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Lecture 1
Foundations of MHD

Overview

In this lecture we build the foundation for the rest of the notes.

1.
It places MHD in its historical and fluid-mechanical setting and states the resistive-MHD system.
2.
It explains where pressure, viscosity, resistivity, and dissipation actually enter, so the equations become an honest transport model rather than a symbolic list.
3.
It rewrites the same dynamics in conservative form, where fluxes through boundaries and global conservation laws become explicit.

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 The Resistive MHD System

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.

Historical Perspective

A short historical note.

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.

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.

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.

Caution

Caution. 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.

The governing equations. 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.

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.

1.2 Stress, Resistivity, and Dissipation

The equations in (1.7)–(1.13) look deceptively compact. Their real content depends on what is meant by stress, conductivity, and heat transport. In liquid metals the closures can be very close to the ordinary Navier–Stokes ones; in plasmas they are often anisotropic and collisionality-dependent. This section keeps the algebra at the single-fluid level but prepares the ground for that later distinction.

Historical Perspective

One of the oldest conceptual moves in continuum mechanics is the distinction between forces that act throughout a volume and forces transmitted through its boundary. In MHD this distinction becomes especially useful because the Lorentz force behaves as a body force in the momentum equation, while viscosity and pressure naturally enter through a stress tensor.

Body and surface forces. Let \(V\) be a material volume bounded by a surface \(S\) with outward area element \[ d\vect {S} = \hat {\vect {n}}\, dS. \] We write the total force density as the sum of a body-force term and a surface-force term.

Body forces. Typical examples are

\[\vect {f}_{\mathrm {body}} = \rho \vect {g} + \J \times \B + \rho _e \E . \tag{1.17}\]
Under the quasi-neutral approximation, \(\rho _e \approx 0\), so the dominant electromagnetic body force is the Lorentz force \(\J \times \B \) already used in (1.8).

Surface forces. Introduce a pressure-stress tensor \(\tens {P}\) so that the force exerted on the fluid by its surroundings is

\[d\vect {F}_{\mathrm {surf}} = - \tens {P}\cdot d\vect {S}. \tag{1.18}\]
The total surface force on the volume is then
\[\vect {F}_{\mathrm {surf}} = - \oint _S \tens {P}\cdot d\vect {S} = - \int _V \divergence \tens {P}\, dV. \tag{1.19}\]

Therefore the local momentum equation is

\[\rho \frac {D\uvec }{Dt} = -\divergence \tens {P} + \vect {f}_{\mathrm {body}}. \tag{1.20}\]

Symmetry and angular momentum. Angular momentum conservation requires the stress tensor to be symmetric. To see this, define

\[\vect {L} = \int _V \rho \, \vect {r}\times \uvec \, dV. \tag{1.21}\]
Using (1.20) and integrating by parts gives a surface torque plus the volume term
\[\int _V \epsilon _{ijk} P_{kj}\, dV. \tag{1.22}\]
This vanishes for arbitrary material volumes only if
\[\boxed { P_{ij}=P_{ji}. } \tag{1.23}\]

That result is worth remembering: the symmetry of the stress tensor is not a decorative assumption but the local statement of angular-momentum conservation.

Pressure and viscous stress. We decompose the pressure-stress tensor as

\[\boxed { \tens {P} = p\,\tens {I} - \tens {\Pi }, } \tag{1.24}\]
where \(p\) is the isotropic pressure and \(\tens {\Pi }\) is the viscous stress tensor.

Substituting (1.24) into (1.20) gives

\[\boxed { \rho \frac {D\uvec }{Dt} = -\grad p + \divergence \tens {\Pi } + \J \times \B + \rho \vect {g}. } \tag{1.25}\]
Equation (1.25) is the more explicit version of (1.8): the pressure term and the viscous term are simply the divergence of the surface stress written in split form.

