Stochastic Lagrangian for the two-dimensional visco-resistive magnetohydrodynamics

Magnetohydrodynamics in two dimensions is of particular use in all those contexts where the plasma is strongly magnetized in a given direction, so that its relevant dynamics takes place almost only on the plane perpendicular to the strong 'guiding' field. This approach, referred to as reduced MHD, is of great use for describing the fusion plasma in tokamaks, due to the presence of a strong toroidal field, or the solar plasma, undergoing the action of strong fields emerging from the photosphere. In turbulent conditions, where the medium properties are expected to be irregular, the dissipative terms of viscosity and resistivity can be interpreted as forcing terms of stochastic nature, so as to mimic the large fluctuations of microscopic quantities. The statistical dynamics of the system can then be encoded in a suitably constructed stochastic Lagrangian. In this paper, we obtain the stochastic Lagrangian for the 2D viscoresistive MHD model, where the scalar vorticity omega of the plasma and the axial component of the vector potential psi are used as dynamical variables. In the case of Gaussian delta-correlated noises the stochastic Lagrangian is completely calculated, and some possible developments of these results are discussed.