1. For most of the basin
Question
What causes the strong narrow currents on the west side of the ocean basin?
The westward Intensification

2. Stommel’s Model Rectangular ocean of constant depth
Surface stress is zonal and varies with latitude only
Steady ocean state
Simple friction term as a drag to current
Vorticity balance: Sverdrup balance +friction
Flow patterns in this ocean for three conditions:
non-rotating ocean (f=0)
f-plane approximation (f=constant)
-plane approximation (f=fo+y)

3. f-plane
-plane

4. Negative vorticity generation  Positive vorticity generation
Wind stress () + friction () =0
Negative vorticity generation
 Positive vorticity generation

5. Westerly winds in north, easterly winds in south
Ekman effect drives the water to the center,
Increase sea level generates anticyclonic geostrophic currents
Internal friction (or bottom Ekman layer) generate downslope cross-isobaric flow, which balance the wind-driven Ekman transport

6. The  effect Generate positive vorticity 
Generate negative vorticity 
Generate positive vorticity 

7. In the west, water flows northward
Wind stress () + Planetary vorticity () + Friction () = 0
In the east, water flows southward
Wind stress () + Planetary vorticity () + Friction () = 0
Friction (W) > Friction (E)

8. Non-rotation Ocean, f=0

10. Quasi-geostrophic vorticity equation
where
Boundary conditions on a solid boundary L
(1) No penetration through the wall (used for the case of no horizontal diffusion)
along the boundary L
(2) No slip at the wall
along the boundary L
n is the unit vector perpendicular to the boundary L

11. Non-dimensionalize Quasi-Geostrophic Vorticity Equation
Define non-dimensional variables based on independent scales L and o
The variables with primes, as well as their derivatives, have no unit and generally have magnitude in the order of 1. e.g.,

12. Note that U has not been decided yet.

15. Non-dmensional vorticity equation
If we choose
we have
Sverdrup relation
Define the following non-dimensional parameters
, nonlinearity. ,
, bottom friction. ,
, lateral friction. ,

16. 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)

17. 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  <<L
In dimensional terms,
The Sverdrup relation is broken down.

18. 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

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

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

21. 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?