Environmental Modelling & Software 21 (2006) 234–242 www.elsevier.com/locate/envsoft
A hybrid approach to particle tracking and Eulerian–Lagrangian models in the simulation of coastal dispersion Seung-Won Suh* Department of Ocean System Engineering, Kunsan National University, Kunsan 573-701, Republic of Korea Received 1 November 2003; received in revised form 1 February 2004; accepted 1 April 2004 Available online 21 December 2004
Abstract A hybrid approach to the problem of predicting the dispersion of contaminants (e.g. suspended solids or heated water) in coastal areas is presented. A neutral random walk particle tracking method is adopted in the vicinity of the source point, where steep concentration gradients occur. In the case of heat dispersion particles are assigned with buoyancy, which leads to additive horizontal diffusivity. In the far field, Eulerian–Lagrangian concentration models are used to represent the advection-dominated characteristics of coastal regions. In two simple model tests, the proposed scheme showed reasonable accuracy compared to the analytical solution, especially around the source point where the concentration gradients were high (VcR0:005). The present results strongly advocate the use of a random walk model for the near field and a Eulerian–Lagrangian concentration model for the entire domain when modeling dispersion of contaminants in coastal waters. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Coastal modeling; Random walk; Near field; Buoyant particle
1. Introduction Lagrangian-based approaches have been used in coastal dispersion analysis because, compared to Eulerian methods (EM), they give less numerical errors and better maintain the physical meaning in the tracking and identification of particle locations. However, Lagrangian methods (LM) require moving coordinate systems, which significantly increases the computational burden. Thus, combined methods that exploit the desirable characteristics of LM and EM, such as Eulerian– Lagrangian methods (ELM), have been developed for coastal dispersion modeling in advection-dominated systems. Dredging or dike construction around this area yields inevitable dispersion of suspended solids, harming the coastal environment. * Fax: C82 63 469 1713. E-mail address:
[email protected] 1364-8152/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.envsoft.2004.04.015
Concentration models such as EM or ELM are still widely applied in coastal areas where the flows are dominated by advection. However, those methods are known to give erroneous results near source points. In response to this problem, random walk particle tracking (RWPT) was developed to avoid the numerical errors produced by fixed grid numerical methods in the modeling of advection-dominated systems. The RWPT method, which explicitly considers particle movements, may be a better approach because the governing equations of the prevailing schemes such as EM and ELM, expressed as partial differential equations of concentrations, are basically denoted as number of particles in a control volume of water mass. An additional shortcoming of concentration models is that they do not directly resolve dispersion phenomena in a physical point. Thus, tracking the movement of each particle undergoing random motion, as is done in RWPT, should be more reasonable. Moreover, advances
S.-W. Suh / Environmental Modelling & Software 21 (2006) 234–242
in computing technology have made this approach potentially viable even on personal computers. A limitation of concentration models is their inability to resolve the concentration field on scales less than the spatial scale of discretization. To resolve this problem, a hybrid model has been developed (Neuman, 1984; Zhang, 1995) in which, near sources or zones of high spatial gradient, mass is represented by a large collection of particles, each of which is assigned a concentration value. This procedure is somewhat awkward and suffers from mass conservation problems that arise from the conversion of particle concentrations to node concentrations. Another limitation of concentration models is that the computational effort is spread evenly across the domain. In contrast, the computational effort in RWPT models is concentrated in the regions containing the most particles (Dimou, 1989). RWPT has been applied in diverse areas such as groundwater transport (Zheng et al., 2000), coastal biophysical problems related to zooplankton or fish larvae (Batchelder, 2002), sediment plume behavior and its effects on phytoplankton growth due to light attenuation (Xie and Yapa, 2002), coastal oceanic fronts of river discharge (Korotenko, 2000), coral reef modeling (Spagnol et al., 2002), and chemical exposure assessment in marine environments (Sabeur et al., 2002). A drawback of RWPT is that, even using highperformance computers, the particle number must be restricted because the saving and tracking of the history of every particle requires large amounts of memory and computing time. Another drawback of hybrid methods is that, when such methods are used to generate concentration data (e.g. predicted environmental concentration) rather than particle locations, numerical errors arise during the conversion from the particle to the concentration representation as the final step of RWPT in a fixed computational grid system. Thus, it is recommended that RWPT be applied only in cases where it is absolutely necessary, such as in the vicinity of source points where high concentration gradients cause concentration models to generate numerical errors. Application of RWPT across an entire region is inappropriate because it would involve the unnecessary tracking of all particles, even those in the far field region. The greater the number of particles used, the greater the difficulty arise in tracking each particle and converting particle into concentration in the far field, where concentration gradients are mild and turbulent dispersion is already included in the concentration partial differential equations. In the model developed here, RWPT with or without buoyancy is exploited in the near field region around a source point for dispersion analysis of heated water and suspended solids in coastal regions. In addition, depth averaged 2-dimensional ELM is applied in the far
235
field for efficient and accurate modeling. These two numerical schemes were combined by creating a simple hybrid method in which RWPT was incorporated into the concentration model by projecting the number of particles into a Eulerian control volume. Two key factors when implementing particle tracking methods are choosing the proper number of particles and developing a formalism for treating the boundary between the particle and concentration representations.
2. Formulation of the random walk model In a random walk model, mass is transported as discrete particles. At each time step the displacement of each particle consists of an advective, deterministic component and an independent, Markovian component. The conservative form of a 2-dimensional depthaveraged mass transport equation when sinks/sources are neglected can be represented as follows: vðchÞ vðuchÞ vðvchÞ v vc v vc C C Z hDxx C hDyy vt vx vy vx vx vy vy ð1Þ where x and y are Cartesian coordinates, t is time, h is depth, c is concentration, u and v are the velocities in the x and y directions, respectively, Dxx and Dyy are the x and y directional diffusivities, respectively. Modification of Eq. (1) yields: vðchÞ v Dxx vh vDxx C C Cu ch vt vx h vx vx v Dyy vh vDyy v2 v2 C C Cv ch Z 2 ðDxx chÞC 2 Dyy ch vy h vy vy vx vy ð2Þ The particle tracking model follows the concept of random movement of the Monte-Carlo method. Specifically, particles are represented as real entities spread across the computational domain rather than as concentration. The particle tracking approach, which is based on statistical concepts, requires a powerful computing environment to track each particle in a collection of particles spreading over the domain. The basic concepts are as follows. The displacement of an arbitrary particle in a 1-dimensional field is comprised of a deterministic part a and a stochastic part b, as in the Langevin equation (Tompson et al., 1988): dx Zaðx; tÞCbðx; tÞxðtÞ dt
ð3Þ
236
S.-W. Suh / Environmental Modelling & Software 21 (2006) 234–242
where x is a random number. Integration of Eq. (3) yields: Z tCdt xðtCdtÞ xðtÞZaðx; tÞdtC bðx; tÞxðtÞdt ð4Þ t
and taking the difference form we obtain, Dxn Zxn xn1 Za xn1 ; tn1 DtCb xn1 ; tn1 DWðtn Þ ð5Þ Insertion of Eq. (5) into the traditional transport equation followed by rearrangement and comparing both equations through similarity theory yields the following relations: aZu
bZ
vDxx vx
pffiffiffiffiffiffiffiffiffiffi 2Dxx
pffiffiffiffiffi DWðtn ÞZZn Dt
ð6Þ
ð7Þ
ð8Þ
where Z is a random number with mean 0 and variance 1. Thus, particle movement consists of a deterministic part, a xn1 ; tn1 Dt, and an independent Markovian part, b xn1 ; tn1 DWðtn Þ. 2.1. Neutral particle random walk model If M is the total mass distributed in a computational domain and r is the density, then Mp ðx; tÞZrc and comparing Eqs. (2) and (5) yields the following expression (Dimou, 1989): vc v 1 vb2 v 1 2 vc Z b a ð9Þ c C vt vx 2 vx vx 2 vx Then, comparing Eq. (9) with the random walk equation yields the relationship: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vDxx Dxx vh DxZ uC C ð10Þ DtC 2Dxx DtZn1 vx h vx
