Evolution of individual snowflakes during metamorphism

Authors


Abstract

[1] The morphological changes of individual snowflakes evolving within a dry snow aggregate have been studied using X-ray computed microtomography (micro-CT). Fresh dry snow was collected during a snowfall, sealed, and stored in a  −5°C cold room between periodic observations using micro-CT. Time series 3-D images were used to examine the structural evolution of an individual snowflake within the aggregate over a 2-month period, after which the snowflake had lost its original dendritic structure. Analysis of the aggregate showed that the fraction of large ice particles increased over this period while the total number of particles decreased, presumably to lower the free energy of the snow specimen. This approach enables the study of metamorphism of individual snowflakes in a local environment close to that found in nature. The evolution of structural parameters, including the volume fraction of ice, the surface-to-volume ratio of the ice matrix, the thickness and separation of the ice structure determined by the distance transform of the ice and pore space, were monitored and analyzed using coarsening theories. The computed growth exponent was smaller than the values obtained in the earlier work by Legagneux et al. (2004) and Kaempfer and Schneebeli (2007), who also interpreted the isothermal metamorphism in terms of coarsening theories.

1. Introduction

[2] Shortly after snow precipitation, every single snowflake experiences unique structural changes determined by its specific geometry, dimensions, orientation, and local environment. Accompanied by the physical rearrangement of snowflakes, these changes result in the structural evolution of the snowpack over time. The process is known as snow metamorphism. The structural evolution during snow metamorphism significantly influences the properties of the snowpack, such as the thermal conductivity [Kossacki et al., 1994; Arons and Colbeck, 1995; Schneebeli and Sokratov, 2004; Satyawali and Singh, 2008; Satyawali et al., 2008], the optical properties [Warren, 1982; Flanner and Zender, 2006], and the mechanical properties [Schweizer et al., 2003], and may produce either negative or positive feedbacks on climate change [Taillandier et al., 2007].

[3] Each snowfall typically contains numerous snowflake morphologies, which adds complexity to the study of snow metamorphism. One way to simplify the problem is to examine individual snowflakes. To that end, several studies have been performed using different techniques.

[4] Early work used optical microscopy for periodic observations of individual snowflakes on a slide or in a shallow container sealed with a cover glass [Bader et al., 1954; Kuroiwa, 1974]. Since these observations do not mimic the natural heat and mass exchange between snowflakes, they probably do not accurately reflect the real behavior of snowflakes within a snowpack. In addition, images obtained using an optical microscope can only provide 2-D information. For instance, even though Kuroiwa [1974] noticed a snowflake growing thicker along the c axis, this feature was hard to show in optical images.

[5] Later on, scanning electron microscopes (SEMs) were used to study snowflakes at a much higher resolution and with a greater depth of field [Rango et al., 1996a, 1996b; Erbe et al., 2003; Dominé et al., 2003; Chen and Baker, 2009; Chen and Baker, 2010]. However, because of the high vacuum in the SEM chamber, mass sublimation, a key process contributing to snow metamorphism, is greatly accelerated. Besides, as Dominé et al. [2003] and Chen and Baker [2009, 2010] pointed out, electron beam damage may introduce artifacts in the snow.

[6] Recently, 3-D characterization of snow structure has been provided using new experimental techniques, such as serial sectioning [Schneebeli, 2000] and X-ray computed microtomography (micro-CT) [Flin et al., 2003; Schneebeli and Sokratov, 2004; Kaempfer et al., 2005; Kaempfer and Schneebeli, 2007; Flin and Brzoska, 2008; Heggli et al., 2009]. Micro-CT enables the internal structure of a snow aggregate to be viewed nondestructively and thus enables its evolution to be modeled both in space and time. For instance, based on micro-CT observations of snow aggregates under isothermal conditions, Legagneux et al. [2004], Legagneux and Dominé [2005], and Kaempfer and Schneebeli [2007] applied a classical sintering process and derived growth parameters; Vetter et al. [2010] used a Monte Carlo algorithm to simulate the structural evolution at the grain and pore scale; and Löwe et al. [2010] discussed the isothermal evolution of snow at various length scales from the collective dynamics point of view.

