Thermal conductivity of snow measured by three independent methods and anisotropy considerations

The thermal conductivity of snow determines the temperature gradient, and by this, it has a direct effect on the rate of snow metamorphism. It is therefore a key property of snow. However, thermal conductivities measured with the transient needle probe and the steady-state, heat flux plate differ. In addition, the anisotropy of thermal conductivity plays an important role in the accuracy of thermal conductivity measurements. In this study, we investigated three independent methods to measure snow thermal conductivity and its anisotropy: a needle probe with a long heating time, a guarded heat flux plate, and direct numerical simulation at the microstructural level of the pore and ice structure. The three methods were applied to identical snow samples. We analyzed the consistency and the difference between these methods. As already shown in former studies, we observed a distinct difference between the anisotropy of thermal conductivity in small rounded grains and in depth hoar. Indeed, the anisotropy between vertical and horizontal thermal conductivity components ranges between 0.5–2. This can cause a difference in thermal conductivity measurements carried out with needle probes of up to −25 % to +25 % if the thermal conductivity is calculated only from a horizontally inserted needle probe. Based on our measurements and the comparison of the three methods studied here, the direct numerical simulation is the most reliable method, as the tensorial components of the thermal conductivity can be calculated and the corresponding microstructure is precisely known.


Introduction
Thermal conductivity is an important physical property of snow.It is especially important to understand heat transfer and mass flux in snow, since snow has a significant effect on the large scale heat budget of the Earth surface.It is a very relevant factor to the Earth's climate system as heat flows through the snow on the ground (Cook et al., 2007).Given a heat flux, thermal conductivity determines the temperature gradient through the snow pack and it is therefore one of the most fundamental properties determining the rate of snow metamorphism.It is therefore not surprising that the first measurements of thermal conductivity of snow were conducted in the late 19th century (for an overview see Sturm et al., 1997).However, science is still struggling to find the most reliable methods to measure effective thermal conductivity, k eff in snow, and, in a next step, to establish a correlation between microstructure and measurements (Calonne et al., 2011).Recent interest in these questions has increased due to the inconsistencies between measurements, their regression curves and simulation models (e.g. between Sturm et al., 1997 andCalonne et al., 2011).
The reason why the measurement of thermal conductivity in snow is difficult is multifaceted.First, snow measurement methods are not standardized, and only recently Calonne et al. (2011) have compared some measurements with different instruments on the same snow samples.This is a serious shortcoming, as it is very difficult (even impossible) to find exactly the same snow structure twice.The often limited number of measurements makes the parameters of the fitting curves difficult to compare.Second, snow is in many cases extremely fragile, causing artifacts during extraction of the sample or during measurement.For this reason usually Published by Copernicus Publications on behalf of the European Geosciences Union.
only one or very few measurements are made within the same snow sample and therefore the intersample variability is not well determined.Third, the microstructural characterization of snow was for a long time semi-quantitative.The recent advent of microtomography and stereological techniques allows for a much more precise quantification beyond porosity.These techniques provide, for example, specific surface area and thickness distributions of ice and air.
Thermal conductivity is important in many different domains, such as food science (Nesvadba, 1982;Cogne, 2003), soil science (Vonherzen and Maxwell, 1959;Abu-Hamdeh and Reeder, 2000) and Earth science (Brigaud and Vasseur, 1989;Popov et al., 1999).In all these areas similar problems in snow are apparent; especially important is the contact problem when measuring with needle probes (Riche and Schneebeli, 2010a).
