1. Water, salt, and heat budget
●Conservation laws
application: box models
●Surface fresh water flux:
evaporation, precipitation, and river runoff
●Surface heat flux components:
sensible, latent, long and shortwave
●Ocean meridional transport

2. Mass Conservation
Continuity equation

3. Mass Conservation Continuity equation
, total mass in a column, we have
Integrating in ocean depth, ,
where .
Vertical boundary conditions:
E-evaporation, P-precipitation, R-river runoff
(measured in m/s, 1mm/day=1.1574x10-8 m/s).
Melting of sea ice may also be a factor
(neglected here)

4. Gaussian formula:
Integrating a two dimensional vector field over an area S with boundary L, we have
Where is a unit vector perpendicular to the boundary L.
Define the mass in a water column of bottom area as S:
Integrating the continuity equation in S with boundary L:
Using Gaussian formula

5. Lateral boundary conditions:
If L is a closed basin (e.g., the coastal line of an ocean domain):
no slip condition:
on L.
free slip condition:
In both cases 
Then

6. Salt Conservation The molecular diffusivity of salt is
, vertical eddy diffusion coefficient.
, horizontal eddy diffusion coefficient.
The molecular diffusivity of salt is
Ratio between eddy and molecular diffusivity:
Integrating for the whole ocean column,

7. We have known that
and
However, both E and P transfer the fresh water with S=0
There is a net salt influx into the oceans from river runoff (R), which is totally about 3 x 1012 kg/year. 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.
The amount is small and negligible for salt budget.
Overall,

8. There is a net vertical salt flux near the sea surface driven by the fresh water flux.
Consider a thin interfacial layer, the balance of fresh water flow is
Where m is the rate of volume of the sea water entrained into the thin layer from its bottom
The corresponding turbulent salt flux is or
Where So is usually chosen as 36‰.
Usually, we neglect the effect of E-P on mass balance (i.e., w(z=0)=0) and take into its effect on salinity as
Apparent salt flux

9. Box Model
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

10. (Vi , Vo) » X. Large exchange with the outside.
Denote excess fresh water as
Since
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.

11. 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

12. Circulation Patterns