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Peer-reviewedCC BYStochastic Partial Differential Equations Analysis and Computations

On the Itô-Stratonovich diffusion limit for the magnetic field in a 3D thin domain

  • Federico Butori1
  • Franco Flandoli1
  • Eliseo Luongo2
  1. 1Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
  2. 2Fakultät für Mathematik, Universität Bielefeld, 33501 Bielefeld, Germany
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Abstract

We introduce a stochastic model for a passive magnetic field in a three dimensional thin domain. The velocity field, white in time and modelling phenomenologically a turbulent fluid, acts on the magnetic field as a transport-stretching noise. We prove, in a quantitative way, that, in the simultaneous scaling limit of the thickness of the thin layer and the separation of scales, the mean in the thin direction of the magnetic field is close to the solution of the equation for the magnetic field with additional dissipation. For certain choices of noises with correlations between their components, without mirror symmetry and with a non zero mean helicity, we identify, in the limit, a first-order term, in addition to the extra dissipation term. However, in the limit system this term does not produce growth of the L2 L 2 norm, which is commonly referred to in the literature as the dynamo effect. Consequently, we extend a no-dynamo theorem to thin layers.

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Cite

BibTeXRIS
BibTeX
@article{butori2026itôstratonovich,
  title = {{On the Itô-Stratonovich diffusion limit for the magnetic field in a 3D thin domain}},
  author = {Federico Butori and Franco Flandoli and Eliseo Luongo},
  journal = {Stochastic Partial Differential Equations Analysis and Computations},
  year = {2026},
  month = may,
  doi = {10.1007/s40072-026-00427-x},
  url = {https://doi.org/10.1007/s40072-026-00427-x},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - On the Itô-Stratonovich diffusion limit for the magnetic field in a 3D thin domain
AU  - Federico Butori
AU  - Franco Flandoli
AU  - Eliseo Luongo
PY  - 2026
DA  - 2026/05/29
JO  - Stochastic Partial Differential Equations Analysis and Computations
DO  - 10.1007/s40072-026-00427-x
UR  - https://doi.org/10.1007/s40072-026-00427-x
AB  - We introduce a stochastic model for a passive magnetic field in a three dimensional thin domain. The velocity field, white in time and modelling phenomenologically a turbulent fluid, acts on the magnetic field as a transport-stretching noise. We prove, in a quantitative way, that, in the simultaneous scaling limit of the thickness of the thin layer and the separation of scales, the mean in the thin direction of the magnetic field is close to the solution of the equation for the magnetic field with additional dissipation. For certain choices of noises with correlations between their components, without mirror symmetry and with a non zero mean helicity, we identify, in the limit, a first-order term, in addition to the extra dissipation term. However, in the limit system this term does not produce growth of the L^2 L 2 norm, which is commonly referred to in the literature as the dynamo effect. Consequently, we extend a no-dynamo theorem to thin layers.
SN  - 2194-0401
SN  - 2194-041X
ER  -