2. 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

3. Quasi-Geostrophic Approximation
Quasi-geostrophic approximation has three components
(1) The β-plane apporximation
(2) Small surface deviation
(3) Geostrophic approximation in terms of fo
Basic condition

4. Quasi-Geostrophic Approximation
Potential Vorticity
Quasi-Geostrophic Potential Vorticity
Quasi-Geostrophic Potential Vorticity Equation
Defines the evolution of geostrophic stream function ψ

5. Quasi-Geostrophic Approximation
If we ignore the surface change (or have a rigid lid), we have the absolute vorticity conservation, i.e.,

6. What does QG momentum equation look like?
Continuity equation
is ageostrophic flow
is responsible for the divergence in the QG system
has a rotational component
is totally determined by geostrophic flow at any given instance

7. Quasi-Geostrophic Approximation
Replace the relative vorticity by its geostrophic value
Approximate the horizontal velocity by geostrophic current in the advection terms
Under -plane approximation, f=fo+y, we have

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

9. Boundary Value Problem
Boundary conditions on a solid boundary L
(1) No penetration through the wall  
(2) No slip at the wall

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. Different terms are important at different places of the basin
f-plane
-plane

12. 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.,

13. Note that U has not been decided yet.

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

17. 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)

18. 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.

19. 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

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

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

22. 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 an eastern boundary layer?