Abstract
Convection in planetary cores can generate fluid flow and magnetic fields, and a number of sophisticated codes exist to simulate the dynamic behaviour of such systems. We report on the first community activity to compare numerical results of computer codes designed to calculate fluid flow within a whole sphere. The flows are incompressible and rapidly rotating and the forcing of the flow is either due to thermal convection or due to moving boundaries. All problems defined have solutions that allow easy comparison, since they are either steady, slowly drifting or perfectly periodic. The first two benchmarks are defined based on uniform internal heating within the sphere under the Boussinesq approximation with boundary conditions that are uniform in temperature and stress-free for the flow. Benchmark 1 is purely hydrodynamic, and has a drifting solution. Benchmark 2 is a magnetohydrodynamic benchmark that can generate oscillatory, purely periodic, flows and magnetic fields. In contrast, Benchmark 3 is a hydrodynamic rotating bubble benchmark using no slip boundary conditions that has a stationary solution. 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 Fourierâfinite element code. There is good agreement between codes. It is found that in Benchmarks 1 and 2, the approximation of a whole sphere problem by a domain that is a spherical shell (a sphere possessing an inner core) does not represent an adequate approximation to the system, since the results differ from whole sphere results.
Highlights
The predominant theory for the generation mechanism of the Earth’s magnetic field is that of magnetic field generation by thermal and compositional convection, creating the so-called self-excited dynamo mechanism
It is over eleven years since the undoubtedly successful numerical dynamo benchmark exercise of Christensen et al (2001); Christensen et al (2009), hereinafter B1. This benchmark exercise was set in the geometry of a spherical shell, with convection driven by a temperature difference between an inner core and the outer boundary of the spherical shell
Three different benchmarks were devised, the first being purely thermal convection, and the second and third being dynamos. The latter two benchmarks differed in the treatment of the inner core: in one case the inner core was taken to be electrically insulating and fixed in the rotating frame, and in the other case the core was taken to be electrically conducting and free to rotate in response to torques that are applied to it, arising from the convection in the outer core
Summary
The predominant theory for the generation mechanism of the Earth’s magnetic field is that of magnetic field generation by thermal and compositional convection, creating the so-called self-excited dynamo mechanism. We note in passing that the whole sphere geometry is relevant to the generation of magnetic fields in the early Earth, prior to the formation of the inner core In this time period the convection in the core was driven by secular cooling (and possibly internal heating), and this is precisely the scenario studied here in benchmarks 1 and 2. The benchmarks 1 and 2 set up here differ from those of B1 in their use of stress-free boundary conditions, rather than non-slip conditions This arose purely as a result of our survey of parameter space while searching for whole-sphere dynamos that possess simple solutions with clear diagnostics suitable for a benchmark.
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