We acquired tomographic time series of two similar snow samples, containing a thin layer of mineral dust and undergoing metamorphism under isothermal conditions and moderate vertical temperature gradient conditions.

The mineral dust used to contaminate the snow samples was Mongolian sand. Its mineralogical composition was 60 % quartz, 20 % albite, 11 % calcite, 9 % illite and traces of kaolinite (∼ 0.1 %), as measured by X-ray diffraction using the Rietveld technique . The elemental concentration for major constituents of mineral dust was determined by wavelength-dispersive X-ray fluorescence following , resulting in concentration ratios Si/Al=4.3 and Fe/Ca=0.55. The total dust mass was estimated from this elemental composition as in , leading to an iron mass fraction in the measured dust of MRFe%=3.6 %. These values are within the typical range of dust composition reported by .

The two samples composed of recent snow (DF/RG), according to the international classification for seasonal snow on the ground , were prepared as follows. Fresh snow was first collected in the field and stored at -20 ∘C for 9 d. It was then processed in a cold room at -5 ∘C: a snow layer was sieved with a sieve size of 1.6 mm on a 40 cm ×60 cm polystyrene plate. Half of its top surface was contaminated by manually sieving dust with a metallic sieve (0.5 mm ×0.5 mm holes). To smoothen this process, a textile mesh (0.5 mm ×1 mm holes) had been previously folded and placed in the sieve before operation. Another snow layer of same properties was then sieved on top of the whole surface. Finally, this preparation was left sintering in a closed insulated box for 24 h. Snow cores with a diameter of 10 mm were vertically extruded from the prepared snow so that only about half of the core horizontal cross sections were contaminated with dust. The top and bottom of the cores were then cut, resulting in a dust surface located in the middle of the core with a total height slightly above 8 mm. The cores were then smoothly inserted into the aluminum sample holders (cylinders of diameter and height of 10 mm) and sealed with copper columns on which 1 mm thickness ice lenses had been previously grown. The snow samples thus contained a thin horizontal dust layer in their middle spreading only one half of their cross section (see Fig. ). The samples were then stored at -20 ∘C for 4 d.

In order to accurately control the temperature boundary conditions of a sample while it is scanned by X-ray tomography, we used a previously developed cryogenic cell , which is based on a thermoelectrical regulation and a vacuum insulation. The temperatures of the Peltier modules are monitored during the whole experiment and controlled with a relative precision of about ±0.01 ∘C. The Pt100 probes were calibrated together in a thermo-regulated bath so that the accuracy of their temperature difference is below ±0.02 ∘C, and their absolute temperature is known at ±0.05 ∘C. The insulation with vacuum is designed to avoid any inadvertent lateral heating of the sample . One sample was subjected to isothermal conditions at -2 ∘C. Due to an imperfect thermal connection in the experimental setup (later detected on X-ray images), a slight temperature gradient was unavoidable but was estimated to be lower than 5 K m-1 during this experiment. The other sample was subjected to a downward vertical temperature gradient of 19 K m-1 (top is cold, bottom is warm) at a mean temperature of -2 ∘C.

After complete setting of the boundary conditions, the snow samples were regularly scanned at a resolution of 7.5 µm via X-ray tomography (EasyTom XL Micro, RX Solutions) at Laboratoire 3SR, Grenoble. The X-ray tube was powered by a current of 100 µA and a voltage of 80 kV. Assuming a reasonable X-ray source efficiency of 0.1 %, i.e., a very small fraction of the energy input is effectively converted into X-ray photon energy, it can be assumed that the absorption of the X-rays by the sample holder, the snow and the dust has no significant impact on the sample temperature. The detector was composed of 1920 pixels ×1536 pixels with a physical pixel size of 127 µm. Each scan consisted of 1440 radiographs covering a 360∘ rotation. One radiograph was computed as the average of five frames captured at one frame per second. A complete scan lasted 2 h and was started every 3 h, on average. The isothermal experiment lasted about 100 h, leading to 32 tomographic images. The temperature experiment lasted about 200 h leading to 75 images. The 3-D image resolution (7.5 µm) and the subsequent tomograph settings were chosen to capture most of the dust particles with a mean mass diameter around 5 µm (according to the Coulter analysis for dust particles with a diameter between 0.6 and 20 µm; see next paragraph) while keeping the scanning time (2 h) short enough to correctly measure the microstructure time evolution. These tomographic data thus comprise the most resolved time series, 7.5 µm spatial resolution and 3 h average temporal resolution, of snow evolution ever measured.

