The drift that turns a current into rolls

Irving Langmuir crossed the Sargasso Sea in 1927 and noticed that the floating weed was not scattered. It lay in long straight lines running with the wind, tens of metres apart, and when the wind shifted the lines re-formed along the new direction within a quarter of an hour. He went home to Lake George and measured what was underneath: the water sank under each line and rose between them, in rolls whose axes pointed downwind.

The Stokes drift essays so far have treated the drift as a transport. The drift in a wave that has none found that parcels creep forward under a wave whose velocity at any fixed point averages to zero, and the drift a rotating planet takes back found that the Coriolis force acting on that transport drives a current that cancels it. Both were horizontally uniform, and so neither could make rolls.

What makes the rolls is the drift doing something other than transporting: acting on the vorticity of the current it moves through.

The force a drift exerts on a sheared current #

Averaged over a wave period, the momentum equation for a current carrying a Stokes drift $\mathbf{u}_s$ acquires an extra term, which Alex Craik and Sidney Leibovich wrote down in 1976:

$$\mathbf{F} = \mathbf{u}_s \times \boldsymbol{\omega},$$

where $\boldsymbol{\omega}$ is the vorticity of the current. It is the drift’s contribution to the wave-averaged momentum, and it has the form of a Coriolis force with the drift standing in for the rotation — or of the Lorentz force on a current in a magnetic field, with the vorticity standing in for the field.

It does nothing to an irrotational current, and the ocean’s surface current is not irrotational. Wind drags the surface layer downwind faster than the water beneath, so the current’s speed falls with depth and its vorticity points horizontally, across the wind. A drift along the wind crossed with a vorticity across it gives a force pointing up or down — and for a current uniform in the horizontal that vertical force is simply absorbed into the pressure. The interesting case is what happens when the current is not quite uniform.

Why a small disturbance grows #

Suppose a strip of the surface layer is moving downwind a little faster than its neighbours. That jet, varying across the wind, has vertical vorticity at its flanks. The drift, which is stronger at the surface than below, tilts that vertical vorticity towards the horizontal — into vorticity pointing downwind, which is a roll. The roll carries surface water towards the jet from both sides and down under it.

The current’s shear then does its part. Water brought down from the surface, where the current is fast, arrives carrying more downwind momentum than the water around it: the jet under the convergence is strengthened. A stronger jet has stronger flanks, the drift tilts them into a stronger roll, and the loop closes.

Writing that loop down for rolls uniform downwind, with a wavenumber $l$ across the wind, a uniform current shear $U'$, a drift $U_s(z)$ decaying over a depth $D = 1/2k$, and an eddy viscosity $\nu$, gives two equations for the downwind velocity $u$ and the overturning streamfunction:

$$\sigma u = U'\,\phi + \nu(\partial_z^2 - l^2)\,u, \qquad \sigma\,\xi = -l^2 U_s'\,u + \nu(\partial_z^2 - l^2)\,\xi,$$

where $\phi$ is the vertical velocity with its sign reversed and $\xi = (\partial_z^2 - l^2)\phi$. Without the viscosity, and treating the depth structure as locally uniform, they multiply into

$$\sigma^2 = U'\,U_s'.$$

The growth rate is the geometric mean of the two shears. That one line carries most of the physics, and the eigenvalue problem solved below is how much of it survives the viscosity, the surface and the depth structure.

Two shears, and nothing without either #

The four bars are the line above made concrete. With the current’s shear and the drift’s shear of the same sign — a wind blowing with the waves — the roll grows. With the current reversed — a wind blowing against a swell, or a tidal current opposing it — the product is negative and the roll decays faster than viscosity alone would make it. With either shear removed, nothing feeds the roll and it decays at the bare viscous rate.

That settles a question which is often answered carelessly. The rolls are sometimes described as driven by the wind and sometimes as driven by the waves, and neither description is complete: a wave field with no wind current makes no rolls, and a wind current with no waves makes none either. Each shear supplies one half of the feedback loop, and the loop is a product.

The roll spacing a Langmuir number chooses #

Measure depth in the drift’s decay depth and time in $(U'U_s'(0))^{-1/2}$, the inverse of the inviscid growth rate at the surface. Then one number is left: the Langmuir number,

$$\mathrm{La} = \frac{(\nu T)^{1/2}}{D},$$

which is the eddy viscosity’s diffusion length over one growth time, compared with the drift’s decay depth. A small Langmuir number means strong forcing against weak mixing.

