Temperature gradient metamorphism in dry snow is driven by heat and water vapor transfer through snow, which includes conduction/diffusion processes in both air and ice phases, as well as sublimation and deposition at the ice–air interface. The latter processes are driven by the condensation coefficient

Natural snowpacks are frequently subjected to temperature gradients induced by meteorological conditions. In the case of temperature gradient in dry snow, heat and water vapor are transported through the snowpack by heat conduction through ice and air and by vapor diffusion in air. These phenomena are coupled by the sublimation–deposition processes at the ice–air interfaces. In practice, such transfer processes can be enhanced by natural air convection induced by the temperature gradient (e.g.,

Models to describe the heat and mass transfer at the pore scale, referred to as the micro-scale, have been proposed (e.g.,

In the last few decades, several models have been presented to describe the heat and mass transfer at the scale of a snow layer, referred to as the macro-scale. At that scale, the snow microstructure is not explicitly represented, and simulations can be carried out on entire snowpacks. The first models assumed saturated vapor conditions in the snow (e.g.,

Other approaches largely rely on the assumption of saturated vapor conditions, which seems valid for faster kinetics and rather high values of

Further uses of the abovementioned models, as well as their implementation in full snow cover models, are limited by some challenges. One is the difficulty in choosing between models, as they differ in many ways. They were derived using different methods; involve different balance equations and effective parameters; and are valid for different, often unclear, domains of validity in terms of

This paper aims (i) to define the heat and mass transport modeling in dry snow for the full range of

We apply the homogenization technique of multiple-scale expansions

As in

Physical phenomena under consideration at the representative elementary volume (REV) scale.

In what follows,

The next step is the normalization of the above pore-scale description in Eqs. (

Dimensionless numbers

The next key step is to estimate the above six dimensionless numbers with respect to the separation of scale parameter

Characteristic values of the properties evaluated at

Estimation of the dimensionless number

Estimations of the dimensionless number

Case A is

Case B is

Case C is

Case D1 is

Case D2 is

The next step is to introduce multiple-scale coordinates

Case A corresponds to the model presented in

According to the order of magnitude of the dimensionless numbers in case B, the asymptotic analysis presented in the Supplement (Sect. S2) shows that the heat transfer and the water vapor diffusion at the macroscopic scale are described by Eqs. (S.B.29) and (S.B.44). Returning dimensional variables, the macroscopic model is written as

According to the order of magnitude of the dimensionless numbers in case C, the asymptotic analysis presented in the Supplement (Sect. S3) shows that the heat transfer and the water vapor diffusion at the macroscopic scale are described by Eqs. (S.C.42) and (S.C.47). Returning dimensional variables, the macroscopic model is written as

According to the order of magnitude of the dimensionless numbers in cases D1 and D2, the asymptotic analysis presented in the Supplement (Sect. S4) shows that these two cases lead to the same macroscopic description. Returning dimensional variables, the macroscopic model (Eqs. S.D1.41–S.D1.45 or Eqs. S.D2.41–S.D2.45) is written as

Figure

Definition of the three different macroscopic models and their domain of validity with respect to

A first important outcome is that the hypothesis

A second point concerns the relative role of the vapor diffusion and of the sublimation–condensation in the vapor transport, which directly impacts the model formulation. In models A and B, the water vapor transfer is mainly limited by the sublimation–deposition at the ice–air interfaces. Model A consists of two equations of temperature and vapor density coupled through source terms that are proportional to

In contrast, in model D, the water vapor transfer is mainly limited by the diffusion process at the micro-scale. In that case, the model consists of a single heat transfer equation in which

The transition between a diffusion-limited and sublimation–deposition-limited vapor transport is captured by model C. This transition appears around a transition value

In this section, two simple snow microstructures, a bilayer snowpack and an assemblage of spherical grains and pores, are first considered to illustrate the influence of the microstructure and of the parameters taken at the pore scale on the macroscopic parameters of models A, B, and D (Sect.

Then, a simplified 2D snow microstructure is considered to evaluate the models by comparing simulation results obtained with the pore-scale description and with the macroscopic modeling (Sect.

Illustration of the bilayer snowpack problem

As a first example, we consider the classical bilayer material problem, and the snowpack is seen as a succession of horizontal layers of pure air and of pure ice, as illustrated in Fig.

Evolution of the SC estimates of the thermal conductivities

The next analytical model is the self-consistent model

Evolution of the normalized SC estimates

Evolution of the thermal conductivity with temperature for

For model A, the SC estimate of the effective thermal conductivity of snow

For thermal conductivity, the SC estimates

Next, we look at the impact of temperature on the properties. The impact is weaker than the one of porosity and more complex to understand, as dependencies are multiple. To help understand, we first break down the dependencies and show in Fig.

Keeping the above considerations in mind, the evolution of

We perform a numerical evaluation of the obtained macroscopic models on a simplified 2D snow microstructure, as in

Finite element numerical simulations were performed using the code from COMSOL Multiphysics software on a 2D vertical snow layer of 10 cm height and 0.5 cm width (Fig.

Illustration of the 2D geometry for the pore-scale modeling and the macroscopic equivalent modeling.

