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Lecture 0
Impedance Matching: The Background You Already Have

MHD looks new mostly because undergraduate physics is taught in compartments.

Magnetohydrodynamics often looks more exotic than it is because vector calculus, electromagnetism, hydrostatics, ideal flow, viscous flow, and continuum mechanics are usually taught in separate compartments. This lecture is meant to reconnect those subjects before the full MHD system appears.

The point is not that MHD is easy. It is not. The point is that most of its first language is already familiar. If you know what a flux is, what a circulation is, what a divergence measures, what pressure does in a fluid, what Maxwell’s equations say, and why the Navier–Stokes equation is a momentum balance, then you already possess most of the mathematical reflexes needed to start reading MHD.

Roadmap

This opening lecture is intentionally preparatory. Part I turns this background into the actual single-fluid equations and their closures. Part III then develops force balance and Grad–Shafranov structure, Part IV turns those equilibria into wave and stability problems, and Part V carries that logic into the classic kink, interchange, internal-kink, TAE, and ballooning calculations. Later still, Part VI revisits many of the same ideas once ideal constraints are relaxed.

Overview

This is not a remedial lecture. It is a reactivation lecture. We are not trying to reteach everything from undergraduate electromagnetism and fluid dynamics. We are trying to remind you that those subjects are assumed background, and that MHD is built from them rather than placed on top of them as a completely separate edifice.

At the same time, there is one honest warning: if Maxwell’s equations in both integral and differential form still feel foreign, or if the idea of a control volume and a stress tensor is entirely new, then this is the moment to pause and repair that background before going deeper. The rest of the book will assume it.

0.1 The shared geometry of fields and flows

The first piece of soft impedance matching is geometric. Long before one writes a plasma momentum equation, one has already learned the two integral theorems that organize the subject:

\[\int _V \nabla \cdot \mathbf {F}\, dV = \oint _{\partial V} \mathbf {F}\cdot \mathbf {n}\, dA, \qquad \int _S (\nabla \times \mathbf {F})\cdot \mathbf {n}\, dA = \oint _{\partial S} \mathbf {F}\cdot d\boldsymbol {\ell }.\]
These are not decorative facts from a vector-calculus course. They are the machinery that turns local statements into global ones, and they are the reason that fluid mechanics and electromagnetism feel like close cousins.

In a fluid course, one learns to think in terms of a control volume: what enters, what leaves, and what accumulates. In electromagnetism, one learns essentially the same geometry in a different vocabulary: how much field threads a surface, how much circulation is measured around a loop, and how local sources appear as divergences or curls.

For example, mass conservation for a fixed control volume \(V\) reads

\[\frac {d}{dt}\int _V \rho \, dV = -\oint _{\partial V} \rho \mathbf {u}\cdot \mathbf {n}\, dA,\]
which becomes, after the divergence theorem,
\[\partial _t \rho + \nabla \cdot (\rho \mathbf {u}) = 0. \tag{0.3}\]
This is already a prototype for how much of MHD works: a conservation law written first as balance over a volume and then as a local differential equation.

You have seen this structure before

At a very high level,

\[\text {accumulation in a region} = -\text {outward flux through the boundary} + \text {sources inside the region}.\]
That sentence organizes the continuity equation, Gauss’s law, the induction equation, charge conservation, heat balance, and much of continuum mechanics.

Fluid language Electromagnetic language
mass flux \(\rho \mathbf {u}\) magnetic flux \(\mathbf {B}\) through a surface
circulation \(\oint \mathbf {u}\cdot d\boldsymbol {\ell }\) emf \(\oint \mathbf {E}\cdot d\boldsymbol {\ell }\)
control volume balance integral Maxwell law
local conservation equation differential field equation

The nouns change, but the geometry does not.

A second important idea is the material derivative

\[\frac {D}{Dt} = \partial _t + \mathbf {u}\cdot \nabla ,\]
which asks for the rate of change seen by a moving parcel of fluid. That same habit of mind—following a moving object rather than staring at a fixed point in space—returns everywhere in MHD. Advection, frozen-in flux, transport of vorticity, and the kinematics of field-line motion all depend on this distinction.

0.2 A one-page identity sheet

There are a handful of vector and tensor identities that get used so often in MHD that it is worth having them in one place. They are not profound by themselves, but they are the algebraic hinges on which many derivations turn.

Overview

The quickest way to get lost in a first reading of MHD is often not physics but algebra. If an expression involving a divergence, curl, dyad, or convective derivative suddenly feels unfamiliar, it is usually one of the identities below.

