Michael Roop, Mathematics, Chalmers University of Technology Title: Lie-Poisson discretization and long time behaviour of ideal magnetohydrodynamics on the sphere

Abstract: An important class of models arising in fluid mechanics represents PDEs formulated as geodesic flows on the group of diffeomorphisms of some manifold. They are usually referred to as Euler-Arnold equations. Their Hamiltonian nature suggests that they have a lot of (infinitely many) conserved quantities, such as energy, and Casimir functions associated to the underlying Lie-Poisson structure. A natural approach to discretization of such equations is to preserve those conservation laws. Preservation of Casimir functions is known to be essential for long time simulations of fluids. In this talk, we will address one example of such equations, the system of incompressible magnetohydrodynamics (MHD) equations on the sphere, and will present its fully discrete analogue that completely preserves the underlying Lie-Poisson structure. In particular, the numerical method exactly preserves the Casimirs and nearly preserves the Hamiltonian in the sense of backward error analysis. The method is applied to two models describing the motion of magnetized ideal fluids, reduced MHD (RMHD) model, and Hazeltine’s model (also called Alfvén wave turbulence equations). Numerical simulations reveal the formation of large scale vortex blob structures in the long time behaviour.

This is a joint work with Klas Modin, arXiv:2311.16045.