David R Fearn, Chris J Lamb, Douglas R McLean and Richard R Ogden.
In rapidly rotating systems, a (and, in certain circumstances, the)
most important nonlinear effect is the geostrophic how VG
1f associated with Taylor's
(1963) constraint. Its role has been extensively studied in the context
of a2 - and aw-dynamos,
and, to a lesser extent in magnetoconvection problems. Here, we investigate
its role in the magnetic stability problem, using a cylindrical geometry.
First, we investigate the influence of a representative variety of arbitrarily
prescribed flows V(s)1f ,
with V(s) = sW(s),
and find that there can be a significant reduction in the critical field
strength for flows having a negative outward gradient (dW/ds
< 0). We then choose a typical such flow (V = -Rm
s2 ) and focus attention on the interaction between the
magnetic instability present (or not) when the flow is absent (Rm
= 0) and the instability driven by differential rotation when the flow
is stronger. It is found that instability (even when driven only by the
differential rotation) exists only above a minimum field strength. Finally,
having gained an understanding of the roles that differential rotation
can play, we investigate the nonlinear magnetic stability problem,
where the nonlinear effect is the geostrophic how. We find cases where
the geostrophic flow has the property of destabilising the system. This
can happen for the most unstable mode, so the nonlinear effect of the geostrophic
flow can be subcritical. Corresponding nonlinear calculations at finite
Ekman number E (Hutcheson and Fearn, 1995a, b) did not find subcriticality
so there must be some value of E < 10-4 below which
the geostrophic how dominates the other nonlinear effects and subcriticality
becomes possible. What that value is may influence how low E must
be taken in full geodynamo simulations to correctly qualitatively
describe the dynamics of the core.