Spatio-temporal Lie-Poisson discretization for incompressible magnetohydrodynamics on the sphere

We give a structure preserving spatio-temporal discretization for incompressible magnetohydrodynamics (MHD) on the sphere. Discretization in space is based on the theory of geometric quantization, which yields a spatially discretized analogue of the MHD equations as a finite-dimensional Lie--Poisson system on the dual of the magnetic extension Lie algebra $\mathfrak{f}=\mathfrak{su}(N)\ltimes\mathfrak{su}(N)^{*}$. We also give accompanying structure preserving time discretizations for Lie--Poisson systems on the dual of semidirect product Lie algebras of the form $\mathfrak{f}=\mathfrak{g}\ltimes\mathfrak{g^{*}}$, where $\mathfrak{g}$ is a $J$-quadratic Lie algebra. Critically, the time integration method is free of computationally costly matrix exponentials. The full method preserves the underlying geometry, namely the Lie--Poisson structure and all the Casimirs, and nearly preserves the Hamiltonian function in the sense of backward error analysis. To showcase the method, we apply it to two models for magnetic fluids: incompressible magnetohydrodynamics and Hazeltine's model. For the latter, our simulations reveal the formation of large scale vortex condensates, indicating a backward energy cascade analogous to two-dimensional turbulence.