Modelling the suspended matter dynamics in a marine environment using a three dimensional σ coordinate model: application to the eastern Irish Sea

Modelling the suspended matter dynamics in a marine environment using a three dimensional r coordinate model: application to the eastern Irish Sea R. Periaan ~nez * Departamento Fı ısica Aplicada 1, E.U. Ingenierı ıa Teecnica Agrı ıcola, Ctra Utrera km 1, 41013 Sevilla, Spain Abstract A numerical model to study the suspended matter dynamics in the sea has been developed. The model solves the three dimensional hydrodynamic equations, that are written in normalized r coordinates so that vertical resolution is not reduced in the shallower regions. Wave–current interaction has also been included. Simultaneously, the model solves the suspended matter equation, also written in normalized coordinates, that includes advection, diffusion and settling of particles. Deposition and erosion of the sediment, that are formulated using threshold velocities, are incorporated into the sea bed boundary condition of the suspended matter equation. A standard formula for flocculation has also been used. The model has been applied to the eastern Irish Sea. Computed and observed suspended matter concentrations have been compared: both set of data are in good agreement. The model has also been tested through the study of the dispersion of radionuclides released from a nuclear fuel reprocessing plant. Sensitivity studies have been carried out. Keywords: Tides; Sediments; Erosion; Deposition; Wind waves; Water quality 1. Introduction Research of suspended sediments in tidal regimes has usually been focused on net rates of scour or accretion. However, in recent years there has been an increasing interest in modelling suspended matter since this is essential for water quality models. First, when sediment is stirred up * Tel.: +34-95-4233669; fax: +34-95-4232644. E-mail address: [email protected] (R. Peri aa~ nnez). nutrients, trace metals and organic contaminants are released into the water column. In turn, as suspended sediment settles out the dissolved chemicals may be scavenged. Secondly, since the light intensity at depth depends inversely on suspended sediment load in the water column above, suspended matter impacts on primary productivity [1]. Thus, mathematical models have been developed to simulate the suspended matter distributions in different aquatic environments and under different approaches. However, although the applications of sediment transport models are wide spread, work on their development is relatively scarce. One reason for this lack of models is that many of the variables associated with sediment transport are not well determined [2]. Abril and Garc ııa-Le oon [3] developed a two dimensional, depth averaged, model to describe the basic aspects of suspended matter dynamics in the sea over large time scales. Thus, the model was formulated in terms of residual (averaged) water circulation. The same approach, but using a three dimensional model, was used [4] to study suspended sediments in the North Sea. Works in [2] and [5] studied the short scale (both spatial and temporal) aspects. These models solve the depth-averaged advection–diffusion equation with the deposition and resuspension terms, both depending on the instantaneous water state. Thus, the models require the simultaneous solution of the hydrodynamic equations under tidal forcing. Both models were applied to study the suspended matter distribution and the sedimentation processes in estuarine environments. A similar approach was adopted [6], but using a three dimensional model, to study a small harbour siltation problem. Holt and James [7] studied, using a 3D baroclinic model, the distribution of suspended matter particles introduced from several individual coastal sources in the southern North Sea. By convention, particulate matter in suspension is defined as the material that is retained on a 0.4–0.5 lm pore size filter. Smaller material is considered to be dissolved. Muddy sediments consists of clays with some variable silt content and particle diameter is [8] <62.5 lm. Larger particles such as sands settle much more rapidly out of suspension in water than mud particles