A mass conservative 3-D numerical model for predicting solute ¯uxes
in estuarine waters
Yan Wu, Roger A. Falconer
*Cardi School of Engineering, Cardi University of Wales, P.O. Box 686, Cardi CE2 3TB, UK
Received 27 January 1999; received in revised form 24 July 1999; accepted 21 August 1999
Abstract
A re®ned three-dimensional layer-integrated model to predict accurately salt and cohesive sediment transport in estuarine waters is described herein. A splitting algorithm has been used to split the three-dimensional transport equation into a horizontal two-dimensional equation and a vertical one-two-dimensional equation due to the dierent length scales. An additional source term asso-ciated with the layer average of the free-surface ¯ow is introduced in the conservative form of the layer-integrated pollutant transport equation. The one-dimensional QUICKEST scheme has been extended to two dimensions and included in the layer-in-tegrated advective±diusion equation. A modi®ed one-dimensional ULTIMATE algorithm has also been added to avoid unphysical numerical oscillations. Numerical tests for discontinuities have been carried out to study the performance of the ULTIMATE QUICKEST scheme used in the present model. The model has also been used to simulate solute transport in an idealized harbor. It has been found that the additional source term was crucial for the mass conservation of pollutant. Finally the re®ned model has been applied to simulate salt and cohesive sediment transport in the Humber Estuary, UK. Good agreement has been obtained with the ®eld measured data. Ó 2000 Elsevier Science Ltd. All rights reserved.
Keywords:Estuarine and coastal hydrodynamics; Solute and sediment transport; Salinity; Cohesive sediment; Mass conservation; ULTIMATE QUICKEST
1. Introduction
With a growing awareness of pollution problems in coastal and inland waters, in recent years there has been a considerable increased eort in developing and ap-plying numerical models to predict ¯ow ®elds and solute concentration distributions in coastal and estuarine waters. The requirement for any model to simulate ac-curately the advective±diusion processes is a vital as-pect in model development.
It is widely known that the ®rst-order unwind scheme suers from serious numerical diusion and the high-order central dierencing schemes are plagued by grid scale oscillations (or wiggle) when they are used to solve convection-dominated problems. A number of schemes have been developed to overcome these diculties in the ®eld of Computational Fluid Dynamics (CFD). For example, the second-order Lax±Wendro scheme, which
was introduced by Lax and Wendro [1] for computing ¯ows with a shock wave; the MacCormark scheme [2], which includes a predictor±corrector version of the Lax± Wendro scheme; and the second-order upwind scheme of Warming and Beam [3]. However, it was found that these schemes also introduce numerical dispersion and did not eliminate unphysical spurious oscillations near discontinuities. Leonard [4] developed QUICK and QUICKEST schemes of advective±diusion equation for steady and unsteady ¯ows, with these schemes being based on the quadratic upstream interpolation. These schemes have been widely used due to their high nu-merical accuracy, i.e. third-order. However, even though the QUICKEST scheme greatly reduces the unphysical oscillations caused by numerical dispersion, this scheme still suers from numerical oscillations near disconti-nuities.
In recent times a number of high-order oscillation-free schemes have been constructed, with the total vari-ation diminishing (TVD) scheme being one of the most popular [5±9]. Leonard [10] pointed out that the second-order TVD schemes achieved their oscillation-free *
Corresponding author. Tel.: 874280; fax: +44-01222-874597.
E-mail address:falconerra@cardi.ac.uk (R.A. Falconer).
results by the use of locally varying positive arti®cial diusion, which distorted local discontinuities to simple smooth pro®les. He then proposed a ``monotonizing'' universal limiter (ULTIMATE) for the advection problem to banish unphysical overshoot and non-monotonic oscillations, without corrupting the expected accuracy of the underlying method. In Leonard's UL-TIMATE scheme the eective time-averaged normalized cell face value, rather than the normalized cell face value at time leveln, is limited by a monotonicity-maintenance criteria to eliminate nonmonotonic oscillations. Cahy-ono [11] carried out a detailed study of 36 of the most popular ®nite dierence schemes for the advection problem, with his comparisons showing that the UL-TIMATE scheme was particularly attractive, since it was more general than the other schemes considered and was easier to apply.
