on d t er m a s ‘tidal

r a d i a ti o n s t r e s s’ – a n al o go u s t o ‘set-up’ i n w in d w av es . In a n a pp lic at io n i n t he

s ou th er n N or th S ea , t he y f ou nd t ha t t he f ir st a nd s ec on d t er ms w er e o f t he s am e

o r de r o f m a gn i tu d e .

Quadratic friction

T h e m od ul us i n t he f ri ct io n t er m p re ve n ts s im p le t ri go n om et ri c e xp an si o n, b u t i t

c a n b e s h o w n ( C a r t w r i g h t , 1968) t h a t

U 2 sin ωtj sin ωtj ¼ 8

3π

U 2 ðsin ωt ð1=5Þ sin3ωt ð1=35Þ sin5ωt . . . :Þ:

(2:56)

T hu s, t he q ua dr at ic f ri ct io n t er m g en er at es o dd h ar mo ni cs ( i. e. M6, M10 e tc f ro m M2).

H o w e ve r, a p p ro x i ma t i ng 1 / (ς + D) i n t he q ua dr at ic b ed s tre ss t erm b y ( 1 − ς / D)/ D,

w e s e e t h a t t h e c o m b i n a t i o n o f t h e ς / D t e r m w i t h t h e M2 c u r re n t i n (2.56) c a n a l sol e a d t o s i g n i f i c a n t c o n t r i b u t i o n s a t t h e f r e q u e n c y M4 f r o m t h e f r i c t i o n t e r m .

Coriolis

W h i l e t h e C o r i o l i s t e r m i s l i n e a r a n d d o e s n o t g e n e r a t e r e s i d u a l f l o w s , i t c a n p l a y a

m a j o r r o l e i n c o n f i g u r i n g t h e r e s i d u a l s p r o d u c e d b y t h e n o n - l i n e a r t e r m s .

2.6.3 Ebb –

flood asymmetry F ri ed ri ch s a n d A u br ey (1988 ) h av e s ho wn h ow e bb a nd f lo od a sy m me tr ie s g en er at e

n e t r e s i d u a l v e l o c i t i e s a n d a s s o c i a t e d n e t d i f f e r e n t i a l e r o s i o n p o t e n t i a l s . T i d a l r e c t i -

f i ca t io n a s so c i at e d w i th t h e p h a se - lo c k ed M2 a n d M4 c o ns t it u en t s i s o f p a rt i cu l ar

interest.

2.6 Higher harmonics and residuals 43



S e d i m e n t e r o s i o n a n d d e p o s i t i o n a r e r e l a t e d m o r e d i r e c t l y t o n e a r - b e d v e l o c i t i e s

t h a n t o t h e m a g n i t u d e a n d d i r e c t i o n o f ( d e p t h - i n t e g r a t e d ) f l o w s . H e n c e , t h e r e l e v a n t

parameter generating non-linearities is current rather than flow. In many estuaries,

t h e c r o s s - s e c t i o n a l a r e a a t l o w w a t e r i s a s m a l l f r a c t i o n o f t h a t a t h i g h w a t e r . T h u s ,

propagation of an oscillatory flow constituent introduces major non-linearities incurrents.

A s su m in g a t r ia n g ul a r c r o s s s e c t io n w i th c o ns t an t s i de s l op e s, t a n α, c on t in u it y o f

a s i n u s o i d a l n e t e b b a n d f l o o d f l o w r e q u i r e s

U 1ðtÞA ¼U 1ðtÞ & 1ðtÞ þ D½ 2

tan α þ U 2ðtÞ þ U 0ðtÞ½ A; (2:57)

w he re t he c ro ss -s ec ti on al a re a a t m ea n w at er l ev el A = D2/tan α. T he c ur re nt s U 2and U 0 a re t h e f i rs t h i gh e r- h ar m on i c a n d r e si d u al c u rr e nt c o mp o n en t s r e qu i re d t o

balance the oscillatory flows associated with the predominant constituent given by

& 1ðtÞ ¼ & 1 cosðωtÞ and U 1ðtÞ ¼ U 1 cosðωt þ θ Þ: (2:58)

T h e n r e t a i n i n g o n l y t e r m s o f O(a) , w h e r e a = ς 1*/ D, w e o b t a i n

U 2ðtÞ ¼ U 1a cosð2ωt þ θ Þ and U 0 ¼ U 1 a cosðθ Þ: (2:59)

E q ua t io n ( 2 .5 9 ) i nd ic at es t ha t a n et d ow ns tr ea m c ur re nt a cc om pa ni es t he p ro -

pagation of the primary tidal constituent. In shallow, macro-tidal estuaries, these

t w o t e r m s a r e l i k e l y t o b e t h e m o s t s i g n i f i c a n t n o n - l i n e a r c u r r e n t c o m p o n e n t s .

2.7 Surge – tide interactions

D e t a i l e d d e s c r i p t i o n s o f t h e g e n e r a t i o n a n d p r o p a g a t i o n o f s t o r m s u r g e s a r e b e y o n d

t he p re se nt s co pe , s ee H ea ps (1967 , 1983) f or f ur th e r i nf or ma ti on . H o we ve r, t hef ol lo wi ng e xa mp le o f s ur ge – t id e i nt er ac ti o n i s i n cl ud ed t o i ll us tr at e t he p o te nt ia l

m a g n i t u d e a n d c o m p l e x i t y o f i n t e r a c t i o n s w h e n t h e c o m p o n e n t t e r m s a r e o f s i m i l a r

m a g n i t u d e a n d ‘ period’.

F lo o di ng o ft en i nv ol ve s n ot o nl y l ar ge b u t ‘ peculiar ’ s ur ge s. T he s ur ge t h re at

t o L on do n a ri se s w he n t he p ea k o f a l ar ge s to rm s ur ge s c oi nc id es w it h t he p ea k

o f h ig h w at er o n a s pr in g t id e. H ow ev er, m ax im um s ur ge p ea ks , d ef in ed a s t he

d i ff e re n ce b e t we e n o b se r ve d a n d ( t id a ll y ) p r ed i ct e d w a te r l e ve l s, i n va r ia b ly o c cu r

o n t he r is in g t id e , a f ew h o ur s b e fo re f or ec as te d h ig h w at er. H ow ev er, r el ia nc e o nt hi s s ta ti st ic al r el at io n sh ip , a lb e it r ob u st , s ee m ed t oo p re ca ri ou s a nd t he T ha me s

F l o o d B a r r i e r w a s c o n s t r u c t e d t o p r o t e c t L o n d o n .

F ig u re 2 .8 s ho ws r es ul ts f ro m a n um er ic al s ol ut io n o f s ur ge – t id e i nt er ac ti on i n t he

T h am es f or t he f lo od in g o f 1 97 0 ( Pr an dl e a nd Wo lf , 1978) . A c o nc e p tu a l d i vi s io n

44 Tidal dynamics



i nt o s ep ar at e c om po n en ts o f i nt er ac ti on w as u s ed t o l in k s im ul ta ne o us ‘ parallel’

s im ul at io ns o f t id e a nd s urg e. D yn am ic al c ou pl in g w as i nt ro du ce d v ia t he n on -

l in ea r t er ms . T hu s, f or e xa mp le U S+U T | U S+U T | i n t he q ua dr at ic f ri ct io n t er mw as r ep re se nt ed b y t he c o mp on en t U S | U S+U T | i n t he s urg e m od el ( su bs cri pt S ) a nd

U T | U S+U T | i n t h e t i d a l m o d e l ( s u b s c r i p t T ) .

U s i n g t h i s ‘ parallel’ m o d e l a p p r o a c h , P r a n d l e a n d W o l f ( 1978) s h o w e d h o w t h i s

s y s t e m a t i c o c c u r r e n c e o f s u r g e p e a k s o n t h e r i s i n g t i d e i s d u e t o t h e a d v a n c e o f t h e

t id al p ha se a s a r es ul t o f t he d is pl ac em en t o f t he M2 a m ph id ro m ic s ys te m i n t he

a d j a c e n t s o u t h e r n N o r t h S e a ( F i g . 1 A . 2) . T h i s d i s p l a c e m e n t i s c a u s e d b y e n h a n c e d

d ep th s d ue t o p os it iv e s ur ge l ev el s t hr ou gh ou t t he N or th S ea . T he s ub se qu en t

r ed uc ti on i n s ur ge a mp li tu de i mm ed ia te ly b ef or e t he h ig h es t w at er l ev el s ho wn i nF ig . 2 .8 i s m ai nl y d ue t o e nh an ce d f ri ct io na l d is si pa ti on . T hi s e nh an ce me nt i s

c o n c e n t r a t e d i n s p a c e a n d t i m e w i t h i n a s h a l l o w c o a s t a l r e g i o n i n t h e o u t e r T h a m e s

a pp ro ac he s a nd o cc ur s w he n m ax im um v al ue s o f b ot h s ur ge a nd t id al c ur re nt s a re

aligned concurrently.

F i g . 2 . 8 . S u r g e - t i d e i n t e r a c t i o n a t T o w e r P i e r i n t h e T h a m e s . S surge, T tide, S 0 andT 0 c o mp o n en t s w i th i n te r ac t i on t e rm s c r os s l i nk e d.

2.7 Surge – tide interactions 45



2.8 Summary of results and guidelines for application

T he p ro pa ga ti on o f t id es g en er at ed i n o ce an b as in i nt o e st ua ri es i s e xa mi ne d,

i ll us tr at in g t he v ar ia ti on s i n e le va ti on a nd c ur re nt r es po ns es . T he c on tr ol li ng

m ec ha ni sm s a re d es cr ib ed , e xp la in in g h o w t he s em i- di ur na l a nd d iu rn al o ce anc o n s t it u e n t s p r o d u c e h i g h e r- h a r m o ni c a n d r e s i d ua l c o m p o n en t s w i t h i n e s t u a r i e s .

T h e l e a d i n g q u e s t i o n s a r e :

How do tides in estuaries respond to shape, length, friction factor and river flow?

Why are some tidal constituents amplified yet others reduced and why does this vary

from one estuary to another?

T he se q ue st io ns a re a dd re ss ed b y s yn th es is in g t he s ys te m d yn am ic s w it hi n

s i m p l i f i e d e q u a t i o n s . T h i s i n v o l v e s l i n e a r i s a t i o n o f f i r s t - o r d e r t e r m s a n d o m i s s i o nof ‘s e c o n d - o r d e r ’ t er ms . O mi ss io n s et s l im it s o f a pp li ca bi li ty o f t he d er iv ed

a n a l y t i c a l s o l u t i o n s t o e s t u a r i n e r e g i o n s w h e r e p r o c e s s e s a r e w i t h i n r e l a t e d s c a l i n g

bound s. Simil arly, linear isati on only remai ns valid withi n restri cted param eter

r a n g e s .

I n m os t e st ua ri es , l at er al v ar ia ti on s i n s ur fa ce e le va ti on s a re r es tr ic te d b y t he l ar ge

l e n g t h t o b r e a d t h r a t i o ; h e n c e , f o r ‘first-order ’ r e s p o n s e s , i t i s s u f f i c i e n t t o c o n c e n -

t ra te o n a xi al v a ri at io n s. A xi al s ea l ev el g ra di en ts c on st it ut e t he e ff ec ti ve d ri vi ng

f o r c e f o r t i d a l p r o p a g a t i o n , a n d c r o s s - s e c t i o n a l l y a v e r a g e d s o l u t i o n s a r e a p p r o p r i a t e

t o d e sc r ib e t i da l e l ev a ti o ns a m pl i tu d es , ς * . B y c o nt ra st , v al ue s o f t he t id al c ur re nt

amplitude, U * , a re s en si ti ve t o l oc al is ed v ar ia ti on s i n b ot h d ep th a nd t he b ed f ri ct io n

c o e f f i c i e n t . H e n c e , v a l u e s o f U * v a r y s i g n i f i c a n t l y , n o t o n l y a x i a l l y b u t t r a n s v e r s e l y

a n d v e r t i ca l l y, C h a p t e r 3 e x p l or e s t h es e v a ri a ti o n s m o re f u ll y.

P r o c e e d i n g f r o m (2.1) t o ( 2 . 1 1 ) , t h e c o n d i t i o n s n e c e s s a r y t o r e d u c e t h e f u l l y 3 D

n on -l in ea r e qu at io ns t o a 1 D l in ea ri se d f or m a re d es cr ib ed . T h e f oc us i s o n t id al ly

d om in at ed e st ua ri es , i .e . m es o - a nd m ac ro -t id al e st ua ri es w h er e ς * at the mou the xc ee ds 1 m . I n s uc h e st ua ri es , t he p ri nc ip al l un ar s em i- di ur na l t id al c on st it ue nt ,

M2, g en e ra ll y p re d om in at es , i .e . h a s a n e le va ti on a mp li tu de a t t he m ou th g re at er

t ha n t he s um o f a ll o th er c on st it ue nt s. T hi s c ha ra ct er is ti c e na bl es t he r el ev an t

e qu at io ns t o b e l in ea ri se d d ir ec tl y, i n t er ms o f t hi s M2 c o n st i t u e nt . R e l a t ed d e t a i ls

o f h ow t he q ua dr at ic b ed f ri ct io n t er m i s l in ea ri se d a re d es cr ib ed i n S e ct i on 2 . 5.

S i mi l ar l y, f o r t h e a x ia l c o nv e ct i ve t e rm U ∂U / ∂ X , S e c ti o n 2 . 6 i l l us t ra t es h o w s u c h

a p r o d u c t o f v e l o c i t i e s a t o n e f r e q u e n c y ω p r o du c es c o n st i tu e nt s a t z e ro f r eq u e nc y

(Zo, residual) and 2ω ( or M4 for ω = M2) . T he s ig ni fic an ce o f t he v ar ia ti oni n c r o s s - s e c t i o n a l a r e a s b e t w e e n h i g h a n d l o w w a t e r s i n t h e g e n e r a t i o n o f b o t h M 4

a n d Z0 c o n s t i t u e n t s i s d e s c r i b e d i n S e c t i o n 2 . 6 . 3.

E q u at i on ( 2 .1 3 ) p r ov id es a n e xp li ci t s ol ut io n f or U * i n t er ms o f t he e le va tio n

g ra di en t. T h is s ol ut io n s im p li fi es f or t he c on tr as ti ng e x tr em e c as es o f e st ua ri es

46 Tidal dynamics



w hi ch a re ( i) s ha ll ow a nd f ri ct io na ll y d om in at ed ( 2 . 1 4 ) a n d ( i i ) d e e p a n d

f r i c t i o n l e s s ( 2 . 1 5 ) . M a n y e a r l i e r s t u d i e s h a v e f o c u s e d o n t h e l a t t e r . T h i s d e m a r -

c at io n b et we en t he r el at iv e i nf lu en ce s o f b ed f ri ct io n a nd t he t em po ra l a cc el -

e r a t io n ( in er ti a) i s i ll us tr at ed i n F i gs . 2 . 1 a n d 2 . 4. T h e g e ne r al i se d s o lu t io n (2.13)

i nd ic at es h ow t he se t wo t er ms a re o rt ho go na l ( i. e. h av e t id al p ha se s s ep ar at ed by 90°) and together balance the sea level gradient. F i gu r e 2 . 2 i l l us t ra t es h o w t h is

balance holds for M2 b u t not for M4 or M6 w h er e t he e ff ec ti ve d ri vi ng f or ce s

i s n o l on ge r s ur fa ce g ra di en t b u t n on -l in ea r t er ms d er iv ed f ro m t h e p ro pa ga ti on

of M2.

E x pl ic it s ol u ti o ns f o r U * in terms of ς * can be ob tain ed by invokin g the

‘synchronous’ e s tu ar y a pp ro xi ma ti on , i .e . w he re s ur fa ce g ra di en ts d ue t o a xi al

v a r i a t i o n s i n t i d a l p h a s e a r e m u c h g r e a t e r t h a n t h o s e d u e t o v a r i a t i o n s i n a m p l i t u d e .

F o r t h is a p pr o xi m at i on , F ig . 2 .3 i n di ca te s h ow i n m os t e st ua ri es , t id al v el oc it ie sg e n e r a l l y r a n g e f r o m 0 . 5 t o 1 . 0 m s−1. S i m i l a r l y , f o r t h e s e ‘synchronous’ solutions,

t h e r a t i o o f t h e f r i c t i o n t o i n e r t i a l t e r m s a p p r o x i m a t e s 1 0 ς *: D, w h e r e D i s t h e w a t e r

d ep th . F ur the r f ea tu re s o f t hi s s ol ut ion , n am el y p ha se l ag b et we en ς * and U *,

e st ua ri ne l en gt h a nd b at hy m et ry, a re e xp lo re d i n C ha pt er 6 while C ha pt er 7 describes

r e l a t e d s o r t i n g a n d t r a p p i n g o f s e d i m e n t s .

A n a ly t ic a l s o lu t io n s f o r w h ol e -e s tu a ry t i da l r e sp o n se s r e qu i re s o m e f u nc t io n al

s p e c i f i c a t i o n o f g e o m e t r y . T w o c a s e s a r e e x a m i n e d w i t h a x i a l v a r i a t i o n s i n b r e a d t h s

a n d d e p t h s d e s c r i b e d b y ( i ) X n and X m (the synchronous approximation is equivalent

to m = n = 0.8) and (ii) exp (α X ) a nd e xp ( β X ) . T h e f ir st o f t he se b at hy me tr ic a pp -

r o x i m a t i o n s p r o v i d e s a g e n e r a l i s e d r e s p o n s e d i a g r a m (F i g . 2 . 5) , s h o w i n g a m p l i t u d e

a nd p ha se v ar ia ti on s a lo ng a ny s uc h e st ua ry a s a f un ct io n o f t he f un nel lin g p ar am ete r

ν = (n + 1 )/ (2 −m) . M a xi m um a m pl i fi ca t io n o c cu rs f o r ν = 1 a nd n od al l en gt hs , a n a -

l o g o u s t o ‘quarter-wave length amplification’ i n a f r i c t i o n l e s s p r i s m a t i c c h a n n e l , a r e

indicated.

T h e d i f f e r e n c e s i n t h e s e r e s p o n s e s f o r a r a n g e o f t i d a l c o n s t i t u e n t s a n d v a r i a t i o n si n t he f ri ct io n c oe ff ic ie n t a re i ll us tr at ed i n F ig . 2 .6 fo r t he B ay o f F un dy. T hi s

e x am p le i l lu s tr a te s h o w, t y p ic a ll y, d i ur n al s s h o w l i tt l e a m pl i fi c at i on , w h e re a s h i gh e r

h a rm o ni c s a r e o f te n g r ea t ly a m p li f ie d .

T he c o rr es p on di ng s o lu t io n s f or t h e s ec o nd c as e o f e xp o ne nt ia l r ep re se nt at io ns o f

bathymetry indicate (F ig . 2 .7), h ow fo r α + 2 β < 4π, e s tu a ri n e r e sp o n se s r e se m bl e

t h a t o f a n u n d e r - d a m p e d o s c i l l a t o r w h i l e f o r α + 2 β > 4π e s t u a r i e s a r e o v e r - d a m p e d

a n d w i l l s h o w l i t t l e c h a n g e i n t i d a l e l e v a t i o n a m p l i t u d e .

T he se g en er al is ed r es po ns e F ra me wo rk s v ar y a cc or di ng t o t he v al ue o f t hel i ne a ri s ed f r ic t io n f a ct o r, i l lu s t ra t in g h o w b a th y m et r y a n d f r ic t io n t o ge t he r d e te r-

m i n e t h e n a t u r e o f t i d a l p r o p a g a t i o n i n e s t u a r i e s . M o r e o v e r , t h r o u g h t h e a d o p t i o n o f

d im en s io n le ss p a ra me te rs , t he F ra me wo rk s c an e xp la in t he t id al r es po n se a t a ny

point, for all tidal constituents, along any funnel-shaped estuary.




F i g . 2 . 8 i l l u s t r a t e s s u r g e – t i d e i n t e r a c t i o n i n t h e T h a m e s w h e r e t h e s u r g e a n d t i d e

c om po ne nt s w er e o f r ou gh ly e qu al m ag ni tu de a nd ‘ periodicities’ – effectively

precluding linearisation against the M2 t i d a l c o n s t i t u e n t . T h i s e x a m p l e e m p h a s i s e s

t h e m a g n i t u d e a n d c o m p l e x i t y o f i n t e r a c t i o n s i n s u c h c a s e s .

References

A u b r e y , D . C . a n d S p e e r , P . E . , 1 9 8 5 . A s t u d y o f n o n l i n e a r t i d a l p r o p a g a t i o n i n s h a l l o w i n l e t / e s t u a r i n e s y s t e m P a r t I : O b s e r v a t i o n s . Estuarine, Coastal and Shelf Science, 21 (2),185 – 205.

B o w d e n , K . F . , 1 9 5 3 . N o t e o n w i n d d r i f t i n a c h a n n e l i n t h e p r e s e n c e o f t i d a l c u r r e n t s . Proceedings of the Royal Society of London, A, 219 , 4 2 6 – 446.

C a r t w r i g h t , D . E . , 1 9 6 8 . A u n i f i e d a n a l y s i s o f t i d e s a n d s u r g e s r o u n d n o r t h a n d e a s t B r i t a i n .

Philosophical Transactions of the Royal Society of London, A, 263 ( 1 1 3 4 ) , 1 –

55.D o r r e s t e i n , R . , 1 9 6 1 . Amplification of Long Waves in Bays. E n g i n e e r i n g p r o g r e s s a t U n i v e rs i t y o f F l o r i da , G a i n es v i l l e , 15 (12).

F r i e d r i c h s , C . T . a n d A u b r e y , D . G . 1 9 8 8 . N o n - l i n e a r d i s t o r t i o n i n s h a l l o ww e l l - mi x e d e s t u a r i e s ; a s y n t h e s i s . Estuarine, Coastal and Shelf Science, 27, 5 2 1 – 545.

F r ie d ri c hs , C . T. a n d A u br e y, D . G. , 1 9 94 . Ti d al p r op a ga t io n i n s t ro n gl y c o nv e rg e nt c h an n el s . Journal of Geophysical Research, 99 ( C 2 ) , 3 3 2 1 – 3336.

G a r r e t t , C . , 1 9 7 2 . T i d a l r e s o n a n c e i n t h e B a y o f F u n d y . Nature, 238, 4 4 1 – 443.G a r r e t t , C . J . R . a n d G r e e n b e r g , D . A . , 1 9 7 7 . P r e d i c t i n g c h a n g e s i n t i d a l r e g i m e : t h e o p e n

boundary problem. Journal of Physical Oceanography, 7, 1 7 1 – 181.

G a r r e t t , C . J . R . a n d M u n k , W . H . , 1 9 7 1 . T h e a g e o f t h e t i d e s a n d t h e Q o f t h e o c e a n s . DeepSea Research, 18, 4 9 3 – 503.G i l l , A . E . , 1 9 8 2 . Atmosphere-Ocean Dynamics. A c a d e m i c P r e s s , N e w Y o r k .G o d i n , G . , 1 9 8 8 . T h e r e s o n a n t p e r i o d o f t h e B a y o f F u n d y . Continental Shelf Research, 8

( 8 ), 1 0 05 – 1010.H e a p s , N . S . , 1 9 6 7 . S t o r m s u r g e s . I n : B a r n e s , H . ( e d . ) , Oceanography and Marine Biology

Annual Review, V o l . 5 . A l l e n a n d U n w i n , L o n d o n , p p . 1 1 – 47.H e a p s , N . S . , 1 9 8 3 . S t o r m s u r g e s , 1 9 6 7 – 1982. Geophysical Journal of the Royal

Astronomical Society, 74, 3 3 1 – 376.H u nt , J . N. , 1 9 64 . T id a l o s ci l la t io n s i n e s tu a ri e s. Geophysical Journal of the Royal

Astronomical Society, 8, 4 4 0 –

455.H u n t e r , J . R . 1 9 7 5 . A n o t e o n q u a d r a t i c f r i c t i o n i n t h e p r e s e n c e o f t i d e s . Estuarine, Coastal

Marine Science, 3, 4 7 3 – 475.I a nn i el l o , J . P. , 1 9 77 . T id a ll y -i n du c ed r e si d u al c u rr e nt s i n e s tu a ri e s o f c o ns t an t b r ea d t h a n d

depth. Journal of Marine Research, 35 ( 4 ) , 7 5 5 – 786.I a n n i e l l o , J . P . , 1 9 7 9 . T i d a l l y - i n d u c e d c u r r e n t s i n e s t u a r i e s o f v a r i a b l e b r e a d t h a n d d e p t h .

Journal of Physical Oceanography, 9 ( 5 ) , 9 6 2 – 974.J e f f r e y s , H . , 1 9 7 0 . The Earth, 5 t h e d n . C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e .L a m b , H . , 1 9 3 2 . Hydrodynamics, 6 t h e d n . C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e .L a r o uc h e , P. , K o u t i to n s k y V. C . , C h a nu t , J . - P. , a n d E l - S ab h , M. I . , 1 9 8 7 . L a t e ra l s t r a t if i c at i o n

a n d d y n a m i c b a l a n c e a t t h e M a t a n e t r a n s e c t i n t h e l o w e r S a i n t L a w r e n c e E s t u a r y . Estuarine and Coastal Shelf Science, 24 ( 6 ) , 8 5 9 – 871.

L e Bl o nd , P. M ., 1 9 78 . O n t i da l p r op a ga t io n i n s h al l o w r i ve r s. Journal of Geophysical Research, 83 ( C 9 ), 4 7 17 – 4721.

M u n k , W . H . a n d C a r t w r i g h t , D . E . , 1 9 6 6 . T i d a l s p e c t r o s c o p y a n d p r e d i c t i o n . Philisophical Transactions of Royal Society of London, A, 259, 5 3 3 – 581.

48 Tidal dynamics



Nihoul, J.C.J. and Ronday, F.C., 1975. The influence of the tidal stress on the residualcirculation. Tellus, 27 , 4 8 4 – 489.

P r a n d l e , D . , 1 9 7 8 . R e s i d u a l f l o w s a n d e l e v a t i o n s i n t h e s o u t h e r n N o r t h S e a . Proceedingsof the Royal Society of London, A, 359 ( 1 6 9 7 ) , 1 8 9 – 228.

P r a n d l e , D . , 1 9 8 0 . M o d e l l i n g o f t i d a l b a r r i e r s c h e m e s : a n a n a l y s i s o f t h e

o p en - bo u nd a ry p r ob l em b y r e fe r en c e t o A C c i rc u it t h eo r y. Estuarine and Coastal Marine Science, 11, 53 – 71.P r an d l e, D . , 1 9 85 . C l as s i fi c at i on o f t i da l r e sp o ns e i n e s tu a ri e s f r om c h an n el g e om e tr y.

Geophysical Journal of the Royal Astronomical Society, 80 ( 1 ) , 2 0 9 – 221.P r a n d l e , D . , 2 0 0 3 . R e l a t i o n s h i p b e t w e e n t i d a l d y n a m i c s a n d b a t h y m e t r y i n s t r o n g l y

c o n v e r g e n t e s t u a r i e s . Journal of Physical Oceanography, 33 , 2 7 3 8 – 2750.P r a n d l e , D . , 2 0 0 4 . H o w t i d e s a n d r i v e r f l o w s d e t e r m i n e e s t u a r i n e b a t h y m e t r i e s . Progress in

Oceanography, 61, 1 – 26.P r a n d l e , D . a n d J . W o l f . , 1 9 7 8 . T h e i n t e r a c t i o n o f s u r g e a n d t i d e i n t h e N o r t h S e a a n d R i v e r

Thames. Geophysical Journal of the Royal Astronomical Society, 55 ( 1 ) , 2 0 3 – 216.

P r a n d l e , D . a n d R a h m a n M . , 1 9 8 0 . T i d a l r e s p o n s e i n e s t u a r i e s . Journal of Physical Oceanography, 10 ( 1 0 ) , 1 5 5 2 – 1573.

P r o u d m a n , J . , 1 9 2 3 . R e p o r t o f B r i t i s h A s s o c i a t i o n f o r t h e A d v a n c e m e n t o f S c i e n c e . Report of the Committee to Assist Work on Tides. p p . 2 9 9 – 304.

S a un d er s , P. H ., 1 9 77 . Av e ra g e dr a g i n a n o s ci l la t or y f lo w. Deep Sea Research, 24, 381 – 384.T a y l o r , G . I . , 1 9 2 1 . T i d e s i n t h e B r i s t o l C h a n n e l . Proceedings of the Cambridge

Philosophical Society/Mathematical and Physical Sciences, 20, 3 2 0 – 325.V a n V e e n , J . , 1 9 4 7 . A n a l o g y b e t w e e n t i d e s a n d A C e l e c t r i c i t y . Engineering , 184, 4 9 8 ,

520 – 544.X i u , R . 1 9 8 3 . A s t u d y o f t h e p r o p a g a t i o n o f t i d e w a v e i n a b a s i n w i t h v a r i a b l e c r o s s - s e c t i o n .

First Institute of Oceanography, National Bureau of Oceanography. Q i n g d a o / S h a n d on g , C h i n a.

Z i mme rma n , J . T. F. , 1 9 7 8 . To p o g r ap h i c g e n e ra t i o n o f r e s i du a l c i r c u la t i o n b y o s c i ll a t o ry( t i d a l) c u r re n t s . Geophysical and Astrophysical Fluid Dynamics, 11, 35 – 47.

References 49



3

Currents

3.1 IntroductionThe factors determining the magnitudes of depth-averaged tidal currents were

described in Chapter 2. Here we explore the vertical structure of both tidal- and

wind-driven currents. The structure of density-driven currents is described in

Chapter 4. These structures are incorporated into subsequent theories relating to

salinity intrusion, Chapter 4; sediment dynamics, Chapter 5 and morphological

equilibrium, Chapters 6 and 7.

Models of tidal propagation involve numerical solutions to the momentum and

continuity equations. In shelf seas, given adequate numerical resolution, the accu-

racy of simulations depends primarily on the specification of open-boundary con-

ditions and water depths. Thus, the early 2D (vertically averaged) shelf-sea models

(Heaps, 1969) paid scant attention to the specification of bed-stress coefficients.

By contrast, applications in estuaries and bays often involve extensive calibration

procedures requiring careful adjustment of bed friction coefficients (McDowell and

Prandle, 1972). This predominant influence of frictional dissipation in shallow

macro-tidal estuaries was illustrated in Chapter 2. In the more-recent 3D models,accurate specification of vertical eddy viscosity, E , is similarly essential to repro-

duce vertical current structure and related temperature and salinity distributions.

Validation of estuarine models of tidal propagation is often limited to com-

parisons against tide gauge recordings of water levels. In large estuaries with

appreciable changes in phase and amplitude, accurate simulation of tidal elevations

implies reasonable reproduction of depth-averaged currents (assuming accurate

bathymetry). However, in small estuaries, tidal elevations often show little variation

in either phase or amplitude from the open-boundary specifications, and so suchvalidation offers little guarantee of accurate representation of tidal currents.

In comparison with tidal elevations, currents are characterised by significant

variability both temporally and spatially. Thus, while observations of water levels in

estuaries typically have a (non-tidal component) noise: tidal signal ratio of O (0.2),

50



for currents these components are often of equal magnitudes. In situ current

measurements can be made via mechanical, electro-magnetic or acoustic sensors,

while surface measurements can be made remotely using H.F. Radar. Such mea-

surements are generally more costly, less accurate and less representative than for

elevations. The spatial inhomogeneity of currents complicates the use of adjacent observations in prescribing ‘related constituents’ for tidal analyses of short-term

recordings. Hence, extended observational periods are necessary to accurately

separate constituents, especially at the surface where contributions from wind-

and wave-driven current components are largest.

Since the tidal solutions described in Sections 3.2 and 3.3 omit wind forcing,

density and convective terms, a brief description of their relative magnitudes is

presented. While the following description provides simple scaling analyses, the

presence of significant wind stress or density gradients may radically change themagnitude and vertical distribution of the vertical eddy viscosity coefficient and

thus, interactively, the tidal current structure. Souza and Simpson (1996) provide a

good example of how the vertical structure of tidal current ellipses can be radically

changed by pronounced stratification.

The depth-averaged axial momentum equation may be written as

dU

dt

¼ @ U

@ t þ U

@ U

@ X

þ V @ U

@ Y

¼ g @&

@ X 0:5D

@ ρ

ρ@ X

fU ðU 2 þ V 2Þ1=2

D

f wW2

D þ ΩV ;

(3:1)

where U and V are velocities along axial and transverse horizontal axes X and Y ;

W is the wind speed; ς is surface elevation; ρ density; D water depth; Ω the

Coroilis parameter; f and f W coefficients linked with bed friction and wind drag,

respectively.

3.1.1 Convective term

It was shown in Section 2.2 that the ratio of the inertial to the axial convective term

can be approximated by

@ U

@ t : U

@ U

@ X c : U ; (3:2)

where c = ( gD)1/2

is the wave celerity. Tidal currents only approach ‘critical’celerity, c = U , in confined sections, suggesting that omission of the convective

terms is generally valid. However, non-linear coupling of tidal constituents intro-

duced by this term may often be significant for higher-harmonic constituents, as

described in Section 2.6.

3.1 Introduction 51



The transverse convective term V ∂U / ∂Y can be significant close to coastline

features and in regions of sharp changes in bathymetry (Pingree and Maddock,

1980; Zimmerman, 1978). Prandle and Ryder (1989) present detailed quantitative

analyses of the roles of convective terms in a fine-resolution numerical model

simulation. The spatial signatures associated with these terms have been mapped both by H.F. Radar (Fig. 3.9; Prandle and Player, 1993) and from ADCP current

observations (Geyer and Signell, 1991).

3.1.2 Density gradients

The influence on current structure of axial density gradients and vertical stratifica-

tion associated with saline intrusion is described in Chapter 4. Density stratification

associated with heat exchange at the water surface may be significant when the (bed)frictional boundary layer does not extend through the whole depth, (Appendix 4A).

This is generally confined to the deepest, micro-tidal estuaries.

From (3.1), the ratios of the surface gradients associated with elevation and saline

intrusion are in the ratio

2π &

λ : 0:5D

0:03

LI

or &

D :

0:002 λ

LI

; (3:3)

where LI is the salinity intrusion length and 0.03 ρ is the additional density of sea water.

Hence, in terms of surface elevation, the saline density gradient will only be

important in micro-tidal, deep estuaries. While salinity intrusion can significantly

change the vertical structure of currents, Prandle (2004) confirmed that it has little

impact on tidal levels in estuaries.

3.1.3 Wind forcing

The wind-induced stress at the sea surface may be approximated by (Flather, 1984)

τ w ¼ 0:0013 W 2 nm2: (3:4)

The equivalent bed-stress term for tidal current U is

τ B ¼ 0:0025 ρ U 2: (3:5)

Thus, for wind stress to exceed the bed stress, the wind speed W > 44U .

3.1.4 Approach

In Section 3.2, we assume that flow is confined to the axial direction, X . Then,

specifying a surface gradient, the effects on tidal current structure of the bed

52 Currents



friction coefficient, f , and vertical eddy viscosity, E , are examined. These solutions

are valid in long narrow estuaries, i.e. Ω B≪ c ( B, breadth; Ianniello, 1977). In such

estuaries, little transverse variation can occur in the axial component of surface

gradients and cross-sectional variations in currents correlate directly with depth

variations (Lane et al ., 1997). This section derives the scaling laws that explain thediversity of tidal current structure.

In wider estuaries, flow is more 3-dimensional, and, in Section 3.3, the impacts of

the Earth’s rotation via the Coriolis force are demonstrated. Since the Coriolis force

depends directly on latitude, the acute sensitivity of tidal currents close to the inertial

latitudes (30° for diurnal and 70° for semi-diurnal) is shown.

Section 3.4 examines time-averaged tidal- and wind-driven currents. Whereas

for elevations, these components are generally small in relation to the predominant

oscillatory tidal variations, for currents these components can be comparable,especially during extreme events.

3.2 Tidal current structure – 2D ( X-Z )

In this section, it is assumed that tidal currents are confined to rectilinear flow along

the X -axis and can be determined from the linearised equations of motion. Theories

of the vertical structure of tidal- and wind-driven currents remain largely dependent

on the concept of a vertical eddy viscosity parameter, E . The analytical solutions

derived here (Prandle, 1982a) assume a constant value of E . Prandle (1982b) extended

the present derivation to the case of E varying linearly with height above the bed.

Throughout this chapter, the ‘ bed’ can be regarded as the interface between

the logarithmic layer and the Ekman layer (Bowden, 1978). Precise simulation of

vertical structure closer (1 m) to the bed involves consideration of boundary layer

theory and variations of E with time and depth. Such simulations must ensure

that this logarithmic velocity profile is smoothly matched with the exterior flow(Liu et al ., 2008). Section 3.3 includes comparisons between the present analytical

solutions and the currents derived from detailed numerical simulations in which E

is calculated from a ‘turbulent energy closure model’ described in Appendix 3B.

3.2.1 2D analytical solution

The momentum equation for tidal flow confined to one horizontal dimension,

neglecting vertical acceleration, wind forcing, convective and density terms, can be expressed as

@ U

@ t ¼ g

@&

@ X þ

1

ρ

@

@ Z F z; (3:6)




where F z is the component of frictional stress exerted at level Z by the water above

that level. Expressing the frictional stress in terms of a (constant) vertical eddy

viscosity, E , gives

F z ¼ ρ E

@ U

@ Z : (3:7)

Limiting consideration to a single tidal constituent of frequency, ω, at any position,

we assume

U ðz; tÞ ¼ Re½U ðZ Þeiωt (3:8)

and

ζ ðtÞ ¼ Re½W eiωt

; (3:9)

where U (Z) and W take a complex form to reflect tidal phase variations.

Substituting (3.7), (3.8) and (3.9) into (3.6), we can eliminate the time variation

eiωt to give

i ωU ¼ g @

@ X W þ E

@ 2

@ Z 2U : (3:10)

Equation (3.10) is satisfied by the solution

U ¼ A1ebz þ A2ebz þ C (3:11)

with

b ¼ iω

E

1=2

and C ¼ g

i ω

@

@ X W : (3:12)

Boundary conditionsAt the surface Z = D, the frictional stress F z = 0, i.e.

A1b ebD A2b ebD ¼ 0

or A1 ¼ A2 e2bD: (3:13)

At the bed Z = 0, we assume the stress described by (3.7) is equal to the stress

described by the linearised quadratic friction law, Section 2.5 (Proudman, 1953,

p. 316),

E ðA1b A2bÞ ¼ 8

3π f U U z¼0 ¼

8

3π f U ðA1 þ A2 þ C Þ; (3:14)

where U * represents the depth-averaged tidal amplitude.

54 Currents



Continuity

The total flow is given by

U D ¼ð D

0

U dZ ¼A1

b ebD 1

A2

b ebD 1

þ C D: (3:15)

Velocity profiles

Combining (3.11) to (3.15), we obtain the following solution for the velocity U at

any height Z ,

U ðzÞU

¼ ðebZ

þ ebZ þ2 y

ÞT

þ Q ; (3:16)

where T ¼ ð1 e2 yÞ j 1

y 1

2 e2 y (3:17)

and Q ¼ j ð1 e2 yÞ 1 e2 y½

T (3:18)

j ¼

3 π E b

8 f jU j (3:19)

y ¼ bD: (3:20)

Letting j = J i½ and y = Y i½, we obtain

J ¼ 3π ðE ωÞ1=2

8 f U (3:21)

and Y ¼ ω

E

1=2

D: (3:22)

3.2.2 Comparisons with observations

The velocity profile described by (3.16) is a function of two variables, J and Y.

Y may be interpreted as a depth parameter converted to a dimensionless form by

Ekman scaling (Faller and Kaylor, 1969; Munk et al ., 1970). J is also dimensionlessand reflects the effect of the quadratic bottom stress through the bed-stress coeffi-

cient f and the depth-averaged velocity amplitude U *. It may be shown from (3.16)

that when J is large, the velocity distribution is always uniform through depth.

For large values of Y , the solution (3.16) approaches an asymptote that is a function




of J only. Thus, the full range of vertical tidal current structure is represented by the

parameter ranges 0 < Y < 5 and 0 < J <10.

Figure 3.1 shows, in contour form, the ratio of the velocity at the bed to thedepth-averaged velocity, i.e. |U Z=0/ U *| plotted as a function of Y and J . It also

shows the phase difference Δθ = θ s− θ b between velocities at the surface and bed,

with Δθ < 0 indicating a phase advance at the bed. Pronounced vertical structure

occurs in the range Y >1 and J < 2. This latter point is emphasised by Fig. 3.2

(Prandle, 1982a) where complete velocity profiles are shown for co-ordinate values,

( J , Y) = (a) (0.5,5); (b) (5,5); (c) (0.5,0.5) and (d) (5,0.5).

The profiles described by the above theory correspond in character with mea-

sured profiles for tides in rivers, estuaries and shallow seas (Van Veen, 1938). Thedepth mean velocity occurs at a fractional height ( z = Z / D), 0.25 < z < 0.42 – validating

the common engineering assumption that average velocity occurs close to 0.4 D

above the bed. Velocities measured at z =0.4 D will provide an estimate of the depth

mean value with a maximum error of 4% in amplitude and 1.5° in phase.

0.45

4

3

2

1

01 2 3 4 5

J

y

6 7 8 9 10

0.50.60.7 0.8 0.9

0.95

Fig. 3.1. Surface to bed differences in amplitude and phase of tidal currents as f ( J, Y ). (Left) Velocity at the bed as a fraction of the depth-averaged value. (Right)Phase differences between velocities at the surface and bed (negative valuesindicate phase advance at the bed).

56 Currents



The maximum velocity generally occurs close to the surface. However, for Y > 4

maximum velocity occurs below the surface, and for Y 5 maximum velocity occurs

at mid-depth.

3.2.3 Eddy viscosity formulation, Strouhal number

A fuller interpretation of the above theory requires some formulation for eddy

viscosity. Two commonly assumed expressions in dimensionally consistent form are

E ¼ αU D (3:23)

E ¼ β U 2

ω : (3:24)

Bowden (1953) suggested a formulation similar to (3.23) while Kraav (1969)

suggested a formulation similar to (3.24). Substituting (3.23) into (3.21) and(3.22) we obtain

Y ¼ 8

3π f

J

α: (3:25)

5 –40°

–30°

–20°

–10°

–5°

–1°

4

3

2

1

01 2 3 4 5

J

y

6 7 8 9 10

Fig. 3.1. (cont.)




Similarly, substituting (3.24) into (3.21) gives

J ¼ β 1=2

8=3π f : (3:26)

McDowell (1966) presents values of the phase difference θ between surface and

bed found for oscillatory flows in a laboratory flume, showing that θ increases

continuously as the parameter f S R decreased, where S R = U P / D is the Strouhalnumber. Prandle (1982a) showed that these data correspond to Y =1.7 J , thus

supporting Bowden’s (1953) formulation for eddy viscosity and from (3.25) sug-

gesting a value of α 0.5 f . Using E = α0 | U z =0 | D, Bowden found values of α0 in

the range 0.0025 – 0.0030, thus agreement with the present value for α requires

–20°1.0

0.5

Z / D

0 0.5(a) (b)

(c) (d)

1.0

–10° 0° 10° 20°

θ

θ

–20°1.0

0.5

0 0.5 1.0

–10° 0° 10° 20°

θ

θ

0.5

–20°1.0

0.5

Z / D

0 1.0

–10° 0° 10° 20°θ

θ

U z = 0 / U

–20°1.0

0.5

0 0.5 1.0

–10° 0° 10° 20°θ

θ

U z = 0 / U

Fig. 3.2. Vertical structure of current amplitude and phase as f ( J, Y ). Profiles for ( J , Y ) = (a) (0.5, 5.0), (b) (5.0, 5.0), (c) (0.5, 0.5) and (d) (5.0, 0.5).

58 Currents



|U z = 0 / U | 0.5. Prandle (1982b) showed that analytical solutions of (3.6) require

E >0.5 fU*D.

Bowden’s (1953) formulation is appropriate in most estuaries when the boundary

layer thickness extends to the surface, whereas in deeper water, Kraav’s (1969)

formulation applies. Davies and Furnes (1980) used the latter formulation inmodelling tidal current structure in depths of up to 200 m over the UK continental

shelf area.

By assuming Bowden’s (1953) formulation for eddy viscosity, with α = f , the

vertical structure (3.16) reduces to a function of the single parameter f S R ,

with Y 0.83J 50/ S R 1/2. Figure 3.3 (Prandle, 1982a) shows the resultant ver-

tical structure as a function of S R for the case of f = 0.0025. For small values of

S R , i.e. S R < 50, the current structure is uniform except for a small phase advance

close to the bed. For larger values of S R , the variation in current amplitudeincreases continuously, approaching an asymptote in the region of S R = 1000.

The phase variation also increases with increasing S R but reaches a maximum

difference between surface and bed in the region of S R = 350. Thereafter, the

phase variation decreases with increasing S R with only 1° difference between

surface and bed for S R = 10 000. The phase variation is generally concentrated

close to the bed except when 100 < S R < 1000.

In a typical strongly tidal estuary with an M2 tidal current amplitude U * 1 ms−1,

S R > 1000 for D < 40 m, and thus the vertical structure will tend towards the asymp-

totic solution for large S R shown in Fig. 3.3.

3.3 Tidal current structure – 3D ( X -Y - Z )

The fundamental difference in extending the above theory for uni-directional to fully

3D currents is the inclusion of the Coriolis force in the momentum equation (3.6).

Although for the major tidal constituents in many estuaries, thedepth-integrated effect of this term may be of second order, the (differential)

influence on vertical structure can be highly significant.

3.3.1 Tidal ellipse resolved into clockwise

and anti-clockwise components

In a horizontal plane, tidal current vectors rotate and, relative to a fixed origin,

describe an elliptical path. A common practice is to resolve the ellipse into twocircular motions, one rotating clockwise ( R2, g 2) and the other anti-clockwise

( R1, g 1). Appendix 3A provides a fuller description of this transformation. In vector

notation with X the real axis and Y the orthogonal complex axis, the tidal vector, R,

is given by

3.3 Tidal current structure – 3D ( X-Y-Z ) 59



R ¼ U þ iV : (3:27)

Resolving R into a clockwise component R2 and an anti-clockwise component

R1 (| R1| and | R2| constant),

R ¼ R1 þ R2: (3:28)

1.0

1.14

1.10

1.07

1.03

1.00

0.90

0.80

0.700.64

1.18

1.01

0.8

0.6

Z / D

Z / D

0.4

0.2

0

1.0

0.8

0.6

0.4

0.2

010 100

s

(b)

1000 10 000

10 100s

(a)

1000

–5°

–4°

–3°

–2°

–1°

1°

2°

3°4°

5°6°

7°8°

0°

10 000

Fig. 3.3. Current profile as a function of the Strouhal number, S R = U * P / D.(a) amplitude U (z)/ U mean; (b) phase structure θ ( z ) – θ mean.

60 Currents



3.3.2 3D analytical solution

The linearised equations of motion for flow in two dimensions may be written as

@ U

@ t ΩV ¼ g

@&

@ X

þ 1

ρ

@

@ Z

F zx (3:29)

and

@ V

@ t þ ΩU ¼ g

@&

@ Y þ

1

ρ

@

@ Z F zy; (3:30)

where Ω is the Coriolis parameter and F zx and F zy are the components of F z along X

and Y . These equations can be transformed using (3.27) and (3.28) into the following

equations for the separate rotational components:

anti-clockwise i ðΩ þ ωÞ R1 ¼ G 1 þ @

@ Z E

@

@ Z R1 (3:31)

clockwise i ðΩ ωÞ R2 ¼ G 2 þ @

@ Z E

@

@ Z R2; (3:32)

where G 1 and G 2 are rotational components of the surface gradient.

Comparing (3.31) and (3.32) with (3.10) we see that for E constant through depth,

the equations are analogous. Thus, the solutions will be equivalent with the impor-

tant difference that for the anti-clockwise component

b ! B1 ¼ i ðΩ þ ωÞ

E

1=2

; (3:33)

while for the clockwise component

b ! B2 ¼ i ðΩ ωÞ

E 1=2

: (3:34)

For Ω > ω, the resulting velocity profiles can be deduced directly from the 1D

profiles described previously by simply replacing ω by ω0 = Ω± ω. For Ω < ω, the

clockwise parameter B2 may be rewritten in the form

B 02 ¼ i

i ðω ΩÞ

E

1=2

: (3:35)

From (3.11), velocity structure is dependent on exp(±b Z ) then noting that

ei1=2q ¼ ðeiq2

Þ1=2; (3:36)

whereas




ei3=2q ¼ ðeiq2Þ1=2

(3:37)

with q a real constant, we see that the introduction of an additional i in (3.37) simply

changes the direction of rotation. In consequence, the sign of the phase difference

between surface and bed changes; however, since the vector is rotating in theopposite direction, there remains a phase advance at the bed.

The current structure for the a – c and c – w components of the tidal ellipse can be

estimated from Fig. 3.3 by calculating their respective Strouhal numbers as follows:

S ac ¼ 2π R1j j

D Ω þ ωð Þð Þ and S cw ¼

2π R2j j

D ω Ωð Þð Þ : (3:38)

Thus, for semi-diurnal constituents in latitudes less than 70°, vertical structure will

be greater for the clockwise component than for the anti-clockwise component.The more pronounced current structure for the clockwise component means that the

tidal current ellipse becomes more positively eccentric towards the bed. (Positive

eccentricity indicates that | R1| > | R2|.) Similarly, it can be deduced that the direction

of the major axis of the ellipse will veer in a clockwise sense towards the bed.

At mid-latitudes, for the other major tidal frequency bands, i.e. diurnal, quarter-

diurnal, the ratio (Ω + ω) : | Ω−ω | is smaller than for the semi-diurnal band. Hence,

the difference between the velocity structures for the two rotational components

should be less than that for M2.

3.3.3 Sensitivity to friction factor and eddy viscosity

Appendix 3A indicates how the ellipse parameters – A amplitude of the major axis,

E C eccentricity, ψ direction and phase – can be calculated from the a – c and c – w

vector components. In summary, the anti-clockwise current vector ( R1, θ 1,) and

clockwise vector ( R2, θ 2) are related to the more conventional parameters as follows:

A ¼ R1 þ R2; E C ¼R1 R2

R1 þ R2

; ψ ¼ θ 2 þ θ 1

2 and

¼ θ 2 θ 1

2 :

(3:39)

Figure 3.4 (Prandle, 1982b) illustrates typical current structures for M2 at latitude

55° N together with the sensitivity to both f and E . These results show how, in the

vicinity of the bed, reducing E enhances vertical current structure, decreasesamplitude, increases eccentricity (in a positive a – c sense) and advances phase.

These trends are similar to those shown for increasing bottom friction, but in the

latter case, there is an additional reduction in the overall current amplitudes (relative

to the frictionless values used in prescribing the surface gradients).

62 Currents



3.3.4 Sensitivity to latitude

Figure 3.5 (Prandle, 1997) shows solutions for (3.31) and (3.32) as functions of

latitude and water depth for surface gradients which correspond to a ‘

free stream’

rectilinear tidal current R* = 0 . 3 2 m s−1. (Free stream corresponds to the solution of

(3.31) and (3.32) with D infinite or f zero, i.e. in deep, frictionless conditions.) The

figure emphasises the enhanced influence of the friction term at latitudes corre-

sponding to the inertial frequency, i.e. for M2: sin−1(24/2/12.42) 75°.

Figure 3.6 (Prandle, 1997) extends these results using simulations by Prandle

(1997), from (A) a 2D vertically averaged model and (B) a 3D model incorporating

a Mellor and Yamada (1974) level 2.5 closure scheme as described in Appendix 3B.

The simulations cover a range of R* = 0.1, 0.32 and 1.0 m s−1

. The contour valuesshown are restricted to | R|=0.9 R*, phase and direction = ±10° and eccentricity =

±0.1. These can be regarded as representative boundaries between conditions where

tidal propagation is little influenced by friction (in deeper water) and conditions

where bottom friction becomes increasingly significant (in shallower water).

(a)

(b)

0° –10°–20°–30°–40°

φ

0.20 0.4

E c

0° –10°–20°–30°–40°

φ

1.0

0.5

Z 0

0

0 0.5

A

A

ε 0 = 0.1

ε 0

= 0.5

ε 0 = 0.05

1.0 0.20 0.4E c

0° –10°–20°–30°

ψ

ε 0 = 0.1

ε 0

= 0.5

ε 0 = 0.05

Z 0

kW 0

= 0.4 kW 0 = 0.1

kW 0 = 0.02

1.0

0.5

00 0.5 1.0 0° –10°–20°–30°

ψ

kW 0 = 0.4

kW 0 = 0.1

kW 0 = 0.02

Fig. 3.4. Sensitivity of current structure to eddy viscosity and bed friction.

A amplitude, E c eccentricity, ψ direction and phase.(a) Eddy viscosity E = ε 0 D

2Ω, ε 0 = 0.05, 0.1 and 0.5 (kW 0 = 0.1).

(b) Bed stress τ = ρ fU *U , fU * = kW 0Ω D, kW 0 = 0.02, 0.1 and 0.4 (ε 0 = 0.1).Results for M2 amplitude of 1 m s−1 at latitude 55° N.




The qualitative agreement for depth-averaged ellipse parameters between the

2D and 3D models indicates how in deeper water these results are very close, while

in shallow water, by suitable adjustment of the bed friction coefficient, 2D model

results can be adjusted to approximate those of 3D models. It was shown in Section 2.5

that for a predominant tidal constituent, i, the quadratic friction term f Ri Ri| can belinearised (in one dimension) to (8/3π) f Ri Ri* (where Ri* is the tidal amplitude). The

equivalent linearisation for other constituents, j, is (4/ π) f R j Ri*. Thus, the frictional

effect is linearly proportional to f Ri* for all constituents, and thus any enhancement

of f to improve reproduction of the primary constituent in a 2D model should not

adversely affect other constituents. The results shown in Fig. 3.6 for f = 0.0125 are

included as an illustration of an enhanced frictional coefficient applicable for a

secondary constituent factored by 3/2 Ri*/ R j* in such simulations.

3.3.5 Surface to bed changes in tidal ellipses

Figure 3.7 (Prandle, 1997) indicates the changes in current ellipse parameters between

the surface and the bed as calculated from models: (B) the level 2.5 closure k – ɛ model

0.5

0.5

60°

30°

10 25 50

Depth (m)

Depth (m)

100

Amplitude

Direction Eccentricity

Latitude

200

10 25 50 100 200

0°

0.6

0.4

0.7

0.3

0.8

0.2

0.9

0.1

0.01

320°

320°

330°

330°

340°

340°

350°

350°

60°

60°

30°

30°

10 25 50

Depth (m)

100

Phase

Latitude

Latitude Latitude

200

10 25 50Depth (m)

100 200

0°

0°

60°

30°

0°

Fig. 3.5. Modulation of depth-averaged M2 ellipse due to bed friction, as f (Depth,Latitude). Contours show changes (fractional for amplitude R =0.32ms−1) relativeto frictionless values of zero eccentricity, phase and direction.

64 Currents



Fig. 3.6. Modulation of depth-averaged M2 ellipse parameters by friction in 2D and3D models.(Left) Bed friction in a 2D model;(Right) bed friction and vertical eddy viscosity (MY 2.5) in a 3D model.Contours show: – amplitude 0.9, - - - - phase 350°, – – – – direction 350°, …….eccentricity 0.1 and –– bed friction coefficient f × 5 = (0.0125). For frictionless R* =0.1(top), 0.32 (middle), 1.0 (bottom) m s−1.



and (C) an analytical solution with E constant (Prandle, 1982a). The close level of

agreement shown between these two approaches results from specifying E E 0 in

the analytical model, where E 0 is the time-averaged value of vertical eddy viscosity

at the bed computed in the k – ɛ model. While it can be shown that such tuning can beused to adjust any specific ellipse parameter, it is not possible to force precise

agreement for all four parameters simultaneously. Thus, if a detailed representation

of current profiles is required, it is necessary to use model (B) to provide a detailed

description of the temporal and vertical variations in E .

0.150.2

0.3

0.4

0.45

0.4

0.2

0.3

0.3

0

–0.2

–0.1

0

–0.2 –0.1

100°

30°

Latitude

60°

25Depth (m)

δA

50 100 200

100°

30°

Latitude

60°

25Depth (m)

δE

50 100 200 100°

30°

Latitude

60°

25Depth (m)

δE

50 100 200

100°

30°

Latitude

60°

25Depth (m)

δA

50 100 200

0.35

Fig. 3.7. Surface to bed changes in M2 amplitude and eccentricity as f (Depth,Latitude) (amplitude as a fraction of frictionless depth-averaged R* = 0 . 3 2 m s−1,

Ec = 0). (Left) Analytical solution (Section 3.2); (right) 3D numerical model(MY 2.5).

66 Currents



The mean depth-averaged value of E calculated from the k – ɛ model (B) is

typically four times larger than the value at the bed E 0, and thus specifying this in

model (C) significantly reduces the vertical structure shown in Fig. 3.7.

3.3.6 Currents across the inter-tidal zone

Successive wetting and drying of inter-tidal areas can generate significant cross-

shore currents. If it is assumed that such currents are the primary source of in-filling

and draining, then tidal current speeds over a uniform bank slope SL will have

an amplitude of ως *. / SL, where ς * is the amplitude of the surface elevation. For a

semi-diurnal current, this corresponds to a maximum cross-shore current amplitude

of 7 cm s−1 for every kilometre of exposed bank. Prandle (1991), shows, from both

modelling simulations and H.F. Radar observations, how such currents over inter-tidal zones modify tidal ellipse characteristics in the near-shore zone.

3.4 Residual currents

Removal of the oscillatory tidal component from current observations leaves

residuals that may include contributions from ‘rectified’ tidal propagation, direct

(localised) wind forcing, indirect (larger scale) wind forcing, surface waves and

horizontal and vertical density gradients. Interaction of any of these components

with the tidal component can generate significant modulation of the latter, contri-

buting an additional non-tidal residual. Selective filtering of the non-tidal currents

into frequency bands can be used to separate many of these components.

Density-driven residual currents are described in Chapter 4. Soulsby et al . (1993)

provide a review of the nature and impact of current interaction with surface waves,

quantifying the effect on the bed friction coefficient. Wolf and Prandle (1999)

indicate how, in shallow water, such wave impacts can substantially reduce tidalcurrents.

3.4.1 Wind-driven currents

Relating observed wind-driven currents to wind forcing is notoriously difficult.

Both the wind itself and the associated currents exhibit appreciable fine-scale

(temporal and spatial) variability. In shallower water, wind forcing may be partially

balanced by surface slopes. In constricted cross sections, these slopes can sub-sequently generate currents which are orders of magnitude greater than indicated

by direct localised surface wind forcing (Prandle and Player, 1993).

The response of the sea surface to surface wind forcing under steady-state

conditions was originally studied by Ekman; a convenient summary of results is

3.4 Residual currents 67



given by Defant (1961). The essential features can be described by considering a

steady-state solution to (3.31) and (3.32) (Prandle and Matthews, 1990). Thus

rewriting these equations in the form

iΩR þ S ¼ E @ 2R

@ Z 2 ; (3:40)

where S is the surface gradient term and E the vertical eddy viscosity is assumed

constant.

Surface ( Z = D) and bed ( Z = 0) boundary conditions are, respectively,

τ w ¼ ρ E @ R

@ Z (3:41)

ρ F Rz ¼0 ¼ ρ E @ R@ Z

; (3:42)

where τ w is the surface wind stress and F Rz = 0 a (linearised) representation of bed

friction.

A solution in the form of an Ekman spiral,

R ¼ A ebz þ C ; (3:43)

is satisfied by

RðzÞ ¼ τ w

ρ E b ebD ebz þ

Eb

F 1

; (3:44)

where b2 = i (Ω/ E ).

In deeper water, bD≫ 1, i.e. D≫ ( E / Ω)½, the first term in (3.44) predomi-

nates and

Rz ¼D ¼ τ w i1=2

ρ ðΩE Þ1=2 ; (3:45)

i.e. a surface current of magnitude dependent on latitude and veering at 45° clock-

wise to the wind stress.

In shallow water, the second term predominates and

R ¼ τ w eb D

ρ F ; (3:46)

i.e. a current of magnitude dependent on the bed friction coefficient and aligned with

the wind.

Figure 3.8 shows these wind-driven surface-current responses derived from

statistical analyses of H.F. Radar observations (Prandle and Matthews, 1990).

68 Currents



Substituting (3.4) for wind stress, τ , yields a response in close agreement with that

shown from the Radar observations in Fig. 3.8. The steady-state surface currents are

typically 1 or 2% of wind speed, increasing in deeper water in both magnitude and

veering towards the theoretical deep water values of 45° to the right of the wind. The

observed veering ranges from 3° to 35° (clockwise).

3.4.2 Tidal current residuals

The mechanisms by which tidal energy propagating into estuaries at semi-diurnal

and diurnal periods generate higher-order harmonics and residual components is

described in Section 2.6. It was shown how determination of residual currents is

complicated by which parameter is defined to be the (linear) controlling factor.In shallow water, calculation of residual current components is further complicated

by the respective reference systems, e.g. fixed distances above the bed or below the

surface, fractional heights. The continuous profile provided by an Acoustic Doppler

Current Profiler (ADCP) allows observational data to be readily interpolated onto

Fig. 3.8. Wind-driven surface current for a wind of 1 m s−1 eastwards. Derived fromH.F. Radar measurements.

3.4 Residual currents 69



the selected vertical framework. However, for residual and higher harmonics,

derived values are especially sensitive to the prescribed framework (Lane et al .,

1997).

An Euler current at a fixed height above the bed corresponds to the velocity

observed by a moored current meter. The tidally averaged, depth-integrated Euler residual is defined as

U E ¼ 1

P

ð P

0

1

D þ &

ð Dþ&

0

U ðZ Þ dZ

0@

1A dt: (3:47)

The tidally averaged, depth-integrated, residual transport current is

U T ¼ 1

DP

ð P

0

ð Dþ&

0

U ðZ Þ dZ

0@

1A dt (3:48)

with U T D = Q, the river flow per unit breadth.

For uniform flow of a single tidal constituent in one direction, the difference

between the above residual current components is

U T ¼ U E þ 0:5 &

U

=D cos ðθ Þ; (3:49)

where θ is the phase difference between ζ and U and the second term is referred to as

the Stokes’ drift. Cheng et al . (1986) provide further details of the difference

between Eulerian and Lagrangian flows in non-uniform flow fields.

Prandle (1975) showed that the depth-integrated net energy propagation can be

approximated by

EN ¼ ρ gD 0:5 & U cosðθ Þ½ : (3:50)

This implies that net propagation of tidal energy will be accompanied by a small

residual Stokes’ drift. In open seas, this can produce a persistent residual circulation.

A modelling study by Prandle (1984) showed typical residual currents of 1 – 3 c m s−1

in the continental shelf around the UK. However, in an enclosed estuary, some

compensating outflow must be present, as noted in Section 2.6.3.

As an example of these persistent tidally driven residual currents, Fig. 3.9 shows a

residual surface current gyre measured in a year-long H.F. Radar deployment in

the Dover Strait (Prandle and Player, 1993). Such gyres are not exceptional but rather a characteristic of most coastlines, although rarely identified with conven-

tional instruments. Geyer and Signell (1991) mapped a similar headland gyre with a

vessel-mounted ADCP.

70 Currents




Despite large ebb and flood tidal excursions within estuaries, longer-term mixing

of salt, sediments and pollutants can remain sensitive to small-scale variability in

currents and persistent residual circulation. Hence, the focus here is on deriving the

scaling factors which determine the vertical structure of currents, illustrating the

sensitivity to tidal current amplitude, tidal period, depth, friction factor and latitude.

The key question is:

How do tidal currents vary with depth, friction, latitude and tidal period?

Equation (2.13) shows how the depth-averaged tidal velocity amplitude, U *, is

proportional to the sea surface gradient, ς x, divided by the sum of the inertial term

ωU and the linearised friction term FU , where ω = 2π/ P , P tidal period. Here

‘single-point ’ analytical solutions for the associated vertical structure are derived

Fig. 3.9. Residual tidal currents observed at the surface by H.F. Radar.




for specified values of ς x. For the case of a constant eddy viscosity, E , this vertical

structure is shown to be determined by two parameters Y = D(ω/ E )1/2 and J = 3π

( E ω)1/2/8 f U * ( D water depth and f bed friction coefficient). Y is analogous to an

Ekman height and J introduces the effect of bed friction.

