Forward simulation of three-dimensional advection-diffusion using a fundamental solution for a linear flow

Abstract

A novel Lagrangian-based algorithm is presented for tracking a patch of a passive tracer that evolves according to the 3D advection diffusion equation with a general steady velocity. The algorithm is based on approximating the flow locally by a linear one in space, relative to the center of a patch moving with the flow. We seek a semi-analytic solution to the corresponding PDE, based on the Anzats that the exact solution assumes a Gaussian form whose contours are ellipsoids whose axes are evolving according to velocity parameters. We assume that the coefficients that govern this evolution are unknown and we prove that they have to satisfy a coupled system of non-linear first order ODEs. The system is solved numerically using an RK4 scheme. We first check the algorithm against special linear flows with known exact analytical solutions. We then apply the algorithm to Ekman flows in two and three dimensions and compare our results in two dimensions to the grid based, finite element solver COMSOL Multiphysics. In a regime where the patch does not grow too much in size, the results of our algorithm, which is significantly faster, show acceptable agreement with COMSOL.