Grain-Size

The grain-size was measured by sieving. It is a fast, efficient method that samples a large number of grains. We used a stack of nine sieves, each 20 cm in diameter. Snow samples were collected monthly from each snow layer during the winter of 1986-87. Each sample (500 cm³) was gently sheared across the mesh of the top sieve to break it apart. In most cases the samples were quite fragile and this happened readily. Once the sample was disaggregated, the sieve stack was agitated by hand for 60 s with a rocking motion. After agitation, the snow fraction in each sieve was weighed with an accuracy of ±0.1 g. The work was done at or below −18 ° C G to prevent snow from sticking to the sieves.

At one time, sieving was widely used in snow studies (Reference Bader,, Haefeli, Bucher,, Neher, Eckel and ThamsBader and others, 1954; Reference Benson,Benson, 1962; Reference Keeler,Keeler, 1969; Reference Fukue.Fukue, 1977; Reference Granberg,Granberg. 1985), but it has seen less use recently because it requires disaggregation of the snow sample, breaking snow grains as well as bonds, creating a bias toward smaller sizes. However, “the bond between individual grains is substantially weaker than the individual grains, so that it is possible to prepare a cohesionless, siftable mixture from cohesive snow without breaking up individual crystals to an appreciable extent” (Reference Bader,, Haefeli, Bucher,, Neher, Eckel and ThamsBader and others, 1954; see also Reference Kry.Kry 1975). Moreover, microscopic examination of snow fractions after sieving (unpublished information from M. Sturm, 1992) confirms that most grains remain intact, particularly for depth hoar where the bonds between grains are small in relation to the grains themselves.

The average grain-size determined by sieving was compared with the average size determined by photogrammetric methods. For two snow layers, samples were collected six to ten times during the winter. Part of each sample was sieved, and part was disaggregated onto a black background and photographed using a Zeiss Tessovar microscope. From the photographs, 50-200 grains were digitized by tracing their outlines with the Stylus of a Zeiss-Plot stereological system. The mean cross-sectional area of the digitized grains was determined and reported as the diameter of a circle with the equivalent area.

Density and Layer Thickness

During 11 winters, between 1966 and 1987. we systematically measured the density profile of snow on the ground and the tables several times a winter. In 1986-87 the density and thickness of ten layers of snow on the ground and the comparable ten layers on the tables were measured approximately twice a month. Since depth-hoar metamorphism obliterated all primary natural stratigraphic markers in the snow on the ground (but not on the tables where depth hoar did not develop), it was necessary to introduce artificial markers to identify these layers. On both the ground and the tables, colored powder was spread on the snow surface after each snowfall. In this way, the upper and lower boundaries of layers were delineated and it was possible to make density and grain-size measurements tied to individual layers. Snow compaction was computed from measured changes in thickness of the snow between layers (accuracy ±3mm). Little or no cross-boundary migration of snow grains was observed.

Density was measured using a 100cm³ cutter and standard techniques (National Research Council of Canada. 1954; unpublished information from U.S.A. Snow, Ice and Permafrost Research Establishment, 1962: Reference Carroll,Carroll, 1977). The cutter could sample individual layers thicker than 3 cm. Multiple measurements were made on each layer, and the results averaged, fine-grained and new snow could be sampled with a reproducibility of ±2%, so the average of several density measurements is estimated to have the same accuracy. For depth hoar, which is fragile and more difficult to sample, reproducibility was ±7%.

Temperature

Individual vertical temperature profiles in the snow were measured hundreds of times from 1963 onwards using dial thermometers or thermistors. Continuous (hourly) profiles were monitored at the experimental site in 1984-87 using a data logger and an array of thermistors described by Reference Sturm,Sturm (1989, Reference Sturm. and Johnson1991) and Reference Sturm. and Johnson.Sturm and Johnson (1992). Additional continuous monitoring was done in 1989-90 and 1991-92 at a site 15 km away (Reference Sturm., Holmgren eu and ListonSturm and others, 1995) using a similar method.

Isotopic Ratios

Stable-isotope ratios of ¹⁸O to ¹⁶O and of deuterium to hydrogen were measured during 11 winters between 1966 and 1987. Snow samples (500 cm³) were collected sequentially from top to bottom at the end of winter, before the thaw. Analysis protocols arc described by Reference Friedman., Benson, Gleason, Taylor,, O’Neill and KaplanFriedman and others (1991). Results are expressed relative to Vienna Standard Mean Ocean Water (V-SMOW) in standard delta units (δD and δO) in parts per thousand (per mil). The isotope ratios were used to track the transport of water vapor from the soil to the snow and upward within the snow. In effect, water vapor from the soil, which is isotopically heavier (higher values of δD and δO) than the overlying snow, was used as a natural tracer. The isotope ratio of the soil moisture was measured by collecting the ice that formed under the impermeable tarpaulin at the end of winter.The vapor flux out of the soil was also measured by collecting this ice from a known area and weighing it.

