The wave that does not know what the gas is made of

The stress a field exerts on every surface ended with a speed nobody had asked for. Dividing the curvature force of a bent field line by the density of the plasma frozen into it gave an acceleration, and the acceleration came out as $v_A^2/R$ — a squared velocity, from a calculation about forces, with no wave anywhere in it.

That is what always happens when a tension is divided by an inertia, and it happens here for the same reason it happens on a violin string. A magnetic field bent sideways pulls back; the plasma stuck to it has mass; a restoring force and an inertia are a wave equation with the speed already written into it. What is peculiar is not the existence of the wave but the speed:

$$v_A = \frac{B}{\sqrt{\mu_0 \rho}}.$$

There is no temperature in it, no pressure, no ratio of specific heats, no sound speed. Every other wave in a fluid asks what the fluid is; this one asks only how much of it there is.

The string, made explicit #

Take a uniform field along $x$ with plasma frozen into it, and displace one field line sideways by a small amount $\xi(x,t)$.

The line is now bent. The stress calculation already gave what a bend costs: a transverse force per unit volume of $B^2/\mu_0 R$, where $R$ is the radius of curvature. For a small displacement the curvature is $\partial^2\xi/\partial x^2$, so the transverse force per unit volume is $(B^2/\mu_0)\,\partial^2\xi/\partial x^2$, and Newton’s law for the plasma of density $\rho$ carried along by that line reads

$$\rho\,\frac{\partial^2 \xi}{\partial t^2} = \frac{B^2}{\mu_0}\,\frac{\partial^2 \xi}{\partial x^2}.$$

That is the wave equation, with the same structure as every other one on this site — an equation that lets a shape travel, with a tension over an inertia in the coefficient. The tension is $B^2/\mu_0$ per unit area and the inertia is $\rho$ per unit volume, so the speed is the square root of their ratio.

Two features of that pulse are worth separating, because they usually arrive together and are independent.

It does not spread. The dispersion relation is $\omega = v_A k$, a straight line, so every component of a packet travels at the same speed and the packet keeps its shape indefinitely. A sound wave in air does the same and a wave on deep water does not; the shear wave is on the non-dispersive side of that divide for a structural reason rather than an approximate one, since nothing in the restoring force refers to a length.

It does not compress anything. The displacement is across the field, so the plasma moves perpendicular to the direction the wave travels, and the divergence of that motion is zero everywhere. The density never changes anywhere at any time. That is the property no sound wave can have and the reason this wave has no thermodynamics in it: a wave that never compresses the gas never asks the gas how it responds to being compressed, so the equation of state does not appear.

One expression, twenty-eight decades #

The speed’s indifference to the medium makes it worth evaluating in places that have nothing else in common.

The Earth’s liquid outer core carries the same wave at two and a half centimetres a second. The corona carries it at seven thousand kilometres a second. Between them are a laboratory tank of liquid sodium at walking pace and a tokamak at two per cent of the speed of light, and the expression is the same expression in all four.

The core’s number is the one that turns into an observation. Torsional oscillations — cylindrical shells of the outer core rotating slightly against one another in the manner of two pendulums swapping their motion, with the field between them supplying the restoring torque — have a period set by the shell separation over the Alfvén speed, which for a field of a few millitesla comes out at a few years to a few decades. Those periods are seen in the length of the day, which varies by a millisecond or so on exactly that timescale as angular momentum passes between the core and the mantle. A wave nobody can look at, in a fluid nobody has sampled, is measured by timing the rotation of the planet.

The ceiling matters too, and it is where the derivation admits its own limits. As the density falls, the expression rises without bound, and above $c$ it is simply wrong — the correct relativistic version saturates at the speed of light, because the field’s own energy contributes to the inertia and a field carrying more energy than the matter it threads is dominated by its own mass. In a magnetar’s magnetosphere that is the ordinary condition, and everything in this essay is replaced there.

Three waves, and which of them the field cares about #

The shear wave is one of three. Letting the gas have a pressure as well restores the sound speed $c_s$, and the three branches come from one quadratic:

$$v^2 = \tfrac{1}{2}\left[(c_s^2 + v_A^2) \pm \sqrt{(c_s^2+v_A^2)^2 - 4c_s^2 v_A^2\cos^2\theta}\right],$$

with the shear branch $v = v_A|\cos\theta|$ sitting between the two roots at every angle.

The three behave very differently at right angles to the field. The slow wave stops existing: its speed goes to zero, because a slow wave needs a component of field along the direction of travel and there is none. The shear wave stops too, for the same reason. Only the fast wave crosses the field, and it does so at $\sqrt{c_s^2 + v_A^2}$ — carrying the gas pressure and the magnetic pressure together, which is exactly what a compressional wave in a magnetised fluid should do, since the two pressures add in the balance.

Comparing the two versions of the diagram shows what is fixed and what is not. Everything about the magnetosonic curves depends on the ratio of the two speeds. The figure-of-eight does not move at all. Its size is set by $B$ and $\rho$ and by nothing else, and the gas can be heated, cooled, or swapped for another gas of the same mass density without the shear branch noticing.

