1. Water, salt, and heat budget
 Conservation laws
application to a limited volume
 Surface fresh water flux:
evaporation, precipitation, and river runoff
 Surface heat flux components:
sensible, latent, long and shortwave
 Ocean meridional transport
Pond and Pickard: Chapter 4
Pickard and Emery: Chapter 5

2. Mass Conservation Continuity equation
There is no source or sink within the box
Mass is conserved

3. Where
Is individual derivative (following a water parcel
is three-dimensional divergence

4. incompressible If
=>
For a parcel of mass m and volume V, mass conservation requires
Incompressible means the parcel’s volume is conserved

5. For a finite volume V (e.g., a box aligned with x y z coordinates)
Rate of mass change
within V
Is net mass transport through the two x-y faces of the box
Mass change in a closed volume is caused by net transport through its surface
At the sea surface,
Is the surface fresh water flux

6. Salt Conservation
where
Molecular diffusivity of salt

7. Equation for Mean Flow
Averging within T:
Turbulent transport

8. Parameterizing Turbulent Transport
Ax, Ay, and  are eddy diffusivity (or Austausch coefficients)
Ax ≈ Ay >> 

9. Salt Conservation
, vertical eddy diffusion coefficient.
, horizontal eddy diffusion coefficient.
The molecular diffusivity of salt is
Ratio between eddy and molecular diffusivity:
The rate of salt change in a finite volume is equal to the net salt transport through its surfaces. The net transport includes both current and turbulent contributions.

10. We have known that, at the sea surface
However, E, P and I transfer the fresh water with S=0
There is a net salt influx into the oceans from river runoff (R), which is globally about 3 x 1012 kg/year total. About 10% of that is recycled sea salt (salt spray deposited on land).
The turbulent salt flux through the surface and at the bottom of the sea are small
(entrainment of salt crystals into atmosphere)
(subsidence at the bottom,
underwater volcano-hydrothermal vents)
Compared to the total salt amount in the ocean: 5 x 1019 kg, the rate of annual salt increase is only one part in 17 million/year. As we know, the accuracy of present salinometer is ± Given average salinity 35 psu, the instrument uncertainty is in the order of ±0.003/35=1500/17 million.
For oceanic circulations on years,this amount is small and negligible for salt budget. Therefore, the total salt amount in the global oceans can be seen as conserved.

11. Application to a Limited Volume
Under steady-state conditions, we apply the conservations of mass and salt to a box of volume V filled with sea water.
Conservation of volume:
Where Vi is inflow, Vo outflow; P precipitation, E evaporation, and R river runoff.
Salt conservation:
influx
outflux

12. (Vi , Vo) » X. Large exchange with the outside.
Denote excess fresh water as
Since
(accurate within 3%)
With
, we have
Knudsen’s relations 
and
Usually when Si and So are large,
If Si≈So,
(Vi , Vo) » X. Large exchange with the outside.
If Si » So,
Vi « X. Vo slightly larger than X. Small exchange.

13. Examples Basin Mediterranean Sea Black Sea Totoal volume (km3)
X=P-E(m3/s)
-7x104
6.5 x 103 Si
36.3 35 So
37.8 17
Vi (m3/s, km3/yr)
1.75x106, 5.5 x 104
6x103, 0.02x104
Vo(m3/s)
1.68 x 106
13x103
Flushing time (yr) 70
3000

14. Annual Mean Precipitation (mm/day)-COADS

15. Mean SST, CPC Analysis, January 1982-December 2001

17. Annual mean evaporation (mm/day)-COADS

20. Annual Mean E-P (mm/day)-COADS

21. Meridional Freshwater Transport
For a steady state,
Integrating throughout an x-y cross-section of an ocean basin,
Mass balance
Salt balance
Freshwater balance
o is freshwater density

22. Meridional freshwater transport (in 109 kg/s) in the ocean, with the quantities FP
and FA referring to the freshwater transport of the Pacific-Indian through flow and that of the Antarctic Circumpolar Current at Drake Passage, respectively (Wijffels et al., 1992).

23. An evaporation rate of 1. 2 m/yr is equivalent to removing about 0
An evaporation rate of 1.2 m/yr is equivalent to removing about 0.03% of the total ocean volume each year. An equivalent amount returns to the ocean each year, about 10% by way of rivers and the remainder by rainfall.
The yearly salt exchange is less than 10-7 of the total salt content of the ocean.