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 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.

Numerical Study of Coupled Water and Vapor Flow, Heat Transfer, and Solute Transport in Variably‐Saturated Deformable Soil During Freeze‐Thaw Cycles

“…According to the energy conservation law and the Fourier heat transfer law based on local thermal equilibrium, the subsurface temperature distribution during the freeze‐thaw period can be calculated using the advection‐conduction equation, including the transient effects of latent heat of fusion that can be considered a source term (e.g., Tubini et al., 2021) as follows (e.g., De Vries, 1958; Zheng et al., 2002): $$${C}_{e}\frac{\partial T}{\partial t}+{L}_{v}\frac{\partial {\theta }_{v}}{\partial t}-{L}_{f}{\rho }_{i}\frac{\partial {\theta }_{i}}{\partial t}=\nabla \cdot \left[{\lambda }_{e}\nabla T-{C}_{w}T{\boldsymbol{u}}_{\boldsymbol{w}}-{D}_{v}\nabla {\rho }_{v}\cdot \left({C}_{v}T+{L}_{v}\right)\right]$$$ $$${C}_{e}={C}_{i}{\theta }_{i}+{C}_{w}{\theta }_{w}+{C}_{\mathrm{s}\mathrm{g}}{\theta }_{\mathrm{s}\mathrm{g}}+{C}_{a}{\theta }_{a}$$$ $$${\lambda }_{e}={{\lambda }_{i}}^{{\theta }_{i}}{{\lambda }_{w}}^{{\theta }_{w}}{{\lambda }_{\mathrm{s}\mathrm{g}}}^{{\theta }_{\mathrm{s}\mathrm{g}}}{{\lambda }_{a}}^{{\theta }_{a}}$$$ where L v (J · m −3 ) is the volumetric latent heat of vaporization of liquid water, which is given by L v = L 0 ρ w , and L 0 (J · kg −3 ) is the latent heat of vaporization, which can be described as a function of temperature L 0 = 2.501 × 10 6 − 2,369.2...…”

“…According to the energy conservation law and the Fourier heat transfer law based on local thermal equilibrium, the subsurface temperature distribution during the freeze-thaw period can be calculated using the advection-conduction equation, including the transient effects of latent heat of fusion that can be considered a source term (e.g., Tubini et al, 2021) as follows (e.g., De Vries, 1958;Zheng et al, 2002):…”

Exploration of Thermal Bridging Through Shrub Branches in Alpine Snow

“…The baseline soil volumetric thermal capacity used was 2.6 × 10 6 J m −3 K −1 , corresponding to a density of 1,300 kg m −3 and a specific heat of 2000 J K −1 kg −1 . To represent the effect of water phase change on temperature, a Soil Freezing Characteristic Curve (SFCC) was used under the form of an increased apparent thermal capacity (Tubini et al., 2021). In practice, the thermal capacity of the soil was increased near the freezing point, accounting for the extra‐energy needed to change the soil temperature just below 0°C, as a fraction of the energy is converted to latent rather than sensible heat.…”

Brief communication: Everest South Col Glacier did not thin during the last three decades

“…Modeling the changes of ice or snow masses in response to atmospheric forcing is a complex task which involves resolving heat transport and conservation, with concurrent phase changes (e.g., Tubini et al, 2021). To model the surface mass balance of the South Col Glacier, Potocki et al (2022) relied on the COSIPY model (Sauter et al, 2020).…”