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 vorticity balance of an homogeneous fluid (=constant) on an f-plane

3. If f is not constant, then

5. Assume geostrophic balance on -plane approximation, i.e.,
( 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

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

7. Posing the gyre problem
Boundary conditions on a solid boundary L
(1) No penetration through the wall  
(2) No slip at the wall

9. Non-dimensional Equation, An Example
Consider a homogeneous fluid on a -plane
Define the following non-dimensional variables:
(geostrophy)
By definition 

10. Taking into the equations, we have

11. Define the non-dimensional parameters
Rossby Number
Horizontal Ekman Number
Ekman depth
Vertical Ekman Number
Then, we have (with prime dropped)
The solution

12. In the interior of the ocean, Eh<<1 and Ez<<1
(geostrophy)
Near the bottom or surface, Ez≈O(1) 
In the surface and bottom boundary layers, the vertical scales are redefined
(shortened, a general character of a boundary layer)

13. Non-dmensional vorticity 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.

14. Interior (Sverdrup) solution
If <<1, S<<1, and M<<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)

15. 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.

16. The Stommel model Bottom Ekman friction becomes important in WBL.
at x=0, 1; y=0, 1. No-normal flow boundary condition
(Since the horizontal friction is neglected, the no-slip condition can not be enforced. No-normal flow condition is used).
Interior solution

17. , we have Let Re-scaling in the boundary layer: Take into
As =0, =0. As ,I

18. can be the interior solution under different winds)
The solution for is ,
.  A=-B
, (
can be the interior solution under different winds)
For , , .
For , , .

20. 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?