A Galerkin Method for Solving the Inertia-Free Navier-Stokes Equation in a Full Sphere

The dynamics of the Earth's interior are thought to be well described by the magnetostrophic approximation, in which both the magnitudes of viscosity and inertia are negligible. The resulting leading-order dynamical balance in the Navier-Stokes equation therefore involves the Coriolis term, pressure, the Lorentz force and buoyancy. For numerical reasons, weak viscosity is often retained although this is not expected to be dynamically important. The system of equations for the unknown flow is then purely linear, and involves an inversion of the discretised Coriolis and viscous terms.

In a full sphere, we present a Galerkin scheme which results in a very low condition numbers for the linear system, when compared to other standard methods, leading to highly accurate solutions for the flow. The excellent properties of our method reflect the facts that: (i) the boundary conditions are handled implicitly, (ii) the correct behaviour at the origin is hard-wired into the scheme and (iii) we do not need to take any curl operations to formulate the discrete equations.