This paper presents a study of high Rayleigh number (up to 200 times supercritical) axisymmetrical convection in a spherical shell with an aspect ratio relevant for the Earth's lower mantle. Both bottom-heated and internal heated cases have been considered. Computations have been carried out for an infinite Prandtl number isoviscous fluid with free slip isothermal boundary conditions. The first part of the paper is devoted to the influence of the resolution on the accuracy of the numerical results. It is shown that the resolution strongly influences the onset of time dependence. Recent methods of non-linear physics have been used to prove that the time dependence and the chaotic behaviors of the solutions are real ones. From these results we can confirm that convection is chaotic, in this particular geometry, even for Rayleigh numbers 200 times critical. Aperiodic boundary layer instabilities are found to be incapable of breaking up the large-scale flow, owing to the shear of the global circulation. Spectral analysis of the power associated with the thermal anomalies shows that there is an upward cascade of energy, due to small-scale chaotic instabilities, from l = 2 to l = 4-6 at the bottom boundary, in agreement with new seismic observations at the core-mantle boundary [1-3].
Bibliographical noteFunding Information:
This research has been supported by a C.N.R.S. grant "A.T.P. dynamique des fluides g6ophysiques et astrophysiques", a N.S.F. grant NSF 8511200 and a grant from the Minnesota Supercomputer Institute. We thank Richard Walsh of the Minnesota Supercomputing Center, Inc.