A coast sends the drift back, or sends it along

The floor that gives the drift back asked what a swell’s Stokes drift does to a shelf sea, where the floor is within reach of the Ekman layer that the Earth’s rotation sets up under the waves. In the open ocean that layer takes the drift’s transport back exactly: the Coriolis force on the drift drives an Eulerian current whose transport is its precise opposite, as the drift a rotating planet takes back found. Over a shelf the floor holds a stress, and whatever it holds is transport that survives. The sea keeps all of it at half an Ekman depth, three-quarters at one, a quarter at two, and the kept transport turns to the right as the sea deepens.

That calculation had no coast in it. Its shelf went on for ever, so the kept transport went on for ever too. Near a shore that cannot be, because water that arrives at a coast has to go somewhere. The essay’s closing question was what: how large a set-up the coast builds, whether the return runs along the bottom or along the shore, and what the swell still delivers to the beach. It has an answer that depends, once again, on one length.

The model: one more unknown #

The coast is straight, the shelf flat, and nothing varies along the shore. Take x onshore and y along the coast. The steady Eulerian current, written as one complex number $W = u + iv$, obeys the same balance as over the open shelf — eddy viscosity against Coriolis force, with the Coriolis force on the drift as the forcing — plus one new term, the push of a sloping sea surface:

$$\nu W'' - i f W = i f\,U_s(z)\,e^{i\theta} + g\,\frac{d\eta}{dx}.$$

Here $U_s$ is the drift’s profile under the swell, θ the angle the waves arrive at, and η the height of the sea surface. With nothing varying alongshore, mass conservation says the cross-shore transport cannot change with distance from the coast, and it is zero at the coast, so it is zero everywhere. That one condition fixes the one new unknown, the slope.

The equation is linear, so its solution is the open shelf’s current plus the slope times the current a unit slope drives, and the latter has a closed form of the same kind as the former — a constant, $i/f$, plus two Ekman exponentials fitting the floor and the surface. The slope is then whatever makes the total cross-shore transport vanish. Every number below uses the earlier essay’s standard case: an eight-second swell of one metre amplitude, an eddy viscosity of 0.01 m²/s and the latitude of the North Sea, 55°, for an Ekman depth of 12.9 metres.

Across the shore, and along it #

The figure draws the Lagrangian mean current — what a marked parcel actually does, the drift plus the Eulerian current — against depth, in seas of 8, 25 and 60 metres, for a swell arriving straight at the coast. Each cross-shore profile carries no net water: what goes onshore in one part of the column comes back offshore in another. The alongshore profiles are where the three seas differ most.

At 8 metres the sea is well inside the Ekman layer, the rotation has little room to act, and the exchange is entirely across the shore: onshore in part of the column, offshore in the rest, and almost nothing along the coast. At 25 metres, two Ekman depths, a current along the shore has appeared, a third as strong as the current across it and carrying 0.18 of the Stokes transport: the sea has turned part of its return into a flow parallel to the coast. At 60 metres the open shelf had already taken most of the transport back before any coast was needed, and the profiles are the open shelf’s own spiral, barely disturbed.

A set-up, and then a set-down #

The slope has a classical value to compare with. In a channel with no rotation, a steady swell against its end is balanced by a slope just steep enough to drive a return current carrying the Stokes transport back. With a uniform eddy viscosity, a stress-free surface and a no-slip floor, that return is parabolic and the slope is $3\nu M_S / H^3$ — steep in shallow water, because a thin layer resists a return flow strongly, and shallow in deep water.

Shallower than an Ekman depth the shelf sets up exactly as the channel does: 18.1 millimetres per kilometre at 5 metres, the channel’s 18.1, and at 10 metres 1.71 against 1.69. The rotation has no room to act on a layer thinner than the one it would build, which is the same statement the earlier essay made about the kept transport, now made about the set-up.

Deeper, the two part company, and in a surprising direction. From about 2.2 Ekman depths — 28 metres here — the shelf’s slope goes negative: the sea sets down against the coast. The kept transport has turned so far to the right that its cross-shore part points offshore, away from the coast the swell is arriving at, and the sea has to slope down towards the land to drive a return onshore. The set-down is small — at most 0.044 millimetres per kilometre, near three Ekman depths — because by then little transport is kept at all; and deeper still the sign alternates as the kept transport spirals inwards, by amounts too small to draw. But its existence says that “waves pile water against a coast” is a statement about shallow water, not about waves.

