Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Ekman layer — where it appears

The thin viscous layer on a boundary of a rotating fluid, in which friction and the Coriolis force balance. It pumps fluid into and out of the interior, which spins a container up in the geometric mean of the rotation and diffusion times rather than the diffusion time.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

The current at the surface is 45° from the wind, and nothing sets that angle. The Ekman spiral drawn as a hodograph: each point is the velocity at one depth, and depth runs along the curve. At the surface the flow is at exactly 45 degrees to the wind that drives it — not approximately, exactly, and independently of the wind, the viscosity and the latitude. By one Ekman depth the flow has turned another radian and lost 1/e of its speed; by three it is a hundredth of the surface value and pointing back the way it came. The angle is a property of the equation having two terms in it, and nothing else.

The layer that stops at a depth

Every other boundary layer grows. This one does not — rotation supplies a frequency, the balance against diffusion supplies a length, and the transport that comes out contains the stress on the surface and not the viscosity underneath it.

viscous · Rotating How long a fluid takes to forget it was not rotating. The fraction of solid-body rotation a container's interior has reached, against time, by the two available routes. The Ekman layers on the end walls pump fluid radially and carry angular momentum inwards in a hundred seconds; diffusion alone would need ten thousand.

How long a fluid takes to forget it was not rotating

Spin a container of water and the fluid inside reaches solid-body rotation in a hundred seconds rather than the three hours diffusion would need. The shortcut is the thin layers on the end walls, and the advantage they give is exactly the reciprocal of the square root of the Ekman number.

viscous · Rotating A float under swell on a rotating planet goes round instead of away. The track of a float at the surface over 1 inertial periods (16.9 hours) after a 8 s swell of amplitude 1 m arrives at latitude 45°, in kilometres, the waves travelling to the right. Without friction the float runs round a circle of radius Uₛ/f = 0.479 km and comes back to where it started every 16.92 hours. With a drag on the Eulerian current it spirals out into a steady drift veered to the right: 24.3 per cent of the drift at 76.0° for a drag of 0.25 f; 70.7 per cent of the drift at 45.0° for a drag of 1 f; 94.9 per cent of the drift at 18.4° for a drag of 3 f. In a non-rotating ocean the same float would have gone 3.0 km straight on.

The drift a rotating planet takes back

In a wave tank the Stokes drift is cancelled by a return current because the tank has walls. The open ocean has none, and the drift is cancelled anyway: the Coriolis force acts on the water's real motion, drives an Eulerian current that answers it, and leaves the depth-integrated transport exactly zero at every viscosity. A float under steady swell with nothing to stop it goes round a circle instead of away.

kinematics · Stokes drift The kept transport spirals into nothing as the sea deepens. The net Lagrangian transport as a vector, scaled on the Stokes transport, traced as the water depth increases from a quarter of an Ekman depth to eight, for an 8-second swell with an eddy viscosity of 0.01 m²/s. Shallow water keeps the whole transport pointing with the waves, at the right-hand end. As the sea deepens the vector shortens and swings to the right, crosses the across-wave axis near two Ekman depths, and winds into the origin, which is the open ocean's exact cancellation.

The floor that gives the drift back

In the open ocean the Coriolis force drives a current that cancels a swell's Stokes transport exactly. Over a continental shelf the sea floor holds a stress, and whatever it holds is transport the rotation does not take back. How much survives depends almost only on the depth in Ekman depths; which way it points depends on the wave.

kinematics · Stokes drift At a coast the swell's water goes back, or goes along. The Lagrangian mean current — the Stokes drift plus the Eulerian current — under an 8-second swell arriving straight at a coast, in seas 8, 25 and 60 metres deep, each as a fraction of its own surface drift against depth as a fraction of the water's. Left, the part across the shore, positive onshore; right, the part along it. The cross-shore parts carry no net water. At 8 metres, inside the Ekman layer, the whole exchange is across the shore; at 25 the return is shared with a current along the coast; at 60 the open shelf's own rotation has already taken the transport back, and the coast has little left to do.

A coast sends the drift back, or sends it along

A shelf sea under a swell keeps part of the Stokes drift's transport, turned to the right by the rotating Earth, and near a coast some of it points at the land. The sea answers with a slope in its surface. In water shallower than an Ekman depth the slope drives the water straight back as an undertow, as it would without rotation; a few Ekman depths down the coast turns the transport into a current along the shore instead, and there the sea can slope down towards the land it faces.

kinematics · Stokes drift The current along the shore runs fastest three Ekman depths down. The depth-averaged Eulerian current along the shore, positive to the left looking shoreward, against the local depth in Ekman depths, for a swell arriving 20° off the shore-normal at the shelf edge and for one arriving head-on. Both make a jet to the right, fastest at 2.9 Ekman depths for the oblique swell (-1.42 mm/s at 37.6 m) and 2.6 for the head-on one (-0.948 mm/s), and a weak current the other way in water shallower than one Ekman depth. Rotation turns the transport the shelf returns, so even a swell with no alongshore component drives one.

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

A swell's Stokes transport arrives at a coast and the sea must send it back or turn it aside. On a flat shelf one depth decided which. On a real shelf, deepening from the beach to its edge, every depth is present at once, and the sea does both: it sets up a few centimetres against the beach, dips a tenth of a millimetre where the depth passes two Ekman depths, and runs a current along the shore that is fastest about three Ekman depths down — wherever the friction puts it.

kinematics · Stokes drift

Named alongside it

The objects these essays reach for when they reach for this one.

TransportCoriolisEddy viscosityStokes driftBoundary conditionMass conservationRotating frameRotationBoundary layerEulerian and LagrangianFree surfaceModel limit

All concepts