Roll waves stop merging at a spacing their fronts choose

When the flux outruns the pressure found the threshold at which a steep chute’s smooth sheet of water breaks into roll waves. A wide channel carries three wave speeds: two dynamic ones, at which pressure travels, and one kinematic one, at which a change in the flux travels. When the kinematic wave outruns the downstream dynamic one — when the Vedernikov number passes one, a Froude number of two for Chézy friction — uniform flow is unstable at every wavelength, and disturbances grow into a train of surges, each a breaking front with a long thinning back.

The essay ended on what it could not decide: the spacing. Its linear calculation grew every wavelength, so it selected none, and a photograph of a chute measures exactly the quantity it left open. The essay named the mechanism that might decide it — surges of different heights travel at different speeds, so the train merges and coarsens — and the two outcomes that mechanism allows. Either the merging never stops, and the spacing is a record of how far the flow has travelled; or it stops at some amplitude, and there is a selected spacing after all.

A chute that can be followed far enough #

The calculation the essay asked for is a march of the full nonlinear shallow-water equations down a long chute from small random disturbances, with the surges counted. In units of the uniform flow — depth one, speed one — the equations are mass and momentum with gravity along the slope and Chézy’s friction against it,

$$h_t + (hu)_x = 0, \qquad (hu)_t + \Big(hu^2 + \frac{h^2}{2F^2}\Big)_x = h - u|u| + \nu\,(h u_x)_x,$$

with $F$ the uniform flow’s Froude number and lengths in the normal depth over the friction coefficient — about 250 depths for a concrete chute. The last term is a small eddy viscosity. The shallow-water equations without it turn each surge’s front into a discontinuity, a bore, whose internal structure they do not describe; the viscosity gives the front a width and a dissipation, and stands in for the turbulence that rounds a real bore. Its size is one of the things the answer turns out to depend on, so it is varied.

The channel is periodic, two hundred units long, which turns the question about distance into a question about time: a train that would coarsen down a long chute coarsens here as it circulates, and the time it has run, times the surges’ speed, is the distance it has travelled. The flow is seeded with depth noise of a thousandth, and the march is second-order — limited reconstruction, approximate Riemann fluxes, friction and viscosity handled implicitly — at two and four hundred points per unit length, and at eight hundred for one check.

The march first reproduces the threshold. At a Froude number of 1.5 the noise decays, more than tenfold in three hundred time units; at 3 it grows a hundredfold and saturates at a depth range of more than the uniform depth itself, the surges’ crests about twice the uniform depth and their troughs about seven tenths.

Merging, then not #

Above the threshold the march shows both halves of the earlier essay’s picture, in order. The noise first grows into many small surges — more than seventy in the two hundred units at time 100, at the finer resolution — and these merge quickly: taller surges, faster, overtake shorter ones ahead, and the two fronts combine into one taller front that runs on faster still. By time 200 there are 55; by 400, 45. Then it stops. From time 400 to time 1000 the count stays at 45. The surges are still running, faster than the water, circulating round the periodic chute again and again; they simply no longer catch each other.

The reason is in the snapshots. Early, the surges have a spread of heights, and a spread of heights is a spread of speeds, so they overtake. Merging is selective: it removes the small surges, each absorbed by a larger neighbour, and so narrows the spread. Once the survivors are nearly equal in height they run at nearly equal speeds and the gaps between them stop closing. Each surge’s back is a stretch of flow running down towards uniform conditions under friction and gravity; a long back gives the following surge time to rebuild before it arrives, and a surge that has rebuilt to its neighbour’s height no longer gains on it. The train settles into nearly identical surges at nearly equal spacing, which is a travelling periodic wave — the kind of solution Dressler wrote down in 1949 for any spacing he chose. The march says which of his spacings a train reaches from noise.

What happens in a merger #

A merger is two fronts becoming one, and the shallow-water equations say exactly what that does. A front is a jump in depth that travels at a speed set by the depths on either side: the larger the jump, the faster it runs relative to the water ahead of it. When a taller surge’s front reaches the back of a shorter one, it is running into water that the shorter surge has already raised, so its own jump is smaller there and it slows a little; but the shorter surge’s front, now carrying the water of both, has a larger jump and speeds up. The two fronts close, and when they meet they are one front whose jump is the sum of both. The merged surge is taller than either, so it runs faster than either did, and it begins to gain on the surge ahead.

