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Lecture 1
Introduction

Overview

This lecture has a narrow but important job.

1.
place MHD in the historical line from nineteenth-century fluid mechanics to twentieth-century plasma physics;
2.
state the resistive-MHD equations in the notation used throughout these notes;
3.
make clear that MHD is a closure problem: the equations are only as honest as the ordering assumptions and constitutive relations behind them.

Magnetohydrodynamics (MHD) is the continuum theory of electrically conducting fluids in magnetic fields. It is simultaneously one of the oldest and one of the most durable approximations in plasma physics. In liquid metals it can be a remarkably direct description of the laboratory system. In plasmas it is a controlled coarse-grained theory, valid only after one averages over kinetic scales and commits to a closure. The reward for this loss of microscopic detail is that one gains a language capable of discussing global equilibrium, waves, stability, induction, topology, and self-organization in a unified way.

1.1 Why MHD Begins with Fluid Mechanics

The natural point of departure is classical fluid dynamics. Nineteenth-century work by Euler, Navier, Stokes, Helmholtz, Kelvin, Rayleigh, and many others established the continuum viewpoint. It clarified the meaning of pressure and stress, the role of vorticity, and the stability questions that still underlie modern plasma theory. MHD does not replace that framework; it extends it by adding the possibility that the fluid carries current and that electromagnetic fields can exchange momentum and energy with the flow.

That historical continuity matters. Many of the balances that dominate MHD—advection against diffusion, pressure against tension, inertia against restoring forces, boundary layers against body forces—are best understood first as fluid-mechanical balances. Magnetic fields do not erase fluid mechanics; they reorganize it.

1.2 A Short Historical Note

Historical Perspective

The recognizable subject of magnetohydrodynamics emerged when several traditions met. Hartmann and Lazarus established canonical liquid-metal experiments; Alfvén identified the wave that now bears his name and pushed cosmic electrodynamics; Elsasser emphasized induction and dynamo action in planetary settings; Cowling produced a concise continuum formulation; Chandrasekhar unified hydrodynamic and hydromagnetic stability; and Roberts, Hide, and Moffatt carried the subject further into geophysical and astrophysical fluid dynamics Hartmann and Lazarus [1937], Alfvén [1950], Elsasser [1956], Cowling [1957], Chandrasekhar [1961], Roberts [1967], Hide and Roberts [1962], Moffatt [1978]. Later texts by Kulsrud, Freidberg, and Schnack extended this tradition for modern astrophysical and fusion-plasma audiences Kulsrud [2005], Freidberg [1987], Schnack [2009].

These notes are written very much in that lineage. The goal is not merely to record equations, but to understand why certain problems became classic, what approximations made them tractable, and how they connect back to experiment.

1.3 From Navier–Stokes to Magnetohydrodynamics

We begin with Newton’s second law applied to a continuum medium. In an inertial frame, the Navier–Stokes equation may be written as

\[\rho \left ( \pp {\uvec }{t} + \uvec \cdot \grad \uvec \right ) = -\grad p + \mu \lap \uvec + \vect {f}_{\text {body}} .\]

Here

For a conducting fluid in electromagnetic fields,

\[\vect {f}_{\text {body}} = \J \times \B + \rho \vect {g} + \rho _e \E .\]

In the quasi-neutral plasma approximation, \(\rho _e \approx 0\), leaving the Lorentz force \(\J \times \B \) as the dominant electromagnetic body force. This is the step that turns ordinary fluid dynamics into magnetohydrodynamics.

For incompressible liquid metals, the continuum description can be extremely accurate. For plasmas, by contrast, the MHD equations are already the result of averaging over a more microscopic kinetic description. One should therefore think of MHD not as the final theory, but as the first large-scale theory that is simple enough to use.

1.4 Material Derivative and Geometry

Define the material (or convective) derivative:

\[\frac {D}{Dt} \equiv \pp {}{t} + \uvec \cdot \grad .\]

The nonlinear term \(\uvec \cdot \grad \uvec \) represents convective acceleration. It reflects the fact that a moving fluid element samples spatial gradients of velocity as it travels.

Vector derivatives in curvilinear coordinates. In cylindrical coordinates \((r,\phi ,z)\), the radial component of the convective term is

\[\er \cdot (\uvec \cdot \grad )\uvec = u_r \pp {u_r}{r} + \frac {u_\phi }{r}\pp {u_r}{\phi } + u_z \pp {u_r}{z} - \frac {u_\phi ^2}{r}.\]

The final term \(-u_\phi ^2/r\) is the familiar centrifugal acceleration. It appears not because a new force has been added, but because geometry is part of the dynamics.

Example: Keplerian rotation. For steady, axisymmetric flow in a gravitational field,

\[\rho u_r \pp {u_r}{r} = -\pp {p}{r} - \rho \frac {GM}{r^2} + \mu \lap u_r + \rho \frac {u_\phi ^2}{r}.\]

For purely azimuthal flow (\(u_r = 0\)),

\[\rho \frac {u_\phi ^2}{r} = -\pp {p}{r} - \rho \frac {GM}{r^2}.\]

Even before magnetic fields enter, the competition among pressure, inertia, and geometry is already doing most of the conceptual work.

