report on the seasonal equilibrium concentration ... · report on the seasonal equilibrium...

Report on the Seasonal Equilibrium Concentration

Enhancement (sECE) model.

Enrique Portilla and Paul Tett

August 31, 2006

Abstract

This report is a contribution to SARF project 012, ’The development of modelling

techniques to improve predictions of assimilative capacity of water bodies utilised for

marine caged fish farming’. The report marks project milestone 3, a description of

the seasonal ECE model and auxiliary models and their implementation as a Matlab

script.

The seasonal Equilibrium Concentration Enhancement (sECE) model is started by

running one of two Matlab scripts depending on the integration method to be used:

sECEeuler(lochname’, year) or sECEsolver(lochname’,year) are respectively the

scripts used. Each script then calls on scripts implementing other parts of the model

system. These parts include the ACExR physical model described for milestone 1

of the project, and a fish nutrient production model, and may include a catchment

runoff model. Data requirements include timeseries of wind, sunshine and rainfall,

sea-boundary conditions for salinity and nutrients, and river nutrient concentrations.

This report describes the ECE, Runoff and Fish models and considers numerical

and statistical methods. It assembles boundary conditions, and shows results from

running the sECE model system, for the Argyll sea-loch, Creran. The data base has,

so far, been populated only with information for this loch and for selected years.

1

Napier University, Edinburgh

CONTENTS CONTENTS

Contents

Table of Contents 2

1 Introduction 4

2 The seasonal ECE model 6

3 Structure of the Matlab scripts 7

4 The physical model 9

5 Numerical methods 10

5.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 The Euler method in tracer.m . . . . . . . . . . . . . . . . . . . . . . . . . 125.3 Problems with the Euler approximation . . . . . . . . . . . . . . . . . . . . 135.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Statistical Methods 15

6.1 Temporal variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.2 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

7 Meteorological data 19

8 The catchment runoff model 20

8.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.2 Water flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.3 Precipitation and evapotranspiration . . . . . . . . . . . . . . . . . . . . . . 238.4 Parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248.5 Model and Comparison with data . . . . . . . . . . . . . . . . . . . . . . . . 248.6 Seasonal variation and long-term trend . . . . . . . . . . . . . . . . . . . . . 25

9 The fishfarm model 26

9.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269.2 Blackfish’ model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

10 Boundary conditions 29

11 Results for Loch Creran 30

2

CONTENTS CONTENTS

11.1 On technical aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3111.2 On the results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

12 List of Parameters 33

13 References 35

14 Figures 37

15 APPENDIX 62

3

1 INTRODUCTION

1 Introduction

An ECE model is one of the simplest solutions obtainable from the following generalizedequation for the rate of change of a water quality state variable Y in the presence offishfarming,

∂Y

∂t= −∇ϕY + βY + ΓY (typical units: amount m−3d−1)(1)

where, on the right hand side:

• the first term deals with physical transports, being the divergence of the flux vector(ϕy) , in which are included advective and diffusive terms in one or more spatialdimensions; the values of the fluxes at the boundary of the modelled domain mustbe specified;

• the second term (βY ) is the sum of the biological and chemical sources and sinks forthe variable;

• the third term (ΓY ) gives the input of the variable by the farm, or its loss to thefarm.

In the present case, the state variable is the concentration of a ’plant nutrient’ such as’Dissolved Available Inorganic Nitrogen’ or DAIN, symbolized by S (typical units, mil-limoles per cubic metre, where a millimole (abbreviation: mmol) of DAIN contains 14 mgof the element nitrogen). The simplest solution arises when the farm is assumed to besited within a well-mixed box of volume V m3, the contents of the box exchanging at aconstant daily rate E with adjacent water containing DAIN at concentration S0. Solvingthe divergence term therefore has only to take account of this single cross-boundary flux(figure 1), and thus:

(2) ∇ϕY = E(S − S0).

A strict definition of the exchange rate E is that it gives the instantaneous probability,averaged over a tidal cycle, and expressed as a daily rate, that a randomly selected smallpacket of water inside the box will be replaced by a small packet from the adjacent water,

4

1 INTRODUCTION

both packets being identical in volume. An exchange rate of 0.10 d−1 implies that about10% of the water in the box will be exchanged each day, the precise proportion being e0.1,or 10.52%.

In this simplest case, there are assumed to be no active biological processes, and henceβS = 0. The farm input of DAIN, totalling si inmmol d−1, is spread throughout the box,and hence ΓS = si/V . Thus, equation (1) becomes:

dS

dt= −E (S − S0) +

si

V(typical units : mmol m−3d−1)(3)

If a steady state is assumed, the solution for DAIN is:

Seq = So +si

EV(typical units : mmol m−3 )(4)

and the second right-hand term, si/(EV ) , is often referred to as the equilibrium concen-

tration enhancement. It provides an indicator of the ecological pressure on a water bodyfrom fishfarm nutrients. Any model using something like equation (4) is therefore referredto as an ECE model. The dynamic equation in (3) is an example of what will henceforthbe called a seasonal ECE model, intended for use in estimating time-changing enhance-ments. This report concerns such a model, and the model’s name will be abbreviated fromnow on as sECE.

ECE models have been in use for some time to rank sea-lochs and voes by their effectivenutrient loading (Gillibrand and Turrell, 1997). Several limitations on existing versionshave however been recognized, and include the need to improve the parameterisation ofexchange and to take account of seasonal variations in boundary concentrations and localand fishfarm inputs of nutrients. A physical basis for better estimation of exchange rateswas described by Gillibrand and Inall (2006) in the first SARF012 milestone report, andpart of the work described in this report for the third milestone is that of adding thisphysical model to a seasonal ECE model. The outcome of this is a set of Matlab scripts.The other part of the work concerned the boundary conditions for nutrients in the ECEmodel. In principle these are provided by:

• a database of regional climatologies for nutrients;

5

2 THE SEASONAL ECE MODEL

• a simple catchment runoff model, driven by daily values of rainfall and air temper-ature which provides time-varying river flow, combined with a data-base of regionalclimatologies for river nutrient concentrations;

• a model for inputs of nitrogen and phosphorous from a fish-farm, drive, by data onfood supply (Black, 2001).

A climatology consists of a typical seasonal cycle of the variable in question, togetherwith estimates of variability about this typical cycle due to various causes. At present wehave focussed on loch Creran, in northern Argyll, as a test site, and have assembled theclimatology only for its relevant boundaries, which are firth of Lorn. The runoff model hasbeen tested against flows measured in the river Creran, and the river nutrient data comefrom samples of river Creran water.

2 The seasonal ECE model

Compared with the ECE model of Gillibrand and Turrell (1997), the novelty of this sECEmodel system is that (i) it allows concentrations of dissolved nutrients to vary throughoutthe year as a result of seasonal variations in physical transport, nutrient boundary condi-tions, and input from fish-farms; and (ii) the use of a three-layer model for sea-lochs andvoes (Gillibrand and Inall, 2006) improves the estimation of physical exchange and allowsfarm inputs to be split between pelagic and benthic mineralization. Therefore, we canrestate equation1 for dissolved nutrients as:

(5)∂Sjij

∂t= −∇ϕSij

[+βSij

]+ ΓSij

where i = {1, 2or3} refers to physical layers in the model and j refers to the nutrientelements nitrogen (N , called DAIN when referring to the sum of dissolved nitrate, nitriteand ammonium), phosphorus (P , called DIP when referring to the main form, dissolvedinorganic phosphate), and silicon (Si, usually in the form of dissolved silica).

The terms here presented are similar to those already presented in the introduction butcan be expanded as follows:

• The physical transport flux divergence ϕSij gives the net results of the movementsof water containing nutrients between the layers in the model, between the model do-

6

3 STRUCTURE OF THE MATLAB SCRIPTS

main and the sea, and into the model’s surface layer from simulated river discharges.A routine provided by the physical model (Gillibrand and Inall, 2006)), calculatesthese movements, and is described in the next section of this report.

• All βSij terms are zero in an ECE model, which predicts nutrient effects in theabsence of phytoplankton uptake or chemical and biological recycling. Hence the’beta-term’ is ignored in the (s)ECE model.

• ΓSij refers to nutrients input by the fish farm: these take the form of phosphateand ammonium excreted by fish through their gills, and phosphate, ammonium andnitrate released into the water column when fish faeces and waste food mineralize onor in the seabed. These inputs are provided by a simple fish model (Black, 2001).Input of nutrient are different among physical layers and it was implemented tohappend on top and bottom layers (layers 1 and 3 defined by the physical model).Because fish farms do not enrich silica, the s(ECE) model deals only with DAIN andDIP as nutrients. The ratio of nutrient nitrogen to silicon is important in controllingphytoplankton floristic composition and silica will be simulated in the subsequent(L)ESV model.

Subsequent sections describe the runoff submodel, the meteorological (rainfall, sunshineand temperature) data used to drive it, and the climatologies for river and sea-boundarynutrients for the loch Creran example.

3 Structure of the Matlab scripts

We start by outlining the main steps that are involved in installing and running the Mat-lab scripts for the sECE model system. Most of the points below will subsequently beconsidered in more detail.

1 The scripts must be installed in a way that preserves the directory structure; thisstructure is designed to allow each part of the model system to be updated indepen-dently. Once the compressed file is ”unzipped” a tree of folders is copied and only onematlab script is available: cdsarfac.m. This script will load all the SARFAC treedirectories into the matlab path directory and functions can be run in any location.

2 One of two main scripts should be run, depending on the choice of numerical inte-gration schemes. These are invoked from the Matlab control window according tothe following pattern: run output=sECExxxxx(’lochname’, year). The character

7

3 STRUCTURE OF THE MATLAB SCRIPTS

string given for ’lochname’ must be one of the names in sea-loch catalogue compiledby A. Edwards. Hypsometric data giving the underwater dimensions of the loch) isavailable for all water bodies in this catalogue. Other data for the sECE model is atpresent only available for the name ’Creran’. The placeholder year must be replaceby an integer value, giving the year for which model output is required. Severaloptions are possible here. At this stage, the sECE model will run for basically for2 years 1975, when no salmonids were farmed in loch Creran, and 2003, when thefish-farm activity was well established and data is available to run all the subsequentmodels. Alternatively the model can be run using climatology for some or all of theforcing variables.

3 River flow and nutrient data are needed. This can be supplied as data files, or theflow data can be generated by the Runoff model if no river flow data is available.This model will be driven with rainfall and solar radiation data for the year specifiedin step 2. A timeseries of daily values of freshwater discharges and nutrient contentswill be made.

4 The script will then run the physical model (ACExR) for the year specified. It canbe forced with freshwater runoff data from the runoff model. It will provide withinformation on exchange fluxes, layer volumes and and layer thickness.

