When the flux outruns the pressure

The kinematic wave is derived from conservation of mass and a rule relating flux to depth, and its derivation has a section headed why the momentum equation is missing. The answer given there is that momentum has been replaced by an assumption — that friction balances gravity at every instant — and an assumption is not a limit. Nothing in that essay says what breaks it.

Something does, and it is visible from a car. A steep concrete spillway does not carry a smooth sheet of water. It carries a train of surges, each with a breaking front, marching down the chute at a fixed spacing, and the pattern is so regular that it looks designed. Those are roll waves, they are what the kinematic description does not have, and their appearance is a threshold rather than a gradual deterioration.

Three speeds in one channel #

Shallow water running down a slope carries information at three speeds and they are not the same kind of information.

Two of them are the dynamic speeds $u \pm \sqrt{gh}$, which are the characteristics of the shallow-water equations. They carry pressure: a change of depth somewhere sends news of itself upstream and downstream at those speeds, and everything about surges, jumps and the hydraulic control of a channel is written in them.

The third is the kinematic speed $c_k = \mathrm{d}q/\mathrm{d}h$, which carries flux. For a Chézy channel $q \propto h^{3/2}$, so $c_k = \tfrac32 u$; for Manning, $q \propto h^{5/3}$ and $c_k = \tfrac53 u$. In both cases $c_k = \gamma u$ with $\gamma > 1$, which is the kinematic theory’s central surprise: a flood crest moves faster than the water it is made of, at five thirds of it for every depth, every roughness and every slope.

Two of those three speeds are therefore proportional to $u$ and the third is not. They must cross.

The distinction between the two kinds of news is worth making carefully, because it is the whole of the argument and it is easy to blur. A dynamic wave is a statement about the momentum equation: a change of depth is a change of pressure gradient, and the fluid responds to it by accelerating. It travels relative to the water, at the speed a gravity wave travels on still water of that depth, and the water then carries it. A kinematic wave is a statement about the mass equation with a constitutive law in it: a change of depth is a change of discharge, and the extra discharge simply goes where the water is going. It has no existence relative to the water at all — nothing in it travels — and its speed is set by the slope of the flux curve.

So the two speeds are built from different equations and would have nothing to do with one another if they did not happen to be measured against the same ground. The comparison that follows is therefore not a comparison of two waves in the usual sense; it is a comparison of two descriptions, asking which of them delivers its information first.

The overtaking #

The crossing is where $\gamma u = u + \sqrt{gh}$, which rearranges to

$$V \equiv (\gamma - 1)\,\mathrm{Fr} = 1,$$

the Vedernikov number. For Chézy that is Froude 2, for Manning Froude 1.5.

What happens at that crossing is easiest to see as a race. A patch of slightly deeper water sends the news of its own depth downstream at $u + \sqrt{gh}$, and the downstream water adjusts. It also sends the news of its own flux downstream at $\gamma u$. Below the threshold the depth news arrives first, the downstream water has already begun to adjust when the extra discharge reaches it, and the disturbance spreads out. Above it the extra discharge arrives before any warning of it, piles into water that has not adjusted, and deepens it further.

That is a positive feedback with no length scale in it, which is why what follows is not a gradual roughening.

The race can be run the other way to see what keeps ordinary flows stable. On a river the Froude number is a fifth, so the downstream dynamic wave travels at $u + 5u = 6u$ against the kinematic wave’s $1.5u$: the depth news arrives four times sooner and the adjustment is comfortably complete before the discharge turns up. On a spillway at Froude 11 the dynamic wave travels at $u + u/11 = 1.09u$ and the kinematic one at $1.5u$ — the flux arrives first, and by a wide margin. The ordering reverses because one speed is anchored to the water and the other to gravity, and gravity stops helping once the water is fast enough.

A threshold that is neutral at every wavelength at once is unusual and it is what makes this a criterion rather than a stability curve. Most instabilities in fluid mechanics — Kelvin–Helmholtz, Rayleigh–Bénard, pipe transition — select a wavenumber, and the selected wavenumber is most of what the analysis is about. Here the whole spectrum changes sign together, and the spacing of real roll waves is decided by what happens after they have grown, by coarsening and merging, rather than by which mode grew fastest.

