The current that runs across the river

The depth that costs least and the section that decides the river are complete descriptions of a reach: a depth, a discharge, a slope, a profile obeying a first-order equation. Both are one-dimensional, both assume the velocity is uniform across the section and the pressure is hydrostatic, and both are excellent.

They also cannot contain the single most consequential thing a river does, which is that it does not run straight.

Two things happen and one of them is visible #

Water going round a bend needs a centripetal force and there is nothing to supply it but a transverse pressure gradient, which in a channel with a free surface means a transverse slope. Taking the velocity as uniform across the section and the pressure as hydrostatic gives the superelevation

$$\Delta z = \frac{V^2 B}{g R_c},$$

with $B$ the width and $R_c$ the centreline radius. Taking the velocity as a free vortex instead — $V \propto 1/r$, which is what an inviscid fluid entering a bend would adopt — gives a different expression with a different coefficient.

A gravel-bed river forty metres wide running at a metre and a half a second round a bend of a hundred and twenty metres tilts its surface by seven and a half centimetres. Nobody standing on the bank sees that. It is real, it can be surveyed, and it is not what makes a river meander.

The comparison between the two models is worth reading as more than a consistency check. What separates them is the width over the radius, and that is the parameter every “gently curving” assumption in hydraulics is implicitly about. At a radius a hundred times the width they agree to five parts in $10^5$; at a radius one and a half times the width — which is a tight meander bend or a canal turning a corner — they differ by twenty-seven per cent, and neither is obviously the right one, because a real bend’s velocity distribution is set by the upstream reach and by friction rather than by either idealisation.

So the superelevation is known to a few per cent on a gentle bend and to twenty-five per cent on a tight one, and it is worth knowing which regime a measurement was made in before quoting it. Where the formula matters most — a canal designed so that the water does not overtop the inner bank on a bend, or a flume built to hold a given discharge — is exactly where it is least reliable.

The part that is not uniform over the depth #

The tilt is fixed by the depth-averaged velocity, through the depth-averaged balance. So it is the same at every level from the bed to the surface: a pressure gradient in a hydrostatic fluid does not know how deep it is.

The centripetal requirement is not the same at every level, because the velocity is not. Near the bed the water is slower, so it needs less centripetal force than the tilt is providing, and the surplus drives it inward. Near the surface it is faster, needs more than it is getting, and is thrown outward. The result is a circulation in the cross-section, superimposed on the downstream flow, which the water follows as a helix.

Solving it takes two integrations. The transverse momentum balance is

$$\frac{\partial}{\partial z}\left(\nu_t \frac{\partial v}{\partial z}\right) = -\frac{u(z)^2 - \bar u^2}{R_c} + G,$$

with no slip at the bed, no stress at the surface, and $G$ — a uniform correction to the depth-averaged pressure gradient — fixed by the one condition the cross-section imposes: no net transverse discharge. A channel cannot pass water sideways, so whatever the profile is, it integrates to zero, and it does here to three parts in $10^{15}$.

That last condition is the one that is easy to get wrong, and getting it wrong is instructive: solve with the surface stress free instead and the profile leans entirely one way, which is a flow carrying water out of the river.

What the condition is doing physically is worth a sentence. The depth-averaged balance says the tilt is set by the mean velocity; that is a statement about the average, and averages do not determine profiles. The correction $G$ is the amount by which the true tilt differs from the depth-averaged estimate, and it is small — but it is the only free constant available, and without it there is one boundary condition too few. A profile problem needs a profile’s worth of conditions, and the transverse mass balance is the one the one-dimensional description had already used up.

The resulting profile crosses zero at about half the depth. That is not exactly half, and where it crosses is set by the streamwise profile’s own shape: a rougher channel has a fuller logarithmic profile, a smaller departure of $u$ from $\bar u$ near the bed, and a crossing slightly higher up. None of that changes the direction of the circulation, which is the only thing the sediment cares about.

The strength is the aspect ratio and nothing else #

$$\frac{v_{\max}}{V} \approx 5.95\,\frac{h}{R_c}.$$

That is a remarkably clean result for a quantity that came out of a two-point boundary-value problem with a logarithmic velocity profile and a parabolic eddy viscosity in it. The proportionality is exact — over a sweep of radii spanning a factor of twenty the constant varies by less than a part in a thousand — and the constant itself depends only on the channel’s roughness, weakly, through the shape of the streamwise profile.

