1. Ekman Flow
September 27, 2006

2. Remember from last time…

3. Define c (current speed) as:
u and v change, but c stays constant: Coriolis force does no work!

4. Flow is in a circle: Inertial or Centripetal force = Coriolis force
where
Inertial radius If
c ~ 0.1 m/s
f ~ 10-4
then
r ~ 1 km

5. Inertial Period is given by T where:If
f ~ 10-4
then
T ~ 6.28x104sec ~ 17.4 hrs

6. r fc c
c2/r

7. Inertial Oscillations
Latitude (j)
Ti (hr)
D (km)
for V = 20 cm/s
90°
11.97
2.7
35°
20.87
4.8
10°
68.93
15.8
Table 9.1 in Stewart
Note: V is equivalent to c from previous slides, D is equal to the diameter

8. Inertial currents in the North Pacific in October 1987
Figure 9.1 in Stewart

9. Ekman flow
Fridtjof Nansen noticed that wind tended to blow ice at an angle of 20°-40° to the right of the wind in the Artic
Nansen hired Ekman (Bjerknes graduate student) to study the influence of the Earth’s rotation on wind-driven currents
Ekman presented the results in his thesis and later expanded the study to include the influence of continents and differences of density of water (Ekman, 1905)

10. Balances of forces acting on an iceberg on a rotating earth
Figure 9.2 in Stewart

11. So again…

12. So again…

13. Balance in the surface boundary layer is between vertical friction and Coriolis – all other terms are neglected

14. At the surface (z=0)
friction
c.f.
45°

15. If we assume the wind is blowing in the x-direction only, we can show:

16. Wind Stress Frictional force acting on the surface skin
ρa: density of air
cD: Drag coefficient – depends on atmospheric conditions, may depend on wind speed itself
W: usually measured at “standard Anemometer height” ~ 10m above the sea surface

17. Ekman Depth (thickness of Ekman layer)
For Mid-latitudes:
Av = 10
ρ = 103
f = 10-4
Plug these into the D equation and:
meters

18. Figure 9.3 in Stewart

19. Typical Ekman Depths Table 9.3 in Stewart U10(m/s) Latitude 15° 45° 520
180m
110m
Table 9.3 in Stewart

20. At z=0 y x v u
45°

21. At z=-D
y,v v
x,u u -π
-π/4

22. Ekman Number
The depth of the Ekman layer is closely related to the depth at which frictional force is equal to the Coriolis force in the momentum equation
The ratio of the forces is known as Ekman depth
Solving for d:

23. Ekman Transport
Transport in the Ekman Layer is 90° to the right of the wind stress in Northern Hemisphere

24. Ekman Pumping

25. By definition, the vertical velocity at the sea surface w(0)=0, and the vertical velocity at the base of the layer wE (-D) is due to divergence of the Ekman flow:
horizontal divergence operator
vector mass transport due to Ekman flow

26. If we use the Ekman mass transports in we can relate Ekman pumping to the wind stress.

27. ME Ek
pile up
of water
anticyclonic wE
Hi P
Lo P

28. Hydrostatic Equilibrium
Typical Scales:
L » 106m
f» 10-4s-1
U» 10-1m/s
g» 10 m/s2
H» 103m
r» 103kg/m3
with these we can “scale” the equations of motion

30. The momentum equation for vertical velocity is therefore:
and the only important balance in the vertical is hydrostatic:
Correct to 1:106

31. The momentum equation for horizontal velocity in the x direction is
The Coriolis balances the pressure gradient, known as the geostrophic balance