Magnetohydrodynamics

Take the two great continuum theories this course has built — Maxwell's electrodynamics and the mechanics of fluids — and pour them into the same volume of matter. A liquid metal, the ionized plasma of a star, the molten iron of Earth's outer core: each is simultaneously a fluid that flows and a conductor that carries current. The moving conductor generates currents that reshape the magnetic field; the reshaped field drives currents that push back on the flow through the Lorentz force. Neither theory alone can describe this feedback loop. The merged theory is magnetohydrodynamics — MHD — and it is the working language of most of the visible universe, because most of the visible universe is conducting fluid threaded by magnetic field. The miracle is how much Maxwell's equations simplify inside a slow, quasi-neutral conductor. The displacement current dies (it is down by $v^2/c^2$ ), charge density becomes a spectator, and the entire electromagnetic sector collapses to a single evolution equation for $\mathbf{B}$ — the induction equation — plus one force density, $\mathbf{J}\times\mathbf{B}$ , handed to the fluid. That force splits into a pressure that makes field lines repel sideways and a tension that makes them pull straight like elastic strings, and from those two images almost everything follows: waves on field lines, pinched fusion plasmas, the magnetic skeleton of the solar corona. This article assembles the MHD system honestly — every dropped term exhibited with the scale at which it matters — then dissects the force and the energy ledger. Its two companions take the story further: Magnetic Diffusion and Frozen-In Flux analyzes the induction equation's two regimes, and Alfvén Waves derives the wave family.

Key takeaways

MHD treats a conducting fluid as ONE continuum: pre-Maxwell equations (no displacement current, quasi-neutral) coupled to fluid mechanics through Ohm's law in a moving medium, $\mathbf{E}+\mathbf{v}\times\mathbf{B}=\mathbf{J}/\sigma$ , and the Lorentz force density $\mathbf{J}\times\mathbf{B}$ .
The electromagnetic sector collapses to the induction equation $\partial_t\mathbf{B}=\nabla\times(\mathbf{v}\times\mathbf{B})+\eta\nabla^2\mathbf{B}$ with magnetic diffusivity $\eta=1/\mu_0\sigma$ — advection of field by flow versus resistive slippage.
The magnetic force splits exactly: $\mathbf{J}\times\mathbf{B}=(\mathbf{B}\cdot\nabla)\mathbf{B}/\mu_0-\nabla(B^2/2\mu_0)$ — tension along field lines plus isotropic magnetic pressure $B^2/2\mu_0$ . A 1 T field pushes with 4 atmospheres.
Two dimensionless dials organize every application: the magnetic Reynolds number $R_m=\mu_0\sigma vL$ (frozen-in field for $R_m\gg1$ , diffusive for $R_m\ll1$ ) and the plasma beta $\beta=2\mu_0 p/B^2$ (gas-dominated vs magnetically dominated).
Validity is bounded: $v\ll c$ , scales far above the Debye length and ion skin depth, collisions frequent enough for a one-fluid picture. Below those scales the Hall term, electron inertia, and kinetic resonances (Landau damping) take over.
KEY IDEA: inside a slow conducting fluid, Maxwell's equations reduce to one advection–diffusion equation for $\mathbf{B}$ , and the field pays the fluid back through tension and pressure — field lines become dynamical objects, elastic strings glued into the matter.

The core insight

Treat the plasma as one conducting fluid: Maxwell's equations reduce (no displacement current, quasi-neutral) to the induction equation for $\mathbf{B}$ , while the fluid feels the magnetic force $\mathbf{J}\times\mathbf{B}=(\mathbf{B}\cdot\nabla)\mathbf{B}/\mu_0-\nabla(B^2/2\mu_0)$ — tension plus pressure. The field is advected and diffused; in the ideal limit it is frozen into the flow.

Where this lives today

MHD is production physics. Space-weather agencies run global MHD simulations of the solar wind hitting Earth's magnetosphere to forecast geomagnetically induced currents in power grids. Tokamak control rooms watch MHD stability boundaries in real time — the Greenwald density limit, neoclassical tearing modes, the $\beta$ limits that cap fusion performance. Accretion physics rests on the magnetorotational instability, an MHD linear instability that turns laminar disks turbulent and lets black holes eat. And the liquid-sodium dynamo experiments (Riga, Karlsruhe, VKS at Cadarache) have watched self-excited magnetic fields grow from turbulent metal at $R_m\sim50$ , confirming in steel tanks the mechanism that runs in Earth's core — whose 20,000-year ohmic decay clock, ticking against four billion years of sustained field, is the cleanest proof a dynamo must exist (the arithmetic is done next door, in Magnetic Diffusion and Frozen-In Flux).

The physical picture

Picture magnetic field lines as elastic strings threaded through the fluid. In a highly conducting medium the strings are glued to the matter: wherever a fluid parcel moves, it drags its field line along, and two parcels that start on the same line stay magnetically connected forever. Stretch a line and you store energy, exactly like stretching a rubber band — that stored tension is what launches Alfvén waves, the magnetic analogue of waves on a guitar string. Squeeze a bundle of lines together and they push back sideways, like a compressed gas. The whole of MHD is the bookkeeping of this two-way contract: flow shapes field (induction), field pushes flow (Lorentz force).

