Modelling the dispersion of radionuclides by a river plume: Application to the Rhone river

Continental Shelf Research 25 (2005) 1583–1603 Modelling the dispersion of radionuclides by a river plume: Application to the Rhone river R. Peria ´n ˜ez Dpto. Fı ´sica Aplicada I, E.U. Ingenierı ´aTe ´cnica Agrı ´cola, Universidad de Sevilla. Ctra. Utrera km 1, 41013 Sevilla, Spain Received 16 January 2004; received in revised form 25 January 2005; accepted 15 April 2005 Available online 14 June 2005 Abstract A numerical model to simulate the transport of radionuclides in a river plume has been developed. The model solves the hydrodynamic equations, including baroclinic terms and a 1-equation turbulence model, suspended matter equations, including several particle sizes and deposition and erosion of sediments, and the radionuclide dispersion equations. The exchanges of radionuclides between water, suspended matter and bed sediments are described in terms of kinetic transfer coefficients. The dependence of these coefficients with water salinity is explicitly included in the model since in a river plume it changes from freshwater to seawater values. The model has been applied to the Rhone River plume, where radionuclides have been mainly discharged from a nuclear fuel reprocessing plant. Four particle sizes have been considered. Computed currents, salinity patterns, suspended sediment distributions and sedimentation rates are, in general, in agreement with observations. The model has been applied to simulate the transport of 125 Sb, 137 Cs and 239;240Pu in the river plume. The model provides realistic radionuclide distributions in water, suspended matter and bottom sediments. Other valuable information, as fractions of radionuclides in suspended particles and distribution coefficients, can also be provided by the model. r2005 Elsevier Ltd. All rights reserved. Keywords: Rhone River; Hydrodynamics; Sediment; Numerical modelling; Dispersion; Radionuclides 1. Introduction In recent times there has been an increasing interest in the development of dynamic radionuclide dispersion models in the marine environment since they are useful instruments that can be used in the assessment of the radiological consequences of existing or planned routine discharges of radionuclides, as well as to determine the potential impacts of accidental discharges (Thiessen et al., 1999). An understanding of the processes that govern the fate of discharged radionuclides is also of interest, both from the political and the environmental point of view, since the knowledge of the dose received by the population in the future may be needed. Finally, useful oceanographic ARTICLE IN PRESS www.elsevier.com/locate/csr 0278-4343/$ - see front matter r2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.csr.2005.04.001 E-mail address: [email protected]. information can be obtained by means of the application of marine radionuclide dispersion models (Prandle, 1984; Salomon et al., 1995). A general discussion on this point may be seen in Scott (2003). Also, some discussion on the interest of radionuclides as indicators of transport, deposition and resuspension processes may be seen in the book of Eisma (1993, p. 65), and references included there. Radionuclide dispersion in the marine environment is governed by advective transport and turbulent diffusion (these processes being induced by tidal, wind and long-term residual currents), but radionuclides also suffer chemical processes like adsortion/desorption reactions between water and solid particles (suspended matter particles and bottom sediments) and are affected by other complex physical processes (like deposition of suspended particles on the bed and erosion of the bottom sediment). Due to computational limitations, the first radionuclide dispersion models were long-term models in which advective and diffusive transport was parameterized in terms of effective transfer coefficients (Abril and Garcı´a-Leo ´n, 1993; Nielsen, 1995; Nielsen et al., 1997; Gurbutt et al., 1988). Also, time steps used in these models were long enough to allow the description of adsorption/desorption reactions in terms of equilibrium distribution coefficients, kds. More recently, dynamic models in which advective transport is explicitly calculated for tidally resolving computed currents have been developed. The treatment of adsorption/desorption has also been improved through the use of kinetic transfer coefficients. Thus, models can be applied to situations other than equilibrium conditions. This kind of model has been applied to estuaries (see, for instance, Peria ´n ˜ez et al., 1996a; Peria ´n ˜ez and Martı´nez-Aguirre, 1997; Margvelashvily et al., 1999; Peria ´n ˜ez, 2002) and the marine environment (for instance Piasecki, 1998; Peria ´n ˜ez, 1999, 2000; Aldridge et al., 2003; Goshawk et al., 2003; Smith et al., 2003; Nakano and Povinec, 2003). However, although some models have already been developed to study the water circulation in a river plume (Ruddick et al., 1995; Luyten et al., 1996; Estournel et al., 1997; Marsaleix et al., 1998; Cugier and Le Hir, 2002), none of them has been applied to simulate the transport of radionuclides from the river to the coastal area. This is an interesting problem since a complex intrusion process occurs when a freshwater input runs into marine waters. It mainly depends upon the interaction between the bouyancy induced momentum fluxes, which spread the freshwater offshore, and the turbulent dilution and dissipation mechanisms, which reduce density differences between the two water masses. Also, adsorption/ desorption reactions are affected by salinity changes. The dependence of kinetic coefficients governing such reactions with salinity must be explicitly included in a model. Finally, the size spectra of suspended particles discharged by the river may be rather wide. A recent sensitivity study (Peria ´n ˜ez, 2004b) has indicated that model output is very sensitive to the suspended particle size used in the model. Thus, more realistic results should be obtained if several particle sizes are considered. In this paper, a model for simulating the transport of radionuclides in a river plume is described and applied to the Rhone River plume, which introduces radionuclides released to the river into the Mediterranean Sea. The model domain covers all the river plume in the coastal area, where salinity changes from freshwater to seawater values. The artificial radionuclides reach the Rhone River through weathering of surface soils contaminated by atmospheric fallout and through the effluents from nuclear facilities: several nuclear power plants located along the river course and, mainly, from Marcoule nuclear fuel reprocessing plant (it ceased operations in 1997 but releases have continued since washing effluents are produced and discharged). In the case of the Rhone, the low mixing of the discharge waters gives rise to a well-identified surface plume in which a thin upper layer (1 or 2 m) is separated from the ambient seawater by a sharp density gradient. This plume extends 20 or 30 km offshore. Numerical modelling of these plumes is a difficult task that requires the inclusion of density differences in the full 3D hydrodynamic equations. The Rhone River gives the opportunity to test the behavior of radionuclide dispersion models, including a kinetic approach for the transfers between the liquid and solid phases, when ARTICLE IN PRESS R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–16031584 integrated with a detailed hydrodynamic model of a complex oceanographic system as a surface river plume. It must be pointed out that some detailed models have already been developed to study the water circulation in the Rhone plume (Estournel et al., 1997, 2001; Marsaleix et al., 1998). Also, some models describe the transport of sediments in the plume (Kondrachoff et al., 1994; Estournel et al., 1997), although in a rather simple way. Indeed, only one particle size was considered in these references and deposition was not included. However, a radionuclide dispersion model for the plume, integrated with a hydrodynamic model and a detailed suspended sediment model, has not been described and tested before. The model solves the full 3D hydrodynamic equations using a one-equation turbulence model for eddy viscosity. The water circulation is used to solve the suspended matter equations (one equation for each particle size), including processes of deposition and sediment erosion, and the radionuclide dispersion equations. The model is described in the following section. Then, results are presented. 2. Model description The hydrodynamic, suspended sediments and radionuclide dispersion sub-models are presented in what follows. 2.1. Hydrodynamic model The full 3D hydrodynamic equations including the terms corresponding to density gradients can be written in the hydrostatic and Boussinesq approximations as (Kowalick and Murty, 1993) qz qtþq qxðhþzÞZz h udz  þq qyðhþzÞZz h vdz  ¼0, ð1Þ qu qtþuqu qxþvqu qyOvþgqz qxþg r0Zz z qrw qxdz ¼q qzKqu qz  þAq2u qx2þq2u qy2  ,ð2Þ qv qtþuqv qxþvqv qyþOuþgqz qyþg r0Zz z qrw qydz ¼q qzKqv qz  þAq2v qx2þq2v qy2  ,ð3Þ where the zcoordinate is measured upwards from the undisturbed sea level, his water depth, zis the displacement of the sea surface from the undisturbed level, uand vare the two components of the water velocity along the x- and y-axis, respectively, Ois the Coriolis parameter, rwis water density, r0 is a reference density, and Kand Aare the vertical and horizontal eddy viscosities, respectively. The vertical component of the water velocity, w, is obtained from the continuity equation qu qxþqv qyþqw qz¼0. (4) The water density is derived from a equation of state relating density to salinity (Kowalick and Murty, 1993) rw¼r0ð1þaSÞ, (5) where Sis salinity and a¼7:45 104.Asin Marsaleix et al. (1998), the effect of temperature on density is omitted for the sake of simplicity. Indeed, temperature differences between the Rhone River and the ambient seawater are small enough to consider that the bouyancy effects in the plume are represented only by the salinity gradients (Marsaleix et al., 1998). The reference salinity is taken as r0¼998:9kg=m3. The salinity is determined from an advection– diffusion equation qS qtþuqS qxþvqS qyþwqS qz ¼Aq2S qx2þq2S qy2  þq qzKqS qz  .