Stability Boundaries and Sufficient Stability Conditions for Stably Stratified, Monotonic Shear Flows

TL;DR

This paper investigates the linear stability of inviscid, stratified shear flows with monotonic profiles, introducing a more efficient method for finding stability boundaries and deriving new sufficient stability conditions that extend classical criteria.

Contribution

It develops a systematic approach to identify marginally unstable modes using Sturm-Liouville problems and generalizes the Rayleigh-Fj{ }o}rtoft criterion for stratified flows.

Findings

Efficient determination of stability boundaries via Sturm-Liouville problems.

Derivation of new sufficient stability conditions for stratified shear flows.

Extension of classical unstratified stability criteria to stratified cases.

Abstract

Linear stability of inviscid, parallel, and stably stratified shear flow is studied under the assumption of smooth strictly monotonic profiles of shear flow and density, so that the local Richardson number is positive everywhere. The marginally unstable modes are systematically found by solving a one-parameter family of regular Sturm-Liouville problems, which can determine the stability boundaries more efficiently than solving the Taylor-Goldstein equation directly. By arguing for the non-existence of a marginally unstable mode, we derive new sufficient conditions for stability, which generalize the Rayleigh-Fj{\o}rtoft criterion for unstratified shear flows.