Toward a Multiscale Approach for Geodynamo Models

Abstract:

The generation of the Earth’s magnetic field by dynamo action in the liquid iron core is modeled by a large set of coupled, non-linear partial differential equations. Numerical models presently involve direct discretization of the geodynamo equations and allow to produce axial dipolar magnetic fields that are qualitatively comparable to the Earth's one, but whose dynamics remain considerably remote from the geophysical regime.

Indeed, due to the extreme values of the dimensionless numbers characterizing the Earth’s core dynamics, the relevant regime remains far beyond the reach of direct numerical simulation - so far that one cannot simply rely on the increase in computational power. Simplification of the governing equations is not straightforward. In particular, the importance of return flow from the thin Ekman layers located at the inner core and core-mantle boundaries into the main flow prevents one from purely suppressing the viscous dissipation term in Navier-Stokes equation even in the limiting case where inertia is neglected. Therefore more advanced models are needed, which require prior mathematical treatment of the equations of magnetohydrodynamics. The one-dimensional structure of most viscous and magnetic layers demonstrates the possibility of huge computational savings by means of multiscale techniques. In our approach, asymptotic matching is applied on simplified problems such as the Proudman-Stewartson flow to solve for the viscous shear layers while keeping the mainstream resolved on a coarse grid.