Eric Held

NIMROD: A computational laboratory for studying nonlinear fusion magnetohydrodynamics

Nonlinear numerical studies of macroscopic modes in a variety of magnetic fusion experiments are made possible by the flexible high-order accurate spatial representation and semi-implicit time advance in the NIMROD simulation code [A. H. Glasser et al., Plasma Phys. Controlled Fusion 41, A747 (1999)]. Simulation of a resistive magnetohydrodynamics mode in a shaped toroidal tokamak equilibrium demonstrates computation with disparate time scales, simulations of discharge 87009 in the DIII-D tokamak [J. L. Luxon et al., Plasma Physics and Controlled Nuclear Fusion Research 1986 (International Atomic Energy Agency, Vienna, 1987), Vol. I, p. 159] confirm an analytic scaling for the temporal evolution of an ideal mode subject to plasma-β increasing beyond marginality, and a spherical torus simulation demonstrates nonlinear free-boundary capabilities. A comparison of numerical results on magnetic relaxation finds the n=1 mode and flux amplification in spheromaks to be very closely related to the m=1 dynamo modes...

Computational modeling of fully ionized magnetized plasmas using the fluid approximation

Strongly magnetized plasmas are rich in spatial and temporal scales, making a computational approach useful for studying these systems. The most accurate model of a magnetized plasma is based on a kinetic equation that describes the evolution of the distribution function for each species in six-dimensional phase space. High dimensionality renders this approach impractical for computations for long time scales. Fluid models are an approximation to the kinetic model. The reduced dimensionality allows a wider range of spatial and∕or temporal scales to be explored. Computational modeling requires understanding the ordering and closure approximations, the fundamental waves supported by the equations, and the numerical properties of the discretization scheme. Several ordering and closure schemes are reviewed and discussed, as are their normal modes, and algorithms that can be applied to obtain a numerical solution.

Nonlocal closures for plasma fluid simulations

The application of fluid models in studies of transport and macroscopic stability of magnetized, nearly collisionless plasmas requires closure relations that are inherently nonlocal. Such closures address the fact that particles are capable of carrying information over macroscopic parallel scale lengths. In this work, generalized closures that embody Landau, collisional and particle-trapping physics are derived and discussed. A gyro/bounce-averaged drift kinetic equation is solved via an expansion in eigenfunctions of the pitch-angle scattering operator and the resulting system of algebraic equations is solved by integrating along characteristics. The desired closure moments take the form of integral equations involving perturbations in the flow and temperature along magnetic field lines. Implementation of the closures in massively parallel plasma fluid simulation codes is also discussed. This implementation includes the use of a semi-implicit time advance of the fluid equations to stabilize the dominant ...

Exact linearized Coulomb collision operator in the moment expansion

In the moment expansion, the Rosenbluth potentials, the linearized Coulomb collision operators, and the moments of the collision operators are analytically calculated for any moment. The explicit calculation of Rosenbluth potentials converts the integro-differential form of the Coulomb collision operator into a differential operator, which enables one to express the collision operator in a simple closed form for any arbitrary mass and temperature ratios. In addition, it is shown that gyrophase averaging the collision operator acting on arbitrary distribution functions is the same as the collision operator acting on the corresponding gyrophase averaged distribution functions. The moments of the collision operator are linear combinations of the fluid moments with collision coefficients parametrized by mass and temperature ratios. Useful forms involving the small mass-ratio approximation are easily found since the collision operators and their moments are expressed in terms of the mass ratio. As an application, the general moment equations are explicitly written and the higher order heat flux equation is derived.

Conductive electron heat flow along magnetic field lines

In this work, a unified closure for the conductive electron heat flux along magnetic field lines is derived and examined. Both free-streaming and collisional pitch-angle scattering of electrons are present in the drift kinetic equation which is solved using an expansion in pitch-angle eigenfunctions (Legendre polynomials). The closure takes the form of a generic integral operator involving the electron temperature variation along a magnetic field line and the electron speed. Derived for arbitrary collisionality, the heat flux closure may be written in forms resembling previous collisional and collisionless expressions. Electrons with two to three times the thermal speed are shown to carry heat for all collisionalities and thermal electrons make an important contribution to the heat flow in regimes of moderate to low collisionality. As a practical application, the flow of electron heat along a chaotic magnetic field is calculated in order to highlight the nonlocal nature of the closure which allows for heat to flow against local temperature gradients.