and bed load transport is the most important mechanism in moving this coarse sediments [1,8]. Usually, only particles with a diameter <62.5 lm are considered when modelling suspended matter dynamics [2,3,5,6,9,10]. Indeed, it has been pointed out [11] that for all practical purposes mud can be regarded as synonymous of suspended matter. Also, contaminants have little affinity for coarse materials and, on the other hand, they have little influence in light intensity through the water column since they are mainly transported as bed load. Thus, the dynamics of muddy sediments is much more interesting for water quality studies. The objective of this paper is to present a three dimensional model that can be applied to study the suspended matter distribution in a large marine environment. Interest will focus on the basic aspects of suspended matter dynamics; aspects that have to be considered when developing a water quality model. The model solves the three dimensional hydrodynamic equations, which are written in normalized rcoordinates so that vertical resolution is not reduced in the shallower areas, and, simultaneously, the suspended matter equation is solved too. This equation, also written in normalized coordinates, includes advection–diffusion processes, settling, deposition and erosion of the sediments. Only muddy sediments are considered in the model. The enhancement in bed stress due to wave–current interaction may be essential for suspended sediment transport, and this is particularly important during major wind events (when wave heights are large and swell is significant). Thus, a wave–current interaction model [12] has been included in the suspended matter transport model. The model has been applied to simulate the suspended matter distribution and the sedimentation rates in the eastern Irish Sea. A nuclear fuel reprocessing plant at Sellafield (see Fig. 1) releases radionuclides to the sea. Some radionuclides, as for instance Pu and Am, have a large affinity to be fixed to the solid phase (suspended matter and bottom sediments). Thus, the information provided by this model is essential if we are interested, for instance, in studying the dispersion of such pollutants. Indeed, this suspended matter model has been applied to simulate the dispersion of 239;240Pu. The comparison between observed and computed plutonium levels in water and suspended matter provides an extra validation of the suspended matter model. The model equations are presented in Section 2. After, results are presented and discussed. Some sensitivity studies have also been carried out. Fig. 1. Map of the computational domain. The star is Sellafield nuclear fuel reprocessing plant and circles are the points where suspended matter concentrations were measured (see Section 3.2). Each unit in the xand yaxis is 5000 m (grid element number). Contours represent the distribution in % of particles with a diameter <62.5 lm in the top sediment layer of the eastern Irish Sea. A and B give the positions of compartments (16, 20) and (10, 12), respectively. 2. The model The marine area under study is represented by a grid containing a certain number of grid cells or compartments. The grid has uniform spacing in the horizontal, Dxand Dy, and spacing in the vertical direction is Dr, as will be shown below. Each compartment contains a certain concentration mof suspended matter in g=m3. 2.1. Hydrodynamics The three dimensional hydrodynamic equations are written using normalized rcoordinates in the vertical. This way a constant number of grid boxes is used in the vertical at each horizontal grid point and resolution is not reduced in the shallower areas. The transformation to rcoordinates is [13] r¼zþf hþf;ð1Þ where his the undisturbed depth of water, fis the sea surface displacement from the mean level due to tidal oscillations and zcoordinate is measured from the mean sea level to the bottom. Thus, the hydrodynamic equations are transformed from the interval f6z6hinto the constant interval 0 6r61. The form of the equations in normalized coordinates has been given previously [13–15] and will not be repeated here. The boundary conditions applied at the sea surface are: qNou or  r¼0 ¼ðhþfÞFs;ð2Þ qNov or  r¼0 ¼ðhþfÞGs;ð3Þ where Fsand Gsdenote the components of the wind stress acting on the sea surface along the xand ydirections, Nis eddy viscosity and qis the water density. Similarly, at the sea bed qNou or  r¼1 ¼ðhþfÞFb;ð4Þ qNov or  r¼1 ¼ðhþfÞGb;ð5Þ where Fband Gbare the two components of the bed stress. A flow dependent eddy viscosity, N, has been used in the model. This formulation has been used previously and has given good results for tidal flow studies [16–18] N¼CNffiffiffiffiffiffiffiffiffiffiffiffiffiffi  uu2þ vv2 ph;ð6Þ where CN¼0:0025 is a dimensionless experimentally measured coefficient and  uu and  vv are depth mean currents along the xand yaxis, respectively. Eddy viscosity has been taken constant in the vertical. This approximation has also been used previously [17,19]. 2.2. Wave–current interaction model The effect of enhanced bed turbulence due to wind waves influences the tidal current by an increase in the current friction factor, fc, thus producing an increase in the bed stress components Fband Gb. These are written using the following linear laws [14,15]: Fb¼1 2fcquðbÞ;ð7Þ Gb¼1 2fcqvðbÞ;ð8Þ where uðbÞand vðbÞare the near bed current components, usually specified 1 m above the bed [12]. It is also usual to define a friction coefficient as kf¼fc=2. Although a quadratic law for bottom friction is now more extended than a linear formulation, the linear law has been used since a faster convergence of the equations is obtained [15]. Also, there are no appreciable differences if the model is calibrated using a quadratic law instead of the linear one [15]. Following [12], the total bed shear stress, st, based upon a current shear stress, sc, and a wave shear stress, sw,is st¼scþswð9Þ with sw¼1 2fwqU2 w;ð10Þ where fwis the wave friction factor and Uwis the maximum wave orbital velocity Uw¼aww sinhðkhÞð11Þ with awwave amplitude, wwave angular speed and kwave number determined from the dispersion relation w2 g¼ktanhðkhÞ;ð12Þ where gis the acceleration due to gravity. It is assumed [12] that the current does not influence the wave field (this is a consistent assumption with the method used in this work in which the model is run with the wave field supplied externally, and thus, there is no dynamic feedback from the hydrodynamic model). The calculation of an effective drag coefficient fctaking into account wave effects is carried out as follows. The wave friction velocity is Uw¼ffiffiffiffiffiffiffiffiffiffi sw=q pð13Þ with swgiven from Eq. (10). At t¼0, an initial fc, not accounting for waves, is computed from fc¼K lnð30zrÞ=kbc  2 ;ð14Þ where K(0.4) is the von Karman’s constant, kbc is taken as the Nikuradse roughness kb (kb¼30z0;z0being the roughness length) and zris a reference level above the bed usually taken [12] as 1 m. Once fcis determined, the current friction velocity is Uc¼ffiffiffiffiffiffiffiffiffi sc=q pð15Þ with scthe vector sum of Fband Gbfrom Eqs. (7) and (8). The combined friction velocity Ucw for waves and currents is Ucw ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U2 cþU2 w q:ð16Þ The apparent bottom roughness kbc felt by the current due to the presence of waves is given by kbc ¼kbC1 UcwAb Uwkb  b ð17Þ with b¼1Uc Ucw ;ð18Þ C1¼24:0ð19Þ and a wave near bottom excursion amplitude Ab¼Uw=w. This value of kbc is used at the next time step to determine fcfrom Eq. (14) and hence the bed stress in the three dimensional hydrodynamic model from Eqs. (7) and (8). A more detailed description of the wave–current interaction model can be seen in [12]. 2.3. Suspended matter equations The three dimensional suspended matter equation has been obtained by transforming the equation proposed by Nicholson and O’Connor [6] into rcoordinates. Such equation includes advective–diffusive transport plus vertical fall (settling) of suspended particles om otþuom oxþvom oyþwom or ¼o oxKh om ox  þo oyKh om oy  þ1 ðhþfÞ2 o orKv om or  1 hþf oðwsmÞ or; ð20Þ where mis the suspended matter concentration in g=m3,wis the vertical water velocity along the raxis, uand vare water velocities along the xand yaxes, Khand Kvare the horizontal and vertical diffusion coefficients, respectively, and wsis the settling velocity of suspended particles. The vertical diffusion coefficient can