Each curve rises and falls, and the reason at each end is different. Very wide rolls grow slowly because the tilting depends on the jet’s cross-wind gradient, which a wide roll hardly has. Very narrow rolls are killed by viscosity, which damps them at a rate proportional to the square of their wavenumber. In between is a spacing that grows fastest, and it is that spacing an ocean selects when rolls start from the random disturbances a real surface always has.

The fastest roll’s wavenumber falls steadily as the Langmuir number rises: 5.2, 2.2, 0.90 and 0.25 decay depths to the minus one for Langmuir numbers of 0.05, 0.13, 0.3 and 0.8. Its spacing, $2\pi/l$, is therefore 1.2, 2.9, 7.0 and 25 decay depths.

The growth rates were found by shift-and-invert iteration on an 80-level discretisation of a layer twelve decay depths deep, and each eigenvalue’s residual is a few parts in $10^{11}$. The same growth rate was found a second way, by marching the equations forward in time from an arbitrary start and measuring how fast the solution grew once the fastest mode had taken over; at La = 0.15 and a wavenumber of one, the two agree to $2 \times 10^{-6}$.

What the rolls look like #

The picture reproduces what Langmuir saw, including the detail that surprised him most: the fastest downwind flow is under the lines, not between them. That is the loop again. The overturning carries fast surface water down at the convergence, so the convergence is where the downwind jet is.

Anything that floats is carried to the convergence by the surface flow and stays there, because it cannot follow the water down. Weed, foam, oil and plastic all collect in the same lines, which is why windrows are visible from a ship’s deck and from a satellite and why slicks on a windy sea are striped.

That the jet and the vertical velocity have almost the same profile is itself the first equation read off. With little viscosity, $\sigma u \approx U'\phi$: the jet is nothing but the current’s shear displaced vertically by the overturning, so it follows the overturning depth by depth. The overturning lives within about two decay depths of the surface, which is where the drift’s shear is; below three decay depths the drift is nearly gone and so is the mode.

How the mixing sets the size #

The growth rate at the peak falls from 0.61 at a Langmuir number of 0.05 to 0.063 at 0.8, and the rolls stop growing at all near a Langmuir number of 1.5 — but by then the fastest roll is a hundred decay depths across in a layer twelve deep, so the threshold found is a threshold for that layer rather than for the ocean. In a real mixed layer the thermocline under it is a lid, rolls cannot be much wider than a few times the layer’s depth, and the depth takes over from the Langmuir number as the thing that picks the spacing. That matches the observation that windrow spacings scale with the mixed layer’s depth, from a few metres in a lake to a few hundred over a deep winter mixed layer.

The same instability wearing three other coats #

The line $\sigma^2 = U'U_s'$ has a shape that recurs across fluid mechanics, and recognising it says what kind of instability this is.

A rotating shear flow. Put a plane shear flow in a frame rotating about the axis across the shear and the Coriolis force acts on the disturbance velocities exactly as the vortex force does here, with twice the rotation rate in the drift’s place. Such a flow is unstable to rolls aligned with the flow when the rotation opposes the shear’s own vorticity and is weaker than it, and the growth rate squared is twice the rotation rate times the shear less twice the rotation rate — again a product of two things that must share a sign. The Stokes drift’s shear is playing the part of a rotation that varies with depth.

A curved flow. Rayleigh’s criterion for flow between rotating cylinders says that a swirling flow is unstable where its angular momentum falls outward, and the Taylor vortices that result are rolls whose growth rate squared is a product of the flow’s shear and its rotation. The centrifugal force on a displaced ring does what the vortex force does to a displaced jet.

A stratified fluid heated from below. A parcel displaced upward in an unstably stratified layer finds itself lighter than its surroundings and keeps going, at a rate whose square is minus the buoyancy frequency squared. The rolls of convection are the same kind of loop — a displacement that creates the force that displaces it further — and they have the same kind of threshold, set by a dimensionless number that weighs the forcing against diffusion.

In each case two things of the same sign multiply into a growth rate, and viscosity picks a size. That is why Langmuir cells look like convection cells turned on their side, and it is why the Langmuir number is the counterpart of a Rayleigh number, read the other way up. The difference is where the destabilising agent comes from: gravity in the convecting layer, rotation in the Taylor–Couette gap, and here a second-order average of a wave that has no mean velocity at any fixed point.

Where the energy goes, which is not where the growth comes from #

The scaling has a consequence that is worth stating by itself. Write the current’s shear as $\beta$ times the drift’s surface shear. Rescaling the downwind velocity by $\sqrt\beta$ removes $\beta$ from both equations, so the growth rate, the preferred spacing and the shape of the overturning do not depend on how the product of the shears is shared out between the current and the waves. A strong current with a weak drift and a weak current with a strong drift, with the same product and the same viscosity, make identical rolls growing identically.

