Dipole Moment Scaling for Convection-Driven Planetary Dynamos

Dipole moments from numerical dynamo models driven by various combinations of thermal and chemical convection are used to derive scaling relationships applicable to dipolar planetary magnetic fields. We compare published values of time-averaged dipole moments from dynamo models covering wide ranges of Ekman, Rayleigh, and Prandtl numbers. Three distinct moment regimes are found. In the first regime, just beyond the onset of dynamo action, the dipole moment increases linearly with Rayleigh number and the non-dipole field is typically weak. Another regime at very large Rayleigh numbers is characterized by small, highly time-variable dipole moments and strong non-dipole fields. Between these is an intermediate Rayleigh number regime in which the dipole moment tends toward a constant value that depends primarily on the core radius, rotation rate, density, and electrical conductivity, and secondarily on the Prandtl numbers of the fluid. Dynamos in this regime with homogeneous boundary conditions have the dipole moments that are relatively insensitive to the vigor of convection, the style of convection (thermal or chemical), and the concentration of internal heat sources. Planetary dynamos in this regime are expected to evolve with relatively minor changes in time-average dipole moment. The geodynamo may have been in this regime over much of Earth's history. Heterogeneous boundary conditions tend to reduce the dipole moment and large amounts of boundary heterogeneity can extinguish dynamo action entirely. The early demise of the Martian dynamo may have occurred in this regime.