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

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10.1063/1.4801022

### Abstract

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 and from the asymptotic solution for large *x*-values. The new fitting formulas are practically exact (less than 1% error) for arbitrary *x* and *Z* (the ion charge number, checked up to *Z* = 100). The ion coefficients for equal electron and ion temperatures are moderately modified by including the ion-electron collision operator. When the ion temperature is higher than the electron temperature, the ion-electron collision and the temperature change terms in the moment equations must be kept. The ion coefficient formulas from 3 moment (Laguerre polynomial) calculations, precise to less than 0.4% error from the convergent values, are explicitly written.

© 2013 AIP Publishing LLC

Received 22 January 2013
Accepted 22 March 2013
Published online 17 April 2013

Acknowledgments: The research was supported by the U.S. DOE under Grant Nos. DE-FG02-04ER54746, DE-FC02-04ER54798, and DE-FC02-05ER54812. This work was performed in conjunction with the Plasma Science and Innovation Center (PSI-Center) and the Center for Extended Magnetohydrodynamics Modeling (CEMM).

Article outline:

I. INTRODUCTION

II. GENERAL MOMENT EQUATIONS

A. Braginskii’s Chapman-Enskog method

B. Closure scheme in the general moment method

C. Calculation of collision matrices

D. Moment equations for closures

III. GEOMETRICAL APPROACH TO SOLVING MOMENT EQUATIONS

A. *l* = 0 (scalar) moments and moments

B. *l* = 1 vector moments

C. *l* = 2 tensor moments

IV. TRANSPORT THEORY FOR ELECTRON VECTOR MOMENTS

V. CONVERGENCE PROPERTY OF ELECTRON COEFFICIENTS

A. Convergence test with increasing moments

B. Asymptotic behavior of the coefficients

C. Lorentz gas model

VI. ELECTRON CLOSURE COEFFICIENTS FROM FITTING

A. Electron heat flow and friction

B. Electron viscosity

VII. ION CLOSURE (TRANSPORT) COEFFICIENTS

A. Effective ion collision operator

B. Ion coefficients from *K* = 2 calculation

C. Ion coefficients from *K* = 3 calculation

VIII. DISCUSSION

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2013-04-17

2014-04-17

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