Alfvén Waves

In 1942, in a two-column note to Nature, Hannes Alfvén proposed something electromagnetic orthodoxy said was impossible: a wave in a conductor. Every result of the waves chapter agreed — electromagnetic waves die within a skin depth of entering a good conductor; a perfectly conducting fluid should be the last place to look for wave motion. Alfvén saw what the objection missed. In a conductor that can move, the frozen-in magnetic field hands the fluid a new elasticity. Field lines under tension $B_0^2/\mu_0$ , loaded with the fluid's inertia, are strings — and strings carry waves. Pluck a field line in the solar corona and a transverse ripple runs along it at the Alfvén speed $v_A=B_0/\sqrt{\mu_0\rho}$ , about a thousand kilometers per second, guided by the field like a pulse down a wire. The claim earned him skepticism (Fermi's nod at a 1948 Chicago seminar famously ended it) and, in 1970, the Nobel Prize. Today Alfvén waves are among the most-observed waves in nature: spacecraft surf them in the solar wind, coronal imagers watch loops ring like plucked strings after flares, and every tokamak carries a family of Alfvén eigenmodes that fusion designers must respect because fast fusion products can drive them unstable. This article closes the MHD trio. The system was assembled in Magnetohydrodynamics; its field-freezing geometry was proved in Magnetic Diffusion and Frozen-In Flux. Here we linearize and harvest the waves: the shear Alfvén wave end-to-end, then the full three-mode family — fast, slow, Alfvén — that ideal MHD supports, with the Friedrichs diagram that organizes them and the numbers that make them astrophysics' working tools.

Key takeaways

Linearizing ideal MHD about a uniform $\mathbf{B}_0$ , $\rho_0$ yields the shear Alfvén wave: transverse, incompressible, with $\omega=k_\parallel v_A$ and $v_A=B_0/\sqrt{\mu_0\rho_0}$ — magnetic tension is the spring, fluid inertia the mass.
The dispersion depends only on $k_\parallel=k\cos\theta$ : energy is channeled strictly along the field (group velocity $=v_A\hat{\mathbf{b}}$ ), no matter which way the phase fronts point.
The wave's fingerprint is the Walén relation $\delta\mathbf{v}=\mp\,\delta\mathbf{B}/\sqrt{\mu_0\rho_0}$ , with exact equipartition between kinetic and magnetic perturbation energy — the test spacecraft apply to solar-wind turbulence daily.
Restoring compressibility completes the family: fast and slow magnetosonic waves with $\omega^2/k^2=\tfrac12[(v_A^2+c_s^2)\pm\sqrt{(v_A^2+c_s^2)^2-4v_A^2c_s^2\cos^2\theta}]$ ; ideal MHD carries exactly 7 modes (2 Alfvén + 2 fast + 2 slow + 1 entropy).
Anchor numbers: solar corona $v_A\approx10^3\,\mathrm{km/s}$ , coronal loops ring with ~3-minute periods (coronal seismology reads $B$ from them); solar wind at 1 AU $v_A\approx50\,\mathrm{km/s}$ ; tokamak TAE frequencies ~70 kHz.
The wave energy flux $\rho\langle\delta v^2\rangle v_A$ from the photosphere is of the right order to heat the corona — the hard part (still open) is dissipating nearly ideal waves.
KEY IDEA: freezing field into fluid gives field lines mass and keeps their tension — so magnetized plasmas are strung instruments, and $v_A=B_0/\sqrt{\mu_0\rho_0}$ is the speed at which magnetic stress travels.

The core insight

In a perfectly conducting fluid, magnetic field lines are frozen into the matter and act as elastic strings under tension $B_0^2/\mu_0$ loaded by fluid of density $\rho$ ; transverse plucks travel along $\mathbf{B}_0$ at $v_A=B_0/\sqrt{\mu_0\rho}$ as incompressible, dispersionless Alfvén waves.

Where this lives today

Alfvén waves are daily observables. Parker Solar Probe's encounters revealed "switchbacks" — kinked, folded Alfvénic structures pouring out of the corona — and the mission's core science is tracking how their energy dissipates into wind heating. Coronal seismology (worked example 1) is now a standard SDO data product, mapping active-region fields no magnetograph can reach. In fusion, toroidicity-induced Alfvén eigenmodes (TAEs) live in frequency gaps of the Alfvén continuum at $f=v_A/4\pi qR$ — about 70 kHz for ITER-grade parameters ( $B=5.3$ T, deuterium at $10^{20}\,\mathrm{m^{-3}}$ , $q=1.5$ , $R=6.2$ m) — and 3.5 MeV alpha particles born faster than $v_A$ can pump them resonantly, ejecting the very particles that must heat the plasma; TAE control is a first-order ITER design constraint. And in the interstellar medium, the Alfvénic turbulent cascade (its rigorous theory lives in the Fluid track under MHD Turbulence) sets how cosmic rays scatter and diffuse through the Galaxy.

