Kinetic theory of two-dimensional point vortices at order $1/N$ and $1/N^{2}$

TL;DR

This paper studies the long-term relaxation dynamics of two-dimensional point vortices, identifying two regimes based on the angular velocity profile and deriving kinetic equations at order 1/N and 1/N^2, validated by simulations.

Contribution

It derives and compares kinetic equations at orders 1/N and 1/N^2 for different velocity profiles, linking relaxation mechanisms to system profiles and validating predictions with simulations.

Findings

Relaxation occurs via two-body resonances for non-monotonic profiles.

Three-body couplings dominate relaxation in monotonic profiles.

Numerical simulations confirm kinetic theory predictions.

Abstract

We investigate the long-term relaxation of a distribution of $N$ point vortices in two-dimensional hydrodynamics. To focus on the regime of weak collective amplification, we embed these point vortices within a static background potential and soften their pairwise interaction on small scales. Placing ourselves within the limit of an average axisymmetric distribution, we stress the connections with generic long-range interacting systems, whose relaxation is described within angle-action coordinates. In particular, we emphasise the existence of two regimes of relaxation, depending on whether the system's profile of mean angular velocity (frequency) is a non-monotonic [resp. monotonic] function of radius, which we refer to as profile (1) [resp. profile (2)]. For profile (1), relaxation occurs through two-body non-local resonant couplings, i.e. $1/N$ effects, as described by the inhomogeneous Landau equation. For profile (2), the impossibility of such two-body resonances submits the system to a ``kinetic blocking''. Relaxation is then driven by three-body couplings, i.e. ${1/N^{2}}$ effects, whose associated kinetic equation has only recently been derived. For both regimes, we compare extensively the kinetic predictions with large ensemble of direct $N$-body simulations. In particular, for profile (1), we explore numerically an effect akin to ``resonance broadening'' close to the extremum of the angular velocity profile. Quantitative description of such subtle nonlinear effects will be the topic of future investigations.