Geometric discretizations in hydrodynamics: from plasma physics to thermal quasi-geostrophy

Many physical processes are modeled by partial differential equations (PDE), and their efficient discretization is a challenging problem and an active field. A common class of models arising in mathematical physics are PDEs formulated in terms of a Lie-Poisson structure on the dual of infinite-dimensional Lie algebras, such as the Lie algebra of vector fields. They are referred to as Euler-Arnold systems. In the present thesis, an important subclass of such equations is addressed, namely equations of incompressible magnetohydrodynamics (MHD) and thermal quasi-geostrophy (TQG) on the sphere. The thesis comprises four papers.In the first paper, a spatio-temporal discretization of MHD on the sphere is developed. The method fully preserves the underlying Lie-Poisson structure. Space discretization is based on truncation of the Lie-Poisson structure and yields a finite-dimensional Lie-Poisson system. Further, a structure preserving time integrator is developed. This integrator exactly preserves all the Casimirs and nearly preserves the Hamiltonian function in the sense of backward error analysis of symplectic integrators.In the second paper, the developed structure preserving discretization is applied to Hazeltine's model of 2D turbulence in magnetized plasma and its two limiting cases, the reduced MHD (RMHD) model and the Charney-Hasegawa-Mima (CHM) model. Simulations reveal the formation of large-scale coherent structures in the long time behavior of some fields, and small scales in other fields, which indicates the presence of both inverse and direct cascades of the conserved quantities.In the third paper, the global model for thermal quasi-geostrophy (TQG) is developed and its Hamiltonian structure is given. Structure preserving spatio-temporal discretization developed for MHD is adapted for TQG, and the long time behavior is studied.In the fourth paper, the reduced model of axially symmetric magnetohydrodynamics on the three-sphere is derived and its Hamiltonian formulation is given. The finite dimensional Zeitlin's matrix model is extended 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.