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| Titel |
Nonlinear development of unstable wave-modes coupled in the critical layer of a weakly supercritical zonal flow |
| VerfasserIn |
Galina Rybushkina, Sergey Shagalov |
| Konferenz |
EGU General Assembly 2010
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 12 (2010) |
| Datensatznummer |
250032294
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| Zusammenfassung |
| This study explores nonlinear development of barotropic instability in a horizontally sheared
quasigeostrophic zonal flow on the beta-plane in the presence of the stable atmospheric
density stratification. The background zonal flow is specified by a free shear layer
velocity profile most typical of the Antarctic polar vortex periphery. According to
the linear inviscid theory based on the Rayleigh-Kuo’s theorem free shear layer
flows are capable of supporting both baroclinic and barotropic unstable modes
whenever the gradient of the Coriolis parameter β is smaller than a marginal value
βm. Supercriticality of the inviscid flow defined as δβ = βm - β is set to be of
the order of a small parameter É. To capture effects arising from small dissipation
an inverse Reynolds number and an Ekman dissipation parameter are also scaled
in terms of É. Extracting the kinetic energy of the basic flow by simultaneously
growing weakly unstable Rossby wave-modes in a weakly dissipative zonal flow
near the onset of instability is confined to their common nonlinear critical layer
(CL) (relatively thin resonant region surrounding the level where the wave speed of
marginal modes matches the mean flow). Only two unstable modes coupled in the CL
(the barotropic mode and the main baroclinic one) are shown to develop in the
flow provided that some restriction is imposed on the internal deformation radius
value.
For this case an asymptotic approach based on the method of the matched asymptotic
expansions in É is employed to derive a closed system of equations governing the dynamics
of the wave-modes amplitudes and evolution of the potential vorticity in the nonlinear CL. If
the supercriticality is small enough for dissipation to dominate inside the CL, the CL-flow
evolves in a quasi-steady weakly nonlinear regime and an approximate solution for the
potential vorticity in the form of an expansion in the small wave amplitudes can be obtained.
In this case simultaneous development of the wave-modes is governed by a reduced
system of two coupled Landau-Stuart equations. In the absence of resonant modes
coupling through the condition of the second harmonic resonance the primary effect of
their nonlinear interaction inside the CL is the suppression of the baroclinic mode.
The physical mechanism of this effect is based on the energy transfer from the
baroclinic mode to the barotropic one caused by the interaction of the wave-modes
with the second-order beat-wave vorticity perturbations in the CL. At the higher
levels of the supercriticality nonlinearity and time dependence play a significant
role in the CL and development of the instability can no longer be described by
the weakly nonlinear equations. For this reason the flow regimes and instability
development scenarios arising successively as the supercriticality is increased are studied
with the aid of a numerical scheme including the Fourier series expansion in zonal
direction and the finite-difference approximation across the CL for the periodic
boundary conditions corresponding to the annular zonal flow. In particular for the
moderate levels of the supercriticality mutual suppression of the barotropic and the
baroclinic modes (modes competition) leading to the modes selection is revealed. |
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