Alfvén Wave

Definition

An Alfvén wave is a transverse oscillation of a magnetised plasma in which the magnetic field lines and the matter frozen to them swing sideways together, propagating along the field at the Alfvén speed:

v_A = B / √(μ₀ · ρ)

Here B is the background magnetic field strength (tesla), μ₀ = 4π×10⁻⁷ T·m/A is the permeability of free space, and ρ is the plasma mass density (kg/m³). The wave was predicted by Swedish physicist Hannes Alfvén in 1942; he received the 1970 Nobel Prize in Physics for founding the field of magnetohydrodynamics (MHD). The single sentence to keep in your head: magnetic field lines under tension behave like vibrating strings, and the frozen-in plasma is the mass they have to drag along.

How it works

Two ingredients make an Alfvén wave possible, and they map one-to-one onto the two ingredients of a vibrating string.

• Restoring force — magnetic tension. A magnetic field exerts a tension along its own lines of force, with magnitude B²/μ₀ per unit cross-sectional area. Bend a field line sideways and this tension pulls it straight again, exactly like the tension in a stretched guitar string.
• Inertia — the frozen-in plasma. In a good conductor the magnetic flux is frozen into the fluid: the field and the matter move together (Alfvén's frozen-flux theorem). So when the field line swings, it must drag the plasma with it. The plasma mass density ρ is the inertia, playing the role of the string's linear mass density.

For a string, the wave speed is v = √(T/μ) — square root of tension over linear density. Substitute the magnetic tension B²/μ₀ for T and the plasma density ρ for μ, and you recover the Alfvén speed directly:

v_A = √( (B²/μ₀) / ρ ) = B / √(μ₀ ρ)

The pure (shear) Alfvén wave is transverse and incompressible: the plasma sloshes perpendicular to the background field, but its density and pressure never change. There is no compression, no sound — only the elastic twang of bent field lines. In ideal MHD it is also dispersionless: every wavelength travels at the same v_A, just like an ideal string, so a complex disturbance keeps its shape as it runs along the field.

Worked example — Alfvén speed in the solar corona

Take a coronal magnetic loop with a typical field strength of 10 gauss and a number density of about 10¹⁵ particles per cubic metre, mostly protons. Let's compute v_A from scratch.

B   = 10 G = 10 × 10⁻⁴ T = 1.0×10⁻³ T
n   = 1×10¹⁵ m⁻³  (protons)
ρ   = n · m_p = 1×10¹⁵ × 1.67×10⁻²⁷ kg ≈ 1.67×10⁻¹² kg/m³
μ₀  = 4π×10⁻⁷ ≈ 1.257×10⁻⁶ T·m/A

v_A = B / √(μ₀ ρ)
    = 1.0×10⁻³ / √(1.257×10⁻⁶ × 1.67×10⁻¹²)
    = 1.0×10⁻³ / √(2.10×10⁻¹⁸)
    = 1.0×10⁻³ / 1.45×10⁻⁹
    ≈ 6.9×10⁵ m/s  ≈  690 km/s

So a kink launched at the base of this loop races up the field at roughly 700–1,000 km/s — about 0.2% of light speed, and far faster than the local sound speed of ~200 km/s. Crank the field up to 100 G in a sunspot's strong-field loop and v_A jumps to ~7,000 km/s; the field's stiffness, not the gas, sets the tempo. This dominance of magnetic over thermal pressure is captured by the plasma beta, which in the corona is only β ≈ 0.01–0.1.

Variants and related MHD modes

The shear Alfvén wave is one of three wave modes that ideal MHD supports. They differ in how they combine magnetic tension, magnetic pressure, and gas pressure.

Mode	Restoring force	Compressible?	Propagation vs B	Speed (β ≪ 1)
Shear Alfvén wave	Magnetic tension	No (incompressible)	Along B	v_A = B/√(μ₀ρ)
Fast magnetosonic	Magnetic + gas pressure	Yes	Any angle, fastest across B	≈ √(v_A² + c_s²)
Slow magnetosonic	Gas pressure (tension limits it)	Yes	Mostly along B	≈ c_s (sound speed)
Torsional Alfvén	Magnetic tension (twist)	No	Along axisymmetric flux tube	v_A
Kinetic Alfvén	Tension + finite-gyroradius effects	Weakly	Nearly across B at small scales	≈ v_A (dispersive)
Alfvén eigenmode (TAE)	Tension in toroidal geometry	No	Standing along field lines	≈ v_A / (2qR)

The torsional variant is a rotational twist running along a cylindrical flux tube — think of wringing a rope rather than waving it. Kinetic Alfvén waves appear when the wavelength shrinks toward the ion gyroradius; they become dispersive and develop a small parallel electric field, which is how they can accelerate particles in the aurora and the magnetosphere. In toroidal fusion devices, reflections turn travelling Alfvén waves into discrete toroidal Alfvén eigenmodes sitting in frequency gaps of the continuous spectrum.

Dispersion relation and energy

For a uniform plasma with the wavevector k at angle θ to the field, the shear Alfvén branch obeys a strikingly simple dispersion relation:

ω = k‖ · v_A = k · v_A · cos θ

Only the field-parallel component of the wavevector matters — energy funnels along the field lines, never across them. The phase and group velocities differ in direction: the phase velocity points along k, but the group velocity (energy flow) is locked to ±v_A along B regardless of θ. That guided behaviour is exactly why convective motions at the Sun's surface can pump energy hundreds of thousands of kilometres up a coronal loop without it leaking sideways.