At the start of the tomographic time series, according to the first tomographic image, the sample of the isothermal experiment was characterized by a density of 215 kg m-3 and a specific surface area of 32 m2 kg-1 and the sample of the temperature gradient experiment by a density of 230 kg m-3 and a specific surface area of 32 m2 kg-1. A post-mortem Coulter analysis of the samples indicated that the dust content was about 0.5 mg g-1 for dust particles with a diameter between 0.6 and 20 µm.

The sets of radiographs were reconstructed into 3-D images of the different attenuation coefficients of the materials composing the samples: air, ice, aluminum sample holder and dust. Due to slight movements of the tomograph elements, the different images of one time series do not necessarily share the same coordinate system. The images were thus registered assuming that the sample holder neither move nor deform with time. To this end, the Python package SimpleITK was used to find the composition of an isotropic scaling and a rigid transformation that best matches the sample holder grayscale image between the considered and the reference images, according to a grayscale correlation metric. The sample holder was not perfectly cylindrical but contained different marks so that the 7 degrees of freedom of the transformation can be unambiguously determined. Translations up to 50 µm were corrected with this procedure, while the other degrees of freedom remained mainly unchanged.

The segmentation of images containing multiple materials is notoriously difficult due to the presence of noise and partial volume effects increased by blur. Following the approach of , we performed a 4-D (3-D space + time) energy-based segmentation. The energy function was composed of a data term representing intensity priors, i.e., the likelihood of a voxel to be composed of one material according to its grayscale value and a regularization term representing the cost for voxels to be segmented differently from their spatial and temporal neighboring voxels. The data term was based on Bayesian classification of material mixture, accounting for partial volume and gradient modeling . The regularization term was set as the area of the 4-D surface separating the different materials in space and time. showed that the regularization using the surface area between air and ice is of particular interest in the segmentation process by mimicking the curvature-driven metamorphism. showed that regularization in time is essential to get a consistent segmentation of time series.

Three main geometrical characteristics were derived from the tomographic data: volumetric contents, surface areas and vertical displacements with time. To avoid edge effects due to the sample holder and the ice lenses, these characteristics were computed on a sub-volume with a 1 mm margin to the sample holder and to the lens initial position, i.e., on a centered cylinder with a diameter of 8 mm and a height of 6 mm. Volumetric contents were computed by voxel counting in the segmented images. The typical size of the dust particles, defined as non-connected dust parts, was computed as the radius of a sphere with the same volume. Surface areas (total area and contact area) were computed on the segmented images by the Crofton approach, which showed good accuracy with anisotropic structures . An anisotropy factor Q was also calculated based on the ice–air stereological surface areas in the vertical direction Sz and horizontal directions Sx and Sy. It was defined as Q=(2/Sz)/(1/Sx+1/Sy) . We also computed the ratio η between the dust–ice contact area and the dust surface area for each particle. A high η value means that the particle is almost completely embedded in the ice matrix. Due to the limited accuracy in computing surface areas of discretized objects, which are small compared to the voxel size, the dust particles with a volume smaller than 53 voxels were not taken into account here. Nevertheless, small contact areas of the order of the resolution between the ice matrix and the particles might still be missed due to segmentation errors. Computation of the displacements of the dust particles and the ice interface was more complex. Almost no dust particle entered or exited the considered volume during the experiments, and the movement of the overall dust center of mass thus provided a robust estimation of the bulk dust movement. It was not possible to measure the total water fluxes with a similar approach since ice and vapor entered and exited the measured volume. The movement of the ice interface between two successive scans was calculated using an implicit representation of the ice surface as the zero-isosurface of a certain level set function ϕ. The function ϕ was here defined as the signed Euclidean distance map of the segmented ice component, smoothed with a Gaussian kernel with a standard deviation of 2 voxels. The level set equation links the interface velocity v to the temporal and spatial derivatives of ϕ: ∂ϕ∂t+v⋅∇ϕ=0. The time derivatives ∂/∂t were computed between two consecutive images generally separated by 3 h. The spatial derivatives ⋅∇ were computed by convolution with a first-order derivative operator. Assuming that v is normal to the interface, it was possible to recover the full velocity vector and, in particular, its vertical component. Lastly, each dust particle was tracked individually between two consecutive grayscale images by volumetric digital image correlation . This technique captures sub-resolution displacement of dust particles but fails to track large dust displacements (e.g., exceeding the particle size). We rejected the displacements when the tracking metric, namely normalized cross correlation, was lower than 0.9 or when they were computed on a particle that was too small (particle volume lower than 53 voxels). Exemplary slices showing the binary segmentation and the ice interface velocity computation are provided in the Appendix (Fig. ).

