CARITRO Project: Snow droughts e green water: how climate change modifies the hydrological cycle in the Alpine Region.

Due to the impact of climate change, the Alpine region is experiencing a dual effect: a decrease in snowfall leading to snow droughts, and an increase in water losses through evapotranspiration, also known as green water. These changes have significant implications for the sustainable management of water resources and the preservation of ecosystems. This project, funded by the CARITRO foundation, aims to address these challenges by developing innovative models to accurately quantify snow melt and evapotranspiration losses. The ultimate goal is to provide practitioners with user-friendly calculation tools that are more advanced than traditional lumped models but less complex than intricate "process-based" 3D models. Initially proposed by Niccolò Tubini, the project has been taken up by John Mohd Wani with minimal modifications.  

The complete project plan can be found here

Theoretical and Numerical Tools for Studying the Critical Zone from Plots to Catchments

 What a valuable work is the thesis by Niccolò Tubini, here presented in its draft.  It covers works in hydrology of the critical zone, numerics, programming, software engineering, open science methods. Having a so wide horizon of interest it could  not be easy to grasp in all of these details, but it is well written and, we hope inspiring. As the Author, Ph.D. candidate says: "In the following we suggest that studying the CZ requires tools that are not yet readily available to researchers; then we propose one of our own. These tools should be flexible enough to allow the quick embedding of advancements in science"

Who wants to access the draft, can click on the figure of the Thesis first page below.

The Thesis included the work present in the two submitted paper by Niccolò,  on The Cryosphere and a second one presented in GMDD regarding WHETGEO-1D.  Whilst a thesis being considered kind of a definitive work, this one remains very much a work in progress with the extensions of the codes foreseen to arrive soon and whose informatics has already been implemented. All the tools developed during the Thesis are open source and freely available both as executable and source codes on Github.  Any comment or suggestion to the  Thesis as well as to the papers are welcomed. 

The Video of the defense is here.

WHETGEO 1D is out

WHETGEO-1D (Water HEat and Transport in GEOframe) is a physically based model simulating the water and energy budgets in a soil column. WHETGEO-1D is developed as an open-source code, adopting the Object-Oriented paradigm and a generic programming approach to improve its usability and expandability. WHETGEO-1D is fully integrated in the GEOframe/OMS3 system allowing the use of the many ancillary tools it provides. It comes on top of several years of work on engineering software, discussing and debating about Richards equation, taking care of getting appropriate integration methods in a travel that crossed hydrology, mathematics, numerics and software engineering.  Click on the figure below to access the paper.

The code is really solid and was throughly tested over the last three year by my students of the Hydrology class and in various applications. Its inputs and outputs can be analyzed by using Python and some standard Notebooks prepared to help the user to do it.  The paper has been submitted to the Geoscientific model development discussions (GMDD) and it is available for discussion to everyone and can be obtained by clicking on the Figure above. Complementary material is present in the GEOframe blog. The paper comes with its software, documentations and test and the GEOframe blog post explains where all of it is. You did a nice job Niccolò Tubini!

Here the paper on GMD.

Chain derivation rule is not valid when you go discrete

 One interesting fact that is not clear to most of people is that the chain derivation rule is not valid when we go from the continuous domain of calculus to the discrete domain of numerics. This means, for instance that:

$$\frac{\partial \theta(\psi(t))}{\partial t} \neq \frac{\partial \theta(\psi(t))}{\partial \psi} \frac{\partial \psi(t)}{\partial t} $$

where, for instance, $\theta$ is the volumetric water contente, $\psi$ is soil suction, $t$ is time.  The arcane is explained because when you go to discretize the differential we have:

$$ \frac{\theta^{n+1}-\theta^n}{\psi^{n+1}-\psi^n} $$

where the upper scripts $n+1$ and $n$ indicate respectively the $n+1$ and $n$ time step.  Both on the numerator and denominator we have a dependence on time step $n+1$ which is unknown and the way they are approximated can bring to an inconsistent estimation of the derivative.  As a result,  conservation equations are said written in conservative form when they contain are written as:

$$ \frac{\partial \# }{\partial t}= - \nabla \cdot * $$

where $\#$ stands for any state variable to be conserved and $*$ for any flux expression. $\nabla \cdot$ is the divergence operator. 

