A spherical shell numerical dynamo benchmark with pseudo-vacuum magnetic boundary conditions

Abstract

SUMMARY It is frequently considered that many planetary magnetic fields originate as a result of convection within planetary cores. Buoyancy forces responsible for driving the convection generate a fluid flow that is able to induce magnetic fields; numerous sophisticated computer codes are able to simulate the dynamic behaviour of such systems. This paper reports the results of a community activity aimed at comparing numerical results of several different types of computer codes that are capable of solving the equations of momentum transfer, magnetic field generation and heat transfer in the setting of a spherical shell, namely a sphere containing an inner core. The electrically conducting fluid is incompressible and rapidly rotating and the forcing of the flow is thermal convection under the Boussinesq approximation. We follow the original specifications and results reported in Harder & Hansen to construct a specific benchmark in which the boundaries of the fluid are taken to be impenetrable, non-slip and isothermal, with the added boundary condition for the magnetic field B that the field must be entirely radial there; this type of boundary condition for B is frequently referred to as ‘pseudovacuum’. This latter condition should be compared with the more frequently used insulating boundary condition. This benchmark is so-defined in order that computer codes based on local methods, such as finite element, finite volume or finite differences, can handle the boundary condition with ease. The defined benchmark, governed by specific choices of the Roberts, magnetic Rossby, Rayleigh and Ekman numbers, possesses a simple solution that is steady in an azimuthally drifting frame of reference, thus allowing easy comparison among results. Results from a variety of types of code are reported, including codes that are fully spectral (based on spherical harmonic expansions in angular coordinates and polynomial expansions in radius), mixed spectral and finite difference, finite volume, finite element and also a mixed Fourier-finite element code. There is good agreement among codes.