ð6Þ Vertical eddy viscosity is determined from the 1- equation turbulence model described in Davies and Hall (2000). This model has also been used in ARTICLE IN PRESS R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–1603 1585 Xing and Davies (1999) to simulate a river plume. The equation for the turbulent kinetic energy Eis qE qt¼Kqu qz  2 þqv qz  2 () þb0 q qzKqE qz  eþg r0 Kqr qz.ð7Þ The first term in the right-hand side of the equation represents generation of turbulence by the vertical shear, the second term is diffusion of turbulence and the last term is loss of turbulence by bouyancy (conversion of kinetic energy into potential energy). erepresents dissipation of turbulence, that is written as e¼C1E3=2‘, (8) where ‘is a mixing length and C1a numeric coefficient. The vertical viscosity is finally written as a function of energy as K¼C0‘E1=2þlt, (9) where C0is a numeric coefficient and ltis a background value of viscosity, that is the minimum possible value that it may have (Ruddick et al., 1995; Cugier and Le Hir, 2002). The values of the numeric constants appearing above are (Davies and Hall, 2000): b0¼0:73, C0¼C1=4, C1¼C3 0and C¼0:046. The background viscosity is fixed as lt¼104m2=s, which is the same value used in Ruddick et al. (1995). The mixing length is derived from an algebraic expression (Davies and Hall, 2000) ‘¼1 1=‘1þ1=‘2 (10) with ‘1¼kðzþz0þhÞeb1ðzþhÞ=h, (11) ‘2¼kðzszÞ, (12) where k¼0:4 is the von Karman’s constant, b1¼ 2:0andzsand z0are the roughness lengths of the sea surface and bottom, respectively. Boundary conditions have also to be provided for sea surface elevation, currents, salinity and kinetic energy. At the sea surface r0Kqu qz  z¼z¼Fs, (13) r0Kqv qz  z¼z¼Gs, (14) where Fsand Gsare the components of the wind stress acting on the sea surface. For salinity, it is assumed that there is no flux through the sea surface and bottom qS qz  z¼h;z¼z¼0. (15) In the case of kinetic energy, the boundary condition at the sea surface is Ez¼z¼tw r0ffiffiffiffiffiffiffi CE p, (16) where twis the wind stress (solved in the components Fsand Gs)andCE¼0:07 (Estournel et al., 1997). If there is not wind, then the condition is (Davies and Hall, 2000) qE qz  z¼z¼0. (17) At the sea bottom, boundary conditions are Ez¼h¼tb r0ffiffiffiffiffiffiffi CE p, (18) where tbis the bottom stress, solved in components Fband Gb, and CE¼0:07 as in Estournel et al. (1997). For the currents, r0Kqu qz  z¼h¼Fb, (19) r0Kqv qz  z¼h¼Gb. (20) The two components of the bottom stress are written using a quadratic friction law Fb¼r0ku ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2þv2 p; Gb¼r0kv ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2þv2 p;(21) where kis a friction coefficient to be obtained after a calibration exercise. A no-flux condition is applied along land boundaries. A radiation condition in the form of ARTICLE IN PRESS R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–16031586 Orlanski (1976) is applied along open boundaries for the water velocity component that is normal to the boundary: qx qtþcqx qn¼0, (22) where nis the direction normal to the boundary, x¼u;vis the water velocity component to which the condition is applied and cis the wave speed calculated as in Orlanski (1976). This condition has also been used in Marsaleix et al. (1998). Surface elevations should be prescribed along open boundaries to propagate tides inside the computational domain. However, as will be shown below, tides are neglected in the present model application and the radiation condition is also applied to surface elevations. Boundary conditions at the river mouth are S¼0 (23) that is the freshwater value and ql¼Q ld , (24) where qlis the water velocity in the direction of the outflow, Qis the river discharge and land dare the width of the river mouth and the thickness of the outflow water layer, respectively. 2.2. Suspended sediment model The suspended sediment dynamic is governed by an advection–diffusion dispersion equation to which the settling, erosion and deposition terms are added. A number Nof particle grain sizes are considered in the model. Each particle size is represented by a concentration mi. The equation for size iis qmi qtþuqmi qxþvqmi qyþðwws;iÞqmi qz ¼Aq2mi qx2þq2mi qy2  þq qzKqmi qz  ,ð25Þ where ws;iis the corresponding settling velocity for particle size i. The deposition and erosion terms are incorporated into the sea bed boundary condition of the equation. The deposition rate is written as in other modelling studies (see, for instance, Nicholson and O’Connor, 1986; Holt and James, 1999; Liu et al., 2002a; Lumborg and Windelin, 2003; Wu et al., 1998; Prandle et