The physical picture

Take the string analogy seriously — it is not an analogy but an identity. A stretched string carries transverse waves at $v=\sqrt{T/\lambda}$ , tension over linear mass density. Now consider a tube of magnetic flux of cross-section $A$ threading a frozen-in fluid. Magnetohydrodynamics (this chapter) dissected the $\mathbf{J}\times\mathbf{B}$ force and found a tension $B_0^2/\mu_0$ per unit area along the field: the tube pulls with $T=(B_0^2/\mu_0)A$ . The fluid glued to the tube (frozen flux — the previous article's theorem) loads it with $\lambda=\rho A$ per unit length. The string formula then predicts

v=\sqrt{\frac{T}{\lambda}}=\sqrt{\frac{B_0^2A/\mu_0}{\rho A}}=\frac{B_0}{\sqrt{\mu_0\rho}}\equiv v_A,

where: $v_A$ is the Alfvén speed, and the honest derivation below (linearized MHD, no analogies) lands on exactly this number — the string picture is quantitatively exact, cross-section canceling and all. Read it physically: stronger field, stiffer string; denser plasma, heavier string. In the corona ( $B$ modest, $\rho$ tiny) $v_A$ reaches thousands of km/s; in the dense solar interior it crawls.

Figure

A plucked field line — the transverse pulse rides along B0 at the Alfvén speed; velocity and field perturbations lock in antiphase (Walén relation)

Three qualitative features make this wave unlike any wave met so far in the track. It is transverse and incompressible — the fluid sloshes sideways, density untouched, so gas pressure never enters and the wave survives in cold and hot plasmas alike. It is one-dimensional in disguise: the frequency depends only on $k_\parallel$ , so the wave's energy travels strictly along $\mathbf{B}_0$ regardless of the phase direction — magnetic field lines are waveguides, piping energy from the photosphere into the corona along magnetic highways. It is dispersionless: all frequencies ride at $v_A$ , so pulses keep their shape across astronomical distances (until kinetic physics at the ion scale finally intervenes — Waves in Magnetized Plasmas picks up that thread).

The mathematics: linearize and harvest

The equilibrium is the simplest ideal-MHD state: uniform $\mathbf{B}_0=B_0\hat{\mathbf{z}}$ , uniform $\rho_0$ and $p_0$ , fluid at rest, $\eta=0$ . Perturb every field — $\rho=\rho_0+\delta\rho$ , $\mathbf{v}=\delta\mathbf{v}$ , $\mathbf{B}=\mathbf{B}_0+\delta\mathbf{B}$ , $p=p_0+c_s^2\delta\rho$ with $c_s^2=\gamma p_0/\rho_0$ the sound speed — keep first order, and Fourier-analyze with $e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}$ .

Derivation AThe shear Alfven wave from linearized ideal MHD

We derive the Alfvén branch end-to-end: eight fields go in, one clean dispersion relation comes out. The strategy is to linearize continuity, momentum, and induction, choose the transverse polarization that decouples from sound, and watch tension emerge as the only restoring force. The step to savor is the last: the perpendicular wavenumber drops out entirely — the wave cares only about distance measured along the field.

Step 1. Linearized continuity ties density to compression:

-i\omega\,\delta\rho+i\rho_0\,\mathbf{k}\cdot\delta\mathbf{v}=0.

Why we watch this equation: if a polarization has $\mathbf{k}\cdot\delta\mathbf{v}=0$ , then $\delta\rho=0$ and the pressure term dies with it — the incompressible branch will be found exactly there.

Step 2. Linearized momentum (the $\mathbf{J}\times\mathbf{B}$ force with $\mathbf{J}=\nabla\times\delta\mathbf{B}/\mu_0$ ; quadratic terms dropped):

-i\omega\rho_0\,\delta\mathbf{v}=-ic_s^2\,\delta\rho\,\mathbf{k}+\frac{1}{\mu_0}(i\mathbf{k}\times\delta\mathbf{B})\times\mathbf{B}_0.

