This post is not self-explanatory and requires digging into other posts and some papers.
Please review Section 2 of Concetta's paper (https://onlinelibrary.wiley.com/doi/10.1002/eco.70009?af=R) and verify the calculations presented there.
The model in D'Amato and Rigon (2025) uses a non-capacitive approach (it doesn't account for the thermal capacity of plants), and so will be if the same derivation is specialized for snow (or soil), which is a limitation. However, this approach is still more physically based than semi-empirical formulations or degree-day methods commonly used. In the literature, these are referred to as "stationary solutions" of the system. Despite the name, these solutions respond instantaneously to changing boundary conditions (radiation, latent and sensible heat fluxes), as evident in equation (10), which varies with radiation, wind velocity, and roughness.
Equation (10) and subsequent equations in Concetta's paper are essentially the Prospero solutions (though exact implementation should be verified in Concetta's code). The time interval of integration is, in principle instantaneous, but eventually you would like to integrate it over a finite time step (a hour, or a day, for instance).
A non negligible aspect is that snow can melt into water and for any temperature you get from the energy budget, you need to partition the water in liquid water and ice. For this reason you probably need a partitioning function, like the one used for partitioning precipitation in rainfall and snowfall or you can simply use a melting law like in simple models but now the temperature used should not be the air tempeature but the snow temperature. See melting in simple models in the links below
for further information.
An important term is missing from the formulation: heat exchange by conduction with the ground, which should be represented as:
G = C_s T_Δg := C_s (T_g - T_s)
Where:
- C_s is an appropriate exchange coefficient (can be taken as C_s = K/L, where K is the bulk thermal conductivity of the layer and L is its depth)
- T_g is the ground temperature (which could be taken as the multi-annual air temperature average)
- T_s is the snow temperature
Since this flux depends on the independent variable, it introduces additional terms that modify solution (10). Please derive these calculations independently.
Other terms that don't depend on the independent state variables can be included in the S_nk term. With these modifications, the Prospero code can effectively simulate the snow energy budget. Similar arguments apply to soil modeling.
A further consideration is the proper parameterization of the conductances C and C_E fluxes in equation (10), which differ from transpiration cases. For soil, according to Lehman-Or theory, evaporation should be modeled as potential until the water storage exceeds a threshold S_T, then decreasing proportionally with storage below this threshold when implementing an integrated model (Details ? I do not know).
I know that there are several missing aspects in this post. Who is interested, please ask.
P.S. - These components have then to be carefully coupled to the other components. With respect to this, please consider the following:
First review the presentation materials I've shared:
Getting new features to the linear systems (Vimeo2025)The topic is that, based on my analysis, the second option is clearly the one should be applied in integrated distributed models (like GEOframe-NewAGE). However, this means we cannot simply subtract ET (or any other sink) from total rainfall - we need to incorporate this directly into the equation solver. While the example in the presentation uses a linear system with an analytical solution, the same principle applies to our non-linear fluxes where we use numerical integration. Therefore appropriate modifications could be necessary to the basic GEOframe-NewAGE codes.
