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Molvig, Kim, 22.616 Plasma Transport Theory, Fall 2003. (Massachusetts Institute of Technology: MIT OpenCourseWare), http://ocw.mit.edu (Accessed 07 Jul, 2010). License: Creative Commons BY-NC-SA
Plasma Transport Theory
Fall 2003
To date, the most effective way to confine a plasma magnetically is to use a toroidal, or doughnut-shaped, device called a tokamak pictured in this schematic. (Image courtesy of the U.S. Department of Energy's Office of Fusion Energy Sciences.)
Course Highlights
This course includes selected lecture notes, links to related resources and a complete set of assignments with solutions.
Course Description
This course describes the processes by which mass, momentum, and energy are transported in plasmas, with special reference to magnetic confinement fusion applications.
The Fokker-Planck collision operator and its limiting forms, as well as collisional relaxation and equilibrium, are considered in detail. Special applications include a Lorentz gas, Brownian motion, alpha particles, and runaway electrons.
The Braginskii formulation of classical collisional transport in general geometry based on the Fokker-Planck equation is presented.
Neoclassical transport in tokamaks, which is sensitive to the details of the magnetic geometry, is considered in the high (Pfirsch-Schluter), low (banana) and intermediate (plateau) regimes of collisionality.
Syllabus
Description
Description of the processes by which mass, momentum, and energy are transported in plasmas, with special reference to magnetic confinement fusion applications.
The Fokker-Planck collision operator and its limiting forms, as well as collisional relaxation and equilibrium, are considered in detail. Special applications include a Lorentz gas, Brownian motion, alpha particles, and runaway electrons.
The Braginskii formulation of classical collisional transport in general geometry based on the Fokker-Planck equation is presented.
Neoclassical transport in tokamaks, which is sensitive to the details of the magnetic geometry, is considered in the high (Pfirsch-Schluter), low (banana) and intermediate (plateau) regimes of collisionality.
Course Prerequisites
22.615
Textbook
Helander, Per, Dieter J. Sigmar. Collisional Transport in Magnetized Plasmas. Cambridge University Press, 2001. ISBN: 9780521807982.
Problem Sets
The problem sets (approximately weekly) are an essential part of the course. Working through these problems is essential to understanding the material. Problem sets will generally be assigned on Tuesday and will be due on the following Tuesday. All problem sets will be posted on the course Web site. Problem set solutions will be posted on the Web site following the due date. No problem sets will be accepted after the solutions have been posted.
Exams
There will be one comprehensive TAKE HOME final exam.
Term Paper
There is no term paper required for this course.
Grading
The final grade for the course will be based on the following:
| ACTIVITIES |
PERCENTAGES |
| Weekly Problem Sets |
60% |
| Final Exam |
40% |
|
Web Site
As the semester progresses, we will post important information and other helpful material on the course Web site. You should check the Web site for announcements (rescheduling etc.) prior to each class. All problem sets and solutions will also be posted on the Web site.
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 |
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