A Depth-Hoar Metamorphic Sequence

Metamorphism in the subarctic snow results in the development of a sequence of five distinct textural layers (Fig.2;Table 1). The uppermost layer (Ml) consists of new or recent snow. The underlying layers exhibit kinetic crystal-growth forms and would be called either “solid” or “skeletal” depth hoar (Reference Akitaya.Akitaya, 1974). Fhc top two depth-hoar layers (M2 and M3) have textures that are common. They appear in the International Classification of Seasonal Snow on the Ground (ICSSG: Reference Colbeck.Colbeck and others, 1990) as types 4a or 4b, or 5a and 5b if they tended to form vertical chains of cups. Layer M4, consisting of squat, eroded prisms of depth hoar, has no textural analog in the ICSSG and does not seem to form in moderate climates. Characteristically, its columnar prisms have sharp, striated lower edges, but smooth, eroded upper edges indicative of growth at the bottom and mass loss by sublimation at the top. It is noticeably the weakest layer in the snow-pack. often collapsing spontaneously or due to a slight jarring. Layer M5 would be classified as 5c in the ICSSG, and the inclusion of this texture and the type example used in the ICSSG comes from observations made during this study.

Fig. 2. The stratigraphic sequence of snow textures (crystal and grain morphologies) typically found in the subarctic snow cover during the winter. It is also a metamorphic sequence for individual layers of snow, with layers progressing from Ml toward M5. See Table I and text for layer descriptions.

Table 1. Five metamorphic textures of subarctic snow

The sequence described above is both stratigraphic and metamorphic. As a layer of snow metamorphoses, it evolves from M1 to M2 and so on. Consequently, layers at the base of the snow exhibit more “advanced” stages of depth-hoar metamorphism than those above. Not only have the lower layers been subjected to temperature gradients longer than the overlying layers, but they have also experienced stronger gradients, since metamorphism early in the winter occurs when the snow pack is thinner (Fig.1).

Changes in grain-size and number

Mean grain-size of a sieve sample was calculated using the following equation (Reference Friedman, and SandersFriedman and Sanders. 1983):

(1)

where d is the mean grain diameter for t he sample, M_j is the Weight fraction in the jh sieve.and D _jj+1 is the average of The sieve-mesh opening of the j th and j+1 th sieves, with a total of L sieves.

From the sieving, grain distributions were also determined. Normally, a size distribution is determined from the weight fraction in each sieve (Reference Royse.Royse, 1970; Reference Blatt., Middleton and MurrayBlatt and others, 1972; Reference Friedman, and SandersFriedman and Sanders, 1983) based on the premise that the size of a grain is related to the size of the sieve-mesh opening through which it passed. For spheres and other simple, regular shapes, this is true (Reference Ludwick, and HendersonLudwick and Henderson, 1968), but for highly convoluted shapes like depth hoar, a simple relationship may not exist, To avoid complicated and possibly arbitrary conversions between grain-size and sieve size, we dealt directly in grain mass. For several representative depth-hoar samples, the individual grains in each sieve were weighed to establish a relationship between the sieve-mesh size and the average mass of the grains in the sieve (Fig.3). Reference Bader,, Haefeli, Bucher,, Neher, Eckel and ThamsBader and others (1954) performed a similar procedure, and their data (Fig.3) agree well with ours.

The data were fit with a polynomial:

(2)

where is the average mass (g) of a grain in sieve, D _jj+1 , in mm. is defined in Equation (1) and α = 0.0001266. If a collection of perfect ice spheres were sieved, the value of α would equal 0.0001698 (dotted line inFig.3). The curve for depth hoar lies surprisingly close to the curve for spheres, and it is likely that more rounded snow textures would lie even closer. Using Equation (2), the average mass of a grain Caught in a particular sieve can be determined. Since the total weight fraction of snow in the sieve was measured, the total number of grains in each sieve could be estimated. Summing for all sieves permitted the total number of grains in a sample to be estimated as well (Appendix A).

Fig. 3. Grain mass as a function of sieve size, n is the number of samples that Were weighed. Sieve size is customarily given as the length of the diagonal of the mesh openings. Data from Reference Bader,, Haefeli, Bucher,, Neher, Eckel and ThamsBader and others (1954) are included for comparison, and are surprisingly consistent given the radical differences in length of time the sieves were agitated (see text). A calculated curve (dotted) is shown for a hypothetical snow consisting of ice spheres.