be written as a function of the eddy viscosity [20] Kv¼N;ð21Þ where the non-dimensional number ranges [21] from 0.1 to 0.5. It is assumed that there is no flux of particles through the sea surface. Deposition and erosion of the sediment are incorporated into the sea bed boundary condition [6] 1 hþfKv om or  r¼1 ðwsmÞr¼1¼EfqMwsðbÞmðbÞ1 q qcd :ð22Þ The first term of the right member of the equation represents erosion: Eis the erosion constant or erodability that also acts as a scaling factor, Mis some power of the water velocity, typically [22] in the range 2–5, and fgives the fraction of particles with a diameter <62.5 lm in the sediment. This way, the formula derived by Fukuda and Lick [23] and Lavelle et al. [24] has been used for the erosion rate, although the factor fhas been included to take into account that only particles with a diameter <62.5 lm can be incorporated as suspended matter into the water column, as has been discussed above. It is considered that erosion occurs only when the near bed current magnitude, q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2ðbÞþv2ðbÞ p, is larger than an erosion threshold velocity qce. Otherwise the erosion term is set to zero. The second term represents deposition of particles on the sea bed, that has been based on the concept by Teisson [25] and also used by Clarke [2]. Note that ðbÞ means that the corresponding magnitude is evaluated at the reference level zr¼1 m above the bed. qcd is the deposition threshold, thus deposition occurs only if q<qcd; otherwise this term is set to zero. Small sediment particles tend to join together to form larger units called flocs. Thus, these particles are termed cohesive sediments [8]. Flocculation affects the settling velocity in such a way that it increases with concentration at low concentrations, then attains a maximum value and thereafter decreases due to hindered settling. A standard formula used to represent the effect of flocculation is [2,6,26–29] ws¼amb;ð23Þ where aand bhave to be obtained from measurements or from model calibration. It must be noted that hindered settling occurs at suspended matter concentrations of the order of [28] 103 ppm, thus Eq. (23) is valid when concentrations are smaller than this value. Since concentrations in the Irish Sea are of the order of 1 ppm (see below) hindered settling effect is not included in the model. Indeed, concentrations of the order of 103ppm are not found in the sea [28]. A source term must be included to the suspended matter equation in those grid cells located along the coastline. This term represents the input of particles from runoff of continental waters [3] and can be written as cIi;j;ð24Þ where crepresents the mass of suspended load per continental water volume unit and Ii;jis the total volume of continental water that enters compartment i;j.Ii;jwill be different from zero when the compartment is situated along the coastline. The solution of the suspended matter equation provides the suspended matter concentration at each position in the model domain and at each time step, but also gives information of the sedimentation processes that take place in the area under study. Thus, the net sedimentation rate, RS, is obtained as the difference between the deposition and erosion terms RS¼wsðbÞmðbÞ1 q qcd EfqM:ð25Þ 2.4. Numerical solution The hydrodynamic equations are solved using a finite difference explicit scheme, although the Saul’ev method [29] is applied to the vertical diffusion term to retain stability [30]. Some boundary conditions must also be given. At closed boundaries the normal component of the current and the flux of suspended particles are set to zero. Along open boundaries, amplitude and phase of the sea surface elevation are specified from observations. Since our objective consists of reproducing the basic features of suspended sediment dynamics in the system, only the main tidal constituent, M2, has been used. Of course, better results would be obtained if other constituents (at least N2and S2) are included, since they will enhance erosion. However, as will be shown below, the basic aspects of suspended matter distribution (and also of plutonium dispersion) can be obtained although only the M2tide is