The reflection that changes the wavelength

Every wave in this collection so far reflects the way a ball bounces: the angle to the boundary is preserved, and a mirror tilted by ten degrees turns a beam by twenty. What happens where the medium changes is a statement about impedance, and the geometry of it never varies, because the speed of the wave is a number and a number has no direction in it.

An internal wave’s speed is not a number. Its dispersion relation fixes the angle its energy travels at, through $\sin\theta = \omega/N$ with $\theta$ measured from the horizontal, and says nothing at all about its wavelength. The restoring force is gravity; gravity points one way; so the fluid has a preferred direction built into it and the angle is a property of the frequency rather than of the geometry.

Reflection has to preserve the frequency, because the boundary is not moving. So it has to preserve the angle to the vertical. And a wall that is not horizontal cannot then send the beam back as a mirror would.

What has to be conserved, and what does not #

The reflection law of an ordinary wave comes from one requirement: the phase along the boundary has to match on both sides, at every point and every instant. That single restriction produces the law of reflection, Snell’s law and the diffraction grating, and it works because the wave’s speed is isotropic — knowing the tangential wavenumber and the speed fixes the normal one, up to sign.

Here the same restriction applies and gives a different answer, because the relation between wavenumber and frequency is anisotropic. Preserving the tangential wavenumber and the frequency leaves the total wavenumber free to change, and it does. The reflected beam has the same angle to the vertical and a different wavelength, related to the incident one by

$$\frac{\lambda_r}{\lambda_i} = \left|\frac{\sin(\theta + \alpha)}{\sin(\theta - \alpha)}\right|$$

with $\alpha$ the slope of the wall. For $\alpha = 0$ that is one, and the ordinary law is recovered — a flat floor reflects an internal wave exactly as a mirror does, which is the check that the strange law contains the familiar one.

For $\alpha \to \theta$ it diverges. That is the critical case: a slope inclined at exactly the ray’s own angle, on which the reflected beam runs along the wall itself with its wavelength squeezed to nothing.

The divergence is a simple pole, which the figures check rather than assert: the ratio times the distance from critical approaches $\sin 2\theta$ as the slope is brought within a hundredth of a degree of the ray. That matters because a divergence and a singularity of some other order would have quite different physical consequences, and only the pole gives the observed concentration.

Critical slopes are not rare #

The obvious objection to all of this is that a slope exactly matching a ray angle sounds like a measure-zero coincidence, and it is not.

The semidiurnal tide has a frequency fixed by the Moon. The buoyancy frequency of the deep ocean is between $10^{-3}$ and $10^{-2}$ per second, and the Coriolis frequency depends on latitude, so the internal tide’s ray angle is a few degrees from the horizontal over most of the ocean — typically between one and five. Continental slopes and mid-ocean ridge flanks are between one and ten degrees over enormous areas.

So the two overlap, and a substantial fraction of the world’s continental slope is at or near critical for the internal tide. The Bay of Biscay slope, the Hawaiian ridge flanks and the Australian North West Shelf are all documented cases where the criticality is measured rather than inferred, and the observed consequence is what the pole predicts: elevated shear, elevated turbulence, and mixing rates one to two orders of magnitude above the ocean interior’s background — an entropy produced by a gradient in a place chosen by geometry.

What happens where the slope is critical #

Between the ordinary case and the divergence there is a regime worth separating, because it is where most of the ocean actually sits.

A slope gentler than the ray angle is subcritical: the reflected beam continues in the same horizontal direction, propagating up the slope, with its wavelength shortened. A slope steeper than the ray angle is supercritical: the beam is turned back the way it came, with its wavelength lengthened. The two behaviours are qualitatively different — one transmits energy shoreward and the other returns it seaward — and the boundary between them is a slope equal to the ray angle, at which the wavelength ratio passes through infinity.

That is not a smooth transition through a large number. It is a change of topological character, and it is why the criticality parameter — the ratio of the seabed’s slope to the ray’s — is the first thing an observational programme measures. A shelf edge whose criticality passes through one somewhere along its length has a place where the internal tide changes from being reflected to being transmitted, and that place is where the energy is deposited.

What is observed at such places is more than shear. The shoaling internal tide steepens as it runs up a subcritical slope, in the same way a front steepens until it cannot, and breaks into a train of nonlinear solitary waves — boluses of dense water that run up the slope, mixing as they go. They are directly observed on the Australian and Portuguese shelves and carry a substantial fraction of the shoreward energy flux. Nothing in the linear ray argument produces them, and the ray argument is what says where to point the instruments.

