Convection in a rapidly rotating system: the smallinclination limit and its planetary applications

An asymptotic method based on a continuous superposition of waves is used to studythe linear stability of convection in a rapidly rotating system. The method gives a uniformrepresentation of the solutions which allows us to impose the boundary conditions and then tominimize the Rayleigh number. This study was done for Prandtl numbers between 0.01 and 100.In the spherical case, for a self-gravitating, internally heated fluid in the small inclination limit,six branches are unveiled. In these branches, infinitesimal amplitude convection takes placepreferentially near the surface of a cylinder coaxial with the axis of rotation in a zone ofthickness ∝ T−1⧸12, T being the Taylor number. The Rayleigh number ofthree of these flows differs at the most by sixty percent; however, in some intervals of the Prandtlnumber the difference is less than ten percent. Since these flows are located at different radialdistances, this method predicts mixed-modes convection in separate zones at slightlysupercritical values of the Rayleigh number for all Prandtl numbers. A solution exhibitingconvection in separate zones at low supercritical Rayleigh numbers is proposed for the first time.Applications to atmospheres and dynamos of the planets and the starts are discussed.