Stability of Thermal Convection in a Rapidly Rotating Sphere

Abstract

Linear stability of thermal convection driven by self-gravity and internal heat source in a rotating sphere is analyzed by a singular perturbation method at large Taylor number T a . The neutral disturbance localized in the radial direction generally tilts from that direction. The radial structure of such oblique disturbance takes a Gaussian form spreading over length-scale of O ( T a -1/12 ). The localness of disturbance becomes weaker as the tilt angle decreases. In the special case of uniform heat source and linear gravity force it is proved that the critical Rayleigh number R a c * should be attained by such disturbances that have vanishing tilt angle (F. H. Busse, J. Fluid Mech. 44 (1970) 441). Since, however, no disturbance without tilting can be localized in the radial direction, this critical Rayleigh number is nominal. At the Rayleigh number slightly greater than the critical ( R a * - R a c * = O ( T a -1/9 )), the disturbance tilts only slightly (by O ( T a -1/18 )) and spreads over length-scale of O ( T a -1/9 ) in the radial direction.