advection, u, an artificial velocity due to the rate of change of the horizontal dispersion coefficient, and another artificial velocity due to uneven bathymetrical changes. Therefore particle movements within the RWPT formalism depend on the actual velocity arising from hydrodynamics and artificial velocities due to gradient of dispersion, uneven depth and the random dispersion mechanism. 2.2. Buoyant particle random walk model Fig. 1 shows a schematic of the movement of neutral and buoyant particles, in which surface discharged heated water is treated as a buoyant mass just in front of the discharging point and the vertical movement of particles due to the imposed buoyancy leads to horizontal spreading. Hence this can be regarded as an additive dispersion until discharged heated water poses buoyancy compared to ambient water. The following steps are carried out in the modeling of buoyant particles (Suh, 1998). The density of discharged heated water (ro ) is less than that of ambient water (ra ), and hence there exists a buoyant density difference, DrZra ro . Thus, heated water can be treated as a collection of particles having mass and buoyancy. Due to the density difference Dr, particles move upward and spread in the horizontal plane, which causes additive horizontal dispersion until the particles lose their buoyant nature (i.e. Dr approaches zero). This additive buoyant dispersion, Db, is a function of space and time. The spatial components Dbx and Dby can be represented as: Dbx ZDmax expð kxÞ; for x%xnf ; Dbx RDxx Dby ZDmax expð kyÞ; for y%ynf ; Dby RDyy
with the boundary conditions Dbx ZDmax at x Z 0 and Dbx Dxx at x Z xnf, and similarly in the y direction. If the coefficient k and the maximum dispersion coefficient Dmax can be determined from field measurements, we can resolve the behavior in the near field. The temporal component, Dbt, has a similar form, but with time as the independent variable. The strength of Db is related to t=0
Applying the same approach for the y-directional field yields, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vDyy Dyy vh DyZ vC C ð11Þ DtC 2Dyy DtZn2 vy h vy Eqs. (10) and (11) behave as the governing equations of 2-dimensional RWPT. From these equations we see that RWPT depends on both a deterministic part a and a random dispersion part b. Moreover we note that advection is comprised of three components: real
ð12Þ
t=t1
t=t2 one particle (conventional no layer)
multiple particles layer concept w/o buoyancy
multiple particles with buoyancy
Fig. 1. Random walk motion of neutral and buoyant particles in near field (tnf %t1 ; t2 %tff ).
S.-W. Suh / Environmental Modelling & Software 21 (2006) 234–242
237
Dr, i.e. Dmax fDr, Dmax Zð1CnDrÞD, where n should be determined by field observations. Thus we can express the random walk relationship in terms of these new buoyant dispersions Dbx, Dby and Dbt which give the following semi-active buoyant particle random walk tracking algorithm: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vDbx Dbx vh C DxZ uC ð13Þ DtC 2Dbt DtZn1 vx h vx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vDby Dby vh C DyZ vC DtC 2Dbt DtZn2 vy h vy
ð14Þ
This scheme has two main components: a deterministic part representing the pseudo-velocity due to spatial variations in depth and dispersion, and a random component for particle movement. Buoyant dispersion is included in both components; it has a greater effect on the movements of particles in the near field compared to the far field. 2.3. RWPT in a simplified case In order to test the feasibility and applicability of the RWPT model, we first applied it to a simple system with known analytical solutions. Under the assumption of uni-directional, uniform flow from left to right, as shown in Fig. 2, the analytical solution for continuous release of mass m can be written as: ( 2 ) xx m cðx; tÞZpffiffiffiffiffiffiffiffiffiffi ð15Þ 2s2x 2psx where s2x Zs20x C2Dt, xZx0 Cut, and subscript 0 means the initial location of the source point. To test the stability of the RWPT model for a high advection system, we considered the very high-speed case where u Z 10 m/s and D Z 10 m2/s. In the comparison we only put 1000 particles in order to see the effect of relative small number of particles. The results are shown in Fig. 3.
