Helical ${\alpha}$-dynamos as twisted magnetic flux tubes in Riemannian space

TL;DR

This paper derives an analytical solution for the alpha-dynamo equation in twisted Riemannian flux tubes, showing how torsion influences magnetic field growth and decay in a highly torsioned helical dynamo model.

Contribution

It provides a new analytical solution for alpha-dynamo equations in twisted flux tubes, highlighting the role of torsion in magnetic field amplification and decay.

Findings

Toroidal magnetic field grows linearly with time.

Poloidal component influenced by torsion and twist.

Magnetic and vorticity components decay spatially as power laws.

Abstract

Analytical solution of ${\alpha}$-dynamo equation representing strongly torsioned helical dynamo is obtained in the thin twisted Riemannian flux tubes approximation. The $\alpha$ factor possesses a fundamental contribution from torsion which is however weaken in the thin tubes approximation. It is shown that assuming that the poloidal component of the magnetic field is in principle time-independent, the toroidal magnetic field component grows very fast in time, actually it possesses a linear time dependence, while the poloidal component grows under the influence of torsion or twist of the flux tube. The toroidal component decays spatially with as $r^{-2}$ while vorticity may decay as $r^{-5}$ (poloidal component) where r represents the radial distance from the magnetic axis of flux tube. Toroidal component of vorticity decays as $r^{-1}$. In turbulent dynamos unbounded magnetic fields may decay at least as $r^{-3}$.