The hydrological cycle is significantly influenced by the presence of water in its condensed states in middle and extreme latitudes. Various hydrological parameters change below 0 Celsius, such as water viscosity, thermal capacity, and hydraulic conductivity. Consequently, mainstream hydrology treatments that neglect freezing provide incorrect results in winter, high elevations, and the far north and south for most of the year. In the current state of global warming that threatens the cryosphere which is progressively disappearing, it is even more crucial to address its dynamics.

A little of-of-date itinerary can be found in a previous post here. To understand our progress, three milestone theses summarize the work done

- Matteo's Together, we worked out the Thermodynamics of non equilibrium for ice-systems and the theory of freezing soils. Matteo implemented also an integrator in GEOtop, not the perfect one, but acceptable. Matteo's 2011 paper is a benchmark paper in the topic.
- Stefano's brought GEOtop to some maturity and especially fine tuned the various tools related to snow and ice. Stefano's 2014 paper remains a landmark in our work.
- Niccolò's pushes forward the previous work. Especially remarkable is his work on re-implementing the informatics according to new (for us) concepts in OO programming and using (finally) safe algorithms for the integration of the equations. His WHETGEO and FreeThaw papers are a must read for completeness and clarity.

Our work's focus was primarily on the critical zone, where we modified the Darcy-Buckingham law to account for freezing and thawing and their related hydrological and mechanical effects. We primarily focused on the hydrological effects neglecting the mechanical ones but not neglecting the energy budget, a common practice in hydrology, which is obviously not possible. Consequently, we faced the necessity to simultaneously solve both the mass budget and the energy budget.

The formulation of the equations can be found in the theses and papers cited above, and you will realize that establishing a correct relation between the Darcy scale energy content and the corresponding water (liquid or solid) is the main challenge. Proper physics requires the consideration of interfaces between the phases: air-water-soil-ice. While a complete understanding of this relation has not been yet achieved, some working approximations have been obtained. Looking at the two compartments, snow and ice in the soil, they differ in many aspects, with snow lacking soil and being affected by its aerial origin. Both snow and ice in the soil have their own complexities, which affect their evolution. They often interact and the fate of the soil with or without snow is quite different.

While determining the correct equations would be satisfactory goal for many, it remains unresolved how to numerically estimate these equations. It turns out that these mildly nonlinear equations pose problems when solved using the usual algorithms based on variations of the Newton method. Convergence of the numerical methods is not guaranteed, and many workarounds have been deployed to overcome these difficulties, often leading to issues with mass and energy conservation principles.

Fortunately, Casulli and Zanolli (2010) found a method to address these challenges. The fundamental paper can be found in bibliography, and a progressive approach to its formulation can be obtained by reading Casulli's lecture notes and completed by watching videos in this blog. Ideally, attending Casulli's annual school in Trento, held every second half of January after our GEOframe winter school, would provide the best understanding. Tubini's recent papers are the result of this approach, and FreeThaw and WHETGEO are concrete implementations of these algorithms in Java/OMS/GEOframe.

Another aspect to consider is the implementation of these algorithms in informatics. Concepts related to this can be found in parts of Tubini's thesis and the related papers, especially Tubini and Rigon, 2022.

Finally, as a source of information, all of these models' open-source codes can be found on GitHub, both for the older GEOtop and the more recent GEOframe model components.

References

Casulli, V. 2017.

*Advanced Numerical Methods for Free-Surface Hydrodynamics*.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.Dall’Amico, Matteo. 2010. “Coupled Water and Heat Transfer in Permafrost Modeling.” Edited by Riccardo Rigon and Stephan Gruber. Phd, University of Trento. http://eprints-phd.biblio.unitn.it/335/.

Dall’Amico, M., S. Endrizzi, S. Gruber, and R. Rigon. 2011. “A Robust and Energy-Conserving Model of Freezing Variably-Saturated Soil.”

*The Cryosphere*. https://tc.copernicus.org/articles/5/469/2011/.Endrizzi, Stefano. 2007. “Snow Cover Modelling at a Local and Distributed Scale over Complex Terrain.”

*Ph.D. Thesis*, January, 1–189.Endrizzi, S., S. Gruber, M. Dall’Amico, and R. Rigon. 2014. “GEOtop 2.0: Simulating the Combined Energy and Water Balance at and below the Land Surface Accounting for Soil Freezing, Snow Cover and Terrain Effects.”

*Geoscientific Model Development*7 (6): 2831–57. https://doi.org/10.5194/gmd-7-2831-2014.Tubini, N. 2021, June. “Theoretical and Numerical Tools for Studying the Critical Zone from Plots to Catchments.” Edited by R. Rigon and S. Gruber. Ph.D., Dipartimento di Ingegneria Civile, Ambientale e Meccanica, Università di Trento.

Tubini, Niccolò, Stephan Gruber, and Riccardo Rigon. 2021. “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*15 (6): 2541–68. https://doi.org/10.5194/tc-15-2541-2021.Tubini, Niccolò, and Riccardo Rigon. 2022. “Implementing the Water, HEat and Transport Model in GEOframe (WHETGEO-1D v.1.0): Algorithms, Informatics, Design Patterns, Open Science Features, and 1D Deployment.”

*Geoscientific Model Development*15 (1): 75–104. https://doi.org/10.5194/gmd-15-75-2022.