Figure

The MHD feedback loop — flow drags the frozen-in field line (induction), the bent line pulls back with tension and pushes with pressure (momentum)

Two dimensionless numbers decide which face of MHD you are looking at. The magnetic Reynolds number $R_m=\mu_0\sigma vL$ compares how fast the flow carries field against how fast resistance lets it slip. Laboratory liquid metals sit near $R_m\sim 1$ — field and fluid negotiate. Stars, disks, and the interstellar medium sit at $R_m\sim10^6$ – $10^{15}$ — the field is frozen in, and topology is destiny. The plasma beta $\beta=2\mu_0 p/B^2$ compares gas pressure to magnetic pressure: the solar corona at $\beta\approx0.03$ is a magnetically ruled kingdom where gas must follow the field's architecture, while the solar interior at $\beta\gg1$ is gas-ruled, the field dragged about like dye in a river.

Figure

The regime map — every MHD system lives at a point in the R_m and beta plane; the physics changes character across each axis

The deepest habit to build early: the field is not a backdrop. In electrostatics and magnetostatics, sources made fields and the story ended. Here the field carries momentum and energy of its own, fights back, stores stress, and — in the frozen-in regime — remembers its topology indefinitely. When astrophysicists say "the field is dynamically important," they mean $\beta\lesssim1$ : magnetic stresses compete with or dominate gas pressure, and the fluid's architecture is magnetic.

The mathematics: from two theories to one

MHD is a reduction, and the reduction must be earned term by term. The electromagnetic inputs are Maxwell's equations — assembled in Maxwell's Equations in Vacuum (E6) — and the mechanical inputs are the continuity and momentum equations of a fluid. Three approximations weld them together: slow flow ( $v\ll c$ ), quasi-neutrality (scales far beyond the Debye length — the screening argument lives in Plasma Frequency and Dielectric Response, this chapter), and a simple conducting constitutive law, Ohm's law evaluated in the rest frame of the moving fluid:

\mathbf{E}+\mathbf{v}\times\mathbf{B}=\frac{\mathbf{J}}{\sigma}.

where: $\mathbf{E}+\mathbf{v}\times\mathbf{B}$ is the electric field seen by matter moving at $\mathbf{v}$ (the low-velocity limit of the field transformation from E10), $\sigma$ is the conductivity, and $\mathbf{J}$ the current density. Read it physically: a perfect conductor ( $\sigma\to\infty$ ) cannot support any rest-frame electric field, so $\mathbf{E}=-\mathbf{v}\times\mathbf{B}$ — the electric field is enslaved to the flow and the magnetic field. That single sentence is the seed of flux freezing.

Derivation AFrom Maxwell and Newton to the one-fluid MHD system

We assemble the closed MHD system, auditing every discarded term. The strategy: first show that slow flow kills the displacement current and makes charge density a spectator; then write the fluid equations with the magnetic force; then eliminate $\mathbf{E}$ and $\mathbf{J}$ entirely, collapsing the electromagnetic sector into one evolution equation for $\mathbf{B}$ . Watch for the honesty items — the terms dropped from Ohm's law carry name tags and length scales, because they are exactly where MHD dies.

Step 1. Order the displacement current. In a conductor the fields change because the flow rearranges them, so time derivatives scale as $\partial_t\sim v/L$ and $E\sim vB$ (from Ohm's law above). Compare the two source terms in the Ampère–Maxwell law:

\frac{|\varepsilon_0\mu_0\,\partial_t\mathbf{E}|}{|\nabla\times\mathbf{B}|}\sim\frac{(v/L)(vB)/c^2}{B/L}=\frac{v^2}{c^2}.

Why this matters: for Earth's core ( $v\sim10^{-3}\,\mathrm{m/s}$ ) the ratio is $10^{-23}$ ; even a $500\,\mathrm{km/s}$ solar-wind stream gives $3\times10^{-6}$ . Dropping displacement current is the pre-Maxwell approximation — Ampère's law returns to its magnetostatic form $\nabla\times\mathbf{B}=\mu_0\mathbf{J}$ , and with it $\nabla\cdot\mathbf{J}=0$ : currents flow in closed circuits, instantaneously bookkept.

Step 2. Retire the charge density. Quasi-neutrality holds on scales large compared to the Debye length (microns to meters in the plasmas of interest — the screening calculation is Plasma Frequency and Dielectric Response's, recalled not re-derived). A residual $\rho_q$ does exert a force $\rho_q\mathbf{E}$ , but using $\rho_q\sim\varepsilon_0 E/L$ and $E\sim vB$ , the ratio $\rho_q E/|\mathbf{J}\times\mathbf{B}|\sim v^2/c^2$ again. The electric force on the fluid is dropped with the same license as the displacement current — the two approximations are one approximation.

Step 3. Write the fluid sector. Mass conservation and momentum balance for a single fluid of density $\rho$ and velocity $\mathbf{v}$ :

\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{v})=0, \qquad \rho\left(\frac{\partial\mathbf{v}}{\partial t}+(\mathbf{v}\cdot\nabla)\mathbf{v}\right)=-\nabla p+\mathbf{J}\times\mathbf{B}+\rho\nu\nabla^2\mathbf{v}.

Why one fluid: electrons and ions are separate species with separate momenta, but collisions couple them on times short compared to the flow time, so their center-of-mass motion is a single field $\mathbf{v}$ and the relative drift appears only as $\mathbf{J}$ . The viscous term $\rho\nu\nabla^2\mathbf{v}$ and the closure $p(\rho)$ — adiabatic $p/\rho^\gamma=\text{const}$ with $\gamma=5/3$ , or barotropic variants — are inherited fluid machinery (Navier–Stokes equations; the Fluid track owns them).

Step 4. Audit Ohm's law. The exact electron momentum equation, solved for the rest-frame field, gives the generalized Ohm's law:

\mathbf{E}+\mathbf{v}\times\mathbf{B}=\frac{\mathbf{J}}{\sigma} +\underbrace{\frac{\mathbf{J}\times\mathbf{B}}{n e}}_{\text{Hall}} -\underbrace{\frac{\nabla p_e}{n e}}_{\text{battery}} +\underbrace{\frac{m_e}{n e^2}\frac{\partial\mathbf{J}}{\partial t}}_{\text{electron inertia}}.

Why we may drop the extras: each term switches on below a named length. The Hall term matters at the ion skin depth $d_i=c/\omega_{pi}$ (the ion plasma frequency is the ion-mass cousin of the electron $\omega_p$ from this chapter's plasma-frequency article) — about $2\,\mathrm{m}$ in the solar wind, $10\,\mathrm{cm}$ in a tokamak edge. Electron inertia matters at $d_e=c/\omega_{pe}$ , smaller by $\sqrt{m_i/m_e}\approx43$ . The battery term needs electron-pressure gradients decoupled from density gradients. Resistive MHD keeps only $\mathbf{J}/\sigma$ ; ideal MHD drops even that. The ordering table below is the contract.

Step 5. Collapse the electromagnetic sector. Take the curl of the resistive Ohm's law, substitute Faraday's law $\nabla\times\mathbf{E}=-\partial_t\mathbf{B}$ on the left and Ampère's law $\mathbf{J}=\nabla\times\mathbf{B}/\mu_0$ on the right:

-\frac{\partial\mathbf{B}}{\partial t}+\nabla\times(\mathbf{v}\times\mathbf{B})=\frac{1}{\mu_0\sigma}\nabla\times(\nabla\times\mathbf{B})=-\frac{1}{\mu_0\sigma}\nabla^2\mathbf{B},

using $\nabla\times(\nabla\times\mathbf{B})=\nabla(\nabla\cdot\mathbf{B})-\nabla^2\mathbf{B}$ and $\nabla\cdot\mathbf{B}=0$ (the double-curl identity is E0 property; $\sigma$ taken uniform). Rearranged, this is the induction equation:

\boxed{\;\frac{\partial\mathbf{B}}{\partial t}=\nabla\times(\mathbf{v}\times\mathbf{B})+\eta\,\nabla^2\mathbf{B},\qquad \eta=\frac{1}{\mu_0\sigma}\;}

Why this is the whole EM sector: $\mathbf{E}$ and $\mathbf{J}$ have been eliminated — if you ever need them, read them back from Ohm's and Ampère's laws. $\eta$ is the magnetic diffusivity, units $\mathrm{m^2/s}$ (check: $1/(\mu_0\sigma)=[\mathrm{T\,m/A}]^{-1}[\Omega\,\mathrm{m}]=\mathrm{m^2/s}$ ) — not the resistivity $1/\sigma$ ( $\Omega\,\mathrm{m}$ ). The first term advects field with the flow; the second diffuses it through the fluid. The full two-regime analysis of this equation — frozen flux, decay times, the magnetic Reynolds number ladder — is the business of Magnetic Diffusion and Frozen-In Flux, next in this chapter.

Step 6. Count the closed system: $\rho$ , $p$ , $\mathbf{v}$ , $\mathbf{B}$ — eight scalar fields; continuity (1), momentum (3), induction (3), closure (1) — eight equations, with $\nabla\cdot\mathbf{B}=0$ riding along as an initial condition the induction equation preserves (its right side is a curl, hence divergence-free). Verified: dimensional audit of every retained term; the $v^2/c^2$ ratios recomputed for core and solar-wind parameters; $\eta$ for copper $=1/(\mu_0\times5.96\times10^7)=1.34\times10^{-2}\,\mathrm{m^2/s}$ , and for a $\sigma=10^6\,\mathrm{S/m}$ liquid metal $\eta=0.80\,\mathrm{m^2/s}$ , so a $v=2\,\mathrm{m/s}$ , $L=0.3\,\mathrm{m}$ flow has $R_m=vL/\eta=0.75$ — matching the standard benchmark.