From both comparisons of observed structure against these solutions and self-consistency of the analytical solutions, the approximation E = f U * D is shown to be

valid in strongly tidal, shallow waters where the influence of the bottom boundary

layer extends to the surface. Figure 3.7 shows a comparison of tidal current structures

determined using this constant value for E versus detailed numerical simulations

employing a turbulence closure module (Appendix 3B). The overall level of agree-

ment indicates the validity of the approximation.

Adopting this approximation for E , the characteristics of vertical structure of tidal

currents can be reduced to dependency on the Strouhal number, S R = UP / D withY J 50/ S R

1/2 for f = 0.0025. The amplitude structure becomes more pronounced

with increasing S R up to an asymptotic limit of S R 350. Accompanying phase

variations are a maximum for this value of S R but decrease for both smaller and

larger values (Fig. 3.3). In meso- and macro-tidal estuaries, the Strouhal number will

be well in excess of 1000.

Commonly observed features explained by this theory include, Figs. 3.1 and 3.3:

(1) depth-mean velocity occurring at fractional height ( z = Z / D) above the bed, z = 0.4;(2) phase advance at the bed relative to the surface of up to 20° (at S R 350). A similar

phase advance at the coastal boundaries relative to the deep mid-section generally

occurs.

(3) maximum velocities occurring at the surface, except for Y > 4 (S R < 300) where sub-

surface maxima occur (although not especially pronounced).

The above solutions ignore the effects of the Earth’s rotation represented by the

Coriolis term in (2.1) and (2.2). Even in long narrow estuaries, the Coriolis term issignificant in determining the details of tidal current structure. This significance can be

understood by noting that tidal currents do not simply ebb and flood along one axis but

rotate in an elliptic pattern. The characteristics of these ellipses are represented by the

parameters: AMAX amplitude along the principal axis, AMIN amplitude along the

(orthogonal) minor axis, ψ direction of the principal axis, φ phase (time of maximum

current). The eccentricity E C = AMIN/ AMAX with the additional convention of positive

for anti-clockwise (a – c) rotation and negative for clockwise (c – w).

Vertical variations in these ellipse parameters calculated from observations canoften appear bewilderingly complex. However, an underlying systematic structure

invariably emerges when the ellipse is separated into its clockwise and anti-

clockwise rotational components. (1D flow corresponds to E C = AMIN =0 and

occurs when the magnitude of the two rotational components are equal). This

72 Currents



separation enables the influence of the Coriolis term to be directly illustrated. Thus,

expanding the above theory to include the Coriolis term introduces separate

Strouhal Numbers for each rotational component as follows:

S ac ¼ 2π U

Dðω þ ΩÞ and S cw ¼ 2π U

Dðω ΩÞ : (3:51)

For a semi-diurnal constituent, ω 1.4×10−4, while at a latitude of 50°, the Coriolis

term Ω 1.1×10−4. Thus, for such conditions, we see that the c – w Strouhal number

is generally an order of magnitude greater than the a – c value. This results in a much

more pronounced vertical structure for the c – w component and directly explains the

commonly observed features of

(1) increasing c – w eccentricity towards the surface(2) major axis veering a – c towards the surface.

The acute sensitivity of tidal currents close to inertial latitudes, where ω=Ω

(70° for semi-diurnal constituents and 30° for diurnal), is shown in Fig. 3.5.

Although this may be qualified by evidence from Csanady (1976) that the thickness

of the boundary layer and in consequence the vertical eddy viscosity is itself related

to the Coriolis parameter.

The above mathematical approach used in determining the details of tidal current structure can be usefully applied to derive the steady-state vertical current structure

associated with wind forcing (3.44). This theory explains the observed ‘Ekman

spiral’ patterns of wind-driven currents shown in Fig. 3.8. Maximum surface

currents, up to a few percent of the wind speed, occur in deep water, veering 45°

to the right of the wind (northern hemisphere). In shallow water, bed friction reduces

these currents and aligns them more closely with the wind direction.

Appendix 3A

3A.1 Tidal current ellipse

Figure 3A.1 (Prandle, 1982b) illustrates how a current ellipse may be resolved into

two rotary components. The current vectors are shown at time t = 0.

On the left, the anti-clockwise rotating component is given by

R1 ¼ R1j jðcos g1 þ i sin g1Þ: (3A:1)

In the middle, the clockwise rotating component is

R2 ¼ R2j jðcos g2 þ i sin g2Þ: (3A:2)

The maximum current occurs at time t MAX when R1 and R2 are aligned, i.e. when

Appendix 3A 73



ωtMAX þ g1 ¼ ωtMAX þ g2

or tMAX ¼ g2 g1

2ω : (3A:3)

The phase, φ, is given by

j ¼ ωtMAX (3A:4)

and the maximum amplitude, AMAX, is

AMAX ¼ R1j j þ R2j j: (3A:5)

The inclination, ψ , of the maximum current to the X -axis is given by

ψ ¼ ωtMAX þ g1 ¼ ωtMAX þ g2 ¼ 1

2ð g2 þ g1Þ: (3A:6)

The minimum amplitude, AMIN, is when R1 and R2 are opposed, hence

AMIN ¼ R1j j R2j jj j: (3A:7)

The eccentricity of the ellipse, E c, is defined as the ratio AMIN: AMAX. However, in

addition, we use the convention of E c positive for anti-clockwise current rotation

| R1| > | R2| and negative for clockwise rotation, | R2| > | R1|. Thus letting μ = | R1|/| R2|,

we define E c as follows:

E c ¼ μ 1

μ þ 1: (3A:8)

Hence, | E c | is a minimum when μ = 1 and | E c | increases as μ increases above this

value.

y

x x x

y A

Ψ

y

R 1

R 2g 1g 2

R 1 R 2 R = R 1 + R 2

wt

wt

Fig. 3A.1. Decomposition of a tidal current ellipse into anti-clockwise and clockwise

rotating components.

74 Currents



Appendix 3B

3B.1 Turbulence model

The Mellor – Yamada level 2.5 mode (MYL 2 5, Mellor and Yamada (1974)) is

widely used to determine the values of the vertical eddy viscosity, K M, and vertical

eddy diffusivity, K q, coefficients. Refinements of this model described by Deleersnijder

and Luyten (1993) are incorporated here. Umlauf and Burchard (2003) describe more

recent developments.

The turbulent kinetic energy (TKE or k ) equation is k = q2/2

@

@ t

q2

2 ¼ @

@ Z K q

@

@ Z

q2

2 ðdiffusionÞ

þK M@ U

@ Z

2

þ @ V

@ Z

2" #

ðshear productionÞ

q3

B1

¼ 0 ðdissipationÞ; (3B:1)

where B1 = 16.6 and the eddy coefficients are proportional to the mixing length l and

associated stability functions S q and S H; thus

K q ¼ l q; S q ¼ 0:2l q (3B:2)

K M ¼ l q; S M ¼ 0:4l q: (3B:3)

The mixing length l is determined from the related equation

@

@ t q

2

l ¼

@

@ Z K q

@

@ Z q

2

l

þ E 1l K M

@ U

@ Z 2

þ

@ V

@ Z 2" #

Wq3

B1 : (3B:4)

The wall proximity function W is defined as

W ¼ 1 þ E 2l 2

K vLð Þ2 ; (3B:5)

where E 1 = 1.8, E 2 = 1.33. K v is the van Karman constant (0.4), and L is a function

of the distance to the seabed d b and to the sea surface d s; thus

L ¼ d s d b

d s þ d b: (3B:6)

The boundary conditions specified at the surface and bed are

Appendix 3B 75



K M@ R

@ Z ¼

τ B

ρ (3B:7)

K M@ R

@ Z ¼

τ 0

ρ ; (3B:8)

where the velocity R = U + iV , τ 0 is the applied wind stress and τ B is the tidal stress at

the bed ( ρ fR| R|).

At the surface and bed q2l = 0.

References

Bowden, K.F., 1953. Note on wind drift in a channel in the presence of tidal currents.

Proceedings of the Royal Society of London, A, 219, 426 – 446.Bowden, K.F., 1978. Physical problems of the Benthic Boundary Layer. Geophysical

Surveys, 3, 255 – 296.Cheng, R.T., Feng, S., and Pangen, X., 1986. On Lagrangian residual ellipse. In: van de

Kreeke, J. (ed.), Physics of Shallow Estuaries and Bays (Lecture Notes on Coastal andEstuarine Studies No. 16) Springer-Verlag, Berlin, pp. 102 – 113.

Csanady, G.T., 1976. Mean circulation in shallow seas. Journal of Geophysical Research,81, 5389 – 5399.

Davies, A.M. and Furnes, G.K., 1980. Observed and computed M2 tidal currents in the North Sea. Journal of Physical Oceanography, 10 (2), 237 – 257.

Defant, A., 1961. Physical Oceanography, Vol. 1. Pergamon Press, London.Deleersnijder, E. and Luyten, P., 1993. On the Practical Advantages of the

Quasi-Equilibrium Version of the Mellor and Yamada Level 2 – 5 Turbulence Closure Applied to Marine Modelling . Contribution No. 69. Institui d’Astronomie et deGeophysique, Universite Catholique de Louvain, Belgium.

Faller, A.J. and Kaylor, R., 1969. Oscillatory and transitory Ekman boundary layers. Deep-Sea Research, Supplement , 16, 45 – 58.

Flather, R.A., 1984. A numerical model investigation of the storm surge of 31 January and1 February 1953 in the North Sea. Quarterly Journal of the Royal Meteorological Society, 110, 591 – 612.

Geyer, W.R. and Signell, R., 1991. Measurements and modelling of the spatial structure of nonlinear tidal flow around a headland. In: Parker, B.B. (ed.), Tidal Hydrodynamics.John Wiley and Sons, New York, pp. 403−418.

Heaps, N.S., 1969. A two-dimensional numerical sea model. Philosophical Transactions Royal Society, London, A, 265, 93 – 137.

Ianniello, J.P., 1977. Tidally-induced residual currents in estuaries of constant breadth anddepth. Journal of Marine Research, 35 (4), 755 – 786.

Kraav, V.K., 1969. Computations of the semi-diurnal tide and turbulence parameters in the North Sea. Oceanology, 9, 332 – 341.

Lane, A., Prandle, D., Harrison, A.J., Jones, P.D., and Jarvis, C.J., 1997. Measuring fluxes inestuaries: sensitivity to instrumentation and associated data analyses. Estuarine,Coastal and Shelf Science, 45 (4), 433 – 451.

Liu, W.C., Chen, W.B., Kuo, J-T., and Wu, C., 2008. Numerical determination of residencetime and age in a partially mixed estuary using a three-dimensional hydrodynamicmodel. Continental Shelf Research, 28 (8), 1068 – 1088.

76 Currents



McDowell, D.M., 1966. Scale effect in hydraulic models with distorted vertical scale.Golden Jubilee Symposia, Vol. 2, Central Water and Power Research Station, India,

pp. 15 – 20.McDowell, D.M. and Prandle, D., 1972. Mathematical model of the River Hooghly.

Proceedings of the American Society of Civil Engineers. Journal of Waterways and

Harbours Division, 98, 225 – 242.Mellor, G.L. and Yamada, T., 1974. A hierarchy of turbulence closure models for planetary boundary layers. Journal of the Atmospheric Science, 31, 1791 – 1806.

Munk, W., Snodgrass, F., and Wimbush, M., 1970. Tides off-shore: Transition fromCalifornia coastal to deep-sea waters. Geophysical Fluid Dynamics, 1, 161 – 235.

Pingree, R.D. and Maddock, L., 1980. Tidally induced residual flows around an island dueto both frictional and rotational effects. Geophysical Journal of the Royal Astronomical Society, 63, 533 – 546.

Prandle, D., 1975. Storm surges in the southern North Sea and River Thames. Proceeding of the Royal Society of London, A, 344, 509 – 539.

Prandle, D., 1982a. The vertical structure of tidal currents and other oscillatory flows.Continental Shelf Research, 1, 191 – 207.

Prandle, D., 1982b. The vertical structure of tidal currents. Geophysical and Astrophysical Fluid Dynamics, 22, 29 – 49.

Prandle, D., 1984. A modelling study of the mixing of 137Cs in the seas of the Europeancontinental shelf. Philosophical Transactions of the Royal Society I of London, A, 310,407 – 436.

Prandle, D., 1991. A new view of near-shore dynamics based on observations fromH.F. Radar. Progress in Oceanography, 27, 403 – 438.

Prandle, D., 1997. The influence of bed friction and vertical eddy viscosity on tidal

propagation. Continental Shelf Research, 17 (11), 1367 – 1374.Prandle, D., 2004. Saline intrusion in partially mixed estuaries. Estuarine, Coastal and Shelf

Science, 59, 385 – 397.Prandle, D. and Ryder, D.K., 1989. Comparison of observed (H.F. radar) and modelled

near-shore velocities. Continental Shelf Research, 9, 941 – 963.Prandle, D. and Matthews, J., 1990. The dynamics of near-shore surface currents generated

by tides, wind and horizontal density gradients. Continental Shelf Research, 10,665 – 681.

Prandle, D. and Player, R., 1993. Residual currents through the Dover Strait measured byH.F. Radar. Estuarine, Coastal and Shelf Science, 37 (6), 635 – 653.

Proudman, J., 1953. Dynamical Oceanography. Methuen and Co. Ltd, London.Soulsby, R.L., Hamm, L., Klopman, G., Myrhaug, D., Simons, R.R., and Thomas, G.P.,

1993. Wave – current interaction within and outside the bottom boundary layer. Coastal Engineering , 21, 41 – 69.

Souza, A.J. and Simpson, J.H., 1996. The modification of tidal ellipses by stratification inthe Rhine ROFI. Continental Shelf Research, 16, 997 – 1008.

Umlauf, L. and Burchard, H., 2003. A generic length-scale equation for geophysicalturbulence models. Journal of Marine Research, 6, 235 – 265.

Van Veen J., 1938. Water movements in the Straits of Dover. Journal du Conseil, Conseil International pour l ’ Exploration de la Mer , 14, 130 – 151.

Wolf, J. and Prandle, D., 1999. Some observations of wave – current interaction. Coastal Engineering , 37, 471 – 485.

Zimmerman, J.T.F., 1978. Topographic generation of residual circulation by oscillatory(tidal) currents. Geophysical and Astrophysical Fluid Dynamics, 11, 35 – 47.

References 77



4

Salin e in tr u sio n

4.1 Introduction

T he n at ur e o f s al in e i nt ru si on i n a n e st ua ry i s g ov er ne d b y t id al a mp li tu de , r iv er f lo w

a n d b a t h y m e t r y . T h e p a t t e r n o f i n t r u s i o n m a y b e a l t e r e d b y ‘interventions’ s u c h a s

d r e d g i n g , b a r r i e r c o n s t r u c t i o n o r f l o w r e g u l a t i o n a l o n g s i d e i m p a c t s f r o m c h a n g e s i n

m sl o r r iv er f lo w s l in ke d t o G lo ba l C li ma te C ha ng e . A d ju st me nt s t o t he i nt ru si on

m ay h av e i mp or ta nt i mp li ca ti o ns f or f ac to rs s uc h a s w at er q ua li ty, s ed im en ta ti on

a nd d is pe rs io n o f p ol lu ta nt s. W he re as t id al p ro pa ga ti on c an b e e xp la in ed f ro m

s im pl e a na ly ti ca l f or mu la a nd a cc u ra te ly m od el le d, i t i s o ft en d if fi cu lt t o e x pl ai no b s e r v e d c h a n g e s i n i n t r u s i o n f r o m s p r i n g t o n e a p o r f l o o d t o d r o u g h t .

T he l a te ra ll y a ve ra g ed m as s c on se r va ti on e q ua ti o n m ay b e w ri t te n a s ( Oe y,

1 9 8 4 )

@ C

@ t þ U

@ C

@ X þ W

@ C

@ Z ¼

1

DB

@

@ X DBK x

@ C

@ X

þ D

@

@ Z BK z

@ C

@ Z

; (4:1)

where C i s c o n c e n t r a t i o n , U and W v e l o c i t i e s i n t h e a x i a l X a nd v e rt i ca l Z directions,

D w a t er d e pt h , B breadth, K x and K z e dd y d i sp e rs i on c o ef f ic i en t s . L e w is ( 1997)

provides detailed descriptions of how advection and dispersion terms in (4.1) interact

t o p r o m o t e e s t u a r i n e m i x i n g .

D is pe rs io n o f s al t i nv ol ve s i nt er ac ti ng 3 D v ar ia ti on s i n p ha se , a mp li tu de a nd m ea n

v al ue s o f b ot h c ur re nt s a n d t he s al in e d is tr ib ut io n. T he se v ar ia ti on s a re s en si ti ve

t o t h e l e ve l o f d e ns i ty s t ra t if i ca t io n w h ic h m a y v a ry a p pr ec i ab l y – t e m p or a l ly a n d

both axially and transversally. The spectrum of such spatial and temporal variations

includes

( 1 ) t h e t i d al c y cl e , w i t h p r o n ou n ce d v e r t ic a l m i x i ng o c cu r ri n g o n o n e o r b o th o f p e a k f l o o d

a n d e b b c u r r e n t s d u e t o b o t t o m f r i c t i o n , o r a t s l a c k t i d e s d u e t o i n t e r n a l f r i c t i o n ( L i n d e n

a n d S i mp s o n , 1988);

( 2) t he n e ap – s p r i n g c y c l e , w i t h m i x i n g o c c u r r i n g m o r e r e a d i l y o n s p r i n g t h a n n e a p t i d e s ;

78



( 3 ) t h e h y d ro l og i ca l c y c l e, w i th v a ri a ti o ns i n b o t h r i v e r f l o w a n d s a l i ni t y o f s e a w a t er a t t h e

mo u t h ( G o d i n , 1985);

( 4) s to rm e ve nt s i nc lu di ng s t or m s ur ge s g en er at ed b ot h i nt e rn al ly a nd e xt er na ll y ( Wa ng

a n d E l l i o t t , 1978) a n d s u r f a c e w a v e m i x i n g ( O l s o n , 1986);

(5 ) v ar ia ti on s i n w at er d en si ty d ue t o o th er p ar am et er s, i n p ar ti cu la r t em pe ra tu re

(A p p e n d i x 4 A ) a n d s u s p e n d e d s e d i m e n t l o a d ( C h a p t e r 5 ).

T his c ha pte r d oe s n ot co ns id er th e i nf lu en ce o f t empe ra tu re on d ens it y. A n

A pp en di x i s i n cl ud ed d es cr ib in g t he s ea so na l c y cl e o f t em p er at ur e v ar ia ti on s a s a

f un ct io n o f w at er d ep th a nd l at it ud e. N un es a nd L en no n (1986) i nd ic at e h o w e va po ra -

t io n a t l ow l at it ud es c an p ro du ce m ax im um s ali ni ti es a t t he h ea d o f a n e st uar y –

i n v e r t i n g t h e c u s t o m a r y p a t t e r n o f s a l i n e i n t r u s i o n .

4.1.1 Classification systems

P r it c ha r d (1955) i nt ro du c ed a c la ss if ic at io n s ch e me f or e st ua ri ne m ix in g. I n c la ss

A, ‘h i g h l y s t r a t if i e d e s t u a ri e s’, t h e d i s p e r s i o n t e r m s o n t h e r i g h t - h a n d s i d e o f (4.1)

a r e n e g l i g i b l e . F o r t y p e B , ‘ partially mixed estuaries’, v e r t i c a l d i s p e r s i o n i s i m p o r -

t a n t . T y p e s C , w i d e , a n d D , n a r r o w , a r e ‘ f u l l y m i x e d e s t u a r i e s’ w i t h d e n s i t y p r o f i l e s

∂C / ∂ Z 0 a n d h e n c e o n l y l o n g i t u d i n a l d i s p e r s i o n i s i n v o l v e d . T h i s l a t t e r a s s e r t i o n

i s i nc on si st en t w it h s ub se qu en t n ot io ns o f m ix in g i n w el l- mi xe d e st ua ri es , f or

e x a m p l e t h e r o l e o f t i d a l s t r a i n i n g d e s c r i b e d i n S e c t i o n 4 . 4.

W h i l e P r i t c ha r d’s c l as s if i ca t io n s y st e m p r ov i de s u s ef u l q u a li t at i ve d e s cr i pt i on s ,

i n m a n y e s tu a ri e s t h e d e gr e e o f s t ra t if i ca t io n v a ri e s a p p re c ia b ly b o th s p at i al l y a n d

temporally. F i gu re 4 .1 s ho ws a n ea p – s pr in g t im e s er ie s o f s al in it y v ar ia ti on s a t t hr ee

d ep th s f or a s in gl e l oc at io n i n t he M er se y E st ua ry. W hi le th e ti da l s ig na ls a re

e v i d e n t , l o n g e r - t e r m ( s u b - t i d a l ) i n f l u e n c e s a r e m u c h g r e a t e r t h a n f o r c o r r e s p o n d i n g

e le va ti on o r c ur re nt t im e s er ie s. F i gu re 4 .2 (Liu et al ., 2008) s ho ws h ow a xi al

d i s t r i b u t i o n s i n t h e D a n s h u e i R i v e r v a r y w i t h c h a n g i n g r i v e r f l o w s .T h e s tr at if ic at io n d ia gr am d er iv ed b y H a ns en a nd R at tr ay (1966) (F ig . 4 .3 ) h a s

been used to classify the nature of mixing in estuaries and the sensitivity of the

s tr at if ic at io n t o c ha ng in g c on di ti on s. T he d ia gr am i s b a se d o n t wo p ar am e te rs : ( i ) δ s/ s,

t he s al in it y d iff er en ce b et we en b ed a nd s ur fa ce d iv id ed b y t he d ep th -a ve ra ge d

s al in it y ( me an v al ue s o ve r a t id al c yc le ), a nd ( ii ) U s= U r es id ua l v el oc it y a t t he

s ur fa ce d iv id ed b y t he d ep th -m ea n v al ue . F ou r e st ua ri ne t yp es a re i de nt if ie d w it h

s ub -d iv is io ns i nt o ( a) m ix ed a nd ( b ) s tr at if ie d a cc or di ng t o w he th er δ s/ s i s l e s s o r

g re at er t ha n 0 .1 . I n e st ua ri es o f t yp e 1 , r es id ua l f lo w i s s ea wa rd s a t a ll d ep th s a ndc on se qu en tl y m ix in g o f s al t i s e nt ire ly b y d iff us io n. F or t yp e 2 , r es id ua l f lo w

r e v e r s e s w i t h d e p t h a n d m i x i n g i s d u e t o b o t h a d v e c t i o n a n d d i f f u s i o n . F o r t y p e 3 ,

t h e v e rt ic a l s t ru c tu r e o f t h e r e si d ua l f l ow i s s o p r on o un c ed t h at a d ve c ti o n a c co u nt s

for over 99% of the mixing process; in type 3(b), mixing is co nfin ed to th e

4.1 Introduction 79



n e ar - su r fa c e r e gi o n . Ty p e 4 e x h ib i t s m a xi m u m s t ra t if i ca t i on a n d a p p ro x i ma t es a

s a lt w e dg e . F i g u re 4 . 11 , d e r i v e d f r o m a ‘s i n g l e - p o i n t ’ n u m e r i c a l m o d e l o f s a l i n e

i n t r u s i o n a n d m i x i n g , e m p h a s i s e s t h e r o l e o f g r a v i t a t i o n a l c i r c u l a t i o n i n e s t u a r i e s

o f t y p e 4 .

P ra nd le (1985 ) showed how the U s= U a xi s c ou ld b e r ep la ce d b y S / F , i .e . t he r ati o

o f t h e r e s i d u a l a c c e l e r a t i o n s a s s o c i a t e d w i t h t h e h o r i z o n t a l d e n s i t y g r a d i e n t a n d b e d

f r i c t i o n d e f i n e d i n (4.42) , y i e l d i n g a m o r e d i r e c t a s s e s s m e n t o f t h e c l a s s i f i c a t i o n o f

a n e s t u a r y b a s e d o n m o r e r e a d i l y a v a i l a b l e p a r a m e t e r s . T h e d e m a r c a t i o n l i n e w h i c h

s e p a r a t e s e s t u a r i e s o f t y p e s 1 a n d 2 c a n t h e n b e e x p l a i n e d b y t h e o c c u r r e n c e o f f l o w

r e v e r s a l f o r S / F > 2 4 o r U s= U 42 (T a b l e 4 . 1).

4.1.2 Mixing in estuaries

D y e r a n d N e w ( 1986) s h o w e d h o w t h e l a y e r R i c h a r d s o n n u m b e r , d e f i n e d a s

Ri ¼

g

ρ

@ ρ

@ Z

@ U @ Z

2 ; (4:2)

28

16

Surface

28

16

Mid-depth

28

16

1 December 1987 1 January 1988

Bed

F ig . 4 .1 . N ea p – S p ri n g c y cl e s o f o b se r ve d s a li n i ty v a ri a ti o ns , ‰, i n t he M ers ey Narrows. Observed values at the surface, mid-depth and bed in water depth of 15 m.

80 Saline intrusion



r ep re se nt in g t he r at io o f b uo y an cy f or ce s t o v er ti ca l t ur bu le nt f or ce , d et er mi ne s

t h e n at u re o f m ix in g i n e st ua ri es . Ve rt ic al m ix in g o cc ur s f or Ri < 0 .2 5 w he n t ur bu le nc e

i s s u ff ic ie nt t o o ve rc om e d e ns it y l ay er in g . I n d et er mi n in g w he t he r a n e s tu ar y i s l ik el yt o b e m i x e d o r s t r a t i f i e d , t h e q u e s t i o n a r i s e s a s t o w h i c h c o m p o n e n t s o f c u r r e n t s h e a r

predominate (tidal, riverine or, circuitously, saline gradient-induced). Linden and

S i m p s o n (1988) a n d S i m p s o n et al. (1990) h a v e e m p h a s i s e d t h a t s t r a t i f i c a t i o n c a n n o t

be simply linked to the ‘gross’ p ar am e te rs u se d i n (4.2). T he va lu e of Ri varies

c ons id era bl y a lon g an e stu ar y an d o ve r bo th t he eb b – f lo od a nd t he s pr in g – neap

c yc le s. We e xpe ct n as ce nt s tr at if ic at io n a t c er ta in t im es a nd l oc at io ns i n a lm os t

a l l e st ua ri es . I n S e ct i on 4 . 5, w e e xa mi ne c on di ti on s l ik el y t o s us ta in , a nd t he re by

c o ns o li d at e, s tr at if ic at io n s h ow in g h ow t he r at io o f r iv e ri ne t o t id al c ur re nt s, U 0/ U *, ist h e c l ea r e st i n di c at o r o f s t r at i fi c a ti o n.

Ti da l a dv ec ti on c an c ar ry a f lu id c ol um n s ev er al k il om et re s f ro m i ts m ea n p os it io n.

T he t ot al u ps tr ea m a nd d ow ns tr ea m e xc ur si on f or a s em i- di ur na l t id e i s a pp ro xi -

m at el y 1 4U * k m, f or t id al c ur re nt a mp li tu de , U * , i n m s−1. H e nc e , s t ra t if i ca t io n

Danshuei river – Tahan stream

–5

–10

–5

–5

0 20 302510 155

–10

–10

D e p

t h ( m )

D e p t h ( m )

D e p t h ( m )

Salinity:ppt

Hsin-Hai Bri.

Hsin-Hai Bri.

Hsin-Hai Bri.

Taipei Bri.

Taipei Bri.

Taipei Bri.

Kuan-Du Bri.

Kuan-Du Bri.

Kuan-Du Bri.

River mouth

River mouth

River mouth

Q75 flo w

Qm flo w

Q10 flo w

Distance from Danshuei River mouth (km)

1 2 3 4

5 6 7 9 1 1

1 3

1 5

1 7

1 9

2 1

2 2 2

3 2 4

2 5

2 7

2 9

3 0 3

1 3 2 3

3

3 2

3 1 2 8

2 4 2 3

2 1 1 9 1 5 1 7

1 3

1 1

9 7 6 5 1 3

3 0

2 8

2 7

2 5

2 2

2 0 1

8

1 5

1 3

1 4

1 2 1 1

1 0

9

7

6 4 3 1 2

F i g . 4 . 2 . A x i a l v a r i a t i o n s i n s a l i n i t y i n t h e D a n s h u e i R i v e r , T a i w a n Q 7 5 , f l o w r a t ee x c e e d e d 7 5 % o f t i m e , Q 1 0 f l o w e x c e e d e d 1 0 % o f t i m e .

4.1 Introduction 81



100

4

0.50 0.75 0.9 0.99 0.999 0.99990.1

0.25

0.50

0.751.0

d

d

1.0

0.1 1 2

3

0.01

0.001

δ s

1 (2)

110(24)

10

100

100

S

U s / U 1000 104 105

1000S / F

104 105 106

F i g . 4 . 3 . H a n s e n a n d R a t t r a y ( 1 9 6 6 ) S t r a t i f i c a t i o n d i a g r a m , m o d i f i e d b y P r a n d l e( 1 9 8 5 ). δ s/ s fr ac ti on al s al i ni t y d if fe re nc e b et we en b ed a nd s u rf ac e, U s= U r e si d u al v e lo c it y a t t h e s u rf a ce : d e pt h -a v er a ge d v a lu e , S / F r at io o f s al in it y t o bed fricti on terms, d f ra ct i on al h ei g ht o f i n te rf ac e a bo ve t h e b ed , ( 4 . 2 8 ) i n as t r a ti f i e d s y s t e m .

Ta b l e 4 . 1 Residual surface gradients and current components at the surface

and bed

Surfacegradient

Surfacevelocity

Bedvelocity

( a ) R i v e r f l o w Q = U 0 D −0.89 F 1.14U 0 0.70U 0( b ) W i n d s t r e s s τ W no net flow 1.15W 0 .3 1 (W / F ) U 0 – 0 . 1 2 (W / F ) U 0( c ) M i x e d d e n s i t y g r a d i e n t – 0.46S 0 . 03 6 (S / F ) U 0 – 0 . 0 2 9 (S / F ) U 0( d ) S t r a t if i e d ‘wedge’

l o w e r - l a y e r d e p t h dH – 1.56 F

/(1−d )21.26U 0

/(1−d ) – 0.18U 0

/(1−d )

Notes: (a) U 0 r i v e r f l o w, (4.11) a n d ( 4 . 1 2 ) , ( b ) W w i n d f o r c i n g , (4.37) a n d ( 4 . 3 8 ) , ( c ) S mi x e d s a l i n it y, (4.15) a n d ( 4 . 1 6 ) , ( d ) s t r a ti f i e d s a l i ni t y, (4,31), (4.32) a n d ( 4 . 3 5 ) . W , S

a n d f r i ct i o n p a r a me t e r, F , d e f i n e d i n (4.42).Source: Prandle, 1985 .

82 Saline intrusion



o b s e r v e d a t o n e l o c a t i o n r e p r e s e n t s a n i n t e g r a t i o n o f p r o c e s s e s a c t i n g o v e r a w i d e

a re a b o th a xi al ly a nd t ra ns ve r sa ll y. A b ra ha m ( 1 9 8 8 ) i nd ic at es t ha t d if fe re nt ia l

a d v ec t io n o v e r d e pt h ∂/ ∂ Z (U (∂ ρ/ ∂ X ) ) i s i m po r ta n t i n d e te r mi n i ng s t ra t if i ca t io n .

T hi s p ro ce ss i s r ef err ed t o a s ‘t i da l s t ra i ni n g’, w he re by l arg er c ur re nt s a t t he

s ur fa ce o n t he f lo od t id e c ar ry d en se r s ea w at er s uf fi ci en tl y f ar u ps tr ea m t ha t m ix in g o cc ur s w he n t he s ur fa ce d en si ty e xc ee ds t ha t a t t he b ed , i s e xp lo re d i n

S e ct i on 4 . 4. C on v er se ly, o n t h e e bb , t he g r ea te r d ow n st re am m ov em en t o f t h e

s u r f a c e w a t e r s e n h a n c e s s t a b i l i t y a n d p r o m o t e s s t r a t i f i c a t i o n . H o w e v e r , i n h i g h l y

s tr at if ie d e st ua ri es , v er ti ca l m ix in g m ay b e a m ax im um o n t he e bb t id e w he n

c u r r e n t s h e a r d u e t o b o t t o m f r i c t i o n a n d b a r o c l i n i c f o r c e s a u g m e n t m i x i n g ( G e y e r

a n d S m i t h , 1 9 8 7 ).

I n a dd it io n t o t id al a dv ec ti on , o b se rv ed s tr at if ic at io n m ay b e i nf lu en ce d b y t he

propagation of internal waves generated elsewhere (New et al ., 1987) . A b ra h am(1988) d e s c r i b e s t h e d i f f e r i n g m i x i n g p r o c e s s e s a s s o c i a t e d w i t h e x t e r n a l ( t i d a l b e d

f ri ct io n ) a nd i nt er na l p ro ce ss es , t h e l at te r a re s h ow n t o b e i mp or ta nt ( ar ou nd t he

t i m e s o f s l a c k w a t e r ) f o r s a l t i n t r u s i o n i n t h e R o t t e r d a m W a t e r w a y .

4.1.3 Present approach

T he p re se nt f oc us i s o n e st ab li sh in g a na ly ti ca l s ol ut io ns t o p ro vi de t he or et ic al

f r a m e w o r k s t o e x p l a i n a n d q u a n t i f y , i n s c a l i n g t e r m s , t h e g o v e r n i n g p r o c e s s e s a n d

t he re by i nt er pr et d et ai le d m od e l s tu di es . T h e e mp ha si s i s o n m ix ed a nd p a rt ia ll y

m ix ed e st ua ri es , t hi s p er mi ts t he a ss um pt io n o f a ( te mp or al ly a nd v er ti ca ll y) c on st an t

r e la t iv e a x ia l d e ns i ty g r ad i en t , S x = (l/ ρ)(∂ ρ/ X ) , w i th d e ns it y l i ne a rl y p r op o rt io n al

t o s al in it y. T he a pp ro ac h f ol lo ws e ar li er s ol ut io ns f or t id al ly a v er ag ed l in ea ri se d

t h e o r i e s d e r i v e d b y O f f i c e r (1976) , B o w d e n (1981) a n d P r a n d l e ( 1985).

S e ct i on 4 . 2 e xa m in e s t h e v e rt i ca l s t ru c tu r e o f r e si d ua l c u rr e nt s a s so c ia t ed w i th

s al in e i nt ru si on ( in cl ud in g t he c as e o f a s al in e‘

wedge’

) , r iv er f lo w a nd w in d f or ci ng .T he v er ti ca l s al in it y p ro fi le , c on si st en t w it h t h is c ur re nt s tr uc tu re f or s al in e i nt ru -

s i o n , i s a l s o d e t e r m i n e d .

S e ct i o n 4 . 3 e x am in e s b o th o bs e rv at i on al a n d t he o re ti ca l a p pr oa ch e s f or p re -

d i c t i n g i n t r u s i o n l e n g t h s . T h e s e i n c l u d e ( i ) f l u m e s t u d i e s , ( i i ) c o u n t e r - b a l a n c i n g a t

t h e l i m it o f i n tr u s io n , u p s tr e a m r e si d u al v e l oc i ty a s so c i at e d w i th S x w i th d o w n-

s t r e a m r i ve r f lo w U 0 a nd ( ii i ) a t h eo r et ic al d er iv at io n f or a s tr at if ie d s al t w ed g e. A

f urt he r e st im at e o f i nt ru si on l en gt h i s o bt ai ne d f rom b al an cin g t he r at e o f m ix in g

a s so c ia t ed w it h v e rt ic al d if fu si on a g ai ns t r iv er i nf lo w. M o st s u ch s tu di es a pp ly t oc ha nn el s o f c on st an t b re ad th a nd d ep th . I t i s s ho wn h ow t he f un ne ll in g s ha pe o f

e s tu a r ie s i n t ro d u ce s a f u rt h er ‘d e g re e o f f r e ed o m’ i n d e te r mi n in g t h e a x ia l l o ca t i on ,

a nd t he re by t he l en gt h, of t he i ntr us io n. T his m ay a cc ou nt fo r w id el y r ep or te d

d i ff i cu l ti e s i n i n te r p re t i ng c h an g es i n i n tr u si o n o v er s p ri n g – n ea p a n d f lo od – drought

4.1 Introduction 83



c y cl es . T h e t im e l ag i nv ol v ed i n s uc h a xi al s hi ft s f ur th er c o mp li ca te s e st im a te s o f

intrusion.

In S e ct i on 4 . 4, t he se t id al ly a ve ra ge d l in ea ri se d t he or ie s a re e xt en de d t o t ak e

a c c o u n t o f t i d a l s t r a i n i n g a n d a s s o c i a t e d c o n v e c t i v e o v e r t u r n i n g . T h e i r a p p l i c a b i l i t y

i s e va lu at ed b y c om pa ri so ns w it h a ‘single-point ’ n u me ri ca l m o de l i n w h ic h b o tht h e d e pt h -a v er a ge d t i da l c u rr e nt a m pl i tu d e, U * , a n d a ( te mp or al ly a nd v er ti ca ll y)

c o n s t an t s a l i n e g r a d i e n t , S x, a re s p ec if ie d. T he m o de l h ig h li gh ts t h e i mp or ta nc e o f

c o nv e ct i ve o v e rt u rn i ng i n c o un t er a ct i ng u n s ta b le d e ns i ty s t ru c tu r es i n tr o du c ed b y

t i d a l s t r a i n i n g . T o e x p l o r e t h e g e n e r a l i t y o f e s t u a r i n e r e s p o n s e s , t h e m o d e l i s r u n f o r

a r a n g e o f v a l u e s o f s a l i n e i n t r u s i o n l e n g t h s , LI, a n d w a t e r d e p t h s , D.

S e ct i on 4 . 5 u se s t he a bo ve r es ul ts t o r e- ex am in e w ha t i nd ic es b es t r ep re se nt

s t r a t if i c a t io n l e v e l s i n e s t u a ri e s .

4.2 Current structure for river flow, mixed

and stratified saline intrusion

H er e w e e xa mi ne t he r es id u al c u rr en t p ro f il es a ss oc ia te d w it h ( i) r iv er f lo w, ( ii ) a

w e ll - mi x ed l o ng i tu d in a l d e ns i ty g r ad i e nt ∂ ρ/ ∂ Z , ( ii i) a w ed ge -t yp e i nt ru si on w it h

d e n s i t y d i f f e r e n c e Δ ρ a n d n o v e r t i c a l m i x i n g a t t h e i n t e r f a c e a n d , f o r c o m p l e t e n e s s ,

( i v ) a s u r f a c e w i n d s t r e s s τ w.

I t i s a ss u me d t h at t he se s ep ar at e m ot io ns m ay b e d e sc ri be d b y l in ea r e qu at io ns ,

a n d h e n c e , a c o m p l e t e f l o w d e s c r i p t i o n c a n b e o b t a i n e d b y s i m p l e a d d i t i o n . F o r t h i s

a s s u m p t i o n t o b e v a l i d ( i ) t h e r e s i d u a l c o m p o n e n t o f t h e b e d f r i c t i o n t e r m m u s t b e

e ffe ct iv el y l in ea ri se d a nd ( ii ) t he r at io o f t id al e le va ti on t o m ea n w at er d ep th m us t b e

s ma ll . C on d it io n ( i) i s s at is fi ed i n m os t t id al e st ua ri es w hi le c on d it io n ( ii ) t en ds

m er el y t o q ua li fy t he a cc ur ac y o f t he r es ul ts i n s ha ll ow, m es o- a nd m ac ro -t id al

estuaries.

T o p r o v i d e u s e f u l q u a n t i t a t i v e r e s u l t s , t w o b a s i c a s s u m p t i o n s a r e m a d e . F i r s t , w ea do pt a v a lu e f or t h e b e d s tr es s c oe ff ic ie n t f = 0 .0 02 5 . S ec on d , t he e dd y v is co s it y

coefficient E i s a s s u m e d t o b e c o n s t a n t a n d g i v e n b y ( P r a n d l e , 1982)

E ¼ fU H ; (4:3)

where U * i s t h e d e p t h - m e a n t i d a l c u r r e n t a m p l i t u d e a n d H i s t o t a l w a t e r d e p t h .

Neglecting convective terms, the axial momentum equation at a point Z a b ov e t he

bed is (3.6 )

@ U

@ t þ g

@&

@ X þ gð& Z Þ

1

ρ

@ ρ

@ X ¼

@

@ Z E @ U

@ Z ; (4:4)

where ς i s t h e s u r f a c e e l e v a t i o n a n d ∂ ρ/ ∂ X t h e d e n s i t y g r a d i e n t , a s s u m e d h e r e t o b e

c o ns t an t o v er b o th t i me a n d v e rt i ca l ly.

84 Saline intrusion



4.2.1 Current structure for river flow Q

O mi tt in g t he d en si ty g ra di en t, f or s te ad y a xi al f lo w ( po si ti ve d ow ns tr ea m) ,

E q u a t i o n ( 4 . 4 ) r e d u c e s t o

g d& dX

¼ g i Q ¼ E @ 2

U @ Z 2

; (4:5)

where iQ i s t h e r e si d u al e l ev a ti o n g r ad i en t a s so c ia t ed w i th Q. B y i nt eg ra ti ng (4.5)

twice w.r.t. Z , t h e c u r r e n t p r o f i l e i s

U Q ¼ g i Q

E

Z 2

2 HZ

EH

β

; (4:6)

w he re th e c on st an ts o f i nt eg ra ti on w er e d et erm in ed f ro m t he t wo b ou nd ar yconditions:

s tr es s a t t he s ur fa ce τ Z ¼H ¼ ρE @ U

@ Z ¼ 0 (4:7)

s tr es s a t t he b ed τ Z ¼0 ¼ ρE @ U

@ Z ¼ ρβ U Z ¼0: (4:8)

Condition (4.8) a pp li es w he n U 44U Q , in w hich case Bowd en (1953 ) s ho we d t ha t

β ¼ 4π

f U :

F in al ly, i nt ro du ci ng t he d ep th -a ve ra ge d v el oc it y U Q ¼ Q

H ¼ 1

H

Ð H

0 U dZ ;

E q u a t i o n ( 4 . 6 ) gives

i Q ¼U Q

g H 2

3E þ

H

β

(4:9)

and

U Q ¼ U Q

z2

2 þ z þ

E

β H

1

3 þ

E

β H

(4:10)

with z = Z / H

B y i n s e r t i n g (4.3), (4.9) a n d ( 4 . 1 0 ) r e d u c e t o

i Q ¼ 0:89 f U U

g H (4:11)

and

4.2 Current structure for river flow, mixed and stratified saline intrusion 85



U Q ¼ 0:89 UQ

z 2

2 þ z þ

π

4 : (4:12)

T h i s r e s i d u a l c u r r e n t p r o f i l e f o r r i v e r f l o w i s i l l u s t r a t e d i n Fig. 4.4(a) (Prandle, 1985).

4.2.2 Current structure for a well-mixed horizontal density gradient

A ga in o m it ti ng t he i ne rt ia l t er m b ef or e a dd in g a w el l- mi xe d l on gi tu di na l d e ns it y

gradient ∂ ρ/ ∂x , t h e s t e a d y s t a t e f o r m o f (4.4) yields

U ¼ g d& dX

H 2

E z

2

2 z E

H β

þ g ρ

@ ρ@ X

H 3

E z

3

6 þ z

2

2 z

2 E

2H β

: (4:13)

T o i s o l a t e t h e i n f l u e n c e o f t h e d e n s i t y g r a d i e n t , s u b s c r i p t M , w e d e f i n e

d&

dX ¼ i Q þ i M: (4:14)

T h en s u bt r ac t in g U Q g iv en b y (4.10) from (4.13) a nd a pp ly in g t he c on di ti onU

M ¼ 0 a nd t he e dd y v is co si ty f or mu la ti on (4.3) y ie ld s t he r es id ua l v el oc it y

s t r u c t u r e f o r ‘mixed’ intrusions:

U M ¼ g

ρ

@ ρ

@ X

H 2

f U U

z3

6 þ 0:269 z2 0:037 z 0:029

(4:15)

1.0 1.0 1.0U

Z Z

0.8

0.6

0.4

0.2

0

U / U

WU / F SU

/

F (c)(b)

U w U m

(a)

Z

0.80.4 1.2 –0.2 0 0.2 –0.02 0 0.02

F i g. 4 . 4. Ve r ti c al s t ru c t ur e f o r r i ve r in e , w i n d- d ri v en a n d d e ns i t y- i nd u ce d r e si d u alc u rr e nt s . ( a ) F r es h wa t er f l ow Q (4.12 ) , p oi nt v al ue s i nd i ca te o bs er ve d d at a f ro mt hr ee p os it io n s; ( b) w in d s tr es s τ w (4.37) ; ( c) w el l- mi xe d l on gi t ud in al d en si t ygradient (4.15). W , F and S d e f i n e d i n (4.42).

86 Saline intrusion



and i M ¼ 0:46 H ρ

@ ρ@ X

: (4:16)

T h e c u r r e n t p r o f i l e (4.15) i s i l l u s t r a t e d i n F i g . 4 . 4 ( c ). T h e a s s o c i a t e d n e t i n c r e a s e i n

m e a n s e a l e v e l o v e r t h e i n t r u s i o n l e n g t h i s t h e n a p p r o x i m a t e l y 0 . 0 1 3 H .

4.2.3 Current structure for a stratified two-layer density regime

U s i n g t h e n o t a t i o n s h o w n i n F i g . 4 . 5 , t h e p r e s s u r e a t h e i g h t Z isi n t he l ay er Z D; p ¼ ρ gð& Z Þ (4:17)

i n t he b ot to m l ay er Z 5D; p ¼ ρ gð& DÞ þ ð ρ þ D ρÞ g ðD Z Þ: (4:18)

I n a dd it io n t o t he e ar li er b o un da ry c on di ti on s (4.7) a n d ( 4. 8) , w e r eq ui re c on -

t i n u i t y o f b o t h v e l o c i t y a n d s t r e s s a t t h e i n t e r f a c e ( s u b s c r i p t I ) ; t h u s a t Z = D,

U I ¼ U B ¼ U T (4:19)

and ρ E T@ U T

@ Z ¼ ð ρ þ D ρÞ E B

@ U B

@ Z ; (4:20)

w h er e s ub s cr ip ts T a nd B d e no te v a lu e s i n t he t op a nd b o tt om l ay er s, r es p ec ti ve ly.

F u r t h e r a s s u m in g :

(a) ρ + Δ ρ ρ i n m a ss c a lc u la t io n s n o t i n v ol v in g b u oy a nc y e f fe c ts ,

( b) z er o n e t f l o w i n t he l ow er l ay er, a nd n et f lo w Q i n t h e t o p l a y e r ,

( c) e dd y v i s co si ty i n t he l ow er l ay er g iv en b y a m od if ie d f o r m o f (4.3), n a me l y ,

E B ¼ fU dH (4:21)

with d =D/H ,

( d) e dd y v i s co si t y i n t he t op l ay er g iv en b y

E T ¼ γE B: (4:22)

Q

ρ

ρ + ∆ρ

Z

D

H

h

x

ς

F i g . 4 . 5 . N o t a t i o n f o r a s t r a t i f i e d s a l i n e w e d g e .




T h e n p r o c e e d i n g a s i n e a r l i e r s e c t i o n s , w e o b t a i n

U T ¼ Q"

Hd

1

γ

z2

2 z þ d

d 2

2

0:308d ð1 d Þ

(4:23)

U B ¼ Q"H

ð1 d Þd 2

ð0:574 z2 þ 0:149 zd þ 0:117d 2Þ (4:24)

i s ¼ d&

dX ¼

Q" f U

gH 2 ; (4:25)

where

" ¼ d

ð1 d Þ2 13γ

ð1 d Þ þ 0:308d (4:26)

a ¼ 0:149 1:149d 1 (4:27)

with a ¼

D ρ

ρ

dh

dX

d&

dX

: (4:28)

F r om e x p re s si o ns (4.27) a nd ( 4. 28 ), t he s lo pe o f t he i nt er fa ce , dh/d X , i s a lw ay s

g r e a t e r a n d o p p o s i t e i n s i g n t o t h e p r o d u c t o f s u r f a c e s l o p e a n d d e n s i t y d i f f e r e n c e ,

(Δ ρ/ ρ dς /d X ) e x c e p t a t t h e s u r f a c e , d = 1 , w he re t he y a re o f e qu a l m a g ni tu de .

I t i s d if fi cu lt t o d e te rm in e a g en er al f or mu la ti on f or γ. P r a n d l e (1985 ) e x am i ne d

v a ri o us p o s si b il i ti e s a g ai n s t o b se r ve d c u rr e nt p r o fi l es a n d s u gg e st e d t h e f o ll o w in g

expedient:

γ ¼ 1 d

d : (4:29)

T h i s p e r m i t s t h e a d d i t i o n a l s i m p l i f i c a t i o n s :

E T ¼ f U ðH DÞ (4:30)

i S ¼ 1:56

ð1 d Þ2

Q f U

g H 2 (4:31)

U T ¼ 1:56 QH

1ð1 d Þ3 z

2

2 z 0:808d 2 þ 1:616d 0:308

(4:32)

U B ¼ 1:56Q

H

1

d 2ð1 d Þ0:574 z2 þ 0:149 zd þ 0:117 d 2

: (4:33)

88 Saline intrusion



4.2.4 Current structure for a constant surface wind stress τw

P r o c e e d i n g a s f o r n e t f l o w Q, b u t w i t h U W ¼ 0 a n d t h e s u r f a c e b o u n d a r y c o n d i t i o n

τ W ¼ ρE @ U

@ Z

; (4:34)

where τ w i s a n i m p o s e d w i n d s t r e s s a n d s u b s c r i p t W i s u s e d t o d e n o t e w i n d - d r i v e n

c o m p o n e n t s , w e o b t a i n

U W ¼ τ W H

ρE

z2

2

1

2 þ

E

β H

z

6

1

6

E

β H

1

3 þ

E

β H

(4:35)

and i W ¼ τ W

ρ gH

0:5 þ E

β H

1

3 þ

E

β H

(4:36)

o r b y i n t r o d u c i n g (4.3)

U W ¼ τ W

ρ fU 0:574 z2 0:149 z 0:117 (4:37)

and i w ¼ 1:15τ w

ρ gH : (4:38)

T he fa ct or 1 .1 5 i n (4.38) l ie s b et we en t he v al ue 1 .5 o bt ain ed f or a n o- sl ip b ed

c o nd it io n a nd 1 .0 f or a f ul l- sl ip c on d it io n b y R os si te r (1954) . T h e c ur re nt p ro fi le

(4.37) i s i l l u s t r a t e d i n F i g . 4 . 4 ( b ).

4.2.5 Time-averaged vertical salinity structure (mixed)S ca li ng a na ly si s c an b e u se d t o j us ti fy th e n eg le ct o f W ∂C / ∂ Z in (4.1), and

o b se rv a ti o ns i nd ic at e t he p re do mi na nc e o f t he v er ti ca l d if fu s io n t er m. T h en ce ,

assuming ∂C / ∂ X = S x, o m i t t i n g t i m e v a r y i n g t e r m s , w e o b t a i n , f o r K z c o n st a n t , t h e

s a l i n i t y s t r u c t u r e :

sðZ Þ ¼

ð ð ρ U M

S x

K z

dZ dZ ; (4:39)

i . e . f r o m (4.15), d e f i n i n g v e r t i c a l v a r i a t i o n s i n s a l i n i t y a s s0 ( Z ) = s( Z ) – S ;

s0 ¼ ρ g S 2x

E z K z

D5

1 0 0 0 0 83 z5 þ 224 z 4 62 z3 146 z2 þ 33

: (4:40)

T h i s t i m e - a v e r a g e d s a l i n i t y p r o f i l e i s s h o w n i n f i g . 4 . 8 .




4.2.6 Summary

F i g u r e 4 . 4 s h o w s t h e r e s i d u a l c u r r e n t p r o f i l e s (4.12), (4.15) a n d ( 4 . 3 7 ) p e r t a i n i n g t o

r iv er f lo w, m ix ed s al in e i nt ru si on a nd w in d f or ci ng . Ta b le 4 . 1 s u m ma r i s e s t h e s e

r es ul ts s ho wi ng c or re sp on di ng v al ue s a t t he s ur fa ce a nd b ed a lo ng w it h r el at ed

g r ad i en t s i n s u rf a ce e l ev a ti o n .

T o i n t e r p r e t t h e m a g n i t u d e s o f t h e s e r e s i d u a l f l o w c o m p o n e n t s , w e i n t r o d u c e t h e

d e p t h - a v e r a g e d e q u a t i o n o f m o t i o n f o r s t e a d y s t a t e r e s i d u a l f l o w :

@ ζ

@ X þ

H

2 ρ

@ ρ

@ X

τ w

ρ gH þ

4

π

fU U 0

gH ¼ 0: (4:41)

T h e n , i n t r o d u c i n g t h e d i m e n s i o n l e s s p a r a m e t e r s

S ¼ H @ ρ ρ@ X

; W ¼ τ W ρ gH

; F ¼ f U

U 0

gH (4:42)

from (4.41), t h e f o r c i n g t e r m s a s s o c i a t e d w i t h d e n s i t y , w i n d a n d b e d f r i c t i o n a r e i n

t h e r a t i o

S

2 : W :

4

π F : (4:43)

From Ta bl e 4 .1, t he d en si ty f or cin g t erm S / 2 i s b al an ce d b y a s ur fa ce g ra di en t – 0.46S

w it h t he r em ai ni ng c om po ne nt d ri vi ng a r es id ua l c ir cu la ti on o f 0 .0 36 SU 0/ F seawards

a t t he s ur fa ce a nd −0.029 SU 0/ F ( l an d wa rd s ) a t t he b e d. S i mi la r ly, t h e w i nd s t re ss

term W i s c ou nt er ac te d b y a s ur fa ce g ra di en t 1 .1 5W w it h t he ‘excess’ b al a n ce d r i vi n g

a c i r c u l a t i o n o f 0 . 3 1 WU 0/ F a t t he s ur fa ce a nd −0.12 WU 0/ F a t t h e b e d . C l e a rl y, w i nd

f or c in g i s m or e e ff e ct iv e i n p r od u ci n g a r e si d ua l c ir c ul at i on t h an l o ng i tu d in a l d e ns it y

g r a d i e n t s a n d b o t h f o r c i n g s i n f l u e n c e e l e v a t i o n s t o a g r e a t e r e x t e n t t h a n c u r r e n t s .

4.3 The length of saline intrusion

4.3.1 Experimental derivation

R ig te r (1973 ) c ar ri ed o ut a n e xt en si ve s tu dy o f i nt ru si on l en gt hs b ot h i n a l ab or at or y

f lu me a nd in t he R ot te rd am Wa te rw ay. T he f lu me w as o f c on sta nt d ep th a nd

breadth. Analysis of the experiments resulted in an expression for saline intrusion

l e n g t h ( P r a n d l e , 1985) a s f o l l o w s :

LI ¼ 0:18 g D ρ= ρð ÞD2

f 0 U U 0

¼ 0:005 D2

f U U 0

; (4:44)

w he re t he l at te r a pp li es f or r ea l e st ua rie s w it h Δ ρ/ ρ 0 .0 27 . P ra nd le (2004b)

i n d i c a t e s t h a t t o m a i n t a i n t u r b u l e n t R e y n o l d s n u m b e r s i n f l u m e s t u d i e s , t h e f r i c t i o n

90 Saline intrusion



f ac to r m us t b e i nc re as ed b y t he s am e a mo un t a s t he v er ti ca l s ca le e xa gg er at io n,

i.e. f 0 = 10 f .

A c om pa ri so n o f o bs er va ti on s f ro m t hi s f lu me s tu d y a nd c om pu te d v al u es f or LI

from (4.44) is s how n in F ig . 4 .6 (Prandle, 1985) f or a r an ge o f t id al a mp li tu de s, d en si ty

differences, bed friction coefficients, river flows, water depths and flume lengths. The

e x ce l le n t a g re e me n t i s c o nf i rm e d b y a c o rr el a ti o n c o ef fi c ie n t c a lc u la t ed a s R = 0 .9 7,

indicating the robustness of (4.44) o v e r a r a n g e o f p a r a m e t e r s e n s i t i v i t y t e s t s .

4.3.2 Derivations from velocity components

A s im pl e h yp ot he si s f or t he l im it o f u ps tr ea m i nt ru si on i s t he p os it io n w he re a b al an ce

e xi st s a t t he b ed b et we en ( i) t he u ps tr ea m v el oc it y a ss oc ia te d w it h a w el l- mi xe d

s a l i n i t y g r a d i e n t a n d ( i i ) t h e d o w n s t r e a m r i v e r v e l o c i t y . F r o m Table 4.1, t h i s r e q u i r e s

0:029 g D2 S x

f U ¼ 0:7 U 0 (4:45)

i:e: LI ¼ 0:011 D2

f U U 0

; (4:46)

where S x = 0 . 0 2 7 / LI.

0 10

10

20

30

40

50

20LR

LC

30 40 50

F i g . 4 . 6 . C o mp u te d v e rs u s o b s er v ed s a l in i t y i n t ru s i on l e n gt h s . C o mp u te d , LC (4.44);observed, LR , f ro m f lu me t es ts ;⊗ r ef er en ce t es t, v ar yi ng : × t i da l a mp li tu de ,♦ densitydifference, ▲ bed friction, ★ river flow, ■ water depth, + channel length.

4.3 The length of saline intrusion 91



4.3.3 Derivation for a stratified ‘ salt wedge’

F r o m i n t e g r a t i o n o f t h e s l o p e o f t h e i n t e r f a c e , P r a n d l e ( 1985) d e r i v e d t h e f o l l o w i n g

e x p r e s s i o n f o r t h e l e n g t h o f i n t r u s i o n o f a s t r a t i f i e d s a l i n e w e d g e

LI ¼ 0:26 g D2

f U U 0

D ρ ρ

¼ 0:07 D2

f U U 0

; (4:47)

where Δ ρ/ ρ 0.027.

T h i s r e s u l t m a y b e c o m p a r e d w i t h t h e f o l l o w i n g e x p r e s s i o n f o r t h e l e n g t h , LA, of

a n a r r e s t e d s a l i n e w e d g e g i v e n b y K e u l e g a n ( I p p e n , 1966, Ch 1):

LA ¼ A g5=4 D9=4

U 5=2

0

D ρ

ρ 5=4

; (4:48)

where A is a p ar am et er w hi ch v ar ie s w it h r iv er c on di ti on s. S in ce K eu le ga n’s

e xp re ss io n w as d er iv ed f ro m o b se rv at io na l d at a, t he c or re sp on d en ce w it h (4.47)

a d d s u s e f u l s u p p o r t t o t h e p r e s e n t t h e o r e t i c a l r e s u l t .

O f f i c e r ( 1 9 7 6 , Ch 4) p r o v i d e s a l t e r n a t i v e d e r i v a t i o n s f o r t h e l e n g t h o f a n a r r e s t e d

s al t w ed ge , t he r es ul ts o bt ai ne d o bv io us ly d ep en d o n t he p ar ti cu la r d yn am ic al

assumptions.

4.3.4 Mean value for saline intrusion length, L I

T h e a b o v e f o r m u l a t i o n s (4.44), (4.46) a n d ( 4 . 4 7 ) a l l s h o w i d e n t i c a l e x p r e s s i o n s f o r

LI b u t w i th q u an t it a ti v e v a lu e s v a ry i ng b y t h e r e sp e ct i ve c o ef fi c ie n ts 0 . 00 5 , 0 . 01 1

a nd 0 .0 7. To r ec on ci le t he se , w e n o te t ha t t he b al an ci ng o f s ea -b e d v el oc it y c om -

ponents in (4.46) s ho u ld s tr ic tl y u se v a lu e s o f D and U a t t he l an dw ar d l im it o f

t he w e dg e, r es ul ti ng i n a d ec re as e i n t he c oe ff ic ie nt s h ow n. L ik ew is e, i t i s w i de ly

o b s e r v e d t h a t i n t r u s i o n l e n g t h s i n s t r a t i f i e d c o n d i t i o n s a r e s i g n i f i c a n t l y l o n g e r t h a ni n m i x e d . T h u s , h e n c e f o r t h w e a d o p t t h e v a l u e (4.44).

A s u bs e qu e n t d e ri v at i on (4.59), b as ed o n b al an ci ng t he m ix in g r at es a ss oc ia te d

w i t h t h e d e n s i t y s t r u c t u r e (4.54) w i t h r i v e r f l o w (4.57), p r o v i d e s a f u r t h e r e s t i m a t e

for LI i n c l o s e a g r e e m e n t w i t h (4.44).

Observed versus computed values

A m ajo r d iff icu lt y i n a ss es si ng t he va li di ty o f (4.44) i s th e p au ci ty o f ac cu ra te

o b se rv at io na l d at a . P ra ct ic al a p pl ic at io n o f t he se e s ti ma te s f or s al in e i n tr us io nsl en gt hs a re c om pl ic at ed b y a w id e s pe ct ru m o f a ss oc ia te d r es po ns e t im es. T he se

r an ge f ro m m in u te s f o r t u rb ul en t i n te ns it y l e v el s t o h ou rs f or e ff ec ti ve v e rt i ca l m ix in g

(S ec ti on 4 .5) a nd d ay s f or e st ua rin e f lu sh in g (4.60). H er e, w e u ti l is e t h e o b se rv at io na l

d at a p r ov id ed b y P ra n dl e (1981) f ro m s ix e st ua ri es ( ei gh t d at a s et s) s um ma ri se d i n

92 Saline intrusion



Table 4.2. T hi s d at a s et p ro vi d es e st im a te s f or i n tr us io n l e ng th , LI, riverflow, Q, a n d

estuarine bathymetry. Values for D and U 0 a t t he c en tr e o f t he i ntr us io n w er e e st im ate d

f r om p o w er s e ri e s a p pr o x im a ti o ns t o b r ea d t h ( xn) a n d d e p t h ( xm) . T h e v a l u e s f o r U *

a r e e s t i m a t e d f r o m (6.9) derived for synchronous estuaries.

F ig u re 4 .7 (Prandle, 2004b ) s ho ws e st im at es o f LI from (4.44) based on values for

U 0

, U * a n d D a t s u cc e ss i ve p o si t io n s a l on g t h e e s tu a ri e s l i st e d i n Ta b l e 4 . 2. T h e s e

v al ue s a re s um ma ri se d i n Ta b l e 4 . 3, i nd ic at in g v al ue s o f U 0, U * and D at the

locations X c, w h e r e t h e v a l u e o f LI from (4.44) e q ua l s t h e o b se r ve d v a lu e L0. I t i s

e nc o ur ag in g t o n o te (Ta b l e 4 . 3 ) t ha t t he v al ue s o f X c a r e i n r e as o n ab l e a g re e me n t

w i t h t h e r e l a t e d o b s e r v a t i o n a l v a l u e s X 0. V a l u e s o f U 0 r an g e f r om 0 .1 7 t o 0 .5 7 c m−1

w i t h t h e e x c e p t i o n o f t h e S t . L a w r e n c e w h e r e U 0 = 1 .4 c m−1.

4.3.5 Axial location of saline intrusion

F i gu r e 4 . 7 a l so i nd ic at es t he l an dw ar d l im it s o f s al in e i nt ru si on X u = ( X c – LI/2)/ L

c o rr e sp o n di n g t o s u c ce s si v e v a lu e s o f X c. We n ot e t ha t, w it h t he e xc ep ti on o f t he

H u d s o n a n d D e l a w a r e , t h e l o c a t i o n s o f X c, w h e r e o b s e r v e d a n d c o m p u t e d v a l u e s o f

LI a r e e q u a l , c o r r e s p o n d w i t h o r a r e s l i g h t l y s e a w a r d s o f m a x i m u m v a l u e s o f X u, i . e .

w h e r e t h e l a n d w a r d l i m i t o f s a l i n e i n t r u s i o n i s a m i n i m u m .

