The geostrophic nonlinearity and its effect on magnetic instability

Abstract

We present a nonlinear hydromagnetic stability analysis in a bounded annular model of the Earth's core. The Ekman number E is the appropriate dimensionless measure of viscosity in the Earth's core. Since E is very small in the core (perhaps as small as 10⁻¹⁵), then either models have to accept unrealistically high values of viscosity, to permit resolution of boundary and internal layers, or they must neglect it in the fluid main body. We make the magnetostrophic approximation and consider an inviscid fluid main body. Although viscous effects are small they become important in the boundary layers. We implicitly incorporate this effect through calculation of the geostrophic flow V _G . For small but finite amplitude solutions, V _G is the dominant nonlinear effect. Neglecting all other nonlinear terms, we execute the stability analysis in a time stepping method where V _G is the only nonlinearity. In the same geometry but at a finite value of E = 10⁻⁴, Hutcheson and Fearn (1995b) executed a similar stability analysis with all nonlinear effects included. They found no subcritical bifurcations. Recent work by Fearn et al. (1997) suggested qualitative differences between the viscous and inviscid approaches when, using the magnetostrophic approximation, they found a subcritical bifurcation. In this work we pursue the inviscid approach and enforce V _G as the only nonlinear effect. We have found that for a variety of s-dependent basic fields and aspect ratios, V _G induces subcriticality for the most unstable mode. For the most unstable modes of other basic fields we show that stable solutions can exist where the role of V _G is to quench exponential field growth. Finally, we present stable and unstable bifurcation diagrams which are consistent with the trend proposed by Malkus and Proctor (1975).

Key Words:

• Earth's core
• geostrophic flow
• magnetostrophic approximation
• stability
• analysis