A sloping shelf puts the swell's current three Ekman depths down

A coast sends the drift back, or sends it along put a straight coast at the edge of a shelf sea crossed by a swell. The swell carries water shorewards in its Stokes drift, the water cannot pile up against the coast for ever, and the sea answers with a surface slope. How depends on one ratio: the depth over the Ekman depth, the thickness of the layer in which the floor’s friction and the Earth’s rotation balance. In water shallower than an Ekman depth the slope drives an undertow along the floor, as in a channel that does not rotate. A few Ekman depths down, rotation turns the returning flow into a current along the shore, and the surface can even slope down towards the land.

Every number in that essay was for a flat shelf, one depth at a time. A real shelf rises from nothing at the beach to a hundred metres or more at its edge, passing through every regime it drew. It asked what the sea does across such a 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 current along the shore concentrates where the depth is two or three Ekman depths — a coastal jet set by the Ekman depth rather than by the wind. Both happen, and the second is the stronger result.

Each station is its own flat shelf #

The shelf here is 50 kilometres wide, its depth rising linearly from the beach at one in five hundred to 100 metres at its edge. The water has an eddy viscosity of 0.01 square metres a second at 55°N, an Ekman depth of 12.9 metres, so the shelf runs from 7.7 Ekman depths at its edge to a sixth of one in the shallowest water followed. An 8-second swell arrives at the edge with an amplitude of half a metre, travelling 20° to the left of straight shoreward.

Nothing varies along the shore, so the water’s net cross-shore transport must be zero at every distance from the beach, not just at the coast, and if the shelf is wide compared with the Ekman depth the current at each distance is decided by the vertical friction there. Each station is then the previous essay’s flat shelf at its own depth, with its own surface slope, and the surface across the shelf is the integral of those slopes. The swell changes on the way in, and is followed: its wavelength from the dispersion relation at each depth, its direction by Snell’s law, and its height by conserving the shoreward flux of its energy. The calculation stops at two metres, short of where the swell would break; the surf zone, with its own physics, is not in it.

The surface across the shelf #

Most of the shelf barely moves. The set-up the undertow needs grows as the inverse cube of the depth, the non-rotating channel’s $3\nu M_S/H^3$, so it is concentrated in the last few kilometres: the surface is 53.7 millimetres above the shelf edge’s a kilometre from the beach, in two metres of water, and almost all of that was gained inside five kilometres.

The outer shelf does what the flat-shelf figures suggested. Between about 33 and 11 kilometres out the surface dips below the edge’s, by at most 0.071 millimetres, at 16.6 kilometres where the depth is 33 metres, 2.6 Ekman depths. The surface therefore does rise towards the coast and dip offshore of that, as the previous essay’s question supposed. But the dip is a thousandth of the set-up. It is a real feature of the balance and no tide gauge would see it.

The current along the shore #

The current along the shore is where the shelf does something the flat shelf could only hint at. Its depth-averaged Eulerian part — the current a moored meter would record — is a jet, running to the right looking shoreward, fastest at 37.6 metres of depth, 2.9 Ekman depths, at 1.42 millimetres a second. In water shallower than about one Ekman depth it reverses and runs weakly the other way, at about half a millimetre a second.

The jet does not need the swell to be oblique. A swell arriving head-on, with no alongshore component of its own, makes one too, fastest at 2.6 Ekman depths and 0.95 millimetres a second. The reason is the rotation. Where the shelf is deeper than an Ekman depth the transport the sea must send back to keep the cross-shore balance is turned by the Coriolis force, and its turned part runs along the shore; the oblique swell’s own alongshore transport adds to it. The current’s direction is the Ekman direction, to the right in the northern hemisphere, whatever side the swell comes from.

The speeds are small for a half-metre swell — the Stokes transport goes as the square of the wave height, so a two-metre swell drives sixteen times as much — but the structure is the result: a coastal jet whose position is set by the friction’s length, not by the coastline or the wind.

