On the computation of steady, self - consistent spherical dynamos

Abstract

In an earlier paper (Fearn and Proctor, 1984) we described results from a preliminary model of a spherical hydromagnetic dynamo driven by convection. An iterative approach was used. Starting from some guess for the mean toroidal field B we solved for the form of the convective instability in the presence of this field. The mean e.m.f. E [defined in (2.13)] associated with the convection was calculated, and from this, anα-effect was constructed (α=E_Φ/B). We then solved a mean fieldαΩ-dynamo model to produce a new “B”. This cycle was repeated until B converged. For a preliminary investigation, there were good reasons for using anα-effect formalism. However, a more straightforward and physically more realistic approach is to use the e.m.f. E_Φ directly to force the mean field dynamo. This “EΩ-dynamo” is used here. The converged results of Fearn and Proctor (1984) are successfully reproduced and in addition we have found converged steady dynamos in the absence of any poloidal flow (cf. Roberts, 1972). Our iterative dynamo is still far from being completely self-consistent since several parameters and the mean fluid flow have had to be arbitrarily prescribed. The next step is to incorporate more of the dynamics. We use the mean momentum equation to determine the mean flow and, in particular, apply Taylor's (1963) constraint to determine the otherwise arbitrary geostrophic flow U_G(s) The EΩ-dynamo permits this to be done with relative ease (see Fearn and Proctor, 1987). No converged results were found. Solutions either became too detailed to resolve, magnetic instabilities became present, or the solution jumped between two different modes of convection.