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Annales Geophysicae An interactive open-access journal of the European Geosciences Union
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Volume 27, issue 11
Ann. Geophys., 27, 4221–4227, 2009
https://doi.org/10.5194/angeo-27-4221-2009
© Author(s) 2009. This work is distributed under
the Creative Commons Attribution 3.0 License.
Ann. Geophys., 27, 4221–4227, 2009
https://doi.org/10.5194/angeo-27-4221-2009
© Author(s) 2009. This work is distributed under
the Creative Commons Attribution 3.0 License.

  09 Nov 2009

09 Nov 2009

Instability of coupled gravity-inertial-Rossby waves on a β-plane in solar system atmospheres

J. F. McKenzie J. F. McKenzie
  • Astrophysics and Cosmology Research Unit, School of Mathematical Sciences, University of KwaZulu-Natal, Durban, 4041, South Africa
  • Department of Physics, CSPAR, University of Alabama, AL, USA
  • Senior Member, King's College, Cambridge, UK

Abstract. This paper provides an analysis of the combined theory of gravity-inertial-Rossby waves on a β-plane in the Boussinesq approximation. The wave equation for the system is fifth order in space and time and demonstrates how gravity-inertial waves on the one hand are coupled to Rossby waves on the other through the combined effects of β, the stratification characterized by the Väisälä-Brunt frequency N, the Coriolis frequency f at a given latitude, and vertical propagation which permits buoyancy modes to interact with westward propagating Rossby waves. The corresponding dispersion equation shows that the frequency of a westward propagating gravity-inertial wave is reduced by the coupling, whereas the frequency of a Rossby wave is increased. If the coupling is sufficiently strong these two modes coalesce giving rise to an instability. The instability condition translates into a curve of critical latitude Θc versus effective equatorial rotational Mach number M, with the region below this curve exhibiting instability. "Supersonic" fast rotators are unstable in a narrow band of latitudes around the equator. For example Θc~12° for Jupiter. On the other hand slow "subsonic" rotators (e.g. Mercury, Venus and the Sun's Corona) are unstable at all latitudes except very close to the poles where the β effect vanishes. "Transonic" rotators, such as the Earth and Mars, exhibit instability within latitudes of 34° and 39°, respectively, around the Equator. Similar results pertain to Oceans. In the case of an Earth's Ocean of depth 4km say, purely westward propagating waves are unstable up to 26° about the Equator. The nonlinear evolution of this instability which feeds off rotational energy and gravitational buoyancy may play an important role in atmospheric dynamics.

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