Step 3. Linearized ideal induction (from $\partial_t\mathbf{B}=\nabla\times(\mathbf{v}\times\mathbf{B})$ , the master equation recalled from Magnetohydrodynamics):

-i\omega\,\delta\mathbf{B}=i\mathbf{k}\times(\delta\mathbf{v}\times\mathbf{B}_0).

Why no $\eta$ term: we work in the ideal limit; resistive damping returns at the end as a correction $\sim\eta k^2$ , negligible when $R_m\gg1$ .

Step 4. Choose the transverse polarization. Put $\mathbf{k}=k_\parallel\hat{\mathbf{z}}+k_\perp\hat{\mathbf{x}}$ (no generality lost) and try $\delta\mathbf{v}=\delta v\,\hat{\mathbf{y}}$ — perpendicular to both $\mathbf{B}_0$ and the plane of $\mathbf{k}$ . Then $\mathbf{k}\cdot\delta\mathbf{v}=0$ : by Step 1, $\delta\rho=0$ , and the pressure force vanishes. Sound has left the problem.

Step 5. Reduce the induction equation. Expand $\mathbf{k}\times(\delta\mathbf{v}\times\mathbf{B}_0)$ with the BAC–CAB rule: $\delta\mathbf{v}(\mathbf{k}\cdot\mathbf{B}_0)-\mathbf{B}_0(\mathbf{k}\cdot\delta\mathbf{v})$ . The second term is zero; hence

\delta\mathbf{B}=-\frac{B_0k_\parallel}{\omega}\,\delta v\,\hat{\mathbf{y}}.

Why this is the Walén relation in embryo: $\delta\mathbf{B}$ is parallel to $\delta\mathbf{v}$ (both along $\hat{\mathbf{y}}$ ), with ratio $-B_0k_\parallel/\omega$ . The perturbed field tips the lines sideways in step with the fluid — the plucked string, drawn by algebra.

Step 6. Substitute into the momentum equation. The magnetic force reduces (same BAC–CAB, using $\mathbf{k}\cdot\delta\mathbf{B}=0$ and keeping the $\hat{\mathbf{y}}$ component) to $(ik_\parallel B_0/\mu_0)\,\delta B_y\,\hat{\mathbf{y}}$ :

-i\omega\rho_0\,\delta v=\frac{ik_\parallel B_0}{\mu_0}\left(-\frac{B_0k_\parallel}{\omega}\,\delta v\right) \quad\Rightarrow\quad \omega^2\rho_0=\frac{k_\parallel^2B_0^2}{\mu_0}.

Why only $k_\parallel$ survives: the restoring force is tension, $(\mathbf{B}\cdot\nabla)\mathbf{B}/\mu_0$ linearized $=ik_\parallel B_0\,\delta\mathbf{B}/\mu_0$ — it involves the gradient along the line only. Bending the line costs energy; the perpendicular arrangement of adjacent lines costs nothing at this order.

Step 7. Collect the dispersion relation:

\boxed{\;\omega^2=k_\parallel^2v_A^2=k^2v_A^2\cos^2\theta,\qquad v_A=\frac{B_0}{\sqrt{\mu_0\rho_0}}\;}

where: $\theta$ is the angle between $\mathbf{k}$ and $\mathbf{B}_0$ . Phase speed $\omega/k=v_A\cos\theta$ dies at perpendicular propagation; but the group velocity $\partial\omega/\partial\mathbf{k}=v_A\hat{\mathbf{z}}$ points along the field at full $v_A$ for every $\mathbf{k}$ — the waveguide property, and the reason Friedrichs diagrams draw the Alfvén energy surface as two dots on the field axis. Verified: the calibration check of the base — $B_0=1\,\mathrm{nT}$ , $n=5\,\mathrm{cm^{-3}}$ protons gives $v_A=10^{-9}/\sqrt{\mu_0\times5\times10^6\times1.673\times10^{-27}}=9.8\,\mathrm{km/s}\approx10\,\mathrm{km/s}$ ✓; dimensions $[B/\sqrt{\mu_0\rho}]=\mathrm{T}/\sqrt{(\mathrm{T\,m/A})(\mathrm{kg/m^3})}=\mathrm{m/s}$ ✓; the string formula $\sqrt{T/\lambda}$ with $T=B_0^2A/\mu_0$ , $\lambda=\rho A$ reproduces $v_A$ identically.

The wave's internal anatomy deserves its own bookkeeping, because it is the observable — the signature spacecraft actually test.

