Ekman Transport - Mathematical Derivation

Mathematical Derivation

Some assumptions of the fluid dynamics involved in the process must be made in order to simplify the process to a point where it is solvable. The assumptions made by Ekman were:

• no boundaries;
• infinitely deep water;
• eddy viscosity, is constant (this is now known not to be totally true);
• the wind forcing is steady and has been blowing for a long time;
• barotropic conditions with no geostrophic flow;
• the Coriolis parameter, is kept constant.

The simplified equations for the Coriolis force in the x and y directions follow from these assumptions:

(1)

(2)

where is the wind stress, is the density, is the East-West velocity, and is the north-south velocity.

Integrating each equation over the entire Ekman layer:

where

Here and represent the zonal and meridional mass transport terms with units of mass per unit time per unit length. Contrarily to common logic, north-south winds cause mass transport in the East-West direction.

In order to understand the vertical velocity structure of the water column, equations 1 and 2 can be rewritten in terms of the vertical eddy viscosity term.

where is the vertical eddy viscosity coefficient.

This gives a set of differential equations of the form

In order to solve this system of two differential equations, two boundary conditions can be applied:

• as
• friction is equal to wind stress at the free surface .

Things can be further simplified by considering wind blowing in the y-direction only. This means is the results will be relative to a north-south wind (although these solutions could be produced relative to wind in any other direction):

(3)

where

• and represent Ekman transport in the u and v direction;
• in equation 3 the plus sign applies to the northern hemisphere and the minus sign to the southern hemisphere;
• is the wind stress on the sea surface;
• is the Ekman depth (depth of Ekman layer).

By solving this at z=0, the surface current is found to be (as expected) 45 degrees to the right (left) of the wind in the Northern (Southern) Hemisphere. This also gives the expected shape of the Ekman spiral, both in magnitude and direction. Integrating these equations over the Ekman layer shows that the net Ekman transport term is 90 degrees to the right (left) of the wind in the Northern (Southern) Hemisphere.