On the It\^o-Stratonovich Diffusion Limit for the Magnetic Field in a 3D Thin Domain

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.