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Earth Science 1905

On the Influence of the Earth's Rotation on Ocean-currents

Vagn Walfrid Ekman

Ekman explained why wind-driven water — and the drift ice on it — slides off at an angle to the wind, not along it: rotation bends each layer, and the whole column spirals.

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In depth · the introduction

An Arctic explorer noticed his ice-locked ship drifting not with the wind but stubbornly off to its side — and a young physicist worked out why the whole ocean does the same thing.

The idea, unpacked

When wind blows across the sea, it doesn't simply shove the water along. The water moves, and on a spinning planet anything that moves gets nudged sideways — to the right in the Northern Hemisphere — by the Coriolis effect. So the thin skin of water at the very top slides off at an angle to the wind, about 45°. That top layer then drags the layer just below it, which also gets bent to the right, and so on down: each layer points a bit more to the side and moves a bit more slowly than the one above. Stack those arrows up and they trace a spiral going down into the dark.

Add up the motion of the entire spiral and something tidy falls out: the water as a whole is carried at a right angle to the wind — fully 90° to the side. Ekman proved that this sideways total doesn't depend on the messy details of friction at all. It is the wind and the planet's spin, and almost nothing else.

Where it came from

In the 1890s the Norwegian explorer Fridtjof Nansen deliberately froze his ship, the Fram, into the Arctic pack ice to drift across the polar sea. He noticed that the ice did not creep along with the wind but always veered 20° to 40° to the right of it. Nansen suspected the Earth's rotation, but he was no theorist, so he brought the puzzle to the physicist Vilhelm Bjerknes.

Bjerknes handed it to a gifted student, Vagn Walfrid Ekman, who in 1902, still in his twenties, sketched the answer and then laid it out in full in this 1905 memoir. From the bare equations of a rotating fluid he produced the spiral and the right-angle transport — a clean piece of theory that explained a real observation from the top of the world, and quietly founded the physics of how winds drive the sea.

Why it mattered

This is the hidden engine behind some of the ocean's most important behaviour. Because wind pushes surface water sideways, a wind blowing along a coastline can drag the surface layer out to sea — and cold, dark, nutrient-loaded water wells up from below to replace it. These upwelling zones, off Peru, California, and West Africa, are tiny in area but feed a huge share of the world's fish. The same sideways push organises the great ocean gyres and helps set the stage for El Niño. Ekman's spiral turned ‘the wind blows and the sea responds' from a vague picture into something you can calculate.

An analogy

Imagine a tall stack of loose playing cards lying flat. Slide your hand across the top card: it moves, and it nudges the card beneath it, which nudges the next — but suppose each card can only slide off slightly turned from the one above. Push the top card north and it goes north-east; the card below it goes more east; the next more south-east; and so on, each one weaker, until far down the cards barely twitch. Seen from above, the directions sweep around in a spiral. Now, the whole deck's net drift? Straight east — at a right angle to your hand. Play with the compass below: the wind points up, and as you sink down the spiral the current swings round and dies away.

An interactive compass: a grey wind arrow points up, a violet surface-current arrow leans 45° right, a green arrow shows net transport 90° to the right, and a blue arrow shows the current at a chosen depth on a drawn spiral. A Depth slider turns the blue arrow clockwise and shrinks it; a button switches hemisphere.

Where it sits

This belongs to the Library's Earth-science thread, the part of physics that plays out on a spinning planet. The sideways nudge it relies on is the Coriolis effect that also curves the winds and steers storms; the same rotating-fluid mathematics shapes the atmosphere just above the waves. Ekman's transport became the foundation that Harald Sverdrup (1947), and later Stommel and Munk, built on to explain the Gulf Stream and the ocean's great circulation — the slow machinery that, alongside work like Milankovitch's orbital cycles in this Library, helps govern the planet's climate.

The original document
Original source text

The problem — Nansen's drifting ice

V. W. Ekman · On the Influence of the Earth's Rotation on Ocean-currents · Arkiv f. Mat., Astr. och Fysik 2 (1905), 1–53
During the Fram expedition Fridtjof Nansen had noticed that the wind-driven drift of the pack ice did not follow the wind, but ran consistently 20°–40° to the right of it, and he guessed the Earth's rotation was the cause. He put the problem to Vilhelm Bjerknes, who handed it to his student Ekman. The memoir sets out to derive, from the equations of motion, exactly what a steady wind does to the water beneath it on a rotating planet.
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The balance — friction vs. Coriolis

Ekman writes the steady balance in each thin horizontal layer: the wind's pull is communicated downward by internal friction (an eddy viscosity), while the Coriolis force, proportional to the velocity and the Earth's rotation, deflects every moving parcel sideways. Solving the resulting equations, the velocity turns and weakens with depth in a fixed, exponential way.
In the idealized deep, steady, uniform case the surface current flows at 45° to the right of the wind (Northern Hemisphere); each deeper layer points a little further right and moves a little slower, the tips of the vectors tracing a logarithmic spiral. At the depth he calls the depth of frictional resistance, the current has reversed direction and faded to a small fraction of the surface speed.
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The consequence — transport at a right angle

Adding up the flow through the whole spiralling layer, the total volume of water transported is directed at 90° to the right of the wind — independent of the (poorly known) friction. This is the result later called Ekman transport. It accounts for Nansen's sideways ice, and implies that wind blowing along a coast will pile water against it or pull it away, forcing water up from below — the upwelling that feeds the richest fisheries.
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Vagn Walfrid Ekman · Sweden, 1905