Go deeper · Derivation B · 1 minWalen relation, equipartition, and field-guided energy flux

Derivation BWalen relation, equipartition, and field-guided energy flux

We extract three exact properties of the shear wave: the locked ratio of velocity to field perturbation (Walén relation), the equal split of wave energy between kinetic and magnetic forms, and the Poynting flux that carries it along $\mathbf{B}_0$ . These three are the working toolkit of solar-wind physics — equipartition and the Walén lock are how one identifies Alfvénic turbulence in data.

Step 1. From Step 5 of the previous derivation, with $\omega=\pm k_\parallel v_A$ (upper sign: propagation along $+\mathbf{B}_0$ ):

\delta\mathbf{v}=\mp\frac{\delta\mathbf{B}}{\sqrt{\mu_0\rho_0}},

the Walén relation. Why the sign matters: a wave running parallel to $\mathbf{B}_0$ anti-correlates $\delta\mathbf{v}$ with $\delta\mathbf{B}$ ; antiparallel propagation correlates them. Spacecraft use the correlation sign to tell sunward from anti-sunward wave populations — how we know the solar wind's Alfvénic turbulence is mostly outward, hence solar-born.

Step 2. Kinetic energy density, time-averaged over a cycle ( $\langle\cos^2\rangle=\tfrac12$ ): $\langle u_{\rm kin}\rangle=\tfrac12\rho_0\langle\delta v^2\rangle=\tfrac14\rho_0\,\delta v_0^2$ .

Step 3. Magnetic energy density: $\langle u_{\rm mag}\rangle=\langle\delta B^2\rangle/2\mu_0$ . By Walén, $\delta B_0^2=\mu_0\rho_0\,\delta v_0^2$ , so

\langle u_{\rm mag}\rangle=\frac{\mu_0\rho_0\,\delta v_0^2}{4\mu_0}=\tfrac14\rho_0\,\delta v_0^2=\langle u_{\rm kin}\rangle.

Why equipartition had to happen: the wave is a harmonic oscillator distributed through space — tension the spring, inertia the mass — and harmonic oscillators time-average their energy half-and-half. Any measured Alfvénic interval violating equipartition is telling you about kinetic-scale physics or compressive contamination.

Step 4. The wave's electric field, from ideal Ohm's law $\mathbf{E}=-\delta\mathbf{v}\times\mathbf{B}_0$ : magnitude $\delta v\,B_0$ , directed in the $\hat{\mathbf{x}}$ sense. The Poynting vector $\mathbf{S}=\mathbf{E}\times\delta\mathbf{B}/\mu_0$ (the E7 machinery, evaluated on the perturbation) time-averages to

\langle\mathbf{S}\rangle=\langle u\rangle\,v_A\,\hat{\mathbf{z}},\qquad \langle u\rangle=\langle u_{\rm kin}\rangle+\langle u_{\rm mag}\rangle=\tfrac12\rho_0\,\delta v_0^2.

Why along $\hat{\mathbf{z}}$ only: energy flux = energy density × group velocity, and the group velocity is $v_A\hat{\mathbf{b}}$ — the waveguide theorem again, now as an energy statement. This flux, evaluated with photospheric numbers, is the coronal-heating budget of worked example 2. Verified: equipartition checked symbolically ( $B_0^2/\mu_0v_A^2=\rho_0$ identically); Poynting reduction audited component-wise ( $\hat{\mathbf{x}}\times\hat{\mathbf{y}}=\hat{\mathbf{z}}$ , cross terms with the static field assigned to the background, standard perturbation bookkeeping); sign convention cross-checked against the Walén relation both propagation directions.

The full family: fast, slow, and the Friedrichs diagram

Keeping compressibility ( $\mathbf{k}\cdot\delta\mathbf{v}\neq0$ ) adds gas pressure to the restoring forces, and the remaining polarizations organize into two more branches.

Go deeper · Derivation C · 2 minThe magnetosonic dispersion relation

Derivation CThe magnetosonic dispersion relation

We complete the mode inventory of ideal MHD. Strategy: project the linearized momentum equation onto the plane containing $\mathbf{k}$ and $\mathbf{B}_0$ (the shear wave already took the perpendicular direction), eliminate $\delta\rho$ and $\delta\mathbf{B}$ , and solve the resulting 2×2 eigenproblem. Watch the discriminant: it never vanishes for $0<\theta<\pi/2$ , so fast and slow are genuinely separate branches with an avoided crossing.

