In this third lecture the ideal and extended magnetohydrodynamics (MHD) fluid moment descriptions of magnetized plasmas are discussed first. The ideal MHD equilibrium in a toroidally axisymmetric tokamak plasma is discussed next. Then, the collisional viscous force closure moments and their effects on the parallel Ohm's law and poloidal flows in the extended MHD model of tokamak plasmas are discussed. Finally, the species fluid moment equations are transformed to magnetic flux coordinates, averaged over a flux surface and used to obtain the tokamak plasma transport equations. These equations describe the transport of the plasma electron density, plasma toroidal angular momentum and pressure of the electron and ion species "radially" across the nested tokamak toroidal magnetic flux surfaces.[-]

In this final, fourth lecture the many effects on radial tokamak plasma transport caused by various physical processes are noted first: transients, collision- and microturbulence-induced transport, sources and sinks, and small three-dimensional (3-D) magnetic field perturbations. The main focus of this lecture is on the various effects of small 3-D fields on plasma transport which is a subject that has come of age over the past decade. Finally, the major themes of these CEMRACS 2014 lectures are summarized and a general framework for combining extended MHD, hybrid kinetic/fluid and transport models of tokamak plasma behavior into unified descriptions and numerical simulations that may be able to provide a "predictive capability" for ITER plasmas is presented.[-]

This series of 4 lectures discusses the key physical processes in fusion-relevant plasmas, the equations used to describe them, and the interrelationships between them. The focus is on developing comprehensive equations and models for magnetically-confined fusion plasmas on a hierarchy of time scales. The relevant plasma equations for inertial fusion are also briefly mentioned. The pedagogical development begins with the very short time scale microscopic charged-particle-based Coulomb collision processes in a plasma. This microscopic description is then used to develop a comprehensive plasma kinetic equation, fluid moment, magnetohydrodynamic (MHD) and hybrid kinetic/fluid moment plasma descriptions, and finally the long time scale equations for plasma transport across the confining magnetic field. The present grand challenge in magnetic fusion is to develop a "predictive capability" for deuteron-triton (D-T) burning plasmas in ITER (http://www.iter.org). Individual .pdf files of the final, corrected sets of viewgraphs are available via http://homepages.cae.wisc.edu/~callen/plasmas.

This initial lecture first discusses the wide range of characteristic length and time scales involved in modeling fusion plasmas. Next, the Coulomb scattering of a charged test particle's velocity and the differences between the ensemble-averaged electron and ion collisional scattering and relaxation rates are discussed. Then, the mathematical properties of these collisional scattering processes are used to develop a Fokker-Planck collision operator. Finally, a general plasma kinetic equation (PKE) is developed and its general properties discussed.[-]

In this second lecture a Green function solution of the perturbed plasma kinetic equation (PKE) that determines the effects of Coulomb collisional scattering on linear Landau damping is presented first. This is followed by the development of the fluid moment equations obtained from the PKE. An extended Chapman-Enskog-type approach is used to determine the needed collisional and fluid moment closures for this comprehensive, hybrid kinetic/fluid model. Finally, closures for collision-dominated unmagnetized and magnetized plasmas are presented and their limitations discussed.[-]

The subject matter of this talk concerns the derivation of the Finite Larmor radius approximation, when collisions are taken into account.
Several studies are performed, corresponding to different collision kernels. The main motivation consists in computing the gyro-average of the Fokker-Planck- Landau operator, which plays a major role in plasma physics. We determine its equilibria and derive the fluid approximation around them, leading to a new Euler type system of conservation laws.
The main technique applies for studying highly anisotropic parabolic problems, for example the heat equation with disparate diffusion coeffcients with respect to the parallel and perpendicular directions.[-]

Gyrokinetic simulation is considered to be an essential tool to study turbulent transport driven by micro-scale instabilities in tokamak plasmas. It is roughly categorized into two approaches; delta-$f$ local and full-$f$ global approaches. In full-$f$ approach, both turbulent transport and profile evolutions are solved self-consistently under the power balance between external heat source/sink. In this talk, we address (A) numerical technique to treat such full-$f$ gyrokinetic Vlasov-Poisson equations [1] and (B) characteristics of global ion-scale turbulence and transport barrier [2]. We also discuss (C) the role of stable modes in collisionless or weakly collisional plasmas [3].[-]

We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond [1, 3]. First we establish the convergence towards the Euler equation under several assumptions on the energy and on the norms of the initial data. Then we analyze the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density of particles with a moving point charge, characterized by a Dirac mass in the phase-space.[-]