## Ekman Motion

In the 1890s. the Norwegian scientist and explorer Fridtjof Nansen led an expedition across the Arctic ice. His specially designed vessel, the Fram, was allowed to freeze into the ice and drift with it for over a year. During this period, Nansen observed that ice movements in response to wind were not parallel to the wind, but at an angle of 20-40° to the right of it. Ekman developed his theory of wind-driven currents in order to explain this observation.

Ekman considered a steady wind blowing over an ocean that was infinitely deep, infinitely wide, and with no variations in density. He also assumed that the surface of the ocean remained horizontal, so that at any given depth the pressure due to the overlying water was constant. This hypothetical ocean may be considered to consist of an infinite number of horizontal layers, of which the topmost is subjected to friction by the wind (wind stress) at its upper surface and to friction (eddy viscosity) with the next layer down at its lower surface; this second layer is acted upon at its upper surface by friction with the top layer, and by friction with layer three at its lower surface; and so on. In addition, because they are moving in relation to the Earth, all layers are affected by the Coriolis force.

By considering the balance of forces - friction and the Coriolis force - on the infinite number of layers making up the water column, Ekman deduced that the speed of the wind-driven current decreases exponentially with depth. He further found that in his hypothetical ocean the direction of the current deviates 45° cum sole from the wind direction at the surface, and that the angle of deviation increases with increasing depth. The current vectors therefore form a spiral pattern (Figure 3.6(a)) and this theoretical current pattern is now known as the Ekman spiral. The layer of ocean under the influence of the wind is often referred to as the Ekman layer.

We do not need to go into the calculations that Ekman used to deduce the spiral current pattern, but we can consider the forces involved in a little more detail. As discussed in Chapters 1 and 2, the Coriolis force acts 90° cum sole of the direction of current flow (i.e. to the right of the flow in the wind wind stress

Figure 3.6 (a) The Ekman spiral current pattern which under ideal conditions would result from the action of wind on surface waters. The lengths and directions of the dark blue arrows represent the speed and direction of the wind-driven current.

(b) For the Ekman layer as a whole, the force resulting from the wind is balanced by the Coriolis force, which in the Northern Hemisphere is 90° to the right of the average motion of the layer (broad blue arrow).

wind

Figure 3.6 (a) The Ekman spiral current pattern which under ideal conditions would result from the action of wind on surface waters. The lengths and directions of the dark blue arrows represent the speed and direction of the wind-driven current.

(b) For the Ekman layer as a whole, the force resulting from the wind is balanced by the Coriolis force, which in the Northern Hemisphere is 90° to the right of the average motion of the layer (broad blue arrow). Ekman layer depth ol —► frictional influence, 0

Ekman layer depth ol —► frictional influence, 0

wind stress

Coriolis farce average motion of the Ekman layer

Northern Hemisphere, to the left in the Southern), and increases with latitude. More specifically, the Coriolis force is proportional to the sine of the latitude, and for a particle of mass m moving with speed u it is given by:

where Q (capital omega) is the angular velocity of the Earth about its axis and <|> (phi) is the latitude. The term 2Q, sin <j> is known as the Coriolis parameter and is often abbreviated to/, so that the expression given above becomes simply:

Coriolis force = mfu (3.2b)

Thus, the Coriolis force - the force experienced by a moving parcel of water by virtue of it being on the rotating Earth - is at right angles to the direction of motion and increases with the speed of the parcel. This is important for the balances of forces that determine motion in the ocean -for example, the balance between frictional forces and the Coriolis force for each of the infinite number of layers of water making up the Ekman spiral.

Ekman deduced that in a homogeneous infinite ocean the speed of the surface current, no, is given by

where x is the surface wind stress, Az is the coefficient of eddy viscosity for vertical mixing, p (rho) is the density of seawater and/is the Coriolis parameter.