The wave that picks an angle

Every wave in this collection so far has answered the same question in the same way: given a wavelength, how fast does it travel and in what direction? A wave in a stratified fluid answers a different question, and the difference is not a refinement.

Where the relation comes from #

A parcel displaced vertically in a stably stratified fluid oscillates at the buoyancy frequency $N$, because gravity supplies a restoring force proportional to the displacement. That is the previous rung and it concerns one parcel moving straight up and down.

Now let the parcel move along a line at angle $\phi$ to the horizontal. Buoyancy still acts vertically, so only its component along that line does any restoring, and the restoring force per unit displacement is reduced by $\cos^2\phi$. The frequency is the square root of that:

$$\omega = N\cos\phi$$

where $\phi$ is the angle the direction of motion — and therefore the wavevector — makes with the horizontal.

That is the entire dispersion relation and it is worth pausing on what is missing from it. There is no $k$. The magnitude of the wavevector does not appear, so a wave of any wavelength whatever, at a given frequency, travels with its wavevector at the same angle. Frequency does not select a speed here; it selects a direction.

The consequence for the beams is immediate. Energy travels with the group velocity, which is the gradient of $\omega$ with respect to $\mathbf{k}$ — and a function of the direction of $\mathbf{k}$ alone has a gradient with no radial component. So the group velocity is exactly perpendicular to the wavevector, and the beams run at $90° - \phi$ to the horizontal, which means their angle to the horizontal has sine $\omega/N$.

Perpendicular, exactly #

The right angle between phase and energy is not an approximation and does not depend on anything.

Watching a laboratory tank makes this vivid and slightly disorienting. Crests appear to march steadily across each beam, from one edge to the other, while the beam itself sits still and carries energy outward along its length. A photographer following a crest is following something that is not going where the wave is going. There is no other classical wave in ordinary experience where the two directions are at right angles rather than parallel or nearly so.

The phase actually moves downward in a beam whose energy is going upward, and vice versa. That is a genuine and checkable prediction: it is how oceanographic records distinguish an internal wave radiating up from one radiating down, and it works because the sign of the vertical phase speed is opposite to the sign of the vertical group speed.

The frequency ceiling #

The relation has no solution above $\omega = N$, and the fluid’s behaviour there is worth stating carefully.

Drive a stratified fluid faster than its own buoyancy frequency and nothing radiates. The response is confined to the neighbourhood of the driver and decays with distance — it is evanescent — and the energy put in has nowhere to go except back into whatever is doing the driving.

This is the same kind of ceiling that a waveguide has a floor of, turned upside down. There, a wave below cutoff has an imaginary wavenumber and does not propagate; here, a frequency above $N$ gives an imaginary angle and does not propagate. Both are cases where a dispersion relation stops having real solutions, and both produce a large local field that goes nowhere.

The ceiling is measurable and it is measured. Ocean spectra show a sharp edge at the local buoyancy frequency; atmospheric ones do too. The edge is not a resonance and not a peak — it is the end of the band, and the reason the spectrum stops there is that above it there is no wave to have any energy in.

The cross, and how to read it #

The four beams of the hero figure have a name — the St Andrew’s cross — and the reason there are four rather than two is worth a sentence, because it says something about the symmetry.

The relation fixes $|\cos\phi|$, not $\cos\phi$. So for a given frequency there are four wavevector directions and four corresponding beam directions: up-left, up-right, down-left, down-right, all at the same angle to the horizontal. A source that pushes fluid up and down symmetrically radiates into all four equally, and the picture is a cross rather than a pair of rays.

That is the visible face of a symmetry the medium has and the source does not break. The fluid is isotropic in the horizontal and stratified in the vertical; nothing distinguishes left from right or, for a symmetric source, up from down. Break either symmetry — put the source near a boundary, or make it oscillate along a slanted line — and the four beams stop being equal, which is a rather sensitive way of reading off what the source was doing.

The cross was first produced in a laboratory in 1963, by Görtler and then more famously by Mowbray and Rarity, who oscillated a cylinder in a tank of stratified brine and photographed the result with a shadowgraph. Measuring the angle against the frequency reproduced $\sin\theta = \omega/N$ over the whole accessible range, and it remains one of the cleanest confirmations of a dispersion relation there is: a single photograph and a protractor.

What that frequency is in real fluids #

The buoyancy frequency of the atmosphere near the ground is about $0.01$ radians per second, giving a period of ten minutes. In the ocean’s thermocline it is a few times larger; in the deep ocean much smaller, with periods of hours. Both are slow compared with anything a person would call a wave, and that slowness is why internal waves went unrecognised for so long: a wave with a ten-minute period and a wavelength of kilometres is not something one notices standing on a beach.

The signature that does get noticed is on the surface. Where an internal wave passes, the horizontal flow it produces converges and diverges, and the convergence gathers up the surface slicks of biological film into bands. Satellite pictures of the sea surface routinely show sets of parallel bands tens of kilometres long marching away from a shelf edge; those are the surface trace of a wave whose amplitude is fifty metres and which lies entirely below the surface.

