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These models and simulations have been tagged “Water-Column”.
This model implements a very simple proxy for vertical dispersion of heat in a lake based on the equation:
dT/dt = 1/A d(EA)/dz (dT/dz)
where: T: temperature (oC); t: time (days); z: depth (m); A: cross-sectional area (m2); E: vertical dispersion coefficient (m2 d-1)
If we consider that E is constant (it is in this model), then the equation becomes dT/dt = (EA/A)(d^2T/dz^2) = E(d^2T/dz^2), the classic diffusion equation
The model is simplified by exchanging temperature as a state variable, rather than executing the full heat balance. This would require a computation of fluxes of atmospheric longwave and shortwave radiation, water longwave radiation, water conduction and convection, and water evaporation and condensation.
The vertical dispersion coefficients are adjusted artificially so that mixing increases at lower temperatures, thus quickly homogenizing the water column in colder months of the year.
Joao G. Ferreira