Spectral stability of nonlinear gravity waves in the atmosphere
Abstract
We apply spectral stability theory to investigate nonlinear gravity waves in the atmosphere. These waves are determined by modulation equations that result from WentzelKramersBrillouin 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 pseudoincompressible regime does account for the altitudinal varying background density. Second, we show for the first time that upwardtraveling wave fronts solving the inviscid modulation equations, that compare to pseudoincompressible theory, are unconditionally unstable. Both inviscid regimes turn out to be illposed 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2018
 arXiv:
 arXiv:1802.06420
 Bibcode:
 2018arXiv180206420S
 Keywords:

 Physics  Fluid Dynamics