Fig. 3. Comparison of random walk particle distribution in flat bed.
The results show that the initial stage of the random walk is almost the same as the analytical solution, but that after this initial stage, the random walk veers slightly away from the analytical solution due to the random nature of the particle movements. These findings suggest that particle tracking can give reliable results close to a source point, where steep concentration gradients exist in normal concentration models. However RWPT is not suitable in the far field where the ambient turbulent field dominates yielding random particle movements and thus even spreading of particles in all directions. The above comparison may be of limited generality because we tested only the idealized flat-bottom case; hence we carried out further tests on a sloping bottom case to represent systems with spatial depth variations, which is commonly encountered in real coastal situations. Now, rewriting the 1-dimensional dispersion equation for the sloping bottom case gives: vc vc D v vc Cu h Z0 ð16Þ vt vx h vx vx Assuming that the depth varies according to the exponential law, hðxÞZh0 expðkxÞ, we can reformulate Eq. (16) as the following modified 1-dimensional transport equation: vc vc v2 c kD D 2 Z0 vt vx vx
Fig. 2. Meshes for RWPT simulation, particles are released at center point.
ð17Þ
In this expression kD behaves as an artificial advection, xZx0 kDt. Numerical tests for this case were performed for the no flow case, using values of u Z 0 m/s and D Z 10 m2/s. Now, assume that the slope of the bed varies according to the expression hðxÞZ5 expð0:0003xÞ; 5000%x%5000, which corresponds to a depth variation of 1.1 m–22.4 m on moving 10 km in the on–off shore direction; this variation is in
238
S.-W. Suh / Environmental Modelling & Software 21 (2006) 234–242
advantage of using an ELM is that it allows the use of large Courant numbers, a feature that is particularly important in surface waters whose flows are dominated by advection. The governing equation of mass transport in a 2dimensional field can be expressed in index notation as: vc vc v2 c Cu)i ZDij CrcGS vt vxi vxi vxj
Fig. 4. Comparison of RWPT in sloping channel, i.e. h Z 5 exp(0.0003x) for 5000%x%5000.
accord with a mild slope of a real coastal zone. The results of these calculations are shown in Fig. 4. The numerical simulation showed satisfactory results, as in the flat-bottom case, although the accuracy decreased with time after particle release. The analysis of the pure dispersion case demonstrates that RWPT shows good agreement with the analytical solution in the vicinity of the source point, despite the steep concentration gradient (i.e. vc=vxR0:005), and shows a slightly skewed result in the far field region characterized by a very mild concentration gradient (vc=vx%0:002). These tests on simple systems thus indicate that RWPT is highly recommendable in the near field region, but it may result in erroneous particle distributions in the far field. 3. Eulerian–Lagrangian simulation 3.1. EL concentration model ELM requires hydrodynamic driving force information, that is, we must carry out hydrodynamic modeling prior to the EL dispersion modeling. The governing equations for hydrodynamic simulation are the depthaveraged continuity and momentum equations. These equations were solved in the present work using Galerkin’s finite element method using linear interpolation. A harmonic approach for hydrodynamic equations is suitable for periodic tidal simulation in this coastal area. This approach has been shown to be an efficient treatment in coastal and estuarine modeling. In these numerical simulations, two kinds of harmonic model were applied: 2-dimensional TEA (Westerink et al., 1992) for surface heat discharge simulation, and the depth-averaged form of 3-dimensional TIDE3D (Walters, 1987) for suspended solids dispersion. For computation of the concentration mode, a 2dimensional finite element ELM was adopted. The main
ð18Þ
where c is the concentration of pollutant material, t is time, xi is the i-th coordinate, Dij is the dispersion coefficient in the x Z 1,2 direction, u)i is the i-directional apparent velocity, r is the reaction coefficient, and S means sink or source. Since the governing equation, Eq. (18), includes both hyperbolic and parabolic operators, an operator split EL scheme may be a plausible choice (Baptista, 1987). Introducing a temporary variable, cf, defined as the concentration at a mid time level between the prior computation level n 1 and the present time level n, yields the following separated equations: n1 cf cn1 vc C u)i Z0 ð19Þ vxi Dt
