The drift in a wave that has none

Watch a gull sitting on a swell. It rises, moves forward a little as the crest arrives, sinks, moves back as the trough passes, and returns to almost where it started. The usual account of that motion is that the wave carries its shape and its energy along while the water stays put, which is the right first thing to say and is not the whole of it.

The gull does not return to where it started. It is a little further forward each time, by the same amount, and the amount is calculable.

Zero at a point, and not zero for a parcel #

The linear deep-water wave has a velocity field that can be written down:

$$u = akc\,e^{kz}\cos(kx - \omega t), \qquad w = akc\,e^{kz}\sin(kx - \omega t),$$

with $c = \omega/k$ the wave speed, $a$ the amplitude and $ak$ the steepness. At any fixed point the average over a cycle of either component is exactly zero. That is the Eulerian mean, it is what a current meter bolted to a mooring reads, and there is no current in this wave.

A parcel is not at a fixed point. It moves, and while it moves it samples the field at different places, and the sampling is biased in two ways that push the same direction:

• Depth. The velocity decays as $e^{kz}$, so the parcel is in faster water at the top of its orbit — where it is moving forward — than at the bottom, where it is moving back.
• Phase. The parcel is carried forward under a crest, which means it stays under the crest a little longer than it stays under the trough, so the forward half of the cycle lasts fractionally longer.

Each bias is first order in the steepness and each multiplies a velocity that is first order, so the result is second order:

$$u_s = (ak)^2 c\,e^{2kz}.$$

That is Stokes’ result, from 1847, and it is the difference between the Lagrangian mean — the average velocity of a parcel — and the Eulerian mean, which is zero.

Measuring it rather than quoting it, and the trap in doing so #

The check made here is direct: integrate the trajectories through the exact linear field by Runge–Kutta, fit a straight line to the displacement, and compare against $(ak)^2c$. The agreement is 0.16 per cent at a steepness of 0.025, 0.65 per cent at 0.05 and 2.6 per cent at 0.1 — and the pattern in those numbers is the check that matters.

The trap is where the parcel is released, and it cost more than the answer is worth to find. The drift depends exponentially on the depth of the centre of the orbit. Release a parcel at $z = 0$ at the phase where the horizontal velocity is greatest and it orbits entirely below the release point — its mean depth is a whole orbit radius down — and the measured drift is short by $e^{-2ak}$, which at a steepness of 0.1 is a fifth of the answer. Released instead at the phase where the vertical velocity is greatest, the orbit is centred on the release depth and the measurement converges.

That is worth recording as a general caution. A second-order quantity measured off a first-order field is sensitive to details of the set-up that are themselves first order, and the way to know whether such a measurement is right is to check its convergence rate rather than its plausibility.

What it does in the ocean #

For a swell of one metre amplitude and eight seconds’ period in deep water: $c = gT/2\pi = 12.5$ m/s, $k = 2\pi/(cT) = 0.063\ \mathrm{m}^{-1}$, so $ak = 0.063$ and the surface drift is $(ak)^2c = 0.049$ m/s. That is five centimetres a second, or four kilometres a day, and it decays with depth over $1/2k = 8$ metres.

The consequences are large and are mostly about things that float:

• Surface pollution and plastic collect where the drift converges, which is why debris follows wave fields rather than currents.
• Search and rescue drift models carry a Stokes term explicitly, because a person in the water is carried by it and a drogued buoy is not — the two separate at a rate that matters within hours.
• Sediment on a beach is carried shoreward by the drift near the bed and seaward by the return flow above it, so the direction a grain travels depends on its size and settling velocity — a question this collection prices in a steady flow and which a wave makes into an argument between two second-order terms.
• Ocean mixed-layer models include Langmuir circulation, which is generated by the interaction of the Stokes drift with the wind-driven current and would not exist without this second-order term.

A real sea is a spectrum, and the drift is dominated by the waves nobody notices #

The arithmetic above is for one wave, and the swell it was evaluated on is the wave a person looking at the sea would describe. That is the wrong wave, and the reason is a weighting that is easy to derive and easy to miss.