included. Moreover, as will also be shown, it seems that suspended matter concentrations are not dominated by erosion events. Thus, it is not expected that the increase in bed stress produced by these other constituents produce a significant change in erosion and in suspended matter concentrations. The surface current component that is normal to the open boundary is obtained from a radiation condition [21,31]. The open boundary condition described in [15] is applied to the suspended matter equation. The advection and diffusion terms in the suspended matter equation are solved using secondorder accuracy explicit schemes [21] and the Saul’ev method is again applied to the vertical diffusion term. A FORTRAN code has been developed to solve the equations involved in our model. Such code was implemented on a HP X-Class computer. 3. Results and discussion 3.1. Model application The computational domain, covering the eastern Irish Sea, is presented in Fig. 1. The model has a horizontal resolution Dx¼Dy¼5000 m and time step is fixed as Dt¼60 s. Ten layers are considered in the vertical, thus Dr¼0:1. All stability conditions [21] are satisfied with these selections. Water depths are introduced from bathymetric maps and range from 55 m in the west to a shallower region around the British coast. The fraction of particles with a diameter <62.5 lmin the sediment, f, has been specified from observations [3,32] and the distribution is also presented in Fig. 1. It can be seen that there is a mud bank that lies along the British coast and a smaller one located between Anglesey and the Isle of Man. Sea surface elevations and phases along the open boundary have also been obtained from observations [33]. The mean value 0.3 has been taken for in Eq. (4). The horizontal diffusion coefficient has been fixed as Kh¼500 m2=s since results in agreement with observations are obtained with this value. It has been suggested [34] that for the Irish Sea 500 <Kh<900 m2=s and, also, this value has been successfully used in previous modelling work [15]. Indeed, the advection–diffusion dispersion equation has been previously calibrated, which consists of selecting the optimum diffusion coef- ficients and boundary conditions, through the numerical study of the dispersion of 137Cs in the eastern Irish Sea [15]. In such paper, observed and computed distributions of dissolved 137Cs were compared, providing a wide validation of the dispersion model. The optimum selections obtained in [15] are now used in this work to simulate the suspended matter dynamics. One of the main difficulties in sediment transport modelling is the method of obtaining the erosion and deposition thresholds and the erodability constant. Thus, they are selected, by trial and error, in such a way that they produce the best fit of model results to observations, although parameters must be physically realistic. A value of 0.18 m/s has been used for the deposition threshold velocity [5,28] and a typical value [8,35] of 0.21 m/s has been taken for the erosion threshold velocity. Indeed, it has already been pointed out [2] that due to the small size of cohesive sediments they tend to be slow falling and thus there is some lag between when deposition ends and erosion begins. During the intermediate time interval, the sediment suspension is maintained by water turbulence with no apparent effect on erosion or deposition. Thus, usually [2] qcd <qce. On the other hand, after some model runs, a value of 2:7104gm 11=2s5=2was selected for the erodability constant and it was taken M¼3:5. In Eq. (23), it was taken a¼1:7106and b¼1:6ifmis given g=m3and wsin m/s. Since suspended matter concentrations in the Irish Sea are of the order of 1 g=m3, settling velocities of the order of 106m/s are obtained, which is the order of magnitude that can be found [22,36] for the settling velocity of cohesive sediments. It can be found in literature [27,28] that branges between 0.5 and 2.0. A value of 1.6 has also been obtained [26] and was used in previous modelling work [5]. It can be found in literature that aranges from 103to 109[2,26]. c(Eq. (24)) was fixed, after a calibration exercise, as 25 g=m3. Although there can be seasonal variations in this parameter, it has been taken constant. Thus, the model gives an estimation of the average suspended matter concentration over the sea. The input of continental waters, Ii;j, was obtained from [37], also presented in [3]. 