al., 2000; Cancino and Neves, 1999) DPi¼ws;imiðbÞ1tb tcd  , (26) where miðbÞis particle concentration of size i evaluated at the sea bottom and tcd is a critical deposition stress above which no deposition occurs since particles are maintained in suspension by water turbulence. The settling velocity for each particle size is determined from Stokes’s law ws;i¼rrw rw gD2 i 18n, (27) where rand Diare suspended particle density and diameter, respectively, and nis kinematic viscosity of water. It is also possible to include the process of aggregation in the model through a classic relationship in the form (Mehta, 1989; Clarke, 1995) ws;i¼a1ma2 i. (28) This approach has been used in some suspended sediment models (Cancino and Neves, 1999; Clarke and Elliott, 1998; Nicholson and O’Con- nor, 1986; Lumborg and Windelin, 2003) and also in radionuclide dispersion models (Peria ´n ˜ez, 1999, 2000). Other authors (Estournel et al., 1997;Liu et al. 2002a, b;Jiang et al., 2000; Segschneider and Sundermann, 1998; Holt and James, 1999) directly calculate the settling velocity from Stokes’s law. Tattersall et al. (2003) have used a mixed approach. At each time step the settling velocity resulting from Eq. (28) is calculated and compared with that obtained from Stokes’s law. Then the higher value is used. Wu et al. (1998) use Eq. (28) or Stokes’s law depending on the suspended matter concentration. Stokes’s law has been used in this work due to the difficulty of finding appropriate values for a1and a2for each particle size. Moreover, Thill et al. (2001) have found that it is not clear that the aggregation process could play a role in enhancing settling at the Rhone River mouth. ARTICLE IN PRESS R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–1603 1587 The erosion rate is written in terms of the erodability constant (Liu et al. 2002a, b;Cancino and Neves, 1999; Prandle et al., 2000; Holt and James, 1999; Nicholson and O’Connor, 1986) ERi¼Ef i tb tce 1  , (29) where Eis the erodability constant, figives the fraction of particles of size iin the bed sediment and tce is a critical erosion stress below which no erosion occurs. The model can also calculate sedimentation rates as the balance between the deposition and erosion terms. Bed-load transport of coarse particles has not been included since radionuclides are essentially adsorbed on the fine particles (McKay and Walker, 1990). This approach has also been used in other models (for instance, Peria ´n ˜ez, 1999). 2.3. Dispersion of radionuclides It is considered that the exchanges of radionuclides between the liquid and solid phases are governed by a single reversible reaction. Thus, the transfer from the liquid to the solid phase is governed by a kinetic coefficient k1and the inverse process by a kinetic coefficient k2. It is known that actinide adsorption tends to be a surface phenomenon and will depend on the surface of particles per water volume unit at each point and time step. This quantity has been denoted as the exchange surface (Peria ´n ˜ez et al., 1996b; Peria ´n ˜ez and Martı´nez-Aguirre, 1997; Peria ´n ˜ez, 1999, 2000, 2002). Thus, the kinetic coefficient k1;iis written as: k1;i¼w1ðSm;iþSs;iÞ¼k11;iþk12;i, (30) where Smand Ssare the exchange surfaces for suspended matter and bottom sediments respectively and w1is a parameter with the dimensions of a velocity denoted as the exchange velocity (Peria ´n ˜ez et al., 1996b; Peria ´n ˜ez and Martı´nez- Aguirre, 1997; Peria ´n ˜ez, 1999, 2000, 2002). The index i¼1...Nis used to represent each one of the particle sizes included in the model. Assuming spherical particles, the exchange surfaces are written as (see references cited above) Sm;i¼3mi rRi (31) and Ss;i¼3Lf if RiU, (32) where Riis particle radius, Lis the sediment mixing depth (the distance to which the dissolved phase penetrates the sediment), Uis the thickness of the water layer above the bed sediments that interacts with them and fis a correction factor that takes into account that part of the sediment particle surface may be hidden by other particles. For simplicity, it has been assumed that Uis equal to the vertical grid size: U¼Dz. This formulation has been successfully used in all modelling works cited above. The exchange velocity depends on salinity as in Laissaoui et al. (1998) w1¼w0 1ð1dÞ, (33) where d¼S SþS0 . (34) In these equations w0 1is the freshwater value of the exchange velocity and S0is the salinity value at which 50% of saturation occurs (Laissaoui et al., 1998). It must be noted that as salinity increases, the transfer of radionuclides to the solid phase decreases due to competition effects of radionuclides with ions dissolved in water. The relations given above have been tested through laboratory experiments (Laissaoui et al., 1998). The kinetic coefficient k2is considered to be constant and to have the same value for all particle sizes. Recent work has shown that a kinetic model involving 2 consecutive reversible reactions (2-step model) may be more appropriate