1. Review of conservation equations State, Mass and Momentum

2. Equation of State (EOS-80)
Determines water density from T, S, and P
A through N are polynomials
T is temperature in oC
S salinity
P pressure in bars
K secant bulk modulus – measure of compresibility

3. Check values: A B C D T0 999.842594 8.24493E-1 -5.72466E-3 4.8314E-4F G
7.944E-2
1.6483E-2
E-2
E-4
E-2
E-5
E-5 H I J
2.2838E-3
E-4
E-3
E-5
E-4
E-6
E-7 M N
E-5
E-7
E-6
2.0816E-8
5.2787E-8
9.1697E-10
Check values:

4. 90 % of Ocean Water Greater influence of salinity on density
Mean T & S for World Ocean
Greater influence of salinity on density

5. Effects of Temperature and Salinity on Density
Thermal Expansion
Saline Contraction
x 10-4 oC-1
x 10-4 S-1
Density changes by 0.2 kg/m3 for a T change of 1oC,
and by 0.8 kg/m3 for a S change of 1.

6. Flux of mass out (kg/s) =
, w
, v
Mass per area per time
(kg/(m2 s))
Flux of mass in (kg/s) =
Flux of mass out (kg/s) =
Net Flux of mass in ‘x’ =
Net Flux of mass in ‘y’ =
Net Flux of mass in ‘z’ =

7. The change of mass per unit time going through the volume element is:
And the change of mass per unit time per unit volume is:
which is the same as: or

8. Boussinesq approximation
This is the Continuity Equation or Equation of Conservation of Mass
How valid is the Boussinesq approximation in the OCEAN?
How would you determine that?
1 sigma-t throughout one day
= 1 / (24*3600.) = 1.1510-5
But

9. z x
Continuity Equation in Bulk Form:

10. Conservation of Salt:
Conservation of Heat:
Equation of State:

11. z Sb S0 x
Continuity Equation in Bulk Form:
Salt Conservation Equation in Bulk Form: VbSb = V0S0

12. Conservation of Momentum (Equations of Motion)

13. Pressure gradient + friction + tides + gravity + Coriolis
Pressure gradient: Barotropic and Baroclinic
Friction: Surface, bottom, internal
Tides: Boundary condition
Gravity: Only in the vertical
Coriolis: Only in the horizontal
REMEMBER, these are FORCES PER UNIT MASS

14. Pressure gradient + friction + tides + gravity + Coriolis
Pressure gradient: Barotropic and Baroclinic
Friction: Surface, bottom, internal
Tides: Boundary condition
Gravity: Only in the vertical
Coriolis: Only in the horizontal
REMEMBER, these are FORCES PER UNIT MASS

15. Hydrostatic Pressure
Total Pressure
Pressure gradient force per unit mass
Barometric Barotropic Baroclinic
Note that even if the density is constant with depth, the horizontal pressure gradient increases with depth if there is a horizontal density gradient

16. Pressure gradient + friction + tides + gravity + Coriolis
Pressure gradient: Barotropic and Baroclinic
Friction: Surface, bottom, internal
Tides: Boundary condition
Gravity: Only in the vertical
Coriolis: Only in the horizontal
REMEMBER, these are FORCES PER UNIT MASS

17. Friction
@ surface:
@ bottom:
@ interior:

18. Pressure gradient + friction + tides + gravity + Coriolis
Pressure gradient: Barotropic and Baroclinic
Friction: Surface, bottom, internal
Tides: Boundary condition
Gravity: Only in the vertical
Coriolis: Only in the horizontal
REMEMBER, these are FORCES PER UNIT MASS

19. Gravity
[0, 0, g] = [0, 0, 9.81]
Coriolis
[-fv, fu, 0]