Plasma Transport Theory

Location and Times

Classes will be held in Room NW16-213 each Tuesday and Thursday from 1:00 PM to 2:30 PM.

September

4

Introduction and Basic Transport Concepts

1. Form of transport equations
2. Random walk picture -- guiding centers
3. Coulomb cross section & estimates
4. Fusion numbers:
..a. Banana diffusion
..b. Bohm & Gyro-Bohm diffusion
5. Transport matrix structure
..a. Onsager symmetry

Sigmar & Helander Chapter 1

9

Diffusion Equation Solutions and Scaling

1. Initial value problem
2. Steady state heating problem (temperature) w/ power source
3. Density behavior
..a. Include pinch effect
4. Magnetic Field diffusion
5. Velocity space diffusion
..a. Relaxation behavior w/o friction
..b. Need for friction in equilibration

11

Coulomb Collision Operator Derivation

1. Written notes for these lectures (2 sets)
2. Fokker-Planck Equation derivation

Sigmar & Helander Chapter 3

Problem Set #1 (9/11 to 9/18)
1. Fusion Transport Estimates
2. Diffusion equation solution and properties
3. Diffusion equation Green's function
4. Metallic Heat Conduction
5. Monte Carlo solution to diffusion equation and demonstration of Central Limit Theorem

16

Coulomb Collision Operator Derivation II

1. Calculation of Fokker-Planck coefficients
2. Debye cutoff
..a. Balescu-Lenard form and completely convergent form
3. Collision operator properties
..a. Conservation laws
..b. Positivity
..c. H-Theorem

Sigmar & Helander Chapter 3

18

Coulomb Collision Operator Derivation III

1. Electron-ion Lorentz operator
2. Energy equilibration terms
3. Electrical Conductivity - the Spitzer-Harm problem
..a. Example of transport theory calculation
4. Runaway electrons

Sigmar & Helander Chapter 3

Problem Set #2 (9/18 to 9/25)
1. Equilibration
2. Fokker-Planck equation accuracy
3. Collision Operator Properties
4. H-theorem
5. Positivity

23, 25

Classical (collisional) Transport in Magnetized Plasma

1. Moment equations
2. Expansion about local thermal equilibrium (electron transport)
3. Linear Force/Flux relations
4. Transport coefficients
..a. Dissipative and non-dissipative terms
5. 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
6. Physical picture of dissipative flows
..a. Guiding center scattering random walk

Sigmar & Helander Chapter 4

Problem Set #3 (9/25 to 10/2)
1. Moment Equation Structure

30

Classical Transport in Guiding Center Picture

1. Alternate formulation displays microscopic physics more clearly (needs gyrofrequency >> collision frequency)
2. Follows hierarchy of relaxation processes - "collisionless relaxation"
3. Transformation to Guiding Center Variables
..a. Physical interpretation
4. Gyro-averaged kinetic equation IS Drift Kinetic Equation
5. Gyro-averaged collision operator
..a. Spatial KINETIC diffusion of guiding center
6. Transport theory ordering

October

2

Classical Transport in Guiding Center Picture II

1. Expansion of distribution function and kinetic equation
..a. Maximal ordering (math & physics)
2. Zero order distribution - local Maxwellian
3. 1st order - Generalized Spitzer problem
..a. Inversion of (velocity space) Collision operator
..b. Integrability conditions
..c. Identification of Thermodynamic Forces
4. 2nd order - Transport Equations
..a. Integrability conditions yield transport equations
..b. And complete specification of zero order f
5. Transport Coefficient Evaluations
..a. Equivalence to prior results
6. Physical picture of flows
..a. Guiding center flows & "magnetization" flows

Problem Set #4: (10/2 to 10/9)
1. Collisional Guiding Center Scattering
2. Diamagnetic Flow (alternately termed “Magnetization” flow)
3. Electron-Ion Temperature Equilibration
4. Flux-Friction Calculation of Radial Flux

7

Random (Stochastic) Processes, Fluctuation, etc. (Intro.)