3.2. Suspended matter distribution in the eastern Irish Sea The hydrodynamic part of the model was tested by comparing observed and computed values of tide amplitudes and phases and semi-major axis magnitude and direction of the M2current ellipse at different depths and locations. After a calibration process, the friction coefficient was initially fixed as kf¼0:011 (without wave–current interaction) for the whole model domain. The difference between computed and observed tidal amplitudes is always <10%, only in one point is 17.4% (comparisons have been made for 12 points). In the case of tidal phases, absolute values of the differences between observed and computed phases range from 0°to 27°, being the mean value 10.5°. The difference between computed and observed semi-major axis magnitude of the M2 current ellipse ranges from 1.5% to 45%. Only in three points errors are above 25% (12 points used for comparison). The absolute value of the difference between observed and computed orientation of the axis ranges from 0.1°to 30.5°, being the mean value 7.6°. Details can be seen in [15] and will not be repeated here. For the M2tide, strong rectilinear flow with current ellipses aligned in a west–east direction occurs in the northern and southern parts of the eastern Irish Sea, separated by an area of significantly weaker, near circular, ellipses to the east of the Isle of Man. With The model response when the settling velocity is changed (increasing and decreasing ain Eq. (23) by a factor 3) is shown in Fig. 9(c). It can be seen that an increase in aleads to lower suspended matter concentrations since settling is enhanced. Alternatively, higher concentrations are obtained when ais decreased. Again, variations in aaffects mainly the regions located close to the coast, where suspended matter concentrations are higher (see Eq. (23)). Finally, it has been found that essentially the same results are obtained if an erosion threshold velocity is not used. Indeed, it has been recently pointed out [22] that the stipulation of a minimum threshold velocity below which no erosion occurs often has little effect in tidal conditions where velocities exceed 0.5 m/s, as is the case [15]. On the other hand, too low suspended matter concentrations are obtained if a deposition threshold is not used. Thus, deposition threshold must be considered in the deposition term. 4. Conclusions A dynamic model to study the suspended matter distribution in the eastern Irish Sea has been developed. The model solves the three dimensional hydrodynamic equations, which are written in Fig. 9. (a) Measured and computed suspended matter profiles (ppm) following the points of [41], shown in Fig. 1, in surface waters. ‘‘computed’’ are the model results when parameters presented in Section 3.1 are used. ‘‘A’’ are results when Eis increased by 3 and ‘‘B’’ are model results when Eis decreased by a factor 3. (b) Same as (a), ‘‘A’’ and ‘‘B’’ are results when cis doubled and halved, respectively. Same as (a), ‘‘A’’ and ‘‘B’’ are results when ais increased and decreased, respectively, by a factor 3. normalized rcoordinates, with a flow dependent formulation of the eddy viscosity. Simultaneously, the model solves the equation that governs the suspended matter dynamics, which is also written in normalized coordinates. The suspended matter equation includes advection, diffusion and settling of particles. Deposition and erosion are included into the sea bed boundary condition of the equation and are formulated using threshold velocities. A standard formula to represent flocculation has also been used. The model includes a source term to take into account the input of particles from runoff of continental waters. The model has shown that suspended particles are well mixed vertically in the eastern Irish Sea and that suspended matter concentrations are not dominated by erosion events. Indeed, small sedimentation rates have been obtained, showing that no major erosion or deposition is taking place at present time. Measured and