Why a positive feedback need not have a scale #

A feedback loop that has no preferred wavelength is worth pausing on, because it is not what most instabilities look like and the reason is visible in the equations.

An instability selects a wavelength when two effects compete at different powers of the wavenumber — a destabilising term growing as $k$ and a stabilising one as $k^2$, say, which gives a maximum in between. Kelvin–Helmholtz has surface tension doing that, Rayleigh–Bénard has viscosity and conduction, and pipe flow has viscosity alone. In each, the stabilising term is a diffusion of something across the wave, and a diffusion always costs more at short wavelengths than at long ones.

Here there is no such term. The only mechanism in the problem beyond advection is friction, and friction acts at a rate rather than over a length: it relaxes the flow towards uniform at $gS_0/u$ per second, whatever the wavelength of the disturbance being relaxed. So the destabilising and stabilising effects scale together, their ratio has no $k$ in it, and the stability boundary is a pure number.

That is why the criterion is a Froude number and not a curve, and it is also why nothing here predicts a spacing. A theory whose boundary has no wavelength in it cannot produce one, and the spacing of real roll waves has to come from somewhere else — which is taken up at the end.

The exact dispersion relation, and its two ends #

The claim above is not a hand-waving race argument; it is the linearised shallow-water equations, which reduce to a single quadratic for the complex phase speed. Its two limits are the two descriptions the essay is about.

As the wavelength goes to infinity, one root approaches

$$c \to c_k - \mathrm{i}\,k\,D, \qquad D = \frac{q\left(1 - V^2\right)}{2 S_0}.$$

The real part is the kinematic speed, so the kinematic wave is the long-wave limit of the full equations rather than a separate model — which is the honest answer to the question the kinematic theory leaves open, and it is unavailable within that theory because it needs the momentum equation the theory discards.

The imaginary part is a diffusion, and the coefficient is the classical flood-routing diffusivity, discharge over twice the slope, multiplied by $1 - V^2$.

At the other end, as the wavelength goes to zero, the two roots approach $u + \sqrt{gh}$ and $u - \sqrt{gh}$: the dynamic waves, with the friction unable to act over so short a time. Computed at a wavenumber of $10^5$ per metre they land on those two values to two parts in $10^{12}$.

So one quadratic contains both theories. Long waves are kinematic and diffusive; short waves are dynamic and non-dispersive; and the transition between them is smooth and happens at a wavelength set by the friction. There is no separate kinematic model to be justified — only a limit to be taken, and a condition under which taking it is safe.

Which channels are on which side #

The sorting is clean and it is worth noting how much room there is. A large river in flood has a Vedernikov number of about 0.1 — a factor of ten below the threshold — and an irrigation canal 0.18. Even a mountain stream at Froude 1.35, which is supercritical and looks violent, is at 0.68. The threshold is reached only on artificial surfaces steep enough that the flow is thin, and there it is exceeded comfortably: a lined chute at 2.7, a spillway at 5.6.

That is the practical content of the whole calculation. Flood routing, which is where kinematic waves earn their living, is done on channels a decade below the threshold, and the kinematic approximation there is not a convenience but very nearly exact — the diffusivity is within one per cent of its zero-Froude value on a river at $V = 0.1$.

It is worth asking why the threshold is so hard to reach in nature, because the answer is not a coincidence. A natural channel adjusts: a river that runs too fast erodes its bed, lowers its slope and roughens itself with the material it has torn up, and the adjustment stops when the flow is carrying about as much sediment as it is given. The Froude numbers that come out of that are consistently below one for sand-bed rivers and between one and two for steep gravel ones, and the absence of natural roll waves is a statement about what a bed made of loose material will tolerate rather than about hydraulics.

Line the channel with concrete and the adjustment is prevented. The slope is whatever the engineer chose, the roughness is whatever the finish gives, and nothing erodes — so the Froude number goes wherever the design puts it, which on a spillway is well past the threshold. Roll waves are essentially an artificial phenomenon, and the mudflows and snow avalanches that also show them are the exceptions, where the material is moving too fast to reshape its own bed.

What a roll wave is, once it exists #

The linear analysis says uniform flow ceases to exist above the threshold and says nothing about what replaces it, and what replaces it is a shock train.

