Plasma Transport Theory

Lecture Notes

The following lecture topics were covered in class. Notes have been posted where available.

Introduction and Basic Transport Concepts

• Form of Transport Equations
• Random Walk Picture – Guiding Centers
• Coulomb Cross Section and Estimate
• Fusion Numbers: (a) Banana Diffusion, (b) Bohm and Gyro-Bohm Diffusion
• Transport Matrix Structure: (a) Onsager Symmetry

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

Coulomb Collision Operator Derivation

• Written Notes for these Lectures (2 sets):
  (1) Collisions and Transport Theory I (PDF)
  (2) Collisions and Transport Theory II: Electrical Conductivity - Spitzer Problem (PDF)
• Fokker-Planck Equation Derivation

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

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

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

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

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

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

Distribution Function of Fluctuations

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

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

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

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

Coulomb Collision Operator Properties

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

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
  Fast ion Collisions, Alpha Slowing Down and Fusion Alpha Distribution

Like-Particle Collisional Transport

• Ion Thermal Conduction Calculation
• Guiding Center Picture Calculation
• Heat Flux - Heat Friction Relation

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
• Analytic Details of Thermal Conduction Calculation Including Complete Expression

Ware Pinch Effect

Magnetization Bootstrap Current

Simplified Implicit Transport Coefficient

Diagonal Transport Coefficients

Onsager Symmetry of Transport Coefficients