The stability of toroidal magnetic fields with equatorial symmetry: implications for the Earth's magnetic field

Abstract

Geophysical observation suggests that the symmetry of the Earth's magnetic field has been predominantly dipolar over the last 2.5 m.y. Given that such an antisymmetric field will vanish at the equatorial plane, we might expect antisymmetry to be a source of magnetic field instability, as the presence of so-called critical surfaces are robust destabilising features, independent of field morphology. To examine the influence of antisymmetry on the stability of magnetic fields a detailed investigation of the stability of model toroidal fields is presented. Using a cylindrical geometry ( s ∗,ϕ,z ∗ ), a basic model field B(s ∗,z ∗) ϕ is subject to an infinitesimal perturbation which is allowed to grow (or decay) in the presence of B ϕ . The addition of an antisymmetric z-dependence to a previously stable s-dependent field, B′(s ∗) ϕ , causes the new field, B(s ∗,z ∗) ϕ , to destabilise; the addition of an antisymmetric z-dependence to an already unstable field significantly lowers Λ c (where Λ c is a measure of the critical field strength). The addition of antisymmetric z-dependence gives an extra critical surface at the equator ( z ∗ = 0 ), which, in turn, gives an extra degree of freedom for field line reconnection, thus leading to a lower Λ c. The use of a cylindrical geometry permits examination of the spatial structure and complex growth rate of the instability in the large field strength, small ohmic diffusion regime (defined by large Λ). In this regime, the most unstable perturbations are found to belong to the resistive class: their mechanism is reconnection and breaking of field lines near a critical surface. Comparisons between results in cylindrical and spherical geometry in the low field strength regime are favourable. Results are also given for the ideal instability which can exist in the absence of diffusion. Our model shows strong localisation of the ideal mode far from the critical surfaces.