Integral constraints on the linear instability of stratified flow with planar shear at an arbitrary angle to the vertical

TL;DR

This paper derives integral constraints and extends classical stability theorems for stratified parallel flows with planar shear at any angle to the vertical, providing new bounds on instability growth rates.

Contribution

It generalizes the Miles-Howard stability criterion and Howard's semicircle theorem to non-vertical shear flows, and introduces a new upper bound for instability growth rates.

Findings

Reproduces Miles-Howard criterion for vertical shear

No stability condition for non-vertical shear flows

Derives a new upper bound for growth rates

Abstract

Integral constraints on the linear instability of stratified parallel flow with planar shear at an arbitrary angle to the vertical are derived using the analytical approach of Miles and Howard, for perturbations with 2D spatial structure, which are thought to be the most unstable. The general stability formulation reproduces the Miles-Howard stability criterion for vertical shear, but yields no stability condition for non-vertical shear, confirming expectations from earlier studies. This study also extends Howard's semicircle theorem to non-vertical planar shear, and derives a new expression for the upper bound of the instability growth rate (extending that obtained by Howard), which is consistent with published numerical results.