c n cf v2 c Z Dij vxi vxj Dt
!n Crcn
ð20Þ
The pure advective equation, Eq. (20), can be rewritten in the following discrete form: Dc vc vc Z Cu)i Z0 Dt vt vxi
ð21Þ
This equation implies that the suspended solid concentration, c, remains constant on the following characteristic lines: dxi Zu)i ðx; y; tÞ dt
ð22Þ
The concentration is traced along the characteristic and interpolated onto finite elements. 3.2. Combining RWPT with the EL concentration model Application of RWPT across an entire region is not an economical approach because it entails tracking the time history of each particle, even in the far field. Thus, it is recommended that RWPT be used only near source points, where high concentration gradients typically occur during drilling, dumping or dike construction, and to use ELM in the far field. It is difficult to convert
S.-W. Suh / Environmental Modelling & Software 21 (2006) 234–242
from a particle representation to a concentration representation on a fixed numerical grid as the final stage of RWPT. Since most numerical models are formulated in terms of concentration rather than the location of each particle, we need a scheme for converting RWPT data to concentration data. Previously, Suh (2001) suggested a simple welding method in which the near field concentrations are assigned values that have been pre-computed by another near field model, CORMIX3 (Jones and Jirka, 1993). And then further far field simulations are continued from near field calculations. The results obtained using this approach showed fairly good agreement with field observations. In most coastal dispersion analyses, the behavior around the source point is not usually the primary concern: the source is looked on as merely releasing mass into a water body and the regions of interest are generally the affected areas in the far field. In addition, most regulations for water quality constituents including suspended solids and heated water are expressed as concentrations or excess temperatures, not in terms of particle locations. Thus if we use RWPT, it is necessary to convert the resulting particle location data into concentration data. Conversion of particle locations into concentrations is straightforward. When we use the finite element approximation in far field ELM, we can write the following expression for the concentration in an arbitrary element, under the assumption that each particle has the same mass, cel Zm
Pel Ael hel
ð23Þ
where m is the mass of a particle (i.e. total mass in the system divided by the number of particles), Pel is the number of particles in an element. A is the area of an element, and h denotes the mean depth of the element. Since this projection involves mapping of particles in an element, if we use nodal concentration values we can calculate the concentration in an element by using a finite element interpolation scheme such as linear interpolation. Thus RWPT satisfies mass preservation perfectly, although during the projection it may cause some error. The error associated with converting from particle locations to concentrations is greater when irregular elements are used. Moeller (1993) studied the determination of the adequate size of a Gaussian plume near a source point in 1-dimensional and 2-dimensional RWPT. He suggested that the Gaussian dispersion height, H, expressed as a function of Gaussian width s, pffiffiffiffiffiffiffiffi which is equal to 2Dt, has some relationship with the number of particles, N, used in tracking. Thus, in the modeling of the near field we use the particle mode. Suspended solids arising from coastal construction are represented by neutral particles. The movements of these particles are represented according
239
to Eqs. (10) and (11). For buoyant particle movement in heat dispersion analysis, random walks are replaced by Eqs. (13) and (14). The advection equations are solved numerically using a fourth order Runge-Kutta method. In either case, when the particles reach the near field boundary, the particle locations are mapped into node concentrations. After this mapping, the model computations continue in the concentration mode. In the case of a continuous source, a certain number of particles are released at every time step. Each unit mass of particles is mapped onto node concentrations if xRxnf ; yRynf .
