Quasigeostrophy

Assuming columnar geometry (quasi-geostrophic hypothesis), we find that the motions depend only on the spatial coordinates in a plane perpendicular to the rotation axis, making the problem two-dimensional and reducing its size significantly.

There is a large body of research on quasi-geostrophic flows that has been developed specifically to better understand motions in thin fluid layers (Earth atmosphere and oceans). We use this approximation to compute dynamical regimes of thermal convection instead in a rapidly rotating sphere.
We have been able to reproduce turbulent convection for very low viscosity and to exhibit a regime where the small diffusivities do not control the dynamics. Interestingly, in this new regime, we discover that thermal convection becomes subcritical and consequently shuts down abruptly, which could explain rapid extinction of some ancient planetary dynamos (Mars, Moon…).

We have also obtained a new set of reduced quasi-geostrophic momentum equations valid in deep shells, such as planetary cores, including their equatorial regions. We generalize these equations to non spherical bodies in order to document the pressure coupling between the fluid and its solid boundaries. We thus obtain very good approximations of the slowest inertial modes. We are also pairing the quasi-geostrophic equation with the induction equation (for the magnetic field) in order to model the secular variation of the Earth’s magnetic field.