A limsup fast dynamo on $\mathbb{T}^3$

TL;DR

This paper constructs a specific velocity field on the 3-torus that causes magnetic energy to grow exponentially along a diverging sequence of times, demonstrating a limsup fast dynamo mechanism for all positive diffusivities.

Contribution

It provides a novel construction of a limsup fast dynamo on $\

Findings

Demonstrates exponential magnetic energy growth along a subsequence of times.

Establishes continuity of growth rate with respect to initial data and diffusivity.

Proves the weak form of the fast dynamo conjecture on $\

Abstract

We construct a time-dependent, incompressible, and uniformly-in-time Lipschitz continuous velocity field on $\mathbb{T}^3$ that produces exponential growth of the magnetic energy along a subsequence of times, for every positive value of the magnetic diffusivity. Because this growth is not uniform in time but occurs only along a diverging sequence of times, we refer to the resulting mechanism as a limsup fast dynamo. Our construction is based on suitably rescaled Arnold-Beltrami-Childress (ABC) flows, each supported on long time intervals. The analysis employs perturbation theory to establish continuity of the exponential growth rate with respect to both the initial data and the diffusivity parameter. This proves the weak form of the fast dynamo conjecture formulated by Childress and Gilbert on $\mathbb{T}^3$, but the considerably more challenging version proposed by Arnold on $\mathbb{T}^3$ remains an open problem.