Natural snow was collected at Chamrousse (1800 m, French Alps) on 22 February 2011 and stored at -20 ∘C for 2 weeks. This snow was then sieved in a cold room at -5 ∘C to obtain a horizontal snow slab of 100 cm length, 50 cm width, and 14 cm height, composed of rounded grains RG; at 300 ± 15 kg m-3 (result from macroscopic density measurements). The snow slab was confined at the base and the top between two copper plates whose temperature was controlled by a thermoregulated fluid circulation. The whole system was insulated with 8 cm thick polystyrene plates. An illustration of the experimental setup is given in Fig. . Isothermal conditions at -5 ∘C were first applied to the snow slab during 24 h. This aimed at sintering snow grains whose bonds may have been destroyed by sieving. During the following 3 weeks, the temperature of the cold room was held at -4 ∘C and the upper and lower copper plates were maintained at -7 and -1 ∘C, respectively, generating a steady vertical temperature gradient of 43 K m-1 through the snow slab.

The snow slab was sampled using a cylindrical core drill approximately every 3 days over the 3 weeks, leading to seven samples in total at the end of the experiment. Macro photographs of snow particles were also taken to characterize snow type. During the sampling operation, the temperature of the cold room was temporarily held at -7 ∘C (temperature of the upper copper plate) in order to minimize the change of boundary conditions of the snow slab. The polystyrene plates and the upper copper plate were then temporarily removed in order to access the snow slab. The samples were taken in the middle height of the layer and at a minimum distance of 5 cm from edges and from regions already sampled. The air gap created by the sampling was systematically refilled with freshly sieved snow to prevent strong modifications of the thermal field of the snow slab. Immediately after sampling, each snow specimen was put in a plastic box and impregnated with 1-chloronaphthalene. This organic product, in liquid state above -15 ∘C, was poured along the box walls, slowly filling the open pores of snow. Then, the sample was frozen in an iso-octane bath cooled by dry ice (-78 ∘C) to allow for the solidification of the 1-chloronaphthalene. The impregnation is required to stop the metamorphism of the snow microstructure and consolidate the snow sample for further machining processes. The absorption properties of the ice, air and 1-chloronaphthalene ensure a good contrast between these three components for the X-ray tomographic acquisition. Cylindrical snow cores were extracted from the samples by machining with a press drill which is mounted on a lathe and operated in a cold room at -30 ∘C. Each snow core was then glued to the upper part of a copper sample holder by a droplet of 1-chloronaphthalene and sealed into a Plexiglas cap. The prepared samples were finally stored at -20 ∘C until the tomographic acquisitions.

Each core was scanned using the conical X-ray microtomograph of the 3SR lab set with an acceleration voltage of 75 kV and a current of 100 µA. As this microtomograph operates in an ambient temperature room, the snow core was placed in a specially designed cryogenic cell composed of a Peltier module, which maintains a regulated temperature of around -30 ∘C at the bottom part of the copper sample holder. Figure  shows a schematic of this cell. A continuous dry and cold air circulation between the sample holder and the double-wall Plexiglas chambers of the cell prevents the deposition or condensation of water vapor on their sides. In addition, the heat generated by the Peltier module is dissipated by water circulation. The whole system is able to rotate 360∘ during the acquisition. Each tomographic acquisition lasted around 2 h during which 1200 radiographs of the entire impregnated snow sample were taken. Horizontal cross sections of the sample were reconstructed from radiographs using DigiXCT

DigiXCT: http://www.digisens3d.com/en/

The full 3-D tensors of tortuosity τ (dimensionless) of effective thermal conductivity k (in W m-1 K-1) and of intrinsic permeability K (in m2) were computed from 3-D images. For that purpose, specific boundary value problems arising from the homogenization process have been numerically solved on representative elementary volumes (REVs) extracted from 3-D images of snow by using the software Geodict

