These notes grew out of graduate lectures on magnetohydrodynamics given without pretending that fusion, space physics, and astrophysics are separate intellectual worlds. One of the pleasures of MHD is that the same equations reappear in liquid-metal experiments, in tokamaks and reversed-field pinches, in the solar wind, in planetary interiors, and in accretion flows.
This web edition keeps that spirit, but adds interactive explorers, movies, and stitched navigation so the classic problems can be approached from several directions at once: derivation, geometry, experiment, and physical intuition.
This edition also comes with a full disclosure. At the beginning of the Spring 2026 semester, I treated the project as an experiment in what was actually possible with ChatGPT: I fed in handwritten lecture notes from roughly fifteen years earlier, and the OCR-plus-editorial pipeline produced a remarkably faithful LaTeX draft of about two hundred pages. From there I filled in missing derivations, repaired notation, expanded the history, and continued developing both the notes and the web edition in active collaboration with ChatGPT.
A theoretical result does not become fully intelligible until it has been reconstructed at least once. These notes are therefore organized around problems rather than formalism alone, and the algebra is carried far enough that the approximations, balances, and experimental consequences stay visible.
They are modular by design. You can read them linearly, but they also work as a problem-driven reference. Readers interested in equilibrium and stability can move quickly into tokamak geometry and kink/ballooning theory; readers drawn to astrophysical applications may prefer dynamos, shocks, solar wind, and reconnection.
The current web edition covers Lectures 0 through 42, from the opening foundations through equilibrium, waves, kinetic MHD, interchange, kinks, TAEs, ballooning modes, tearing, helicity, reconnection, shocks, and the late-book appendices.
If you want a printable version, use the PDF edition linked above. For a long-lived archive record, use the Zenodo project DOI 10.5281/zenodo.20140830.
Preferred citation: Cary Forest, Classic Problems in Magnetohydrodynamics: Interactive Web Edition (University of Wisconsin--Madison, 2026), magnetohydrodynamics.physics.wisc.edu, project DOI 10.5281/zenodo.20140830.
Zenodo DOI: 10.5281/zenodo.20140830
Source / release record: github.com/cbforestWI/Classic-Problems-in-MHD
These notes grew out of graduate lectures organized around the idea that fusion, space physics, astrophysics, and liquid-metal MHD are not separate intellectual worlds. The same equations reappear in all of them; what changes is the dominant balance, the closure, and the physical intuition needed to use them well.
The current manuscript also has an unusual provenance. It began this semester as an experiment in whether handwritten lecture notes from about fifteen years ago could be recovered through OCR and turned into a serious LaTeX draft with ChatGPT. That first transcription was unexpectedly successful, and the present edition was then expanded, corrected, and reorganized through a continuing human-AI collaboration.
I have also come to believe that a subject becomes memorable when it is organized around problems rather than around formalism alone. That is why these notes are built around flux freezing and its failure, equilibrium and force balance, waves, stability, dynamos, self-organization, reconnection, shocks, and the geometric and topological constraints that make magnetized fluids both elegant and dangerous.
These lectures lean heavily on the classical literature and on a long line of excellent books and notes, especially Cowling, Chandrasekhar, Roberts, Hide, Moffatt, Kulsrud, Miyamoto, Freidberg, Schnack, Bellan, Wesson, Krall and Trivelpiece, and Zohm.
Russell Kulsrud deserves a special note here. I first learned MHD in my first year at Princeton, in the spring of 1987 at the Graduate Program in Plasma Physics, from Russell himself. His kinetic-MHD notes were very much on his mind at the time, and they helped shape the way I first encountered the subject. Later I learned a great deal more from him as I moved into experimental dynamo work, and then again from his textbook, which remains one of the clearest guides to how plasma physics, kinetic thinking, and MHD fit together.
These notes also reflect material learned from conversations, classes, blackboards, and draft notes, including ideas and derivations I learned from Dalton Schnack, Steve Cowley, Alex Schekochihin, Dmitri Ryutov, and many others.
These notes are still evolving. Some lectures are more polished than others, some topics are treated in more depth than others, and there will certainly be mistakes, omissions, awkward explanations, and places where a clearer path through the physics is still needed.
If you notice an error, a weak derivation, a broken reference, or have a suggestion for improving the notes or the web edition, please contact Cary Forest or open an issue through the GitHub site repository. Corrections and suggestions are genuinely welcome.
Equilibrium and shapingAnalytic Grad-Shafranov structure, shaping, reconstruction, and the geometry of confined plasmas.
Waves and continuaFrom basic Alfvén-wave geometry to toroidal continua, gaps, and energetic-particle-driven mode structure.
Relaxation and reconnectionTearing, helicity, self-organization, and the ways magnetic topology changes when ideal constraints fail.Begin with impedance matching, one-fluid MHD, Braginskii closure, and CGL if you want the assumptions and limits made explicit from the start.
Jump to the Grad-Shafranov sequence for force balance, tokamak shaping, reconstruction, and the equilibrium logic behind later stability theory.
Follow the ladder from interchange and kinks through internal kink, TAE, ballooning, and resistive instabilities, with explorers attached to many of the central cases.
Liquid-metal flows, solar wind, disc winds, dynamos, shocks, and reconnection show how the same equations reorganize themselves in very different physical settings.