A drift made of two things that average to zero

The drift in a wave that has none establishes the fact: the velocity at a fixed point under a passing wave averages to exactly zero, every parcel of water moves steadily forward, and the drift is the square of the steepness times the wave speed. It explains the mechanism the way it is always explained — a parcel is slightly higher and slightly further into the crest during the forward half of each orbit than during the backward half, so the orbit does not close.

That explanation is correct and it is a picture rather than a formula. This essay is the formula, and it turns out to be a statement about memory: the drift is the average of the parcel’s displacement — which is a time integral of everything it has met — dotted into the gradient of the velocity where it is now.

Two quantities, each of which averages to exactly zero, producing something that does not.

Where the formula comes from #

Seven depths, one wave: the measured drift reads 0.007126 at 0.05 wavelengths down against a predicted 0.007085, 0.005272 against 0.005249 at 0.2, and 0.001061 against 0.001060 at a full wavelength — an error falling from 0.58 per cent to 0.089 as the orbits shrink, which is the second-order theory’s own truncation showing.

A parcel released in an oscillating flow sits at some position and moves about it. Write its position as an orbit centre plus a displacement from that centre. Then the velocity it experiences is not the velocity at the centre; it is the velocity a displacement away, which to first order is the velocity at the centre plus the displacement dotted into the velocity gradient.

Averaging over a cycle, the first term is the Eulerian mean, which for a linear wave is exactly zero. What is left is the average of the second term:

$$u_S = \langle \boldsymbol{\xi}\cdot\nabla\mathbf{u}\rangle,$$

where the displacement is itself the time integral of the velocity. The displacement is a memory — it is where the parcel has got to, which is a running total of where it has been — and the gradient is a property of the present. The drift is the correlation between the two.

That is a general statement and it is worth noticing how few of the wave’s properties it uses. It does not use the dispersion relation, the free surface, gravity or the fact that the disturbance is a wave at all. Any oscillation with a spatial gradient produces a mean transport of this form.

Each half is exactly a half #

For a deep-water wave the displacement and the gradient are both known in closed form, so the average splits into two terms and each can be computed on its own.

The horizontal term is the parcel’s fore-and-aft displacement correlated with the fore-and-aft gradient of the horizontal velocity. The vertical term is its up-and-down displacement correlated with the way horizontal velocity grows towards the surface. Each carries 0.500000000000 of the total, and their sum is $a^2\omega k\,e^{2kz}$ to twelve figures.

The two halves come out at 0.00262438 each, summing to 0.00524876 against a predicted 0.00524876 — the same digits, because the decomposition is exact rather than fitted, and the horizontal share is 0.500000000.

The equality is not a coincidence, and it is not an assumption either. Both terms come out as the average of the square of a sinusoid, because the wave’s displacement and its gradient are in quadrature in exactly the way that makes the two products equal. In shallower water, where the orbits are ellipses rather than circles, they are no longer equal — the horizontal term grows and the vertical one shrinks, and their sum is the shallow-water drift. The half-and-half is a property of deep water, and it is the reason the classical result has the tidy form it does.

The control, which is where the argument is actually won #

The formula says the drift needs a gradient. That is a falsifiable statement and it is cheap to falsify, so it is worth doing rather than asserting.

Take the same integrator, the same fit, the same amplitude and the same frequency, and feed it a velocity field that oscillates identically everywhere — no dependence on position at all. Every parcel moves through exactly the same excursion as before. The measured drift is 2·10⁻¹⁵ of the orbital speed, which is the accumulated rounding of a quarter of a million Runge-Kutta steps and is zero.

Then feed it a field carrying only the horizontal part of the wave’s gradient. It drifts at 0.502 of the wave’s own drift, which is the horizontal half and nothing else, to within the half per cent the second-order theory itself is good to.

So the gradient is not a detail of the derivation. It is the whole mechanism, and an oscillation without one transports nothing however violent it is.

What “second order” means here, and why it is not a small correction #

The drift is the product of two first-order quantities, so it is second order in the wave amplitude. Measured over amplitudes spanning a factor of sixteen, the exponent is 2.0002.

That has a consequence which is easy to state and often missed. A first-order description of the flow — the linear wave, the mode shape, the eigenfunction — contains the drift nowhere. It is not that the drift is small in the linear solution; it is that it is absent from it, exactly, and appears only when two first-order quantities are multiplied together and averaged. Halving the wave amplitude divides the drift by four and the orbit by two: 0.01663, 0.004117, 0.001027, 0.0002565, 0.00006411 against orbits of 0.0724, 0.0362, 0.0181, 0.00905, 0.00452. The fitted exponent is 2.00024, and the four local exponents run 2.014, 2.004, 2.001, 2.000 as the amplitude falls.

