Quasi-geostrophic Flow in the Earth's core in the Presence of Topography at the Boundaries

Quasi-geostrophic flows have been shown to arise naturally in simulations of the geodynamo outside the cylindrical surface tangent to the inner core. They can be continued from the Earth's core surface to its interior and are a natural ingredient of secular variation models. In the context of rotating shallow water models for oceans and atmospheres, dynamically consistent depth-averaged equations have been derived from a three-dimensional Lagrangian by restricting the fluid elements to move in columns. We follow the same strategy to generalize a recent model for quasi-geostrophic flows in deep and magnetized planetary cores (Labbé & al, 2015) to non-axisymmetric geometries: the reduced equations directly follow from an approximation of the Lagrangian. The model incorporates the torsional Alfvén waves that carry angular momentum and can be adapted to time variable topographies. For axisymmetric bodies, conservation of angular momentum (when the boundary is electrically insulating and stress-free) results from the invariance of the Lagrangian under rotational symmetry. We rely on our new equations to investigate the breaking of this symmetry in magnetized deep planetary cores. Although the Lagrangian formulation is particularly appropriate to derive non linear equations, we first check the accuracy of the approximation by comparing the inertial modes obtained numerically respectively from our model, which employs non orthogonal curvilinear coordinates, and from 3D calculations.