computed suspended matter concentrations in the sea have also been compared. Model results are in agreement with observations, thus, it seems that the model gives a realistic representation of the suspended matter distribution in the eastern Irish Sea. It has been found that wave–current interaction affects the erosion rate of the muddy area located between Anglesey and the Isle of Man, but erosion is not significantly altered in the rest of the sea. However, there is advection of suspended particles due to the wind induced current. The suspended matter model has been used to simulate the dispersion of 239;240Pu released to the sea from Sellafield nuclear fuel reprocessing plant. It has been found that there are significant differences between the surface and bottom Pu concentrations in suspended matter. Thus, the use of a 3D model is relevant for water quality studies, although there are not important differences between the surface and bottom suspended matter concentrations. Acknowledgements Work partially supported by EU Contract FI4PCT960046, Spanish CICYT Contract AMB971720-CE and ENRESA. References [1] A.W. Morris, M.J. Howarth, Bed stress induced resuspension (SERE 88/89), Cont. Shelf Res. 18 (1998) 1203–1213. [2] S. Clarke, Advective/Diffusive processes in the firth of forth, Ph.D. Thesis, University of Wales, Bangor, 1995. [3] J.M. Abril, M. Garc ııa-Le oon, Modelling the distribution of suspended matter and the sedimentation process in a marine environment, Ecol. Modelling 71 (1994) 197–219. [4] W. Puls, J. Sundermann, Simulation of suspended sediment dispersion in the North Sea, in: R.T. Cheng (Ed.), Residual Currents and Long Term Transport, Springer, Berlin, 1990, pp. 356–372. [5] R. Peri aa~ nnez, J.M. Abril, M. Garc ııa-Le oon, Modelling the suspended matter distribution in an estuarine system: application to the Odiel river in southwest Spain, Ecol. Modelling 87 (1996) 169–179. [6] J. Nicholson, B.A. O’Connor, Cohesive sediment transport model, J. Hydraul. Eng. 112 (1986) 621–640. [7] J.T. Holt, I.D. James, A simulation of the southern North Sea in comparison with measurements from the North Sea Project. Part 2: suspended particulate matter, Cont. Shelf Res. 19 (1999) 1617–1642. [8] D.T. Pugh, Tides, Surges and Mean Sea Level, Wiley, Chichester, 1987. [9] R.H. Belderson, Holocene sedimentation in the western half of the Irish Sea, Mar. Geol. 2 (1964) 147–163. [10] P.A. Gurbutt, P.J. Kershaw, J.A. Durance, Modelling the distribution of soluble and particle absorbed radionuclides in the Irish Sea, in: J.C. Guary, P. Guegueniat, R.J. Pentreath (Eds.), Radionuclides. A Tool for Oceanography, Elsevier, Amsterdam, 1987, pp. 395–407. [11] D. Eisma, Supply and deposition of suspended matter in the North Sea, Spec. Publ. Int. Ass. Sediment 5 (1981) 415–428. [12] A.M. Davies, J. Lawrence, Modelling the effect of wave current interaction on the three dimensional wind driven circulation of the eastern Irish Sea, J. Phys. Oceanogr. 25 (1995) 29–45. [13] A.M. Davies, A three dimensional modal model of wind induced flow in a sea region, Prog. Oceanogr. 15 (1985) 71–128. [14] A.M. Davies, C.V. Stephens, Comparison of the finite difference and Galerkin methods as applied to the solution of the hydrodynamic equations, Appl. Math. Modelling 7 (1983) 226–240. [15] R. Peri aa~ nnez, A three dimensional rcoordinate model to simulate the dispersion of radionuclides in the marine environment: application to the Irish Sea, Ecol. Modelling 114 (1998) 59–70. [16] J.E. Jones, A.M. Davies, A high resolution three dimensional model of the M2,M4,M6,S2,N2,K1and O1tides in the eastern Irish Sea, Estuarine Coastal Shelf Sci. 42 (1996) 311–346. [17] A.M. Davies, J. Lawrence, A three dimensional model of the M4tide in the Irish Sea: the importance of open boundary conditions and the influence of wind, J. Geophys. Res. 99 (C8) (1994) 16 197–16 227. [18] A.M. Davies, S.C.M. Kwong, R.A. Flather, Formulation of a variable function three dimensional model, with applications to the M2and M4tide on the north west European Continental Shelf, Cont. Shelf Res. 17 (1997) 165–204. [19] A.M. Davies, S.C.M. Kwong, R.A. Flather, A three dimensional model of diurnal and semidiurnal tides on the European