Barotropic instability of shear flows

TL;DR

This paper investigates the stability of shear flows under Coriolis effects, developing a new method to determine sharp stability conditions, especially for sinusoidal profiles, and explores related dynamical behaviors.

Contribution

It introduces a novel approach to identify precise stability boundaries for shear flows with Coriolis effects, correcting previous results and analyzing bifurcations and damping phenomena.

Findings

Coriolis force significantly alters shear flow stability.

The stability boundary for sinus profiles is precisely determined.

Numerical results confirm the theoretical stability conditions.

Abstract

We consider barotropic instability of shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability conditions. We study the flow with Sinus profile in details and obtain the sharp stability boundary in the whole parameter space, which corrects previous results in the fluid literature. Our new results are confirmed by more accurate numerical computation. The addition of the Coriolis force is found to bring fundamental changes to the stability of shear flows. Moreover, we study dynamical behaviors near the shear flows, including the bifurcation of nontrivial traveling wave solutions and the linear inviscid damping. The first ingredient of our proof is a careful classification of the neutral modes. The second one is to write the linearized fluid equation in a Hamiltonian form and then use an instability index theory for general Hamiltonian PDEs. The last one is to study the singular and non-resonant neutral modes using Sturm-Liouville theory and hypergeometric functions.