Wind, friction and the Ekman spiral
When wind blows over the sea, friction drags the top layer. Because Earth spins, moving water is turned. In the northern half it turns to the right; in the southern half to the left.
The top layer moves about 45 degrees from the wind direction. It drags the layer below, which is turned a little more and moves slower. Layer after layer, the arrows form a shrinking spiral: the Ekman spiral. The motion fades out at about 100 metres. This top zone is the Ekman layer.
Ekman transport and upwelling
Add up the movement of all layers in the Ekman layer. The net water movement is 90 degrees to the right of the wind in the north (90 degrees to the left in the south). This is Ekman transport.
The amount of water moved per metre of width is M = τ / (ρ f). Here τ is the wind stress (the drag of wind per square metre), ρ is the density of sea water (about 1025 kg/m³) and f is the Coriolis number (about 0.0001 per second in mid-latitudes).
Coastal upwelling: if the wind blows along a coast in the north with the coast on its left, Ekman transport moves surface water away from the coast. Cold water from below rises to fill the gap. This water is full of nutrients, so fishing is rich there.
The water hill and gyres
Across a whole ocean, the trade winds blow westward near the tropics and the westerlies blow eastward further north. Both give Ekman transport towards the middle of the basin. Water piles up there in a gentle hill about one metre high, spread over thousands of kilometres.
The hill gives a pressure force that pushes water down its slope. As the water moves, the Coriolis effect turns it to the right until the two forces balance. The water then flows along the contours of the hill, not down it. This is a geostrophic current. In the northern half it goes clockwise round the hill, forming a gyre. In the southern half gyres go anticlockwise.
It is the same idea as geostrophic wind, with a water hill in place of a high pressure zone.
Speed of a geostrophic current
The current speed is v = (g / f) × slope, where slope is the height difference divided by the distance, and g = 9.8 m/s². A steeper hill gives a faster current. Gyres are not even: the water flows faster and narrower on the west side of each ocean, as in the Gulf Stream and the Kuroshio. These are western boundary currents.
Try it: blow on a tray of water
Fill a shallow tray with water and sprinkle a few tiny paper bits. Blow gently along the tray with a straw. See how the water piles at the far end and the bits move with the wind. In a real ocean Earth's spin adds a turn. Now stand a bowl on a slowly turning stool and stir in one direction: notice the water slides to the outside. Guess what happens to the surface level at the wall and in the middle, then look.
Key formulas and definitions
- Surface current: about 45° right of the wind (north)
- Net Ekman transport: 90° right of the wind (north)
- Ekman transport per metre: M = τ / (ρ f)
- Geostrophic current: v = (g / f) × (Δh / d)
Worked examples
1. Wind stress is 0.1 N/m². Find the Ekman transport per metre of width. (ρ = 1025 kg/m³, f = 1 × 10⁻⁴ s⁻¹)
M = τ / (ρ f) = 0.1 / (1025 × 0.0001) = 0.1 / 0.1025 ≈ 0.98 m²/s. That is about 1 cubic metre of water per second for each metre of coast.
2. A sea-surface hill is 1 m high over 1000 km. Find the geostrophic current. (f = 1 × 10⁻⁴ s⁻¹, g = 9.8 m/s²)
Slope = 1 / 1 000 000 = 1 × 10⁻⁶. v = (9.8 / 0.0001) × 10⁻⁶ = 98 000 × 10⁻⁶ ≈ 0.098 m/s, about 10 cm/s.
3. In the northern half, a wind blows towards the east. In which direction does the net water movement go?
Turn 90° right of east: towards the south.
4. In the northern half, wind blows towards the south. Which way is the Ekman transport, and what happens if the coast lies to the west?
Right of south is west. If the coast is to the west, water is pushed towards the coast. If the coast is to the east, water leaves the coast and deeper water rises (upwelling). Always turn right of the wind direction and check whether it points to or away from the coast.
Common mistakes
- Thinking the water moves in the same direction as the wind. The net movement is 90° to the side.
- Saying the surface moves 90° from the wind. The surface is about 45°; the 90° is the net of all layers.
- Forgetting the southern half turns the other way (left).
- Thinking the current flows down the water hill. It flows along the hill, balanced by Coriolis.