Modulational stability of nonlinear saturated gravity waves

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 nondimensional 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, that is, 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.