Electron Parallel Closures for Arbitrary Collisionality

Electron parallel closures for heat flow, viscosity, and friction force are expressed as kernel-weighted integrals of thermodynamic drives, the temperature gradient, relative electron-ion flow velocity, and flow-velocity gradient. Simple, fitted kernel functions are obtained for arbitrary collisionality from the 6400 moment solution and the asymptotic behavior in the collisionless limit. The fitted kernels circumvent having to solve higher order moment equations in order to close the electron fluid equations. For this reason, the electron parallel closures provide a useful and general tool for theoretical and computational models of astrophysical and laboratory plasmas

Computational Methods in Modeling Fusion Plasmas

Fusion provides an attractive potential alternative to using fossil fuels for energy. Fusion requires vastly less fuel resources than does current non-renewable energy processes (virtually a 100% reduction in the required mass of fuel needed). The fuel sources needed (mainly deuterium and lithium) are also highly abundant on the Earth and fusion generates minimal waste products. One of the biggest obstacles to practical fusion energy is how to contain the reactants long enough for energy output to significantly exceed energy input. The equations governing plasma dynamics and confinement are highly nonlinear and do not admit simple analytic solutions in realistic situations. To obtain predictions of various plasma confinement scenarios, it is often necessary to turn to other means, such as computational modeling, to simulate the relevant plasma dynamics. Evaluating the effectiveness and reliability of the computational methods used for simulation then becomes extremely important, especially when subsequently using your code to predict new physics to the scientific community. In this work, we present an effort to analyze the effectiveness of one of the computational techniques used in the NIMROD code, which code Eric Held (USU) and others in the scientific community have helped to develop. This method involves resolving something called the Grad-Shafranov equation, which governs the potential plasma equilibria that can exist in tokamak plasmas. Here we evaluate the effectiveness of the method and discuss the potential implications resulting from this analysis

Ion Parallel Closures

Ion parallel closures are obtained for arbitrary atomic weights and charge numbers. For arbitrary collisionality, the heat flow and viscosity are expressed as kernel-weighted integrals of the temperature and flow-velocity gradients. Simple, fitted kernel functions are obtained from the 1600 parallel moment solution and the asymptotic behavior in the collisionless limit. The fitted kernel parameters are tabulated for various temperature ratios of ions to electrons. The closures can be used conveniently without solving the kinetic equation or higher order moment equations in closing ion fluid equations

Electron Heat Flow Due to Magnetic Field Fluctuations

Radial heat transport induced by magnetic field line fluctuations is obtained from the integral parallel heat flow closure for arbitrary collisionality. The parallel heat flow and its radial component are computed for a single harmonic sinusoidal field line perturbation. In the collisional and collisionless limits, averaging the heat flow over an unperturbed surface yields Rechester-Rosenbluth like formulae with quantitative factors. The single harmonic result is generalized to multiple harmonics given a spectrum of small magnetic perturbations. In the collisionless limit, the heat and particle transport relations are also derived. © 2016 IOP Publishing Ltd

Moment-Fourier approach to ion parallel fluid closures and transport for a toroidally confined plasma

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 are 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 density, temperature, and flow velocity perturbations in terms of radial gradients of equilibrium 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.Comment: 25 pages, 9 figure

Electron Parallel Transport for Arbitrary Collisionality

Integral (nonlocal) closures [J.-Y. Ji and E. D. Held, Phys. Plasmas 21, 122116 (2014)] are combined with the momentum balance equation to derive electron parallel transport relations. For a single harmonic fluctuation, the relations take the same form as the classical Spitzer theory (with possible additional terms): the electric current and heat flux densities are connected to the modified electric field and temperature gradient by transport coefficients. In contrast to the classical theory, the dimensionless coefficients depend on the collisionality quantified by a Knudsen number, the ratio of the collision length to the angular wavelength. The key difference comes from the proper treatment of the viscosity and friction terms in the momentum balance equation, accurately reflecting the free streaming and collision terms in the kinetic equation. For an arbitrary fluctuation, the transport relations may be expressed by a Fourier series or transform. For low collisionality, the electric resistivity can be significantly larger than that of classical theory and may predict the correct timescale for fast magnetic reconnection