5 Information will then be loaded for the boundary conditions for the two state variablesconsidered in the sECE model: DAIN and DIP.

6 The Blackfish submodel will be run to convert information on the amount of foodused by the farm, into nutrient loading to layers 1 and 3 of the model.

7 The script will then step through the year, carrying out numerical integration of thedifferential balance equation (5). The available options are to use a forward Eulerapproximation (in (sECEeuler), requiring many interations within each day, or oneof the Matlab solver routines (called by (sECEsolver), with larger and adaptivetimesteps.

8 Finally, the script outputs the results from the numerical similations as structure ar-rays (See matlab manual). Graphs for the evaluation of the output can be performedwith the function sECEplot(’year’,ouput), where yearis the year used in the pre-vious analysis, and ouput is an structure array containing solution of the numericalintegration of DAIN and DIP in each simulated layer.

8

4 THE PHYSICAL MODEL

4 The physical model

An important part of this milestone was to include time-varying exchange with adjacentsea-water in the sECE model. This was done using the ACExR physical model developed formilestone 1 (Gillibrand and Inall, 2006). Many of the equations in this model were takenfrom the FjordEnv model of Stigebrandt (2001) with improvements for tidal exchange inScottish lochs. The conceptual framework is a three layer system simulating horizontal andvertical exchange, as shown in figure 3, which is updated from (Gillibrand and Inall, 2006).The FjordEnv equations provide steady-state solutions for given boundary conditions; inACExR they are solved each day for new values of the boundary conditions, thus allowingsimulation of a seasonal cycle. ACExR includes one tracer (salinity) and can be used tosimulate layer-dependent temporal evolution of any scalar property. In the case of thesECE model, the scalars of interest are the concentrations of nutrients in each layer. Inessence, The ACExR model provides information on the term (−∇ϕY ) from the equation 1.It considers this term as including 9 components of exchange, 6 of which act on the surfacelayer.

The ACExR model can be used either in a diagnostic mode, where the enhanced FjordEnvequations are used to estimate the surface layer salinity, or in fully prognostic mode, wherethe salinity is treated as a scalar property and calculated by solving an advection-diffusionequation implemented in the routine tracer. The prognostic mode is used in the sECEmodel.

The diagnostic equations provide ’steady-state’ solutions of layer thickness and water flow,depending on loch hypsometry, wind speed, freshwater input, and tidal amplitude. Thesesolutions include ’steady–state’ values of surface salinity under the given conditions; salini-ties in layers 2 and 3 remain at their initial values. Used in this mode, the Matlab functionCalcE steps through a year one day at a time, saving daily values of volume (V , m3),thickness (H, m) and salinity (S, on the practical scale) for each of the three layers, andof the daily water fluxes (E, m3 d−1) for each componenent of exchange.

The prognostic equations allow salinities in each layer to change dynamically, as a resultof the effects of water fluxes calculated in the Euler forward-difference routine tracer,which is called repeatedly during each day by CalcE. These changes in salinity have aninfluence on some of the FjordEnv-derived solutions, and hence the prognostic equationsoutput values of V , H and E, as well as those of S, that are different from those calculatedin the diagnostic mode.

9

5 NUMERICAL METHODS

The boundary data needed to run the physical model are freshwater runoff, wind forcingand vertical distributions of salinity in the adjacent sea. If no data are provided, sinusoidalcurve of annual period can be used to simulate the 3 driving functions. For the purposes ofthis work, we drove the physical model using fresh water runoff simulated by the sRunOff

model (see section 8). The wind forcing was done by daily mean wind speed climatologyfor Tiree between 1957 and 83. Salinity forcing was obtained from UKHO climatology.These boundary conditions are shown in figure 3.

A characteristic of the model framework described by Gillibrand and Inall (2006) is thechange in volume in layers 1 and 2 due to estuarine circulation and water entrainment(labelled E.Qest and E.Qh in figure 3). These are volume fluxes resulting from the deep-ening or shallowing of the surface layer, with consequent entrainment of layer 2 water ordetrainment of layer 1 water. Such changes in volume add some difficulties to the numericalintegration of the equation (5). They were foreseen in the framework delivered in Gillibrandand Inall (2006) with a routine called tracer.m. Tracer contains a set of Euler forwarddifference equations, to estimate concentration of any variable in 3 layers. In the nextsection, we will consider some of the issues involved in such numerical integration.

5 Numerical methods

The purpose of this section is to discuss some of the problems that arise during numericalintegration of the equations for the chemical and biological variables in this sECE model(and subsequent more complex models). We consider the effect of changing volume on thenumerical integration, the adequacy of the use of the Euler integration method, and analternative method using a standard Matlab routine.

5.1 Conservation of mass

To begin with the discussion of the numerical methods we will try to explain the allowancewe have to make in the implementation of the models for changes in volume during anumerical time-step in order to conserve the total mass of the simulated scalar variable.Such conservation is vital when making a nutrient budget for a loch. We require that thenumerical routines that move quantities of nutrients between layers, do not add or subtractany nutrient to the total in the loch. We need to be assured of this in order to explainsimulated changes as the result (in the sECE model) only of inputs of nutrients fromrivers, farms and the sea, and losses to the sea. In subsequent models, simulated biological

10

5.1 Conservation of mass 5 NUMERICAL METHODS

processes will provide nonconservative gains and losses; again, we must be confident thatthe simulated additional changes result from these processes and not from numerical errorsin the schemes for exchanging scalars between layers.

To clarify this problem, consider the expansion of a differential term when both the concen-tration of a state variable (Y ), and the volume (V ) of a layer, change with time (t):

(6)d (V (t)Y (t))

dt= V (t)

dY (t)dt

+ C(t)dV (t)

dt

We will merge now equation (5) and equation (6) into a single function that describeschanges in the property of any scalar due to physical exchanges, changes in volume of thedifferent layers, biological reactions and external inputs. The equation uses the symbol φ

of Gillibrand and Inall (2006) for a conservative scalar such as salinity or (in the sECEmodel) nutrient concentration in layer j:

(7)dφj

dt=

1Vj

(∇(Qjφj) + βφj + Γφj) +φj

Vj

dVj

dt

The term dVj/dt accounts for changes in volume in the different layers of water due to waterentrainment and estuarine circulation. The equation described in (7) provides the rationaleused to integrate the sECE model equations using the ode23 solver of matlab.

Equation 6 can be re-written in its discrete form:

(8)∆ (V Y )

∆t= V0

∆Y

∆t+ Y0

∆V

∆t

Working out this expresion leads to the following expresion which is used in the tracer.m

function in ACExR.

(9)∆ (V Y )

∆t=

YtVt − Y0V0

∆t

For a tracer such as salinity which is not changed by biological processes, nor are there

11

5.2 The Euler method in tracer.m 5 NUMERICAL METHODS

fish-farm inputs, so that the terms βφj and Γφj are zero, the discrete equation can bewritten as:

(10)YtVt − Y0V0

∆t= ∇(Qjφj)

and the concentration of salt at any time can be computed as follows:

(11) Yt = ∆t1V0∇(Qjφj) + Y0

V0

Vt

The last term V0/Vt will account for changes in volume.

5.2 The Euler method in tracer.m

In order to explain simply the effect on changes in volume only 3 components of the 9components of the physical fluxes will be considered in this section (figure 4). The first ofthese is the estuarine circulation, labelled E.Qest in the Matlab script. This moves waterfrom layer 2 into layer 1, displacing the same volume from layer 1 into the sea, and hencedoes not change layer volume. The second and third examples, also shown below, are ofentrainment processes which do change layer volume.E.Qh is the volume flux resulting fromthe deepening or shallowing of the surface layer, with consequent entrainment of layer 2water or detrainment of layer 1 water. In the present version of the model, only entrainmentis considered, and the thickness of the surface layer is prevented from increasing indefinitelyby the volume flux E.Qent caused by the entrainment of layer 1 water into layer 2 due totidal flow in layer 2.

The function tracer.m contains a set of Euler forward difference equations generalisatedin equation 11 and exemplified here for a generalized concentration variable in the surfacelayer extracted from tracer with only 3 components:

Ynew[1] = Y[1] * (Vp[1] / V[1])+(deltaT / V[1]) ...

* (E.Qest * (Y[2] - Y[1])...

+ E.Qh*Y[2] + E.Qent.Y[1]);

Y is the concentration of the tracer at the start of the time-step deltat, which is a smallfraction (typically, 0.01) of a day. Ynew is the concentration at the end of the time step.

12

5.3 Problems with the Euler approximation 5 NUMERICAL METHODS

[1] and [2] specify the layer concerned 1. Vp is the volume of the layer at the start of thetime-step and V that at the end of the time-step. Volumes are changed by:

V[1] = Vp[1] + deltat * (E.Qh + E.Qent);

Upwards fluxes are positive, and thus, in the present model, E.Qh ≤ 0 and E.Qent ≥0.

5.3 Problems with the Euler approximation

The Euler method for numerical solution tries to estimate the value of a differential equationby incrementing time steps (∆t) that are sufficiently small. Let us assume that a statevariable (Y ) changes with time, taking the value Y0 at time t = t0 and the value Yt at timet = t0 +∆t. That is to say, Y0 = f(t0) and Yt = f(t0 +∆t), where f is the known functionused to model the relationship between Y and t.

The equation that describe the value of the state variable at any time-point will have thisform:

(12) Yt = Y0 + ∆t∆Y

∆t

What is the problem with Euler? Its error can be explained using the Taylor series expan-sion of a function in the neighbourhood of a point. In the present context, the expansionis:

(13) Yt0+∆t = Y0 + Y ′0∆t +

Y ′′0

2∆t + · · ·

where Y ′0 and Y ′′

0 are the first and second time deviates of Y respectively at time (t =t0).

The Euler forward difference method uses the approximation from equation (12) that canbe rewritten as:

1For clarity, Pascal-type (square) array brackets have been used to enclose layer numbers; replace themby round brackets to get valid Matlab statements.

13

5.3 Problems with the Euler approximation 5 NUMERICAL METHODS

(14) Yt0+∆t − Y0 = ∆Y = Y ′0∆t

The error of this approximation is described by the remainder of the Taylor series, usuallytaken as dominated by the second order term when ∆t is small. This can be examinedeasily in the case of exponential growth at rate µ (d−1), for which the exact solutionis:

Yt0+∆t = Y0eµ∆t i.e. ∆Y = Y0

(eµ∆t − 1

)(15)

The first order term in the Taylor series is

Y ′0∆t = Y0µ∆t giving the euler approximation(16)

∆Y = Y0µ∆t

and the second order term is

Y ′′0

2∆t = Y2µ

2∆t2(17)

Thus, the ratio of the second order to the first order term is µ2 ∆t, and the relative value

of the progressive inaccuracy in the simulation of Y clearly decreases as the time stepsare made smaller. At first sight, the Euler method should work if ∆t is made sufficientlysmall. However, repeated iteration of calculations may increase the errors that arise fromrounding of numbers during digital processing, especially in cases in which the value of aterm is calculated from a small difference between two large numbers, in in which ratiosare taken of numbers close to zero.

