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

TL;DR

The paper develops an extended Miles–Howard framework for inviscid, stratified parallel flows with planar shear at an angle $\theta$ to the vertical, focusing on 2D perturbations. It derives integral stability relations, identifies when a classical stability criterion applies, and provides a new upper bound for the instability growth rate that generalizes Howard’s result to angled shear. Additionally, it generalizes Howard’s semicircle theorem to non-vertical planar shear, offering a geometric constraint on the phase speed and growth rate. The findings clarify why vertical shear admits a stability threshold ($Ri>1/4$) while non-vertical shear does not, and they connect with prior numerical studies, reinforcing the relevance of 2D perturbations in assessing linear instability of stratified shear flows.

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.

Figures (1)

• Figure 1: Upper bound $\omega_{imax} L/U_0$ of the growth rate of the linear instability of planar stratified flow (\ref{['growth5']}) with shear at angle $\theta$ to the vertical as a function of $F/\cos \theta$, for different values of $\theta$ (see line labels for details).