Stochastic differential equation models of vortex merging and reconnection

We show that the stochastic differential equation sSDEd model for the merger of two identical two-dimensional vortices proposed by Agullo and Verga f“Exact two vortices solution of Navier– Stokes equation,” Phys. Rev. Lett. 78, 2361 s1997dg is a special case of a more general class of SDE models for N interacting vortex filaments. These toy models include vorticity diffusion via a white noise forcing of the inviscid equations, and thus extend inviscid models to include core dynamics and topology change se.g., merger in two dimensions and vortex reconnection in three dimensionsd. We demonstrate that although the N=2 two-dimensional model is qualitatively and quantitatively incorrect, it can be dramatically improved by accounting for self-advection. We then extend the two-dimensional SDE model to three dimensions using the semi-inviscid asymptotic approximation of Klein et al. f“Simplified equations for the interactions of nearly parallel vortex filaments,” J. Fluid Mech. 288, 201 s1995dg for nearly parallel vortices. This model is nonsingular and is shown to give qualitatively reasonable results until the approximation of nearly parallel vortices fails. We hope these simple toy models of vortex reconnection will eventually provide an alternative perspective on the essential physical processes involved in vortex merging and reconnection. © 2005 American Institute of Physics. fDOI: 10.1063/1.1932310g