Moment approach to deriving parallel heat flow for general collisionality

In the moment approach, a parallel electron heat flux density is obtained for arbitrary collisionality in the transport ordering. The parallel moment equations are derived from the drift kinetic equation with the exact linearized Landau operator and analytically solved for the heat flux in integral form. Quantitative analysis of the integral heat flux for sinusoidal temperature profiles shows that the number of moments required for convergence increases as collision length increases. The integral heat flux well agrees with the Braginskii heat flux and the collisionless heat flux for high and low collisionalities, respectively. Incorrect application of the Braginskii heat flux in moderately collisional or nearly collisionless plasmas often leads higher heat flux than the integral closure. This fact is consistent with numerical studies of electron heat confinement in the sustained spheromak physics experiment [Hooper et al., Nucl. Fusion 39, 863 (1999)] which show a modest (6%) increase in core temperature ...

Closure and transport theory for high-collisionality electron-ion plasmas

Systems of algebraic equations for a high-collisionality electron-ion plasma are constructed from the general moment equations with linearized collision operators [J.-Y. Ji and E. D. Held, Phys. Plasmas 13, 102103 (2006) and J.-Y. Ji and E. D. Held, Phys. Plasmas 15, 102101 (2008)]. A systematic geometric method is invented and applied to solve the system of equations to find closure and transport relations. It is known that some closure coefficients of Braginskii [S. I. Braginskii, Reviews of Plasma Physics (Consultants Bureau, New York, 1965), Vol. 1] are in error up to 65% for some finite values of x (cyclotron frequency × electron-ion collision time) and have significant error in the large-x limit [E. M. Epperlein and M. G. Haines, Phys. Fluids 29, 1029 (1986)]. In this work, fitting formulas for electron coefficients are obtained from the 160 moment (Laguerre polynomial) solution, which converges with increasing moments for x≤100 and from the asymptotic solution for large x-values. The new fitting fo...

Landau collision operators and general moment equations for an electron-ion plasma

The general moment equations for an electron-ion plasma are established. The distribution functions for electrons and ions are expanded in terms of orthogonal polynomials of random velocity variables in contrast to the total velocity variables [J.-Y. Ji and E. D. Held, Phys. Plasmas 13, 102103 (2006)]. The moments of the streaming part of the kinetic equation are explicitly written with simple formulas. A simple version of the exact linearized Coulomb collision integrals is presented for like species. The electron-ion and ion-electron operators that conserve momentum and energy are also calculated with a small mass-ratio approximation. It is shown in the relaxation theory that the Lorentz operator, as a replacement of the like-species operator, is acceptable only for the high-order harmonic moments.

Conductive electron heat flow along an inhomogeneous magnetic field

In this work a general closure for the conductive electron heat flux parallel to an inhomogeneous magnetic field is derived and examined. Free-streaming and collisional effects are present in the drift kinetic equation which is solved using an expansion of eigenfunctions of the bounce-averaged, pitch-angle scattering operator. For bounce times short compared to transit and collision times, the subsequent heat flow closure takes the form of an integral operator acting on electron temperature variations along magnetic field lines. The general closure agrees qualitatively with previous forms for the heat flux in homogeneous magnetic geometry and importantly, predicts a substantial reduction in the heat flow due to the trapping of particles in magnetic wells and the enhanced effect of collisions near the trapped/passing particle boundary.

Unified form for parallel ion viscous stress in magnetized plasmas

In this work a unified form for the parallel ion viscous stress in a magnetized plasma is presented. Approximately valid for arbitrary collisionality, the integral nature of this generalized closure results from assuming the maximal ordering between collisional pitch-angle scattering and free streaming effects and from taking a Chapman–Enskog-type approach which includes the parallel ion viscous stress itself as a drive. The ion drift kinetic equation is solved in a sheared slab using an expansion in eigenfunctions of the Lorentz scattering operator. Integrating the coefficient equations in space and taking the proper velocity space moments couples the parallel viscous stress closure to an integral momentum restoring term, thus generalizing the concept of momentum conservation for simplified Coulomb collision operators. The integral closure involves following ions along magnetic field lines which are the ideal, time-independent characteristics of the perturbed distribution function. The fact that the visc...