Quasi-geostrophic model of the instabilities of the Stewartson layer

TL;DR

This paper develops a quasi-geostrophic two-dimensional model to analyze the instabilities of Stewartson layers in rotating shear flows, deriving new scaling laws and validating findings with experiments.

Contribution

It introduces a novel quasi-geostrophic model that accurately handles mass conservation and extends instability analysis to large slopes, providing new asymptotic laws and thresholds.

Findings

Critical Rossby number scales as E^{3/4} for flat containers.

Instability in sloping containers is a Rossby wave with threshold proportional to β E^{1/2}.

Model validation with experimental results confirms its accuracy and limitations.

Abstract

We study the destabilization of a shear layer, produced by differential rotation of a rotating axisymmetric container. For small forcing, this produces a shear layer, which has been studied by Stewartson and is almost invariant along the rotation axis. When the forcing increases, instabilities develop. To study the asymptotic regime (very low Ekman number $E$), we develop a quasi-geostrophic two-dimensional model, whose main original feature is to handle the mass conservation correctly, resulting in a divergent two-dimensional flow, and valid for any container provided that the top and bottom have finite slopes. We use it to derive scalings and asymptotic laws by a simple linear theory, extending the previous analyses to large slopes (as in a sphere), for which we find different scaling laws. For a flat container, the critical Rossby number for the onset of instability evolves as $E^{3/4}$ and may be understood as a Kelvin-Helmoltz shear instability. For a sloping container, the instability is a Rossby wave with a critical Rossby number proportional to $\beta E^{1/2}$, where $\beta$ is related to the slope. We also investigate the asymmetry between positive and negative differential rotation and propose corrections for finite Ekman and Rossby numbers. Implemented in a numerical code, our model allows us to study the onset over a broad range of parameters, determining the threshold but also other features such as the spatial structure. We also present a few experimental results, validating our model and showing its limits.