Schlutow, M. and Wahlén, E. and Birken, P.
(2019)
*Spectral stability of nonlinear gravity waves in the atmosphere.*
Mathematics of Climate and Weather Forecasting, 5
(1).
pp. 12-33.
ISSN 2353-6438

Full text not available from this repository.

Official URL: https://doi.org/10.1515/mcwf-2019-0002

## 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 fully account for the altitudinal varying background density. Second,we show for the first time that upward-traveling non-plane wave fronts solving the inviscid nonlinear 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 nonlinear traveling wave solutions have localized amplitude. As a consequence of the nonlinearity, envelope and linear group velocity, as given by the derivative of the frequency with respect to wavenumber, do not coincide anymore. These waves blow up unconditionally by embedded eigenvalue instabilities but the instability growth rate is bounded from above and can be computed analytically. Additionally, all three types of nonlinear modulation equations are solved numerically to further investigate and illustrate the nature of the analytic stability results.

Item Type: | Article |
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Subjects: | Mathematical and Computer Sciences Mathematical and Computer Sciences > Mathematics Mathematical and Computer Sciences > Mathematics > Applied Mathematics |

Divisions: | Department of Mathematics and Computer Science > Institute of Mathematics > Geophysical Fluid Dynamics Group |

ID Code: | 2339 |

Deposited By: | Ulrike Eickers |

Deposited On: | 09 Apr 2019 11:55 |

Last Modified: | 09 Apr 2019 11:55 |

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