Herein, we formulate, analyze, and apply a numerical method for solving a Chapman-Enskog-like (CEL) continuum kinetic model for plasmas. It is shown that centering the heat flux at the beginning of the time step and the ion temperature at the end of the time step in the kinetic equation allows for a numerically-stable time advance of the coupled fluid-kinetic system. In addition, it is shown that numerical stability is impossible to achieve without explicitly enforcing key tenets of the CEL closure approach, in particular, that the number density (n), flow (u), and temperature (T) moments of the kinetic distortion remain small in time. We show that with a method to constrain these moments, it is possible to remove both the numerical growth and numerical damping from the linear modes. We apply the results from the linear stability analysis to allow for a numerically-stable fully nonlinear axisymmetric evolution of profiles in NIMROD, wherein we observe the asymptotic evolution of the flow in a DIII-D tokamak equilibrium (based on DIII-D ITER Baseline Scenario (IBS) discharge 174446 at 3390 ms). We compare the self-consistently computed results to analytics and to results from a previously benchmarked fixed-background delta f implementation in NIMROD. Agreement with prediction is found for both the dynamics and asymptotics of the flow. This work demonstrates the first successful published benchmarking of the full CEL approach in a plasma fluid code.

A general method of solving the drift kinetic equation is developed for an axisymmetric magnetic field. Expanding a distribution function in general moments, a set of ordinary differential equations is obtained. Successively expanding the moments and magnetic-field involved quantities in Fourier series, a set of linear algebraic equations is obtained. The set of full (Maxwellian and non-Maxwellian) moment equations is solved to express the first-order density, temperature, and flow velocity in terms of radial gradients of the zeroth-order pressure and temperature. Closure relations that connect parallel heat flux density and viscosity to the radial gradients and parallel gradients of temperature and flow velocity are also obtained by solving the non-Maxwellian moment equations. The closure relations combined with the linearized fluid equations reproduce the same solution obtained directly from the full moment equations. The method can be generalized to derive closures and transport for an electron-ion plasma and a multi-ion plasma in a general magnetic field.

Exact moments of the Boltzmann collision operator are calculated in the irreducible Hermitian moment expansion written in terms of the random-velocity variable of each species. The formulas are presented in closed, algebraic form and can be straightforwardly implemented in computer algebra systems. They are valid for two arbitrary masses, temperatures, and flow velocities, and hence include all other existing results derived for distribution functions expanded with respect to reference states of one temperature and flow velocity. In comparison, the Landau collisional moments are good approximations for large Coulomb logarithm and small relative flow velocity, but they fail to predict the correct behavior of most collisional moments for large relative flow even for weakly coupled plasmas.

Nuclear fusion converts the rest mass energy of ions like the deuteron and triton into kinetic energy. In theory, this energy can be harvested from a thermonuclear reactor to provide power outputs on the scale of 500 MW. High temperatures are needed for significant fusion to occur, hence the tokamak (a modern fusion confinement device) employs strong magnetic fields to keep the ionized gas (plasma) away from the tokamak wall. However, a problem that occasionally arises in a tokamak is that during a disruption an inductive electric field is created which can accelerate electrons to relativistic speeds (these electrons are called runaway electrons (RE’s)). The magnetic field lines also become stochastic (volume-filling) and can intersect with the wall. Following magnetic field lines, RE’s are led to and can obliterate the expensive plasma facing components (PFC’s). This work involves getting the physics of RE’s into NIMROD, a plasma-fluid code. Overall, this is a very large and complex problem so we will focus on seeing if NIMROD’s relativistic electron model agrees with other 2D phase-space codes in terms of how the electrons get accelerated. This involves looking at the different processes: acceleration by the inductive electric field, the drag force associated with colliding off the background plasma, and the release of energy through synchrotron radiation. Considering all of this we want to see if NIMROD can predict/calculate a balance of these forces to lead to a steady state vortex pattern in the relativistic phase-space.

We show that the energy, force, and torque between two spherically symmetric multipole density distributions are identical to those between two point multipoles, and apply this point-sphere equivalence to coated spherical dipole magnets. We also show that the potential and field of such a distribution are equivalent to those due to point multipoles located at the center of the distribution. We expand the inverse-distance potential in terms of harmonic (Hermite irreducible) tensors, whose properties enable us to express the potential energy, force, and torque for two arbitrary source distributions in a series of point-multipole interactions. This work generalizes recent work on interactions between uniformly magnetized dipole spheres [B. F. Edwards, D. M. Riffe, J.-Y. Ji, and W. A. Booth, Am. J. Phys. 85, 130 (2017)] to interactions between spherically-symmetric multipole spheres.

