# Modulational stability of nonlinear saturated gravity waves

## TL;DR

This paper investigates the stability of nonlinear saturated gravity waves using a theoretical framework, introducing a wave-Reynolds number to analyze wave behavior and potential secondary wave generation.

## Contribution

It introduces a wave-Reynolds number to analyze the stability of nonlinear gravity waves and demonstrates their destabilization in the saturation region through analytic and numerical methods.

## Key findings

- Lindzen-type waves destabilize when wave-Reynolds number is around unity.
- Secondary waves may be generated via wave-mean-flow interactions.
- Implications for wave breaking heights and mean-flow acceleration are discussed.

## Abstract

Stationary gravity waves, such as mountain lee waves, are effectively described by Grimshaw's dissipative modulation equations even in high altitudes where they become nonlinear due to their large amplitudes. In this theoretical study, a wave-Reynolds number is introduced to characterize general solutions to these modulation equations. This non-dimensional number relates the vertical linear group velocity with wavenumber, pressure scale height and kinematic molecular/eddy viscosity. It is demonstrated by analytic and numerical methods that Lindzen-type waves in the saturation region, i.e. where the wave-Reynolds number is of order unity, destabilize by transient perturbations. It is proposed that this mechanism may be a generator for secondary waves due to direct wave-mean-flow interaction. By assumption the primary waves are exactly such that altitudinal amplitude growth and viscous damping are balanced and by that the amplitude is maximized. Implications of these results on the relation between mean-flow acceleration and wave breaking heights are discussed.