Magnetostrophic magnetoconvection

Abstract

Perhaps the most fundamental problem in dynamo theory is the smallness of the Ekman number E which is O(10 −15) based on estimates of the molecular diffusivity of the Earth's core. It is difficult to achieve values much lower than O(10 −4) in numerical models. The “headline” value of E can be reduced using hyperviscosity but it is becoming clear that this is not helping our aim of moving towards the geophysically relevant parameter regime of low viscosity, and several factors indicate that E= O(10 −4) is not small enough. The magnetostrophic approximation is an alternative approach that neglects viscous effects altogether ( E=0) in the interior of the core. A consequence of this is Taylor's constraint [Taylor, J.B., 1963. The magnetohydrodynamics of a rotating fluid and the Earth's dynamo problem. Proc. R. Soc. London, Ser. A 274, 274–283]. So far, fundamental problems have prevented the successful working of a convection-driven dynamo model using this approach [Walker, M.R., Barenghi, C.F., Jones, C.A., 1998. A note on dynamo action at asymptotically small Ekman number. Geophys. Astrophys. Fluid Dyn. 88, 261–275]. Nevertheless, the difficulties with hyperviscosity mean that it is important to continue to learn as much as possible about the E→0 limit through the ( E=0) magnetostrophic approximation. If viscous effects are retained only in Ekman boundary layers, a standard analysis modifies Taylor's constraint to give a prescription for the mean azimuthal flow. This so-called “geostrophic flow” V G( s, t)1 φ is the dominant nonlinear effect when the magnetic field is weak and can act to equilibrate an otherwise exponentially growing linear solution in an “Ekman state”. Two variants of this approach have been studied: mean-field dynamos and magnetoconvection. Here, we consider the latter. We use a cylindrical geometry [adopting cylindrical polars ( s, φ, z)]. In the presence of an imposed magnetic field B 0, we time-step the governing equations to find the dependence of the nonlinear (Ekman state) solutions as a function of the Rayleigh number Ra. We investigate the role of alternative magnetic boundary conditions and of whether there is an inner cylindrical boundary or not. These introduce modest quantitative differences. The major qualitative difference arises from the orientation of gravity g and the imposed temperature gradient ∇ T. When in the axial ( z) direction, a transition towards a Taylor state (along the lines envisaged by Malkus and Proctor [Malkus, W.V.R., Proctor, M.R.E., 1975. The macrodynamics of α-effect dynamos in rotating fluids. J. Fluid Mech. 67, 417–443.]) is found at Ra close to Ra c. When g and ∇ T 0 are aligned in the radial ( s) direction, supercritical Ekman states are found for a large range of values of Ra, and there is no evidence of the system moving towards a Taylor state. These results are consistent with the earlier work of Skinner and Soward [Skinner, P.H., Soward, A.M., 1990. Convection in a rotating magnetic system and Taylor's constraint II. Geophys. Astrophys. Fluid Dyn. 60, 335–356.] and Jones and Roberts [Jones, C.A., Roberts, P.H., 1990. Magnetoconvection in rapidly rotating Boussinesq and compressible fluids. Geophys. Astrophys. Fluid Dyn. 55, 263–308]. The expected cancellation in the Taylor integral is clearly observed as a Taylor state is approached. This emphasises the inability of the standard prescription for V G to be useful other than for an Ekman state.