22.615 | Spring 2007 | Graduate

MHD Theory of Fusion Systems


Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session


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 case-by-case basis.


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.


There will be a take home midterm and a take home final.


The final grade for the course will be based on the following:

Homework 20%
Midterm exam 40%
Final exam 40%


1 Derivation of the Boltzmann equation  

The moment equations

Derivation of ideal MHD equation


MHD equilibrium

Validity of MHD

4 Toroidal equilibrium and radial pressure balance  
5 The screw pinch and the Grad-Shafranov equation Homework 1 handed out
6 The safety factor and the ohmic tokamak  
7 The first order Grad-Shafranov 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

Formulation of the stability problem

Real tokamaks (with Bob Granetz)


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  

Course Info

As Taught In
Spring 2007
Learning Resource Types
Lecture Notes
Problem Sets