The section that decides the river

A long reach of open channel carrying a steady discharge has two depths that can be computed from the channel alone.

The normal depth is where the bed slope and the friction slope balance, so that gravity’s pull along the channel exactly matches the resistance and nothing accelerates. The critical depth is where the Froude number is one, so that the water moves exactly as fast as a long wave on it.

Neither of them is the depth the water actually has.

The equation, and how many conditions it needs #

What the water has is a profile obeying

$$\frac{dy}{dx} = \frac{S_0 - S_f(y)}{1 - \mathrm{Fr}(y)^2},$$

which is a first-order ordinary differential equation and therefore needs exactly one condition.

Its two special depths appear in the two halves of the fraction. At normal depth the numerator vanishes, so the surface is flat and a reach that reaches normal depth stays there. At critical depth the denominator vanishes, so the surface would have to be vertical — which is the equation announcing that it has stopped describing anything, since a vertical water surface is not a gradually varied flow.

One condition is one number. Which leaves the question of where to impose it, and that question turns out not to be a matter of taste.

Both depths deserve a sentence about what they are, since a reader will meet them as formulas.

The normal depth is the answer to “how deep would this channel be if it went on for ever?” — it is the equilibrium of a balance between two slopes, and a reach long enough approaches it whatever happens at its ends. It depends on the discharge, the roughness and the bed slope, and on nothing about the reach’s length or its boundaries.

The critical depth is the answer to a different question: at what depth does the water move as fast as a long wave on it? That is the one Froude number that really is one, and unlike the normal depth it does not involve the roughness or the bed slope at all. It is $(q^2/g)^{1/3}$ and nothing else, so two channels of quite different character carrying the same discharge per unit width have the same critical depth.

Which of the two is larger decides everything that follows. If the normal depth is above the critical one, a long reach settles into subcritical flow and the reach is called mild; if below, it settles into supercritical flow and the reach is steep.

The one that is not a choice #

Put a weir in a mild reach and the depth at the weir is fixed by the weir. Integrating upstream from there, the profile relaxes onto the normal depth over about seven hundred metres and stays there for as far as the calculation is carried.

The same equation, the same starting depth, the same integrator, with the sign of the step reversed: the departure from normal depth grows instead of decaying, and fifty kilometres later the calculation has water fifty metres deep in a channel two metres deep.

That is not a numerical failure. It is the correct solution of the equation from that condition in that direction, and it is meaningless as a river.

Why the directions differ #

The reason is in the linearisation. Write $y = y_n(1 + e)$ and expand: to first order

$$\frac{de}{dx} = -\frac{10}{3}\,\frac{S_0}{y_n}\,\frac{e}{1 - \mathrm{Fr}_n^2},$$

so the departure decays going one way and grows going the other, exponentially, at a rate the channel sets.

The relaxation length that implies is 665 metres for this channel, and the computed profile decays at 668 — four parts in a thousand apart, which is the linearisation’s own error rather than the integrator’s.

In a subcritical reach the sign is such that the decay is upstream. That is the mathematical statement of something physical: the Froude number is below one, so long waves travel upstream faster than the water travels down, and a disturbance at the downstream end can be felt above it. The condition belongs where the influence comes from.

On a bed twenty-five times steeper the normal depth falls below the critical one, the flow is supercritical, the denominator changes sign, and every statement above reverses. Now the profile relaxes going downstream and runs away going upstream, and the condition belongs at the top of the reach.

There is a stronger way to say all of this, and it is the one worth keeping.

The reach forgets its control. Whatever depth is imposed at the weir, seven hundred metres above it the profile is at normal depth to within a per cent, and everything about the weir has gone. So the information in a boundary condition has a range, and beyond that range the flow is determined by the channel alone.

That is why the same computation can be run from three quite different conditions and give three profiles that are the same profile a kilometre away.

The relaxation length is a property of the reach — it goes as the normal depth over the bed slope, divided by one minus the square of the Froude number — and it is a useful number in its own right. A steep, rough channel forgets quickly; a flat, smooth one remembers for a very long way, which is why a weir on a nearly level river can raise the water level for kilometres and why the term “backwater” means what it does.

What a control section is #

A control is any place in a reach where the depth is fixed by something other than this equation: a weir, a sluice gate’s vena contracta, a lake at the downstream end, the crest of a spillway, a free overfall. And the Froude number decides which way its influence runs.

A weir controls upstream. A sluice gate controls downstream, because the flow beneath it is supercritical. A free overfall and a spillway crest sit at critical depth and control both, which is what makes them useful as gauging structures: the discharge follows from the depth at the crest and from nothing else.

