Bounds on the phase velocity in the linear instability of viscous shear flow problem in the $\beta$-plane

TL;DR

This paper extends classical results on viscous shear flow stability to the beta-plane context, providing new bounds on phase velocity and a sufficient stability condition for geophysical flows.

Contribution

It introduces bounds on phase velocity and a stability criterion for viscous shear flows in the beta-plane, expanding upon previous work in a more geophysically relevant setting.

Findings

Derived bounds on phase velocity in beta-plane shear flows

Established a sufficient stability condition for these flows

Extended classical viscous flow results to geophysical fluid dynamics

Abstract

Results obtained by Joseph ({\it J. Fluid Mech.} {\bf 33} (1968) 617) for the viscous parallel shear flow problem are extended to the problem of viscous parallel, shear flow problem in the beta plane and a sufficient condition for stability has also been derived.