22.616 | Fall 2003 | Graduate
Plasma Transport Theory

## Calendar

LEC # TOPICS KEY DATES
1 Introduction and Basic Transport Concepts

Form of Transport Equations

Random Walk Picture – Guiding Centers

Coulomb Cross Section and Estimates

Fusion Numbers: (a) Banana Diffusion, (b) Bohm and Gyro-Bohm Diffusion

Transport Matrix Structure: (a) Onsager Symmetry

2 Diffusion Equation Solutions and Scaling

Initial Value Problem

Steady State Heating Problem (temperature) w/ Power Source

Density Behavior: (a) Include Pinch Effect

Magnetic Field Diffusion

Velocity Space Diffusion: (a) Relaxation Behavior w/o Friction, (b) Need for Friction in Equilibration

3 _Coulomb Collision Operator Derivation

_Written Notes for these Lectures (2 sets)

Fokker-Planck Equation Derivation

Problem Set #1

Fusion Transport Estimates

Diffusion Equation Solution and Properties

Diffusion Equation Green’s Function

Metallic Heat Conduction

Monte Carlo Solution to Diffusion Equation and Demonstration of Central Limit Theorem

4 Coulomb Collision Operator Derivation II

Calculation of Fokker-Planck Coefficients

Debye Cutoff: (a) Balescu-Lenard form and (b) Completely Convergent Form

Collision Operator Properties: (a) Conservation Laws, (b) Positivity, (c) H-Theorem

5 Coulomb Collision Operator Derivation III

Electron-ion Lorentz Operator

Energy Equilibration Terms

Electrical Conductivity - The Spitzer-Harm Problem: (a) Example of Transport Theory Calculation

Runaway Electrons

Problem Set #2

Equilibration

Fokker-Planck Equation Accuracy

Collision Operator Properties

H-theorem

Positivity

6-7 Classical (collisional) Transport in Magnetized Plasma

Moment Equations

Expansion About Local Thermal Equilibrium (Electron Transport)

Linear Force/Flux Relations

Transport Coefficients: Dissipative and Non-dissipative Terms

Physical Picture of Non-dissipative Terms: (a) “Diamagnetic” Flow Terminology and Physics from Pressure Balance and Show that Bin<Bout, (b) “Magnetization” Flow Terminology from FLR, J=Curl M

Physical Picture of Dissipative Flows: (a) Guiding Center Scattering, (b) Random Walk

Problem Set #3

Moment Equation Structure

8 Classical Transport in Guiding Center Picture

Alternate Formulation Displays Microscopic Physics more clearly (needs Gyrofrequency » Collision Frequency)

Follows Hierarchy of Relaxation Processes - “Collisionless Relaxation”

Transformation to Guiding Center Variables: (a) Physical Interpretation

Gyro-averaged Kinetic Equation IS Drift Kinetic Equation

Gyro-averaged Collision Operator: Spatial KINETIC Diffusion of Guiding Center

Transport Theory Ordering

9 Classical Transport in Guiding Center Picture II

Expansion of Distribution Function and Kinetic Equation: (a) Maximal Ordering (Math and Physics)

Zero Order Distribution - Local Maxwellian

1st order - Generalized Spitzer problem: (a) Inversion of (Velocity Space) (b) Collision Operator, (c)Integrability Conditions and Identification of Thermodynamic Forces

2nd order - Transport Equations: (a) Integrability Conditions Yield Transport Equations, (c) Complete Specification of Zero Order f

Transport Coefficient Evaluations: (a) Equivalence to Prior Results

Physical Picture of Flows: (a) Guiding Center Flows and “Magnetization” Flows

Problem Set #4

Collisional Guiding Center Scattering

Diamagnetic Flow (alternately termed “Magnetization” flow)

Electron-Ion Temperature Equilibration

10 Random (Stochastic) Processes, Fluctuation, etc. (Intro.)

Probability and Random Variables

Ensemble Averages

Stochastic Processes: (a) Fluctuating Electric Fields, (b) Correlation Functions, (c) Stationary Random Process

Integrated Stochastic Process - Diffusion: (a) Example of Integral of Electric Field Fluctuations giving Velocity Diffusion, (b) Integrated Diffusion Process

