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

An asymptotic method based on a continuous superposition of waves is used t o study the linear stability of convection in a rapidly rotating system. Th e method gives a uniform representation of the solutions which allows us to impose the boundary conditions and then to minimize the Rayleigh number. T his study was done for Prandtl numbers between 0.01 and 100. In the spheric al case, for a self-gravitating, internally heated fluid in the small incli nation limit, six branches are unveiled. In these branches, infinitesimal a mplitude convection takes place preferentially near the surface of a cylind er coaxial with the axis of rotation in a zone of thickness cc T-1/12, T be ing the Taylor number. The Rayleigh number of three of these flows differs at the most by sixty percent; however, in some intervals of the Prandtl num ber the difference is less than ten percent. Since these flows are located at different radial distances, this method predicts mixed-modes convection in separate zones at slightly supercritical values of the Rayleigh number f or all Prandtl numbers. A solution exhibiting convection in separate zones at low supercritical Rayleigh numbers is proposed for the first time. Appli cations to atmospheres and dynamos of the planets and the starts are discus sed. (C) 1999 Elsevier Science Ltd. All rights reserved.