14

5.4 Conclusion 6 STATISTICAL METHODS

5.4 Conclusion

A forward Euler approximation is generally regarded (Press et al., 1989) as both inaccu-rate (in terms of the final result), and inefficient (requiring mainy small-time steps). Theremay also be stability issues (Press et al., 1989) It is likely that the incorporation of dy-namic biological terms will increase these difficulties. Press et al. recommend a 4th-orderRunga-Kutta method with an adaptive step size. Therefore it seemed desirable to useone of the Matlab standard numerical integration procedures to calculate dynamic changesin conservative and nonconservative tracers during a day. Thus, the sECEsolver scriptuses the Matlab ode23 numerical integration routine. For comparison purposes the Eulernumerical integration was also implemented, and is called by using the (sECEeuler script.Running two sets of simulations will allow a comparison of numerical methods.

6 Statistical Methods

In this section we will describe the different statistical methods used in this work. Statis-tical modeling was used in order to estimate seasonal variability and long term trends inboundary conditions used by the sECE model system. Statistical methods were used tocompare model output (i.e., simulations) with observations of state variables.

6.1 Temporal variation

The sECE model (and any subsequent models that builds on it) is intended to be forcedeither by time-series of boundary conditions for a particular year or years, or by climatology,or by a mixture of the two. Boundary variables include concentrations in the sea and river,river flows, and meteorological variables such as sunshine and wind speed. A climatologyfor a variable is a time-series of values, or an equation allowing such values to be computed,that are supposed to represent (a) the typical ornormal conditions on a particular day ina standard year, and (b) the typical or usual range of variation of values on that day. Anobvious climatology is that for sunshine, with low values in midWinter and high values inmidSummer, and variability about these typical values that arises because on some daysthe sky is cloudy and on other days it is clear. However, we can also refer to a nutrientclimatology, which has, in most Scottish waters, peak values in late Winter and lowestvalues following the Spring bloom of phytoplankton.

15

6.1 Temporal variation 6 STATISTICAL METHODS

For all the forcing variables and the boundary conditions the same approach was taken.We tried to estimate the parameters of a model composed by a long term trend, a seasonalterm and some noise, such as:

(18) Yt = Tt + St + εt

where Yt is the dependent variable, for example freshwater runoff at time t. Tt representsthe trend, St represents the seasonality and εt random fluctuations. If St corresponds tosystematic variability at a frequency of 1 yr−1 or higher, then a climatology is composed ofa description of St and some statistics of εt (such as the limits including 95% of its values.Models of this sort were fitted to available data sets using two techniques.

In the first technique, trend was assumed to be linear, i.e. of the form:

(19) Tt = α + γt

at time t, and seasonality was modelled using a series of trigonometric functions. Theseasonal component at time t can be represented by:

(20) St =∑

k

(β2k−1 sin ((2πk/s) t) + β2k cos ((2πk/s) t))

where s is the number of samples in one year cycle (typically 365 if daily samples areavailable) and k corresponds to the kth harmonic, 2πk/s. The most appropriate numberof harmonics was chosen using the AIC selection criteria (Venables and Ripley, 1994).

For some variables (such as Solar radiation), the time-series available from a nationaldatabase was long enough to allow changes in the seasonal pattern to be analysed usingGeneralized Additive Models (Venables and Ripley, 1994). A Generalized Additive Model(GAM) uses additive terms (non linear terms) to estimate a dependent variable Y from aset of predictor variables X1, · · · , XP with the following formula:

16

6.2 Model Validation 6 STATISTICAL METHODS

(21) Yt = α +p∑

k=1

fi(Xj) + ε

where α is the intercept and fj is an smooth function which curvature is set by cross-validation (Ref). For the exploration of changing seasonality the following model wasfitted to the data:

(22) Yt = +s(jul) + s(abs jul) + s(jul, abs jul)

where s(jul) stands for an smooth function which summarises pattern for Julian days (be-tween 1 and 365), s(abs jul) stands for an smooth function which summarises pattern forJulian days from the beginning of first year of study (1 and 7300) , and s(jul ,abs jul)

stands for the interaction between both patterns. River discharge provides an example ofa variable for which seasonal patterns might be changing as a result of climate change:as the climate warms, precipitation that once fell as winter snow may now fall as rain.In earlier times, rivers in the Scottish highlands might therefore show low flow in Winter,with a peak in Spring as the snow melted. Now, with warmer winters, they might show aflow peak in Winter.

6.2 Model Validation

Model output needs to be compared with field observations in order to evaluate and validatea model. It is proposed here that Linear Regression be used for this purpose, with modelprediction (Y ) being displayed on the x-axis while field data(Y ) is displayed on the y axis.2 Linear regression is used extensively in model validation because of the expectationof a linear relationship between model predictions and field data. Also, the statisticaltheory associated to linear regression is well understood and interpretable. The main

2The arrangement with Y on the horizontal axis is counter to what some would consider standardpractice, based on the argument that regression implies causation of the y-axis (the dependent variable)by the x-axis (the independent variable). The arrangement in which the model output is taken for theindependent variable (Y ) is made because Y is assumed to be produced without error, whereas the fieldobservations are assigned to the dependent variable (Y ) because they are not under the control of theinvestigator and include an error (stochastic variability) component (Sokal and Rohlf, 1995). Finally,having observed values on the y-axis is also done in order to allow for variability in sampling, which can bedealt with by weighted regression.

17

6.2 Model Validation 6 STATISTICAL METHODS

disadvantages are that the relationship between Y and Y might, in fact, not be linear.Moreover, the least squares procedures is very sensitive to out-liers or to errors differentfrom a statistical normal distribution. In those case intervention from the modeller isrequired to deal with poor testing data.

The procedure is summarized in figure (5). The key parameter to observe in the regressionline is the slope parameter. This should be close to 1 to show good agreement. Values closeto zero would mean no relationship amongst data, and therefore that the best estimateof Y is, merely, the mean of the measured data. Conversely, for slopes greater thanone, the closer to infinity, the lower the relationship between model predictions and fieldobservations.

The regression intercept is also important, and deviations from a zero intercept mean biasin the predictions of the model.

The third parameter to evaluate in exploring the association between modelled and ob-served values is r2, the proportion of variance in Y that is explained by Y . If the rela-tionship is linear, then r2 is a measure of goodness of fit, and estimates the proportionof variation in the observed values explained by the model. However its interpretationmust be done carefully, particularly when comparing regression lines based on differenttransformations of observations and simulations.

Once the regression line has been fitted to the relationship between the model output andthe field data is performed, it is possible to use the resulting statistics to compute a rangeof reliability for the model output (Flavelle, 1992) for any time xp.

(23) syp =

√√√√MSresidual

(1 +

1n

+(xi − xp)

2∑ni=1 (xi − x)2

)

where n is the number of pairs, MSresidual stands for the mean squared error of residuals,which is the sum of squared differences of observed and predicted values divided by thedegrees of freedom ( i.e. n − 2). The bound can be dwrown by adding or substractingthe cuatinty, tn−2syp to the predicted value in the regression line (yp). Therefore, yp =yp ± tn−2syp

This range is said to be empirical, because no direct causal relationship exists between fielddata and model outputs, and the regression error is only to be interpreted as proportion ofthe variance in the linear regression which is not accounted for our model output (Loehle,

18

7 METEOROLOGICAL DATA

1997).

The computation of the different parameters of the regression needed for model validationcan be made by any standard statistical software. We used a free statistical packagecalled R (R Development Core Team, 2005). Examples of the application of the proceduresuggested here, are given later in this report (section ??, figures 12 and 13).

7 Meteorological data

In order to run some of the sub-models, meteorological data were needed. Here we sum-marise the data available from the British Atmospheric Data Centre (BADC) database3

on Air temperature, Solar radiation and Rainfall as recorded at the Dunstaffnage meteo-rological station. With some corrections for the effects of mountains, these data will betreated as representative of the northern part of Argyll, including loch Creran. There isaround 30 years worth of data from the seventies until the late nineties. For some variablesthe time-series are more extensive, but generally good quality data after 2000 is less easyto get from BADC.

Environmental variables can be differentiated into two groups as a function of the seasonalpattern (figure 7). (1) Relative stable seasonal pattern from year to year (Air Temperatureand Solar radiation) and (2) Changing seasonal pattern from year to year (Rainfall). Thisdistinction is useful in establishing a typical pattern to force models.

Equally important for the design of a typical temporal pattern is the long term trend(figure 8) . Over the past 3 decades, rainfall and temperature have increased significantly(just significant, p = 0.096, for rainfall and clearly significant, p < 0.001, for temperature)at a rate of 0.037 and 0.002 per year respectively. However, an insignificant decrease(p = 0.133) was observed for Solar radiation over that period of time.

Finally, an analysis of the residuals from the time series decomposition was performed inorder to determine any any changes on the long term other than a simple linear increase.This was done by means of the cumsum and is shown in figure (9). On the one hand, solarradiation and temperature seem to follow similar patterns. On the other hand, rainfallpattern seem to follow an opposite pattern. Several periods are highlighted in the graph

3Meteorological data is collected by the UK Meteorological Office, who make some of it available toBADC by arrangement with NERC. We got our data from BADC under a licence that restricts its use toresearch.

19

8 THE CATCHMENT RUNOFF MODEL

and they will be discussed in the context of the outputs from the runoff model (section 8),which uses at least 2 of these 3 meteorological variables.

8 The catchment runoff model

Freshwater runoff from a catchment (deemed, here, to mean the same as river discharge’from the rivers draining the catchment) is important in the sECE model because it drivessome components of physical exchange and helps control surface layer thickness and alsobecause it can be (although is not necessarily) a major source of nutrient for a loch or voe.Therefore, we have given it some attention in the present study, and in particular havedeveloped a simple run-off model for a two-segment catchment.