1.5 The Resistive-MHD System

Caution

Conceptual warning. The equations below are already an approximation. They assume a single-fluid description, quasi-neutrality, and the neglect of displacement current in Ampère’s law. They do not become predictive until the viscous stress, heat flux, and electrical response have been closed.

We now state the governing equations of resistive MHD in SI units.

Continuity

\[\pp {\rho }{t} + \divergence (\rho \uvec ) = 0. \tag{1.7}\]

Momentum

\[\rho \left ( \pp {\uvec }{t} + \uvec \cdot \grad \uvec \right ) = -\grad p + \J \times \B + \divergence \tens {\Pi }. \tag{1.8}\]

Ohm’s law

\[\E + \uvec \times \B = \eta \J . \tag{1.9}\]

Faraday’s law

\[\pp {\B }{t} = -\curl \E . \tag{1.10}\]

Ampère’s law (MHD limit)

\[\muo \J = \curl \B . \tag{1.11}\]

Solenoidal constraint

\[\divergence \B = 0. \tag{1.12}\]
Combining Ohm’s law with Faraday’s law gives the induction equation,
\[\pp {\B }{t} = \curl (\uvec \times \B ) - \curl (\eta \J ). \tag{1.13}\]

This is the core system to which the remainder of these notes repeatedly returns.

1.6 Compressibility, Closure, and Thermodynamics

Closure requires an additional thermodynamic relation. For an ideal gas, one convenient adiabatic form is

\[\frac {D}{Dt} \left ( \frac {p}{\rho ^\gamma } \right ) = 0, \qquad \gamma = \frac {C_p}{C_v}. \tag{1.14}\]

Equivalently, one may evolve the internal energy per unit mass \(e\), with

\[e = \frac {p}{(\gamma - 1)\rho },\]
and write a schematic internal-energy equation as
\[\rho \left ( \pp {e}{t} + \uvec \cdot \grad e \right ) + p\,\divergence \uvec = \Phi _{\mathrm {visc}} - \divergence \vect {q} + \eta J^2 .\]

Here \(\Phi _{\mathrm {visc}}\) denotes viscous heating and \(\vect {q}\) is the heat flux. In a simple isotropic model one often takes \(\vect {q} = -\kappa \grad T\), but in a magnetized plasma the heat flux can be highly anisotropic. This is an early reminder that the phrase “the MHD equations” hides several different closure choices.

Takeaways
  • MHD is fluid mechanics plus electromagnetic induction; it is not a replacement for continuum mechanics but an extension of it.
  • The model is only as good as the scale separation and closures behind it.
  • The rest of these notes will repeatedly return to a small set of competing effects: advection, pressure gradients, magnetic tension, diffusion, and geometry.

Bibliography

    Julius Hartmann and Freimut Lazarus. Hg-dynamics II: Experimental investigations on the flow of mercury in a homogeneous magnetic field". Matematisk-fysiske Meddelelser, 15(7):1–45, 1937.

    Hannes Alfvén. Cosmical Electrodynamics. Clarendon Press, Oxford, 1950.

    Walter M. Elsasser. Hydromagnetic dynamo theory. Reviews of Modern Physics, 28(2):135–163, 1956. doi:10.1103/RevModPhys.28.135.

    T. G. Cowling. Magnetohydrodynamics. Interscience Publishers, New York, 1957.

    S. Chandrasekhar. Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford, 1961.

    P. H. Roberts. An Introduction to Magnetohydrodynamics. Longmans, London, UK, 1967. ISBN 9780582447288.

    Raymond Hide and Paul H. Roberts. Some elementary problems in magneto-hydrodynamics. Advances in Applied Mechanics, 7:215–316, 1962. doi:10.1016/s0065-2156(08)70123-6.

    H. K. Moffatt. Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press, Cambridge, England; New York, NY, USA, 1978. ISBN 9780521216401. Cambridge Monographs on Mechanics and Applied Mathematics.

    Russell M. Kulsrud. Plasma Physics for Astrophysics. Princeton University Press, Princeton, NJ, 2005. ISBN 9780691120737.

    Jeffrey P. Freidberg. Ideal Magnetohydrodynamics. Plenum Press, New York, NY, USA, 1987. ISBN 9780306425127.

    Dalton D. Schnack. Lectures in Magnetohydrodynamics: With an Appendix on Extended MHD, volume 780 of Lecture Notes in Physics. Springer, Berlin, Heidelberg, 2009. ISBN 9783642006876. doi:10.1007/978-3-642-00688-3.

Problems

Problem 1.1. The MHD Equations and Their Closure

(a)
Starting from the single-fluid MHD equations, write down the full set of resistive, viscous MHD equations for:

Clearly identify which equations are:

(b)
Count the number of scalar unknowns and scalar equations. Explain why constitutive relations (closures) for the viscous stress tensor \(\tens {\Pi }\) and heat flux \(\vect {q}\) are required.
(c)
In ideal MHD, Ohm’s law is \[ \E + \uvec \times \B = 0. \] Explain physically what assumptions are required to neglect:

State clearly the ordering assumptions (spatial and temporal) underlying ideal MHD.