Generalized modulation theory for strongly nonlinear gravity waves in a compressible atmosphere

TL;DR

This paper develops a generalized modulation theory for strongly nonlinear gravity waves in the Earth's atmosphere, analyzing their stationary solutions, stability, and conditions leading to wave overturning or destabilization.

Contribution

It extends Grimshaw's modulation equations to include realistic boundary conditions and background fields, providing new insights into wave stability and overturning in a compressible atmosphere.

Findings

Wave-Reynolds number must be less than one above a certain height.

Resonance occurs at wave-Froude number of 1/√2, leading to destabilization.

Large horizontal wavelengths cause waves to overturn before instability can develop.

Abstract

This study investigates nonlinear gravity waves in the compressible atmosphere from the Earth's surface to the deep atmosphere. These waves are effectively described by Grimshaw's dissipative modulation equations which provide the basis for finding stationary solutions such as mountain lee waves and testing their stability in an analytic fashion. Assuming energetically consistent boundary and far-field conditions, that is no energy flux through the surface, free-slip boundary, and finite total energy, general wave solutions are derived and illustrated in terms of realistic background fields. These assumptions also imply that the wave-Reynolds number must become less than unity above a certain height. The modulational stability of admissible, both non-hydrostatic and hydrostatic, waves is examined. It turns out that, when accounting for the self-induced mean flow, the wave-Froude number has a resonance condition. If it becomes $1/\sqrt{2}$, then the wave destabilizes due to perturbations from the essential spectrum of the linearized modulation equations. However, if the horizontal wavelength is large enough, waves overturn before they can reach the modulational stability condition.