For an isotropic Newtonian fluid,

\[\boxed { \tens {\Pi } = \mu \left [ \grad \uvec + (\grad \uvec )^T - \frac {2}{3} (\divergence \uvec )\tens {I} \right ] + \zeta (\divergence \uvec )\tens {I}, } \tag{1.26}\]
where \(\mu \) and \(\zeta \) are the shear and bulk viscosities. If the flow is incompressible and \(\mu \) is uniform, then
\[\divergence \tens {\Pi } = \mu \lap \uvec . \tag{1.27}\]

Caution

The isotropic Newtonian closure is not the last word in plasma MHD. It is a very useful model for liquid metals and for introductory derivations, but in a magnetized plasma the true viscous stress is generally anisotropic. That is exactly the issue revisited in Lecture 3.

Resistive induction. The magnetic field evolves according to the induction equation (1.13). For uniform resistivity, using Ampère’s law (1.11) and \(\divergence \B =0\) from (1.12), one finds

\[\boxed { \pp {\B }{t} = \curl (\uvec \times \B ) + \eta _{\mathrm {m}} \lap \B , \qquad \eta _{\mathrm {m}} \equiv \frac {\eta }{\muo }. } \tag{1.28}\]

The first term on the right describes advection and stretching of magnetic field, while the second describes diffusion of magnetic structure. In later lectures this competition will reappear as the magnetic Reynolds number and the frozen-in limit.

Dissipation and heating. Two positive-definite sinks of ordered mechanical or magnetic energy appear naturally:

\[\boxed { Q_{\mathrm {visc}} = \tens {\Pi }:\grad \uvec , \qquad Q_{\mathrm {ohm}} = \eta J^2. } \tag{1.29}\]
These are the terms that feed the internal-energy equation (1.14). Viscosity damps velocity gradients; resistivity damps current gradients. Both convert organized motion or field structure into heat.

Experimental perspective. This is one of the places where theory first starts to feel like hardware. In liquid-metal MHD, (1.25) and (1.28) can be almost embarrassingly literal: pressure gradients drive the flow, the Lorentz force brakes or redirects it, and the diffusive coefficients can often be measured independently. In laboratory plasmas the same equations still organize the dynamics, but the closure hidden in \(\tens {\Pi }\) and in the conductivity is much more delicate. That is why the question “what is the right MHD model?” is often really a question about transport.

Takeaways
  • Pressure and viscosity are best understood as parts of a stress tensor acting across surfaces.
  • The explicit resistive–viscous momentum equation is (1.25); the corresponding induction equation is (1.28).
  • Viscous and Ohmic dissipation, (1.29), are not afterthoughts. They are the channels through which ordered flow and magnetic structure are irreversibly degraded.

1.3 Conservative Form

The point of conservative form is not aesthetic. It is the form in which one can integrate over a control volume and read off exactly what crosses the boundary. Starting from the continuity equation (1.7), the explicit momentum equation (1.25), and the induction equation (1.28), we now reorganize the same physics into flux form.

Historical Perspective

The conservative rewriting of continuum equations is older than MHD itself, but in magnetized fluids it acquires special force because the magnetic field transports momentum and energy through stresses and fluxes that are easy to obscure in non-divergence form. The Maxwell-stress interpretation, the Poynting-flux interpretation, and the later development of finite-volume numerical methods all reinforce the same lesson: if one wants to understand shocks, jump conditions, and global conservation laws, the divergence form is the natural language.

Mass conservation.

\[\boxed { \pp {\rho }{t} + \divergence (\rho \uvec ) = 0. } \tag{1.30}\]
This is already conservative. It says that mass changes inside a volume only because the mass flux \(\rho \uvec \) crosses its boundary.

Momentum flux. Using (1.25) together with (1.30), one obtains

\[\boxed { \pp {(\rho \uvec )}{t} + \divergence \left ( \rho \uvec \uvec + p\,\tens {I} - \tens {\Pi } \right ) = \J \times \B . } \tag{1.31}\]
The dyadic term \(\rho \uvec \uvec \) is advective momentum transport; \(p\,\tens {I}-\tens {\Pi }\) is the mechanical stress from Section 1.2.

