Violent relaxation in two-dimensional flows with varying interaction range

TL;DR

This study investigates how varying the interaction range in two-dimensional flows influences the system's relaxation to equilibrium, revealing different regimes such as coarsening and dipolar structures depending on the interaction parameter.

Contribution

It demonstrates that changing the interaction range parameter $oldsymbol{ extit{ extalpha}}$ in 2D flows reproduces various quasi-stationary states, extending understanding beyond the classical Euler case.

Findings

Small $ extalpha$ leads to coarsening and sharp interfaces.

Large $ extalpha$ results in stable dipolar structures.

Varying $ extalpha$ recovers regimes seen in previous Euler studies.

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.