The number that stops the mixing

Every flow on this site so far has treated up and down as the same direction. Stratify the fluid — let its density fall with height, as the ocean’s and the atmosphere’s do — and that stops being true: lifting a parcel now costs energy, and the fluid has a frequency of its own.

$$N^2 = -\frac{g}{\rho}\frac{\mathrm{d}\rho}{\mathrm{d}z}$$

is that frequency squared, and everything below is a consequence of a fluid having one.

A wave whose frequency picks its direction #

The dispersion relation for an internal gravity wave is

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

with φ the angle of the wavevector from the horizontal. There is no wavelength in it.

That is unlike anything else in this subject. A sound wave’s frequency fixes its wavelength; a water wave’s fixes its speed and its wavelength together; a shock’s speed is fixed by its strength. Here the frequency fixes an angle, and the wave is free to have any wavelength at all while travelling in exactly the same direction.

The experimental consequence is the picture everybody has seen: a cylinder oscillated in a tank of salt-stratified water radiates four straight beams, the St Andrew’s cross, and raising the frequency closes the cross towards the vertical until at ω = N it shuts.

The energy goes sideways #

Every other wave carries its energy in the direction the phase advances. An internal wave carries it at ninety degrees to that, so the crests move across the beam while the energy runs along it, and a float in the path of a beam moves along the crests rather than with them.

This is asserted rather than stated: the group velocity is obtained by differentiating ω(k) numerically, and its dot product with the wavevector must vanish to the precision of the differencing. A figure drawn with the energy going the usual way would look entirely ordinary and be wrong about the one thing this subject is famous for.

The reason is visible in the relation. ω depends only on the direction of k, so the gradient of ω with respect to k — which is the group velocity — has no component along k at all. Anything whose frequency depends on direction alone has this property; internal waves are the case where it matters.

A quarter, from an energy balance #

Now the other half of stratification: it resists being stirred.

Take a shear flow with $\mathrm{d}u/\mathrm{d}z$ and swap two parcels a distance δ apart. Sharing their momenta releases kinetic energy $(\mathrm{d}u/\mathrm{d}z)^2\delta^2/4$ per unit mass; lifting the heavier one costs $N^2\delta^2$. The exchange is possible when the first exceeds the second, and the ratio of the two is exactly the reciprocal of four times the Richardson number

$$\mathrm{Ri} = \frac{N^2}{(\mathrm{d}u/\mathrm{d}z)^2}.$$

A quarter, from an argument with no fluid mechanics in it beyond bookkeeping. And it is the same quarter that Miles and Howard proved rigorously in 1961 — by an analysis of the Taylor–Goldstein equation that is genuinely hard — to be sufficient for stability.

The direction of the theorem, which is routinely reversed #

The Miles–Howard theorem says: if Ri > 1/4 everywhere, the flow is stable. It says nothing about Ri < 1/4.

That distinction is the most-abused result in geophysical fluid mechanics. A measured profile with a minimum Richardson number of 0.1 is not a turbulent profile; it is a profile with no guarantee. Flows with much smaller minimum Richardson numbers are observed to be stable, because stability depends on the whole profile rather than on its worst point, and because a sufficient condition being violated is not evidence of anything.

The parcel-exchange argument makes the asymmetry obvious: it establishes that below a quarter the exchange is energetically possible, which is a necessary condition for an instability and not a sufficient one. Energy being available says nothing about whether a growing mode exists to take it.

The site has met this shape before. The inflection criterion is necessary for inviscid instability and not sufficient; every eigenvalue can say stable while the energy grows by a factor of Re². A criterion is not a threshold, and the two are conflated whenever a number is quoted without its quantifier.

What the numbers are for #

The buoyancy period is a floor on how fast anything can oscillate as a wave, and the numbers explain several things that look unrelated.

Lee waves have kilometre wavelengths. Air crossing a mountain is displaced and oscillates at N, so the wave it leaves downstream has a wavelength of $2\pi U/N$ — about 6 km at 10 m/s in a stratosphere-like N — which is why they show as regularly spaced cloud bands with a spacing anyone can measure from an aeroplane window.

The ocean’s internal tide is slow. The deep ocean’s buoyancy period is about fifty minutes, so internal waves there have periods from an hour upwards, and the semi-diurnal tide is well within that band. Internal tides carry a substantial fraction of the energy that mixes the deep ocean.