The model, described in this section, was first made in order to simulate flows in the riverCreran during years for which no flow measurements were available. The flow part wassimplifed from the HYSLOPE routine in the CHUM hydrochemical model as applied to anupland catchment in Cumbria (Tipping, 1996). The model is forced by precipitation andevapotranspiration, calculation of the latter being improved for the SARF012 project by useof a Makkink equation described by Allen et al. (1998). Although sophisticated hydrologicalmodels are available, such as the LF2k model used by SEPA, or the MIKE SHE of DHI,and might be used to build up a database of flows in all rivers supplying lochs or voes withfishfarms, the simplified approach adopted here has allowed us to investigate sensitivities tometeorological forcing, and permits coupling (not yet attempted) to a catchment nutrientmodel.

8.1 Principles

Processes operate on catchment segments of downslope length L (on axis l) and alongslopewidth W , at mean elevation E (on axis e) represented in figure (10). Space units aremetres and time units days. Soil (including subsoil) contains a single water-absorbinglayer of thickness h and porosity p; the water level may exceed this thickess, resultingin surface water which runs off more quickly than does water flowing through the soil.Downslope flow is described by Darcy’s Law:

flow = Kh∂e

∂t(typical units are m−2d−1 per cross slope metre)(24)

20

8.2 Water flow 8 THE CATCHMENT RUNOFF MODEL

where K is a hydraulic conductivity with a value that depends on the soil material. Wateris input from precipitation (snow is assumed not to lie) at R (m d−1) to each squaremetre, and lost by evapotranspiration T (m d−1), driven only by solar radiation energy.Other sources of heat (such as from air) for converting liquid water to vapour are ignored.As implemented, the model deals with two segments, 1 being the lower. The model isappropriate for the small and mountainous catchments of rivers draining the west coastof Scotland and some of the islands, where there is a relatively thin layer of soil andsubsoil over rock, and rainfall converts into runoff with only a small delay. Neglecting thestorage of water in long-lying Winter snow might have led to errors for climate conditionsa generation ago, but seems a realistic simplification for present-day conditions.

8.2 Water flow

Water flow is dealt with mainly in the Matlab routine RunOffFun.m. The rate of changeof water volume in the higher Segment 2 is:

∆V2

∆t= −ϕ2W2 + (R2 − T2) L2W2 (typical units are m3 d−1)(25)

Similarly rate of change of volume in the lowewr segment 1 is:

∆V1

∆t= −ϕ1W1 + (R1 − T1) L1W1 (typical units are m3 d−1)(26)

In the case of segment 1, the term ϕ1W1 gives the river flow at the mouth of the catchment.This is the value needed as a boundary input to the loch models.

Flow in any segment n is derived from Darcy’s law (equation 24):

ϕn =(

Khyd,nf1 (hn) + Ksurff2 (hn))(

∆e

∆l

)n

( m2 d−1 per cross-slope metre)(27)

Where the gradients are:

21

8.2 Water flow 8 THE CATCHMENT RUNOFF MODEL

(∆e

∆l

)2

=2(E2 − E1)L2 + L1

(28) (∆e

∆l

)1

=2(E1)L1

(29)

The two functions introduced in equation (27) can be expanded:

(30) f1(h) =

0 if h ≤ 0h if 0 < h < hmax

hmax if h ≥ hmax

and:

(31) f2(h) = p

{0 if h ≤ 0

h− hmax if h ≥ hmax

Water level h (m) in the segment is given by:

h = (V/Vmax) ∗ hmax where Vmax = L ∗W ∗ p ∗ hmax(32)

where the dimensions L and W are those of the segment in question, and hmax is thethickness of the layer of soil and rock, porosity p, that holds the water.

Finally, the gradient conductivity stated in equation (27) is:

(33) Khyd,n = Khyd,max

(f1 (hn)hmax,n

)m

where m can take values from 0 to 2 but is typically nearer to 0 (Tipping, 1996).

22

8.3 Precipitation and evapotranspiration 8 THE CATCHMENT RUNOFF MODEL

8.3 Precipitation and evapotranspiration

According to equations 25 and 26), the volume of water stored in a catchment is alsochanged as a result of the balance between rainfall (Rn) and evapotranspiration (En).

Precipitation (hereafter called, for simplicity, rainfall’) into any segment is used directlyby drawing on time-series from meteorological stations. There may be a need to cor-rect the rainfall measured at coastal stations to the higher values that typically occur inmountainous regions.

Evapotranspiration (Tn) is estimated indirectly. Various approaches were explored, rangingfrom the simple conversion of all absorbed, direct, solar radiation into water evaporation,to complex models that take account of secondary radiation (e.g. from clouds) and type ofground cover. We decided to use an hybrid equation in part simplified from the Penman-Menteith equation as described by and in part using the Makkink equation given by Allenet al. (1998).

As programmed in the function Evapotrans, net evapotranspiration is assumed to resultfrom the sum of: (1) a Radiation term and (2) an Aerodynamic term.

The Radiation term follows the Makkink equation:

EMK = CMK1λ

Rsσ

σ + γ(typical units are mm d−1)(34)

where CMK is the Makkink coefficient (0.63), Rs is the downward flux of short-wave’ (ordirect solar) radiation; λ is the latent heat of evaporation of water; σ is the slope of the satu-rated vapor pressure curve at the ambient temperature; and γ is the psychometric constantand depends on atmospheric pressure, i.e. on altitude. Evapotrans calculates the variablesfrom temperature, altitude, fundamental parameters, and empirical functions.

This equation does not use explicitly include a value for the albedo (or reflectivity) of theland surface, and it also contrasts with the full Penman-Menteith equation, which addslong-wave’ output and input to the short-wave’ solar input. Short-wave’ is otherwise knownas total solar radiation, including the short-wave infra-red, visible light, and ultravioletlight reaching the Earth’s surface from the Sun. Long-wave’ is the heat radiation of longerwavelength emitted by the ground or clouds, the latter forming a secondary source ofheating for the ground.

23

8.4 Parameter values 8 THE CATCHMENT RUNOFF MODEL

The Aerodynamic term can be calculated from wind speed and saturation vapour pressureat the current temperature, using parameter values that may be related to the type ofvegetation cover. In the present version of Evapotrans, however, only the Radiation termis used. This results in satisfactory runoffs for the Creran catchment, and Allen et al. (1998)show that the, averaged over a season, the radiation term is several times larger than theaerodynamic term. Therefore, temperature and Solar radiation data are the only timeseriesneeded to drive Evapotrans.

Finally, it may be noted that no allowance is made in the present model for reducedevaporation from snow. This parallels our discounting of water-storage in snow underpresent climatic conditions.

8.4 Parameter values

The parameter values used for this particular study, based on (Tipping, 1996) and Allenet al. (1998) are displayed in table 1.

8.5 Model and Comparison with data

In order to test the Runoff model, we compared its simulation of water discharge intoloch Creran with guage measurements of the River Creran flowfrom years 1977 to 1978,available in Tayler (1983). In his thesis, Tayler (1983) gives a conversion factor for thefraction that river Creran accounts for the total fresh water input into loch Creran, andthis was used to scale the output from Runoff model as shown in Figure 11.

Prior to the model evaluation, both observed and model data were transformed onto alogarithmic scale. This was done in order to reduce hetreroscedasticity (heterogeneity ofthe variances) and improve the score on validation. It is clearly visible that flow data arehighly volatile and the model was not able to predict such volatility with either accuracy orprecision. However when the data were log-transformed, the fit improved considerably, andalthough there exists a consistent underestimation of river flow (Intercept and the slopeare statistically different from zero and one respectively), they have a good correlation andthe slope is not very different from 1.

However, the Runoff model might be missing an underlying process involved in the RiverCreran Flow. Just notice that the agreement between data and field data improved whenmodel data was moved one step backwards respectively to field data (figure ??). This is

24

8.6 Seasonal variation and long-term trend 8 THE CATCHMENT RUNOFF MODEL

figure highlights the incapacity of our model to translate, sufficiently quickly, rainfall intoriver flow. Changing soil porosity p or thickness h might improved this response. Theadvantage of a (comparatively) simple model is that such factors can easily be identifiedand explored.

Finally, an envelope was drawn over the predicted data in order to identify the possiblevalues for prediction of river discharge(Figure 13)

8.6 Seasonal variation and long-term trend

Because multi-year series of sunshine and rainfall data are available from the British Atmol-spheric Data Centre (BADC), the runoff model could be used to simulate long timeseriesof discharge from the Creran catchment. These timeseries were then subject to analysisfor seasonal patterns and long term trends (see section 6.1)

We fitted a GAM model composed of a long term trend and a seasonal term. The resultof this procedure is shown in figure 14, and identifies a slight change in the seasonal cyclepatern from year to year. However, this change was not major, and we so opted to usea simpler analytical method, which involves estimation of the parameters of a harmonicseries to better explain seasonal change on output of runoff time-series. These models werefitted using Least-squares regression which inference of these models is well documented(Sokal and Rohlf, 1995). Therefore, if the estimation of prediction intervals containing95% of possible response for runoff necessary, they can be easily computed (see Sokal andRohlf (1995) for details on estimation of confidence intervals).

The result of seasonal pattern estimated from the harmonic analysis did not differ muchfrom that estimated with the GAM approach. Simulated Creran catchment flows for years1977 and 1978 are displyed together with the seasonal pattern estimated by harmonicanalysis in figure 15. To force the physical model (Gillibrand and Inall, 2006), a time-series of predicted Runoff values was used using the harmonics regression parameters if nodata is specified.

Also we were interested to see if there was any long term change in the runoff estimates,which would show as a significant trend term in equation (21). Seasonal cycle and linearlong term trend were extracted from the timeseries in several steps (figure 16), followingthe approach defined in section (6.1): Firstly a linear long term trend was extracted;secondly later the seasonal trend was estimated; and finally the remaining variation wasanalysed.

25

9 THE FISHFARM MODEL

The remaining variation, what we have called ε in equation (21) has a mean 0, but smallchanges in this value will mean that a simple model of linear increase in runoff does notadequately capture the long term trend. In order to highlight changes in the mean errorterm, we applied a cumsum over the residuals (figure 17).

Changes in the long term trend for runoff in loch Creran were spotted. Firstly a significantincrease in freshwater runoff was observed during the years of analysis (p < 0.001), andsecondly, variation about that linear trend also were clearly visible (figure 17) and theycan be related with positive and negative events of NAO (North Atlantic Oscillation 4)This result is not surprising due to the main driver of the run off model, rainfall. Indeed,Positive NAO years will increase rainfalls in northern Europe and will be related withincreases in the slope of the cumsum function (figure 17). Alternatively, negative NAOyears, will produce drier years and will be related to decrease on the cumsum function(figure 17).

