Stretch-Twist torus dynamo in compact Riemannian manifolds

TL;DR

This paper derives eigenvalues for a Riemann twisted torus dynamo flow metric, revealing stability properties and growth rates of flow components, extending previous dynamo models with new geometric insights.

Contribution

It provides a novel eigenvalue analysis of the Riemann twisted torus dynamo metric, connecting flow stability and growth rates to geometric properties and previous dynamo theories.

Findings

Eigenvalues of the Riemann twisted torus dynamo are ${m}_{ m{ iny ext{±}}}=rac{1 ext{±} ext{ extsqrt{5}}}{2}$.

Flow becomes unstable near the torus axis as Re approaches infinity.

The ${ m ext{ extalpha}}$-effect is a second-order effect in curvature and flow velocity.

Abstract

Earlier Arnold, Zeldovich, Ruzmaikin and Sokoloff [\textbf{JETP (1982)}] have computed the eigenvalue of a uniform stretching torus transformation which result on the first Riemann metric solution of the dynamo action problem. Recently some other attempts to obtain Riemann metrics representing dynamo action through conformal maps have been undertaken [{\textbf{Phys. Plasmas 14 (2007)}]. Earlier, Gilbert [\textbf{Proc. Roy. Soc. London A(2003)}] has investigated a more realistic dynamo map solution than the one presented by Arnold et al by producing a shearing of the Arnold's cat map, by eigenvalue problem of a dynamo operator. In this paper, the eigenvalue of the Riemann twisted torus dynamo flow metric is obtained as the ratio between the poloidal and toroidal components of the flow. This result is obtained from the Euler equation. In the twisted torus, the eigenvalue of the Riemann metric is ${m}_{\pm}=\frac{1\pm{\sqrt{5}}}{2}$, which is quite close to the value obtained by Arnold. In the case the viscosity Reynolds number $Re\to{\infty}$, the torus flow is unstable as one approaches the torus axis. In Arnold's dynamo metric the eigenvalues are ${\chi}_{\pm}=\frac{3\pm{\sqrt{5}}}{2}$ which are very close to the above value. Eigenvalues determine the growth rates of the velocity ratio between poloidal and toroidal components of the flow. The curved flow in torus follow previous work by Chen et al [\textbf{Phys Fluids (2006)}]. The ${\alpha}$-effect dynamo is shown to be a second-order effect in the torus curvature and velocity flow. Loop dynamo flows and maps are also discussed.