1. Probability and Random Variables
2. Ensemble averages
3. Stochastic processes
..a. Fluctuating electric fields
..b. Correlation functions
..c. Stationary random process
4. Integrated Stochastic process - Diffusion
..a. Example of integral of Electric field fluctuations giving velocity diffusion
..b. Integrated Diffusion process

9

Distribution Function of Fluctuations

1. Central Limit Theorem
2. "Normal Process" definition
..a. Cumulant expansion mentioned. . .
..b. Example of Guiding center diffusion coefficient

14

Fluctuation Spectra – Representation of Fields

1. 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"
2. Physical interpretation in terms of waves
3. Definition of Spectrum as FT of correlation function
4. Generalize to Space & Time dependent fields
..a. Statistical "homogeneity"
5. Continuum Limit rules

16

Diffusion Coefficient from Fluctuation Spectrum

1. Stochastic process evaluation of particle velocity diffusion coefficient from homogeneous, stationary electric field fluctuation spectrum
2. Physical Interpretation via resonant waves
3. Superposition of Dressed Test Particles - field fluctuations
4. Diffusion (tensor) from discreteness fluctuations - Collision Operator
5. Correlation time estimates

21

Turbulent Transport – Drift Waves

1. Space diffusion of guiding center from potential fluctuations and ExB drift
2. Estimates and scalings from drift wave characteristics
..a. Bohm scaling
..b. Gyro-Bohm scaling from realistic saturated turbulence level

Problem Set #5: (10/21 to 11/4)
1. Fluctuation origin of U tensor
2. Diffusion from plasma waves
3. Correlation times
4. Turbulent Drift wave transport

23

Coulomb Collision Operator Properties

1. Correct details of electron-ion operator expansion including small v behavior
2. Energy scattering
3. Fast ion collisions, alpha slowing down and fusion alpha distribution

28,30

November

4, 6

Full Classical Transport in Magnetized Plasma Cylinder

1. Includes ion and impurity transport
2. Estimates and orderings for electron and ion processes
3. 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
4. Moment equation expressions for perpendicular flows
..a. Flux-friction relations
..b. Leading order approximations
5. Particle flux relations
6. Non-ambipolar fluxes, Viscosity, Plasma Rotation
..a. Limits to mantra
..b. Calculation of ambipolar field
..c. Impurity transport
..d. Steady state profiles

Sigmar & Helander Chapter 5

11

Veterans Day

13, 18

Like-Particle Collisional Transport

1. Ion thermal conduction calculation
2. Guiding center picture calculation
3. Heat flux - heat friction relation
4. Analytic details of thermal conduction calculation including complete expression

Problem Set #6
1. Ambipolar Potential in a Magnetized Plasma Column
2. Self-Adjoint Property of Collision Operator
3. Conservation Laws for Linearized Collision Operator
4. Ambipolarity and Impurity Diffusion
5. Diamagnetic Fluxes
6. Generalized Flux-Friction Relations
7. Like-Particle (Ion) Collision Fluxes

20, 25

Neoclassical Transport

1. Introductory concepts:
* Particle orbits & Magnetic geometry
* Particle mean flux surface, moments, flows & currents
2. Tokamak orbit properties
..a. Trapped particle fraction
..b. Bounce time (circulation time)
3. Bounce averages
4. 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)
5. Moment Relations and Definitions
6. Bounce Average Kinetic Equation Derivation
7. Perturbation theory for the "Banana" Regime
8. Banana Regime Transport Theory
..a. Particle moment
..b. Energy moment
..c. Toroidal current
..d. Transport Coefficient formalism
9. Structure of the Transport Matrix
..a. Onsager Symmetry
10. Evaluation of Neoclassical Transport

27

Thanksgiving

December

2, 4

Neoclassical Transport (cont.)

9, 11

Neoclassical Transport (cont.)

13 - 17

TAKE HOME FINAL EXAM

1. Ware Pinch Effect
2. Magnetization Bootstrap Current
3. Simplified Implicit Transport Coefficient
4. Diagonal Transport Coefficients
5. Onsager Symmetry of Transport Coefficients