Measurement techniques of thermal conductivity fall in three main classes: steady-state methods, transient methods, and direct numerical simulations.The steady-state method uses the heat flux plate.The heat flux plate measures the heat flux through two plates placed at each side of the measured sample.The method is standardized in ASTM (2008).This method was used e.g. by Izumi and Huzioka (1975) and Yamada et al. (1974).Needle probe instruments are based on the transient line heat source method (De Vries, 1952;Blackwell, 1954).They are widely used in snow science (Lange, 1985;Sturm and Johnson, 1992;Morin et al., 2010).A variant of these two methods that is applicable to smooth surfaces, is the divided bar method generally used to measure the thermal conductivity of rocks and sediments (Pribnow et al., 2000).The Fourier method, as it was first used by Abels (1892) and recently by Brandt and Warren (1997), measures the approximately sinusoidal temperature change in the ground, and gives a thermal conductivity averaged over the depth of the temperature sensor.During the past decade, the direct numerical simulation of the thermal conductivity was developed (Ams et al., 2001;Kaempfer et al., 2005;Petrasch et al., 2008;Calonne et al., 2011).This computational method has the advantage that the 3-D microstructure is known.Together with the well-known thermal conductivity of ice and air, the effective thermal conductivity of the snow can be calculated.
A problem which has received relatively little attention is the thermal anisotropy in snow.Izumi and Huzioka (1975) made a detailed study on the anisotropy of the snow thermal conductivity, and its evolution during temperature gradient metamorphism.More recently, some studies pointed out the thermal anisotropy of snow (Calonne et al., 2011;Shertzer and Adams, 2011).The determination of anisotropy is an important part of this study.
The thermal conductivity of porous media can be anisotropic.In this case, the effective thermal conductivity k eff is a tensor (Incropera et al., 2006) (in the following, we write for simplicity of notation: k instead of k eff ) with −j = k∇T , where j is the heat flux in x, y, z, k is the ther-mal conductivity tensor with the diagonal components k x , k y , k z (orthorhombic system) and ∇T is the temperature gradient in x, y, z in a Cartesian coordinate system.In the following, we assume k x = k y , because a snowpack usually has a horizontal layering.Arons and Colbeck (1995) refer to Izumi and Huzioka (1975) for measurements of anisotropy.However, there are in fact very few experiments measuring tensorial components.Riche and Schneebeli (2010b) measured two components (horizontal k x and vertical k z ), assuming k x = k y , with the same computer simulation as used in the present study.Calonne et al. (2011) computed all tensorial components of thermal conductivity.
The determination of thermal conductivity is simplified in almost all regression equations (for an overview see Armstrong and Brun, 2008) to 1-D form, and k = k z , with k z being the vertical component.If heat flow is measured parallel to the bedding of the snowpack, and the isotherm at the top and the bottom are parallel, then heat flow will be perpendicular to the bedding, independent of an anisotropic structure.Therefore, anisotropy is irrelevant if regression equations of thermal conductivity are used under such conditions.However, as we show in this paper, this bias was not always corrected for all instruments, especially in the case of the needle probe.The theory for this case will be introduced in Sect.2.2.2.
Heat flux in snow is not purely conductive: it has also a latent heat component.Latent heat is generated through the phase change induced by recrystallization (Pinzer and Schneebeli, 2009b).The ratio between conductive heat flux and latent heat flux depends on the absolute temperature and the temperature gradient.We address this phenomenon in Sect.2.2.4.
We focus on three fundamentally different methods to measure the thermal conductivity of snow: guarded heat flux plate (HFP), single needle probe (NP) and direct numerical simulation (SIM).The main objective is to understand the accuracy and the limitations of these three methods with consideration of thermal anisotropy.To compare the methods, the same snow samples were used for each method, although each method has a different characteristic measurement volume.We analyzed the effect of thermal anisotropy of snow on NP measurements and the effect of latent heat flux in HFP measurements.We corrected the effective thermal conductivity measured with the NP, k NP eff for anisotropy.To exclude effects specific to snow, we compared HFP and NP with inert materials.These materials were chosen to cover the range of thermal conductivity typical for snow.
We do not consider the effect of convection processes on heat transport because convective processes depend not only on the conductive properties of the material, but also on its permeability (Sturm and Johnson, 1991;Arakawa et al., 2009;Zermatten et al., 2011).Additional snow samples were used to calculate the anisotropy of the thermal conductivity with the SIM.The properties of these samples are shown in Table 2.

