Abstract
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.
Highlights
Gravity waves are an omnipresent oscillation mode in the atmosphere
This study investigates strongly nonlinear gravity waves in the compressible atmosphere from the Earth’s surface to the deep atmosphere
Before we focus on a specific solution and its stability we want to discuss how a general solution to the modulation equations with boundary, far-field and antitriptic-flow conditions may look like in the original variables of the Navier-Stokes equations
Summary
Gravity waves are an omnipresent oscillation mode in the atmosphere. They redistribute energy vertically and laterally and thereby affect the dynamics relevant for weather and climate prediction [1, 2]. The natural extension to linear wave theory is to incorporate weakly nonlinear effects which remains asymptotically valid as long as the amplitudes are relatively small This approach works exceedingly well for the description of oceanic internal waves. That weakly nonlinear theory is in certain cases not sufficient for atmospheric gravity waves These cases comprise situations when the amplitudes come close to the regime where the horizontal wind perturbation due to the waves is of the same order of. Nonlinear theory for gravity waves was studied in [12, 13, 14, 15, 16] Nonlinear effects such as Doppler shift of the frequency and interaction with the mean flow appear as higher-order corrections to the linear model.
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