Vector identities used constantly

\[\begin{aligned}\nabla \cdot (\phi \mathbf {A}) &= \phi \, \nabla \cdot \mathbf {A} + \mathbf {A}\cdot \nabla \phi ,\\ \nabla \times (\phi \mathbf {A}) &= \nabla \phi \times \mathbf {A} + \phi \, \nabla \times \mathbf {A},\\ \nabla \cdot (\mathbf {A}\times \mathbf {B}) &= \mathbf {B}\cdot (\nabla \times \mathbf {A}) - \mathbf {A}\cdot (\nabla \times \mathbf {B}),\\ \nabla \times (\mathbf {A}\times \mathbf {B}) &= (\mathbf {B}\cdot \nabla )\mathbf {A} - (\mathbf {A}\cdot \nabla )\mathbf {B} + \mathbf {A}\,\nabla \cdot \mathbf {B} - \mathbf {B}\,\nabla \cdot \mathbf {A},\\ \nabla (\mathbf {A}\cdot \mathbf {B}) &= (\mathbf {A}\cdot \nabla )\mathbf {B} + (\mathbf {B}\cdot \nabla )\mathbf {A} + \mathbf {A}\times (\nabla \times \mathbf {B}) + \mathbf {B}\times (\nabla \times \mathbf {A}),\\ (\mathbf {u}\cdot \nabla )\mathbf {u} &= \nabla \!\left (\frac {u^2}{2}\right ) - \mathbf {u}\times (\nabla \times \mathbf {u}).\end{aligned} \tag{0.11}\]

The last identity is especially useful because it rewrites the nonlinear advection term in terms of a gradient plus a vorticity piece. This is one reason vorticity appears so naturally in both fluid mechanics and MHD.

Tensor and dyadic identities used repeatedly

Write the dyad \(\mathbf {A}\mathbf {B}\) componentwise as \((\mathbf {A}\mathbf {B})_{ij}=A_i B_j\). Then

\[\begin{aligned}\nabla \cdot (\mathbf {A}\mathbf {B}) &= (\mathbf {B}\cdot \nabla )\mathbf {A} + \mathbf {A}\,\nabla \cdot \mathbf {B},\\ \nabla \cdot (\phi \mathbf {I}) &= \nabla \phi ,\\ \nabla \cdot (\nabla \mathbf {u}) &= \nabla ^2\mathbf {u},\\ \nabla \cdot \!\left [(\nabla \mathbf {u})^{T}\right ] &= \nabla (\nabla \cdot \mathbf {u}),\\ \nabla \cdot \!\left [\nabla \mathbf {u}+(\nabla \mathbf {u})^{T}\right ] &= \nabla ^2\mathbf {u} + \nabla (\nabla \cdot \mathbf {u}).\end{aligned} \tag{0.16}\]

These identities are the reason the divergence of the Newtonian stress tensor turns into the familiar viscous operators in Navier–Stokes and MHD. In the incompressible limit, \(\nabla \cdot \mathbf {u}=0\), the last line collapses immediately to a Laplacian.

Why this page matters later

These formulas will be used, often without ceremony, in the induction equation, in the conversion between conservative and advective forms of the momentum equation, in the derivation of vorticity equations, in stress-tensor manipulations, and in the magnetic-force rewrite \(\mathbf {J}\times \mathbf {B}=\nabla \cdot \mathbf {T}_{\rm M}\) using the Maxwell stress tensor.

0.3 Electromagnetism, reactivated

The Maxwell equations are the electromagnetic side of the bridge. In SI units they are

\[\begin{aligned}\nabla \cdot \mathbf {E} &= \frac {\rho _e}{\varepsilon _0}, \\ \nabla \cdot \mathbf {B} &= 0, \\ \nabla \times \mathbf {E} &= -\partial _t \mathbf {B}, \\ \nabla \times \mathbf {B} &= \mu _0 \mathbf {J} + \mu _0\varepsilon _0 \partial _t \mathbf {E}.\end{aligned} \tag{0.17}\]

For first-course MHD, the most important equations are usually (0.18) and (0.19), together with a reduced form of Ampère’s law in which the displacement current is neglected:

\[\nabla \times \mathbf {B} \approx \mu _0 \mathbf {J}. \tag{0.21}\]
This is the standard non-relativistic, low-frequency limit in which the characteristic speeds are small compared with the speed of light and the electromagnetic evolution is slaved to the conducting medium.