9 The fishfarm model

This part of the sECE model system deals with direct anthropogenic inputs of nutrientsto a loch, those represented by the ΓS term in equation (5) - in contrast to nutrient fluxesresulting from exchange with the external waters (boundary conditions) or input in thefreshwater runoff. These latter inputs are represented by the ∇ϕSij term in equation (5)and dealt with, as already discussed, by numerical solutions of the transport equations. Ofcourse, the boundary conditions and river contents may be anthropogenically enhanced,but this takes place on the CSTT’s ’C’ scale, and not within the ’B’ scale of the ACmodels.

9.1 Principles

This part is the only place in the sECE model system where a biological process is described.The process is the conversion by farmed fish of organic matter in their food into excretedwaste. This is an undesired byproduct of fish growth, since the farmer wishes to maximizethe conversion of feed into marketable fish, but it is unavoidable. The excreted nutrients arethose referred to here as DAIN (dissolved available inorganic nitrogen’, including nitrogenin the form of is ionized compounds: nitrate, nitrite and ammonium) and DIP ( dissolved

4information about this climatology phenomenon can be found in the following adress:htpp://www.metoffice.com/research//seasonal/regional/nao/index.html

26

9.2 Blackfish’ model 9 THE FISHFARM MODEL

inorganic phosphorus’, mainly in the form of phosphate). Fish excrete dissolved productsof metabolism chiefly through their gills; the relevant compounds of P and N are mainlyphosphate and ammonia. The latter is rapidly ionized to ammonium, plus a little ureawhich is likely to be transformed microbially into ammonium, and so included here inDAIN. In addition, uneaten food, fish faeces, shed scales, etc, sink to the seabed, andare there broken down by (mainly) microbial processes, releasing more phosphate and amixture (depending on conditions) of ammonium and nitrate. Finally, when dissolvedoxygen is available, dissolved ammonium is used by micro-organisms as a source of energyand is converted to nitrate. It is because of these interconversions that the acronym DAINwas introduced, the point being that all the included forms of nitrogen are potentiallyavailable to phytoplankton.

9.2 Blackfish’ model

The model used for conversion follows Black (2001), who summarized data on the use bysalmonids of typical feed. In summary,

• DAIN excreted by fish accounts for 3.83% (by weight) of the food supplied, andexcreted DIP accounts for 0.408%; this is assumed to enter layer 1 of the simulatedloch or voe;

• DAIN mineralized at the seabed from particulate waste accounts for 1.83% of thefood mass, and mineralized DIP accounts for 0.792%; this is assumed to enter layer3.

These values are assumed to be constant throughout the year, the effects of water tem-perature on fish metabolic processes being taken into account because of higher feedingrates when fish are growing faster. This allows a very simple algorithm to be implementedin the blackfish routine, and the only forcing data required are those in time-series offood supplied to the fish (in kg d−1), information that fish farmers are required to makeavailable5.

Finally, because seawater nutrient concentrations are commonly measured in moles, theblackfish code coverts the mass of released nutrient into moles by dividing the mass of

5According to the data assembled by Black 2001 1.2 kg of dry fish food result in the production of 1 kgof marketable fish. If this appears to be a very high efficiency, it should be appreciated that this is wet’fish mass and includes parts such as the head, fins and bones, that are not usually eaten. The 1.2 kg offood contains 96 g of nitrogen and 18 g of phosphorus, of which 26 g of nitrogen and 3.6 g of phosphorusfinds its way into the harvested fish.

27

9.3 Discussion 9 THE FISHFARM MODEL

the nutrient element by its atomic weight - i.e. mass of nitrogen is divided by 14 andmass of phosphorus is divided by 31. Thus, 14 kilograms of DAIN gives 1 kilomole (kmol),which diluted with a million cubic metres of seawater, gives a concentration increase of 1millimole (mmol) per cubic metre.

Most farmed salmon in Scottish waters are grown on a 2 year cycle. Each cycle commenceswhen the fish cages are moved to a new location within a loch, or to a location whichhas been allowed to lie fallow during the previous 2 years, in order to allow the benthiccommunity to recover from the input of particulate waste. The cages are stocked withsmall fish, and food supply is increased as they grow during the first year. Food inputtypically reaches a peak during Summer of the second year, and during this period fish areharvested, usually on several occasions. Fish-farming locations are consented by SEPA,which places a limit to annual fish production. In the present work we have used, asan example, a 2-year time-series of monthly mass (in kg) of food supplied to a farm inloch Creran consented for 1000 tonnes annual production. The timeseries is contained in.../data/FOOD/food.export.dat6, and can be edited or replaced to simulate the impactof different sizes of fish farm.

The evolution of a 2 year harvesting farm is shown in figure (18). We also tried to modelthe farm food supply as a function of two combined models relating to the growth andharvesting of the fish: (1) a logistic model for growth over two years and (2) an annualsinusoidal cycle for temperature-related growth. There are some differences with observedvalues, but this formula can be used in order to simulate fish farm impacts in a sea lochgiven no more information than a consented annual production of fish.

9.3 Discussion

The EU ECASA project is currently evaulating models for the release of nutrient by farmedfish. Complex models are available, but require more types of forcing data and values formore parameters. The current view is that the approach here (Black, 2001) is probablyas reliable as any of the more complex models. However, it may in future versions benecessary to reconsider the fate of the particulate waste. As presently described, it sinksinstantly to the the seabed and immediately mineralises realising nutrient into layer 3. Infuture versions it may be necessary to take account of:

• accumulation of particulate waste and delayed mineralization;6In the incomplete file address or path given here, ... refers to the directory or folder containing the

parts of the sECE model system.

28

10 BOUNDARY CONDITIONS

• losses of nutrients, for example through dentrification (the conversion of nitrate intodissolve nitrogen gas, which is not available to most phytoplankters);

• partition of sinking particulate waste between layers 2 and 3, depending on thetopography of the seabed and the siting of the farm.

10 Boundary conditions

In order to run the sECE the models and the state variables contained in this framework,three different sets of external influences are required. Forcing variables are those variableswhich cannot be dynamically modeled and represent an important part of the change ofthe state variables. These forcing variables have been introduced in the context of themeteorological data (section 7). Likewise, initial conditions are determined by the pre-existing state of the system, and so we need some information about historical values (andtrends) to start a simulation. Finally, the boundary conditions describe how the statevariable are influenced from outside the model domain, in the case of the sECE model byway of nutrient input from rivers or open ocean, and are the object of this section.

The sECE model simulates Disolved Available Inorganic Nitrogen (DAIN) and Disolvedinorganic Phosphorous (DIP), and has been implemented so far for these nutrients inLoch Creran. The loch was chosen because of the great quantity of data describing itshydrography and bio-geochemistry from the early 1970s. Loch Creran connects to thelarger fjord Loch Linnhe, the Firth of Lorn, and eventually to the open sea (figure 19).Most of the boundary conditions come from the station situated in the Greag Islands ( seestation LY1 in figure (19) ) as a proxy of open ocean value for nutrients.

Following the same approach as for meteorological data, we were interested in seasonaland long term patterns for concentration of DAIN and DIP at the Greag Islands samplingsite. The available data for these variables derives from seasonal samples between 1979and 1983 and more frequent samples from 2000 (Grantham, 1983a,b,c; Grantham et al.,1983) (figure 20)7. Because of the changing frequency, any attempt at estimating a longterm trend have to be made with caution. For example, a significant increase in the yearto year concentration of DIP was observed (figure 20b) if only data from 2000 onwardswere considered. This increase was not significant if all the data were used.

7Brian Grantham (SMA) was responsible for chemical measurements in 1979-1983, Johanna Fehling(UHI/SAMS during PhD studies in 2000-2003, and Celine Laurent (Napier) during PhD studies in 2003-2005

29

11 RESULTS FOR LOCH CRERAN

Consequently, we were not able to distinguish any long term variation on the concentrationof nutrients in the Greag Islands, and the seasonal pattern displayed as a blue thick line infigure (20), can be used as forcing data when observations are not available. In figure (21)we represent all the data available combined together with a 95 % envelope containing anypossible value at any point of the season.

For land derived nutrients, we used an average value of measured concentration of nutrientsin river Creran between 2003 and 2004 for DAIN and DIP (unpublished data from PhDstudy by C.Laurent, data not shown here) . This constant value was multiplied to thefreshwater input in order to account for land nutrient loading. The value used was of10 mmol m−3 for DAIN and 0.03 mmol m−3. These concentrations would be consideredtypical of highland rivers with little or no enrichment.

11 Results for Loch Creran

Figures 22 and 23 show the outputs of the sECE mode in its sECEsolver versionl. TheMatlab ode23 numerical integration function was used to obtain the output shown in thissection. The result for the euler numerical integration will be compared in the Discussionsection.

Two years were under study, 1975, when no fish-farm activity was established in LochCreran and 2003, when a well developed industry was producing 1000 tons salmons peryear.

River discharge was simulated by sRunoff for each of these years, and used as part ofthe forcing for ACExR, the remainder of the physical forcing (wind) being provided byclimatology. The boundary conditions for nutrients were also provided by the climatologydiscussed in the previous section. Finally, blackfish was driven by feed data from thesecond year of the management cycle.

Simulation for 1975 seem have a better agreement between observed and predicted valuefor DAIN and DIP in layer 1 than those observed in 2003. However, differences betweensimulations and observations can be partly explained in Summer by the absence of phyto-plankton from the sECE model: these would have been present in the real world, removingnutrients. Differences between boundary conditions and layer 2 were less than 100%. thedifferences were greater when comparing boundary conditions and layer 1. (Note thatsimulation for DAIN in 2003 does not completely comply with these statements, mostly

30

11.1 On technical aspects 11 RESULTS FOR LOCH CRERAN

due to the fact that observed DAIN in the boundary conditions was 0 and so estimates ofrelative difference can become large.

Differences between layers for both years are best assessed in relation to boundary condi-tions or as a difference between the upper 2 layers - as shown in the right hand panels ofthe diagrams. For both years of simulation, layers 1 and 2 differed in both state variables.However, the differences occurred only once per year for DIP (figure 23), whereas they oc-curred several times for DAIN (figure 22b). The ’% change related to BD’ peak occurringin Spring in both years and nutrients is simply a reaction to decreasing external nutri-ents (forced by the imposed boundary conditions). Summer peaks in these graphs may becaused by river discharge (in both years) or fish-farm inputs (in 2003). Relatively smallinputs can have large proportionate effect during Summer, when sea-boundary nutrientconcentrations are low.

Finally a surprising result obtained was the amount of nutrients predicted by the sECEmodel in layer 3 (Figure not shown). It represents until 40 times more nutrient that theother 2 layers. the increase in layer 3 occurred only in 2003, result of the input of particulatefarm waste into layer 3, combined with too slow a rate of mixing between layer 3 and layer2.

