Define

• Current decreases exponentially with depth. At the same time, its direction changes clockwise with depth (The Ekman spiral).

we have

,

.

and

• At the sea surface (z=0), the surface current flows at 45o to the right of the wind direction

• At DE, the current magnitude is 4% of the surface current and its direction is opposite to that of the surface current.

• 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.

Depen

ds o

n co

nsta

nt A z

=>

( , , )

(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.

(1) Relationship between surface wind speed W and (Vo, DE).

Wind stress magnitude

has an error range of 2-5%.(4)

Other properties

(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.

More comments

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, 12727-12748.)

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

Ekman Transport

Integrating from surface z= to z=-2DE (e-2DE=0.002), we have

by the Ekman current. Since , we have

Starting from a more general form of the Ekman equation

(without assuming AZ or even a specific form for vertical turbulent flux)

where and are the zonal and meridional mass transports by the

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

Ekman pumpingIntegrating the continuity equation

is transport into or out of the bottom of the Ekman layer to the ocean’s interior (Ekman pumping).

through the layer:

Where and are volume transports. Assume and let , we have

, upwelling

Water pumped into the Ekman layer by the surface wind induced upwelling is from 200-300 meters, which is colder and reduces SST.

, downwelling

Upwelling/downwelling are generated by curls of wind stress

Coastal and equatorial upwellingCoastal 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 200 - 400 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 150 - 200 m).

An example of coastal 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.

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.

Cold SST associated

with the coastal and equatorial upwelling

Ekman layer at the bottom of the seaFor 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 conditionsZ=0 (bottom of the sea)

As z-(into the interior)

or

or

General solution:

If z, VE0, i.e., A=0

If z=0, VE=-Vg=B

Let

We have

Let

Solution

For z0,

The direction of the total currents

where

The near bottom the total current is 45o to the left of the geostrophic current.

Ekman Transport

Transport at the top of the bottom Ekman layer

Assume , the solution can be written as

Using the continuity equation

We have

Since

Given the integral

i.e.,

The vertical velocity at the top of the bottom boundary layer

Ekman pumping at the bottom.