Average grain-sizes determined by sieving and from photographs are compared in Figure 4. Sieve measurements tend to be biased toward smaller grain-sizes as a result of breakage of single grains. Stereological measurements tend to be biased toward larger grains because two touching grains may be digitized as a single larger grain. Reference Dullien, and MehtaDullien and Mehla (1972). working with salt grain-size distributions, found the same bias. In general, we observed differences to be less than 1 mm. However, for snow that had metamorphosed into large, complicated grains, the agreement between the two methods was not as good. For these samples, differences in size arose because the large, ornate grains were poorly suited to being characterized by an equivalent circle or measured by sieving.

Fig. 4. A comparison of average grain-size determined by sieving with size determined by photographic methods for loose grains. The agreement is good at 10 cm height. At 4 cm height, the agreement is acceptable except for large, ornate grains.

Metamorphosis of the snow into depth hoar was accompanied by a two- to three-fold increase in the grain-size. In Figure 5, the average size, determined by sieving and calculated using Equation (1), has been plotted for four layers of snow on the ground and two layers on the tables. At 4 cm height, and to a lesser extent for the other layers, some growth had already taken place before the first measurements were made, so the initial grain-size is not known. As suggested by the dotted lines in Figure 5, most grains wore less than 1 mm in diameter at the start. They grew quickly after deposition, but growth rates soon decreased, as can be seen in the curves for the layers at 20 and 30 cm. Grains in layers of snow on the tables, not subjected to strong temperature gradients, maintained nearly constant size during the same period that grains on the ground were growing.

Fig. 5. The increase in mean grain diameter of four layers of snow on the ground and two layers on the tables during the winter of 1986-87. Some growth(indicated by dotted lines) had occurred before the first measurement. The initial size was about 1 mm. Compare the lack of change in size for layers on the tables with layers on the ground.

As grain-size increased, the grain-size distributions changed (Fig.6). The shift in the distribution curves indicates a the steady increase in the average grain-size, (b) the steady increase in standard deviation, evidenced by widening of the curves, and (c) a gradual shift from unimodal to bimodal then back to unimodal distributions. This evolution can be seen most clearly for the snow layers at 10 and 20 cm, where a strong mode at 1.44 mm gradually-disappeared while a second mode at 2.68 mm built, shifted to the right and broadened as the snow metamorphosed into depth hoar. At 4 cm height, the layer was already depth hoar when it was first sieved, so its distribution curves remained constant throughout the winter. Again, the large changes observed for snow layers on the ground contrast sharply with the limited changes observed for layers on the tables.

Fig. 6. Cumulative grain-size distribution curves for four layers on the ground and one layer on the tables. The snow at 4 cm had already metamorphosed into depth hoar before it was first sieved. A dotted line indicating the initial distribution has been added to each set for reference. Heavy solid lines are the final distribution of the winter.

The evolution of grain-size was accomplished through an order-pf-magnitude decrease in the number of grains per unit mass (Fig.7: Appendix A). Data from four layers of snow have been plotted as a function of time since deposition to highlight the trend. For these layers, most of the reduction in number occurred in the first few weeks. For example, the number of grains in the snow layer at 30 cm changed in the first 40 d from approximately 65 x 10⁶ to less than 10 X 10⁶ grains kg⁻¹, but in the following 110 d the number of grains hardly changed at all. The decrease was accomplished by a drop in the number of small grains and a rise in the number of large grains. When first measured, there were no grains in the four largest sieves, but more than 50 x 10⁶ grains kg⁻¹ in the smallest sieve. By the end of the winter, the four largest sieves contained over 2 x 10⁴ grains kg⁻¹, all of them depth hoar, and the number in the smallest sieve had been reduced by a factor of 5.

Fig. 7. The evolution of the number of snow grains (per unit mass) for snow layers at 4, 10, 20 and 30 cm, winter of 1986-87.

The data in Figure 7 can be fit with an exponential curve:

(3)

where g is the number of grains per unit mass at time t. g₀ is the initial number of grains in the sample, ≈47.6 x 10⁶ kg⁻¹, and β₁ and β₂ are constants equal to 42.7 x 10⁶. and 0.0524, respectively, when time (t) is measured in days.Though we do not use the results of Equation (3) directly, we present it here because the redaction in the number of snow grains during depth-hoar metamorphism has been noted by many observers (Paulcke, 1934: Reference Seligman.Seligman, 1936; Reference Bader,, Haefeli, Bucher,, Neher, Eckel and ThamsBader and others, 1954: Reference Benson,Benson, 1962: Reference Akitaya.Akitaya, 1974) but not previously quantified.