This issue was brought to light by the mathematician P.L. Roe in his 1981 paper, and to us by colleagues Vincenzo Casulli and Michael Dumbser and is clearly illustrated in Tubini et al. (2021) below.  

References

Roe, P. L.: Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, 43, 357–372, 1981

Tubini, N., Gruber, S., and Rigon, R.: A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation, The Cryosphere Discuss. [preprint], https://doi.org/10.5194/tc-2020-293, in review, 2020.

Freezing soil requires new algorithms

Assuming you have got the physics right and you wrote the right equations, which is not given for granted when you deal with freezing soils, you have to solve the equations. This paper, deals with this last topic: given the freezing soil equation it implements a new algorithm to solve it. This algorithm was invented by Casulli and Zanolli in their 2010 paper. They called it nested Newton, we renamed it NCZ from Newton-Casulli-Zanolli. It was implemented for solving Richards equation which present a spiky term called hydraulic capacity, that poses serous challenges to the convergence of the solver. We extended here to a new equation with the same type of terms. In fact all equations that involve phase transitions have terms of this type. 

To someone a new algorithm for integration of some equation can seem a minor achievement but, while in some type of simulation, the numerical errors of traditional methods can be somewhat constrained, in most of the simulation they do not and tend to increase up to a point that any prediction either quantitative or qualitative becomes useless. Obviously our case has an enormous effect when dealing with simulations of permafrost areas under the threat of climate change. If this introduction makes you curious, you can find the the preprint at The Cryosphere page, by clicking on the Figure above.

Reference

Casulli, Vincenzo, and ZANOLLI. 2010. “A Nested Newton-Type Algorithm for Finite Colume Methods Solving Richards’ Equation in Mixed Form.” SIAM Journal of Scientific Computing 32 (4): 2225–73.

Tubini, Niccolò, Stephan Gruber, and Riccardo Rigon. “A Method for Solving Heat Transfer with Phase Change in Ice or Soil That Allows for Large Time Steps While Guaranteeing Energy Conservation.”

Advances in Richards 2D presentation at the Italian Hydrological Society meeting

The work of Niccolò Tubini is going who already developed a very solid Richards1D (with ponding),  code coupled with the energy budget is proceeding towards a 2D version on unstructured grids coupled, at present with a 1D de Saint-Venant equation. This is the summary of the work done so far given at the Italian Hydrological society meeting in Bologna.

Actually the de Saint-Venant coupling is not yet ready but it will be very soon. Stay tuned. Click on the figure for getting the presentation.

EGU Wien 2019: Two numerical models to solve Richards and energy equations

This contribution discusses two methods of integration for Richards equation and the heat equation (for T > 0 centigrades).  Results are encouraging and show that temperature could be important to get the right runoff production.

The model uses new numerics based on work by Vincenzo Casulli and Paola Zanolli, called nested Newton. The original poster is obtained by clicking on the figure.

Theoretical progress in freezing-thawing processes studies

This and a related post reports about the Master thesis by Niccolò Tubini and Stefano Tasin. It was a couple years ago that I graduate my last Master guy, and I am happy with these two graduations.
Niccolò thesis is about modelling permafrost.

I already worked on it during the Ph.D. thesis by Matteo Dall'Amico, obtaining interesting results, which were published in this 2011 paper. From it we built. Initially the idea was that the work by Matteo dall’Amico was clear enough to go directly to a full three-dimensional implementation of 3D algorithms on a unstructured grid. That was actually not the case an we had to rework all the theory. I do not want to waste its reading. So, if you want to know the story, please click here. 

The presentation Niccolò gave for his Master (Laurea) degree is here.