The wave that dies with nothing to rub against

Damping in a wave normally means something is rubbing. A resistance in a circuit, a viscosity in a fluid, a radiated field carrying energy off. In each case the wave’s energy ends up somewhere warmer, and the process cannot be undone.

The equation integrated for that figure contains no collisions. It is the Vlasov equation for the distribution of particle velocities, coupled to Poisson’s equation for the field the particles themselves make, and the only operator acting on the distribution is a rotation of phase whose rate depends on how fast a particle is moving. Nothing in it can produce heat.

The wave nevertheless dies, and it dies exponentially.

The measurement, and what it is checked against #

Two numbers make the result more than a plausible picture.

The rate. Fitted to the envelope of the field over twenty plasma periods, it is $0.1534$ per period at a wavenumber of half a reciprocal Debye length. The published root of the kinetic dispersion relation at that wavenumber is $0.1534$. Those are two independent computations of the same thing and they agree to four figures.

The accounting. The linearised system has an exactly conserved quantity — the weighted norm of the perturbation plus the field energy, which is the perturbation’s free energy — and across the whole run it moves by $2.4\times10^{-10}$. The field loses four decades of amplitude and every bit of it is still there, in the distribution.

So the energy has not gone anywhere warm. It has gone into the ordered motion of the particles, and the books balance to a part in ten thousand million.

What the particles are actually doing #

The mechanism is a resonance between the wave and the particles moving at its own speed, and the sign of the effect is decided by a slope.

A particle moving at very nearly the wave’s phase speed sees a field that is almost steady rather than oscillating. Instead of being pushed one way and then the other and ending up where it started, it is pushed one way for a long time. Such a particle exchanges energy with the wave in earnest.

Which way the exchange goes depends on which side of the phase speed the particle is on. Slightly slower particles are picked up and accelerated — they gain, and the wave loses. Slightly faster ones are held back — they lose, and the wave gains. The image usually offered is surfing, and it is a good one as long as it is remembered that the surfer who is going too fast is being slowed down by the wave rather than falling off it.

The net direction is therefore decided by which kind is more numerous, which is a statement about the slope of the distribution at the phase speed. On a Maxwellian the distribution falls, so there are more slow ones — $29$ per cent more within a narrow window in the figure — and the wave loses.

This is the sense in which the effect is not about dissipation. It is a transfer between the wave and one narrow class of particles, decided by a derivative, and reversing the sign of that derivative reverses the transfer.

Where the wave goes #

If nothing is dissipated, the wave’s amplitude must still be recorded somewhere. It is.

Particles free-stream: a particle at velocity $v$ carries its bit of the perturbation along at $v$, so a perturbation that starts smooth in velocity acquires a phase $kvt$ that grows with time and differently for each velocity. After a while the perturbation oscillates rapidly as a function of velocity, and it goes on getting finer for ever.

The field does not care about the perturbation itself. The field comes from Poisson’s equation, and Poisson’s equation sees only the integral of the perturbation over velocity — the charge density. An integral over a rapidly oscillating function is nearly nothing.

So the field vanishes while the perturbation does not. The number of sign changes in the figure is $10$ at ten plasma periods, $24$ at twenty-four and $40$ at forty, which is the winding rate $k\Delta v$ made visible. Nothing has been destroyed and nothing has been randomised; the information has been moved into a place the field cannot look.

This is a mechanical version of a much more general observation about how a reversible microscopic law produces an irreversible-looking average. The averaging is doing the work, and the equation underneath is untouched.

The experiment that gave it back #

The claim that nothing is lost has an obvious test: undo the winding and the wave should return.

It was done, in 1968, and the answer is the plasma wave echo. Launch a wave at one wavenumber, let it damp away to nothing, and launch a second wave at a different wavenumber somewhere further along. Neither is present any longer as a field. But the two have left their imprints in the distribution’s velocity structure, and at a particular later time and place those two imprints come back into phase with each other — and a third wave appears, out of a plasma with no measurable field in it.

The echo is not a small effect and it is not ambiguous. It exists only if the information about both original waves survived their apparent disappearance, which is precisely what the free-energy conservation says.

That also identifies exactly what would destroy the recoverability. The echo is killed by collisions, however rare, because a collision moves a particle in velocity and smears the fine structure. So the observable lifetime of the echo measures the collision rate — and it does so far more sensitively than any transport measurement, because the structure being smeared is finer than anything else in the problem.

The argument that took thirty years to settle #

Landau’s paper of 1946 is a short one and it was not believed for a long time, which is worth recounting because the objection was a good one.

The quantity the damping rate depends on is how many particles move at the wave’s own phase speed, and that number is read off the exponential tail of a thermal distribution. The tail falls fast enough that a factor of two in the phase speed is orders of magnitude in the rate — which is why the damping is negligible for long wavelengths, where the phase speed is far out on the tail, and overwhelming for short ones, where it sits among the bulk of the particles.