A d op ti ng t hi s l at te r r es ul t a s a c ri te ri on t o d et er mi ne t he p o si ti o n, xi, w h e r e t h e

s a l i n e i n t r u s i o n w i l l b e c e n t r e d , r e q u i r e s i n d i m e n s i o n l e s s t e r m s , w i t h x = X / L:

@

@ xðx 0:5l iÞ ¼ 0; (4:49)

where l i = LI/ L. U t il i si n g (4.44) a n d i n tr o du c i ng t h e s h al l ow w a te r a p pr o x im a ti o n,

(6.9), f o r c u r r e n t a m p l i t u d e i n r e l a t i o n t o e l e v a t i o n a m p l i t u d e , ς *,

T a b l e 4 . 2 Estuarine parameters

n m L(km) LI(km) D(m) B(km) Q(m3 s1) ς *(m)

(A) Huds on 0.7 0.4 248 99 11.6 3.7 99 0.8

(B) Pot omac 1.0 0.4 184 74 8.4 18 112 0.7(C) Delaware 2.2 0.3 214 43 4.4 28 300 0.6(D) Brist ol Ch. 1.7 1.2 138 55 29.3 20 80 4.0(E) Brist ol Ch. 138 8500 4.0(F) Thames 2.3 0.7 95 76 12.6 7 480 2.0(G) Thames 38 19 2.0(H) St. Lawrence 1.5 1.9 48 167 74 48 210 1.5

Notes: L e s t u a r in e l e n g th , LI o b s e r v e d i n t r u s i o n l e n g t h ; n, m b r e a d t h a n d d e p t h v a r i a t i o n s( xn, xm), Q r i v e r f l o w, D depth, B b r e a d t h a n d ς * t i d a l a m p l i t u d e a t t h e m o u t h .Source: Prandle, 1981 .




U 2 ¼ & oð2 gDÞ1=2

1:33 f : (4:50)

T h e n f u r t h e r a s s u m i n g Q =U 0 Di

2

/tan α, w h e r e t a n α i s t h e s i d e s l o p e f o r a t r i a n g u l a r c r o s s s e c t i o n , w e o b t a i n

x2i ¼

333 Q tan α

D5=20

: (4:51)

1.0(a) (e)

(f)

(g)

(h)

(b)

(c)

(d)

0.5

0.0

–0.5

–1.0

1.0

0.5

0.0

–0.5

–1.0

1.0

0.5

0.0

–0.5

–1.0

1.0

0.5

0.0

–0.5

–1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

1.0

0.5

0.0

1.0

0.5

0.0

–0.5

–1.00.0 0.2 0.4 0.6 0.8 1.0

1.0

0.5

0.0

1.0

0.5

0.0

–0.5

–1.0

0.0 0.2 0.4 0.6 0.8 1.0

1.0

0.5

0.0

1.0

0.5

0.0

–0.5

–1.00.0 0.2 0.4 0.6 0.8 1.0

1.0

0.5

0.0

1.0

0.5

0.0

–0.5

–1.00.0 0.2 0.4 0.6 0.8 1.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

1.0

0.5

0.0

F i g . 4 . 7 . R a t i o o f c o m p u t e d , Lc, t o o b s e r v e d , L0, s a l i n e i n t r u s i o n l e n g t h s a t v a r y i n glocations, X c. R es ul ts f or ( a) H ud so n ( b) P ot om ac ( c) D el aw ar e ( d) B ri st ol C ha nn el ,Q = 8 0 m3 s−1 ( e ) B ri s to l C h an n el , Q = 4 8 0 m3 s−1 ( f ) T h am e s, Q = 1 9 m3 s−1 (g)

Thames, Q = 2 1 0 m3 s−1 a n d ( h ) S t . L a w r e n c e . H o r i z o n t a l a x i s , X C / L. P o i n t v a l u e X

i s t h e c e n t r e o f o b s e r v e d i n t r u s i o n . V e r t i c a l a x i s , ( L H S ) l o g 10 LC/ L0, f u l l l i n e , ( R H S )l a n d w a r d l i mi t o f i n t r u s i o n ; X U = ( X C – LI/2)/ L, d a s h e d l i n e ; LC from (4.44).

94 Saline intrusion



B y f ur th er i nt ro d uc in g t h e d e ri va ti o n f or a s yn ch ro n ou s e st ua ry i n C h ap t er 6 that

d ep th s a nd b re ad th s v ar y w it h x 0.8, t he d ep th , Di at xi is D0 xi0.8 g iv in g a r es id ua l

velocity U 0 a t t h e c e n t r e o f t h e i n t r u s i o n :

U 0 ¼ D

1=2

i

333 m s1: (4:52)

T h is e x pr e ss i on y i el d s v a lu e s o f U 0 = 0 . 0 0 6 m s−1 for D = 4 m , 0 . 0 1 2 m s−1 for

D = 1 6 m .

Noti ng that ( 4 . 5 1 ) c o r re s p on d s t o l i = 2 / 3 xi, t he se v al ue s f or U 0 w i l l i n c re a s e

by a fact or of 2 at the upst ream limit and decre ase by 40% at the down strea m

l i mi t . N o ti ng a l so t h e i na cc u ra cy i nh e re nt i n m e as ur e me n ts o f U 0, w e c on cl ud e

t ha t t h es e e s ti m at e s o f r es i du a l v el o ci t y a s so c ia te d w i th r i ve r f l ow i n t he s a li n e

i nt ru si on r eg io n a re r ea so na bl y c on si st en t w it h t he o bs er ve d v al ue s s ho wn i n

T a b l e 4 . 3 .I f i n p ro ce ed in g f ro m (4.49) to (4.51) w e i n tr o du c e e s tu a ri n e b a th y me t ry w i th

breadth B0 xn a n d d e p t h D0 x

m, w e o b t a i n t h e f o l l o w i n g a l t e r n a t i v e f o r m f o r (4.51)

xi ¼ 855 Q

D3=20 B0 11m=4 þ n 1ð Þ

!1=ð11m=4Þþn1

: (4:53)

A n e sp ec ia ll y in ter es ti ng f ea tu re o f t he r es ult s f or t he a xia l l oc at io n o f s al in e

intrusion (4.51) a nd (4 .5 3) a nd t he e xp re ss io n fo r re si du al r iv er f lo w c ur re nt (4.52) i s t h ei r i nd e pe nd en ce o f b ot h t id al a mp li tu de a nd b ed f ri ct io n c oe ff ic ie nt .

( Al th ou gh t he re i s a n i mp li ci t r eq ui re me nt t ha t t id al a mp li tu de i s s uf fi ci en t t o

m a in t ai n p a rt i al l y m i xe d c o n di t io n s .) E q ua t io n s (4.51) a n d ( 4. 53 ) e mp ha s is e h ow

t h e c e n t r e o f t h e i n t r u s i o n a d j u s t s f o r c h a n g e s i n r i v e r f l o w Q. T h i s ‘a x i a l m i g r a t i o n’

T a b l e 4 . 3 Observed and computed estuarine parameters for saline intrusion

X 0 X c U 0 ( c m s−1) U* (m s−1) D (m)

(A) Huds on 0.8 0 0.70 0.35 0.59 9.7

(B) Pot omac 0.6 0 0.50 0.20 0.54 6.3(C) Delaware 0.8 0 1.0 0.29 0.47 4.4(D) Brist ol Ch. 0.3 0 0.40 0.22 1.53 9.5(E) Brist ol Ch. 0.5 5 0.65 0.28 1.70 17.3(F) Thames 0.6 0 0.55 0.17 1.01 8.0(G) Thames 0.7 5 0.50 0.57 1.06 10.2(H) St. Lawrence 0.6 0 0.65 1.4 0.80 30.0

Notes: X 0 ( f r a c t i o n o f L ) c e n t r e o f o b s e r v e d i n t r u s i o n , X c c e n t r e o f i n t r u s i o n w h e n LI ( 4 . 4 4 ) = L0, o b s e r v e d ; D , U 0 = Q/ a r e a , U * d e p t h , r e s i d u a l a n d t i d a l c u r r e n t s a t X c.Source: Prandle, 2004a.




c an s ev er el y c om pl ic at e t he s en si ti vi ty o f s al in e i nt ru si on b ey o nd t he a n ti ci pa te d

d i r e c t r e s p o n s e s a p p a r e n t f r o m t h e e x p r e s s i o n (4.44) f o r t h e l e n g t h o f i n t r u s i o n , LI.

4.4 Tidal straining and convective overturning

T h e p r e c e d i n g s t u d i e s , i n v o l v i n g t i d a l l y a v e r a g e d l i n e a r i s e d t h e o r i e s r e l a t i n g t o t h e

v e r t i c a l s t r u c t u r e o f s a l i n i t y a n d v e l o c i t i e s , a r e n o w e x t e n d e d b y n u m e r i c a l s i m u l a -

t i o n s i n c o r p o r a t i n g t i d a l s t r a i n i n g a n d a s s o c i a t e d c o n v e c t i v e o v e r t u r n i n g .

4.4.1 Rates of mixing

B e f o r e d e s c r i b i n g t h e m o d e l l i n g s t u d y , w e f i r s t e x a m i n e r a t e s o f m i x i n g a s s o c i a t e d

w it h ( i) t im e -a ve ra ge d d en s it y s tr uc tu re , ( i i) t id al s tr ai n in g a nd ( i ii ) s u pp ly o f f re sh -water velocity.

From (4.1), the m ean rate of mixing M K a sso ciated with the

t i m e - a v e r a g e d d e n s i t y s t r u c t u r e (4.40) i s ( S im p so n et al ., 1990)

M K ¼ ρ

ð D

0

@

@ Z K z

@ s

@ Z

dZ ¼ 0:02 ρ gS 2x D4

E : (4:54)

T he m ea n r at e o f o ve rt ur ni ng t o c om pe ns at e f or t id al s tr ai ni ng o n t he f lo od p ha se M 0 is

M 0 ¼ ρ 2

π U S x D

ð 1

0

0:7 þ 0:9z 0:45z2 1

dz (4:55)

¼ 0:12 ρS x DU ; (4:56)

w h e r e t h e a p p r o x i m a t i o n f o r t h e v e r t i c a l s t r u c t u r e o f t i d a l c u r r e n t s i s s i m i l a r t o t h a t

o f B o w d e n a n d F a i r b a i r n ( 1952) . N u n e s V a z a n d S i m p s o n ( 1994) d e r i v e a d d i t i o n a l

c on tr ib ut io n s t o v er ti ca l m ix in g a ss o ci at ed w it h w in d , s ur fa ce h ea t e xc ha ng e a nd

evapo-transpiration.

F o r a s ta ti o na ry s al in it y d is tr ib u ti on , t he m ix in g r at e M Q t o b a la n ce f r es h wa t er

velocity U 0 is

M Q ¼ ρU 0 S x D: (4:57)

T h e n t o b a l a n c e M Q, w e e x p e c t v a l u e s o f U 0 of

tidal straining on flood tide 0:12U (4:58)

vertical salinity difference 0:02 gS x D3

E : (4:59)

96 Saline intrusion



For U * = 0 . 5 m s−1, a ss um in g n o m ix in g o n t he e bb t id e, (4.58) i n di ca te s a v al ue

of U 0 = 3 c m s−1. S in ce t hi s v al u e g e ne ra ll y e xc ee ds t ha t a ss oc ia te d w it h v e rt ic al

s a l i n i t y d i f f e r e n c e s (4.59), w e e x p e c t t h a t t i d a l s t r a i n i n g w i l l o f t e n e l i m i n a t e v e r t i c a l

s al in it y d iff er en ce s b y t he e nd o f th e f lo od t id e, s ee F ig . 4 .1 0. E q u at i on ( 4 .5 9 )

provides an expression for saline intrusion length LI a l m o s t i d e n t i c a l t o (4.44).F o r a p ri s ma ti c c h an n el , t h e e s tu a ri n e f l us hi n g t i me , T F c an b e a p pr o xi ma t ed

by the time taken to replace half of the fresh water in the intrusion zone by river

f l o w, i . e .

T F ¼ 0:5ðLI=2Þ

U 0

: (4:60)

F o r t y p i c a l v a l u e s o f T F = 2 – 1 0 d a y s , t h i s r e q u i r e s U 0 v ar yi ng f ro m 1 . 5 t o 3 c m s−1

o ve r t he r an ge o f i nt ru si on l en gt hs LI = 12.5 to 1 00 km considered in F i gs 4 .11a n d 4 . 1 2 .

4.4.2 Modelling approach (Prandle, 2004a )

T he s ig ni fi ca nc e o f t id al s tr ai ni ng i s e va lu at ed b y r ef er en ce t o a ‘single-point ’

n um e ri ca l m o de l i n w hi ch t h e t im e- va ry in g c yc le o f d ep th -a ve ra ge d t id a l c ur re nt

amplitude, U * , a nd a ( te mp o ra ll y a nd v er ti ca ll y) c on st an t s al in e g ra di en t, S x, a r e

s p e c i f i e d ( P r a n d l e , 2004a). T h e m o d e l i l l u s t r a t e s t h e r o l e o f c o n v e c t i v e o v e r t u r n i n g

i n c ou nt er ac ti ng u ns ta bl e d en si ty s tr uc tu re s i nt ro du ce d b y t id al s tr ai ni ng . T he

v al id it y o f t he m od el i s e va lu at ed b y s im ul at io n o f m ea su re me nt s b y R ip pe th

et al . (2001).

To e xp lo re t he g en er al it y o f e st ua ri ne r es po ns es , t he m od el i s r un f or a w id e r an ge

o f v a l u e s o f s a l i n e i n t r u s i o n l e n g t h s , LI , a n d w a t e r d e p t h s , D. A d d i t i o n a l s e n s i t i v i t y

a n a l y s e s a r e m a d e f o r c h a n g e s i n U * a n d t h e b e d s t r e s s c o e f f i c i e n t , f . T h e m o d e l i s

u se d t o c al cu la te t he f ol lo wi ng p ar am et er s: s ur fa ce -t o- be d d if fe re nc es i n b ot hr e s i d ua l v e l o c it y, δu, a n d s a li n it y, δ s; t h e p ot en ti al e n er gy a no ma ly M (4.65); t h e

r a ti o o f m i xi n g b y d i ff u s iv i ty : o v er t ur n in g ; t h e e f fi c ie n cy o f m i x in g ε a n d t h e r i v e r

flow U 0 ( r e q u i r e d t o b a l a n c e m i x i n g r a t e s ) . T h u s , t h e m o d e l s i m u l a t i o n s i l l u s t r a t e ,

o v e r a w i d e r a n g e o f e s t u a r i n e c o n d i t i o n s , t h e i m p a c t o f t i d a l s t r a i n i n g a n d , t h e r e b y ,

t h e s i g n i f i c a n c e o f i t s o m i s s i o n i n t h e a n a l y t i c a l f o r m u l a t i o n s .

4.4.3 Model formulation

W e a s s u m e ( v e r t i c a l l y a n d t e m p o r a l l y ) c o n s t a n t c o e f f i c i e n t s f o r e d d y v i s c o s i t y a n d

d if fu si vi ty ; h o we ve r, a p ar ti al a ss es s me nt o f b uo y an cy m o du la ti on o f e d dy v is co s it y

a n d d i f f u s i v i t y i s e x a m i n e d b y i n t r o d u c i n g t h e f o r m u l a t i o n s o f M u n k a n d A n d e r s o n

(1948) :

4.4 Tidal straining and convective overturning 97



ed d y v i scosi ty E z ¼ f U D ð1 þ 10RiÞ1=2

(4:61)

ed d y d i ffusi v i ty K z ¼ f U D=S c ð1 þ 3:33RiÞ3=2; (4:62)

w h er e t h e R i c ha r ds o n n u mb e r, Ri

, i s g i v e n b y (4.2) and S c

i s t he S ch mi dt n um be r

r e p r e s e n t i n g t h e r a t i o E : K z. N u n e s V a z a n d S i m p s o n ( 1994) d e s c r i b e a w i d e r a n g e

o f s u c h f o r m u l a t i o n s .

A s i n g l e - p o i n t n u m e r i c a l m o d e l w a s f o r m u l a t e d p r o v i d i n g s o l u t i o n s t o (4.1) and

( 4. 4) . T he m od el i s f or ce d v ia s pe ci fi ca ti on o f t he d ep th -a ve ra ge d s em i- di ur na l

t i da l c u rr e nt a m pl i tu d e, U * , a nd t he ( te mp o ra ll y a nd v er ti ca ll y c on st an t) s al in i ty

gradient, S x. T he m od el w as r un fo r a ( sm al l) n um be r o f t id al c yc le s to a ch ie ve

c y c l i c c o n v e rg e n c e .

B ou nd ar y c on di ti on s we re s pe ci fi ed as z er o s tr es s a t t he s ur fa ce an d b eds t r e s s

τ Z ¼ ρ f U 2z ¼ 0: (4:63)

T h e m o d el i n co r po r at e s c o nv e ct i ve o v e rt u rn i n g w h en e v er d e ns i ty d e cr e as e s w i th

depth.

To d is ti n gu is h t he e ff ec t o f ( i) b uo ya nc y m od ul at io n o f E and K z (via (4.61) and

( 4. 62 ) ) a nd ( ii ) c o nv ec ti ve o v er tu rn in g, t h re e s ep ar at e s im ul at io ns w e re m ad e . T he f ir st

s im ul at io n u s ed c on st an t v a lu e s o f E = fU * D and K z = S c E . T he s ec ond m od if ie d t he se

values according to (4.61) a n d ( 4 . 6 2 ) a n d t h e t h i r d i n t r o d u c e d c o n v e c t i v e o v e r t u r n i n g

( i n a d d i t i o n t o (4.61) a n d ( 4 . 6 2 ) ) . R e s u l t s f r o m t h e f i r s t t w o s i m u l a t i o n s s h o w e d l i t t l e

d i ff e re n ce , e x ce p t f o r a p pl i c at i on s i n v ol v i ng p r on o u nc e d s t ra t if i c at i o n w h e re (4.1) and

( 4.4 ) b eco me i nva li d. H en ce , r es ult s a re o nl y s ho wn f or ( i) S im ul at io n 1 w he re

E = fU * D, K z = S c E w i t h o u t c o n v e c t i v e o v e r t u r n i n g a n d ( i i ) S i m u l a t i o n 3 w i t h E and

K z from (4.61) a n d ( 4 . 6 2 ) a n d c o n v e c t i v e o v e r t u r n i n g i n c l u d e d .

4.4.4 Model results

Currents

F i gu r e 4 . 8( a ) sh o ws t id al ly a ve ra ge d ( re si du al ) c ur re nt p ro fi le s o bt ai ne d f ro m

S im ul at io ns 1 a nd 3 t og et he r w it h t he t he or et ic al p ro fi le (4.15). T he p ar am et er s,

U * = 0 . 6 m s−1 f = 0 .0 02 5, D = 3 2 m , L = 4 0 0 k m , S c = 0 .1 , w e r e u s e d t o c or re sp on d

t o t he o bs er va ti on s b y R ip pe th et al . (2001). Wi th ou t o ve rt ur ni ng , t he m od el

a c cu r at e ly r e pr o du c ed t h e e a rl i er t h eo r et i ca l v e rt i ca l p r of i le s o f b o t h t i da l c u rr e nt a mp li tu de a nd p ha se . T he r es ul ts f or m od el S im ul at io n 1 a nd t he t he or et ic al s ol ut io n

(4.15) ar e n ea r- id en ti ca l. B y c on tr as t, t h e r es ul ts f or S im ul at io n 3 s ho w s im il ar

profiles but with the magnitude doubled.

98 Saline intrusion



S e ns it iv i ty a na ly se s o f S im ul at io n 3 s ho w ed l it tl e v ar ia ti on w it h c ha n ge s i n t he

S c h m i d n u m b e r. H o w e v e r, d o u b l i ng U * p r o d u c e s a o n e - t h i r d r e d u c t i o n i n r e s i d u a l

v e l o c i t i e s w h i l e q u a d r u p l i n g f p r o d uc e s a t w o -t h ir d r e du c ti o n.

Salinity

F i g ur e 4 . 8 (Prandle, 2004a) s h ow s t id al ly a ve ra ge d s al in it y p ro fi le s ( co nv er te d t o

psu) for these same simulations compared with the theoretical profile from (4.40).

A g a i n , t h e t h e o r e t i c a l r e s u l t i s n e a r - i d e n t i c a l t o S i m u l a t i o n 1 . H o w e v e r , S i m u l a t i o n3 s ho ws r es ul ts f ou r t im es l arg er w it h a n et s ur fa ce -t o- be d v al ue o f δ s = 0 .2 5‰

s i m i l a r t o t h e o b s e r v e d v a l u e s i n d i c a t e d b y R i p p e t h et al . (2001).

F i gu r e 4 . 9 (Prandle, 2004a) s h o w s a t i d a l c y c l e o f d i s s i p a t i o n d u e t o ( i ) b o t t o m

friction ρ fU 3/ D a nd ( ii ) d ep t h- av er ag ed v er ti ca l s he ar ρ E (∂U / ∂ Z )2. The former

1.0

0.8

0.6

0.4

0.2

0.0 –0.04 –0.02 0.00 0.02 0.04

1.0

0.8

0.6

0.4

0.2

0.0 –0.15 –0.075 0.0 0.075 0.15

1.0

0.8

0.6

0.4

0.2

0.0 –0.15 –0.075 0.0 0.075 0.15

1.0

0.8

0.6

0.4

0.2

0.0 –0.04 –0.02 0.00 0.02 0.04

(b)

(a)

F ig . 4 .8 . Ve rt ic al p ro fi l es o f ( a) r es id ua l v el oc i ty (4.15), m s−1, a nd ( b) s al in it ydifferences (4.40), ‰. L e f t : d a s h e d l i n e s , t h e o r y ; f u l l l i n e s , n u m e r i c a l m o d e l ; t h i n ,S i m u l a t i o n 1 ; t h i c k , S i m u l a t i o n 3 . R i g h t : S i m u l a t i o n 3 w i t h t h i n , f = 0 . 0 1 ; d a s h e d ,U * = 1.2 m s−1; t hi ck S c = 1 . Va lu es c or re sp on d t o U * = 0 .6 m s−1, f = 0 . 0 0 2 5 ,

D = 3 2 m , L = 4 0 0 k m , S c = 1 0 .




s h o w s a p a t t e r n d i r e c t l y l i n k e d t o t i d a l c u r r e n t s p e e d . T h e v e r t i c a l s h e a r s h o w s p e a k

v al ue s o n t he d e ce le ra ti ng p ha se s o f b o th e bb a nd f lo od f lo w s b u t w it h m ar k ed l ys tr on ge r v al ue s o n t he e bb . I nt er es ti ng ly, t he p ea k v al ue s o f d is si pa ti on , o rd er o f ( 2 ×

10−2 W m−3) , a r e s i m i l a r f o r b o t h v a l u e s o f t h e S c= 1 0 a n d 1 . R i p p e t h et al . i n d i c a t e d

c o m p a r a b l e r a t e s o f d i s s i p a t i o n u p t o 0 . 5 × 1 0−2 W m−3.

Tidal cycle and depth profiles

F ig ur e 4 .1 0 (Prandle, 2004a) s ho ws t ha t t id al c yc le s o f v er ti ca l d if fe re nc es f or

s al in it y f or t he S im ul at io n 3 r un s w it h S c = 10 and 1.0. Fo r S c = 1 .0 , c om pl et e

v er ti ca l m ix in g, d ue t o c on ve ct iv e o ve rt ur ni ng , i s s ho wn t o s ta rt a bo ut 1 .5 h a ft er t he

s t a r t o f t h e f l o o d t i d e a n d t o c o n t i n u e f o r a b o u t o n e - t h i r d o f t h e c y c l e . C o n v e r s e l y ,

for S c = 1 0 , c o n v e c t i v e o v e r t u r n i n g o n l y o c c u r s a t t h e e n d o f t h e f l o o d t i d e a n d i s

l i m i t e d i n e x t e n t t o n e a r - b e d a n d n e a r - s u r f a c e .

4.4.5 Model applications for a range of estuarine conditions

F o l l o w i n g t h e a b o v e f o r m u l a t i o n a n d e v a l u a t i o n o f t h e s i n g l e - p o i n t m o d e l , a r a n g e

o f s i mu l at i on s w e re c o mp l et e d c o rr e sp o nd i n g t o

depths D = 4 , 5 .7 , 8 , 11 .3 , 1 6, 2 2. 6, 3 2, 4 5. 3, 6 4 m

density gradients S x = (Δ ρ/ ρ)/ LI, LI = 12.5, 17.7, 25, 35.5, 50, 70.7, 100, 141.4, 200 km

t i d a l v e l o ci t y U * = 0 . 5 , b e d f r i c t i o n f = 0 . 0 0 2 5 , S c h mi d n u mb e r S c = 10

5 × 10 –6

4 × 10 –6

3 × 10 –6

2 × 10 –6

1 × 10 –6

0

0.025

0.020

0.015

0.010

0.005

0.000

π 2πMaxflood

Maxebb

0

π 2πMaxflood

Maxebb

0

F i g . 4 . 9 . T id a l c y cl e s o f mi x in g a n d d i s si p at i on r a te s . To p : d e p th - av e ra g ed mi x in grates (kg m−3 s−1) . T hin , d iff us io n S c = 0 .1 ; d as he d, d iff us io n S c = 1; thick,o ve rt ur ni ng . B ot to m: d is s ip at io n r at es ( W m−3) . F ul l, i nt er na l s he ar, d as he d, b edfriction; thin S c = 1 .0 , t h i ck S c = 1 0 .

100 Saline intrusion



Δ ρ = 0 .0 27 ρ i s o ce an s a li n it y a n d LI t he l e ng t h o f s a li n e i n tr u s io n . S i mu l at i on sw er e l im it ed t o c o nd it io ns w he re t h e b ed -t o- su rf ac e s al in it y d if fe re n ce , δ s, e s ti m at e d

from (4.40) w a s l e s s t h a n 1 0‰. P r a n d l e ( 2004a) provides full details of these model

simulations.

Tidal velocities

F i gu r e 3 . 3 i nd ic at es t he f or m o f t he s o lu ti on (3.16) f o r t id al c ur re nt p ro fi le s. T he

d if fe re nc e i n a mp li tu de b e tw ee n s ur fa ce a nd b ed i nc re as es m o no to ni ca ll y w it h

l arg er v al ue s o f t he S tr ou ha l n um be r (S R = U * P / D) , a pp ro ac hi n g a n a sy mp to tec lo se t o S R = 3 50 . S in ce S tr ou ha l n um be rs i n a lm os t a ll m es o- a nd m ac ro -t id al

e s t u a r i e s w i l l b e w e l l i n e x c e s s o f t h i s v a l u e , t h e e f f e c t s o f t i d a l s t r a i n i n g a r e l i k e l y

t o b e b ro ad ly s im il ar a cr os s a w id e r an g e o f s u ch e st ua ri es . I n s im ul at io ns e x te nd in g

from S R = 2 00 t o 1 0 0 00 , t he p re se nt a pp li ca ti on s s ho w a c lo se f it b et we en t he or y

– 4 0

– 2 0

2 0

0

0

1.0

0.8

0.6

0.4

0.2

0.00 Max

ebbπ 2πMax

flood

1.0

0.8 – 1

2 0 – 8

0

– 1 0 0

– 6 0

–20

20

6 0

6 0

– 1 4 0 – 1 2 0

– 1 0 0

– 8 0

– 4 0

0

4 0

8 0

1 0 0 1 2 0 1 4 0

1 6 0

8 0 10 0 12 0

0.6

0.4

0.2

0.00 Max

ebbπ 2πMax

flood

F ig . 4 .1 0. D ep th p ro fi le s o f s al in it y d if fe re nc es o v er a t id al c yc le . To p S c = 10, bottom S c = 1 .0 ; c o nt ou rs 0 .1‰.




a nd m od el f or b ot h a mp li tu de a nd p ha se o f t id al c ur re nt s. M or eo ve r, t he se r es ul ts f or

v er ti ca l p ro fi le s o f t id al c ur re nt a mp li tu de a nd p ha se a re l ar ge ly i ns en si ti ve t o

s a li n it y g r ad i en t s o v e r t h e r a ng e c o n si d er e d. H e n ce , f u rt h er c o n si d er a ti o n o f t i da l

c u r r e n t s i s o m i t t e d .

S o u z a a n d S i m p s o n ( 1997) u s e d H . F . R a d a r o b s e r v a t i o n s o f s u r f a c e t i d a l e l l i p s e si n a c oa st al ( fr es h wa te r) p lu m e r eg io n t o i nd ic at e h ow a t wo -l ay er t id al r es po n se

c an d ev el op . I n t he l im it , t he s ur fa ce l ay er r es po n se i s e ff ec ti ve ly f ri ct io n le ss , a nd

t he lo we r l ay er r es po nd s l ik e a w ate r c ol um n r ed uc ed i n d ep th to t ha t a t t he ir

i nt er fa ce . W h il e s u ch r es ul ts m ay b e e nc ou n te re d i n e st ua ri es w it h w ea k t id e s, t he

a s s o c i a t e d d y n a m i c s i n v o l v e c o n v e c t i v e a n d C o r i o l i s t e r m s o m i t t e d f r o m (4.4).

Bed-to-surface residual current differences δu

and sea level gradient ∂ς / ∂X

Figure 4.11(a) (Prandle, 2004a) s h o w s t h e v a l u e s o f δu, the bed-to-surface difference

i n t i d a l l y a v e r a g e d v e l o c i t i e s d e r i v e d f r o m (4.15) a n d S i m u l a t i o n s 1 a n d 3 ( u s i n g t h e

m o de l d e sc r ib e d i n Section 4.4.3) . T he f ig ur e i n di ca te s v a lu es o f δu throughout the

r an ge o f b o th i nt r us io n l e ng th s, LI, a nd w at er d ep th s, D, n ot ed a bo ve . T he r es ul ts

s h o w n a r e f o r U * = 0 .5 m s−1, f = 0 . 0 0 2 5 a n d S c = 1 0 . T h e d o t s i n Figs 4.11 a n d 4 . 1 2

r e p r e s e n t v a l u e s o f ( D, LI) f r o m t h e e i g h t o b s e r v a t i o n a l d a t a s e t s l i s t e d i n Tables 4.2

a n d 4 . 3 .The us e of axes with scales based o n lo g2(200/ LI) and log2( D/ 4) r es ul ts i n c on to ur

d is tr ib ut io ns w it h s lo pe a = n/ m fo r a p ar am et er d ep en de nt o n LIm Dn. Thu s,

for (4.15), w he re δu∝ D2/ LI fU * , t he s lo pe i s − 2 a n d t h e s p a c i n g i s l o g 2 R1/ n on

the D- a x i s a n d l o g2 R1/ m o n t h e LI a x i s , w h e r e R i s t h e r a t i o b e t w e e n c o n t o u r v a l u e s .

12.5(a) (b)

25

L 1

( k m )

50

100

2004 8

1

5

5

10.1

16

D (m)

32 64

12.5

25

L 1

( k m )

50

100

2004 8

0.1

1.01.0

10

0.01

0.1

16

D (m)

32 64

Û = 0.5 m s –1

S c = 10

k = 0.0025Û = 0.5 m s

–1

S c = 10

ν = 1

ν = 0

k = 0.0025

F ig . 4 .11 . B ed t o s ur fa ce c ha ng es a s f (depth, D, a n d s a li n e i n t ru s io n l e ng t h, LI).( a ) r e s id u al v e lo c it y ( c m s−1) , ( b ) s a l i n i t y (‰) . D a s h e d l i n e s , n u m e r i c a l m o d e l , f u l ll i ne s , t h eo r y (4.15) a nd ( 4 .4 0 ). D o ts i n di c at e o b s er v ed v a lu e s ( P ra n dl e , 2004b).S h a d e d a r e a i n ( b ) i n d i c a t e s 0 < υ < 1 , w h e r e υ ( H a n se n a n d R a t t ra y, 1966) , i s t h er at io o f s al t f lu x a ss oc ia te d w it h v er ti ca l d if fu si on t o t ha t d ue t o g ra vi ta ti on alconvection.




T he v al ue s o f δu f ro m S im ul at io n 1 (E = f U * D, n o o ve rt ur ni ng ) a re i n c lo se

a gr ee me nt w it h t he d er iv ed e xp re ss io n (4.15), a s i nd ic at ed p re v io us ly. H o we ve r, t he

i n tr o du c ti o n o f o v er t ur n in g i n to S i mu l at i on 3 s i gn i fi c an t ly e n ha n c es v a lu e s o f δu

r e l a t i v e t o (4.15) w h i l e m a i n t a i n i n g t h e t h e o r e t i c a l d e p e n d e n c e o n D2/ LI.

A s s o c i a t e d r e s u l t s f o r t h e m a g n i t u d e o f t h e t i d a l l y a v e r a g e d s e a s u r f a c e g r a d i e n t ∂ς / ∂ X fr om b ot h S im ul at io ns 1 a nd 3 a re i n p re ci se a gr ee me nt w it h t he d er iv ed

v a lu es f r om (4.16). O ve r t he r an ge s o f D and LI considered, ∂ς / ∂ X is s ho wn t o

increase up to 10 × 10−6 w hi ch c or re sp on ds to a n et i nc rea se i n s ea l eve l o f 1 c m k m−1

o f i n t ru s i on l e n gt h .

Bed-to-surface density differences, δs

F i g u r e 4 . 11 ( b ) s ho ws v al ue s o f δ s, t h e t i da l ly a v er a ge d b e d- t o- s ur f ac e d i ff e re n ce

i n s a li ni t y. A g ai n , r es u lt s f ro m S i mu l at i on 1 ( wi t ho ut o v er tu r ni ng ) a re i n c l os ea g r e e m e n t w i t h t h e d e r i v e d e x p r e s s i o n f r o m ( 4 . 4 0 ) , m a i nt a in i n g t h e d e pe n de n c y

on D3/ LI2.

12.5

25

L 1 ( k

m )

50

1

0.1

0.110

1

10

0.015

0.005

0.001

100

2004 8 16

D (m)

32 64

12.5

25

L 1 ( k m )

50

100

200

0.4

0.25

0.1

Mixingoverturning:dispersion

U o cm s –1

Φ j m –3 ε

2.5

0.12.0

1.5

1.0

4 8 16

D (m)

32 64

12.5

25

L 1 ( k m )

50

100

2004 8 16

D (m)

32 64

12.5

25

L 1 ( k

m )

50

100

2004 8 16

D (m)

32 64

F i g . 4 . 1 2 . P o t e n t i a l e n e r g y a n o ma l y , φM; mi x i ng e f fi c i en c y, ε ; o v e r tu r n in g :d i f f u s i v i t y ; r i v e r f l o w . C o n t o u r s a s f o r F i g . 4 . 1 1.( T o p l e f t ) T i d a l me a n p o t e n t i a l e n e r g y a n o ma l y φM, (4.64), i n J m−3.( T o p r i g h t ) E f f i c i e n c y o f mi x i n g , ε .(Bottom left) Tidal mean ratio of mixing by convective overturning : vertical diffusivity.(Bottom right) Balancing river flows, U 0 ( c m s−1).




H ow e ve r, a s p re vi ou s ly i nd ic at ed i n F ig . 4 .8, r es ul ts f ro m S im ul at io n 3 ( wi th

o ve rt ur ni ng ) a re r ad ic al ly d if fe re nt w it h e nh a nc ed v al ue s o f δ s. M or eo v er, t he se

c on to ur s s ug g es t a d ep en d en cy o n D/ LI. T he t id al c yc le s s ho wn i n F ig . 4 .1 0 indicate

h ow o v er tu r ni ng e li mi na te s t he b a la n ci ng o f e bb – f l oo d t i da l s t ra i ni n g m a in t ai n ed

i n S i m u l a t i o n 1 .

4.4.6 Mixing processes

T h e p o t e n t i a l e n e r g y a n o m a l y φE, i s d e f i n e d a s

φE ¼ 1

D ð D

0

s0 gðZ DÞdZ ; (4:64)

where s0 i s d e t e r m i n e d f r o m t h e t i d a l c u r r e n t s t r u c t u r e i n d i c a t e d i n (4.40).

T hi s r ep re se nt s t he a mo un t o f e ne rg y r eq ui re d t o m ix th e w ate r c ol um n t o a

u n if o rm d e ns i ty a n d h e nc e , i n ve r se l y, t h e e f fe c ti v en e ss o f v e rt i ca l m i xi n g .

Substituting (4.40) in (4.64), w e o bt ai n a t im e- av er ag ed v al ue o f φE associated

w i t h t h e t i d a l l y a v e r a g e d d e n s i t y s t r u c t u r e :

M ¼ 0:0007 ρ g2 S 2x D6

EK z

: (4:65)

T he t id al ly a v er ag e d r es ul ts f or φM s ho w n i n F i g. 4 .1 2 c o rr es po nd t o t he d e ns it y

d i ff e re n ce s d e s cr i be d i n t h e p r ev i ou s s e ct i on . F o r S i mu l at i on 1 , w i th o ut o v e rt u rn -

i ng , c al cu la te d v al u es o f φM ( n ot s ho w n) a re i n c lo se c or re sp on d en ce w it h (4.65)

m a in t ai n in g a d e pe n d en c y o n D2/ LI. W h er ea s, f or S im ul at io n 3 w it h o ve rt ur ni ng ,

r a d i c a l l y e n h a n c e d v a l u e s o f φM a r e o b t a i n e d w i t h a d e p e n d e n c y c l o s e r t o D5/3 / LI.

T y p i c a l l y , l a r g e r v a l u e s o f φE a r e f o u n d o v e r m u c h o f t h e t i d a l c y c l e , b u t t h e s e a r e

r e d u c e d t o z e r o b y o v e r t u r n i n g o v e r a s m u c h a s o n e - t h i r d o f t h e t i d a l c y c l e .D e t a i l s o f t h e s e p a r a t e m i x i n g p r o c e s s e s w i t h i n t h e s i m u l a t i o n s c a n b e q u a n t i -

f i e d . F i g ur e 4 . 12 ( P r a n d l e , 2 0 0 4 a ) s ho ws , f or S im ul a ti on 3 , t he r at i o o f ( ti da ll y

a ve ra g ed ) m ix i ng b y c on ve c ti ve o ve rt ur n in g t o t ha t b y v e rt ic a l d if fu si vi t y. T he

m a x i m u m v a l u e o f t h i s r a t i o i s 0 . 4 a n d o c c u r s i n s h a l l o w w a t e r f o r l a r g e v a l u e s o f

LI. T h i s r a t i o d e c r e a s e s t o 0 . 1 i n d e e p e r m o r e s t r a t i f i e d c o n d i t i o n s .

T h e e f f e c t i v e n e s s o f m i x i n g , ε , i s d e f i n e d b y S i m p s o n a n d B o w e r s ( 1981) a s t h e

r a t i o o f w o r k d o n e b y m i x i n g t o t h a t b y t i d a l f r i c t i o n a t t h e b e d . F i g u r e 4 . 1 2 shows,

f or S i mu l at io n 3 , v a lu es o f ε r an g in g f r om l es s t ha n 0 .0 01 i n w el l- mi x ed c o nd i-t i o n s u p t o 0 . 0 1 5 i n m o r e s t r a t i f i e d c o n d i t i o n s . H e r e , t h e t o t a l w o r k d o n e i n c l u d e s

both that by bed fricti on ( ρ fU 3 bed) a nd t ha t b y i nt er na l s he ar ∫ ρ E z(dU /d Z )2 d Z .

I nt er es ti n gl y, t he S i mp so n a nd B o we r s ( 1 9 8 1 ) e st ima te o f ε ≈ 0 .0 04 l ie s c lo se

t o t he c e nt re o f t he c al cu la t ed d is t ri bu ti o ns . H ea r n ( 1 9 8 5 ) n ot ed t he u bi qu it y o f




ε 0 .0 03 i n s he lf s ea s. H e a ls o s ho ws v al ue s o f ε r an g in g f r om 0 .0 0 3 t o 0 . 01 6

based on observ ation s at three sites (90 – 1 00 m d ee p) i n t he C el ti c S ea .

F i gu r e 4 . 12 a ls o s ho ws t he v al ue o f U 0 r eq ui re d t o b al an ce t he t ot al r at e o f

m ix in g. I n S im ul at io n 3 , t he se v al ue s r an ge f ro m a m ax im um o f 2 .5 c m s−1 in

s ha ll ow w at er t o 1 .5 c m s−1

i n d e e p e r w a t e r. A v a l u e o f 3 c m s−1

i s s u gg e st e df r o m ( 4 . 5 8 ) d ue t o t i d a ll y a v e r ag e d t i d al s t ra i ni n g a n d b e t w ee n 0 . 1 a n d 1 . 0 c m s−1

f r o m ( 4 . 5 9 ) d ue t o v er ti ca l d en si ty d iff er en ce s. T he l at te r v al ue s a re c lo se r t o

t yp ic al o bs er ve d v al ue s s ho wn i n Ta b le s 4 . 2 a nd 4 .3 a nd s u gg es t t ha t s o me

r e la x at i on o f c o ns t an t S x m ay o cc ur t ha t r ed uc es m ix in g a ss oc ia te d w it h t id al

s t r a i n i n g .

4.5 Stratification4.5.1 Flow ratio, F R

I t i s g en er al ly a cc ep te d t ha t s tr at if ic at io n s tr en gt he ns w it h i nc re as ed v al ue s o f

depth D a n d r i v e r f l o w Q a n d w i t h d e c r e a s e d v a l u e s o f b r e a d t h B a n d t i d a l c u r r e n t

amplitude U * . T h e f l o w r a t i o , F R , i s d e f i n e d a s t h e n e t f r e s h w a t e r f l o w o v e r a t i d a l

c yc le d iv id ed b y t he t id al p ri sm , T P, e nte ri ng t he e stu ary e ac h f lo od t id e, i .e .

a p p r o x i m a t e l y t h e v o l u m e b e t w e e n h i g h a n d l o w w a t e r s :

F R ¼ QPT P

Qπ U BD

¼ U 0π U

: (4:66)

T h e a b o v e a s s u m e s t h a t T P c a n b e a p p r o x i m a t e d b y U * A P / π , w h e r e A = BD i s t h e

c ro ss -s ec ti on al a re a. S ch ul tz a nd S im mo ns (1957 ) s ug ge st ed t ha t e st ua ri es a re

w e l l - m i x e d f o r F R < 0 . 1 , i . e . U 0 < 0 . 0 3 U *.

T h e p a r a m e t e r d e p e n d e n c i e s i n (4.66) c o n f ir m t h e t e nd e nc i es t o w ar d s s t ra t if i ca -

t io n n ot ed a b ov e – w it h t he e xc ep ti on o f t he i nf lu en ce o f d ep th D. T he f ol lo wi ng

s e c t i o n e x p l a i n s t h i s a n o m a l y , i l l u s t r a t i n g h o w t h e k i n e t i c r e l a t i o n s h i p (4.66) needst o b e e xt en de d t o c on s id e r t h e b al an ce b e tw ee n t he r at e o f m ix in g ( of te n c on ce n -

t ra te d n ea r t he s ur fa ce i n s tr at if ie d f lo w) a nd t he t id a l d is si pa ti on ( co n ce nt ra te d

c l o s e t o t h e b e d ) . C h ap t er 3 s h o w s t h a t f o r (ω D/ fU * ) > 0 . 2 5 , i . e . D/ U * > 5 s f o r a

s e mi - di u rn a l t i de , m a x im u m v e lo c it i es o c cu r a r ou n d m i d- d ep t h , w i th l i tt l e c u rr e nt

s he ar i n t he t op h al f o f t he w at er. S ub se qu en t s ec ti on s c on fi rm t ha t, e xc ep t f or

d e e p e r m i c r o - t i d a l e s t u a r i e s w h e r e m i x i n g s c a r c e l y e x t e n d s t o t h e s u r f a c e , (4.66) is

t h e p r i m a r y i n d i c a t o r o f s t r a t i f i c a t i o n .

4.5.2 Energy and time requirements

T h e S i mp s o n – H un te r (1974 ) c ri te ri a f or s tr at if ie d c on di ti on s i s b as ed o n t he r at io

betw een the inc reas e in pote ntia l energ y due to mixi ng and the asso ciat ed work




d o n e b y t i d a l f r i c t i o n . T h e q u a n t i t a t i v e v a l u e i n d i c a t i v e o f m i x i n g t h r o u g h o u t t h e

d e p t h ,

H =U 3555 m2 s3; (4:67)

i s b a s e d o n o b s e r v a t i o n a l e v i d e n c e o f t h e r m a l s t r a t i f i c a t i o n i n s h e l f s e a s .P r a n d l e (1997 ) i n d i c a t e d t h a t s t r a t i f i c a t i o n l e v e l s c a n b e c a l c u l a t e d f r o m t h e t i m e ,

D2/ K z, f o r c o m p l e t e v e r t i c a l m i x i n g b y d i f f u s i o n o f a p o i n t s o u r c e i n t r o d u c e d a t t h e

s ur fa ce o r b ed . I ns er ti ng (4.3) a nd s pe ci fy in g th e t im e l im it a s t he d ur at io n o f t he e bb

o r f l o o d p h a s e s , s t r a t i f i c a t i o n r e q u i r e s

D2

K z

¼ D

S c f U 46 h o r

D

U 456 S c s: (4:68)

Since U * i s u s u a l l y i n t h e r a n g e 0 . 5 – 1 . 0 m s−1, a n d Sc b e t w e e n 0 . 1 a n d 1 , w e s e e t h e

c o rr e sp o nd e n ce w i th t h e S i mp s on – H u n te r c r it e ri o n. M o st e s tu a ri e s h a ve v a lu e s o f

D2/ K z > 1 h , i nd i ca ti ng t ha t s om e d eg re e o f i nt ra -t id a l s tr at if ic at io n w il l o cc ur. I n

C ha pt er 6, i t i s s ho wn h ow, f or a s yn ch ro no us e st ua ry, b ot h c ri te ri a c or re sp on d

c l o s e l y t o s t r a t i f i c a t i o n o c c u r r i n g f o r t i d a l e l e v a t i o n a m p l i t u d e s ς * ≤ 1 m.

4.5.3 Richardson number

Ti me - a n d d ep th -a ve ra ge d v al u es f or Ri (4.2) c an b e c al cu la te d u si ng (4.40) for

∂ ρ/ ∂ Z and (4.55) for ∂U / ∂ Z , g i v i n g

Ri ¼

68 gS 2x D4

104 E K z

0:45U

D 2

¼ 100 S cU 0

U

2

: (4:69)

T h u s , w i t h E from (4.3), S x = 0 .0 27 / LI, w h e r e LI i s t h e s a l i n e i n t r u s i o n l e n g t h f r o m

(4.44). T h e c o n d i t i o n , Ri < 0 . 2 5 , f o r m i x i n g t o o c c u r , c o r r e s p o n d s t o U 0 < 0 . 0 5 U *

for S c = 1 a n d U 0 < 0 . 0 1 6 U * f o r S c = 1 0 .

4.5.4 Stratification number

I pp en a nd H ar le ma n ( 1 9 6 1 ) d e mo ns tr a te d t ha t v er t ic al m ix in g c o ul d b e r el a te d

t o t h e b a l a n c e b e t w e e n t u r b u l e n c e a s s o c i a t e d w i t h t h e r a t e o f d i s s i p a t i o n o f t i d a le ne rg y b y b ed s tr es s, G = ε ( 4 / 3π) f ρU *3 LI, a nd t he e ne rgy r eq ui re d t o i nc re as e

t he p o te n ti al e ne rg y l ev e l b y v er t ic al m ix i ng , J = ½ Δ ρ g H 2U 0. T h e p ar am et er ε

r e pr e se n ti n g t h e e f fi c i en c y o f m i xi n g, t y p ic a l ly 0 . 0 0 4 ( S e c t i on 4 . 4 . 6) , i s i n t r o -

d u c e d h e r e t o g i v e a m o d i f i e d s t r a t i f i c a t i o n n u m b e r , d e f i n e d a s




S 0T ¼ S T" ¼ "G

J ¼ 0:85 " f U LI

ρ

ρ gH 2 U 0

¼ 0:017 " U

U 0

2

: (4:70)

T hu s, t he s tr at if ic at io n n um be r e xt en ds t he c on ce pt o f t he S im ps on – H u n te r c r it e ri o n

d e s c r i b e d i n S e c t i o n 4 . 5 . 2 t o b a l a n c e n e t m i x i n g o f f r e s h w a t e r i n p u t o v e r t h e s a l i n e

i n t r u s i o n l e n g t h . P r a n d l e (1985) s h o w e d t h a t e s t u a r i e s c h a n g e f r o m m i x e d t o s t r a t i -

f i e d a s t h e s t r a t i f i c a t i o n n u m b e r d e c r e a s e s f r o m a b o v e 4 0 0 t o b e l o w 1 0 0 ( F i g . 4 . 1 3;

Prandle, 1985) . A d o p t i n g t h e l i m i t S T = 2 50, t hen f or ε = 0 . 00 4 , t h e b o un d ar y c o rr e -

s p o n d s t o S T

` = 1 a n d U 0

< 0 . 0 1 U *.

4.5.5 Vertical salinity difference

H an se n a nd R at tra y (1966 ) i nd ic at ed t ha t a n or ma li se d s ur fa ce t o b ed s al in it y

d i f f e r e n c e s o f δ s/ s < 0 . 1 c o u l d b e a d o p t e d a s t h e b o u n d a r y f o r m i x e d a n d s t r a t i f i e d

e s t u a ri e s . F r o m F i g . 4 . 1 3 a n d ( 4.70),

δs

s ¼ 4 S 0:5T ¼ 31

U 0

U

: (4:71)

Thus, δ s/ s < 0 . 1 i m p l i e s t h a t e s t u a r i e s w i l l b e m i x e d f o r U 0 < 0 . 0 0 3 U * . U s i n g t h e

s ug g es te d r ev is ed d em ar ca ti on , b as ed o n F ig . 4 .1 3, of S T = 2 50 , s tr at if ic at io n i s

d e f i n e d a s δs/ s > 0 . 2 5 , U 0 > 0 . 0 1 U *.

10

5

2

1.0

0.1

0.011 2 5 10 100

S t

1000 10000

Well-mixedStratified

δ s / s

F i g. 4 . 13 . L e ve l o f s t ra t if i ca t i on δ s/ s v e rs u s S t r a ti f i c a ti o n n u mb e r S T. S T = 0 .0 17ε (U */ U 0)2, (4.70). • Rigter ’s (1973 ) flume tests, + WE S flume test s, ×observations.





Wi t hi n e s tu a ri e s, s a li n e i n tr u si o n c a n i m pa c t s i gn i fi c an t ly o n f l or a , f a un a , b i ol o g y,

c h e m i s t r y , s e d i m e n t s , m o r p h o l o g y , f l u s h i n g t i m e , c o n t a m i n a n t p a t h w a y s , e t c . H e r e

w e e xa mi ne t he d yn am ic s o f m ix in g , i nd ic at in g h o w b at hy m et ry a nd b ot h m ar in ea n d f l u v i a l c o n d i t i o n s d e t e r m i n e t h e d e g r e e o f s t r a t i f i c a t i o n a n d t h e e x t e n t o f a x i a l

penetration. A p p e n d i x 4 A o u t l i n e s t h e c h a r a c t e r i s t i c s o f t h e r e l a t e d d e p e n d e n c y o f

w at er d e ns it y o n t em pe ra tu re , i ll us tr at in g h ow t he s ea so n al c yc le v ar ie s a cc o rd in g t o

t h e l e v e l o f s t r a t i f i c a t i o n a n d l a t i t u d e .

T h e l e a d i n g q u e s t i o n i s :

How does salt water intrude and mix and how does this change over cycles of

spring – neap tides and flood-to-drought river flows?

A p a r t f r o m a n o c c a s i o n a l s u r f a c e s c u m l i n e , t h e e b b a n d f l o o d o f s a l i n e i n t r u s i o n

i n e st ua ri es p as se s l arg el y u nn ot ic ed . Ye t, t he e xt en t o f s al in e i nt ru si on w as o ft en t he

d et er mi ni ng f ac to r i n t he s it in g o f t ow ns o r i nd u st ri es r el ia nt o n f re sh r iv er w at er.

M o re o v er, t h e e x te n t a n d n a tu r e o f i n tr u si o n s d e te r mi n es n e t e s tu a ri n e c o nc e n tr a -

t io ns o f b o th d is so lv e d m ar in e t ra ce rs a n d f lu v ia l c o nt am in an ts . A t t he i nt er fa ce w i th

r iv er w at er, s al t w at er p ro du ce s e le ct ro ly ti c a tt ra ct io n b et we en f in e s us pe nd e d

s ed im en ts r es u lt in g i n r ap id ly s et tl in g ‘flocs’ w h ic h a cc um ul at e a t t he s ea wa rd

( e b b ) a n d l a n d w a r d ( f l o o d ) l i m i t s o f i n t r u s i o n .

I n s t ro n gl y t i da l e s tu a ri e s, s a li n e i n tr u si o n h a s l i tt l e i m pa c t o n t i da l p r o pa g at i on

(Prandle, 2004a) . C on ve rs el y, t he n at ur e o f s al in e i nt ru si on i s o ve rw he lm in gl y

d et er mi ne d b y t he c o mb in at io n o f t id al m ot io ns a lo ng si de t he f lo w o f r iv er w at er.

P r it c ha r d (1955) i nt ro du ce d a g en er al is ed c la ss if ic at io n o f e st ua ri es a cc or di ng t o

t he ir s al in it y i nt ru si on , v ar yi ng f ro m f ul ly m ix ed ( ve rt ic al ly ) i n s tr on gl y t id al ,

s ha ll ow e st ua ri es w it h s ma ll r iv er f lo ws t h ro u gh t o ‘a r re s te d s a li n e w e dg e’ i n d e e pe r,

m i c r o - t i d a l e s t u a r i e s w i t h l a r g e r i v e r f l o w s . H a n s e n a n d R a t t r a y ( 1966 ) p r o d u c e d a

g e ne r al i se d s t ra t if i ca t io n d i ag r am (F ig . 4 .3) , c on ve rt ed b y P ra nd le (1985) in terms of

m o r e r e a d i l y a v a i l a b l e p a r a m e t e r s . H o w e v e r , t h i s d i a g r a m d o e s n o t d i r e c t l y a n s w e r

t h e a b o v e q u e s t i o n c o n c e r n i n g t h e r a n g e o f s a l i n e i n t r u s i o n i n e s t u a r i e s .

S tr at if ic at io n i s c om pl ic at ed b y c on ti nu ou s i nt er ac ti on s b et we en t he d yn am ic s

promoting mixing and associated adjustments to the background salinity gradients.

The Richardson number (4.2), q u a n t i f i e s t h e r a t i o o f ‘stabilising’ buoyancy inflow to

t h e r a te o f t ur b ul en t m i xi ng . I t h a s p r ov en a r o bu st i n di c at or o f s t ra t if ic a ti o n a c ro ss a

w id e r an ge o f f lu id d yn am ic s. H ow ev er, t hi s n um be r c an v ar y a pp re ci ab ly o ve r t he t id ala nd f lu v ia l c yc l es n o te d a b ov e a n d f ro m m ou t h t o h e ad a n d l a t er a ll y w it hi n a n e s tu a ry.

S e c t i o n 4 . 3 d e s c r i b e s h o w v a r i o u s a p p r o a c h e s f o r e s t i m a t i n g t h e l e n g t h o f s a l i n e

intrusion, LI, all sugges t a d ependence on D2/ f U *U 0 ( D depth f b e d f ri ct io n

coefficient, U * t id al c ur re nt a mp li tu de a nd U 0 r e si du al c ur re nt a ss oc ia te d w it h




r iv er f lo w) . A ss es sm en ts a ga in st a r an ge o f f lu me t es ts i nd ic at e t he v al id it y a nd

r o b u s t n e s s o f t h i s f o r m u l a t i o n . H o w e v e r , t h i s f o r m u l a f a i l s t o a c c o u n t f o r o b s e r v e d

v a ri a ti o ns i n i n tr u si o n l e n gt h s ( U nc l es et al ., 2002) . T he a dd it io n al f ac to r, w hi ch

m u s t b e i n c o r p o r a t e d i n f u n n e l - s h a p e d e s t u a r i e s , i s a x i a l m i g r a t i o n o f t h e i n t r u s i o n .

T h is i n tr o du c es a c o mp l ex i n te r -d ep e nd e nc y b e tw e en t h e l e n g th a n d t h e l o ca t io n o f t h e i n t r u s i o n . A n a l y s i s o f o b s e r v a t i o n s s u g g e s t s t h a t t h i s a x i a l m i g r a t i o n a d j u s t s t o

e n a b l e m i x i n g t o o c c u r a s f a r s e a w a r d s a s i s c o n s i s t e n t w i t h c o n t a i n i n g t h e m i x i n g

w it hi n t he e st ua ry. T he t im e d el ay f or s uc h a xia l m ig ra ti on o f th e in tr us io n i s r el at ed

t o t he e s tu a ri n e f l us h in g t i me T F. I t i s s h o w n i n C h ap t er 6 a n d f r o m o b s e rv a t io n s

t h a t T F i s i n t h e r a n g e o f d a y s , a n d h e n c e c h a n g i n g t i d a l a n d r i v e r f l o w c o n d i t i o n s

o v er s u ch p e ri o ds f u rt he r c o mp l ic a te s t h e c a lc u la t io n o f LI. C l ea r ly, m o de l s i m-

u l at i on s n e ed t o e x te n d a d eq u at e ly i n t i me t o a c co m mo d at e s u ch m i gr at i on s a n d

i n c o rp o r a te a p p ro p r ia t e ‘r e l a x a t i o n’ o f b ot h s ea wa rd a nd la nd wa rd b ou nd ar yc o n d i t i o n s .

A b r a h a m (1988) n o t e d t h e s i g n i f i c a n c e o f ‘t i d a l s t r a i n i n g’, w h e r e b y o n t h e f l o o d

t i de , l a rg e r n e ar -s u r fa c e t id a l v e lo c it i es a d v ec t d e ns e r m o re s a l in e w a te r o v er f r es h er

l ow er l ay er s l ea di ng t o m ix in g b y c on ve ct iv e o v er tu rn in g. S im p so n et al . (1990)

provided both theoretical and observational quantification of this phenomenon.

H ow ev er, d es pi te t he n eg le ct o f t hi s l at te r e ffe ct , ‘t i da l ly a v er a ge d’ analytical

s o lu t io n s f o r s a li n it y -i n du c e d v e rt i ca l p r of i le s o f r e si d ua l v e lo c it i es a n d s a li n it i es

a re f ou nd t o b e w id el y a pp li ca bl e. T he s um ma ry o f t he se a na ly ti ca l s ol ut io ns i n

S e c t i on 4 . 2 . 6 and Ta b l e 4 . 1 e m p ha s is e s h o w, f o r s t ea d y- s ta t e e q ui l ib r ia , t h e d y na -

m i ca l a d ju s tm e n ts t o s a li n e i n tr u si o n i n vo l v e s m al l ( r el a ti v e t o t i da l c o m po n en t s) ,

r e s i d u a l c u r r e n t s a n d s u r f a c e g r a d i e n t s .

T h e se s o lu t io n s p r o vi d e e s ti m at e s f o r r e si d u al f l ow c o m po n en t s a s so c ia t ed w i th

( i) r iv er f lo w Q (4.12), ( ii ) w in d s tr es s τ w (4.37), ( ii i) a w el l- mi xe d l on gi tu di na l

d e n s i ty g r a d i en t (4.15), a nd ( iv ) a f ul ly s tr at if ie d s al in e w ed ge (4.32) a n d ( 4 .3 3 ).

T he re la ti ve m ag ni tu de s o f e ac h o f t he se c om po ne nt s a re d ef in ed i n t er ms o f d i m e n s io n l e s s p a r a m e te r s (4.42). T h e r es id u al c ur re n t p ro fi le s ( i) , ( ii ) a nd ( ii i) a re

s h o w n i n F i g . 4 . 4 .

F r o m Ta b l e 4 . 1, t he d en si ty f or ci ng t er m 0 .5S i s b a la n ce d b y a s u rf ac e g ra di -

ent −0 . 4 6S w i t h t h e r e m a i n i n g c o m p o n e n t d r i v i n g a r e s i d u a l c i r c u l a t i o n o f 0 . 0 3 6 S

U 0/ F s e a w a r d s a t t h e s u r f a c e a n d −0 . 0 2 9S U 0/ F ( l a n d w a r d s ) a t t h e b e d . S i m i l a r l y ,

t he w in d s tr es s t er m −W i s c ou nt er ac te d b y a s ur fa ce g ra di en t 1 .1 5W w i th t he

‘e x c e s s’ b a la nc e d ri vi n g a c ir cu la ti o n o f 0 .3 1W U 0/ F a t t he s ur fa ce a nd −0 . 1 2W

U 0/ F a t t h e b e d (W , S and F a re d ef in ed i n ( 4 . 4 2 ) ) . T hu s, f or s te ad y- st at ec on d it i on s, b o th w i nd a n d d e ns i ty f o rc i ng a re m a in l y b a la n ce d b y s u rf a ce g ra d i-

e nt s w it h o nl y a s ma ll f ra ct io n o f t he f or ci ng e ff ec ti ve i n m ai nt ai ni ng a v er ti ca l

circulation.




S e ct i on 4 . 4 u s es t he se a na ly ti ca ll y d er iv ed c u rr en t a nd s al in it y p ro fi le s t o e st im at e

t he ( lo ca li se d) r at es o f v er ti ca l m ix in g a nd c om p ar es t h es e a lo n gs id e t he v al u es

a s s o c i a t e d w i t h t i d a l s t r a i n i n g t o t h e r a t e o f s u p p l y o f r i v e r w a t e r . T h e s e a n a l y t i c a l

s ol ut io ns a re f ur th er c o mp ar ed w it h r es ul ts f ro m a ‘single-point ’ n u m e ri c a l m o d e l

w hi ch i nc o rp o ra te s t id al s tr ai ni ng a nd c o nv ec ti ve o ve rt ur ni n g w he ne ve r d e ns it yd ec re as es w it h d ep th . T he m od el a ls o i nc lu de s t he M un k a nd A nd er so n ( 1948)

m od if ic at io ns o f t he e dd y d if fu s iv it y a n d e dd y v i sc os it y c oe ff ic ie nt s, E and K z,

based on Richardson number representations of buoyancy effects.

T he a pp li ca ti on o f t hi s n um er ic al m od el e xt en ds t o a r an ge o f v al ue s o f b ot h

i n t r u si o n l e n g t h s, LI , a nd w at er d ep th s, D. S en si ti vi ti es t o t he S ch m id t n um be r,

e xp re ss in g t he r at io o f E : K z a re a ls o e xp lo re d. R es ul ts a re s ho wn f or s ur fa ce -

t o -b e d d i ff e re n c es i n r e si d ua l v e lo c it y, δu, a nd s al in it y, δ s; p o t en t ia l e n er g y a n om a ly,

φE; t he r at io o f m ix in g b y d i ff us iv it y: o ve rt ur ni ng ; t he e ff ic ie nc y o f m ix in g ε andr e s i d u a l r i v e r f l o w U 0 ( r e q u i r e d t o b a l a n c e m i x i n g r a t e s ) .

Q u al i ta t iv e ly, b o th n u me r ic a l m o de l a n d a n al y ti c al s o lu t io n s f o r t h e r e si d ua l

c ur re nt s a nd s al in it y p ro fi le s a re i n c lo s e a gr ee me nt . H ow ev er, t he m od el s ug -

g e s t s t h a t t i d a l s t r a i n i n g c a n i n c r e a s e t h e m a g n i t u d e s o f t h e s e c u r r e n t s a n d s a l i n i t y

s tr uc tu re s ev er al f ol d. Av er ag ed o ve r a t id al c yc le , t he c om po ne nt o f m ix in g

a s s o c i a t e d w i t h o v e r t u r n i n g i s s h o w n t o b e g e n e r a l l y s i g n i f i c a n t l y l e s s t h a n m i x i n g

by diffusion, typically 0.1−0 .4 o f t he l at te r. H ow ev er, t he p ro n ou nc ed i mp ac t o f

o ve rt ur ni ng i n r em ov in g s tr at if ic at io n o n t he f lo od t id e i s e vi de n t. T he m od el a ls o

q ua nt if ie s m ix in g r at es r el at ed t o b ed f ri ct io n a nd i nt er na l s he ar. T he r es ul ts

c on fi rm S im p so n a nd B o we r ’s (1981) o bs er va ti on al f in di ng s t ha t l es s t ha n 1 %

o f t h e e n e r g y i n v o l v e d i n t i d a l d i s s i p a t i o n i s e f f e c t i v e i n p r o m o t i n g v e r t i c a l m i x i n g .