That is the coarsening’s engine and also, eventually, its limit. Every merger produces a surge taller than its neighbours and so restarts the race locally; but each merger also removes a short surge, and short surges are the only ones that can be caught. As the train loses its short members the heights converge, and a train of equal heights has no engine left. The merging does not stop because something resists it; it stops because it has used up the differences it ran on. The settled spacing is the one at which the friction and slope, rebuilding each surge’s back over its length, leave every surge the same height as the one ahead.

The surges travel as the flux does #

The surges travel faster than the water, and an earlier essay said why a disturbance on a chute should: a wave nothing in it travels with carries changes in the flux at the kinematic speed, which for Chézy friction is one and a half times the water’s. Roll waves are that kinematic wave grown large — the instability is the kinematic wave outrunning the pressure that would smooth it — and their fronts move at about the speed the flux’s changes do. So the train circulates round the periodic channel, or runs down a real chute, faster than any parcel of water in it, and a parcel is overtaken by surge after surge on its way down, lifted at each front and let down along each back. Over a settled train, each parcel spends most of its time on the long thin backs and a short time in the fronts. The train’s mean depth is the uniform flow’s — the march conserves the water exactly, redistributing the same volume into crests and troughs — but its mean flux need not be, since friction is not linear in the depth, and a depth gauge on a roll-wave chute that reports a time average is averaging a sawtooth, not a level.

The spacing the fronts choose #

What sets the settled spacing is not the noise or the channel. Two noise seeds on a channel twice as long give 64 and 66 surges in 400 units, spacings of 6.25 and 6.06, against 6.67 for the 200-unit channel at the same resolution: the length and the noise move it by a few per cent. The eddy viscosity moves it more. At the same resolution, a viscosity of 0.02 settles the train at a spacing of 4.65, 0.05 at 6.67 and 0.1 at 7.41: more dissipation in the fronts, longer waves. And the resolution moves it as much as the viscosity does: at the same viscosity, 6.67 at 200 points a length, 4.44 at 400, 5.56 at 800. The grid’s own dissipation, which the coarser march adds to the viscosity’s, changes the fronts and so the spacing.

That is the essay’s second finding, and it is the one to take seriously. The merging stops in every march — at every resolution, viscosity, seed and channel length tried, the count stops changing by time 400 or so — but the spacing at which it stops depends on how the surge fronts dissipate energy, by about twenty per cent across the resolutions available here. The selection is real, and what selects is the inside of the bores. A wave whose spacing is chosen by its shocks is a familiar thing in this collection’s other subjects: the column the chord rule cannot settle found a kinematic wave’s shocks deciding a sedimenting column’s structure, and the same logic of fronts that dissipate and so decide holds here.

The physical eddy viscosity of a chute’s flow, scaled as the march scales it, is smaller than any value marched here, by a factor of a few hundred, and resolving fronts that thin needs grids far finer than these. The trend in the figure says a real chute’s train would settle at a shorter spacing than the march’s; how much shorter is an extrapolation this calculation does not make. In units of the depth, the marched spacings of four to seven units are a thousand to two thousand depths, longer than most measured roll waves, which is the direction the trend says the extrapolation would correct.

A chute short enough to be still merging #

The answer to that question has a second part, about chutes rather than trains. The merging stops after the train has run for about four hundred time units, which at the surges’ speed is several hundred units of distance — of the order of a hundred thousand depths. That is a very long chute: a spillway carrying a centimetre of water would need the better part of a kilometre. Starting from larger disturbances than a thousandth of the depth shortens the approach, since less distance is spent growing the first surges, but the merging itself takes a few hundred units on its own.