Vlasov had written the collisionless equation and found the oscillation, and had treated the singular integral over the resonant velocity by taking its principal value — which throws away the imaginary part and gives an undamped wave. Landau pointed out that the problem is an initial-value problem rather than a normal-mode one, solved it by Laplace transform, and found that the contour has to pass under the pole. The imaginary part that results is the damping.

The objection was that the operator is anti-Hermitian and its spectrum is therefore real, so a genuinely decaying solution should not exist. That objection is correct. Van Kampen and Case exhibited the actual eigenmodes a decade later: a continuum, one for every velocity, each of them undamped and singular, and any physically sensible initial condition is a superposition of them. The apparent decay is the superposition dephasing, exactly as the winding figure shows, and the “Landau mode” is not an eigenmode at all but the pole of the analytically continued response function.

Both descriptions give the same field, and each makes something obvious that the other hides. Landau’s makes the exponential rate obvious. Van Kampen’s makes it obvious that nothing has been lost — which is what the echo went on to demonstrate in a laboratory.

Turn the slope over and it grows #

Because the rate is proportional to a derivative, an experiment that changes the sign of that derivative changes damping into growth with no other alteration.

Fire a beam of fast electrons through a plasma. The combined distribution now has a bump on its tail, and on the rising side of that bump there are more fast particles than slow ones at the relevant speed. Waves whose phase speed falls there are amplified rather than damped, and they grow until the beam is smeared out.

That is the bump-on-tail instability, and it is the same expression with one sign changed. Its industrial version is the travelling-wave tube, in which a beam is deliberately run alongside a slow electromagnetic wave to amplify it, and its astrophysical version is the mechanism behind solar radio bursts. It is the same bargain a resonance always offers: a small coupling, applied at exactly the right rate for long enough.

The generality is worth stating plainly. Resonance between a wave and the particles moving at its speed transfers energy in whichever direction the distribution’s slope dictates, and damping and instability are the two signs of one calculation.

The scale that decides how much #

The size of the effect is dominated by an exponential, and the exponential is a counting argument.

The wave’s phase speed, for a long wave, is many thermal speeds — the dispersion relation puts it at $\omega_p/k$, which is large when $k$ is small. The number of particles moving that fast is the tail of a Maxwellian, which is exponentially small. So a long wave has almost nobody to resonate with and is barely damped.

As the wavelength shrinks towards the Debye length, the phase speed falls towards the thermal speed, the resonant population becomes the bulk of the distribution, and the damping becomes catastrophic: a wave at $k\lambda_D = 0.5$ loses its amplitude in about six oscillations.

That is why plasma oscillations exist as oscillations at all. The short ones are gone before they have completed a period, so the only ones that survive to be observed are those long compared with the Debye length — which is the same statement as the Debye length being the distance beyond which a charge is not felt, read in the frequency domain instead.

The asymptotic formula for the rate is the one usually quoted, and the figure shows how well it does. At $k\lambda_D = 0.5$ it is within one and a half per cent of the exact root. At $0.3$ it is sixty per cent high. An approximation whose error varies by that much across the range it is quoted for is one worth solving the exact problem beside.

The same mechanism where there is no charge at all #

The argument used nothing about electricity except that the force is long-ranged and computed from the distribution itself. Replace the electric field by a gravitational one and it runs unchanged.

The same resonance operates where there is no charge at all. A self-gravitating system responds to a perturbation through the stars moving at the wave’s own pattern speed, exactly as a plasma responds through the electrons moving at its phase speed — with one sign changed, because gravity is attractive where the electrostatic force between like charges is not. The mechanism is a resonance between a wave and the particles that keep pace with it, and it does not care what the force is.

A star cluster or a galactic disc is a collisionless system in exactly the same sense: the time for individual encounters to matter is longer than the age of the universe, and the dynamics is the same kinetic equation with gravity in place of the Coulomb force. Perturbations to such a system are damped by resonance with the stars moving at the pattern’s own speed, and the effect is central to how a spiral arm transfers angular momentum and how a bar slows down.

The sign is what changes. Gravity is attractive, so the analogue of the Debye shielding is an instability rather than a screening: a long-wavelength perturbation grows instead of oscillating. What survives unchanged is the resonance, and the fact that a smooth distribution can damp a wave while producing no entropy at all.

The astronomical version has a name of its own — it is one half of what makes dynamical friction work, the other half being the wake a massive body drags behind it. A satellite galaxy spiralling into a larger one is being slowed by exactly this transfer, and the calculation is Landau’s with $G$ where $e^2/\varepsilon_0$ was.

Damping used on purpose #

A mechanism that moves energy from a wave into one narrow class of particles, at a velocity the experimenter chooses, is a heater with a dial on it. That is how the largest plasma experiments in the world put energy into their fuel.

A tokamak has to reach temperatures at which the current driven through it can no longer heat it, because a hot plasma’s resistance falls as the temperature to the power of minus three halves and ohmic heating gives out. What is used instead is radio-frequency power, launched from an antenna as a wave whose phase velocity along the magnetic field is chosen. Particles moving at that velocity resonate with it and take its energy, by exactly the transfer this essay computes, and the wave is absorbed.