Each surge is a small hydraulic jump travelling downstream faster than the water beneath it, with a smooth backwater profile behind it and a breaking front. The profile between fronts is a solution of the steady equations in the frame of the wave; the front is a jump satisfying the momentum condition in that same frame; and the whole periodic structure is an exact solution which Dressler wrote down in 1949.

What the analysis cannot give is the spacing, because the linear problem has no preferred wavelength. A train of roll waves coarsens: a faster surge overtakes a slower one, they merge, and the spacing grows down the chute until something stops it. What stops it is the chute ending.

The amplitude, unlike the spacing, is bounded and by something identifiable. A surge grows until its front is a jump strong enough to dissipate, per wave, exactly the excess energy the slope supplies over what the bed friction can absorb — so the wave’s height is set by an energy balance in which the jump is the only sink. That is why roll waves on a steeper chute are taller rather than more frequent, and why they appear at all only where the slope is delivering more energy than friction can take: below the threshold, friction alone keeps up.

It also explains the one consequence an engineer cares about. A chute carrying roll waves has to have walls high enough for the surges rather than for the mean depth, and the surge can be twice it. A spillway designed on the uniform-flow depth overtops, intermittently, at a frequency nobody predicted because nothing in the steady design contains it.

What the picture cannot show #

The friction law is Chézy’s and the threshold depends on it. The exponent $\gamma$ is 3/2 for Chézy and 5/3 for Manning, and that difference moves the threshold from Froude 2 to Froude 1.5, which is a quarter. A real chute is somewhere between the two, and the threshold is therefore known to about that precision and no better.

The channel is wide and rectangular. A trapezoidal or triangular section has a different relation between area and depth, so a different $\gamma$ and a different threshold; the effect is substantial in a narrow section, where the wetted perimeter changes fast with depth.

The analysis is linear, and a chute is short. A disturbance growing at the computed rate needs a certain distance to become visible, so a flow past the threshold on a chute of a few metres may reach the bottom before anything has grown. The threshold predicts instability; it does not predict roll waves at a given place.

And the flow is taken as one-dimensional. Real roll waves on a wide chute are not straight: they form cells across the flow, merge laterally as well as streamwise, and the three-dimensional structure is what a photograph of a spillway shows and what none of this contains.

Who found it, and when #

Jeffreys computed the stability of uniform flow down a slope in 1925 and got the Froude-number criterion. Vedernikov’s papers, giving the number that carries his name, are of the 1940s. Dressler’s periodic shock-train solution is 1949, and Lighthill and Whitham’s 1955 paper — the one that names kinematic waves and derives the flood wave — states the overtaking condition explicitly and is where the two halves of this subject were joined.

The surprising connection is with a threshold in a completely different medium, computed in this collection already. A nozzle chokes when the flow speed reaches the speed at which pressure news travels, because past that point nothing downstream can inform anything upstream. Here uniform flow fails when the speed at which flux news travels reaches the speed at which pressure news travels downstream. Both are a signal being overtaken, and the difference is instructive: choking is a fluid outrunning its own pressure waves, and this is one carried signal outrunning another. A channel at Froude 2 is not choked — its flow is supercritical well below that and behaves perfectly well — it is a channel in which the two kinds of news have changed places.

Still open: what sets the spacing a chute actually shows #

The threshold is sharp and the wavelength is not decided by anything in this calculation, which leaves the one quantity a photograph measures unexplained.

Roll waves coarsen. Two adjacent surges travel at slightly different speeds, because a taller one moves faster, so the train merges and the spacing grows with distance down the chute. If nothing stops that, the observed spacing is simply a record of how far the flow has travelled since the disturbance began, and a longer chute should show longer waves — which is roughly what is measured and is not obviously the whole story, since reported spacings cluster more tightly than a pure coarsening argument predicts.

The calculation that would settle it is a march of the full nonlinear shallow-water equations down a long chute from small random disturbances, with the surge positions tracked and the mean spacing recorded against distance. Two outcomes are possible and they say different things: a spacing growing without limit, which makes the pattern a transient that never finishes, or a spacing that saturates, which would mean the merging shuts off at some amplitude and there is a selected state after all. Which of the two happens is the difference between roll waves being a coarsening process and being a pattern, and the same question has been asked — and answered differently — about the vortex street and about the spacing of dunes.