Where the beams go #

The energy in a beam travels at a fixed angle, and that has consequences that a wave travelling in circles would not have.

A beam reflects strangely. When a beam hits a sloping bottom it must leave at an angle set by its frequency, not by the surface’s normal — because the frequency is conserved and the frequency fixes the angle to the horizontal, not to the boundary. So the ordinary law of reflection fails entirely: a beam meeting a slope is compressed or expanded rather than mirrored, and repeated reflections in a wedge focus energy into a smaller and smaller region.

That focusing is called an internal wave attractor and it is a real phenomenon in enclosed basins and in laboratory tanks: energy put in over a wide region concentrates onto a closed path, and the amplitude there grows until something breaks it. It is the exact opposite of what happens to sound in an enclosure, where reflections build a mode filling the whole volume.

Critical slopes are dangerous. A slope whose angle equals the beam angle takes an incident beam and returns it along itself with the wavelength going to zero, which means the shear goes to infinity. Real continental slopes have angles comparable with typical internal-tide beam angles, and that coincidence is where a large fraction of the ocean’s tidal energy is thought to be dissipated.

A beam is not a mode. The whole apparatus of standing waves, harmonics and quantised wavelengths that a bounded medium produces has no counterpart here, because there is no relation between frequency and wavelength to quantise. A stratified basin driven at one frequency does not select a wavelength; it selects a direction, and whatever wavelength was put in travels along it.

And a beam does not spread. Nothing in the relation gives a beam a diffraction angle, because the angle is fixed by the frequency rather than by the aperture. What limits a beam is viscosity, and since the wavelength across it is not fixed by anything either, the thin parts decay first and the beam gets smoother as it goes.

For an ordinary dispersive wave the phase and group speeds differ in magnitude and share a direction — the envelope and the crests travel the same way at different rates. The stratified case is off that chart entirely: the two differ in direction by ninety degrees, and their magnitudes are related by the geometry of the angle rather than by any curve of speed against wavelength. There is no dispersion relation of the usual kind because the frequency does not depend on the wavelength at all.

The same relation elsewhere #

Two other systems have a dispersion relation of this shape, and the family resemblance is exact rather than poetic.

Inertial waves in a rotating fluid. Replace buoyancy with the Coriolis force and the relation becomes $\omega = 2\Omega\cos\theta$, with $\theta$ the angle between the wavevector and the rotation axis. Same form, same perpendicularity of group and phase, same ceiling at $2\Omega$, same St Andrew’s cross in a laboratory tank. The restoring force is entirely different and the mathematics is identical, because in both cases the restoring force acts along a fixed direction in space rather than along the direction of motion.

Magnetic waves along a field line. An Alfvén wave travels along the field and not across it, for the same structural reason: the tension that restores it lies along the field, so the group velocity is fixed in direction regardless of the wavevector’s magnitude. A field line in a conducting fluid behaves like a string under tension, and a string carries waves along itself.

The general principle is worth extracting because it is not confined to fluids. When the restoring force has a fixed direction in space rather than along the wave’s own motion, the medium becomes anisotropic in the strongest possible way: the frequency depends on the direction of travel and not on the scale. Everything peculiar in this essay follows from that one sentence.

An ordinary wave decides where to go next by rebuilding its front from the wavelets its own points emit, and the envelope of a family of spheres is a surface parallel to the original. A stratified fluid’s front cannot be constructed that way, because its wavelets are not spheres — the medium’s response depends on direction — and the envelope of a family of anisotropic wavelets is what produces a beam rather than a spreading front.

Why the atmosphere cares #

Internal waves in the atmosphere carry momentum from where they are made to where they break, and that transport is not optional in a climate model.

Air flowing over a mountain range launches waves that travel upward at the angle this essay is about. The angle depends on the wind speed and the stratification, so the waves propagate along a path that curves as those change with height. They carry with them the momentum extracted from the flow at the ground, and they deposit it wherever they finally break — which is typically tens of kilometres up, in the stratosphere or the mesosphere.

The result is a force applied to the upper atmosphere by a mountain range beneath it. It is large enough to matter: the mean circulation of the mesosphere is driven substantially by this deposited momentum rather than by anything local, and a model that leaves it out gets the temperature structure of the upper atmosphere badly wrong. Since the waves are far smaller than any model’s grid, the effect has to be inserted as a parameterisation, and gravity-wave drag schemes are among the more delicate parts of a weather model.

A packet built from a band of wavelengths has its envelope travelling at the group velocity while its crests travel at the phase velocity. In an ordinary medium those point the same way and differ only in speed. In a stratified one the envelope and the crests move at right angles to each other, and it is the same construction with a dispersion relation that depends on direction instead of on scale — which is the one substitution that turns a familiar picture into an unfamiliar one.

