1. Temperature in a free surface flow
TEM – Lecture 27
Temperature in a free surface flow

2. Free surface boundary condition
Depend on the property being studied
In case of gases and vapours requires the knowledge of the partial pressure in the atmosphere.
In case of heat requires the incident radiation (solar and diffuse) and the heat flux radiated by the water. Latent and sensitive heat fluxes require the knowledge of atmospheric moisture and temperature.

3. Sensitive and Latent Heat Fluxes across the Free Surface
At the interface the concentration is the same.
The flux depends on the vertical gradient and on diffusivity, i.e. on the capacity to remove the exchanged material from the interface zone.
Profile of a property being transferred from the air into the water
Air
Water
In this case, the vertical flux is downwards. Inside the water the property located at the interface is removed downwards, decreasing the concentration. The atmosphere replies locating more property there.
Seeing the way around, the Atmosphere transports property to the interface, increasing its value and consequently the gradient in the water increasing the flux.
The “slower” flux controls the process.

4. Sensible and Latent Heat Fluxes
Depend on the convection coefficient. Several have been publish in the literature, e.g.
The Bowen’s coefficient relates the Sensitive and the Latent convection coefficients:

5. Sensible and Latent fluxes
(See Notes at FENIX for more info)

6. Long Wave radiation exchange
Swinbank (1963) proposes the following equation for
the net long wave radiation at the water surface:
Where is the net longwave radiation at the free surface, ε is the water emissivity (0.97)
σ is the Stefan-Boltzman constant (5.67*10-8 W/m2/K4) and C is the cloud fraction (=0 if clean sky
and =1 if completely covered).

7. Solar Radiation
Where tss is the sunset time and tsr is the sunrise time. The maximum radiance hitting the hearth surface is also supposed to vary as a co-sinus with the latitude, being maximum at the equator (1370 W/m2). The actual radiation entering in the water is reduced due to the albedo (surface reflection, α) and cloud cover (nc). They can be both considered zero.
Consult (e.g.):
Brock, T. D. (1981) - Calculating solar radiation for ecological studies. Ecological Modelling.