The wave that is required to stand still

Photographs of lens-shaped clouds standing motionless over a mountain range are common enough to be unremarkable, and what they show is genuinely odd. The wind is blowing at twenty or thirty metres a second, and a wave is a shape that travels while nothing else does. The cloud is not moving at all. Air enters its upwind edge, becomes cloud, crosses it, and evaporates out of the downwind edge, continuously, for hours.

The cloud is not an object being carried along. It is a place where the air happens to be high enough and cold enough to condense, and the place stays put because the wave that put it there is standing still. The rung below this one established what an internal wave is and how strangely it behaves: its frequency fixes the direction its energy travels and says nothing about its wavelength. This rung is what happens when a boundary insists that the whole pattern be stationary.

The relation that has no wavelength in it #

The rung below found the dispersion relation of an internal wave and it is worth restating, because everything here is a consequence of its peculiar shape.

$$\omega = N \cos\varphi,$$

with $\varphi$ the angle the wavevector makes with the horizontal. The frequency depends on a direction and not on a magnitude, so a wave of any wavelength whatever can have any frequency below $N$, provided it travels at the right angle.

That underdetermination is normally a nuisance and here it is the mechanism. A source of one frequency in an unbounded stratified fluid radiates at one angle and at every wavelength at once. Fix a second condition and the wavelength becomes determined — and a mountain in a steady wind is exactly a second condition.

What standing still requires #

Put the ridge in a wind of speed $U$. In the ground frame the pattern is steady, so its frequency is zero; in the frame of the air, which is moving at $U$, a stationary pattern of horizontal wavenumber $k$ is oscillating at $Uk$. That is the intrinsic frequency, and it has to satisfy the dispersion relation.

Working the algebra through gives the vertical wavenumber directly:

$$m^2 = \frac{N^2}{U^2} - k^2.$$

Two things fall out. Waves with $k > N/U$ have $m^2 < 0$ and do not propagate at all — they decay with height, in the same way a pipe below its cutoff carries no note, and for the same algebraic reason — a squared wavenumber going negative is only some values fitting written as an inequality. And in the limit of a broad ridge, where $k$ is small, $m \to N/U$ and the vertical wavelength is

$$\lambda_z = \frac{2\pi U}{N}.$$

The absence is worth dwelling on. No dimension of the ridge is in that expression. A one-kilometre hill and a ten-kilometre massif in the same wind, in the same air, produce waves with the same vertical wavelength. What the ridge’s shape decides is how much — which wavenumbers are excited strongly, and therefore how much of the response propagates rather than sitting locally over the terrain.

Glider pilots use this in the other direction. The horizontal spacing of a wave train is measured off the clouds, and with the sounding known it gives the wind aloft; with the wind known it gives the stability. A cloud photograph is an instrument.

The tilt, and what it is a picture of #

The single most informative feature of the streamline figure is that the crests lean upstream with height. It is also the feature that hand-drawn versions almost always reverse, and getting it right is a check on the whole solution.

An internal wave’s energy travels perpendicular to its wavevector. For a stationary mountain wave the energy must travel upward, because that is where it came from and nothing above is supplying any. The vertical group velocity is positive when the vertical phase velocity is negative, so phase moves downward while energy moves up — and a pattern whose phase descends is a pattern whose lines of constant phase lean upstream as they rise.

So the tilt is a direct observation of a group velocity opposed to a phase velocity, visible in a photograph. It is the same relationship a packet moving at another speed than its own crests describes, in a case where the opposition is complete rather than partial: the two velocities are not merely different, they are perpendicular, and their vertical components have opposite signs.

The consequence is practical. A wave leaning upstream is carrying energy away from the mountain and upward, permanently — an arrival that keeps arriving at every level it passes. A wave whose crests were vertical would be carrying nothing, and one leaning downstream would be receiving energy from above.

What the ridge does decide #

Saying that the wavelength contains nothing about the mountain invites the obvious objection: something about the mountain must matter, or every hill would produce the same wave. It does, and separating what it decides from what it does not is the useful part.

A ridge of width $a$ forces a spread of horizontal wavenumbers centred near $1/a$. Which of those propagate is decided by the inequality above — $k < N/U$ — so the ridge’s width decides what fraction of the forcing goes into propagating waves rather than into a disturbance that sits over the terrain and decays. A broad ridge, with $a \gg U/N$, puts nearly all of its forcing into the propagating band and produces a clean vertically propagating wave. A narrow one, with $a \ll U/N$, puts nearly all of it above the cutoff, and the flow simply goes over the hill and settles down again.