What $\beta$ does decide is the energy. Multiplying each equation by its own variable and integrating over depth gives two separate budgets. The downwind jet gains energy from the current’s shear, at a rate $-U'\langle uw\rangle$, and loses it to viscosity. The overturning gains energy from the vortex force, at $-U_s'\langle uw\rangle$, and loses it to viscosity. Each closes on its own.

The split runs all the way from waves to current across the plausible range. At β = 0.1 the current supplies 20 per cent of the roll’s energy; at β = 1, seven-tenths of it; at β = 10, 96 per cent. None of that changes how fast the roll grows.

For the moderate-wind case the current supplies 85 per cent. That runs against the most common short description of Langmuir circulation, which is that the waves power it, and it is the eddy viscosity that makes the difference: the current’s shear is $u_*^2/\nu$, so a larger eddy viscosity means a weaker shear, a smaller β and a larger share for the waves. Measurements and simulations of the turbulent surface layer usually find the wave’s share dominant, and the reason is consistent with this figure — the turbulence the rolls themselves create mixes the current’s shear away and lowers β. The linear calculation cannot say what β a fully developed layer settles at. It can say that the growth rate would not notice.

In metres and minutes #

The spacing of 23 metres is in the range Langmuir and everyone since have measured under moderate winds, and an e-folding time of four minutes is consistent with his observation that the lines re-form within a quarter of an hour of a wind shift — three or four e-foldings take a disturbance to a visible amplitude. Neither agreement is strong evidence, because the eddy viscosity is borrowed and uncertain by a factor of several and both numbers move with it: doubling it raises the Langmuir number to 0.22 — it goes as the viscosity to the three-quarters, since the time scale itself contains the viscosity — widens the fastest roll to 39 metres and doubles its e-folding time to eight minutes.

What the linear rolls cannot show #

Amplitude. Every mode here is infinitesimal. A real roll grows until it has mixed the shear that feeds it, and its final strength, its spacing after nonlinear adjustment, and the merging of neighbouring rolls into wider ones are outside a linear calculation entirely.

A constant eddy viscosity. The mixing in a real surface layer is itself turbulent, varies with depth, and is partly made by the rolls. A constant is the simplest closure that lets the problem be posed, not a description of the layer.

The wave field’s own response. The drift is prescribed and is not depleted by what it drives, and the waves are a single monochromatic swell. A real sea has a spectrum, and its drift profile decays faster near the surface and more slowly below than a single exponential does.

Rotation and stratification. The ocean case has neither in it. The Coriolis force turns the current with depth, as the drift a rotating planet takes back showed, so the current’s shear is not aligned with the drift at every depth. And a stable thermocline beneath the layer suppresses rolls that reach it.

Downwind variation. The rolls are exactly uniform downwind. Real windrows break, join and wander, and the instability that lets a roll vary along its axis is a different eigenvalue problem.

Who found it #

Irving Langmuir described the lines and the rolls under them in 1938, eleven years after the voyage, having measured them on Lake George with floats and umbrellas weighted to hang at different depths. For almost forty years afterwards there were many candidate explanations and none that produced rolls from the equations. Alex Craik and Sidney Leibovich derived the wave-averaged vortex force in 1976 and showed that it made the wind-driven layer unstable; Leibovich’s review of 1983 set out the instability described here and the Langmuir number. James McWilliams, Peter Sullivan and Chin-Hoh Moeng’s large-eddy simulations of 1997 took the same force into fully turbulent surface layers and introduced the turbulent Langmuir number used to classify them.

The observation that the growth rate cannot see how its forcing is divided, while the energy budget can, is a direct reading of the scaling rather than anything new. It is, though, the most economical way to understand why the literature has said both that the waves drive the rolls and that the wind does.

Still open: what a mixed layer of finite depth does to the drift’s transport #

The rolls here live in a deep ocean, and their onset was set by the depth of water the calculation allowed. The same is true of the transport argument that preceded them. The drift a rotating planet takes back found the Eulerian current exactly cancelling the drift’s transport at every eddy viscosity, in water deep enough that nothing below the Ekman layer mattered.

On a continental shelf the water is not that deep, and the bottom holds a stress. The next calculation puts a floor under the same Coriolis–Stokes problem and asks how much of the cancellation survives it: whether a shelf sea shallower than its own Ekman depth keeps the drift’s transport, which way the surviving transport points, and at what depth the planet’s rotation stops being able to take the drift back.