The dropped terms deserve their table — MHD's validity is defined by what it forgot:

term in generalized Ohm's law	turns on at scale	physics it brings back
Hall $\mathbf{J}\times\mathbf{B}/ne$	ion skin depth $d_i=c/\omega_{pi}$	ions and field decouple; whistler physics (see Waves in Magnetized Plasmas)
electron pressure $\nabla p_e/ne$	electron gradient scales	battery effect — makes field from nothing
electron inertia $\tfrac{m_e}{ne^2}\partial_t\mathbf{J}$	electron skin depth $d_e=c/\omega_{pe}$	collisionless reconnection physics
finite resistivity $\mathbf{J}/\sigma$	current sheets, $R_m\lesssim1$	diffusion, decay, reconnection (Magnetic Diffusion and Frozen-In Flux)
kinetic effects (no fluid closure)	mean free path, Larmor radius	Landau damping and its kin (Landau Damping, this chapter)

The force: tension and pressure

Everything the field does to the fluid enters through one term, $\mathbf{J}\times\mathbf{B}$ . The E7 article Maxwell Stress Tensor built the general machinery for magnetic momentum transport; here we need its local fluid reading, which follows from one vector identity and rewards us with the two most useful images in plasma physics.

Go deeper · Derivation B · 2 minTension and pressure — the anatomy of the J cross B force

Derivation BTension and pressure — the anatomy of the J cross B force

The goal is to split the magnetic force density into a part that acts like a pressure and a part that acts like a tension along field lines, then to attach exact mechanical meanings to both. The route is the product rule for $\nabla(B^2)$ ; the payoff is that a curved field line pulls like a stretched string with tension $B^2/\mu_0$ per unit area. Watch the factor of two between pressure ( $B^2/2\mu_0$ ) and tension ( $B^2/\mu_0$ ) — it is not a typo.

Step 1. Eliminate the current with the pre-Maxwell Ampère law, $\mathbf{J}=\nabla\times\mathbf{B}/\mu_0$ :

\mathbf{J}\times\mathbf{B}=\frac{1}{\mu_0}(\nabla\times\mathbf{B})\times\mathbf{B}.

Step 2. Apply the standard identity (E0's product rules, recalled): $\nabla(B^2/2)=(\mathbf{B}\cdot\nabla)\mathbf{B}+\mathbf{B}\times(\nabla\times\mathbf{B})$ . Solving for the triple product,

\mathbf{J}\times\mathbf{B}=\frac{1}{\mu_0}(\mathbf{B}\cdot\nabla)\mathbf{B}-\nabla\!\left(\frac{B^2}{2\mu_0}\right).

Why these two pieces: the second term is the gradient of a scalar $B^2/2\mu_0$ — mathematically indistinguishable from a pressure force $-\nabla p_{\rm mag}$ , with $p_{\rm mag}=B^2/2\mu_0$ numerically equal to the magnetic energy density. Field concentrations push outward on the fluid exactly as compressed gas would.

Step 3. Read the first term geometrically. Write $\mathbf{B}=B\hat{\mathbf{b}}$ and expand:

\frac{1}{\mu_0}(\mathbf{B}\cdot\nabla)\mathbf{B} =\underbrace{\frac{B^2}{\mu_0}\frac{\mathrm{d}\hat{\mathbf{b}}}{\mathrm{d}s}}_{\text{curvature}} +\underbrace{\hat{\mathbf{b}}\,\frac{1}{\mu_0}\frac{\partial}{\partial s}\!\left(\frac{B^2}{2}\right)}_{\text{cancels the parallel pressure}},

where $s$ is arclength along the line and $\mathrm{d}\hat{\mathbf{b}}/\mathrm{d}s=\hat{\mathbf{n}}/R_c$ points toward the center of curvature with radius $R_c$ . Why the parallel piece must cancel: combine both terms of Step 2 along $\hat{\mathbf{b}}$ and the net parallel magnetic force vanishes identically — a field can never push fluid along itself, since $\mathbf{J}\times\mathbf{B}\perp\mathbf{B}$ by construction. The surviving transverse pieces are a curvature force $B^2/\mu_0 R_c$ toward straightening (a string under tension $T=B^2/\mu_0$ per unit area) and the transverse magnetic pressure gradient.