Where the energy actually goes #

A polar diagram of phase speed is a picture of how fast crests move in each direction, and it is routinely misread as a picture of where the wave goes. For the shear wave the two are as different as they can be.

The group velocity in polar coordinates is $v_g = v_{\text{ph}}\hat{k} + (\mathrm{d}v_{\text{ph}}/\mathrm{d}\theta)\,\hat{\theta}$. Put $v_{\text{ph}} = v_A\cos\theta$ into that and the two terms combine into $(v_A, 0)$ — a constant vector along the field, independent of $\theta$ entirely.

Send a shear wave into a plasma at eighty degrees to the field and its crests crawl along at $v_A\cos 80° = 0.17 v_A$ in the direction they were sent, while the energy goes straight down the field line at the full $v_A$. The wave is a disturbance running along a wire, and the wire is the field line.

That is the single most consequential fact in the subject, and it is a statement about anisotropy rather than about magnetism. A magnetised plasma is not a medium with a preferred direction in the mild sense that light in a crystal has one. It is a bundle of channels. Energy put in at one point travels along the line through that point and reaches no other line, which is why a coronal loop is a lit object with a dark neighbour a thousand kilometres away — a channel with no walls made of nothing but a direction, why heat conduction in a fusion device is ten orders of magnitude worse across the field than along it, which is the drift that does not care what the charge is seen as a transport coefficient, and why solar energetic particles arriving at the Earth are guided by the interplanetary field rather than by geometry.

It also explains why the corona has structure at all. A photograph of the Sun in extreme ultraviolet shows threads. A medium that transported energy isotropically would have no threads, because everything would smear; a medium of independent channels has as many threads as it has field lines that are doing something different from their neighbours.

The wave that had to be produced before it was believed #

Hannes Alfvén published the derivation in 1942, and it was not accepted for the better part of a decade, for a reason worth stating because it is a good objection rather than a bad one.

The objection was that Maxwell’s equations already say what waves a magnetic field supports, and they travel at $c$. A disturbance of a magnetic field propagating at a hundredth of a per cent of the speed of light appeared to be a claim that light had been slowed down by a factor of ten thousand, and nothing in the equations does that.

The answer is that the wave is not a wave in the field alone. It is a wave in the field and the matter frozen to it, and the matter’s inertia is the entire reason for the slowness — exactly as light in glass is slowed by the electrons it drives rather than by anything happening to the vacuum. Setting $\rho \to 0$ in the expression returns the speed to infinity, which is the non-relativistic limit’s way of saying $c$; the whole of the finite answer comes from the mass being carried.

What settled it was making one. Lundquist ran a current sheet through liquid mercury in a field in 1949 and found a disturbance travelling at the predicted speed; Lehnert repeated it in liquid sodium, whose conductivity is high enough for the wave to survive several wavelengths. The laboratory versions are heavily damped, because a liquid metal’s magnetic Reynolds number is not the astrophysical one, and the experiments are difficult for exactly that reason. The modern instrument is a linear plasma device tens of metres long in which the dispersion relation is mapped directly by driving an antenna at one end and measuring the phase at the other, and the measured $\omega(k)$ is a straight line whose slope is $B/\sqrt{\mu_0\rho}$ with both quantities independently known.

Alfvén received a Nobel prize in 1970 for the subject the wave started. The delay between the derivation and the acceptance is the useful part of the story: the objection was that a known equation gave a known answer, and the resolution was that a different question had been asked.

The lever the Sun spins down with #

The wave has a consequence far from any corona, and it is the reason the Sun turns as slowly as it does.

A star’s wind carries mass away, and mass carried away carries angular momentum. If the wind simply left, each parcel would take the angular momentum it had at the surface, and the star would lose spin very slowly indeed — the Sun’s whole mass loss over its lifetime is a hundredth of a per cent.

But the wind is a plasma and the field is frozen into it, so the field goes with it while remaining attached to the star. Out to some distance the field is strong enough to enforce co-rotation: the plasma is dragged round with the star as though on a rigid spoke, and its specific angular momentum grows as the square of its distance. Past that distance the wind’s inertia wins and it leaves on a straight line.

The distance where the changeover happens is where the wind speed equals the Alfvén speed. Inside it, a disturbance can run back down the field to the star faster than the flow carries it out, so the two ends stay in communication and torque is transmitted; outside, no signal can get back, and the connection is severed. The Alfvén radius is a causal boundary, and the speed that defines it is the speed of this essay.

For the Sun that radius is between ten and thirty solar radii, so every gram of wind leaves with the angular momentum of material a few hundred times further out than the surface — a lever arm of a few hundred, and a torque several hundred times what the bare mass loss would give. Integrated over four and a half billion years it is the difference between the Sun’s present twenty-five-day rotation and the day or two a young star has. The braking is why old stars turn slowly, why rotation is usable as an age indicator, and why a star’s magnetic activity — which is driven by rotation — declines over its life.

