Stretch fast dynamo mechanism via conformal mapping in Riemannian manifolds

TL;DR

This paper presents two new analytical solutions for self-induction equations in Riemannian manifolds, demonstrating how conformal mappings and geometric properties can generate and enhance dynamo action in plasma physics.

Contribution

It introduces novel analytical solutions for dynamos in Riemannian manifolds, highlighting the role of conformal mappings and geometric effects in magnetic field amplification.

Findings

Conformal effects enhance the Zeldovich stretch in dynamos.

A new dynamo solution is obtained via conformal mapping of Arnold's dynamo.

Magnetic flux tube amplification depends on Frenet torsion.

Abstract

Two new analytical solutions of self-induction equation, in Riemannian manifolds are presented. The first represents a twisted magnetic flux tube or flux rope in plasma astrophysics, which shows that the depending on rotation of the flow the poloidal field is amplified from toroidal field which represents a dynamo. The value of the amplification depends on the Frenet torsion of the magnetic axis of the tube. Actually this result illustrates the Zeldovich stretch, twist and fold (STF) method to generate dynamos from straight and untwisted ropes. Motivated by the fact that this problem was treated using a Riemannian geometry of twisted magnetic flux ropes recently developed (Phys Plasmas (2006)), we investigated a second dynamo solution which is conformally related to the Arnold kinematic fast dynamo. In this solution it is shown that the conformal effect on the fast dynamo metric only enhances the Zeldovich stretch, and therefore a new dynamo solution is obtained. When a conformal mapping is performed in Arnold fast dynamo line element a uniform stretch is obtained in the original line element.