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Alpha-unstable flows and the fast dynamo problem
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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.
Forward citations
Cited by 4 Pith papers
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A piecewise-affine, time-periodic shear flow on the 3-torus is a universal ideal dynamo: every non-zero divergence-free L^p initial field grows exponentially for large shear amplitude.
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One fixed Lipschitz, divergence-free, autonomous velocity field on the flat three-torus is a fast dynamo: for every small diffusivity an amplifying magnetic eigenmode exists with uniformly positive growth, while the p...
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A subsequentially fast dynamo on $\mathbb{T}^3$
A smooth flow on T^3 is built so that the induction equation grows magnetic energy exponentially at rate at least 1/4, for any prescribed countable set of diffusivities accumulating at zero.
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Long-wave instability of periodic shear flows with constant magnetic field for the 2D resistive MHD equations
For 2D resistive MHD on a periodic strip, a sufficiently strong shear flow with a horizontal magnetic field is linearly unstable at small aspect ratio when resistivity dominates viscosity, and stable otherwise.
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