Conformal maps in periodic flows and in suppression of stretch-twist and fold on Riemannian manifolds

TL;DR

This paper explores how conformal maps influence dynamo action in Riemannian manifolds, showing that certain conformal conditions suppress stretching and twist, thereby preventing fast dynamo generation in magnetic flux tubes.

Contribution

It demonstrates that conformal maps under Floquet conditions relate exponential stretching to the conformal factor, and establishes conditions under which non-stretching conformal maps cannot support fast dynamos.

Findings

Conformal maps relate Lyapunov exponents to conformal factors.

Non-stretching conformal flux tubes cannot support fast dynamos.

Riemann curvature influences the suppression of dynamo action.

Abstract

Examples of conformal dynamo maps have been presented earlier [Phys Plasmas \textbf{14}(2007)] where fast dynamos in twisted magnetic flux tubes in Riemannian manifolds were obtained. This paper shows that conformal maps, under the Floquet condition, leads to coincidence between exponential stretching or Lyapunov exponent, conformal factor of fast dynamos. Unfolding conformal dynamo maps can be obtained in Riemann-flat manifolds since here, Riemann curvature plays the role of folding. Previously, Oseledts [Geophys Astrophys Fluid Dyn \textbf{73} (1993)] has shown that the number of twisted and untwisting orbits in a two torus on a compact Riemannian manifold induces a growth of fast dynamo action. In this paper, the stretching of conformal thin magnetic flux tubes is constrained to vanish, in order to obtain the conformal factor for non-stretching non-dynamos. Since thin flux tube can be considered as a twisted or untwisting two-torus map, it is shown that the untwisting, weakly torsion, and non-stretching conformal torus map cannot support a fast dynamo action, a marginal dynamo being obtained. This is an example of an anti-fast dynamo theorem besides the ones given by Vishik and Klapper and Young [Comm Math Phys \textbf{173}(1996)] in ideally high conductive flow. From the Riemann curvature tensor it is shown that new conformal non-dynamo, is actually singular as one approaches the magnetic flux tube axis. Thus conformal map suppresses the stretching directions and twist, leading to the absence of fast dynamo action while Riemann-flat unfolding manifolds favors non-fast dynamos.