11 Distribution Function of Fluctuations

Central Limit Theorem

“Normal Process” Definition: (a) Cumulant Expansion Mentioned, (b) Example of Guiding Center Diffusion Coefficient

12 Fluctuation Spectra – Representation of Fields

Fourier Representation of Random Variable: (a) Mapping of “All Curves” to Set of All Fourier Coefficients, (b) Fourier Spectral Properties for Stationary Process, (c) Equivalence of “Random Phase Approximation”

Physical Interpretation in Terms of Waves

Definition of Spectrum as FT of Correlation Function

Generalize to Space & Time Dependent Fields: (a) Statistical “Homogeneity”

Continuum Limit Rules

13 Diffusion Coefficient from Fluctuation Spectrum

Stochastic Process Evaluation of Particle Velocity Diffusion Coefficient from Homogeneous, Stationary Electric Field Fluctuation Spectrum

Physical Interpretation via Resonant Waves

Superposition of Dressed Test Particles - Field Fluctuations

Diffusion (Tensor) from Discreteness Fluctuations - Collision Operator

Correlation Time Estimates

14 Turbulent Transport – Drift Waves

Space Diffusion of Guiding Center from Potential Fluctuations and ExB Drift

Estimates and Scalings from Drift Wave Characteristics: (a) Bohm scaling, (b) Gyro-Bohm Scaling from Realistic Saturated Turbulence Level

Problem Set #5

Fluctuation Origin of U tensor

Diffusion from Plasma Waves

Correlation Times

Turbulent Drift Wave Transport

15 Coulomb Collision Operator Properties

Correct Details of Electron-ion Operator Expansion Including Small v Behavior

Energy Scattering

Fast ion Collisions, Alpha Slowing Down and Fusion Alpha Distribution

16-17 _Full Classical Transport in Magnetized Plasma Cylinder
_
Includes Ion and Impurity Transport

Estimates and Orderings for Electron and Ion Processes

Ambipolarity and Two “Mantra” of Classical Transport: (a) “Like Particle Collisions Produce no Particle Flux”, (b) “Collisional Transport is Intrinsically Ambipolar”, (c) Microscopic Proof of Mantra for Binary Collisions

Moment Equation Expressions for Perpendicular Flows: (a) Flux-Friction Relations, (b) Leading Order Approximations

Particle Flux Relations

Non-Ambipolar Fluxes, Viscosity, Plasma Rotation: (a) Limits to Mantra, Calculation of Ambipolar Field, (b) Impurity Transport, and Steady State Profiles

18-19 Like-Particle Collisional Transport

Ion Thermal Conduction Calculation

Guiding Center Picture Calculation

Heat Flux - Heat Friction Relation

Analytic Dtails of Thermal Conduction Calculation Including Complete Expression

Problem Set #6

Ambipolar Potential in a Magnetized Plasma Column

Conservation Laws for Linearized Collision Operator

Ambipolarity and Impurity Diffusion

Diamagnetic Fluxes

Generalized Flux-Friction Relations

Like-Particle (Ion) Collision Fluxes

20-21 Neoclassical Transport

Introductory concepts: (a) Particle orbits and Magnetic Geometry, (b) Particle Mean Flux Surface, Moments, Flows and Currents

Tokamak Orbit Properties: (a) Trapped Particle Fraction, (b) Bounce Time (Circulation Time)

Bounce Averages

Tokamak Moments and Flux-Surface averages: (a) Constant of Motion variables, (b) Moments @ Fixed Space Position, (c) Flux-Surface Averaged Moments, (d) Bootstrap Current (Magnetization Piece)

Moment Relations and Definitions

Bounce Average Kinetic Equation Derivation

Perturbation Theory for The “Banana” Regime

Banana Regime Transport Theory: (a) Particle Moment, (b) Energy Moment, (c) Toroidal Current, (d) Transport Coefficient Formalism

Structure of the Transport Matrix: (a) Onsager Symmetry

Evaluation of Neoclassical Transport

22-25 Neoclassical Transport (cont.)
26-30 _TAKE HOME FINAL EXAM

_Ware Pinch Effect

Magnetization Bootstrap Current

Simplified Implicit Transport Coefficient

Diagonal Transport Coefficients

Onsager Symmetry of Transport Coefficients