The same law, in two fluids that are not stratified #

The reflection law is strange because the dispersion relation ties frequency to direction, and stratification is not the only way to arrange that.

A rotating fluid does it too. Inertial waves in a fluid rotating at $\Omega$ obey $\omega = 2\Omega\cos\varphi$ — the same form, with the rotation axis playing the part of gravity — so their energy also travels at an angle fixed by frequency, they also form St Andrew’s crosses from an oscillating source, and they also produce attractors in a tilted container. The laboratory demonstrations are done in rotating tanks and give the same limit cycles, and the astrophysical application is to the fluid interiors of rotating stars and planets, where tidal forcing at an orbital frequency is thought to concentrate onto exactly such paths.

A magnetised plasma does a third version. A shear Alfvén wave’s energy travels along the field whatever direction its wavevector points, which is the same relation taken to its limit: the angle is not merely fixed by the frequency but fixed at zero. The reflection law for a field line meeting a boundary at an angle is correspondingly extreme, and the phase-mixing that heats a corona is the same wavelength-contraction mechanism as this essay’s attractor, driven by a gradient in the wave speed rather than by a tilted wall.

Three fluids with nothing in common — a salt-stratified tank, a rotating one, and a plasma — share the reflection law because they share the shape of the dispersion relation. That is the strongest form of the habit this collection keeps finding: the behaviour belongs to the equation and not to the substance.

What repeated reflections do #

A single reflection changes a wavelength by a factor. A closed basin applies the factor over and over.

Consider a basin with a flat floor, a flat surface, one vertical wall and one sloping wall — a trapezoid, which is roughly the cross-section of a great many real basins. A ray bounces off the floor and the surface without changing wavelength, off the vertical wall without changing it either, and off the sloping wall by a factor that is not one. So each complete circuit multiplies the wavelength by something, and if that something is less than one, the ray shrinks.

The result is a wave attractor, and it has no analogue in any of the wave systems this collection has treated. A rectangular basin has modes: standing patterns filling the whole domain, at a discrete set of frequencies, exactly as only some notes fit a pipe. Tilt one wall and the modes are gone. What replaces them is a limit cycle — a single closed path, generally with an irrational relationship to the basin’s dimensions, onto which all rays converge.

That the attractor is generic rather than special is the part worth stating carefully. It is not that some basins have attractors; it is that any basin without the reflectional symmetry of a rectangle has one, for almost every frequency, and the exceptions form a set of measure zero. The rectangular case with its clean modes is the special one.

Leo Maas and Frans-Peter Lam derived this in 1995 from exactly the ray argument above. Maas and colleagues then built a trapezoidal tank, filled it with salt-stratified water, oscillated it gently at one frequency, and photographed the result: the energy in the tank sat on a single line, in the place the ray tracing said it would.

What the concentration destroys #

The obvious reading of an attractor is that energy piles up there, and the useful reading is almost the opposite.

What contracts is the wavelength, not primarily the energy. A beam whose wavelength has been squeezed by a factor of a thousand has its shear multiplied by a thousand at the same energy, because shear is a velocity over a length. Viscosity acts on shear. So the attractor is not a place where energy accumulates; it is a place where energy is removed, at a rate that rises as the contraction proceeds, until dissipation balances the supply.

An attractor is therefore a drain rather than a reservoir, and that is what makes it interesting to oceanography rather than merely to geometry. The ocean’s problem, sketched at the end of the rung below, is that the deep water formed at high latitudes has to be brought back up against its own stratification, and nothing at the surface can reach it. The mixing has to happen in the interior, and it has to happen somewhere specific, because measurements show the interior is mostly extraordinarily quiet — turbulent diffusivities near $10^{-5}$ square metres per second, an order of magnitude below what the overturning circulation requires on average.

The resolution is that the mixing is not spread out. It is concentrated where internal waves break, which is above rough topography, at critical slopes, and on attractors — and there the measured diffusivities are $10^{-3}$ and above. An ocean that mixes in a few places at a hundred times the average rate and nowhere else is a different ocean, dynamically, from one that mixes uniformly, and which of the two the real one is has been settled in favour of the first largely by measurements made because this geometry predicted where to look.

