Landau damping and mode-selective instability of a vortex gas

Abstract

We show that a collisionless point-vortex gas behaves as a genuine Vlasov system. Starting from the non-canonical Hamiltonian structure of point-vortex dynamics, we derive a reduced Klimontovich equation and the corresponding Vlasov description directly on the physical plane, without introducing auxiliary momentum variables. The theory predicts a rich spectrum of collective phenomena, including vortex Rossby-like waves, global oscillations, resonant critical layers, and a dielectric response operator analogous to that of plasmas and self-gravitating systems. For smooth vortex distributions, differential rotation gives rise to a vortex analogue of Landau damping, whereby collective modes transfer energy and angular momentum to resonant vortex annuli in the absence of collisions or dissipation. Remarkably, differential rotation acts as a stringent mode-selection mechanism: for a Gaussian vortex cloud, only a single quadrupolar quasi-mode survives, while all higher multipoles are destroyed by phase mixing. We further predict a vortex counterpart of the beam--plasma instability, in which a thin vortex ring orbiting an Onsager patch selectively excites Kelvin modes with an azimuthal symmetry controlled by the ring population. These results establish kinetic theory as a natural framework for collective vortex dynamics and suggest new mechanisms for vortex transport, vortex avalanches, and rotational glitches in neutron-star superfluids.