Dispersion of particles in two-dimensional circular vortices

The spreading of n particles simultaneously released is modeled numerically and analytically using a Langevin's equation. In the numerical experiments, particles are dispersed utilizing the random walk technique plus advection caused by a vortex with rigid-body rotation (RV) or irrotational (IV). In each vortex, the dispersion is described analyzing the statistical behavior of the particles' position as a function of time t. In the RV case, analytical expressions for the statistics indicate (like the numerical experiments) that the dispersion or variance is linear in t and the diffusion coefficient D depends on the angular speed Omega of the vortex. If the time scale of the random walk is much smaller than the rotation period (i.e., Delta t << 2 pi/Omega), then D decreases lightly if Omega increases relative to a pure two-dimensional (2D) random walk. In the experiments of the IV in which all particles start at the same point, we observe an asymmetric dispersion (the cloud becomes a comet and then a spiral), which leads to a rapid growth in the variance proportional to t(3) in the first time steps. This anomalous dispersion occurs while the particles distribute around the origin and the spiral reaches to form a belt. Afterward, the variance grows linearly in t as in standard dispersion, but D is in general larger than the case without the vortex, even if initially the particles formed a ring around the origin. D tends to increase with the circulation for more intense vortices.