Inert samples for comparison between needle probe and heat flux plate
The results of NP and HFP measurements were compared using several common porous and non-porous materials.Snow thermal conductivity ranges from 0.024 (k air ) to 0.8 W m −1 K −1 .We selected materials that cover this range, and are suitable for both HFP and NP.We used polystyrene foam, plasticine (Fimo), paraffin wax (Glorex), and water with 5 % agar-agar (Morga AG).Similar materials were used by Pinzer and Schneebeli (2009a).In addition, we used granular and sintered coarse-grained sea salt (grain size 2 mm ± 1 mm) as substitute for snow.These materials do not undergo a phase change at the temperature range of the experiments and give minimal contact resistance for both HFP and NP.

Transient measurement with the needle probe
The theory for transient measurements with a NP for an ideally conducting material is well known (Blackwell, 1954;Carslaw and Jaeger, 1959;Hartley and Black, 1976).The method was applied for thermal conductivity measurements in snow by different groups (Sturm and Johnson, 1992;Morin et al., 2010).The thermal conductivity, k NP (W m −1 K −1 ), is calculated as where t 2 (s) and t 1 (s) are the linear time region of the measurement, q (W m −1 ) the heating power, T (K) the temper-  (Fierz et al., 2009), Density: gravimetrically determined (correspond to the local density of the computed volumes within 15 %), SSA: specific surface area from tomography, i.th: thickness of ice structures by tomography, i.sp: pore size by tomography.ature difference.Note that k NP corresponds to k z in case of an isotropic medium.
In this study, we used an NP (TP02, Hukseflux Thermal Sensors) as used by Morin et al. (2010) and Calonne et al. (2011).The needle is 1.2 mm in diameter and 150 mm long.The applied heating power was 0.07 W m −1 in order to have a total temperature increase smaller than 2.5 • C in the snow.This corresponds to a maximal temperature increase of 1 • C in the time interval t 1 −t 2 used in Eq. ( 1).In order to avoid the www.the-cryosphere.net/7/217/2013/The Cryosphere , 7, 217-227, 2013 F. Riche and M. Schneebeli: Thermal anisotropy of snow effect of a high contact resistance, the first 30 s of the measurement were discarded.Convection effects occurred after 100 s.For these reasons, the thermal conductivity was calculated between t 1 = 30 s and t 2 = 100 s (Morin et al., 2010;Sturm et al., 1997), which corresponds to the linear portion of Eq. ( 1), when plotting the temperature in function of time on logarithmic scale.Before use, the accuracy of the NP was checked with measurements in glycerin 80 % (Glycerin CAS 56-81-5).
In a homogenous block of snow (45 cm × 45 cm × 45 cm), with the sides parallel and orthogonal to the layers, two horizontal and two vertical measurements were performed.The needle was inserted such that the temperature sensor of the needle was always positioned at the same height of the block.From these four measurements, k z , k x and the anisotropy (α, see Sect.2.2.2) were calculated.Grubbe et al. (1983) analyzed the case of line source measurements in anisotropic rock.He developed the general case, when the bedding plane is inclined to the surface of the Earth.Assuming that NP measurements in snow are always parallel or orthogonal to the layering of the snowpack, the following formulas can be used.If the z-axis of the coordinate system is orthogonal to the snow surface, then k x = k y in the horizontal plane and k z in the vertical plane.If the needle is horizontally inserted, the following components are measured (see also Fig. 1):

Anisotropy
with a vertically inserted needle: The vertical component k z can then be extracted from We define the anisotropy factor α = k z /k x , which is the inverse of the factor A used by Grubbe et al. (1983) but the same as the anisotropy coefficient used by Calonne et al. (2011).α can be better compared qualitatively by the anisotropy factor γ of Izumi and Huzioka (1975), which is defined as γ Therefore, k z , if the measurement is done with a horizontal NP, is (5)