Step 1. With $\delta\mathbf{v}$ in the $(\hat{\mathbf{x}},\hat{\mathbf{z}})$ plane, eliminate $\delta\rho$ via continuity and $\delta\mathbf{B}$ via induction, exactly as before but keeping all terms. The momentum equation becomes a linear system for $(\delta v_x,\delta v_z)$ :

\omega^2\delta v_x=\left[(v_A^2+c_s^2)k_\perp^2+v_A^2k_\parallel^2\right]\delta v_x+c_s^2k_\perp k_\parallel\,\delta v_z, \qquad \omega^2\delta v_z=c_s^2k_\perp k_\parallel\,\delta v_x+c_s^2k_\parallel^2\,\delta v_z.

Step 2. Set the determinant to zero. With $k_\perp=k\sin\theta$ , $k_\parallel=k\cos\theta$ , the characteristic equation factors into the biquadratic

\left(\frac{\omega^2}{k^2}\right)^2-(v_A^2+c_s^2)\frac{\omega^2}{k^2}+v_A^2c_s^2\cos^2\theta=0.

Step 3. Solve:

\boxed{\;\frac{\omega^2}{k^2}=\frac{1}{2}\left[(v_A^2+c_s^2)\pm\sqrt{(v_A^2+c_s^2)^2-4v_A^2c_s^2\cos^2\theta}\,\right]\;}

where: the $+$ root is the fast magnetosonic wave (pressure and tension pushing together), the $-$ root the slow wave (the two restoring forces working against each other along the field). Neither depends on the perturbation amplitude — linear theory — and both go compressive except at special angles.

Step 4. Audit the limits. At $\theta=0$ (along the field): the discriminant is $|v_A^2-c_s^2|$ , and the roots are exactly $\{v_A^2,\,c_s^2\}$ — a pure sound wave and a wave degenerate with the shear Alfvén branch. At $\theta=\pi/2$ : roots are $\{v_A^2+c_s^2,\,0\}$ — the fast wave becomes the "magnetosound" of combined pressures ( $p+B^2/2\mu_0$ both resisting compression), and the slow wave cannot propagate perpendicular at all. In the cold limit $c_s\to0$ (the $\beta\ll1$ corona): fast $\to v_A$ at all angles (isotropic), slow $\to0$ — a cold magnetized plasma carries only field-shaped waves. In $B_0\to0$ : fast $\to c_s$ , ordinary sound, and the magnetic family evaporates.

Step 5. Count the modes honestly. Ideal MHD evolves 8 fields ( $\rho,p,\mathbf{v},\mathbf{B}$ ) with one constraint ( $\nabla\cdot\mathbf{B}=0$ ): 7 dynamical degrees of freedom = 7 wave modes at each $\mathbf{k}$ : shear Alfvén ( $\pm$ ), fast ( $\pm$ ), slow ( $\pm$ ), and one zero-frequency entropy mode (a static pressure-balanced density ripple advected with the fluid). No mode is missing and none is spurious. Verified: the biquadratic's two roots recomputed numerically for $c_s/v_A=0.6$ : at $\theta=0$ , fast $=1.000\,v_A$ , slow $=0.600\,v_A$ ; at $\theta=90°$ , fast $=1.166\,v_A=\sqrt{1+0.36}\,v_A$ ✓, slow $=0$ ✓; discriminant positivity checked across $\theta$ (avoided crossing), and the $\theta=0$ degeneracy fast/Alfvén confirmed.

Figure

Friedrichs diagram for cs equal to 0.6 vA — polar plot of phase speed versus angle to B0: fast envelope outside, Alfvén lobes cos-theta, slow lobes hugging the axis; computed curves, not sketches

Limiting cases

case	result	meaning
$\theta=0$	Alfvén and fast degenerate at $v_A$ ; slow $=c_s$	along the field: a string wave and a sound wave
$\theta\to\pi/2$	Alfvén and slow phase speeds $\to0$ ; fast $\to\sqrt{v_A^2+c_s^2}$	only combined-pressure sound crosses field lines
$c_s\to0$ ( $\beta\ll1$ )	fast isotropic at $v_A$ , slow extinct	cold corona: field-shaped waves only
$B_0\to0$	fast $\to$ sound; Alfvén, slow extinct	ordinary hydrodynamics recovered
$c_s=v_A$ ( $\beta\approx1$ )	maximal fast/slow splitting at oblique $\theta$	solar wind at 1 AU lives here
$\omega\to\Omega_{ci}$	dispersion appears; branch splits	ion-cyclotron physics — Waves in Magnetized Plasmas
finite $\eta$	damping rate $\sim\eta k^2/2$	negligible at $R_m\gg1$ ; why coronal waves refuse to die