A stratified layer stops turbulence dead. In the night-time atmosphere, radiative cooling makes the surface layer strongly stable, and the Richardson number rises above a quarter within a few metres of the ground. Above that the turbulence that mixed the daytime boundary layer simply stops, which is why pollutants accumulate near the ground overnight and why the night-time wind decouples from the surface.

The Froude number, which asks the same question differently #

There is a second dimensionless group for a stratified flow and it is worth having beside the Richardson number, because the two ask about different things and are often confused.

The stratified Froude number U/NH compares the time a flow takes to cross an obstacle of height H with the buoyancy period. Below about one the fluid does not have time to be lifted over: it goes round instead, and the flow is said to be blocked. Above one it goes over and radiates lee waves downstream.

That single number explains a set of familiar things. Fog sits in a valley rather than being scoured out by the wind above it, because the cold air is dense and the local Froude number is far below one. A wind splits round a mountain and reconverges rather than climbing it, for the same reason, and the split is visible in satellite cloud pictures as a wake. A stably stratified plume from a chimney spreads sideways at its equilibrium height rather than continuing up.

The distinction from the Richardson number is that Ri is about a shear and Fr is about an obstacle: one asks whether a velocity gradient can overturn the density gradient, the other whether a flow can climb over something. They are related — both are ratios of the buoyancy’s strength to the flow’s — and neither is a substitute for the other.

Why an internal wave beam does not spread #

One more consequence of a frequency that fixes an angle, and it is the reason these beams are so striking in a laboratory.

An ordinary wave packet spreads as it travels, because its components have different frequencies and therefore different speeds and directions. An internal wave packet at a single frequency has components with different wavelengths — all travelling in exactly the same direction, because the direction depends only on the frequency. So the packet does not disperse in angle at all: it stays a straight, sharp-edged beam for as far as viscosity allows.

The beams in a stratified tank are consequently among the cleanest wave pictures in fluid mechanics, and what makes them clean is precisely the property that makes the dispersion relation strange.

The same two pictures in a laboratory tank #

It is worth drawing both key figures again at laboratory numbers, because the contrast in scale is part of the argument: the physics is identical and the clocks differ by three orders of magnitude.

That invariance is the reason a tank experiment says anything about an ocean. Both are governed by the same two dimensionless numbers, and a laboratory that matches them reproduces the behaviour with a buoyancy period of four seconds instead of fifty minutes — which is the whole basis of modelling by similarity, and one of the rare cases where the match is easy because there are only two numbers to hit — no Reynolds number appears in any of the results above, so the usual impossibility of matching it does not arise.

The number in a measured profile #

The Richardson number is nearly always computed from a measured profile rather than from known gradients, and the measurement introduces a difficulty worth naming.

Both N² and the shear are derivatives, and a derivative estimated from data depends on the spacing over which it is taken. A profile sampled every metre gives one Richardson number; the same profile sampled every ten centimetres gives another, usually smaller, because fine structure that averages out over a metre appears as a large local shear over ten centimetres.

That dependence is not a nuisance to be averaged away: it is physical. A layer can be stable at metre scale and unstable at centimetre scale, and the turbulence that results is the fine-scale instability, which is exactly the case the theorem’s “everywhere” quantifier is about.

So a reported Richardson number is meaningless without its resolution, and the literature is careful about this in a way that a casual reading of the quarter is not. It is the same difficulty as measuring the stretching of a material line: the quantity converges only as fast as the instrument resolves the structure that produces it.

The waves carry momentum, and put it down somewhere else #

The lee waves above are treated as a pattern, and they are also a transport. An internal wave carries a vertical flux of horizontal momentum, and it carries it without depositing any along the way — so the drag the air exerts on a mountain range is not felt by the air just above the mountain. It is carried upward by the wave and delivered wherever the wave finally stops.

Where that is depends on two things, and both of them amplify the effect.

The wave grows as it climbs. The energy flux is conserved, and the density falls exponentially with height, so the velocity amplitude rises as $\rho^{-1/2}$ — a factor of ten between the ground and the mid-stratosphere and far more above. A wave of a metre a second over a mountain is tens of metres a second by the time it reaches the mesosphere, at which point it overturns and breaks whatever its amplitude at the source.

And it can be absorbed sooner. A wave with horizontal phase speed $c$ propagating into a shear reaches a critical level wherever the mean wind equals $c$. Approaching it the vertical wavelength collapses, the wave slows to a stop relative to the flow, and it is absorbed rather than transmitted — depositing all of its momentum at that level.