The same mechanism operates in the ocean with the tide instead of the wind and a ridge instead of a mountain, and it is the leading candidate for how the abyssal ocean is stirred. In both cases the essential fact is the one at the top of this page: the energy leaves along a fixed angle, so what is launched at the bottom arrives at a predictable place far away, and the source and the sink are not in the same neighbourhood.

The ship that could not get under way #

The most direct evidence that these waves carry away real momentum is a nuisance that sailors reported for centuries before anybody could explain it.

Nansen, working the Fram along the Siberian coast in 1893, found the ship slowing to a crawl in calm water with the engine running normally — barely a knot and a half, with a strange resistance and an unsteady wake. The condition had a name in Norwegian already, and it happened where melting ice had laid a few metres of fresh water over the salt water beneath.

Ekman worked it out in 1904, in a tank, as his doctoral problem. A layer of fresh water over salt is a stratification with its buoyancy concentrated at one interface, and a hull whose draft reaches down to that interface pushes on it as it moves. The interface responds by radiating waves — internal waves, of the kind this essay is about, propagating along and below the boundary rather than on the surface — and the energy carried off has to come from the engine.

The drag can be several times the ordinary hull resistance, which is enough to stop a small ship. And it comes and goes: the effect is largest when the ship’s speed is near the speed of the interfacial wave, so that the hull sits in the trough it is itself generating, and a vessel that can accelerate past that speed escapes. Nansen’s Fram could not.

Nothing about the surface shows it. The water is flat, the engine is turning, and the power is going into a wave nobody can see, propagating along a boundary tens of metres down.

The spectrum that is the same everywhere #

If a frequency selects a direction rather than a wavelength, then an ocean full of internal waves at many frequencies is an ocean full of beams at many angles, superposed. What that adds up to is a question with a surprising answer.

Garrett and Munk assembled the available measurements in 1972 and found that the internal wave field of the open ocean has very nearly the same spectrum everywhere — the same distribution of energy across frequency and across vertical scale, at the same overall level, to within a factor of two or so, from one ocean to another and from one decade to another. The band runs from a low-frequency limit that belongs to a later rung up to the buoyancy frequency, where this essay’s ceiling ends it.

That universality is peculiar. The waves are generated by winds and tides, which are neither uniform nor steady, and they are dissipated by breaking, which happens in particular places. There is no obvious reason why the field between the two should look the same in the Pacific and the Atlantic. The accepted explanation is that the nonlinear transfer of energy between waves is fast enough to fill out a characteristic shape whatever the input was — a saturated state, in which putting more in mostly means taking more out at the small-scale end.

The practical use is the departure rather than the agreement. Where a measured shear or strain variance runs above the universal level, the wave field is more energetic than usual and the rate at which it is breaking is higher; where it runs below, less. That relation is now the standard way of estimating the ocean’s turbulent mixing rate from a profiling instrument that measures nothing turbulent at all — a mixing rate inferred from how far a wave spectrum sits above a reference curve.

What the picture cannot show #

The beams are drawn as lines and are really bands. The relation fixes the direction and not the width, so a real beam’s cross-section is whatever the source imposed, and the drawing’s clean rays stand for something with structure across them. Nothing in the linear theory says how wide.

The amplitude is nowhere. Everything here is kinematics — where the energy goes — and none of it says how much. That requires solving for the source’s coupling to the wave field, which depends on its size relative to the wavelength and is a genuinely harder problem.

The relation is linear and the interesting parts are not. Internal waves steepen, break, and generate turbulence, and the breaking is what actually mixes the ocean. The linear relation says where the energy is delivered; it says nothing about what happens when it arrives, and the mixing rate of the deep ocean — a number that sets the timescale of the whole circulation — is still argued about.

And the buoyancy frequency is treated as constant. In every real fluid it varies with height, so beams refract, turn, and can be reflected at a level where $N$ falls to the wave’s own frequency. A beam launched in the ocean’s thermocline and travelling downward reaches a depth where $N = \omega$ and turns back, which means the pretty straight lines are locally straight and globally curved.

The ladder from here #

Later rungs on this anchor: reflection from a sloping boundary and the attractors that repeated reflections produce; the internal tide, generated where the barotropic tide runs over topography, which carries a terawatt of power in the world ocean and is the leading candidate for what mixes it; wave breaking and the parametric subharmonic instability that transfers energy to small scales; lee waves, the standing case, where a steady wind over a ridge produces a stationary pattern whose wavelength the wind speed fixes; and the rotating case, where buoyancy and Coriolis act together and the relation acquires both a ceiling and a floor.

The neighbouring ladders are the parcel that cannot leave its layer, which supplies the frequency everything here is measured in, and the medium deciding the speed, which is the ordinary case this one is the exception to. A guide with a cutoff is where the same “no real solution” argument produces a floor instead of a ceiling.