Lorentz force as a divergence. Using Ampère’s law (1.11) and \(\divergence \B =0\) from (1.12),

\[\J \times \B = \divergence \left [ \left ( \frac {B^2}{2\muo } \right )\tens {I} - \frac {\B \B }{\muo } \right ]. \tag{1.32}\]
Substituting (1.32) into (1.31) gives the full conservative momentum equation
\[\boxed { \pp {(\rho \uvec )}{t} + \divergence \tens {T} = 0, } \tag{1.33}\]
with total momentum-flux tensor
\[\boxed { \tens {T} = \rho \uvec \uvec + \left ( p+\frac {B^2}{2\muo } \right )\tens {I} - \frac {\B \B }{\muo } - \tens {\Pi }. } \tag{1.34}\]

Caution

The isotropic term \((B^2/2\muo )\tens {I}\) looks like a pressure and \(-\B \B /\muo \) looks like a tension, but they should really be interpreted together through (1.32) and (1.34). The field transports momentum as a tensor, not as a scalar pressure with an optional correction.

Energy densities. Define the total energy density

\[\mathcal {E} = \frac {1}{2}\rho u^2 + \rho e + \frac {B^2}{2\muo }. \tag{1.35}\]
Its three pieces are kinetic, internal, and magnetic energy.

Kinetic energy equation. Dotting (1.25) with \(\uvec \) and using (1.30) gives

\[\pp {}{t} \left ( \frac {1}{2}\rho u^2 \right ) + \divergence \left [ \left ( \frac {1}{2}\rho u^2 \right )\uvec + \left ( p\,\tens {I} - \tens {\Pi } \right )\cdot \uvec \right ] = p\,\divergence \uvec - \tens {\Pi }:\grad \uvec + \J \cdot \E - \eta J^2. \tag{1.36}\]

Magnetic energy equation. Starting from Faraday’s law (1.10) and Ampère’s law (1.11),

\[\pp {}{t} \left ( \frac {B^2}{2\muo } \right ) + \divergence \left ( \frac {\E \times \B }{\muo } \right ) = - \J \cdot \E . \tag{1.37}\]
The flux \(\E \times \B /\muo \) is the Poynting flux.

Internal energy equation. A convenient conservative form for the internal energy is

\[\pp {(\rho e)}{t} + \divergence (\rho e\,\uvec ) = - p\,\divergence \uvec - \divergence \vect {q} + \tens {\Pi }:\grad \uvec + \eta J^2. \tag{1.38}\]
For an ideal gas,
\[e=\frac {p}{(\gamma -1)\rho }. \tag{1.39}\]
If, in addition, the flow is adiabatic and we neglect \(\tens {\Pi }\), \(\vect {q}\), and \(\eta \), then (1.38) reduces to
\[\boxed { \frac {Dp}{Dt} = -\gamma p\,\divergence \uvec , } \tag{1.40}\]
which is equivalent to the adiabatic law (1.14).

Total energy conservation. Adding (1.36), (1.37), and (1.38) gives

\[\boxed { \pp {\mathcal {E}}{t} + \divergence \left [ \left ( \frac {1}{2}\rho u^2 + \rho e + p \right )\uvec - \tens {\Pi }\cdot \uvec + \vect {q} + \frac {\E \times \B }{\muo } \right ] = 0. } \tag{1.41}\]

Why this form matters. For shocks, discontinuities, and numerical algorithms, (1.30), (1.33), and (1.41) are the natural starting point because they survive volume integration without hidden integration-by-parts steps. Conservative form is where the geometry of fluxes becomes algebraically explicit.

Takeaways
  • Magnetic stresses belong in the same flux bookkeeping as fluid stresses.
  • The Maxwell stress in (1.32) is the cleanest way to see how magnetic fields carry momentum.
  • The conservative pressure law (1.40) is not fundamental by itself; it is the reduced form that survives after a closure has been chosen.

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.

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

    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.

    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.

    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.

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

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

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

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

    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.

    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.

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.