Fast dynamo action of the unit ABC flow
On $\mathbb{T}^3 = (\mathbb{R}/2\pi\mathbb{Z})^3$ let $$u = (\sin z + \cos y,\ \sin x + \cos z,\ \sin y + \cos x),$$ the ABC flow with $A = B = C = 1$. For magnetic diffusivity $\eta > 0$ let $L_\eta B = \eta \Delta B + \nabla \times (u \times B)$ be the kinematic induction operator on complex, mean-zero, divergence-free $L^2$ vector fields, with domain the $H^2$ fields in that space, and let $s(\eta) = \sup\{\operatorname{Re} \lambda : \lambda \in \sigma(L_\eta)\}$. Arnold and Korkina found numerically that this flow makes magnetic fields grow exponentially in a range of magnetic Reynolds numbers $1/\eta$ (1983). Bouya and Dormy computed the dominant eigenvalue up to $1/\eta = 25000$ and found that it still varies significantly with $\eta$ there, without having reached an asymptotic behaviour (2013). Sorella and Villringer constructed a time-dependent, uniformly Lipschitz, divergence-free velocity field on $\mathbb{T}^3$, built from rescaled ABC flows, for which the magnetic energy grows exponentially along a diverging sequence of times at a rate bounded below uniformly as the diffusivity tends to zero (2025). In preprints, Coti Zelati, Sorella and Villringer constructed fast dynamos on $\mathbb{T}^3$ from a stretch-fold-shear mechanism: a time-periodic Lipschitz velocity field (2026), and an open set, in a $C^k$ topology, of smooth time-independent velocity fields (2026).
Problem. Is $\liminf_{\eta \downarrow 0} s(\eta) > 0$? The velocity field and its three coefficients stay fixed as $\eta \to 0$.
Source: Stated in this spectral form, for the 1:1:1 ABC flow, by AIM — Open Applied Problems, after the fast dynamo conjecture of Zeldovich and Sakharov (Arnold's Problems, Problem 1994-28); the flow's dynamo action has been studied numerically since Arnold and Korkina (1983), see Bouya and Dormy (2013).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP, SP
- Keywords
- ABC flows, AIM Open Applied Problems, fast dynamos
References
@article{arnold1983,
author = {Arnol'd, V. I. and Korkina, E. I.},
title = {Growth of the magnetic field in three-dimensional steady flow of incompressible fluid},
journal = {Vestnik Moskov. Univ. Ser. I Mat. Mekh.},
number = {3},
year = {1983},
pages = {43--46},
note = {In Russian},
}
@article{bouya2013,
author = {Bouya, Isma{\"e}l and Dormy, Emmanuel},
title = {Revisiting the {ABC} flow dynamo},
journal = {Phys. Fluids},
volume = {25},
number = {3},
year = {2013},
pages = {037103},
doi = {10.1063/1.4795546},
eprint = {1206.5186},
archivePrefix = {arXiv},
}
@misc{cotizelati2026periodic,
author = {Coti Zelati, Michele and Sorella, Massimo and Villringer, David},
title = {A fast dynamo on the three-torus},
year = {2026},
eprint = {2603.09861},
archivePrefix = {arXiv},
}
@misc{cotizelati2026autonomous,
author = {Coti Zelati, Michele and Sorella, Massimo and Villringer, David},
title = {Smooth autonomous fast dynamo action on the three-torus},
year = {2026},
eprint = {2609.04153},
archivePrefix = {arXiv},
}
@misc{sorella2025,
author = {Sorella, Massimo and Villringer, David},
title = {A limsup fast dynamo on $\mathbb{T}^3$},
year = {2025},
eprint = {2511.23024},
archivePrefix = {arXiv},
}
@misc{aim-abc-fast-dynamo,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Fast dynamo action of the unit ABC flow}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/218-abc-fast-dynamo.md},
note = {Entry 218 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00572, "Fast dynamo action of the unit ABC flow", clopen.solutions.