Common trapThe shear Alfvén wave is not a magnetosonic wave — and magnetic pressure plays no role in it

Two standard confusions. First: the pure Alfvén wave is incompressible ( $\delta\rho=0$ , $\mathbf{k}\cdot\delta\mathbf{v}=0$ ) and restored by tension only; the fast and slow modes are compressive and involve gas plus magnetic pressure. Calling every MHD wave "an Alfvén wave" erases the physics — only the shear mode has the waveguide property and the Walén lock; only the fast mode crosses field lines in a cold plasma. Second: the formula $v_A=B_0/\sqrt{\mu_0\rho}$ is SI; the Gaussian form $B_0/\sqrt{4\pi\rho}$ differs by the usual $\sqrt{\mu_0/4\pi}$ bookkeeping, and mixing unit systems here is a classic factor-of- $\sqrt{4\pi}$ wound. Bonus trap: $\omega=k_\parallel v_A$ means phase fronts can run obliquely while energy runs along $\mathbf{B}_0$ — pointing a "wave direction" arrow along $\mathbf{k}$ will mislead you about where the power goes.

Worked examples

1 · Coronal seismology — reading the Sun's field from a ringing loop (confidence builder)

Problem. A coronal loop of length $L=10^8\,\mathrm{m}$ (100 Mm), electron density $n_e=5\times10^{14}\,\mathrm{m^{-3}}$ (hydrogen plasma), carries $B_0=10^{-3}\,\mathrm{T}$ (10 G). Find $v_A$ , the fundamental standing-wave period, and invert: what does an observed 5-minute period say about $B_0$ ?

Step 1. Mass density: $\rho=n_e m_p=5\times10^{14}\times1.673\times10^{-27}=8.4\times10^{-13}\,\mathrm{kg/m^3}$ .

Step 2. Alfvén speed: $v_A=B_0/\sqrt{\mu_0\rho}=10^{-3}/\sqrt{1.2566\times10^{-6}\times8.4\times10^{-13}}=9.8\times10^{5}\,\mathrm{m/s}$ — a thousand kilometers per second, thirty times the sound speed there ( $c_s\approx1.7\times10^5\,\mathrm{m/s}$ at $10^6$ K): deep low- $\beta$ territory.

Step 3. A loop anchored at both footpoints rings like a string clamped at both ends: fundamental period $P=2L/v_A=2\times10^8/9.8\times10^5=204\,\mathrm{s}\approx3.4\,\mathrm{min}$ . TRACE and SDO see exactly such post-flare loop oscillations, with periods of minutes, decaying over a few cycles.

Step 4. Invert for the field: $B_0=(2L/P)\sqrt{\mu_0\rho}$ . An observed $P=300\,\mathrm{s}$ on the same loop gives $B_0=(2\times10^8/300)\times1.03\times10^{-9}=6.9\times10^{-4}\,\mathrm{T}\approx7\,\mathrm{G}$ . Answer: $v_A\approx980\,\mathrm{km/s}$ , $P\approx3.4$ min; a 5-minute ringer implies $B\approx7$ G. This is coronal seismology — the only practical way to measure coronal magnetic fields, and it is nothing but the string formula run backwards.

2 · The coronal heating budget — enough power, wrong bottleneck (exam level)

Problem. Chromospheric observations show transverse motions $\delta v\approx30\,\mathrm{km/s}$ at the coronal base where $n\approx10^{15}\,\mathrm{m^{-3}}$ and $B_0\approx10^{-3}\,\mathrm{T}$ . Compare the upward Alfvén-wave energy flux with the ~300 W/m² needed to sustain the quiet corona (~10⁴ W/m² for active regions), and identify the real difficulty.

Step 1. Base density $\rho=10^{15}\times1.673\times10^{-27}=1.7\times10^{-12}\,\mathrm{kg/m^3}$ ; local $v_A=10^{-3}/\sqrt{\mu_0\rho}=6.9\times10^5\,\mathrm{m/s}$ .

Step 2. The flux derived above: $F=\langle u\rangle v_A=\tfrac12\rho\,\delta v^2\,v_A$ (using $\delta v_0^2=2\langle\delta v^2\rangle$ bookkeeping folded in — take $F\approx\rho\langle\delta v^2\rangle v_A$ as the standard estimate). Numerically: $F=1.7\times10^{-12}\times(3\times10^4)^2\times6.9\times10^5\approx1.0\times10^{3}\,\mathrm{W/m^2}$ .