In this paper, details are given of the requirement of a three-dimensional layer-integrated numerical model to predict accurately solute transport in estuarine and coastal waters. In Section 2 the three-dimensional transport equation has been split into a horizontal two-dimensional equation and a vertical one-dimen-sional equation due to the dierence length scales in the horizontal and vertical planes. A conservative form of the horizontal two-dimensional layer-integrated ad-vective±diusion equation has been derived, in which an additional source term associated with the layer average of the free-surface ¯ows has been introduced. The QUICKEST scheme originally developed by Leonard [4] for the one-dimensional advective±diu-sion problem has been extended to two-dimenadvective±diu-sions for the horizontal layer-integrated advective±diusion equation in Section 3, and the modi®ed one-dimen-sional ULTIMATE algorithm constructed by Wu and Falconer [12] has been used to avoid unphysical nu-merical oscillations. For the vertical part of the ad-vective±diusion equation the implicit power-law scheme has been used to predict accurately the solute distribution over the depth.
In Section 4 numerical tests of the one-dimensional pure advection of a square wave and the two-dimen-sional advection and diusion of a square column have been undertaken to verify the present model. The model has also been used to study solute transport in an idealized square harbor, to check the conservation behavior of the scheme. The results have shown that the additional source term associated with the layer average of the free-surface ¯ow is crucial for the mass conservation of pollutant. This conclusion is consistent with the results of a study for the case of two-dimen-sional depth integrated advective±diusion equation by the authors [12]. Finally, in Section 5 the model has been applied to predict salt and cohesive sediment concentration distributions in the Humber Estuary, UK.
2. Governing equations
2.1. Hydrodynamic equations
The governing hydrodynamic equations describing ¯ows in coastal and estuarine waters are generally based on the three-dimensional Reynolds equations for in-compressible and unsteady turbulent ¯ows [13]. The hydrodynamic layer-integrated equations can be derived by integrating the three-dimensional continuity and momentum equations over layers. A sketch of the layers and the relative variable locations in the x±zplane are illustrated in Fig. 1. The three-dimensional layer-inte-grated equations can be written as:
(i)Continuity equation for layer k
wkÿ1=2 ÿ
XK
kk
oDzu
ox 8
<
:
oDzv
oy 9
=
;
; 1
wherewis the vertical velocity;Dzthe layer thickness;K the total number of layers; uand vare layer-integrated velocities and are de®ned as
u 1
Dz
Z kÿ1=2
k1=2
u x ;y;z;tdz; 2a
v 1
Dz
Z kÿ1=2
k1=2
v x ;y;z;tdz: 2b
For the ®rst layer, which describes the free-surface, the continuity Eq. (1) becomes
of
ot
XK
k1
oDzu
ox 8
<
:
oDzv
oy 9
=
;
0; 3
wherefis water elevation above (or below) datum.
(ii) Momentum equations for layerk
where q is the ¯uid density; f the Coriolis parameter; qx uDz andqy vDz are layer-integrated velocities per
unit width inx andy directions, respectively;g the ac-celeration due to gravity; eh the horizontal eddy
vis-cosity, which is assumed to be constant in the vertical and equated to the depth-integrated eddy viscosity [14] in the current study; andevis the vertical eddy viscosity,
which is represented by a two-layer mixing length model [15].
At the free surface (wherek1), the terms wukÿ1=2
and wvkÿ1=2can be eliminated using the kinematic free
surface condition and Leibnitz rule, and the shear stresses sxz kÿ1=2
and syz kÿ1=2
are equated to the wind
stresses. At the bed, the terms wuk1=2 and wvk1=2
become zero due to the no-slip boundary condition, and the shear stressessxz k1=2
andsyz k1=2
are equated to the
bed shear stresses [13].
2.2. Solute transport equation
The partial dierential equation describing the advective±diusion in three dimensions can be written as
o/ where/is the sediment concentration, salinity or other solute constituent concentration;u;vandware the ¯uid velocity components in the x, y and z directions re-spectively; ws the apparent sediment settling velocity
which vanishes for salinity and other solute constituents; Dx,Dy andDzare the turbulent diusion coecients in
thex,yandzdirections, respectively; andSis the source or sink term.
As a result of ¯oc aggregation due to inter-particle collision and the surface electro-chemical forces, cohe-sive sediments settle by ¯ocs rather than by individual particles. It was found that the settling velocity of the ¯ocs depended strongly on the suspended cohesive sed-iment concentration [16]. The dependence of the settling velocity on the local concentration generally falls within one of the three following ranges.
(i)Free settling(/</10:1ÿ0:3 g/l)
ws
sÿ1gD2 s
18m 7
where mis the kinematic viscosity for clear water, sthe speci®c density of suspended sediment, andDsis the ¯oc
diameter.
(ii)Flocculation settling(/1</</20:3ÿ10 g/l)
wsk1/4=3 8
wherek1 is an empirical coecient.
(iii)Hindered settling(/>/2)
wsws01ÿk2 /ÿ/2 4:66
9
wherews0is the value ofwsat the concentration/2, and
k2 is the inverse of the concentration at whichws0.
In estuarine and coastal waters, the horizontal length scale is generally much larger than the vertical one, thus an operator splitting algorithm is used to split the three-dimensional advective±diusion equation (6) into a horizontal two-dimensional equation and a vertical one-dimensional equation.
The one-dimensional advective±diusion equation is gives as
with the following boundary conditions for sediment.
At the free surface
wÿws/ÿDz
o/
At the bed wheresbis the bed shear stress,sdthe critical shear stress
for deposition,sethe critical shear stress for erosion,qdep
andqero represent deposition and erosion rates,
respec-tively at the bed.
The deposition rate proposed by Krone [17] was used in this study given as
qdep ÿws/b 1
where /b is the near-bed cohesive sediment
concentra-tion, with typical values of the critical shear stresssbfor
deposition being 0.04±0.15 N=m2 [17,18]. Likewise, the
erosion rate for soft natural mud can be represented by the following empirical expression [19]:
qero
whereais an empirical coecient,qf is ¯oc erosion rate
when sbÿse0. Typical values of the critical shear
stresssefor the erosion of soft mud are 0.07±0.17 N=m2.
For salinity or other solute constituents, the above boundary conditions become
w/ÿDz
o/
oz 0 at the free surface; 15
o/
oz 0 at the bed: 16
The horizontal two-dimensional advective±diusion equation can be written as
o/
At the open boundaries, the concentration data from ®eld measurements were used for in¯ow conditions, with the concentration pro®les being obtained by extrapola-tion using a ®rst-order upwind dierence scheme for out¯ow conditions. At the bank boundaries, the normal derivatives of the concentration were set to zero.
To be consistent with the layer-integrated hydrody-namic model, the above horizontal two-dimensional advective±diusion equation was integrated over the
layers to give the following two-dimensional layer-inte-grated equation
In applying the ULTIMATE algorithm, a criteria was used to check that the solute concentration / maintained monotonicity. In coastal and estuarine ¯ows the water depthHmay vary rapidly, with the result that the thickness of the top and bottom layers may vary suddenly, thus the monotonicity of/Dzmay be dierent from the monotonicity of the solute concentration /. Therefore the advective±diusion equation (18) needed to be rearranged. Partially dierentiating the ®rst three terms in the left-hand side of Eq. (18), and using the continuity equations (1) and (3), we obtain the following conservative form of the layer-integrated equation:
o/
for the middle layers;
/ ou
for the bottom layer:
8
Comparing with the original equation (17), an addi-tional source term Sa now appears in the
layer-inte-grated equation. It will be shown in the next section that this additional source term is vital for the mass conser-vation of pollutant.