Where the jet sits #

That claim can be tested by changing the friction. As the eddy viscosity is varied from 0.002 to 0.05 square metres a second, the Ekman depth goes from 5.8 to 28.9 metres and the jet moves from 24 to 69 metres of water. In Ekman depths it moves much less, from 4.2 to 2.4: the jet stays within a factor of two of three Ekman depths while its absolute depth triples. On a real shelf, where the eddy viscosity changes with the weather, the swell’s jet moves across the shelf with it.

The shore set-up moves in proportion to the eddy viscosity — 16.6, 53.7 and 160 millimetres at 0.003, 0.01 and 0.03 — because it is the channel’s balance, and the channel’s slope is the friction’s.

What floats along the shore #

The jet is a current. What carries a floating object, a patch of oil or a larva is the water’s own motion, the Lagrangian velocity: the current plus the swell’s Stokes drift, whose alongshore part is the oblique swell’s own. Over the outer shelf the two nearly cancel, because the sea’s answer to the swell’s transport is mostly the current that removes it. Near the beach the drift wins outright. Divided by a small depth, the swell’s alongshore transport gives a depth-averaged drift of 24 millimetres a second at two metres, against a current of half a millimetre: 24.4 in all, almost all of it the waves.

So the two things one might call “the longshore current” in this model are in different places and go different ways. The water drifts along the shore with the swell, fastest at the beach, carried by the waves. The current that the sea itself makes runs the other way near the beach and has its jet twenty kilometres out. A drifter released near the shore measures the first; a current meter on a mooring at the jet’s depth measures the second.

Two releases, a few kilometres apart #

The difference between the two velocities shows up in what a released object does. A patch of floating material released five kilometres from the beach, in ten metres of water, moves along the shore with the water’s depth-averaged velocity of 3.1 millimetres a second — about a quarter of a kilometre a day — in the swell’s direction, and almost all of that is the swell’s own drift; the current there adds a tenth. The same patch released nineteen kilometres out, in the jet, moves the other way at 0.58 millimetres a second, about 350 metres a week, because there the current the sea makes is larger than the drift it answers. Two releases fourteen kilometres apart on the same shelf, under the same swell, travel in opposite directions.

Both speeds are small against the tide and the wind, which is why this circulation is hard to see. But the tide reverses twice a day and its mean over many cycles is small, while the swell’s circulation is steady for as long as the swell lasts; over a week of persistent swell the water in the jet moves a third of a kilometre one way and the water five kilometres from the beach nearly two kilometres the other. For anything that drifts for a season — larvae, plastic, a slick — the residual is what matters, and the residual has this structure.

The slope, station by station #

The surface’s shape comes from one curve: each station’s slope as a fraction of the channel’s. In water shallower than an Ekman depth the fraction is one, and the shelf is a channel. Deeper, rotation turns part of the return flow into the alongshore current and less slope is needed; beyond about two Ekman depths the fraction changes sign, and the surface slopes the other way. The dip in the surface is the integral of that negative stretch, and it is small because the channel’s slope it is a fraction of has itself fallen by the cube of the depth.

The swell on its way in #

The swell is not the same wave across the shelf. Its amplitude falls slightly over the outer shelf, where the group speed rises in intermediate depths, then rises as the water shoals, to 0.6 metres at two metres of depth. Refraction turns it from 20° at the edge to 6.8° near the beach. Both matter to the transport the sea answers: the shoreward Stokes transport grows as the swell steepens, and its alongshore part shrinks as the swell turns. The steepening is what concentrates the set-up so sharply at the shore.

Checks across the shelf #

The swell’s shoreward energy flux and Snell’s invariant are the same at all 161 stations to rounding. At every station the depth-integrated balance — Coriolis force on the Lagrangian transport, pressure gradient and floor stress — holds to $10^{-18}$, as the flat-shelf calculation guaranteed it would. Quadrupling the number of stations moves the set-up at the beach by 0.09 per cent, the error of integrating a slope that varies as the inverse cube of the depth. And the set-up is proportional to the eddy viscosity, as the channel’s balance says it must be. The tests also refuse a flat floor, a swell travelling away from the shore and a shelf whose edge is shallower than its beach.