So both of the earlier essay’s outcomes describe real chutes. A long chute — a dam’s spillway, a steep road gutter in a storm, a long irrigation flume — carries a train that has settled, and its spacing is a property of the flow. A short one — most laboratory flumes, a short concrete channel — carries a train still merging, and its spacing does record how far the water has come. The observation the essay began from, that reported spacings cluster more tightly than pure coarsening predicts, is the settled trains showing through: the longer the chutes in a survey, the more of them have reached the spacing their fronts choose.

Why the selection is in the fronts #

The shallow-water equations have no length of their own at the surge scale: they have the depth, and lengths along the chute scaled on the depth over the friction coefficient, and the travelling-wave solutions exist at every spacing. Something has to break that freedom, and in these equations the only thing that can is the dissipation at the fronts. Each front dissipates energy at a rate set by its height jump, and a train settles where the energy the slope feeds into the flow between fronts balances what the fronts dissipate. Change how the fronts dissipate — more viscosity, a coarser grid — and the balance moves to a different spacing. That is why the spacing depends on what the march does at the fronts and not on how long the chute is or how it was disturbed.

In a real chute the fronts are turbulent hydraulic jumps, and their dissipation is set by the turbulence in their rollers, which the shallow-water equations replace by a jump condition or a viscosity. The selected spacing is therefore a statement about the rollers. Measuring it, on a long enough chute, is in effect measuring how a roll wave’s front dissipates, which is one reason the spacing is worth photographing.

The convention, and what the march leaves out #

Depths are over the uniform depth, speeds over the uniform speed, lengths over the uniform depth divided by the Chézy friction coefficient, and times over that length divided by the uniform speed. The eddy viscosity is in the same units. The surges are counted as local maxima of the depth above the mean by a quarter of the range; the spacing is the channel’s length over the count. The channel is periodic, so the train is followed in time rather than down a real chute’s length, and a spatially developing chute — with its train born at one end and carried out at the other — would show the same merging and settling at a distance equal to the time times the surges’ speed, but would also select among the noise the inlet provides. Friction is Chézy’s with a constant coefficient; a real chute’s friction depends on the depth, which changes the surges’ shapes. The flow is two-dimensional in the vertical plane: no cross-channel structure, no air entrainment at the fronts, both of which a real chute has.

A chute engineer has one more reason to care which regime a chute is in. The surges’ crests, twice the uniform depth, set the freeboard a chute’s walls need, and a train still merging has crests that grow down the chute while a settled one does not; the wall height a long spillway needs at its foot is set by the settled train’s crest, not by a growing one extrapolated.

Who saw roll waves settle #

Cornish photographed roll waves on Alpine channels in the 1900s and recorded their spacing; Jeffreys found the threshold in 1925 and Dressler the travelling periodic solutions in 1949. Brock’s laboratory chute measurements of 1969 followed roll waves growing and merging along a flume. Balmforth and Mandre in 2004 studied roll-wave dynamics and coarsening in the shallow-water equations with a viscous term, which is the setting this march takes up; the arrest and its dependence on the fronts’ dissipation are what this march finds, with the numbers stated here.

How it was checked #

The finite-volume march conserves the water on the periodic channel to rounding over fifty time units. Below the threshold, at Froude 1.5, noise of a hundredth decays to under two thousandths in two hundred time units. The arrest of the merging was checked at three resolutions, three viscosities, two channel lengths and two noise seeds, and in every case the count stopped changing by about time 400 and held to time 1000; the spacing at which it stopped varied between 4.4 and 7.4 across them, and the figures say which case each number belongs to. The viscosity was moved from an explicit to an implicit update during the work, and the two agree at the resolution where both were run, 4.35 and 4.44. The march refuses a non-positive Froude number, too few cells and a negative viscosity.

Still open: a chute with an inlet #

The periodic channel replaces a chute’s length with time, and a chute is not periodic: its train is born from the inlet’s disturbances and carried out at the outlet, and only the surges that are born and survive are seen. The next calculation would march a chute with a fixed inflow, noisy at a stated level and spectrum, and an outflow that lets the surges leave, and ask whether the settled spacing at the chute’s far end is the periodic channel’s, or whether a steady inflow disturbance at one frequency — a gate’s vibration, say — can lock the train to its own spacing before the fronts’ selection takes over.