On finite amplitude convection in a rotating magnetic system

Abstract

The instabilities that can arise in a stratified, rapidly rotating, magnetohydrodynamic system such as the Earth’s core are often thought to play a key role in dynamo theory — that is, in the study of how the magnetic field in the system is maintained in the face of ohmic dissipation. An account of such instabilities is to be found in the M.A.C.- wave theory of Braginsky (1967), who, however, laid his greatest emphasis on the dissipationless modes, an idealization which leads to difficulties described below, ohmic and thermal diffusion is therefore restored, and three key dimensionless parameters are isolated: q , the ratio of thermal to ohmic diffusivities; A, a measure of the relative importance of Coriolis and magnetic forces; and R , a Rayleigh number, which is here the ratio of buoyancy to Coriolis forces. This study concentrates on a particular M.A.C.-wave model originally proposed by Braginsky. It consists of a horizontal layer containing a uniform horizontal magnetic field, B0, and rotated about the vertial, an adverse temperature gradient being maintained on the horizontal boundaries to provide the unstable density stratification. In the rotationally dominant case of large A, the principle of the exchange of stabilities holds, and the motions that arise in the marginal state are steady. The planform of the convection is in rolls orthogonal to B0. If q and A are sufficiently small the principle of the exchange of stabilities remains valid, but the planform consists of one or other of two families of rolls oblique to B0, or a combination of each. If q is large but A is small, the modes are again oblique, but overstability occurs, a type of oscillation which also arises when q is large and A takes intermediate values, although the motion is then in rolls transverse to B0. A theory is developed for the weakly nonlinear convection that arises when R exceeds only slightly the critical value Rc{q, A) at which marginal convection occurs. A critical curve q = ^D(A) is located which roughly divides the (qA) plane into regions of small q and of large q , although when q is large it separates the largeAq from the small. On the one side of the curve, where q or Xq are sufficiently small, it is concluded that, starting from an arbitrary initial perturbation, the convection that arises when R exceeds Rc will ultimately become a completely regular tesselated pattern filling the horizontal plane. On the other side of the curve the situation is considerably more complicated but it is argued that, for sufficiently large q and Aq, subcritical instabilities can occur and that supercritical bursting is likely; that is, the instability that arises from the general initial perturbation will focus into a small spot in a finite time. The relevance of the theory to sunspot formation is discussed. In an appendix, the form of the weakly nonlinear convection that arises when q differs only slightly from qB, and R only slightly from R c, is considered in situations in which the exchange of stabilities holds.