4. Application of the proposed models 4.1. Suspended solids dispersion In the coastal dike construction near the Keum river estuary located in the west coast of Korea, suspended solids spread to the environment with high advection due to strong tidal currents which reach maximum velocities of around 1.2 m/s for the flood and ebb tides. In order to simulate such dispersion, we must use at least ELM because advection dominates the transport. However, near the source point there may be very steep concentration gradients; hence we applied RWPT in that region following the approach described above. In the simulation, the background concentration of suspended solids was set to zero. The source strength was assumed to be 15 kg/day, which was evenly represented as 5000 particles released continuously over 10 h to mimic daytime production. Particles were tracked for 500 m of near field, in which the concentration gradient was VcR0:005, as in the model tests, and then mapped onto the nodes and operator-splitting ELM was applied. The horizontal dispersion coefficients were assigned values of 10 m2/s. Sample results obtained using the hybrid method of RWPT and ELM are plotted in Fig. 5. According to the model results, additive suspended solids levels of 5 mg/l are found within 2 km of the source points and the region of interest is confined to 5 km in the ebb tide. 4.2. Heated water dispersion To test the ability of buoyant particle tracking to model cooling water discharged from a coastal thermal power plant, we consider the situation at Boryong power plant in Korea. This power plant has 6 units, each producing 500 MW of electricity. Heated water is discharged into the sea, where currents of maximum velocity 1 m/s dominate in the ambient field. Each unit discharges water at a rate of 13 m3/s with an excess temperature of 7 C. In RWPT modeling, 5000 particles were used. Applied parameters were DrZ1:04 kg=m3 , Dmax Z 100 m2/s, Dxx Z 5 m2/s and Dyy Z 10 m2/s.
240
S.-W. Suh / Environmental Modelling & Software 21 (2006) 234–242
Fig. 5. Map showing study area for coastal construction sites near in Kunsan, west coast of Korea (upper panel) and simulated SS distribution in flood and ebb tides (lower panel).
In addition, initial temperature fields were set to zero and then computation begins as cold start. From field measurements, the influence zone of the near field was set to 300 m (Suh, 2001). This distance was chosen because, beyond this zone, the jet momentum loses its nature and the temperature was constant in a vertical direction. The computed results are shown in Fig. 6, along with the results obtained using the previously developed concentration model that employs ELM in conjunction with a patching technique in the near field by CORMIX3 (Jones and Jirka, 1993). The two approaches give substantially different results. The results obtained using the previously developed approach indicate that the regions of excess temperature fields are not scattered but rather are attached to the shoreline. In contrast, the hybrid method proposed here predicts scattering of excess temperature fields to regions further offshore which is more consistent with the trend observed in the field.
5. Conclusion To maintain the conservation law of material transport for advection-dominated coastal regions such as those found on the Korean Peninsula, we applied operator-splitting 2-dimensional ELM. However, in the near field the initial concentration gradient is very steep, which causes numerical errors in concentration models. Thus, to reduce these errors while maintaining consistent modeling across the whole region, we developed an RWPT scheme for use in the near field only. RWPT was applied only in the near field because its application in the far field entails much unnecessary computation and can give unphysical results, and our concern was predicting environmental concentration not a history of particle movements. Thus, the following scheme was devised to achieve effective and accurate dispersion modeling: RWPT is applied only in the near field region, where concentration gradients are steep; at the edge of this near
S.-W. Suh / Environmental Modelling & Software 21 (2006) 234–242
241
Fig. 6. Map showing heat dispersion study area and finite element meshes (upper panel) and computed excessive temperature using RWPT in near field (lower left) and ELM with patching technique (lower right).
