Heat transport in snowpacks is understood to occur through the two processes of heat conduction and latent heat transport carried by water vapor, which are generally treated as decoupled from one another. This paper investigates the coupling between both these processes in snow, with an emphasis on the impacts of the kinetics of the sublimation and deposition of water vapor onto ice. In the case when kinetics is fast, latent heat exchanges at ice surfaces modify their temperature and therefore the thermal gradient within ice crystals and the heat conduction through the entire microstructure. Furthermore, in this case, the effective thermal conductivity of snow can be expressed by a purely conductive term complemented by a term directly proportional to the effective diffusion coefficient of water vapor in snow, which illustrates the inextricable coupling between heat conduction and water vapor transport. Numerical simulations on measured three-dimensional snow microstructures reveal that the effective thermal conductivity of snow can be significantly larger, by up to about

Thermal conductivity is one of the major physical properties of snow. It governs the magnitude of the thermal energy flux through the snowpack when subjected to a thermal gradient, and it thus plays an integral role in the energy budgets of the ground

One of the peculiarities of snow is that energy transport does not solely occur through heat conduction. Indeed, when a snowpack is subjected to a thermal gradient, a macroscopic water vapor flux is also present

The aim of this article is to provide a simplified analysis of the contribution of latent heat to the thermal energy flux in snow and notably to quantify the coupling between these processes at the macroscopic scale. For this we focus on two limiting cases, considering the kinetics of deposition and sublimation of water vapor to be either very fast or very slow. We start by providing theoretical considerations on the relationship between water vapor transport and the effective thermal conductivity. We then perform numerical simulations to quantify the contribution of latent heat to the effective thermal conductivity.

Let us consider a snow sample of volume

Illustration of the microscopic and macroscopic points of view of snow. At the microscopic scale, snow is composed of an ice space (

The effective thermal conductivity

Finally, the effective thermal conductivity is represented by a

In this article, the effective thermal conductivity of snow will be obtained starting from the physics at the microscopic scale. The relevant microscopic physical mechanisms for heat transport are (i) heat conduction in the ice, (ii) heat conduction in the air, (iii) vapor diffusion in the air, and (iv) vapor deposition/sublimation at ice surfaces

The system of Eq. (

In the slow kinetics case, we consider that

In the fast kinetics case,

Using the chain rule, one has

Multiplying the third line by

An equivalent demonstration of this result was proposed by

As this system of equations is equivalent to the one of an inert medium with an increased air thermal conductivity, one can show using methods of homogenization

We now investigate the individual contributions of conduction and vapor transport to

The effective thermal conductivity of snow can thus be decomposed in

It is important to note that

Finally, we want to point out that in the fast kinetics case, the effective thermal conductivity and the effective water vapor diffusion coefficient are linearly related. Indeed, starting from the fact that the effective diffusion coefficient is given by the ratio of the magnitude of the vapor flux over the magnitude of the vapor concentration gradient, one has

Finally, injecting Eq. (

Since the effective thermal conductivity is larger than the conductivity of the least conducting phase, i.e.,

Normalized effective water vapor diffusion coefficient as a function of the effective thermal conductivity under the fast kinetics hypothesis at

Numerous works indicate that

To exemplify and quantify the points raised in Sect.

In each simulation, the temperatures of two opposite sides of the microstructure were imposed in order to obtain a thermal gradient of

In order to test the influence of temperature on the effective thermal conductivity of snow, the simulations were run for different mean temperatures, ranging from

For the different microstructures and mean temperatures, two types of simulations were performed, one in which we assumed no impact of latent heat on the heat conduction (thus obtaining

In total we used

In this section we analyze the influence of the mean temperature on the effective thermal conductivity. For simplicity, we limit ourselves to vertical temperature gradients and thus only deal with vertical effective thermal conductivities and vertical diffusion coefficients of water vapor. As all scalar components are vertical, we do not use the subscript

The temperature dependence of

Furthermore, we define for our analysis

Temperature dependence of the thermal conductivity of ice (

We first focus on only two snow samples: a low-density sample and a high-density sample. The low-density sample is composed of decomposing and fragmented precipitation particles (DF) with a density of

Tetrahedral meshes (ice phase only) of the low-density DF sample

Vertical effective thermal conductivity (

We start by analyzing the low-density sample (left column of Fig.

Contrary to the low-density sample, the high-density sample (right column of Fig.

