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On the Itô-Stratonovich Diffusion Limit for the Magnetic Field in a 3D Thin Domain

Federico Butori, Franco Flandoli, Eliseo Luongo

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 on the thin direction of the magnetic field is close to the solution of the equation of the magnetic field with additional dissipation. In certain choice of noises with correlation between their components, without mirror symmetry and with a non zero mean helicity, we identify an alpha-term, in addition to the extra dissipation term. However, it does not produce dynamo; consequently, we extend a no-dynamo theorem to thin layers.

On the Itô-Stratonovich Diffusion Limit for the Magnetic Field in a 3D Thin Domain

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 on the thin direction of the magnetic field is close to the solution of the equation of the magnetic field with additional dissipation. In certain choice of noises with correlation between their components, without mirror symmetry and with a non zero mean helicity, we identify an alpha-term, in addition to the extra dissipation term. However, it does not produce dynamo; consequently, we extend a no-dynamo theorem to thin layers.
Paper Structure (14 sections, 20 theorems, 260 equations)

This paper contains 14 sections, 20 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Comments on the Physical motivation
  3. Functional Setting and Main Results
  4. Notation
  5. Preliminaries
  6. Function Spaces in the Periodic Setting
  7. Preliminaries on the Thin Layer
  8. Description of the Stochastic System
  9. Main Results
  10. Strategy of the Proof
  11. A Priori Estimates
  12. Proof of \ref{['main Theorem']}
  13. Proof of \ref{['lemma ito strat corrector']}
  14. Some physical remarks

Key Result

Lemma 2.1

Let $q\in \Dot{H}^{\alpha}(\mathcal{D}),\ \alpha\in \mathbb{R}$. Then:

Theorems & Definitions (44)

  • Lemma 2.1
  • Definition 2.2
  • Lemma 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • proof
  • ...and 34 more