Step 3. Verdict: $10^3\,\mathrm{W/m^2}$ comfortably exceeds the quiet-Sun requirement (300) and is within an order of magnitude of active-region demand. Power is not the problem. The problem is dissipation: these are nearly ideal waves in a plasma with $R_m\sim10^{12}$ — ohmic damping $\sim\eta k^2$ is astronomically slow at observable wavelengths. Heating requires manufacturing small perpendicular scales (phase mixing between neighboring field lines with different $v_A$ , turbulent cascade of counter-propagating wave packets) until kinetic dissipation — the physics of Landau Damping and its magnetized kin — can finally cash the energy. Answer: flux ≈ 10³ W/m², sufficient; the open question is the last millimeter of the journey, not the first hundred megameters.

The physicist

Origins · 1 minHannes Alfvén and how this was discovered

Hannes Alfvén

1908–1995 · Stockholm — the electrical engineer who gave plasmas their waves

Alfvén trained in Uppsala and worked at Stockholm's Royal Institute of Technology with an engineer's instincts: circuits, currents, and a stubborn distrust of theory that ignored them. The 1942 wave paper was a page and a half in Nature; the objection — conductors kill waves — he answered by showing frozen-in field lines carry their own elasticity. Acceptance was slow until a 1948 seminar at Chicago, after which Fermi's assent ("of course") reportedly turned the tide overnight. Alfvén's larger program — that cosmic electrodynamics is about currents and double layers, not just fields — made him a lifelong contrarian (he fought the Big Bang, and reconnection language, to the end), but the 1970 Nobel Prize in Physics cited exactly what this article derives: "fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics." The waves, the speed, the theorem of the previous article, and the radius unit of magnetospheric physics all bear his name.

History

1942

— Alfvén's Nature note predicts magnetohydrodynamic waves in conducting fluids

1948

— Chicago seminar and Fermi's assent mainstream the idea; Lundquist seeks the waves in mercury

1949

— Lundquist observes torsional MHD oscillations in liquid mercury at ~1 gauss-scaled speeds

1959

— Explorer and early satellites find Alfvénic fluctuations filling the magnetosphere

1971

— Belcher and Davis show the solar wind beyond 0.3 AU is dominated by outward Alfvénic turbulence

1970

— Nobel Prize in Physics to Alfvén for magnetohydrodynamics

Going further

References. Jackson, Classical Electrodynamics, 3rd ed., §7.7 (magnetohydrodynamic waves) · Zangwill, Modern Electrodynamics, Ch. 14 (magnetohydrodynamics) · Alfvén, H., "Existence of Electromagnetic-Hydrodynamic Waves," Nature 150, 405 (1942) · Boyd & Sanderson, The Physics of Plasmas, Ch. 4 · Kulsrud, Plasma Physics for Astrophysics, Ch. 3.

Provenance. Difficulty/level set to advanced as specified. Equations cross-checked against Jackson 3rd ed. §7.7 and Zangwill Ch. 14: the linearized MHD system, the shear-Alfvén dispersion $\omega^2=k_\parallel^2 v_A^2$ , the magnetosonic relation, and energy equipartition. Verifier should recompute: (1) the linearization steps yielding $\omega^2=k_\parallel^2 v_A^2$ for transverse polarization; (2) equipartition $\langle u_{kin}\rangle=\langle u_{mag}\rangle$ using $\delta B=(B_0/v_A)\delta v$ and $B_0^2/(\mu_0 v_A^2)=\rho_0$ ; (3) the numerical check $v_A\approx10\,\mathrm{km/s}$ for $B_0=1\,\mathrm{nT}$ , $n=5\,\mathrm{cm^{-3}}$ using $m_p=1.67\times10^{-27}\,\mathrm{kg}$ , $\mu_0=4\pi\times10^{-7}\,\mathrm{H/m}$ . Historical attribution (Alfvén 1942, Nobel 1970) is standard. No fabricated constants used; all numbers are textbook/standard values. Lane addendum (2026-07-06): worked numbers (corona $v_A=9.8\times10^5$ m/s and 3.4-min loop period, heating flux $10^3$ W/m², TAE 70 kHz, solar-wind $v_A$ anchors) and the Friedrichs-diagram curves ( $c_s=0.6\,v_A$ , plotted from the computed biquadratic roots) were generated in python during the bulk upgrade; the magnetosonic completion and mode count were added to the base per the lane ownership map.

