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

Abstract: We investigate the long-term relaxation of a distribution of $N$ point vortices in two-dimensional hydrodynamics, in the limit of weak collective amplification. 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 toresonance broadening'' close to the extremum of the angular velocity profile. Quantitative description of such subtle nonlinear effects will be the topic of future investigations.