Whether it heats anything #

The corona is two hundred times hotter than the surface below it, which is a violation of nothing but is certainly a puzzle, and the energy has to come up from below. Waves are one of the two standing candidates — the other is the reconnection of stressed fields braided by footpoint motion — and the wave in question is this one.

The arithmetic of the supply is simple. A shear wave of transverse velocity amplitude $\delta v$ carries an energy flux $\rho \langle \delta v^2\rangle v_A$, so the measurement that decides is a velocity.

Transverse motions of that size are measured, and were the harder half of the problem for thirty years. The Hinode spacecraft resolved swaying in spicules; the CoMP coronagraph found propagating transverse waves throughout the corona; and spectral line widths in coronal holes require unresolved motions of the same order. So the flux is present at roughly the right size for the quiet Sun.

The supply is not the difficulty. The difficulty is that an Alfvén wave is almost impossible to damp. Its dissipation length at coronal resistivity is longer than the solar system, precisely because it never compresses the plasma and so never does any of the things — shocking, heating on compression — that dispose of a sound wave’s energy. A wave that reaches the corona and passes straight through it has heated nothing.

Every proposed mechanism is therefore a way of getting the energy out of a wave that will not give it up on its own. Phase mixing exploits the fact above: neighbouring field lines with different densities carry the wave at different speeds, so an initially smooth wavefront is sheared into ever finer transverse structure until the gradients are steep enough for even a tiny resistivity to act. Resonant absorption does the same at a particular surface where the local Alfvén frequency matches the driving one. Turbulent cascade relies on counter-propagating waves interacting non-linearly, which is the only way two shear waves talk to each other at all. Each makes different predictions about where in a loop the heating occurs, and separating them needs spatial resolution in the corona that is only now arriving.

The honest position is that the energy is there, the mechanism is not settled, and the two candidates — waves and braiding — are not exclusive and may not be distinguishable in principle, since a sufficiently tangled field driven at its footpoints is being shaken and stressed at the same time.

Where this stops being right #

Everything here is linear. The displacement was assumed small enough that the restoring force is proportional to it, which is what made the equation a wave equation. At large amplitude a shear wave steepens the field it rides on, generates compressional components it did not start with, and stops being any of the three clean branches. The solar wind is full of large-amplitude Alfvénic fluctuations to which none of the polar diagrams apply.

The plasma has been one fluid with one temperature. Below the ion gyrofrequency that is a good description and above it there is no such thing as a shear wave: the ions stop following the field, the electrons continue to, and the branch splits into kinetic Alfvén and whistler modes with quite different properties. The frequency at which that happens is a few hertz in the solar wind, which is inside the range spacecraft measure.

Resistivity was set to zero throughout. That is what made the wave lossless and is also what makes the heating problem hard; the two are the same assumption, and it is very nearly true. Restoring a finite resistivity adds a damping term whose size is set by the magnetic Reynolds number, and the numbers involved are the same enormous ones that make reconnection slow.

And the speed can exceed $c$. The expression has no ceiling and physics does. Where the field’s energy density approaches $\rho c^2$ the correct answer saturates below the speed of light, and the figure draws that boundary rather than pretending it is far away, because for the objects with the strongest fields it is not.

What the drawings leave out #

The polar diagrams draw a speed against a direction and show no wave. What is actually happening at each point of the shear-wave curve is that plasma is moving perpendicular to the plane of the paper, which cannot be drawn on the paper — so the branch that the essay is about is the one whose motion is the one direction the figure does not have.

The pulse figure draws a displacement against distance and hides the fact that the field line is doing the displacing. There is no string in the picture and no string in the plasma; there is a field whose curvature has a force associated with it, and calling the arrangement a string is a claim about the form of an equation rather than about the presence of an object.

And no still picture can show the difference between phase and group velocity, which is the subject of two of these figures. The distinction only exists in time: the crests move one way and the envelope moves another, and a drawing of either is a drawing of one instant in which they are indistinguishable.

What the ladder has and has not settled #

Four rungs stand on flux-freezing. The first is a conservation law, the second the prohibition it hides, the third the stress that makes the field mechanical, and this one the wave that stress supports.

The habit worth carrying away is what a missing quantity means. When a result contains fewer variables than the situation does, the ones that are absent are absent for a reason worth finding. The shear wave’s speed contains no thermodynamics because the wave never compresses anything, and tracing the absence back to that is more informative than deriving the presence of the ones that are there. The same test applied to the buoyancy frequency of a stratified fluid finds that it contains no wavelength, for the same kind of reason.

What is left on this ladder is the state a stressed field relaxes into once it has been allowed to release everything it can. A field that has given up all the energy it is able to, while keeping the flux through its boundary and the linkage it started with, is a force-free field with a particular constant — and the fact that laboratory plasmas and the corona both find it is a result about which invariants survive a violent process and which do not.