[7] To date, the 3-D structural evolution of individual snowflakes within a freshly fallen snowpack has not been studied. Our present work aims to fill this gap. We performed periodic observations on an undisturbed fresh dry snow specimen over a period of 70 days using a micro-CT and monitored the changes occurring in an individual snowflake. The metamorphism was followed using a time series of 3-D images of the snowflake as well analyzing the structural parameters of the larger volume of snow enclosing this snowflake. In order to reduce the complexity of the problem, in this work we did not impose any temperature gradient on the specimen and considered the whole process under nearly isothermal conditions.

2. Experimental Procedure

[8] A Skyscan 1172 micro-CT scanner in a −10°C cold room was used to study the internal structure of snow specimens. A cylindrical 15 mm diameter by 53 mm high plastic tube with a 20 mm thick styrofoam outer ring for insulation was used as a sample holder. The sample holder was easily differentiated from the snow itself in micro-CT images, because both the plastic and styrofoam have low X-ray attenuation coefficients compared to ice.

[9] In order to eliminate unnecessary specimen transfers and to maintain the initial snow structure, a dry snow specimen was collected directly in the sample holder during a snowfall in Hanover, NH. The temperature of the snow surface was −7°C. After being filled with snow, the holder was transferred within an insulated box to a cold room held at −5°C. Before being examined, the specimen was sealed in the holder and kept in the cold room for 2 h to equilibrate.

[10] During a micro-CT scan, the snow specimen was fixed on the rotation stage at the center of the chamber between the X-ray source and the 2-D detector. As X-rays traveled through the snow, they were attenuated mainly by the ice matrix, and a shadow X-ray projection image was obtained on the 2-D detector. We used a 40 kV accelerating voltage at a current of 250 μA, with a 15 μm image pixel size and viewed 15 mm of the specimen. The specimen was rotated 0.7° between images and rotated through a total of 180° for a total of 265 images, which took ∼18 min to acquire.

[11] Twenty scans were performed on the snow specimen over 2 months. During the entire period, except for the short times when being examined using the micro-CT, the specimen was stored in the −5°C cold room, using an insulated box in order to reduced temperature fluctuations. A thermocouple was used to monitor the temperature in the box close to the location of the specimen, which detected temperature fluctuations of only ±0.2°C.

[12] Projection images were processed using the software NRecon and CTAn, supplied with the Skyscan 1172 scanner. Gray-scale cross-sectional images were reconstructed from projections using NRecon. Five parameters were adjusted to minimize reconstruction artifacts, including beam-hardening correction, minimal and maximal density values, misalignment compensation, ring artifact correction, and Gaussian smoothing. Usually, as a polychromatic X-ray beam is attenuated, the energy spectrum shifts toward higher energies. This results in a dishing effect in the specimen, which appears to be less dense at the center than at the periphery. Beam-hardening correction was used to minimize this dishing effect and a 100% correction was applied in this work. Minimal and maximal density values were determined based on the X-ray attenuation coefficient histogram. Usually, there are two well-separated peaks in the histogram for a typical two-phase material [Lindquist et al., 1996]. As for snow, the peak at a lower attenuation coefficient is associated with pore space, while the peak at a higher value is associated with the ice. To achieve the best image contrast, the minimal value was determined as the local minimum between these two peaks; and the maximal value was the end of the tail of the entire histogram. A typical plot used for this purpose is shown in Figure 1. The other parameters were determined through visual inspection (misalignment compensation: 2.5, ring artifact correction: 5, and Gaussian smoothing: 2). Once determined for the first reconstruction, the parameters were kept constant for the entire experiment in order to maintain consistency.

Figure 1.

Typical plot of the X-ray attenuation coefficient histogram in NRecon used for determining the minimal and maximal density values for cross-sectional image reconstruction.

