Finite-amplitude thermal convection in fast rotating spherical shell

The study of thermal convection in rotating spherical geometry is fundamental to explain many geophysical and astrophysical phenomena such as the generation of the magnetic fields, or the differential rotation observed in the atmospheres of the major planets. The difficulties associated with the expe- rimental studies enhance the importance of the three-dimensional numerical simulations, such as those presented in this talk.

In order to obtain the evolution equations, the Boussinesq approximation is applied to the mass, momentum and energy conservation equations, which are rewritten in terms of toroidal and poloidal potentials. Together with the temperature field, they are expanded in spherical harmonics over the sphere, and in the radial direction a collocation method is used.

In our own time evolution codes we apply the IMEX-BDF formulae with an explicit treatment of the nonlinear terms of the equations. The use of matrix- free methods allows the implicit treatment of the Coriolis term, and makes the implementation of a step and order control easier. The results show that the use of high order methods, especially those with time-step and order control, increase the efficiency of the time integration, and allows to obtain more accurate solutions.

At low Prandtl number (ratio between the thermal diffusive and the viscous time scales), and with non-slip boundary conditions, the nonlinear dynamics is explored by means of temporal evolutions. Far away from the stable flows (periodic orbits, tori, and connecting orbits), the physical properties of the tur- bulent solutions are studied. Using parameters as close as possible to those of the Earth’s outer core, the numerical simulations are compared with laboratory experiments and realistic measurements.