Diffusion Limit for a Stochastic Model of a Magnetic Field in a 3D Thin Domain

Butori Federico, Scuola Normale Superiore
Luongo Eliseo, Scuola Normale Superiore

We deal with a model for a passive magnetic field generated by the turbulent motion of a conducting fluid in a 3D thin layer. This model is a perturbation of a large scale 2D flow which incorporates 3D turbulent features at scales much smaller (the thin direction) than the typical one for the system. By idealizing the dynamics of the turbulent fluid with a white-in-time noise, the inductive equation for the passive magnetic field reduces to a linear stochastic PDE with noise in form of transport plus stretching. We investigate the behavior of the system as the noise concentrates at higher and higher frequencies and the layer thickness goes to zero. In particular, we consider the average in the thin direction of the solutions and prove (quantitatively) that they converge, in a suitable topology and under a suitable scaling of the noise, to a 3D vector field over a 2D layer, which solves a PDE with additional viscosity and possibly an additional first order term related to the so called AKA (anisotropic kinetic alpha) effect. The presence of the stretching part of the noise, which would be zero in the 2D case, is a typical feature of the 3D geometry, and requires a careful control since it provokes larger and larger stress as the frequency of the noise increases. Finally, since this stochastic inductive equation for the magnetic field, correspond to the linear part of the 3D Navier-Stokes equation in vorticity form perturbed by transport and stretching noise, our work is a a preliminar step towards the investigation of a more realistic description of small scales turbulence in almost 2D fluids, such as those arising in the study of the atmosphere or the oceans. This talk is based on a joint work with F. Flandoli and E. Luongo.

Area: CS11 - Stochastic Geophysical Fluid Dynamics (Antonio Agresti and Giulia Carigi)

Keywords: SPDEs, Magnetic Field, Eddy Viscosity, Turbulence, Scaling Limit

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