1. Basic dynamics The equation of motion Scale Analysis
Boussinesq approximation
Geostrophic balance
(Reading: Pond and Pickard, Chapters 6-8)

2. The Equation of Motion
Newton’s second law in a rotating frame. (Navier-Stokes equation)
Force per unit mass
: Acceleration relative to axis fixed to the earth.
1sidereal day =86164s
1solar day = 86400s
: Pressure gradient force.
: Coriolis force, where
: Effective (apparent) gravity.
g0=9.80m/s2
: Friction molecular kinematic viscosity.

3. Gravitation and gravity

4. Gravity: Equal Potential Surfaces
g changes about 5%
9.78m/s2 at the equator
(centrifugal acceleration 0.034m/s2,
radius 22 km longer)
9.83m/s2 at the poles)
• equal potential surface
normal to the gravitational vector
constant potential energy
the largest departure of the mean sea surface from the “level” surface is about 2m (slope 10-5)
The mean ocean surface is not flat and smooth
earth is not homogeneous

6. Coriolis Force

7. In Cartesian Coordinates:
where

8. Accounting for the turbulence and averaging within T:

9. Given the zonal momentum equation
If we assume the turbulent perturbation of density is small
i.e.,
The mean zonal momentum equation is
Where Fx is the turbulent (eddy) dissipation
If the turbulent flow is incompressible, i.e.,

10. Eddy Dissipation Then Ax=Ay~102-105 m2/s >> Az ~10-4-10-2 m2/s
Reynolds stress tensor and eddy viscosity: ,
Then
Where the turbulent viscosity coefficients are anisotropic.
Ax=Ay~ m2/s
Az ~ m2/s
>>

11. Reynolds stress has no symmetry:
A more general definition:   if
(incompressible)

12. Scaling of the equation of motion
Consider mid-latitude (≈45o) open ocean
away from strong current and below sea surface.
The basic scales and constants:
L=1000 km = 106 m
H=103 m
U= 0.1 m/s
T=106 s (~ 10 days)
2sin45o=2cos45o≈2x7.3x10-5x0.71=10-4s-1
g≈10 m/s2
≈103 kg/m3
Ax=Ay=105 m2/s
Az=10-1 m2/s
Derived scale from the continuity equation
W=UH/L=10-4 m/s
Incompressible  

13. Scaling the vertical component of the equation of motion 
Hydrostatic Equation
accuracy 1 part in 106

14. Boussinesq approximation
Density variations can be neglected for its effect on mass but not on weight (or buoyancy).
Assume that
where
, we have 
neglected

15. Geostrophic balance in ocean’s interior

16. Scaling of the horizontal components
(accuracy, 1% ~ 1‰)
Zero order (Geostrophic) balance
Pressure gradient force = Coriolis force

17. Re-scaling the vertical momentum equation
Since the density and pressure perturbation is not negligible in the vertical momentum equation, i.e.,
, and ,
The vertical pressure gradient force becomes

18. Taking into the vertical momentum equation, we have
If we scale ,
and assume
then 
and
(accuracy ~ 1‰)