Prerequisites. Maxwell's Equations in Vacuum (E6) · Magnetohydrodynamics (MHD) · Lorentz Force and Maxwell Stress Tensor (E7) · Fluid Dynamics (Euler equations).

Leads to. Magnetosonic Waves · MHD Turbulence · Solar Coronal Heating · Magnetorotational Instability.

If a perfect conductor excludes electromagnetic waves, how can a perfectly conducting fluid carry one?

The skin-depth argument assumes the conductor stands still: induced currents cancel the incoming wave's field, and the wave dies in a skin depth (E8's conductor-wave machinery). An MHD fluid moves. The wave here is not an EM wave propagating through the conductor — it is a joint oscillation of the conductor and its frozen-in field, with most of the energy in fluid motion and bent field lines, traveling at $v_A\ll c$ . The induced E-field ( $-\mathbf{v}\times\mathbf{B}$ ) is part of the wave, not its enemy. Different beast, different rules.

Is the string analogy exact or just pedagogical?

Quantitatively exact, which is rare for analogies. Tension per area $B_0^2/\mu_0$ (derived from the stress decomposition in Magnetohydrodynamics) times cross-section, over mass per length $\rho A$ , gives $v_A$ with every factor in place — the honest linearization lands on the same number. Where the analogy needs care: a real string's tension is fixed by external pulls, while the field's tension is self-maintained; and adjacent "strings" here interact through magnetic pressure the moment you go beyond the linear shear wave. The fast and slow modes are precisely the physics the lone-string picture cannot see.

Why does only k-parallel appear in the dispersion relation — what happened to k-perpendicular?

The restoring force is tension, which is the field-aligned second derivative: linearized, $(\mathbf{B}_0\cdot\nabla)\delta\mathbf{B}\propto k_\parallel$ . Perpendicular structure costs nothing at linear order for the shear polarization — you can corrugate the wave across the field arbitrarily finely without changing its frequency. This degeneracy is physical and consequential: it lets neighboring field lines oscillate independently (the seed of phase mixing) and permits $k_\perp\to\infty$ cascades that end in kinetic dissipation — the corona's suspected heating pathway. The degeneracy finally breaks at the ion scale, where the kinetic Alfvén wave picks up $k_\perp$ dispersion (Waves in Magnetized Plasmas territory).

Do Alfvén waves exist in liquid metals, or only in plasmas?

They exist wherever conductivity and field are sufficient, and liquid metal is where they were first seen: Lundquist (1949) drove torsional oscillations in mercury, and later sodium experiments cleaned the result up. The practical difficulty is damping — laboratory $R_m$ is modest, so the waves are heavily diffused (quality factor near unity) unless the field is strong. In plasmas, $R_m$ is enormous and the waves are essentially undamped at MHD scales — which is why the corona's problem is dissipating them, not keeping them alive.

What happens to an Alfvén wave at very high frequency?

The MHD contract expires. As $\omega$ approaches the ion cyclotron frequency $\Omega_{ci}=eB/m_i$ , ions can no longer follow the field-line oscillation rigidly; the left-circular branch resonates at $\Omega_{ci}$ (ion-cyclotron waves) and the right-circular branch continues up as the whistler. Below that, at perpendicular scales near the ion skin depth or gyroradius, the shear wave becomes the dispersive kinetic Alfvén wave with a parallel electric field — able to accelerate electrons (a leading account of auroral electron beams). All of this two-fluid richness is the next article's subject, Waves in Magnetized Plasmas.

Are Elsässer variables worth learning at this level?

Yes, one identity's worth. Define $\mathbf{z}^\pm=\delta\mathbf{v}\mp\delta\mathbf{B}/\sqrt{\mu_0\rho_0}$ . In incompressible ideal MHD these satisfy $(\partial_t\pm\mathbf{v}_A\cdot\nabla)\mathbf{z}^\pm=-(\mathbf{z}^\mp\cdot\nabla)\mathbf{z}^\pm$ (pressure aside): a $\mathbf{z}^+$ packet alone is an exact nonlinear solution — arbitrary-amplitude one-way Alfvénic wave trains do not self-destruct. Turbulence requires counter-propagating packets to collide ( $\mathbf{z}^+$ against $\mathbf{z}^-$ ), which is why solar-wind turbulence theory is written entirely in Elsässer language and why the (mostly one-way) inner heliosphere is less turbulent than naive estimates suggest. The full formalism belongs to MHD Turbulence in the Fluid track.