Theoretical Plasma Physics

Abstract

Lattice Boltzmann algorithms are a mesoscopic method to solve problems in nonlinear physics which are highly parallelized – unlike the direction solution of the original problem. These methods are applied to both fluid and magnetohydrodynamic turbulence. By introducing entropic constraints one can enforce the positive definiteness of the distribution functions and so be able to simulate fluids at high Reynolds numbers without numerical instabilities. By introducing a vector distribution function for the magnetic field one can enforce the divergence free condition on the magnetic field automatically, without the need of divergence cleaning as needed in most direct numerical solutions of the resistive magnetohydrodynamic equations. The principal reason for the high parallelization of lattice Boltzmann codes is that they consist of a kinetic collisional relaxation step (which is purely local) followed by a simple shift of the relaxed data to neighboring lattice sites. In large eddy simulations, the closure schemes are highly nonlocal – the most famous of these schemes is that due to Smagorinsky. Under a lattice Boltzmann representation the Smagorinsky closure is purely local – being simply a particular moment on the perturbed distribution fucntions. After nonlocal fluid moment models were discovered to represent Landau damping, it was found possible to model these fluid models using an appropriate lattice Boltzmann algorithm. The close to ideal parallelization of the lattice Boltzmann codes permitted us to be Gordon Bell finalists on using the Earth Simulation in Japan. We have also been involved in the radio frequency propagation of waves into a tokamak and into a spherical overdense tokamak plasma. Initially we investigated the use of a quasi-optical grill for the launching of lower hybrid waves into a tokamak. It was found that the conducting walls do not prevent the rods from being properly irradiated, the overloading of the quasi-optical grill is not severe with the electric field only being about three times higher than in the ideal case. Moreover, the quasi-optical grill was significantly fewer structural elements that the multijunction grill. Nevertheless there has not been much interest from experimental fusion groups to implementing these structures. Hence we have returned to optimizing the multijunction grill so that the large number of coupling matrix elements can be efficiently evaluated using symmetry arguments. In overdense plasmas, the standard electromagnetic waves cannot propagate into the plasma center, but are reflected at the plasma edge. By optimizing mode conversion processes (in particular, the O-X-B wave propagation of Ordinary Mode converting to an Extraordinary mode which then converts into an electrostatic Bernstein wave) one can excite within the plasma an electrostatic Bernstein wave that does not suffer density cutoffs and is absorbed on the electron cyclotron harmonics. Finally we have started looking at other mesoscopic lattice algorithms that involve unitary collision and streaming steps. Because these algorithms are unitary they can be run on quantum computers when they become available – unlike their computational cousin of lattice Boltzmann which is a purely classical code. These quantum lattice gas algorithms have been tested successfully on exact analytic soliton collision solution. These calculations are hoped to be able to study Bose Einstein condensed atomic gases and their ground states in an optical lattice.