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