Changes in Density and Layer Thickness

The end-of-winter density profile of the subarctic snow-pack was usually uniform with height (Fig.8). In contrast, the profile for snow on the tables showed a decrease in density with height (Fig.8) more typical of snow packs found in temperate climates. The contrast is due to differences in densification rates and in the way vapor transport redistributes mass in the pack. Layers on the ground density more slowly than equivalent layers on the tables, while at the same time they lose mass due to an upward-directed vapor flux.

Fig. 8. Composite end-of-winter density profiles for snow on the ground and tables near Fairbanks. 1966-87. Snow on the tables was not subjected to strong temperature gradients, while snow on the ground was. The ground snow develops a profile that is nearly constant with height, while the table snow develops a profile that decreases with height. The profiles have been calculated from density profiles from 11 winters. Data for table snow are shown by the + symbols. Density profiles from the end of winter have been non-dimensionalized by dividing the snow depth (h) and snow density (ρ) by the total depth (h_total) and the mean density (ρ_ave). Using these nondimensional depth-density curves, the mean profile has been determined.

To illustrate, we note that all ten layers comprising the 1986-87 ground pack densified at a lower rate than the equivalent layers on the tables (Table 2;Fig.9). On both table and ground, layers experienced an initial period of rapid densification followed by a secondary period of slower densification, but during both periods the ground snow densified more slowly than the table snow. In fact, during the secondary period, snow layers on the ground experienced little or no change in density. while equivalent layers on the tables continued to densify.

Fig. 9. Densificalion of snow layer 3 on the ground and the tables, showing typical rapid densification at first, and slower densification later.

Table 2. Density (1 kg m⁻³) of ten snow layers during the winter of 1986-87; layer thicknesses are given in Table 3

As the snow layers densified. they settled and compressed, leading to changes in thickness (Table 3; Fig.10). Commensurate with changes in density, layers on the ground compressed more slowly than layers on the tables. Rates in the lowest lasers on the ground ranged from 0.5 to 1.0 mm d⁻¹, then decreased to less than 0.1 mm d⁻¹ several weeks after the snow was deposited. Layers on the tables compressed at rates twice as fast, and continued to compress after layers on the ground had virtually ceased changing thickness. Basically, the depth hoar on the ground is more resistant to compaction than the fine-grained snow on the tables, a fact that has been noted by Reference Kojima,Kojima (1959), Reference Akitaya.Akitaya (1974) and Reference Armstrong,Armstrong (1985). This “stiffness” arises because the depth hoar forms vertical grain structures, described by Reference Trabant. and BensonTrabant and Benson (1972), Reference Akitaya.Akitaya (1974), Reference Colbeck,Colbeck (1986) and Reference Sturm,Sturm (1989, Reference Sturm.1991), that function as stiff but brittle columns.

Table 3. Thicknesses (cm) of ten snow layers during the winter of 1986-87; densities are given Table 2

Fig. 10. Compaction of snow layer 3 on the ground and the tables, showing typical rapid compaction at first, and dower compaction later.

Vertical Temperature Profiles in the Snow

Vertical temperature profiles in the subarctic snow-pack show strong concave-downward curvature (Fig.11)Footnote ¹ , a fact we have verified many times during the past 30 years (unpublished information from C S. Benson, 1972: Reference Sturm,Sturm. 1989. Reference Sturm.1991). from one winter to the next, despite substantial differences in weather, snow depth and stratigraphy, temperature profiles are similar. In a survey of Over 12000 measured temperature profiles taken between 1987 and 1992. over 75% exhibited concave-downward curvature.

Fig. 11. A typical temperature profile from the subarctic snow showing strong concave-downward curvature.

Isotopes and the Height of Water-Vapor Mixing

About 5% of the total snow water equivalent at the experimental location was derived from the upward flux of water vapor out of the soil. The flux averaged 2.6 x 0⁻⁷ kg m⁻² s⁻¹ (Table 4) , sufficient to deposit a layer ice about 0.5 cm thick beneath an impermeable tarpaulin. Ice beneath the tarpaulin was enriched in deuterium and ¹⁸O. compared to normal winter snowfall, because it was derived from rainwater (Fried man and others, 1991. This enrichment amounted to 40-50 per thousand δD when measured in 1975,1985 and 1986 (Fig.12).Where no tarpaulin was present, the water vapor from the soil entered the snow pack, enriching the basal layer of snow in deuterium and ¹⁸. Based on ten sets of δD profiles taken between 1966 and 1987. this basal enrichment ranged from 12 to 30 per thousand δD when compared to equivalent snow layers on the tables (Fig.12). The enrichment extended up through in cm in the snow, corresponding roughly with the M5 layer (Fig.2).