11.1 On technical aspects

This report described the progress in making and implementing a seasonally varying version(called sECE) of an Equilibrium Concentration Enhancement model using time-varyingwater fluxes in a sea loch described as a 3-layer system. We created a framework thatwas tested for Loch Creran. We have given an account of all the data used to run themodels, i.e. the forcing variables and the boundary conditions for loch Creran. The lochwas studied between 1972 and 1982, before the develop of a fish farm, and intensivelysince 2000, so adequate data were available. Also, meteorological measurements made atDunstaffnage laboratory for the UK Meteorological Office, providing (by way of BADC)good daily values of some forcing variables since the early seventies.

A great amount of work was done collecting data and homogenizing data formats. Excelspread-sheets and flat text files separated either by coma or by tab, were routinely used.It was found that codes for missing values were not consistent across the range of datacollected. Changes in formats were made in some cases, trying to preserve the originalfrom as much as possible. However, a common task performed for all the historic data was

31

11.2 On the results 11 RESULTS FOR LOCH CRERAN

to convert proprietary software spreadsheets, into flat text files. All the data available forLoch Creran was combined together in a single folder.

We want to stress here the need for universal (and simple) format for storing, collecting,and using data to drive the models. Handling data for Loch Creran was time-consuming;the process might became very tedious if numerous lochs are to be studied.

Matlab posses different functions to import data into its working space. However, someare more complicated than others and need the interaction with the programmer to verifythe import. We can distinguish between data that contains information about modelparameters, which needs to be re-assembled infrequently, and and time-series of data neededto force the models. Generally, data used to force daily varying models is of two types:(1) julian days 8 (2) values of the variable on these julian days. We therefore suggest touse flat text files, ’tab’ separated, composed of 2 columns, the first one column giving thejulian day and the second the value of the variable on that day.

The first few rows of the file can contain some identificatory metadata, including columnheadings, but each of these rows should start with a percentage symbol (%), which Matlabidentifies as the start of a comment line. Such files can be loaded quickly and reliably intoMatlab.

Finally, a key problem faced during the implementation of this framework, was the choiceof methods of numerical integration. A classical eulerian and an adaptive integrationmethod were used. Differences started to be remarkable, when models were forced with realdata (figure 24) Therefore, although a simple routine (sECEeuler) is written to couple thephysical model and the sECE model using the function tracer.m, an different function wasintroduced in this framework in order to allow for adaptive integration method sECEsolver.This matter needs further study.

11.2 On the results

Seasonal variation in Dissolved Available Inorganic Nitrogen (DAIN) and Dissolved In-organic Phosphorus (DIP) has been simulated for loch Creran. The seasonal version ofthe ECE model was used coupled with a three-layer physical model (Gillibrand and In-all, 2006), and without biological process descriptions except for those contain in the fish

8Here, ’julian day’ is used correctly, as the the number of days after the start of the Julian calendar;elsewhere in this report we have used it loosely to mean ’day of the year’ in an annual cycle, or ’day sincethe start of a particular year’.

32

12 LIST OF PARAMETERS

model. Therefore, changes in DAIN and DIP are not due to biological interaction, onlyexternal inputs.

The framework here presented seemed satisfactory. It combines the efforts of modelingwater exchanges (within the loch and with the open ocean and from the freshwater runoff)and nutrient external input (from a fish farm and land derived nutrients). At this stage,different strategies might be taken in order to test different case-scenarios. For instance,models could be run under extreme conditions, to reflect possible responses of the system.Alternatively, averaged values can be used to pinpoint deviations from a normalised situ-ation. Note that some variables present changes in the long term. Therefore, comparisonbetween output from different years should be done carefully.

Two scenarios were presented, and both simulations gave reasonably agreement with theobserved values of the loch. Biological process described in the CSTT model, will refine theresponse of the system. A surprising result obtained was the predicted input of nutrients inthe bottom layer by the fish-farm. At the moment this compartment behaves as a dump ofmaterial, and only renewed cyclically. The effect in the biology will be considerable whenremineralisation will lead to local phytoplankton growth. This effect will be investigatedin detail in the sCSTT model. Similarly, oxygen diminution and other side effects to thatof direct nutrient input by the fish-farm will be worth analyzing within the context of theL-ESV model.

12 List of Parameters

33

12 LIST OF PARAMETERS

Tab

le1:

Lis

tof

para

met

ers

and

stat

eva

riab

les

used

inth

isw

ork.

Her

ew

eco

mbi

ned

para

mte

rsus

edto

for

the

sEC

Em

odel

and

para

met

ers

ofth

ech

ildm

odel

sca

lled

wit

hin

the

sEC

Eru

tine

.Par

amet

erD

escr

ipti

onU

nits

Val

ueM

ain

stat

eva

riab

les

DA

IND

isso

lved

avai

labl

ein

orga

nic

Nit

roge

nm

mol

m−

3st

ate

vari

able

DIP

Dis

solv

edin

orga

nic

phos

phor

ous

mm

olm

−3

stat

eva

riab

lem

ain

subm

odel

sFO

OD

load

ing

kgd−

1fo

rcin

gva

riab

leRun

Off

mod

elϕ

nD

owns

lope

flow

m2

d−1

stat

eva

riab

leR

Pre

cipi

tati

onm

d−

1fo

rcin

gva

riab

leT

Los

sby

Eva

potr

ansp

irat

ion

md−

1fo

rcin

gva

riab

leL

dow

nsl

ope

leng

thm

NA

EM

ean

elev

atio

nm

NA

hla

yer

thik

ness

mfr

om1.

4to

2.1

pla

yer

poro

sity

NA

from

0.21

to0.

28W

Alo

ngsl

ope

wid

thm

NA

Khy

drau

licco

nduc

tivi

tym

d−1

NA

(dep

ends

onso

ilm

ater

ial)

Khyd,m

ax

max

imum

hydr

aulic

cond

ucti

vity

md−

120

(bed

roc)

to12

5(S

uper

ficia

lla

yers

)K

surf

Khyd,n

mN

Afr

om0

to2

an

Alb

edo

c TE

nerg

yne

eded

toev

apor

ate

1m3

ofw

ater

KJ

m3

2.5·1

06

evap

otra

nspi

ration

sub

mod

ule

CM

Kth

eM

akki

nkco

effici

ent

0.63

Rs

the

dow

nwar

dflu

xof

shor

t-w

ave

radi

atio

nJ

m−

2d−

1Fo

rcin

gva

riab

leλ

isth

ela

tent

heat

ofev

apor

atio

nof

wat

erM

Jkg

−1

2.45

σth

esl

ope

satu

rate

dva

por

pres

sure

curv

ekP

aoC−

1fu

ncti

onof

Tem

pera

ture

δth

eps

ycho

met

ric

cons

tant

kPa

oC−

1fu

ncti

onof

atm

osph

eric

pres

sure

,(

alti

tude

)

34

REFERENCES

13 References

References

Allen, R. G., Pereira, L. S., Raes, D., Smith, M., 1998. Crop evapotranspiration: guidelinesfor computing crop water requirements). Tech. Rep. 56, FAO, Rome.URL http://www.fao.org/docrep/X0490E/x0490e07.htm

Black, K. D., 2001. Mariculture, environmental, economic and social impacts of. In: Steele,J., Thorpe, S., Turekian, K. (Eds.), Encyclopeida of Ocean Sciences. Academic Press,London, pp. 1578–1584.

Flavelle, P., 1992. A quantitative measure of model validation and its potential use forregulatory purposes. Adv. Water Resour. 15, 5–13.

Gillibrand, P. A., Inall, M. E., March 2006. Improving assimilative capacity modelling forscottish coastal waters: I. a model of physical exchange in scottish sea lochs. Tech. Rep.167, Scottish Association for Marine Science, Dunstaffnage Marine Laboratory, Oban.

Gillibrand, P. A., Turrell, W. R., 1997. The use of simple models in the regulation of theimpact of fish farms on water quality in scottish sea lochs. Aquaculture 159, 33–46.

Grantham, B., 1983a. Firth of lorne study: report no. 1. introduction and details of pro-gramme, with data for the period february 1979 to august 1981. Tech. Rep. 86, ScottishMarine Biological Association., Oban.

Grantham, B., 1983b. Firth of lorne study: report no. 2. hydrography, nutrients and chloro-phyll in the firth of lorne and its associated sea lochs, 3rd-19th february 1982. Tech.Rep. 87, Scottish Marine Biological Association., Oban.

Grantham, B., 1983c. Firth of lorne study: report no. 3. hydrography, nutrients and chloro-phyll in the firth of lorne and its associated sea lochs, 19th-23rd july 1982. Tech. Rep. 88,Scottish Marine Biological Association., Oban.

Grantham, B., Chadwick, A., Shaw, J., 1983. Firth of lorne study: report no. 4. hydrog-raphy, nutrients and chlorophyll in the firth of lorne and its associated sea lochs, 7th -11th february 1983. Tech. Rep. 89, Scottish Marine Biological Association., Oban.

Loehle, C., 1997. A hypothesis testing framework for evaluating ecosystem model perfor-mance. Ecol. model. 97, 153–165.

35

REFERENCES REFERENCES

Press, W. H., Flannery, B. P., Teukolsky, S. A., Vetterling, W. T., 1989. Numerical Recipesin Pascal. Cambridge University Press, New York.

R Development Core Team, 2005. R: A language and environment for statistical computing.R Foundation for Statistical Computing, Vienna, Austria, ISBN 3-900051-07-0.URL http://www.R-project.org

Sokal, R., Rohlf, F., 1995. Biometry: The Principles and Practice of Statistics in BiologicalResearch., 3rd Edition. Freeman and Co., New York.

Stigebrandt, A., 2001. Fjordenv - a water quality model for fjords and other inshore waters.Tech. Rep. C40, Earth Sciences Centre, Goteborg University, Goteborg.

Tayler, I. D., 1983. A carbon budget for creran, a scottish sea-loch. Phd, University ofStrathclyde,Glasgow.

Tipping, E., 1996. Chum: a hydrochemical model for upland catchments. Journal of Hy-drology 174, 305–330.

Venables, W., Ripley, B., 1994. Modern Applied Statistics with S-Plus. Springer-Velag,New York.

