Fifteen Eighty Four

Summary: A First Course in Magnetohydrodynamics offers a much-needed resource for undergraduate physics education.  Despite the fact that magneto-hydrodynamics (MHD) can be used to describe more than 99.99% of the visible universe, it is usually relegated to graduate programmes in plasma physics and almost never taught at the undergraduate level.  In this blog post, I argue that MHD taught from the perspective of fluid dynamics and not plasma physics is what is needed to bring MHD into the undergraduate curriculum, and that this text – one of a very few that provide a rigorous introduction to the basics of MHD with more than 130 worked problems – is an important step in that direction.

The vast majority of mainstream undergraduate physics, whether it be classical mech-anics, electrodynamics, quantum mechanics, statistical mechanics, solid state physics, and so it goes, is generally applied to the solid, liquid, and gaseous states of “ordinary matter”, those we find in abundance on Earth.  However, more than 99.99% of the baryonic or visible universe is in the plasma state, where temperatures are such that particle-particle collisions ionise some or all of the gas.  For this hugely dominant portion of the visible universe, a description other than or at least in addition to those subjects listed above is needed.

Plasma physics and, where isotropy can be assumed, magnetohydrodynamics (MHD) is that description.  Yet neither subject is central to most undergraduate physics curricula and from my perspective, this implies a certain anthropocentricity still remains in how physics is taught.  That a typical university curriculum might devote none of its offerings to a branch of physics describing >99.99% of the visible universe, and virtually all of its offerings to branches of physics describing the remaining <0.01% that happens to correspond to our experience on Earth ought to, in my view, be a concern to address.

And so why aren’t plasma physics and MHD taught more widely in an undergraduate degree programme?  First, the mathematics of plasma physics is daunting at best.  Based on the Vlasov-Boltzmann equation, few if any undergraduates are prepared to delve into the mathematics of a six-dimensional inhomogeneous partial differential equation, never mind the velocity moments required to obtain the equations of MHD.  As such, approaching MHD as the isotropic limit of plasma physics is normally part of a graduate programme in physics or engineering where applications include fusion physics, laboratory plasmas, star formation, planetary discs, and the like.

However, MHD can also be thought of as the “magnetised version” of ordinary fluid dy-namics where the basic premises include conservation of mass, energy, magnetic flux, and Newton’s second law.  Unlike velocity moments of a 6-D inhomogeneous PDE, con-servation laws are entirely intuitive to a senior undergraduate physics student.  Further, the fluid approach to MHD leads one quickly to four wave types and seven wave families making it an exemplar of wave mechanics, a subject also familiar to senior undergraduate students given their exposure to classical mechanics, electrodynamics, and quantum mechanics.  It is therefore my considered opinion that the “fluid dynamics approach” is the only way to make MHD accessible to senior undergraduate students.

Yet, as evidenced by the paucity of fluids-based MHD textbooks, this is rarely done.  While scores of plasma physics textbooks include a significant portion dedicated to MHD, I am aware of two, maybe three MHD textbooks written from a fluid dynamics perspective at the undergraduate level. I note in haste that one of these – Kendall and Plumpton’s Magnetohydrodynamics with Hydrodynamics – is from 1964!

A First Course in Magnetohydrodynamics (AfciMHD) offers a third, maybe fourth.  As a text designed specifically for the senior undergraduate student, it is not a textbook about advanced and specialised applications of MHD such as fusion and stellar physics; plenty of those already exist.  Instead and as a good undergraduate text should do, AfciMHD provides a thorough and mathematically rigorous presentation of the very basics of MHD and then uses these ideas to develop a few fundamental applications and generalisations.

Assuming no background in fluid dynamics, the first three chapters focus on ordinary hydrodynamics beginning with the derivation of the continuity, energy, and momentum equations.  These are then used to demonstrate some basic fluid phenomena such as sound waves, shocks, hydraulic bores, Bernoulli’s principle, and rarefaction fans.  Magnetism isn’t mentioned until the fourth chapter where the ideal induction equation, Alfvén’s theorem, and magnetic helicity are introduced.  The remaining chapters in Part I examine the full suite of 1-D MHD equations revealing all MHD waves including Alfvén waves, tangential discontinuities, fast and slow waves, shocks (fast, slow, and inter-mediate), and rarefaction fans culminating in the most general (semi-)analytical solution to the MHD equations one can find: that of the Riemann problem.  Part II of the text applies the fundamental ideas from Part I to fluid instabilities and the onset of turbulence, viscid hydrodynamics (the Navier-Stokes equation), steady-state MHD with applications to stellar winds and astrophysical jets, and non-ideal MHD including resistive MHD, Hall MHD, and ambipolar diffusion.

AfciMHD further distinguishes itself from most other plasma/MHD texts by including more than 130 tested problems and projects in the chapters’ problem sets.  Worked sol-utions to all problems and projects are available to the instructor at the text’s CUP website (https://www.cambridge.org/9781009381475) under “Resources”.  Lastly, and what I think may be unprecedented for any upper-year physics text, a complete set of 36 “flipped-style” lesson plans is available to the instructor from the CUP website in a format easily modified to suit the instructor’s lecturing style and emphasis.

This is the textbook I wish I had when I was first introduced to MHD in the late 1980s.  Two decades in the making, AfciMHD provides the student with the background nec-essary to explore all avenues of graduate MHD, whether it be laboratory MHD, fusion physics, stellar astrophysics, or the path I chose: computational MHD and the develop-ment with my then-advisor Michael Norman of the public-domain code ZEUS-3D (http://www.ap.smu.ca/~dclarke/zeus3d).

I believe MHD could and should become a standard offering in all fourth-year university physics curricula and be on the same footing as solid-state physics, nuclear physics, general relativity, and the like.  It is my hope that AfciMHD could contribute to this end and may even inspire others to contribute their own undergraduate-level fluids-based MHD textbook.  Not only would this help move the subject from specialised graduate studies to the mainstream undergraduate curriculum where it belongs, it would also address a remaining vestige of anthropocentrism in physics.

Title: A First Course in Magnetohydrodynamics

ISBN: 9781009381475

Author: David Alan Clarke