[13] The determination of a threshold in CTAn for converting gray-scale images to binary images is one of the most critical aspects of image processing because both the appearance of images and structural parameters are highly dependent on the threshold value. The threshold was often determined via a histogram and visual estimation. However, this method can lead to problems when the specimen is composed of several materials with overlapping threshold ranges. Therefore, to assess the threshold more accurately, we used high-resolution SEM images as references. After the entire micro-CT observation, the specimen was sectioned and examined in an SEM. The threshold was validated by comparing binarized micro-CT images and mosaic SEM images of the same area. This method was described in detail in the work of Lomonaco et al. [2008]. Again, we kept the threshold consistent for the entire image processing once it was determined.

[14] The 3-D images of the snow specimen were usually obtained by sequentially stacking binarized cross-sectional images. In order to extract 3-D images of an individual snowflake from the snow aggregate, we first reduced the volume of interest (VOI) by locating the center and edges of the snowflake among the cross-sectional image set. In this case, the snowflake was found in about 100 cross-sectional images, though this changed with the structural evolution over time. After binarizing these images, we sorted through the binaries, determined and manually selected only the areas belonging to the snowflake. The final step was creating 3-D snowflake images by stacking together the area-selected images with clear backgrounds. The 3-D images of a larger volume were also created showing the snowflake as well as its neighbors in order to record the development of bonds among adjacent snow crystals.

[15] In order to quantify the structural changes, several structural parameters were calculated using CTAn, including the volume fraction of ice, the surface-to-volume ratio of the ice matrix, the thickness and separation of ice structure determined by the distance transform of the ice and pore space [Kaempfer and Schneebeli, 2007]. Since these parameters are statistical results, the calculations were conducted based on the 4 × 4 × 4 mm3 cubic VOI, which enclosed the snowflake and was considered as a sintered material. It is important to note that since grain boundaries are not detectable using the micro-CT, a particle was used as a component unit, which could connect with its neighbors through smalls bonds but mostly was isolated from them and can be differentiated from them easily. A particle could be composed of single or multiple ice crystals.

3. Results and Discussion

3.1. Structural Evolution of an Individual Snowflake

[16] Twenty scans were performed on the snow specimen over a period of 70 days. Selected time series images of the lower surface of the snowflake are shown in Figure 2. Initially, the snowflake was a stellar dendrite, ∼3 mm in width, with a hexagonal plate in the middle (Figures 2a and 3a). Six dendritic branches had grown from the corners of the central plate. On the tip of each branch, the crystal grew into a thin hexagonal plate with a raised rim. With increasing time, the snowflake gradually lost its fine structure by mass transport mostly from small protrusions, e.g. secondary branches to hollow depressions, e.g. the spaces between the secondary branches (Figures 2 and 4). In addition, a hole appeared in the center of the snowflake and grew larger at later times (Figure 2).

Figure 2.

3-D micro-CT images showing the structural evolution of the snowflake over a 70 day period: (a) the initial stage, (b) after 10 days, (c) after 20 days, (d) after 30 days, (e) after 48 days, and (f) after 70 days.

Figure 3.

3-D micro-CT images of the snowflake showing the crystal growth on the edge of an arm: (a and b) the initial stage and (c) after 30 days.

Figure 4.

3-D micro-CT images of the snowflake showing its structural changes from different directions: (a) the bottom views, (b) the side views, and (c) the top views.

[17] Instead of sublimating away as the small protrusions did, the raised rims at the branch edges accumulated extra ice and developed steps, as shown in Figure 3. These steps then propagated toward the center of each branch. This observation indicates that the concave corners created by the raised rims supplied sites of high accommodation factors for water molecules. Therefore, water vapor was more likely to condense in these sites than in other places such as a flat surface, leading to local crystal growth.

[18] The advantage of the micro-CT imaging is the ability to examine the snowflake in different orientations (Figure 4). By comparing the bottom views (Figure 4a) and the top views (Figure 4c), we noticed that while steps developed on the lower surfaces of the end plates, on the upper surfaces the plates became rounded from their edges. One possibility is that vapor transferred upward locally, which drove mass to sublimate from the upper surface of the ice crystal. Thus, on the upper surface, it is more difficult for water molecules to accumulate. Instead, sublimation dominated and resulted in rounded edges. The side views (Figure 4b) reveal that instead of being a single layer, the snowflake was capped with two small plates on each side through a narrowed neck. This central part was observed to lose mass at later times, and probably fell through the main body, thus leaving the hole noted earlier.

