Violent relaxation in two-dimensional flows with varying interaction range

Abstract

Understanding the relaxation of a system towards equilibrium is a longstanding problem in statistical mechanics. Here we address the role of long-range interactions in this process by considering a class of two-dimensional or geophysical flows where the interaction between fluid particles varies with the distance as $\sim$ r^($\alpha$--2) with $\alpha$ \textgreater{} 0. Previous studies in the Euler case $\alpha$ = 2 had shown convergence towards a variety of quasi-stationary states by changing the initial state. Unexpectedly, all those regimes are recovered by changing $\alpha$ with a prescribed initial state. For small $\alpha$, a coarsening process leads to the formation of a sharp interface between two regions of homogenized $\alpha$-vorticity; for large $\alpha$, the flow is attracted to a stable dipolar structure through a filamentation process.