1. Class Meeting Nov. 26, 2:00pm-4:45pm
Research I, 162

2. Geostrophic Approximations
Hydrostatic primitive equations
Geostrophic approximation (diagnostic)
In between (approximately geostrophic models)
prognostic
inertial-gravity waves filtered
Planetary geostrophic approximation
Quasi-geostrophic approximation ……

3. Planetary Geostrophic Approximation
Momentum equation
where
Continuity Equation
Vorticity Equation
where
Potential Vorticity Equation
Basic condition
L must be large, L~R, basin-scale movement

4. 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

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

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

7. Quasi-Geostrophic Approximation
Momentum equation
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

9. 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

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

11. 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

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

14. 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

15. 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.,

16. Note that U has not been decided yet.

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

20. 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)

21. 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.

22. 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

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

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

25. 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?