Inertia-less convectively-driven dynamo models in the limit of low Rossby number and large Prandtl number

TL;DR

This paper develops asymptotic dynamo models for planetary magnetic field generation in the low Rossby number and high Prandtl number regime, showing inertia is unnecessary for large-scale dynamos and analyzing the role of diffusivities.

Contribution

It introduces new inertia-less dynamo models for high Prandtl and low Rossby number conditions, expanding understanding of planetary magnetic field generation.

Findings

Magnetic to kinetic energy ratio M is large and not indicative of force balance.

Inertia is not essential for driving large-scale dynamos at low q.

Models show different asymptotic M limits without altering geostrophic balance.

Abstract

Compositional convection is thought to be an important energy source for magnetic field generation within planetary interiors. The Prandtl number, $Pr$, characterizing compositional convection is significantly larger than unity, suggesting that the inertial force may not be important on the small scales of convection as long as the buoyancy force is not too strong. We develop asymptotic dynamo models for the case of small Rossby number and large Prandtl number in which inertia is absent on the convective scale. The relevant diffusivity parameter for this limit is the compositional Roberts number, $q = D/\eta$, which is the ratio of compositional and magnetic diffusivities. Dynamo models are developed for both order one $q$ and the more geophysically relevant low $q$ limit. For both cases the ratio of magnetic to kinetic energy densities, $M$, is asymptotically large and reflects the fact that Alfv\'en waves have been filtered from the dynamics. Along with previous investigations of asymptotic dynamo models for $Pr=O(1)$, our results show that the ratio $M$ is not a useful indicator of dominant force balances in the momentum equation since many different asymptotic limits of $M$ can be obtained without changing the leading order geostrophic balance. Furthermore, the present models show that inertia is not a requirement for driving low $q$, large-scale dynamos.