Energy is shared equally between the magnetic perturbation and the kinetic motion of the plasma — equipartition, as in any harmonic oscillator. The time-averaged energy flux (Poynting plus kinetic) carried by an Alfvén wave of velocity amplitude δv is:

F = ρ · ⟨δv²⟩ · v_A

Plug in coronal numbers — ρ ≈ 1.7×10⁻¹² kg/m³, observed δv ≈ 20 km/s, v_A ≈ 700 km/s — and you get a flux of order 5×10² W/m², comfortably above the ~300 W/m² needed to heat the quiet corona. The energy budget closes; the open question is the dissipation, not the supply.

Where Alfvén waves show up

• Coronal heating. The leading puzzle in solar physics: why is the corona 1–3 million K when the surface is 5,800 K? Alfvén waves launched by photospheric jostling carry energy upward and dissipate it through phase mixing, resonant absorption, and turbulent cascade.
• Parker Solar Probe. NASA's probe flew through the corona and measured the tight velocity–magnetic-field correlation that is the signature of Alfvénic fluctuations — including the famous magnetic "switchbacks" in the solar wind.
• Fusion devices. In tokamaks (ITER, JET) fusion-born alpha particles can resonantly destabilise Alfvén eigenmodes and eject fast ions before they deposit their heat — a key stability concern.
• Earth's magnetosphere and aurora. Field-line resonances and kinetic Alfvén waves accelerate electrons that slam into the upper atmosphere and light the aurora.
• Solar wind acceleration. Alfvén-wave pressure helps push the fast solar wind out past the planets, shaping the heliosphere.
• Laboratory plasma physics. The LArge Plasma Device (LAPD) at UCLA launches and measures Alfvén waves directly, turning the Sun's strings into a benchtop experiment.

Common pitfalls and misconceptions

• Thinking Alfvén waves compress the plasma. The pure shear mode is incompressible — no density change at all. If you see compression, you are looking at a magnetosonic mode, not a shear Alfvén wave.
• Assuming they travel in any direction. Energy flows along the field at v_A; the dispersion ω = k‖v_A means a wave with k purely across B carries no energy. Alfvén waves are guided, not isotropic.
• Confusing Alfvén speed with the speed of light. v_A is a mechanical wave speed set by tension and inertia. Coronal values of ~1,000 km/s are only ~0.3% of c; v_A only approaches c in extreme objects like magnetar magnetospheres.
• Ignoring the frozen-in assumption. If resistivity is high (low magnetic Reynolds number), the field slips through the plasma and the wave is heavily damped. Frozen-in flux (Rm ≫ 1) is what makes the string elastic in the first place.
• Expecting strong dispersion. In ideal MHD the shear wave is dispersionless. Dispersion only enters at kinetic scales (the kinetic Alfvén wave near the ion gyroradius), not in the textbook fluid picture.
• Treating heating as automatic. Launching Alfvén waves is easy; dissipating them in a nearly collisionless corona is hard. The energy supply is sufficient — the research frontier is the mechanism that converts wave energy to heat.

JavaScript — computing Alfvén speed across environments

const MU0 = 4 * Math.PI * 1e-7;      // T·m/A
const M_P = 1.6726e-27;              // proton mass, kg

// Alfvén speed from field strength (T) and proton number density (m^-3)
function alfvenSpeed(B, n) {
  const rho = n * M_P;               // mass density (kg/m^3)
  return B / Math.sqrt(MU0 * rho);   // m/s
}

const envs = [
  { name: "Solar corona",      B: 1.0e-3, n: 1e15 },
  { name: "Sunspot loop",      B: 1.0e-2, n: 1e15 },
  { name: "Solar wind @ 1 AU", B: 5.0e-9, n: 5e6  },
  { name: "Tokamak (ITER)",    B: 5.3,    n: 1e20 },
  { name: "Magnetar surface",  B: 1e10,   n: 1e30 },
];

for (const e of envs) {
  const v = alfvenSpeed(e.B, e.n);
  console.log(`${e.name.padEnd(20)} v_A = ${(v / 1e3).toExponential(2)} km/s`);
}
// Solar corona        v_A = 6.90e+2 km/s
// Sunspot loop        v_A = 6.90e+3 km/s
// Solar wind @ 1 AU   v_A = 4.88e+1 km/s
// Tokamak (ITER)      v_A = 1.16e+4 km/s
// Magnetar surface    v_A ~ c  (relativistic correction needed!)

// Travel time for a wave to cross a coronal loop of length L (m)
function crossingTime(B, n, L) {
  return L / alfvenSpeed(B, n);      // seconds
}
console.log(crossingTime(1e-3, 1e15, 1e8).toFixed(0), "s to cross a 100,000-km loop");
// ~145 s

Note the magnetar line: when v_A computed this way exceeds c, the non-relativistic formula has broken down and you must cap the speed at the light speed using the relativistic Alfvén expression. In every ordinary plasma — corona, solar wind, tokamak — the classical B/√(μ₀ρ) is exact.

A note on history and naming

Hannes Alfvén submitted his prediction to Nature in 1942, and it was initially met with skepticism — the idea that a magnetic field could carry mechanical waves was alien to physicists raised on Maxwell's electromagnetic waves. The tide turned after a famous seminar at the University of Chicago in 1948, when Enrico Fermi, hearing Alfvén speak, simply said "Of course," and the community fell in line behind him. The waves were confirmed in liquid mercury and later in laboratory plasmas, and today the Alfvén speed is one of the first numbers any plasma physicist computes about a new system — because, as Alfvén himself put it, the magnetised plasma is a medium that can sing.