1. Vorticity Measure of angular momentum for a fluid
Tendency of a parcel to rotate
Two components of vorticity
relative (angular momentum in rotating frame)
planetary (rotation of the frame)
Important for understanding western boundary currents

2. Relative Vorticity Positive Negative
Relative vorticity, z, is driven by shears in the flow field

3. Relative Vorticity Positive Negative Negative Positive
anti-cyclonic cyclonic

4. The Sign of Vorticity
Negative Positive
anti-cyclonic cyclonic

5. Relative Vorticity Positive Negative
North
y or v direction
Relative Vorticity
Positive
Negative
East
x or u direction
Relative vorticity is defined as z = Dv/Dx - Du/Dy

6. Example of Relative Vorticity
Northward velocity increases as a function of x distance 34oN)
Relative vorticity is positive
North
y or v direction
10 cm/s
East
x or u direction
500 km

7. Relative Vorticity
Relative vorticity is defined as z = Dv/Dx - Du/Dy = Dv/Dx
Change in Dv is 0.1 m/s for Dx = 500 km
Relative vorticity (z) = Dv/Dx = (0.1 m / s) / (500x103 m) = 2x10-7 s-1

8. Another Example
Eastward velocity decreases as a function of y (north) distance
10 cm/s
North
y or v direction
500 km
East
x or u direction

9. Relative Vorticity
Relative vorticity is defined as z = Dv/Dx - Du/Dy = - Du/Dy
Change in Du is 0.1 m/s for Dy = 500 km
Relative vorticity (z) = - Du/Dy = (- 0.1 m / s) / (500x103 m) = 2x10-7 s-1

10. Relative Vorticity + + Relative vorticity, z = Dv/Dx - Du/Dy
Dv/Dx > 0 -> z > 0
Du/Dy < 0 -> z > 0
cyclonic vorticity

11. Relative Vorticity - - Relative vorticity, z = Dv/Dx - Du/Dy
Dv/Dx < 0 -> z < 0
Du/Dy > 0 -> z < 0
anti-cyclonic vorticity

12. Planetary Vorticity The planet also rotates about its axis
Objects are affected by both planetary & relative vorticity components
Planetary vorticity = 2 W sin f (= f)
2 north pole
0 on equator
- 2 south pole

13. Example for Planetary Vorticity
Planetary vorticity = 2 W sin f (= f)
At 34oN, f = 2 W sin 34o = 8.2x10-5 s-1
Previous examples -> z = 2x10-7 s-1
Ratio of |z| / f = (2x10-7 s-1)/(8.2x10-5 s-1) =
Relative vorticity is small compared with f except near equator (Rossby number)

14. Total Vorticity Only the total vorticity (f + z) is significant
For flat bottom ocean with uniform r & no friction, total vorticity (f + z) is conserved
Coffee cup example…
Water transported north will decrease its z to compensate for changes in f
Water advected south will increase its z

15. Potential Vorticity
Potential vorticity = (f + z) / D

16. Potential Vorticity Potential vorticity = (f + z) / D
PV is conserved except for friction
If f increases, a water mass can spin slower (reduce z) or increase its thickness
Typically, PV is approximated as f/D (z << f)
Used to map water mass distributions & assess topographic steering

17. Potential Vorticity
WOCE
Salinity
P16
150oW

18. Potential Vorticity
WOCE PV
P16
150oW
PV~f/D

19. Potential Vorticity
PV on sq = 25.2

20. Topographic Steering Potential vorticity = (f + z) / D ~ f / D
Uniform zonal flow over a ridge
Let D decreases from to 2000 m
If PV = constant, f must decrease by 2, leading to a equatorward deflection of current
This is topographic steering U D

21. Topographic Steering
Plan view (NH)

22. Topographic Steering A factor of two reduction in f
For 30oN, f = 7.29x10-5 s-1
f/2 = 3.6x10-5 s-1 which corresponds to a latitude of 14.5oN
Displacement = ( o)*(111 km/olat) = 1700 km
Water column is really stratified which reduces the changes of D & thereby f

23. Topographic Steering
Bascially f/H

24. Vorticity Measure of the tendency of a parcel to rotate
Relative (= z rotation viewed from Earth frame)
Planetary (= f rotation of the frame)
Total (z + f) & potential vorticity (z + f) / D are relevant dynamically
Important for diagnosing water mass transport & western intensificaiton...

25. Western Intensification
Subtropical gyres are asymmetric & have intense WBC’s
Western intensification is created by the conservation of angular momentum in gyre
Friction driven boundary current is formed along the western sidewall
Maintains the total vorticity of a circulating water parcel

26. Wind Driven Gyres

27. Wind Driven Gyres
Symmetric gyre

28. Wind Torque in Gyres
Need process to balance the constant addition of negative wind torque
Curl of the wind stress…

29. Stommel’s Experiments
Model of steady subtropical gyre
Includes rotation and horizontal friction
f = constant
f = 2W sinf

30. Stommel’s Experiments
Stommel showed combination of horizontal friction & changes in Coriolis parameter lead to a WBC
Need to incorporate both ideas into an explanation of western intensification

31. Western Intensification
Imagine a parcel circuiting a subtropical gyre
As a parcel moves, it gains negative vorticity (wind stress curl)
Gyre cannot keep gaining vorticity or it will spin faster and faster
Need process to counteract the input of negative vorticity from wind stress curl

32. Western Intensification
Conservation of potential vorticity (f + z)/D
Assume depth D is constant (barotropic ocean)
Friction (i.e., wind stress curl) can alter (f + z)
In the absence of friction
Southward parcels gain z to compensate reduction in f
Northward parcels lose z to compensate increase in f

33. Symmetric Gyre

34. Western Intensification
Friction plays a role due to
wind stress curl (input of -z)
sidewall friction (input of +z) + +
WBC EBC

35. Western Intensification
In a symmetric gyre,
Southward: wind stress input of -z is balanced +z inputs by D’s in latitude & sidewall friction
Northward: D’s in latitude result in an input of - z along with the wind stress input of -z
This is NOT balanced by + z by sidewall friction
Need an asymmetric gyre to increase sidewall friction in the northward flow!!

36. Symmetric Gyre

37. Western Intensification
In a symmetric gyre,
Southward: wind stress input of -z is balanced +z inputs by D’s in latitude & sidewall friction
Northward: D’s in latitude result in an input of - z along with the wind stress input of -z
This is NOT balanced by + z by sidewall friction
Need an asymmetric gyre to increase sidewall friction in the northward flow!!

38. Potential Vorticity

39. Western Intensification
In a asymmetric gyre,
Southward: wind stress input of -z is balanced +z inputs by D’s in latitude & sidewall friction
Northward: D’s in latitude result in an input of -z along with the wind stress input of -z
This IS balanced by LARGE +z from sidewall friction
Total vorticity balance is satisfied & we have an asymetric gyre

40. Potential Vorticity

41. Role of Wind Stress Curl
Spatial D’s in wind stress control where Ekman transports converge
Where changes in tw = 0, the convergence of Ekman transports = 0
This sets the boundaries of gyres
My = 1/(Df/Dy) curl tw = (1/b) curl tw
-> Sverdrup dynamics

42. Munk’s Solution

43. Currents

44. Western Intensification
Intense WBC’s create a source of positive vorticity that maintains total vorticity balance
Creates asymmetric gyres & WBC’s
Boundary currents are like boundary layers
Wind stress curl & D’s in Coriolis parameter with latitude are critical elements
Can be extended to quantitatively predict water mass transport (Sverdrup theory)