Which reach a channel is #

Here is the part that makes the classification less comfortable than it looks. The normal depth grows with discharge as the three-tenths power and the critical depth as the two-thirds, so on one bed slope they cross. Below a certain discharge the reach is mild and above it the reach is steep, on the same channel, with the same roughness.

So whether the boundary condition belongs upstream or downstream is not a property of the channel at all. It is a property of the flow in it that day, and a river in flood can be a different kind of reach from the same river in summer.

What the runaway looks like from inside #

It is worth describing the failed direction carefully, because in a real computation it does not announce itself.

Integrated the wrong way from a depth close to normal, the profile stays close to normal for a long time. The growth is exponential, so from a departure of one part in a thousand it takes seven relaxation lengths to reach a departure of one — five kilometres in this channel. Over the first kilometre or two the profile is indistinguishable from a correct one.

So a modeller integrating the wrong way on a short reach gets a plausible answer, and the same modeller on a long reach gets water fifty metres deep. The failure is not gradual in the useful sense: it is invisible and then catastrophic, and where the boundary between the two lies depends on how accurately the starting depth happened to match the normal depth.

The other tell is the direction of the error. A profile that is running away always departs away from normal depth, in whichever direction it started, so it either climbs indefinitely or falls towards critical depth and the equation’s vertical gradient. Reaching critical depth from above, in a mild reach, is a particularly confusing symptom, because a hydraulic jump would produce exactly that transition physically — and the calculation has arrived there for a purely numerical reason.

Which gives the practical check. Integrate towards normal depth, and if the profile is moving away from it, the direction is wrong. That test costs nothing and does not require knowing the Froude number in advance.

The general statement underneath #

This is a particular case of something more general and worth naming.

An equation whose solutions carry information in a direction has a domain of dependence, and a boundary condition imposed outside that domain is not a boundary condition — it is an over-specified one at one end and a missing one at the other, and the numerical symptom is exponential growth. The gradually varied flow equation is the simplest example in this subject: one equation, one condition, one direction, decided by a single dimensionless number.

The same statement in compressible flow is the warning that cannot arrive: supersonic flow has no upstream influence, so a downstream condition on it is not admissible, and a subsonic flow needs one. The number that decides it is the Mach number rather than the Froude number, and everything else is the same.

And the hydraulic jump is the same analogy’s other half: a supercritical reach that has to become subcritical does it discontinuously, exactly as a supersonic flow that has to become subsonic does, because there is no continuous profile connecting the two through the critical depth that the equation permits.

What the classification is actually for #

The textbook table of profile types — M1, M2, M3, S1, S2, S3 and the rest — reads like taxonomy and is really the enumeration of a small number of cases.

There are two reaches, mild and steep. In each there are three regions: above both depths, between them, and below both. The sign of the numerator and the sign of the denominator are each fixed within a region, so the sign of the gradient is fixed, and the profile in that region can only rise or only fall. Six regions, six shapes, and the letters are a label for which one a stretch of river is in.

What the enumeration is for is knowing which way a profile runs before computing it. A backwater curve behind a weir on a mild reach is above both depths, so the numerator is negative and the denominator positive, so the depth falls going upstream — towards normal depth, asymptotically, which is the relaxation. A drawdown towards a free overfall is between the depths, so the depth falls going downstream towards critical.

None of that requires the equation to be solved. It requires the two depths and a sign, which is why the classification survives as a working tool in a subject that has had numerical solutions for sixty years.

What this means for a computation #

Three consequences, and all three are about setting up rather than about solving.

Count the conditions from the Froude number, not from the geometry. A reach that is subcritical takes one downstream condition and no upstream one; a supercritical reach takes one upstream and none downstream; and a reach that changes between them takes one at the control and none at either end.

A profile that grows exponentially is a sign the condition is at the wrong end, not a sign the step is too large. Refining it makes the growth smoother and does not remove it, which is the diagnostic: a numerical instability improves with refinement and this does not.

And a long computational reach does not need an accurate upstream condition. The relaxation length is a few hundred metres in the case here, and beyond it the profile has forgotten the condition entirely. Where a reach is many relaxation lengths long, the upstream boundary can be anything admissible, which is a considerable practical freedom and is why river models are run from far upstream of the region of interest.

Where the analogy with compressible flow runs out #

The correspondence with gas dynamics is close enough to be worth using and close enough to be worth bounding, so it is worth saying which parts carry over.