Every quantity of this kind behaves the same way, and there are more of them in fluid mechanics than the list usually admits. Acoustic streaming is a correlation between an acoustic displacement and an acoustic gradient. The radiation stress that raises mean sea level inside the surf zone is a correlation between a wave velocity and itself. The Reynolds stress in the mean is not the flow is a correlation between two components of a fluctuation that individually average to zero. Taylor dispersion, which is the subject of two slow things make a fast one, is a correlation between a radial excursion and an axial velocity gradient — and its formula is this one with the wave replaced by a cross-stream wander.

They are all the same object: an average of a memory against a gradient.

What the solver computed, and how it was checked #

The velocity field is the exact potential solution of the linearised deep-water problem. Trajectories are integrated with fourth-order Runge-Kutta at 240 steps a period, and the drift is the slope of a straight line fitted through the parcel’s position at the end of each period.

Two things about that measurement are traps, and this collection has now walked into one of them twice, which is why they are stated here rather than left in a comment.

The release phase. A parcel let go at the moment the horizontal velocity is greatest orbits entirely below its release depth: its orbit centre sits a whole orbit radius down, and since the drift falls exponentially with depth the measured value comes out short by roughly twice the steepness. At a steepness of 0.05 that is 7.4 per cent, which is large enough to look like a genuine higher-order correction and was mistaken for one while this module was being written. Releasing a quarter period earlier centres the orbit on the release depth, and the check now refuses an orbit whose centre has moved by more than a seventh of a radius.

The length of the run. The residual orbit is not small beside the drift per cycle, so a short run fits a line through a signal that is mostly wobble. The window here is long enough for the parcel to slip a whole wavelength relative to the wave — 597 periods at this steepness — after which the residual has been through a full period of its own and averages out of the fit. Shortening it makes the error grow, from 0.43 per cent to 2.6 per cent, which is the signature of a systematic residual rather than of noise.

With both handled, the measured drift agrees with $a^2\omega k\,e^{2kz}$ to between 0.09 and 0.58 per cent over seven depths, and the residual falls as the square of the amplitude — which is the order the closed form was truncated at, so it is the next term of the expansion rather than an error in the arithmetic.

Why the two averages disagree, and which instrument reads which #

The whole subject exists because two reasonable definitions of “the mean velocity” differ, and it is worth being exact about which instrument returns which, because the difference is the drift itself.

A moored instrument — a current meter on a chain, a hot wire in a tunnel, a pressure tapping — sits at a fixed point and averages what passes it. Under a linear wave that average is exactly zero, at every depth, for ever. A drifting instrument — a float, a dye patch, a neutrally buoyant particle — goes where the water goes and returns the drift.

Neither is wrong and neither is a correction to the other. They are averages of the same field over different sets: one over a fixed point and a stretch of time, the other over a moving parcel and the same stretch of time. Two averages of one flow makes the same distinction with density rather than position doing the weighting, and reaches the same conclusion — that the two definitions are each correct for their own purpose and that quoting one where the other is meant is a factor-of-two error rather than a rounding one.

The practical rule follows from the mechanism. A moored instrument measures the field; a drifting one measures the correlation. So anything that depends on where the water ends up — an oil slick, a larva, a plastic bottle, a nutrient — is governed by the drifting answer, and anything that depends on what passes a place — a load on a pile, a heat flux through a section, a sediment stress on a bed — is governed by the moored one.

Why a correlation and not an accumulation #

There is a way of thinking about drift that sounds equivalent and is not, and separating them is worth a section because it decides what happens in flows that are not waves.

The accumulation picture says the parcel gets a small forward push each cycle and the pushes add up. That is a description of the answer, not of the mechanism, and it makes a prediction that is wrong: it suggests that a parcel which is stopped and restarted each cycle would drift at the same rate. It would not. The displacement it is being correlated against is the running integral of the velocities it has already met, and resetting that integral each cycle sets the correlation to zero.

The correlation picture says the parcel’s excursion and the field’s gradient conspire, and it predicts exactly what happens when the conspiracy is broken. The uniform-oscillation control above is that experiment. So is a wave in which the phase between the two velocity components is altered: in an oscillation with somewhere to go, the steady streaming produced near a wall is the same correlation with the phase changed by the viscous boundary layer — and its sign changes with the phase, which no accumulation picture can produce.

