THREE-DIMENSIONAL FINITE DIFFERENCE MODEL FOR TRANSPORT OF CONSERVATIVE POLLUTANTS

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1 Pergamon Ocean Engng, Vol. 25, No. 6, pp , Elsevier Science Ltd. All rights reserved Printed in Great Britain /98 $ PII: S (97) THREE-DIMENSIONAL FINITE DIFFERENCE MODEL FOR TRANSPORT OF CONSERVATIVE POLLUTANTS S. Sankaranarayanan, N. J. Shankar* and H. F. Cheong Department of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore Abstract A three-dimensional finite difference transport model appropriate for the coastal environment is developed for the solution of the three-dimensional convection diffusion equation. A higher order upwind scheme is used for the convective terms of the convection diffusion equation, to minimise the numerical diffusion. The validity of the numerical model is verified through five test problems, whose exact solutions are known Elsevier Science Ltd. NOMENCLATURE C(x, y, z, t) pollutant concentration at location (x, y, z) at any time t C 1, C 2, C 3, constants C i,j,k concentration at grid location i, j, k G(x, y, z, t) point source or sink K scaling constant K x, K y, K z diffusion coefficients in the x, y and z directions L x, L y, L z length of the basin in x, y and z directions r(t), s(t), f(t) expression [see Equation (26)] T p period of rotation U, V, W flow velocities in the x, y and z directions x 0, y 0, z 0 coordinates of the centre of Gaussian pulse dimensionless parameter t time step x, y, z grid spacing in x, y and z directions free surface elevation 1. INTRODUCTION Mathematical modelling of the transport of salinity, pollutants and suspended matter in shallow waters involves the numerical solution of a convection diffusion equation. Many popular finite difference methods, such as the upwind scheme of Spalding (1972) and the flux-corrected scheme (Boris and Book, 1973) are available for the solution of the depthintegrated form of the convection diffusion equation. Another widely used approach is the split-operator approach (Sobey, 1983; Li and Chen, 1989), in which the convection and diffusion terms are solved by two different numerical methods. Noye and Tan (1988) used a weighted discretisation with the modified equivalent partial differential equation *Author to whom correspondence should be addressed. 425

2 426 S. Sankaranarayanan et al. approach for solving one-dimensional convection diffusion equations. Later, Noye and Tan (1989) extended this technique to solve two-dimensional convection diffusion equations. But the above mentioned techniques have difficulty in solving three-dimensional problems, because of extensive matrix inversions at each time step. Numerical studies show that the use of central differencing for the convective terms of the convection diffusion equation results in negative species concentration. Lam (1975) points out that the central difference approximation will overestimate the advective flux so much that it often causes a negative concentration to appear in the neighbouring cell. To circumvent such a shortcoming of central differencing, the upwind or donor cell method introduced by Gentry et al. (1966) is generally used. To overcome the shortcomings of numerical dispersion, Leonard (1979) introduced an upstream interpolation method, namely QUICK (Quadratic Upstream Interpolation Convective Kinematics) for one-dimensional unsteady flow. Later, Leonard (1988) gave an improved version of the QUICK scheme, eliminating the wiggles completely by introducing exponential integration into regions with sharp fronts. Chen and Falconer (1994) showed that the QUICK scheme is only second-order accurate in space and presented different forms of the third-order convection, second-order diffusion for the solution of the convection diffusion equation. A relatively recent phenomenon is the development of three-dimensional transport models. Lardner and Song (1991) used an algorithm splitting the horizontal convection and diffusion by an implicit finite element method, treating the horizontal convection diffusion explicitly and vertical convection diffusion by an implicit finite element method. It is to be noted that Lardner and Song (1991) used a first-order upwind scheme for the convection terms of the convection diffusion equations. Sommeijer and Kok (1995) made a detailed study on the use of various time-integration techniques for the numerical solution of threedimensional convection diffusion equations using finite differences. The numerical model was validated by comparing the results obtained with analytical solutions for the case of transport of a Gaussian pulse in unsteady and non-uniform flow. In the present study, a third-order upwind difference scheme as given in Kowalik and Murty (1993) has been used for the convection terms of the convection diffusion equation. Earlier, the authors Shankar et al. (1996) used a third-order upwind scheme for the convective terms of the shallow water momentum equations. 2. MATHEMATICAL MODELLING OF CONVECTION DIFFUSION PROCESSES 2.1. Governing equations The mathematical model describing the transport processes in three dimensions is given by the convection diffusion equation t + (CU) + (CV) + (CW) 2 C K x y z x x K 2 C 2 y y K 2 C 2 z z 2 = G(x, y, z, t) (1) subject to the initial conditions, C(x, y, z, 0), and the boundary conditions, Equations (2) (4). (x, y, t) at z = 0 on the top surface (2) z