Geodict: http://www.geodict.de

The anisotropy coefficient A(⋆) was computed for each of the microstructural and physical properties mentioned above, except for density and curvatures. This coefficient is defined as the ratio between the vertical component over the horizontal one, such as A(⋆) = ⋆z/⋆xy, where ⋆ = lSSA, lc, τi, τa, k or K. The property is considered isotropic if it exhibits a coefficient A(⋆) close to 1, otherwise the property is anisotropic. Note that, since SSAxy characterizes the vertical surfaces while SSAz describes the horizontal ones, we study the anisotropy coefficient of the vector lSSA = (1/SSAx, 1/SSAy, 1/SSAz) to be consistent with the other coefficients.

After sieving, the density of the snow slab exhibited slight spatial inhomogeneities (300 ± 15 kg m-3, from macroscopic measurements with a corer). In order to focus only on the evolution of snow properties driven by the temperature gradient, readjusted values of effective thermal conductivity and permeability, kr and Kr, were computed as if the density was homogeneous in the snow slab and equal to 294 kg m-3 (average of the density values computed from 3-D images) using the regression proposed in , respectively, as follows: kr=k×kfitρskfitρ294,Kr=K×KfitρsKfitρ294, with ρs as the computed snow density, ρ294 = 294 kg m-3, kfit(ρs) = 2.5 × 10-6 ρs2 - 1.23 × 10-4ρs + 0.024 and Kfit(ρs) = 3.0 × res2exp⁡(-0.0130 × ρs) where the equivalent sphere radius res = 3/(SSA × ρi). In this way, we obtained readjusted thermal conductivity (kxr, kyr, kzr) and permeability (Kxr, Kyr, Kzr) values for a density of 294 kg m-3.

We used two analytical estimates based on ellipsoidal inclusions to estimate the physical properties of snow in the x, y and z directions: the self-consistent estimates and the dilute beds of spheroids. These estimates require basic microstructural information, which are the volume fraction of each phase (ϕ, 1 - ϕ) and the inclusion aspect ratio and size. These latter were obtained from the 3-D images, and we chose to describe the inclusion characteristics using the correlation lengths (lcx, lcy, lcz).

Figure  illustrates the time evolution of the snow microstructure during the experiment of temperature gradient metamorphism. 3-D images obtained from X-ray tomography are presented together with the corresponding vertical cross section and photograph. The color coding of the 3-D images corresponds to the mean curvature field. For a better visualization of the faceted shapes, the images are presented “upside down”: the top of the images corresponds to the lowest and warmest side of the physical sample. We observe qualitatively that the initial rounded grains become bigger and more angular and faceted with time. After about 200 h, depth hoar is obtained showing characteristic striations on the surface of grains (see photographs in Fig. ). At the end of the experiment, the ice structure is preferentially arranged along the vertical direction, i.e., the direction of the temperature gradient, as shown by the cross sections.

All the snow properties computed based on the 3-D images are summarized in Table . The time evolution of snow density, specific surface area, correlation length, tortuosity of ice and air, thermal conductivity and permeability are depicted in Fig. . The blue, green and red symbols represent the x, y and z values of the considered property, respectively. One can observe the following:

The snow density shows no significant evolution with time and the average value over the experiment is 294 kg m-3 (porosity of 0.68). In detail, low variations between 275 and 315 kg m-3 are observed from one image to another (Fig. a). As explained in Sect. , these variations reflect the spatial heterogeneity initially present in the sieved snow layer, and not a real-time evolution generated by the temperature gradient conditions.