S ec ti on 4 .5 r ev is it s t he i ss ue o f i nd ic at or s f or e st ua ri ne s tr at if ic at io n. N ew

t h e o r i e s o n s t r a t i f i c a t i o n a r e r e c o n c i l e d w i t h h i s t o r i c a l i n d i c a t o r s , e m p h a s i s i n g t h a t

U 0/ U * > 0 .0 1 i s t he c om mo n k ey i nd ic at or o f s tr at if ic at io n. C al cu la ti on s a re m ad e o f

l i k e l y v a l u e s o f U 0 i n t h e s a l i ne i n tr u si o n z o n e b a s e d o n ( i ) m o s t s e a w ar d l o c a t io n o f mixing (4.52); ( i i ) a f l o w r a t i o , F R = U 0 π/ U * = 0 . 1 (4.66); ( i i i ) a R i c h a r d s o n n u m b e r

RI = 0 .2 5 (4.69); ( iv ) a b al an ce b et we en g ai n i n p ot en ti al e ne rg y a nd t id al d is si pa ti on

(4.70); a nd ( v) o bs er va ti on s o f s tr at if ic at io n (F ig . 4 .1 3) . A ll f iv e a pp ro ac he s i nd ic at e

v a lu e s o f U 0 c l o s e t o 1 c m s−1 i n t h e i n t r u s i o n z o n e o f ‘mixed’ estuaries – a r es ul t

s u b s e q u e n t l y f u r t h e r c o n f i r m e d i n C h a p t e r 6. T h i s i n d i c a t o r c o n f i r m s t h a t s t r a t i f i c a -

t i on g e n er a ll y i n cr e as e s w i th l a rg e r r i ve r f l ow, n a rr o we r b r ea d th s a n d w e ak e r t i da l

c ur re nt s. H ow ev e r, t he i nd ic at io n t ha t s tr at if ic at io n i nc re as es w it h s h al lo w er w at er i s

c o u n t e r a c t e d b y r e d u c e d t i d a l c u r r e n t s h e a r n e a r e r t h e s u r f a c e i n d e e p w a t e r a n d t h er e l a t e d r e d u c e d r o l e o f t i d a l s t r a i n i n g .

A p p e n d i x 4 A p r o v i d e s e x p r e s s i o n s f o r t h e m e a n a n d s e a s o n a l v a r i a t i o n s o f b o t h

t h e w a t e r s u r f a c e a n d t h e a m b i e n t a i r a s f u n c t i o n s o f d e p t h , l a t i t u d e a n d d e g r e e o f

stratification.




Appendix 4A

4A.1 The annual temperature cycle

A g en er al is ed t he or y i s d ev el op ed t o d es cr ib e t he a nn ua l t em pe ra tu re c yc le i n

e s t u a r i e s a n d a d j a c e n t s e a s . A s i n u s o i d a l a p p r o x i m a t i o n t o t h e a n n u a l s o l a r h e a t i n g

component, S , i s a ss um ed a nd t he s ur fa ce l os s t er m i s e xp re ss ed a s a c on st an t, k ,

t i me s t h e a i r- s ea t e mp e ra t ur e d i ff e re n ce (T a – T s) . F o r v e rt i ca l ly m i xe d c o n di t io n s,

a na ly ti ca l s ol ut io ns s ho w t ha t i n s ha ll ow d ep th s, t he w at er t em p er at u re f ol lo ws

c lo se ly t ha t o f t he a mb ie nt a ir t em pe ra tu re w it h l im it ed s ep ar at e e ff ec t o f s ol ar

h e a t i n g . C o n v e r s e l y i n g r e a t e r d e p t h s , t h e w a t e r s u r f a c e t e m p e r a t u r e v a r i a t i o n s w i l l

be reduced relative to those of the ambient air. Providing such deep water remains

m i x e d v e r t i c a l l y , t h e s e a s o n a l v a r i a t i o n w i l l b e i n v e r s e l y p r o p o r t i o n a l t o d e p t h a n d

m ax im um t em pe ra tu re s w il l o cc ur u p t o 3 m on th s a ft er t he m ax im um o f s ol ar h ea ti ng .

T h e a n n u a l m e a n w a t e r t e m p e r a t u r e w i l l e x c e e d t h e a n n u a l m e a n a i r t e m p e r a t u r e b y

t h e a n n u a l m e a n o f S d i v i d e d b y k . F o r w i d e r a p p l i c a t i o n s , a n u m e r i c a l m o d e l i s u s e d

t o d er iv e g en e ra li s ed e x pr es si on s f o r t h e m e an a n d a mp li t ud e o f b ot h a ir a nd w at er

s u r f a c e t e m p e r a t u r e s a s f u n c t i o n s o f l a t i t u d e , d e p t h a n d t i d a l c u r r e n t s p e e d .

T h e d es cr ip ti on o f t em pe ra tu re i n m ar in e e co lo g ic al m od el s i s i mp or ta nt b ot h

d i r e c t l y f o r i t s i n f l u e n c e o n s p e c i f i c p a r a m e t e r s a n d i n d i r e c t l y v i a i t s c o n t r i b u t i o n t o

v e r t i c a l d e n s i t y v a r i a t i o n s .Wa te r d en si ty c an b e a pp ro xi ma te d b y ρ = 1 000 + 0 . 7S – 0.2 T ( k g m−3) , w he re S is

salinity in ‰ and T i s t e m p e r a t u r e i n ° C . T h u s , s t r a t i f i c a t i o n a t t h e d e m a r c a t i o n l i m i t ,

δS 0.25‰, i s e qu iv al en t t o a s ur fa ce -t o -b e d t em p er at u re d if fe re nc e o f 0 .8 7 5 ° C .

S im pl if ied a nd g en er ali se d a pp ro ac he s ar e u se d t o e xa mi ne h ow t he s ea so na l

t em pe ra tu re s v ar y w it h ( i) t h e l ev el o f s ol ar h ea ti ng ( i. e. c lo u d c ov er a nd l at it ud e) , ( ii )

a mb ie nt a ir t em pe ra tu re , ( ii i) w in d s pe ed ( st ro ng ly c on tr ol li ng t he r at e o f a ir -s ea h ea t

e xc ha ng e) , ( iv ) w at er d ep th a nd ( v) t he d eg re e o f v er ti ca l s ti rr in g. R es u lt s f ro m t he se s

studies (Prandle and Lane, 1995; Prandle, 1998) c a n b e u s e d t o e x a m i n e t h e i m p a c t o f c ha ng es i n a ny o f t he se p ar am et er s t h at m ig h t a ri se f ro m c li ma te t re nd s, e tc .

T h e f u n da m en t al s i mp l if i ca t io n s i n tr o du c ed a r e a s f o ll o ws :

( 1) a pp ro xi ma t io n o f t he a nn ua l s ol ar h ea t i np ut b y a m ea n v al ue p l us a s in us o id al t er m :

S 0 – S * c o s ωt ;

( 2) r ep re se nt at io n o f t he h ea t l os se s b y a t er m k (T s – T a ) , w he re T s i s t he w at er s ur fa ce

temperature, T a t h e a i r t e m p e r a t u r e a n d k a c o n s t an t c o e f fi c i e nt ;

( 3 ) a s su m pt i on o f a l o ca l is e d e q u i l ib r i um , i . e. n e gl e ct o f h o ri z o nt a l c o m p on e n ts o f a d ve c -

t i o n a n d d i s p e r s i o n ;

( 4 ) r ep r es e nt at i on o f v e rt i ca l m i xi ng p r oc e ss e s b y a n e d dy d i sp e rs i on c o ef fi ci en t e n co m -

passing the effects of vertical mixing processes by both vertical advection and dispersion;

( 5) n eg le ct o f b ot h t he ( so la r) d iu rn a l a n d t he n ea p – s p r i n g t i d a l c y c l e s o f v a r i a b i l i t y .

Appendix 4A 111



T he a do pti on o f a s in us oid al a pp ro xi ma ti on t o t he a nn ua l t em pe ra tu re c yc le

i nv ol ve s p ha se v al ue s w it h 3 60 ° c or re sp on di ng t o 1 y ea r. F or c on ve ni en ce , t he

‘year ’ i s h er e s ho rt en ed t o 3 60 d ay s s o t ha t 1 ° c or re sp on ds t o 1 d ay. W he re ‘day

numbers’ a r e c i t e d , s t r i c t l y , t h e s e s t a r t a t t h e w i n t e r s o l s t i c e .

4A.2 Components of heat exchange at the sea surface

G ol ds mi th a nd B un ke r (1979) d es cr ib e f ou r c om po ne nt s i nv ol ve d i n t he h ea t e xc ha ng e

between the atmosphere and the sea, namely

(1) SR – s o l a r r a d i a t i o n , i . e . t h e p r i m a r y e n e r g y s o u r c e ;

(2) LH – l a te nt h ea t f lu x, h ea t l os t f ro m t he s ea b y e va po ra ti on ( oc ca si on al ly g ai ne d b y

condensation);(3) IR – i n f r a r e d b a c k r a d i a t i o n , t h e n e t e f f e c t i v e ‘reflection’ f r o m t h e s e a ;

(4) LS – s e n s i b l e h e a t f l u x , h e a t e x c h a n g e b e t w e e n a i r a n d s e a b y ‘conduction’.

F ig ur e 4 A. 1 s h ow s t he a nn ua l v ar ia ti on o f SR, LH , IR and LS a t 5 5° N, 3 °E ( no rt he rn

North Sea) based on data for 1989 supplied by the UK. Meteorological Office, Cave

(1990) a nd h ea t e xc ha ng e t er ms c al cu la te d f ro m G ol ds mi th a nd B un ke r (1979). T he

e f fe c ti v e v a lu e o f SR a t t h e s e a s u r f a c e i s r e d u c e d b y c l o u d c o v e r , t h i s r e d u c t i o n i s

a b o u t 1 / 8 f o r 2 5 % c o v e r , 1 / 3 f o r 5 0 % c o v e r a n d 4 / 5 f o r 1 0 0 % c o v e r . H o w e v e r , t h e

a n n u a l c y c l e ( f r e q u e n c y ω) a t a l m o s t a l l l a t i t u d e s c a n b e a p p r o x i m a t e d b y

SR ¼ S ð1 β cosotÞ: (4A:1)

T he l at en t h ea t f lu x LH i s a f un cti on o f w in d s pe ed W , the air – s e a t e mp e ra t ur e

d i f fe r e n c e (T a – T s) a n d r e la t iv e h u mi d it y.

T he e ff ec ti ve b ac k r ad ia ti on IR d ep en ds o n t he a bs or pt iv e p ro pe rt ie s o f t he

a t m o s p h e r e a n d b o t h T a and T s; h o w e v e r , i t i s m o s t s e n s i t i v e t o T a .

T h e s e n s i b l e h e a t f l u x LS i s p r o p o r t i o n a l t o b o t h w i n d s p e e d W a n d (T s – T a ) . A i r

g a i n s h e a t q u i c k e r t h a n w a t e r a s a c o n s e q u e n c e o f t h e i r r e l a t i v e t h e r m a l c o n d u c t i v -

i ti es , i .e . 0 .0 22 6 a nd 0 .6 0 W m−1 °C−1, r e s p e c t i v e l y , a n d h e n c e t h e a s s u m p t i o n o f a

c o n s t a n t v a l u e f o r k m a y b e i n v a l i d i n s e a s w h e r e T a > T s f o r e x te n de d p e ri o ds .

F o r s i mp l ic i ty, w e l u mp t o ge t he r s o l ar r a di a ti o n SR a n d i n fr a re d b a ck r a di a ti o n

IR i n t o a s i n g l e h e a t g a i n t e r m S a p p r o x i m a t e d b y

S ðtÞ ¼ S 0 S cosot: (4A:2)

L i k e w i s e , w e l u m p t o g e t h e r t h e LH and LS i n t o t h e s i n g l e h e a t l o s s t e r m L where

LðtÞ ¼ kðT s T aÞ: (4A:3)

A n n u a l m e a n s o l a r r a d i a t i o n , S 0 in (4A.2) i n W m−2, i n t h e a b s e n c e o f c l o u d c o v e r ,

ranges from 317 at the eq uator to 23 4 at 45 ° latitud e an d 13 3 at the p oles. In




C o rr e sp o n di n g v a lu e s f o r S * range from 0 at th e eq uator, to 1 47 at 45 ° and 209 at the

poles. A second harmonic, with frequency 2ω, a mo un ts t o 1 6 W m−2 a t 6 0 ° N , r i s i n g

t o 4 1 a t 7 0° a nd 8 7 a t 9 0° . T hu s, t he p re se nt th eo re tic al a pp ro ac h w hi ch a ss um es j us t

t h e f i r s t h a r m o n i c i n (4A.1) i s r e a s o n a b l e i n l a t i t u d e s u p t o 7 0 ° . F i g . 4 A . 1 (Prandle

a n d L a n e , 1995) s h o w s a t y p i c a l a n n u a l c y c l e i n t h e N o r t h S e a .

4A.3 Analytical solutions in vertically mixed water

F r o m t h e a s s u m p t i o n s (4A.2) a n d ( 4 A . 3 ) , t h e r a t e o f c h a n g e o f t h e s e a t e m p e r a t u r e

T s is

–200

–100

0 Mean

0 2 4 6 8 10 12

100

200 E ( W / M * * 2 )

F i g . 4 A . 1 . A n n u a l c y c l e o f SR, = = s o la r r a di at i on ; – – – – LH , l a t e n t h e a t f l u x ; ——

IR, i n f r a r e d b a c k r a d i a t i o n a n d LS , - - - - s e n s i b l e h e a t f l u x , = = = n e t . V a l u e s f o r 1 9 8 9a t p o s it io ns ( 55 ° N , 3 ° E ) i n t he N or th S ea ( Un it s W m−2).

Appendix 4A 113



@ Ts

@ t ¼

S ðtÞ þ LðT Þ

αD ; (4A:4)

where α i s t h e t h e r m a l c a p a c i t y o f w a t e r ( = 3 . 9 × 1 0 6 J m−3 °C−1).

Assuming

T a ¼ T a T̂ a cos ðot gaÞ; (4A:5)

s i n c e t h e e q u a t i o n i s l i n e a r , T S m u s t t a k e t h e f o r m

T s ¼ T s T̂ s cos ðot gsÞ; (4A:6)

where g a and g s a r e t h e p h a s e l a g s o f a i r a n d s e a s u r f a c e t e m p e r a t u r e r e l a t i v e t o t h e

s o la r r a di a ti o n c y c le (4A.1).

Substituting (4A.1), (4A.3), (4A.5) a n d ( 4 A . 6 ) i n t o ( 4 A . 4 ) g i v e s

αDo T̂ s sin ðot gsÞ ¼ S 0 S cosot

þ kð T a T̂ a cos ðot gaÞ T s þ T̂ s cos ðot gaÞÞ:

(4A:7)

T h u s , f o r t h e t i m e i n v a r i a n t t e r m s ,

T s ¼ T a þS 0

k (4A:8)

a n d f o r t h e s e a s o n a l c y c l e ,

T̂ s cos ðot gs BÞ ¼ S cosot þ kT̂ a cos ðot gaÞ

ðk2 þ α2 D2o

2Þ1=2 ; (4A:9)

where B = a rc ta n (−α Dω, k ).

Figure A4.2 (Prandle and Lane, 1995 ) s ho ws t he v al ue s of T̂ s= T̂ a f or a ra ng e

of values of T̂ a and D. T hese resu lts correspond to S 0 = S * = 1 0 0 W m−2

and k = 5 0 W m

−2 o

C

−1

with g a = 3 0° . T he se r es ul ts a re e ss en ti al ly s im il ar f or 0 ° < g a < 9 0 ° a n d f o r e i t h e r S * or k c h a n g e d b y a f a c t o r o f 2 .

R e f e r r i n g t o (4A.9), f o u r q u a d r a n t s c a n b e d i s t i n g u i s h e d i n F i g . 4 A . 2 differentiat-

i n g s h a l l o w a n d d e e p w a t e r a n d s m a l l a n d l a r g e T̂ a:

I n q u a d r a n t Q1 , D≪ k/ αω and T̂ a≫ S */ k , t h u s T̂ s→ T̂ a and g s→ g a .

I n q u a d r a n t Q2 , D≫ k / αω and T̂ a≫ S */ k , t h u s T̂ s→ T̂ a k / Dαω and g s→ g a + 9 0 ° .

I n q u a d r a n t Q3 , D≪ k / αω and T̂ a≪ S */ k , t h u s T̂ s→S */ k and g s→0°.

I n q u a d r a n t Q4 , D≫ k / αω and T̂ a≪S*/ k , t h u s T̂ s→S */ Dαω and g s→ 90°.

4A.4 Coupled atmosphere-marine model

A m aj or l im it at io n i n u s in g t he a b ov e a na ly ti ca l r es ul ts f o r s en s it iv it y t es ts a ri se s

f ro m t he n e ed t o p r es cr ib e a ( fi xe d) a ir t em p er at ur e, t hu s p re cl ud in g t he f ee db a ck




i mp ac t o f c h an ge s i n s ea t em pe ra tu re o n a mb ie nt a ir t em p er at ur e. C le ar ly, t he re i s

a c lo se t he rm al c ou p li ng b e tw ee n t he a tm os ph er e a nd t he s ea w it h c ha ra ct er is ti c

c y c l e s a t d a i l y , m o n t h l y a n d s e m i - m o n t h l y ( s p r i n g –

n e a p t i d e s ) a n d a n n u a l f r e q u e n -c i e s . P r a n d l e (1998 ) f o r m u l a t e d a ‘s i n gl e p o i nt ’ c o u pl e d a i r – s e a t h er m al e x ch a ng e

m od e l c on ce nt ra ti ng o n r ep ro du ct io n o f t he a nn u al t em p er at ur e c yc le o f t he s ea

s u r f a c e a n d a m b i e n t a i r . T h e f o l l o w i n g i s a b r i e f d e s c r i p t i o n o f t h i s m o d e l .

A m ar in e m od el i s l in ke d t o a n a tm os ph er ic m od el w it h a ir – s e a e x c ha n ge r a te s

f r o m G i l l (1982 ) , c o mp r is i ng l o ng w a ve r a di a ti o n , e v ap o ra t io n a n d c o nv e ct i on ( o r

s e n s i b l e h e a t f l u x ) . V e r t i c a l e x c h a n g e w i t h i n t h e w a t e r c o l u m n i s g o v e r n e d b y b o t h

t id al - a nd w in d -f o rc ed t u rb u le nt i nt en si ty l ev el s, m od u la te d b y v er ti ca l d en si ty

g ra di en ts . I nc id en t s o la r r ad ia ti on i s m od u la te d b y a r ef le ct io n c oe ff ic ie n t a t t heo u t e r e d g e o f t h e a t m o s p h e r e a n d t h e n c e b y i n t e r n a l a b s o r p t i o n . T h e h e a t l o s t b y t h e

s ea s ur fa ce i s s ub s eq ue nt ly a b so rb ed b y t he a tm os p he re , e x ce pt f or 0 .3 QL, w h i c h

i s a ss um e d t o r ad i at e d ir ec tl y t o s pa ce . T he a tm os ph e ri c m od e l e ff ec ti ve ly f or ms

a ‘s u r fa c e l a ye r ’ t o t h e m a ri n e m o de l w i th e x te r na l b o u nd a ry c o n di t io n s ( i nc i de n t

F ig . 4 A. 2 . T he a nn ua l t em pe ra t ur e c yc le i n w el l -m i xe d w at er s. ( a) To p T̂ s / T̂ a

( b) b ot to m g s – g a a s f un ct io ns o f T̂ a a nd d ep th r es ul ts c orr es po nd t o S 0 = S * =1 0 0 W m−2, k = 5 0 W m−2 °C −1, g a = 3 0 ° C .

Appendix 4A 115



s ol ar e ne rg y, r ad ia te d a nd r ef le ct ed t he rm al e ne rg y) s pe ci fi ed a t t he o ut er e dg e o f t hea t mo s ph e re . T h e e x t er n al h e a t l o s s r a te f r om t h e a t mo s p he r e f o ll o ws S t ef a n’s l a w .

T h e s a l i e n t f e a t u r e s a r e i n d i c a t e d i n F i g . 4 A . 3 and Ta b l e 4 A . 1 (Prandle, 1998).

T h e m o d e l p a r a m e t e r s , l i s t e d i n T a b l e 4 A . 1 , w e r e a d j u s t e d t o r e p r o d u c e s e a s o n a l

t em pe ra tu re c yc le s o f t he s ea s ur fa ce a nd a mb ie nt a ir o bs er ve d i n t he N or th A tl an ti c,

Reflectedradiation

Stefan’sradiation

T ao

T a

T s

T ao

4

0.3Q L

Absorption

Absorption

z exp(–z ½)

Air–seathermalexchanges S e a

N e t

i n p u t ( 1 –

r ) ( 1 –A ) S

Solarradiation

S

rs

Q s Q E

0.7Q L

θ σ

A(1–r)S

At m o s p h e r e

S p a c e

F ig . 4 A. 3 . S ch em at ic r ep re se nt at i on o f a tm o sp he re – m a ri n e t h er m al e x ch a ng emodel.

Ta b l e 4 A . 1 Model parameters

A t mo s p h e r i c r e f l e c t i o n r = – 0 . 4 7 + 0 . 8 6 c o s λ ½ ( λ latitude)A t mo s ph e r i c a b s o rp t i o n c o e ff i c i en t A = 0 .11A t mo s ph e r i c t e mp e ra t u r e g r a d ie n t ΔT = 4 2 . 5 ° C

C l o u d c o v e r C = 0 . 5Mi n i mu m e d d y d i f f us i v i t y K z = 10−5 (m2 s−1)A t mo s p h e r i c h e i g h t ( w a t e r e q u i v a l e n t ) d = 2 .5 ( m)R e l at i v e h u mi d it y R = 0 . 8Wi nd s pe ed ( m s−1) W = 6 (1 +( λ/65)2) ( 1 + 0 . 5 c o sωa t )

Number vertical grid n = 10 to 100T ime s t e p Δt = 9 00 s




e xt ra ct ed b y I se me r a nd H as se (1983 ) o v e r t h e l a t i t u d e r a n g e 0 – 6 5° N. I n s ha ll ow

w at er ( <2 00 m ), t he a mp l it u de o f t hi s s ea so na l c y cl e i s m od u la te d b y b o th t h e w at er

d e p th a n d t h e t i da l c u rr e nt a m pl i tu d e w i th l a rg e t i da l c u rr e nt s d e cr e as i ng s e as o na l

amplitudes.

The marine model

Te m p er a tu r es w i th i n t h e w a te r c o lu m n a r e c a lc u la t ed f r om t h e v e rt i ca l d i sp e rs i on

equation:

@

@ tT ¼

@

@ Z K Z

@ T

@ Z þ

QðZ Þ

α ; (4A:10)

where T i s t e m p e ra t u r e , K Z v e r t i c al e d d y d i f f us i v i t y, Q( Z ) t h e a t mo s p he r ic t h er m al

i np ut p er u ni t d ep th a t l ev el Z , α t h e t he rm al c ap ac it y o f w at er, t t im e a nd Z thev e rt i ca l a x is . Th e d e te r mi n at i on o f K Z i s v ia t he M el lo r – Ya m a d a ( 1974) l ev el 2 .5

t u rb u le n c e c l o su r e m o de l (A p p en d ix 3 B) . T h e t id al - a nd w in d -d ri ve n c ur re nt s a re

c a l c u l a t e d f r o m a s o l u t i o n o f t h e m o m e n t u m e q u a t i o n ( C h a p t e r 3).

4A.5 The atmospheric model – solar energy input, reflection,

absorption and radiation

U s i n g t h e n o t a t i o n f r o m F i g . 4 A . 3, t h e n e t s o l a r r a d i a t i o n i n t o t h e s e a i s

SR ¼ S ðcos θ ð1 rÞ AÞ; (4A:11)

w h e r e S = 1 3 5 3 W m2 a nd t he i nc li na ti on o f t he s un r el at iv e t o t he v er ti ca l i s

g i v e n b y

cos θ ¼ sin β sin λ cos β cos λ cosð þ od tÞ; (4A:12)

where ωd

i s t h e f r e q u e n c y o f t h e E a r t h’s d a i l y r o t a t i o n , λ l a t i t u d e a n d χ longitude:

sin β ¼ sin δ sinðoa t oÞ; (4A:13)

where ωa i s t h e f r e q u e n c y o f t h e E a r t h’s a n n ua l r o ta t io n , d e cl i na t io n δ = 2 3. 5° a nd

the ‘reference’ longitude χ 0 8 0 ° f o r t m e a s u r e d r e l a t i v e t o J a n u a r y 1 .

T h e f o ll o wi ng e xp re ss io n w as f ou nd n e ce ss ar y t o r ep ro du ce o b se rv ed ( No rt h

A t la n ti c ) a i r a n d s e a s u rf a ce t e mp e r at u re s :

r ¼ 0:47 þ 0:86ðcos λÞ1=2: (4A:14)

T he g lo ba l a ve ra ge o f s ol ar e ne rg y p er u ni t s ur fa ce a re a i s e qu al t o 0 .2 5 S as

c om p ut ed d ir ec tl y f ro m t he r at io o f a sp e ct π R2 t o s ur fa ce a re a 4π R2. T he g lo ba l

m ea n v a lu e f or t he p re se nt f or mu la ti on f or t he a tm os ph er ic r ef le ct io n f ac to r, r , is

0 .3 0, i n p re ci se a g re em en t w it h G il l (1982 ) . T he f or mu l at io n f or t he a tm os ph e ri c

Appendix 4A 117



a b s o r pt i o n c o e f fi c i e n t, A = 0 .11 , c o rr es p on ds t o a g lo ba l m ea n a bs o rp ti o n f ac to r o f

0 .1 5 w hi le G il l e st im at ed 0 .1 9. S om e f le xi bi li ty e xi st s b et we en t he r es pe ct iv e

f o r m u la t i o n s o f A and r . A l at it ud in al v ar ia bi li ty i n A c o ul d b e i nt ro du ce d i n t he

s am e m an ne r a s s p ec if ie d f or r . A f i xe d t e mp e ra t ur e d i ff e re n ce , ΔT , o f 4 2 . 5 ° C i s

a ss um ed b etw ee n t he s ea s urf ac e a nd t he o ut er e dg e o f t he a tm os ph er e. T hec o mp u te d m e an s e a s u rf a ce t e mp e ra t u re d e pe n d s p r im a ri l y o n t h e p r es c ri p ti o ns o f

A, r and ΔT .

4A.6 Global expressions

T he f ol lo wi ng g e ne ra li se d e xp r es s io n s w e re d er iv ed f ro m l ea st s q ua re s f it ti ng t o

m o d e l r e s u l t s :

T s ¼ 4 0 c os λ 12:5

T a ¼ 3 5 c os λ 10:0

T̂ s ¼ 0:080 λ

1 expðD=50Þ for U 50:2 m s1

T̂ s ¼ 0:064 λ

1 expðD=50Þ

for U 40:2 m s1

T̂ a ¼ 0:086 λ

1 expðD=50Þ for U 50:2 m s1

T̂ a ¼ 0:067 λ

1 expðD=50Þ for U 40:2 m s1: (4A:15)

Te m p e r a tu r e s i n d e g r e es c e n t i g ra d e , d e p t h s D i n m e t r e s a n d U * i s t h e t i d a l c u r r e n t

amplitude.

F i gu r e 4 A .4 (Prandle, 1998) s h o w s s a l i e n t r e s u l t s f o r t h e a n n u a l m e a n t e m p e r a -

tures, T s and T a . M e an t e mp e ra t ur e s a r e o v e rw h e lm i ng l y d e te r mi n ed b y l a ti t ud e ,

w i t h l i t t l e i n f l u e n c e o f w a t e r d e p t h o r t i d a l c u r r e n t a m p l i t u d e .

F i gu r e 4 A .5 (Prandle, 1998 ) s ho ws t he v al ue s f or s ea so na l a mp li tu de s, T̂ s and T̂ a ,

i l l u s t r a t i n g t h e d e p e n d e n c e o n l a t i t u d e i n c o m b i n a t i o n w i t h a n e x p o n e n t i a l f u n c t i o n

o f d e p t h . T h e l a tt er d e pe n de n cy i n cr e as e s T̂ s and T̂ a dramatically in shallow water.

T h i s d e p e n d e n c y o n w a t e r d e p t h r e a c h e s a n a s y m p t o t i c l i m i t f o r D≫1 00 m . I n w at er

depths D > 40 m, th e v alu es of bo th ^

T s and ^

T a increase significantly for the smallest tidal current amplitude U* = 0 . 1 m s−1. H ow ev er, f or U * > 0 . 2 m s−1, t he v al ue s o f b oth

T̂ s and T̂ a converge asymptotically towards the reduced values shown in Fig. 4A.5.

T he p ha se s f or t he s ea so na l c yc le s i nd ic at e t ha t m ax im um n or th er n h em i-

s ph er e s ea s ur fa ce t em pe ra tu re s a lm os t a lw ay s o cc ur b et we en J ul ia n d ay s




2 3 0 a n d 2 5 5 – g e n e r a l l y e a r l i e r i n d e e p e r w a t e r . T h e p h a s e s f o r a i r t e m p e r a t u r e s

g e n e r a l l y f o l l o w t h o s e o f t h e s e a s u r f a c e w i t h m a x i m u m t e m p e r a t u r e s t y p i c a l l y

5 – 1 0 d a y s l a t e r .

T h e c o u p l e d m o d e l i s a p p l i c a b l e f o r l a t i t u d e s b e t w e e n 0 ° a n d 6 5 ° N ; a r e a s f u r t h e r

polewards introduce particular difficulties with surface icing and with anomalous

temperature – d en si ty f un ct io n s. T he m od el c an b e u se d f or s ma ll -a mp li tu de s en s i-

t i v i t y a n a l y s e s t o e x a m i n e , e . g . t h e i m p a c t o f c h a n g e s i n c l o u d c o v e r , w i n d s p e e d , o r

r e l a t i v e h u m i d i t y o r t h e m a g n i t u d e a n d p e r s i s t e n c e o f t h e i m p a c t o f a s i n g l e m a j o r

s to rm . T he g en er al it y o f t he d er iv ed e xp re ss io n s f o r T s and T a p r ov id es a s im pl eu nd er st an di ng o f l ik el y c on di ti on s o ve r a r an ge o f v ar yi ng l at it ud es a nd w it h

v ar yi ng w at er d ep th s a nd t id al c ur re nt a mp li tu de s. L ik ew is e, t he m od el m ay b e

u se fu ll y a d op te d f or i nt er di sc ip li na ry s tu d ie s i nc lu d in g f ee d ba ck m ec ha ni sm s

w h er e by b i ol o gi c al a n d c h em i ca l p a ra m et e rs i m p ac t o n a b so r pt i on a n d r e fl e ct i on

coefficients.

I n s u m m a r y , t h e c o u p l e d o c e a n – a t m o s p h e r e n u m e r i c a l s i m u l a t i o n s s h o w t h a t t h e

m e a n v a l u e s o f b o t h a i r a n d w a t e r t e m p e r a t u r e s a r e o v e r w h e l m i n g l y d e t e r m i n e d b y

( th e c o si ne o f) l at it ud e, w it h l it tl e i n fl ue n ce o f w at er d ep t h o r t id al c ur re nt a mp li -t ud e. B y c on t ra st , c o rr es p on di ng s ea so n al a mp l it u de s v ar y d i re ct ly w it h l at it ud e

a lo ng si de a n e xp on en ti al f un ct io n o f d ep th w it h m uc h l ar ge r v al ue s i n s ha ll ow

w e a k l y m i x e d w a t e r s . I n c r e a s e d s t r a t i f i c a t i o n i n s u l a t e s t h e s e a , e s p e c i a l l y a t g r e a t e r

d ep th s , f ro m b o th s ol ar h ea ti ng a nd s u rf ac e h ea t e xc h an ge s. T h e e ff ec t i s t o l ow er

00

20

40

Latitude λ

60

Temperature °C 20

T a

T s

F i g. 4 A .4 . C o mp u t ed a n nu a l m e an t e mp e ra t ur e s: T s s e a s u r f ac e , T a a m b i en t a i r (4A.15) B u n k er A t l a s O b s e rv a t i o ns : ⊗ a i r , + s e a s u r f a c e .

Appendix 4A 119



both the mean and the variability of deeper water temperatures, especially when

a u t u m n a l o v e r t u r n i n g o c c u r s .

References

A b r a h a m , C . , 1 9 8 8 . T u r b u l e n c e a n d m i x i n g i n s t r a t i f i e d t i d a l f l o w s . I n : D r o n k e r s , J . a n d

v a n L eu s s e n , W. ( e d s ), Physical Processes in Estuaries. S p r i n g e r V e r l a g , B e r l i n , pp. 149 – 180.

B o w d e n , K . F . , 1 9 5 3 . N o t e o n w i n d d r i f t i n a c h a n n e l i n t h e p r e s e n c e o f t i d a l c u r r e n t s . Proceedings of the Royal Society of London, A, 219 , 4 2 6 – 446.

B o w d e n , K . F . , 1 9 8 1 . T u r b u l e n t m i x i n g i n e s t u a r i e s . Ocean Management , 6 (2 – 3),117 – 1 3 5 .

F i g . 4 A . 5 . S e a s o n a l a mp l i t u d e s f o r ( a ) s e a - s u f a c e t e mp e r a t u r e T̂ s ( b ) a mb i e n t a i r T̂ a

(4A .1 5). D as he d c on to ur s a re for t id al c urrent amp li tu de s U * ≥ 0 . 2 m s−1.C o nt i nu o u s c o nt o ur s a r e f o r U * < 0 . 2 m s−1. D ig it al v al u es o n t h e r ig ht -h an d a xi s

i n d i c a t e o b s e r v e d v a l u e s f r o m t h e B u n k e r A t l a s .




B o w d e n , K . F . a n d F a i r b a i r n , L . A . , 1 9 5 2 . A d e t e r m i n a t i o n o f t h e f r i c t i o n a l f o r c e s i n a t i d a lcurrent. Proceedings of the Royal Society of London, S e r i e s A , 214, 3 7 1 – 392.

C a v e W. R . , 1 9 9 0 . Re format Procedures and Software for Meteorological Office Data.U n p u b li s h e d R e p o rt . B r i t is h O c e a no g r a ph i c D a t a C e n t re , B i d s t on O b s e r va t o r y,B i r k en h e a d , U K .

D y er , K . R. a n d N e w, A . L. , 1 9 86 . I n te r mi t te n cy i n e s tu a ri n e m i x in g . I n : Wo l fe , D . A. ( e d. ) , Estuarine Variability. Proceedings of the Eighth Biennial International Estuarine Research Conference, U n i v e r s i t y o f N e w H a m p s h i r e , D u r h a m , 2 8 J u l y – 2 A u g u s t ,

1 9 8 5 . A c a d e m i c P r e s s , O r l a n d o , p p . 3 2 1 – 339.G e y e r, W. R . a n d S mi t h, J . D . , 1 9 8 7 . S h e a r i n s t a b i l i ty i n a h i g h l y s t r a t if i e d e s t u a r y. Journal of

Physical Oceanography, 17 ( 1 0 ) , 1 6 6 8 – 1679.G i l l A . E . , 1 9 8 2 . Atmosphere – Ocean Dynamics. A c a d e m i c P r e s s , O x f o r d .G o d i n , G . , 1 9 8 5 . M o d i f i c a t i o n o f r i v e r t i d e s b y t h e d i s c h a r g e . Journal of Waterway Port

Coastal and Ocean Engineering , 111 ( 2 ) , 2 5 7 – 274.G o l d s m i t h R . A . a n d B u n k e r , A . F . , 1 9 7 9 . WHOI Collection of Climatology and Air – Sea

Interaction (CASI) Data, T e c h n i c a l R e p o r t .H a ns e n, D . V. a n d R at t r ay, M . J. , 1 9 66 . N e w d i me n si o n s i n e s tu a ry c l as s i fi c at i o n.

Limonology and Oceanography, 11, 3 1 9 – 326.H e a r n , C . J . , 1 9 8 5 . O n t h e v a l u e o f t h e m i x i n g e f f i c i e n c y i n t h e S i m p s o n – H u n t e r H / U3

criterion. Deutsche Hydrographische Zeitschrift , 38, 1 3 3 – 145.I p pe n , A . T. ( e d. ) , 1 9 6 6. Estuary and Coastline Hydrodynamics. Mc G r a w - H i l l , N e w Y o r k .I p pe n , A . T. a n d H a rl e ma n , D . R. F. , 1 9 61 . O n e- d i me n si o n al a n al y s is o f s a li n it y i n tr u si o n i n

estuaries. Technical Bulletin N o . 5 , C o mm i t te e o n T id a l H y dr a ul i c s Wa t er w ay sE x p e r ime n t S t a t io n , Vi c k s b ur g , MS .

I s e m e r H . J . a n d H a s s e , L . , 1 9 8 3 . The Bunker Climate Atlas of the North Atlantic Ocean ,

Vo l . 1 , O b s e rv a t i o n s. S p r i ng e r - Ve r l ag , p . 2 1 8 .L e wi s , R ., 1 9 97 . Dispersion in Estuaries and Coastal Waters. J o hn Wi l ey a nd S o ns , C hi ch e st er.L i n d e n , P . P . a n d S i m p s o n , J . E . , 1 9 8 8 . M o d u l a t e d m i x i n g a n d f r o n t o g e n e s i s i n s h a l l o w s e a s

a n d e s t u a r i e s . Continental Shelf Research, 8 ( 1 0 ) , 1 1 0 7 – 1127.L i u , W . C . , C h e n , W . B . , K u o , J - T . a n d W u , C . , 2 0 0 8 . N u m e r i c a l d e t e r m i n a t i o n o f r e s i d e n c e

t i m e a n d a g e i n a p a r t i a l l y m i x e d e s t u a r y u s i n g a t h r e e - d i m e n s i o n a l h y d r o d y n a m i cmodel. Continental Shelf Research, 28 ( 8 ), 1 0 68 – 1088.

M e l l o r O . L . a n d Y a m a d a , T . , 1 9 7 4 . A h i e r a r c h y o f t u r b u l e n c e c l o s u r e m o d e l s f o r p l a n e t a r y boundary layers. Journal of the Atmospheric Science, 31 ( 7 ) 1 7 9 1 – 1806.

M u n k , W . H . a n d A n d e r s o n , E . R . , 1 9 4 8 . N o t e s o n a t h e o r y o f t h e t h e r m o c l i n e . Journal of

Marine Research, 7, 2 7 6 – 295. New, A.L., Dyer, K.R. and Lewis, R.E., 1987. Internal waves and intense mixing periods in

a p a r t i a l l y s t r a t i f i e d e s t u a r y . Estuarine, Coastal and Shelf Science, 24 ( 1 ) , 1 5 – 34. Nunes, R.A. and Lennon, G.W., 1986. Physical property distributions and seasonal trends in

S p e n c e r G u l f , S o u t h A u s t r a l i a : a n i n v e r s e e s t u a r y . Australian Journal of Marine and Freshwater Research, 37 ( 1 ) , 3 9 – 53.

Nunes Vaz, R.A. and Simpson, J.H., 1994. Turbulence closure modelling of estuarinestratification. Journal of Geophysical Research, 95 ( C 8 ) , 1 6 1 4 3 – 16160.

O e y , L . Y . , 1 9 8 4 . O n s t e a d y s a l i n i t y d i s t r i b u t i o n a n d c i r c u l a t i o n i n p a r t i a l l y m i x e d a n d w e l lmi x e d e s t u a r i e s . Journal of Physical Oceanography, 14 ( 3 ) , 6 2 9 – 645.

O f f i c e r , C . B . , 1 9 7 6 . Physical Oceanography of Estuaries. J o h n W i l e y a n d S o n s , N e w Y o r k .O l so n , P. , 1 9 86 . T h e s p ec t ru m o f s u b- t id a l v a ri a bi l it y i n C h es a pe ak e B a y c i rc u l at i on .

Estuarine, Coastal and Shelf Science, 23 ( 4 ) , 5 2 7 – 550.P r a n d l e , D . , 1 9 8 1 . S a l i n i t y i n t r u s i o n i n e s t u a r i e s . Journal of Physical Oceanography, 11,

1311 – 1324.

References 121



P r a n d l e , D . , 1 9 8 2 . T h e v e r t i c a l s t r u c t u r e o f t i d a l c u r r e n t s a n d o t h e r o s c i l l a t o r y f l o w s .Continental Shelf Research, 1 ( 2 ) , 1 9 1 – 207.

P r a n d l e , D . , 1 9 8 5 . O n s a l i n i t y r e g i m e s a n d t h e v e r t i c a l s t r u c t u r e o f r e s i d u a l f l o w s i n n a r r o wt i d a l e s t u a r i e s . Estuarine Coastal and Shelf Science, 20, 6 1 5 – 633.

P r a n d l e , D . , 1 9 9 7 . T h e d y n a m i c s o f s u s p e n d e d s e d i m e n t s i n t i d a l w a t e r s . Journal of Coastal

Research ( S p e c i a l I s s u e N o . 2 5 ) , 7 5 – 86.P r a n d l e , D . , 1 9 9 8 . G l o b a l e x p r e s s i o n s f o r s e a s o n a l t e m p e r a t u r e s o f t h e s e a s u r f a c e a n da m b i e n t a i r : t h e i n f l u e n c e o f t i d a l c u r r e n t s a n d w a t e r d e p t h . Oceanologica Acta, 21 (3),419 – 428.

P r a n d l e , D . , 2 0 0 4 a . S a l i n e i n t r u s i o n i n p a r t i a l l y m i x e d e s t u a r i e s . Estuarine, Coastal and Shelf Science, 59, 3 8 5 – 397.

P r a n d l e , D . , 2 0 0 4 b . H o w t i d e s a n d r i v e r f l o w d e t e r m i n e e s t u a r i n e b a t h y m e t r y . Progress inOceanography, 61, 1 – 26.

P ra nd le D . a nd L an e, A ., 1 99 5. T he a nn ua l t em pe ra tu re c yc le i n s he lf s ea s. Continental Shelf Research, 15 ( 6 ) , 6 8 1 – 704.

P r i t c h a r d , D . W . , 1 9 5 5 . E s t u a r i n e c i r c u l a t i o n p a t t e r n s . Proceedings of the American Societyof Civil Engineers, 81 ( 7 1 7 ) , 1 – 11.

R i gt e r, B . P. , 1 9 73 . M i ni m u m l e ng t h o f s a lt i n tr u si o n i n e s tu a ri e s. Proceedings of the American Society of Civil Engineers Journal of the Hydraulics Division, 99, ( H Y 9 ) ,

1475 – 1496.R i pp e th , T. P. , F i sh e r, N . R. , a n d S i mp s on , J . H. , 2 0 01 . T h e c y cl e o f t u rb u l en t d i ss i p at i on i n

t h e p r e s e n c e o f t i d a l s t r a i n i n g . Journal of Physical Oceanography, 31, 2 4 5 8 – 2471.R o s s i t e r , J . R . , 1 9 5 4 . T h e N o r t h S e a S t o r m S u r g e o f 3 1 J a n u a r y a n d 1 F e b r u a r y 1 9 5 3 .

Philosophical Transactions of the Royal Society of London, A, 246, 3 1 7 – 400.S c h u l t z , E . A . a n d S i m m o n s , H . B . , 1 9 5 7 . F r e s h w a t e r – s a l t w a t e r d e n s i t y c u r r e n t s , a m a j o r

c a u s e o f s i l t a t i o n i n e s t u a r i e s . Technical Bulletin, N o . 2 , Communication on Tidal Hydraulics, U . S . A r m y , C o r p s o f E n g i n e e r s .

S i m p s o n , J . H . a n d B o w e r s , D . G . , 1 9 8 1 . M o d e l s o f s t r a t i f i c a t i o n a n d f r o n t a l m o v e m e n t i ns h e l f s e a s . Deep-Sea Research, 28, 7 2 7 – 73 8.

S i m p s o n , J . H . a n d H u n t e r , J . R . , 1 9 7 4 . F r o n t s i n t h e I r i s h S e a . Nature, 250, 4 0 4 – 406.S i m p s o n , J . H . , B r o w n , J . , M a t t h e w s , J . , a n d A l l e n , G . , 1 9 9 0 . T i d a l s t r a i n i n g , d e n s i t y

c u rr e nt s a n d s t i rr i ng i n t h e c o nt r o l o f e s tu a ri n e s t ra t if i ca t io n . Estuaries, 13 (2),125 – 132.

S o u z a , A . J . a n d S i m p s o n , J . H . , 1 9 9 7 . C o n t r o l s o n s t r a t i f i c a t i o n i n t h e R h i n e R O F I s y s t e m . Journal of Marine Systems, 12, 3 1 1 – 323.

U nc le s, R .J . , S te ph en s , J .A . a nd S mi th , R .E ., 2 0 02 . T he d ep en de nc e o f e st ua ri ne t ur bi d it y o nt i d a l i n t r u s i o n l e n g t h , t i d a l r a n g e a n d r e s i d e n c e t i m e . Continental Shelf Research, 22,1835 – 1856.

Wa n g, D . P. a n d E l li o t t, A . J. , 1 9 78 . N o n- t i da l v a ri a bi l i ty i n C h es a pe a ke B a y a n d P o to m acR i v e r : e v i d e n c e f o r n o n - l o c a l f o r c i n g . Journal of Physical Oceanography, 8 (2),225 – 232.




5

Sediment regimes

5.1 Introduction

Understanding and predicting concentrations of suspended particulate matter

(SPM) in estuaries are important because of their impact on (i) light occlusion and

thereby primary production, (ii) pathways for adsorbed contaminants and (iii) rates

of accretion and erosion and associated bathymetric evolution.

Sediments are traditionally classified according to their (mass equivalent) particle

diameter, d , with sizes ranging from clay <4 μ, 4 μ < silt < 63 μ, 63 μ < sand < 1000 μ

through to gravel and rocks. Importantly, the silt – sand demarcation separates

cohesive (mutual attraction by electro-chemical forces) from non-cohesive sedi-ments. In higher concentrations, the former tend to flocculate into multiple assem-

blages which can both settle more rapidly and inhibit the upward flux of turbulent

energy from the sea bed (Krone, 1962). In extreme cases, a layer of liquid mud may

form a two-phase flow continuum. Moreover, once deposited, consolidation of

cohesive material can radically change re-erosion rates. Only a few percent of ‘mud’

content may strongly influence what might appear to be a cohesionless sandy bed

(Winterwerp and van Kesteren, 2004).

Over millennia, the inter-glacial rise and fall of sea level effectively determinesestuarine morphology. Over shorter time scales, of interest to coastal engineers and

coastal planners, some quasi-equilibria develop encompassing variations in bathy-

metry over ebb to flood and spring to neap tides, alongside seasonal cycles, random

storms and episodic extreme events. For efficient management of estuaries, we need

to understand the associated patterns of sediment movement in order to harmonise

development with natural trends and to mitigate related hazards such as flooding or

silting of navigation channels.

5.1.1 Sediment dynamics

At first sight, sediment dynamics appear deceptively simple – a sequence of erosion,

suspension, transport and deposition. However, severe complications can arise

123



when encompassing a mixture of sediment sizes influenced by past and present

dynamics, modulated by chemical and biological processes. The limits of observa-

tional technology in measuring parameters such as concentrations, size-spectra,

fluxes or net scour/accretion exacerbate these complications.

The predominant influences on sediment regimes in estuaries are tidal and stormcurrents, enhanced in exposed shallow water by wave stirring. Detailed accounts of

the mechanics of sediment motion associated with tidal currents and waves can be

found in Grant and Madsen (1979), Van Rijn (1993) and Soulsby (1997). Postma

(1967) describes general features of the erosion, deposition and intervening trans-

port of SPM in tidal regimes. For all but the coarsest grain sediment, several cycles

of ebb and flood movement may occur between erosion and subsequent deposition.

Hence, deposition can occur over a wide region beyond the source. Since time in

suspension increases for finer, slowly settling material, such mechanisms maycontribute to a residue of fine materials on tidal flats and to trapping of coarser

material in deeper channels.

5.1.2 Modelling

Reproducing these characteristics in models remains sensitive to largely empirical

formulae used in prescribing erosion and deposition rates. Bed roughness strongly

influences these rates; it is largely determined by the composition (fine to coarse)

and form (ripples, waves) of the bed. Sediment processes are complicated by the

continuous dynamical feedback between this roughness and the overlying vertical

structure of tidal currents and waves and their associated turbulence regimes

(Fig. 5.1). Bed roughness can change significantly over both the ebb to flood and

the neap to spring tidal cycles. Associated erosion and deposition rates may then

Eddy

diffusivity

Waves

Flocculation –

accelerated settling

Turbulence damping –

hindered settlingTidal

currents

Effective bed

roughness

f (grain size, bed form)

Consolidation

Bioturbation

Eddy

viscosity

Fig. 5.1. Processes determining sediment erosion, transport and deposition.

124 Sediment regimes



vary considerably over these cycles and dramatically over seasons or in the course

of a major event.

Conventionally, erosion is assumed to occur when the bed shear forces exceed the

resistance of the bed sediment, characterised by a ‘critical shear stress for erosion’.

The rate of erosion is generally assumed to be proportional to the excess of theapplied shear stress over the critical shear stress (Partheniades, 1965). Associated

formulae vary widely, with consequent erosion rates sometimes varying by factors

of 10 or more, emphasising the difficulty in formulating robust portable models.

In nature, this threshold depends on particle size distribution and both chemical

and biological modulation, including effects of bioturbation and biological binding.

Bioturbation of the top metre or so of surficial sediment may significantly reduce

erosion thresholds. Conversely, (surficial) biological binding can have the opposite

effect – especially in inter-tidal zones (Romano et al ., 2003). Erosion depends not only on the prevailing physical, chemical and biological composition but also on the

conditions at the time of deposition and the intervening historical sequence.

Subsequent settlement of particles depends on their size, density and the ambient

regime of turbulence and chemical forces in the surrounding water. Sedimentation

is usually assumed to occur when quiescent dynamical conditions are below some

threshold for erosion at a rate equal to the product of the near-bed concentration

and the ‘settling velocity’. The latter can be estimated from the density, size and

shape of the sediment or determined in laboratory settling tubes.

Accurate simulation of sediment fluxes requires an initial prescription of the

distribution of surficial sediments. Simulations over larger and longer space and

time scales need to incorporate sequential changes in these surficial sediments as

these adjust to variations in tidal and wave conditions resulting from trends and cycles

in the inter-related evolving bathymetry. On even longer time scales, likely changes in

both msl and sources of marine and fluvial sediments need to be incorporated.

5.1.3 Approach

Here we focus on tractable elements, namely regimes dominated by tidal forcing.

Concentrating on tidal components provides inter-comparisons between results from

theory, models and observations in terms of robustly determined constituents. The

analytical solutions obtained are intended to provide insight into mechanisms and

thereby complement rather than substitute for existing complex numerical models.

SPM concentration models involve the solution of the ‘conservation of mass’equation:

dC

dt ¼ @ C

@ t þ U

@ C

@ X þ V

@ C

@ Y þ W

@ C

@ Z ¼ @

@ Z K z

@ C

@ Z sinks þ sources; (5:1)




where C is concentration, U , V and W are velocities along orthogonal axes X , Y

and Z (vertical) and K z is a vertical eddy dispersion coefficient.

The horizontal advective velocities U and V can be accurately calculated in tidal

models, as described in Chapters 2 and 3. The omission of axial and transverse

dispersion terms in (5.1) is generally valid in models with sufficiently fine resolutionin time and space.

Here we assume that erosion is proportional to some power of velocity and no

thresholds are introduced for either erosion or deposition. Moreover, erosion

and deposition are allowed to occur simultaneously. Analytical solutions of (5.1)

show how suspension is determined by three parameters, namely the settling or

fall velocity W s, vertical turbulent displacements characterised by a vertical eddy

dispersion coefficient K z and water depth D. These solutions provide Theoretical

Frameworks illustrating the nature and scaling parameters that determine SPMconcentrations, times in suspension, vertical profiles and the predominant tidal

constituents in time series of SPM.

In strongly tidal conditions without pronounced stratification, the magnitude of

the vertical diffusivity may be approximated by that of the vertical eddy viscosity

coefficient. For consistency of notation with the original papers by Prandle (1997a

and 1997b), henceforth we replace K z by E . In order to derive simplified analytical

solutions, it is assumed that E is constant both temporally and vertically.

The analytical expressions for erosion and deposition of SPM are presented in

Sections 5.2 and 5.3. The mathematical details involved in combining these into

expressions for resulting suspended concentrations are shown in Appendix 5A and

outlined in Section 5.4. Section 5.5 describes the effects of integrating continuous

cycles of erosion, modulated by an exponential settling rate to determine character-

istic tidal spectra of SPM concentrations. Applications of these theories are exam-

ined in Section 5.6 against both modelled and observed results. Section 5.7 reflects

on related progress towards predicting bathymetric evolution in estuaries.

5.2 Erosion

Erosion ER(t ) due to a tidal current U (t ) is generally assumed to be of the form

ERðtÞ ¼ γ ρ f jU ðtÞjN ; for jU ðtÞj 4U c (5:2)

ER

ðt

Þ ¼0; for

jU

ðt

Þj U c; (5:3)

where γ is an empirical coefficient, ρ the density of water, f the bed friction

coefficient and N is some power of velocity typically in the range of 2 – 5.

The stipulation of a minimum threshold velocity, U c, below which no erosion

occurs, often has surprisingly little effect in strong tidal conditions where




velocities exceed 0.5 m s−1. As an example, setting U c =0.5U * for a tidal velocity

U (t ) = U * cos ωt reduces the net erosion by only 10% for N = 4. Thus, for simplicity,

it is convenient henceforth to set U c = 0. Lavelle et al . (1984) carried out similar

analyses of (5.2) and (5.3) and inferred a value of N = 8 for fine sediment; they also

omitted U c. Van Rijn (1993) found transport rates of fine sands proportional to powers of U between 2.5 and 4, dependent on wave conditions.

Lane and Prandle (2006) found, for N =2, a value of γ = 0.0001 (m−1 s) best

reproduced sediment concentrations in the Mersey Estuary.

5.2.2 Spring – neap cycle

A characteristic of tidal currents in mid-latitudes is the predominance of the

semi-diurnal lunar M2 and solar S2 constituents with periods 12.42 h and 12 h,respectively (see Appendix 1A). Although the ratio of their tidal potentials is

1 : 0.46, their observed ratio at the coast is generally closer to 1 : 0.33 (related to

the proportionately greater frictional dissipation effective for all constituents other

than M2). The small difference in their periods produces the widely observed

15-day spring (in phase) and neap (out of phase) MS f tidal constituent alongside

the related quarter-diurnal constituent MS4.

For the case of N =2and U c = 0, the erosional time series for a combination of M2 and

S2 tidal currents of amplitude U M and U s, alongside a constant residual velocity U 0, is

ðU M cos M2t þ U S cos S2t þ U 0Þ2 ¼ 0:5U M2ð1 þ cosM4tÞ

þ 0:5U S2ð1 þ cosS4tÞ þ U 0

2

þ 2U 0ðU M cos M2t þ U S cos S2tÞþ U MU Sðcos MS f t þ cos MS4tÞ: (5:4)

The frequencies M4 = 2M2, S4 = 2S2 and MS4 = M2 + S2 represent quarter-diurnal

constituents while MS f = S2−

M2 has a period of 15 days.Table 5.1 lists the tidal constituents corresponding to (5.4). To illustrate typical

relative magnitudes of these constituents, it is assumed that U M = 3U S = 1 (arbitrary

units). Further assuming U 0≪U M, the largest erosional constituents are Z0, 0.55 M4,

0.5 and both MS4 and MS f , 0.33.

Thus, while tidal currents predominantly involve M2 and S2 constituents, these

translate into time series of erosion characterised in descending order by Z0, M4,

MS4 and MS f . However, this erosional time series is subsequently modulated by the

deposition phase – resulting in further modulation of the SPM concentrations asdescribed in Section 5.5.

The ratio of the erosional amplitudes of M4: MS4 is U M: 2U S; thus, the relative

phasing of these constituents over the spring – neap cycle can strongly influence the

apparent characteristics of the time series. The two constituents are in phase at

5.2 Erosion 127



spring tides – indicating a strong quarter-diurnal variability. Conversely, they are out

of phase at neap tides, often resulting in a predominant semi-diurnal constituent that

can be incorrectly interpreted as suggesting either predominant diurnal currents or

horizontal advective influences.

Influence of U0

From Table 5.1, the two largest constituents associated with U 0 will be M2 and S2.

For these constituents to be equal to the M4 constituent requires U 0 to be 0.25 of the

M2 current amplitude or 0.75 of the S2 amplitude.

5.2.3 Erosion in shallow cross sections

Section 2.6.3 shows that to maintain continuity of an M2 flux U *cos(ωt ) through a

triangular cross section of mean depth D and elevation, ς *cos(ωt −θ ), requires

currents at M4 and Z0 frequencies given by

U 2

¼ a U cos

ð2ωt

θ

Þ and

U 0 ¼ a U cos θ ; (5:5)

where a = ς*/ D.

Assuming erosion proportional to velocity squared, this yields erosional compo-

nents at the following frequencies:

½ U cosðωtÞ a U cosð2ωt θ Þ a U cos θ 2 ¼U 2½ð0:5ð1 þ a2Þ þ a2 cos2 θ Þ

a ðcosðωt θ Þ 2cos θ cos ωtÞþ 0:5cosð2ωtÞ þ 2a2 cos θ cos ð2ωt θ Þ a cosð3ωt θ Þþ 0:5a2 cosð4ωt 2θ Þ: (5:6)

Table 5.1 Amplitudes of erosional constituents

corresponding to (5.4) for N = 2 and U M = 3 US = 1

N = 2 U M = 3U S = 1

Z0 0.5(U M2 + U S2) + U 02 0.55 + U 02

MSf U MU S 0.33M2 2U MU 0 2U 0S2 2U SU 0 0.67U 0M4 0.5U M

2 0.50MS4 U MU S 0.33S4 0.5U S

2 0.05




Since the value of ‘a’ can approach 1 in shallow, strongly tidal estuaries, significant

current components can occur at all of the above frequencies, i.e. Z0, M2, M4, M6

and M8.

5.2.4 Advective component

At any fixed position, advection may constitute a source or sink of sediments. The

differing tidal characteristics of advection in comparison with localised resuspen-

sion are described below.

For a purely advective source, (5.1) reduces to

dC

dt ¼ U

@ C

@ X

(5:7)

with the solution

C ¼ U

ω sin ωt

@ C

@ X (5:8)

for the current component U *cos ωt and a constant horizontal gradient ∂C / ∂ X .

Significant values of ∂C / ∂ X are likely where spatial inhomogeneity in tidal

currents, sediment supply or water depths occur. The resulting tidal constituents

for suspended concentrations are proportional to the product of tidal current ampli-tude and period (of the constituent concerned) and show a 90° phase shift relative to

the current.

5.3 Deposition

5.3.1 Advective settlement versus turbulent suspension (Peclet number)

Siltation or deposition occurs both by steady advective settlement at the fallvelocity W S and by intermittent contacts with the bed via vertical ‘turbulent

excursions’ characterised by the vertical dispersion coefficient, K z, here repre-

sented by E . Specific near-bed conditions determine entrainment or ‘capture’

rates.

The relative importance of advection via the fall velocity W S, compared with

dispersion via E , may be estimated from their associated time constants. The time T Ato fall from the surface to the bed via advection is D/ W S while the time T E to mix

vertically following erosion at the bed is D2

/ E (Lane and Prandle, 1996). Thus, theratio of the time scales for advective settling to turbulent suspension is E / DW S,

i.e. the Peclet number for vertical mixing. The Peclet number is inversely propor-

tional to the familiar Rouse number. In shallow, strongly tidal waters with an

absence of pronounced stratification, a spatially and temporally constant

5.3 Deposition 129



approximation to the vertical eddy viscosity and eddy dispersion coefficients can

conveniently be represented by (Prandle, 1982):

K z ¼ E ¼ fU D; (5:9)

where f , the bed friction coefficient, is approximately 0.0025 and U * is the tidal

current amplitude. The Peclet number is then f U */ W S.

Assuming axial and lateral homogeneity allows the terms involving horizontal

advection in (5.1) to be neglected. A ‘single-point ’ vertical distribution of sus-

pended sediments can then be described by the dispersion equation:

@ C

@ t þ W s

@ C

@ Z ¼ E

@ 2C

@ Z 2 sinks þ sources; (5:10)

where the sediment fall velocity W s replaces the (small) vertical velocity W .

5.3.2 Bottom boundary conditions

A major difficulty in numerical simulations of suspended sediments based on (5.10)

is the representation of conditions close to the bed, where sharp gradients can exist

for both E and W s (flocculation) and hence in concentration. Appendix 5A shows

that the assumption of settlement at the rate W s C o is generally twice the value

pertaining with a Gaussian vertical sediment distribution where dispersion counter-

acts half of this advective settlement.

The use here of an analytical solution avoids the sensitivity, encountered in

numerical schemes, to the precise discretisation of the near-bed region. However,

questions still arise as to the extent to which suspended particles rebound or settle on

collision with the bed. Sanford and Halka (1993) provide a comprehensive analysis

of alternative bottom boundary conditions.Three conditions are examined in Appendix 5A:

[A] A fully reflective bed. Settlement occurs via advection at the rate 0.5W SC o. Dispersive

collisions with the bed rebound.

[B] A fully absorptive bed. Like condition [A], except that all dispersive collisions settle.

[C] Like [A], except that there is additional settlement, by dispersion, of those particles

which have previously been reflected from the surface. This solution is an expedient

approximation, providing solutions intermediate between [A] and [B].

The effect of these three boundary conditions on the rate of deposition is illustrated

in Fig. 5.2 (Prandle, 1997a).

For E / DW S < 1, Fig. 5.2 and Appendix 5A show that the deposition rate, of

0.5W SC 0, is the same for all three conditions.




For E / DW S > 1, the deposition rate varies widely for the three conditions. For

conditions [B] and [C], maximum concentrations occur in the range 1 < E / W S D < 10.

Here, we adopt the intermediate approximation, bottom boundary condition [C].

5.4 Suspended concentrations

5.4.1 Time series of sediment concentration profiles

Appendix 5A shows that, by adopting the boundary condition [C], time series of

sediment concentration profiles associated with each erosional event of magnitude

M take the form

C ðZ ; tÞ ¼ M

ð4π E tÞ1=2 exp ðZ þ W s tÞ2

4E t þ exp ð2D þ W s t Z Þ2

4E t

" #: (5:11)

The observed time series represents a time integration of all such preceding ‘events’

as described in Section 5.5.

In developing generalised theory, it is advantageous to derive expressions for

deposition in terms of depth –

mean concentrations. From (5A.4) and (5A.7),

C z ¼ 0

C D ffiffiffiffiffiffiffiffi

πEtp exp W s

2t2=4Et

1 W 2s t2=4Et 1=2

h i D ffiffiffiffiffiffiffiffiπEt

p : (5:12)

W st / D

0.1

0.5

0.9

100

10

1.0

0.1

0.1 1.0 10 100E / W sD

Fig. 5.2. Sediment fraction remaining in suspension at time W S t / D as a functionof E / W S D. Bottom boundary condition [A] is dashed, [B] is dotted and [C] issolid.




Thus, a deposition rate expressed as W sC z= 0 transforms to 0:56W s D=E ð Þ1=2 C =t1=2

when related to depth-averaged concentrations.

5.4.2 Exponential deposition, half-lives in suspension, t50

Approximating deposition of an initial suspended concentration C 0 by an exponen-

tial loss rate C 0e−αt enables simple analytical expressions to be derived for the time

series of combined erosion and deposition over successive tidal cycles. Such

an exponential decay rate corresponds to a half-life in suspension t 50 = 0.693/ α.

For E/W s D < 1.

It is shown in Appendix 5A, (5A.7) and from Fig. 5.2, that the fraction of

sediment remaining in suspension FR approximates

FR ¼ 1 0:5 W 2s t

E

1=2

: (5:13)

Equating (5.13) to e−αt for FR = 0.5 requires

α ¼ 0:693 W 2s

E :

For E =W sD41

(5:14)

Adopting condition [C], Appendix 5A indicates an expression for the

half-life t 50 = 0.693/ α, i.e. the time required for 50% deposition is equivalent to

α ¼ 0:1E

D2 : (5:15)

For implementation of (5.14) and (5.15), a continuous transition can be obtained by

simple curve fitting of the results shown in Fig. 5.2. Thus, the parameter α can be

closely approximated by

α ¼ 0:693 W s=D

10x ; (5:16)

where x is the root of the equation:

x2 0:79 x þ j ð0:79 j Þ 0:144 ¼ 0 (5:17)

with j =log10 E / W s D.