The snow under temperature gradient conditions evolved into faceted crystals and depth hoar (FC and DH; , Fig. a) while the snow under isothermal conditions remained recent snow (DF and RG; decomposing and fragmented precipitation particles, and rounded grains), , Fig. b). Movies of the microstructure evolution are provided as Supplement (Movies S1 and S2). The time evolution of the snow microstructural properties is shown in Fig. . Snow density remained almost constant with time, with a standard deviation of 0.7 kg m-3 under temperature gradient conditions and 0.6 kg m-3 under isothermal conditions (Fig. a). Temperature gradient metamorphism creates an upward water vapor flux and an apparent downward solid ice flux e.g.,. The constant density under these conditions thus indicates that the water fluxes (vapor and solid) at the sample top and bottom were nearly equal, without any apparent edge effect on the measured snow volume. The specific surface area decreased in a linear fashion with time at a mean rate of -0.043 m2 kg-1 h-1 under temperature gradient conditions and -0.066 m2 kg-1 h-1 under isothermal conditions (Fig. b), which is in line with previous studies e.g.,. The anisotropy factor Q (see Sect. ) remained constant under isothermal conditions whereas temperature gradient metamorphism led to an increase in Q between 50 and 150 h due to the formation of ice structures that tend to be more aligned with the water vapor fluxes (Fig. c). This is in agreement with previous studies e.g.,.

The dust volume estimated from the tomographic images was on average equal to (2.3±0.05)×10-9 m3 (±1σ) in the temperature gradient experiment and (2.6±0.08)×10-9 m3 in the isothermal experiment. The particles which can be detected both by tomography and Coulter analysis, only consists of particles with a typical size in the range 5 to 20 µm. The particles within this size range represented a dust content of 0.2 mg g-1 (40 % of total) according to the Coulter analysis, and more than half of this content was detected by tomography. Almost none of the dust particles, mainly located in the middle of the sample, could escape the volume analyzed by image processing (see Sect. ). The total dust content detected by tomography is therefore expected to be constant with time. The relatively low standard deviation of the dust volume thus indicates that the detection of dust particles in the tomographic data was consistent.

The relative position of dust particles in the ice matrix is affected by snow metamorphism. Some dust particles are obviously progressively embedded in the ice matrix during temperature gradient metamorphism (see Movie S1). To quantify this process, we computed the ratio η between the dust–ice contact area and the dust surface area for each particle (see Sect. ). At the beginning of the experiment, both samples exhibited similar distributions of η characterized by low values. Under temperature gradient conditions, the number of dust particles with a high η value clearly increased with time (Fig. a), while this trend was not present under isothermal conditions (Fig.  b). The relative volume of dust that is more connected to ice than air (η>0.5) increased from about 2 % to 20 % under temperature gradient conditions and remained stable around 2 % under isothermal conditions (Fig. c). This volume decreased slowly under temperature gradient conditions after 100 h, which could correspond to the progressive release by the ice matrix of dust particles that were lying on the ice surface at the beginning of the experiment. We observed that the downward ice growth can completely surround dust particles from the top (Fig. d). It is not the dust particles that enter the ice matrix but the ice that absorbs the particles. Note that our observations may be impacted by the nature of the LAPs, e.g., by its conductivity or water affinity.

