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arXiv · 2504.00855

Alpha-unstable flows and the fast dynamo problem

Michele Coti Zelati·Massimo Sorella·David Villringer
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Abstract

We construct a time-independent, incompressible, and Lipschitz-continuous velocity field in $\mathbb{R}^3$ that generates a fast kinematic dynamo - an instability characterized by exponential growth of magnetic energy, independent of diffusivity. Specifically, we show that the associated vector transport-diffusion equation admits solutions that grow exponentially fast, uniformly in the vanishing diffusivity limit $\varepsilon\to 0$. Our construction is based on a periodic velocity field $U$ on $\mathbb{T}^3$, such as an Arnold-Beltrami-Childress flow, which satisfies a generic spectral instability property called alpha-instability, established via perturbation theory. This provides a rigorous mathematical framework for the alpha-effect, a mechanism conjectured in the late 1960s to drive large-scale magnetic field generation. By rescaling with respect to $\varepsilon$ and employing a Bloch-type theorem, we extend the solution to the whole space. Finally, through a gluing procedure that spatially localizes the instability, we construct a globally defined velocity field $u$ in $\mathbb{R}^3$ that drives the dynamo instability.

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Michele Coti Zelati, Massimo Sorella, David Villringer. 2025-04-01. Alpha-unstable flows and the fast dynamo problem. https://arxiv.org/abs/2504.00855

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