than one consisting of a single (1-step) reversible reaction, specially for long (months) water–sediment contact times (Peria ´n ˜ez, 2004a). However, given the time scale of simulations that are carried out (several days), it is not expected that significant ARTICLE IN PRESS R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–16031588 differences between the 1-step and 2-step model appear in this case. The equation that gives the time evolution of specific activity in the dissolved phase, Cd,is qCd qtþuqCd qxþvqCd qyþwqCd qz ¼Aq2Cd qx2þq2Cd qy2  þq qzKqCd qz  X N i¼1 k11;iCdþk2X N i¼1 miCs;i þpk2 LrsfPN i¼1fiAs;i UX N i¼1 k12;iCd ! , ð35Þ where Cs;iand As;iare, respectively, the concentration of radionuclides in suspended matter and bottom sediments with size iand rsis the sediment bulk density. p¼0 unless we are solving the equation for the water layer in contact with the sediment. In this case p¼1 to allow interactions between water and sediments. The equation that gives the time evolution of specific activity in each of the suspended matter sizes is qðmiCs;iÞ qtþuqðmiCs;iÞ qxþvqðmiCs;iÞ qy þðwws;iÞqðmiCs;iÞ qz ¼Aq2ðmiCs;iÞ qx2þq2ðmiCs;iÞ qy2  þq qzKqðmiCs;iÞ qz  þk11;iCdk2miCs;i þpðDPRiþERRiÞ,ð36Þ where phas the same meaning as above and DPRi is the deposition of radionuclides from the deepest water layer to the sediment evaluated according to DPRi¼ws;imiðbÞCs;iðbÞ U1tb tcd  . (37) Note that (b) means that the corresponding magnitude is evaluated at the deepest water layer. ERRiis the resuspension of radionuclides due to sediment erosion ERRi¼Ef iAs;i U tb tce 1  . (38) Of course, deposition of radionuclides is calculated only if tbotcd and resuspension only if tb4tce. Specific activity in the total suspended phase is calculated as Ctotal s¼PN i¼1miCs;i PN i¼1mi . (39) The equation for the temporal evolution of specific activity in each bottom sediment particle size is qAs;i qt¼k12;i CdU Lrsfik2As;ifþDPRiERRi, (40) where it is remembered that U¼Dzand the deposition and erosion terms are calculated as DPRi¼ws;imiðbÞCs;iðbÞ Lrsfi 1tb tcd  , (41) ERRi¼EAs;i Lrs tb tce 1  . (42) Total specific activity in bed sediments is calculated from specific activity in each particle size Atotal s¼X N i¼1 fiAs;i. (43) 2.4. Numerical solution All the equations are solved using explicit finite difference schemes. The scheme described in Flather and Heaps (1975) is used to solve the hydrodynamic equations. For non-linear terms, Scheme 2 described in such reference is used. It consists of calculating changes in components of the horizontal current due to advection alone and adding them to the changes caused by other physical processes computed from the hydrodynamic equations without the advective terms. Although the scheme in Flather and Heaps ARTICLE IN PRESS R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–1603 1589 (1975) is described for a 2D depth-averaged model, it has been applied to each vertical layer. Additionally, a second-order scheme (see, for instance, Kowalick and Murty, 1993) is used to solve horizontal and vertical diffusion, and the scheme also described in this reference (p. 245) is used for the internal pressure terms. The MSOU second-order scheme (Vested et al., 1996) is used to solve the advection terms in all the dispersion equations (salinity, suspended matter, radionuclides) and a second-order scheme (Kowalick and Murty, 1993) is also used for the diffusion terms. Density gradients are calculated using sy¼ rw1000 instead of the density (Kowalick and Murty, 1993). Also, the stability of the water column is checked each time step and over all the model domain. Mixing is carried out, using the algorithm described in Kowalick and Murty (1993), when it becomes unstable. The hydrodynamic calculations are started from rest and an uniform salinity of 38 g/L is assumed. The effect of the large-scale circulation can be neglected in the area under study (Marsaleix et al., 1998), as well as tidal currents, that are practically non-existent (Marsaleix et al., 1998; Tsimplis et al., 1995). Indeed, M2currents are below 1 cm/s in most parts of the Mediterranean and S2currents have smaller amplitudes. Open boundary conditions must also be given for suspended matter particles and radionuclides. The open boundary condition described in Peria ´- n ˜ez (1999) and Peria ´n ˜ez (2000) has been adopted Ci¼cCi1, (44) where Cirepresents the concentration of suspended matter and radionuclides along the boundary and Ci1represents the concentration just inside the computational domain. The nondimensional value of cis obtained from a calibration exercise. The hydrodynamic model is run for a given water discharge from the Rhone until a steady salinity and current amplitude pattern is obtained. Then the suspended matter model is run using such water circulation. The suspended matter model (without the erosion term) is started from a sea bottom containing no sediments. Then the accumulation of particles of each size is calculated to have a first