On Formulating Kinetic Models (and Solvers) for Plasmas on Manifolds and in Strong Magnetic Fields

Many fusion and laboratory plasmas are magnetized. The presence of the magnetic field qualitatively changes the physics as the field introduces a prefered direction along which the particles can flow. Historically, a large number of models, the most famous being gyrokinetics, have been developed that exploit the magnetized nature of the plasma. We will show an approach to formulating such models, starting with an Hamiltonian forumlation of gyrokinetics. We will describe a model in which the kinetic equations are transformed to a coordinate frame moving with the mean velocity of the particles and aligned with the local, time-evolving, magnetic field. The distribution function in this frame is then expanded in a Fourier-Laguerre series, reducing the 6D Vlasov equation to a sequence of 4D equations for the Fourier-Laguerre coefficients. We call this spectrally expanded set of 4D equations the Parallel Kinetic Perpendicular Moment model. The physics content of this system of equations will be presented, specially when truncated at a few Fourier and Laguerre modes. An alternative closure-based approach to incorporating effects of agyrotopy will be present. Finally, a formulation of kinetic plasma flows on manifolds will be presented. This formulation enables new numerics to study detailed kinetic physics around compact objects like black-holes and neutron stars.