Figure  shows the time evolution of the anisotropy coefficient for five snow properties. Overall, the evolution is the same for all properties: the coefficient increases from values lower than one at the beginning of the experiment to values greater than one at the end. In detail, the magnitude of the coefficients is strongly different from one variable to another. The largest evolution is shown by the coefficient of ice tortuosity A(τi), which increases from 0.86 to 1.90 between 0 and 144 h and then slightly decreases to reach 1.70 at the end of the metamorphism. The anisotropy coefficient of the thermal conductivity A(k) increases from 0.90 to 1.34. The largest increases of A(τi) and A(k) are observed between 0 and 144 h, showing that changes are almost concentrated during this period for both of the properties, as already pointed out above and shown in Fig. . The correlation length and permeability show similar values of anisotropy coefficients, which evolve from 0.96 to 1.25 and from 0.91 to 1.23, respectively. Finally, anisotropy coefficients of air tortuosity and of lSSA are the smallest and evolve during the experiment from 0.97 to 1.13 and from 0.92 to 1.01, respectively.

The distributions of mean curvature of the upward (left panel) and downward (right panel) surfaces of ice are presented in Fig. . They are expressed in terms of occurrence ratio, which gives the ratio in percentage of the ice surface area that exhibits a mean curvature located in a particular range of values over the total ice surface area. The time evolution of these distributions is shown by the plots of different colors. The area averaged and the standard deviation of the mean curvature values are given in Table . Initially (red color), the distribution of upward and downward surfaces are similar, with a peak of mean curvature located around 6 mm-1 and an occurrence ratio of ∼ 3 %. With time, the area-averaged mean curvature decreases gradually, meaning that ice structures tend to become larger. At the end of the experiment (purple color), the upward and downward surfaces exhibit clearly distinct distributions: the peak of mean curvature is now located at 1 mm-1 (occurrence ratio of 4.2 %) for the upward ones, and at 0 mm-1 (occurrence ratio of 4.8 %) for the downward ones.

Using the same representation, Fig.  shows the Gaussian curvature distributions computed from the entire ice–air interface. A log scale was used to allow for a better visualization of the curves. Here, we do not discern between the upward and downward surfaces of ice because the Gaussian curvature distributions of these two cases are similar. Table  provides the time evolution of the area-averaged and standard deviation values. All the distributions are centered at 0 mm-2, but become sharper over time: the maximum occurrence ratio evolves from 4.5 to 17.3 % between 0 and 500 h and the standard deviation continuously decreases. This implies that the proportion of large, flat or cylindrical structures of ice increases. The initial distribution (red color) stands out from the others: it exhibits occurrence ratios which are small between around -50 and 50 mm-2 and large between around -500 and -50 mm-2, compared to those of other distributions. The most important changes in Gaussian curvatures of ice surfaces appear thus early after the beginning of the temperature gradient, where high negative values corresponding to small saddle-shaped structures disappear in favor of low values close to 0 mm-2, which reflect large and less curved structures. Qualitatively, the same observations can be done from the 3-D images of snow at 0 (left panel) and 500 h (right panel) of metamorphism represented in Fig.  by looking at the color maps of Gaussian curvature. In the initial image, we see a lot of green color (negative Gaussian curvature) at bonds between grains, while the final image exhibits mainly yellow-based colors (nearly zero Gaussian curvatures) representing flat and large structures. The seven images of snow represented with the color map of the Gaussian curvature, as well as those of the mean curvature, are available in the auxiliary materials.

Figure  shows the comparison between the time evolution of computed values (dashed lines) and values given by analytical estimates (solid lines) of air tortuosity, effective thermal conductivity and permeability. Computed values are given in the x, y and z directions while the ones given by estimates are described in the z and xy directions. In order to compare estimates and experiments quantitatively, we use the mean of relative differences E(⋆) defined as E(⋆)=⋆from estimate-⋆computed⋆computed, where ⋆ is a component or the anisotropy coefficient of τa, k and K. E(⋆) is given with the associated standard deviation. For the vertical values of properties, E(τaz) = 6.3 ± 4.5 %, E(kz) = -22.6 ± 4.32 % and E(Kz) = -9.6 ± 10.9 %, whereas for the horizontal ones, E(τaxy) = 8.1 ± 5.2 %, E(kxy) = -15.6 ± 6.0 % and E(Kxy) = -8.1 ± 14.7 %.

