1. Define
and
we have
• At the sea surface (z=0), the surface current flows at 45o to the right of the wind direction
Depends on constant Az =>
• Current decreases exponentially with depth. At the same time, its direction changes clockwise with depth (The Ekman spiral).
• DE (≈100 m in mid-latitude) is regarded as the depth of the Ekman layer. DE is not the mixed layer depth (hm). The latter also depends on past history, surface heat flux (heat balance) and the stability of the underlying water. In reality, DE < hm because hm can be affected by strong wind burst of short period. ,
• At DE, the current magnitude is 4% of the surface current and its direction is opposite to that of the surface current. .

2. Other properties
(1) Relationship between surface wind speed W and (Vo, DE).
Wind stress magnitude (
,  ) ,
(2) Relationship between W and DE.
Ekman’s empirical formula between W and Vo.
, outside ±10o latitude
(3) There is large uncertainty in CD (1.3 to 1.5 x 10-3 ±20% for wind speed up to about 15 m/s). CD itself is actually a function of W.
(4)
has an error range of 2-5%.

3. More comments
(1) DE is not the mixed layer depth (hm). The latter also depends on past history, surface heat flux (heat balance) and the stability of the underlying water. In reality, DE < hm because hm can be affected by strong wind burst of short period.
(2) Az = const and steady state assumptions are questionable.
(3) Lack of data to test the theory. (The Ekman spiral has been observed in laboratory but difficult to observe in fields).
(4) Vertically integrated Ekman transport does not strongly depend on the specific form of Az.

4. Progressive vector diagram, using daily averaged currents relative to the flow at 48 m, at a subset of depths from a moored ADCP at 37.1°N, 127.6°W in the California Current, deployed as part of the Eastern Boundary Currents experiment. Daily averaged wind vectors are plotted at midnight UT along the 8-m relative to 48-m displacement curve. Wind velocity scale is shown at bottom left. (Chereskin, T. K., 1995: Evidence for an Ekman balance in the California Current. J. Geophys. Res., 100, )

7. Surface Drifter Current Measurements
a platform designed to move with the ocean current

9. Ekman Transport
Starting from a more general form of the Ekman equation
(without assuming AZ or even a specific form for vertical turbulent flux)
Integrating from surface z= to z=-2DE (e-2DE=0.002), we have
where
and
are the zonal and meridional mass transports by the
by the Ekman current. Since
, we have 

10. Ekman transport is to the right of the direction of the surface winds

12. Ekman pumping Integrating the continuity equation through the layer:
Where and are volume transports.
Assume and let , we have
is transport into or out of the bottom of the Ekman layer to the ocean’s interior (Ekman pumping).
, upwelling
, downwelling
Water pumped into the Ekman layer by the surface wind induced upwelling is from meters, which is colder and reduces SST.

13. Upwelling/downwelling are generated by curls of wind stress

14. Coastal and equatorial upwelling
Coastal upwelling: Along the eastern coasts of the Pacific and Atlantic Oceans the Trade Winds blow nearly parallel to the coast towards the Doldrums. The Ekman transport is therefore directed offshore, forcing water up from below (usually from m depth).
Equatorial Upwelling: In the Pacific and Atlantic Oceans the Doldrums are located at 5°N, so the southern hemisphere Trade Winds are present on either side of the equator. The Ekman layer transport is directed to the south in the southern hemisphere, to the north in the northern hemisphere. This causes a surface divergence at the equator and forces water to upwell (from about  m).

15. An example of coastal upwelling
Note how all contours rise towards the surface as the coast is approached; they rise steeply in the last 200 km. On the shelf the water is colder, less saline and richer in nutrients as a result of upwelling.
Water property sections in a coastal upwelling region, indicating upward water movement within about 200 km from the coast. (This particular example comes from the Benguela Current upwelling region, off the coast of Namibia.) The coast is on the right, outside the graphs; the edge of the shelf can just be seen rising to about 200 m depth at the right of each graph.

16. Cold SST associated with the coastal and equatorial upwelling

18. Ekman layer at the bottom of the sea
For convenience, assume the bottom of the sea is flat and located at z=0, the governing equation and its general solution are the same as the surface case.
Boundary conditions
Z=0 (bottom of the sea) or
As z-(into the interior) or

19. General solution:
If z, VE0, i.e., A=0
If z=0, VE=-Vg=B
We have
Let

20. Let

21. Solution
For z0,

22. The direction of the total currents
where
The near bottom the total current is 45o to the left of the geostrophic current.

24. Ekman Transport

25. Transport at the top of the bottom Ekman layer
Assume
, the solution can be written as
Using the continuity equation

26. We have
Since

27. Ekman pumping at the bottom.
Given the integral
i.e., 
The vertical velocity at the top of the bottom boundary layer
Ekman pumping at the bottom.