4.9 Numerical simulations of solar dynamo action

Ultimately, the solar dynamo problem should be tackled as a (numerical) solution of the complete set of MHD partial differential equations in a rotating, spherical domain undergoing thermally-driven turbulent convection in its outer 30% in radius. It is a peculiar fact that the last full-fledged attempts to do so go back some some twenty years, to the simulations of Gilman and Miller (1981 ); Gilman (1983

); Glatzmaier (1985a ,b ), despite the remarkable advances in computational capabilities having taken place in the intervening years.

These epoch-making simulations did produce cyclic dynamo action and latitudinal migratory patterns suggestive of the dynamo waves of mean-field theory. However, the associated differential rotation profile turned out non-solar, as did the magnetic field’s spatio-temporal evolution. In retrospect this is perhaps not surprising, as limitations in computing resources forced the simulations to be carried out in a parameter regime far removed from solar interior conditions. Since then and until recently, efforts on the full-sphere simulation front have gone mostly into the purely hydrodynamical problem of reproducing the large-scale flows in the solar convective envelope, as inferred by helioseismology (see, e.g., Miesch, 2005 , and references therein). However, the recent numerical simulations of Brun et al. (2004 ) have reached a strongly turbulent regime, and have managed to produce a reasonably strong mean magnetic field, but without equatorward migration or polarity reversals. These authors suggest that these failings can be traced to the absence of a tachocline-like stable region of strong shear at the bottom of their simulation domain. If this is the case, then direct numerical simulation of the solar cycle will turn out to be even more demanding computationally than hitherto believed.

A number of attempts have also been made to reproduce some salient features of the solar cycle by carrying out high-resolution, local simulations in parameters regimes closer to solar-like (see, e.g., Nordlund et al., 1992 ; Tobias et al., 2001 ; Ossendrijver et al., 2002 ). One issue that has received much attention in the past decade is the interaction of a turbulent, convecting fluid (“convection zone”) with a shear flow concentrated in an underlying stably stratified “tachocline” region (see, e.g., Brummell et al., 2002 ). Such simulations have shown that magnetic fields can be pumped into the stable regions by convective downdrafts, and produce elongated structures reminiscent of magnetic flux tubes aligned with the flow. The recent simulations of Cline et al. (2003 ) are particularly interesting in this respect, as they have managed to produce sustained, reasonably regular cyclic activity with occasional polarity reversals for extended periods of simulation time. At first sight this may look superficially like an $a_O_$ dynamo, but the turbulent flow has no net helicity since rotation is not included in the simulations; the poloidal field is not being regenerated by a mean-field-like turbulent $a$ -effect, but rather by the interaction between buoyancy and the Kelvin-Helmholtz instability in the shear layer. Such fascinating numerical experiments must clearly be pursued.