What the coast cannot return goes along it #

The second question was whether the return runs along the bottom or along the shore, and the figure answers it as a fraction. Where the open shelf’s kept transport points straight at the coast — shallower than an Ekman depth — the coast returns it as an undertow, and the current along the shore is a few per cent of the Stokes transport. Where the kept transport has turned, the coast does not so much return it as redirect it: the alongshore transport grows to 0.36 of the Stokes transport near 2.8 Ekman depths, flowing to the right of the waves, and it exceeds the transport the open shelf would have sent at the coast at all.

That excess is the pressure gradient’s doing. The set-up or set-down is a force across the shore, and on a rotating shelf a force across the shore drives a current along it, held in balance by the floor’s friction — a geostrophic current with a bottom Ekman layer under it, of the kind the layer that stops at a depth describes. So the coast’s answer to the swell is partly a slope and partly a river along the shore, and which dominates is set, once more, by the depth in Ekman depths.

Whether a float reaches the beach #

The last question was what fraction of the drift delivers floating material to the beach rather than circulating back out. A float cannot go down with the return, so it goes where the surface water goes, and the figure plots the surface’s cross-shore Lagrangian current. In seas deeper than about half an Ekman depth the surface moves onshore — at about half the surface drift beyond two Ekman depths — and a float arrives at the coast, however little net water does.

In the shallowest seas the model says something else, and it is worth being exact about why. With a uniform eddy viscosity and a stress-free surface, the return current’s parabolic profile is strongest at the surface, and in water shallower than about half an Ekman depth it exceeds the surface drift there: the surface moves offshore, the onshore flow is at the bottom, and a float would be carried out. That is a consequence of the uniform mixing, not a measured fact. Measurements of the inner shelf, notably Lentz and colleagues’ of 2008, found the offshore Eulerian flow under swell close to the Stokes drift’s mirror image at every depth, so that the Lagrangian current nearly vanishes throughout the column; a mixing that weakens towards the surface, or a surface stress from breaking, moves the return current down and gives the surface back to the waves. Where the float goes, in water that shallow, depends on the one ingredient the model has least grip on.

An oblique swell #

Swells rarely arrive straight on, and a swell at an angle carries an alongshore transport of its own that no coast can stop. In the shallowest sea the figure shows the alongshore transport following that share almost exactly — 0.72 of the Stokes transport for a swell at 45°, against the sine of 45°, 0.71 — because the rotation has no room to add anything. Deeper, the rotation’s contribution, always to the right of the incoming waves, is added. At 20 metres a swell must arrive more than 4.3° to the left of straight onshore before the current along the coast runs to the left; at 40 metres every swell in the range drawn, from 45° on one side to 45° on the other, drives a current to the right.

That asymmetry is the practical content for anything carried along a coast. A coastline facing the prevailing swell on a shelf a couple of Ekman depths deep sees a drift along its shore to the right of the waves whatever their exact angle, and a mirror-image coast in the southern hemisphere sees it to the left.

Where the switch is, in metres #

Every result above is a function of the depth in Ekman depths, so where a particular coast sits depends on its mixing and its latitude. The Ekman depth $\sqrt{2\nu/f}$ is 7 metres for a weakly mixed sea at 55° with an eddy viscosity of 0.003 m²/s, 13 metres for the standard case, and 22 metres for a well-mixed one at 0.03; at 30° the same three are 9, 17 and 29 metres, because the Coriolis parameter is smaller there and the Ekman layer thicker. So the undertow regime — the coast behaving like a tank’s end wall — covers the inner shelf out to depths of roughly 7 to 30 metres, and the alongshore regime the mid-shelf beyond it, 15 to 90 metres. The switch is therefore on the part of the shelf where coastal plastic, larvae and sediment spend most of their time, and where it falls on a given day moves with the wind’s mixing. A storm that stirs the water column deepens the Ekman layer and pushes the undertow regime out to sea; a calm, stratified spell pulls it in.

The rotation’s timescale sets how quickly any of this can happen. The Ekman layer is spun up in about an inertial period, fourteen hours at 55° — how long a fluid takes to forget it was not rotating is that time for a different flow — and a swell that lasts a day has time to build it. The set-up in shallow water is faster still, set by the viscous time $H^2/\nu$, which for 5 metres and 0.01 m²/s is under an hour.