3 3-D finite difference transport model 427 (x, y, t) at z = h on the bottom surface (3) z = 0 along the lateral boundaries (4) n where C is the unknown pollutant concentration at location (x, y, z), U, V and W are the flow velocities in the x, y and z directions, respectively, K x, K y and K z are the diffusion coefficients in the x, y and z directions, respectively, G represents any source or sink. x, y and z are the Cartesian coordinates with the xy plane horizontal and occupying the undisturbed position of the water surface, and the z-axis pointing vertically upwards, and n is the outward drawn normal in the xy plane. The position of the free surface is denoted by z = (x, y, t) as shown in Fig. 1 and that of the bottom by z = h(x, y). The values of, U, V and W are to be obtained from a hydrodynamic model such as that of Shankar et al. (1996). Equation (1) reduces to t + U x + V y + W z K 2 C x x K 2 C 2 y y K 2 C 2 z z = G(x, y, z, t) 2 (5) since U x + V y + W = 0 z 2.2. Finite difference formulation of the convection term using first-order upwinding The values of C, U, V and W are defined at all the grid points i, j and k, respectively. A typical convection term / x is evaluated using first-order differencing as x = C i,j,k C i 1,j,k x C i +1,j,k C i,j,k x + O( x) if U(i, j, k) 0 + O( x) if U(i, j, k) 0 (6) Fig. 1. Definition sketch.

4 428 S. Sankaranarayanan et al. The first-order upwind scheme as given by Equation (6) is very popular in the numerical solution of the convection diffusion equation (Lardner and Song, 1991) and shallow water hydrodynamic equations (Shankar et al., 1995; Horie, 1980). However, the use of a firstorder upwind scheme is strongly discouraged by numerical modellers, since it introduces numerical diffusion. In other words, when a first-order upwind scheme is used for the convection terms, it amounts to introducing one additional term in the governing equations (O Brien, 1986) Finite difference formulation of the convection term using third-order upwinding In the present study, a third-order upwind differencing has been used as given in Kowalik and Murty (1993). A typical convective term / x is discretised using a third-order differencing as x = (C i 2,j,k 6C i 1,j,k +3C i,j,k +2C i +1,j,k )/6( x) +o( x) 3 (7) for U i,j,k 0 and i = 3,nx 2. x = ( 2C i 1,j,k 3C i,j,k +6C i +1,j,k C i +2,j,k )/6( x) +o( x) 3 (8) for U i,j,k 0 and i = 3,nx 2. However, for the boundary points, a four-point upstream formula can be written such that either points to the left or to the right are considered in the finite difference approximation. Near the left boundary, i.e. for i = 2 x = ( 11C i,j,k +18C i +1,j,k 9C i +2,j,k +2C i +3,j,k )/6( x) +o( x) 3 (9) Near the right boundary, i.e. for i = nx 1 x = ( 2C i 3,j,k +9C i 2,j,k 18C i 1,j,k +11C i,j,k )/6( x) +o( x) 3 (10) where nx is the number of grid points in the x direction. The second derivatives occurring in diffusion terms are evaluated using a central difference. A typical second derivative 2 C/ x 2 is evaluated using 2 C x = C i +1,j,k 2C i,j,k + C i 1,j,k (11) 2 ( x) Stability criterion Hindmarsh et al. (1984) established the stability criterion for the multi-dimensional advection diffusion equation for an explicit scheme as t 2 K x ( x) 2 + K y ( y) K z ( z) 2 + U ( x) + V ( y) + W ( z) (12)

