Wind Driven Circulation III

1. Wind Driven Circulation III • Closed Gyre Circulation • Quasi-Geostrophic Vorticity Equation • Westward intensification • Stommel Model • Munk Model • Inertia boundary layer • Numerical results • Observations

2. Consider the balance on an f-plane

3. If f is not constant, then

4. Assume geostrophic balance on β-plane approximation, i.e., (β is a constant) Vertically integrating the vorticity equation barotropic we have The entrainment from bottom boundary layer The entrainment from surface boundary layer We have where

5. Quasi-geostrophic vorticity equation For , we have and and where (Ekman transport is negligible) Moreover, We have where

6. Non-dmensional equation Non-dimensionalize all the dependent and independent variables in the quasi-geostrophic equation as where For example, The non-dmensional equation where , nonlinearity. , , , bottom friction. , lateral friction. ,

7. Interior (Sverdrup) solution If ε<<1, εS<<1, and εM<<1, we have the interior (Sverdrup) equation: (satistfying eastern boundary condition)  (satistfying western boundary condition) Example: Let , . Over a rectangular basin (x=0,1; y=0,1)

8. Westward Intensification It is apparent that the Sverdrup balance can not satisfy the mass conservation and vorticity balance for a closed basin. Therefore, it is expected that there exists a “boundary layer” where other terms in the quasi-geostrophic vorticity is important. This layer is located near the western boundary of the basin. Within the western boundary layer (WBL), , for mass balance The non-dimensionalized distance is , the length of the layer δ <<L In dimensional terms, The Sverdrup relation is broken down.

9. The Stommel model Bottom Ekman friction becomes important in WBL. , εS<<1. at x=0, 1; y=0, 1. free-slip boundary condition (Since the horizontal friction is neglected, the no-slip condition can not be enforced. No-normal flow condition is used). Interior solution Re-scaling: In the boundary layer, let ( ), we have

10. The solution for is , .  A=-B ξ→∞, ( can be the interior solution under different winds) For , , . For , , .

12. The dynamical balance in the Stommel model In the interior,   Vorticity input by wind stress curl is balanced by a change in the planetary vorticity f of a fluid column.(In the northern hemisphere, clockwise wind stress curl induces equatorward flow).  In WBL,    , Since v>0 and is maximum at the western boundary, the bottom friction damps out the clockwise vorticity. Question: Does this mechanism work in a eastern boundary layer?

13. Munk model Lateral friction becomes important in WBL. Within the boundary layer, let , we have , Wind stress curl is the same as in the interior, becomes negligible in the boundary layer. For the lowest order, . If we let , we have . And for , . . The general solution is Since , C1=C2=0.

14. Total solution Using the no-slip boundary condition at x=0, (K is a constant).  .to , Considering mass conservation K=0  Western boundary current

17. Scaling Given The cross-stream distance from boundary to maximum velocity is The ratio between the nonlinear and dominant viscous terms is where The continuity relation is also used: Using U=O(2 cm/s), ß=O(10-13 cm-1 s-1), AH=4×106 cm2/s, we have R=4. i.e., the nonlinear terms neglected are larger than the retained viscous terms, which causes an internal inconsistency within the frictional boundary layer.

18. Inertial Boundary Layer If ε>>εI and εM, Given a boundary layer exists in the west where Re-scaling with , we have Conservation of potential vorticity. or

19. The conservation equation may be integrated to yield is an arbitrary function of where This equation states that the total vorticity is constant following a specific streamline.

20. Let (interior stream function plus a boundary layer correction), must satisfy Now consider the region of large ξ, where into equation Take

21. Retain only linear term in (and neglect some other small terms), we have Integrate once and use the boundary condition , we have

22. If , will be oscillatory and not satisfy the boundary condition. A necessary condition for the existence of a pure inertial boundary current is The decaying solution is of the form

23. The dimensional width of the inertial boundary layer is At those y’s where U is on shore and small, the width of the inertial current is small. As the point y0 is approached where U=0, δ will shrink and finally be swallowed up within the thickness of a frictional layer.

24. Since equation is symmetric under transformation A similar inertial boundary layer can exist at the eastern boundary.

26. Inertial Currents with Small Friction In the presence of a small lateral friction, we can derive the perturbation equation as which makes the boundary layer possible only in the western ocean. Moreover, it can be shown that a inertial-vicious boundary layer can be generated in the northern part of the basin where characterized by a standing Rossby wave.

27. Assume the simple balance A parcel coming into the boundary layer has The effect of friction is reduced and the boundary layer is broadened.

29. Bryan (1963) integrates the vorticity equation with nonlinear term and lateral friction. The Reynolds number is define as And δI/δM ranges from 0.56 for Re=5 to 1.29 for Re=60.

30. Veronis (1966), nonlinear Stommel Model

32. Western Boundary current: Gulf Stream

34. Gulf Stream Transport