The Froude number is the Mach number. Both are a flow speed over a signal speed, both are one at the point where the equation changes character, and both decide which way information runs. The mathematics of the two problems is the same to a considerable depth: the shallow-water equations are the gas dynamics equations with a ratio of specific heats of two.

The hydraulic jump is the shock, as this collection has already worked out, and its conjugate depths follow from momentum exactly as a shock’s jump conditions follow from conservation — with the same structure of a total that is fixed and an interior that is not.

And the control section is the sonic throat. The throat that stops listening is a control in exactly this sense: a place where the flow passes through the critical condition, where the discharge is fixed by the local geometry, and where nothing downstream can be felt above it.

What does not carry over is the friction. A gas dynamics problem usually has a bed slope of zero and no resistance, so it has no normal depth and no relaxation; the shallow-water problem’s profiles are shaped by the balance between two slopes, and the exponential forgetting on this page has no counterpart in an inviscid nozzle. That is why open-channel flow has a classification of profile types and compressible duct flow has a classification of choking mechanisms instead.

The reach’s other exact statements #

Two more results about open channels are worth putting beside this one, because together they show which questions have exact answers and which do not.

The conjugate depths of a jump are exact, from momentum alone, with no model of the turbulence inside the jump — which is why a shock in a river can be computed without knowing anything about what is happening in the white water. The energy lost follows as the difference between two known states rather than being modelled.

The most efficient section is exact too. The depth that costs least comes out of minimising the wetted perimeter at a fixed area, which is geometry, and the answer does not depend on the roughness or the discharge.

The profile between them is not exact in the same way, because it depends on Manning’s law, which is a fit. That is the honest division: the jump conditions and the optimal section are conservation and geometry, and the backwater curve is a friction correlation integrated.

What this page adds is that the direction of the profile calculation is exact even though the profile is not. Which end the condition belongs at follows from the sign of $1 - \mathrm{Fr}^2$ and would follow from any friction law whatever — so it is a statement of the same kind as the jump conditions, and it is the one most easily got wrong in practice.

What a gauging structure is for #

The classification has one application that is worth spelling out, because it is the reason control sections are built deliberately rather than merely identified.

A weir, a flume or a spillway crest forces the flow through critical depth. At that depth the discharge and the depth are related by a formula with no roughness, no slope and no downstream condition in it — the flow is passing through the one state where the equation’s denominator vanishes and the local geometry decides everything.

So measuring the depth at such a structure measures the discharge, with an accuracy set by the structure’s geometry rather than by anything about the channel. That is how nearly every river gauging station works, and it is why the structures are built to standard shapes and calibrated once rather than per installation.

The failure mode follows from the same argument. If the downstream water level rises high enough to drown the control, the flow no longer passes through critical there, the downstream condition reaches back through the structure, and the depth-discharge relation is no longer valid. That is called submergence, it is checked for by measuring the downstream level as well, and it is the gradually varied flow equation’s directionality showing up as an operating limit on an instrument.

A last note on the two depths as a diagnostic. Both are computable from the channel and the discharge before any profile is integrated, and the comparison between them is the single most informative thing about a reach: it settles which kind of profile is possible, which end the condition belongs at, whether a hydraulic jump can occur, and which way a control’s influence runs. Two formulas and a comparison, before any equation is solved.

And a note on where the equation comes from. It is the energy equation for a gradually varied free surface, with the friction slope supplied by a uniform-flow correlation evaluated locally. That second step is the approximation: it assumes the flow is locally in equilibrium with its own boundary layer, which is why the equation is called gradually varied and why it fails where the surface curves sharply.

What is not claimed #

The channel is wide and rectangular. That makes the hydraulic radius equal to the depth and gives Manning’s friction slope a closed form, which is what allows the relaxation length to be predicted analytically. A real section’s geometry changes the exponents and not the argument.

Manning’s law is empirical. The friction slope used throughout is $n^2q^2/y^{10/3}$, whose constant carries dimensions and whose exponent is a fit. Every number here inherits that; the structure of the result — a decay one way and a growth the other, at a rate set by the linearisation — does not.

The critical depth is where the model stops rather than where anything is infinite. A real flow approaching critical depth develops a short region of rapidly varied flow with surface curvature in it, which the gradually varied equation neglects by construction. What the vertical gradient marks is the edge of the approximation.

And the relaxation length is the linearised one. It is accurate near normal depth, which is where the profile spends most of its length, and it under-estimates the distance from a condition far from normal depth — as the computed profile shows by taking longer than the exponential to arrive.