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MAGNETOHYDRODYNAMICS
 ECE/NE/Physics 726-Fall 2007
1-2:15 PM
2116 Chamberlin
Prof. Dalton Schnack
3213 Chamberlin
PLEASE TURN OFF ALL CELL PHONES BEFORE ENTERING CLASS!
SYLLABUS (PDF File)
Lecture Notes & Homework Assignments (Link | Directory Listing)
Lectures: 1:00-2:15 PM Tues-Thurs
Location: 2116 Chamberlin Hall
Office Hours: TBD, but I have an open door policy. It would be (usually) most convenient to meet after class.
Homework: Problem sets will be assigned approximately weekly, and will be due approximately 2 weeks after assignment. The problems will amplify points in the lectures, and treat some topics not discussed in detail in class. Please submit assignments as handwritten, and not as MSWord, LaTeX, pdf, postscript, or other computer generated format. ALL HOMEWORK WILL BE GRADED.
Grading Policy: The grade will be determined COMPLETELY BY THE PROBLEM SETS. (So, there are either no exams, or weekly exams, depending on your point of view.) It is therefore important that you work hard on these assignments, and show all your work. In these days of the internet, I’m sure that the solution to most problems I could think of are available somewhere out there. That’s not important; I’m not trying to test or trick you. (By this time in your academic careers you have proven that you are good test takers, and that you can’t be easily fooled. I know that already!) What IS important is that you go through the solutions carefully and understand them, and use them as a means to learn a little MHD. You are encouraged to work together in small or large groups. This often facilitates the learning process.
Meaning of grades: A = exceeds my expectations; B = meets my expectations; C = below my expectations
Other Rules for Class:
- Show up
- Pay attention
- Work hard
If you need to miss class because of religious or personal reasons, please let me know ahead of time and we can work together to make up the material.
I want to work with you to make this a rewarding experience for all of us.
Course Outline: I hope to cover the following topics. We’ll see how the time goes.
1. Physical description of electrically conducting fluids
2. Scalars, vectors, tensors, dyads, etc.
3. Derivation of basic MHD equations
- Continuity
- Equation of motion
- Energy flow
4. The low frequency dynamics of the electromagnetic field
5. Closures
6. Conservation Laws for MHD:
7. Some properties of MHD:
- Ideal MHD equations
- The Frozen Flux theorem
- The effect of resistivity
- Similarity scaling
- The Wöltjer invariants and helicity
8. Equilibrium
- General considerations
- The Virial Theorem
- Examples of simple equilibria: q-pinch, Z-pinch, screw pinch
- Poloidal b, paramagnetic and diamagnetic states
- Force-free fields
- Toroidal equilibrium: the Grad-Shafranov equation, nonlinearity
9. Dynamics of small displacements in ideal MHD
- Linearized equations and the ideal MHD force operator
- Boundary conditions in ideal MHD
- Self-adjointness of the force operator
10. MHD waves in uniform media
- Parallel and perpendicular propagation
- Propagation at arbitrary angle
11. Ideal MHD stability
- The calculus of variations
- The ideal MHD Energy Principle
- The Rayleigh-Ritz technique
- Gravitational interchange mode
- Minimization with respect to the parallel displacement
- Examples: q-pinch, Z-pinch, screw pinch, cylindrical tokamak, etc.
- Suydam/Mercier stability
11. Magnetic reconnection
- Steady state: Sweet-Parker scaling
- Resistive instabilities: tearing modes
12. Lagrange multipliers
13. Taylor relaxation/dynamo
14. Turbulence/dynamo
15. Extended MHD
Note that MHD flows and shock waves will not be covered, because I don’t know enough about theses topics.
Text: There is no appropriate single text for the planned course material. In preparing the lectures I have drawn freely from the following sources:
- George Arfkin, Mathematical Methods for Physicists, 2nd Ed., Academic Press, New York (1970).
- G. K. Batchelor, The Theory of Homogenous Turbulence, Cambridge University Press, Cambridge, UK (1953).
- Dieter Biskamp, Nonlinear Magnetohydrodynamics, Cambridge University Press, Cambridge, UK (1993).
- Dieter Biskamp, Magnetic Reconnection in Plasmas, Cambridge University Press, Cambridge, UK (2000).
- Dieter Biskamp, Magnetohydrodynamic Turbulence, Cambridge University Press, Cambridge, UK (2003).
- S. Chandrasekhar, Hydromagnetic and Hydrodynamic Stability, Dover Publications, New York (1981).
- R. Courant and D. Hilbert, Methods of Mathematic Physics, Vol. 1, Interscience, New York (1953).
- Jeffrey P. Freidberg, Ideal Magnetohydrodynamics, Plenum Press, New York (1987).
- B. B. Kadomtsev, “Hydromagnetic Stability of a Plasma”, in Reviews of Plasma Physics, Vol. 2, p. 153, Consultants Bureau, New York (1966).
- L. D. Landau and E. M. Lifschitz, Fluid Mechanics, Pergamon Press, London, UK (1959).
- L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, UK (1960).
- Wallace M. Manheimer and Chris Lashmore-Davies, MHD Instabilities in Simple Plasma Configurations, Naval Research Laboratory, Washington, DC (1984).
- Donald H. Menzel, Mathematic Physics, Dover Publications, New York (1961).
- H. K. Moffatt, Magnetic Field Generation in Electrically Conducting Fluids, Cambridge University Press, Cambridge, UK (1978).
- Sergio Ortolani and Dalton D. Schnack, Magnetohydrodynamics of Plasma Relaxation, World Scientific, Singapore (1993).
- Eric R. Priest, Solar Magnetohydrodynamics, D. Reidel, Dortrecht (1982).
- V. D. Shafranov, “Plasma Equilibrium in a Magnetic Field”, in Reviews of Plasma Physics, Vol. 2, p. 103, Consultants Bureau, New York (1966).
- J. B. Taylor, Rev. Modern Physics 58, 741 (1986).
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