The Lorentz force density,

\[\mathbf {f}_{\mathrm L} = \rho _e \mathbf {E} + \mathbf {J}\times \mathbf {B},\]
is the term that eventually enters the momentum equation. In quasineutral single-fluid MHD, the \(\rho _e\mathbf {E}\) part is often negligible at leading order, leaving the celebrated \(\mathbf {J}\times \mathbf {B}\) force.

A simple resistive form of Ohm’s law then reads

\[\mathbf {J} = \sigma \left (\mathbf {E} + \mathbf {u}\times \mathbf {B}\right ), \tag{0.23}\]
or, equivalently,
\[\mathbf {E} + \mathbf {u}\times \mathbf {B} = \eta \mathbf {J}, \qquad \eta = \frac {1}{\sigma }.\]
Combining (0.19), (0.21), and (0.23) yields the induction equation,
\[\partial _t \mathbf {B} = \nabla \times (\mathbf {u}\times \mathbf {B}) + \eta _m \nabla ^2 \mathbf {B}, \qquad \eta _m = \frac {1}{\mu _0 \sigma }, \tag{0.25}\]
when \(\eta _m\) is uniform and one uses \(\nabla \cdot \mathbf {B}=0\).

Electromagnetic core ideas

For the purposes of early MHD, you do not need every corner of undergraduate electrodynamics equally.

You do need to be comfortable with the following ideas:

1.
magnetic fields are divergence-free, so magnetic flux is constrained rather than sourced locally;
2.
time-varying magnetic fields generate electric fields through Faraday’s law;
3.
currents generate magnetic fields through Ampère’s law;
4.
the Lorentz force couples the field back to the material motion.

Everything in elementary MHD is a disciplined exploitation of those four facts.

Historical Perspective

Faraday taught physicists to think in terms of lines of force and field structure; Maxwell converted that physical picture into a mathematical field theory whose mature statement appeared in the Treatise of 1873 Maxwell [1873]. For our purposes, the important lesson is not merely historical reverence. It is methodological. MHD inherits from Maxwell a willingness to treat fields as dynamical objects, not just as bookkeeping devices attached to particles.

0.4 Fluid mechanics, reactivated

On the fluid side, the most economical statement is that momentum balance for a continuum can be written as

\[\rho \frac {D\mathbf {u}}{Dt} = \nabla \cdot \boldsymbol {\sigma } + \rho \mathbf {g} + \mathbf {f}_{\mathrm {other}}, \tag{0.26}\]
where \(\boldsymbol {\sigma }\) is the stress tensor. The familiar Navier–Stokes equation is simply a special constitutive choice for \(\boldsymbol {\sigma }\).

For a Newtonian fluid,

\[\boldsymbol {\sigma } = -p\mathbf {I} + \boldsymbol {\tau },\]
with viscous part
\[\boldsymbol {\tau } = 2\mu \mathbf {D} + \lambda (\nabla \cdot \mathbf {u})\mathbf {I}, \tag{0.28}\]
where \(\mu \) is the dynamic viscosity, \(\lambda \) is the second viscosity coefficient, and
\[\mathbf {D} = \frac {1}{2}\left (\nabla \mathbf {u} + (\nabla \mathbf {u})^{T}\right ) \tag{0.29}\]
is the symmetric rate-of-strain tensor.

If the flow is incompressible and \(\mu \) is constant, then \(\nabla \cdot \mathbf {u}=0\) and (0.26) becomes the familiar form

\[\rho \left (\partial _t\mathbf {u}+\mathbf {u}\cdot \nabla \mathbf {u}\right ) = -\nabla p + \mu \nabla ^2\mathbf {u} + \rho \mathbf {g}. \tag{0.30}\]
Without the viscous term one recovers Euler’s equation,
\[\rho \frac {D\mathbf {u}}{Dt} = -\nabla p + \rho \mathbf {g}. \tag{0.31}\]

Hydrostatics and Archimedes

Hydrostatics is the equilibrium limit \(\mathbf {u}=0\). Then the momentum equation reduces to

\[\nabla p = \rho \mathbf {g}. \tag{0.32}\]
This one line contains the usual “pressure increasing with depth” intuition, and it also contains Archimedes’ principle.

To see that, let a body occupy a volume \(V\) with boundary \(\partial V\). The net force exerted on it by the surrounding fluid pressure is

\[\mathbf {F}_b = -\oint _{\partial V} p\, \mathbf {n}\, dA.\]
Applying the divergence theorem componentwise gives
\[\mathbf {F}_b = -\int _V \nabla p\, dV = -\int _V \rho \mathbf {g}\, dV.\]
If the ambient fluid density is uniform, this becomes
\[\mathbf {F}_b = -\rho V \mathbf {g},\]
which is precisely the weight of the displaced fluid. That is Archimedes’ principle in continuum-mechanics form.

