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

TL;DR

This paper constructs a smooth velocity field on the three-dimensional torus that demonstrates a subsequentially fast dynamo effect, leading to exponential magnetic field growth independent of diffusivity, bridging the gap between slow and fast dynamos.

Contribution

The paper introduces the concept of a subsequentially fast dynamo and constructs an explicit example on b4^3, advancing understanding of dynamo mechanisms in magnetohydrodynamics.

Findings

Exhibits exponential magnetic field growth with uniform rate in diffusivity sequences

Provides a new intermediate dynamo classification between slow and fast

Constructs explicit smooth velocity fields demonstrating this behavior

Abstract

We construct a smooth velocity field $u$ on $\mathbb{R}_+ \times \mathbb{T}^3$ that exhibits kinematic dynamo action, causing exponential growth in solutions to the magnetohydrodynamic induction equation, with a rate that is uniform in diffusivity, for suitable sequences of diffusivity $\kappa_j \to 0.$ We call this a subsequentially fast dynamo, giving dynamo behavior intermediate between a truly slow dynamo and a truly fast dynamo.