Dynamo simulations in a spherical shell of ideal gas using a high-order cartesian magnetohydrodynamics code

Most numerical simulations of the geodynamo are cast in spherical geometry, using a spherical harmonic representation for lateral variations, and an expansion in Chebyshev polynomials, or discretisation, in radius. A number of research groups have produced three-dimensional, time-dependent, self-consistent simulations of the geodynamo using this pseudospectral methodology. Computational limitations place a practical bound on the parameter regime that can be explored in this context, however, and Earth-like values currently remain several orders of magnitude out of reach. For the spherically pseudospectral codes, the absence of a fast Legendre transform is a strong factor contributing to this limitation. As a first step toward alternative computational methods for geodynamo modelling, we adapt a pre-existing, efficiently parallelised magnetohydrodynamics (MHD) code, to model spherical shell problems. The Pencil-Code is a Cartesian code that uses sixth-order finite differences, applied to 'pencils' (i.e. array sections) in the x direction in a cache-efficient way. The domain is tiled in the y and z directions, with the communication of boundary elements handled by Message Passing Interface (MPI). Time stepping is via a third order Runge-Kutta method. The code's modular structure allows a flexible selection of various physical processes and variables, making it easily adaptable for many types of compressible MHD problems. We demonstrate dynamo action driven by thermal convection in a spherical shell of compressible ideal gas, for comparison with the ideal gas dynamos of Kageyama and Sato [Kageyama, A., Sato, T., the Complexity Simulation Group, 1995. Computer simulation of a magnetohydrodynamic dynamo. II. Phys. Fluids, B 2, 2793-2805]. As with the previous results, our solutions are characterised by steady state thermal convection with axial columnar cells, and exponential growth and subsequent saturation of magnetic energy. Minor differences in the details of the benchmark and current simulations are attributed to differences in the computational methods and geometry of the domains involved. We discuss the further modifications still needed for a fuller adaptation to geodynamo problems.