Radial collocation methods for the onset of convection in rotating spheres

The viability of using collocation methods in radius and spherical harmonics in the angular variables to calculate convective flows in full spherical geometry is examined. As a test problem the stability of the conductive state of a self-gravitating fluid sphere subject to rotation and internal heating is considered. A study of the behavior of different radial meshes previously used by several authors in polar coordinates, including or not the origin, is first performed. The presence of spurious modes due to the treatment of the singularity at the origin, to the spherical harmonics truncation, and to the initialization of the eigenvalue solver is shown, and ways to eliminate them are presented. Finally, to show the usefulness of the method, the neutral stability curves at very high Taylor and moderate and small Prandtl numbers are calculated and shown.