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 we have The entrainment from bottom boundary layer The entrainment from surface boundary layer We have where barotropic

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

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

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 In dimensional terms, The Sverdrup relation is broken down., the length of the layer δ <<LThe non-dimensionalized distance is

9. The Stommel model Bottom Ekman friction becomes important in WBL., ε S <<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). In the boundary layer, let ( ), we have Interior solution Re-scaling:

10. The solution foris,.  A=-B ξ→∞, ( can be the interior solution under different winds) 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,, C 1 =C 2 =0..

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

17. Scaling The cross-stream distance from boundary to maximum velocity is Given 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 ), A H =4×10 6 cm 2 /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>>ε I and ε M, Given a boundary layer exists in the west where Re-scaling with Conservation of potential vorticity., we have or

19. The conservation equation may be integrated to yield where is an arbitrary function of 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 Take into equation

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

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 y 0 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 δ I /δ M 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