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Spectral stability of nonlinear gravity waves in the atmosphere

Schlutow, Mark LU ; Wahlén, Erik LU and Birken, Philipp LU (2019) In Mathematics of Climate and Weather Forecasting 5(1). p.12-33
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... (More)
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. (Less)
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author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Mathematics of Climate and Weather Forecasting
volume
5
issue
1
pages
12 - 33
publisher
De Gruyter
ISSN
2353-6438
DOI
10.1515/mcwf-2019-0002
language
English
LU publication?
yes
id
130facd2-ff33-4f2c-a85b-de6dd8c69a87
date added to LUP
2020-06-11 17:19:02
date last changed
2021-11-30 13:42:52
@article{130facd2-ff33-4f2c-a85b-de6dd8c69a87,
  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.}},
  author       = {{Schlutow, Mark and Wahlén, Erik and Birken, Philipp}},
  issn         = {{2353-6438}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{12--33}},
  publisher    = {{De Gruyter}},
  series       = {{Mathematics of Climate and Weather Forecasting}},
  title        = {{Spectral stability of nonlinear gravity waves in the atmosphere}},
  url          = {{http://dx.doi.org/10.1515/mcwf-2019-0002}},
  doi          = {{10.1515/mcwf-2019-0002}},
  volume       = {{5}},
  year         = {{2019}},
}