Ekman spiral

The Ekman spiral is an arrangement of ocean currents: the directions of horizontal current appear to twist as the depth changes.

[1] The oceanic wind driven Ekman spiral is the result of a force balance created by a shear stress force, Coriolis force and the water drag.

This force balance gives a resulting current of the water different from the winds.

In the ocean, there are two places where the Ekman spiral can be observed.

This phenomenon was first observed at the surface by the Norwegian oceanographer Fridtjof Nansen during his Fram expedition.

He noticed that icebergs did not drift in the same direction as the wind.

His student, the Swedish oceanographer Vagn Walfrid Ekman, was the first person to physically explain this process.

[2] In order to derive the properties of an Ekman spiral a look is taken at a uniform, horizontal geostrophic interior flow in a homogeneous fluid.

Another result of this property is that the horizontal gradients will equal zero.

In this case, the Navier-Stokes momentum equations, governing geophysical motion can now be reduced to:[3] Where

These parameters have a small variance on the scale of an Ekman spiral, thus this approximation will hold.

, in the equations above, the following is obtained: Using the last of the three equations at the top of this section, yields that the pressure is independent of depth.

will suffice as a solution to the differential equations above.

has the following possible outcomes: Because of the no-slip condition at the bottom and the constant interior flow for

Note that the velocity vector will approach the values of the interior flow, when the

is defined as the thickness of the Ekman layer.

A number of important properties of the Ekman spiral will follow from this solution: The solution for the flow forming the bottom Ekman spiral was a result of the shear stress exerted on the flow by the bottom.

Logically, wherever shear stress can be exerted on a flow, Ekman spirals will form.

Again, the flow is uniform, has a geostrophic interior and is homogeneous fluid.

The equations of motion for a geostrophic flow, which are the same as stated in the bottom spiral section, can be reduced to:[3] The boundary conditions for this case are as follows: With these conditions, the solution can be determined:[3] Some differences with respect to the bottom Ekman spiral emerge.

Whereas in the case of the bottom Ekman spiral, the deviation is determined by the interior flow.

The wind-driven component of the flow is inversely proportional with respect to the Ekman-layer thickness

At last, the flow at the surface is 45 degrees to the right on the northern hemisphere and 45 degrees to the left on the southern hemisphere with respect to the wind-direction.

In case of the bottom Ekman spiral, this is the other way around.

The equations and assumptions above are not representative for the actual observations of the Ekman spiral.

The differences between the theory and the observations are that the angle is between 5–20 degrees instead of the 45 degrees as expected[4] and that the Ekman layer depth and thus the Ekman spiral is less deep than expected.

There are three main factors which contribute to the reason why this is, stratification,[5] turbulence and horizontal gradients.

[3] Other less important factors which play a role in this are the Stokes drift,[6] waves and the Stokes-Coriolis force.