36

14 FIGURES

14 Figures

37

14 FIGURES

S_0Volume VDAIN S

ExchangeE

Figure 1: Schematic representation of the cross boundary flux at rate E of an state variable(DAIN) in box with volume V with the boundary conditions S0

38

14 FIGURES

S0

h1

h2

h3

Qf

V3, S3

V2, S2Qest

(Qest + Qf)

QestQent

∂h1/ ∂t

Qh

V1, S1

Kz23

(Qt + Qi)2

Uw

Kz12

Figure 2: Schematic diagram of the three layer model. Volumes, fluxes and layer paramtersare also represented. Extracted and upgraded from Gillibrand and Inall (2006)

39

14 FIGURES

(a)

(b)

(c)

Figure 3: Boundary data to force the physical model for 1977. (a) Salinity forcing wasobtained from UKHO climatology; the lines show several depths. (b) The wind forcingused daily mean wind speed climatology for Tiree between 1957 and 83. (c) Freshwaterrunoff from runoff model for year 1977

40

14 FIGURES

(a)

Figure 4: Graphycal representation of changes in volume due to estuarine circulation fora 2 box model. Fluxes and principles are extracted from Gillibrand and Inall (2006).

41

14 FIGURES

Yes No

Is β_0=0 ?

Model fitModel #2

Y=β_1X + E

Model fitModel #1

Y=β_0 +β_1X + E

Is β_1=1 ? Is β_1=1 ?

Model

Obs

Figure 5: Schematic clasificationof relationship between models prediction(X) and expectedvalues from direct observation ([E(Y)]) using linear regression. Models will be clasifiedusing a neutral language: Excelent, Good, Fair and Bad models

42

14 FIGURES

sin(2!xT ) xT

cos(2!xT ) x

sin(2!xT 2

) x

cos(2!xT 2

) x

sin(2!xT 3

) x

xcos(2!xT 3

)

"i=1

3(cos(i

2!T ) + sin(i

2!T )) x

Figure 6: Schematic representation of harmonics needed to simulate a seasonal pattern. Trepresent the time period, in this case the seasonal cycle composed by 365 days. X is theaxis representing time. The final line represent a theoretical seasonal cycle composed bythe first 6 harmonics represented in the figure.

43

14 FIGURES

0 100 300

030

001970

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1971

grid$x[xx]

grid

$pre

d[xx

]0 100 300

030

00

1972

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1973

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1974

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1975

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

001976

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1977

grid$x[xx]gr

id$p

red[

xx]

0 100 300

030

00

1978

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1979

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1980

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1981

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1982

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1983

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1984

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1985

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1986

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1987

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1988

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1989

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1990

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1991

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1992

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1993

grid$x[xx]

grid

$pre

d[xx

]

0 100 3000

3000

1994

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1995

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1996

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

030

00

1997

grid$x[xx]

grid

$pre

d[xx

]

Julian day

Sol

ar r

adia

tion

(KJo

ules

m−

2)

(a)

0 100 300

05

10

1972

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1973

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1974

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1975

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1976

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1977

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1978

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1979

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1980

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1981

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1982

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1983

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1984

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1985

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1986

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1987

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1988

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1989

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1990

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1991

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1992

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1993

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1994

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1995

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1996

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1997

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1998

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

05

10

1999

grid$x[xx]

grid

$pre

d[xx

]

Julian day

Mea

n da

yly

tem

pera

ture

(oC

)

(b)

0 100 300

04

8

1972

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1973

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1974

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1975

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1976

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1977

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1978

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1979

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1980

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1981

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1982

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1983

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1984

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1985

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1986

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1987

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1988

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1989

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1990

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1991

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1992

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1993

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1994

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1995

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1996

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1997

grid$x[xx]

grid

$pre

d[xx

]

0 100 300

04

8

1998

grid$x[xx]

grid

$pre

d[xx

]

Julian day

Mea

n da

yly

rain

fall

(mm

)

(c)

Figure 7: Changes in the seasonal cycle in the long term for (a) solar radiation, (b) airtemperature, and (c) rainfall at the Dunstaffnage meteorological station. The red linesshow the long-term mean annual cycle, extracted by time-series decomposition. The blacklines show smoothed values in each year.

44

14 FIGURES

Time

log(

Dat

a)

02

46

8

Time

Sea

sona

l

−1.

5−

0.5

0.5

1.0

Time

Long

Ter

m T

rend

7.23

7.25

7.27

7.29

Rem

aind

er

1970 1975 1980 1985 1990 1995

−8

−6

−4

−2

02

(a)

Time

Dat

a

−5

05

1015

20

Time

Sea

sona

l

−4

−2

02

46

Time

Long

Ter

m T

rend

8.6

8.8

9.0

9.2

9.4

9.6

Rem

aind

er

1975 1980 1985 1990 1995 2000

−10

−5

05

(b)

Time

log(

Dat

a)

01

23

4

Time

Sea

sona

l

−0.

4−

0.2

0.0

0.2

Time

Long

Ter

m T

rend

1.05

1.07

1.09

Rem

aind

er

1970 1975 1980 1985 1990 1995

−1

01

23

(c)

Figure 8: Time series decomposition of log-transformed values of (a) solar radiation, (b) airtemperature, and (c) rainfall (properly, precipitation) at the Dunstaffnage meteorologicalstation.

45

14 FIGURES

1970 1975 1980 1985 1990 1995 2000

−20

0−

100

010

020

0

time

Cum

sum

of t

he r

esid

uals

19721977

19821983

19891993

(a)

1970 1975 1980 1985 1990 1995 2000

−40

0−

200

020

040

060

080

0

time

Cum

sum

of t

he r

esid

uals

19721977

19821983

19891993

(b)

1970 1975 1980 1985 1990 1995 2000

−10

0−

500

5010

015

020

0

time

Cum

sum

of t

he r

esid

uals 1972

19771982

19831989

1993

(c)

Figure 9: Cumulative sum of residuals from time series decomposition (See figure 8) for(a) solar radiation, and (b) Air temperature.

46

14 FIGURES

Figure 10: Schematic representaiton of the equations governing the run-off model for GlenCreran

47

14 FIGURES

020

040

060

0

051015202530

Tim

e (d

ays)

R creran flow m3/s

Fig

ure

11:

Riv

ercr

eran

Flo

wfo

rye

ars

1977

and

1978

esti

mat

edby

dire

ctm

easu

rem

ents

(bro

ken

line)

and

mod

elfr

om2

runo

ffm

odel

(thi

ckbl

ack

line)

.N

ote

that

tota

lfr

eshw

ater

runn

offin

tolo

chcr

eran

is2.

65ti

mes

Riv

erC

rera

ndi

scha

rge

(Tay

ler,

1983

).T

his

conv

ersi

onfa

ctor

was

used

totr

ansf

orm

Fres

hwat

erru

noff

into

rive

rcr

eran

Dis

char

ges.

48

14 FIGURES

●

●●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●●

●●●●

●

●

●●

●●

●●

●

●

●●

●●

●

●

●●●

●

●●●

●●●●●●●

●

●

●

●

●●

●●

●●

●●

●

●

●

●●

●

●

●●

●

● ●

●

●

●

●●

●

●

●

●●

●●

●

●●

●

●●

●

●

●

●●

●●

●●●●●●●●●●

●●●●●●●●

●

●●●

●●●●

●●

●●●●

●●

●

●●● ●●●

●

●

●

●●

●●

●

●●●●●●●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●●●●

●

●●

●

●

●

●

●

●●●

●●●

●●

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−2 0 2 4

−2

02

4

Lag 5

Slope = 0.64R2 = 0.26

Model

Obs

erve

d

Figure 12: Evaluation of modelled River Creran flow againgst field data for years 1977 and1978. the propouse of this figure is to highlight how the model seem to take 1 or 2 days tocapture the

49

14 FIGURES

010

020

030

0

010203040

Julia

n da

y

Creran flow m3s−1

Fig

ure

13:

Riv

erC

rera

nflo

w(d

otte

dlin

es)

iwit

hru

noff

pred

icti

on(c

onti

nuos

line)

in19

77.

Ove

r-im

pose

d(g

ray)

anen

velo

pco

ntai

ning

95%

ofin

terv

aldr

own

from

the

rela

tion

ship

sbe

twee

nob

serv

edan

dm

odel

edda

ta.

50

14 FIGURES

0 100 300

05

1525

35

grid$x[xx]

grid

$pre

d[xx

]/(36

00 *

24)

1977

0 100 3000

515

2535

grid$x[xx]

grid

$pre

d[xx

]/(36

00 *

24)

1978

0 100 300

05

1525

35

grid$x[xx]

grid

$pre

d[xx

]/(36

00 *

24)

1979

0 100 300

05

1525

35

grid$x[xx]

grid

$pre

d[xx

]/(36

00 *

24)

1980

0 100 300

05

1525

35

grid$x[xx]

grid

$pre

d[xx

]/(36

00 *

24)

1981

0 100 300

05

1525

35

grid$x[xx]

grid

$pre

d[xx

]/(36

00 *

24)

1982

0 100 300

05

1525

35

grid$x[xx]

grid

$pre

d[xx

]/(36

00 *

24)

1983

0 100 300

05

1525

35

grid$x[xx]

grid

$pre

d[xx

]/(36

00 *

24)

1984

0 100 300

05

1525

35

grid$x[xx]

grid

$pre

d[xx

]/(36

00 *

24)

1985

0 100 300

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1525

35

grid$x[xx]

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$pre

d[xx

]/(36

00 *

24)

1986

0 100 300

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1525

35

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grid

$pre

d[xx

]/(36

00 *

24)

1987

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1525

35

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d[xx

]/(36

00 *

24)

1988

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1525

35

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]/(36

00 *

24)

1989

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1525

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]/(36

00 *

24)

1990

0 100 300

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1525

35

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id$p

red[

xx]/(

3600

* 2

4)

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0 100 300

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1525

35

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grid

$pre

d[xx

]/(36

00 *

24)

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00 *

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1525

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00 *

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1525

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d[xx

]/(36

00 *

24)

1995

0 100 300

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1525

35

grid$x[xx]

grid

$pre

d[xx

]/(36

00 *

24)

1996

Julian day

Inpu

t Flo

w (

m3s

−1)

Figure 14: Estimated changes on seasonal pattern of Runoff In loch creran from years 1977to 1998. The thik red line represents the overall average seasonal pattern. overinposed asa thin black line are seasonal pattern estimated in each year.

51

14 FIGURES

0 200 400 600

010

2030

4050

6070

Julian Day

loch

Cre

ran

Flo

w m

3s−

1

+++++

+

+

+++++++

+

+

++++++++++++++

+

+

++

+++++

+++++++++++++++++++++

+

+

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++

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Figure 15: Outputs from the Runoff model representing flows (continuos black line) in lochCreran for yeas 1977 and 1978. Also estimated seasonal cycle of Runoff using harmonicregression. The thik blue line represent the averaged predicted value. The broken red linerepresents the envelop containing 95 % of possible values of the whole series of Runoff data.Values are converted to be compared with River Creran fresh water input (broken blackline)

52

14 FIGURES

Time

Dat

a

020

4060

80

Time

Sea

sona

l

−5

05

Time

Long

Ter

m T

rend

14.0

14.5

15.0

15.5

16.0

Rem

aind

er

1980 1985 1990 1995

−20

020

4060

Figure 16: Seasonal decomposition of runoff

53

14 FIGURES

time

Cum

sum

of t

he r

esid

uals

1980 1985 1990 1995

−1e

+08

0e+

001e

+08

2e+

08

1977

1982

1983

1989

1993

Figure 17: Cumsum plot of residuals from seasonal decomposition. Years relate to periodswhen NAO index suffered a mayor change and is related with the shape of the cumsumseries.