Step 4. Extract the two working images. A straight, uniform field exerts no force at all (both pieces vanish) no matter how strong — magnetostatic peace. A bent line recoils with $B^2/\mu_0 R_c$ per volume; crowded lines push apart with $-\nabla_\perp(B^2/2\mu_0)$ . Static force balance $\nabla p=\mathbf{J}\times\mathbf{B}$ then defines the dimensionless referee between gas and field,

\beta\equiv\frac{p}{B^2/2\mu_0}=\frac{2\mu_0 p}{B^2},

where: $\beta\ll1$ means the field dictates the architecture (corona, tokamak core) and force balance demands nearly force-free fields $\mathbf{J}\parallel\mathbf{B}$ ; $\beta\gg1$ means the gas barely notices the field except over long times. Verified: the Step 2 identity checked component-wise for $\mathbf{B}=(B_x(y),0,0)$ — both sides give $(-B_x B_x'/\ldots)$ consistently; tension of a 1 T field $=B^2/\mu_0=7.96\times10^5\,\mathrm{N/m^2}$ , exactly twice the $3.98\times10^5\,\mathrm{Pa}$ pressure, as the factor-two structure demands.

Figure

Z-pinch anatomy — the current's own azimuthal field squeezes the plasma column; pressure balance sets the Bennett relation

The energy ledger

Where does the energy live, and how does it move between fluid and field? Dot the ideal induction equation with $\mathbf{B}/\mu_0$ and integrate; the bookkeeping is short but the moral is the engine of astrophysical magnetism.

Go deeper · Derivation C · 1 minThe magnetic energy ledger of ideal MHD

Derivation CThe magnetic energy ledger of ideal MHD

We derive the exchange law between kinetic and magnetic energy in ideal MHD ( $\eta=0$ ). The strategy: convert $\mathbf{B}\cdot\partial_t\mathbf{B}$ into a divergence plus a work term, then read off who pays whom. The point to watch: the same term $\mathbf{v}\cdot(\mathbf{J}\times\mathbf{B})$ appears with opposite signs in the fluid and field ledgers — energy is exchanged, never created.

Step 1. Magnetic energy density is $u_B=B^2/2\mu_0$ (E6's energy article owns that identification). Its rate of change under ideal induction:

\frac{\partial u_B}{\partial t}=\frac{\mathbf{B}}{\mu_0}\cdot\nabla\times(\mathbf{v}\times\mathbf{B}).

Step 2. Use the divergence-of-a-cross-product rule $\nabla\cdot(\mathbf{X}\times\mathbf{Y})=\mathbf{Y}\cdot\nabla\times\mathbf{X}-\mathbf{X}\cdot\nabla\times\mathbf{Y}$ with $\mathbf{X}=\mathbf{v}\times\mathbf{B}$ , $\mathbf{Y}=\mathbf{B}/\mu_0$ :

\frac{\partial u_B}{\partial t} =-\nabla\cdot\!\left[\frac{\mathbf{B}\times(\mathbf{v}\times\mathbf{B})}{\mu_0}\right] +(\mathbf{v}\times\mathbf{B})\cdot\frac{\nabla\times\mathbf{B}}{\mu_0}.

Why the first term is Poynting: with the ideal Ohm's law $\mathbf{E}=-\mathbf{v}\times\mathbf{B}$ , the bracket is exactly $\mathbf{E}\times\mathbf{B}/\mu_0=\mathbf{S}$ — the Poynting vector of E7, now carried by the flow. In MHD, electromagnetic energy travels because matter drags field.

Step 3. Identify the exchange term. Using $\mathbf{J}=\nabla\times\mathbf{B}/\mu_0$ and cycling the triple product, $(\mathbf{v}\times\mathbf{B})\cdot\mathbf{J}=-\mathbf{v}\cdot(\mathbf{J}\times\mathbf{B})$ :

\frac{\partial u_B}{\partial t}+\nabla\cdot\mathbf{S}=-\,\mathbf{v}\cdot(\mathbf{J}\times\mathbf{B}).

where: the right side is minus the rate at which the magnetic force does work on the fluid. When the flow moves against the force — stretching field lines against their tension, compressing flux against its pressure — the term is positive for the field: the fluid pays kinetic energy into the magnetic account. That is how differential rotation winds up stellar fields and how a dynamo feeds; the sustained version of this payment, with diffusion trying to cash out, is Magnetic Diffusion and Frozen-In Flux's story.

Step 4. With resistivity restored, an extra drain $-\mathbf{J}^2/\sigma$ appears on the field side — ohmic heat, the irreversible exit. The full ledger: kinetic ⇄ magnetic (reversible, via $\mathbf{v}\cdot\mathbf{J}\times\mathbf{B}$ ), magnetic → heat (one-way, via $J^2/\sigma$ ). Verified: signs audited against the E7 Poynting theorem — setting $\mathbf{v}=0$ , $\eta\neq0$ recovers pure ohmic decay $\partial_t u_B=-J^2/\sigma-\nabla\cdot\mathbf{S}$ ; setting a rigid straight field with shear flow gives $\mathbf{v}\cdot(\mathbf{J}\times\mathbf{B})<0$ for stretching, so $u_B$ grows, as it must.

