Kinetic theory of point vortices in two dimensions: analytical results and numerical simulations

TL;DR

This paper develops a kinetic theory for two-dimensional point vortices, combining analytical results with numerical simulations to describe their evolution, relaxation, and metastable states, revealing resonance-driven dynamics and algebraic decay behaviors.

Contribution

It introduces a comprehensive kinetic framework for point vortices, including resonance effects, metastable states, and detailed test vortex relaxation analysis, which advances understanding of 2D hydrodynamic turbulence.

Findings

Vorticity profile evolution driven by resonances between vortex orbits.

System can become trapped in metastable states with non-Boltzmann distributions.

Test vortex relaxation described by a Fokker-Planck equation with algebraic auto-correlation decay.

Abstract

We develop the kinetic theory of point vortices in two-dimensional hydrodynamics and illustrate the main results of the theory with numerical simulations. We first consider the evolution of the system "as a whole" and show that the evolution of the vorticity profile is due to resonances between different orbits of the point vortices. The evolution stops when the profile of angular velocity becomes monotonic even if the system has not reached the statistical equilibrium state (Boltzmann distribution). In that case, the system remains blocked in a sort of metastable state with a non standard distribution. We also study the relaxation of a test vortex in a steady bath of field vortices. The relaxation of the test vortex is described by a Fokker-Planck equation involving a diffusion term and a drift term. The diffusion coefficient, which is proportional to the density of field vortices and inversely proportional to the shear, usually decreases rapidly with the distance. The drift is proportional to the gradient of the density profile of the field vortices and is connected to the diffusion coefficient by a generalized Einstein relation. We study the evolution of the tail of the distribution function of the test vortex and show that it has a front structure. We also study how the temporal auto-correlation function of the position of the test vortex decreases with time and find that it usually exhibits an algebraic behavior with an exponent that we compute analytically. We mention analogies with other systems with long-range interactions.