On the Rayleigh-Taylor instability in presence of a background shear

TL;DR

This paper analyzes the Rayleigh-Taylor instability with background shear, deriving a spectral radius formula, showing velocity variations can neutralize shortwave instabilities, and constructing steady states with specific spectral properties.

Contribution

It generalizes existing spectral radius results to include background shear and introduces a novel compactness criterion for pseudo-differential operators.

Findings

Velocity variations neutralize shortwave instabilities.

Derived a generalized spectral radius formula.

Constructed steady states with unstable discrete spectrum.

Abstract

In this note we revisit the classical subject of the Rayleigh-Taylor instability in presence of an incompressible background shear flow. We derive a formula for the essential spectral radius of the evolution group generated by the linearization near the steady state and reveal that the velocity variations neutralize shortwave instabilities. The formula is a direct generalization of the result of H. J. Hwang and Y. Guo in the hydrostatic case \cite{HG}. Furthermore, we construct a class of steady states which posses unstable discrete spectrum with neutral essential spectrum. The technique involves the WKB analysis of the evolution equation and contains novel compactness criterion for pseudo-differential operators on unbounded domains.