Resonant oscillations in ${\alpha}^{2}$-dynamos on a closed, twisted Riemannian 2D flux tubes

TL;DR

This paper demonstrates that oscillatory ${ m oldsymbol{ extalpha}}^{2}$-dynamos can exist on twisted Riemannian flux tubes with positive curvature due to resonance modes, expanding the understanding of dynamo support beyond negative curvature regions.

Contribution

It shows that resonance modes enable ${ m oldsymbol{ extalpha}}^{2}$-dynamos on positively curved Riemannian manifolds, challenging previous results limited to negative curvature.

Findings

Resonance modes facilitate dynamo action in positive curvature regions.

Dynamo support is possible on 2D manifolds with mixed curvature.

Resonance tuning can induce dynamo behavior in twisted flux tubes.

Abstract

Chicone et al [CMP (1995)] have shown that, kinematic fast dynamos in diffusive media, could exist only on a closed, 2D Riemannian manifold of constant negative curvature. This report, shows that their result cannot be extended to oscillatory ${\alpha}^{2}$-dynamos, when there are resonance modes, between toroidal and poloidal frequencies of twisted magnetic flux tubes. Thus, dynamo action can be supported in regions, where Riemannian curvature is positive. For turbulent dynamos, this seems physically reasonable, since recently, [Shukurov et al PRE (2008)] have obtained a Moebius flow strip in sodium liquid, torus Perm dynamo where curvature is also connected to the magnetic fields via diffusion. This could be done, by adjusting the corresponding frequencies till they achieved resonance. Actually 2D torus, is a manifold of zero mean curvature, where regions of positive and negative curvatures exist. It is shown that, Riemannian solitonic surface, endowed with a steady ${\alpha}^{2}$-dynamo from magnetic filamentary structures [Wilkin et al,PRL (2007)].