For both samples, the difference between

In order to better quantify the difference between the fast and slow kinetics cases, we computed the vertical effective thermal conductivity for the totality of our

Ratio of the fast kinetics over the slow kinetics vertical effective thermal conductivity for various snow samples as a function of density. Computations performed at

Finally, Fig.

Vertical normalized effective diffusion coefficient

The slow kinetics effective thermal conductivities of snow samples covering a broad range of densities and microstructures have been reported by

The thermal conductivities computed at

Effective thermal conductivity of snow as a function of density under the fast kinetics assumption at

We adjusted second-order polynomial functions to derive parameterizations of thermal conductivity as a function of density and this for each of the five temperatures studied. Our parametrization for the vertical effective thermal conductivity at

Temperature dependence of the vertical effective thermal conductivity parameterizations under the fast kinetics hypothesis.

These vertical effective thermal conductivity parameterizations, displayed in Fig.

Normalized effective diffusion coefficient as a function of density under the fast kinetics assumption at

Finally, the estimated normalized effective diffusion coefficients of water vapor are displayed in Fig.

This paper studied two limiting cases, considering either that the kinetics of water vapor deposition/sublimation is sufficiently fast to impose saturated water vapor at the ice interface (very large

As seen in Sect.

The differences between the slow and fast kinetics cases on the effective diffusion coefficient of water vapor were also studied by

All the above reasons suggest that the effective thermal conductivity and diffusion coefficient of water vapor in snow could be well represented under the fast kinetics hypothesis, at least during temperature gradient metamorphism. Further experimental work should be performed to confirm that the fast kinetics assumption generally applies for modeling mass and heat transport in snow and to highlight its potential limitations. Also, the derivation of a theoretical model able to describe heat and mass transfer with arbitrary surface kinetics would allow one to investigate intermediate kinetics in an effort to ultimately select the best modeling assumptions for snow. At the same time, this model could be formulated to explicitly take into account macroscopic convection as this phenomenon has been observed in sub-arctic shallow snowpacks

We showed that in the fast kinetics case, the pure conduction part

This paper investigates the effective thermal conductivity of snow and its relationship to the diffusion of water vapor and its associated latent heat. Using theory, we show that the kinetics of the sublimation and deposition processes at the ice surfaces plays a significant role on the transport of heat in snow. In particular, if the kinetics is slow, we recall that snow can be treated as an inert medium and that heat transport only occurs through conduction in the ice and in the air. In contrast, if the kinetics is fast, vapor transport and latent heat effects become an integral part of heat transport, and the effective thermal conductivity of snow is composed of a purely conductive term and a term proportional to the water vapor diffusivity. Moreover, we show that under the latter hypothesis there is a simple linear relationship between the effective diffusion coefficient of water vapor in snow and the effective thermal conductivity. Since the effective thermal conductivity of snow rarely exceeds

We complemented this theoretical work by finite element simulations of heat conduction through snow microstructures obtained with computed tomography. The simulations were performed on a total of

Using this new set of numerical simulations, we show that the influence of vapor transport in the fast kinetics case can lead to a significant increase in the effective thermal conductivity compared to the slow kinetics case, up to

The codes for the numerical simulations and their analysis will be provided upon direct request to the corresponding author.

The computed values of effective thermal conductivity of snow and of effective diffusion coefficient of water vapor in snow are available in the Supplement of the article.

The supplement related to this article is available online at:

The research was designed by FD, KF, and PH. FD obtained funding. KF performed research and wrote the paper with inputs from FD and PH.

The authors declare that they have no conflict of interest.

We acknowledge Marion Reveillet, Marie Dumont, François Tuzet, Neige Calonne, Anne Dufour, and the ANR JCJC EBONI (grant no. ANR-16-CE01-006) for providing the tomography scan of a melt forms sample. The rest of the unpublished samples were obtained thanks to the help of Jacques Roulle and the tomography apparatus with funding from the INSU-LEFE, the LabEx OSUG, and the CNRM. We thank Neige Calonne and Marie Dumont for their valuable inputs for the article. We are thankful to Charles Fierz and the two anonymous reviewers for reviewing the manuscript and to Carrie Vuyovich for editing it.

This research has been supported by the Climate Initiative program of the BNP-Paribas Fondation (Acceleration of Permafrost Thaw program grant).

This paper was edited by Carrie Vuyovich and reviewed by Charles Fierz and two anonymous referees.