The drift is quadratic in amplitude, so for a spectrum of waves the contributions simply add:

$$u_s(z) = 2\int \sigma\,k\,E(\sigma)\,e^{2kz}\,d\sigma ,$$

with $k = \sigma^2/g$ in deep water. So each component is weighted by $\sigma k = \sigma^3/g$. The integrand is weighted by the cube of the frequency, which means the surface drift is dominated by the short, steep, wind-driven waves — the chop, the ripples, the part of the sea state a significant-wave-height number does not describe — and not by the long swell that carries almost all of the energy.

The depth profile inherits the same bias and is deformed by it. A single wave decays as $e^{2kz}$ with one length in it; a spectrum decays with a different length for every component, and the components that dominate the surface value are precisely the ones with the largest $k$ and therefore the shortest reach. A one-second chop has $k = 4\ \mathrm{m}^{-1}$ and is gone within a few tens of centimetres; the eight-second swell of the section above reaches eight metres. The real profile is therefore extremely steep in the top decimetre and has a long weak tail below it, and neither half is well represented by the single exponential that the one-wave formula draws.

That has a blunt practical consequence, and it is the reason drift modelling is a specialism. The surface value comes out at roughly one per cent of the wind speed for a developed sea, which for a fresh breeze is more than the swell estimate above and is delivered by waves nobody would think to measure. Meanwhile anything floating a metre down — a drogue, a partly submerged container, a person in a lifejacket — is in a completely different part of the profile, and two objects a metre apart in depth separate at a rate comparable to the drift itself. The question is never what is the Stokes drift but at what depth, and the answer changes by more over the first metre than over the next twenty.

It also completes the Langmuir mechanism the earlier list named without explaining. Wind blowing over water leaves a sheared current, and a sheared current has vorticity lying horizontally across the wind. The Stokes drift acts on that vorticity through a term of the form $\mathbf{u}_s\times\boldsymbol{\omega}$ — a genuine force in the mean equations, arising because the Lagrangian and Eulerian averages differ — which tilts the crosswind vortex lines into the wind direction and rolls them up. The result is the pair of counter-rotating helices that gather foam and seaweed into the long windrows visible on any lake in a steady breeze.

Those lines on the water are a second-order term made visible. They exist because the mean of a product is not the product of the means, they require both the shear and the waves, and neither the wind alone nor the waves alone would produce anything like them. It is the same residue this essay measures as a gull’s failure to return to where it started, acting on a current instead of on a bird.

What carries the momentum, and what does not #

There is a book-keeping question hiding under the arithmetic, and it is worth settling because it is where most of the confusion about wave transport lives.

A linear wave carries momentum. Its momentum per unit area of surface is $E/c$, with $E$ the wave energy density — a relation this collection meets again in the account of what a wave system costs a ship. Where is that momentum? It is not in the Eulerian mean velocity, which is zero. It is in the Stokes drift: multiply the drift profile $(ak)^2ce^{2kz}$ by the density and integrate over depth, and the result is exactly $E/c$.

So the drift is not an incidental second-order curiosity attached to a wave. It is the wave’s momentum, expressed as a velocity, and a wave that transports momentum must transport water — which is the answer to the question of how a wave can push a beach without carrying anything to it. When the wave breaks, that momentum has to go somewhere, and where it goes is a longshore current.

The same statement, in three other places #

In the atmosphere, where the same integral applied to internal gravity waves gives a mean flow that the waves deposit where they break. That is the mechanism behind the quasi-biennial oscillation, a reversal of the equatorial stratospheric winds driven entirely by wave momentum.

In a Kelvin wake, where the same drift accompanies every one of the crests whose angle does not depend on speed.

In acoustics, where the drift of a parcel in a sound field is the reason dust collects at the nodes of a standing wave, and where the second-order steady flow generated in a boundary layer is Rayleigh’s streaming — the viscous member of the same family.

And in a wing’s wake, where a material loop drawn round a shed vortex drifts with it, and where the distinction between what the field does at a point and what the fluid does is exactly the distinction Kelvin’s theorem is stated in.

Why an average that follows the fluid is the honest one #

There is a general point here about which average to take, and it is not a matter of taste.