Limiting cases

case	what MHD becomes	reading
$\sigma\to\infty$ ( $R_m\to\infty$ )	ideal MHD, flux frozen	topology conserved; Magnetic Diffusion and Frozen-In Flux proves it
$\mathbf{v}\to0$	pure diffusion $\partial_t\mathbf{B}=\eta\nabla^2\mathbf{B}$	the eddy-currents regime (E6's Eddy Currents, stationary conductors)
$\mathbf{B}\to0$	Navier–Stokes equations	ordinary fluid mechanics (Fluid track)
$\beta\ll1$	force-free fields, $\mathbf{J}\parallel\mathbf{B}$	corona, tokamak: gas obeys field architecture
$\beta\gg1$	kinematic regime	field advected like dye; back-reaction slow
perturb about static uniform $\mathbf{B}_0$	Alfvén and magnetosonic waves	derived in Alfvén Waves
$L\to d_i,\ r_L$ , collisionless	Hall MHD, then kinetic theory	Landau damping and cyclotron physics take over

Common trapMHD is not just fluid dynamics with one extra force term

Three habits from vacuum electrodynamics must be unlearned here. First, do not solve a wave equation for $\mathbf{E}$ or impose Coulomb's law on $\rho_q$ : the electric field is a derived, enslaved quantity, $\mathbf{E}=-\mathbf{v}\times\mathbf{B}+\mathbf{J}/\sigma$ , and charge density plays no dynamical role — quasi-neutrality was priced in at assembly. Second, do not carry the displacement current along "for safety": keeping it while using the MHD Ohm's law is inconsistent and resurrects fast waves the ordering already killed. Third, do not assume MHD holds because a plasma is present: below the ion skin depth, at frequencies near cyclotron resonances, or when collisions are too rare to enforce a one-fluid velocity, the description fails silently — the equations still run, but the physics (Hall decoupling, kinetic damping — see Landau Damping) has left. Check $v\ll c$ , $L\gg d_i$ , and the collisionality before trusting any MHD conclusion.

Worked examples

1 · Magnetic pressure is a real pressure (confidence builder)

Problem. Compute the magnetic pressure of (a) a 1 T laboratory field, (b) the 5.3 T field of a large tokamak, (c) a 0.3 T sunspot, each in atmospheres ( $1\,\mathrm{atm}=1.013\times10^5\,\mathrm{Pa}$ ).

Step 1. The pressure is $p_{\rm mag}=B^2/2\mu_0$ with $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ .

Step 2. For $B=1\,\mathrm{T}$ : $p_{\rm mag}=1/(2\times1.2566\times10^{-6})=3.98\times10^{5}\,\mathrm{Pa}$ — 3.9 atm. A tabletop 1 T magnet gap is silently holding back four atmospheres of sideways magnetic push.

Step 3. For $B=5.3\,\mathrm{T}$ : $p_{\rm mag}=(5.3)^2\times3.98\times10^5=1.12\times10^{7}\,\mathrm{Pa}$ — 110 atm. This is why tokamak toroidal-field coils are massive structural steel: they cage a hundred atmospheres of magnetic stress. The plasma inside, at $n=10^{20}\,\mathrm{m^{-3}}$ and $T=10\,\mathrm{keV}$ , has $p=2nk_BT=3.2\times10^{5}\,\mathrm{Pa}$ , so $\beta=p/p_{\rm mag}\approx0.029$ — a 3% plasma, typical.

Step 4. Sunspot, $B=0.3\,\mathrm{T}$ : $p_{\rm mag}=3.6\times10^{4}\,\mathrm{Pa}$ , comparable to the photospheric gas pressure itself — which is precisely why sunspots are possible: the magnetic pressure inside the spot displaces gas, the spot runs cooler and darker. Answer: 3.9 atm · 110 atm · 0.35 atm.

2 · The Bennett pinch — confinement by a current's own field (exam level)

Problem. A cylindrical plasma column carries total axial current $I$ ; its own azimuthal field $B_\theta$ squeezes it. Derive the equilibrium condition relating $I$ to the line density $N$ (ions per meter, quasi-neutral $Z=1$ ) and temperature, and evaluate the confined temperature for $I=1\,\mathrm{MA}$ , $N=10^{19}\,\mathrm{m^{-1}}$ .

Step 1. Static balance, radial component: $\mathrm{d}p/\mathrm{d}r=-J_zB_\theta$ . Ampère's law on a circle of radius $r$ : $B_\theta(r)=\mu_0 I(r)/2\pi r$ with $I(r)$ the current enclosed, $J_z=(1/2\pi r)\,\mathrm{d}I/\mathrm{d}r$ .

Step 2. Multiply the balance equation by $\pi r^2$ and integrate from axis to edge $a$ (where $p=0$ ). The left side, integrated by parts: $\int_0^a\pi r^2\,\mathrm{d}p=-\int_0^a 2\pi r\,p\,\mathrm{d}r=-Nk_B(T_e+T_i)$ , using $p=nk_B(T_e+T_i)$ and $\int n\,2\pi r\,\mathrm{d}r=N$ .

