Differential rotation of stretched and twisted thick magnetic flux tube dynamos in Riemannian spaces

Abstract

The topological mapping between a torus of big radius and a sphere is applied to the Riemannian geometry of a stretched and twisted very thick magnetic flux tube, to obtain spherical dynamos solving the magnetohydrodynamics (MHD) self-induction equation for the magnetic flux tubes undergoing differential (non-uniform) rotation along the tube magnetic axis. Constraints on the shear is also computed. It is shown that when the hypothesis of the convective cyclonic dynamo is used the rotation is constant and a solid rotational body is obtained. As usual toroidal fields are obtained from poloidal magnetic field and these fields may be expressed in terms of the differential rotation $Ω(r,θ(s))$. In the case of non-cyclonic dynamos the torsion in the Frenet frame is compute in terms of the dynamo constant. The flux tube shear $\frac{\partial}{{\partial}r}Ω$ is also computed. The untwisted tube case is shown to be trivial in the sense that does not support any dynamo action. This case is in agreement with Cowling antidynamo theorem, since in the untwisted case the tube becomes axially symmetric which the refereed theorem rules out. We also show that it is consistent with the Zeldovich antidynamo theorem which rules out planar dynamos. Knowledge of the differential rotation of the Earth, for example, allows one to place limits on the curvature and torsion of the flux tube axis and vice-versa, knowledge of the topology permit us to infer differential rotation and other physical parameters of the stars and planets.