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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.
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ESTUARIES
Dynamics, Mixing, Sedimentation and Morphology
DAVID PRA NDLEUniversity of Wales, UK
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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
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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
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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
4.6 Summary of results and guidelines for application 108
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
5.7 Summary of results and guidelines for application 142
Appendix 5A 145
References 149
6 Synchronous estuaries: dynamics, saline intrusion and bathymetry 151
6.1 Introduction 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
7.1 Introduction 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
7.8 Summary of results and guidelines for application 199
References 202
vi Contents
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8 Strategies for sustainability 205
8.1 Introduction 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
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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
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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
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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
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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
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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
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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
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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
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(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
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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
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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
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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
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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
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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
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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.
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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
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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
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(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
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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)
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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
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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.
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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.
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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.
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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
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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).
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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.
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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)
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( 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
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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,
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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 .
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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 .
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@&
@ 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:
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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 ).
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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.
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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
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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 .
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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
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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
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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)
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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) .
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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)
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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 .
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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
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‘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)
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@
@ 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).
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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.
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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
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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.
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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
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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.
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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 .
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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.
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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.
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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),
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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.
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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
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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)
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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.
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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
3.2 Tidal current structure – 2D ( X-Z ) 55
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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).
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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.)
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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).
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|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
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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.
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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
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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).
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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.
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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.
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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.
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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).
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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
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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).
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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.
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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.
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3.5 Summary of results and guidelines for application
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.
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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
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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
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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.
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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
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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
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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
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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 ;
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( 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
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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
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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
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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 .
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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
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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.
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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
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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).
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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 .
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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)
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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 .
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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
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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.
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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
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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 .
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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).
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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.
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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)
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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) :
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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.
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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 .
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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 .
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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‰.
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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.
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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).
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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
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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
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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
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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.
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4.6 Summary of results and guidelines for application
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
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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.
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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.
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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
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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
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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).
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@ 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
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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 .
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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
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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
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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
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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
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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 .
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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 .
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(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
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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
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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,
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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.
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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
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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.
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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)
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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
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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
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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
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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
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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.
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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.
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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)
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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).
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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 .
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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.
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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).
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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.
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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.
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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.
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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.
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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
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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.
5.7 Summary of results and guidelines for application
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
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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
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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.
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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:
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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.
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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
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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
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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
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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
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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
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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
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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
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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.
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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.
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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
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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).
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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).
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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).
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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,
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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.
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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).
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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).
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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.
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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)
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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.
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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.
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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.
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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.
6.5 Estuarine bathymetry: assessment of theory against observations 169
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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.
6.6 Summary of results and guidelines for application
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
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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.
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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
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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.
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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.
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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
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( 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 .
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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.
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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 .
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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
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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)
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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
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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)
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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
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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)
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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).
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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 .
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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.
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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.
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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 ) .
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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).
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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 .
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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).
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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.
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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 .
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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
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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.
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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 .
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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.
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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 .
7.8 Summary of results and guidelines for application
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
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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
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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
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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
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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.
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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.
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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.
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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
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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
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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.
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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.
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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
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(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.
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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 )
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.
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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.
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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
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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 corres- ponding 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.
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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.
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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
Semi-diurnal tidal cycles
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
Semi-diurnal tidal cycles
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).
8.2 Model study of the Mersey Estuary 213
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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 – – – – – –
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(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.
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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.
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(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
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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.
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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
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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.
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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
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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.
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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.
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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.
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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.
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8.5 Summary of results and guidelines for application
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,
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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.
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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.
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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
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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.
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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 exam- ple, 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).
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232 Strategies for sustainability
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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
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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
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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
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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
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