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 Flux-Friction 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) 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 Self-Adjoint Property of Collision Operator 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