The Rayleigh number for convection in the Earth’s core

Abstract

Geodynamo models depend on two important parameters: the Ekman and Rayleigh numbers. While the difficulty of simulating a geodynamo with very small Ekman number has been widely discussed, the Rayleigh number has received relatively little attention. The energy budget is rather tight, giving hope for a low Rayleigh number and a tractable problem. Entropy considerations constrain the Rayleigh number to be large and positive. In the special case when the conduction profile in the absence of convection has the same functional dependence on radius as the adiabatic temperature, the Rayleigh number is simply related to the entropies of magnetic and thermal diffusion. In general, the Earth’s core may be stirred by both thermal and compositional convection, and may contain distinct stable regions where no convection occurs, but the same entropy argument holds and at least one of the Rayleigh numbers must be large and positive in at least one of the convecting regions. For molecular diffusivities, the Rayleigh number is enormous, 10 29 for thermal convection and 10 38 for compositional convection. Turbulent diffusivities give much lower values, 10 12 for thermal and 10 15 for compositional convection, or 10 3 and 10 6 times the critical Rayleigh number for magnetoconvection in a strong magnetic field. These values exceed the critical Rayleigh number for non-magnetic convection, which is higher than for magnetoconvection by a factor of 10 3 for turbulent diffusivities and 10 5 for molecular values. Convection in the Earth’s core could persist even when the geomagnetic field becomes weak, for example during excursions or polarity reversal. The turbulent value for thermal convection is low enough to be reached in numerical simulations in the near future.