A method for calculating neoclassical transport coefficients with momentum conserving collision operator

1992 ◽  
Vol 4 (11) ◽  
pp. 3638-3643 ◽  
Author(s):  
M. Taguchi
Keyword(s):  
Transport Coefficients ◽  
Collision Operator ◽  
Neoclassical Transport

Anomalous transport coefficients in a magnetized turbulent plasma

1982 ◽  
Vol 28 (2) ◽  
pp. 193-214 ◽  
Author(s):  
Qiu Xiaoming ◽  
R. Balescu
Keyword(s):  
Transport Coefficients ◽  
Collision Operator ◽  
Turbulent Plasma ◽  
Anomalous Transport ◽  
Weak Turbulence ◽  
Dissipative Effects ◽  
Dependent Transport ◽  
Two Fluid ◽  
Two Fluid Plasma

In this paper we generalize the formalism developed by Balescu and Paiva-Veretennicoff, valid for any kind of weak turbulence, for the determination of all the transport coefficients of an unmagnetized turbulent plasma, to the case of a magnetized one, and suggest a technique to avoid finding the inverse of the turbulent collision operator. The implicit plasmadynamical equations of a two-fluid plasma are presented by means of plasmadynamical variables. The anomalous transport coefficients appear in their natural places in these equations. It is shown that the necessary number of transport coefficients for describing macroscopically the magnetized turbulent plasma does not exceed the number for the unmagnetized one. The typical turbulent and gyromotion terms, representing dissipative effects peculiar to the magnetized system, which contribute to the frequency-dependent transport coefficients are clearly exhibited.

scholarly journals Evolution Equation and Transport Coefficients of Swarms in Initial Relaxation Processes

1987 ◽  
Vol 40 (3) ◽  
pp. 367 ◽  
Author(s):  
Keiichi Kondo
Keyword(s):  
Evolution Equation ◽  
Transport Coefficients ◽  
Collision Operator ◽  
Operator Method ◽  
Time Dependent ◽  
Relaxation Processes ◽  
Projection Operator Method ◽  
Model Collision ◽  
The Difference ◽  
Dependent Transport

The problem of a swarm approaching the hydrodynamic regime is studied by using the projection operator method. An evolution equation for the density and the related time-dependent transport coefficient are derived. The effects of the initial condition on the transport characteristics of a swarm are separated from the intrinsic evolution of the swarms, and the difference from the continuity equation with time-dependent transport coefficients introduced by Tagashira et al. (1977, 1978) is discussed. To illustrate this method, calculations on the relaxation model collision operator have been carried out. The results are found to agree with the analysis by Robson (1975).

scholarly journals Projection-operator methods for classical transport in magnetized plasmas. Part 1. Linear response, the Braginskii equations and fluctuating hydrodynamics

2018 ◽  
Vol 84 (4) ◽  
Author(s):  
John A. Krommes
Keyword(s):  
Distribution Function ◽  
Projection Operator ◽  
Linear Response ◽  
Transport Coefficients ◽  
Collision Operator ◽  
Transport Equations ◽  
Transport Processes ◽  
Projection Operators ◽  
Covariant Representation ◽  
Classical Transport

An introduction to the use of projection-operator methods for the derivation of classical fluid transport equations for weakly coupled, magnetised, multispecies plasmas is given. In the present work, linear response (small perturbations from an absolute Maxwellian) is addressed. In the Schrödinger representation, projection onto the hydrodynamic subspace leads to the conventional linearized Braginskii fluid equations when one restricts attention to fluxes of first order in the gradients, while the orthogonal projection leads to an alternative derivation of the Braginskii correction equations for the non-hydrodynamic part of the one-particle distribution function. The projection-operator approach provides an appealingly intuitive way of discussing the derivation of transport equations and interpreting the significance of the various parts of the perturbed distribution function; it is also technically more concise. A special case of the Weinhold metric is used to provide a covariant representation of the formalism; this allows a succinct demonstration of the Onsager symmetries for classical transport. The Heisenberg representation is used to derive a generalized Langevin system whose mean recovers the linearized Braginskii equations but that also includes fluctuating forces. Transport coefficients are simply related to the two-time correlation functions of those forces, and physical pictures of the various transport processes are naturally couched in terms of them. A number of appendices review the traditional Chapman–Enskog procedure; record some properties of the linearized Landau collision operator; discuss the covariant representation of the hydrodynamic projection; provide an example of the calculation of some transport effects; describe the decomposition of the stress tensor for magnetised plasma; introduce the linear eigenmodes of the Braginskii equations; and, with the aid of several examples, mention some caveats for the use of projection operators.

