A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation

Abstract

Abstract. The accurate simulation of heat transfer with phase change is a central problem in cryosphere studies. This is because the nonlinear 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 withing 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 modeling 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 one hour, one day, and ten days. 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 time scale of the processes investigated.

Implementing the Water, HEat and Transport model in GEOframe WHETGEO-1D v.1.0: algorithms, informatics, design patterns, open science features, and 1D deployment

“…equation is fully coupled with the R 2 equation, as in (Dall'Amico et al, 2011) for instance, and the enthalpy function becomes nonlinear. At this point, since the enthalpy function is nonlinear the NCZ algorithm is required to linearise it, as shown in (Tubini et al, 2020). So far, we have not considered the problem of water flow in freezing soils, however being aware of this issue is important for the future developments and code design.…”

“…This aspect, the use of two different equation states, and the fact that the NCZ algorithm can be successfully reused to solve other problems (Casulli and Zanolli, 2012;Tubini et al, 2020), requires a careful design of its implementation, as discussed in the following sections.…”

“…The choice of including the Buffer component in the workflow of the modelling solution is motivated by earlier experiences of the GEOtop community with the GEOtop model, and by more recent experience with the FreeThaw1D model (Tubini et al, 2020), where long spin-ups of the model (∼ 1500 years) were required with consequent large output files (∼ GB). Furthermore, in anticipation of the 2D and 3D developments, the netCDF-3 format is probably not the most appropriate and could be abandoned in favour of more performing file formats Unidata (a, b), such as netCDF-4 or HDF5.…”

“…This method is relevant for those models where the ψ * cannot be computed analytically but requires the application of a root finding method such as the bisection method. An example is the soil internal energy function when considering the phase change of water Tubini et al (2020).…”

“…By including the closure equation of the SFC presented by Dall'Amico et al (2011), it is possible to consider the phase change of water. Moreover, compared to the FreeThaw-1D model Tubini et al (2020), in WHETGEO-1D it is possible to drive the simulation of the soil thermal regime by using the surface energy budget. This aspect represent a novelty with respect to Competing interests.…”