A bend’s secondary flow is therefore decided by how deep the river is relative to how tightly it turns, and not at all by how fast it is going. A faster river has a stronger helix in absolute terms and exactly the same helix in proportion.

The reason the speed drops out is worth following, because it is not obvious and it is the same cancellation that made the superelevation independent of depth. The forcing on the transverse flow is $(u^2 - \bar u^2)/R_c$, which is quadratic in the velocity scale. The resistance to it is the eddy viscosity of a turbulent layer — the eddy viscosity times a second derivative, and the eddy viscosity is $\kappa u_* z(1-z/h)$ — proportional to the friction velocity, which is itself proportional to the mean velocity. So the forcing goes as $V^2$ and the resistance as $V$ times the transverse velocity, and one power of $V$ cancels from each side.

Both the forcing and the resistance are set by the same flow, and a ratio of two things that scale together is a pure number. That is why the result is a clean proportionality rather than a correlation, and why the constant in front of it can be quoted at all.

It also says what would break it. Anything that fixes the transverse resistance independently of the streamwise flow — a viscous channel rather than a turbulent one, a stratified river, a bed of vegetation with its own drag law — removes the cancellation and puts the velocity back into the answer. The result is a statement about a rough turbulent channel and nothing wider.

That last line is worth dwelling on. A flume is the instrument this subject is studied with, and a flume’s bends have a depth-to-radius ratio five times a real river’s, so the effect under study is five times stronger in the apparatus than in the thing the apparatus stands for. That is not an error; it is why a flume shows the effect at all.

The same table contains a second warning that is easier to miss. The irrigation canal has a helix of seventeen per cent — three times the lowland river’s — and the canal is shallower. Its radius is sixty metres against the river’s nine hundred, and depth over radius is what decides, so a small channel turning a sharp corner has a stronger secondary flow than a great river turning a slow one. Scale does not enter, and the intuition that big rivers do big things is simply wrong here.

What the helix costs, which is less than it seems #

A circulation is a flow, a flow dissipates, and it is fair to ask what a bend costs a river in head.

The answer is small and the reason is the same aspect ratio. The secondary flow’s kinetic energy is of order $\rho v_{\max}^2$ per unit volume against the main flow’s $\rho V^2$, and since $v_{\max}/V \approx 6h/R_c$, the ratio of the two is $36(h/R_c)^2$ — a thousandth for a gravel-bed river and a part in ten thousand for a lowland one. Even allowing that the secondary flow is dissipated and regenerated continuously along the bend, a bend’s extra head loss is a small multiple of the straight-reach loss over the same length rather than a new term.

That is the reverse of the impression the figures give, and it is worth stating because it explains a puzzle in the practice. Handbooks for pipes and ducts treat a bend as a substantial local loss, with a coefficient of a few tenths of a velocity head; handbooks for rivers barely treat it at all. The difference is not carelessness. A duct’s bend has a radius of a few diameters and a river’s has a radius of tens of widths, so the duct is at an aspect ratio where the secondary flow is a substantial fraction of the main one and the river is not.

A river’s bend is therefore energetically negligible and morphologically decisive, and those two sentences are about the same few per cent.

Why a river will not stay straight #

The helix is a few per cent of the downstream flow and it decides the river’s shape, because of what it carries rather than how fast it goes.

Sediment moving as bed load travels in the bottom few centimetres of the flow, and the bottom few centimetres are exactly where the helix runs inward. So a bend gathers sediment against its inner bank — a point bar — while the fastest surface water, thrown outward, is pressed against the outer bank and scours it.

A bend therefore migrates sideways, outward, at a rate set by the bank’s erodibility. The migration deepens the bend, which raises $h/R_c$, which strengthens the helix, which migrates it faster: a straight channel in erodible material is unstable, and the instability’s saturated state is a meander train.

Nothing in that argument needs the flow to be turbulent, or the sediment to be of any particular size, or the banks to be of anything in particular. What it needs is a velocity that varies over the depth and a bed load that does not — which is to say, it needs the one thing the one-dimensional description throws away.

