Spectral stability of nonlinear gravity waves in the atmosphere

TL;DR

This paper analyzes the spectral stability of nonlinear gravity waves in the atmosphere using modulation equations, revealing stability of plane waves, instability of wave fronts, and the stabilizing effect of dissipation.

Contribution

It provides the first analysis of the spectral stability of upward-traveling wave fronts and introduces a regularization method with dissipation to control instabilities.

Findings

Plane waves are spectrally stable under modulation equations.

Upward-traveling wave fronts are unconditionally unstable.

Dissipative effects stabilize localized wave solutions.

Abstract

We apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from Wentzel-Kramers-Brillouin theory. First, we establish that plane waves, which represent exact solutions to the inviscid Boussinesq equations, are spectrally stable with respect to their nonlinear modulation equations under the same conditions as what is known as modulational stability from weakly nonlinear theory. In contrast to Boussinesq, the pseudo-incompressible regime does account for the altitudinal varying background density. Second, we show for the first time that upward-traveling wave fronts solving the inviscid modulation equations, that compare to pseudo-incompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be ill-posed as the spectra allow for arbitrarily large instability growth rates. Third, a regularization is found by including dissipative effects. The corresponding traveling wave solutions have localized amplitude and blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results.