Efficiency gains of a multi-scale integration method applied to a scale-separated model for rapidly rotating dynamos

Numerical geodynamo simulations with parameters close to an Earth-like regime would be of great interest for understanding the dynamics of the Earth's liquid outer core and the associated geomagnetic field. Such simulations are far too computationally demanding owing to the large range in spatiotemporal scales. This paper explores the application of a multi-scale timestepping method to an asymptotic model for the generation of magnetic field in the fluid outer core of the Earth. The method is based on the heterogeneous multiscale modelling (HMM) strategy, which incorporates scale separation and utilizes several integrating models for the fast and slow fields. Quantitative comparisons between the multi-scale simulations and direct solution of the asymptotic model in the limit of rapid rotation and low Ekman number are performed. The multi-scale method accurately captures the varying temporal and spatial dynamics of the mean magnetic field at lower computational costs compared to the direct solution of the asymptotic model.