Steady-state measurement with the guarded heat flux plate
The guarded HFP applies a constant heat flux at the bottom of the snow sample.After equilibrium is reached, the heat fluxes Schematic representation of a possible corresponding structure:  et al. (1983).The range of the anisotropy factor between 0.5 and 2 is typical for natural snow.The horizontal line at 0.2 W m −1 K −1 indicates the measured thermal conductivity with the NP horizontally inserted.For an anisotropy factor, α < 1, k z is underestimated by up to 25 %.In case of α > 1, k z is underestimated by up to 25 %.Also shown is the horizontal component of the thermal conductivity, k x = k NP v , which corresponds to the value measured with a vertically inserted NP.
at the top and bottom of the sample are equal.The difference between the top and bottom temperatures is measured.As the geometry of the sample is known, the thermal conductivity k HFP (W m −1 K −1 ) is where q f (W m −2 ) is the heat flux across the sample, h (m) is the height of the sample and T (K) is the temperature difference.In our measurements, the layering was parallel to the plates and therefore k z = k HFP .In order to avoid heat convection and snow metamorphism, the applied heat flux and the resulting temperature gradient were kept at ∇T = 10 − 15 K m −1 and the temperature at −16 • C.
The design of the HFP is based on the work of Köchle (2009) (Fig. 2).The geometrical design and the main characteristics of the HFP are: (i) the heat flux is constant, (ii) the sample is insulated and guarded such that the heat flow is perpendicular to the plates, (iii) the heat flux sensors are guarded by additional surrounding material of equal conductivity, (iv) the temperature difference is measured as precisely as possible, and (v) the size of the sample chamber must be sufficiently large so that an undisturbed snow sample can be used.We built an apparatus with a plate size of 210 mm × 210 mm and an adjustable height between 5 and 80 mm.The base of the chamber consisted of 30 mm-thick styrofoam insulation.On top of the insulation, a film heater (Minco, Type

Latent heat flux
Pinzer and Schneebeli (2009b) and Pinzer (2009) made direct measurements of the recrystallization rate of snow, and by this the effective latent heat flux.They could show that the vapor diffusivity in snow, D eff , is close to the diffusivity in air (Giddings and LaChapelle, 1962;Sokratov and Maeno, 2000).
The latent heat flux is calculated using the effective vapor flux present in the snow sample during a HFP experiment.The vapor flux is defined as where J z (kg m −2 s −1 ) is the heat flux in the z-direction, The latent heat flux is then defined as where q L (W m −2 ) is the latent heat flux, L (J kg −1 ) is the latent heat of sublimation of ice.q t (W m −2 ) is the measured heat flow, which includes both the sensible (q f ) and the latent (q L ) heat fluxes.
The values used for these equations and the calculation of the latent heat flux are in Appendix A.

Tomography and direct numerical simulation
Each sample was directly measured, without further preparation, in the µ-CT (Scanco micro-CT80).The temperature during a measurement was controlled and kept at a constant temperature of −15 • C. The nominal resolution of the images was 18 µm.The size of the original image was 2048 × 2048 × 416 voxels, the scanned volume had a diameter of 36 mm and a height of 7.5 mm.

Image processing
From the µ-CT measurements, a volume of 400 × 400 × 400 voxels (7.2 mm ×7.2 mm ×7.2 mm) was extracted from the original data.This volume was then segmented to a binary image.A Gauss filter (σ = 1, support = 2) was applied, then the ice structure was segmented using an adaptive threshold.The obtained density of the ice volume was compared to the measured density of the snow sample in order to control for the segmentation procedure.If the difference in density was higher than 12 %, the threshold in the segmentation process was manually corrected to obtain the right density within 12 %.Depth hoar samples were up-scaled to a nominal resolution of 36 µm.This was necessary to conduct the SIM within a reasonable time and with a larger representative elementary volume (REV) of a sufficient size (volume of 10.8 mm ×10.8 mm ×10.8 mm.From each sample, except depth hoar, two sub-samples where taken to estimate intrasample variability.The numerical simulation uses the code described Kaempfer and Plapp (2009) and Pinzer (2009).It is a linear steadystate thermal finite element simulation.The finite element mesh consists of hexahedral elements, which correspond to the voxels of the µ-CT measurements.As an input the simulation requires the temperature gradient in the snow and the thermal conductivity of air and ice.The boundary conditions are constant top and bottom temperatures and insulated sidewalls.No phase change is implemented in the model.We used the following values: Slack, 1980) and k air = 0.024 W m −1 K −1 .Based on the simulated heat flux and the prescribed temperature gradient the thermal conductivity is calculated as For each k z and k x an independent simulation is necessary.

