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

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.