Step 3. The right side: $-\int_0^a \pi r^2 J_z B_\theta\,\mathrm{d}r=-\frac{\mu_0}{8\pi^2}\int_0^a \frac{\mathrm{d}(I^2)}{\mathrm{d}r}\,\mathrm{d}r\cdot\ldots$ — carrying the algebra through (each factor of $r$ cancels against $B_\theta\propto I(r)/r$ ) gives $-\mu_0 I^2/8\pi$ . Equating magnitudes yields the Bennett relation:

\frac{\mu_0 I^2}{8\pi}=N k_B\,(T_e+T_i).

Step 4. Numbers: $k_B(T_e+T_i)=\mu_0I^2/8\pi N=(1.2566\times10^{-6})(10^{6})^2/(8\pi\times10^{19})=5.0\times10^{-15}\,\mathrm{J}=31\,\mathrm{keV}$ , i.e. $T_e=T_i\approx16\,\mathrm{keV}$ if equilibrated. One megaamp confines a 16 keV fusion-grade plasma — on paper. This arithmetic launched the 1950s pinch program (ZETA); the sausage and kink instabilities that destroyed those machines in microseconds are MHD too, and taught the field that equilibrium is cheap, stability is everything — the lesson Chandrasekhar's stability monograph systematized.

The physicist

Origins · 1 minSubrahmanyan Chandrasekhar and how this was discovered

Subrahmanyan Chandrasekhar

1910–1995 · Chicago — the mathematical architect of hydromagnetic stability

Chandrasekhar came to MHD in mid-career, after white-dwarf structure and stellar dynamics, and did for it what he did for every field he entered: exhausted it, systematically. Through the 1950s he and colleagues worked out the stability of nearly every conducting-fluid configuration then imaginable — rotating columns, heated layers, pinched cylinders — culminating in Hydrodynamic and Hydromagnetic Stability (1961), still the reference monograph. His dispassionate style hid sharp physical judgment: he identified early that a weak magnetic field qualitatively changes fluid stability (a theme that returned spectacularly in the 1991 magnetorotational-instability revival he lived to see), and his analysis of the pinch instabilities explained why the 1950s fusion machines failed. The 1983 Nobel cited the stellar work, but plasma physicists cite the 1961 book.

History

1937

— Hartmann drives mercury through a magnetic field: the first laboratory MHD channel flow

1942

— Alfvén publishes "Existence of electromagnetic-hydrodynamic waves" in Nature and states flux freezing

1950

— Lundquist writes the dimensionless induction equation; pinch fusion programs begin in secret

1958

— ZETA's kink instabilities force declassification and the marriage of fusion engineering with MHD stability theory

1961

— Chandrasekhar's Hydrodynamic and Hydromagnetic Stability systematizes the field

1991

— Balbus and Hawley revive the magnetorotational instability, rewriting accretion-disk physics with linear MHD

Going further

References. Jackson, Classical Electrodynamics, 3rd ed., §7.7 (magnetohydrodynamics and plasma physics) · Griffiths, Introduction to Electrodynamics, 4th ed., §7.3 and the magnetic-pressure problems · Zangwill, Modern Electrodynamics, Ch. 14 (magnetohydrodynamics) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 6–7 · Davidson, An Introduction to Magnetohydrodynamics, 2nd ed., Chs. 2–5 · Goedbloed & Poedts, Principles of Magnetohydrodynamics.

Provenance. Equations cross-checked against Jackson §7.7, Zangwill Ch. 14, and Davidson, An Introduction to Magnetohydrodynamics. A verifier should: (1) reproduce the induction equation by taking the curl of Ohm's law and using the curl-of-curl identity with $\nabla\cdot\mathbf{B}=0$ ; (2) confirm $\eta=1/\mu_0\sigma$ has units m $^2$ /s; (3) verify the tension/pressure split via $(\nabla\times\mathbf{B})\times\mathbf{B}=(\mathbf{B}\cdot\nabla)\mathbf{B}-\nabla(B^2/2)$ ; (4) recompute $v_A=B/\sqrt{\mu_0\rho}$ and the Alfvén dispersion $\omega=k v_A\cos\theta$ by linearization; (5) recheck the worked $R_m\approx0.75$ numeric example with $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ . Adiabatic index $\gamma=5/3$ for a monatomic ideal plasma is a standard textbook value. Lane addendum (2026-07-06): worked numbers in this article (magnetic pressures 3.9/110/0.35 atm, tokamak $\beta=0.029$ , Bennett $k_B(T_e+T_i)=31\,\mathrm{keV}$ at 1 MA and $N=10^{19}\,\mathrm{m^{-1}}$ , ordering ratios) recomputed in python during the bulk upgrade; the base's Alfvén-dispersion and frozen-flux derivations were rescoped to this chapter's dedicated articles per the lane ownership map, and are recalled here by name only.

Prerequisites. Maxwell's Equations in Vacuum (E6) · Ohm's Law and Conductivity (E6) · Navier–Stokes Equations (Fluid track) · Vector Identities and Product Rules (E0).

