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

Abstract

This study investigates strongly 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/ 1 / 2 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.

Topological invariant and anomalous edge modes of strongly nonlinear systems

“…( 2 ) correctly states the periodicity of nonlinear bulk modes based on the following reasons. First, existing works 58 manifest this form of nonlinear plane waves, such as classical Minkowskian Yang-Mills theory 59 , compressible atmosphere 60 , porous media 61 , and mechanical lattices 32 , 54 . Second, typical studies on weakly nonlinear bulk modes 11 , 25 , 31 – 33 , 54 , 65 reveal that the dynamics of all high-order harmonics are controlled by the single variable θ = ωt – qn : , where is the l -th Fourier component of .…”

Waves in the gas centrifuge: asymptotic theory and similarities with the atmosphere

“…Modulational instabilities of a primary wave may even cause the excitation of new, secondary waves, which was proved to be theoretically possible in Schlutow (2019). A generalized modulation theory studying the stability of almost unconstrained stationary gravity waves resembling mountain lee waves was presented in Schlutow & Wahlén (2020).…”

Waves in the gas centrifuge: asymptotic theory and similarities with the atmosphere

“…Modulational instabilities of a primary wave may even cause the excitation of new, secondary waves which was proven to be theoretically possible in Schlutow [2019]. A generalized modulation theory studying the stability of almost unconstrained stationary gravity waves resembling mountain lee waves was presented in Schlutow and Wahlén [2020].…”