22.616 | Fall 2003 | Graduate
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

Course Info
As Taught In
Fall 2003
Learning Resource Types
assignment Problem Sets
grading Exams with Solutions
notes Lecture Notes