Deterministic and Stochastic Geometric Mechanics for Hall MHD

We derive new models of stochastic Hall magnetohydrodynamics (MHD) by using a symmetry-reduced stochastic Euler-Poincar\'e variational principle. The new stochastic Hall MHD theory has potential applications for uncertainty quantification and data assimilation in space plasma (space weather) and solar physics. The stochastic geometric mechanics approach we take here produces coordinate-free results which may then be applied in a variety of spatial configurations.