For a wind of twenty metres a second in ordinary air, $U/N$ is about two kilometres. So a ridge two hundred metres across barely radiates at all, and one twenty kilometres across radiates almost everything. That is why mountain waves are associated with ranges rather than with hills, and why terrain smoothing in a model destroys the waves along with the peaks.

The ridge’s height decides the amplitude, and it does so linearly in the linear regime: the displacement at the ground is the terrain, and every streamline above carries the same displacement pattern scaled by nothing. The momentum flux, being quadratic in the amplitude, then goes as the square — which is the shape of the drag figure below.

So the division is clean and worth stating in one line. The air decides the wavelength; the mountain’s width decides how much of the response is a wave; the mountain’s height decides how big. Nothing crosses between those three, which is unusual enough to be worth noticing — most problems in fluid mechanics do not separate.

Whether it leaves, or stays behind the hill #

Not every mountain wave escapes upward, and which happens is decided by the air rather than the mountain.

The quantity that decides is the Scorer parameter, $\ell^2 = N^2/U^2$, which is the largest squared horizontal wavenumber that can propagate at a given height. A wave with $k^2$ below the local value travels vertically; above it, it decays.

A profile whose $\ell^2$ falls with height — because the wind increases, or the stability decreases, or both — traps waves in the band between its value aloft and its value below. Those waves bounce between the ground and the level where they become evanescent, and propagate horizontally instead: a train of crests extending a hundred kilometres or more downwind from a single ridge, which is a channel with no walls made by a wind profile, each crest marked by its own lens-shaped cloud.

That is the difference between the two things a photograph can show. A single stationary cloud above and slightly upwind of a summit is a vertically propagating wave. A row of ten of them stretching to the horizon is a trapped one, and its existence is a statement about the wind profile several kilometres up.

The distinction has a cost attached. A vertically propagating wave takes its momentum upward and out of the troposphere. A trapped one keeps it, and deposits it where the train decays — which is why rotor turbulence beneath a trapped wave train is a well-known hazard to aircraft and vertically propagating waves are not.

When the linear answer stops applying, violently #

Everything drawn so far assumes small displacements, and the case where that fails is the reason mountain waves are studied by people who are not gliding.

If the wave amplitude grows until a streamline becomes vertical, the flow overturns. Above that point the wave is breaking, and beneath a breaking wave the flow reorganises into something that has more in common with water going over a weir than with a wave at all: the air accelerates down the lee slope, thins, reaches speeds several times the upstream wind, and then jumps abruptly back to a slow deep state — the atmospheric version of a front that steepens until it cannot.

Boulder, Colorado sits at the foot of such a slope, and on 11 January 1972 it recorded gusts above sixty metres a second — hurricane force, from a synoptic wind aloft of perhaps twenty-five. Roofs were removed across the city and an instrumented aircraft flew through the event, which is why it is the reference case. The energy came from the wave, and the wave came from the terrain.

The transition between the gentle picture and the violent one is not gradual, and predicting which will occur is a genuinely open problem. Two ingredients are known to matter — a layer aloft where the wind reverses or the stability changes sharply, which reflects the wave back down and lets the amplitude build, and a ridge steep enough for the linear response to be large in the first place. Neither is a sufficient condition. Forecasts of downslope windstorms are issued as probabilities, and the events are missed often enough that the towns beneath the big slopes treat the warnings accordingly.

The clouds distinguish the two regimes for anyone looking. A smooth lens over the summit is a linear wave. A ragged, boiling roll of cloud sitting horizontally a few kilometres downwind, beneath the crest of the first wave, is a rotor — a closed circulation with reversed flow at its base, and one of the most dangerous things a light aircraft can meet.

The same wave in the ocean, driven by the Moon #

The mountain is one way of forcing a stationary internal wave. Moving the fluid past a fixed obstacle is another way of saying that the obstacle moves through the fluid, and the ocean does exactly that twice a day.

The tide is a horizontal sloshing of the whole ocean — a barotropic motion, in which the whole column moves together. Where that flow crosses a ridge, a seamount or a continental shelf edge, the same argument as this essay’s applies with the tidal frequency in place of zero: the topography forces internal waves at the frequency of the forcing, and they radiate away into the interior.

The internal tide is the result, and its energy budget is large. Something like a terawatt of the tide’s three and a half terawatts is converted this way, at a handful of places — the Hawaiian ridge, the Luzon Strait, the Mid-Atlantic Ridge — rather than spread evenly. The Luzon Strait alone generates internal waves whose surface signature is visible from orbit, with amplitudes of a hundred metres in the thermocline.