The dial is the useful part. Choosing the phase velocity chooses which part of the distribution is heated, and choosing the wave’s frequency chooses whether the resonance falls in the electron population or the ion one. Because the absorption is strong only where the resonance condition is met, the power can be deposited in a chosen region of the machine rather than dumped uniformly — which matters, since where the heat goes decides whether the discharge is stable.

The more striking use is to drive a current rather than to heat. Launch waves travelling preferentially one way round the torus and they damp on the electrons moving that way, pushing them faster; the distribution acquires a lopsided tail; and a lopsided distribution is a current. That is lower-hybrid current drive, and it is how a tokamak is to be run continuously rather than as a series of transformer pulses — a steady current maintained by an asymmetry deliberately impressed on a velocity distribution, using a wave that disappears while doing it.

What was proved, and what was measured #

Two things happened to this subject long after the physics was settled, and both are worth knowing because each closed a gap the essay has left open.

The first is mathematical. Everything above is linear, and the objection to the linear result is not that it is a poor approximation but that the mechanism which would undo it — the echo — is itself nonlinear. The distribution’s fine structure is not lost, and nonlinear terms can bring it back into phase; nothing in the linear calculation says that the field, having fallen four decades, stays down.

Mouhot and Villani proved in 2010 that it does, for the full nonlinear equation, provided the initial perturbation is small and analytic. The hard part of the proof is exactly the physical effect described above: echoes at successively later times, each capable of returning some energy to the field, and the work is in showing that the returns are summable rather than accumulating. Villani received a Fields Medal that year, and the theorem is one of the few cases where a piece of plasma physics from 1946 turned out to require a genuinely new analytic technique to justify.

The second is experimental, and it took until 2019. Everything before then was inference: a wave was seen to damp at the predicted rate, which is consistent with the mechanism and does not display it. The signature the mechanism actually predicts is in velocity space — a particular pattern of energy transfer, positive on one side of the resonant velocity and negative on the other, integrating to a net gain by the particles. Spacecraft measuring both the field and the full particle distribution fast enough can correlate the two directly, and in the turbulent plasma downstream of the Earth’s bow shock that pattern was found, sitting where the resonant velocity says it should.

So the damping is now a measured mechanism rather than a measured rate, in a plasma nobody built, seven decades after it was deduced from a contour integral.

Where the model runs out #

Everything is linear. The perturbation is small enough that a particle’s orbit is not altered by the wave, which fails as soon as the wave can trap a particle in one of its troughs. A trapped particle bounces back and forth rather than resonating, the exchange averages to zero, and the damping stops — the field settles onto a plateau instead of decaying. That is nonlinear Landau damping, and it happens at amplitudes far below anything that would be called strong.

Phase mixing is the same arithmetic performed on a distribution rather than on a wave. A packet spreads because its parts travel at different speeds: many components, each rotating at its own rate, adding to less and less as they fall out of step. Landau damping is that process with velocity as the label instead of wavenumber — and the crucial difference is that nothing has been lost, which is why the echo experiments can bring the signal back.

The plasma is unmagnetised and one-dimensional. A magnetic field changes the resonance condition entirely, because particles gyrate, and the analogous effect involves a resonance between the wave and a harmonic of the gyrofrequency. That is cyclotron damping and it has a different geometry and a different exponential.

The distribution is Maxwellian. Real space plasmas are not: they have suprathermal tails, which put many more particles at the phase speed than a Maxwellian would and increase the damping by orders of magnitude at long wavelengths.

And the reversibility is exact only for a collisionless plasma, which does not exist. Every real plasma has some collision rate, and the fine velocity structure is what a collision operator destroys first, precisely because it acts on the second derivative in velocity. A structure of scale $\Delta v$ is erased in a time proportional to $\Delta v^2$, and since the structure gets finer as time passes, there is always a moment at which the recoverable becomes unrecoverable. The apparent irreversibility is genuine at long times and is a consequence of the smallest imaginable dissipation acting on something arbitrarily fine.

A plasma is called collisionless when the mean free path exceeds the system, and the term is always a comparison rather than a property. Over a range of densities that length spans many orders of magnitude, and the same plasma is collisionless for a fast process and collisional for a slow one — which is exactly the caveat the previous paragraph turns into a time: the damping has to happen before the collisions do.

The ladder from here #

Later rungs on this anchor: nonlinear Landau damping and the trapping amplitude at which it takes over; the plasma echo as a quantitative measurement of a collision rate; cyclotron damping in a magnetised plasma, where the resonance is with a gyroharmonic; and the same mathematics in a self-gravitating system, where a star cluster’s response to a perturbation is damped by exactly this mechanism with gravity in place of the electric field.

The neighbouring ladders are the frequency below which nothing gets in, which is the dispersion relation this corrects, the long-range force that does not reach, where the Debye length is derived, and the equation that only runs forwards, which asks the same question about irreversibility with molecules instead of a field.