Magnetic dynamo C-flows in Riemannian manifold as generalized Arnold's map

TL;DR

This paper explores generalized magnetic dynamo models on Riemannian manifolds, extending Arnold's map by allowing stretching in multiple directions, and analyzes their mathematical properties and solutions.

Contribution

It introduces a generalized dynamo model on Riemannian manifolds that permits multi-directional stretching, expanding the theoretical framework of magnetic dynamos.

Findings

Generalized solutions allow stretching in multiple directions.

Curvature tensor components are explicitly computed.

The model extends Arnold's dynamo to more complex flow configurations.

Abstract

It is shown that C-flows in Riemannian three-dimensional compact manifold can be naturally considered as generalized dynamo Arnold's metric in compact manifolds, the so-called cat map dynamo. The generalized solution of self-induction equation in the background of this metric shows that one is allowed to consider stretching along both directions of the flow, instead of compressed in one direction and stretched in the other such as in Arnold's dynamo. Though this solution can be considered as unrealistic,at least for incompressible flows, there is another generalized solution which considers distinct stretch and compression intensities along distinct directions. Curvature tensor components are computed by making use of calculus of differential forms.