The consequence is that a mountain range exerts a force on the atmosphere tens of kilometres above itself, and the force is not small. Gravity-wave drag is what decelerates the winter mesospheric jet, and a circulation model that omits it produces stratospheric and mesospheric winds far stronger than the observed ones — one of the clearest cases anywhere of a parameterised sub-grid process controlling a resolved large-scale flow. The equatorial stratosphere’s slow reversal of wind direction, on a period of a little over two years, is driven by the same mechanism with upward-propagating equatorial waves supplying the momentum.

Which puts a different reading on the beams at the top of this essay. In a tank they are a picture of a dispersion relation. In the atmosphere and the ocean they are a conveyor: a mechanism for taking momentum from where a flow meets a boundary and delivering it, undiminished, to a place with no boundary anywhere near it. Nothing else in this collection moves a force that far from the surface that produced it.

What the model does not contain #

No rotation. At the scales where stratification matters most — ocean basins, the atmosphere — the Coriolis force matters too, and the correct dispersion relation for internal waves in a rotating stratified fluid runs from f to N rather than from 0 to N. Everything here is the non-rotating limit, and the site’s rotating-frame machinery is not coupled to it.

Linear waves only. The amplitude never appears. Real internal waves break, and their breaking is one of the main mixing mechanisms in the ocean — a subject this essay’s linear relation cannot touch.

No stability analysis. The quarter is computed from an energy balance and quoted from Miles and Howard as a theorem. Nothing here solves the Taylor–Goldstein equation, so no growth rate is computed and the essay does not claim one.

Constant N. A real stratification varies with height, which refracts internal wave beams, traps them in layers and turns them back at levels where ω = N. Beams in the ocean are curved for exactly this reason and the figures here are straight.

No diffusion of the stratifying agent. Heat and salt diffuse at very different rates, which produces double-diffusive instabilities — salt fingers — in fluids that are stable by every measure in this essay. That is a genuinely different mechanism and none of it is here.

What a stratified fluid does to turbulence #

The last thing worth setting out is what the Richardson number is used for, since the essay has spent most of its length on what it does not mean.

Turbulence in a stratified fluid has to do work against buoyancy to move anything vertically, and the fraction of its energy that goes into that work rather than into more turbulence is the flux Richardson number — a different quantity from the gradient one above, and the one that actually appears in a mixing calculation. Measurements put its ceiling at about 0.2, which is to say that a stratified turbulent flow can spend at most a fifth of its energy budget on mixing and dissipates the rest.

That number is a measurement rather than anything computed here, and it is drawn nowhere in this essay’s figures. What the site can say from the arithmetic above is the reason such a ceiling has to exist: every parcel exchange costs potential energy that is not recoverable, so a stratified fluid converts turbulent kinetic energy into potential energy at a rate that competes with dissipation, and the competition is bounded.

The practical consequence is the one that closes the loop with the previous essay. A strongly stratified fluid cannot sustain vertical motion, so its turbulence becomes layered and quasi-two-dimensional — and once it is two-dimensional, its cascade runs backwards and it organises into large horizontal structures. Stratification is one of the three ways nature makes a flow two-dimensional, and the atmosphere and the ocean are the largest examples of it there are.

Who found it, and when #

Väisälä and Brunt gave the frequency its name in the 1920s, independently and in different fields; the ratio that carries Richardson’s name is from his 1920 paper on atmospheric turbulence, which also contains the rhyme about whorls and littler whorls that the cascade essay quotes.

The rigorous theorem took another forty years. John Miles proved the sufficiency of Ri > 1/4 in 1961 and Louis Howard supplied a much shorter proof in the same volume — the pair of papers is a nice example of a result arriving twice in one issue — and the semicircle theorem that came with them bounds where the unstable modes can be in the complex plane.

What is worth noticing is how long the crude argument stood alone. The parcel-exchange calculation is a first-year exercise and gives the right number; whether the right number meant anything took four decades to settle, and the answer turned out to be “in one direction only”.

Where the ladder goes next #

Two essays in this field have now been about flows constrained by something other than viscosity — a missing dimension, a stable density gradient. The next question comes from the other end of the subject entirely: what shape must a nozzle be so that the waves it makes cancel each other exactly, leaving a supersonic stream that is parallel and uniform? The area ratio decides the Mach number and says nothing at all about the shape.