1. Wind Driven Circulation I: Planetary boundary Layer near the sea surface

2. Monin-Obukhoff Similarity Theory
Surface layer (several tens of meters above surface, 10-15% of the planetary boundary layer) in nearly steady condition
Vertical turbulent flux is nearly constant
horizontal homogeneity (the scale of vertical variation is much smaller than horizontal)
The turbulent mixing length l=z
=0.4±0.01 is von Karman constant
Momentum flux
u* is frictional velocity
() is a universal function of
L is the Monin-Obukhoff length
At altitudes below L, shear production of turbulent kinetic energy dominates over buoyant production of turbulence.

3. In neutral condition, ()=1
von Karman logarithmic law of wall
The surface momentum flux is
zo is aerodynamic roughness length
If we choose wind measurement at a certain height, e.g., 10m above the sea surface, the bulk formula is
is 10m neutral drag coefficient

4. Surface wind stress
Approaching sea surface, the geostrophic balance is broken, even for large scales.
The major reason is the influences of the winds blowing over the sea surface, which causes the transfer of momentum (and energy) into the ocean through turbulent processes.
The surface momentum flux into ocean is called the surface wind stress ( ), which is the tangential force (in the direction of the wind) exerting on the ocean per unit area (Unit: Newton per square meter)
The wind stress effect can be constructed as a boundary condition to the equation of motion as

5. Wind stress Calculation
Direct measurement of wind stress is difficult.
Wind stress is mostly derived from meteorological observations near the sea surface using the bulk formula with empirical parameters.
The bulk formula for wind stress has the form
Where is air density (about 1.2 kg/m3 at mid-latitudes), V (m/s), the wind speed at 10 meters above the sea surface, Cd, the empirical determined drag coefficient

6. Drag Coefficient Cd
Cd is dimensionless, ranging from to (A median value is about ). Its magnitude mainly depends on local wind stress and local stability.
• Cd Dependence on stability (air-sea temperature difference).
More important for light wind situation
For mid-latitude, the stability effect is usually small but in tropical and subtropical regions, it should be included.
• Cd Dependence on wind speed.

7. Cd dependence on wind speed in neutral condition
Large uncertainty between estimates
(especially in low
wind speed).
Lack data in high wind

8. Observations of the drag coefficient as a function of wind speed U10 ten meters above the sea. The solid line is from the formula proposed by Yelland and Taylor (1996). The dotted line is 1000CD= U10 proposed by Smith (1980) and the dashed line follows from Charnock(1955). Triangles are values measured by Powell, Vickery, and Reinhold (2003). From Stetart (2008).
The more recently published formula by Yelland and Taylor (1996) for neutrally stable boundary layer:
(3 ≤ U10 ≤ 6 m/s)
(6 ≤ U10 ≤ 26 m/s)

9. Annual Mean surface wind stress
Unit: N/m2, from Surface Marine Data (NODC)

10. December-January-February mean wind stress
Unit: N/m2, from Surface Marine Data (NODC)

11. December-January-February mean wind stress
Unit: N/m2, from Scatterometer data from ERS1 and 2

12. June-July-August mean wind stress
Unit: N/m2, from Surface Marine Data (NODC)

13. June-July-August mean wind stress
Unit: N/m2, Scatterometers from ERS1 and 2

14. Force per unit mass

15. The primitive equation
(1)
(2)
(3)
(4)
Since the turbulent momentum transports are , ,
etc
We can also write the momentum equations in more general forms
At the sea surface (z=0),
turbulent transport is wind stress. ,

16. Assumption for the Ekman layer near the surface
Az=const
Steady state (steady wind forcing for long time)
Small Rossby number
Large vertical Ekman Number
Homogeneous water (=const)
f-plane (f=const)
no lateral boundaries (1-d problem)
infinitely deep water below the sea surface

17. Ekman layer Near the surface, there is a three-way force balance
Coriolis force+vertical dissipation+pressure gradient force=0
Geostrophic current
Take
and let
Ageostrophic (Ekman) current
then
(note that VE is not small in comparison to Vg in this region)

18. The Ekman problem Boundary conditions At z=0, As z- ,. , . Let
(1)
(2)
Boundary conditions
At z=0,
As z- ,.
, .
(3)
(5)
(6)
(4)
Let
(complex variable), take (1) + i(2), we have
Since
(7)

19. Group equations (7), (8), and (9) together, we have
Take (3) + i (4), we have
(3)
z=0,
(4)
Define
(8)
(5)
Take (5) + i (6), we have
As z-,
(6)
(9)
Group equations (7), (8), and (9) together, we have
(7)
At z=0,
(8)
As z-
(9)

20. Assume the solution for (7) has the following form
Take into
We have
If f > 0,
If f < 0,
In above derivations, we have used the following equality:
For f > 0, the general solution of (7) can be written as

21. As z- At z=0, (8) Therefore, B=0 because grow exponentially as z-
Then
and
At z=0,
(9)
then
and
The final solution to (7), (8), (9) is

22. Set , where Also note that Given We have Current Speed:
and
Also note that
We have
Current Speed:
(=0, eastward)
Phase (direction):