5 3-D finite difference transport model VALIDATION OF THE NUMERICAL MODEL It is necessary to compare the model results with analytical solutions, to validate the developed numerical model. The simplified idealised test cases, for which analytical solutions are available, present a valuable means to debug the developed computer program. Two such numerical examples used by Noye and Tan (1988, 1989) involving transport of a one-dimensional and two-dimensional Gaussian pulse, respectively, and two test cases used by Lardner and Song (1991), involving two-dimensional and three-dimensional transport of a point source pollutant are used to validate the model developed in the present study. The test cases involving the transport of point source pollutant concentration by convection and diffusion are taken, since they represent an important practical case of a sudden influx of a pollutant, such as due to oil spill. The four test cases used in the present study have analytical solutions, which makes it possible to compare the results obtained in the numerical solution. MKS units are used throughout the paper Test case involving one-dimensional convection and diffusion For one-dimensional convection diffusion, Equation (1) reduces to t + U x K x 2 C = G(x, y, z, t) (13) 2 x The analytical solution to the one-dimensional advection and diffusion of a Gaussian pulse of unit height, centred at x = 1 in a region bounded by 0 x 9 as given by Noye and Tan (1988) is C(x, t) = 1 4t +1 exp (x x 0 Ut) 2 K x (4t +1) (14) where U is the velocity in the x direction, x 0 is the centre of the initial Gaussian pulse, K x is the diffusion coefficient in the x direction and t is the time coordinate. The initial condition is given by C(x, t) = 1 4t +1 exp (x 1)2 K x (15) and the boundary condition at the two ends at any time t is obtained by substituting x = 0 and x = 9, respectively, in Equation (14). The values of the various parameters used are K x = m 2 /s, U = 0.8 m/s, and the Gaussian pulse of unit height centred at x = 1. The distribution of the Gaussian pulse at t = 5 s is computed using the analytical solution and compared with the concentration distribution obtained using the numerical solution as shown in Fig. 2. The space step and time step are taken to be m and s, respectively. The Courant number (U t/ x) works out to be 0.2 and the cell Reynolds number (U x/k x ) works out as Test case involving two-dimensional convection and diffusion of a Gaussian pulse For two-dimensional convection diffusion, Equation (1) reduces to t + U x + V y K 2 C x x K 2 y 2 C = G(x, y, z, t) (16) 2 y

6 430 S. Sankaranarayanan et al. Fig. 2. Comparison of analytical and numerical solutions for transport of one-dimensional Gaussian pulse for K x = m 2 /s, U = 0.8 m/s. An analytical solution for solving the two-dimensional convection diffusion equation in a rectangular region bounded by 0 x 2 and 0 y 2 for the transport of a Gaussian pulse of unit height (Fig. 3) centred at (0.5, 0.5) is given by C(x, y, t) = 1 4t +1 exp (x 0.5 Ut)2 (y 0.5 Vt)2 K x (4t +1) K y (4t +1) (17) Fig. 3. Three-dimensional perspective view of the initial Gaussian pulse.

7 3-D finite difference transport model 431 with the initial condition C(x, y,0)= exp (x 0.5)2 K x (y 0.5)2 K y (18) and the values of the concentrations along the boundaries at any time t are obtained by substituting the values of their coordinates in Equation (17). The parameters used in the simulation are U = V = 0.8 m/s, K x = K y = 0.01 m 2 /s. The space and time steps are taken to be m and s and hence the Courant number and the cell Reynolds number are calculated to be 0.4 and 2.0, respectively. According to the analytical solution (Fig. 4), the pulse has moved to x = 1.5 and y = 1.5 with a pulse height of at time t = 5 s, while the numerical solution predicts the pulse height as The concentration distributions computed using numerical and analytical solutions are given in Figs 4a and 4b, respectively. Fig. 4. Concentration distributions by analytical and numerical solutions after 5 s, x = m, t = s, K x = K y = 0.01 m 2 /s, U = V = 0.8 m/s: (a) analytical solution; (b) numerical solution.

