Random walk in case of iso- and diapycnal diffusion
Spivakovskaya, D.; Deleersnijder, E.; Heemink, A.W. (2006). Random walk in case of iso- and diapycnal diffusion, in: Wesseling, P. et al. (Ed.) European Conference on Computational Fluid Dynamics (ECCOMAS CFD 2006): Proceedings.
In: Wesseling, P.; Onate, E.; Périaux, J. (Ed.) (2006). European Conference on Computational Fluid Dynamics (ECCOMAS CFD 2006): Proceedings. TU Delft: Delft. ISBN 90-9020970-0. , more
Large scale diffusion processes in the ocean occur mostly along isopycnal surfaces, i.e. surfaces of equal density. However, there is also diapycnal diffusion, which is associated with a diffusion flux orthogonal to isopycnal surfaces. The diapycnal and isopycnal diffusion fluxes are commonly parameterized a la Fourier-Fick, a formulation involving a diffusion tensor that is not isotropic. In this case Eulerian discretizations of the isopycnal diffusion term yield discrete operators that are not monotonic. So, the Eulerian approach may not always be the best option. Another way is to use the Lagrangian approach, that follows the particle through space at every time step. The movement of particle is modeled with the help of stochastic differential equation, which is consistent with the advection-diffusion equation. By simulating the positions of many particles the diffusion processes can be described. These random walk models allow to avoid a lot of problems, connected with the Eulerian approach, and this makes them very attractive in a number of applications. In this paper the random walk model for the simulation of diffusion processes with space-varying, general positive definite diffusivity, particularly for iso and dia-pycnal diffusion, is established and analyzed. The Lagrangian approach is applied for linear idealized test problems, for which the exact solution is known. The random walk model is also tested for a sinking-diffusion model and it is shown that the Lagrangian approach can also be used for the solution of the adjoint problem. i.e. computing the residence time. The results obtained show that this random walk model may be a good alternative to commonly-used Eulerian models.
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