Assuming the following relationship between fall velocity W S and particle

diameter d

W s ðms1Þ ¼ 106 d 2 ðμmÞ: (5:18)




Figure 5.3 illustrates corresponding half-lives in suspension for W S = 0.005 and

0.0005 m s−1, i.e. d = 71 and 22 μ, as functions of tidal current amplitude and depth.

Thus, for sand, half-lives range from a minute in sluggish shallow water to 2 h in

deep strongly tidal waters. For fine silt, these values range from hours in shallow

water to 2 days in deep water.

More generally, using dynamical solutions for a ‘

synchronous’

estuary, Fig. 7.6indicates values of α ranging for sand from l0−3 to 10−2 s−1 corresponding to values

for t 50 of 10 – 1 m. The equivalent values for fine silt and clay are α from 10−6 to

10−4 s−1 for t 50 of 200 – 2 h.

An important result is that for the finer sediment, the half-life in suspension is

almost always of order (6 h) or greater. This implies that where estuarine sediment

regimes are maintained by the influx of fine marine sediments, these will remain in

near-continuous suspension and so their distributions will share features of a

continuously suspended conservative tracer such as salt.To further illustrate the characteristics and significance of this parameter E / W S D, we

use (2.19) to relate tidal current amplitudes U * to elevation amplitude ζ *. Figure 5.4

(Prandle, 2004) then shows, for D = 8 m, the relationship between W S and E / W S D

for ζ * = 1, 2 and 3 m, i.e. almost the complete range of tidal conditions encountered.

2 h

1 h

10 h

20 h

50 h

1 h

15 m

5 m

1 m

Fall velocity w s = 0.005 m s –1 Fall velocity w s = 0.0005 m s –1

T i d a

l c u r r e n t a m p l i t u d e ( m s – 1 )

T i d a

l c u r r e n t a m p l i t u d e ( m s – 1 )2.0

1.0

00 10 20 30 40

Depth (m)

2.0

1.0

00 10 20 30 40

Depth (m)

Fig. 5.3. Half-lives, t 50, of SPM in suspension as f ( D,U *). (Left) Sand,W

S=0.005ms−1, (right) silt, W

S=0.0005ms−1. t

50= 0.693/ α, α from (5.16).




This illustrates how the demarcation line of E / W S D ≈ 1 coincides with fall velo-

cities of order of (1 mm s−1) or, from (5.18), d = 32 μ. It also shows how the value

of E / W S D lies in the range from 0.1 to 10 for sediment diameters in the range

20 – 200 μ. Hence, the approximation, (5.16), for α, based on Fig. 5.2, should bewidely applicable.

5.4.3 Vertical profile of suspended sediments

It is shown in Appendix 5A that for E / DW S < 0.3, less than 1% of particles reach the

sea surface, whereas for E / DW S > 10, suspended sediments are well mixed verti-

cally. Thus, throughout the range of conditions considered, clay is always likely

to be well mixed vertically, whereas sand will be confined to near-bed regions for

tidal velocity amplitudes U *≤ 0. 5ms−1. Likewise, pronounced vertical structure

throughout the water column is likely for silt when U * ≤ 0.5 m s−1.

A continuous functional description of the vertical profiles of suspended sedi-

ment concentrations was calculated by Prandle (2004) by numerical fitting of a

profile e− β z (where z is the fractional height above the bed) to simulations based on

(5.12). The following expression for β was derived:

β ¼ 0:91 log10 6:3 E

DW s

1:71: (5:19)

Figure 5.5(a) (Prandle, 2004) shows values of β over the range E / DW S from 0 to 2.

Figure 5.5(b) shows corresponding values of sediment profiles e− β z (1−e− β )/ β ,

0.05

0.005

0.00050 1 2 3 4 5

E / W sD

ζ = 2 mζ = 1 m

ζ = 3 m

W s

∧ ∧

∧

Fig. 5.4. Fall velocity W S (m s−1) as a function of E / W s D tidal elevation amplitudesζ* = 1, 2 and 3 m; results are for depth D = 8 m .




illustrating how effective complete vertical mixing is achieved for E / DW S > 2, and

‘ bed load’ only occurs for E / DW S < 0.1. (Concentration profiles are normalised

relative to the depth-averaged value.)

5.5 SPM time series for continuous tidal cycles

The concentration C (t ) associated with erosion varying sinusoidally at a rate of

cos ωt subject to an exponential decay rate – αC involves integration over all

preceding time t 0 from – ∞ to t , that is

C ðtÞ ¼ ð t1

cos ωt0eα

ðt

t0Þdt0 ¼

α cos ωt

þω sin ωt

α2 þ ω2 : (5:20)

Hence, from (5.2), the concentration, C ω, for any erosional constituent, ω, is

given by

C ω ¼ γ f ρ ½U N ω

Dðω2 þ α2Þ1=2 ; (5:21)

where [U N

]ω is the erosional amplitude at frequency ω as shown in Table 5.1for N = 2.

Thus, erosion generated at each of the tidal constituents in the expansion of

source terms is subsequently modulated by an exponential decay rate that involves

an amplitude reduction by the factor (α2 + ω 2)−1/2 and a phase-lag of arctan (ω/ α).

0.1

0.9

1

1.1

2

10

0.5

1

00 1 2

z β

5(a) (b)

4

3

2

1

00 1 2

K z / W sD K z / W sD

Fig. 5.5. Concentration profiles and values of β described by sediment profile e− β z ,(5.19). (a) Values of β as a function of K z / W S D. (b) Corresponding profiles;contours are fractions of depth-mean concentrations, surface is z = 1.

5.5 SPM time series for continuous tidal cycles 135



The response function represented by (5.20) is shown in Fig. 5.6 (Prandle, 1997b)fora

range of values of α and ω from 10−7 t o 1 s−1. For α≫ω, the amplitude of the response

is proportional only to 1/ α with zero phase lag. Conversely, for ω≫α, the amplitude

of the response is proportional only to the period concerned with a 90° phase lag.

Thus, for equal amounts of eroded sand (α 10

−3

s

−1

) and silt/clay (α 10

−6

s

−1

),away from the immediate vicinity of the bed, the amplitude ratios of the suspended

sediment tidal signal for sand are always much smaller than for silt/clay.

Within this silt/clay fraction, the ratio of the SPM tidal constituent amplitudes is

determined by the product of (i) the ratios shown in Table 5.1 and (ii) the duration of

the tidal periods concerned. These amplitudes are largely independent of the specific

value of α (within the range concerned). Likewise, for this silt/clay fraction, the

phases are lagged by 90° in the quater-diurnal to diurnal tidal band. Thus, factoring

the amplitudes in Table 5.1 by tidal period, we expect the MS f , 15 days cycle to predominate followed by M4 and MS4. Phase values of up to 90° for MS f indicate

associated maximum suspended sediment concentrations occurring up to 3.5 days

after maximum currents. Conversely, phase values approaching 0° for M2 and S2

indicate associated maximum suspensions in phase with maximum currents.

10 –1 101

102

103

5°

45°

85°

104

105

106

11

10 –2

10 –3

10 –4

10

–5

10 –6

ω (s –1)

α (s –1)

10 –6 10 –5 10 –3 10 –2 10 –1 110s1D15D1Y D/2 D/4

SAND

S

ILT&CLAY

Fig. 5.6. Concentration amplitudes for unit erosion at cyclic frequencies ω for deposition rate e−αt . Solid contours indicate relative amplitudes and dashed lines phase lag between SPM and erosion (5.20).




For ω≫α, the Z 0 constituent is factored by ω/ α relative to the tidal frequency.

Thus, by using the Z 0/MS f ratio derived from observations alongside the expansions

shown in Table 5.1 and (5.20), a direct estimate of α can be obtained.

5.6 Observed and modelled SPM time series

5.6.1 Observational technologies

Observations are crucial to developing and assessing SPM models. In situ con-

centrations are routinely monitored acoustically, optically and mechanically.

Acoustic backscatter (ABS) probes provide vertical profiles of concentration,

multi-frequency probes provide information on grain size – usually at a single

point. Pumped samples, bottles and traps are used in mechanical devices. Recent

developments of in situ laser particle sizers provide valuable information on particle spectra non-invasively (mechanical samplers can corrupt these spectra).

Available observations suffer from fundamental shortcomings, namely (i) calibra-

tion from sensor units to concentration involving complex sensitivity to particle size

spectra in optical and acoustic instruments and to atmospheric corrections and sun

angle effects in remote sensing; (ii) unresolved particle-size spectra and (iii) limited

spatial and temporal coverage relative to the inhomogeneity of sediment distributions.

The spatial resolution of in situ concentration measurements is generally limited

to single points (or limited profiles) in optical backscatter (OBS) and ABS sensors

and to surface values from satellite or aircraft sensors. Techniques to circumvent

these shortcomings are described by Gerritsen et al . (2000) where the spatial

patterns of surface imagery are used to validate models.

Each instrument has its own calibration peculiarities. Moreover, all of these

calibrations vary as the mean particle size changes. Optical devices rely on occlu-

sion of light – transmittance or reflectance (OBS). Since this is dependent on

the surface area of the particle, recordings are more sensitive to finer scale particles.Hence, observed concentrations need to be calibrated by reference to some represen-

tative particle radius. The plate-like character of flocs complicates such calibrations.

Conversely, ABS (in the range of frequencies used in ABS instruments) increases

with particle volume, and hence, these instruments are more sensitive to coarse

particles. The optical instruments also experience fouling, and all of the instruments

can be swamped above certain concentrations.

Satellite images of (surface) SPM concentrations can be used in conjunction with

model simulations to infer the magnitude of discrete sediment sources. Aircraft surveillance using multi-wavelength imagery can differentiate between the reflec-

tance from SPM associated with chlorophyll and that from various sediment frac-

tions. However, the need for atmospheric corrections introduces some reliance on in

situ calibrations.




On the longer time scale, information in sediment cores (judicious choice of

location is crucial) may be dated using seasonal striations, specific contaminants

(radio nuclides, Pb-210, etc.) and various natural chemical signals or biological

fossils (Hutchinson and Prandle, 1994). The range of such techniques is expanding

rapidly providing opportunities to derive both geographic provenance and asso-ciated age. Both light detection and ranging (LIDAR) and synthetic aperture radar

(SAR) surveys can be used to determine sequences of bathymetric evolution.

5.6.2 Observed time series

Figure 5.7 (Prandle, 1997b) shows three examples of simultaneous time-series recordings

of suspended sediment and tidal velocity. Table 5.2 lists results from tidal analyses of

200

0

30

20

10

0

Day number

Sediment concentration

Velocity

0 5 10 15

0

1 m s –1

0

1 m s –1

5 10 15 20

8

6

4

2

0 10 15 20

0

2 m s –1

(a)

mg I –1

mg I –1

mg I –1

(b)

(c)

Fig. 5.7. Observed SPM and current time series in (a) Dover Straits, (b) MerseyEstuary and (c) Holderness Coast.




these observations. These examples were selected as illustrative of tidally domi-

nated conditions and correspond to tranquil weather conditions. The Dover Strait is

a highly (tidally) energetic zone, 30 km wide and up to 60 m deep, linking the North

Sea to the English Channel with currents exceeding 1 m s−1. The Mersey Estuary is

a shallow (<20 m deep) estuary with tidal range up to 10 m; the measurements

shown were taken in the narrow entrance channel, 1-km wide, 10-km long

(Prandle et al ., 1990). The Holderness measurements were taken some 4 km

offshore of a long, rapidly eroding coastline (glacial till). The Dover Strait and

Mersey sediment recordings used transmissometers; the Holderness recordingswere by an OBS.

In all three cases, the MSf constituent is largest – as anticipated from combining

(5.4) and (5.21). Using the above theory to interpret both the current-SPM ampli-

tude ratios and the phase lags for MSf , Prandle (1997b) derived the following values

of α: Dover Strait – 3 × 1 0−6 s−1, Holderness – 2 × 1 0−5 s−1 and Mersey – 2 × 1 0−5 s−1.

Jones et al . (1994) show that the spectral peak in the sediment distribution in the

Dover Strait corresponds to a settling velocity of 10−4 m s−1. The Mersey and

Holderness are likely to contain more coarse grained components.In all three cases, the M2 or M4 constituent is next largest as anticipated earlier.

Likewise, the phase values for all constituents (relative to the associated current

values) are generally in the range of 0−90°. However, precise correspondence

between these observed results and the theory developed here is complicated by

Table 5.2 Tidal constituents of observed tidal currents and SPM ( Fig. 5.7) , currents

in m s−1 , SPM in mg l −1. Phases adjusted to 0 o for M 2 and S 2

Amplitude (m s−1); (mg l−1) Phase (degrees)

Constant Mean MSf M2 S2 M4 MS4 MSf M2 S2 M4 MS4

Dover U* −8.1 10.2 92.2 24.7 10.8 3.2 90 0 0 76 81SPM 4.21 0.98 0.25 0.27 0.39 0.23 41 96 158 352 354% 100 23 6 6 9 5

HoldernessU * −0.6 1.2 49.6 17.6 0.9 1.5 62 0 0 357 341SPM 8.75 3.39 2.03 1.26 0.69 0.78 16 44 70 118 121% 100 39 23 14 8 9

MerseyU * −0.8 1.8 60.6 19.9 11.5 7.0 32 0 0 234 274SPM 61.5 40.6 12.9 6.3 13.4 10.7 19 28 25 20 11% 100 66 21 10 22 17

Source: Prandle, 1997b.




such factors as the spectral width of settling velocities of the particles involved,

finite supply, advection and vertical variations in concentration.

The mean concentration in the Mersey is an order of magnitude higher than in the

Dover Strait. Concentrations at Holderness lie between these two. Thus, there is a

suggestion of limited supply in the Dover Strait; moreover, the phase relationshipfor the M2 constituent is indicative of a significant advective component.

5.6.3 Modelled time series

Figure 5.8 (Prandle, 2004) shows characteristic model-generated spring – neap cycles

of SPM associated with localised resuspension in a tidally dominated estuary (Prandle,

1997a) for E = K z = 0.1 and 10 W S D. For E =0.1 W S D, particles scarcely reach the

surface and with a short half-life in suspension, the quarter-diurnal constituent predominates. Conversely for E = 10 W S D, particles are evenly distributed through

= 0

= 0

= 0

= 0

= 0

E / DW s = 0.1

E / DW s = 10

C o n c e n t r

a t i o n

C o n c e n t r a t i o n

C10

C9

C8

C7

C6

C5

C4

C3

C2

C1

Days

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Fine particle

Coarse particle

C10

C9

C8

C7

C6

C5

C4

C3

C2

C1

Fig. 5.8. Model simulations of SPM over a spring – neap tidal cycle. C1, C2, … , C10concentrations at fractional heights z 1/2 =0.05 – 0.95. (Top) K z / DW S = 0.1; (bottom)

K z / DW S =10.




the water column and the extended half-life in suspension amplifies the MSf

constituent relative to M4.

Fine resolution 3D estuarine models of tidal propagation can provide reliable and

accurate simulations of water levels and flows, often limited only by inadequacies in

the accuracy and resolution of the bathymetry. Likewise, recent developments incoupling these tidal dynamics with turbulence-closure modules provide good,

detailed descriptions of current structure. However, fundamental uncertainties in

modelling SPM fluxes arise from (i) insufficient information on surficial sediment

distributions; (ii) the inherent complexities of erosion and deposition processes –

especially regarding the influence of flocculation and (iii) the paucity of observa-

tional data for calibration/evaluation. Algorithms relating bed stresses with rates of

erosion are plentiful but cannot easily embrace the full range and mixture of bed

materials or the chemical and biological influences. Likewise, in conditions of high(near-bed) concentrations, sediment particles interact (flocculation and hindered

settlement) modifying both the dynamics and the wider sediment regime.

The biggest difficulty in long-term simulations of SPM in estuaries is the

specification of the available sources of sediments – internal, coastal, riverine and

organic. Lack of information on particle size distributions of suspended sediments is

a major deficiency along with ignorance of the nature and distribution for surficial

sediments. The use of aircraft surveillance to provide SPM distributions (sea sur-

face) may, via assimilation techniques, help to address this issue. While large area,

long-term model simulations can link these sources, specific sequences of events

need to be reproduced – resulting in practice in accumulation of errors.

The ultimate goal is to understand the evolving morphological equilibrium

between bathymetry and the prevailing tidal currents, waves and mean sea level.

Simulating net long-term bathymetric evolution is especially difficult, since it

involves the net temporal integration of the spatial divergence of all simulations.

The requirement for models to reproduce near-zero net accretion and erosion provides a demanding criterion. It is generally concluded that extrapolation using

primitive equation (bottom-up) models is severely limited. Arguably, such models

should only be used to indicate pathways and likelihoods of erosion/accretion for

periods of a few years ahead. Further extrapolation often makes use of geomorpho-

logical (top-down) models which derive stability rules based on geological evidence

of coastal movements. Bottom-up models can be used to test the validity of these

top-down rules in any particular estuary.

The scope of monitoring, modelling and theoretical developments describedhighlights the need for collaboration interdisciplinary and internationally. The

value of basic process studies in near-full scale flumes alongside extended estuarial

observational campaigns is evident. Such data need to be quality assured and stored

systematically for accessible dissemination. Permanent comprehensive observational




networks exploiting synergistic aspects of a range of instruments/platforms inte-

grally linked to modelling requirements/capabilities are especially valuable. The

ultimate derivation of robust, portable SPM models depends on the availability of

a range of such data sets from a wide range of estuaries, i.e. parallel programmes

across estuaries with varying latitudes, scales, geological types and environmentalexposures.


Focusing on strongly tidal estuaries, by integrating erosion, suspension and deposition,

analytical solutions are derived describing suspended sediment time series andtheir

vertical structure. The related scaling parameters indicate the sensitivities to sediment

type, tidal current speed and water depth, providing insight into and interpretation of sediment regimes obtained from observations or numerical simulations.

The leading question addressed is:

How are the spectra of suspended sediments determined by estuarine dynamics?

The major interest with sediment regimes in estuaries generally concerns

bathymetric evolution. Additional ecological interests include (i) the transport of

contaminants by adsorption onto fine sediments and (ii) light occlusion from high

concentrations of suspended sediments. Geologists examine morphological evolution

over millions of years and geomorphologists over millennia. Conversely, coastal

engineers are concerned with relatively immediate impacts following extreme

‘events’ or ‘interventions’ and decadal adjustments to changes in msl and other

long-term trends. Chapter 7 introduces a link between present-day morphological

equilibria and historical changes in msl since the last ice age.

The earliest engineering studies of sediment transport were concerned with

maintaining bathymetric equilibrium in the unidirectional flow regimes of irrigation

and navigation canals. By contrast, one of the notable features in strongly tidal

oscillatory regimes is the persistence over decades and re-emergence (after extreme

events or ‘intervention’) of bathymetric features such as spits, pits and systematic

inter-tidal forms.

Compared with tidal elevations, currents and salinity, suspended sediments

exhibit extreme variability. This embraces (i) size, ranging from fine to coarse

with particle diameters ranging from clay < 4 μ, silt < 60 μ, sand < 1000 μ to

gravels; (ii) temporal, spanning flood to ebb, neap to spring tides, from tranquil tostorm conditions with related extreme values of water levels, velocities, waves and

river flows; (iii) spatial, extending vertically, axially and transversally. Even without

the technical difficulties of measuring concentrations of SPM, conditions just

metres apart can show dramatic differences, reflecting the sensitivity to water




depth, current speed, waves, turbulence intensity and surficial sediment distribu-

tions with their attendant bed features.

Recognising such intrinsic variability, the present approach is limited in scope,

concentrating on first-order effects in strongly tidal regimes. This focus represents a

tractable component of sediment dynamics, involving variability in concentrationson logarithmic scales and exploiting robustly determinable tidal characteristics.

For a wider perspective, many additional features need to be considered, including

the chemistry of fine particles (Partheniades, 1965), near-bed interactions between

flow dynamics and sediments for coarse particles (Soulsby, 1997), wave – current

interactions (Grant and Madsen, 1979) and the impacts of bed features (Van Rijn,

1993). Useful summaries are described by Davies and Thorne (2008) and Van Rijn

(2007(a),(b) and (c)).

Postma (1967) emphasised the significance, in estuaries, of the separation between erosion at one location and the subsequent ‘delayed’ settlement at positions

as far apart as the tidal excursion. Here, we aim to determine representative values

of these delays, encapsulated by the ‘half-life’ of sediment in suspension, t 50.

Rejecting the traditional demarcation into periods of erosion and deposition sepa-

rated by thresholds of velocity or bed stress, ‘single-point ’ analytical solutions for

the simultaneous processes of erosion, suspension and deposition are derived.

Similarly, the plethora of complicated formulae for erosion and settlement are

simplified, with erosion related to some power of tidal velocity and deposition to

the depth – mean concentration multiplied by an exponent based on the half-life in

suspension.

By assuming that eddy diffusivity K z and eddy viscosity E can be approximated by

K z = E = fU * D, (3.23), these solutions indicate how the essential scaling of sediment

motion is synthesised in the dimensionless parameter E / W S D as shown in Figs (5.4),

(5.5) and (5.8), (where f is the bed friction coefficient, D is water depth, U * tidal

current amplitude and W S sediment fall velocity). Turbulent diffusion, parameterised by the coefficient E , promotes the suspension of particles by random vertical oscilla-

tions, whereas the fall velocity W S represents steady advective settlement. The time

taken for a particle to mix vertically by dispersion is D2/ E whereas settlement by

vertical advection occurs within D/ W S. Thus, the ratio of E : W S D reflects the relative

times of deposition by advective settlement to that by diffusive vertical excursions.

5.7.1 Sediment suspensionFor E < 0.1 W S D, particles are confined to the near-bed region, whereas for E >

10W S D, particles are evenly distributed throughout the water column (Fig. 5.8).

Approximating settling velocities by sand, W S = 10−2 m s−l; silt, W S = 10−4 m s−1;

and clay, W S = 10−6 m s−1, it is shown that generally, sand is concentrated near the




bed, clay is well mixed vertically and silt shows significant vertical structure.

Assuming a sediment concentration profile e− β z , an expression, (5.19), for β in

terms of E / W S D is derived. Corresponding vertical profiles are shown in Fig. 5.5.

5.7.2 Deposition

The rate of deposition is expressed by the function C e−αt , where α = 0.693/ t 50represents the exponential settling rate.

For E ≪W S D, deposition is by advective settlement at a rate (1/2)W SC 0and α 0.7 W S

2/ E , (5.14). The fractional rate of deposition is determined by (l/2)

[(W S2t )/ E ]1/2, i.e. 10% after 0.04 E / W S

2, 50% after E / W S2 and 90% after 3.2 E / W S

2

(Appendix 5A).

For E ≫W S D, deposition is independent of W S but dependent on boththe magnitude of the vertical dispersion coefficient and the precise near-bed condi-

tions. Using an expedient bottom boundary condition, the value α 0.1 E / D2 is

derived (5.15).

As E →W S D, the mean time in suspension approaches a maximum and hence

both mean concentration and net transport will increase. Section 7.3.1 shows that

this condition occurs for W S 1 m m s−1, i.e. particle diameter d 30 μ.

Equation (5.17) provides a continuous expression for α across the full range of

E / W S D.

5.7.3 Tidal spectra of SPM

By integrating the above analytical solutions for SPM concentrations over succes-

sive tidal cycles, the spectrum of suspended sediments is calculated, (5.20). The

characteristics of this spectrum are determined by the ratio of the exponential

settlement rate, α, to the frequency of the erosional tidal currents, ω (Fig. 5.6). For α > 10ω, the suspended sediment tidal amplitude is proportional to 1/ α with zero

phase lag of SPM relative to current. Whereas, for ω > l0α, the amplitude is

proportional to tidal period (1/ ω) with a 90° phase lag.

From (5.18) and Fig. 5.3, for sand 10−2 > α > 10−3 s−1 and 1 m < t 50 < 10 m, hence

the former condition applies and the amplitude response at all tidal frequencies is

much reduced.

For silt and clay, 10−4 > α > 10−6 s−1 and 2 h < t 50 < 200 h. Since the major tidal

constituents lie in the range ω ≥ 10−4

s−1

, the cyclical amplitude response is rela-tively independent of α but proportional to tidal period, resulting in an enhancement

of longer period constituents. Wherever there is a plentiful supply of erodible

sediment of a wide size distribution, the resulting suspended sediment time series

away from the immediate near-bed area is likely to be dominated by silt – clay.




In the absence of significant residual currents, the erosional time series for an

M2−S2-dominated tidal regime will show pronounced components at M4, MS4, MSf

and Z0 frequencies, (5.4) and Table 5.1. These latter components are generated by

non-linear combinations of M2 and S2 currents and not by any (usually small) tidal

current amplitudes at these emergent frequencies. The similarity in amplitudes of the M4 and MS4 constituents may reduce the quarter-diurnal signal at neap tides

(when their phases are opposed) and thereby suggest an enhanced semi-diurnal

sediment signal that might be wrongly interpreted as indicating horizontal advection

or a large diurnal current component. When ‘residual currents’ increase to a level of

order 10% of the M2 amplitude (as in strongly tidal shallow waters), the suspended

sediment time series will include M2 and S2 constituents of comparable magnitude

to those described for M4, MS4 and MSf . Over a spring – neap tidal cycle, for E > 10

W S D, the peak in suspended sediment concentration will generally occur 2 – 3 daysafter the occurrence of maximum tidal currents.

In summary, the theory (Table 5.1 and Fig. 5.6) explains how tidal spectra

shown in SPM observations (Table 5.2 and Fig. 5.7) are modulated relative to that

of the generating current spectra. The associated amplitude ratios and phase lags

shown in such observations can be used to infer both the nature of the predominant

sediment type and the roles of localised resuspension versus advection.

Appendix 5A

5A.1 Analytical expression for sediment suspension

An expression is derived for the time sequence of sediment suspension, linking

erosion through to deposition as a function of three variables, namely settling

velocity W S, vertical dispersion coefficient E and water depth D.

5A.2 Dispersion equation

It is assumed that the distribution of suspended sediments can be described by the

dispersion equation; moreover, in the present application, consideration is limited to

motion in one horizontal direction, X . Thence, the version of (5.1) representing the

rate of change in concentration C is

dC

dt ¼

@ C

@ t þ

U @ C

@ X þ

W @ C

@ Z ¼

@

@ Z E

@ C

@ Z

sinks

þsources; (5A:1)

where t is time, U horizontal velocity, W vertical velocity, Z the vertical axis measured

upwards from the bed and E a vertical dispersion coefficient. Horizontal dispersion is

not considered, and E is assumed to remain constant both temporally and vertically.

Equation (5A.1) can be converted to convenient variables as follows:

Appendix 5A 145



z ¼ Z

D; w ¼ W

D ; e ¼ E

D2 ;

where D is the water depth, z = 0 is the bed and z = 1 the surface. By assuming

horizontal homogeneity, i.e. ∂C / ∂ X = 0, (5A.1) may be rewritten:

@ C

@ t þ w

@ C

@ z ¼ e

@ 2C

@ z2 sinks þ sources: (5A:2)

Equation (5A.2) can then be used to describe the transport of particulate material by

setting the vertical velocity W in (5A.1) equal to the settling velocity −W s.

5A.3 Analytical solutions

Fischer et al . (1979) show that the general solution for dispersion of a tracer of mass

M (confined to 1D Z ) released at Z = 0, t = 0 in a fluid moving at a velocity W is

C ðZ ; tÞ ¼ M ffiffiffiffiffiffiffiffiffiffi4πEt

p exp ðZ WtÞ2

4Et : (5A:3)

It is convenient to introduce variables t 0 = ws t and z 0 = z + ws t . Rewriting (5A.3) in

these variables, the concentration at z 0 is

C ðz0; tÞ ¼ M

D ffiffiffiffiffiffiffiffiffiffiffiffiffi

4π ews

t0q exp

z02

4 ews

t0

!: (5A:4)

Since (Fischer et al ., 1979),

ð z

0

ez2dz

¼ ffiffiffiπp

2

ERF

ðz

Þ; (5A:5)

the net amount of sediment in suspension between z 0 = 0 and z 0 is

TS ¼ D

ð z0

0

C ðzÞ dz ¼ M 0 ERF z0 ffiffiffiffiffiffiffiffiffiffiffi

4 ews

t0q

0B@

1CA; (5A:6)

where M 0= M /2 corresponds to the amount of eroded suspended above the bedat t = 0.

The value of the error function in (5A.6) must always be less than 1. Then, since

for 0 < x < 1, ERF( x) x, the net amount deposited at z = 0 after time t 0 is given by

substituting z 0 = t 0 in (5A.6), i.e.




TS M 0

2

t0

ð ews

t0Þ1=2 : (5A:7)

From (5A.7), after t 0 = 0.04 e/ ws, TS =0.1 M 0; after t 0 = e/ ws, TS = 0.5 M 0 while after

t 0 = 3.2 e/ ws, TS =0.9 M 0. These deposits can be separated into an advective compo-

nent due to W s in (5A.3) and a dispersive component associated with E .

By successively differentiating the corresponding top and bottom terms in (5A.7)

w.r.t. t , it follows that the advective deposition is a factor of 2 greater and of opposite

sign to the dispersive ‘erosion’. From differentiation of (5A.6) w.r.t. z , the rate of

advective deposition is W S C (z′=0), i.e. the anticipated net deposition for an absorp-

tive bed that retains all particles making contact. The 50% return rate, implicit in

(5A.3), infers a continuing dynamic relationship at the bed that persists up to some

4e/ ws after erosion. Wiberg and Smith (1985) used a ‘rebound coefficient ’ of

around 0.5 to account for partially elastic collisions between moving grains and

the bed.

5A.4 Boundary condition at the surface z = 1

At the surface, a simple reflective boundary condition is introduced involving a

ghost source (of equal magnitude to the original source) located at z = 2 + wst , i.e.equidistant from the original source at z =−ws t . The ‘first ’ (l% of M 0) particles

passing the surface corresponds to

1

M 0

ð Dð1þwstÞ

0

C ðZ Þ dZ ¼ 0:99 ¼ ERF

1 þ t0 ffiffiffiffiffiffiffiffiffiffiffiffi4

e

ws

t0r

0B@

1CA: (5A:8)

Equation (5A.8) requires the argument of the error function to equal 1.83, hence real

solutions for t 0 require e/ ws > 0.3. Thus, for smaller values of e/ ws, less than 1% of

particles reach the surface.

5A.5 Boundary condition at the bed z = 2, 4, 6, etc.

Sediments ‘reflected’ from the surface arrive back at the bed where they may be

deposited or resuspended so long as, by analogy with (5A.8), ( 2 + t 0) / ( 4et 0/ ws)1/2

<1.83, i.e. e/ ws > 0.6.

The ratio, R, of suspended sediments reflected from the surface towards the

bed (i.e. between z 0 = 1 + wst and z 0 = 2 + wst ) to those yet to reach the surface

(i.e. between z 0 = wst and z 0 = 1 + wst ) is

Appendix 5A 147



R ¼ Rs

RB

¼ERF

2 þ wst ffiffiffiffiffiffiffi4et

p

ERF 1 þ wst ffiffiffiffiffiffiffi

4etp

ERF 1 þ wst

ffiffiffiffiffiffiffi4etp ERF

wst

ffiffiffiffiffiffiffi4etp

: (5A:9)

This ratio shows that for e/ ws < 0.1 almost no particles reach the surface.

Conversely, for e/ ws > 10, there will be little vertical variation in the suspended

sediment concentrations.

Three approximations are considered for the bottom boundary condition as

described below.

[A] Fully reflective bed

Sources of equal magnitude at

INITIALSOURCE

REFLECTIONFROM BED

REFLECTIONFROM SURFACE

z ¼ wst (1) initial sourceð2 þ wstÞ (2) reflection of (1)

ð2 þ wstÞ (3) reflection of (2)

||

ð2

ðn

1

Þ þwst

Þ (2n− 1) reflection of

(2n−

2)ð2n þ wstÞ (2n) reflection of (2n − 1)

Numerically, this series continues until the contribution from the last two terms is

negligible. The condition implies that deposition only occurs at z 0 = 0, hence the

solutions to (5A.6) shown by the dashed line in Fig. 5.2 indicate the rate of deposition.

[B] Fully absorptive bed

This requires zero effective concentration at the bed for those sediments arriving after

reflection from the surface, i.e. sources of equal magnitude but opposite sign. This

results in a series of sources/sinks as in boundary condition [A], combinations 3, 4; 7, 8,etc. representing sinks.

[C] Termination of the series after two terms

Sources at

INITIALSOURCE

REFLECTIONFROM BED

REFLECTION FROMSURFACE

z ¼ wst (1) initial sourceð2 þ wstÞ (2) reflection of (1)

Figure 5.2 illustrates the fraction of sediments remaining in suspension at time,wst , after erosion for these three boundary conditions. The three lines that correspond

to the bottom boundary conditions [A] is dashed, [B] is dotted and [C] is solid.

For values of e/ ws < 1.0, the results show little difference since it was shown

earlier that little sediment passes beyond z = 2 for this condition. However, for




e/ ws > 10, the results differ widely and, for the latter two boundary conditions,

suspension times decrease with increasing values of e/ ws. This implies a greater

‘capture rate’ as a result of increased collisions with the bed.

As a convenient expedient, we subsequently adopt the third boundary condition

as being someway intermediate between the other two extremes. However, it isrecognised that this expedient reduces the validity of results presented for the

range e/ ws≫ 1. In practice, the appropriate boundary condition will be a function

of the benthic boundary layer dynamics and of the sediment characteristics and

concentration.

Thus, subsequently, we assume sediment concentrations given by

C

ðz; t

Þ ¼

M

Dð4πetÞ1=2

exp

ðz þ wstÞ2

4et þexp

ð2 þ wst zÞ2

4et" #: (5A:10)

References

Davies, A.G. and Thorne, P.D., 2008. Advances in the study of moving sediments andevolving seabeds. Surveys in Geophysics, doi:10.1007/S10712-008-9039-X, 36pp.

Fischer, H.B., List, E.J., Koh, R.C.Y., Imberger, J., and Brocks, N.K., 1979. Mixing in Inland and Coastal Waters. Academic Press, New York.

Gerritsen, H., Vos, R.J., van der Kaaij, T., Lane, A., and Boon, J.G., 2000. Suspendedsediment modelling in a shelf sea (North Sea). Coastal Engineering , 41, 317 – 352.Grant, W.D. and Madsen, O.S., 1979. Combined wave and current interaction with a rough

bottom. Journal of Geophysical Research, 84 (C4), 1797 – 1808.Hutchinson, S.M. and Prandle, D., 1994. Siltation in the salt marsh of the Dee estuary

derived from 137Cs analyses of short cores. Estuarine, Coastal and Shelf Science,38 (5), 471 – 478.

Jones, S.E., Jago, C.F., Prandle, D., and Flatt, D., 1994. Suspended sediment dynamics, their measurement and modelling in the Dover Strait. In: (Beven, K.J., Chatwin, P.C.,and Millbank, J.H. (eds), Mixing and Transport in the Environment , John Wiley and

Sons, New York, pp. 183 – 202.Krone, R.B., 1962. Flume Studies on the Transport of Sediments in Estuarine Shoaling

Processes. Hydraulic Engineering Laboratories, University of Berkeley, CA.Lane, A. and Prandle, D., 1996. Inter-annual variability in the temperature of the North Sea.

Continental Shelf Research, 16 (11), 1489 – 1507.Lane, A. and Prandle, D., 2006. Random-walk particle modelling for estimating bathymetric

evolution of an estuary. Estuarine, Coastal and Shelf Science, 68 (1 – 2), 175 – 187.Lavelle, J.W., Mojfeld, H.O., and Baker, E.T., 1984. An in-situ erosion rate for a

fine-grained marine sediment. Journal of Geophysical Research, 89 (4), 6543 – 6552.Partheniades, E., 1965. Erosion and deposition of cohesive soils. Journal of Hydraulics

Division ASCE , 91, 469 – 481.Postma, H., 1967. Sediment transport and sedimentation in the estuarine environment. In:

Lauff, G.H. (ed.), Estuaries. American Association for the Advancement of Science,Washington, DC, pp. 158 – 180.

Prandle, D., 1982. The vertical structure of tidal currents and other oscillatory flows.Continental Shelf Research, 1 (2), 191 – 207.

References 149



Prandle, D., 1997a. The dynamics of suspended sediments in tidal waters. Journal of Coastal Research, 40 (Special Issue No 25), 75 – 86.

Prandle, D., 1997b. Tidal characteristics of suspended sediment concentration. Journal of Hydraulic Engineering , ASCE, 123 (4), 341 – 350.

Prandle, D., 2004. Sediment trapping, turbidity maxima and bathymetric stability in tidal

estuaries. Journal of Geophysical Research, 109 (C8).Prandle, D., Murray, A. and Johnson, R., 1990. Analysis of flux measurements inthe Mersey River. In: Cheng, R.T. (ed.), Residual currents and Long TermTransport. Coastal and Estuarine Studies, Vol. 38. Springer-Verlag, New York,

pp. 413 – 430.Romano, C., Widdows, J., Brimley, M.D., and Staff, F.J., 2003. Impact of Enteromorpha on

near-bed currents and sediment dynamics: flume studies. Marine Ecology ProgressSeries, 256, 63 – 74.

Sanford, L.P. and Halka, J.P., 1993. Assessing the paradigm of mutually exclusive erosionand deposition of mud, with examples from upper Chesapeake Bay. Marine

Geotechnics, 114 (1 – 2), 37 – 57.Soulsby, R.L., 1997. Dynamics of Marine Sands: a Manual for Practical Applications.

Telford, London.Van Rijn, L.C., 1993. Principles of Sediment Transport in Rivers, Estuaries and Coastal

Seas. Aqua Publications, Amsterdam.Van Rijn, L.C., 2007a. Unified view of sediment transport by currents and waves. 1:

Initiation of motion, bed roughness and bed-load transport. Journal of Hydraulic Engineering , 133 (6), 649 – 667, doi:10.1061/(ASCE)0733-9429(2007)133:6(649).

Van Rijn, L.C., 2007b. Unified view of sediment transport by currents and waves. 2:Suspended transport. Journal of Hydraulic Engineering , 133 (6), 668 – 689,

doi:10.1061/(ASCE)0733-9429(2007)133:6(668).Van Rijn, L.C., 2007c. Unified view of sediment transport by currents and waves. 3: Graded

beds. Journal of Hydraulic Engineering , 133 (7), 761 – 775, doi:10.1061/(ASCE)0733-9429(2007)133:7(761).

Wiberg, P.L. and Smith, J.D., 1985. A theoretical model for saltating grains in water. Journal of Geophysical Research, 90 (4), 7341 – 7354.

Winterwerp, J.C. and van Kesteren, W.G.M., 2004. Introduction to the Physics of CohesiveSediments in the Marine Environment. Developments in Sedimentology, Vol. 56.Elsevier, Amsterdam, p. 466.




6

Synchronous estuaries: dynamics, saline intrusion

and bathymetry

6.1 IntroductionPrevious chapters illustrated the nature of tidal elevations, currents, saline intrusion

and sediment regimes in estuaries of varying sizes and shapes over a range of tidal,

alluvial and river flow conditions. Here, we address the more fundamental question

of how morphology is itself determined and maintained by the combined actions of

tidal dynamics and the mixing of river and salt waters.

As in these earlier chapters, considerable simplifications are necessary to obtain

generic analytical solutions to the governing equations. Here, we restrict consid-

eration to strongly tidal, funnel-shaped ‘synchronous’ estuaries. Adopting the line-

arised 1D momentum and continuity equations, the focus is on propagation of a

single predominant (M2) tidal constituent. Since estuaries often involve significant

differences in surface areas between low and high water, a triangular cross section is

assumed.

Section 6.2 indicates how these approximations enable localised values for the

amplitude and phase of tidal current U * to be determined in terms of depth, D and

elevation amplitude, ς *. A further expression for the slope of the sea bed, SL,enables both the shape and the length, L, of an estuary to be similarly determined.

Various derivations for the length of saline intrusion, LI, were discussed in

Chapter 4, based on the expedient assumption of a constant (in time and depth)

axial density gradient S x. All indicated a dependency on D2/ fU * U 0, where U 0 is the

residual river flow velocity and f is the bed friction coefficient. Moreover, axial

migration of the intrusion zone was recognised as vital in explaining observed

variations in LI over the cycles of spring to neap tides and flood to drought river

flows. The deduction, introduced in Section 6.3, that mixing occurs at a minimum inlandward intrusion of salt is used here alongside expressions for L and LI to derive

an expression for Di, the depth at the centre of the intrusion, in terms of U 0. This

analysis indicates that U 0 is always close to 1 cm s−1, as commonly observed.

Linking U 0 to river flow, Q, provides a morphological expression for depth at

151



the mouth of an estuary as a function of Q (with an additional dependence on side

slope gradients).

In Section 6.4, the above results are converted into bathymetric Frameworks,

mapping gross estuarine characteristics against ς * and Q (or D). In this way,

a ‘ bathymetric zone’

is postulated bounded by three conditions: LI /L < 1, E x/ L < 1( E x is the tidal excursion) alongside the Simpson-Hunter (1974) criteria for mixed

estuaries D/ U *3 < 5 0 m2 s−3.

In Section 6.5, the validity of these new theories is assessed by comparison against

observed bathymetries for 80 estuaries in England and Wales.

Peculiarly, the results derived in this chapter take no account of the associated

sediment regimes. Chapter 7 considers the consequent implications for the prevail-

ing sediment regime and rates of morphological adjustment.

6.2 Tidal dynamics

6.2.1 Synchronous estuary approach

In Chapter 2, it was shown how the essential characteristics of tidal dynamics in

estuaries can be readily explained from analytical solutions. These dynamics are

almost entirely determined by a combination of tides at the mouth and estuarine

bathymetry with some modulation by bed roughness and, close to the head, river

flows.

Dyer (1997) describes how frictional and energy conservation effects can com-

bine in funnel-shaped bathymetry to produce a ‘synchronous estuary’ with constant

tidal elevation amplitudes ς *. Prandle (2003) assessed the validity of the ‘synchro-

nous’ solutions using numerical simulations. It was shown that for many convergent

bathymetries, the assumption that ∂ς */ ∂ X →0 remains valid except for a small

section at the tidal limit. Moreover, it was shown in Fig. 2.5 that the shape andlengths derived for a synchronous estuary are in the centre of the observed range for

funnel-shaped estuaries.

The present approach explores localised dynamics in terms of tidal elevation

amplitude and water depth. The ‘synchronous’ assumption enables elevation

gradients to be determined directly from tidal elevation amplitude. The solutions

obtained are for a triangular-shaped estuary. Equivalent results for a rectangular

cross section involve substitution of celerity c = ( gD)1/2 for the present result

c = (0.5 gD)1/2

. The dynamical solutions reduce to explicit functions of ς *, D and bed friction coefficient f . The solutions are independent of the actual value of the

side slopes; while their inclination can be asymmetric, they must remain (locally)

constant. The analytical solutions assume that (i) tidal forcing predominates and can

be approximated by the cross-sectionally averaged axial propagation of a single

152 Synchronous estuaries



semi-diurnal tidal constituent; (ii) convective and density gradient terms can be

neglected and (iii) the friction term can be sensibly linearised.

6.2.2 Analytical solution for 1D momentum and continuity equations (Prandle, 2003 )

Omitting the convective term from the momentum equation, we can describe tidal

propagation in an estuary by:

@ U

@ t þ g

@ B

@ X þ f

U jU j

H ¼ 0 (6:1)

B

@ B

@ t þ

@

@ X A U ¼ 0; (6:2)

where U is velocity in the X -direction, ς is the water level, D is the water depth, H is

the total water depth ( H = D + ς ), f is the bed friction coefficient (~0.0025), B is

the channel breadth, A is the cross-sectional area, g is the gravitational acceleration

and t is the time.

Concentrating on the propagation of one predominant tidal constituent ( M 2), the

solutions for U and ς at any location can be expressed as

& ¼ & cos ðK 1X ωtÞ (6:3)

U ¼ U cos ðK 2 X ωt þ θ Þ; (6:4)

where K 1 and K 2 are the wave numbers, ω is the tidal frequency and θ is the phase

lag of U relative to ς . The synchronous estuary assumption is that axial variations

in ς * are small. In deriving solutions to (6.1) and (6.2), a similar approximation is

assumed to apply to U *. The resulting solutions for U * (Fig. 6.1) indicate that this

additional assumption is valid, except in the shallowest waters. Further assuming a

triangular cross section with constant side slopes, (6.2) reduces to

@ B

@ t þ U

@ B

@ X þ

@ D

@ X

þ

1

2

@ U

@ X & þ DÞð ¼ 0: (6:5)

Friedrichs and Aubrey (1994) indicate that U *(∂ A/ ∂ X )≫ A(∂U */ ∂ X ) in convergent

channels. Likewise assuming (∂ D/ ∂ X )≫ (∂ς*/ ∂ X ), we adopt the following form of

the continuity equation:

@&

@ tþ U

@ D

@ X þ

D

2

@ U

@ X ¼ 0: (6:6)

The component of f U |U | / H at the predominant tidal frequency M 2 may be approxi-

mated by




8

3π

25

16 f

jU jU

D ¼ FU (6:7)

with F = 1.33 fU */ D, where 8/3π derives from the linearisation of the quadratic

friction term (Section 2.5). The factor 25/16 is derived by assuming that the tidalvelocity at any transverse location is given by a balance between the quadratic

friction term (with localised depth) and a (transversally constant) surface slope –

yielding a cross-sectionally averaged velocity of 4/5 of the velocity at the deepest

section.

Substituting solutions (6.3) and (6.4) into (6.1) and (6.6) with the frictional

representation (6.7), four equations (pertaining at any specific location along an

estuary) are obtained, representing components of cos (ωt ) and sin (ωt ) in (6.1) and

(6.6). By specifying the synchronous estuary condition that the spatial gradient intidal elevation amplitude is zero, we derive K 1 = K 2 = k , i.e. identical wave numbers

for axial propagation of ς and U , thence

tan θ ¼ F

ω ¼

SL

0:5 D k; (6:8)

where SL = ∂ D/ ∂ X

U ¼ & g k

ðω2 þ F 2Þ1=2 (6:9)

k ¼ ω

ð0:5DgÞ1=2 : (6:10)

4

3

2 ( m )

1

00 10 20

D (m)

30 40

1.5

1.0

0.5

Fig. 6.1. Tidal Current amplitude, U *(ms−1), as f ( D, ς *). From (6.9) with f = 0.0025.




6.2.3 Explicit formulae for tidal currents, estuarine

length and depth profile

A particular advantage of the above solutions is that they enable the values of a

wide range of estuarine parameters to be calculated and illustrated as direct functions

of D and ς *. The ranges selected for illustration here are ς * (0 – 4 m) and D (0 – 40 m);

these represent all but the deepest of estuaries.

Current amplitudes U*

Figure 6.1 (Prandle, 2004) shows the solution (6.9) with current amplitudes

extending to 1.5 m s−1. As subsequently shown from Fig. 6.3, for ς *≪ D/10,

these currents are insensitive to f . For ς *≫ D/10, these currents change by a factor

of 2 over the range f = 0.001 –

0.004. Noting that for F ≫

ω, (6.9) indicates that U *∝ ς *½ D¼ f −½, thus illustrating why observed variations in U * are generally

smaller than for ς *. Conversely, for F ≪ω, (6.9) indicates that U *∝ ς * D−½. The

contours show that maximum values of U * occur at approximately D = 5 + 1 0 ς *;

however, these are not pronounced maxima.

Depth profile and estuarine length L

In Fig. 6.2 (Prandle, 2004), utilising the values of SL from (6.8), the length, L, of an

estuary is calculated numerically by successively updating SL as D reduces alongthe estuary (assuming a constant value of ς *). By assuming F ≫ω, an equivalent

simple analytical solution can be determined

4

1025

50

100

250

1000

3

Tidalelevationamplitude

ζ(m)

2

1

0 10 20

Depth (m)

30 40

Fig. 6.2. Estuarine length, L (km), as a f ( D, ς *), (6.12), with f = 0.0025.




D ¼ 5

4

ð1:33 & f ωÞ1=2

ð2 gÞ1=4

!4=5

x0 4=5; (6:11)

where x0 = L − X .

Substituting X = 0 and D = D0 at the mouth, estuarine lengths are given by

L ¼ D

5=40

B0

1=2

4

5

ð2 gÞ1=4

ð1:33 f ωÞ1=2 2460

D5=40

& 01=2

for f ¼ 0:0025 (6:12)

(units m, subscripts 0 denote values at the mouth).

The dependency on D5/4/ ς *1/2 in (6.12) and Fig. 6.2 indicates that estuarine lengths

are significantly more sensitive to D than to ς *. Prandle (2003) shows that this

expression for estuarine length is in broad agreement with data from some 50estuaries (randomly selected using previously published data) located around the

coasts of the UK and the eastern USA. For the UK estuaries, estimates of mud

content were available, enabling some of the discrepancies between observed and

estimated values of L to be reconciled by introducing an expression for f based on

relative mud content.

6.2.4 Sensitivity to bed friction coefficient (f)

As indicated by (6.8) and (6.9), the sensitivities of the three parameters shown in

Table 6.1 to the value of the bed friction coefficient f depend on the value of F / ω.

The ratios of the change in parameters correspond to changes in bed friction

coefficient f 0 = ε f . The effective extreme range of ε is from 0.2 to 5. Prandle et al .

(2001) showed how increased bed friction due to wave – current interaction reduced

tidal current amplitudes by up to 70%.

Figure 6.3 (Prandle, 2004) illustrates that F / ω, the ratio of the friction to inertial terms

determined from (6.8), is approximately equal to unity for ς * = D/10. For values of

ς *≫ D/10, tidal dynamics become frictionally dominated, whereas for ς *≪ D/10

friction becomes insignificant. Friedrichs and Aubrey (1994) showed the predominance

Table 6.1 Sensitivity to bed friction coefficient f 0 = ε f

F / ω≫ 1 or F / ω≪ 1 or ς *≫ D/10 ς *≫ D/10

Current amplitude U ε −1/2 1Seabed slope SL ε 1/2 ε Estuarine length L ε −1/2 ε −1




of the friction term in convergent channels, irrespective of depth – this accords with

large values of SL in (6.8).

6.2.5 Rate of funnelling in a synchronous estuary

Chapter 2 described generalised frameworks for tidal response in funnel-shaped

estuaries based on analytical solutions for (6.1) and (6.2). Using the bathymetric

approximations of depth proportional to X m and breadth to X n, Prandle and Rahman

(1980) produced a response framework (Fig. 2.5), showing relative tidal amplitude

response and associated phases across a wide range of estuarine bathymetries. Here,

we calculate how the estuarine lengths and shapes derived for synchronous estuaries

fit within this framework. The ‘funnelling factor ’ ν is given by

ν ¼ n þ 1

2 m: (6:13)

Since (6.11) corresponds to m = n = 0.8, the synchronous estuary solution corre-

sponds to ν = 1.5, i.e. close to the centre of values encountered.

The vertical axis in Fig. 2.5 represents a transformation of the distance from the

head of the estuary given by

y ¼ 4π

2 m

X

gDð Þ1=2P

!ð2mÞ=2

; (6:14)

where P is the tidal period. Hence for X = L, m = 0.8, from (6.12)

410 5

2

1

0.5

0.1

3

2

1

00

10 20

D (m)

30 40

( m )

Fig. 6.3. Ratio of the friction term F to the inertial term ω as a f ( D, ς *), (6.8).




y ¼ 0:9 D3=4

& 1=2

0:6

: (6:15)

Taking D = 5, ς * = 4 (m) together with D =20, ς * = 2 (m) as representative of

minimum and maximum values of y in ‘mixed’

estuaries, these correspond to y = 1.22 and y = 2.8. Figure 2.5 shows that this range of values for y is also represen-

tative of observed lengths, extending from a small fraction to almost a quarter

wavelength for the M2 constituent.

6.3 Saline intrusion

We restrict interest to mixed or partially mixed estuaries and assume a, temporally

and vertically, constant relative axial density gradient, S x = (l/ ρ)(∂ ρ/ ∂ X ), withdensity linearly proportional to salinity. In Chapter 4, the following expression,

(4.44), for saline intrusion length, LI, in mixed estuaries was derived (Prandle,

1985):

LI ¼ 0:005 D2

f U U 0: (6:16)

Here we link the above to a further expression determining the location of the

intrusion along the estuary to derive salient characteristics within the intrusion zone.

In conjunction with the results from Section 6.2, this provides an expression for the

depth at the mouth of an estuary as a function of river flow.

6.3.1 Stratification levels and flushing times

The earlier derivation of tidal current amplitudes (6.9) enables direct estimation of

the Simpson and Hunter (1974) criterion D/ U

*3

> 5 0 m

2

s

−3

for stratification to persist. The results, shown in Fig. 6.4 (Prandle, 2004), indicate that this implies

that estuaries with tidal elevation amplitudes ς * > 1 m will generally be mixed.

An additional indication of stratification levels can be calculated from the time,

T K , for complete vertical mixing by diffusion of a point source – estimated by

Prandle (1997) as T K = D2/ K Z (K Z is the vertical diffusivity). Approximating K Z =

f U * D yields T K = D/ fU *, estimates of which are shown in Fig. 6.5 (Prandle, 2004).

This approach produces results consistent with those in Fig. 6.4. Noting that, for

semi-diurnal constituents, whenever T K > P /2 ~ 6 h, stratification is likely to persist beyond consecutive peaks of mixing on flood and ebb tides. Conversely for

T K < 1 h, little stratification is likely. For intermediate values, 1 h < T K < 6 h,

intra-tidal stratification is likely – especially via tidal straining on the flood tide

(Simpson et al ., 1990).




An estimate of salinity flushing rates based on the time for river water to replacehalf of the salinity content of an estuary is given by

T F ¼ 0:5 ðLI =2Þ

U 0¼

0:0013 D2

f U U 20: (6:17)

Fig. 6.4. Simpson – Hunter stratification parameter D/ U *3 m2 s−3.

Fig. 6.5. Time for vertical mixing by diffusion D2/ K z, K z = fU * D with U * from(6.9).

6.3 Saline intrusion 159



6.3.2 Location of mixing zone, residual current

associated with river flow

In Chapter 4, estimates of the landward limits of saline intrusion xu = ( X i − LI/2)/ L

corresponding to successive values of xi = ( X i/ L), the centre of the intrusion, were

compared with observations in eight estuaries. The best agreement occurred when

the landward limit of saline intrusion was a minimum.

Adopting this latter result as a criterion to determine the position, xi, where the

saline intrusion will be centred, requires in dimensionless terms

@

@ xðx 0:5 l i Þ ¼ 0: (6:18)

Substituting l i = LI/ L, utilising (6.16) and (6.12) and introducing the shallow water approximation to (6.9), yields

U 2 ¼B ω ð 2 g D Þ1=2

1:33 f : (6:19)

Then assuming Q = U 0 Di2/tanα, where tan α is the side slope of the triangular cross

section, we obtain

x2i ¼ 333 Q tan α

D5=20

: (6:20)

Noting that the depth, Di, at xi is D0 xi0.8, we obtain

U 0 ¼ D

1=2

i

333 m s1: (6:21)

For depths ranging from a few metres to tens of metres, (6.21) yields values of

U 0 close to 1cms−1, as commonly observed. Noting that (6.20) corresponds to

l i =2/3 xi, these values for U 0 will increase by a factor of 2 at the upstream limit and

decrease by 40% at the downstream limit.

If we introduce estuarine bathymetry of the form breadth B0 xn and depth D0 x

m,

we obtain the following alternative form for (6.20):

xi ¼ 855 Q

D3=20 B0 ð11m=4 þ n 1Þ !

1=ð11m=4þn1Þ

: (6:22)

An especially interesting feature of the results for the axial location of saline

intrusion, (6.20) and (6.22), and the expression for residual river flow current,

(6.21), is their independence of both tidal amplitude and bed friction coefficient,




although there is an implicit requirement that tidal amplitude is sufficient

to maintain partially mixed conditions. Equations (6.20) and (6.22) emphasise

how the centre of the intrusion adjusts for changes in river flow Q. This ‘axial

migration’ can severely complicate the sensitivity of saline intrusion beyond the

anticipated direct responses apparent from the expression (6.16) for the length of intrusion, LI.

6.4 Estuarine bathymetry: theory

6.4.1 Morphological zone determined by tidal

dynamics and stratification

Using the above result that the riverine component of velocity in the saline intrusionregion approximates 1 cm s−1, Fig. 6.6 (Prandle, 2004) illustrates typical values of the

lengths of saline intrusion obtained from (6.16). Moreover, combining this latter result

with that for estuarine length, L, (6.12), Fig. 6.7 (Prandle, 2004) shows the ratio LI/ L.

Similarly, Fig. 6.8 (Prandle, 2004) shows the ratio of tidal excursion E x, (6.24), as

a fraction of L for a tracer released at the mouth on the flood tide. These values for E xinclude compensation for the reduction in tidal velocity with decreasing (upstream

depths) but ignore axial variations in phase.

By introducing requirements for saline mixing to be contained within an estuary,we can define a ‘ bathymetric zone’ bounded by

Fig. 6.6. Saline intrusion length, LI (km), (6.16). Values scale by 0.01/ U 0, U 0in m s−1.




ðiÞ LI

L

51;

ðiiÞ E x

L51 and

ðiiiÞ D

U 3550 m2 s3; (6:23)

Fig. 6.8. Ratios of tidal excursion to estuarine length, E x/ L, (6.24) and (6.12).

Fig. 6.7. Ratios of saline intrusion to estuarine length, LI/ L, (6.16) and (6.12).




where the tidal excursion

E x ¼ 2

π

U P: (6:24)

This zone, shown in Fig. 6.9 (Prandle, 2004), is reasonably consistent with the

superimposed distribution of ( D, ς *) values from 25 UK estuaries (Prandle, 2003).

6.4.2 Estuarine depths as a function of river flow

The results for xi and Di, (6.21) and (6.22) in Section 6.3.2, can be used to obtain

estimates of the depth D0 at the mouth of the estuary. Noting that with l i =2/3 xi, for

the intrusion to be confined to the estuary, the maximum value for xi = 0.75.

Inserting this value for xi into (6.21), we obtain

D0 ¼ 12:8 ðQ tanαÞ0:4: (6:25)

Combining this result with (6.12), the estuarine length, L, is given by

L ¼ 2980

Q tanα

f & 1=2

: (6:26)

Where estuarine bathymetries were established under historical conditions with

much larger (glacial melt) values of Q, we might expect saline mixing to start

landwards of the mouth. Conversely, where saline mixing involves an offshore

Fig. 6.9. Zone of estuarine bathymetries. Bounded by E x < L, LI < L and D/ U *3 < 5 0 m2 s−3, (6.23).




plume, we postulate either exceptionally large values of Q or that bathymetric

erosion to balance existing river flow is hindered.

The results for U 0, (6.21), and D0, (6.25), are independent of both the friction

coefficient, f , and the tidal amplitude, ς *. O’Brien (1969) noted that the minimum

flow area of tidal inlets was effectively independent of the type of bed material.However, the two expressions for estuarine length, (6.12) and (6.26), are dependent

on the inverse square root of both f and ς *.

Observed versus computed estuarine bathymetries

Examination of a range of UK estuaries indicated that in general, 0.02 > tanα > 0.002,

(6.25) then corresponds to

2:68 Q0:4

4D04 1:07 Q0:4

: (6:27)Figure 6.10 shows results from UK, USA and European estuaries (Prandle, 2004).

For the steeper side slope, (6.27) yields values for D0 of 2.7 m for Q = 1 m3 s−1;6. 7m

for Q = 1 0 m3 s−1; 16.9 m for Q =100m3 s−1 and 42.4 m for Q =1000m3 s−1.

Comparable figures for the smaller side slope are depths of 1.1, 2.7, 6.7 and

16.9 m. Figure 6.10 shows that the envelope described by (6.27) encompasses

almost all of the observed estuarine co-ordinates of (Q, D).

The mean discharge of the world’s largest river, the Amazon, is 200 000 m3 s−1,

representing 20% of net global freshwater flow. Moreover, the cumulative discharge

10

10

78

3

918

20 19

16

21

2212

24

5

X

X XX

X

x

xx

x

e

f

h

j

Xi

g

d b

a

c

20

Depth(m)

10 100

River flow Q (m3 s –1)

1000 10 000

Fig. 6.10. Depth at the mouth as a function of river flow. Theoretical envelope(6.27), observed values from Prandle (2004). UK estuaries are labelled by numbers,others by letters.




of the next nine largest rivers amounts to a similar total (Schubel and Hirschberg,

1982). Outside of these ten largest rivers, Q <15000m3 s−1, from (6.25) this corre-

sponds to D = 50 – 125 m. Thus, the range of values shown in Fig. 6.10 clearly

represents the vast majority of estuaries. Moreover, we note from Fig. 6.7 that the

larger estuaries with D >10 m will often involve freshwater plumes extendingseawards.

6.5 Estuarine bathymetry: assessment of theory

against observations

The above theories for estuarine bathymetry provide formulations for

(1) depth at the mouth, D versus river flow, Q, (6.25);(2) tidal length L versus D and ς * (tidal amplitude), (6.12);

(3) a bathymetric zone, delineated on a framework of D and ς* (Fig. 6.9).

A morphological data set (Future-Coast, Burgess et al ., 2002) for 80 estuaries in

England and Wales, Fig. 6.11(Buck and Davidson, 1997), was used to assess the

validity of the above theories. The UK estuaries include large inter-tidal zones, with

breadths at high tide typically three or more times low tide values. Hence, as for the

earlier theoretical developments, the following analyses of observed morphologies

assume triangular cross sections with side-slope tan α = 2 D/ B. To separate results for

differing geomorphological types in this assessment, we use the same morpholo-

gical classifications as Buck and Davidson (1997), i.e. Rias, Coastal Plain and

Bar-Built. For these estuarine types, mean observed values of the estuarine para-

meters – D, ς *, Q, breadth B and side-slope tan α – are shown in Table 6.2.

6.5.1 Statistical analyses

The observational data set extends over a diverse range of estuaries with attendant

uncertainties and inaccuracies introduced both during the original data collection

and in the subsequent syntheses of the data. All statistical fits removed two outliers

as determined by iterative calculations. The subsequent analyses are sub-divided

into four categories: (i) all estuaries, (ii) Rias, (iii) Coastal Plain and (iv) Bar-Built.

The numbers, N , in each category are as follows: 80(28), 15(61), 30(45) and 35(42)

with the bracketed figures indicating percentage significant correlation at the

99% level.An estimate of percentage of variance (PVA) accounted for is used where

PVA ¼ 100 1 X

O2i X 2i

XO2

i

(6:28)

6.5 Estuarine bathymetry: assessment of theory against observations 165



with Oi observed and X i the ‘ best ’ statistical fit over the range of estuaries i = 1 – N .

The use of normalised PVA is appropriate for the large parameter ranges involved.

The statistical fit is assumed to take the form y = Ax P , with A calculated from least

squares over the range −3 < P < 3 (in increments of 0.01) and with P chosen to

reflect the maximum value for PVA.

Fig. 6.11. Estuaries of England and Wales.




The numerical optimisation calculates r 1/ q versus sq, where r and s are any fitted

parameter and q = P ½, this ensures that the fit applies when the order of the parameters

is reversed.

Mean values for river flow velocity, U 0, were calculated from the values for

Q tan α/ D2 shown in Table 6.2. These are as follows (in cm s−1): all estuaries, 0.5;

Rias, 0.04; Coastal Plain, 0.3 and Bar-Built, 1.0, i.e. consistent with estimatesfrom (6.21). Prandle (2004) showed that values of U 0 derived both from observa-

tions worldwide and from numerical model calculations are generally in the range

0.2 – 1.5cms−1. The much lower values for Rias reflect their peculiar morpholo-

gical development.