Fig. 12. Profiles of δD values (per thousand) for snow on the tables, bare ground and a tarpaulin on the ground. Note that the basal 10 cm of snow on the ground is consistently heavier than the equivalent layer on the tables, while the opposite is true for the top 10 cm.

We have calculated the isotopic composition of the basal layer of the snow-pack for the hypothetical case where soil moisture was simply mixed into the layer. This was done by-adding the mass of soil moisture (listed in Table 4), times the measured δD values of that moisture, to the product of the initial snow mass times its measured δD value. For the years for which we have data, mixing gives calculated values of −193 vs −184 (measured) for 1975; −164 vs −155 for 1985; and −180 vs −177 for 1986. In all cases, the measured value is “heavier” (depleted in lighter isotopes) than the value calculated by mixing. For 1985 and 1986, we can also calculate the δ¹⁸O values for the case of simple mixing and compare them to the measured values.These are plotted in Figure 13. both measured δD and δ¹⁸O values are “heavier” than the hypothetical values due to mixing alone. We conclude, as did Reference Friedman., Benson, Gleason, Taylor,, O’Neill and KaplanFriedman and others (1991). that while mixing is important, it is not the only process affecting the isotopic composition of the basal layers of snow. The slope of the lines connecting the isotopic compositions calculated for pure mixing to the measured compositions (Fig.13) suggests that isotopic fractionation during sublimation and diffusion was also operating.

Fig. 13. Isotope trajectories for basal snow samples, 1985 and 1986. Numbers indicate the slope of the trajectories.

Table 4. Upward-directed soil water-vapor flux

The snow on the ground, but above the impermeable tarpaulin, also showed a slight basal enrichment in δD (Fig.12). This could not have been the result of the migration of soil moisture into the snow, so it must have been due to the preferential removal of water vapor enriched in light isotopes. It confirms our conclusion that the final isotopic composition of the basal snow layer was the result of two processes: enrichment due to the migration of “heavy” soil moisture into the layer, and fractionation of vapor within the layer. resulting in the loss of “light” vapor upward under the influence of temperature gradients.

The “light” vapor appears to have traveled up to the top of the pack where some or all of it condensed. A consistent feature observed from 1966 to 1987 was a 5-20 cm thick layer at the top of the pack that was enriched in light isotopes (hydrogen and ¹⁶O) (Fig.12). The observed enrichment varied from 10 to 30 per thousand δD. In 1975. new snow had fallen just before the samples were collected, so the enriched zone was just below the new snow layer. we know of no mechanism in dry snow that could preferentially remove “heavy” isotopes at the top, so we conclude lite isotopic shift was the result of deposition of light isotopes moving up from below.

Estimate of Layer-to-Layer Vapor Transport

The growth of large depth-hoar crystals (Figs 2 and 5) is proof of the existence of net water-vapor fluxesFootnote ² between show grains. Reference Gubler,Gubler (1985) referred to these as “inter-particle vapor fluxes”. Such local net fluxes do not require the movement of vapor from one snow layer to another, nor do they necessarily result in a change in layer density. Net fluxes between layers, on the other hand, change the layer density and cause grain growth or shrinkage.

The existence of net layer-to-layer vapor fluxes has been questioned. Reference Marbouty.Marbouty (1980) and Reference Armstrong,Armstrong (1985) found that the mass of a layer of Snow remained constant during the growth of faceted depth-hoar grains, suggesting that the net layer-to-layer transpon was zero. Reference Trabant. and BensonTrabant and Benson (1972), working in snow subjected to stronger temperature gradients for longer periods of lime, found measurable amounts of mass transferred from one layer to another. Reference Alley., Saltzman, Cuffey and Fitzpatrick.Alley and others (1990). using measurements of snow density and the concentration of methanesulfonic acid in thin layers of near-surface snow undergoing depth-hoar metamorphism. also concluded that there was significant mass transfer. The present study, conducted at the same location used by Trabant and Benson, reaches die same conclusion.

The magnitude of the net layer-to-layer flux can be estimated by comparing end-of-winter density profiles from the ground and the tables (Fig.8). Since the net layer-to-layer flux on the tables was negligible, we assume that any difference in the density profile on the ground compared to that on the tables is the result of a net upward transport of vapor on the ground. For ground snow, the loss of mass from the bottom of the pack was roughly equal to the gain at the top (Fig.8). If in the course of a winter (≈l70d) the mass (≈14 kg m⁻²) moved up and none moved out of the top of the snow cover, it would require a net vapor flux of 9.5 X 10⁻⁷ kg m⁻² s⁻¹. If this came from a lower layer 0.20 m thick, and went into an overlying layer 0.30 m thick (Fig.8), the average vapor-flux gradient would have been 47.6 x 10⁻⁷kg m⁻² s⁻¹. This is an upper limit since it fails to account for the fact that snow on the tables compacts more rapidly than snow on the ground (Fig.10), a point we return to later.

