On the Late-Time Behaviour of a Bounded, Inviscid Two-Dimensional Flow

TL;DR

This study uses high-resolution numerical simulations to investigate the long-term dynamics of inviscid, incompressible 2D flow on a sphere, revealing persistent unsteadiness and complex vortex interactions contrary to equilibrium predictions.

Contribution

It demonstrates that 2D inviscid flows on a sphere do not settle into steady states but exhibit sustained unsteadiness and complex vortex structures, challenging existing equilibrium theories.

Findings

Flow remains unsteady with persistent small-scale vortices.

Vorticity forms a staircase distribution with sharp gradients.

Interactions of main vortices explain unsteadiness.

Abstract

Using complementary numerical approaches at high resolution, we study the late-time behaviour of an inviscid, incompressible two-dimensional flow on the surface of a sphere. Starting from a random initial vorticity field comprised of a small set of intermediate wavenumber spherical harmonics, we find that -- contrary to the predictions of equilibrium statistical mechanics -- the flow does not evolve into a large-scale steady state. Instead, significant unsteadiness persists, characterised by a population of persistent small-scale vortices interacting with a large-scale oscillating quadrupolar vorticity field. Moreover, the vorticity develops a stepped, staircase distribution, consisting of nearly homogeneous regions separated by sharp gradients. The persistence of unsteadiness is explained by a simple point vortex model characterising the interactions between the four main vortices which emerge.