Why a float does not follow the water #

The distinction between the surface current and the net transport is the distinction every essay on this drift has been built on. A float follows the Lagrangian velocity at the surface: the Eulerian current there plus the Stokes drift, which is the velocity a marked parcel actually has, averaged over a wave period. The net transport is an integral over the column, and it can be zero while the surface moves onshore at half the drift — the water column overturning, onshore at the top and offshore below. The mean is not the flow made the general point that an average of velocity at a fixed point says nothing reliable about where fluid goes; here the average at a point is the Eulerian current, which runs offshore through almost the whole column in the deeper seas, while every float at the surface is carried in. A current meter moored off a beach and a plastic bottle floating past it disagree about which way the water is going, and both are right about what they measure.

What was checked #

Three checks. Integrated over depth, the steady balance says the Coriolis force on the net transport, plus the slope’s push on the whole column, is the floor’s stress, and the computed current satisfies it, with its net cross-shore transport zero, to rounding error at three depths and two angles. In a sea a seventh of an Ekman depth deep the slope agrees with the non-rotating channel’s $3\nu M_S/H^3$ to four parts in a hundred thousand. And the whole problem — the current and the slope together — solved again by finite differences on four thousand levels, with the slope found from the transport condition, agrees with the closed form to three parts in a hundred thousand in the slope, the alongshore transport and the profile. The tests refuse a swell travelling away from the coast, a negative viscosity and a tolerance of zero.

What the picture cannot show #

A sloping floor. Real shelves deepen offshore, so the depth in Ekman depths changes across the shelf, and the balance drawn here for one depth becomes a function of distance from the coast. The cross-shore transport must still be zero, but the slope and the alongshore current vary, and a current along the shore over a deepening floor is steered by it.

The surf zone. Every wave here is unbroken. Within a few wave heights of the shore the waves break, and their loss of momentum drives set-up, undertow and longshore currents of a different size altogether, which is the surf-zone problem rather than the shelf one.

The wave’s own bottom boundary layer. As in the earlier essay, the drift of inviscid wave theory does not vanish at the floor, and the thin oscillatory layer there — whose streaming drives a mean current of its own — is left out.

Uniform mixing, whose consequence for floats in the shallowest water is described above; and a steady state, which a real swell, arriving and dying away over a day or two, has barely time to reach in water much deeper than an Ekman depth.

The same wall, three times #

The coast is the third wall to be put in the drift’s way. The drift a closed box will not allow was a wave tank, whose end wall returns the transport as a current underneath, exactly and with no rotation. The rotating ocean with no walls returned it with the Coriolis force instead. The coast is both at once, and the result is not a mixture of the two but a switch between them: shallower than an Ekman depth the coast behaves like the tank’s end wall; a few Ekman depths down it behaves like a boundary on a rotating plane, where a pressure gradient against a wall drives a current along it. The switch is the same one the drift that turns a current into rolls met in another guise: what the drift does to a sea depends less on the drift than on what the sea is allowed to do back.

Who worked it out #

Wave set-up and set-down at a coast are Longuet-Higgins and Stewart’s, from the radiation-stress theory of the 1960s, and the mean return flow under waves over a flat bottom is Longuet-Higgins’s of 1953. The Coriolis–Stokes force is Hasselmann’s, from 1970. Its combination with a coast, a floor and a set-up on the inner shelf was measured and modelled by Lentz, Fewings and colleagues in the 2000s, who found the offshore Eulerian flow close to minus the Stokes drift over the inner shelf of Martha’s Vineyard. The transport the coast turns alongshore is the same geostrophic adjustment that makes coastal currents follow coastlines everywhere.

Still open: a floor that deepens offshore #

Every figure here is for a flat shelf, so the depth in Ekman depths is one number. A real shelf runs from nothing at the beach to a hundred metres or more at its edge, passing through every regime drawn above. The next calculation lets the depth increase linearly offshore, keeps the cross-shore transport zero at every distance, and asks how the set-up and the alongshore current vary across the shelf: whether the shallow part’s set-up and the deeper part’s set-down meet in a surface that rises towards the coast and dips a few kilometres out, and whether the alongshore current concentrates where the depth is two or three Ekman depths, as the flat-shelf figures suggest it should — a coastal jet set by the Ekman depth rather than by the wind.