Where the memory reading pays #

The case where the correlation is cancelled rather than merely computed is the drift a closed box will not allow: put walls on the tank and the transport integrates to zero, with the profile left free.

The case where the correlation is cancelled rather than merely computed is the drift a closed box will not allow: put walls on the tank and the transport integrates to zero, with the profile left free.

The framing matters because it says which flows have a drift and which do not, without doing the calculation.

A flow needs an excursion and a gradient, correlated. Turbulence has both, which is why a turbulent flow transports a scalar at an effective diffusivity set by the correlation between a velocity and a displacement — the mixing-length argument, stated honestly.

A flow with a short memory has less of it. If a parcel loses its displacement — through diffusion, through a collision, through being scattered — before the gradient has changed sign, the correlation is cut short and the transport falls. That is the whole of why molecular diffusion suppresses Taylor dispersion rather than adding to it, which is one of the more counter-intuitive results in transport theory and is a one-line consequence here.

A scalar released in it is carried by the drift and not by the field. A dye patch under waves moves forward at the drift speed while the water at any fixed point averages to nothing, which is the same statement a scalar is a record of where its fluid was makes in general: the scalar follows the map, and the map is not the field.

And a flow whose oscillation is in phase with its gradient transports more than one whose is not. The correlation is a projection, so the phase is as important as the amplitude — and the phase is the one thing an amplitude spectrum does not carry. What a mean profile cannot tell anybody is that statement made about a shear flow rather than a wave.

What this shares with the ordering of a strain history #

There is a family resemblance between this result and two strainings, and the order they came in, and it is worth naming because the two essays were written from opposite ends.

In both cases a quantity that looks as though it should be an integral of instantaneous rates turns out not to be. There, the stretch of a material line is a product of deformation gradients and the order of the factors matters. Here, the drift is a product of a displacement and a gradient and the phase between them matters. In both cases every instantaneous measure of the flow is identical between the case that produces a large answer and the case that produces none.

The common structure is that both quantities are bilinear in the history rather than linear in the state. A linear functional of the present field can be measured with one instrument at one moment. A bilinear one needs two things measured together, and it is exactly the class of quantity that survives averaging when each factor does not.

That is also why both are invisible to a first-order theory and why both are second order in an amplitude — and it is the reason a collection about what a flow remembers keeps arriving at products of two small things.

Who found it, and when #

Stokes derived the drift in 1847, in the paper that also gave the third-order wave and the dispersion relation’s amplitude correction. His derivation is the expansion above, done by hand: the position is written as a centre plus a displacement, the velocity is expanded about the centre, and the second-order term is averaged.

The general form of the result — that a mean transport arises from the correlation of a displacement with a gradient — was recognised much later and separately in several fields at once. Rayleigh had it for acoustic streaming in 1884. Taylor had it for dispersion in a pipe in 1953. Longuet-Higgins built the wave version into the radiation-stress theory in the early 1960s and used the same correlation to compute the streaming under a wave over a bed. None of them cites the others, and the four results are one calculation.

What the picture cannot show #

The orbits drawn above are the first six of five hundred and ninety-seven. At this steepness the drift per cycle is about a thousandth of the orbit radius, so a picture showing enough cycles for the drift to be visible would show orbits too small to see, and a picture showing the orbits clearly cannot show the drift. The two scales are separated by exactly the factor the whole argument is about, which is why the drift is quoted from a fit rather than pointed at.

Limits recorded rather than smoothed over #

The field is linear, and so is the drift computed from it. The measured values here are the exact Lagrangian drift of the linearised Eulerian field, which is not the same as the drift under a real wave of that steepness: a real wave has a third-order field of its own, and its correction to the drift is a separate term this calculation does not carry.

Deep water only. The two halves being equal is a deep-water statement. In finite depth the orbits are ellipses and the split moves; the essay’s general formula is unaffected and its tidy half is.

No viscosity, no rotation, no wind. The drift measured here is the inviscid, non-rotating, unforced one. A real ocean adds a viscous streaming at the surface and the bed, a Coriolis turning that makes the drift veer with depth, and a wind stress that drives its own current — all of which are comparable in size to this and none of which is in the figures.

And the control is a construction. The uniformly oscillating field is not a solution of anything; it is a velocity field written down to have an excursion and no gradient, which is exactly what is needed to isolate the term. It satisfies continuity trivially and nothing else, and its only job is to be the case the mechanism says must not drift.