What a station-by-station shelf leaves out #

Cross-shore friction and advection. Each station is solved as if its neighbours were the same depth. That needs the shelf to be wide against the Ekman depth, which it is here by a factor of four thousand, and it neglects the momentum the current carries across the shelf; a steep shelf, or a jet narrow against its own width, would need both.

The surf zone. The calculation stops before the waves break. Inside the breakers the radiation stress of breaking waves drives a longshore current of decimetres a second, far larger than anything here, and the set-up there is of a different origin.

A uniform eddy viscosity. Real shelf turbulence increases away from the floor and is stirred by the wind and the tide. The Ekman depth is then not one number, and the jet’s position becomes a statement about an effective one.

Steady, uniform swell. The swell is one period from one direction, steady long enough for the Ekman layer to form, which takes a pendulum day.

What an instrument on the shelf would record #

The structure also says where to look. A current meter moored in thirty or forty metres of water would record the swell’s jet as a steady alongshore flow of a millimetre or two a second, to the right of shoreward, rising and falling with the swell’s energy over days — tiny, but coherent with the swell and not with the tide, which is how such a signal is extracted from a record dominated by everything else. A drifter released inside ten metres would go the other way, with the swell. And a pressure gauge on the beach would see the set-up of a few centimetres rising with the square of the swell height; the dip offshore is below any gauge’s resolution and would have to be inferred. Each instrument answers a different question, and the essay’s figures say which: Eulerian at the jet, Lagrangian at the beach, elevation at the shore. The lesson that a drift made of two things that average to zero drew for a single wave — that the drift is not a current and a current meter does not see it — is here a matter of where on the shelf each lives.

Where this sits among the drifts #

This is the eighth calculation of one quantity, the Stokes drift, and the place to see how the pieces fit. The drift in a wave that has none found that a wave whose water oscillates still carries it forward, at second order. The drift a closed box will not allow found that in a basin with ends the sea must send that transport back, and the drift a rotating planet takes back that on a rotating Earth with no floor the Coriolis force on the drift cancels it — Ursell’s paradox. The floor that gives the drift back showed that a shelf’s friction undoes Ursell’s cancellation in proportion to how shallow the shelf is, and the coast essay bounded it with a coastline. The shelf here puts all of those depths side by side: Ursell’s regime at its edge, the floor’s regime in its shallows, and the coast’s constraint everywhere.

The same structure — a drift, a return flow it forces, and a rotation that turns the return — reappears at the surface of a wind-blown sea in the drift that turns a current into rolls, where the drift’s interaction with the current’s shear makes Langmuir cells, and in a river bend in a river drifts right, and only a reach can show it, where the same Coriolis turning is a few per cent of a much larger flow.

The convention: Eulerian and Lagrangian, and to the right #

The alongshore current is positive to the left looking shoreward; the jet is negative, to the right, which is the Ekman direction in the northern hemisphere. Eulerian means the velocity at a fixed point averaged over the waves, what a current meter measures; Lagrangian means the average velocity of the water itself, which adds the Stokes drift and is what a drifter measures. Depths are given in Ekman depths, $\sqrt{2\nu/f}$, 12.9 metres here. The set-up is measured from the surface at the shelf edge.

Longuet-Higgins, Ekman, and the shelf #

Longuet-Higgins derived the mass transport of waves over a viscous floor in 1953 and, with Stewart, the radiation stress that drives currents inside the surf zone in the 1960s. Hasselmann in 1970 and Xu and Bowen in 1994 added rotation to the wave-driven mean flow over a shelf, which is where the turning of the return flow into an alongshore current first appears. The station-by-station shelf, the dip’s size against the set-up, and the jet’s position held at about three Ekman depths while the friction changes, are what this calculation adds.

Still open: the jet against a wind #

The shelf here has a swell and no wind. A wind blowing along the shore drives its own Ekman transport and its own coastal jet, typically centimetres a second — ten times the swell’s — and its direction depends on which way the wind blows. The next calculation adds a steady wind stress at the surface, along and across the shore, and asks when the swell’s jet three Ekman depths down survives as a distinct feature beside the wind’s, and whether a swell arriving under an opposing wind can reverse the near-shore current — which would say whether the swell-driven circulation is ever what a mooring on a real shelf records.