6.5.2 Theory versus observed morphologies

The summary of observed estuarine bathymetries shown in Table 6.2 usefully

encapsulates the descriptions of estuarine types outlined by Davidson and Buck

(1997). Thus, in general, Rias are short, deep and steep-sided with small river flows.

Coastal Plain estuaries are long and funnel-shaped with gently sloping triangular

cross sections providing extensive inter-tidal zones. Bar-Built estuaries are short

and shallow with small values of both river flow and tidal range. Prandle (2003)

noted that sandy estuaries tend to be short and muddy estuaries tend to be long. In

sedimentary terms, Bar-Built estuaries are located along coasts with plentifulsupplies of marine sediments and, consequently, are close to present-day equili-

brium. Coastal Plain estuaries are continuing to infill following ‘over-deepening’

via post-glacial river flows while Rias are drowned river valleys (with related cross

sections) as a consequence of (relative) sea level rise.

Table 6.2 Observed estuarine lengths, L , depths, D , river flows, Q , and

side-slope, tan α

Type No. L ~ A D p R (PVA)Mean

L (km) D~AQ p R (PVA)Mean

D (m)MeanQ (m3s−1)

Mean(tan α)

All 80 1.28 D1.24 0.69 20 3.3Q0.47 0.55 6.5 14.9 0.013Ria 15 0.99 D1.10 0.89 12 5.1Q0.32 0.74 9.3 6.3 0.037Coastal

Plain30 1.95 D1.12 0.69 33 3.0Q0.38 0.67 8.1 17.9 0.011

Bar-built 35 1.92 D1.15 0.66 9 2.4Q0.35 0.72 3.6 9.5 0.014Theory 1.83 D1.25 2.3Q0.40

Note: Best statistical Fits and correlation (PVA, percentage of variance accounted for).

Theoretical values are from (6.12) and (6.25) with mean values ς * = 1.8 m and tan α = 0.013.Source: Prandle et al ., 2006.




Length ( L ~ AD P ) and depth (D)

The powers, 1.12 and 1.15 of D, shown in Table 6.2 for Coastal Plain and

Bar-Built estuaries lie close to the theoretical value of 1.25 as do the coefficients

1.95 and 1.92 to the theoretical 1.83. The power, 1.10, of D for Rias is in reasonable

agreement with the theoretical value of 1.25, but the reduced value 0.99 of thecoefficient reflects the shorter lengths of these estuaries.

Overall, we note statistically significant relationships between all of these estuar-

ine parameters in all types of estuaries indicating the tendency for estuarine

morphologies to be confined within restricted parameter ranges.

Depth ( D ~ AQ P ) and river flow (Q)

The powers of Q, 0.32 for Rias and 0.38 for both Coastal Plain and Bar-Built

estuaries, are all close to the theoretical value of 0.40. Likewise, the related values

for the coefficient A are close to the theory except for Rias where the higher

coefficient reflects their greater depths.

The relationships between L and D shown in Table 6.2 can be used to estimate the

power m for the Prandle and Rahman (1980) estuarine response, Fig. 2.5, in which

D∝ xm. Prandle (2006) extended these statistical analyses to calculate corresponding

relationships between L and breadth, B, thereby providing estimates of n for B∝ xn.

Together, from (6.13), these values of m and n indicate the following values of the‘funelling factor ’ ν : All, 1.85; Ria, 1.72; Coastal Plain, 2.07 and Bar-Built, 2.25.

Maximum tidal amplification occurs for ν = 1 with considerable reduction of this

peak for ν > 2. Thus, tidal elevations and currents are likely to be more spatially

homogeneous in Bar-Built estuaries reflecting conditions closer to equilibrium.

Bathymetric zone

Figure 6.12 (Prandle et al ., 2005) shows the ‘zone of estuarine bathymetries’ bounded

by (6.23). This encompasses most of the observed estuaries. For ς * > 3 m, Fig. 6.1shows that tidal current amplitudes can exceed 1.5 m s−1 and the axial slope of the bed

increases significantly. Consequently, such estuaries are generally in the form of deep

contiguous bay-estuary systems such as the Bristol Channel or Bay of Fundy.

6.5.3 Minimum depths and flows for estuarine functioning

Figures 6.4 and 6.5 indicate that ς * >1 m for a ‘mixed’ estuary, this translates to a

minimum value of D = 1 m and from (6.12) L 2.5 km, for an estuary to functionover a complete tidal cycle.

This minimum length requirement corresponds to D > ς *0.4. Then, by substitut-

ing the values for transverse slope, tan α, from Table 6.1 into (6.25) we derive

minimum values of Q in m3 s−1 for ς * = 1, 2 and 4 m as shown in Table 6.3.




6.5.4 Spacing between estuaries

Having postulated that estuarine bathymetry is determined by ς * and Q, the question

arises as to how estuaries adjust over geological time scales to climate change. In particular, what are the consequences of changes in rainfall and catchment areas as

coasts advance or retreat under falling or rising msl. These questions are addressed

further in Chapter 8; here we distil these questions into estimates of the spacing

between estuaries.

Table 6.3 Minimum values of river flow, Q (m3 s−1 ), for estuaries

to function over a complete tidal cycle of amplitude ς

ς * = 1 m ς * = 2 m ς * = 4 m

All estuaries 0.13 0.75 4.2Rias 0.05 0.26 1.5Coastal Plain 0.15 0.88 5.0Bar-Built 0.12 0.69 3.9

Fig. 6.12. Bathymetric zone. Bounded by: E x < L, LI < L and D/ U * 3 < 5 0 m2 s−3.




For a long straight coastline with spacing, SP (km), between estuaries and a

rectangular catchment of landward extent CT (km), the river flow Q is given by

Q ¼ 0:032 SP CT R; (6:29)

where R is the annual rainfall reaching the river (m a−1

). Thus, typical UK values of SP =10 km, CT =50km and Q = 1 5 m3 s−1 indicate that R ~ 0 . 9 m a−1, which is in

broad agreement with observations.

By introducing (6.25) with tan α = 0.013 (Table 6.2), we obtain the following

expression for spacing between estuaries:

SP ¼ 41 D5=2

R CT : (6:30)

We note from Table 6.2 and from global observational data shown by Prandle(2004) that few estuaries have values of D > 20 m. Hence, to avoid small values

of SP for continental land masses with large values of CT , we anticipate the

formation of deltas or multiple ‘sub-estuaries’ linked to the sea by tidal basins

such as in Chesapeake Bay.


By introducing the assumption of a ‘synchronous’ estuary (where the surface slopedue to the gradient in phase of tidal elevations significantly exceeds that from

changes in amplitude ς *), explicit localised expressions are obtained for both

the amplitude and phase of tidal currents and the slope of the sea bed, SL, in terms

of ( D, ς *) where D is water depth. Integration of the expression for SL provides an

estimate of the shape and length, L, of an estuary. By further combining these results

with (4.44) for the length of saline intrusion, LI, an expression linking the depth at the

estuarine mouth with river flow, Q, is derived, i.e. a morphological framework linking

the size and shape of any estuary to river flow and tidal amplitude.

Recent attempts to forecast how estuarine bathymetries may evolve as a con-

sequence of Global Climate Change have prompted the more fundamental

question:

What determines estuarine shape, length and depth?

Estuaries form an interface between the rise and fall of the tide at the coast and theriver discharges. Hence, we expect bathymetries must reflect a combination of tidal

amplitude, ς *, and river flow, Q, alongside some representation of the alluvium.

However, geological adjustment rates are extremely slow in comparison with

the (relatively) rapid changes in msl (and hence water depth) and Q. Thus, we




anticipate that bathymetries reflect some intermediate adjustment between ante-

cedent formative conditions and subsequent present-day dynamic equilibrium.

This adjustment rate will depend on both the supply of sediments for deposition

and the ‘hardness’ of the geology for erosion.

Tidal currents and frictional influence

By introducing the ‘synchronous’ estuary assumption, Section 6.2 derives localised

expressions, (6.8) and (6.9), for the amplitude and phase of tidal currents in terms

of ς * and D, illustrating why U * is invariably in the range 0.5 < U * < 1 . 5 m s−1

(Fig. 6.1). These synchronous solutions also explicitly quantify the issue raised in

Chapter 2 concerning the ratio of the frictional to inertial terms. Figure 6.3 shows

how this ratio approximates 10 ζ * : D.

Lengths and shape

The expression (6.8) for the slope of the sea bed, SL, provides an estimate of the

shape of an estuary (6.11). By subsequent integration of these slopes, Fig. 6.2 shows

estuarine lengths, L (6.12), as a function of ς * and D and the bed friction coefficient,

f , where the latter introduces some representation of the alluvium.

Range of estuarine morphologies

By combining these results for U * and L with the expression (6.16) for the length

of saline intrusion, LI, a ‘ bathymetric zone’ for ‘mixed’ estuaries is determined

enveloped by the conditions, LI/ L < 1, E x/ L < 1 ( E x tidal excursion length) and the

Simpson-Hunter (1974) criterion for vertical mixing ( D/ U *3< 5 0 m−2 s3). Figures 6.5

and 6.6 use the expression (6.9) for U * to indicate how the latter criterion for

determining mixed versus stratified estuaries closely approximate the condition

ς * 1 m.

By introducing the observation that mixing occurs at a minimum in landwardintrusion of salt, an expression linking the depth at the mouth of an estuary D0 with

river flow, Q, is derived, (6.21) and (6.25). Interestingly, this expression is inde-

pendent of both ς * and f . Assuming a triangular cross section with side slopes, tanα,

between 0.02 and 0.002, Fig. 6.10 illustrates the resultant envelope of D0 as a

function of Q. This envelope supports the conclusion, derived in Chapter 4, that the

river flow velocity within the saline intrusion zone in mixed estuaries is invariably

of the order of (1 cm s−1).

In Section 6.5, it is shown that for ‘mixed’ estuaries to function over the completetidal cycle requires minimum values of D ≈ ς *0.4 and Q ≈ 0.25 m3 s−1. Typical

maximum values of around D ≈ 20 m and ς * ≈ 3 m can be postulated and used to

suggest the formation of deltas and composite estuaries in the draining of large

continental land masses.




Assessment of bathymetric framework

Using the above relationship between D0 and Q, the earlier ‘ bathymetric zone’ is

converted into a framework for estuarine morphology in terms of the ‘ boundary

conditions’ ς * and Q, quantifying how tides and river flows determine estuarine size

and shape. This framework, Fig. 6.12, is assessed against an observational data set extending to 80 UK estuaries.

While individual estuaries exhibit localised features (related to underlying geol-

ogy, flora and fauna, historical development and ‘intervention’), the overall values

of depth, length and (funnelling) shape are shown to be consistent with the new

dynamical theories. Moreover, the morphological features which characterise Rias,

Coastal Plain and Bar-Built estuaries can be rationalised by reference to these

theories.

The derived theoretical proportionality of L to D01.25 was substantiated for all

three estuarine types. For both Coastal Plain and Bar-Built estuaries, magnitudes of

L are in close agreement with the theory while Rias indicate significantly reduced

values for L.

Similarly, the theoretical proportionality of D0 to Q0.4 was substantiated for all

three estuarine types, with good magnitude agreement for Coastal Plain and Bar-

Built estuaries while Rias indicated significantly larger values of D.

Theoretical Frameworks

Results derived in this chapter are synthesised into comprehensive Theoretical

Frameworks describing estuarine dynamics, salinity intrusion and bathymetry in

terms of the ‘natural’ boundary conditions Q, ς * and f . The underlying relationships

are as follows.

(a) Current amplitude U *∝ ς *½ D¼ f −½ Shallow water (6.9)

∝ς * D−1/2 Deep water (b) Estuarine length L∝ D5/4/ ς *½ f ½ (6.12)(c) Depth variation D( x)∝ x0.8 (6.11)(d) Depth at the mouth D0∝ (aQ)0.4 (6.25)(e) Ratio of friction: inertia F / ω∝ 10ς */ D (6.8)(f) Stratification limit D/ U *3 ~ ς * = 1 m (6.24)(g) Salinity intrusion L

i,∝ D2/ fU 0U * (6.16)

(h) Bathymetric zone Bounded by LI < L, E x < Land D/ U *3 < 5 0 m−2 s3

(6.23)

(i) Flushing time T F∝ LI/ U 0 (6.17)

These Frameworks provide perspectives against which to assess the morphology

of any particular estuary. By identifying ‘anomalous’ estuaries, possible causes can

be explored such as distinct regional patterns of historical evolution, engineer-

ing ‘interventions’, ‘hard’ geology, dynamics or mixing inconsistent with the




theoretical assumptions, non-representative observational data, vagaries of sedi-

ment supply and wave impact.

The impact of flora and fauna has long been recognised as influencing the supply,

consolidation and bio-turbation of bed sediments. Section 6.2 shows how associated

variations in the effective bed friction factor, f , have a major impact on dynamics andthence on both morphology and the sediment regime.

Corollary

The validation of these new theories provokes a paradigm shift, explored in Chapter 7,

suggesting that prevailing estuarine sediment regimes are the consequence of rather than

the determinant for bathymetries.

References

Buck, A.L. and Davidson, N.C., 1997. An Inventory of UK Estuaries. Vol. 1. Introduction and Methodology. Joint Nature Conservancy Committee,Peterborough, UK.

Burgess, K.A., Balson, P., Dyer, K.R., Orford, J., and Townend, I.H., 2002. Futurecoast the integration of knowledge to assess future coastal evolution at a national scale. In:28th International Conference on Coastal Engineering. American Society of Civil

Engineering , Vol. 3, Cardiff, UK, pp. 3221 –

3233.Davidson, N.C. and Buck, A.L., 1997. An inventory of UK estuaries. Vol. 1. Introductionand Methodology. Joint Nature Conservation Committee, Peterborough, UK.

Dyer, K.R., 1997. Estuaries: a Physical Introduction, 2nd ed. John Wiley, Hoboken, NJ.Friedrichs, C.T. and Aubrey, D.G., 1994. Tidal propagation in strongly convergent channels.

Journal of Geophysical Research, 99 (C2), 3321 – 3336.O’Brien, M.P., 1969. Equilibrium flow area of inlets and sandy coasts. Journal of

Waterways and Coastal Engineering Division ASCE , 95, 43 – 52.Prandle, D., 1985. On salinity regimes and the vertical structure of residual flows in narrow

tidal estuaries. Estuarine Coastal and Shelf Science, 20, 615 – 633.

Prandle, D., 1997. The dynamics of suspended sediments in tidal waters. Journal of Coastal Research, 25, 75 – 86.

Prandle, D., 2003. Relationships between tidal dynamics and bathymetry in stronglyconvergent estuaries. Journal of Physical Oceanography, 33 (12), 2738 – 2750.

Prandle, D., 2004. How tides and river flows determine estuarine bathymetries. Progress inOceanography, 61, 1 – 26.

Prandle, D., 2006. Dynamical controls on estuarine bathymetry: assessment against UK data base. Estuarine Coastal and Shelf Science, 68 (1 – 2), 282 – 288.

Prandle, D. and Rahman, M., 1980. Tidal response in estuaries. Journal of Physical Oceanography, 70 (10), 1552 – 1573.

Prandle, D., Lane, A., and Manning, A.J., 2006. New typologies for estuarine morphology.Geomorphology, 81 (3 – 4), 309 – 315.

Prandle, D., Lane, A., and Wolf, J., 2001. Holderness coastal erosion – Offshore movement by tides and waves. In: Huntley, D.A., Leeks, G.J.J., and Walling, D.E. (eds), Land – Ocean Interaction, Measuring and Modelling Fluxes from River Basins toCoastal Seas. IWA publishing London, pp. 209 – 240.

References 173



Simpson, J.H. and Hunter, J.R., 1974. Fronts in the Irish Sea. Nature, 250, 404 – 406.Simpson, J.H., Brown, J., Matthews, J., and Allen, G., 1990. Tidal straining, density

currents and stirring in the control of estuarine stratification. Estuaries, 13 (2),125 – 132.

Schubel, J.R. and Hirschberg, D.J., 1982. The Chang Jiang (Yangtze) estuary: establishing

its place in the community of estuaries. In: Kennedy, V.S. (ed.), EstuarineComparisons. Academic Press, New York, pp. 649 – 654.




7

Sy n ch r o no u s estu ar ies: sed imen t tr app in g

a n d s o r t i n g – stab le m o r ph o log y

7.1 Introduction

S u s p e n d e d s e d i m e n t s i n e s t u a r i e s g e n e r a l l y i n c r e a s e u p s t r e a m t o p r o d u c e a ‘turbid-

i t y m a xi m u m’ ( TM ) i n t he v ic in it y o f t he u ps tr ea m l im it o f s al in e i nt ru si o n, w it h

c o nc e n tr a ti o n s h u g el y i n cr e as e d r e la t iv e t o o p en - se a c o n di t io n s. U n c le s et al . ( 2002)

s u mm a ri s e o b s er v at i on a l s t ud i e s o f t h es e TM a n d d i sc u s s m e ch a ni s ms r e sp o n si b le

f o r t h e i r f o r m a t i o n . H e r e , f o r t h e c a s e o f s t r o n g l y t i d a l e s t u a r i e s , w e d e v e l o p g e n e r i c

q u an t it at iv e e xp re ss io n s t o r ep re se nt t he m ec h an is ms p ro d uc in g t h es e h ig h s ed i-

m e n t c o n c e n t r a t i o n s . T h e a i m i s t o i d e n t i f y t h e s c a l i n g p a r a m e t e r s w h i c h d e t e r m i n et h e s e n s i t i v i t i e s t o s e d i m e n t t y p e ( s a n d t o c l a y ) , s p r i n g t o n e a p t i d e s a n d d r o u g h t t o

f lo od r iv e r f lo w s. R ec og n is in g t he l on g- te rm s ta bi li ty o f e st ua ri ne b at hy m et ry,

d e s p i t e t h e c o n t i n u o u s l a r g e e b b a n d f l o o d s e d i m e n t f l u x e s , a n a d d i t i o n a l a i m i s t o

i d en t if y f e ed b a ck p r oc e ss e s t h at m a in t ai n t h is s t ab i li t y.

P o s t m a (1967) f i r s t d e s c r i b e d t h e m e c h a n i s m s r e s p o n s i b l e f o r e s t u a r i n e t r a p p i n g

o f f in e s ed im en ts , n am el y g ra vi ta ti on al c ir cu la ti on , n on -l in ea ri ti es i n t he t id al

d y n a m i c s a n d d e l a y s b e t w e e n r e s u s p e n s i o n a n d s e t t l e m e n t . H o w e v e r , t h e s e r e s p e c -

t i v e r o l e s a r e d i f f i c u l t t o i s o l a t e e i t h e r f r o m o b s e r v a t i o n s o r f r o m m o d e l s . H e n c e , i t h as b ee n d if fi cu lt t o g ai n a c le ar i ns ig ht i nt o t he s ca li ng o f t he se p ro ce ss es a nd t o

e st im at e t he ir s en si ti vi ti es t o c ha ng es i n e it he r m ar in e o r f lu vi al f or ci ng o r t o i nt er na l

parameters.

7.1.1 Earlier studies

P o s t m a (1967 ) d e s c r i b e d t h e n a t u r e o f s e d i m e n t d i s t r i b u t i o n s i n t i d a l e s t u a r i e s a n d

i nd ic at ed l ik e ly c on t ro l li ng m ec ha n is m s. S al ie nt c ha ra ct er is ti cs i de nt if ie d b y

P o s t m a i n c l u d e

( 1) c on ce nt ra ti o ns i n s us pe ns io n o f f in e s ed im en t s t ha t a re m uc h h ig he r t ha n i n r el at ed

m a ri n e o r f l uv i al s o ur c es ;

175



( 2) p ro ve na nc e, b ot h o f s us pe nd ed a nd o f s ur fi ci al b ed s ed im e nt s, i s o ve rw he lm i ng ly o f

ma r i n e o r i g i n;

( 3) e vi de nc e o f l ag s b et we en p ea ks i n c ur re nt a nd c on ce nt ra ti on , u p t o 4 d ay s f or t he s pr in g –

n e a p c y c l e ;

( 4 ) l a gs t h at a r e a s s o ci a te d w i th c u mu l at i v e e r o si o n a n d d e l a ye d s e tt l i ng f o r f i n e r p a r t ic l es ;

( 5) l ag s t ha t a re n eg li gi bl e f or p ar ti cl es w it h d ia me te rs d ≥ 100 μm ( se tt l in g v el oc it y

W s ≥ 0 . 0 1 m s−1) a n d

( 6 ) p e ak s u sp e n de d c o n c en t r at i on s (TM ) a r e w i d e l y o b s e r v e d , o f t e n r e l a t e d t o t h e u p s t r e a m

l im it o f g ra vi t at io na l c ir cu l at io n ( sa li ne i n tr us io n) a nd w it h p ar ti cl e s i ze s t yp ic al l y

1 0 0 > d > 8 μm.

P os tm a s ta te s t h at w hi le e st ua ri es m ay c on ta in b ot h c oa rs e a nd f in e m at er ia l, i t i s t he

c h a r a c t e r i s t i c s o f t h e l a t t e r w h i c h g e n e r a l l y p r e d o m i n a t e i n d e t e r m i n i n g b a t h y m e t r y

i n c o n j u n c t i o n w i t h t i d a l a m p l i t u d e , r i v e r f l o w s a n d s e d i m e n t s u p p l i e s . P o s t m a a l s on o t e s t h e i m p a c t s o f b o t h f l o c c u l a t i o n a n d w a v e s o n s e d i m e n t r e g i m e s . W h i l e t h e s e

l at te r i mp ac ts a re w id el y a ck no wl ed ge d, t he d et ai ls o f t he se p ro ce ss es a re n ot

c o n s i d e r e d h e r e .

D ro nk ers a nd Van d e K re ek e (1986) o bs er ved th at e ve n in t ida l ba ys w ith n o

e ff ect iv e r iv er f lo w, a l an dw ar d i nc re as e i n S PM i s o ft en o bs er ve d. T he y s how ed h ow

Postma’s (1967) c on ce pt s of e bb – f lo od a sy mm et ry c an b e q ua nt if ied t o d et er mi ne t he

n e t t r a n s p o r t o f f i n e s e d i m e n t s .

F es ta a nd H an se n (1978 ) , u si ng a 2 D l at e ra ll y a ve r ag ed m od e l, i nd i ca te d h o w

TM c ou ld b e g en er at ed b y g ra vi ta ti on al c ir cu la ti on . T he ir s tu dy e ff ec ti ve ly

a ss um ed a r ec ta n gu la r c ro s s s ec t io n a nd i nv ol v ed t id al l y a ve r ag ed d yn a mi cs .

U si ng a s t ea dy - st at e n um er ic a l m o de l w it h r iv er f l ow U R = 0 . 0 2 m s−1 ( o m i t t i n g

both erosi on and depo sition ), they found that the stren gth and locat ion of the

TM d e pe nd ed c ri ti ca ll y o n s et tl in g v el oc it y, w it h p ro no un ce d TM for W s >

5 × 1 0−6 m s−1. D y e r a n d E v a n s ( 1 9 8 9 ) i nc or po ra te d s eq ue nt ia l e ro si on a nd

d e p o s i t i o n i n t o a v e r t i c a l l y a v e r a g e d e s t u a r i n e m o d e l a n d s h o w e d h o w t h e s p e c i -f ic a ti on o f t he e r os i on t h re sh o ld i n fl u en c ed t h e b al a nc e o f i mp or t a nd e xp o rt o f

s u s p e n d e d s e d i m e n t s .

T he w id el y o bs er ve d o cc ur re nc e o f TM cl os e t o t he u ps tr ea m l im it o f s al in e

i nt ru si on l ed t o a ss um pt io ns t ha t g ra vi ta ti on al c ir cu la ti on w as t he g en er at in g

m ec ha ni sm . J ay a nd M us ia k ( 1994) e mp ha si se d t he r ol es o f b ot h b ar ot ro pi c a nd

baroclinic mechanisms and the need to incorporate axial and vertical components of

t he se i n e st im at in g n et s ed im en t f lu xe s. J ay a nd K uk ul ka (2003) q u an ti fi ed n et

r es id ua l s ed im en t f lu xe s a ss o ci at ed w it h c ou pl in g b et we en t id al v ar ia ti on s i n c ur re nt a nd S PM . T he y s ho w h ow t he se f lu xe s c an f ar e xc ee d t he f lu xe s d er iv ed f ro m a

product of net residual current and tidally averaged concentration. Their analysis is

s im il ar t o t he p re se nt o ne , e xc ep t t ha t f or p a rt ia ll y s tr at if ie d s y st em s, b ar oc li ni c

m o di f ic a ti o ns t o t i da l c u rr e nt p r o fi l es a r e m o re s i gn i fi c an t .




Markofsky e t a l . (1986 ) s ho we d t ha t t he TM i n t he We se r t ra ps s il t, a ct in g a s a

f il te r w it hi n t he e st ua ry. U ti li si ng o b se rv ed t im e s er ie s o f s u sp en de d s ed im en ts i n

t he We se r, G ra be ma n n a nd K ra us e ( 1989) s ho w ed t ha t l oc al is ed d ep os it io n a nd

r es us pe ns io n a re d om in an t p ro ce ss es i n t he TM , w it h s ys te ma ti c r ep et it io n o f

c o n d i t i o n s o v e r d i s p a r a t e c y c l e s o f t i d e s a n d r i v e r f l o w s . L a n g e t a l . ( 1989) d e v e l -o pe d a 3 D n um er ic al m od el t o r ep r od uc e t he se o bs er v at io ns , i nc or po ra ti ng a s et tl in g

velocity W s = 5 × 1 0 – 4 m s−1. T he y s ug ge st ed t ha t s om e c or re la ti on m ig ht e xi st

between patterns of energy dissipation and source distributions of suspended

sediments.

H am b li n (1989 ) i nd ic at ed t ha t l oc al r es us pe ns io n, e bb – f lo o d a sy mm et ry a nd

s a l i n e i n t r u s i o n a l l c o n t r i b u t e t o TM i n t h e S t . L a w r e n c e R i v e r . A m o d e l s i m u l a t i o n

specified W s = 3 × 1 0−4 m s −l, c on si st en t w it h o bs er ve d p ar ti cl e d ia me te rs i n t he

r a n g e 1 0 < d < 20 μm.U n c l e s a n d S t e p h e n s ( 1989) d e s c r i b e e x t e n s i v e m e a s u r e m e n t o f TM i n t h e T a m a r

R iv er, w he re t id al r an ge v a ri es b e tw ee n 2 a nd 6 m . T he y n ot ed a n o rd e r o f m ag ni -

t u d e d i f f e r e n c e i n TM c o n c e n t r a t i o n s o v e r t h e s p r i n g – n e a p c y c l e . T h e y a l s o n o t e d a

preponderance of silt 20 < d < 40 μm , b o t h s u s p e n d e d a n d d e p o s i t e d w i t h i n t h e TM

r e gi o n . T h e y s u g ge s te d t h at r e su s pe n si o n, t i da l p u mp i ng a n d g r av i ta t io n al c i rc u la -

t i o n m a y a l l b e c o n t r i b u t i n g m e c h a n i s m s . T h e a u t h o r s a l s o n o t e t h e i m p o r t a n c e o f

f lo cc ul at io n o f f in e s ed im en ts i n s us pe ns io n a nd o f t he r ap id c ha ng e i n d en si ty /

porosity of recently deposited sediments.

I n a s ub se qu en t i nv es ti ga ti on o f TM i n t he Ta ma r R iv er, F ri ed ri ch s e t a l .

( 1 9 9 8 ) d et er mi ne d t ha t TM w er e p rim ar il y l in ke d t o i nt er na ll y g en er at ed

n o n -l i ne a r it i es w i th t h r ee d o m in a n t e ff e c ts : ( i ) f l oo d -d o m in a n t a s y mm e tr y, ( i i)

r iv er f lo w a n d ( ii i) s e tt li n g l a g e ff ec te d v i a b re a dt h c on ve rg en c e. C on v er s el y,

a x i a l v a r i a t i o n s i n d e p t h w e r e n o t f o u n d t o b e i m p o r t a n t . A n a x i a l l y v a r y i n g b e d

e r od ab i li ty w a s f ou n d t o b e n e ce ss a ry t o m ai n ta in a n o v er al l s e di me nt b u dg e t.

T he ir m od el i nt ro du ce s a t im e l ag b et we en p ea ks i n c ur re nt s pe ed a nd c on ce n-t ra ti on o f 4 5 m in .

A ub re y ( 1 9 8 6 ) e m p ha s i s e d t h e r o le o f t i d a l d i s t o rt i o ns a s s o c i at e d w i t h

c ha nn el g eo me tr y a nd b ed f ri ct io n i n p ro du ci ng f lo od o r e bb d omi na nc e i n

s h a l l o w e s t u a ri e s . A u br e y n ot e d t he d ep en d en ce o f t h e m ag ni t ud e o f t he se d is to r-

t ion s o n ς * /D . H er e, w e c on ce nt ra te o n e stu ar ie s w it h l arge ti dal a mp li tu de –

d ep th r at io s i n s hal lo w, t ri an gu la r c ro ss s ect io ns w it h a ss oc ia te d s tr on g v er ti ca l

mixing.

7.1.2 Approach

The present approach in volves direct integration of expressions fo r tidal

d yn a mi cs , s al in it y i nt ru si on a nd s ed im en t m ot io ns i nt o a n a na ly ti ca l e mu la to r.

7 . 1 I n tr od u ct i o n 177



T he e mu la to r i s a pp li ca bl e w it hi n s tr on gl y t id al ( he nc e m ix ed ) f un ne l- sh ap ed

e st ua ri es a nd i nc or po ra te s p ro ce ss es t ha t a re p ro no un ce d i n s ha ll ow e st ua ri es

w it h t r ia n g ul a r c r os s s e ct i o ns . I t p r o vi d es c l ea r i l lu s tr a ti o n s o f p a r am e te r d e p en -

d e n c i e s a n d e n a b l e s c o n d i t i o n s o f z e r o n e t s e d i m e n t f l u x t o b e d e t e r m i n e d . W h i l e

t h e s e r e s u l t s a r e , t o s o m e d e g r e e , d e p e n d e n t o n a p r i o r i a s s u m p t i o n s , t h e a p p r o a c hh as t he a dv an ta ge o f ( i) r el at iv el y s tr ai gh tf or wa rd m at he ma ti cs , ( ii ) g en er ic

a pp li c at io n o ve r a w id e r an ge o f p a ra me te rs , n am e ly t id al e le va t io n a m pl it u de

ς * , w at er d ep th D , r iv er f lo w Q , s ed im en t t yp e o r f al l v el oc it y W s a n d f r ic t io n

c o e f f i c i e n t f .

T he p ri ma ry o bj ec ti ve i s t o i nt eg ra te i nt o a n ‘a n a l y ti c a l e m u l a to r ’ simplified

d e s c r i p t i o n s o f t h e s a l i e n t m e c h a n i s m s o f

( 1) t id al a nd r es id ua l c ur re nt s a ss oc i at ed w i th s al i ne i nt ru si on , r iv er f lo w a nd t id al n on -linearities;

( 2 ) s e di m e nt e r os i o n, s u sp e ns i on a n d d e p o s it i on

t o p ro du ce e xp re ss io ns f or m ea n c on ce nt ra ti on s a nd n et c ro ss -s ec ti on al f lu xe s

o f S P M .

Section 7.2 s u mm a ri s es re s ul t s f o r (1) as described previou sly in Chapters 4

and 6. Similarly, Section 7.3 s um ma ri se s r es ul ts f o r (2), based on C hapter 5,

i n t r o du c i n g c o n t i nu o u s f u n c t io n a l r e p r e se n t a t io n s (7.22) f or t he h al f- li fe o f s ed i-m e n t s i n s u s p e n s i o n , t 50, a n d f o r t h e i r v e r t i c a l p r o f i l e e− β z , (7.24). In S e c t i o n 7 . 4 , t h e

v a l i d i t y o f t h e r e s u l t i n g e x p r e s s i o n s f o r s e d i m e n t c o n c e n t r a t i o n s a n d n e t f l u x e s a r e

a s s e s s e d a g a i n s t n u m e r i c a l m o d e l s i m u l a t i o n s (F i g . 7 . 1; P r an d l e, 2 0 0 4c ) .

S e ct i on 7 . 5 e xa m in e s c o mp o n en t c o nt r ib u ti o n s t o n e t s e d im e n t f l ux w i th a t te n -

d an t s en si ti vi ty a na ly se s. I t i s s ho wn h ow s ep ar at e c ur re nt c om po ne nt s f ro m

t h e t i da l n o n -l i ne a ri t ie s , i n vo l vi n g c o s θ a nd s in θ , l ar ge ly d et er mi ne t he b al an ce

between import and export, where θ i s t h e p h a s e d i f f e r e n c e b e t w e e n t i d a l e l e v a t i o n ,

ς * a n d c u r r e n t , U * . T h e c o m b i n a t i o n s o f p a r a m e t e r v a l u e s w h i c h p r o d u c e z e r o n e t s e d i m e n t f l u x e s a r e t h e n d e t e r m i n e d .

S e ct i on 7 . 6 s u mm a ri s es t h es e e a rl i er r e su l ts i n di c at i ng i n F ig . 7 .8 h o w, f or f al l

v el oc it ie s o f 0 .0 00 1, 0 .0 01 a nd 0 .0 1 m s−1, t he b al an ce b et we en n et i mp or t o r

e xp or t c ha n ge s a s t id al a mp li tu de s v ar y f ro m 1 t o 4 m ( in di ca ti ve o f n ea p t o s pr in g

v a r i a t i o n s ) . P r o c e e d i n g u p s t r e a m f r o m d e e p t o s h a l l o w w a t e r , t h e b a l a n c e b e t w e e n

i mp or t o f f in e s ed im en t s a nd e xp o rt o f c o ar se r o ne s b ec om es f in er, i .e . s el ec ti ve

‘sorting’ a nd t ra pp in g . L ik e wi se , m or e i mp or ts , e x te nd in g t o a c oa rs er f ra ct io n,

o c c u r o n s p r i n g t h a n o n n e a p t i d e s .

In S e ct i on 7 . 7, t he a bo ve r es ul ts a re e nc ap su la te d i nt o n ew e st ua ri ne t yp ol o gy

f r am e wo r k (F i g s 7 . 9 – 7 . 11 ) c o mp a ri n g o b s er v ed v e rs u s t h e or e t ic a l d y na m ic s , b a t hy -

m e t r y a n d s e d i m e n t r e g i m e s f o r a w i d e r a n g e o f e s t u a r i e s .




7.2 Tidal dynamics, saline intrusion and river flow

E s tu a ri n e d y na m ic s a r e l a rg e ly d e te r mi n ed b y t i de s a n d e s tu a ri n e b a t hy m et r y w i th

s om e m od ul at io n b y r iv er f lo w e vi de nt c lo se t o t he t id al l im it . F or t he c as e o f a

s y n c h r on o u s e s t u a ry, C h ap t er 6 p r es e nt s l o ca l is e d s o lu t io n s f o r t h e a m p li t ud e a n d

phase of tidal currents expressed in terms of tidal elevation amplitude and water

d e p t h . F o r c o m p l e t e n e s s , t h e s e s o l u t i o n s a r e s u m m a r i s e d h e r e . S i m i l a r l y , t h e s a l i e n t f ea tu re s o f t he d yn am ic s a ss oc ia te d w it h s al in e i nt ru si on a nd r iv er f lo w, d es cr ib ed i n

C h a p t e r 4, a r e s u m m a r i s e d .

T h e s e s o l u t i o n s a p p l y t o t h e c a s e o f a s i n g l e ( p r e d o m i n a n t ) t i d a l c o n s t i t u e n t i n a

t r i a n g u l a r - s h a p e d e s t u a r y . C o n v e c t i v e a n d d e n s i t y g r a d i e n t t e r m s a r e n e g l e c t e d , a n d

0.05(a) Sand

Silt

Clay

1

10

100

400

0.005

0.0005

D (m)

4 8 16

W s(m s –1)

(b) –10

–100

–500

0

500

1000

0.05 Sand

Silt

Clay

0.005

0.0005

D (m)

4 8 16

W s(m s –1)

(c)

–10

–100

–500

0

1000

0.05 Sand

Silt

Clay

0.005

0.0005

D (m)

4 8 16

W s(m s –1)

F ig . 7 .1 . A na ly ti ca l E mu la to r v er su s N um er ic al M od el v al ue s o f s us pe nd edc o n c e n t r a t i o n s a n d n e t f l u x e s .Mo d e l ( s o l i d c o n t o u r) , a n a l yt i c a l e mu l a t o r (7.29a) (dashed);( a ) D e pt h - a n d t i me - av e ra g ed c o nc e nt r at i on s ( m g l−1),(b) net depth-integrated sediment fluxes from the numerical model;(c) net depth-integrated sediment fluxes from the analytical emulator (7.31).Va lu e s a re f o r ζ* = 2 m a n d f = 0. 002 5. Fl ux es a re in t m−1 w i dt h p er y ea r, p o si t iv elandward.

7 . 2 Ti d al d y na m i cs , s a l i n e i n t r us i on a n d r i v e r f l o w 179



t h e f r i c t i o n t e r m i s l i n e a r i s e d . I t c a n b e s h o w n ( P r a n d l e , 2004b) t h a t t i d a l p r o p a g a -

t i o n i n ‘mixed’ e s t ua r ie s i s a l mo s t e n ti r el y u n a ff e ct e d b y s a li n e i n tr u si o n.

7.2.1 Analytical solution for 1D tidal propagation

( Chapter 6 ; Prandle, 2003 )

O m i t t i n g t h e c o n v e c t i v e t e r m f r o m t h e m o m e n t u m e q u a t i o n , w e c a n d e s c r i b e t i d a l

propagation in an estuary by

@ U

@ t þ g

@&

@ X þ f

U jU j

H ¼ 0 (7:1)

B @ B

@ t þ

@

@ X

A U ¼ 0; (7:2)

where U i s v el oc it y i n t he X direction, ς i s w at er l ev el , D i s w at er d ep t h, H i s t h e

t ot al w at er d ep th ( H = D + ς ), f i s t h e b ed f ri ct io n c oe ff ic ie nt ( ~0 .0 02 5) , B i s t he

c h a n n e l b r e a d t h , A i s t h e c r o s s - s e c t i o n a l a r e a , g i s g r a v i t a t i o n a l a c c e l e r a t i o n a n d t is

time.

C o n ce n tr a ti n g o n t h e p r op a ga t io n o f o n e p r ed o m in a nt t i da l c o n st i tu e nt , M2, t h e

s o l u t i o n s f o r U and ς a t a n y l o c a t i o n c a n b e e x p r e s s e d a s

& ¼ & cos ðK 1 X ω tÞ (7:3)

U ¼ U cos ðK 2 X ω t þ θ Þ; (7:4)

where K 1 and K 2 a r e t h e w a v e n u m b e r s , ω i s t h e t i d a l f r e q u e n c y a n d θ t h e p h a s e l a g

of U* r e l a t i v e t o ς *.

F ur th er a ss um in g a t ri an gu la r c ro ss s ec ti on w it h c on st an t s id e s lo pe s, w he re

U ∂ A/ ∂ X ≫ ∂U / ∂ X and ∂ D/ ∂ X ≫∂ς / ∂ X , w e s i m p l i f y t h e c o n t i n u i t y e q u a t i o n t o

@& @ t þ U @ D

@ X þD

2 @ U

@ X ¼ 0: (7:5)

T h e c o m p o n e n t o f f U |U |/ H a t t h e p r e d o m i n a n t t i d a l f r e q u e n c y M2 m a y b e a p p r o x i -

m a t e d b y ( P r a n d l e , 2004a):

8

3π

25

16 f

jU jU

D ¼ FU ; (7:6)

with F = 1 .3 3 f U */ D. B y s pe ci fy in g t he s yn ch ro no us e st ua ry c on di ti on t ha t

t he s pa ti al g ra di en t i n t id al e le va ti on a mp li tu de i s z er o, w e d er iv e K 1 = K 2 = k ,i . e . i d e n t i c a l w a v e n u m b e r s f o r a x i a l p r o p a g a t i o n o f ς and U . T h u s ,

tan θ ¼ F

ω ¼

SL

0:5 D K (7:7)




U ¼ & g k

ðω2 þ F 2Þ1=2 ; (7:8)

where SL= ∂ D/ ∂ X , a n d

k ¼ ω

ð0:5DgÞ1=2 : (7:9)

T hu s , i n s h al lo w w a te r, v a lu es o f U* a r e p r op o rt i on a l t o D1/4(ς */ f )1/2 a nd t an θ is

proportional to D−3/4(ς * f )1/2.

7.2.2 Saline intrusion ( Chapter 4 )

F o r t h e c a s e o f a w e l l - m i x e d e s t u a r y w i t h a c o n s t a n t a x i a l s a l i n i t y g r a d i e n t , S x, t h ef o ll o wi n g e x pr e ss i on f o r r e si d ua l v e lo c it i es a s so c i at e d w i th s a li n e i n tr u si o n, U s, at

f r a c t io n a l h e i g h t z (= Z / D ) a b o v e t h e b e d w a s d e r i v e d u s i n g (4.15), P r a n d l e ( 1985):

U s ¼ g S xD3

K z

z3

6 þ 0:269z2 0:0373z 0:0293

; (7:10)

where K z i s t h e v e r t i c a l e d d y d i f f u s i v i t y .

F o r w e l l - m i x e d c o n d i t i o n s , t h e l e n g t h o f s a l i n e i n t r u s i o n LI i s g i v e n b y (4.44):

LI ¼ 0:005 D

f U U R; (7:11)

where U R i s t h e r e s i d u a l v e l o c i t y c o m p o n e n t a s s o c i a t e d w i t h r i v e r f l o w a n d S x may

be approximated by the ‘relative’ s a l i n i t y g r a d i e n t :

S x ¼ 0:027

LI

: (7:12)

H er e w e a ss um e t ha t t he e dd y d iff us iv it y c oe ff ic ie nt e qu al s t he e dd y v is co si ty

c o e f f i c i e n t a n d i s g i v e n b y

K Z ¼ E ¼ f U D: (7:13)

T h e n f r o m (7.10), (7.11), (7.12) a n d ( 7 . 1 3 ) , w e e s t i m a t e U s a t t h e b e d a s

U S ¼ 1:55 U R; (7:14)

i n d e p e n d e n t o f f , U * or D. M o r e o v e r , i t w a s s h o w n i n C h a p t e r 5 that U R i s g e n e r a l l yorder of (1 cm s−1) ; h e n c e , w e a n t i c i p a t e v a l u e s o f r e s i d u a l c u r r e n t s a s s o c i a t e d w i t h

a n a x i al s al in it y g r a di en t o f t yp ic al ly 1 o r 2 c m s−1. P r a n d l e ( 2004b) s h o w e d t h a t t h e

n e g l e c t o f c o n v e c t i v e o v e r t u r n i n g i n d e r i v i n g (7.10) r e s ul ts i n a n u nd er es ti ma ti on o f

U s b y a facto r of up to 2 . F i gu r es 6 . 4 a n d 6 .5 ( Pr an dl e, 2004a) s how that for a

7 . 2 Ti d al d y na m i cs , s a l i n e i n t r us i on a n d r i v e r f l o w 181



s y nc h ro n o us e s tu a ry, s tr a ti f ic a ti o n i s g e n er a ll y c o nf i ne d t o ς * < 1 m. However, some

e le me nt o f i nt r a- ti da l s tr at if ic at io n w il l o c cu r f or l ar ge r v a lu es o f ς * . S u ch s t ra t if i c at i o n

c a n s i g ni f ic a nt l y i n cr e as e v a lu e s o f U s; f or e xa mp le , D ro nk er s a nd Va n d e K re ek e

(1986) s h o w v a l u e s o f U s u p t o 1 0 c m s−1 in the partially stratified Volkerak estuary.

7.2.3 River flow

E q ua t io n ( 4 .1 2 ) p ro v id es t he f ol lo wi ng a pp ro xi ma ti on f or t he v e rt ic al p ro fi le o f

r i v e r f l o w i n a s t r o n g l y t i d a l e s t u a r y :

U R ¼ 0:89 U R z2

2 þ z þ

π

4

: (7:15)

7.3 Sediment dynamics

C ha pt er 5 de s cr i be s l o ca l is e d s o lu t io n s f o r s u s pe n de d s e d im e n t c o n ce n tr a ti o n s.

T h es e a r e s u m ma r is e d h e re p r io r t o i n co r po r at i on a l on g s id e s y n ch r on o u s e s tu a ry

s o l u t i o n s i n S e c t i o n 7 . 4.

7.3.1 Suspension, erosion, deposition and vertical profiles

Neglecting horizontal components of advection and dispersion, a localised distribu-

t i o n o f s u s p e n d e d s e d i m e n t s c a n b e d e s c r i b e d b y t h e d i s p e r s i o n e q u a t i o n

@ C

@ t W S

@ C

@ Z ¼ K Z

@ 2C

@ Z 2 S k þ S c; (7:16)

where S k and S c a r e s i n k s a n d s o u r c e s o f s e d i m e n t . F o l l o w i n g P r a n d l e ( 1997a and

1997b) , w e a ss u me s ed im en t c on ce nt ra ti on t im e s er ie s a ss oc ia te d w it h e ac h e ro -

s i o n a l e v e n t o f m a g n i t u d e M , (5.11):

C ðZ ; tÞ ¼ M

ð4π K ztÞ1=2 exp

ðZ þ W stÞ2

4K zt þ exp

ð2D þ W st Z Þ2

4K zt

" #: (7:17)

T hi s a ss u me s d ep os it io n a t t he b ed b ot h b y a dv ec ti on a nd d is pe rs io n f or p ar ti cl es

t h a t r e t u r n t o t h e b e d f o l l o w i n g ‘reflection’ a t t h e s u r f a c e . A n o b s e r v e d t i m e s e r i e s

r e p r e s e n t s a t i m e i n t e g r a t i o n o f a l l s u c h p r e c e d i n g ‘events’.

Erosion

Erosion ER(t ) , d u e t o a t i d a l c u r r e n t U (t ) , i s a s s u m e d t o b e o f t h e f o r m

ERðtÞ ¼ f γ ρ U j jN ; (7:18)




where N i s s om e p ow er o f v el oc it y ty pi ca ll y i n t he r an ge o f 2 – 5 and ρ i s w at er

d e n s i ty. F o r N = 2 , P ra nd le e t a l . (2001 ) d e r i v e d γ = 0.0 001 m−1 s ; f o r c o n ve n ie n ce ,

t h i s f o r m u l a t i o n i s a d o p t e d h e r e . T h i s i m p l i e s a r a t e o f e r o s i o n d i r e c t l y p r o p o r t i o n a l

t o f r i c t i o n a l b e d s t r e s s .

Deposition

C h ap te r 5 d e sc r ib e s s o lu t io n s o f (7.16), s ho w in g t ha t d ep os it io n c an b e a pp ro x i-

m a t e d b y a n e x p o n e n t i a l f u n c t i o n e−αt with ‘half-life’ t 50 = 0 .6 9 3/ α, w h e r e α i s g i v e n

by the larger of (5.14) a n d ( 5 . 1 5 ) , i . e .

α ¼ 0:693 W 2sK Z

(7:19)

α ¼ 0:1 K zD2

: (7:20)

T h e p a ra m et e r K z/ W s D, w h ic h c h ar a ct e ri s es t h e g o v er n in g m e ch a n ic s , i s i n v er -

s el y p ro po rt io na l t o t he f am il ia r R ou se n um be r. I t i s s ho wn i n C ha pt er 5 that

m a x i m u m h a l f - l i v e s o c c u r a t K z/ W s D = 2 .5 . F o r K z/ W s D≫1 , d i s p e r s i o n p r e d o m i -

n a t e s , a n d s e d i m e n t s a r e w e l l m i x e d v e r t i c a l l y . C o n v e r s e l y , f o r K z/ W s D≪ 1 , a d v e c -

t i ve s e tt l in g p r ed o m in a te s , a n d s e di m en t s r e ma i n c l os e t o t h e b e d . F i gu r e 5 . 4 s h o w s

t h e r e l a t i o n s h i p b e t w e e n W s and K z / W s D f o r a r a n g e o f v a l u e s o f ς * ( u t i l i s i n g ( 7 . 8 )

a nd ( 7. 13 )) . T hi s i ll us tr at es h o w t he d em ar ca ti on l in e o f K z/ W s D ~1 c oi nc id e s w it h

f al l v el oc it ie s o f o rd er o f ( 1 m m s−1) o r , f o r

W sðms1Þ ¼ 106 d 2ð μÞ (7:21)

a p a r t i c l e d i a m e t e r d o f o rd er o f ( 3 0 μm ) . I t i s s h o w n i n S e c t i o n 7 . 5 t h a t t h e s e d i m e n t

balance in estuaries is especially sensitive to sediments in this range.

To c on st ru ct a n a na ly ti ca l e mu la to r, w e n ee d a c on ti nu o us t ra ns it io n b et we en

conditions (7.19) a n d ( 7 . 2 0 ) . B y s i m p l e c u r v e f i t t i n g , t h e r e s u l t s s h o w n i n F i g . 5 . 2c a n b e c l o s e l y a p p r o x i m a t e d b y (5.17):

α ¼ 0:693 W s=D

10x ; (7:22)

where x i s t h e r o o t o f t h e e q u a t i o n

x2 0:79x þ j ð0:79 j Þ 0:144 ¼ 0 (7:23)

with j = l o g10 K z/ W s D.

V e r t i c a l p r o f i l e o f s u s p e n d e d s e d i m e n t s

I n s u b se q ue n t e s ti m at e s o f n e t t i da l f l ux e s , c o n t in u o us f u n ct i on a l d e sc r ip t io n s o f t h e

v e rt i ca l p r of i le s o f s u s pe n de d s e d im e nt c o n ce n tr a ti o n s a r e r e qu i re d . B y n u me r ic a l

7 . 3 S e di m en t d y n am i cs 183



f itt in g o f a p ro fi le e− β z t o s im ul at io ns b a se d o n (7.17), t he f ol lo wi ng e xp re ss io n f or β

w a s d e r i v e d u s i n g (5.19):

β ¼ 0:91 log10 6:3 K z

W sD

1:7

1: (7:24)

F ig ur e 5 .5 s ho w s , fo r a r a n ge o f v a l u e s of K z/ W s D , v a l u e s o f β f r o m ( 7 . 2 4 )

a lo ng s id e r el at e d s ed im e nt p ro fi l es . T h es e r es u lt s i l lu s tr at e h o w e ff ec ti ve c o m-

plete verti cal mixing is achie ved for K z/ W s D > 2 , a n d ‘ bed load’ o n ly o c cu rs f or

K z/ W s D < 0 .1 .

7.4 Analytical emulator for sediment concentrations and fluxes

H er e, t he d yn am ic al r es ul ts d es cr ib ed i n S e ct i on 7 . 2 a re i nt eg ra te d w it h t he s ed im en t

s o lu t io n s f r om S e ct i on 7 . 3 to fo rm an ‘a n a l y ti c a l e m u l a to r ’ f o r d et er mi ni ng n et

s ed im en t f lu xe s ( Pr an dl e, 2004c) . E ro si on , a ss um ed p ro po rt io na l t o v el oc it y

s qu ar ed , i s m od ul at ed b y a n e xp on en ti al s et tl in g r at e t o y ie ld m ea n a nd t id al ly

v a r y i n g c o m p o n e n t s o f s e d i m e n t c o n c e n t r a t i o n . T h e a p p r o a c h a s s u m e s t h a t c o n t i n -

u o us c y cl i ca l e r os i on a n d d e p os i ti o n c o ex i s t w i th o u t t h re s ho l d v a lu e s. F i gu r e 7 . 7

s um ma ri se s h o w t he s e f lu x c om p on en ts a re d et er mi ne d f ro m t he p ro du ct o f t id al

a nd r es id u al v el oc it ie s ( mo di fi ed f or v e rt ic al s tr uc tu re c o mp on en ts ) w it h t he sec o n s t i t u e n t s o f s e d i m e n t c o n c e n t r a t i o n ( l i k e w i s e m o d i f i e d f o r v e r t i c a l s t r u c t u r e ) .

7.4.1 Erosional velocity components

S e ct i on 2 . 6. 3 s ho w s t ha t t o m ai nt ai n c on ti nu it y i n a t id al ly v ar yi ng c ro s s s ec ti on

w he re n et fl ux i s U 1 t im es t he m ea n c ro ss -s ec ti on al a re a, t he M2 current U 1* c o s (ωt )

g i v e n b y (7.4) m u s t b e a c c o m p a n i e d b y b o t h M4 a n d Z0 ( r e s i d u a l) c o m p o ne n t s :

U 2 ¼ aU 1 cos 2ωt θð Þ½

U 00 ¼ aU 1 cos θ;(7:25)

where a = ς */ D f o r a t r i a n g u l a r c h a n n e l a n d a = 0.5 ς */ D f o r a r e c t a n g u l a r c h a n n e l .

Mean concentration

H en ce fo rt h fo r c la ri ty o f o th er s ym bo ls , th e a st er is ks u se d t o d es ig na te t id al

amplitudes, ς * a n d U * , a r e o m i t t e d .A s s u m i n g e r o s i o n i s p r o p o r t i o n a l t o v e l o c i t y V s q u a r e d , t h e e x p a n s i o n o f M2, M4

a n d Z0 v e l o ci t y c o n s t it u e n t s g i v e s

V 2 ¼ ½U 1 cos ωt þ U 2 cos ð2ωt θ Þ þ U 02; (7:26)




where U 0 = U 00 + U R + U s, w i t h U 2 and U 0

0 from (7.25) and U R and U s from (7.15)

a n d ( 7 .1 0 ), r e sp e ct i ve l y. T h is y i el d s s u s pe n de d m a ss c o mp o n en t s a t t h e f o ll o wi n g

t i d a l f r e q u e n c i e s ( s c a l e d b y f γρ from (7.18)):

ðU 1 cos ωt þ U 2 cosð2ωt θ Þ þ U 0Þ

2

¼ V

2

0 þ V

2

ω þ V

2

2ω þ V

2

3ω þ V

2

4ω

with V 20 ¼ 0:5ðU 21 þ U 22Þ þ U 20

V 2ω ¼ U 1U 2 cos ðωt θ Þ þ 2 U 0U 1 cos ωt

V 22ω¼ 2U 0U 2 cos ð2ωt θ Þ þ 0:5 U 21 cos 2ωt

V 23ω¼ U 1U 2 cos ð3ωt θ Þ

V 24ω¼ 0:5U 22 cos ð4ωt 2θ Þ:

(7:27)

7.4.2 Mass of sediments in suspension

From S e c ti o n 5 . 5, t he m as s i n s us pe ns io n f or e ac h c yc li ca l e ro s io n c om po n en t a t

frequency ω, w h e n m o d u l a t e d b y d e p o s i t i o n a t t h e r a t e e−αt , i s g i v e n b y (5.20):

C ðtÞ ¼ ð t

1

cos ωt0eαðtt0Þ dt0 ¼ α cos ωt þ ω sin ωt

α2 þ ω2

; (7:28)

w he re t he i nt eg ra l s um s b ac kw ar d i n t im e t o r ep re se nt a ll r em ai ni ng c on tr ib ut io ns t o

s u s p e n de d c o n c e nt r a t i on .

Thus, (7.28) m od ul at es t he e ro si on a t a ny f re qu en cy t o y ie ld a c on ce nt ra ti on

proportional to 1/(α2 + ω2)1/2. F r o m (7.27), (7.28) a n d ( 7 . 1 8 ) , w e n o t e t h a t t h e n e t

m a s s i n s u s p e n s i o n i n c l u d e s c o m p o n e n t s a t f r e q u e n c i e s σ a s f o l l o w s :

σ ¼ 0; mass MCI CD ¼ f γρ U

2

0 þ 0:5 U

2

1 þ U

2

2 α

(7:29a)

σ ¼ ω; mass MC2 CD ¼ f γρ U 1U 2

α2 þ ω2 ½α cosðωtθ Þ þ ω sinðωtθ Þ (7:29b)

σ ¼ ω; mass MC3 CD ¼ f γρ2 U 0 U 2

α2 þ ω2 ½α cos ωt þ ω sin ωt: (7:29c)

A dd it io n al c om po n en ts c an b e s ho w n n o t t o c on tr ib ut e t o n e t s ed im en t f lu x a nd ,

h e nc e , a re o m i tt e d i n (7.29).F ig u re 7 .1 a (Prandle, 2004c) shows the corresponding mean concentrations obtained

from (i) a cyclical tidal numerical model simulation of (7.17) a n d ( i i ) t h e σ = 0 c omp o-

n en t o f (7.29). Go od a gr ee me nt i s s how n ov er a ra nge o f v alue s o f b ot h W s and D. T he se

results are for ς * = 2 m , f = 0 .0 02 5 a n d U R = 0 . 0 1 m s−1, w i t h U 1 calculated from (7.8).

7 . 4 A n al y t ic a l e m ul a to r f o r s e di m en t c o nc e nt r a ti o ns a n d f l ux e s 185



T he f ul l c on ce nt ra ti on t im e s er ie s c an b e c al cu la te d b y s um mi ng e ac h o f t he

components, σ = 0, ω to 4ω, in (7.27) w it h t he ir m o du la ti on b y d ep os it io n a t t he

r a t e e−αt a s i n d i c a t e d i n (7.28).

7.4.3 Net sediment fluxes

S ed im en t f lu xe s i nv ol ve t he p ro du ct o f e ro si on , f ro m ( 7 . 2 7 ) , m od ul at ed b y

d e p os i ti o n, v i a ( 7 . 2 8 ) , m ul ti pl ie d b y t he v el oc it y c om po ne nt a pp ro pr ia te t o a n

( a s s u m e d ) c o n s t a n t d e p t h i . e . U 1 cos ωt + U RS, w h e r e U RS = U R + U S. T h e v e r t i c a l

s t r u c t u r e o f c o n c e n t r a t i o n s i s a p p r o x i m a t e d b y e− β z , w i t h β e s t i m a t e d f r o m ( 7 . 2 4 ).

T h e r e s u l t i n g c o n c e n t r a t i o n t i m e s e r i e s c a n b e c a l c u l a t e d b y s u m m i n g e a c h o f t h e

f i v e c o m p o n e n t s i n ( 7 . 2 7 ) t o g e t h e r w i t h t h e i r m o d u l a t i o n b y d e p o s i t i o n a t t h e r a t ee−αt ( S e c t i o n 7 . 3 ).

T h e v e rt i ca l s t ru c tu r e o f t h e v e lo c it y c o m po n en t s a s so c ia t ed w i th s a li n it y i n tr u -

s i o n a n d r i v e r f l o w a r e s p e c i f i e d f r o m (7.10) and (7.15), r e sp e c ti v el y. T h e v e rt i ca l

s t r u c t u r e f o r t h e M2 t i d a l c o n s t i t u e n t i s a p p r o x i m a t e d b y

U ðzÞ ¼ U 0:7 þ 0:9 z 0:45 z2

; (7:30)

where U i s t h e d e p th - av e ra g ed t i da l v e lo c i ty a m p li t ud e .

No account of vertical phase variations is considered. For the M4 constituent,

it can b e shown that only the ratio of bed to dep th mean value is required; for

s i mp l ic i ty, a u n if o rm v e rt i ca l s t ru c tu r e i s a s su m ed .

T h e p a r a m e t e r s U 1, U 2 and U 0, d e s c r i b i n g e r o s i o n i n (7.26), r e p r e s e n t v e l o c i t i e s

a t t he b ed . We i nt ro du ce t he c oe ff ic ie nt s P and Q t o r ep re se nt d ep th v ar ia ti o ns

( r e l a t i v e t o t h e b e d ) i n U 1 and U RS, r e s p e c t i v e l y . F r o m (7.30) t h e v a l u e o f P ~ 1 / 0. 7,

w h i l e f r o m (7.14) a n d ( 7 . 1 5 ) Q ~ 1/ − 0 . 6 9 . N e t s e d i m e n t f l u x e s a r e t h e n o b t a i n e d b y

m u l t i p l y i n g t h e e r o s i o n a l c o m p o n e n t s (7.27), m o d u l a t e d v i a t h e s u s p e n s i o n e x p r e s -sion (7.28), a n d t h e v e r t i c a l p r o f i l e (7.24) b y t h e v e l o c i t y c o m p o n e n t ( PU 1 cos ωt +

QU RS) . T he l at te r i s c on si st en t w it h t he a ss um pt io n o f a n et s em i- di ur na l w at er

f l u x p l u s r i v e r a n d s a l i n i t y c o m p o n e n t s . T h i s y i e l d s r e s i d u a l s e d i m e n t f l u x e s ( F ) as

follows:

F1 ¼ 1

α Q U RS U 20 þ 0:5 ðU 21 þ U 22Þ

(7:31a)

f o r m a s s M C 1 a n d f l o w QU RS.

F2 ¼ 1

α2 þ ω2 P U 21 U 2 ð0:5 α cos θ 0:5 ω sin θ Þ (7:31b)

f o r m a s s M C 2 a n d f l o w PU 1 cos ωt .




F3 ¼ 1

α2 þ ω2 P U 21 U 0 α (7:31c)

f o r m a s s M C 3 a n d f l o w PU 1 cos ωt .

F or ( 7. 31 ), f lu x m ag ni tu de s a re a ga in s ca le d b y f γρ, a nd c on ce nt ra ti on s a re

m o d u l a t e d b y t h e v e r t i c a l s t r u c t u r e o f t h e c o n c e n t r a t i o n β e− β z /(l − e−

β ).

T h e f ir st c om p on en t, F l ( ma ss c o mp on en t M C1 ), r ep re se nt s t h e f lu x a ss oc ia te d

w it h t he p ro du ct o f ( i) t im e- av er ag ed S PM c on ce nt ra ti on , M C1 i n ( 7. 29 ), a nd

( ii ) r es id ua l c ur re n ts a ss oc ia te d w it h r iv er f lo w a nd s al in e i nt ru si on . T he s ec on d

c om p on en t, F 2 ( ma ss c om po n en t M C2 ), r es u lt s f ro m ( ii i) t h e s em i- di ur na l S PM

c om p on en t g en er at ed b y t he c om b in a ti on o f U 1 cos ωt and U 2 c os ( 2ωt − θ ) in (7.26)

a dv e ct ed b y ( iv ) t he s em i- di ur na l c ur re nt U 1 cos ωt . T he f ir st ( co s θ ) part of the

r es ul ti ng f lu x m ay b e i nt er pr et ed a s r ep re se nt in g t he d ow n st re am (U 2 i s n e g at i vefrom (7.25)) expo rt o f coarser (large α) s ed im en ts . E qu iv al en tl y, t he s ec on d

part (sin θ ) r ep re se nt s u ps tr ea m i mp or t o f f in er s ed im en ts ( sm al l α) . T he t hi rd

c om po ne nt , F 3 ( ma ss c om po ne nt M C3 ), i s s im il ar t o th e s ec on d, e xc ep t t ha t

t he s em i- di ur na l S PM c om po ne nt ( v) a ri se s f ro m t he c om bi na ti on o f U 0 and

U 1 cos ωt in (7.26), i .e . M C3 i n ( 7. 29 ). S in ce U 0 ~ U 00 ~ U 2 cos θ , t hi s t hi rd f lu x

c o mp o n en t e f fe c ti v el y i n cr e as e s t h e ( c os θ ) t e r m i n c o m p o n e n t ( i i ) b y a f a c t o r o f 3

(see S e c t i o n 7 . 5).

D e pt h- in te gr at ed f lu x es , c o rr es p on di ng t o t he c on d it io n s f or F i g. 7 .1 (a ), aresho wn (i) in F ig . 7 .1 (b ) fo r a c yc li ca l t id al n um er ic al s im ul at io n a nd ( ii ) i n

F i g. 7 . 1( c ) f o r t he a b ov e a na ly ti ca l e xp an s io n ( 7. 31 ). F o r t he se f lu xe s, a gr ee me nt

i s n ot a s c lo se a s f or c on ce nt ra ti on s. H ow ev er, t he e ss en ti al p at te rn s a nd g en e ra l

m ag n it u de s a re s u ff ic ie nt ly s im il ar t o w ar ra n t a do p ti on o f t he a na ly ti ca l e mu la to r f or

s u b s e q u e n t a n a l y s e s o f t h e c h a r a c t e r i s t i c s a n d s e n s i t i v i t i e s o f n e t s e d i m e n t f l u x e s i n

t i d a l e s t u a ri e s .

