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Physics of the Ekman Spiral
The effect is a consequence of the Coriolis effect which subjects moving objects to a force to the right of their direction of motion in the northern hemisphere (and to the left in the Southern Hemisphere). Thus, when a persistent wind blows over an extended area of the ocean surface in the northern hemisphere, it causes a surface current which accelerates in that direction, which then experiences a Coriolis force and acceleration to the right of the wind: the current will turn gradually to the right as it gains speed. As the flow is now somewhat right of the wind, the Coriolis force perpendicular to the flow's motion is now partly directed against the wind. Eventually, the current will reach a top speed when the force of the wind, of the Coriolis effect, and the resistant drag of the subsurface water balance, and the current will flow at a constant speed and direction as long as the wind persists. This surface current drags on the water layer below it, applying a force in its own direction of motion to that layer, repeating the process whereby that layer eventually becomes a steady current even further to the right of the wind, and so on for deeper layers of water, resulting in a continuous rotation (or spiraling) of current direction with changing depth. As depth increases, the force transmitted from the driving wind declines and thus the speed of the resultant steady current decreases, hence the tapered spiral representation in the accompanying diagram. The depth to which the Ekman spiral penetrates is determined by how far turbulent mixing can penetrate over the course of a pendulum day.
The diagram above shows the the Ekman spiral in the Northern hemisphere. The force from above a layer is in red (beginning with the wind blowing over the water surface), the ultimate Coriolis force is in dark yellow, and the net resultant water movement is in pink, which then becomes the force from above for the layer below it, accounting for the gradual clockwise spiral motion as you move down.