Historical Perspective

Archimedes before the modern language. Archimedes’ On Floating Bodies is vastly older than the vector-calculus notation we now use, but the physical content is already recognizable: equilibrium, buoyancy, and the relation between force balance and displaced volume Archimedes [1897]. It is worth recalling this because MHD often feels modern in notation while being classical in structure.

Bernoulli and ideal motion

For steady, incompressible, inviscid flow with conservative body force \(\mathbf {g}=-\nabla \Phi \), Euler’s equation implies Bernoulli’s relation along a streamline:

\[p + \frac {1}{2}\rho u^2 + \rho \Phi = \text {constant along a streamline}. \tag{0.36}\]
Bernoulli’s law is a first example of a more general idea: under appropriate assumptions, momentum balance can be integrated into a useful scalar invariant. MHD returns to this repeatedly, sometimes in familiar form and sometimes in far less obvious geometries.

Historical Perspective

Bernoulli, Euler, Navier, and Stokes. Bernoulli’s Hydrodynamica of 1738 is one of the great founding texts of fluid mechanics Bernoulli [1738]. Euler’s 1757 memoir gave a systematic differential formulation for ideal-fluid motion Euler [1757]. In the nineteenth century, Navier and Stokes supplied what became the standard viscous extension Navier [1823], Stokes [1845]. If you want a historical narrative that treats these developments as a connected story rather than a list of surnames attached to equations, Darrigol’s history is especially useful Darrigol [2005]; Tamburrino’s bicentenary article is also handy for the Navier-to-Stokes arc Tamburrino [2024].

Vorticity and circulation

Another fluid idea that will reappear in MHD is vorticity,

\[\boldsymbol {\omega } = \nabla \times \mathbf {u},\]
and its integral partner, circulation,
\[\Gamma = \oint _C \mathbf {u}\cdot d\boldsymbol {\ell }.\]
These are not just diagnostics for swirl. They are examples of how curl, line integrals, and transport of structure naturally organize dynamics. The same geometric habits reappear in magnetic induction, flux conservation, and the role of topology in plasma dynamics.

0.5 Stress, shear, strain, and tensors without apology

This part is often compressed in earlier courses, but it matters enough in MHD that it should be stated plainly.

The displacement field \(\boldsymbol {\xi }\) of a solid gives the small-strain tensor

\[\boldsymbol {\varepsilon } = \frac {1}{2}\left (\nabla \boldsymbol {\xi } + (\nabla \boldsymbol {\xi })^{T}\right ).\]
In a fluid, where the material keeps deforming, one usually works instead with the rate-of-strain tensor already introduced in (0.29),
\[\mathbf {D} = \frac {1}{2}\left (\nabla \mathbf {u} + (\nabla \mathbf {u})^{T}\right ).\]
The antisymmetric part of the velocity gradient,
\[\mathbf {W} = \frac {1}{2}\left (\nabla \mathbf {u} - (\nabla \mathbf {u})^{T}\right ),\]
represents local rigid-body rotation. Thus
\[\nabla \mathbf {u} = \mathbf {D} + \mathbf {W}.\]
The symmetric part deforms a parcel; the antisymmetric part rotates it.

For a simple shear flow

\[\mathbf {u} = \dot \gamma y\, \hat {\mathbf {x}},\]
one finds
\[\nabla \mathbf {u}= \begin {pmatrix} 0 & \dot \gamma & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end {pmatrix}, \quad \mathbf {D}= \frac {\dot \gamma }{2} \begin {pmatrix} 0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0 \end {pmatrix}, \quad \mathbf {W}= \frac {\dot \gamma }{2} \begin {pmatrix} 0 & 1 & 0\\ -1 & 0 & 0\\ 0 & 0 & 0 \end {pmatrix}.\]
This is a useful reminder that shear is partly deformation and partly local rotation. In MHD, where shear layers, current sheets, and anisotropic transport matter, those distinctions are not optional niceties.

The stress tensor \(\boldsymbol {\sigma }\) is equally central. Its meaning is simple even if the notation initially looks heavy: it is the linear map that turns a surface normal into the traction acting on that surface. Once written that way, the momentum equation (0.26) becomes the natural continuum version of Newton’s second law.