F i g ur e 7 . 1( b ) a nd ( c) s ho ws m ax im um u ps tr ea m s ed im en t f lu xe s e xc ee di ng

1000 t m−1 w id t h p er y e ar o cc ur ri ng f o r t h e f in es t s ed im e nt i n t h e s ha ll o we s t w at e r.

T hi s v al ue r ed uc es f or c o ar se r m at er ia l a nd i n d e ep e r w a te r, r ev er si n g d ir e ct io n f o r

W s > 0 . 0 0 2 m s−1, w i t h m a x i m u m d o w n s t r e a m f l u x f o r W s ~ 0 . 0 0 5 m s−1. B y c o m p a r -

i so n, e st im at es o f s ed im en t d ep os it io n i n t he M ers ey R iv er o ve r t he l as t c en tu ry

i n v o l v e n e t s e d i m e n t f l u x e s o f o r d e r o f ( 1 0 0 0 t year −1 m−1) ( L a n e , 2004).

7.5 Component contributions to net sediment flux

H a v i n g d e m o n s t r a t e d t h e v a l i d i t y o f t h e a n a l y t i c a l e m u l a t o r i n S e c t i o n 7 . 4, w e n o w

a p p l y t h e e m u l a t o r t o e x a m i n e t h e c o m p o n e n t c o n t r i b u t i o n s t o n e t s e d i m e n t f l u x e s

a n d , s p e ci f ic a ll y, t o i d en t if y c o nd i ti o n s c o n si s te n t w i th z e ro n e t f l ux e s, i . e. b a th y -

metric stability.

7 . 5 C o mp o n en t c o nt r i bu t i on s t o n e t s e d i me n t f l u x 187



7.5.1 Components of sediment flux: river flow, salinity and tidal

current constituents

We n ot e, f ro m S e ct i on 7 . 2, t ha t t he c o mp on en ts o f r es id u al v el oc it ie s U R and U sa ss oc ia te d w it h b ot h r iv e r f lo w a nd s al in it y a re t yp ic al ly t wo o rd er s o f m ag ni tu de

l e s s t h a n U 1. P r a n d l e ( 2003) c a l c u l a t e d , f o r a r a n d o m s e l e c t i o n o f 2 5 U K e s t u a r i e s ,

a n a ve ra ge v al ue o n e xt re me s pr in g t id es o f a = ς / D = 2 /3 . T he n f ro m (7.25), we

e s t i m a t e t h a t t h e c o m p o n e n t s o f U 0 and U 2 r e l a t e d t o v a r y i n g e s t u a r i n e c r o s s s e c t i o n

a r e t y p i c a l l y a s f o l l o w s : ( i ) f o r U 2, c om pa ra b le b u t l es s t ha n U 1 a n d ( i i ) f o r U 0, an

o r d e r o f m a g n i t u d e l e s s t h a n U 1.

For ς * = 2 m a n d f = 0 .0 02 5 , F i g . 7 . 2( a ) a n d ( b ) ( P ra n d le , 2 0 0 4c ) s h o ws r e si d ua l

s ed im en t f lu xe s a ss o ci at ed w it h ( i) r iv er f lo w U R = − 0 . 0 1 m s−1 a nd ( ii ) s al in it y

v e l o c i t y U S f r o m ( 7 . 1 0 ) . R iv er f lo w p ro du ces n et d ow ns tre am f lu xe s w hi ch

i nc re as e w it h g re at er w at er d ep th s a nd w it h f in er s ed im en ts . S al in e i nt ru si on

produ ces upst ream fluxe s incre asing with great er depth s and shows a maxim um

for W s ~ 0 .0 0 2 m s− 1. We s e e t h a t t h e n e t f l u x e s a s s o c i a t e d w i t h F 1 i n ( 7 . 3 1 ) ar e

s i g n i f i c a n t l y l e s s t h a n f o r F 2 a n d F 3 .

F i gu r e 7 . 2( c ) a nd ( d) s ho ws n et f lu xe s a ss oc ia te d w it h t he t he t id al c ou pl in g

t e r m s , F 2 a n d F 3 . T h e r e l a t i v e m a g n i t u d e o f t h e s e c o u p l i n g t e r m s i s d i r e c t l y r e l a t e d

0.05(a) (c)

(d)(b)

–10

10

10

100

1000

3000

100

250

–10

–100 –1000

–2000

–100

–500

0.005

0.0005

4 8 16

D (m)

W s(m s –1)

0.05

0.005

0.0005

W s(m s –1)

4 8 16

D (m)

F i g. 7 . 2. C o mp o ne n ts o f n e t d e pt h - in t eg r at e d s e di m en t f l ux f r om ( 7 .3 1 ).( a ) F 1 , r i v e r f l o w U R = − 0 . 0 1 m s−1;( b ) F 1 , s a l i n e i n t r u s i o n v e l o c i t i e s , (7.10);( c ) c o s i n e t e r m s i n c o m p o n e n t s F 2 a n d F 3 o f ( 7 . 3 1 ) ;( d ) s i n e t e r m s i n c o m p o n e n t F 2 .C o n d i t i o n s , c o n v e n t i o n s a n d u n i t s a s f o r F i g . 7 . 1.




t o t he p ha se d iff er en ce θ b e tw ee n M2 t i da l e le va ti on a nd c ur re nt . T he g en er al

n at ur e o f t hi s b al an ce b et we en t he ‘cosine’ and ‘sine’ t er ms i s s ho wn i n F i g. 7 .3o v e r a r a n g e o f v a l u e s o f h a l f - l i f e i n s u s p e n s i o n , t 50, a n d θ . T h e i r c o n t r i b u t i o n s v a r y

i n a c o m p l e x f a s h i o n g i v e n t h e c o u n t e r a c t i n g s e n s i t i v i t i e s o f U 1 and θ t o c h a n g e s i n

depth, (7.7) a n d ( 7. 8) . L ik e wi se , t he ir s en si ti vi ty t o h al f- li ve s i n s us p en si on , t 50(or W s) , i s r e f l e c t e d i n t h e d e p e n d e n c e o f t h e c o s i n e a n d s i n e t e r m s o n α/(α2 + ω2)

and ω/(α2 + ω2) , r e s p e c t i v e l y . C o m b i n i n g t h e s e n s i t i v i t i e s o f t h e s e t e r m s i n ( 7 . 3 1 ) t o

both α and θ , w e a n t i c i p a t e t h a t f o r t h e f i n e s t s e d i m e n t i n s h a l l o w w a t e r ( θ →90°),

t he s in θ t er m w il l p re do mi na te , w hi le f or c oa rs er s ed im en ts i n d ee pe r w at er s

(θ →0 ° ) , t h e c o s θ t e rm w i l l p r e d o mi n a t e .

F ro m ( 7. 31 ), a ss um in g t he t er m i nv ol vi ng U 1 p re do mi na te s i n F 1 a nd t ha t

U 0 ~ U 00 = − aU 1 cos θ , t h e n t h e r a t i o o f F 1 t o F 2 + F 3 t e r m s c a n b e s i m p l i f i e d t o

0:5 U 21 U RS=α

0:5 a U 31 ð3 α cos θ ω sin θ Þ=ðα2 þ ω2Þ(7:32)

( n e g l ec t i n g t h e p a r a m et e r s Q and P a n d t h e e f f e c t o f v e r t i c a l p r o f i l e s i n c o n c e n t r a -

t i o n ) ; n o t i n g t h a t U RS ~ − 0 . 0 1 m s−1

a n d l e t t i n g r = ω/ α, (7.32), s i m p l i f i e s t o

0:01

aU 13cos θ

1 þ r2

r sin θ

1 þ r2

: (7:33)

–0.01

–0.5

–1

–2

–3(a)

100

1

0.01

t 50(h)

0.5

0.25

0.1

0.01(b)

90° 60° 30° 0°90° 60° 30° 0°

90°

0.1

0.5

–0.1 –1

–2 –3

0

100

1

0.01(c)

t 50(h)

60° 30° 0°θ

θ θ

F i g. 7 .3 . Va l ue s o f n e t i m p or t ( + ve ) a n d ex p or t t e rm s i n (7.31) as f (θ , t 50) . ( a) C os in et e r m s , ( b ) s i n e t e r m a n d ( c ) s u m m a t i o n o f ( a ) a n d ( b ) .




F i g u r e 7 . 3 ( a ) ( P ra n d le , 2 0 04 c ) r e pr e se n ts (−3 cos θ) / ( l + r 2), F i g . 7 . 3( b ) represents

(r sin θ)/ (l + r 2) and F i g. 7 . 3( c ) r ep re se nt s t he s um ma ti on o f t he se . We n ot e t ha t t he se

t e r m s b a l a n c e i n t h e r e g i o n 1 < t 50 < 1 0 h w h e n t 50 ~ 1 0( 1 − θ / 9 0) h . F u r th e r d e s c ri p -

t i o n o f t h e s i g n i f i c a n c e o f t h e p a r a m e t e r r f o l l o w s i n S e c t i o n 7 . 7.

7.5.2 Conditions for zero net flux of sediments

To convert from the generalised scaling results shown above to more specific estuarine

conditions, we adopt the dynamical solutions for synchronous estuaries. This enables

r es ul ts t o b e p re se nt ed i n t er ms o f t he i mm ed ia te ly f am il ia r p ar am et er s, ς * (tidal elevation

amplitude) and D ( wa te r d ep th ), p er ta in in g a t a ny s el ec te d s ec ti on o f s uc h e st ua ri es .

T he n f or a ny c om bi na ti on o f ( D, ς * ), b y s ca nn in g a cr os s a r an ge o f s ett li ng

velocities, W s, a b a l a n c e b e t w e e n t h e p r e d o m i n a n t l a n d w a r d a n d t h e s e a w a r d t i d a lc o up l in g t e rm s i n (7.33) c a n b e f o u n d . Figure 7.4 (Prandle, 2004c) shows these ‘zero

flux’ v al ue s f or a ( i) h al f -l if e t 50; ( i i) p a rt ic l e d ia m et er d , (7.21); ( ii i ) f al l v e lo ci t y W s,

(7.22) a n d ( i v) m ea n c o nc en t ra ti o n, ( 7. 29 ) . T h e i n di c at e d r a ng e o f v al u es f or p a rt ic l e

d ia m et e r, 3 0 < d < 50 μm , i s r ea s on ab l y c o ns is t en t w it h r es u lt s d es c ri b ed i n Section

7.1.1 f ro m P os tm a (1967) , U nc le s a nd S te ph en s (1989) a nd H amb lin (1989). Likewise,

t he r an ge o f v al ue s f or s et tl in g v el oc it y, 1 < W s < 2 . 5 m m s−1, is close to the value

4(a) (c)

(d)(b)

3

2

1

0

1.5

1.0

2.02.5

4

3

2

1

04 8

D (m)

ζ(m)^

ζ(m)^

D (m)

16 32

30

40

200

0.51

23

100

4 8 16 32

150 20050

F i g . 7 . 4 . C o n d i t i o n s f o r z e r o n e t s e d i m e n t f l u x a s f ( D, ς* ) , w i t h U 0 = − 0 . 0 1 m s−1

and f = 0 . 0 0 2 5 .( a ) H a l f - l i f e t 50 (h),( b ) p a r t i c l e d i a m e t e r (μm),( c ) f a l l v e l o c i t y W s ( m m s−1) a n d( d ) m e an s e di m e nt c o nc e nt r at i on ( m g l−1).




a do pt ed f or t he We se r R iv er b y L an g e t al . (1989) . M or eo ve r, t he re i s r ec en t

e v i d e n ce ( M a n n i ng , 2004) f ro m a n u mb er o f e st u ar ie s o f p re do m in a nt s et tl em e nt ,

v i a f l o c c u l a t i o n , a t v a l u e s o f W s ~ 1 m m s−1. T he m e an d e pt h a nd t id a ll y a v er a ge d

s u sp en d ed c on ce n tr at i on s, 5 0 < C < 2 0 0 m g l−1, a r e w i t h i n t h e l o w e r r a n g e o f

o b se rv e d m ax im um c on c en tr a ti on s, 1 0 –

1 0 0 0 0 m g l−1

, f o u n d in TM ( U n c l e se t a l ., 2 0 0 2 ) . H ow ev er, t he se e st ima te s m ay b e s ub st an ti al ly i nc re as ed b y

v a r ia t io n s i n t h e e f fe c ti v e f r i ct i o n c o e ff i ci e nt , h e re t a ke n a s f = 0 .0 0 25 . A n i nt er -

e s t i n g f e a t u r e o f t h e s e r e s u l t s i s t h e n e a r c o n s t a n c y o f t h e p a r a m e t e r K z/ W s D ~ 1 . 6

t hr ou gh ou t t he r an ge o f ( D , ς * ). I t w as n ot ed e ar li er t ha t m ax im um h al f- li ve s

c o r r e s p o n d t o K z/ W s D = 2 .5 , w hi c h c o in ci d es w i th W s ~ 1 m m s −1. T h i s i n d i c a t e s

t h e c o ex i st en ce o f m ax i mu m c on ce n tr at io n s w it h c on d it io n s o f m or ph o lo gi c al

stability.

7.5.3 Sensitivity to fall velocity, W s

F i gu r e 7 . 5( a ) a nd ( b) s h ow s n et s ed im en t f lu x es a n d m ea n s us p en de d c on c en tr a-

tions, calculated from (7.31) a n d ( 7. 26 ), f or t he v al ue s W s = 1 and 2 m m s−1,

s u g ge s te d f r om F i g . 7 . 4( c ) t o b e c on si st en t w it h s ta bl e m or ph o lo gy. F or t he f in er

m at er ia l, n et f lu xe s a re u ps tr ea m f or a ll b ut t he l ow es t t id es a nd d ee pe st w at er.

M o re o v er, t h es e u p s tr e am f l ux e s i n cr e as e r o ug h ly i n p r o po r ti o n t o ς 3. C o n v e r se l y,

4

(a) (b)

3

2

1

0

0

4 8 16 4 8 16

10

50

100

150 200

–100

–500

–100

010002000

D (m) D (m)

–100

1000

10,000

500

400

250

100

5000

ζ(m)^

4

3

2

1

0

ζ(m)^

3000

Fig. 7.5. Sediment fluxe s (top) in t year −1 m−1 a n d c o nc en t r at i o ns ( b ot t om )i n m g l−1 as f ( D, ς* ). ( a) L ef t, W S = 0 . 0 0 1 m s−1 a nd ( b) r ig ht , W S= 0 . 0 0 2 m s−1.

D depth, ς t i d a l e l e v a t i o n a mp l i t u d e .




f o r t h e c o a r s e r m a t e r i a l , n e t f l u x e s a r e d o w n s t r e a m f o r a l l b u t t h e l a r g e s t t i d e s a n d

d ee pe st w ate r. T he se r es ul ts c le ar ly il lu str at e h ow b ot h th e d ir ec ti on a nd t he

m ag ni tu d e o f t he n et s ed im en t f lu x es c an v ar y a br up tl y b et we en s pr in g a nd n e ap

t i d e s , f r o m m o u t h t o h e a d , a n d f o r d i f f e r e n t s e d i m e n t s i z e s .

7.5.4 Sensitivity to bed friction, f , and fall velocity, W s

C or re sp on d in g c al cu la ti on s w er e m ad e f or t he c om po n en t c on tr ib ut io n s t o t he se

n e t f l u x e s . H o w e v e r , t h e s c a l i n g o f t h e c o m p o n e n t s s h o w s l e s s v a r i a t i o n w i t h e i t h e r

ς or D t h a n f o r v a r i a t i o n s i n W s a n d t h e b e d f r i c t i o n a l c o e f f i c i e n t f . T o i l l u s t r a t e t h i s ,

F i g. 7 . 6. S en s it iv i ty o f d y na mi cs , s ed im en t c on ce nt ra ti on s a nd n et f lu xe s t o f al lvelocity W S a n d f r i c t i o n f a c t o r f . R e s u l t s f o r ς * = 3 m a nd D = 8 m .(a) – ( c ) . P e r c e n t a g e o f n e t s e d i me n t f l u x i n ( 7 . 3 1 ) .( a ) F 1 ,

( b ) c o s i n e f l u x t e r m i n F 2 a n d F 3 ,( c ) s i n e t e r m i n F 2 ,( d ) t i d al v e lo c it y U * a n d p h a s e θ ,( e ) n e t f l u x F l + F 2 + F 3 ( t y e a r −1 m−1),( f ) c o n c e n t r a t i o n s ( mg l−1) a n d(g) half-life t 50 (h).




F i g . 7 . 6 ( P r a n d l e , 2 0 0 4 c ) s h o w s , f o r t h e c a s e o f ς = 3 m a n d D = 8 m , t h es e p r op or-

t i o n a l c o m p o n e n t s ( i n p e r c e n t a g e s ) a s s o c i a t e d w i t h ( a ) U RS ( b ) c o s θ a n d ( c ) s i n θ in

(7.33) f or 0 .0 00 5 < W s < 0 . 0 5 m s−1 a nd 0 .0 00 5 < f < 0 .0 12 5. T h es e r e su lt s s h ow,

c o n si s te n t w i th F ig s 7 .4 a nd 7 .5 a nd ( 7. 28 ), t ha t t he c os in e t er m p re do mi na te s

o v e ra l l, p r od u ci n g a m a x im u m ( p r op o r ti o na l ) s e a wa r d f l u x f o r c o ar s e s e d im e n tso v e r t h e s m o o t h e s t b e d s . C o n v e r s e l y , t h e s i n e t e r m p r o d u c e s n e t u p s t r e a m f l u x e s

f or t he f in es t s ed im en ts o ve r t he r ou gh es t b ed s. B ot h o f t he se r es ul ts e mp ha si se t he

r e in f or c em e nt o f a x ia l s e d i me n t s o r t in g n o t e d i n S e c t i o n 7 . 5. B y c o n t r a s t , t h e r o l e o f

t he r iv er a nd s al in e v el oc it y c om po ne nt s i s g en er al ly n eg li gi bl e i n a ll b ut t he d ee pe st

w at er s. M or eo ve r, w h il e t he s al in it y c o mp on en t p re do mi na te s o ve r t he r iv er in e

c o mp o ne n t i n t r an s po r ti n g ( n ea r -b e d ) c o ar s er m a te r ia l l a nd w ar d , t h e o p po s it e o c cu r s

f o r ( m o r e u n i f o r m l y d i s t r i b u t e d ) f i n e r s e d i m e n t s .

T h e s i g n i f i c a n t s e n s i t i v i t y o f b o t h U * a n d θ t o c h a n g e s i n b e d f r i c t i o n i s e v i d e n t i nF i g . 7 . 6( d ) w i t h c o ns e q ue n t i n fl u e nc e s o n c o nc e n tr a t io n s a n d f l u xe s . F i g ur e 7 . 6 (e ) – (g)

i ndi ca te s t he r el at ed s en si ti vi ti es o f n et f lu x, m ea n c on ce nt ra ti on a nd h al f- li fe i n

suspension.

7.6 Import or export of sediments?

T h i s s e c t i o n s u m m a r i s e s a n d i n t e r p r e t s t h e a p p l i c a t i o n o f e a r l i e r r e s u l t s f o r z e r o n e t

s e d i m e n t f l u x e s . F i g u r e 7 . 7 p r e s e n t s a s c h e m a t i c o f t h e d y n a m i c a l a n d s e d i m e n t a r y

c o m p o n e n t s i n t e g r a t e d i n t o t h e a n a l y t i c a l e m u l a t o r u s e d a b o v e t o d e r i v e c o n d i t i o n s

c o r r e s p o n d i n g t o z e r o n e t f l u x o f s e d i m e n t s , i . e . s t a b l e b a t h y m e t r y .

F i g. 7 . 7. D y na m ic a l a n d s e di m e nt a ry c o mp o ne n ts i n t eg r at e d i n to t h e a n a l yt i ca lemulator.

7 .6 I mp o rt o r e x po rt o f s ed i me nt s? 193



7.6.1 Sensitivity to half-life, t50 , and phase difference, θ ,

between tidal elevation and current

T he s en si ti vi ty o f c on di ti on s c or re sp on di ng t o z er o n et f lu x o f s ed im en ts a re

i l lu s tr a te d i n F ig . 7 .8 f or ( i) c ha ng es i n s ed im en t s iz e, ( ii ) o ve r t he l en gth o f a n

e st ua ry ( in d ic at ed b y c ha n gi ng d ep th s) a n d ( ii i) o ve r t he s p ri ng – n ea p t id al c yc le . N et

t r a n s p o r t s o f f i n e s e d i m e n t s a r e q u a n t i f i e d i n (7.33) a n d i l l u s t r a t e d i n F i g . 7 . 3 ( c ). In

F ig . 7 .8 (Prandle et al . , 2 00 5) , l oc i o f s pr in g – n ea p v ar ia ti o ns f or t id al e le va ti on

amplitudes ς = 1, 2, 3 and 4 m are superimposed onto these results. These lo ci are for

depths D = 4 and 16 m an d for f all velocities W S = 0.01, 0.001 and 0.000 1 m s−1. Th e

v al ue s f or θ a re f ro m (7.7) a d op ti ng t he a ss um p ti on f or b e d f ri ct io n c oe ff ic ie nt

f = 0.001 (d /10)½, w he re d i s p ar ti cl e d ia me te r i n m ic ro me tr e a nd (7.21) relates W S to

d . C o r r e s p o n d i n g v a l u e s f o r t 50 a r e f r o m (7.22).

P ro gr es si ng f ro m n e ap t o s pr in g t id es i nc re as es ( th e a bs ol ut e v al ue o f) θ and

h en ce r ed uc es t h e e x po rt o r i nc re as es t h e i mp or t o f s ed im en ts . T hi s s am e t re nd i s

f ou nd i n s ha ll ow er d ep th s, e mp ha si si ng h ow e st ua ri es c an t ra p s ed im en ts i n

u ps tr ea m s ec ti on s. H o we ve r, a s m or e f in er s ed i me nt s a re t ra pp ed , t he e ff ec ti ve

v al ue o f f d e c r e a se s , r e s u l ti n g , (6.12), i n a t en de nc y t o i nc re as e e st ua ri ne l en gt h.

H en ce , s om e e qu il ib ri um w il l p re va il , g ov er ne d b y t he b al an ce b et we en t he t yp e a nd

–90° –45° 0°100

1

0.01 0.1 0.5 0.9 0.99

0

–0.01

–0.1

Import

Exp ort

–0.5

Sand

Silt

10

0.1

4 m

16 m

4 m

16 m

4 m

= 4 3 2 1 mTidal amplitude ζˆ

W s = 0.01 0.001 0.0001 m s 1

16 m

H a l f l i f e t 5 0

( h )

Phase advance, θ , of ζ with respect to Û ˆ

F ig . 7 .8 . S pr in g – n ea p v ar ia b il it y i n n et i mp o rt v er su s e xp or t o f s ed im en t s (7.33), asa f (t 50, θ ) . V a r i a b i l i t y o v e r e l e v a t i o n a mp l i t u d e ς* = 1 , 2 , 3 an d 4 m , f al l v el oci ti esW S = 0 .0 00 1, 0 .0 01 a nd 0 .0 1 m s−1 f o r d e p t hs D = 4 a n d 1 6 m .




t h e q u a n t i t y o f ( m a r i n e ) s e d i m e n t s u p p l y , e n h a n c e d t r a p p i n g o f f i n e s e d i m e n t s a n d

c o n s e q u e n t m o r e e n e r g e t i c d y n a m i c s a t t e m p t i n g t o i n c r e a s e d e p t h s .

From Fig. 5.2, for K z/ W s D < 2, we can approximate α = W s2/ K z, i.e.

r = ω/ α = 3 . 5 × 1 0−7U * D/ W s2. F o r t y p i c a l v a l u e s o f U * ~ 1 m s−1 and D ~ 1 0 m ,

t h e n r < 1 f o r W s > 2 × 1 0−3

m s−1

. H en ce f ro m ( 7 . 3 3 ) , r iv er f lo w a n d a ss o ci at eds a l i n i t y i n t r u s i o n ( i n v o l v i n g v e l o c i t i e s o f a f e w c e n t i m e t r e s p e r s e c o n d ) w i l l h a v e

l i t t l e i m p a c t f o r s e d i m e n t s c o a r s e r t h a n t h e a b o v e a n d a s l o n g a s ( aU *)≫0 . 0 1 m s−1,

i.e. a = ς / D≫0 . 0 1 ( f o r t h e M2 constituent).

Wi th in t he se r eg i on s w he re t id al i nf lu en ce s p re do m in a te , t he r at io o f s ed im en t

i m p o r t ( I M ) t o e x p o r t ( E X ) i s g i v e n b y t h e d e n o m i n a t o r t e r m s i n b r a c k e t s i n (7.33):

IM

EX

¼ r

3

tan θ ¼ 0:4 f U

W s

2

(7:34)

o n t h e b a s i s o f t h e a b o v e a p p r o x i m a t i o n f o r α a n d s u b s t i t u t i n g f o r t a n θ from (7.7).

Z e r o n e t f l u x t h e n c o r r e s p o n d s t o

W s ¼ 0:6 f U 0:0015 U (7:35)

(for f = 0 . 00 2 5 ). T h is e s ti m at e f o r W s

i s i n c l o s e a g r e e m e n t w i t h t h e v a l u e s s h o w n i n

F i g . 7 . 4 ( c ).

T he d is tr ib ut io n f or m ea n s ed im en t c on ce nt ra ti on m ay b e a pp ro xi ma te d b yinserting W s from (7.35) i n ( 7 .2 9 ), n e gl e ct i ng U 0 a nd u si ng t he a bo ve v al ue s f or

α, t h u s

C ¼ γ ρ f U 2=Dα γ ρ U ð1 þ a2Þ: (7:36)

Noting, as indicated in S e ct i on 7 . 2, t ha t i n s ha ll ow w at er, U * i s p ro po rt io na l t o

(ς / f )½ D1/4, w e s ee t ha t f or c on d it io n s o f s ta bl e m or p ho lo gy, m ax im um c on ce n tr a-

t io ns w il l g en er al ly o cc ur f or l ar ge v al ue s o f a = ς / D a nd s ma ll v al ue s o f f . T hi s

s e n s i ti v i t y t o a i s b r oa d ly s u bs t an t ia t ed i n t h e c o n ce n tr a ti o ns s h ow n i n F i g . 7 . 5 ( a ) .H o w ev e r, t h e c o mp a ra b l e s e ns i ti v it i es t o f s ho wn f or t he s pe ci fi c c on di ti on s o f

F i g . 7 . 6 i n d ic a te a m o re c o mp l ex r e la t io n sh i p.

7.6.2 Sediment trapping and turbidity maxima

F ro m t he a bo ve , w e c on cl ud e t ha t s tro ng ly t id al e st ua ri es o f t he s ha pe c on si de re d h er e

w i ll t r an s po r t f i ne s e d im e n ts u p s tr e am v i a n o n- l in e ar t i da l r e ct i fi c a ti o n t e rm s . T h i s

process will be limited either by the absence of corresponding bed sediments for r e s u s p e n s i o n o r b y t h e c o u n t e r a c t i n g i n f l u e n c e o f r i v e r f l o w c l o s e t o t h e t i d a l l i m i t .

A ss um in g t ha t n ea p t o s pr in g t id al v ar ia ti on s c an b e c ha ra ct er is ed a s a s in gl e

s e mi - di u rn a l c o n st i tu e n t o f v a ry i n g a m pl i tu d e, t h en s p ri n gs w i ll t r an s po r t s i gn i fi -

c an tl y m or e s ed im en ts ; m ore ov er, t he m ax im um s ed im en t s iz e t o b e m ov ed

7 .6 I mp o rt o r e x po rt o f s ed i me nt s? 195



l an dw ar d w il l b e l arg er t ha n a t n ea ps (F ig . 7 .8) . To m a in t ai n m o rp h o lo g ic a l s t ab i li t y,

t he se n et s pr in g a nd n ea p fl ux es m us t b al an ce . H en ce , w e a nt ic ip at e a w id e

d is tr ib ut io n a nd m o re c on ti nu o us s us p en si on o f s ed im en t s iz es c or re sp on d in g t o

t h is s p ri n g – n e a p v a ry i ng d e m ar c at i on , i . e. f r om F i g . 7 . 4( b ) s e d im e n t d i am e te r s o f

t y pi c al l y 3 0 –

50 μm . T h us , i n c om b in at io n, t he s e p ro ce ss es w il l p ro du ce t ra p pi ngand ‘sorting’ o f s ed im en ts w it h a ss oc ia te d p at te rn s v ar yi ng o ve r s pr in g – neap

a n d ( c l o s e t o t h e t i d a l l i m i t ) d r o u g h t – f l o o d c y c l e s . H o w e v e r , t h i s s i m p l i f i e d a p p r o x -

i m a t i o n o f t h e s p r i n g – n e a p c y c l e o v e r l o o k s t h e r o l e o f a c c o m p a n y i n g c o n s t i t u e n t s ,

i n p a r t i c u l a r , t h e M Sf c o m p o n e n t , w h i c h c a n s i g n i f i c a n t l y e n h a n c e t h e ( a p p a r e n t ) z 0c o n s t i t u e n t i n t h i s s i m p l i f i e d e x a m i n a t i o n o f n e a p – spring variability.

T he a cu te s en si ti vi ty o f n et f lu xe s t o b ed r ou gh ne ss s ho wn i n F i g. 7 .6 a l s o

e m p h a s i s e s t h e f e e d b a c k l i n k s t o b e d s e d i m e n t d i s t r i b u t i o n s . T h e s e c a n o c c u r o v e r

c y c l e s o f e b b a n d f l o o d a n d s p r i n g a n d n e a p s t i d e s a s w e l l a s w i t h g r a d u a l l o n g -t e r m c h a n g e s a n d e p i s o d i c e x t r e m e e v e n t s .

T he s or ti ng m ec ha ni sm s a nd a xi al a cc um ul at io n o f t ra pp in g o ve r t he c yc le s

d es cr ib ed w il l d et er mi ne t he s iz e f ra ct io n a nd h en ce c on ce nt ra ti on s w it hi n t he

TM r eg io n. H ow ev er, t he l oc at io n i s l ik el y t o c oi nc id e w it h a m ax im um o f ς / D,

m o d u l a t e d b y t h e e v e n t u a l p r e d o m i n a n c e o f r i v e r f l o w o v e r t i d a l a c t i o n t o w a r d s t h e

t i d a l l i m i t .

7.6.3 Link to tidal energetics

H a v i n g s h o w n t h a t t h e b a l a n c e o f i m p o r t t o e x p o r t o f s e d i m e n t s d e p e n d s d i r e c t l y o n

t h e p h a s e l a g , θ , b et we en t id al e le v at io n a n d c ur re nt , w e n ot e t he d ir ec t c or rs po n -

d e n c e w i t h n e t t i d a l e n e r g y d i s s i p a t i o n w h i c h i s p r o p o r t i o n a l t o U * c o s θ . T h u s , w e

i de nt if y a r el at io ns hi p b et we en t he w ho le e st ua ry t id al e ne rg y b al an ce a nd t he

l o ca l is e d c r o ss - se c ti o n al s e di m en t f l ux b a la n ce . S u c h r e la t io n s hi p s h a v e b e en s u g-

g e s t e d p r e v i ou s l y ( B a g n o ld , 1963) , a n d m in im is in g n e t t id a l e ne rg y d is si pa ti on i su s e d a s a s t a b i l i t y c o n d i t i o n i n s o m e m o r p h o l o g i c a l m o d e l s .

7.7 Estuarine typologies

7.7.1 Bathymetry

Figures 7.9 and 7.10 (Prandle e t a l ., 2005) show observed lengths, L, a n d d e p t h s a t t h e

mouth, D, f ro m 5 0 U K e st ua ri es p lo tt ed a s f u n ct io ns o f ( Q, ς * ), i .e . t h e m ea n r iv er f lo wand the M2 t id al a mp li tu de a t t he m ou th . T he e st ua ri es a re r es tr ic te d t o t ho se c la ss if ie d a s

either bar-built or coastal plain (Davidson and Buck, 1997). Corresponding theoretical

values for L, (6.12), and D, (6.25) a r e s ho wn f or c om pa ri so n. O ve ra ll , t he o bs er ve d

v al ue s o f d ep th s a nd l en gt hs a re b ro ad ly c on si st en t w it h t he n ew d yn am ic al t he or ie s.




T he s ma ll er d ep th s i n b ar -b ui lt e st ua ri es a re c le ar ly d em on st ra te d. B y i de nt if yi nge st ua ri es w he re d ep th s d iv er ge s ig ni fi ca nt ly f ro m t he t he or y, e st im at es c an b e m ad e

o f t he m uc h l ar ge r f lo ws e xi st en t i n t he ir p os t- gl ac ia ti on f or ma ti on . R eg io na l d is cr e-

pancies can also be used for inferring coasts with scarcity or plentiful supplies of

sediment for infilling.

F i g . 7 . 9 . O b s e r v e d v e r s u s t h e o r e t i c a l e s t u a r i n e l e n g t h s , L( k m ) , a s f (Q, ς ) . C o n t o u r ss ho w t he or et ic al v al ue s f or L (6.12). O bs er ve d d at a f ro m e st ua ri es s ho wn i nF i g . 6 . 1 2 .

F i g. 7 . 10 . O b se r ve d v e rs u s t h eo r et i ca l e s tu a ri n e d e pt h s , D( m) , ( at t he m ou th ) a s f (Q, ς ) . T o p a x i s s h o w s t h e o r e t i c a l v a l u e s f o r D (6.25), f o r s i d e s l o p e t a n α = 0 .0 13 .

O b s e r v e d d a t a f r o m e s t u a r i e s s h o w n i n F i g . 6 . 1 2 .

7 . 7 E s tu a ri n e t y p ol o gi e s 197



A ss um in g n o i n- fi ll b ut d ee pe ni ng v ia m sl r is e, P ra nd le et al . (2005) e st im at ed t he

‘age’, N y ea rs , o f e st ua ri es , b y l ea st s qu ar es f it ti ng o f t he e xp re ss io n DT(n ) =

DO(n ) − N S (n) , w he re s ub sc ri pt T d en o te s t he or et ic al a nd s u bs cr ip t O o b se rv ed

depths. S ( n) i s t h e e s t i m a t e , f o r e a c h e s t u a r y , o f t h e a n n u a l r a t e o f r e l a t i v e m s l r i s e ,

e x t r a c t e d f o r t h e e s t u a r i e s s h o w n i n F i g . 6 . 1 2 f r o m S h e n n a n (1989 ) . T h e m s l t r e n d sv a ry f ro m −5 to 2. 0 m m p er y ea r; ov er 1 0 0 00 y ear s, t hi s i s e qui va le nt t o c ha ng es

of −5 to 2 0 m in es tu arin e de pt hs . T he r es ul ting v alue s w ere a s fo ll ow s: Ri as ,

N ~ 11 0 00 y ea rs ; f or c oa st al p la in e st ua ri es , N ~ 1 5 00 0 y ea rs w hi le f or b ar- bu il t

estuaries, N ~ 1 00 y ea rs . T he s e r e su lt s b ro ad ly c o nf ir m t h e m or ph ol og ic al d es cr ip -

t i o n s o f t h e i r d e v e l o p m e n t , (Section 6.5.2).

7.7.2 Sediment regimesT hi s b at hy me tr ic t yp ol og y i s e xt en de d i n F ig . 7 .11 (Prandle et al ., 2005) to

r ep re se nt at io ns o f t he ( de pt h a nd t id al ) m e an S PM c on ce nt ra ti on s (7.36) a nd t he

f al l v el oc it ie s c on si st en t w it h z er o n et s ed im en t f lu x (7.35). A d di t io n al o b s er v a-

t i o n a l d a t a f r o m t h r e e E u r o p e a n e s t u a r i e s a r e a l s o i n c l u d e d ( M a n n i n g , 2004).

C = 500 mg l

–1

C = 250 mg l –1

W s = 3 mm s –1

W s = 1 mm s –1

Fall vel(mm s –1

)

Sed conc(mg l –1

)

t f = 2 days t f = 10 days5.6, 1652

4.7, 7844.5, 657

2.4, 380

4.9, 2695.5, 136

3.3, 146

0,0

3.7, 64

3.8, 102Tamar

(UK) macro

Tamar(UK) meso

Medway (UK)

2.1, 32

1.6, 10

Dollard (NL/D)

3.3, 43

1.8, 18

2.2, 49

Schelde (B/NL)

Severn(UK)

Gironde(F)

Bar-BuiltCoastal Plain

River flow, Q (m3 s –1)

T i d a l e l e v a

t i o n a m p l i t u d e , ( m )

0.10

1

2

3

4

1 10 100

F i g. 7 . 11 . ‘Equilibrium’ v a lu es o f s ed im e nt c on ce nt ra ti o ns a nd f al l v el oc it i es .E st ua ri ne f lu sh in g t im es . O bs er ve d v er su s e qu il ib ri um t he or y f or : s ed im en t concentrations C (7.36) ( d as he d c on to ur s) a nd f al l v el oc it ie s W S (7.35) (fullc o n t ou r s ) . O b s e rv e d v a l u es f o r W S and C i n D o l l ar d , G i r o n de , Me d w ay, S c h el d e ,S e ve r n a n d Ta m ar ( M an n in g , 2004). • spring, ° n e ap t i de s. F lu sh i ng t i me s f ro m(7.37), d o tt e d c o nt o ur.




Note from (7.35) and (7.36) t h a t t h e c u r v e s f o r f a l l v e l o c i t y , W S, a n d c o n c e n t r a -

tion, C , a l i g n d i r e c t l y w i t h t h o s e f o r t i d a l c u r r e n t a m p l i t u d e , U * (7.8). T h i s t y p o l o g y

i ll us tr at es w h y m a ny e st ua ri es s h ow h ig h l ev el s o f f in e s us pe n de d s ed im en ts . T he

r e su l ts f o r W s s u gg es t a n ar ro w r an ge o f f al l v el oc it ie s, t yp ic al ly b et we en 1 a nd

3 m m s−1

.T he r es ul ts f ro m M an ni ng (2004 ) sh own in F ig . 7 .11 a re r ep re se nt at iv e o f

o bs erv ed s et tl em en t o f f in e s ed im en ts i n a w id e r an ge o f E ur op ea n e st ua ri es .

T h e s e s t u d i e s i n d i c a t e d t h a t s e t t l i n g w a s p r i m a r i l y v i a t h e f o r m a t i o n o f m i c r o - a n d

m ac ro -f lo cs , i nv ar ia bl y c lo s e t o t he r an ge s u gg es te d b y t he p re se nt t he or y. L ik e wi se ,

prevailing observed suspended sediment concentrations are in good agreement with

t h e o r et i c a l v a l u e s.

F i g u r e 7 . 1 1 a l s o s h o w s l o c i o f r e p r e s e n t a t i v e f l u s h i n g t i m e s , T F , (4.60):

T F ¼ 0:5 LI =2

U 0(7:37)

with LI from (7.11).

F o r r i v e r - b o r n e d i s s o l v e d o r s u s p e n d e d s e d i m e n t s , t h e i n d i c a t e d v a l u e s g e n e r a l l y

l ie b et we en 2 a nd 1 0 d ay s ( fo r r es id ua l c ur re nt U 0 = 1 c m s−1) . T he se v al ue s a re

c on s is te n t w it h t he r an g e i nd ic at ed f ro m o bs er va ti on s b y B al ls (1994 ) and Dyer

(1997) . F l us h in g t i me s g r ea t er t h a n t h e p r in c ip a l s e mi - di u rn a l t i da l p e ri o d p r ov i d e

v a lu a b le l o ng e r- t er m p e rs i st e nc e o f m a ri n e- d er i ve d n u tr i en t s, w h il e f l us h i ng t i me sl es s t ha n t he 1 5- da y s pr in g – n ea p c yc le y ie ld e ff ec ti ve f lu sh in g o f c on ta mi n an ts .

H en ce , t he re m ig ht b e s om e e co lo gi ca l a dv an ta ge t o t he b at hy me tr ic e nv el op e

d e f i n e d b y t h e s e t w o f l u s h i n g t i m e s .

A ss um in g t ha t f in e m ar in e s ed im en t e nt er s a n e st ua ry i n a lm os t c on ti nu ou s

s us pe ns io n ( li ke s al t) a nd n on e l ea ve s, a n e st im at e o f m in im um i n- fi ll r at es , I F,

c a n b e o b t a i n e d f r o m ( P r a n d l e , 2004a):

I F ¼ ρsT F0:69 C

; (7:38)

where ρs i s t he d en si ty o f t he d ep os it ed s ed im en t a nd C i s t he m ea n s us pe nd ed

c o n c e n tr a t i on f r o m (7.36).

P ra nd le (2003) sho wed that for U 0 ~ 0 . 0 1 m s−1, (7.38) i n di ca te s i n- fi ll t im es

r an gi ng f ro m 2 5 t o 5 00 0 y ea rs f or d ep th s f ro m 5 t o 3 0 m . N ot in g t ha t ‘capture rates’

a re t yp ic al ly o nl y a f ew p er c en t o f e nt ry r at es ( L an e a nd P ra nd le , 2006) , w e e xp ec t

t h e s e m i n i m u m t i m e s t o i n c r e a s e b y o n e o r t w o o r d e r s o f m a g n i t u d e .


S y n ch r on o us e s tu a ry s o lu t io n s f o r t i da l d y n am i cs a n d s a li n it y i n tr u si o n d e ri v ed i n

C h a p t e r 6 a r e e x t e n d e d t o i n c l u d e e r o s i o n , s u s p e n s i o n a n d d e p o s i t i o n o f s e d i m e n t s .

I n t e g r a t i n g t h e s e p r o c e s s e s i n t o a n ‘a n a l y t i c a l e m u l a t o r ’ y i e l d s e x p l i c i t e x p r e s s i o n s

7 . 8 S u mm a ry o f re s ul t s a n d g u id e li n es f o r a p p l i ca t io n 199



f o r c r o s s - s e c t i o n a l f l u x e s o f s u s p e n d e d s e d i m e n t s – e n a b l i n g c o n d i t i o n s o f z e r o n e t

f l u x ( b a t h y m e t r i c s t a b i l i t y ) t o b e i d e n t i f i e d . T h u s , i t i s s h o w n h o w t h e e x c h a n g e o f

s ed im en ts s wi tc he s f ro m e xp o rt t ow ar ds i mp o rt a s t he r at io o f t id al a mp li tu de t o

d e p t h i n c r e a s e s a n d a s s e d i m e n t s i z e d e c r e a s e s , p r o v i d i n g q u a n t i t a t i v e e x p l a n a t i o n s

f o r t r a p p i n g , s o r t i n g a n d TM i n e s t u a ri e s .T h e l e a d i n g q u e s t i o n s a r e :

What causes trapping, sorting and high concentrations of suspended

sediments?

How does the balance of ebb and flood sediment fluxes adjust to maintain

bathymetric stability?

S u sp en de d c on ce n tr at io ns o f f in e s ed im en ts i n t id al e st ua ri es t yp ic al ly r an g e f ro m

1 00 t o m o re th an 1 00 0 m g 1−1, w h er e as c o nc e nt r at i on s i n s h el f s e as a r e i n va r ia b ly

less than 10 mg l−1. M or eo ve r, o b se rv a ti o na l a nd n um er ic al m o de ll in g s tu di es ,

C ha pt er 8, i nd ic at e th at o nl y a s ma ll fr ac ti on o f t he n et t id al f lu x o f s ed im en ts

i s p er ma ne nt ly d ep os it ed . B y i nt ro du ci ng a ‘s y nc h r on o us e s tu a ry’ assumption,

C h a p t e r 6 s h o w e d h o w e s t u a r i n e b a t h y m e t r i e s a r e d e t e r m i n e d b y t h e t i d a l e l e v a t i o n

amplitude, ζ* , a nd r iv er f lo w, Q, a lo n gs id e t he b ed f ri ct io n c oe ff ic ie nt f ( a p ro xy

r e p r e s e n t a t i o n o f t h e a l l u v i u m ) . S i n c e t h i s t h e o r y t a k e s n o a c c o u n t o f t h e p r e v a i l i n g

s e d i m e n t r e g i m e , a p a r a d i g m r e v e r s a l e m e r g e d – s u g g e s t i n g t h a t s e d i m e n t r e g i m e s

a r e a c o n s e q u e n c e o f , r a t h e r t h a n a d e t e r m i n a n t f o r b a t h y m e t r y .

T o e x p l a i n a n d e x p l o r e t h i s n e w p a r a d i g m a n d a d d r e s s t h e a b o v e q u e s t i o n s , h e r e

w e e x t e n d t h e s e ‘synchronous’ s o l u t i o n s f o r t i d a l d y n a m i c s a n d s a l i n i t y i n t r u s i o n t o

i nc lu de s ed i me nt d y na mi cs . ( A s y nc hr on o us e st ua ry i s o ne w he re a xi al s ur fa ce

g ra di en ts a ss oc ia te d w it h p ha se c ha ng es g re at ly e xc ee d t ho se f ro m a mp li tu de

v a ri a ti o ns .) S e d im e nt e r os i on i s r e du c ed t o i t s s i mp l es t f o rm a t, i . e. p r o po r ti o na l t o

v el oc it y a t t he b ed s qu ar ed . P os tm a’s (1967) d es cr ip ti on o f d el ay ed s et tl em en t i n

s us p en si on i s i nt ro du ce d b y t he a do pt io n o f e x po ne nt ia l s et tl in g r at es , w it h a ss o-c i a t e d h a l f - li v e s , t 50, b as ed o n a l oc al is ed ‘v e r t i ca l d i s p e rs i o n - ad v e c t iv e s e t t l in g’

m o de l d e sc r ib e d i n C h ap t er 5 . T hi s s ed im en t m od u le i s c om bi n ed w it h t he d yn a-

m i c a l s o l u t i o n s f r o m C h a p t e r 6 t o f o r m a n ‘a n a l y ti c a l e m u l a to r ’, p r o v i d i n g e x p l i c i t

e x p r e s s i o n s f o r c o n c e n t r a t i o n s a n d c r o s s - s e c t i o n a l f l u x e s o f s e d i m e n t . T h e v a l i d i t y

o f t h es e a n al y ti c al e x p re s si o ns i s a s se s se d b y c o mp a ri s on w i th d e ta i le d n u me r ic a l

m o d e l s i m u l at i o n s . F i g ur e 7 . 1 s h ow s g o o d a g re e me n t f o r b o t h c o nc e nt r at i on s a n d

n e t f l u x e s o v e r a w i d e r a n g e o f f a l l v e l o c i t i e s , W s, a n d w a t e r d e p t h s , D.

7.8.1 Component contributions and sensitivity analyses

Net contributions to sediment fluxes from residual currents associated with salinity

i n t r u s i o n , r i v e r f l o w a n d t i d a l n o n - l i n e a r i t i e s a r e s h o w n i n F i g . 7 . 2. S e d i m e n t f l u x e s




w e r e e s t i m a t e d b y t h e p r o d u c t o f t h e s e r e s i d u a l c u r r e n t s , U s + U R + U M2 (as functions

o f d ep th ) w it h r el at e d c o ns ti t ue n ts o f s ed im e nt c o nc en tr at io n a nd t h ei r r es pe ct iv e

e x po n en ti al d ep th v ar ia ti o ns . T he r es ul ti ng f lu xe s i n di ca te c om p le x p a tt er ns o f n et

u p s tr e am o r d o wn s t re a m m o ve m en t a c u te l y s e ns i ti v e t o W s and f . F o r e s t u a r i e s w i t h

s i gn i fi c an t r a ti o s o f ζ* : D, i . e. s u b st a n ti a l v a r ia t io n s i n c r o ss - se c ti o na l a r e a b e t we e nh ig h a nd l ow t id al l ev el s, t he c on tr ib ut io n f ro m t he r el ate d g en er at io n o f h ig he r

h a rm o n ic a n d r e si d ua l c u rr e n t c o mp o ne n t s, (7.25), f a r e x c e e d s t h e o t h e r c o m p o n e n t s .

7.8.2 Conditions for zero net flux of sediments (i.e. bathymetric stability)

F ig u re 7 .3 i l lu st ra te s h o w s ep ar at e c om p on en ts f ro m t he se t id al n o n- li n ea ri ti es ,

i n v ol v in g c o s θ a nd s in θ (where θ i s t he p h as e d i ff er en ce b et we en ζ* a nd c ur re nt

U * ) , d e t e r m i n e t h e b a l a n c e b e t w e e n i m p o r t a n d e x p o r t . F r o m (7.32), c o m b i n a t i o n sof θ and t 50 c or re sp on di ng t o z er o n et fl ux o f s ed im en ts c an b e d et er mi ne d.

F ig u re 7 .4 shows ‘z er o f lu x’ v a lu es f or t 50, p a rt ic le d ia me te r d a n d f al l v el oc it y

W s a l o ng si de t he r el at ed s u sp en de d s ed i me nt c on ce nt ra ti on s C a s f un ct io ns o f

ζ* a n d D .

To i l lu s tr a te t h e a c ut e s e ns i ti v it y t o W s, F i g. 7 .5 s h ow s h ow t he se f lu x b al an ce s

v ar y fo r a d ou bl in g o f W s from 0.001 to 0.002 m s−1. T hi s s en si ti vi ty i nd i ca te s

h o w s e l e c t i v e s o r t i n g o c c u r s a n d e x p l a i n s h o w e s t u a r i e s t r a p m a t e r i a l i n a s i z e r a n g e

30 – 50 μm. It is sho wn that for zero net flu x, W s ~ f U * . T h is l at te r r el at io n sh ip

c oi nc id es w it h v al u es o f K Z/ W S D ( t he b a si c s c al i ng p a ra m et e r c h ar a ct e ri s in g s u s -

pended sediments), in the range 0.1 – 2 .0 , i .e . c lo se t o c on di ti on s c or re sp on d in g t o

m a x i m u m s u s p e n d e d s e d i m e n t c o n c e n t r a t i o n s .

F i g u r e 7 . 6 i l l u s t r a t e s t h e s e n s i t i v i t y t o t h e b e d f r i c t i o n c o e f f i c i e n t , f , o f t h e c o m p o -

n en ts o f n e t f lu x ; s u s pe nd ed c on ce nt ra ti on s; t id al c ur re nt a mp li t ud e a nd p ha se (ς *, θ );

a nd t he h al f- li fe i n s us pe ns io n, t 50. S in ce n et t id al e ne rg y d is si pa ti on i s d ir ec tl y

proportional to cos θ , t hi s s en si t iv it y o f θ to f h i gh li g ht s a f e ed b ac k l i nk b et w ee nt h e s ta bi li ty o f e st u ar in e- wi d e d yn am ic s a nd b ot h s us pe nd e d a nd d ep os i te d s ed im e nt s .

7.8.3 Neap to spring, coarse to fine, mouth to head variations

in net import versus export

F ig u re 7 .8 s um ma ri se s t he se e ar li er r es ul ts , i ll us tr at in g h o w t h e b a la n ce b et we en n et

i mp or t o r e xp or t v ar ie s f or d ep th s f ro m 4 t o 1 6 m ; f al l v el oc it ie s o f 0 .0 00 1, 0 .0 01

a n d 0 . 0 1 m s−1

a nd t id al a mp li tu de s f ro m 1 t o 4 m ( re pr es en ta ti ve o f n ea p t o s pr in gv a r i a t i o n s ) . T h i s i n d i c a t e s h o w i n p r o c e e d i n g u p s t r e a m f r o m d e e p t o s h a l l o w w a t e r ,

t he b al an ce b et we en i mp o rt o f f in e s ed im en t s a n d e xp or t o f c oa rs er o n es b ec om es

f i ne r, i . e. s e le c ti v e ‘sorting’ a n d t ra pp in g . L ik ew is e, m or e i mp or ts , e xt en di ng t o a

c o a r s e r f r a c t i o n , o c c u r o n s p r i n g t h a n o n n e a p t i d e s .

7 . 8 S u mm a ry o f re s ul t s a n d g u id e li n es f o r a p p l i ca t io n 201



O v e r a l l , t h e e m u l a t o r p r o v i d e s n e w i n s i g h t s i n t o t h e b a l a n c e b e t w e e n t i d e s , r i v e r

f l o w a n d b a t h y m e t r y a n d o n t h e r e l a t i o n s h i p o f t h e s e w i t h t h e p r e v a i l i n g s e d i m e n t

r eg im e. T h e c on d it io n s d e ri v ed f or m a in t ai ni ng s ta bl e b at hy me tr y e xt en d e ar li er

c on ce pt s o f f lo od - a n d e b b- do mi na te d r eg im es , w it h s ed im en t i mp o rt a nd e xp o rt

s h o w n t o v a r y a x i a l l y , w i t h s e d i m e n t t y p e , a n d o v e r t h e s p r i n g t o n e a p t i d a l c y c l e .T he se s ta bl e c on di ti on s a re s ho wn t o c or re sp on d b ot h w it h c on di ti on s f or m ax im um

S PM c on ce n tr at io ns a nd w it h o bs er va ti on s o f t he p re d om in an t s et tl in g r at es i n m an y

estuaries.

7.8.4 Typological frameworks

T h es e d e ve l o pm e nt s e n ab l e t h e b a th y me t ri c f r am e wo r ks , d e ve l op e d f o r d y n am i cs

a nd s al in e i nt ru si on i n s y nc hr on ou s e st ua ri es i n C ha pt er 6 , t o b e e xt en de d t o i nd ic at et he n at ur e o f t he s ed im en ta ry r eg im es c on si st en t w it h b at hy me tr ic s ta bi li ty.

C o mp a ri s o ns o f t h es e e x te n d ed t y po l og i ca l f r am e wo r k s a g a in s t o b se r va t io n a l d a ta

a r e s h o w n i n F i g s 7 . 9 – 7.11.

B y r e l a t i n g t h e d i f f e r e n c e b e t w e e n o b s e r v e d a n d t h e o r e t i c a l d e p t h s t o t h e y e a r s o f

( re la ti ve ) s ea l ev el c ha ng e s in ce a n e st ua ry w as f or me d, r ep re se nt at iv e a ge s f or R ia s,

C o as t al P l ai n a n d B a r- B ui l t e s tu a ri e s a r e c a lc u la t ed .

T h e c h a r a c t e r i s t i c s o f d y n a m i c a l a n d b a t h y m e t r i c e s t u a r i n e p a r a m e t e r s f o r ‘mixed’

e st ua rie s, l is te d a s ( a) t o ( i) i n S e c ti o n 6 . 6, c an b e f ur th er e xte nd ed t o i nc lu de t he

following sedimentary parameters:

ð jÞ Suspended concentration C / f U (7:39)

ðkÞ Equilibrium fall velocity W s / f U (7:40)

O v er a ll , t h e f i t f o u nd b e tw e en o b se r ve d a n d t h eo r et i ca l b a th y m et r ie s s u bs t an t ia t es

t he e ar li er p ar ad ig m r ev er sa l (C h ap t er 6 ) t ha t p re va il in g s ed i me nt d y na mi cs a re a

c o n s e q u e n c e o f s e l e c t i v e s o r t i n g a n d t r a p p i n g b y t h e e x i s t i n g d y n a m i c s a n d b a t h y -m e t r y . I t i s t h e n p o s t u l a t e d t h a t a s s o c i a t e d b a t h y m e t r i c e v o l u t i o n w i l l b e d e t e r m i n e d

by changes in tidal amplitude, river flow and bed roughness. The rate of change will

be modulated, or even superseded, by the supply of sediments and regional changes

i n r e l a t i v e s e a l e v e l .

References

A u br ey, D . G. , 1 9 86 . H y dr o dy n am i c c o nt r ol s o n s e di m e nt t r an s p or t i n w e ll - mi x e d b a ysa n d e s t u a r i e s . I n : V a n d e K r e e k e , J . ( e d . ) , P h y s i c s o f S h a l l o w E s t u a r i e s a n d B a y s .S p r i ng e r - Ve r l a g , N e w Yo r k , p p . 2 4 5 – 285.

B a g n o l d , P . A . , 1 9 6 3 . M e c h a n i c s o f m a r i n e s e d i m e n t a t i o n . I n : H i l l , M . N . ( e d . ) , T h e S e a : Ideas and Observations on Progress in the Study of the Seas, Vo l. 3 , T he E ar th B en ea th

t h e S e a : H i s t o r y. J o h n W i l e y , H o b o k e n , N J , p p . 5 0 7 – 582.




B a ll s , P. W. , 1 9 94 . N u tr i en t i n pu t s t o e s tu a ri e s f r om n i ne S c ot t is h E a st C o as t R i ve r s:I n f l u e n c e o f e s t u a r i n e p r o c e s s e s o n i n p u t s t o t h e N o r t h S e a . E s t u a r i n e , C o a s t a l a n d S h e l f S c i e n c e, 39 , 3 2 9 – 352.

D a v i d s o n , N . C . a n d B u c k , A . L . , 1 9 9 7 . A n I n v e n t o r y o f U K E s t u a r i e s . V o l . 1 . I n t r o d u c t i o na n d M e t h o d o l o g y. J o i n t N a t u r e C o n s e r v a t o r y C o mmu n i c a t i o n , P e t e r b o r o u g h , U K .

D r on k er s , J . a n d Va n d e K r ee k , J . , 1 9 86 . E x pe r im e nt a l d e te r mi n at i o n o f s a lt i n tr u si o nme c h a n i s ms i n t h e V o l k e r a k e s t u a r y , N e t h e r l a n d s . J o u r n a l S e a R e s e a r c h , 20 ( 1 ) , 1 – 19.D y e r , K . R . , 1 9 9 7 . E s t u a r ie s : A P h y s i ca l I n t ro d u c ti o n, 2 n d e d n . J o h n W i l e y , H o b o k e n , N J .D y e r , K . a n d E v a n s , E . M . , 1 9 8 9 . D y n a m i c s o f t u r b i d i t y m a x i m u m i n a h o m o g e n e o u s t i d a l

channel. J o u r na l o f C o as t al R e se a rc h, 5, 23 – 30.F e st a , J . F. a n d H a ns e n, D . V. C ., 1 9 7 8. T u rb i d it y ma x ima i n p a rt i al l y mi x ed e s t ua r ie s : a

two-dimensional numerical model. Estuarine, Coastal and Marine Science, 7, 3 4 7 – 359.F r i e d r i c h s , C . T , A r mb r u s t , B . D . , a n d d e S w a r t , H . E . , 1 9 9 8 . H y d r o d y n a mi c s a n d e q u i l i b r i u m

s e d i me n t d y n a mi c s o f s h a l l o w , f u n n e l - s h a p e d t i d a l e s t u a r i e s . I n : D r o n k e r s , J .a n d S c h a f f e r s , M. ( e d s ) , P h y s i c s o f E s t u a r i e s a n d C o a s t a l S e a s, A . A . B a l k e m a ,

B r o o k f i e l d , V t . p p . 3 1 5 –

328.G r a b e ma n n , I . a n d K r a u s e , G . , 1 9 8 9 . T r a n s p o r t p r o c e s s e s o f s u s p e n d e d ma t t e r d e r i v e d f r o m

t i m e s e r i e s i n a t i d a l e s t u a r y . J o u r n a l o f G e o p h y s i c a l R e s e a r c h , 94, 1 4 3 7 3 – 14380.H a m b l i n , P . F . , 1 9 8 9 . O b s e r v a t i o n s a n d m o d e l o f s e d i m e n t t r a n s p o r t n e a r t h e t u r b i d i t y

m a x i m u m o f t h e u p p e r S a i n t L a w r e n c e e s t u a r y . J o u r n a l o f G e o p h y s i c a l R e s e a r c h , 94,14419 – 14428.

J a y , D . A . a n d K u k u l k a , T . , 2 0 0 3 . R e v i s i n g t h e p a r a d i g m o f t i d a l a n a l y s i s : T h e u s e s o f n o n - s ta t i o n ar y d a t a . O c e a n D y n a m i c s , 53, 1 1 0 – 123.

J ay, D .A . a nd M us i ak , J .D ., 1 99 4. P ar ti cl e t ra pp i ng i n e st ua ri ne t ur bi di ty m ax i ma . Journal of Geophyical Research, 99, 2 0 4 4 6 – 20461.

L a n e , A . , 2 0 0 4 . M o r p h o l o g i c a l e v o l u t i o n i n t h e M e r s e y e s t u a r y , U K 1 9 0 6 – 1 9 9 7 : C a u s esa n d e f f ec t s . E s t u a r i n e C o a s t a l a n d S h e l f S c i e n c e , 59, 2 4 9 – 263.

L an e, A . a nd P ra nd l e, D ., 2 00 6. R an d om -w al k p ar ti c le m od el l in g f or e st im at i ng b at hy m et ri ce v o l u t i o n o f a n e s t u a r y . E s t u a r i n e , C o a s t a l a n d S h e l f S c i e n c e , 68 (1 – 2 ) , 1 7 5 – 187.

L a n g , G . , S c h u b e r t , R . , M a r k o f s k y , M . , F a n g e r , H - U . , G r a b e m a n n , I . , K r a s e m a n n , H . L . , Neumann, L.J.R., and Riethmuller, R., 1989. Data interpretation and numericalm o d e l i n g o f t h e m u d a n d s u s p e n d e d s e d i m e n t e x p e r i m e n t 1 9 8 5 . J o u r n a l o f Geophysical Research, 94, 1 4 3 8 1 – 14393.

M a n n i n g , A . J . , 2 0 0 4 . O b s e r v a t i o n s o f t h e p r o p e r t i e s o f f l o c c u l a t e d c o h e s i v e s e d i m e n t s i nt h re e w e st e rn E u ro p ea n e s tu a ri e s. I n S e di m e nt T ra n sp o rt i n E u ro p ea n E s tu a ri e s.

Journal of Coastal Research, SI 41, 70 – 81.M a r k o f s k y , M . , L a n g , G . , a n d S c h u b e r t , R . , 1 9 8 6 . S u s p e n d e d s e d i m e n t t r a n s p o r t i n r i v e r s

a n d e s t u a r i e s . I n : V a n d e K r e e k e , J . ( e d . ) P h y s i c s o f S h a l l o w E s t u a r i e s a n d B a y s .S p r i ng e r - Ve r l a g , N e w Yo r k , p p . 2 1 0 – 227.

P o s t m a , H . , 1 9 6 7 . S e d i m e n t t r a n s p o r t a n d s e d i m e n t a t i o n i n t h e e s t u a r i n e e n v i r o n m e n t . I n :L a u f f , G . H . ( e d . ) , Estuaries , P u b l i c a t i o n N o . 8 3 . A m e r i c a n A s s o c i a t i o n f o r t h eA d va n ce m en t o f S c ie n ce , Wa s hi n g to n , D C , p p . 1 5 8 – 179.

P r a n d l e , D . , 1 9 8 5 . O n s a l i n i t y r e g i m e s a n d t h e v e r t i c a l s t r u c t u r e o f r e s i d u a l f l o w s i n n a r r o wt i d a l e s t u a r i e s . E s t u ar i n e, C o as t al a n d S h el f S c ie n ce, 20 , 6 1 5 – 633.

P r a n d l e , D . , 1 9 9 7 a . T h e d y n a m i c s o f s u s p e n d e d s e d i m e n t s i n t i d a l w a t e r s . J o u r n a l o f C o a s t al R e s e ar c h, 25, 75 – 86.

P r a nd l e . D . , 1 9 9 7 b . T i d a l c h a ra c t e ri s t i c s o f s u s p e nd e d s e d i me n t c o n c en t r a t io n s . J o u r n a l o f Hydraulic Engineering , 123 ( 4 ) , 3 4 1 – 350.

P r an d l e, D . , 2 0 03 . R e la t i on s hi p s b e tw e en t i da l d y na m i cs a n d b a th y me t ry i n s t ro n g lyc o n v er g e n t e s t u a ri e s . J o ur n a l o f P h y s i ca l O c e a no g r a p h y, 33 ( 1 2 ) , 2 7 3 8 – 2750.

References 203



P ra nd l e, D ., 2 00 4a . H ow t id es a nd r iv er f lo ws d et er mi ne e st ua ri ne b at hy m et ri es . P ro g r es s i nOceanography, 61, 1 – 26.

P r a n d l e , D . , 2 0 0 4 b . S a l i n e i n t r u s i o n i n p a r t i a l l y m i x e d e s t u a r i e s . Estuarine, C o a s t a l a n d S h e l f S c i e nc e, 59 ( 3 ) , 3 8 5 – 397.