field region, where the concentration gradient becomes mild, the particle locations are mapped onto a Eulerian grid; in the far field region, the ELM model should be exploited. Simple model tests of the proposed scheme demonstrated its feasibility in regions with high concentration gradients (VcR0:005) in modeling dispersion of neutral particles, and the applicability of the newly suggested additive horizontal dispersion due to buoyant particles in heated fluid discharge. Simulations of the dispersion of suspended solids and heated water using the proposed schemes showed trends in reasonable agreement with field observations and more realistic spreading than the near field patching technique developed previously (Suh, 2001). Thus, the proposed hybrid method can be applied in coastal regions with satisfactory results.
Acknowledgment This work is supported by Coastal Research Center in Kunsan National University.
References Baptista, A.E., 1987. Solution of advection dominated transport by Eulerian Lagrangian methods using the backwards method of characteristics, PhD thesis, MIT. Batchelder, H.P., 2002. Spatial and temporal distributions of mesozooplankton in idealized models using individual based particle tracking simulations. ICES/GLOBEC workshop on transport of cod larvae, Hillerod, Denmark.
242
S.-W. Suh / Environmental Modelling & Software 21 (2006) 234–242
Dimou, K., 1989. Simulation of estuary mixing using a 2-dimensional random walk model, MS thesis, MIT. Jones, G.R., Jirka, G.H., 1993. CORMIX3: an expert system for the analysis of and prediction of buoyant surface discharges, Technical report, Cornell University. Korotenko, K.A., 2000. Matter transport in meso-scale oceanic fronts of river discharge type. Journal of Marine Systems 24, 85–95. Moeller, J.C., 1993. Comparison of Eulerian–Lagrangian random walk and hybrid methods of modeling pollution transport, MS thesis, MIT. Neuman, S.P., 1984. Adaptive Eulerian–Lagrangian finite element method for advection dispersion. International Journal of Numerical Methods in Engineering 20, 317–337. Sabeur, Z.A., Hockley, M.C., Barker, R.G., 2002. Use of PROTEUS for advanced environmental impact assessment and management of marine industrial chemicals. Proceedings of IMEMS2002. Spagnol, S., et al., 2002. An error frequently made in the evaluation of advective transport in two-dimensional Lagrangian models of advection–diffusion in coral reef waters. Marine Ecology Progress Series 235, 299–302. Suh, S.W., 1998. Thermal dispersion analysis using semi-active particle tracking in near field combined with two-dimensional Eulerian–
Lagrangian far field model (in Korean). Journal of Korean Coastal Engineers 10 (2), 73–82. Suh, S.W., 2001. A hybrid near-field/far-field thermal discharge model for coastal area. Marine Pollution Bulletin 43 (7–12), 225–233. Tompson, A.F., Vomvoris, E.G., Gelhar, L.W., 1988. Numerical simulation of solute transport in randomly heterogeneous porous media: motivation, model development and application, Tech. Rep. 316, R.M. Parsons Laboratory, MIT. Walters, R.A., 1987. A model for tides and currents in the English Channel and southern North Sea. Advances in Water Resources 10, 138–148. Westerink, J.J., et al., 1992. Tide and storm surge predictions using finite element model. Journal of Hydraulic Engineering, ASCE 118 (10), 1373–1390. Xie, H., Yapa, P.D., 2002. Simulating the behavior and the environmental effect of sediment plumes from manganese nodule mining. Proceedings of IMEMS2002, pp. 115–136. Zhang, X.Y., 1995. Ocean outfall modeling – interfacing near and far field models with particle tracking method, PhD thesis, MIT. Zheng, C., Wang, P.P., Dortch, M.S., 2000. Subsurface contaminant transport modeling: challenges and resolutions. Proceedings of 16th IMACS World Congress.
