Annales Geophysicae www.ann-geophys.net 0992-7689 1432-0576 27 11 2009 10.5194/angeo-27-4221-2009 http://www.ann-geophys.net/27/4221/2009/ http://www.ann-geophys.net/27/4221/2009/angeo-27-4221-2009.html http://www.ann-geophys.net/27/4221/2009/angeo-27-4221-2009.pdf 4221 4227 2009-11-09 Instability of coupled gravity-inertial-Rossby waves on a β-plane in solar system atmospheres J. F. McKenzie mckenziej@ukzn.ac.za 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 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. Cane, M A. and Sarachik, E S.: Forced baroclinic ocean motions: I. The linear equatorial unbounded case, J Mar Res., 34, 629–665, 1976. Charney, J G. and Drazin, P G.: Propagation of planetary-scale disturbances from the lower into the upper atmosphere, \jgr, 66, 83–109, \doi10.1029/JZ066i001p00083, 1961. Eckart, C.: Hydrodynamics of Oceans and Atmospheres, Pergamon Press, 1960. Gill, A E.: Atmosphere Ocean Dynamics, vol 30 of International Geophysics Series, 1982. Lighthill, J.: Waves in Fluids, Cambridge University Press, 1980. Mekki, O M. and McKenzie, J F.: The Propagation of Atmospheric Rossby Gravity Waves in Latitudinally Sheared Zonal Flows, Roy. Soc. London Phil. T. Ser. A, 287, 115–143, 1977. Moore, D W. and Philander, S G H.: The Sea – Modelling of the Tropical Oceanic Circulation, vol 6, Wiley – Interscience, 1977. Pedlosky, J P.: Geophysical Fluid Dynamics, Springer Verlag, 1987. Sakai, S.: Rossby-Kelvin instability: a new type of ageostrophic instability caused by a resonance between Rossby waves and gravity waves, J. Fluid Mech., 202, 149–176, \doi10.1017/S0022112089001138, 1989. Stoker, J J.: Water Waves: The Mathematical Theory with Applications, Interscience Publishers, Inc. New York, 1965.