[19] The 3-D images of a larger volume were acquired to examine the interactions between the snowflake and surrounding snow crystals. We did not observe the mechanical breakdown of snow crystals into smaller seed crystals, which has been reported occurring at the early stage of snow metamorphism [Colbeck, 1983]. Instead, more connections developed between the snowflake and adjacent particles (Figure 5). Two possible reasons account for this difference. First, the snow specimen was well protected from movements in the holder and sintering dominated the evolution over the entire observation period. Second, the pressure in the specimen was very low compared to that in the field snowpack, which is caused by the accumulation above. Therefore, as seen in Figure 5, the overall result of mass transport over this period is an increase in individual particle sizes at the expense of the number of particles in this VOI.

Figure 5.

3-D micro-CT images of a volume enclosing the snowflake showing the bonds developed between particles over time and the reduction in the number of particles: (a) the initial stage and (b) the final stage.

3.2. Theoretical Considerations

[20] As discussed by Legagneux and coworkers [Legagneux et al., 2004; Legagneux and Dominé, 2005], the physics of snow metamorphism under isothermal conditions can be viewed as being very similar to that of Ostwald ripening, with the water vapor pressure at the interface between ice and air replacing the solute concentration. Thus, the evolution of the mean radius of snow particles can be described as

equation image

where R(0) is the initial value at t = 0; n is the growth exponent, which depends on the dominant mechanism during the process, for example, n was suggested to be 2 or 3 when growth is limited by lattice or surface diffusion, respectively [Greskovitch and Lay, 1971; Lange and Kellett, 1989]; and K is the growth rate, which is affected by the solid fraction and the particle shape [DeHoff, 1984; 1991].

[21] The driving force for snow coarsening is the variation of vapor pressure over the ice particle surfaces [Colbeck, 1998; Blackford, 2007], which is essentially caused by the difference of local surface curvature according to Kelvin's equation

equation image

where P(r) is the saturated vapor pressure over a surface of radius of curvature r at temperature T, P0 is the saturated vapor pressure over a flat surface, γ is the surface tension, Vm is the molar volume, and R is the universal gas constant. Therefore, for a spherical particle, the smaller the radius of curvature, the greater the vapor pressure over its surface. Since the geometry of a natural snowflake is more complex, locally, the mean radius of curvature is given by

equation image

where r1 and r2 are the two principal radii of curvature. For example, the saddle shape between two raised protrusions, with r1 and r2 of opposite sign, has a much larger mean radius of curvature and sometimes behaves as a flat surface with respect to the protrusions themselves.

3.3. Structural Parameters Evolution

[22] Analyses on structural parameters were conducted based on a 4 × 4 × 4 mm3 cubic VOI, surrounding the snowflake. Selected time series images of this VOI are shown in Figure 6, from which the images of the snowflake were extracted. The images of the cubic VOI are oriented with the vertical direction as the specimen was in the micro-CT in order to illustrate the relationship between the snowflake and the aggregate. Even though the snowflake was embedded within the aggregate, the end of one arm, which corresponded to the lowest arm of the snowflake shown in Figures 2 and 4, could be identified on one side of these cubic VOI images.

Figure 6.

3-D micro-CT images showing the structural evolution of a 4 × 4 × 4 mm3 cubic volume of interest (VOI) over a 70 day period: (a) the initial stage, (b) after 10 days, (c) after 20 days, (d) after 30 days, (e) after 48 days, and (f) after 70 days. The cubic volume encloses the snowflake, with the squares highlighting the end of one of its arms.

3.3.1. Solid Volume Fraction

[23] The solid volume fraction, one of the most important structural parameters, is defined as the object volume (Obj.V), which is the volume occupied by ice, divided by the total volume (TV) of snow, i.e., Obj.V/TV. The initial value of Obj.V/TV was 7.7%. This increased continuously to 12.9% after 70 days (Figure 7). This change cannot be completely attributed to the sintering of ice particles. Another cause was the settling and rearrangement of particles in the snow specimen. To examine this, we monitored the position of the snowflake itself and found that its center was ∼1.5 mm lower after 70 days. Because the cubic VOI, which we used for quantifying the structure, was smaller than the entire specimen, particle settling resulted in more ice being added to this volume, thus contributing to the increase in Obj.V/TV.