8 432 S. Sankaranarayanan et al Test case involving two-dimensional convection and diffusion of a point source For an infinite region, the exact solution of the two-dimensional convection diffusion Equation (16), the concentration C(x, y, t) at a given time t for a point source pollutant placed at the centre of the domain (x 0, y 0 ) is given by C(x, y, t) = K 4 K x t 4 K y t exp (x x 0 Ut) 2 (y y 0 Vt) 2 4K x t 4K y t (19) where (x 0, y 0 ) is the location of the point source and K is a scaling factor, taken to be for this test case. The transport of an instantaneous point source pollutant, initially placed at the centre of a domain of size 10,000 10,000 m 2, by convection and diffusion is simulated using the model. Uniform horizontal velocities of U = V = 0.5 m/s, grid spacing x = y = 5000 m, time step t = 50 s and K x = K y = 10,000 m 2 /s are used. The solution for the bounded region is taken large enough so that the concentrations remain essentially zero Fig. 5. Concentration distributions for two-dimensional convection and diffusion by analytical and numerical solutions after 36,000 s, x = y = 5000 m, t = 50 s, K x = K y = 10,000 m 2 /s, U = V = 0.5 m/s: (a) analytical solution; (b) numerical solution.

9 3-D finite difference transport model 433 Fig. 6. Concentration distributions for three-dimensional convection and diffusion by analytical and numerical solutions, at z = 13 m from the surface, after 23,000 s, x = y = 2000 m, z = 6.5 m, t = 50 s, K x = K y = K y = 2000 m 2 /s, K z = 0.01 m 2 /s, U = V = 0.2 m/s, W = 0.0 m/s, t 0 = 5000 s: (a) analytical solution; (b) numerical solution. at the boundaries for the time interval of the computation (Lardner and Song, 1991). The concentration distribution obtained after t = 36,000 s using the model developed in the present study compares well with that given by the analytical solution, Equation (19), as shown in Fig. 5aFig. 5b, respectively. The maximum relative error between the analytical and numerical solutions is about 10% Test case involving three-dimensional convection and diffusion in uniform flow For an infinite region, the exact solution C(x, y, z, t) at a given time t for a point source placed at the centre of the domain is given by

10 434 S. Sankaranarayanan et al. Fig. 7. Concentration distributions for three-dimensional convection and diffusion by analytical and numerical solutions, at z = 32.5 m from the surface, after 23,000 s, x = y = 2000 m, z = 6.5 m, t = 50 s, K x = K y = 2000 m 2 /s, K z = 0.01 m 2 /s, U = V = 0.2 m/s, W = 0.0 m/s, t 0 = 5000 s: (a) analytical solution; (b) numerical solution. where C(x, y, z, t) = C x (x, t, U)C y (y, t, V)C z (z, t, W) (20) 1 C x (x, U, t) = 4 K x t exp (x x 0 Ut) 2 4K x t (21) C y (y, V, t) = C x (z, W, t) = 1 4 K y t exp (y y 0 Vt) 2 4K y t (22) 1 4 K z t exp (z z 0 Wt) 2 4K z t (23)

11 3-D finite difference transport model 435 Fig. 8. Concentration distributions for three-dimensional convection and diffusion by analytical and numerical solutions, at z = 52 m from the surface, after 23,000 s, x = y = 2000 m, z = 6.5 m, t = 50 s, K x = K y = 2000 m 2 /s, K z = 0.01 m 2 /s, U = V = 0.2 m/s, W = 0.0 m/s, t 0 = 5000 s. where (x 0, y 0 ) is the location of the point source and K is a scaling factor, taken to be for this test case. For applying the boundary conditions, Equations (2) and (3), the derivative of C(x, y, z, t) is evaluated as [C(x, y, z, t)] z = C x (x, t, U)C y (y, t, V)C z (z, t, W) [z 0 + w(t + t 0 ) z] 2[K z (t + t 0 )] Substituting the values of z = 0 and z = h in Equation (18), the boundary condition Equations (2) and (3) at the top and bottom surfaces are satisfied in the numerical solution. The initial condition is given by C(x, y, z, t 0 ), where if t 0 0, then the problem has a smooth initial condition. The transport of an instantaneous point source pollutant, initially placed at the centre of a domain of size 40,000 40, m 3, by convection and diffusion is simulated (24)

