On the Itô-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 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 $L^2$ norm, which is commonly referred to in the literature as the dynamo effect. Consequently, we extend a no-dynamo theorem to thin layers.