1. Schedule for final exam
Last day of class for CLIM712 (Dec. 6, 06)
Exam period: Dec., 12 - Dec., 19
Two choices:
a) close-book exam (12/11 or 12/13)
b) take home

2. Consider the vorticity balance of an homogeneous fluid (ρ=constant) 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
and
, we have
and
where
(Ekman transport is negligible)
Moreover,
We have
where

6. Quasi-Geostrophic Vorticity Equation
Boundary conditions on a solid boundary L
(1) No penetration through the wall  
(2) No slip at the wall

7. 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.

8. 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)

9. 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.

10. The Stommel model Bottom Ekman friction becomes important in WBL.
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

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

13. 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?