Historical Perspective

Cauchy and the modern continuum viewpoint. The modern tensorial description of stress is due in large part to Cauchy in the early 1820s Cauchy [1823]. This is one reason continuum mechanics belongs in the prologue to MHD. The tensor language is not an ornamental modernization added after the fact; it is the natural language for local force transmission in continuous media.

Why tensor language appears here

Tensor notation often appears abruptly, but the underlying idea is simple: a second-rank tensor is the natural way to encode a linear map between directions and forces, between normals and fluxes, or between gradients and responses.

In MHD this becomes unavoidable. Even when one begins with a scalar pressure, the electromagnetic stress already has tensor structure, and kinetic plasma theory later promotes pressure itself to a tensor in many important limits.

0.6 The first MHD bridge

Now we can say in one line what idealized single-fluid MHD is trying to do. It couples the fluid balances to the electromagnetic field equations. A minimal system is

\[\begin{aligned}\partial _t \rho + \nabla \cdot (\rho \mathbf {u}) &= 0, \\ \rho \frac {D\mathbf {u}}{Dt} &= -\nabla p + \mathbf {J}\times \mathbf {B} + \mu \nabla ^2\mathbf {u} + \rho \mathbf {g}, \\ \partial _t \mathbf {B} &= \nabla \times (\mathbf {u}\times \mathbf {B}) + \eta _m \nabla ^2\mathbf {B}, \\ \nabla \cdot \mathbf {B} &= 0, \\ \nabla \times \mathbf {B} &= \mu _0\mathbf {J}.\end{aligned}\]

There are many refinements—energy closure, Hall terms, anisotropic viscosity, anisotropic conductivity, two-fluid effects, kinetic corrections—but this much is enough to make the conceptual point.

MHD is not “electromagnetism plus a fluid glued on.” Nor is it “fluid mechanics with a magnetic forcing term tacked onto the right-hand side.” It is a coupled field theory in which the medium shapes the field and the field pushes back on the medium.

There is a particularly nice analogy, emphasized already in classic equilibrium work, between hydromagnetic equilibria and incompressible fluid motion. Indeed, Grad and Rubin explicitly noted that their equilibrium equations are formally identical to those of incompressible flow if one identifies the magnetic field with a fluid velocity and the pressure term with a Bernoulli-like quantity Grad and Rubin [1958]. Shafranov likewise made the connection between magnetohydrodynamic equilibria and hydrodynamical vortices in his 1958 paper Shafranov [1958]. It is worth stating this bridge early because it clarifies how naturally equilibrium theory fits into the broader continuum framework.

Static equilibrium is the cleanest first example. If the velocity vanishes and gravity is ignored, then the momentum equation reduces to

\[\nabla p = \mathbf {J}\times \mathbf {B}. \tag{0.50}\]
This is already a profoundly instructive equation. It says that pressure gradients are balanced not by inertia or viscosity but by magnetic stresses. Later, under axisymmetry, this balance becomes the Grad–Shafranov equation for a flux function \(\psi \). The important point here is continuity of structure: the Grad–Shafranov operator is a more specialized descendant of familiar mathematical habits—flux functions, elliptic operators, balance laws, and careful boundary conditions—not an alien object dropped from orbit.

Historical Perspective

Hartmann’s 1937 mercury-flow papers are among the clearest early demonstrations that electrically conducting fluids in magnetic fields really do constitute a distinct physical regime Hartmann and Lazarus [1937]. In fact, Hartmann explicitly wrote of the emergence of a “new field of research,” the field of “Hg-Dynamics.” A few years later Alfvén identified the wave mode that now bears his name Alfvén [1942]. By 1958, equilibrium theory had reached the landmark papers of Grad–Rubin and Shafranov Grad and Rubin [1958], Shafranov [1958]. This is a good miniature history of the subject itself: laboratory flows, waves, and equilibrium theory grew together rather than in isolation.

0.7 A roadmap for the rest of the book

A compact historical sequence is:

Takeaways
  • Archimedes: hydrostatics, buoyancy, floating bodies Archimedes [1897].
  • Bernoulli and Euler: ideal flow, pressure, energy-like invariants, differential equations of motion Bernoulli [1738], Euler [1757].
  • Cauchy, Navier, and Stokes: stress, continuum force balance, viscosity, and the modern momentum equation Cauchy [1823], Navier [1823], Stokes [1845].
  • Faraday and Maxwell: field language and electromagnetic dynamics, mathematically consolidated by Maxwell Maxwell [1873].
  • Hartmann: conducting-liquid flow in magnetic fields as a distinct laboratory subject Hartmann and Lazarus [1937], Hartmann [1937].
  • Alfvén: wave dynamics specific to conducting media Alfvén [1942].
  • Grad, Rubin, and Shafranov: the equilibrium mathematics central to fusion and plasma confinement Grad and Rubin [1958], Shafranov [1958].