Why that matters is a question about the whole circulation. The deep ocean is filled with cold dense water that sinks at high latitudes, and it has to come back up, which means something has to mix it against its own stratification. Nothing at the surface reaches deep enough. The leading candidate for the missing energy is the internal tide, breaking in the interior after propagating away from the topography that made it — and how much reaches where is exactly the question the next rung’s reflection law bears on.

The frequency here is not zero and the geometry is not a ridge in a wind, and the arithmetic is the same arithmetic. What is being forced is set by the flow past a shape; what propagates is set by the dispersion relation; and where the energy ends up is set by the boundaries it meets afterwards.

The force delivered somewhere else #

The last thing a mountain wave does is the one that took longest to appreciate, and it is the reason this subject is in every weather model rather than only in gliding manuals.

A wave carrying energy upward carries momentum with it. For a stationary wave the momentum flux is horizontal momentum transported vertically, which is a stress: the mountain pushes back on the air, the air’s reaction is carried upward by the wave, and it is delivered wherever the wave finally breaks.

The square is what makes this a matter of a few places rather than an average. Doubling the height of a ridge quadruples what it exerts, so the Himalaya, the Andes, the Rockies and the Antarctic Peninsula between them account for most of the world’s mountain-wave drag, and a smoothed model terrain that flattens them loses nearly all of it.

The delivery point is the surprising part. The wave does not slow the air passing over the mountain by much; it carries the momentum up, without depositing any, until it reaches a level where it breaks — for the largest waves that is the stratosphere or the mesosphere, tens of kilometres above and often hundreds of kilometres downwind. A mountain therefore exerts a force on air that never came near it, at an altitude where nothing else could have.

Global forecasting models did not include this until the 1980s, and the symptom was systematic: the modelled westerlies in the winter stratosphere were far too strong, and the polar vortex too cold and too stable. Adding a parameterisation of orographic gravity-wave drag fixed both. Every operational model now carries one, and its tuning is one of the larger uncertainties in the whole system, because the drag depends on terrain features smaller than a grid box and on a breaking level nobody observes directly.

Where this stops being right #

The solution drawn is linear and hydrostatic. It assumes the displacement is small compared with the vertical wavelength and that the ridge is broad compared with $U/N$. Real mountain waves routinely break both: a wave whose amplitude approaches $1/m$ overturns, which is what a rotor is, and a narrow ridge produces a non-hydrostatic response with a different structure.

The wind and stability have been uniform with height, except where the figure explicitly varies them. Real soundings have layers, and a layer boundary partially reflects the wave, so the vertical structure is an interference pattern between upgoing and reflected waves rather than a single clean wave.

Nothing here is three-dimensional. A real mountain is not an infinite ridge, so the response includes a ship-wave pattern spreading downstream as well as the trapped and vertically propagating parts, and the drag is smaller than the two-dimensional estimate by a factor that depends on the shape.

And the trapping criterion neglected the curvature of the wind profile. The full Scorer parameter is $N^2/U^2 - U''/U$, and the second term matters in a jet, where it can create or destroy a trapping layer that the first term alone does not predict.

What the pictures cannot show #

The streamline figures are steady, which is the whole subject, and steadiness is exactly what a still picture cannot distinguish from stillness. Every line drawn is a path air travels along at twenty metres a second; the drawing shows the path and not the traffic on it, and a reader has to supply the fact that the pattern is being continuously rebuilt out of new air.

Nor can any of them show the cloud. Condensation happens where a streamline is near its highest point and the air has cooled below its dew point, which depends on the humidity — a quantity that appears in no equation here. The same wave in dry air is invisible, and most mountain waves are.

Where this ladder goes next #

Three rungs stand on stratification. The first put one parcel in a column and found a frequency. The second let many parcels make a wave and found a relation that fixes an angle and not a wavelength. This one adds a boundary that requires the pattern to stand still, and the wavelength the dispersion relation left free is decided by the wind.

The habit worth carrying away is about relations that under-determine. When a dispersion relation leaves a quantity free, look for the boundary condition that will fix it, because something will. Here the freedom is the wavelength and the boundary is a mountain in a wind. The same shape appears whenever a medium supports a continuum and a geometry selects from it — a laser cavity picking modes from a gain curve, a crystal picking wavevectors from a lattice.

What is left on this ladder is the other kind of boundary. A mountain sets a wave standing; a sloping wall sends one back. And because an internal wave’s angle is fixed by its frequency rather than by anything about the wall, reflection cannot be the mirror operation it is for every other wave — the wavelength changes on reflection, and in a closed basin the changes accumulate.