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 GyroBohm 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) FokkerPlanck 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 FokkerPlanck Coefficients Debye Cutoff: (a) BalescuLenard form and (b) Completely Convergent Form Collision Operator Properties: (a) Conservation Laws, (b) Positivity, (c) HTheorem 

5 
Coulomb Collision Operator Derivation III
Electronion Lorentz Operator Energy Equilibration Terms Electrical Conductivity  The SpitzerHarm Problem: (a) Example of Transport Theory Calculation Runaway Electrons 
Problem Set #2
Equilibration FokkerPlanck Equation Accuracy Collision Operator Properties Htheorem Positivity 
67 
Classical (collisional) Transport in Magnetized Plasma
Moment Equations Expansion About Local Thermal Equilibrium (Electron Transport) Linear Force/Flux Relations Transport Coefficients: Dissipative and Nondissipative Terms Physical Picture of Nondissipative 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 Gyroaveraged Kinetic Equation IS Drift Kinetic Equation Gyroaveraged 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) ElectronIon Temperature Equilibration FluxFriction Calculation of Radial Flux 
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) GyroBohm 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 Electronion Operator Expansion Including Small v Behavior Energy Scattering Fast ion Collisions, Alpha Slowing Down and Fusion Alpha Distribution 

1617 
_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) FluxFriction Relations, (b) Leading Order Approximations Particle Flux Relations NonAmbipolar Fluxes, Viscosity, Plasma Rotation: (a) Limits to Mantra, Calculation of Ambipolar Field, (b) Impurity Transport, and Steady State Profiles 

1819 
LikeParticle 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 SelfAdjoint Property of Collision Operator Conservation Laws for Linearized Collision Operator Ambipolarity and Impurity Diffusion Diamagnetic Fluxes Generalized FluxFriction Relations LikeParticle (Ion) Collision Fluxes 
2021 
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 FluxSurface averages: (a) Constant of Motion variables, (b) Moments @ Fixed Space Position, (c) FluxSurface 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 

2225  Neoclassical Transport (cont.)  
2630 
_TAKE HOME FINAL EXAM
_Ware Pinch Effect Magnetization Bootstrap Current Simplified Implicit Transport Coefficient Diagonal Transport Coefficients Onsager Symmetry of Transport Coefficients 
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