Vorticity

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. 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 W @ north pole 0 on equator - 2 W @ 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) = 0.0025 • 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. D U Topographic Steering • Potential vorticity = (f + z) / D ~ f / D • Uniform zonal flow over a ridge • Let D decreases from 4000 to 2000 m • If PV = constant, f must decrease by 2, leading to a equatorward deflection of current • This is topographic steering

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 = (30-14.5o)*(111 km/olat) = 1700 km • Water column is really stratified which reduces the changes of D & thereby f

23. Topographic Steering Basically 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 intensification...

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)