Symmetries of and entropy production by transport in toroidal confinement systems

1998 ◽  
Vol 59 (4) ◽  
pp. 695-706 ◽  
Author(s):  
H. SUGAMA ◽  
W. HORTON
Keyword(s):  
Entropy Production ◽  
Transport Process ◽  
Collision Operator ◽  
Positive Definite ◽  
Anomalous Transport ◽  
The Self ◽  
Entropy Production Rate ◽  
Onsager Symmetry ◽  
Neoclassical Transport ◽  
Linearized Collision Operator

A synthesized formulation of classical, neoclassical and anomalous transport in toroidal confinement systems with electromagnetic fluctuations and large mean flows is presented. The positive-definite entropy production rate and the conjugate flux–force pairs are rigorously defined for each transport process. The Onsager symmetries of the classical and neoclassical transport matrices are derived from the self-adjointness of the linearized collision operator. The linear gyrokinetic equation with given electromagnetic fluctuations determines the anomalous fluxes with the quasilinear anomalous transport matrix, which satisfies the Onsager symmetry.

Monte Carlo Simulation of Neoclassical Transport in Axisymmetric and Ripple Tokamaks

1982 ◽  
Vol 37 (8) ◽  
pp. 899-905 ◽  
Author(s):  
W. Lötz ◽  
J. Nührenberg
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Electric Field ◽  
Pitch Angle ◽  
Particle Distribution ◽  
Transport Coefficients ◽  
Angle Scattering ◽  
Wide Range ◽  
Neoclassical Transport

Simple axisymmetric and ripple tokamak model fields are used to compute neoclassical trans-port coefficients by Monte Carlo simulation over a wide range of mean free paths in the approximation of small gyroradius. Further assumptions are a monoenergetic particle distribution which is only subject to pitch angle scattering and a vanishing electric field. Pfirsch-Schlüter, plateau, banana and ripple transport coefficients are obtained. In the ripple regime the description is unified by introducing the concept of an effective ripple. Cases in which ripple transport is diminished due to collisionless detrapping are observed

scholarly journals Gyrokinetic simulations of neoclassical transport using a minimal collision operator

2008 ◽  
Author(s):  
G. Dif-Pradalier ◽  
V. Grandgirard ◽  
Y. Sarazin ◽  
X. Garbet ◽  
Ph. Ghendrih ◽  
...  
Keyword(s):  
Collision Operator ◽  
Neoclassical Transport ◽  
Gyrokinetic Simulations

scholarly journals Projection-operator methods for classical transport in magnetized plasmas. Part 2. Nonlinear response and the Burnett equations

2018 ◽  
Vol 84 (6) ◽  
Author(s):  
John A. Krommes
Keyword(s):  
Correlation Functions ◽  
Projection Operator ◽  
Transport Coefficients ◽  
Collision Operator ◽  
Time Correlation ◽  
Chem Phys ◽  
Magnetized Plasmas ◽  
Burnett Equations ◽  
Momentum Exchange ◽  
Time Formalism

The time-independent projection-operator formalism of Breyet al. (PhysicaA, vol. 109, 1981, pp. 425–444) for the derivation of Burnett equations is extended and considered in the context of multispecies and magnetized plasmas. The procedure provides specific formulas for transport coefficients in terms of two-time correlation functions involving both two and three phase-space points. It is shown how to calculate those correlation functions in the limit of weak coupling. The results are used to demonstrate, with the aid of a particular non-trivial example, that the Chapman–Enskog methodology employed by Catto & Simakov (CS) (Phys. Plasmas, vol. 11, 2004, pp. 90–102) to calculate the contributions to the parallel viscosity driven by temperature gradients is consistent with formulas previously derived from the two-time formalism by Brey (J. Chem. Phys., vol. 79, 1983, pp. 4585–4598). The work serves to unify previous work on plasma kinetic theory with formalism usually applied to turbulence. Additional contributions include discussions of (i) Braginskii-order interspecies momentum exchange from the point of view of two-time correlations; and (ii) a simple stochastic model, unrelated to many-body theory, that exhibits Burnett effects. Insights from that model emphasize the role of non-Gaussian statistics in the evaluation of Burnett transport coefficients, including the effects calculated by CS that stem from the nonlinear collision operator. Together, Parts 1 and 2 of this series provide an introduction to projection-operator methods that should be broadly useful in theoretical plasma physics.

Neoclassical transport coefficients for finite-aspect-ratio and bean-shaped tokamak plasmas

1987 ◽  
Vol 30 (4) ◽  
pp. 1152 ◽  
Author(s):  
E. C. Crume ◽  
C. O. Beasley ◽  
S. P. Hirshman ◽  
W. I. van Rij
Keyword(s):  
Aspect Ratio ◽  
Transport Coefficients ◽  
Tokamak Plasmas ◽  
Neoclassical Transport
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