Calibration of the needle probe and the heat flux plate
We measured the thermal conductivity of the inert materials (polystyrene foam, plasticine, wax at room temperature and at cold lab temperature (−20 • C), agar gel (5 %), and salt in granular form and as a sintered block) with the NP and the HFP (Fig. 3).In homogenous, isotropic and non-porous materials (plasticine, polystyrene, wax), k NP and k HFP were similar.k NP was slightly higher than k HFP for agar, but k NP was 30-40 % lower than k HFP for the granular salt.Similarly, a long heating time (100 s) resulted in a 15 % lower k NP than k HFP for the sintered salt block.A short heating time (30 s) resulted in a k NP 70 % lower than the two other measurements.
A linear regression was calculated for the short and the long heating time NP versus HFP, forcing the curve through the origin.The measurement of the salt block at short heating time was not considered.The correlation between NP measurements and HFP measurements was high: we found that the regression equation for both the short and long heating time was k NP = 0.96k HFP with a root mean square error of 0.07, and all points within 99 % significance.

Numerical simulation
The thermal conductivity was calculated for all samples listed in Table 1 (all measured and calculated thermal conductivities are listed in the supplementary material).Two independent volumes of each sample were calculated, except for the depth hoar sample, where the size of the REV was too large to extract two independent volumes from the scanned Measurements with NP in agar are known to be too high due to convection (Boumaza and Redgrove, 2003).The NP measures a too low a value compared to the HFP for the porous materials (granular salt grains and sintered sea salt).The k NP resulting from a 100 s instead of a 30 s measurement increases thermal conductivity almost threefold for the sintered salt block.
volume.Density and thermal conductivity of these samples are shown in Supplementary Table 1.The density of the two paired samples varied between 0.1 % and 3.2 %, (average 1.3 %), the thermal conductivity between 0.5 % and 34 % (average 6.1 %).The outlier with 34 % was for the faceted snow with a density of 261 kg m −3 .The result of the simulation depends on the threshold for the segmentation of the ice structure.To test for the sensitivity, we increased the threshold by 10 % for one sample.This increased the volumetric density by 4 % and the thermal conductivity by 8 %.

Comparison of the needle probe and direct numerical simulation
The vertical and horizontal thermal conductivities measured with long heating-time NP, corrected for anisotropy, were systematically lower by 10-35 % compared to the values calculated by the SIM (Fig. 4).We found that the trend in anisotropy is in most cases the same . The correlation between k SIM and k NP was k NP = 0.79k SIM − 0.021 (root mean square error 0.023).The difference between the null hypothesis (the 1 : 1 line) and the correlation was significant at the 99 % level.

Comparison of the three methods
The vertical component k z measured by the NP were in all samples lower than the ones measured by HFP and calculated by SIM (Fig. 5).k SIM are systematically lower than k SIM x,z .Colors correspond to the International classification for seasonal snow on the ground (Fierz et al., 2009).Light pink: rounded grains, light blue: faceted crystals, blue: depth hoar.than k HFP z (Fig. 6).The differences between NP compared to SIM and HFP were highly significant, and the difference between SIM and HFP was significant.
The comparison of the relative change of k NP h between the two NP measurements before and after the HFP measurements showed a mean increase of 6.3 % ± 8.4 % (minimum −4.9 %, maximum +18.5 %).

