Scaling laws for planetary dynamos

Abstract

We propose new scaling laws for the properties of planetary dynamos. In particular, the Rossby number, the magnetic Reynolds number, the ratio of magnetic to kinetic energy, the Ohmic dissipation timescale and the characteristic aspect ratio of the columnar convection cells are all predicted to be power-law functions of two observable quantities: the magnetic dipole moment and the planetary rotation rate. The resulting scaling laws constitute a somewhat modified version of the scalings proposed by Christensen and Aubert. The main difference is that, in view of the small value of the Rossby number in planetary cores, we insist that the non-linear inertial term, $ ${\boldsymbol u} \cdot \nabla {\boldsymbol u}$$, is negligible. This changes the exponents in the power-laws which relate the various properties of the fluid dynamo to the planetary dipole moment and rotation rate. Our scaling laws are consistent with the available numerical evidence.