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

Mass Conservation Continuity equation

Mass Conservation Continuity equation Integrating in ocean depth,, total mass in a column, we have., 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) where Vertical boundary conditions:

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

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

Salt Conservation where Molecular diffusivity of salt

Equation for Mean Flow Averging within T: Turbulent transport

Parameterizing Turbulent Transport A x, A y, and  are eddy diffusivity (or Austausch coefficients) A x ≈ A y >> 

Salt Conservation, 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,

We have known thatand 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 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) the amount is small and negligible for salt budget. (subsidence at the bottom, underwater volcano-hydrothermal vents) Overall, Compared to the total salt amount in the ocean: 5 x 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,

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 S o 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

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 V i is inflow, V o outflow; P precipitation, E evaporation, and R river runoff. Salt conservation: influxoutflux

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

BasinMediterranean SeaBlack Sea Totoal volume (km 3 )3.8 x x 10 6 X=P-E(m 3 /s)-7x x 10 3 SiSi SoSo V i (m 3 /s, km 3 /yr)1.75x10 6, 5.5 x x10 3, 0.02x10 4 V o (m 3 /s)1.68 x x10 3 Flushing time (yr) Examples 

Circulation Patterns O 2 > 160  mol/kg (4ml/l) Hydrogen Sulphide H 2 S~6ml/l

Annual Mean Precipitation (mm/day)-COADS

Annual mean evaporation (mm/day)-COADS

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

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 of the total salt content of the ocean.

Wijffels, 2001 Where transport increases northward, freshwater is being added to the ocean. Freshwater added 80 o S-40 o S 10 o S-10 o N 40 o N-80 o N Freshwater removed 40 o S-10 o S 10 o N-40 o N Transport should be zero at the poles for global balance The fresh water transport is small compared to the total circulation