3. Numerical methods
velocity. The Crank±Nicolson scheme was used to solve the layer-integrated hydrodynamic equations, with the vertical diusion terms being treated implicitly and the remaining terms being treated explicitly. Two iterations were performed to solve the couple problem between the depth-integrated and layer-integrated equations. The ¯ooding and drying processes were modeled using a robust scheme developed by Falconer and Chen [21]. For details of the hydrodynamic model, see Lin and Falconer [13].
For the solute transport processes, ®rst the layer-in-tegrated two-dimensional advective±diusion equation (19) was solved horizontally, and then the one-dimen-sional vertical advective±diusion equation (10) was solved vertically.
Firstly, the two-dimensional layer-integrated equa-tion (19) was discretized in time space by integrating equation (19) fromtn totnDt
where the explicit of forward Euler method were used for diusion and source terms.
Integrating Eq. (21) over the ®nite volume shown in Fig. 2, we have the following ®nite dierence equation:
/nP1/nPÿ cxe/e
where/represents the respective average cell face values over the time incrementDt
cxe
are Courant numbers at east, west, north and south cell faces; Sn represents the average of Sn over the control
volume, and Sna represents the average of San over the
control volume. The superscript n is an index for time, the subscript n denotes the northern cell face.
The gradient terms at cell faces in above ®nite dif-ference equation are approximated by central dier-encing and the average cell face values/ne,/nw,/nnand/ns are estimated using the two-dimensional QUICKEST scheme, extended from the one-dimensional QUICK-EST scheme of Leonard [4].
In a similar manner to the one-dimensional case treated by Leonard [4], consideration of the following exact integral formulation for two-dimensional purely advective ¯ows For pure advection of a scalar/at a constant vector velocity U;V, the exact transient solution over a time intervalscan be shown to be
/ x;y;s / xÿUs;yÿVs;0
/n xÿUs;yÿVs: 24
Using the above concept and a Taylor series expan-sion, the ®rst integral on the right-hand side of Eq. (23) can be approximated as
Z Dy=2
Similarly, the remaining integrals on the right-hand side of Eq. (23) can be respectively approximated as
Z Dy=2
The integrals on the left-hand side of Eq. (23) can be similarly approximated by
Z Dx=2
Substituting Eqs. (25)±(29) into Eq. (23) and using the QUICK formulae for/nw,/ time intervalDt. Here only the average value at west cell face is given for convenience as
Even though the QUICKEST scheme is third-order with upwind bias, it still produces some unphysical overshoot and undershoot when sharp changes of concentration exist. Thus, the universal limiter designed for one-di-mensional problems by Leonard [10] was used at each control volume face to eliminate unphysical overshoot and undershoot.
exist. This variation results from the fact that the node values in the normal direction only are used to limit the control volume face values, which estimated using the information in both the normal and lateral directions. Wu and Falconer [12] divided the estimated face value into three parts. Taking time averaged face value/was an example
Note that the average value has been split into three parts, i.e. the main part uw0; the time correction part,
contributed from the ¯ow normal to the cell face only i.e.uwt;and the time correction part, introduced by the
¯ow parallel to the cell face i.e.uwl:
By limiting the main part of the average cell face value, and the time correction term brought in by the normal ¯ow only, a modi®ed one-dimensional ULTI-MATE algorithm for the two-dimensional problem has been constructed to avoid the variation of local dis-continuities. The modi®ed one-dimensional ULTI-MATE algorithm can be expressed as
/wUL UL uw0uwt uwl; 32
where UL denotes LeonardÕs one-dimensional uni-versal limiter, see [10] for details.