The magnetized resistivity and electrothermal tensors when substituted into the induction equation lead to electrothermal magnetic field generation, resistive magnetic diffusion, and magnetic field advection due to resistivity gradients, temperature gradients, and currents. The advection terms driven by the temperature gradient and current have cross field components (perpendicular to both the magnetic field and the driving term) that depend on significantly modified versions of Braginskii's transport coefficients [S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontovich (Consultants Bureau, New York, 1965), Vol. 1, p. 205]. The improved fits to Braginskii's coefficients given by Epperlein and Haines [Phys. Fluids 29, 1029 (1986)] and Ji and Held [Phys. Plasmas 13, 042114 (2013)] give physically incorrect results for cross field advection at small Hall parameters (product of cyclotron frequency and collision time). The errors in Epperlein and Haines' fits are particularly severe, giving increasing advection velocities below a Hall parameter of one when they should decrease linearly to zero. Epperlein and Haines' fits can also give erroneous advection terms due to variations in the effective atomic number. The only serious error in Braginskii's fits is an overestimate in advection due to perpendicular resistivity. New fits for the cross field advection terms are obtained from a direct numerical solution of the Fokker–Planck equation and Ji and Held's higher order expansion approach that are continuous functions of the effective atomic number.

In this work, continuum kinetic formulations are employed as a mechanism to include closure physics in an extended magnetohydrodynamics model. Two continuum kinetic approaches have been implemented in the plasma fluid code NIMROD [Sovinec et al., “Nonlinear magnetohydrodynamics with high-order finite elements,” J. Comput. Phys. 195, 355 (2004)] including a Chapman–Enskog-like (CEL) formulation and a more conventional δf approach. Ion kinetic closure schemes are employed to describe the neoclassical flow properties in axisymmetric toroidal geometry. In particular, predictions for steady-state values of poloidal flow profiles in tokamak geometry are provided using both the δf formulation and two different solution techniques for the CEL approach. These results are benchmarked against analytic theory predictions as well as results from the drift kinetic code DK4D. The continuum kinetic formulations employed here show agreement with both the analytic theory and DK4D results, and offer a novel velocity space representation involving higher-order finite elements in pitch angle.

The Maxwell-Boltzmann statistics of the quantum ideal gas is studied through the canonical partition function by exactly counting discrete quantum states without the continuum approximation. Analytic expressions for energy, pressure, entropy, and heat capacity are expressed in terms of Jacobi theta functions and complete elliptic integrals. The results show typical effects of discrete energy levels in the low temperature limit while they reproduce thermodynamics of the classical ideal gas in the high temperature limit.

Exact moments of the Landau collision operator are calculated for the irreducible Hermite polynomials written in terms of the random-velocity variable. We present closed, algebraic formulas which can be implemented in computer algebra systems. The formulas reproduce the results for the total-velocity moment expansion (J-Y Ji and E D Held, 2006 Phys. Plasmas 13 102 103) and for the random-velocity moment expansion with the small mass-ratio approximation (J-Y Ji and E D Held, 2008 Phys. Plasmas 15 102 101). For verification of the formulas, example calculations for several lowest order moments are presented. The collisional moments can be applied in the derivations of Braginskii and integral (nonlocal) closures for arbitrary relative flow velocity between electrons and ions.

A novel numerical method is employed to compute the integral form of the axi-symmetric Trubnikov-Rosenbluth potentials. Two methods for quadrature in pitch-angle are described and their convergence properties are studied. Careful attention is given to quadrature over a singular Green's function. It is shown that an infinite series representation of the Green's function can be used more efficiently than its closed form involving complete elliptic integrals. Then a collocation method in speed, with its associated quadrature scheme, is laid out and its convergence properties are studied. Using the proposed scheme, accurate low-order moments of the field collision operator are obtained using relatively few velocity space degrees of freedom. The scheme is showcased by solving for the equilibrium, axi-symmetric bootstrap current in tokamaks. A C0 Gauss-Lobatto-Legendre finite element pitch-angle basis with vertex nodes at the trapped/passing boundary is shown, in the context of the integral methods used, to be much more efficient than the more common Legendre polynomial expansion.

This final report pertains to work performed at Utah State University (USU) on the subject of kinetic closures for fluid simulations of high-energy-density (HED) plasmas. This report was written following the end date of the three year grant that started August 1, 2016, and ended July 31, 2019. The key aspect of the work at USU was to develop and implement kinetic closures for fluid simulations of more strongly coupled plasmas. Funds for this project provided some summer salary for the PI at USU, Dr. Eric Held, but mainly provided a robust stipend for his graduate student, David Hansen, who has worked on the project since the start date. Mr. Hansen is nearing completion of his PhD thesis which he will defend either in December of 2019 or the spring of 2020. With this award Dr. Eric Held and his PhD student David Hansen developed a computer code called {\bf X-section} that can be used to generate collisional scattering cross sections which summarize particle interactions in high-energy-density plasmas.