Using the same representation, the time evolution of anisotropy coefficients from computed values and values given by estimates of the three properties above is shown in Fig. . For the air tortuosity, permeability and thermal conductivity, E(A(τa)) = -1.6 ± 0.9 %, E(A(k)) = -7.9 ± 7.4 %, and E(A(K)) = -1.1 ± 4.2 %, respectively.

From the mean curvature distributions (Fig. ) and the 3-D images (Fig. ), we observe that the snow microstructure undergoes a strong directional faceting under a temperature gradient. At the beginning of the experiment, the top and base of the ice grains are identical in shape and exhibit mainly convex surfaces, characteristic of rounded grains. With time, these regions become strongly different, showing mostly convex surfaces at the top and faceted surfaces at the bottom of the grains, which is typical of faceted crystals and depth hoar. This observation is in agreement with the results of and , who show asymmetries in the mean curvature distributions of snow samples submitted to low TGs (3 and 16 K m-1 at -3 ∘C) during 3 weeks. This asymmetric behavior can be attributed to the growth and decay properties of the ice crystal at the molecular level, where some sites are energetically more stable for the deposition or sublimation of a water molecule (see TLK or Kossel crystal models in ). Thus, the convex surfaces that undergo vapor deposition grow by a layer by layer process resulting in the production of facets. Conversely, the sublimation of convex crystals preferentially leads to the generation of kink and step sites, which results in the rounding of the shapes (see e.g., Knight, 1966, and Flin and Brzoska, 2008, for more details). Our mean curvature results confirm the above considerations: during the experiment, the upward-oriented grain surfaces are warmer than the surrounding air and sublimate, inducing a rounding of the interface. Symmetrically, the downward-oriented surfaces of the ice microstructure, colder than air, undergo vapor deposition and generate facets.

From the temporal evolution of the snow (e.g., Fig. ), the metamorphism can be divided in two periods: (i) between 0 and 73 h, the microstructure, resulting from equitemperature conditions, recrystallizes toward a new pattern of structure to adapt to the temperature gradient conditions. In particular, the small but numerous connections between grains sublimate and favor the growth of large structures (see Gaussian curvature distributions in Fig. ). During this transitional state, the ice structure becomes more tortuous (minimum value of tortuosity over the whole experiment reached at 73 h in Fig. d) because of the decrease of the links between grains. (ii) After 73 h, the structure evolves in the continuity of the pattern developed during the first stage (consolidation): grains and connections become increasingly larger especially in the gradient direction, and the ice network becomes gradually less tortuous. Our observations confirm the previous studies of , and concerning the evolution of bonds under TG.

Because the air tortuosity is about 5 times higher than the ice tortuosity (see Fig. d), our snow samples can be considered as being comprised of a tortuous skeleton of ice surrounded by large channels of air in the three directions. Our initial z value of ice tortuosity of 0.19 (referred as 0A in Table ) is quantitatively in good agreement with the work of , who computed from a 3-D image of rounded grains at 268 kg m-3 a vertical ice tortuosity of τf2 = 4.4, which corresponds to 0.23 according to our tortuosity definition. In accordance with the observations of , , and , the snow density remains constant over the experiment, while grains and pores grow continuously as shown by correlation lengths and curvature distributions. It means that the microstructure is rebuilt without a supply or loss of mass. Moreover, the snow becomes more and more anisotropic with a structure elongated in the vertical direction, as illustrated by the anisotropy coefficients of the correlation length and of the ice tortuosity. From a thermodynamic point of view, this microstructural anisotropy can be seen as a way to decrease the vertical temperature gradient by enhancing the heat flow in that direction .

Heat conduction in snow is mostly due to the conduction of ice which conducts 100 times better than the air. Consequently, the effective thermal conductivity of snow is strongly linked to its density and to the ice tortuosity, as illustrated in Fig. . Anisotropy of thermal conductivity is lower than that of ice tortuosity because the air phase taken into account in the computation of conduction reduces this anisotropy. Air conduction cannot be neglected in the computation of snow effective conductivity even if the ratio ki/ka is about 88 at 271 K, as already shown by e.g., .