Calculation of Layer-to-Layer Vapor Transport

To calculate the net layer-to-layer flux, it is necessary to solve the continuity equation for a compacting layer of snow:

(4)

where ρ_s is the layer density. t is time. h is layer thickness, z is a vertical coordinate, and J_v is a vertical mass flux, here limited to water vapor. Both J_v and z are positive upward, and the minus sign accounts for the fact that when more vapor enters through the bottom than exits through the top of the layer (a negative flux gradient) in increases the mass of the layer. The overbar denotes spatial averages across the layer.

We can solve Equation (4) using the density and thickness measurements made in 1986-87 (Tables 2 and 3). though large uncertainties in thickness (±3mm) and density (±14 kg m⁻³) result in uncertainties in flux almost as large as the flux values themselves. Also, data from a crucial period between October and November, when the two basal layers of snow were probably experiencing vigorous vapor transport, are missing. Combined, the problems make our data insufficient to do a full seasonal vapor mass balance. However, they are sufficient to indicate the general time evolution of layer-to-layer fluxes in the snow, our main interest.

For each snow layer, linear compaction and densification curves of the form:

(5)

(6)

were fit lo the data in Tables 3 and 4. Compaction and densification rates were relatively high immediately after a snow layer was deposited, but quickly dropped to lower rates (Figs 9 and 10), so the initial period of rapid change and the final period of slower change were fit separately. For initial and final line segments, derivatives were approximated by differentiating Equations (5) and (6). ρ_s and h were evaluated at the midpoint of each line segment, and Equation (4) was then solved, giving an early and late-winter value of the vapor flux gradient. The results are listed in Table 5, with subscript “i” referring to the initial and “f” to the final line segments. The results are also depicted graphically in Figure 14. The flux gradient was multiplied by the average layer thickness to determine the net flux from the layer (also in Table 5). Based on an error analysis (Appendix B), we list significant values in bold type. Initial values were not available for layers 1. 2 and 6, nor final values for layers 9 and 10.

Table 5. Calculated Layer-to -layer water flux gradients, net vapor fluxes and integrated fluxes, 1986 87

The mass redistribution took place in a complex and episodic manner. Peak flux gradients, generally observed immediately after a layer was deposited, were an order of magnitude greater than average values. These peak values, however, were maintained for relatively short periods of time, suggesting that much of the mass redistribution in the snow took place in short, intense bursts.

A more puzzling result was that the snow layers than gained mass (+ in Fig.14) were not always near the surface of the snow. Stable-isotope data indicate an accumulation of vapor in the upper layers of snow (Fig.12), and these layers are preferential sites for condensation of water vapor because they are colder than the snow lower in the pack (Reference Nyberg,Nyberg, 1938). Condensation of vapor from the air results in the development of surface hoar (Reference Lang,, Leo and BrownLang and others, 1985: Reference Colbeck.Colbeck. 1988). We expected that condensation of Vapor moving up from the base of the pack would result in increased mass just below the surface. Instead, the calculations indicate that the mass often increased deeper in the pack. The coarse time resolution (two values per winter for each layer) make it possible that near-surface layers accumulated vapor when they were at the surface. but lost vapor later, when they were buried by new snow, thus ending in a net deficit over the period of calculation. It is also possible that we are mistaken in thinking the snow/air interface is the locus for vapor condensation.

Calculated flux gradients and net fluxes for the bottom two snow layers were relatively low (Table 5:Fig.14), but the measured flux of vapor from the soil (Table 4) and the stable-isotope data (Fig.12) suggest that these layers should have experienced some of the highest fluxes in the pack. They are the warmest layers, and are subjected to strong temperature gradients in early season (Fig.1). The most likely explanation for the low values is that we missed a key period of high flux early in the w inter just after the two layers were deposited. They had been on the ground nearly 40 d at the time that they were first measured in November 1986, and had already metamorphosed into depth hoar. This interpretation is even more plausible if we compare flux gradients and net fluxes based on measurements of individual layers (1986-87) with the values calculated from end-of-winter density profiles (1966-87; Table 6) The average flux gradient for 1986-87 is about half the value calculated from a comparison of table and ground density profiles (Fig.8), and the net flux in 1986-87 is about ten times lower. The values calculated from density-profile comparisons include early winter fluxes that are missing from the 1986-87 data set. and they agree with the values published by Reference Trabant. and BensonTrabant and Benson (1972) (Table 6).