Comparison of thermal conductivity with previous studies
The effective thermal conductivities, calculated by the SIM (k eff = (k z + k x )/2), corresponds well to the data set and the fitting equation presented in Calonne et al. (2011) (Fig. 7).However, the snow samples of Calonne et al. (2011) all have a relatively small anisotropy, except for four samples that are either faceted snow or depth hoar.By taking into account only depth hoar (DH) and faceted crystals (FC) samples, which both have a higher conductivity in the vertical direction, it appears that the effective thermal conductivity of these samples is much more different from k z than for the other samples, such as small rounded grains.By fitting k z of depth hoar and facets samples, the following regression equation is obtained: with a correlation coefficient of 0.986 (for 17 samples, 13 samples from this study and 4 from the study of Calonne et al., 2011) and with k z corresponding to the thermal conductivity of air when snow density is zero, as suggested in Calonne et al. (2011) (Fig. 7).

Anisotropy factor
k SIM z,y was calculated for 35 samples in total (Fig. 8).The temperature of ice and air was always set to −20 • C. We found that the majority of samples showed a certain degree of anisotropy, and isotropic snow seemed to be rather the exception than the rule.The anisotropy factor α was calculated from the data of Izumi and Huzioka (1975) for comparison.The anisotropy of our snow samples with a similar snow structure and density showed approximately the same anisotropy values as those of Izumi and Huzioka (1975).The anisotropy factor was also compared to Calonne et al. (2011)   conduct heat better in the vertical direction.Rounded grains are almost isotropic or conduct heat better in the horizontal direction (Fig. 8).

Latent heat flux in the heat flux plate
All HFP measurements were carried out at a temperature of −16 • C ±1 • C. For a temperature gradient of 11 K m −1 , the calculated latent heat flux was 0.01 W m −2 .In the HFP measurement, the measured heat flux was approximately 2 W m −2 , therefore the latent heat flux (calculated) represented less than 1 % of the measured heat flux (Eq.8).The latent heat flux had therefore no significant effect on the heat flux at this temperature.However, at higher temperatures, latent heat flux becomes significant.At a temperature of −5 • C and with the same temperature gradient, the latent heat flux would increase to 0.14 W m −2 , corresponding to a contribution of 14 %.

Discussion
The thermal conductivities of NP and HFP in the inert, isotropic and solid reference materials lie almost perfectly on the 1 : 1 line (Fig. 3).Both methods, NP and HFP, gave the same absolute values of thermal conductivity in these materials.Both instruments showed no temperature dependence, tested with wax at 20 • C and −20 • C. The higher k NP with agar was caused by convection, and it was also observed by Boumaza and Redgrove (2003).The NP measurement in sintered salt with a short heating time (30 s) is a reminder that NPs with short heating time do not correctly measure thermal conductivity due to contact resistance problems present in porous materials (Riche and Schneebeli, 2010a).The long heating time NP measurements in granular and sintered salt resulted in significantly lower values (50 % for granular, and 20 % for sintered salt grains respectively) than the HFP measurements.We think that there are two possible reasons.The first is that the high contact resistance influenced linearity within the evaluated time span.However, a longer measurement time (longer than 100 s) was not possible, because of the clearly visible onset of convective processes (Sturm and Johnson, 1992).The second reason can be the heterogeneity of the temperature field, as already pointed out in Calonne et al. (2011).It is well known from the measurement of the dielectric properties that the radius of curvature of the electrode must be much larger than the snow grain diameter (Mätzler, 1996).The size of the salt grains was about twice the size of the diameter of the NP, so this empirical rule was clearly violated.The HFP always gave values higher than the SIM and the NP (Fig. 5).The difference between HFP and SIM can mainly be explained by the sensitivity of the SIM to the small variations of snow density.It happened that the densities of the µ-CT-measurements were lower than the measured density of the snow.This effect will cause simulated values lower than the real thermal conductivity value.HFP methods are considered less reliable than NP, because they are more susceptible to measurement-induced metamorphism (Sturm and Johnson, 1992).We observed such a predicted increase of k on the order of 5 % (see Sect. 3.4).However, this increase did not explain the much larger difference between NP compared to SIM and HFP.
The difference between HFP and SIM could be caused by two factors.The thermal grease applied could increase the thermal conductivity.Therefore a correction in the effective height would be necessary.A 2 mm thick layer of thermal grease on both plates would lead to an increase of about 5 % in k HFP .Such a correction was not applied.Another reason could be that the finite resolution of the µ-CT and the threshold eliminated heat-conducting bonds.This leads to a maximal absolute uncertainty of k SIM by 10-15 %.
k NP compared to k SIM gave consistent results.They show the same anisotropy trend in most of the measurements (Fig. 4).However, k NP corrected for anisotropy was significantly lower than k SIM and k HFP .Riche and Schneebeli (2010a) showed that significant microstructural changes around the needle occur, but suggested that this effect was only important for short heating times.Measurements in glycerin show no effect of an air gap on the thermal conductivity (M.Sturm and J. B. Johnson, personal communication, 2010).Calonne et al. (2011) suggested that the measured volume around the NP is too small.Another explanation could be that the thermal field is too far from homogenous conditions for such a thin NP to apply the theory developed for transient methods (Blackwell, 1954;Mätzler, 1996).
The thermal conductivity values obtained with SIM correspond well to other studies (shown in Fig. 7).A regression equation was calculated for k z for depth hoar and faceted crystals samples.This fit shows that the anisotropy of the snow samples has to be taken into account.Thermal conductivity components depend not only on the density but also on the snow type.It is therefore difficult to give only one empirical relation between thermal conductivity and structural properties (such as density) that would be correct for all snow samples.Izumi and Huzioka (1975) and Yamada et al. (1974) showed the existence of anisotropic thermal conductivity in snow.We found that a considerable anisotropy exists for most snow types.The single new snow sample that we measured was almost isotropic.Snow samples with small rounded grains had an α < 1, and faceted snow and depth hoar had in general α > 1, which is similar to the results of Izumi and Huzioka (1975) and Calonne et al. (2011).All depth hoar samples with a density above 200 kg m −3 had an α between 1.25 and 1.75.Melt forms covered a very broad range of α between 0.75 and 1.75, probably depending on the anisotropy before wet snow metamorphism started.Our measurements corroborate that k NP h must be corrected by Eq. ( 5) to give a correct k z .This correction, as proposed by Grubbe et al. (1983), clearly reduced the difference between NP and SIM.The most comprehensive data sets of thermal conductivity (Sturm et al., 1997(Sturm et al., , 2002;;Sturm and Johnson, 1992) and their fitting equations should be corrected for anisotropy, depending on grain type and density.Such a correction is also necessary in homogenous layers of certain snow types, and should be done in future measurements, especially for longterm measurements with needle-probes (Morin et al., 2010).However, NP still shows a systematic bias to SIM and HFP, even after correction for anisotropy.
Snow pack models are currently one-dimensional (Etchevers et al., 2004).They use a single thermal conductivity component, k z .Anisotropy of snow thermal conductivity could therefore have important consequences for heat flow and energy balance in complex terrain.
Based on the results of Sokratov and Maeno (2000) and Pinzer (2009), the latent heat flux was estimated within a factor of two by Pinzer et al. (2012).So far, the latent heat flux was considered as the major weak point of the HFP.However, at low temperatures (as used in this study), the calculated latent heat flux is less than 1 %; it is therefore negligible.The higher thermal conductivity of the HFP measurements compared to SIM can not be explained on this ground.The thermal grease applied between the plates could cause about 5 % increase, but does not explain completely the difference between the measured values of HFP and SIM.

Conclusions
We compared three different methods to measure the thermal conductivity of snow using identical snow samples.This study analyzed the systematic differences found between methods.HFP gave consistently higher results compared to SIM.The NP systematically underestimated the thermal conductivity of snow, even after correction for anisotropy.
We confirm that many snow types are anisotropic regarding thermal conductivity.This implies that at least one horizontal and one vertical measurement with the NP is required, and the individual layers must be sufficiently (at least a few cm) thick.Natural anisotropy can cause an variation of ±25 %.We also found that anisotropy is not always correlated with grain shape (Fig. 8).However, the scatter is very large, and a correction for anisotropy is difficult.
We found that the reproducibility of all methods was similar, with relative differences between the methods of about ±10 %.The HFP proved to be reliable.HFP are very timeconsuming, and best suited for laboratory experiments.In combination with SIM, it could also be used to investigate the effect of latent heat flux on the microstructure.
The current design of the NP produces two problems for snow thermal conductivity measurements.The first is systematically too low a value of the measured thermal conductivity, and the second is that at least two measurements are necessary to determine the anisotropy of the thermal conductivity.HFP and SIM gave similar results.However, HFP is a very time-consuming method and measures only k z .Direct numerical simulations seems to be the most precise method to obtain reliable values of the effective thermal conductivity www.the-cryosphere.net/7/217/2013/The Cryosphere, 7, 217-227, 2013

