The influence of deferential rotation on magnetic instability, and nonlinear magnetic instability in the magnetostrophic limit

Abstract

In rapidly rotating systems, a (and, in certain circumstances, the) most important nonlinear effect is the geostrophic flow V_g (s)1φ associated with Taylor's (1963) constraint. Its role has been extensively studied in the context of α ² − and αω-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_g (s)1φ, with V(s) = sω(s), and find that there can be a significant reduction in the critical field strength for flows having a negative outward gradient (dω/ds < 0). We then choose a typical such flow (V= -R_ms² ) and focus attention on the interaction between the magnetic instability present (or not) when the flow is absent (R_m = 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 flow. 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 Feam, 1995a, b) did not find subcriticality so there must be some value of E < 10 ⁻⁴ below which the geostrophic flow 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.

Key Words:

• Earth's core
• magnetic instability
• geostrophic flow
• Taylor's constraint