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

TL;DR

The paper develops a kinetic-theory framework for the long-term relaxation of axisymmetric distributions of 2D point vortices, identifying two distinct relaxation routes governed by the mean angular-velocity profile: $1/N$ two-body resonances in non-monotonic profiles and $1/N^{2}$ three-body resonances under kinetic blocking for monotonic profiles. It leverages angle–action coordinates and a softened, static background potential to study both diffusion and flux, comparing analytical predictions from the inhomogeneous Landau and Landau–lenard formalisms with extensive $N$-body simulations. A key finding is the role of resonance broadening near extrema of the frequency profile, which regularises divergences and yields quantitative agreement with simulations, while also highlighting saturation of broadening at large nonlinear times. The work connects vortex dynamics to broader long-range interacting systems and points to future directions including a clean derivation of the $1/N^{2}$ Balescu–Lenard equation, deeper understanding of resonance broadening, and extensions to more general distributions and collective effects.

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.

Figures (19)

• Figure 1: Left: Illustration in configuration space, ${ (x,y) }$, of the dynamics of a few point vortices following the DF from Eq. \ref{['eq:DF_LOC']}. Right: Same, but in action space ${ (\theta , J) }$. In that space, the vortices follow (perturbed) straight-line trajectories.
• Figure 2: Illustration of the DF from Eq. \ref{['eq:DF_LOC']} (full lines) and the considered frequency profiles (dashed lines). Profile (1) (top panel) is non-monotonic and has an extremum in ${ J_{\star} \!=\! 1.03 \, J_{0} }$, while profile (2) (bottom panel) is monotonic.
• Figure 3: Nyquist contour for profile (1) for various values of the active fraction $q$, for the resonance ${ k \!=\! 1 }$. We refer to Appendix \ref{['app:LinearResponse']} for the definition of the "dielectric matrix", ${ \mathbf{E}_{k} (\omega) }$. For ${ q \!\gtrsim\! 0.0010 }$, the system is found to be linearly unstable.
• Figure 4: Contribution to the relaxation rate, ${ \mathcal{R}_{1} (J , k) }$ (Eq. \ref{['eq:def_R1k']}, top panel), and diffusion coefficient, ${ \mathcal{D} (J , k) }$ (Eq. \ref{['eq:def_mDk']}, bottom panel), evaluated in ${ J / J_{0} \!=\! 1.1 }$, as a function of the resonance number, $k$. The different colours correspond to different values of the softening length, $\epsilon$. For small $k$, we recover the expected scaling in $k^{-1}$, before it being damped by softening.
• Figure 5: Initial relaxation rate in action space, ${ \mathcal{R}_{1} (J) }$ (Eq. \ref{['eq:def_R1']}, top panel), and diffusion coefficient, ${ \mathcal{D} (J) }$ (Eq. \ref{['eq:def_mD']}, bottom panel), for profile (1). For the simulations, the uncertainty is illustrated with the 16% and 84% contours obtained by bootstrap over the available realisations. We refer to Appendix \ref{['app:NumericalSimulations']} for details on the numerical setup. For the prediction, the full line corresponds to the ${ 1/N }$ Landau equation (Eqs. \ref{['eq:Landau_1N']} and \ref{['eq:def_D2']}), while the dashed line is from the "regularised" ${ 1/N }$ Landau equation (Eqs. \ref{['eq:Landau_1N_REG']} and \ref{['eq:D2_1N_REG']}), with the regularised time, ${ T_{\mathrm{reg}} / T_{\mathrm{dyn}} \!=\! 307 }$, adjusted to the slope around the sign switch of ${ \mathcal{R}_{1} (J) }$. Close to the extremum of the frequency profile, broadening effects play a crucial role.