Fig. 14. Layer-to-layer vapor-flux gradients for the Snow -pack of 1986-87. Plus signs indicate a layer that was gaining mass, minus signs one that was losing mass. The size of the symbol (big or little) suggests if the rate was high or low. The vertical dashed lines in layers. 3,4,5,7 and 8 indicate the initial and final periods (see Figs 9 and 10).

Table 6. Comparison of net fluxes and flux gradients from this and other.studies

For a true comparison, however, it is necessary to correct both our 1966-87 density profile values and those of Reference Trabant. and BensonTrabant and Benson (1972) for differential snow settlement on the tables and the ground. Both sets (Table 6) have been calculated without measuring changes in the thicknesses of snow layers. The calculations are based on two assumptions: (1) there is no net vapor flux in snow on the tables ( in Equation (4) is zero), and (2) the compaction rate, , is the same on the tables as on the ground. Under these assumptions, the flux gradient on the ground can he evaluated by subtracting the densification rate For snow on the tables from the densification rate for snow on the ground. Unfortunately, the second assumption is wrong. Figure 10 shows that for stratigraphically equivalent layers, the layer on the table compacts more quickly than the layer on the ground, particularly immediately after deposition. Later in the winter the compaction rates are more nearly equal. In 1980-87, the difference between table and ground compaction term averaged 1.2 x 10⁶ kg m⁻³s⁻¹. Using this value, we have corrected the flux gradient and net flux values in Table 6, Even with the correction, flux gradients computed from the snow-layer data of 1986-87 are still 50% lower than the flux gradients computed from density profiles for 1966-87. If we assume that flux gradients for the 40 d period at the beginning of winter missing from the 1986-87 data were approximately equal to the maximum observed flux gradient (≥200 x 10⁷ kg m⁻²s⁻¹m⁻¹), then the recalculated values for 1986-87 are consistent with the values computed from other data. Given that flux measurements Cannot be made with great accuracy, we consider the agreement between the two methods quite good. Also, despite missing the flux from the two basal layers in 1986-87, the data for higher layers are still valid and are useful for showing the magnitude of the layer-to-layer fluxes and the complexity of the evolution of the layer-to-layer vapor-flux system.

A Model of Layer-to-Layer Vapor Transport

we now develop a simple model in order to examine the time-dependent transport of water vapor. It is a one-dimensional vapor-diffusion model, based on observed vertical temperature profiles in the snow, and the dependence of the equilibrium water-vapor density on temperature. The temperature profiles (Fig.11), and the dependence of the water-vapor density on temperature, are both non-linear functions. In combination, they produce an equilibrium water-vapor density profile in the snow that has two inflection points. As a consequence, the model predicts that there will be a semi-stationary zone low in the pack where the snow density will decrease due to the loss of water vapor, and a complementary zone high in the pack where the density will increase due lo the accumulation of vapor.

To run the model, we assume vertical heat and vapor How. We further assume that the thermal conductivity is constant with height. In fact, thermal conductivity increases with height (Reference Sturm. and Johnson.Sturm and Johnson, 1992), which would result In vertical temperature profiles that had concave- upward curvature, if conductivity were the only variable. Because this is the opposite of what is normally observed (Fig.11), our assumption of constant conductivity results in conservative estimates of vapor-flux gradients.

We also assume that the water-vapor density is in thermal equilibrium with the local snow and can be predicted by the Clapeyron equation, a common assumption (Reference Bader,, Haefeli, Bucher,, Neher, Eckel and ThamsBader and others, 1954; Reference Colbeck,Colbeck, 1982). The equilibrium vapor density (ρ_v) is:

(7)

where T1₀, and ρ_v0 are the temperature and vapor density at the melting point 0 ° C and 4.847 gm⁻³), L is the latent heat of sublimation (2838 Jg⁻¹), R is the gas constant for Water vapor (0.4619 Jg⁻¹ C⁻² and T is temperature (see Reference Colbeck.Colbeck (1990) for the derivation of Equation (7)).

To each measured vertical temperature profile, we fit a smooth non-linear function of the form:

(8)

using a least-squares regression that minimizes the residuals. Equation (8) was chosen because it fits the data well, but Otherwise has no physical significance.The fitting process is required to smooth the data in order to compensate for slight errors in temperature or the thermistor position. We substitute Equation (8) into Equation (7) to get a relationship between the equilibrium water-vapor density and height in the snow (ρ_v = ρ_v(z)) for each temperature profile.