What an attractor does to a measurement #

There is a practical consequence of all this that is worth separating from the physics, because it changes how the ocean is sampled.

An instrument lowered on a wire measures a profile at one place. If the energy in a basin is spread across modes, one profile at a random place is a fair sample of the whole and the average of several is a good estimate. If it is concentrated onto a line occupying a thousandth of the cross-section, a random profile misses it almost always, and the few that hit it look like outliers to be discarded.

That is not hypothetical. Microstructure profilers — instruments that resolve turbulent dissipation directly, at centimetre scales — return distributions of dissipation rate spanning four or five orders of magnitude, with the mean dominated by a handful of profiles. For twenty years the question of whether those were real or instrumental was genuinely open, and the geometry in this essay is part of why they are now taken seriously: a medium that focuses its own energy onto thin sheets is expected to produce exactly that distribution.

The sampling problem has an arithmetic consequence too. An average of a heavy-tailed distribution converges slowly, so estimating a basin’s dissipation from profiles requires far more of them than a normal distribution would — and the number needed depends on how thin the sheets are, which is the quantity the ray argument predicts and the viscous problem sets. Ocean mixing budgets carry uncertainty factors of two or three for this reason, and the uncertainty is about where to look rather than about the instruments.

Where this stops being right #

The ray description ignores the wavelength it is tracking. A ray is a valid description while the wavelength is small compared with the basin, and the whole point of the attractor is that it stops being so. Near the limit cycle the ray argument predicts its own failure, and what happens instead — how wide the attractor actually is, and at what scale it saturates — requires the viscous problem rather than the geometric one.

Everything here is linear and two-dimensional. Real basins are three-dimensional, and a ray in three dimensions has a whole cone of directions available at each frequency rather than four. Attractors survive in three dimensions and are harder to characterise, and whether the ocean’s geometry produces them at the relevant frequencies is not fully settled.

Rotation has been left out entirely. At the scales of an ocean basin the Coriolis force matters, and it changes the dispersion relation to $\omega^2 = N^2\cos^2\varphi + f^2\sin^2\varphi$, which puts a floor under the frequency as well as a ceiling. Rays still have fixed angles, so attractors still exist, but the range of frequencies that have them is bounded on both sides.

And the tank experiments are small. A laboratory basin has a Reynolds number many orders of magnitude below an ocean’s, so the attractor there is broad and viscous where an ocean’s would be narrow and turbulent. That the geometry is confirmed does not confirm the fate of the energy at oceanic scales, which is inferred rather than observed.

What the pictures cannot show #

The ray figures draw lines, and a ray is not a thing. What is travelling is a beam of finite width carrying a wave of finite wavelength, and the entire content of the attractor result is about what happens to that wavelength — which the lines, having no width and no wavelength, cannot display. The contraction is drawn as a convergence of paths, and it is really a compression of a structure the drawing does not contain.

Nor can any still figure show the direction of travel along the attractor. A limit cycle is a closed path traversed in a definite sense, and reversing that sense gives a different physical solution: the ray tracing run backwards from the attractor diverges, filling the basin, which is the statement that the attractor is attracting. A drawing of the path alone is symmetric under a reversal that the physics is not.

What the ladder has reached #

Four rungs stand on stratification. One parcel gave a frequency; many parcels gave a relation that fixes an angle and leaves the wavelength free; a mountain in a wind fixed the wavelength by requiring the pattern to stand still; and a sloping wall changes it on every reflection until the changes accumulate into a single line.

The habit worth carrying away is about what a boundary condition acts on. A reflection law is a statement about which quantities the boundary is allowed to change, and that list depends on the medium rather than on the boundary. For an isotropic medium the wavelength is not on the list and the angle is; for this one it is the other way round, and every consequence in the essay follows from the swap. The test to apply elsewhere is to ask what the dispersion relation ties to what — a medium whose frequency fixes a direction will always have a strange reflection law, which is why the same phenomenon appears for inertial waves in a rotating fluid and for magnetohydrodynamic waves along a field.

What is left on this ladder is what the attractor does once it is no longer a ray: the parametric subharmonic instability that transfers energy from the tidal frequency to half of it, the breaking that follows, and the question of how much of the terawatt entering the internal tide is dissipated near where it was generated rather than in the far field. That last number is the one an ocean model most wants and the one least well measured.