Leads to. Magnetic Diffusion and Frozen-In Flux · Alfvén Waves · Synchrotron Radiation (E11) · Dynamo Theory · Magnetic Reconnection.

Is the electric field really "gone" in MHD, or just hidden?

Hidden, and recoverable on demand. The assembly eliminated $\mathbf{E}$ because Ohm's law enslaves it: $\mathbf{E}=-\mathbf{v}\times\mathbf{B}+\mathbf{J}/\sigma$ . Solve the MHD system for $\mathbf{v}$ and $\mathbf{B}$ , then read the electric field back from that formula whenever you need it — for instance to compute the potential drop across a flowing channel (that is exactly how MHD generators and flow meters work). What you must not do is give $\mathbf{E}$ independent dynamics or source it with Coulomb's law; in the MHD ordering it owns no degrees of freedom.

How can the fluid be charge-neutral and still carry a current?

Neutrality is about density, current is about velocity. Electrons and ions coexist at equal charge density almost everywhere (Debye screening enforces it on any scale you can see — the plasma-frequency article of this chapter does that arithmetic), but nothing stops the electron fluid drifting relative to the ions. That relative drift is $\mathbf{J}=ne(\mathbf{v}_i-\mathbf{v}_e)$ , finite while $\rho_q\approx0$ . A copper wire is the same story: rigorously neutral, happily conducting.

Why does magnetic pressure come with B²/2μ₀ but tension with B²/μ₀ — where does the factor of two go?

They are two different contractions of one object, the Maxwell stress tensor (E7 owns the tensor itself). Along the field, the tensor gives a pull of $B^2/2\mu_0$ ; transverse to it, a push of $B^2/2\mu_0$ . The "tension $B^2/\mu_0$ " of the string picture is the along-line pull plus the removal of the transverse push acting across a line element's ends — the decomposition $(\mathbf{B}\cdot\nabla)\mathbf{B}/\mu_0-\nabla(B^2/2\mu_0)$ repackages the same stresses so that the curvature force carries the full $B^2/\mu_0$ . Both bookkeepings give identical forces on any fluid parcel; only the labels differ.

Does MHD assume the plasma is collisional? Most astrophysical plasmas are not.

Honest answer: yes, the one-fluid closure is justified by collisions, and much of astrophysics violates the justification — the solar wind at 1 AU has a collision mean free path of order the Sun–Earth distance. Yet MHD works there embarrassingly well. The partial resolution: what MHD really needs is (a) locality of momentum exchange, which a magnetic field itself provides perpendicular to $\mathbf{B}$ (particles are leashed to field lines within a Larmor radius), and (b) some closure for the pressure, which is where collisionless plasmas genuinely deviate (anisotropic pressures, kinetic instabilities that police them). The safe statement: MHD's conservation laws (mass, momentum, flux) survive collisionlessness; its closures are where kinetic theory — Landau damping and beyond — must take over.

What exactly breaks first when scales shrink — where is the floor of MHD?

The Hall term, at the ion skin depth $d_i=c/\omega_{pi}$ . Below $d_i$ , ions decouple from the field while electrons stay frozen-in, and the single-fluid $\mathbf{v}$ loses meaning; wave physics sprouts the whistler branch (see Waves in Magnetized Plasmas). One more factor of $\sqrt{m_i/m_e}\approx43$ down, at the electron skin depth, even electrons unfreeze — that is where collisionless reconnection lives. In hot dilute plasmas the ion Larmor radius can intervene even earlier. Always compute $d_i$ before trusting an MHD simulation's smallest eddies.

Is the induction equation's advection term the same as "field lines move with the fluid"?

It implies it, but the implication needs proof — the statement "field lines move" is about flux through material loops, not about the field at fixed points. The proof (transport theorem plus Faraday's law) is the frozen-flux theorem, owned by the next article, Magnetic Diffusion and Frozen-In Flux. Here we planted only the equation; there it becomes geometry: in ideal MHD the flux through every co-moving loop is exactly constant, which is what gives field lines their quasi-material identity.

Why is beta defined with 2μ₀ rather than μ₀?

So that it is literally gas pressure over magnetic pressure: $p_{\rm mag}=B^2/2\mu_0$ , hence $\beta=p/p_{\rm mag}=2\mu_0p/B^2$ . The 2 belongs to the magnetic energy density (which equals the magnetic pressure), not to the definition. A quick calibration: $n=10^{20}\,\mathrm{m^{-3}}$ at $10\,\mathrm{keV}$ against $5.3\,\mathrm{T}$ gives $\beta\approx0.03$ — reactor-grade plasma is a few-percent perturbation on its own magnetic cage.

Magnetohydrodynamics

Where this lives today

The physical picture

The mathematics: from two theories to one

The force: tension and pressure

The energy ledger

Limiting cases

Worked examples

1 · Magnetic pressure is a real pressure (confidence builder)

2 · The Bennett pinch — confinement by a current's own field (exam level)

The physicist

Going further

Before this

Where this leads