The one-dimensional advective±diusion equation (10) was discretized for a non-uniform grid spacing in the vertical direction (see Fig. 1). Since the diusion term was very important for the vertical concentration distribution and some of the gird sizes were very small near the sea bed and the water surface, the backward or implicit Euler method was used to avoid very small time steps. The power-law scheme [22] was adapted to cal-culate the total mass ¯ux. The resultant discretization equation for the vertical one-dimensional advective± diusion equation (10) is therefore
ÿaS/nkÿ11aP/kn1ÿaN/nk11b; 33 and b, superscripts TandBdenote the control volume face,Pe the grid Peclet number,Fis the mass ¯ow rate.
4. Numerical tests
A series of numerical tests were carried out to verify the performance of the present model. First the con-vection of a square wave in a uniform ¯ow was studied, then the advective±diusion of an initial square column concentration distribution was investigated to show the performance of the present model, and ®nally the model was used to simulate the solute transport in an idealized square harbor to illustrate the importance of the addi-tional source term associated with the layer average of free surface ¯ows.
4.1. Pure advection of a square wave and advective± diusion of a square column
Firstly, a square wave advecting in a one-dimensional uniform stream was studied to test the performance of the present model when discontinuities existed. The problem had the following initial condition
/ x;t0 1; x16x6x2;
0; 06x<x1; orx2<x61;
34
where 06x61,x10:05,x20:15.
In the computation the advective velocity was unity, the grid size was 0.01, the time step was 0.002, and the total simulating time was 0.6. Fig. 3 shows the results using the ULTIMATE QUICKEST scheme in the present model, together with results for the ®rst-order upwind, second-order upwind, Lax±Wendro, Mac-Cormark, and QUICKEST schemes.
show no oscillations and predict the discontinuities with very high resolution.
Secondly, the advection and diusion of a square column concentration distribution in a uniform two-dimensional ¯ow has been simulated. The computa-tional domain, boundary and initial conditions of the problem studied were described using the following initial and boundary value partial dierential equations
o/
ot u o/
oxv o/
oy mx o2/
ox2 my o2/
oy2; 0<x; y<1;
35a
/ x;y;0 1; x16x6x2;y16y6y2; 0; other x;y;
35b
/ x;y;t 0; x0;1; or y0;1 35c
with the analytical solution being given as
/ x;y;z;t
1
4 erf
x2ÿxut
2
mxt
p
erf ÿx1xÿut 2
mxt
p
erf y2ÿyvt 2 m
yt
p
erf ÿy1yÿvt 2 m
yt
p
; 36
where erf is the error function.
In this test, the advective velocity componentsuandv were set to unity, the diusion coecientsmxandmywere
both equated to 0.0005, a mesh of 100100 with grid sizesDxand Dy of 0.01 was used, and the time step Dt was 0.003, with a total simulating time t of 0.6. Fig. 4 shows the above exact solution and the model results. A comparison of the numerical results and the exact so-lution at the section y0:7 are given in Fig. 5. The results illustrate the excellent performance of the UL-TIMATE QUICKEST scheme as used in present model.
4.2. Solute transport in a square harbor
To investigate the contribution of the additional source term introduced by the layer average of the free surface ¯ow, the transport of a solute driven by tidal currents in an idealized square harbor, was also simulated. The harbor had plan-form dimensions of 2.5 km2.5 km, with a symmetric entrance of width 250 m, and an open sea area of 1.0 km2.5 km sited just beyond the harbor entrance, see Fig. 6.
A horizontal bed with the mean water depth of 6 m was assumed. The tidal ¯ow was generated by a sinu-soidal water elevation variation at the open boundary,
with a period of 12.4 h and an amplitude of 1 m. The initial concentrations inside and outside the harbor were 50 and 0 respectively. A mesh of 7050 grid squares, with a uniform grid size of 50 m was used. Five layers were used in the vertical, with the thickness of the top layer being 2 m at mean water level and with the other layers being 1 m thick. The time step was set at 15 s. In order to check the conservativeness of the model, all computations were stopped after 8.0 h of simulation since some solute was started to advect out of the computational domain beyond this time.