That compressed sequence is not the whole history, but it already shows why MHD deserves to be called a classic subject. It sits at the confluence of two of the great nineteenth-century field theories—continuum mechanics and electromagnetism—and it becomes experimentally urgent as soon as one cares about liquid metals, plasma discharges, astrophysical flows, or magnetic confinement.

Core ideas carried forward

By the end of this lecture, the following claims should feel reasonable:

1.
MHD begins from mathematics you already know: divergences, curls, fluxes, circulation, balance laws, and constitutive closure.
2.
Maxwell’s equations are assumed background, not optional enrichment.
3.
Navier–Stokes is not a distraction from MHD; it is one of its parents.
4.
Hydrostatics, Bernoulli, shear, strain, and stress tensors are not side topics. They are part of the conceptual grammar needed later.
5.
When the book turns to magnetic induction, equilibrium, waves, instabilities, and experiments, the right response is not “this is a new universe.” It is “these are familiar structures in a stronger coupling.”

For readers who want a little more background before moving on, Batchelor remains an excellent fluid-mechanics anchor Batchelor [1967]; Chandrasekhar is a classic for stability and the fluid–magnetic interface Chandrasekhar [1961]; Davidson is a clear entry point for modern MHD Davidson [2001]; and the edited volume by Molokov, Moreau, and Moffatt is useful for historical perspective across several branches of the subject Molokov et al. [2007].

The rest of the book can now move without apology into MHD proper. We have not proven everything we will need, and we have certainly not reviewed every detail. But we have done the more important thing: we have reminded ourselves that the subject begins on familiar ground.

Bibliography

    James Clerk Maxwell. A Treatise on Electricity and Magnetism. Clarendon Press, Oxford, 1873. 2 vols.

    Archimedes. The Works of Archimedes. Cambridge University Press, Cambridge, 1897. Contains On Floating Bodies, Books I–II.

    Daniel Bernoulli. Hydrodynamica, sive de viribus et motibus fluidorum commentarii. Johann Reinhold Dulsecker, Argentorati, 1738.

    Leonhard Euler. Principes généraux du mouvement des fluides. Mémoires de l’Académie Royale des Sciences et des Belles-Lettres de Berlin, 11:274–315, 1757.

    Claude-Louis Navier. Mémoire sur les lois du mouvement des fluides. Mémoires de l’Académie des Sciences de l’Institut de France, 6, 1823. Read in 1822; the volume is commonly cited as 1823 and was issued in 1827.

    George Gabriel Stokes. On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. Transactions of the Cambridge Philosophical Society, 8: 287–319, 1845.

    Olivier Darrigol. Worlds of Flow: A History of Hydrodynamics from the Bernoullis to Prandtl. Oxford University Press, Oxford, 2005.

    A. Tamburrino. From navier to stokes: Commemorating the bicentenary of navier’s equation on the lay of fluid motion. Fluids, 9(1):15, 2024. doi:10.3390/fluids9010015.

    Augustin-Louis Cauchy. Recherches sur l’équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques. Bulletin de la Société Philomathique de Paris, pages 9–13, 1823.

    Harold Grad and H. Rubin. Hydromagnetic equilibria and force-free fields. Technical report, New York University, Institute of Mathematical Sciences, October 1958. Also circulated through the 2nd International Conference on the Peaceful Uses of Atomic Energy, Geneva.

    V. D. Shafranov. On magnetohydrodynamical equilibrium configurations. Soviet Physics JETP, 6:545–554, 1958.

    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. Existence of electromagnetic-hydrodynamic waves. Nature, 150(3805):405–406, 1942.

    J. Hartmann. Hg Dynamics I: Theory of the laminar flow of an electrically conducting liquid in a homogeneous magnetic field". Mathematisk-fysiske Meddelelser, 15(6), 1937.

    G. K. Batchelor. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 1967.

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

    P. A. Davidson. An Introduction to Magnetohydrodynamics. Cambridge University Press, Cambridge, 2001.

    S. Molokov, R. Moreau, and H. K. Moffatt, editors. Magnetohydrodynamics: Historical Evolution and Trends. Springer, Dordrecht, 2007.