Shelf, J. Geophys. Res. 102 (C4) (1997) 8625–8656. [20] B.E. Launder, D.B. Spalding, Mathematical Models of Turbulence, Academic Press, London, 1972. [21] Z. Kowalick, T.S. Murty, Numerical Modelling of Ocean Dynamics, World Scientific, Singapore, 1993. [22] D. Prandle, Tidal characteristics of suspended sediment concentrations, J. Hydraul. Eng. 123 (1997) 341– 350. [23] M.K. Fukuda, W. Lick, The entrainment of cohesive sediments in freshwater, J. Geophys. Res. 85 (C5) (1980) 2813–2824. [24] J.W. Lavelle, H.O. Mojfeld, E.T. Baker, An in situ erosion rate for a fine grain marine sediment, J. Geophys. Res. 89 (C4) (1984) 6543–6552. [25] C. Teisson, Cohesive suspended sediment transport: feasibility and limitations of numerical modelling, J. Hydraul. Res. 29 (1991) 755–769. [26] A.J. Mehta, On estuarine cohesive sediment suspension behaviour, J. Geophys. Res. 94 (C10) (1989) 14 303–14 314. [27] M. Pejrup, Suspended sediment transport across a tidal flat, Mar. Geol. 82 (1988) 187–198. [28] D. Eisma, Suspended Matter in the Aquatic Environment, Springer, Berlin, 1993. [29] V.K. Saul’ev, On a method of numerical integration of the equation of diffusion, Dokl. Acad. Nauk. USSR 185 (1957) 1077–1083. [30] A.M. Davies, Application of the Dufort–Frankel and Saul’ev methods with time splitting to the formulation of a three dimensional hydrodynamic sea model, Int. J. Numer. Methods Fluids 3 (1985) 33–60. [31] P.D. Glorioso, A.M. Davies, The influence of eddy viscosity formulation, bottom topography and wind wave effects upon the circulation and flushing time of a shallow estuarine region, J. Phys. Oceanogr. 25 (1995) 1243–1264. [32] R.J. Pentreath, Behaviour of radionuclides released into coastal waters, IAEA TECDOC 329, Vienna, 1985. [33] M.J. Howarth, Atlas on tidal elevations and currents around the British Isles, Department of Energy, London OTR 89, 1990, p. 293. [34] K.F. Bowden, Processes affecting the salinity of the Irish Sea, Mon. Not. R Astron. Soc. Geophys. 6 (Suppl.) (1950) 63–90. [35] A.D. Heathershaw, Comparison of measured and predicted sediment transport rates in tidal currents, Mar. Geol. 42 (1981) 75–104. [36] D. Prandle, C.F. Jago, S.E. Jones, D.A. Purdie, A. Tappin, The influence of horizontal circulation on the supply and distribution of tracers, Philos. Trans. R Soc. Lond. A 343 (1993) 405–421. [37] MAFF, Aquatic environment monitoring, Report 17, Lowestoft, 1987. [38] J.J. Williams, J.D. Humphery, P.J. Hardcastle, D.J. Wilson, Field observations of hydrodynamic conditions and suspended particulate matter in the southern North Sea, Cont. Shelf Res. 18 (1998) 1215–1233. [39] R. Kirby, W.R. Parker, R.J. Pentreath, M.B. Lovett, Sedimentation studies relevant to low level radioactive effluent dispersal in the Irish Sea. Part 3: an evaluation of possible mechanisms for the incorporation of radionuclides into marine sediments, Institute of Oceanographic Sciences, Report 178, Wormley, 1983. [40] P.J. Kershaw, D.J. Swift, D.C. Denoon, Evidence of recent sedimentation in the eastern Irish Sea, Mar. Geol. 85 (1988) 1–14. [41] P.J. Kershaw, A. Young, Scavenging of 234Th in the eastern Irish Sea, J. Environ. Radioactivity 6 (1988) 1–23. [42] R. Peri aa~ nnez, J.M. Abril, M. Garc ııa-Le oon, Modelling the dispersion of non conservative radionuclides in tidal waters. Part 1: conceptual and mathematical model, J. Environ. Radioactivity 31 (1996) 127–141. [43] R. Peri aa~ nnez, J.M. Abril, M. Garc ııa-Le oon, Modelling the dispersion of non conservative radionuclides in tidal waters. Part 2: application to 226Ra dispersion in an estuarine system, J. Environ. Radioactivity 31 (1996) 253–272. [44] R. Peri aa~ nnez, A. Mart ıınez-Aguirre, U and Th concentrations in an estuary affected by phosphate rock processing: experimental results and a modelling study, J. Environ. Radioactivity 35 (1997) 281–304. [45] R. Peri aa~ nnez, Modelling the tidal dispersion of 137Cs and 239;240Pu in the English Channel, J. Environ. Radioactivity 49 (2000) 259–277. [46] D. Boust, P.I. Mitchell, K. Garcia, O. Condren, L. Le oon-Vintro, G. Leclerc, A comparative study of the speciation and behaviour of plutonium in the marine environment of two reprocessing plants, Radiochim. Acta 74 (1996) 203–210.