1. The β-spiral Determining absolute velocity from density field
Assumptions:
1) Geostrophic
2) incompressible
3) steady state
2) + 3) 

2. Use the thermal wind equation with Boussinesq approximation
Take into
Write u and v to polar format as
In the northern hemisphere,
if w > 0, the current rotates to the right as we go upward (or to the left as we go downward) 

3. Take geostrophic equation
If v≠0, w changes with z and can not be zero everywhere. Thus the β effect makes the rotation of the geostrophic flow with depth likely, hence the term “β spiral”
use

5. Consider an isopycnal surface at
If we go along this surface in the x-direction 
Similarly
Take into  

6. Rewrite the thermal wind relation 
Suppose we have derived u’ and v’ based on some reference level
If the observations should be error free, two levels would be sufficient
Considering the observation errors, particularly noise from time-dependent motions, this equation will not be exactly satisfied. Computationally, a least-square technique is used.

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

8. 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

9. 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

10. 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.

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

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

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

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

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
Small Rossby number
Large vertical Ekman Number
Homogeneous water (ρ=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 three-way force balance
Coriolis force+vertical dissipation+pressure gradient force=0
Take
and let
( , ageostrophic (Ekman) current, note that is not small in comparison to in this region)
then

18. The Ekman problem Boundary conditions At z=0, As z→-∞, ,. , . Let
, .
Let
(complex variable)
At z=0,
As z→-∞,

19. The solution Assuming f > 0, the general solution is
Using the boundary conditions, we have
Set
, where
and
note that