1. TEM – Lectures 21 to 23

2. Upwind for concentration on the face
If the velocity is positive (rightwards flow):
In particular case of constant volume, discharge and cross section area (1D channel with the same cross section and flow everywhere:

3. Rearranging the equation (upwind, Positive Velocity)
Using 3 vectors to store the coefficients that relate the new (at time t+dt) with values at time t in the point and in the neighbouring points one can write:

4. Rearranging the equation (upwind, Negative Velocity)
Using 3 vectors to store the coefficients that relate the new (at time t+dt) with values at time t in the point and in the neighbouring points one can write:

5. Average Values would be another option to calculate the concentration on the volume faces => This would generate a method named Central Differences

6. Explicit Central Differences
The equation can again be written in the form:

7. Implicit calculation
The equation can again be written in the form:

8. Decay term How to discretise the decay term?
Let us consider a case where there is no flow and no diffusion:

9. Stability Coefficients have to be positive.
The explicit method has stability conditions.
The implicit method has no stability conditions. As a consequence sink terms should always be computed implicitly.

10. How to implement
Possibility 1: to add the sink term to the coef.CENTER
Possibility 2: To use the fractional time step and to update the new concentration value

11. Atmospheric Boundary Condition
Heat Exchange: Convection,
Temporal discretisation:

12. Atmospheric Boundary Condition
Heat Exchange: Latent Heat Flux (Bowen’s Law)
is the water evaporated in m/s
is the Bowen’s ratio (0.62mb/K)
is the saturation vapor pressure at Water temperature (bar)
is the saturation vapor pressure at Atmospheric Temperature (bar)
is the relative humidity.

14. Latent Heat Flux
latent heat flux can create stability problems only when the system gets dry.

15. Radiative heat Heat Flux
Computed on the traditional way would be:
Swinbank (1963) proposes the following equation of the net long wave radiation at the water surface:
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).
Swinbank, W.C., Long-wave radiation from clear skies. Quarterly Journal of the Royal Meteorological Society 89, 339–348.

16. Solar Radiation
Depends on the daily hour, day of the year, latitude and cloud cover.
It has to be measured or forecasted
The clear sky radiation can be computed using a set of sinus. The cloud cover can be provided as a probability.
The detailed calculation of the actual solar radiation is not complicated, but is beyhond the objective of this course.

17. Solar radiation
In this course the solar radiation will be assumed to vary as a sinus between sunrise and sunset.
The maximum radiation is assumed to be the solar constant (1370 W/m2) times the cosinus of the Latitude.
Texts to be consulted:

18. Albedo and cloud cover
The actual radiation is a reduced due to the albedo and cloud cover
The albedo (part of the energy reflected) of water depends on the incidence angle, being small for small angles. It can be assume as zero, as well as the cloud cover.
Descriptive text: