The thermo-mechanical cell was designed for micro-CT measurements of snow settlement under controlled load and temperature conditions. To this end, we drew inspiration from the Snowbreeder 5 cell developed at the WSL-SLF and adapted the setup of for settlement analysis. In particular, we controlled the sample temperature independently of the cold-room temperature, and we monitored the applied load to ensure that it remained constant throughout the tests.

The thermo-mechanical cell (Fig. ) consists of an aluminium cylindrical tube with a diameter of 20 mm and a height of 90 mm. The wall thickness of the aluminium cell (0.5 mm) was chosen to allow X-rays to penetrate the cell and image the sample. A piston transmits the load to the sample. The mass of the piston and crown alone is m=28.79g. Because the piston is hollow, additional mass can be added to increase the applied load.

Two Peltier thermoelectric modules (TEC-14,7-3,9-34,0-70-30/0-R from Arctic TEC), referred to simply as Peltier units, are placed at the lower and upper ends of the cell, providing precise control of the sample temperature (Fig. ). The Peltier modules are driven by a controller unit (TEC-1161-4A from MEERSTETTER). These Peltier modules ensure that temperature differences within the cell remain on the order of 10-3K and provide an absolute temperature accuracy better than ±0.5K. Heat sinks and fans dissipate the heat produced by the Peltier units. The cell can be operated at temperatures up to ±10K relative to the ambient temperature. A thick insulating styrofoam ring is added around the cell to ensure isothermal conditions. The Peltier units maintain temperature control during the experiment and prevent sample heating during scans. Temperature is measured at each Peltier module at 0.5 Hz.

Two additional measurements are integrated into the system:

Sample top displacement: A laser sensor (OM30-P0100.HV.YUN from Baumer) mounted on the supporting frame measures the distance to a crown fixed to the piston applying the load to the snow sample. It is used to track the vertical displacement of the top of the snow sample. The crown design enables displacement tracking regardless of the cell orientation. The laser signal is transmitted via a 16-bit ADC acquisition card (MCCUSB234 from Digilent) to a computer located outside the cold room. The displacement data are recorded at a frequency of 0.2 Hz, with a measurement range of 15 mm and a resolution of 1 µm. The vertical strain εzz is defined as:3εzz=h-hiLwhere L is the sample height, and h and hi are the current and initial laser distances, respectively.

The evolution of the snow microstructure was monitored during the compression tests using an X-ray tomograph (DeskTom130, RX Solutions). Every 8 h, a series of scans was acquired; each series consisted of one high-resolution scan (HR) and one low-resolution scan (LR). The HR scans captured an inner region (diameter of 10 mm and height of 10 mm) with a nominal voxel size of 11.23 µm, and the LR scans covered the whole sample at a nominal voxel size of 27.44 µm. The scan settings for the different modalities are summarized in Table . The greyscale attenuation images were segmented into pore space and a continuous ice matrix .

To assess the consistency between the bulk deformation measured by laser and the deformation within the sample, axial strain was computed using Digital Volume Correlation (DVC) applied to low-resolution (LR) images. The analysis was conducted over a single region of interest (ROI) with dimensions of 10.3×10.3×10.3mm3.

Snow samples were prepared as follows. Natural snow was collected in the field and stored in styrofoam boxes in a cold room at a constant temperature of -20°C for several months, where it evolved into Decomposing and Fragmented precipitation particles (DF) through isothermal metamorphism. The snow particles were then sieved into a sample holder using a 1.6 mm sieve. The sample holder was cylindrical, with a diameter of 17 mm and a height of 17 mm, and was used to mould the snow samples. The diameter of the sample holder, 17 mm, was smaller than the 20 mm diameter of the cell tube, thereby creating unconfined conditions during the creep test and avoiding uncontrolled sidewall friction (Fig. ). The samples were left to sinter for at least two weeks at -18°C before being placed into the cell tube.

We conducted creep tests with stepwise temperature variations to determine the temperature sensitivity of snow viscoplasticity. Temperature control was achieved using the Peltier modules. A stress of 1.25 kPa was applied to the snow sample, corresponding to the overburden stress exerted by a 0.45 m thick snow layer at the density used in the experiment. To limit lateral heat transfer, the cold room temperature was set to the average temperature of the experimental series. Each experiment lasted 5 d and consisted of 5 temperature steps, each lasting 1 d, with temperatures ranging from -6 to -18°C in 3 °C increments. The first 8 h of each experiment, corresponding to sample stabilisation (i.e., densification of the layer at the sample-piston interface and the establishment of steady-state conditions), were excluded from the analysis. This protocol was repeated on five snow samples with initial densities ranging from 278–320 kgm-3

Note that, due to a technical issue, samples 1 and 3 do not include a temperature step at -18°C.

The temporal evolution of the thickness e of a layer subjected to an overburden stress σ is related to the vertical strain rate through ε˙zz=e˙/e (taken to be positive in compression). In its generic form , the densification law can be expressed as a function of the solid fraction Φ=ρ/ρs, where ρ is the snow density and ρs=917.15kgm-3 is the density of ice: 4ε˙zz(T,σ)=ε˙0(T)f(σ,Φ),f(σ,Φ)=Sνσσ0,sΦmn, where Sν is the boundary condition prefactor, equal to 1 here for free boundary conditions, n is the stress exponent, m is the solid-fraction exponent characterising the microstructure, and ε˙0(T) is the temperature-dependent prefactor. The latter is set to 1 s-1 at the reference temperature T0=263.15K, for which the reference stress of ice is σ0,s=272MPa . The stress exponent was set to n=2.2 and the solid-fraction exponent to m=4, based on experimental and modelling studies .

The objective of the present study is to determine the temperature dependence of the prefactor ε˙0(T). To quantify the influence of temperature on strain rate, the sample must undergo measurable deformation; this inevitably induces densification and thus geometric hardening, i.e., an increase in the reference stress σ0=σ0,sΦm. As a result, even at constant temperature, the strain rate decreases during the test. To minimise geometric hardening, we apply a low overburden stress so that the microstructure remains only weakly deformed. Moreover, we use Eq. () to isolate the temperature prefactor ε˙0(T) by dividing the measured vertical strain rate ε˙zz by its mechanical component f(σ,Φ).

We first show the results obtained on one sample (sample 1). The corresponding data for the other experimental series are provided in Appendix .

Figure  shows the temporal evolution of the sample height, applied load, and temperature. At each temperature step, an apparent change in the slope of the displacement curve is observed, reflecting a change in the strain rate. In addition, small displacement jumps occur at temperature transitions and are attributed to thermal expansion of the experimental cell. These jumps do not affect the subsequent analysis, as their magnitude is small and the analysis focuses on the derivative of the displacement (strain rate) obtained outside the scan periods. Because the applied load is small compared with the sensor measurement range (0–1 kg), the transmitted signal is weak and highly sensitive to the contact quality in the slip ring as well as to external perturbations. Nevertheless, we observe that the applied load remains nearly constant throughout the experiment, varying by less than 0.17 kPa, and is close to the nominal imposed load, which indicates that parasitic friction from the piston or the washer is negligible. In the remainder of this study, the load applied to the sample is therefore assumed to be constant at 1.25 kPa (mass of the piston and crown divided by the sample cross-section).

To quantify the observed slope changes, the effects of densification and geometric hardening are first removed to isolate the temperature dependence of the deformation rate. Starting from the measured strain rate ε˙, the temperature-dependent prefactor ε˙0 is computed and shown in Fig. . Despite significant fluctuations associated with experimental measurement noise, a systematic decrease in ε˙0 is observed at each temperature step. Figures  and show the temporal evolution of the strain rate prefactor ε˙0, computed from both laser displacement measurements and DVC for all samples. The measurements are consistent with each other, indicating that the laser-based deformation accurately reflects the sample response despite the sample heterogeneities.

For each temperature plateau, the mean value of ε˙0 is calculated and plotted as a function of inverse temperature 1/T (Fig. ). On a logarithmic scale, ε˙0 exhibits a linear dependence on 1/T, consistent with an Arrhenius-type relationship, for which the slope equals -Q/R. The apparent activation energy of sample 1 is thus estimated to be QS1=134±30kJmol-1. Uncertainties here represent one standard deviation (68 % confidence interval) derived from the covariance matrix, assuming Gaussian uncertainties in both temperature and strain rate.

We can compare the densification model (Eq. ) with the measured vertical deformations using the previously calculated apparent activation energy (Fig. ). If temperature variations are neglected and the temperature is assumed to remain constant at its initial value, the deformation decreases more slowly and deviates from the experimental measurements. This comparison shows that the slowdown in deformation rate is primarily due to the decrease in temperature, as geometric hardening alone cannot account for the observed behaviour.

In this section, we analyse the full set of experimental results. At a given temperature, the temperature prefactor ε˙0 varies by approximately 40 % across the different experimental samples (Fig. ). This dispersion is likely due to differences in second-order microstructural features that the mechanical model does not explicitly capture. In particular, samples 3 and 5 consistently exhibit higher values of ε˙0 across the entire temperature range, whereas samples 1 and 4 show systematically lower values. Since our focus is on the temperature dependence, each sample series was first corrected for these second-order microstructural effects by subtracting the mean offset. Figure  shows the corrected scaled strain-rate prefactor as a function of inverse temperature for each sample. The data collapse onto a single master curve, indicating that all samples share a similar temperature dependence. We therefore use the complete dataset to characterise the viscoplastic response, accounting for the associated uncertainties.

Fitting a single Arrhenius law to all data yields an apparent activation energy of Q=105±6kJmol-1. However, a simple Arrhenius formulation fails to capture the change in slope observed around -12°C, particularly for series 4 and 5. By fitting a two-regime Arrhenius law (Eq. ) to the data, two apparent activation energies are identified: Qh=126±6kJmol-1 and Ql=51±18kJmol-1, with a transition temperature of Tc=-13±1°C. The apparent activation energy associated with the low-temperature regime is subject to substantial uncertainty, owing to the large experimental scatter at low temperatures, where deformation rates are low. The two-regime Arrhenius law is statistically preferred according to the Akaike Information Criterion (ΔAIC=-8), indicating that the data support a transition between two temperature-dependent regimes. 5ε˙0(T)=exp⁡-QhR1T-1T0,T>Tc,exp⁡-QlR1T-1T0+Ql-QhR1Tc-1T0,T≤Tc.

Tomographic scans were performed every 8 h during the tests to monitor the evolution of the microstructure. Figure  shows the evolution of the density and specific surface area (SSA) calculated over a 5.6×5.6×5.6mm3 volume. An increase in density was observed, whereas the SSA showed only a slight decrease. The microstructural variations are minimal during the experiment, with relative changes in density below 4 % (less than 10 kgm-3) and in SSA below 7 % (less than 1.4 m2kg-1). The observed variation of the SSA is within the margin of error of tomographic image analysis e.g.,. It confirms that the microstructure remained essentially unchanged during the compression tests, indicating a minimal effect on the mechanical properties.

In this study, we isolated the effect of temperature on the viscoplastic response by (i) using the same snow sample throughout each test, (ii) limiting microstructural variations across temperature steps, and (iii) accounting for geometric hardening using the model proposed by . This procedure enables robust identification of the temperature sensitivity of snow viscoplasticity and disentangles it from microstructural effects. We discuss here how the choice of geometric hardening model and sample heterogeneity may affect the results.

Figure  illustrates the sensitivity of the identified apparent activation energy to the parameters of the constitutive law. If the apparent activation energy were estimated without accounting for geometric hardening (n=0 and m=0), a value of Q=116±7kJmol-1 would be obtained, corresponding to a deviation of approximately 10 %. This deviation remains relatively small because microstructural changes are intentionally limited. Nevertheless, it highlights the importance of decoupling the effects of geometric hardening and temperature. Furthermore, within the range of n and m typically reported for snow , the sensitivity of Q to these parameters is low, resulting in a maximum deviation of 3 %.

In our model, each sample was assumed homogeneous and characterised by its mean density. However, the tomographic scans reveal spatial heterogeneities within the samples (Fig. ), which can lead to stress concentrations and higher strain rates (Fig. ). Differences in microstructural heterogeneity between samples, as well as second-order microstructural parameters (Fig. ), affect the mean strain-rate but not its temperature dependence. These effects can therefore be readily accounted for, allowing the influence of temperature to be isolated (Fig. ). These results clearly demonstrate the necessity of using the same snow sample to identify the temperature sensitivity of snow viscoplasticity. If different samples are used at each temperature step, microstructural effects can no longer be disentangled from temperature dependence e.g.,.

Our results (Qh=126±6kJmol-1 above Tc=-13±1°C and Ql=51±18kJmol-1 below) fall within the wide range of activation energies reported in the literature (Fig. ) and confirm previous reports of two temperature regimes in snow . For example, observed an activation-energy transition near -6°C, with Q=56kJmol-1 below this threshold and Q=260kJmol-1 above. reported an increase in apparent activation energy from approximately 69kJmol-1 between -18 and -11°C to about 120kJmol-1 between -12 and -5°C for snow with a density of 430 kgm-3. However, our apparent activation energy estimates are substantially higher than those reported by , who found an apparent activation energy of Q=54kJmol-1 between -13 and -3°C. The very high apparent activation energy values reported by and are likely due to microstructural variability, because only two temperatures were used to determine the apparent activation energy, which makes the estimated activation energy highly sensitive to differences in microstructure between samples.

At first glance, the viscoplastic response of snow exhibits a temperature sensitivity similar to that reported for polycrystalline ice. In particular, the bilinear temperature dependence inferred from our data yields apparent activation energies consistent with commonly reported values for ice creep. The high-temperature regime is characterised by an apparent activation energy of Qh=126±6kJmol-1, in good agreement with values reported for polycrystalline ice, typically around 130 kJmol-1 e.g.,. At lower temperatures, a second regime is observed, with Ql=51±18kJmol-1, close to the commonly used value of 68.8kJmol-1 for polycrystalline ice below -10°C .