We observe two distinct stages in the evolution of the effective thermal conductivity (Fig. e): a decrease between 0 and 73 h, followed by an increase until the end of the experiment. This observation is in agreement with the result of , who also noticed a two-stage evolution of this property in roughly similar conditions (average temperature of -8 ∘C, temperature gradient of 100 K m-1, snow density of 268 kg m-3). This behavior can be related to the evolution of the microstructure explained in Sect.  since (i) the destruction of connections between grains during the first period of the temperature gradient limits the heat conduction in snow and (ii) the growth of large ice structures in the second period, in particular in the z direction, enhances the heat transfer.

As already mentioned, the intrinsic permeability is proportional to h(ϕ)τadc2, where dc is a characteristic length of the microstructure which could be linked to the correlation length or to the inverse of the specific surface area . Consequently, the anisotropy of the permeability depends on both anisotropies of the air tortuosity and of the chosen characteristic length. Since the anisotropy coefficient of permeability, air tortuosity, and correlation length present similar time evolutions, while that of lSSA stands out by remaining close to one (see Fig. ), our results seem to show that the anisotropy of the permeability is mainly related to that of the air tortuosity and of the correlation length for snow. The fact that the permeability is related to the density and a characteristic length can explain that this property is less influenced by the density variations than the thermal conductivity as described in Sect. , but undergoes the same continuous increase over the experiment that is observed for the characteristic length.

Figure  shows an overall increase of the effective thermal conductivity and permeability at a constant density, underlying the role of the microstructure rearrangement . Moreover, a significant vertical anisotropy is observed for both variables with coefficients until ∼ 1.3. Thus, for snow subjected to a temperature gradient, the first approximation of expressing the physical properties depending on the density and on a single characteristic length for the permeability e.g., should be improved by introducing for instance new microstructural parameters which could reflect the anisotropy phenomena.

The computed effective properties of snow were compared with analytical estimates based on basic information of the microstructure such as the snow density and the anisotropy of heterogeneities (air and ice) through the correlation lengths. Even if the analytical estimates fail to reflect all the details of the microstructure (e.g., bonds between grains), our results clearly show that they capture the overall evolution of the considered properties with a mean of relative differences ranging from -22.6 to 6.3 % (see Sect. ). These estimates do not include any fitting parameter; in contrast to the parameterization of where the lower bounds equation (see Fig. ) was adjusted to the computed data of thermal conductivity by introducing two parameters. The fitting parameters are necessary since the lower bounds equation does not take into account the connectivity of both phases, in contrast to the self-consistent estimates. We compared the self-consistent estimates and the parameterization of quantitatively; overall, both estimates are in agreement for the present data.

In both of the presently proposed estimates, the time evolution of physical properties during the metamorphism mainly depends on one parameter Q which is associated with the shape heterogeneity (pore/grain). In our case, the use of the correlation length to define Q seems relevant. Of course, future comparisons between these estimates and other sets of data are needed in order to validate this approach in a large density range and other values of temperature gradients. Let us remark that these self-consistent estimates can be improved by introducing a finer description of the snow microstructure, i.e., by considering the ice skeleton or the pore network as a complex assemblage of several classes of ellipsoidal inclusions with a given orientation, a given aspect ratio and a given volume fraction. However, these improvements will require developing specific algorithms in order to compute, from the 3-D images, the geometric pore size/shape distribution using an anisotropic structural element like an ellipsoid. These computations are not straightforward. Finally, other relevant estimates of these effective properties can be probably obtained by introducing (i) other microstructural information in bounds or exact contrast expansions by computing three- or four-point probability functions on 3-D images or (ii) refined variables allowing us to describe the evolution of grain shapes and contacts between grains (mean and Gaussian curvatures, fabric tensor) during the metamorphism.