Fig. 1 .
Fig. 1.The effect of anisotropy of snow on the vertical component of thermal conductivity (k z ) if thermal conductivity is measured with a horizontally inserted NP.Calculations are based on Grubbeet al. (1983).The range of the anisotropy factor between 0.5 and 2 is typical for natural snow.The horizontal line at 0.2 W m −1 K −1 indicates the measured thermal conductivity with the NP horizontally inserted.For an anisotropy factor, α < 1, k z is underestimated by up to 25 %.In case of α > 1, k z is underestimated by up to 25 %.Also shown is the horizontal component of the thermal conductivity, k x = k NP v , which corresponds to the value measured with a vertically inserted NP.
D H 2 O (m 2 s −1 ) is the water vapor diffusivity, C z (Pa m −1 ) is the gradient of vapor pressure, M H 2 O (kg) is the mass of a water molecule, e b and e t (Pa) are the bottom and top pressures according to Murphy and Koop (2005) respectively, h (m) is the height of the sample, κ B (J K −1 ) is the Boltzmann constant and T m (K) is the mean temperature of the sample.

Fig. 3 .
Fig. 3. Thermal conductivity of several inert, isotropic materials measured with the heat flux plate (HFP) and the needle probe (NP).HFP and NP have similar values for inert, non-porous materials.Measurements with NP in agar are known to be too high due to convection(Boumaza and Redgrove, 2003).The NP measures a too low a value compared to the HFP for the porous materials (granular salt grains and sintered sea salt).The k NP resulting from a 100 s instead of a 30 s measurement increases thermal conductivity almost threefold for the sintered salt block.

Fig. 4 .
Fig. 4. Comparison of the horizontal and vertical component of the thermal conductivity from direct numerical simulation, k SIM and needle probe measurement, k NP .All measurements, except two, show the same trend in anisotropy, however all components of k NP x,z

Fig. 5 .
Fig. 5. Comparison of the vertical components of the thermal conductivity, k z for the three methods.HFP gave always the highest value and the NP the lowest one, the SIM is close to HFP.The bars indicate estimated error.Colors correspond to the International classification for seasonal snow on the ground(Fierz et al., 2009).Light pink: rounded grains, light blue: faceted crystals, blue: depth hoar.

Fig. 6 .
Fig. 6.Relative differences of k z for the 8 samples used for both HFP and NP with SIM as reference.HFP values are 20 % higher, NP values are 35 % lower than those obtained by SIM.The median of the relative differences is the line inside the box; the top and the bottom of the boxes are the quartiles.

Fig. 8 .
Fig. 8. Anisotropy factor α of different snow types as a function of density (local density of the volume used to compute the thermal conductivity).The anisotropy shown here was calculated from SIM.The 8 samples measured with SIM, HFP and NP are indicated with error bars.Colors, shapes and abbreviations correspond to the International classification for seasonal snow on the ground(Fierz et al., 2009).Lime: precipitation particles (PP), forest green: decomposing and fragmented precipitation particles (DF), light pink: rounded grains (RG, small rounded particles: RGsr, large rounded particles: RGlr), light blue: faceted crystals (FC), blue: depth hoar (DH), red: melt forms (MF).

Table 1 .
Structural properties of the snow samples used for the intercomparison of the different measurement methods * .

Table 2 .
Structural properties of the additional snow samples used for the simulation of thermal conductivity anisotropy * .
* Snow type: description, ISC: International classification for seasonal snow on the ground