Anything that is carried — a pollutant, a temperature, a parcel of dyed water, a floating object — moves with the Lagrangian mean, because that is the definition of being carried. Anything measured by a fixed instrument is an Eulerian mean. The two differ by exactly the Stokes drift, and the difference has a name — the quasi-Stokes velocity — and appears in every ocean model that has to transport a tracer.

The distinction is the same one this collection makes about streamlines and particle paths, and it is the same size of error: the pattern at an instant is not what parcels do, and the average at a point is not what parcels do either.

It also explains a familiar difficulty with photographs of flows. A long exposure of particles in a wave gives streaks that are nearly closed loops, and reading a mean transport off them requires resolving the gap where the loop fails to close — which is a fraction $ak$ of the loop’s own size. What a photograph of a flow shows is exactly the wrong instrument for a second-order quantity, and the reason is arithmetic rather than optical.

How it is measured at sea, and why the two instruments disagree #

The drift is small and the wave orbits are not, so measuring it means extracting a few centimetres a second of mean from a signal whose fluctuation is a metre a second. Two approaches are used and they answer different questions.

A drifter — a float that follows the surface — moves with the Lagrangian mean and therefore carries the drift with it, by construction. A current meter on a fixed mooring reads the Eulerian mean and does not. The difference between a drifter track and a moored record, taken in the same water on the same day, is one of the standard ways of measuring the Stokes drift, and it is the cleanest demonstration that the two averages are different physical quantities rather than two estimates of one.

The complication is that a drifter is not a fluid parcel. It has windage, it has inertia, and if it is drogued at depth it samples a different part of the profile — so the difference between the two records contains the drift and a correction that depends on the hardware. Every ocean-drift dataset carries a slip correction for exactly this reason, and it is of the same order as the quantity being measured.

What the picture cannot show #

The field is linear and the drift is second order. Integrating trajectories through a first-order field is exactly what Stokes did, and the answer is correct to second order; the next correction requires the second-order wave field, which has a different profile shape and its own mean-level set-down. Everything here is truncated at the same order as the formula it checks, which is why the errors fall as the steepness squared.

Deep water only. In finite depth the drift profile changes shape — it does not decay to zero at the bottom but to a finite value, and mass conservation in a closed basin then demands a return flow that this analysis says nothing about. In a laboratory flume the return flow is comparable to the drift and the measured profile looks nothing like the figure above.

The wave train is infinite and steady. A real group of waves has a beginning and an end, and the drift under a group is accompanied by a return flow beneath it — a set-down under the group and a compensating current — that keeps the mass balanced. That is a first-order effect on the answer for a finite group, and it is one of the several places where a result derived for an infinite wave train needs care before it is applied to the sea.

And viscosity is absent. A real wave has a thin oscillatory boundary layer at the surface and another at the bottom — each the same layer whose thickness a frequency sets — and each generates its own second-order streaming; the classical result is that the drift profile in a viscous fluid differs substantially from Stokes’ near both boundaries. That is the same physics as the streaming essay and it is not in these figures.

Who found it, and when #

Stokes derived the drift in 1847 in the same paper that established the finite-amplitude wave theory carrying his name, and the derivation is a Taylor expansion of the velocity about the parcel’s mean position — the standard route ever since. The Lagrangian-versus-Eulerian distinction was Longuet-Higgins’ subject a century later, and his 1953 paper on mass transport in water waves is where the viscous corrections and the finite-depth return flow are worked out.

The surprising connection is with the mean of an oscillating stream in this same field. There, a flow whose mean velocity is zero has a mean pressure that is not, because pressure is quadratic in velocity. Here, a flow whose mean velocity is zero has a mean displacement that is not, because displacement is the integral of velocity along a path that the velocity itself decides. Both are the same failure of averaging to commute with a nonlinear operation, and in both the second-order residue is the entire physical effect rather than a correction to it.

Where the ladder goes next #

Beside this rung sits the viscous version, where the same second-order term drives a steady flow with a coefficient of $-3/4$ that comes out of an integral. Above it lies wave-induced transport in a stratified or rotating fluid, where the drift interacts with the background rotation and produces the mean flows that dominate the middle atmosphere — which this collection has the machinery for and has not yet written.