Scalability of pseudospectral methods for geodynamo simulations

The problem of understanding how Earth's magnetic field is generated is one of the foremost challenges in modern science. It is believed to be generated by a dynamo process, where the complex motions of an electrically conducting fluid provide the inductive action to sustain the field against the effects of dissipation. Current dynamo simulations, based on the numerical approximation to the governing equations of magnetohydrodynamics, cannot reach the very rapid rotation rates and low viscosities (i.e. low Ekman number) of Earth due to limitations in available computing power. Using a pseudospectral method, the most widely used method for simulating the geodynamo, computational requirements needed to run simulations in an 'Earth-like' parameter regime are explored theoretically by approximating operation counts, memory requirements and communication costs in the asymptotic limit of large problem size. Theoretical scalings are tested using numerical calculations. For asymptotically large problems the spherical transform is shown to be the limiting step within the pseudospectral method; memory requirements and communication costs are asymptotically negligible. Another limitation comes from the parallel implementation, however, this is unlikely to be threatened soon and we conclude that the pseudospectral method will remain competitive for the next decade. Extrapolating numerical results based upon the code analysis shows that simulating a problem characterizing the Earth with Ekman number E = 10(-9) would require at least 13 000 days per magnetic diffusion time with 54 000 available processors, a formidable computational challenge. At E = 10(-8) an allocation of around 350 million CPU hours would compute a single diffusion time, many more CPU hours than are available in current supercomputing allocations but potentially reachable in the next decade. Exploration of the 10(-6) <= E <= 10(-7) regime could be performed at the present time using a substantial share of national supercomputing facilities or a dedicated cluster. Copyright (C) 2010 John Wiley & Sons, Ltd.