Figure 7.

Evolution of the proportion of the total snow volume occupied by ice (Obj.V/TV), with error bars showing the variation of the area fraction of ice in the 2-D cross-sectional images.

3.3.2. Specific Surface Area

[24] Specific surface area (SSA), i.e., the surface area of snow per unit volume, measures the complexity of structures and is often used to quantify gas exchanges between snowpack and the lower atmosphere. We applied the approach of Legagneux et al. [2004] and Legagneux and Dominé [2005], in which SSA was linked to the radius of curvature using

equation image

where f is a geometry factor that accounts for nonspherical structures. By substituting R in equation (1), the form

equation image

was obtained to describe the evolution of SSA over time, with

equation image

where SSA(0) is the initial value at t = 0. We fitted our experimental measurements using the above form, as shown in Figure 8. In the best fit, where correlation coefficient (R2) is greater than 0.98, the computed SSA(0) is 2.5% larger than our first data point, which indicates that changes occurred before our first observation. The computed n is 1.45, which is lower than the values reported by Legagneux et al. [2004] and Kaempfer and Schneebeli [2007]. As being discussed by Legagneux et al. [2004], coarsening theories are based on a steady state prediction, but this prediction is not appropriate for snow metamorphism. Therefore, the computed n value suggests an evolution rate but cannot be associated with any mass diffusion mechanism at this point.

Figure 8.

Evolution of the surface-to-volume ratio (SSA) of the ice matrix over time. The circles are the experimental measurements. The curve fitted to the data using equation (5). The fit yields τ = 22.7 and n = 1.45, with R2 > 0.98.

3.3.3. Structure Thickness and Structure Separation

[25] The structure thickness (Sr.Th) of the ice matrix was addressed using the mean structure thickness and thickness distribution of the ice matrix in the cubic VOI. The local Sr.Th of a point was defined as the diameter of the largest sphere that encloses the point and is bounded within solid surfaces [Hildebrand and Rüegsegger, 1997a]. For the snow specimen, containing only ice matrix and pore space, the structure separation (Sr.Sp) of ice matrix is essentially the Sr.Th of the pore space and the latter can be interpreted as the distance between ice particles.

[26] The evolution of mean Sr.Th and mean Sr.Sp over the observation period are presented in Figure 9. We follow the approach of Kaempfer and Schneebeli [2007], who applied Sr.Th into the coarsening form, i.e., equation (1). However, instead of replacing R in equation (1) directly using Sr.Th as in their work, we considered Sr.Th/2 as a more accurate alternative to R according to the definition of Sr.Th [Hildebrand and Rüegsegger, 1997a]. The evolution of Sr.Th over time is, therefore, expressed as

equation image

where the Sr.Th(0) is the initial value at t = 0. We fitted our experimental measurements using the above form (Figure 9). The computed parameters of the best fit are n = 1.95, K = 3.2 × 10−4. The discrepancies between the values of n computed from equations (5) and (7) are probably caused by equating Sr.Th/2 to the radius of curvature, since Sr.Th/2 is only identical to the radius of curvature for an aggregate of nonoverlapping spheres [Hildebrand and Rüegsegger, 1997a]. The mean Sr.Sp was analyzed using the same approach. The measurements were fitted using the form expressed as

equation image

The computed parameters are n = 6.34 and K = 2.6 × 10−5, which are remarkably different from the evaluations for Sr.Th. This difference suggests that the behavior of pore space is not identical to that of the ice matrix, although in dry snow, pore space is the volumetric complement to the ice matrix.

Figure 9.

Evolutions of structure thickness (Sr.Th) and structure separation (Sr.Sp) of the ice matrix over time. The circles and squares are experimental measurements of Sr.Th and Sr.Sp, respectively. The curves fitted to the data using equations (7) and (8).