estimation of the distribution of sediment particle sizes over the model domain. Next, the suspended matter model is started (with erosion) from the estimated particle size distribution until a steady state is reached. This way a selfconsistent distribution of different particle sizes on the sea bed over the model domain can be obtained. It is assumed that all suspended and surficial sediments originate from the Rhone. This distribution is relevant to calculate the adsorption of radionuclides by the bed sediments. Once the water circulation, salinity distribution, suspended matter distribution for each particle size and distribution of particles in bed sediments are known and stored in files, the radionuclide dispersion model may be run using these files as input data. The model domain is presented in Fig. 1. The model resolution is Dx¼Dy¼1000 m. A variable grid is used in the vertical to have enough resolution to solve the salinity gradients in the surface plume. Thus, 10 layers with Dz¼1 m are used followed by thicker layers increasing to Dz¼15 m. Time step is fixed as 5 s to solve the hydrodynamic equations due to the CFL stability condition. It is increased to 60 s to solve the suspended matter and radionuclide equations. However, when the radionuclide dispersion model was applied to plutonium, time step had to be decreased to 10 s since the following stability condition is introduced by the terms describing ARTICLE IN PRESS 5 152025303540455055 5 10 15 20 25 30 35 Rhone Mediterranean Sea FRANCE 10 Fig. 1. Model domain. Each unit in the x- and y-axis is 1000 m. Depths are given in m. R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–16031590 the transfers between the liquid and the solid phases Dto1 kmax , (45) where kmax is the maximum kinetic rate involved in the equations. It means that the activity transferred from one phase to another in a time step must be smaller than the activity content in the origin phase. This condition is more restrictive for a highly reactive radionuclide as Pu than for more conservative ones as, for instance, Cs. 3. Model application and results Values to the different parameters in the model must be given, as well as the appropriate source terms for suspended matter particles and radionuclides. The bed friction coefficient has been fixed as k¼0:015 and the horizontal diffusivity as A¼20 m2=s. The roughness lengths of the sea surface and bottom have been defined as zs¼z0¼0:0034 m. These values are similar to those used by Xing and Davies (1999). The thickness of the freshwater layer discharged from the Rhone is 2 m (Broche et al., 1998) and the width of the river mouth has been fixed as d¼Dx¼1000 m, which is very close to reality. Indeed, this approach has already been used in Marsaleix et al. (1998). As commented before, large-scale circulation and tides can be neglected in the area. The hydrodynamic model has been run for the average discharge from the Rhone River, 1700 m3=s(Broche et al., 1998; Thill et al., 2001), and without wind. The computed water circulation and salinity and viscosity distributions are stored in files that will be used to calculate the suspended matter distribution, deposition rates and the dispersion of radionuclides. Wind conditions affect the extension and shape of the plume (Estournel et al., 1997, 2001). Two types of winds are predominant in the region: northwest winds (representing 45% of winds exceeding 10 m/s on average over the year) and southeast winds (20% of winds exceeding 10 m/s). The plume response to winds has been studied in the references given above. Although wind stress is formally included in the hydrodynamic equations, the main difficulty in the present work is that observed sedimentation rates, the distribution of different particle classes over the bed and measured radionuclide concentrations in bed sediments integrate many different wind and river discharge conditions. On the other hand, measured radionuclide concentrations in the water column correspond to the particular conditions during sampling, that, on the other hand, took place over a period of several months (Martin and Thomas, 1990). As a consequence, it was decided to run the hydrodynamic model for average discharge from the Rhone River and without wind. This may be a too simple approach, but can give a realistic view of the main dispersion processes in the plume, given the generally good agreement obtained between computations and measurements. Nevertheless, the sensitivity of radionuclide transport in the plume to changing winds and river discharge is an interesting problem that has to be addressed in the next future. The computed currents in the domain for these conditions are presented in Fig. 2. These currents were obtained after a simulation time of 48 h. This pattern is in agreement with the earlier calculations in Estournel et al. (1997),Marsaleix et al. (1998) and Arnoux-Chiavassa (1998). Water discharged by the river moves in a southeast direction and then rotates towards