12 436 S. Sankaranarayanan et al. using the model. Uniform horizontal velocities of U = V = 0.2 m/s and W = 0 m/s, grid spacing x = y = 2000 m, time step t = 50 s and K x = K y = 2000 m 2 /s, K z = 0.01 m 2 /s are used. For a smooth initial condition, the value of t 0 is taken to be 5000 s. The solution for the bounded region is taken large enough so that the solution remains essentially zero at the boundaries for the time interval of the computation (Lardner and Song, 1991). The concentration distribution obtained after t = 18,000 s using the model developed in the present study compares well with that given by the analytical solution Equation (20) at various levels, showing good comparison as in Figs 6 8, respectively. The maximum relative error between the analytical and numerical solutions is about 13% Test case involving three-dimensional convection and diffusion in unsteady non-uniform flow The analytical solution for the transport of conservative pollutant in unsteady, nonuniform flow as given by Sommeijer and Kok (1995) is C(x, y, z, t) = exp z L z f(t) x L x r(t) 2 y L y s(t) 2 (25) To stress the local nature of the concentration, the value of is taken to be 30. r(t) = cos 2 t T p ; s(t) = sin 2 t T p and f(t) = 4t (T b + t) (26) The flow fields are given by U(x, y, z, t) = C 1 sin x L x + y L y sin z L z d(t) (27) Fig. 9. Concentration distribution C(x, y, z, t) for t = 0 at the surface (z = 0).

13 3-D finite difference transport model 437 Fig. 10. Concentration distribution C(x, y, z, t) for t = 0 at z = 50 m from the top. Fig. 11. Concentration distribution C(x, y, z, t) for t = 0 at z = 100 m from the top: (a) analytical solution; (b) numerical solution.

14 438 S. Sankaranarayanan et al. Fig. 12. Concentration distributions for three-dimensional convection and diffusion by analytical and numerical solutions, at the surface (z = 0) after 10,800 s, x = y = 2000 m, z = 10 m, t = 50 s, K x = K y = K z = 0.5 m 2 /s: (a) analytical solution; (b) numerical solution. where V(x, y, z, t) = C 1 cos x L x + y L y sin z L z d(t) (28) W(x, y, z, t) = C 1 cos x L x + y L y + C 2 sin x L x + y L y L z cos z L z d(t) d(t) = cos 2 t T p C 1 = 3, C 2 = 4, and = 0.05 The flow velocities given by Equations (27) (29) satisfy the continuity relation for the mass balance given by (29)

15 3-D finite difference transport model 439 Fig. 13. Concentration distributions for three-dimensional convection and diffusion by analytical and numerical solutions, at z = 50 m from the surface, after 10,800 s, x = y = 2000 m, z = 10 m, t = 50 s, K x = K y = K z = 0.5 m 2 /s: (a) analytical solution; (b) numerical solution. U x + V y + W z = 0 (30) The values of the various parameters are given by L x = L y = 20,000 m, L z = 100 m, K x = K y = K z = 0.5 m 2 /s, T p = 43,200 s, T b = 32,400 s. It is to be noted that the source term, G(x, y, z, t), is evaluated at each time step from Equation (1), as given in Appendix A. The initial concentration distribution at t = 0 is given by substituting t = 0 in Equation (25). The typical concentration distributions at the surface, 50 m from the top and at the

16 440 S. Sankaranarayanan et al. Fig. 14. Concentration distributions for three-dimensional convection and diffusion by analytical and numerical solutions, at the bottom (z = 100 m) after 10,800 s, x = y = 2000 m, z = 10 m, t = 50 s, K x = K y = K z = 0.5 m 2 /s. bottom of the basin, given as initial conditions are shown in Figs The space and time steps are taken to be 2000 m and 50 s. The concentration distribution at time t = 10,800 s obtained at depths of 0, 50 and 100 m from the top obtained in the present study compares well with the analytical solution as shown in Figs The maximum percentage error between analytical and numerical solutions is found to be 11%.