However, this apparent similarity should be interpreted with caution. demonstrated that deformation in snow is primarily governed by basal slip in the ice matrix. This result suggests that the temperature sensitivity of snow viscoplasticity may primarily be inherited from that of basal slip. Yet, the temperature sensitivity of basal slip at the single-crystal scale remains poorly constrained. The apparent activation energy associated with basal slip has been reported to lie in the range 60-90kJmol-1 at low temperatures (≤-10°C). investigated the temperature sensitivity of ice single crystals over the range -20 to -0.2°C and reported an activation energy of Q=70±2kJmol-1, independent of temperature. However, they observed that the stress exponent of basal slip, nb, varied linearly with temperature, decreasing from nb=2.07 at -20°C to nb=1.95 at -0.2°C. Such temperature dependence of the stress exponent is commonly reported in single crystals e.g.,.

In our model, n was fixed at 2.2. However, when this relative temperature dependence of the exponent is taken into account, using the temperature dependence of n reported for basal slip and normalizing its value at -10°C to recover the stress exponent of 2.2 used for snow, we obtained n(T)=2.14-6×10-3T. This yields a single effective activation energy of Q=78±6kJmol-1 (Fig. ) over the full temperature range. This value is consistent with the activation energy for basal slip determined by . This result suggests that part of the apparent bilinear temperature dependence observed at the macroscopic scale may not reflect a true transition in activation energy, but rather the effect of a temperature-dependent stress exponent. These results highlight the need for dedicated experimental and modelling studies to better constrain the temperature dependence of the stress exponent and its contribution to the apparent macroscopic temperature sensitivity of snow viscoplasticity.

In this study, we did not investigate the behaviour of snow above -6°C. Additional experiments in this temperature range could provide further insight into the active deformation mechanisms. In polycrystalline ice, a strongly nonlinear increase in creep rate as the melting point is approached has often been reported, with grain-boundary processes suggested as a possible explanation e.g.,. In contrast, showed that a system deforming predominantly by basal slip, as may be the case for snow , does not necessarily exhibit such an acceleration of creep close to the melting point. Whether the temperature dependence observed here persists at higher temperatures or whether additional mechanisms lead to a nonlinear increase in creep rate therefore remains an open question.

We compare our model with the parametrisation of (BR92), implemented in the detailed snowpack model Crocus, and with the parametrisation of (K75) for fresh snow, implemented in the snowpack model Snowpack . In the BR92 parametrisation, the temperature dependence is expressed as eaη(Tm-T), with aη=0.1K-1, where Tm is the melting temperature and T is the snow temperature in °C. This expression corresponds to the first-order expansion (close to 0 °C) of the Arrhenius law with an apparent activation energy of 62 kJmol-1, which is close to the value obtained for the low temperature regime but significantly lower than that obtained for the high temperature regime. In the K75 parametrisation, the effects of density and temperature are coupled through a power law in the expression of the compactive viscosity, ηK75=7×10-9ρ(4.75-T/40), where ρ is the snow density and T is the temperature in °C.

Figure  shows the strain rate predicted by the detailed snowpack models and our parametrisation for different densities: 100, 200, and 400 kgm-3. The BR92 model exhibits the weakest temperature sensitivity. At low densities, the K75 model is close to BR92, whereas at a density of 400 kg m-3 it shows a variation in strain rate between -30 and 0°C that is similar to our parametrisation. Finally, only our parametrisation exhibits two distinct temperature regimes.

Let us consider an isothermal snowpack 1 m thick with an initial density of ρ=100kgm-3. Figure  shows the evolution of snow height predicted by the different models at different temperatures. At low density and across all temperatures, our model predicts significantly higher strain rates than those obtained with the BR92 and K75 parametrisations, which yield nearly identical results in this regime. This behaviour is consistent with the commonly reported limitations of the BR92 formulation for fresh snow . A clear sensitivity of snow compaction to temperature also emerges. After one hour, compared to a snowpack at -6°C, the snow height at -20°C decreases by 21 %, 23 %, and 21 % for the BR92, K75, and our parametrisation, respectively. After one day, the influence of temperature and the differences in temperature sensitivity among the models become less pronounced. Overall, temperature mainly affects the characteristic timescale of compaction and, to a lesser extent, the long-term snowpack height. For example, after seven days, an isothermal snowpack at -20°C exhibits a snow height that is 20 %, 34 %, and 27 % higher than that predicted at -6°C for the BR92, K75, and our parametrisation, respectively. These results underline that accurately capturing the temperature sensitivity of snow viscoplasticity is essential for predicting snowpack evolution.