the west due to Coriolis acceleration and moves along the coast. Broche et al. (1998) released a surface drifter 2 km south ARTICLE IN PRESS 0 5 10 15 20 25 30 35 0 10 20 30 40 50 20 cm/s Fig. 2. Computed currents for average water discharge ð1700 m3=sÞand no wind. R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–1603 1591 seems clear that the model produces rather realistic activity levels in the area of the river mouth. Indeed, the distribution map in Fig. 11 is very similar to that presented in Charmasson (2003), where a sharp decrease in inventories with distance from the river mouth can be seen. Finally, it must be taken into account that measured bed concentrations include deposition from global fallout. Only sampling stations close to the river mouth in Charmasson (2003) present 137 Cs inventories clearly in excess with respect the cumulative deposit due to global fallout (Chernobyl is not considered since occurred later than the time of our simulations). Thus, most of 137 Cs coming from the river appears in sediments of a well-delimited zone in the close vicinity of the river mouth. Our simulations are in agreement with these findings. Nevertheless, it seems clear that the transport of radionuclides under extreme and changing wind conditions, as well as peak river discharge, needs to be investigated. The model can also give a wide amount of information, as fractions of radionuclides in suspended matter (for total suspended load and each particle size) and distribution coefficients for suspended matter–water and bed sediment–water systems (also for the total solid phase and each particle size). It is interesting to point out that the model can calculate distribution coefficients instead of requiring them as input data. As an example, the computed fraction of 137 Cs fixed to suspended matter particles in surface waters of the plume is presented in Fig. 12. It can be seen that about 60% is fixed to solid particles. In Eyrolle and Charmasson (2001) it is pointed out that 30% of 137 Cs is transported by the Rhone River in the particulate phase, although this percentage is increased to 83% in Martin and Thomas (1990). Indeed, these authors have found that the particulate phase is the major vector for most of the radionuclides. Calmet and Fernandez (1990) have also found that 137 Cs associated with particles represents 68% of the Rhone input. Thomas (1997) has also found that radionuclide transport in the particulate phase is specially relevant during flood events, when the fraction of radionuclides in particles reaches 0.75–0.99. Computed suspended matter–water distribution coefficients (total suspended matter) for surface waters are also presented in Fig. 12. They are of the order of 104L=kg in the area of the river plume, decreasing in one order of magnitude out of it and approaching the average value recommended by IAEA (1985) for Cs (3 103). This behavior may be similar to that observed (Mitchell et al., 1995)in the Irish Sea for Pu [and also reproduced by a model (Peria ´n ˜ez, 1999)], where distribution coefficients decrease when increasing distance from Sellafield. It was attributed to the nature of discharges since virtually all plutonium was released in particulate form (Mitchell et al., 1995). The same situation is presented in the plume since most 137 Cs is released in particulate form. However, the decrease in the exchange velocity with salinity included in the model (Eq. (33)) will also produce a decrease in the ARTICLE IN PRESS 5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10152025303540455055 5 10152025303540455055 Fig. 12. Computed 137 Cs fraction fixed to suspended particles in surface waters (up) and computed suspended matter–water kd(L/kg) in surface waters (down). R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–16031598 distribution coefficients as distance from the river mouth increases and, consequently, salinity increases too. South–north vertical profiles of specific activity in suspended matter in front of the river mouth and for several particle sizes are presented in Fig. 13, together with specific activity in total suspended load. It can be seen that the radionuclide content in the plume suspended load is essentially controlled by the smallest particles, since Figs. 13a and b are very similar. Coarse particles quickly sink to the bottom and, as a consequence, radionuclides are only present in a small region in front of the river mouth as can be seen in Figs. 13c and d. This is in agreement with the results of Martin and Thomas (1990), who found a deposition belt close to the river mouth. It seems that this deposition belt is produced by the coarsest particles in the plume. Calmet and Fernandez (1990) have measured a decrease in 137 Cs activity with increasing depth in the water column. Such decrease has also been produced by the model although numbers cannot be directly compared since measurements include the input of radionuclides from the Rhone plus ARTICLE IN PRESS Distance (km) Depth (m) 2 8 10 12 14 16 18 20 22 24 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 64 2 8 10 12 14 16 18 20 22 24 64 2 8 10 12 14 16 18 20 22 2464 2 8 10 12 14 16 18 20 22 2464 (A) (B) (C) (D) Fig. 13. Computed 137 Cs activity in suspended matter in Bq=m3for a south–north vertical profile in front of the river mouth. (A) total suspended matter. (B), (C) and (D) 3, 20 and 40 mm particles, respectively. Distances are measured from the south of the model domain. The river mouth is at km 24. 