Quasi-geostrophic MHD equations: Hamiltonian formulation and nonlinear stability

Magnetic fields in stars and planets are generated by a dynamo process that results from multi-scale interactions of the flows in conducting fluids. On the large scales, these flows are dominated by a strong zonal component, while the magnetic fields exhibit a strong toroidal/zonal character. Although dissipation certainly acts on these flows, the kinematic and magnetic viscosities associated with these large-scale flows are small, so that, over the timescale of several years and beyond, the system may be modelled as a conservative one. In this context, the Hamiltonian formulation may give several insights, providing a systematic way to relate the symmetries of the system with conservation laws. In the present article, we introduce the Hamiltonian formulation for a model that reasonably describes the dynamics of large-scale flows in stars and planets: the two-dimensional magnetohydrodynamic quasi-geostrophic equations. In this context, we find the invariants of the system, which are of two kinds: the Casimirs, related to the particle relabelling symmetry, and the zonal momentum, which is related to the translational invariance in the zonal direction. We then use these invariants to study the stability of some stationary solutions that are relevant for geophysical and astrophysical applications.