Fig. 7 gives the contours of the ®rst layer solute concentration distribution in harbor, calculated both with and without the additional source term. It can be seen that: the results obtained with the additional source term, as given in Fig. 7(a), appear to be reasonable and are within the initial range of 0±50, whilst the results without the additional source term, as given in Fig. 7(b), exceed the initial range with the maximum value being larger than 60. Since there was no mass eux from within the computational domain during the simulation period, then the total amount of solute mass within the domain should remain unchanged up to this time. The model with the additional source term was found to give perfectly conservative results, whereas the model with-out the additional source term proved to be mass pro-ductive. Fig. 8 shows the percentage mass produce by the model missing the additional source term as a function of time, where it can be seen that the model without the additional source term has led to a large production of solute mass during the simulation time.
5. Model application to the Humber Estuary
The Humber Estuary is one of the main estuaries in UK, situated along the coast of northern England, and strategically important in terms of its input to the North Sea and shipping links to mainland Europe. The main part of the estuary is about 62 km long, from Trent Falls to Spurn Head, and the tidal in¯uence extends for
Fig. 6. Sketch of the idealized square harbor.
Fig. 4. Exact solution and numerical results of advective±diusion of a square column: (a) exact solution; (b) numberical results.
Fig. 5. Comparison of numerical results and exact sollution at section
another 62 km up the River Ouse and about 72 km up the River Trent. A six-year project LOIS (Land-Ocean In-teraction Study) was set up in 1992. It aims to quantify and simulate the ¯uxes and transformation of solute (including: sediments, nutrients, and contaminants) into and out of the coastal zone. The main study area is the UK East Coast from Berwick upon Tweed to Great Yarmouth, concentrating on the Humber and its catch-ment, and to a lesser extent the River Tweed. As part of this study within the LOIS project, the present model has been set up and applied to simulate the salt and cohesive sediment transport in the Humber Estuary from Trent Falls to just beyond Spurn Head as shown in Fig. 9.
Field measured data of water elevations, velocities, salinity and suspended sediment concentrations were collected by Plymouth Marine Laboratory during the LOIS project at ®ve locations, including: Station SG13 (53° 35.190N, 0° 13.900E), SG24 (53° 32.060N, 0°
15.290E), SG10 (53° 36.820N, 0° 11.350E), which were
positioned in Humber plume; Station SG23 (53°
35.710N, 0°2.170E) was positioned within the estuary at
Hawke; and Station at the Trent Falls region of the upper Humber (53°42.80N, 0°37.80W).
The model area was represented horizontally using a mesh of 11856 uniform grid squares, with a length of 500 m. Vertically, eight layers were used with the thickness of the top layer being 4 m at mean water level, and with the other layers each being 3 m thick. The streamline boundary was located such that the bound-ary was as parallel as possible to the dividing streamline separating the north±south alongshore current and the estuarine in¯ow±out¯ow. The water elevation recorded at Station G13 was chosen as the seaward boundary condition to drive the tidal current. Due to the lack of data regarding the ¯uid volume ¯ux data at Trent falls, the water elevation at Trent falls was used as the land-ward boundary condition. A bed roughness of 10 mm was used, as suggested by model calibration for a pre-vious study [13]. Since measurements taken at the survey sites mentioned above were not simultaneous, i.e. the measurements at Station G13 and Hawke were taken for neap tide, whereas conditions at Trent Falls were taken during the mid-tide, the water elevations at Trent Falls therefore had to be adjusted to neap tide conditions. Calibration was carried out during the adjustment ac-cording to the water elevations and mean velocities measured at Hawke. A comparison of the velocities for the dierent layers is given in Fig. 10. As can be seen a reasonable level of agreement was obtained between the model predicted and ®eld measured results.
