Kinematic dynamo models

Abstract

Spherical kinematic dynamo models with axisymmetric magnetic fields are examined, which arise from the mean field electrodynamics of Steenbeck and Krause, and also from the nearly axisymmetric limit of Braginskii. Four main cases are considered: (i) there is no mean flow, but the dynamo is maintained by microscale motions which create a mean electromotive force, ( E ) , proportional to the mean magnetic field, B (the α effect); (ii) in addition to an α effect which creates poloidal mean field from toroidal, a mean toroidal shearing flow (angular velocity w ) is present which creates toroidal mean field from poloidal more efficiently than by the α effect; (iii) in addition to the processes operative in (ii), a mean meridional circulation, m , is present; (iv) ( E ) is produced by a second order inductive process first isolated by Radler. When these processes are sufficiently strong, they can maintain magnetic fields. The resulting situations are known as (i) α 2 dynamos, (ii) αω dynamos, (iii) αω dynamos with meridional circulations, and (iv) Rädler dynamos. Models of each type are considered below, but cases (ii) and (iii) give rise to particularly interesting results. If | m | is sufficiently small, or zero [case (ii)], the most easily excited dynamo is oscillatory and is of dipole type if α ω ′ < 0 in the northern hemisphere (and negative in the southern); here ω ′ denotes the outward gradient of ω . The oscillation resembles a Parker dynamo wave, generated at the poles, absorbed at the equator and always moving towards lower latitudes, as for the butterfly diagrams of sunspots. If α ω ′ > 0 in the northern hemisphere, the direction of wave motion is reversed, and also the quadrupolar solution is more readily excited than the dipolar. If | m | is sufficiently large, and of the right magnitude and sense (which is model dependent), it is found that the dynamo which regenerates most easily is steady. It is of dipolar form if α ω ′ > 0 but quadrupolar if α ω ′ < 0 . These models appear to be relevant to the Earth, where meridional circulations might be provided by, for example, Ekman pumping. Evidence for a remarkable symmetry property is adduced. If m and α ω ′ are reversed everywhere in the state in which the dipole (say) is most readily excited, it is found that the state in which a quadrupole is most easily regenerated is recovered, almost precisely. Moreover, the critical magnetic Reynolds number for each is closely similar. As a corollary, the critical Reynolds numbers for dipolar and quadrupolar solutions of opposite α ω ′ are nearly identical for the α ω dynamo (m = 0).