The self-reinforcement has a limit and it is worth naming, because an instability with no limit predicts nothing. A bend that migrates outward lengthens the channel, which lowers its slope, which lowers the velocity; and a bend tight enough eventually intersects itself, cuts off, and leaves an oxbow. So the meander train is a saturated state rather than a runaway, and its characteristic wavelength — measured at about eleven channel widths across five decades of river size, from a laboratory rivulet to the Mississippi — is one of the most robust empirical relations in the subject and has no derivation from the argument above.

That is an honest gap rather than a small one. The helix explains why a bend migrates; it does not explain why the bends that survive are spaced at eleven widths rather than at five or at forty. Every attempt to get that number has needed a stability calculation on the coupled flow-and-bed system rather than a statement about the flow alone.

What the picture cannot show #

The bend is circular and of constant curvature, and a real one is not. A river’s curvature grows and decays along the bend, and the helix takes a distance to develop — several channel widths — so a short bend never reaches the strength computed here and a long one overshoots at the exit.

The eddy viscosity is a parabola and the streamwise profile is logarithmic. Both are fits to straight-channel measurements, applied in a bend where the flow is not a straight-channel flow; the constant in front of $h/R_c$ inherits whatever error they carry, which is why it is quoted in the literature anywhere between five and eleven.

The banks are vertical and the bed is flat. A real bend has a deep outer pool and a shallow inner bar, so the depth varies across the section by a factor of several — and since the helix’s strength goes as the depth, the helix is strongest exactly where the bed is lowest, which reinforces the pattern this calculation predicts and cannot contain.

And nothing here moves any sediment. The argument that the helix carries bed load inward is a statement about the direction of the near-bed flow, not a transport calculation, and what a bed carries begins with the weight it has to lift. How much sediment moves, and whether the bar and the pool reach a steady shape, is a different subject with a different set of empirical laws in it.

And the flow is steady, where a real one passes a flood wave that travels faster than its water. A river’s discharge varies by an order of magnitude between a drought and a flood, the depth with it, and the helix’s strength goes as the depth — so the circulation is strongest in flood, which is also when the bank material is weakest. Almost all of a meander’s migration happens in a few days a year, and a calculation done at the mean discharge is describing the wrong flow.

Who found it, and when #

Thomson described the secondary circulation in 1876 and got the direction right from exactly the argument above — the tilt is set by the mean and the requirement is not. Rozovskii’s 1957 monograph is the standard solution and the source of the coefficient; Engelund’s 1974 treatment is the one most often used in morphological models. That a bend migrates outward and that this is why rivers meander was argued through the 1960s and 70s and is now the standard account.

The surprising connection is with a flow that has no bend in it at all. An Ekman layer forms because a pressure gradient balances the Coriolis force in the interior and cannot balance it near a boundary, where friction has reduced the velocity — so the near-wall fluid turns across the isobars, and the result is a spiral. This is the same structure with the centrifugal force in place of the Coriolis one: a body force that is balanced on the average and not locally, in a fluid whose velocity varies over the depth, always produces a cross-flow. The two are so closely analogous that the secondary flow in a bend is sometimes solved by borrowing the Ekman solution outright, and the reason the analogy is exact is that neither force depends on the direction of the velocity in the plane, only on its magnitude.

Still open: what the helix does when the bed is not flat #

Every number above is computed on a rectangular section, and the section a bend actually has is the one the helix made.

A developed bend has a pool against the outer bank two or three times as deep as the inner bar, and the helix’s strength goes as the local depth over the radius — so the circulation is strongest at the outer bank and nearly absent over the bar. That is a feedback the calculation above cannot represent, because it assumes a depth. Whether the feedback saturates, and at what cross-sectional shape, is the question morphological models answer numerically and no closed argument settles.

The calculation that would say something general is the same two-point problem solved with the depth a function of the transverse coordinate, and the transverse mass balance imposed across the whole section rather than at each station — which couples the profiles at different radii and turns two integrations into an integral equation. The quantity to ask it for is the shape at which the inward bed-load transport exactly balances the outward slope-driven transport at every point, because that shape is the equilibrium cross-section of a bend, and it is a prediction with no fitted sediment constant in it that nobody appears to have made.