Abstract
Numerical solutions of the incompressible magnetohydrodynamic (MHD) equations are reported for the interior of a rotating, perfectly-conducting, rigid spherical shell that is insulator-coated on the inside. A previously-reported spectral method is used which relies on a Galerkin expansion in Chandrasekhar–Kendall vector eigenfunctions of the curl. The new ingredient in this set of computations is the rigid rotation of the sphere. After a few purely hydrodynamic examples are sampled (spin down, Ekman pumping, inertial waves), attention is focused on selective decay and the MHD dynamo problem. In dynamo runs, prescribed mechanical forcing excites a persistent velocity field, usually turbulent at modest Reynolds numbers, which in turn amplifies a small seed magnetic field that is introduced. A wide variety of dynamo activity is observed, all at unit magnetic Prandtl number. The code lacks the resolution to probe high Reynolds numbers, but nevertheless interesting dynamo regimes turn out to be plentiful in those parts of parameter space in which the code is accurate. The key control parameters seem to be mechanical and magnetic Reynolds numbers, the Rossby and Ekman numbers (which in our computations are varied mostly by varying the rate of rotation of the sphere) and the amount of mechanical helicity injected. Magnetic energy levels and magnetic dipole behaviour are exhibited which fluctuate strongly on a timescale of a few eddy turnover times. These seem to stabilize as the rotation rate is increased until the limit of the code resolution is reached.
Highlights
In a previous paper, a spectral method for computing incompressible fluid and magnetohydrodynamic (MHD) behavior inside a sphere was introduced (Ref. [1], hereafter referred to as “MM”)
The computations were limited to moderate Reynolds numbers, with boundary conditions in which the normal components of the velocity field, magnetic field, vorticity, and electric current density were required to vanish at a rigid, spherical, insulator-lined, perfectly-conducting shell, and the three components of the velocity and magnetic field were regular at the origin
The main changes reported here are: (1) the introduction of a Coriolis term in the equation of motion; and (2) the velocity field v and magnetic field B are expanded in orthonormal Chandrasekhar-Kendall (“C-K”) vector eigenfunctions of the curl [3, 4, 5, 6, 1]
Summary
A spectral method for computing incompressible fluid and magnetohydrodynamic (MHD) behavior inside a sphere was introduced (Ref. [1], hereafter referred to as “MM”). Included are examples [7, 8] of: (i) spin down, or decay of relatively rotating kinetic energy due to the action of viscosity; (ii) Ekman pumping with flow patterns that result from rotating boundaries; (iii) internal waves, threedimensional relatives of meteorological Rossby waves, that depend on the stabilization introduced by rotation for their oscillatory features; and (iv) some mechanically-forced runs with a finite angle between the symmetry axis of the forcing and the axis of rotation This fourth case results in columnar vortices due to the effect of rotation [8, 9] (note convection is absent in our present formulation).
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