Elliptical instability of a rapidly rotating, strongly stratified fluid

TL;DR

This paper analyzes the elliptical instability in rapidly rotating, strongly stratified fluids, revealing that both cyclonic and anticyclonic flows are unstable with growth rates exponentially small in the Rossby number, and provides explicit formulas for these growth rates.

Contribution

It offers an analytical expression for the instability growth rate in stratified rotating fluids, highlighting the dependence on flow eccentricity and stratification parameters.

Findings

Both cyclonic and anticyclonic flows are unstable.

Growth rates are exponentially small in Rossby number.

Analytical results are confirmed by numerical solutions.

Abstract

The elliptical instability of a rotating stratified fluid is examined in the regime of small Rossby number and order-one Burger number corresponding to rapid rotation and strong stratification. The Floquet problem describing the linear growth of disturbances to an unbounded, uniform-vorticity elliptical flow is solved using exponential asymptotics. The results demonstrate that the flow is unstable for arbitrarily strong rotation and stratification; in particular, both cyclonic and anticyclonic flows are unstable. The instability is weak, however, with growth rates that are exponentially small in the Rossby number. The analytic expression obtained for the growth rate elucidates its dependence on the Burger number and on the eccentricity of the elliptical flow. It explains in particular the weakness of the instability of cyclonic flows, with growth rates that are only a small fraction of those obtained for the corresponding anticyclonic flows. The asymptotic results are confirmed by numerical solutions of Floquet problem.