Abstract. The accurate simulation of heat transfer with phase change is a central problem in cryosphere studies. This is because the non-linear behaviour of enthalpy as function of temperature can prevent thermal models of snow, ice, and frozen soil from converging to the correct solution. Existing numerical techniques rely on increased temporal resolution in trying to keep corresponding errors within acceptable bounds. Here, we propose an algorithm, originally applied to solve water flow in soils, as a method to solve these integration issues with guaranteed convergence and conservation of energy for any time step size. We review common modelling approaches, focusing on the fixed-grid method and on frozen soil. Based on this, we develop a conservative formulation of the governing equation and outline problems of alternative formulations in discretized form. Then, we apply the nested Newton–Casulli–Zanolli (NCZ) algorithm to a one-dimensional finite-volume discretization of the energy–enthalpy formulation. Model performance is demonstrated against the Neumann and Lunardini analytical solutions and by comparing results from numerical experiments with integration time steps of 1 h, 1 d, and 10 d. Using our formulation and the NCZ algorithm, the convergence of the solver is guaranteed for any time step size. With this approach, the integration time step can be chosen to match the timescale of the processes investigated.

Abstract. The “Digital Earth” (DE) metaphor is very useful for both end users and hydrological modelers (i.e., the coders). In this opinion paper, we analyze different categories of models with the view of making them part of Digital eARth Twin Hydrology systems (DARTHs). We stress the idea that DARTHs are not models, rather they are an appropriate infrastructure that hosts (certain types of) models and provides some basic services for connecting to input data. We also argue that a modeling-by-component strategy is the right one for accomplishing the requirements of the DE. Five technological steps are envisioned to move from the current state of the art of modeling. In step 1, models are decomposed into interacting modules with, for instance, the agnostic parts dealing with inputs and outputs separated from the model-specific parts that contain the algorithms. In steps 2 to 4, the appropriate software layers are added to gain transparent model execution in the cloud, independently of the hardware and the operating system of computer, without human intervention. Finally, step 5 allows models to be selected as if they were interchangeable with others without giving deceptive answers. This step includes the use of hypothesis testing, the inclusion of error of estimates, the adoption of literate programming and guidelines to obtain informative clean code. The urgency for DARTHs to be open source is supported here in light of the open-science movement and its ideas. Therefore, it is argued that DARTHs must promote a new participatory way of performing hydrological science, in which researchers can contribute cooperatively to characterize and control model outcomes in various territories. Finally, three enabling technologies are also discussed in the context of DARTHs – Earth observations (EOs), high-performance computing (HPC) and machine learning (ML) – as well as how these technologies can be integrated in the overall system to both boost the research activity of scientists and generate knowledge.

Abstract. This paper presents WHETGEO and its 1D deployment: a new physically based model simulating the water and energy budgets in a soil column. The purpose of this contribution is twofold. First, we discuss the mathematical and numerical issues involved in solving the Richardson–Richards equation, conventionally known as the Richards equation, and the heat equation in heterogeneous soils. In particular, for the Richardson–Richards equation (R2) we take advantage of the nested Newton–Casulli–Zanolli (NCZ) algorithm that ensures the convergence of the numerical solution in any condition. Second, starting from numerical and modelling needs, we present the design of software that is intended to be the first building block of a new customizable land-surface model that is integrated with process-based hydrology. WHETGEO is developed as an open-source code, adopting the object-oriented paradigm and a generic programming approach in order to improve its usability and expandability. WHETGEO is fully integrated into the GEOframe/OMS3 system, allowing the use of the many ancillary tools it provides. Finally, the paper presents the 1D deployment of WHETGEO, WHETGEO-1D, which has been tested against the available analytical solutions presented in the Appendix.

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Abstract. This paper presents WHETGEO and its 1D deployment, a new, physically based model simulating the water and energy budgets in a soil column. The purpose of this contribution is twofold. First, we discuss the mathematical and numerical issues involved in solving the Richardson-Richards equation, conventionally known as Richards' equation, and the heat equation in heterogeneous soils. In particular, for the Richardson-Richards equation (R2) we take advantage of the nested Newton-Casulli-Zanolli (NCZ) algorithm that ensures the convergence of the numerical solution in any condition. Second, starting from numerical and modelling needs, we present the design of a software that is intended to be the first building block of a new, customisable, land-surface model that is integrated with process-based hydrology. WHETGEO is developed as an open-source code, adopting the Object-Oriented paradigm and a generic programming approach in order to improve its usability and expandability. WHETGEO is fully integrated in the GEOframe/OMS3 system allowing the use of the many ancillary tools it provides. Finally the paper presents the 1D deployment of WHETGEO, WHETGEO-1D, which has been tested against the available analytical solutions presented in Appendix.