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Numerical calculations of fluid dynamos powered by thermal convection in a rotating, electrically conducting spherical shell are analyzed. We find two regimes of nonreversing, strong field dynamos at Ekman number 10 -4 and Rayleigh numbers up to 11 times critical. In the strongly columnar regime, convection occurs only in the fluid exterior to the inner core tangent cylinder, in the form of narrow columnar vortices elongated parallel to the spin axis. Columnar convection contains large amounts of negative helicity in the northern hemisphere and positive helicity in the southern hemisphere and results in dynamo action above a certain Rayleigh number, through a macroscopic a 2 mechanism. These dynamos equilibrate by generating concentrated magnetic flux bundles that limit the kinetic energy of the convection columns. The dipole-dominated external field is formed by superposition of several flux bundles at middle and high latitudes. At low latitudes a pattern of reversed flux patches propagates in the retrograde direction, resulting in an apparent westward drift of the field in the equatorial region. At higher Rayleigh number we find a fully developed regime with convection inside the tangent cylinder consisting of polar upwelling and azimuthal thermal wind flows. These motions modify the dynamo by expelling poloidal flux from the poles and generating intense toroidal fields in the polar regions near the inner core. Convective dynamos in the fully developed regime exhibit characteristics that can be compared with the geomagnetic field, including concentrated flux bundles on the core-mantle boundary, polar minima in field intensity, and episodes of westward drift. Christensen et al., 1998.] Perhaps the most fimdamental result from these models is the demonstration that rotating convection in a conducting spherical shell produces an external magnetic field dominated by the axial dipole component. The intensity of the axial dipole field in the Glatzmaier-Roberts and Kuang-Bloxham models in particular is comparable to the geomagnetic dipole. The success of numerical dynamo models in reproducing the gross features of the geomagnetic field is encouraging. It suggests that the fluid mechanical processes responsible for generation of the magnetic field in Earth are governed mostly by the fundamental elements in the models, namely convection, rotation, the spherical shell geometry and simple boundary conditions. The broad-scale field structure of convection-driven dynamos appears to be less sensi-10,383 10,384 OLSON ET AL.: NUMERICAL DYNAMO MODELS rive to other model parameters, for example, the way subgrid-scale processes such as turbulence and mixing are parameterized.In this paper we analyze the results of numerical calculations of rotating magnetoconvection leading to dynamos that are dynamically similar to the dynamos referred to above, except that we restrict attention to more modest values of some key dimensionless parameters, specifically the Ekman and Rayleigh numbers, and we treat the inner core in a more simplifie...

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S U M M A RYWe analyse *50 3-D numerical calculations of hydrodynamic dynamos driven by convection in a spherical shell. We examine rigid and stress-free boundaries, with Prandtl number 1, magnetic Prandtl numbers in the range 0.5^5, Ekman numbers E~10 {3^1 0 {4 and Rayleigh numbers to 15 times critical. No parametrizations such as hyperviscosities are used. Successful dynamos are compared with non-magnetic convection solutions. Results for various spectral truncations suggest that the calculations are well resolved when the kinetic and magnetic energy drops by more than two orders of magnitude from the spectral peak to the cut-o¡, although the basic features are still captured at lower resolution. With few exceptions we obtain dipole-dominated magnetic ¢elds. The dynamos operate in the strong-¢eld regime where Lorentz and Coriolis forces are of similar order. The critical magnetic Reynolds number for selfsustained dynamos is of order Rm~50. However, we also ¢nd that the ¢eld can die away when Rm is too large. The minimum magnetic Prandtl number at which we ¢nd dynamo action depends on the Ekman number to the 3/4 power. Dynamos at E~10{3 are subcritical whereas those at E~10 {4 are generally supercritical. The presence of the magnetic ¢eld tends to break the equatorial symmetry of the £ow and favours convection inside the inner core tangent cylinder. With stress-free boundaries, dynamo action suppresses the axisymmetric azimuthal wind that dominates in non-magnetic convection. The ¢eld morphology is broadly similar for both kinds of boundary conditions. When low-pass ¢ltered, several models exhibit ¢eld structures that resemble the geomagnetic ¢eld at the core^mantle boundary to a surprising degree.

80 years ago, Joseph Larmor planted the seed that grew into today's imposing body of knowledge about how the Earth's magnetic field is created. His simple idea, that the geomagnetic field is the result of dynamo action in the Earth's electrically conducting, fluid core, encountered many difficulties, but these have by now been largely overcome, while alternative proposals have been found to be untenable. The development of the theory and its current status are reviewed below. The basic electrodynamics are summarized, but the main focus is on dynamical questions. A special study is made of the energy and entropy requirements of the dynamo and in particular of how efficient it is, considered as a heat engine. Particular attention is paid to modeling core magnetohydrodynamics in a way that is tractable but nevertheless incorporates the dynamical effects of core turbulence in an approximate way. This theory has been tested by numerical integrations, some results from which are presented. The success of these simulations seems to be considerable, when measured against the known geomagnetic facts summarized here. Obstacles that still remain to be overcome are discussed, and some other future challenges are described. CONTENTS