A Mathematical Introduction to Magnetohydrodynamics

The course provides a basic introduction to magnetohydrodynamics (MHD), with emphasis on its mathematical aspects (as opposite to physical phenomena). Essentially, MHD is the theory of electrically conducting fluids in presence of a magnetic field. Since MHD is one of the two building blocks (together with kinetic theory) of theoretical plasma physics, its understanding is of paramount importance for applied mathematicians who deal with plasma physics and nuclear fusion applications. The basic "milestones" along the path are:
- Basic concepts and quantities of fluid dynamics.
- Reynolds transport theorem and the equation of fluid dynamics.
- Relation to kinetic theory.
- Multi-fluid description of plasmas and quasi-neutral limit.
- Derivation of MHD equations from multi-fluid theories.
- Global conservation theorems for MHD.
- Topology of the magnetic field lines.
- Conservation of the magnetic flux.
- Qualitative aspects of the solutions of MHD equations.
- Reduced MHD equations and conservation theorems.
- Variational formulation of MHD.
- Hamiltonian formulation of MHD and reduced MHD.

Upon completion of the course, students will be able to understand the basic elements and methods of magnetohydrodynamics and its most important mathematical aspects. On the basis of the acquired familiarity with basic methods and concepts, the student will be able to assess the technical literature on the subject and ready to apply the basic methods presented in the course to new problems, e.g., in a Master thesis work.

I have tried to design the course in a reasonably self-contained way. No previous knowledge is assumed about plasma physics of fluid dynamics, as the necessary concepts will be introduced during the course. On the other hand, the students are expected to know basic calculus, basic mathematical analysis, and the theory of ordinary differential equations. Understanding of partial differential equations and basic numerical methods for their solution are an advantage but not a prerequisite.

The basic teaching method is class lectures on the board. Students are welcome to participate actively with questions and comments. The presentation of numerical results (and the code used to produce them) will be given in form of a computer presentation when needed. Lecture notes will be available and a specific list of references will be given for self-study. Discussions even outside the classroom are encouraged. Students however will need to study the lecture notes and the suggested references in details.

Blackboard

- O. Maj, lecture notes of the course.
- A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Dynamics}, Springer-Verlag (1993).
- D. D. Schnack, Lectures on Magnetohydrodynamics}, Springer (2009).
- E. Priest, Magnetohydrodynamics of the Sun}, Cambridge University Press (2014).
Further readings will be suggested during the classes with comments and indications as appropriate to the specific topic. For the final examination the material covered by the lecture notes is more than sufficient.

An oral examination of about 30 minutes is envisaged. This examination consists of two parts. In the first part the student presents a short lecture on a topic of his/her choice within the scope of the classes. This short lecture should be contained within 10-15 minutes, indicatively. Questions are asked during and after the presentation, strictly on the chosen topic. In the second part two short questions on other topics covered in classes are asked to probe the student overall understanding. Personal notes are not allowed. Reference material can be used for helping memory with non essential results such as vector calculus identities, special definitions, etc... This examination format should allow an assessment of both the knowledge and the ability of the student to develop a mathematically precise argument on the considered topics.