A Tokamak is a toroidal plasma heated to over 50 million degrees Celsius. The plasma current serves to generate the helical component of the magnetic field necessary for equilibrium. (Image courtesy of NASA.)
This course discusses MHD equilibria in cylindrical, toroidal, and noncircular tokamaks. It covers derivation of the basic MHD model from the Boltzmann equation, use of MHD equilibrium theory in poloidal field design, MHD stability theory including the Energy Principle, interchange instability, ballooning modes, second region of stability, and external kink modes. Emphasis is on discovering configurations capable of achieving good confinement at high beta.
Syllabus
Description
This course discusses MHD equilibria in cylindrical, toroidal, and noncircular tokamaks. It covers derivation of the basic MHD model from the Boltzmann equation, use of MHD equilibrium theory in poloidal field design, MHD stability theory including the Energy Principle, interchange instability, ballooning modes, second region of stability, and external kink modes. Emphasis is on discovering configurations capable of achieving good confinement at high beta.
Course Prerequisites
In order to register for 22.615, you should have previously completed 22.611J/8.613J/6.651J, with a grade of C or higher. Exceptions to this policy will require the permission of the instructor, and will be granted on a casebycase basis.
Textbooks
Freidberg, J. P. Ideal Magnetohydrodynamics. This is out of print but Xerox copies will be available to registered students shortly after the start of classes.
Goedbloed, Hans, and Stefaan Poedts. Principles of Magnetohydrodynamics. Cambridge, UK: Cambridge University Press, 2004. ISBN: 9780521626071.
Wesson, John. Tokamaks. 3rd ed. Oxford, UK: Oxford University Press, 1987. ISBN: 9780198563280.
Problem Sets
The weekly problem sets are an essential part of the course. Working through these problems is crucial to understanding the material.
Problem sets will generally be assigned at Tuesday's lecture and will be due at start of class on the following Thursday.
Exams
There will be a take home midterm and a take home final.
Grading
The final grade for the course will be based on the following:
Grading criteria.
ACTIVITIES 
PERCENTAGES 
Homework 
20% 
Midterm exam 
40% 
Final exam 
40% 
Calendar
Course calendar.
LEC # 
TOPICS 
KEY DATES 
1 
Derivation of the Boltzmann equation 

2 
The moment equations
Derivation of ideal MHD equation


3 
MHD equilibrium
Validity of MHD


4 
Toroidal equilibrium and radial pressure balance 

5 
The screw pinch and the GradShafranov equation 
Homework 1 handed out 
6 
The safety factor and the ohmic tokamak 

7 
The first order GradShafranov equation 
Homework 2 handed out 
8 
Effect of a vertical field on tokamak equilibrium 
Homework 1 handed in 
9 
The high beta tokamak 

10 
The high beta tokamak (cont.) and the high flux conserving tokamak 
Homework 2 handed in
Homework 3 handed out

11 
Flux conserving tokamak (cont.) 

12 
PF design I  the plasma 

13 
PF design II  the coil solver 
Homework 3 handed in 
14 
Formulation of the stability problem
Real tokamaks (with Bob Granetz)


15 
Variational techniques
Alternate concepts (with Darren Sarmer)


16 
Variational principle 

17 
Stability of simple function 
Homework 4 handed out 

Midterm exam 

18 
Lecture 18 

19 
Lecture 19 
Homework 4 handed in
Homework 5 handed out

20 
Lecture 20 

21 
Lecture 21 
Homework 5 handed in 
22 
Lecture 22 


Final exam 
