A probabilistic method for Navier-Stokes vortices

Abstract

Consider a Navier-Stokes incompressible turbulent fluid in R². Let x(t) denote the position coordinate of a moving vortex with initial circulation Γ₀ > 0 in the fluid, subject to a force F. Define x(t) as a stochastic process with continuous sample paths described by a stochastic differential equation. Assuming a suitable notion of weak rotationality, it is shown that the stochastic equation is equivalent to a linear partial differential equation for the complex function ψ, i∂ψ/∂t = [-Γ᎔+F]ψ, where |ψ|² = ρ(x,t), ρ being the probability density function of finding the vortex centre in position x at time t.