17 3-D finite difference transport model CONCLUSIONS A three-dimensional finite difference transport model appropriate for the coastal environment is developed for the numerical solution of the three-dimensional convection diffusion equation. A higher order (third-order) upwind scheme is used for the convective terms of the transport equation, to minimise the numerical diffusion. The validity of the numerical model is verified through four test problems, for the transport of point source pollutants in uniform flow, by comparing the results obtained in the present study with those of analytical solutions. The validity of the numerical solution is also tested against analytical solutions for the case of transport of a Gaussian pulse in unsteady and nonuniform flow. However, for problems involving transport of pollutants in coastal waters, the velocity field is to be obtained a priori from the solution of a compatible hydrodynamic model developed by the authors (Shankar et al., 1996). Work is currently in progress to couple the transport model developed in the present study with the multi-level three-dimensional hydrodynamic model. REFERENCES Boris, J. B. and Book, D. L. (1973) Flux corrected for transport algorithm that works. Journal of Computational Physics 11, Chen, Y. and Falconer, R. A. (1994) Modified forms of the third-order convection, second-order diffusion equation. Advances in Water Resources 17, Gentry, R. A., Martin, R. E. and Daly, B. J. (1966) An Eulerian differencing method for unsteady compressible flow problems. Journal of Computational Physics 8, Hindmarsh, A. C., Gresho, P. and Griffiths, D. F. (1984) The stability of explicit Euler integration for certain finite difference approximations of the multidimensional advection diffusion equation. International Journal for Numerical Methods in Fluids 4, Horie, T. (1980) Hydraulic investigation on sea flow and substance dispersion in estuarine and coastal region. Technical Note 360, Port and Harbour Research Institute, Japan. Kowalik, Z. and Murty, T. S. (1993) Numerical Modeling of Ocean Dynamics. World Scientific, Singapore. Lam, D. C. L. (1975) Computer modeling of pollutant transport in Lake Erie. Water Pollution 25, Lardner, R. W. and Song, Y. (1991) An algorithm for three-dimensional convection and diffusion with very different horizontal and vertical scales. International Journal for Numerical Methods in Engineering 32, Leonard, B. P. (1979) A stable and accurate convective modeling procedure based on upstream formulation. Computer Methods in Applied Mechanics and Engineering 19, Leonard, B. P. (1988) Simple high accuracy resolution program for convective modeling of discontinuities. International Journal for Numerical Methods in Fluids 8, Li, Y. S. and Chen, C. P. (1989) An efficient split operator scheme for 2D advection diffusion equation using finite elements and characteristics. Applied Mathematical Modeling 13, Noye, B. J. and Tan, H. H. (1988) A third-order semi-implicit finite difference method for solving the onedimensional convection diffusion equation. International Journal for Numerical Methods in Engineering 26, Noye, B. J. and Tan, H. H. (1989) Finite difference methods for the two-dimensional advection diffusion equation. International Journal for Numerical Methods in Fluids 9, O Brien, J. J. (1986) Advanced Physical Oceanographic Modeling, p Riedel, Boston. Shankar, N. J., Cheong, H. F. and Sankaranarayanan, S. (1995) Multilevel finite difference model for threedimensional hydrodynamic circulation. Internal Report, Department of Civil Engineering, National University of Singapore, p. 55. Shankar, N. J., Cheong, H. F. and Sankaranarayanan, S. (1996) Multilevel finite difference model for threedimensional hydrodynamic circulation. International Journal of Ocean Engineering, 24(9), Sobey, R. J. (1983) Fractional step algorithm for estuarine mass transport. International Journal for Numerical Methods in Fluids 3, Sommeijer, B. P. and Kok, J. (1995) Implementation and performance of the time integration of a 3D numerical transport model. International Journal for Numerical Methods in Fluids 21, Spalding, D. B. (1972) A novel finite difference formulation for differential expressions involving both first and second derivatives. International Journal for Numerical Methods in Engineering 4,

18 442 S. Sankaranarayanan et al. APPENDIX A From Equation (1) the value of G(x, y, z, t) is evaluated to be where G(x, y, z, t) = t 2 C K x x K 2 y + U(x, y, z, t) x 2 C y K 2 C 2 z z 2 + V(x, y, z, t) + W(x, y, z, t) y z (A1) C(x, y, z, t) = exp z L z f(t) x L x r(t) 2 y L y s(t) 2 (A2) r(t) = cos 2 t s(t) = 1 T p ; sin 2 t 4t T p and f(t) = (A3) (T b + t) t = C(x, y, z, t) 4t + T b + t x L x r(t) sin 2 t T p (A4) + T p y L y s(t) cos 2 t T p 4t (T b + t) + 2 T p x L x r(t) (A5) ( 2 ) = C(x, y, z, t) x L x ( 2 ) = C(x, y, z, t) y L y x L y s(t) (A6) z = C(x, y, z, t) 1 (A7) L z 2 C x = C(x, y, z, t) ( 2 ) (L x ) 2 (L x ) x 2 L x r(t) 2 (A8) 2 C y = C(x, y, z, t) ( 2 ) (L y ) 2 (L y ) y 2 L y s(t) 2 (A9) 2 C = C(x, y, z, t) (A10) 2 z (L z ) 2