Hamiltonian formulation and matrix discretization for axisymmetric magnetohydrodynamics

Equations of ideal magnetohydrodynamics (MHD) play an important role in the studies of turbulence, astrophysics, and plasma physics. These equations possess remarkable geometric structures and symmetries. Indeed, they admit a geodesic formulation in the sense of Arnold, as a Lie--Poisson flow on the dual of an infinite-dimensional Lie algebra. Zeitlin's model, previously developed for MHD on the flat torus and the two-sphere, is a matrix approximation of MHD consistent with the underlying geometric structures. In this paper, we derive the reduced model of axially symmetric magnetohydrodynamics on the three-sphere and give its Hamiltonian formulation. We further extend finite dimensional Zeitlin's matrix model for MHD from 2D to axially symmetric 3D flows of magnetized fluids, yielding the first discrete model for 3D magnetohydrodynamics compatible with the underlying Lie--Poisson structure.