0.001 0.01 0.1 1 0 5 10 15 20 25 239,240-Pu distance (km) Fig. 14. Measured (points) and computed (lines) south–north profiles of 239,240 Pu in surface water (solid line, boxes), in Bq=m3, and suspended matter (dashed line, crosses), in Bq/g, in front of the river mouth. Distances are measured from the south of the model domain (the river mouth is at km 24). R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–1603 1599 Chernobyl fallout and the model does not account for the last. A comparison of model results with measurements for the application to 239,240 Pu is presented in Fig. 14, where south–north plutonium profiles in water and suspended matter in front of the river mouth are shown. Plutonium fallout backgrounds have been added to the specific activities calculated by the model, which correspond to the input from the river only. Specific activity in water due to fallout plutonium is 0:016 Bq=m3(Martin and Thomas, 1990). Using the Pu kdvalue measured in the northwestern Mediterranean (Molero et al., 1995), 1:4105L=kg, which is also in good agreement with the average value of the Pu distribution coefficient in coastal waters recognized in IAEA (1985),1:0105L=kg, the specific activity in suspended matter particles due to fallout plutonium (again assuming equilibrium for fallout radionuclides) is 0.0024 Bq/g. It can be seen that the model gives a correct estimation of plutonium levels in both phases in the river plume. These activity levels are two orders of magnitude lower than those of 137 Cs due to the smaller Pu input rate. The fraction of plutonium at the surface fixed to the suspended particulate phase is given in Fig. 15. It can be seen that over 90% of Pu in the plume is attached to particles. This is in agreement with Thomas (1997), who found that plutonium predominates in particulate form in most hydrological conditions, and with Eyrolle and Charmasson (2004), who have reported that 85% of Pu isotopes are bound to particles in the river. Computed distribution coefficients in the plume are of the order of 3 105L=kg, decreasing out of it and approaching the value found by Molero et al. (1995). The general behavior is similar to that of 137 Cs. 4. Conclusions A full 3D model has been developed to simulate the transport of radionuclides in a river plume. Currents are almost entirely baroclinic (induced by water density differences). Thus, the hydrodynamic model must include the terms accounting for such density differences in the equations and must also have a vertical resolution high enough to solve the pycnocline. A turbulence model has also been coupled to provide the vertical eddy viscosity. Together with the hydrodynamic equations, the model solves the suspended matter dynamics. Several suspended particle sizes have been included, since model output is rather sensitive to this parameter and the size spectra of particles discharged by a river is generally wide. The effect of salinity on uptake kinetics has finally been included. The model provides a wide amount of information, as horizontal distributions and vertical profiles of specific activities in water, total suspended matter and each suspended particle size, percentages of radionuclides in total suspended matter and each particle size, distribution coefficients for total suspended matter and each particle size and distribution of radionuclides in bed sediments, for the total sediment and each particle size. The model has been applied to simulate the transport of 137 Cs, 125 Sb and 239+240 Pu in the Rhone River plume. The computed water circulation and salinity pattern is, in general, in agreement with observations and earlier hydrodynamic models. Four suspended matter particle sizes have been considered in the model. It gives a realistic representation of the behavior of the different particle sizes in the plume, as well as of the sedimentation processes. Model results for radionuclide dispersion are, in general, in good agreement with observations. Thus, it seems that the ARTICLE IN PRESS 5 10152025303540455055 5 10 15 20 25 30 35 Fig. 15. Computed 239,240 Pu fraction fixed to suspended particles in surface waters. R. Peria ´n˜ez / Continental Shelf Research 25 (2005) 1583–16031600 processes included in the model have been adequately described. Nevertheless, as already commented, the effect of dynamic winds and changing river flows on radionuclide transport in the plume has to be addressed in the next future. 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