The ice interface grows in the direction of the temperature gradient, while the vapor flux is in the opposite direction, which results in an overall positive upward water mass flux e.g.,. In contrast, dust does not sublimate and is thus necessarily characterized by a downward movement driven by gravity. We classified three main types of dust movement by order of displacement magnitude and continuity in time: (type 1) small movements due to the gravity-driven settlement of the ice matrix, (type 2) moderate movements of dust lying on a sublimating ice surface and (type 3) large and sudden movements due to particles falling into the pore space after loss of contact with the ice matrix.

Three different techniques were used to compute dust and ice movements from the tomographic images (see Sect. ). The apparent movement of the ice structure is mainly due to sublimation–condensation at its surface and possibly slight gravity-driven settlement of its volume. This movement was characterized by the displacement of the ice surface via a level set approach. In contrast, dust particles move as solid bodies. The overall dust movement was characterized by the displacement of its center of mass, and the individual movement of each particle was computed by digital image correlation. The latter technique does not capture large movements, and we can thus assume reasonably that only movements of types 1 and 2 are captured by this method.

Figure a and b show the distribution of dust and ice surface vertical speeds for the two samples. The movements in the isothermal experiment (typically less than 4 µm between two successive images) were on the order of magnitude of the image analysis resolution. In contrast, non-negligible movements of dust and ice were present in the temperature gradient experiment. The vertical speeds of ice and dust (not including type 3 movements) appear to be very close. Differences may occur by a preferential deposition of dust particles on flat upward-looking ice interfaces that evolve more slowly than the highly convex interfaces aligned with the gradient.

Figure c shows the cumulated displacement of the ice interface and of the dust particles. Again, vertical movements were only detectable for the temperature gradient experiment, in which a -200 µm mean displacement of the dust center of mass was observed. The mean displacement of all dust particles is very close to the mean displacement of the ice interface. However, even if the dust movement is driven by the ice movement and gravity, this resemblance seems coincidental since type 1 movements are significantly smaller than the ice ones, and type 3 movements are significantly faster. Based on linear regressions between the cumulated displacements and time in the permanent regime (time longer than 140 h; Pearson correlation coefficient R2>0.98), we estimated a total vertical speed of dust of -1.49 µm h-1, with the falling dust (type 3) contributing to 55 % of this movement. In addition, this global downward movement is accompanied by a vertical spread of the dust mass: +100 µm for the whole temperature gradient experiment in contrast to +5 µm for the whole isothermal experiment.

The downward movement of dust is linked to its relative position to the ice matrix. Indeed, Fig.  shows that dust particles highly connected to the ice matrix (typically η>0.4) do not move significantly. In contrast, particles slightly connected to the ice matrix exhibited larger mean downward movements. Particles embedded in the ice matrix can be considered as markers of the ice matrix settlement (type 1). Particles slightly connected to the ice matrix may follow the ice surface movement (type 2) or fall in the pore space (type 3). Indeed, a slight connection to the ice matrix implies that the connecting ice is also close to air and may therefore sublimate. In addition, the dust velocity varies with particles size: smaller particles are slightly faster than the larger ones for movement of type 1 and 2 (Fig. ). This trend with particle size is important since smaller particles have more impact on albedo reduction e.g.,. It can be speculated that this trend is due to the fact that large particles have a larger absolute surface connected to the ice, thus reducing the speed of ice sublimation below the particle. For movements of type 3, we also expect the smaller particles to move faster than the large ones. Indeed, their fall into the pore space might be stopped later by the ice matrix, compared to the fall of larger particles (see Fig. ). The distribution of η did not exhibit any dependency to particle size (not shown).

