Kinetic theory of two-dimensional point vortices with collective effects

TL;DR

This paper develops a kinetic theory for two-dimensional point vortices incorporating collective effects, deriving equations that describe their relaxation dynamics and stochastic motion, and analyzing how these processes scale with system size.

Contribution

It introduces a comprehensive kinetic framework that includes collective effects for point vortices, extending previous models and providing explicit expressions for diffusion and drift.

Findings

Derived a Lenard-Balescu-type kinetic equation for axisymmetric flows.

Calculated diffusion and drift coefficients including collective effects.

Analyzed the N-scaling of relaxation times for vortices.

Abstract

We develop a kinetic theory of point vortices in two-dimensional hydrodynamics taking collective effects into account. We first recall the approach of Dubin & O'Neil [Phys. Rev. Lett. 60, 1286 (1988)] that leads to a Lenard-Balescu-type kinetic equation for axisymmetric flows. When collective effects are neglected, it reduces to the Landau-type kinetic equation obtained independently in our previous papers [P.H. Chavanis, Phys. Rev. E 64, 026309 (2001); Physica A 387, 1123 (2008)]. We also consider the relaxation of a test vortex in a "sea" (bath) of field vortices. Its stochastic motion is described in terms of a Fokker-Planck equation. We determine the diffusion coefficient and the drift term by explicitly calculating the first and second order moments of the radial displacement of the test vortex from its equations of motion, taking collective effects into account. This generalizes the expressions obtained in our previous works. We discuss the scaling with N of the relaxation time for the system as a whole and for a test vortex in a bath.