Kinematic dynamos in spheroidal geometries

Abstract

The kinematic dynamo problem is solved numerically for a spheroidal conducting fluid of possibly large aspect ratio with an insulating exterior. The solution method uses solenoidal representations of the magnetic field and the velocity by spheroidal toroidal and poloidal fields in a non-orthogonal coordinate system. Scaling of coordinates and fields to a spherical geometry leads to a modified form of the kinematic dynamo problem with a geometric anisotropic diffusion and an anisotropic current-free condition in the exterior, which is solved explicitly. The scaling allows the use of well-developed spherical harmonic techniques in angle. Dynamo solutions are found for three axisymmetric flows in oblate spheroids with semi-axis ratios 1≤/≤25. For larger aspect ratios strong magnetic fields may occur in any region of the spheroid, depending on the flow, but the external fields for all three flows are weak and concentrated near the axis or periphery of the spheroid.

Turbulent Kinematic Dynamos in Ellipsoids Driven by Mechanical Forcing

“…There have been several attempts using spectral methods for purely hydrodynamical cases (see, e.g., Lorenzani & Tilgner, 2001;Schmitt & Jault, 2004) but very few results for the magnetohydrodynamical case (see Ivers, 2017). There have been several attempts using spectral methods for purely hydrodynamical cases (see, e.g., Lorenzani & Tilgner, 2001;Schmitt & Jault, 2004) but very few results for the magnetohydrodynamical case (see Ivers, 2017).…”

The Origin of Large-Scale Magnetic Fields in Low-Mass Galaxies

“…The magnitude of the critical dynamo number D c increases with the layer thickness but the dynamo number increases even faster, so that the ratio D/D c , a measure of the dynamo efficiency, is higher in a thicker disc (with all other parameters being the same). Similarly to this tendency, the dynamo efficiency in spheroids increases along the sequence from a flattened spheroid to the sphere [30].…”

Kinematic dynamos in triaxial ellipsoids

“…Other numerical strategies have been proposed to explore the effect of IN BCs in non-spherical geometries (e.g. using the boundary-element method [40], or non-orthogonal coordinates [17]). Here, we can compare the two BCs in ellipsoids by accurately approximating IN BCs using the finite-element method (see details in appendix A).…”

“…To sidestep the known difficulties of the ellipsoidal coordinate system, we employ the Cartesian coordinates and expand the magnetic field onto global Cartesian polynomial elements satisfying local boundary conditions (BCs), namely pseudo-vacuum or perfectly conducting BCs. Using such BCs in non-spherical geometries is indeed simpler than using insulating BCs, which are not straightforward to implement in non-spherical dynamo codes [17,40]. The paper is divided as follows.…”