At the pore scale, the snow layer consists of 200 periodic cells of 0.5

At the macroscopic scale, the snow layer is seen as a continuous equivalent medium. The heat and the mass transfer is described by the homogenized equations Eqs. (

Evolution of the thermal conductivities

Figure

Evolution of the dimensionless diffusion coefficients

Results between pore-scale and macro-scale simulations are compared in terms of temperature, vapor density, and mass change rate. At the pore scale, the average values of each variable were taken over the cell and computed as follows:

Simplified example of the transition from the

Vertical profiles of

Temperature and water vapor density in the middle of the snow layer as a function of

We describe first the main features observed in the pore-scale simulations. All the variables show an impact of the

Next we compare the different macroscopic models to the pore-scale simulations. In both Figs.

This section presents the evaluation of the macroscopic models A, B, and D, based on observations of natural snow evolution from three cold-laboratory TGM experiments. We first introduce the experimental data (Sect.

Overview of the experimental settings used in the simulations.

We used the datasets provided by

Next we study the estimates of the effective properties and other input parameters required to run models A, B, and D. For the sake of simplicity, model C is not systematically shown. For each model, these properties are computed from the 3D images of snow of the experiment Bouvet A and Bouvet B. Those values are then compared to different parameterizations from the literature or fitted regressions, and we select the more suited ones to be used later in the models.

Model A involves three effective parameters that are the effective thermal conductivity

Average values of effective conductivity and normalized effective diffusivity as a function of density and SSA as a function of time, as computed from the tomography images of Bouvet A (symbols;

Figure

Given the above considerations, we selected two sets of parameters to simulate Bouvet A, Bouvet B, and Kamata with model A, which are summarized in Table

Summary of the effective parameters used in the simulations.

Models B and D only involve the apparent thermal conductivities of snow

The thermal conductivity estimates for model B (

For model D,

In this section, we compare simulations from models A, B, and D with the measurements from the three experiments Bouvet A, Bouvet B, and Kamata. The simulations were performed with the software COMSOL Multiphysics by resolving the homogenized equations on a 1D geometry that corresponds to the snow layer of the experiments. We used Eqs. (

Figure

Vertical steady-state profiles of

Looking at the models, the main observation is that they all underestimate

Models B and D show a unique

Next, we evaluate the models regarding mass changes across the vertical dimension of the snow layer. We look at the vertical density profile of snow, as well as the height of the air gap formed at the base of the layer at the end of the experiments, caused by an upward mass transfer during the TGM. For model A, the air gap height is defined as the highest height value at which the density is zero. For models B, C, and D, the vertical profile of density cannot be evaluated because they only predict deposition and thus density increase, due to the boundary condition issue already described in Sect.

Figure

Looking at the experiments Bouvet A and Bouvet B, a first description of model A is given, with

Simulations of the Kamata experiment with model A differ from the ones of Bouvet A and Bouvet B. Indeed, a mass gain is not predicted in the upper part of the snow layer but instead in a zone right above the mass loss region. This is particularly visible for

In the present work, macroscopic models for heat and mass transfer in dry snow have been derived by homogenization from the physics at the pore scale for different values of the condensation coefficient

At the macroscopic scale, model A

According to their definition, models B and D do not depend on

When comparing experiments and simulations with the three models, it appears that they are able to reproduce the main features of the heat and mass transport during the TGM, including the non-linear temperature profile and, for model A, the upward vapor transport with, eventually, the formation of a millimeter-scale basal air gap. However, a major discrepancy lies in the fact that temperature values are underestimated by all the models. More precisely, the heat source inducing the non-linearity in the temperature profile seems underestimated. The best predictions of the temperature deviation

Temperature measurements in Bouvet A were performed with Pt100 sensors with an accuracy of

The macroscopic modeling of heat and mass transport in dry snow relies on the effective parameters of

As the points raised above do not seem sufficient to explain the model errors, a plausible cause remains to be investigated and is the definition of the model itself, i.e., the definition of the physics at the pore scale considered for the homogenization. A first element concerns the source terms in model A, which are derived from the Hertz–Knudsen equation and rely on a condensation coefficient

Another point is that the natural convection was not taken into account at the pore scale. This process was, however, hypothesized to be key for heat and mass transport of snow under strong temperature gradients, such as Arctic and sub-Arctic ones (e.g.,

Finally, the cross-coupling effects between the temperature and water vapor density, such as the Soret and Dufour effects, were not considered in the physics at the pore scale. The effect of the vapor density gradient on the heat flux, called the Dufour effect, is characterized by the thermodiffusion coefficient

This paper presents the definition and evaluation of the equivalent macroscopic modeling of heat and mass transport during the TGM in dry snow. First, we applied the homogenization process to retrieve the macroscopic models valid for condensation coefficients

In the second part of the paper, we evaluated the homogenized models A, B, C, and D by comparison with three laboratory experiments of the TGM of snow

The theoretical development of the macroscopic equivalent descriptions are available in the Supplement.

The supplement related to this article is available online at:

CG, NC, and LB conducted the derivation of the macroscopic models. LB applied the models to the experimental data. The analyses and interpretations were carried out by LB, FF, NC, and CG. LB and CG prepared the paper, with contributions from all co-authors.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

We warmly thank the editor, Jürg Schweizer, and the reviewers, Tien Dung Le and the anonymous reviewer, for their insightful comments which helped to significantly improve the quality of the paper.

The 3SR lab is part of the Labex Tec 21 (Investissements d’Avenir; grant no. ANR-11-LABX-0030). CNRM/CEN is part of Labex OSUG@2020 (Investissements d’Avenir; grant no. ANR-10-LABX-0056). This research has been supported by the Agence Nationale de la Recherche through the MiMESis-3D ANR project (grant no. ANR-19-CE01-0009).

This paper was edited by Jürg Schweizer and reviewed by Tien Dung Le and one anonymous referee.