The findings presented are especially important for arctic and sub-arctic snowpacks which often undergo strong temperature gradient metamorphism over several months. showed that in low-arctic shrub tundra (Quebec, Canada), the basal snow layer is exposed to a temperature gradient mostly above 20 K m-1 between mid November and early February. Assuming that the dust downward velocity remains constant and equal to -1.49 µm h-1 for 3 months, or that the dust downward acceleration remains constant and equal to -0.0049 µm h-2 (parabolic fit) during this period (Fig. c), the dust center of mass would move down by 3.2 mm or 22.9 mm, respectively. The particles would also spread vertically, as shown in the temperature gradient experiment by the increase in the standard deviation of dust position. hypothesized that, for BC particles with a typical mass median diameter of 0.4 µm, such a downward movement would lead to a cleaning of the snow layer because the particles would end up on the ground. The vertical displacement measured in this study is rather small compared to the typical thickness of depth hoar layers (5–30 cm) found in arctic regions. However, the main driver of dust motion in dry snow is the presence of intensive sublimation and condensation. The movements of type 2 are directly linked to the local ice interface velocity, and particles embedded within the ice matrix (characterized by the movement of type 1) can escape and enter movement of types 2 or 3 at a rate linked to the ice residence time . Therefore, dust motion may be faster for larger temperature gradients that are often found in arctic and sub-arctic snowpacks, e.g., roughly 50 K m-1 on average over winter and up to 300 K m-1, as reported by . Moreover, small particles tend to move faster than the big ones (see Fig. ), and particles whose sizes were lower than the image resolution (7.5 µm) were not detected here. We could thus hypothesize that BC particles would move downward faster than the dust particles analyzed here.

Eventually, we numerically evaluated the potential impact of the observed dust motion on the solar energy absorbed by the snowpack if the layer containing dust is at the snow surface. To do this, we used the Two-streAm Radiative TransfEr in Snow model TARTES . We considered a semi-infinite snowpack composed of dry snow with a density of 230 kg m-3 and a specific surface area of 32 m2 kg-1, and four idealized hypothetical cases: (a) clean snowpack, (b) contamination with a 1 mm thick layer of dust (concentration of 1 mg g-1) on top, (c) same contamination but the dust is moved 10 mm downwards and (d) same contamination but the amount of dust is spread over the top 21 mm. Case (d) roughly corresponds to the overall displacement extrapolated from our data and estimated for 3 months of temperature gradient (see calculation above). We assumed that the mixing state of dust in snow is the same in all cases as it appears to stabilize after a few days (Fig. ). The dust particles are all assumed to be outside the ice matrix (external mixture). We considered typical mass-absorption efficiency of mineral dust as measured by (sample Lybia PM_2.5 therein) for the mass fraction of dust particles of aerodynamic diameters lower than 2.5 µm. The simulated particles are thus smaller than those followed with the tomographic images. This choice was made to be consistent with the common atmospheric dust found in snow e.g.,. The impact on albedo simulated here will also vary significantly for other types of dust with different chemical composition (e.g., Table 4 in ). The albedo in the visible wavelengths is affected by the presence and location of the dust (Fig. a). Exposed to a typical winter incoming solar radiation with a solar zenith angle of 70∘ (instantaneous incoming energy assumed to be 376 W m-2), the instantaneous energy absorbed by the snowpack is 66, 76, 73 and 74 W m-2 for cases a, b, c and d, respectively (Fig. a). Figure b displays the instantaneous absorbed energy for this incoming radiation, the four idealized snowpacks and varying dust content. It shows that when dust content reaches 0.5 mg g-1, differences larger than 1 W m-2 can be found between the different dust locations. Note that this impact remains relatively small compared to the impact due to dust burial by new snow deposition, which may episodically occur during the winter season. Moreover, showed that the sublimation of the snow surface to the atmosphere can enhance the concentration of LAPs at the topmost layer, which may counterbalance the optical impact of the observed downward movement.