The value of the Prandtl number P exerts a strong influence on convection driven dynamos in rotating spherical shells filled with electrically conducting fluids. Low Prandtl numbers promote dynamo action through the shear provided by differential rotation, while the generation of magnetic fields is more difficult to sustain in high Prandtl number fluids where higher values of the magnetic Prandtl number P m are required. The magnetostrophic approximation often used in dynamo theory appears to be valid only for relatively high values of P and P m . Dynamos with a minimum value of P m seem to be most readily realizable in the presence of convection columns at moderately low values of P . The structure of the magnetic field varies strongly with P in that dynamos with a strong axial dipole field are found for high values of P while the energy of this component is exceeded by that of the axisymmetric toroidal field and by that of the non-axisymmetric components at low values of P . Some conclusions are discussed in relation to the problem of the generation of planetary magnetic fields by motions in their electrically conducting liquid cores.

PACS 91.25.Cw -Origins and models of the magnetic field; dynamo theories PACS 92.60.hk -Convection, turbulence, and diffusion PACS 47.20.Ky -Nonlinearity, bifurcation, and symmetry breaking Abstract. -Bistability and hysteresis of magnetohydrodynamic dipolar dynamos generated by turbulent convection in rotating spherical fluid shells is demonstrated. Hysteresis appears as a transition between two distinct regimes of dipolar dynamos with rather different properties including a pronounced difference in the amplitude of the axisymmetric poloidal field component and in the form of the differential rotation. The bistability occurs from the onset of dynamo action up to about 9 times the critical value of the Rayleigh number for onset of convection and over a wide range of values of the ordinary and the magnetic Prandtl numbers including the value unity.

For the understanding of planetary and stellar dynamos an overview of the major parameter dependences of convection driven dynamos in rotating spherical fluid shells is desirable. Although the computationally accessible parameter space is limited, earlier work is extended with emphasis on higher Prandtl numbers and uniform heat flux condition at the outer boundary. The transition from dynamos dominated by non-axisymmetric components of the magnetic field to those dominated by the axisymmetric components depends on the magnetic Prandtl number as well as on the ordinary Prandtl number for higher values of the rotation parameter τ . The dependence of the transition on the latter parameter is also discussed. A variety of oscillating dynamos is presented and interpreted in terms of dynamo waves, standing oscillation or modified relaxation oscillations.

Abstract.Numerical simulations of the geodynamo have successfully represented many observable characteristics of the geomagnetic field, yielding insight into the fundamental processes that generate magnetic fields in the Earth's core. Because of limited spatial resolution, however, the diffusivities in numerical dynamo models are much larger than those in the Earth's core, and consequently, questions remain about how realistic these models are. The typical strategy used to address this issue has been to continue to increase the resolution of these quasi-laminar models with increasing computational resources, thus pushing them toward more realistic parameter regimes. We assess which methods are most promising for the next generation of supercomputers, which will offer access to O(10 6 ) processor cores for large problems. Here we report performance and accuracy benchmarks from 15 dynamo codes that employ a range of numerical and parallelization methods. Computational performance is assessed on the basis of weak and strong scaling behavior up to 16,384 processor cores. Extrapolations of our weak scaling results indicate that dynamo codes that employ two-or three-dimensional domain decompositions can perform efficiently on up to ∼ 10 6 processor cores, paving the way for more realistic simulations in the next model generation.