P r an d l e, D . , 2 0 04 c . S e di m en t t r ap p i ng , t u rb i di t y m a xi m a a n d b a th y m et r ic s t ab i li t y i n

ma c r o- t i d al e s t u ar i e s . J o u r n a l o f G e o p h y s i c a l R e s e a r c h , 109 ( C 0 8 0 01 ) , 1 3 p p .P r a n d l e , D . , L a n e , A . a n d M a n n i n g , A . J . , 2 0 0 5 . E s t u a r i e s a r e n o t s o u n i q u e . Geophysical Research Letters, 32 ( 2 3 ) , L 2 3 6 1 4 .

P r a n d l e , D . , L a n e , A . , a n d W o l f , J . , 2 0 0 1 . H o l d e r n e s s c o a s t a l e r o s i o n – O f f s h o r e mo v e me n t by tides and waves. In: Huntley, D.A., Leeks, G.J.J., and Walling, D.E. (ed), Land-O c e a n I n t e r a c t i o n , M e a s u r i n g a n d M o d e l l i n g F l u x e s f r o m R i v e r B a s i n s t o C o a s t a l Seas. I W A p u b l i s h i n g , L o n d o n , p p . 2 0 9 – 240.

S h e n n a n , I . , 1 9 8 9 . H o l o c e n e c r u s t a l m o v e m e n t s a n d s e a - l e v e l c h a n g e s i n G r e a t B r i t a i n . Journal of Quaternary Science, 4 ( 1 ) , 7 7 – 89.

U n c l e s , R . J . a n d S t e p h e n s , J . A . , 1 9 8 9 . D i s t r i b u t i o n s o f s u s p e n d e d s e d i m e n t a t h i g h w a t e r i n

a ma c r o ti d a l e s t u ar y. J o ur n a l o f G e o p h ys i c a l R e s e ar c h, 94, 1 4 3 9 5 –

14405.U n c l e s , R . J . , S t e p h e n s , J . A . , a n d S m i t h , R . E . , 2 0 0 2 . T h e d e p e n d e n c e o f e s t u a r i n e t u r b i d i t y

o n t i d a l i n t r u s i o n l e n g t h , t i d a l r a n g e a n d r e s i d e n c e t i m e s . C o n t i n en t a l S h e l f R e s e a rc h,22, 1 8 3 5 – 1856.




8

Strategies for sustainability

8.1 Introduction

Rising sea levels and enhanced storminess, resulting from Global Climate Change

(GCC), pose serious concerns about the viability of estuaries worldwide. This book

has shown how related impacts are manifested via evolving interactions between

tidal dynamics, salinity intrusion, sedimentation and morphology.

Drawing on a case study of the Mersey Estuary, Section 8.2 assesses the cap-

abilities of a fine-resolution 3D model against the perspective of a 100-year record

of changes in tides, sediments and estuarine bathymetries. Both historical and recent

observations are used alongside the Theoretical Frameworks, described in preced-

ing chapters, to interpret ensemble simulations of parameter sensitivities and so

reduce levels of uncertainty surrounding future forecasts.

There is an urgent need to develop models that can indicate the possible nature,

extent and rate of morphological changes. With accurate information on bathymetry

and surficial sediment distribution, ‘Bottom-Up’ numerical models (i.e. solving

momentum and continuity equations as in the Mersey study) can accurately repro-duce water levels and currents. However, simulation of sediment regimes involves

net fluxes generally determined by non-linear coupling between flow and sediment

suspension, i.e. processes over much wider spectral scales. Future forecasts must

encapsulate a broad spectrum of changing conditions, consequently the range of

possible future morphologies widens sharply. While ‘Bottom-Up’ numerical models

can be used for sensitivity analyses to identify areas susceptible to erosion or deposi-

tion, longer-term extrapolations become increasingly chaotic. As an alternative to

‘Bottom-Up

’ models for forecasting bathymetric evolution, geomorphologists use

‘Top-Down’, ‘rule-based’ models (Pethick, 1984). A number of ‘stability criteria’ are

used, based generally on fitting observed bathymetry to simplified dynamical criteria.

Since this fitting often extends over millennia, these ‘Top-Down’ approaches provide

valuable long-term perspectives.

205



In Section 8.3, the explicit formulae and Theoretical Frameworks are used to make

future predictions regarding likely impacts from GCC. Quantitative estimates are

made of possible changes in estuarine bathymetries up to 2100. Attendant conse-

quences for tides, storms, salinity intrusion and sediment regimes are discussed.

Section 8.4 indicates strategies for long-term management of estuaries, withsections on modelling, observations, monitoring and forecasting. Appendix 8A

emphasises the potential of operational modelling and monitoring in global-scale

efforts to address GCC.

8.2 Model study of the Mersey Estuary

This case study illustrates how the formulation and validation of a detailed numerical

model can utilise observational data ranging from process ‘measurements’, extended‘observations’ to permanent ‘monitoring’. Likewise, it shows how the Theoretical

Frameworks, described in earlier chapters, are used to interpret ensemble sensitivity

simulations.

8.2.1 Tidal dynamics, sediment regime and bathymetric

evolution of the Mersey

Tidal ranges in the Mersey vary from 4 to 10 m over the extremes from neap to spring.

The estuary has been widely studied because of its vital role in shipping. The

‘ Narrows’ at the mouth of the 45-km long estuary is approximately 1.5-km wide

with a mean depth (below chart datum) of 15 m (Fig. 8.1; Lane and Prandle, 2006).

Tidal currents through this section can exceed 2 m s−1. Further upstream in the inner

estuary basin, the width can be as much as 5 km, and extensive areas are exposed at

low water. Freshwater flow into the estuary, Q, varies from 25 to 300 m3 s−1 with a

mean‘flow ratio

’(Q × 12.42 h/volume between high and low water) of approximately

0.01. Flow ratios of less than 0.1 usually indicate well-mixed conditions (4.66), though

in certain sections during part of the tidal cycle, the Mersey is only partially mixed.

Suspended sediments and net deposition

Figure 8.2 ( Lane and Prandle, 2006) shows observed suspended sediment time series

from locations in the Narrows recorded in 1986 and 1992; Table 8.1 summarises these

results. The 1986 observations included five simultaneous moorings across the

Narrows, providing estimates of net spring and neap tidal fluxes of sediments.Prandle et al . (1990) analysed four sets of observations of SPM indicating tidally

averaged cross-sectional mean concentrations varying as a function of tidal ampli-

tudes, ς *, as follows: 32 mg l−1 for ς *= 2.6 m, 100 mg l−1 for ς *=3.1m,200mgl−1 for

ς *=3.6m and 213mgl−1 for ς * = 4.0 m. These values correspond to a tidal flux

206 Strategies for sustainability



(on ebb or flood) of 40 000 t on a mean tide, reducing to as little as 2500 t at neap and

increasing by up to 200 000 t on springs – in reasonable agreement with earlier

estimates from a hydraulic model study by Price and Kendrick (1963).

Hutchinson and Prandle (1994) used contaminant sequences in sediment cores

(analogous to tree-ringing) to estimate net accretion rates in the adjacent and the

similarly sized Dee Estuary. These amounted to 0.3 Mt a

−1

between 1970 and 1990and 0.6 Mt a−1 between 1950 and 1970. Using in situ bottle samples, Hill et al .

(2003) derived settling velocities, W S, of 0.0035 m s−1 for spring tidal conditions

and 0.008 m s−1 for neaps. Noting that particle diameter d (μm) 1000 W S½ (m s−1),

these correspond to d = 59 and 89 μm, respectively.

Bathymetry

Data were available from surveys carried out by the Mersey Docks and Harbour

Company in 1906, 1936, 1956, 1977 and 1997. Differences in net volume within the Narrows are of the order of a few percent from one data set to the next. The largest

changes appear in the inter-tidal regions of the inner estuary basin, particularly from

Hale and Stanlow to Runcorn where the low water channel positions change readily,

and volume differences between successive surveys exceed 10%. The overall

Mersey

Narrows

Inner estuary

basin

Upperestuary

50

Kilometres

GladstoneLock

Rock lighthouse(New Brighton)

PRINCES PIER

RockFerry

Garston

Eastham

StanlowInce

Hale

Fiddler’sFerry

Widnes

Runcorn

Warrington

5

2

0

2

2

0

0 0

00

– 2

5

5

2

0

2

– 2

– 2

0

- 2

– 2

–2 – 5

– 5

– 5

–5

2

2

1 0

1 0

1 0

1 0

2 0

2 0

2 0

1 5

1 5

Transect line

Alfred Lock

Wallasey

Birkenhead

R i v e r D

e e

R i v e r

R i b b l e

R i v e r

Mer sey

Liverpool

W i r r a l

Warrington

Runcorn

Birkenhead

England

Wales

Liverpool Bay

Kilometres

1003°W

53.5°N

3°W

N

LiverpoolGreatSankey

Estuary model boundary

P2 P3

Fig. 8.1. Liverpool Bay and the Mersey Estuary location map. The 1992 transect line corresponds to positions P2, P3; tide gauges are marked with dots. Depths(1997 bathymetry) are in metres below Ordnance Datum Newlyn (ODN). Chart datum is approximately the lowest astronomical tide level and is 4.93 m below

ODN.




pattern is a decrease in estuary volume of about 60 mm3 or 8% between 1906

and 1977 despite sea level rise averaging 1.23 mm per year during the past

century (Woodworth et al ., 1999). After this period, there is a small increase of

10 mm3.

8.2.2 Modelling approach

Here we illustrate the capabilities and limitations of a 3D Eulerian hydrody-

namic model coupled with a Lagrangian sediment module (Lane and Prandle,

(a)

(b)

(c)

(d)

S

e d i m e n t c o n c . ( m g l – 1 )

S e d i m e n t c o n c . ( m g l – 1 )



2000

1500

1000

500

0

3

April1986

April1986

July

1992

July1992

4 5 6 7 8 9 10 11 12 13 14 15 16

3 4 5 6 7 8 9 10 11 12 13 14 15 16

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

2000

1500

1000

500

0

1200

1000

800

600

400

200

0

1200

1000

800

600

400

200

0

Fig. 8.2. Observed sediment concentrations in the Narrows. (a) surface, (b) mid-depthin 1986, (c) 280 m from Wirral shore and (d) 290 m from Liverpool shore in 1992.




Table 8.1 Suspended sediment concentrations, net deposition and net tidal fluxes

Suspended sediment concentration(mgl−1) at position (2)

Net sediment deposited Net tid

Mean Max Min (103 t a−1) Spring

Sediment settling velocity W s (ms−1)0.005 25 67 0 1800 46.50.0005 213 442 0 4900 306.0

Observed in 1986 300 1100 0 200.0 b

500 1500 0 1992 (1) 53 115d 0

(2) 250 1500d 02300a,c

a Lane (2004). bPrandle et al . (1990).cThomas et al . (2002).d90% of sediments have concentrations less than this value.

Notes: Observed Values: Transect line P2−P3 (Fig. 8.1): (1) 280 m from Wirral; (2) 290 m froSource: Lane and Prandle, 2006.



2006) to quantify impacts on the estuarine sediment regime and indicate the

rate and nature of bathymetric evolution. Particular emphasis is on quantifying

the variations in sediment concentrations and fluxes in sensitivity tests of bed

roughness, eddy viscosity, sediment supply (particle sizes 10 – 100 μm), salinity

intrusion and 2D versus 3D formulations of the hydrodynamic model. Themodel was not intended to reproduce bed-load transport associated with coarser

sediments.

Recognising the limited capabilities to monitor the often extremely heteroge-

neous SPM, a wide range of observational data were used for assessing model

performance. These include suspended concentrations (axial profiles of mean and

‘90th percentile’), tidal and residual fluxes at cross sections, estuary-wide net

suspension and deposition on spring and neap tides, surficial sediment distributions

and sequences of bathymetric evolution.The Eulerian hydrodynamic model provides velocities, elevations and diffusivity

coefficients for the Lagrangian ‘random-walk ’ particle model in which up to a

million particles represent the sediment movements. It includes a wetting-and-drying

scheme to account for the extensive inter-tidal areas. Forcing involved specifying

tidal elevation constituents at the seaward limit in the Mersey Narrows, and river

flow at the head. The model uses a 120-m rectangular grid horizontally and a

10-level sigma-coordinate scheme in the vertical.

Calibration of the model (Lane, 2004) involved simulating effects of ‘ perturba-

tions’, i.e. varying the msl, bed friction coefficients, vertical eddy viscosity and the

river flow. The optimum combination to minimise differences between observed

and modelled tidal elevation constituents was then determined. The model was most

sensitive to changes in bathymetries and bed friction coefficients, particularly in the

inner basin. River flow only has an appreciable effect for discharges significantly

higher than those usually encountered.

8.2.3 Lagrangian, random-walk particle module

for non-cohesive sediment

Random-walk particle models replicate solutions of the Eulerian advection – diffusion

equation by calculating, for successive time steps Δt , the height above the bed, Z ,

and horizontal location of each particle following

(1) a vertical advective movement – W S Δt (downwards),

(2) a diffusive displacement l (up or down),

(3) horizontal advection.

The displacement length l = √ (2 K zΔt ) (Fischer et al ., 1979), with the vertical

eddy viscosity coefficient K z approximated by f U * D, where f is the bed friction




coefficient, U * the tidal current amplitude and D depth. Contacts with the surface

and bed during this diffusion step are reflected elastically. Deposition occurs

when the particle reaches the bed during a discrete advective settlement step – W SΔt . For any grid square containing deposited particles, new particles are released

into suspension by time-integration of the erosion potential.A simple algorithm for the erosion source was adopted:

ER ¼ γρ fU P; (8:1)

where ρ is water density and a value of P = 2 was assumed. Having specified P , all

subsequent calculations of concentration, flux and sedimentation rates are linearly

proportional to the coefficient γ. A value of γ =0.0001ms−1 was found to produce

suspended sediment concentrations comparable with those in Fig. 8.2. The corresponding values of tidal and residual cross-sectional fluxes were also in reasonable

agreement with observed values shown in Table 8.1.

8.2.4 Simulations for fall velocities, W S = 0.005 and 0.0005 m s−1

Figure 8.3 (Lane and Prandle, 2006) shows cross-sectional mean suspended sediment

concentrations, at successive locations landwards from the mouth, over two spring –

neap cycles commencing from the initial introduction of sediments. The examples

chosen are for sediment fall velocities, W S of 0.005ms−1 (coarse sediment d = 70μm,

black lines) and 0.0005 m s−1 (finer sediment, d = 22μm, grey lines), respectively.

Starting with no sediment in the estuary, all particles are introduced at the seaward

boundary of the model using the erosion formula (8.1). An unlimited supply is

assumed together with zero axial concentration gradient (∂C / ∂ X = 0) for inflow

conditions. To reflect the effect of changing distributions of surficial sediments on

the bed friction coefficient, this was specified as f = 0.0158 W S¼

.For W S =0.0005ms−1 (grey lines), the suspended sediment time series change

from predominantly semi-diurnal (linked to advection) at the mouth to quarter-

diurnal (linked to localised resuspension) further upstream. Even close to the mouth,

a significant quarter-diurnal component is generated at spring tides. Close to the

mouth, peak concentrations occur some three tidal cycles after maximum spring

tides, while further upstream this lag extends up to seven cycles.

For the coarser sediment, W S =0.005ms−1 (black lines), Fig. 8.3 shows much

reduced concentrations largely confined to the seaward region, although the slower ‘adjustment ’ rate suggests that a longer simulation is required to introduce the

coarser sediments further upstream. The time series is predominantly quarter-diurnal

and peak concentrations coincide with peak tides; the sediments have a much shorter

half-life in suspension as described in Section 5.5.




Figure 8.4(a) (Lane and Prandle, 2006) shows corresponding time series of

cumulative inflow and outflow of sediments across the mouth of the estuary model.

Differences between inflow and outflow, in Fig. 8.4(b), indicate net suspension (highfrequency) and net deposition (low frequency). For W S = 0.0005 m s−1, the mean tidal

exchange of sediments is around 110 000 t per tide, of which approximately 6% is

retained amounting to 7000 t per tide. For W S =0.005ms−1, the mean exchange is

22 000 tonnes of which approximately 12% is retained or about 3000 t per tide.

Neap

1000

500

01000

500

01000

500

01000

500

0

1000500

0500

0500

0500

500

500

500

500

0

0

0

0

0

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58

Spring Neap

1

2

3

4

5

6

7

8

9

10

11

12

Semi-diurnal tidal cycles

Spring NeapNeap

S u s p e n d e d s e d i m e n t c o n c e n t r a t i o n s ( m g I – 1 )

Fig. 8.3. Suspended sediment concentrations at 12 positions along the Mersey1 the mouth, 12 the head; grey lines settling velocity W S = 0.0005 m s−1; black linesW S = 0.005 m s−1.




8.2.5 Sensitivity to sediment size

Full details of the sensitivity tests are shown by Lane and Prandle (2006), these are

summarised in Tables 8.2 and 8.3. For a more extensive quantitative evaluation of

the model, single neap – spring tidal cycle simulations were used. Results are sum-

marised in Table 8.2 for particle diameters d from 10 to 100μm.The model reveals that mean suspended sediment concentrations vary approxi-

mately with d −2. Equation (7.29a) indicates variability ranging from d 0 to d −4 for

finer to coarser sediments. The extent of landward intrusion increases progressively

for finer sediments. A minimum capture rate of 2.8% occurs for d = 30 μm with

10(a)

(b)1.0

N e t d e p o s i t e d s e d i m e n t ( m i l l i o n t o n n e s )

0.8

0.6

0.4

0.2

0

8

6

4

C u m u l a t i v e s e d i m e n t

i n f l o w / o u t f l o w

( m i l l i o n t o n n e s )

2

00 2 4 6 8 10 12 14 16 18 20 22 24 26 28


SpringNeap Neap

30323436

w s = 0.0005 m s –1

w s = 0.0005 m s –1

w s = 0.005 m s –1

w s = 0.005 m s –1

Inflow

Inflow

Outflow

outflow

Running-averageover tidal cycle

3840424446485052545658

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28


SpringNeap Neap

303234363840424446485052545658

Fig. 8.4. Sediment fluxes at the mouth and net estuarine suspension and erosion.(a) Cumulative inflow and outflow at the mouth of the Mersey. (b) Net suspension(high frequency) and deposition (low frequency).




Table 8.2 Sensitivity of modelled sediments to particle diameters d from 10 to 100 μm for (a

concentrations; (b) mean concentrations; (c) estuarine-wide mean suspended and net depo

tides

(a)d (μm) 90th percentile suspended sediment concentrations (mg 1−1) at 2 km intervals upstream from

10 4186 3896 3804 3366 3274 3167 2471 1525 1052 714 560 420 369 344 325 297 258 240 218 183 140 84 52 30 169 148 133 101 89 67 47 29 12 6 3 40 140 118 103 75 64 45 26 12 5 3 2 50 108 95 83 62 53 35 16 7 4 3 2 60 83 72 65 47 39 22 8 3 1 – –

70 63 56 50 38 29 16 5 1 – – –

80 50 45 40 30 22 11 2 – – – –

90 39 35 29 22 16 7 – – – – –

100 31 29 25 17 12 4 – – – – –

(b)d (μm) Mean suspended sediment concentrations (mg 1−1) at 2 km intervals upstream from the mo

10 1645 1465 1423 1183 1118 1049 875 669 562 418 252 20 176 157 151 128 120 108 93 75 60 43 27

30 68 56 47 34 27 20 14 8 4 2 1 40 51 41 33 22 17 11 6 3 1 1 –

50 41 33 27 19 14 8 4 2 1 – –

60 32 26 20 14 10 5 2 – – – –

70 23 19 15 10 7 3 – – – – –

80 18 14 11 7 5 2 – – – – –

90 13 10 8 5 3 1 – – – – –

100 10 8 6 3 2 – – – – – –



(c)Suspended Deposited Exchange

d (μm) Neap Spring Neap Spring Neap Spring

10 156.21 1531.21 28.40 420.84 57.53 2508.81 20 2.15 206.55 −0.99 61.07 9.80 490.07 30 3.97 34.77 −0.67 4.93 5.86 106.96 40 2.31 22.62 −0.54 5.06 5.51 87.48 50 1.77 18.07 −0.05 6.12 4.14 71.20 60 1.13 12.97 −0.04 7.15 3.21 58.85 70 0.79 9.28 0.56 6.49 2.93 43.83 80 0.58 7.36 0.23 7.64 2.02 36.91 90 0.41 5.48 0.14 7.54 1.70 30.55

100 0.26 4.48 0.10 5.62 1.15 25.09

Notes: Units: 103 tonnes.Source: Lane and Prandle, 2006.



a corresponding deposition rate of 1 Mt per year. While capture rates increase

progressively with increasing sediment size (above d = 30 μm), corresponding

decreases in concentration yield a maximum deposition at 60 μm of 2 Mt per year.

This maximum is close to the preponderance of sediments with W S =0.003ms−1

Table 8.3 Sensitivity of modelled sediments, R1 – R8, for WS =0.0005ms −1 ( d = 22 μ )

(a)

RunMean suspended sediment concentrations (mg 1−1) at 2 km intervals upstream fromthe mouth

(1) 127 109 98 77 67 58 47 34 25 16 10 6 4 2 2 1 1(2) 333 304 301 263 253 234 205 171 141 106 71 50 49 35 29 23 25(3) 132 117 112 95 89 79 69 58 48 36 26 17 17 8 7 5 5(4) 127 112 104 84 76 69 56 43 32 21 12 7 5 2 2 1 1(5) 48 42 40 33 31 28 22 17 14 10 7 5 4 1 1 1 1(6) 196 171 155 127 116 105 86 60 42 24 12 7 5 2 1 1 1(7) 73 65 60 49 44 38 30 23 16 10 5 2 2 – 1 – –

(8) 125 109 102 84 76 67 55 44 34 23 14 8 7 3 2 2 1

(b)

Run90th percentile suspended sediment concentrations (mg 1−1) at 2 km intervalsupstream from the mouth

(1) 290 250 229 182 166 144 118 86 63 40 24 16 16 10 9 7 9(2) 807 783 799 742 703 615 534 420 336 206 147 111 98 72 63 55 45(3) 279 259 257 218 208 186 171 148 114 81 63 48 52 31 24 22 18(4) 277 249 237 188 177 165 141 115 77 44 24 17 15 8 6 4 3(5) 90 84 82 70 67 62 53 44 37 28 18 16 16 8 6 7 5(6) 388 347 328 265 254 240 210 151 90 55 33 22 17 8 7 5 1

(7) 149 137 133 107 99 93 81 59 36 18 11 8 7 4 4 1 1(8) 278 250 245 199 183 163 148 121 82 46 29 21 19 10 7 8 5

(c)Suspended Deposited Exchange

Deposited per year

% deposit exchange

AveragesuspendedRun Neap Spring Neap Spring Neap Spring

(1) 8.58 70.21 −0.88 10.73 9.14 182.69 2200 3.8 31.03(2) 21.91 476.73 −1.20 118.22 11.49 1020.43 9300 5.9 167.55(3) 8.60 124.63 −1.17 30.70 8.41 289.48 3200 5.1 47.21

(4) 7.78 121.18 −0.99 26.61 7.72 285.81 2700 4.3 45.21(5) 4.08 23.55 −0.58 2.83 4.44 60.80 800 3.5 11.84(6) 19.11 134.84 −1.51 23.94 17.92 310.08 3000 3.6 62.73(7) 6.62 61.39 −0.48 6.31 5.28 134.45 800 2.6 26.10(8) 8.32 116.75 −1.59 24.30 6.71 271.36 2500 4.2 44.54

Note: Units: 103 tonnesParameters as in Table 8.2.Diameter 22 μm, Ws = 0.0005 m s−1.Source: Lane and Prandle, 2006.




(d = 54 μm) found by Hill et al . (2003). In Section 7.5, it was shown that the size of

suspended sediments corresponding to ‘equilibrium’ conditions of zero net deposi-

tion or erosion is in the range 20 – 50 µm. Throughout the range of d = 30 – 100 μm,

net sedimentation remains surprisingly constant at between 1 and 2 Mt per year.

This sedimentation rate is in close agreement with observational evidence(Table 8.1).

8.2.6 Sensitivity to model parameters

The model’s responses to the following parameters were quantified: vertical structure

of currents, eddy diffusivity, salinity, the bed friction coefficient and sediment supply.

Table 8.3 shows, for W S =0.0005ms−1 (d = 22 μm), the sensitivity to Run

numbers:

(R1) – No vertical current shear, i.e. a 2D hydrodynamic model.

(R2) – Depth-varying eddy diffusivity , K z( z ) = K z (−3 z 2 + 2 z + 1), i.e. depth-mean value K zat the bed, 1.33 K z at z = Z / D = 0.33 and 0 at the surface

(R3) – A time varying value of K z(t ), with a quarter-diurnal variation of amplitude 0.25 K z

producing a peak value 1 h after peak currents.

(R4) – Mean salinity-driven residual current profile (4.15) U z = g S x D3/ K z (−0.1667 z 3 +

0.2687 z 2− 0.0373 z − 0.0293), where the salinity gradient S x was specified over a

40 km axial length.

(R5) – Bed friction coefficient halved, f = 0.5 × 0.0158W s¼.

(R6) – Bed friction coefficient doubled, f = 2.0 × 0.0158W s¼.

(R7) – Erosion rate at the mouth 0.5γ, i.e. halving the rate of supply of marine sediments.

(R8) – Baseline simulation.

While the calculated values of sediment concentration and net fluxes varied widely

and irregularly, the net deposition remained much more constant. The acute and

complex sensitivity to bed roughness and related levels of eddy diffusivity andviscosity is evident from Table 8.3. The acute sensitivity to bed roughness

and sediment supply leads to concern that migration of new flora and fauna might

lead to ‘modal shifts’ with potentially dramatic consequences.

To comprehend these sensitivities, in shallow water we can approximate, from

Prandle (2004), the following dependencies on the friction factor f :

Tidal velocity amplitude U * f −1/2

Sediment concentration C

f 1/2

Tidal sediment flux U *C f 0

Residual sediment flux <UC >U *C cos θ f 1/2,

where θ is the phase lag of tidal elevation relative to currents and residual sediment

flux corresponds to net upstream deposition. These theoretical results are consistent




with the increases in concentration and residual fluxes for larger values of f shown

by the model for both sediment types.

8.2.7 Summary

A century of bathymetric surveys indicate a net loss of estuarine volume of about

0.1%, or 1 million cubic metres, per year. Similar percentage losses are found in

many of the large estuaries of NW Europe. Sea level rise of 1.2 mm a−1 represents

only a 0.02% annual increase. This relative stability persists in a highly dynamic

regime with suspended sediment concentrations exceeding 2000 mg 1−1 and spring

tide fluxes of order 200 000 t. Detailed analyses of the bathymetric sequences

indicate that most significant changes occur in the upper estuary and in inter-tidal

zones. Long-period, up to 63 years, tidal elevation records in the lower estuary showalmost no changes to the predominant M2 and S2 constituents.

A 3D Eulerian fine-resolution hydrodynamic model coupled with a Lagrangian,

random-walk sediment module was used to show how the dominant fluxes involve

fine (silt) sediments on spring tides. The closest agreement between observed and

model estimates of net imports of sediments occurs for sediments of diameter of

approximately 50 μm – both dredging records and in situ observations indicate

that sediments of this kind predominate. The model showed little influence of river

flow, saline intrusion or channel deepening on the sediment regime. Conversely, the

net fluxes were sensitive to both the bed friction coefficient, f , and the phase

difference, θ , between elevation tidal velocity and elevation.

Upper-bound rates of infill of up to 10 Mt a−1 are indicated by the model,

comparable with annual dredging rates of up to 5 Mt. The limited mobility of

coarse sediments was contrasted with the near-continuously suspended nature of

the finest clay. While the model indicated that sedimentation rates might increase

significantly for much finer particles, this is likely to be restricted by the limitedavailability of such material in the adjacent coastal zone. The present approach

can be readily extended to study changes in biological mediation of bottom

sediments, impacts of waves, consolidation and the interactions between mixed

sediments.

8.3 Impacts of GCC

By 2050, GCC could significantly change mean sea levels, storminess, river flowsand, hence, sediment supply in estuaries (IPCC, 2001). The tidal and surge response

within any estuary will be further modified by accompanying natural morphological

(post-Holocene) adjustments alongside impacts from past and present ‘interventions’.

Generally, relatively small and gradual morphological adjustments are expected.




As an illustration, deposition per tide of a depth-mean concentration of 100 mg l−1

in 10 m water depth amounts to about 0.35 mm, or 25 cm per year. In reality, as

shown in the Mersey, ‘capture rates’ (upstream deposition as a proportion of the

net tidal inflow of suspended sediments) are typically only a few percent. Thus,

simulations need to extend over decades to embrace responses over the full range of forcing cycles involved. However, as noted previously, longer-term extrapolations

with ‘Bottom-Up’ models become increasingly chaotic, and hence, here we exam-

ine impacts from GCC by using the Theoretical Frameworks developed in earlier

chapters.

8.3.1 Impacts on tide and surge heights

The response Framework, Fig. 2.5, based on analytical expressions derived byPrandle and Rahman (1980), provides immediate indications of likely changes in

the estuarine response of tides and surges. Figure 2.5 shows that amplification of

tides (and surges) between the first ‘node’ and the head of the estuary can be up to a

factor of 2.5. Concern focuses on conditions in estuaries where, for the excitation

‘ period’, P , the bathymetric dimensions (length, depth and shape) result in the

estuarine mouth coinciding with this node with consequent resonant amplification.

This occurs when, (2.26),

y ¼ 0:75 ν þ 1:25; (8:2)

where y ¼ 4π L

Pð2mÞð gDÞ1=2 and v ¼

nþ 1

2m:

L and D are the estuarine length and depth (at the mouth), and m is the power of axial

depth variation and n of breadth variation.

The estuarine length, LR , for maximum amplification is then

LR ¼ ð2mÞð0:75ν þ 1:25Þ g1=2D1=2 P

ð4π Þ: (8:3)

The Framework extends from 0 < ν < 5 encompassing the following range of

shapes and associated resonant lengths:

ðaÞ canal m¼n¼ 0; ν ¼ 0:5 Lc ¼ 0:25ð gDÞ1=2PðbÞ embayment m¼n¼ 0:5; ν ¼ 1 L¼ 3=3:25Lc

ðcÞ linear m¼n¼ 1; ν ¼ 2 L¼ 2:75=3:25Lc

ðdÞ funnel m¼n¼ 1:5; ν ¼ 5 L¼ 2:5=3:25Lc:

(8:4)

Thus, as shown in Section 2.4.1, the range of funnelling (b) to (d) results in a

relatively small reduction in the ‘quarter-wavelength’ resonant length applicable for

a prismatic channel. Figure 8.5 indicates corresponding resonant periods for a




semi-diurnal tidal period for a ‘linear ’ estuary with m = n = 1. From (8.4), these

results are broadly applicable for a wide range of estuarine shapes. The figure

shows that even for a depth at the mouth of 4 m, resonance at semi-diurnal

frequencies will only occur for LR > 60 km, while for D = 1 6 m , LR >100km.

From (8.3), for a synchronous estuary (m = n = 0.8, ν = 1.5), LR = 37 D1/2 (km) or

94 km for the mean observed depth, D = 6.5 m. This emphasises that onlythe longest of UK estuaries, such as the Bristol Channel, are likely to exhibit

significant tidal amplification.

Using the expression (6.12) for the length of a synchronous estuary

L ¼120D5=4

ð f & Þ1=2 (8:5)

with the bed friction coefficient f = 0.0025 and ς * tidal elevation amplitude. By

inserting (8.3) into (8.5), we derive the following expressions for resonant values of

LR and DR in terms of tidal amplitude, ς *:

LRðkmÞ 180 & 1=3 (8:6)

and

DRðmÞ 31& 2=3: (8:7)

Hence, for ς * = 1 m, LR = 180 km and DR = 31 m while for ς * = 4 m, LR = 285 kmand DR = 78 m.

Thus, we only anticipate resonance at the semi-diurnal frequency in deep systems

such as the Bristol Channel where the estuarine ‘resonance’ extends to the adjacent

shelf sea. Hence, we do not expect dramatic changes in tidal or surge responses in

12.5

25

50

100

200

4 8D (m)

L I ( k m )

16 32 64

24

12

6

3

1.5

Fig. 8.5. Resonant periods (h) as a function of depth (at the mouth) and length.Results are for linear axial variations in depth and breadth, but from (8.4) are more

widely applicable.




estuaries for anticipated changes in sea level of up to 1 m. Thus, increases in flood

levels due to rises in msl are likely to be of the same order as the respective increases

in adjacent open-sea conditions. Exceptions to the above are possible for surge

response to secondary depressions which can have effective periodicities of sig-

nificantly less than 12 h and hence corresponding reductions in ‘resonant ’ estuarine

lengths.

8.3.2 Bathymetric adjustments

Chapter 6 shows how, for ‘synchronous’ estuaries, a ‘zone of estuarine bathymetry’

can be determined bounded by

E x

L51; L

1

L51and D

U 3550m2s3; (8:8)

corresponding to both tidal excursion, E x, and salinity intrusion length, LI, being

less than estuarine length L and the Simpson – Hunter (1974) criterion, D/ U 3, for a

‘mixed’ estuary.

Figure 6.12 shows how Bar-Built and Coastal Plain estuaries in the UK generally

fit within this bathymetric zone. By introducing the expression (6.25) linking depth

at the mouth to river flow and side slope gradient tan α

D0 ¼ 12:8ðQ tanαÞ0:4: (8:9)

Figures 7.9 and 7.10 show comparisons between observed lengths and depths (at the

mouth) against the theoretical values (8.5) and (8.9). The observed values were

extracted by Prandle et al . (2006) from the ‘FutureCoast ’ database (Burgess et al .,

2002).

These Frameworks, Figs 6.12, 7.9 and 7.10, then provide immediate visualindications of the likely stability and sensitivity of any particular estuary to changes

in D, Q or ς *. Estuaries located within the bathymetric zone in Fig. 6.12, with depths

and lengths in broad agreement with the theoretical values in Figs 7.9 and 7.10,

might be regarded as in present-day dynamic equilibrium. Consequently, future

morphological adjustments might be expected to remain consistent with these

theories and follow relatively rapidly. By contrast, estuaries outside the zone or

where either depth or length is inconsistent with the theories might suggest anom-

alous characteristics. By identifying the bases of such anomalies, the implicationsfor future morphology can be assessed.

Clearly a sea level rise, of say 1 m, will have a much bigger impact on shallow

estuaries than deep. Prandle (1989) examined the change in tidal response in

estuaries due to variations in msl, where the locations of the coastal boundaries




remained fixed (i.e. construction of flood protection walls). The results showed the

largest impacts in long, shallow estuaries.

The theories synthesised in Figs 6.12, 7.9 and 7.10 do not consider sedimentation.

Changes in the nature and supply of marine sediments can lead to abrupt changes in

estuarine morphology. This supply can directly determine the nature of the surficialsediments and thereby bed roughness. Changing flora and fauna can, via their

effects on sea-bed roughness and associated erosion and deposition rates, have

abrupt and substantial impacts on dynamics and bathymetry. Peculiarly, the relation-

ship (8.9) between depth at the mouth and river flow is independent of both tidal

amplitude and bed roughness. However, from (8.5), the associated estuarine length

will shorten as sediments become coarser.

8.3.3 Depth, breadth and length changes for 2100 variations

in msl and river flow

Estimates of ‘ precautionary’ changes in msl by 2100 (Defra/Environment Agency

Technical Summaries, 2003 and 2004) amount to an increase of 50 cm. Corresponding

estimates for river fluxes include both increases and decreases of up to 25%.

Inserting these changes in river flow, Q, into (8.9) and the resulting changes in

depth, δ D, into (8.5), we can estimate the changes in length, δ L. Likewise, the

changes in breadth, δ B, associated with the changes in D can be estimated by

assuming the side-slope gradients, tan α, are unchanged. Table 8.4 provides quanti-

tative indications of the resultant changes. The representative values of D, L and B

over the range of estuarine geomorphologies were calculated from the FutureCoast

data set (Prandle, 2006).

The changes δ D correspond to δQ0.4, changes δ L to (δQ0.4)1.25 and δ B to 2δ D/

tan α. The results show that, on average, the ‘dynamical’ adjustment to a 25%

change in river flows may change depths by as much as the projected sea level

Table 8.4 Changes in depth, length and breadth for a 25% change in river flow

and 0.5 m increase in msl

Estuary type D (m) δ DQ (±) L (km) δ LQ (±) δ Lmsl (+) B (m) δ BQ (±) δ Bmsl (+)

All minimum 2.5 0.25 5 0.62 1.28 130 38 77Mean 6.5 0.65 20 2.50 1.94 970 100 77

All maximum 17.3 1.73 41 5.12 1.49 3800 266 77Coastal Plain 8.1 0.81 33 4.12 2.57 1500 147 91Bar-Built 3.6 0.36 9 1.12 1.59 510 51 71

Notes: Change in river flow – subscript ‘Q’, 0.5 m increase in msl – subscript ‘msl’.Source: Prandle, 2006.




rise – with this effect reduced in smaller estuaries and significantly increased in

larger ones. The resulting changes in estuarine lengths and breadths follow similar

patterns with the bigger ‘dynamical’ changes occurring in the larger estuaries where

they are significantly greater than those due to the specified sea level rise. Overall,

we anticipate changes in estuarine lengths of the order of 0.5 –

5 km and breadths of the order 50 – 250 m due to the 25% change in river flow. Corresponding changes

due a sea level rise of the order 50 cm involve increases in both lengths of order

1 – 2.5 km and breadths of order 70 – 100 m.

8.3.4 Impacts on currents, stratification, salinity,

flushing and sediments

Indicative impacts of GCC on the above parameters can be similarly calculatedusing the respective parameter dependencies and Theoretical Frameworks sum-

marised in Section 1.5. While the peculiar conditions in any specific estuary will

determine the actual response, such immediate indications can provide useful

perspectives.

8.4 Strategies for modelling, observations and monitoring

8.4.1 Modelling

Coupled hydrodynamic and mixing models are required as the basis for transport-

ing and mixing contaminants both horizontally and vertically. The dynamical

processes involved occur over time scales of seconds (turbulent motions), to hours

(tidal oscillations), to months (seasonal variations) with corresponding space scales

from millimetres to kilometres. In addition to these hydrodynamic and mixing

models, sediment and ecological models are required with robust algorithms

for sources, sinks and biological/chemical reactive exchanges for longer-termsimulations.

Both proprietary and public-domain model codes typically involve investment

of tens of years in software development and continued maintenance by sizeable

teams. Such effort is increasingly beyond the resources of most modelling groups,

and standardised, generic modules in readily available public-domain codes are

likely to be widely adopted. The development of such modules has removed much

of the mystique that traditionally surrounded modelling of marine processes. The

diversity of estuaries makes it unlikely that a single integrated model will evolve.Moreover, retention of flexibility at the module level is both necessary and desirable

to accommodate a wide range of applications and to provide ensemble forecasts.

Further developments of Theoretical Frameworks are important to interpret such

ensemble simulations.

8.4 Strategies for modelling, observations and monitoring 223



To understand and quantify the full range of threats from GCC, whole-system

models are required – incorporating the impacts on marine biota and their potential

biogeographic consequences. The introduction of various ‘Water Framework

Directives’ for governance of regional seas and coasts emphasises the need for

development of well-validated, reliable models for simulating water quality, ecologyand, ultimately, fisheries. A systems approach is needed, capable of integrating marine

modules and linking these into holistic simulators (geological, socio-economic, etc.).

Rationalisation of modules to ensure consistency with the latter is an important

goal, together with standardisation of prescribed inputs such as bathymetry and

tidal boundary conditions. Such enhanced rationalisation will enable the essential

characteristics of various types of models to be elucidated including the inherent

limits to predictability.

In practice, coupling might be limited to sub-set representations (statistical emu-lators) encapsulating integrated parameters such as stratification levels or flushing

times. To overcome the limitations of individual modules in such total-system

simulations, methodologies are required both to quantify and to incorporate the

range of uncertainties associated with model set-up, parameterisation and (future

scenario) forcing. This requirement can be achieved by ensemble simulations pro-

viding relative probabilities of various outcomes linked to specific estimates of risk.

Model simulations and assessments should extend beyond a single ebb and

flood cycle to include the spring – neap tidal cycle and seasonal variations in river

flows and related density structures. Clearer insights and understanding of scaling

issues should emerge by comparing modelling results with the new Theoretical

Frameworks and against as wide a range of observational data as can be obtained.

8.4.2 Observations

Successful applications of models are generally limited by the paucity of resolutionin observational data (especially bathymetry) used for setting-up, initialising, for-

cing (meteorological and along model boundaries), assimilation and validation.

This paucity of data is a critical constraint in environmental applications. More

and better observational data, extending over longer periods, are essential if model-

ling accuracy and capabilities are to be enhanced.

Instrumentation is lagging seriously behind model development and application,

and this gap is expected to widen. A new generation of instrumentation is needed for

the validation of species-resolving ecosystem models. Despite recent advances, the

range of marine parameters that can be accurately measured is severely restricted

and the cost of observations is orders of magnitude greater than that associated with

models.




Comprehensive observational networks are needed exploiting synergistic aspects

of the complete range of instruments and platforms and integrally linked to model-

ling requirements. Permanent in situ monitoring is likely to be the most expensive

component of any observational network, and it is important to optimise such

networks in relation to the modelling system for the requisite forecasts.To define estuarine boundary conditions, there is a related requirement for

accurate (model) descriptions of the state of adjacent shelf seas. Permanent coastal

monitoring networks have been established in coastal seas and estuaries measuring

water levels, currents profiles, surface winds, waves, temperature, SPM, salinity,

nutrients, etc. using tide gauges, mooring and drifting buoys, platforms, ferries

alongside remote sensing from satellites, radar and aircraft. Regional monitoring

networks are being established via the Global Oceanographic Observing System

(GOOS) networks, (UNESCO, 2003).Up-scaling of knowledge from small-scale experimental measurements is required

to provide larger- and longer-term algorithms employed in numerical models.

Test-bed observational programmes are needed to assess model developments,

these should ideally extend to water levels, currents, temperature and salinity,

waves, turbulence, bed features, sedimentary, botanical, biological and chemical

constituents. To maximise the value of such observations to the wider community,

results should be made available in complete, consistent, documented and accessible

formats.

8.4.3 Monitoring

A basic monitoring strategy for studying bathymetric changes, capable of better

resolving processes operating in estuaries, should include the following:

(1) shore-based tide gauges throughout the length of an estuary, supplemented by water level recorders in the deeper channels;

(2) regular bathymetric surveys, e.g. 10-year intervals with more frequent re-surveying in

regions of the estuary where low water channels are mobile;

(3) a network of moored platforms with instruments for measuring currents, waves, sedi-

ment concentrations, temperature and salinity.

Maximum use should be made of the synergy between satellite, aircraft, ship, sea

surface, seabed and coastal (radar) instrumentation (Prandle and Flemming, 1998).

Likewise, new assimilation techniques should be used for bridging gaps in monitor-

ing capabilities. Observer systems sensitivity experiments can be used to determine

the value of the existence or omission of specific components in a new or existing

monitoring system.

8.4 Strategies for modelling, observations and monitoring 225




Strategic planning to address long-term sustainability of estuaries needs to make full

use of developments in modelling, monitoring and theory. New Theoretical

Frameworks provide a perspective on the threat from GCC.

The leading question is:

How will estuaries adapt to GCC?

8.5.1 Challenges

Management challenges include

(1) promoting sustainable exploitation, i.e. permitting commercial and industrial development

subject to assessment of associated impacts, e.g. dredging, reclamation and fish-farming

(2) satisfying national and international legislation and protocols relating to discharges;

(3) improving and promoting the marine environment, monitoring water quality, support-

ing diverse habitats and expanding recreational facilities;

(4) reducing risks in relation to flooding, navigation and industrial accidents;

(5) long-term strategic planning to address future trends including GCC.

A major difficulty in estuarine management is the general uncertainty in linking

specific actions to subsequent responses over the local to wider scales and from the

immediate to longer time scales. For example, it has generally proved difficult to

predict improvements to estuarine water quality following clean-up campaigns due

to leaching of contaminants from historical residues in bed sediments. Similarly, the

full impacts from ‘interventions’ may manifest themselves in unforeseen ways at

remote sites at a much later time. While such uncertainties can never be entirely

overcome, the pragmatic objective is to arrive at a balanced perspective. This

perspective should provide indications of the scale of vulnerability based on anensemble of ‘approaches’ using theory, measurements and modelling to draw on

present and past behaviour of the estuary concerned and on related experiences in

adjacent systems and in similar estuaries elsewhere.

8.5.2 Modelling case study

Section 8.2 describes a case study of a modelling simulation of the Mersey Estuary,

indicating how theory and observational data are used to assess the model resultsand interpret parameter sensitivity tests. Figures 8.2 and Table 8.1 illustrate the use

of results from earlier modelling and observational studies. Figures 8.3 and 8.4

show results from a random-walk particle representation of sediment transport.

This study highlights the value of long-term observational data sets. Unfortunately,




such data sets are overwhelmingly from large (navigable) estuaries and, as such, can

be misleading in relation to experience in smaller, shallower estuaries. Thus, in

Section 8.3, the acute sensitivity of ‘near-resonant ’ tidal responses in the Bay

of Fundy (Garrett, 1972) and the Bristol Channel (Prandle, 1980) are shown to

be exceptional. Figure 8.5 indicates that resonant response for semi-diurnal tidalconstituents only occurs in estuaries greater than 60 km in length. The related

demarcation between ‘inertially dominated’ systems and the much more commonly

encountered, shorter and shallower ‘frictionally dominated’ systems was shown in

Fig. 6.3.

8.5.3 Strategic planning

Section 8.4 considers future modelling and observational strategies. Strategic planning for estuarine sustainability must encompass the wide spectra of tem-

poral, spatial and parameter scales encompassing physics to ecology and

micro-turbulence to whole estuary circulation. Advantage must be taken of the

rapid advances in numerical modelling with the associated growth in computa-

tional power, monitoring technologies and scientific understanding. However,

securing investment in such advances generally requires demonstrable benefits

to end users.

Faced with specific planning issues such as a proposed engineering ‘intervention’

or the need to improve flood protection, managers will often commission a model-

ling study. The range of models available and their requirements was outlined in

Section 1.4. It was indicated that the selection of an appropriate model depends on

the availability of observational data to set-up, initialise, force, validate and assess

simulations. Obtaining data is almost always much more expensive than a model

study.

It is important to distinguish between model studies which involve‘interpolation

’

as opposed to ‘extrapolation’. The results illustrated in Tables 8.2 – 8.4 are essen-

tially ‘interpolation’, i.e. examining small perturbations close to existing parameter

ranges. By contrast, ‘extrapolation’ involves larger perturbations which can change

the ranking of controlling processes and introduce new elements outside of the

range of validity of the model.

Ideally, an estuarine manager should have access to a range of modelling cap-

abilities, routinely assessed by a wide range of continuously monitored data.

Confidence in future predictions then rests on the degree to which such modellingsystems can reproduce observed cycles, patterns and trends and interpret these

against Theoretical Frameworks. Development of a strategic programme needs to

exploit all such technologies to provide the robust perspective required both for

long-term strategic planning and addressing specific day-to-day issues.




8.5.4 Impacts of GCC

The success of new theories in explaining the evolution of morphologies over the past

10 000 years of Holocene adjustments lends confidence for their use in extrapolation

over the next few decades. The explicit analytical formulae and Theoretical Frameworks,

summarised in Section 1.5, can provide guidance on the relative sensitivity of an estuary

to both local ‘intervention’ and wider-scale impacts such as GCC. Figures 6.12, 7.9 and

7.10 represent new morphological frameworks. For any particular estuary, examining the

loci on such frameworks from mouth to head and over the range of prevailing conditions

can provide a perspective on the relative stability. Where these loci extend outside of the

theoretical zones, the possibility of anomalous responses can be anticipated.

We do not expect drastic changes in estuarine responses to tides or surges from

the projected impacts of GCC over the next few decades. Some enhanced sensitivity

might be found in relation to shorter ‘ period’ (6 h) surges associated with secondary

depressions, particularly in larger estuaries. Maintaining fixed boundaries in the

face of continuous increases in msl may enhance surge response in the shallowest

estuaries (Prandle, 1989).

In the absence of ‘hard geology’, enhanced river flows may result in small increases

in estuarine lengths and depths, developing over decades. By 2100, we anticipate

changes in UK estuaries due to (precautionary) projected 25% changes in river flow: of

order 0.5 –

5 km in lengths and of order 50 –

250 m in breadths. Corresponding changesdue to a projected sea level rise of 50 cm are increases in lengths of order 1 – 2.5 km and

breadths of order 70 – 100 m. In both cases, bigger changes will occur in larger

estuaries. Although we do not expect dramatic impacts on sediment regimes, changing

flora and fauna could, through their effect on sea-bed roughness and associated erosion

and deposition rates, have abrupt and substantial impacts on dynamics and bathymetry.

Ultimately, an international approach is necessary to quantify the contribution to

and effect from GCC. This extends to development of models and instruments (and

their platforms), planning of monitoring strategies, exchange of data, etc. The paceof progress will depend on successful collaboration in developing structured

research, development and evaluation programmes. The ultimate goal is a fusion

of environmental data and knowledge, utilising fully the continuous development of

communications and computational capacities. Appendix 8A indicates technologies

likely to be widely available to estuarine managers in the next decade or so.

Appendix 8A

8A.1 Operational oceanography

Operational oceanography is defined as the activity of routinely making, dissemi-

nating and interpreting measurements of coasts, seas, oceans and the atmosphere

to provide forecasts, nowcasts and hindcasts.




8A.2 Forecasting, nowcasting, hindcasting and assimilation

Forecasting

Forecasting includes real-time numerical prediction of processes such as storm

surges, wave spectra and sea ice occurrence. Forecasts on a climatic or statistical

basis may extend forward for hours days, months, years or even decades.

Accumulation of errors, both from model inaccuracies and from uncertainties in

forcing, limits realistic future extrapolations.

Nowcasting

In nowcasting, observations are assimilated in numerical models and the results are

used to create the best estimates of fields at the present time, without forecasting.

These observations may involve daily or monthly descriptions of sea ice, sea surfacetemperature, toxic algal blooms, state of stratification, depth of the mixed layer or

wind – wave data.

Hindcasting

Observational data for hindcasting are assimilated into models to compile sets of

historic fields and distributions (typically monthly or annually) of variables such as

sea surface elevation, water temperature, salinity, nutrients, radio-nuclides, metals

and fish stock assessments.

Assimilation

Data assimilation forms the interface between models, observation and theory and,

thus, is an essential component in simulation systems (Fig. 8A.1). Assimilation

is used to transfer observed information to update the model state, the model

forcing and/or the model coefficients. The challenge is to take advantage of the

complementary character of models and observations, i.e. the generic, dynamically

continuous character of process knowledge embedded in models alongside thespecific character of observed data.

Coupled models

ASSI

MI

LATION

Set-up dataOperational

model

Application End

user

Atmosphere

Ocean

Coast

Monitoringnetwork

Bathymetry

Satellite

Ships

Buoys, etc.

Initial conditions

Fig. 8A.1. Components of an operational modelling system.

Appendix 8A 229



8A.3 Model generations

Numerical modelling has been used in marine science for almost 50 years. A

convenient distinction is as follows:

Generation 1: exploratory models where algorithms, numerical grids and schemesare being developed often utilising specific measurements focused

on process studies.

Generation 2: pre-operational models with (effectively) fully developed codes

undergoing appraisal and development, generally against temporary

observational measurements or test-bed data sets.

Generation 3: operational models in routine use and generally supported by a

permanent monitoring network, such as that shown in Fig. 8A.2.

A cascade time of approximately 10 years is typically required to migrate between

each generation.

Real-time operational uses include tidal predictions and hazard warning for storm

surges, oil or chemical spill movement, search and rescue, eutrophication, toxic algal

blooms, etc. Pre-operational simulations often involve assessing and understanding

the health of marine ecosystems and resources and their likely sensitivity to changing

D a t a

m a n a g e

m e n t

C e n t r e –

d a t a

a r c h i v e

M o d e l l i n

g

a n d p r o

d u c t

d e v e l o p m

e n t

Fig. 8A.2. Coastal observatory.




conditions. These are typically concerned with assessment of absorptive capacity

for licensing of discharges, evaluating environmental impacts of intervention (recla-

mation, dredging, etc.) and climate change. Exploratory applications extend from

formulation of environmental management policies to developing the underpinning

science and technology to address both anthropogenic influences and natural trends.

8A.4 Forecasting

Even though the immediate issues of concern may seem far-removed from real-time

‘operational’ forecasting, recognition of global-scale developments in modelling

of marine systems is important. Ultimately, estuarine research needs to be linked to

the parallel progress in operational oceanography on both regional and global scales

directed by GOOS (UNESCO, 2003). Effective operation of real-time forecastsrequires the resources of a meteorological agency for communications, processing

and dissemination of forcing data, alongside oceanographic data centres responsible

for dissemination of quality-controlled marine data.

A major objective of operational oceanography is to minimise damage from future

events by reducing uncertainties in forecasting, ranging from storms on a short

time scale to rising sea levels and temperatures over the longer term. Operational

oceanography is central to sustainable exploitation and management of our marine

resources.

Progress in aircraft and satellite remote sensing (Johannessen et al ., 2000) will

dictate the rate of development of operational oceanography for many parameters.

Lead times of a decade or more are required for development of new sensors,

commercial production of prototype instruments and international agreement on

new satellite programmes. Remotely sensed data must be processed in hours if it is

to be useful in operational forecasting. The need for enhanced information from

atmospheric models is a high-priority item in operational forecasting. As an example, accuracy and extent, in time ahead, of wind forecasts are the primary limiting

factors for wave and surge forecasting. The ultimate goal is dynamical coupling of

estuaries – seas – ocean marine models through to terrestial and atmospheric modules,

i.e. global integration of water, thermal and chemical budgets (Prandle et al ., 2005).

References

Burgess, K.A., Balson, P., Dyer, K.R., Orford, J., and Townend, I.H., 2002. FutureCoast – The integration of knowledge to assess future coastal evolution at a national scale. In:The 28th International Conference on Coastal Engineering. American Society of Civil

Engineering , Vol. 3. Cardiff, UK, New York, pp. 3221 – 3233.Defra/Environment Agency, 2003. Climate Change Scenarios UKCIP02: Implementation

for Flood and Coastal Defence. R&D Technical Summary W5B-029/TS.

References 231



Defra/Environment Agency, 2004. Impact of Climate Change on Flood Flows in River Catchments. Technical Summary W5-032/TS.

Fischer, H.B., List, E.J., Koh, R.C.Y., Imberger, J., and Brooks, N.H., 1979. Mixing in Inland and Coastal Waters. Academic Press, New York.

Garrett, C., 1972. Tidal resonance in the Bay of Fundy. Nature, 238, 441 – 443.

Hill, D.C., Jones, S.E., and Prandle, D., 2003. Dervivation of sediment resuspension ratesfrom acoustic backscatter time-series in tidal waters. Continental Shelf Research,23 (1), 19 – 40, doi:10.1016/S0278-4343(02) 00170-X.

Hutchinson, S.M. and Prandle, D., 1994. Siltation in the saltmarsh of the Dee Estuaryderived from 137Cs analysis of shallow cores. Estuarine, Coastal and Shelf Science,38 (5), 471 – 478.

IPCC, 2001. Edited by Watson, R.T. and Core Writing Team. Climate Change 2001:Synthesis Report. A Contribution of Working Groups I, II, and III to the Third

Assessment Report of the Intergovernmental Panel on Climate Change. CambridgeUniversity Press, Cambridge, United Kingdom, and New York.

Johannessen, O.M., Sandven, S., Jenkins, A.D., Durand, D., Petterson, L.H., Espedal, H.,Evensen, G., and Hamre, T., 2000. Satellite earth observations in OperationalOceanography. Coastal Engineering , 41 (1 – 3), 125 – 154.

Lane, A., 2004. Bathymetric evolution of the Mersey Estuary, UK, 1906 – 1997: causes andeffects. Estuarine, Coastal and Shelf Science, 59 (2), 249 – 263.

Lane, A. and Prandle, D., 2006. Random-walk particle modelling for estimating bathymetricevolution of an estuary. Estuarine, Coastal and Shelf Science, 68 (1 – 2), 175 – 187,doi:10.1016/j.ecss.2006.01.016.

Pethick, J.S., 1984. An Introduction to Coastal Geomorphology. Arnold, London.Prandle, D., 1980. Modelling of tidal barrier schemes: an analysis of the open-boundary

problem by reference to AC circuit theory. Estuarine and Coastal Marine Science,11, 53 – 71.

Prandle, D., 1989. The Impact of Mean Sea Level Change on Estuarine Dynamics .C7-C14 in Hydraulics and the environment, Technical Section C: MaritimeHydraulics. Proceedings of the 23rd Congress of the IAHR, Ottawa, Canada.

Prandle, D., 2004. How tides and river flows determine estuarine bathymetries. Progress inOceanography, 61, 1 – 26, doi:10.1016/j.pocean.2004.03.001.

Prandle, D., 2006. Dynamical controls on estuarine bathymetries: assessment against UK database. Estuarine, Coastal and Shelf Science, 68 (1 – 2), 282 – 288, doi:10.1016/j.ecss.2006.02.009.

Prandle, D. and Flemming, N.C. (eds.), 1998. The Science Base of EuroGOOS. EuroGOOS ,Publication, No. 6. Southampton Oceanography Centre, Southampton.

Prandle, D. and Rahman, M., 1980. Tidal response in estuaries. Journal of Physical Oceanography, 10 (10), 1522 – 1573.

Prandle, D., Lane, A., and Manning, A.J., 2006, New typologies for estuarine morphology.Geomorphology, 81 (3 – 4), 309 – 315.

Prandle, D., Los, H., Pohlmann, T., de Roeck Y-H., and Stipa, T., 2005. Modelling inCoa stal and Shelf Seas – European Challenge. ESF Marine Board Postion Paper 7 .European Science Foundation, Marine Board.

Prandle, D., Murray, A., and Johnson, R., 1990. Analyses of flux measurements in the River Mersey. pp 413 – 430 In: Cheng, R.T. (ed.), Residual Currents and Long TermTransport, Coastal and Estuarine Studies, Vol. 38. Springer-Verlag, New York.

Price, W.A. and Kendrick, M.P., 1963. Field and model investigation into the reasons for siltation in the Mersey Estuary. Proceedings of the Institute of Civil Engineers, 24,473 – 517.




Simpson, J.H. and Hunter, J.R., 1974. Fronts in the Irish Sea. Nature, 250, 404 – 406.Thomas, C.G., Spearman, J.R., and Turnbull, M.J., 2002. Historical morphological change

in the Mersey Estuary. Continental Shelf Research, 22 (11 – 13), 1775 – 1794,doi:10.1016/S0278-4343(02)00037-7.

UNESCO, 2003. The Integrated Strategic Design Plan for the Coastal Ocean Observation

Module of the Global Ocean Observation System. GOOS Report No. 125. IOCInformation Documents Series No. 1183.Woodworth, P.L., Tsimplis, M.N., Flather, R.A., and Shennan, I., 1999. A review of the

trends observed in British Isles mean sea level data measured by tide gauges.Geophysical Journal International , 136 (3), 651 – 670.

References 233



Index

Annual Temperature Cycle 111 – 120global expressions air and sea surface

118 – 120heat-exchange components:

solar radiation 111, 112, 114, 117,latent heat flux 112infra-red back-radiation 112sensible heat flux 112

Analytical emulator 9, 10, 177 – 178, 184 – 187Asymmetry ebb-flood 40 – 44

Bathymetric: stability 10,

Zone 16Bathymetry

exponential 35 – 38, 47 power law 32 – 35, 47synchronous 27 – 31under-damped 36, 38over-damped 36, 38critical-damping 36, 38

Bed roughness 124 – 125Bessel Function 33

Circulation 2

Climate Change 3, 12, 13, 16Coastal Observatory 16Convective overturning 84, 96 – 105,Coriolis 24Current components

density-driven 52depth-mean 56maximum 57residual 69 – 70surface-bed changes 64 – 67tidal 53

amplitude 16

related constituents 51structure 17

wind-driven 52, 67 – 69, 73Current ellipse 7, 59 – 62, 73 – 74

a-c and c-w 61 – 62maximum amplitude 74minimum amplitude 74

direction 74 phase 74eccentricity 74

Currents, sensitivity toeddy viscosity 62friction 62inter-tidal zone 67latitude 63 – 64tidal constituent 64

Decay time 2Declination 19

Deposition/settlement rate 125, 144 boundary conditions: 130 – 131

reflective bed 130, 143, 148absorptive bed 130, 148

delayed 143, 148exponential 132 – 134

Differential advection 83Dissipation

bottom friction 99vertical shear 99

Eddy diffusivity 97

Eddy viscosity: 97model 75 – 76formulation 57 – 59

Ekman Spiral 68, 73,Elliptic orbit 19Episodic events 2Equations of motion 24 – 25

1D momentum equation 1531D continuity equation 153

linearised friction term 153electrical-analogy 40,momentum 51, 53, 84

steady state 90terms:

convective 42 – 43, 51 – 52Coriolis’ 72, 73density gradient 52inertial 42friction 43

234



non-linear 41 – 43surface-gradient 43

Equilibrium tide 20Erosion rate 125, 126 – 129

advective component 129in shallow water 128 – 129spring-neap cycle 127 – 128

Estuaries:age 11definition 1depth 16,length 16minimum depth and flow 168tidal dynamics 152 – 158

Estuary Response Diagram 33Estuary typologies: 196 – 199

bathymetry 196 – 198in-fill rates 199sediment regimes 198 – 199spacing between 169 – 170Theoretical Frameworks 205

Eulerian currents 70

Fall velocity 132Feed-back mechanism 11Flow Ratio 105 – 106Flushing time 2, 16, 97Friction 16, 47

linearization 38 – 39Froude Number 25

Gaussian distribution 130Global Climate Change 228

bathymetric adjustments 221 – 222changes to 222 – 223environmental effects 218impacts on tides and surges

219 – 221quarter-wavelength resonance 219

H F Radar 15, 68, 70Higher harmonics 3Hydraulic Scale Model 3

Inertia 16Inertial latitudes 53, 73,Internal waves 83

Kelvin Number 25

Lagrangian currents 70

Management Strategies 231modelling 223 – 224monitoring 225observations 224 – 225

Mean Sea Level 2, 11Mersey Estuary Study: 206 – 218

bathymetry 207 – 208hydrodynamic model 208 – 210Lagrangian sediment model 210 – 211

sediment regime: 206 – 207concentration & fluxes 211 – 212

sensitivity to : particle size 213 – 217model parameters 217 – 218

tidal dynamics 206Mixing 2

classification:Hansen-Rattray 79, 108Pritchard 79

effectiveness 104 – 105rates: 96 – 97

mixed salinity 96tidal straining 96river flow 96convective overturning : diffusivity 104

Models: 97 – 105forcing 15global 16resolution 13, 14, 15scope 13set-up 15validity 13development stage:

exploratory 230operational 228 – 231,

pre-operational 230modes of application

assimilation 229forecasting 229, 231hindcasting 229nowcasting 229

types:Bottom-up 205Top-down 205

Monitoring : 16 programme 12, 15sensors 15strategy 15, 16

Morphology 2

Net energy propagation 70 Noise:Signal ratio 50

Observations 15Overturning 2

Peclet Number 129Potential Energy Anomaly 104Proxy data 4Phase difference

(currents : elevation) 11Process measurements 15

Q Factor 40Quarter wavelength resonance 26, 31, 34, 35

Residuals 3Remote sensing 15Richardson Number 80, 90, 106, 108Rouse Number 183Rotterdam Waterway 83

Index 235



Salinity intrusion 2, 16axial location 93 – 96, 160 – 161flushing times 159length 83, 90 – 96, 108, 158

derivations – experimental 90 – 91velocity components 91salt-wedge 92observations 93

residual current 86 – 88stratification 158

Schmidt Number 98Sea level gradient:

Stokes’ drift 70Strouhal Number 58 – 59, 62, 72,

73, 101Surface-to-bed differences:

velocity 99, 102 – 103, 107salinity 103 – 104

Sustainable exploitation 4Synchronous estuary: 9 14

bathymetric determinants 163 – 164definition 152Morphological Zone 161 – 163, 168observation vs theory 164–168

236 Index