[27] The evolution of Sr.Th distribution and Sr.Sp distribution were illustrated using four sets of data, which were obtained from the first observation and the observations after 20, 30, and 70 days successively. Figure 10 shows the Sr.Th distributions from these four observations. As time increases, the entire histogram shifts to larger values and becomes wider. We fitted each data set with a lognormal distribution given by

equation image

where ln(x) is assumed to follow a normal distribution with mean μ and standard deviation σ. The values of these two parameters were computed for the best fits, as shown in Table 1. Although the computed curves fit the measurements well with all the R2 greater than 0.9, we noticed a slight decrease in R2 over time, which indicates that the histograms plotted based on the measurements become less skewed as the structure evolves. We calculated the mean (or expected value) of the variable X, which is E(X) = equation image, and the variation of X, which is V(X) = exp(2μ + σ2)[exp(σ2) − 1], based on the fitting curves. As is evident from the values listed in Table 2, E(X) and V(X) continuously increase over the observation period, with V(X) increasing more rapidly at shorter times. Correspondingly, the distribution broadens with a decrease in its height and an increase in its mean value, a change which decreases with increasing time. The evolution of Sr.Sp distribution is presented in Figure 11. In addition to the shift of the entire histogram to a larger value over time as it occurred to the Sr.Th distribution, a second peak tended to develop in the larger-value range at later times, which confirms that the pore space behaved differently to the ice matrix. A possible explanation for the development of a second peak in Sr.Sp distributions is that as adjacent ice particles sintered into larger ones, some of the pores experienced fast expansion.

Figure 10.

Evolution of the structure thickness distribution of the ice matrix over time. Histograms are four sets of experimental measurements, which are fitted using a lognormal distribution expressed as equation (9), with two parameters μ and σ computed for each best fit. The mean E and the variation V of the variable X were calculated showing the characteristics of the distributions and are shown in Table 2.

Figure 11.

Evolution of the structure separation distribution of the ice matrix over time. Histograms are four sets of experimental measurements.

Table 1. Values of the Parameters μ and σ and the R2 of the Lognormal Distribution Fitted to the Structure Thickness Distribution of the Ice Matrix
ParametersTime (days)
Initial203070
μ (ln(mm))1.051.491.772.04
σ (ln(mm))0.270.260.350.30
R20.9950.9820.9600.933
Table 2. Values of the Mean E and the Variation V of the Variable X in the Lognormal Distribution Fitted to the Structure Thickness Distribution of the Ice Matrix
CharacteristicsTime (days)
Initial203070
E(X) (mm)0.0890.140.190.24
V(X) (mm2)0.691.55.26.0

4. Summary

[28] Periodic nondestructive observations of the metamorphism of an individual snowflake within a dry snow aggregate held at −5°C were performed using a micro-CT over a 2-month period. 3-D images of the snowflake and surrounding snow crystals were obtained showing the structural changes. The structural evolution was also quantified by monitoring the evolution of several structural parameters within a 4 × 4 × 4 mm3 cubic VOI, enclosing the snowflake, and interpreted using coarsening theories. The increase in the ice volume fraction was attributed to both the sintering and the settling of snow crystals. The specific surface area, the structure thickness and the structure separation of the ice matrix can be well fitted using coarsening formulae, with the calculated growth exponent n lower than 2. The behaviors of the ice matrix and pore space were different. During the snow metamorphism, while the structure thickness distributions of the ice matrix were well fitted by lognormal distributions with the central peak shifting to larger values and the entire histogram becoming wider with increasing time; the structure thickness distributions of pore space developed a second peak at later times.

[29] In summary, observing the snowflake using the micro-CT not only provides 3-D information of its structure but enables the study of its metamorphism in a more natural environment.

Acknowledgments

[30] This work is sponsored by Army Research Office contract 51065-EV. The micro-CT was acquired through National Science Foundation grant OPP-0821056. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing official policies, either expressed or implied of the NSF, the ARO or the U. S. Government. The authors are grateful to the reviewers for helpful comments.

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