A fast dynamo on the three-torus

TL;DR

This paper proves the existence of fast dynamo action on the three-torus using a hyperbolic flow mechanism, demonstrating exponential magnetic field growth even with diffusion.

Contribution

It introduces a rigorous proof of fast dynamo action for a specific hyperbolic flow on the three-torus, employing anisotropic Banach spaces for spectral analysis.

Findings

Existence of an eigenvalue with modulus greater than one in the ideal dynamo operator.

Persistence of magnetic field growth under diffusion in the vanishing resistivity limit.

Development of anisotropic Banach spaces to analyze hyperbolic dynamo dynamics.

Abstract

We study the kinematic dynamo equation on the three-torus and provide a rigorous proof of fast dynamo action for a time-periodic, divergence-free, Lipschitz velocity field. Our construction is based on a stretch-fold-shear mechanism generating a uniformly hyperbolic flow. To analyze the associated dynamics, we develop anisotropic Banach spaces adapted to the underlying hyperbolic structure, allowing us to recover a discrete spectral picture for the ideal dynamo operator. In the strong-chaos regime, we show that this operator admits an eigenvalue with modulus strictly larger than one. We then prove that this instability persists under the singular perturbation induced by diffusion, yielding exponential growth of the magnetic field uniformly in the vanishing resistivity limit.