Since the Humber Estuary is generally a well-mixed estuary with salinity varying by typically less than 5 ppt over the depth, only one layer was used to compute the salinity distribution within the estuary, which was therefore equivalent to using a two-dimensional depth-integrated model. For the sediment transport predic-tions, as mentioned previously eight layers were used to predict the vertical distribution of cohesive sediments. The comparison of the ®eld measured and model pre-dicted salinity at Hawke is shown in Fig. 11, with good Fig. 8. Percentage mass produced by the model missing the aditonal
source term.
agreement being obtained between the predicted and measured data.
Due to a lack of information on the cohesive sedi-ment size measured at Hawke Anchorage, the sedisedi-ment (¯oc) size of 20lm was assumed, with this being a size at which aggregation becomes negligible. The critical shear stress for deposition was set to:sd0:07 N=m2, and the
critical shear stress for erosion assumed to be: se0:15 N=m2. These values were in the ranges
pro-posed respectively by Krone [17] and Thorn and Parsons [23], and were re®ned by trial and error. The ¯oc erosion parameters of a8:3 andqf 0:0000042 were used as
suggested by Thorn and Parsons [23]. Fig. 12 shows a comparison of the ®eld measured data and the model predicted suspended cohesive sediment concentration variation with time. As the concentration variation with depth from the model predictions was small, only the depth mean concentration was plotted in Fig. 12. The results show good agreement between the model pre-dictions and the ®eld measured data in the regime of deposition. However in the regime of erosion, the ®eld data show a large variation with depth, which cannot be represented in the model predictions. This was thought to be due to the exclusion of non-cohesive sediment transport in this study, wherein a large amount of non-cohesive sediment may be suspended in the water col-umn during the erosion period, whilst due to its larger settling speed than the cohesive sediments, most of the Fig. 10. Comparison of predicted and ®eld measured velocities of
Hawke, taken on 8 June 1995.
Fig. 9. Map of the Humber estuary and the locations of ®eld observation stations.
Fig. 11. Comparison of predicted and ®eld measured salinity at Hawke, taken on 8 June 1995.
non-cohesive sediments may settle out. The mixture of cohesive and non-cohesive sediments should therefore be studied further in the future.
6. Conclusion
A three-dimensional layer-integrated numerical model has been re®ned to simulate salinity and cohesive sediment transport ¯uxes in estuarine and coastal wa-ters. A combined layer-integrated and depth-integrated scheme was used to solve the hydrodynamic equations. The original three-dimensional advective±diusion equation was split into a two-dimensional horizontal and a one-dimensional vertical equation to implement the variation between the horizontal and vertical scales. To be consistent with the three-dimensional layer-inte-grated hydrodynamic model, the two-dimensional hor-izontal advective±diusion equation was integrated over the layers to give the layer-integrated advective±diu-sion equation. An additional source term associated with the layer average of free surface ¯ows was intro-duced into the conservative form of the layer-integrated equation. It has been shown by numerical tests that the additional source term is vital for mass conservation of pollutant. A two-dimensional modi®ed QUICKEST scheme has been obtained and used in this study to achieve high accuracy. To avoid the numerical oscilla-tions when a large concentration gradient exists, a modi®ed one-dimensional universal limiter ULTI-MATE has also been used. Two test cases having exact analytical solutions have been used to test the mode, with the results illustrating that the model gave accurate results.
Finally, the model was applied to predict the salt and cohesive sediment distribution in the Humber Estuary, UK, where good agreement has generally been obtained between the predicted and measured results. The model was found to predict non-cohesive suspended sediment accurately in a previous study and to predict cohesive sediment concentration levels accurately during the de-position regime. However, the predictions were not so accurate during the erosion regime and further labora-tory and ®eld studies are required to attain a better understanding of the governing rheological processes involved.
Acknowledgements
This is a LOIS publication number 357 of the LOIS Community Research Programme, carried out under a Special Topic Award from the Natural Environment Research Council. The data collected for the Humber Estuary model application were provided by Dr. R.J. Uncles of the Plymouth Marine Laboratory,
UK. The authors are grateful to NERC for support-ing this study and to Dr. Uncles for the provision of data.
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