54

14 FIGURES

● ●●

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5 10 15 20

050

100

150

200

250

300

month

food

Figure 18: fish fed in a two year harvesting cycle (open circles) and model trying to predictthe behavior in a fish farm plant

55

14 FIGURES

Figure 19: Location of Loch Creran showing the position of the boundary-conditions station(Creag Isles, LY1)

56

14 FIGURES

●●

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7000 8000 9000 10000

02

46

810

Julian day since 1978

DIN

con

cent

ratio

n

●

●

19781979198019811982198320002001200220032004

1978 1979 1980 1981 1982 1983 2000 2001 2002 2003 2004

(a)

●

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7000 8000 9000 10000

0.0

0.2

0.4

0.6

0.8

1.0

Julian day since 1978

DIP

con

cent

ratio

n

●

●

19781979198019811982198320002001200220032004

1978 1979 1980 1981 1982 1983 2000 2001 2002 2003 2004

(b)

Figure 20: Time series of Available Boundary conditions for Disolved Available InorganicNitrogen (a) and Disolved inorganic phosphorous (b) in Loch creran from 1978 to 2004.Blue line represent seasonal cycle modelled by using all the data available. In red is seaonalcycle modelled just for recent years (from 2000). note that for the case of DIP, a significantincrease in the year to year trend is observed, but it is not consistent with values in thelate 70’s- early 80’s

57

14 FIGURES

●●

●

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●

0 50 100 150 200 250 300 350

02

46

8

Julian Days

DIN

con

cent

ratio

n

●

●

19781979198019811982198320002001200220032004Seasonal cycleUprLwr

(a)

●

●

●

●

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●

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0 50 100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0

Julian days

DIP

con

cent

ratio

n

●

●

19781979198019811982198320002001200220032004Seasonal cycleUprLwr

(b)

Figure 21: Seasonal change on Disolved Available Inorganic Nitrogen (a) and Disolvedinorganic phosphorous (b) in Loch creran from 1978 to 2004. The legend code the datafor different years with different colors and points and the seasonal pattern (continuos redline ) with envelope (broken red line) containing 95% of the data

58

14 FIGURES

(a)

(b)

Figure 22: Output from sECEsolver for DAIN in loch Creran for years 1975 (a) and 2003(b). Panels from top to bottom and from left to right: Prediction for layer 1; differencesin percentage between layer 1 and boundary conditions; prediction for layer 3; differencesbetween layer 2 and boundary conditions (percentage); differences between layer 1 and 2,differences beteen layer 1 and 2 in percentages.

59

14 FIGURES

(a)

(b)

Figure 23: Output from sECEsolver for DIP in loch Creran for years 1975 (a) and 2003(b). DIstribution of each panel is similar to figure (22)

60

14 FIGURES

(a)

(b)

Figure 24: Differences in the output by using euler numerical integration methods (con-tinuous line) and adaptive integration (broken line) over the first two layers of loch Creran(layer 1 blue, layer 2 green) in 2003: a) using smooth forcing data to drive all the models,b) using real data to force models.

61

A APPENDIX

A Structure of the Matlab script

A.1 Folder structure

In this appendix we will explain the steps required to run the sECE model in Matlab.In order to run the model we need to unzip the compressed file with all the informationnecessary for the model to work. we name it SARFAC.zip. Once the file is unziped adirectory tree structure will be installed into your machine (Figure 1).

Figure 25: Folders structure after the SARFAC.zip has been decompressed

6 folders are created in your directory tree. Also a matlab file cdsarfac.m is present inthe root of the SARFC directory. The other Folders present are:

• ACExR contains ? (?) model.

• data contains the data needed to run the models. At the moment of the creation ofthis report, only data for Creran regioin was available and the information containedis:

– Air temperature,

62

A.2 Set the working path A APPENDIX

– Chlorophyll CHL

– Solar radiation SolarRadiation

– Rainfall,

– Food loading,

– Runoff

– Dissolved Available Inorganic Nitrogen (DIN)

– Dissolved Inorganic Phosphorous (DIP)

• Runoff contains all the routines needed to run the runoff model (section 8)

• sECE contains all the rutines to run the sECE model. It also include the rutineblackfish.m commented in the document (section 9).

• The folder called Validation possesses data of DIP and DAIN collected in lochCreran in 1975 and 2003. These data is used in one of the plotting routines tocompare with model outputs.

In order to run the sECE models, a valid verion of Matlab is needed. We coded the scriptsusing version 7.1 of matlab. Functions and scripts here written, where not tested in earlierversions, so some crashes might occur.

A.2 Set the working path

Once the matlab program is up and running, we have to place our workspace in the samedirectory the zip file was installed. To do so, we have 2 options (figure 2): (1) any of themenues in matlab, (2) type the path for this folder in the command window. Suppose wehave installed the SARFAC.zip in the folder called /Users/kike/Hdrive/models/Coupled/. We have to give the full path in order to place ourselves into the right directory. ForUnix machines the command to type is

>> cd (’/Users/kike/Hdrive/models/Coupled/SARFAC/’)

>>

If we are using a windows machine the command should use is slightly different (note theuse of c: at the beginning of the path)

63

A.2 Set the working path A APPENDIX

Figure 26: Snapshot from matlab window indicating where working path can be changed

>> cd (’c:/Users/kike/Hdrive/models/Coupled/SARFAC/’)

>>

Once we have set the working directory then type in the command cdsarfac to run thescript which will load all the path infromation into your matlab path. the output issomehting like this:

64

A.3 Run the program A APPENDIX

>> cdsarfac

Subfolders has been added to your path

When your session is closed, the configuration is reset

to your original path setup. the list of folders added are:

/Users/kike/Hdrive/models/Coupled/SARFAC

/Users/kike/Hdrive/models/Coupled/SARFAC/ACExR

/Users/kike/Hdrive/models/Coupled/SARFAC/Figures

/Users/kike/Hdrive/models/Coupled/SARFAC/RunOff

/Users/kike/Hdrive/models/Coupled/SARFAC/data

/Users/kike/Hdrive/models/Coupled/SARFAC/data/AirTemp

/Users/kike/Hdrive/models/Coupled/SARFAC/data/CHL

/Users/kike/Hdrive/models/Coupled/SARFAC/data/DIN

/Users/kike/Hdrive/models/Coupled/SARFAC/data/DIP

/Users/kike/Hdrive/models/Coupled/SARFAC/data/FOOD

/Users/kike/Hdrive/models/Coupled/SARFAC/data/Rain

/Users/kike/Hdrive/models/Coupled/SARFAC/data/RunOff

/Users/kike/Hdrive/models/Coupled/SARFAC/data/SolarRadiation

/Users/kike/Hdrive/models/Coupled/SARFAC/sECE

/Users/kike/Hdrive/models/Coupled/SARFAC/validation

>>

A.3 Run the program

There are two scripts in the sECE folder which possess the algorithms to run the sECEmodel: (1) sECEeuler, which uses Euler numerical integration, and (2) sECEsolver, whichuses Runge-Kutta numerical integration. We will use the first function for year 2003 toobtain predictions of DAIN and DIP from the sECE model.

We will use the function called sECEeuler, and the output wil be an structure array calledout.

>> out=sECEeuler(’Creran’,2003)

65


Once this has been typed the following messages appear in the command line providingand it will ask eventually the user to choose running other subrutines. By default therunoff model is run firstly and the output used to force the physical model. This is themessage we obtain in the command line:

------------------------------------------

------------------------------------------

sECEeuler started at: 04-Sep-2006 16:45:12

abort with ctrl-C, ctrl-break or <apple>-<.>

when prompted, you may select [default] with <RET>

=== Do you want to run the Runoff model [Y]? Y/N y

The first option we have is to run the Runoff model. If we press yes, or enter, it willrun the sRunOff routine for the year selected. Eventually it will output the followingmessage.

Calling RunOff; may take some time: please wait for next prompt

RunOffCall is calculating river flow for : 2003

Once the runoff model has been run, we are given the option to run the physical model

=== Do you want to run the Physical model [Y]? Y/N y

The output that appear in the screen is directly displayed from the ACExR routine.

66


Calling ACExR model; may take some time: please wait.

Searching sea loch database for Loch Creran

Deriving hypsographic function

Reading boundary forcing data

Initialising variables

Modified Const.ce = 1.71e-09

Calculating exchange rates

H1 min and max: 2.026 24.312

V1 min and max: 27492359.98474 192697562.9869

Initial Layer thicknesses : 20.464 20.056 10.13

Initial Layer salinities : 33.1056 33.4366 33.7732

Physical model has been run and data are now in memory

Finally, without any other user requirement, the function steps through the code andproduce the following messages until the numerical integration finishes.

67


Calling sECE model itself, with Euler integration;

-- may take some time: please wait for script to end.

Run off data for 2003 loaded

Nutrients (DIN) for Year 2003used a la place!

Sorry but Nutrient forcing data for 2003 end on day 150

year 2002 used instead

Nutrients for Year 2003



Chlorophyll for Year 2003used!



The fish model has been run.

******************************

End of sECEeuler at: 04-Sep-2006 16:49:13

******************************

We now have an structure array composed by 3 parts. H3, containing 356 values of layerthikness for the 3 different layers assumed in the ACExR model, P3, containing DIP con-centration for 365 days and 3 layers, and S3 containing DAIN concentration for 365 daysand 3 different layers.

68


>> out

out =

H3: [365x3 double]

P3: [365x3 double]

S3: [365x3 double]

An script has been written in order to assess the output of the sECE model. the routine iscalled sECEplot, and requires the information of the year when the simulation is run (inthis case 2003) and the output from the numerical integration (out)

>> fig=sECEplot(2003,out)

We have assign the name fig because the function sECEplot gives as an output the in-formation of the 3 graphic windows created. They can be called later in order to savethe windows, change axis or any other manipulation. For instance, if we want to save thesecond window containing information on DAIP as a .png file called toto.png, we couldtype the following command into the command line

>> saveas(fig.f2,’toto.png’,’png’)

69

report on the seasonal equilibrium concentration ... · report on the seasonal equilibrium...

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