The magnetoconvection problem under the magnetostrophic approximation
is investigated as the nonlinear regime is entered. The model consists
of a fluid filled sphere, internally heated, and rapidly rotating in
the presence of a prescribed, axisymmetric, toroidal magnetic field. F
or simplicity only a dipole parity and a single azimuthal wavenumber (
m = 2) is considered here. The leading order nonlinearity at small amp
litude is the geostrophic flow U-g which is introduced to the previous
ly linear model (Walker and Barenghi, 1997a, b). Walker and Barenghi (
1997c) considered parameter space above critical and found that U-g ac
ts as an equilibration mechanism for moderately supercritical solution
s. However, for solutions well above critical a Taylor state is approa
ched and the system can no longer equilibrate. More importantly though
, in the context of this paper, is that subcritical solutions were fou
nd. Here subcritical solutions are considered in more detail. A was fo
und that, at (Ra) over tilde = (Ra) over tilde(c), R-m is strongly dep
endent on Lambda. ((Ra) over tilde(c) is the critical value of the mod
ified Rayleigh number (Ra) over tilde, Rm is a measure of the maximum
amplitude of the generated geostrophic flow while Lambda, the Elsasser
number, defines the strength of the prescribed toroidal field.) Rm at
(Ra) over tilde = (Ra) over tilde(c) proves to be the key measure in
determining how far into the subcritical regime the system can advance
.