Next we assume that the vapor transport can be described by:

(9)

where J_v , is the water-vapor flux, Do is the water-vapor diffusion coefficient in air, ø is the porosity, is the vertical convective Velocity, and F is an enhancement factor. Equation (9) is similar to Reference Colbeck,Colbeck’s (1982) Equation (1). Reference Sturm,Sturm (1989, Reference Sturm.1991) and Reference Sturm. and JohnsonSturm and Johnson (1991) have shown that convection is common in the subarctic snow, so the second term on the right in Equation (9) may be important. But to simplify, we accommodate the convection in a heuristic fashion: we allow F, the enhancement factor, to increase if there is convection, but we remove the explicit convection term. Reference Colbeck.Colbeck (1993) suggests F ranges From 4 to 6, but admits we know little about its line range of values. If convection were present it would be higher than 6, and the magnitude of our flux gradients would increase. Since our main interest is the pattern of vapor transport, this simplification is unlikely to be severe.

without an explicit term For convection, Equation (9) reduces to Fick’s law. Differentiating with respect to height in the snow (z)

(10)

Reference De Quervain.De Quervain (1973) derives a similar result.

We now assume that F is constant with height. In reality, it probably varies. Reference Colbeck.Colbeck’s (1993) work suggests it should be a function of pore size in the snow. As shown in Figures 2 and 5, grain-size decreases with height, so pore size probably does too. and F must vary. However, we have no data on how F varies, so we leave it a constant. if we varied, it would change the magnitude but not the sense of our results.

Combining Equations (10) and (4), we get:

(11)

in which the flux gradient is given in terms of the second derivative of the water-vapor density, which can be approximated by numerically differentiating the combined Equations (7) and (8) twice, using a central difference scheme. Equation (11) also describes the rate at which a layer of snow will density due to vapor condensation or sublimation, and we refer to it this way in our results.

Using Equations (7) – (11) and measured temperature profiles (taken every 6 th in 1986-87 and every 1 h in 1989-90 and 1991-92), we have calculated the vapor-flux gradient (or densification rate resulting From vapor condensation) as a function of height in the snow-pack. Tb illustrate the procedure, the temperature profile in Figure 11, typical of many profiles, has been fit with Equation (8). The densification-rate profile corresponding to the temperature profile, calculated using F= 4, is shown in Figure 15. It has a minimum about one-third of the way up through the pack, and a maximum just below the top. These, respectively, are the locations of the most vigorous mass loss and gain. For each temperature profile, the same procedure was followed and the magnitudes of the maximum and minimum densification rates were recorded. They are plotted as a function of time in Figure 16. The height of the minimum (as a fraction of total depth) is also plotted; the height of the maximum is not shown since it was always slightly below the snow surface.

Fig. 15. The densification rate as a function of height (see text for method of calculation). calculated for the temperature profile shown in Figure 11. The densification rate maximum is just below the snow surface, and the minimum is about halfway between the base and the top of the snow.

Densification rates were generally about +50 x 10⁻⁷ kg m⁻² s⁻¹ m⁻¹ , though peak values ranged as high as ±500 X 10⁻⁷ kg m⁻² s⁻¹ m⁻¹. This range of values is similar to that calculated from changes in snow-layer density (Table 6). Most peaks were transient spikes associated with rapid changes in air temperature. Maximum and minimum densification rates were coupled, but not identical. When the pack began to lose mass from the lower part at a high rate, it began to accumulate at a high rate in the upper part. In the model, we do not force the integrated value to be zero (i.e., our system is “Open”), so these two rates do not have to be equal.

Several large events have been marked in Figure 16. All of these events occurred when the air temperature was low (T_air < −25 ° C) and the snow relatively thin (<30 cm).The events have the same form: densification rates increase rapidly when the air temperature initially drops, but decay as the cold weather continues. Unlike the transient spikes, these events last several days and indicate an appreciable amount of transport. The events show marked differences in magnitude from one winter to the next. In 1986-87, one event had densification rates in excess of 500 x 10⁻⁷ kg m⁻² s⁻¹ m⁻¹ and lasted 10 d. This occurred while the snow was less than 20 cm deep. In the other winters, the snow-pack built up in the Fall before the air temperature decreased, so calculated densification rates were about one-fifth of the magnitude observed in 1986-87.

The height of the minimum densification rate (top panels in Figure 16) tended lo be low in the pack, particularly early in the winter when loss rates were highest. The result suggests that there would be a zone of net mass loss, potentially a weak layer, low in the snow pack, Its predicted location coincides nicely with the location of layer M4. which is noticeably weak (Fig.4) and has textural features that suggest mass loss by sublimation. In 1989-90, the location of the minimum moved slowly but progressively higher in the snow during the season. Early in the winter when transport rates were highest, it tended to be located 20-30% of the way up from the base of the snow. By the end of the winter it was 70-80% up from the base.

Fig. 16. The minimum and maximum densification rates (see Figs 11 and 15) for the winters of 1986-87, 1989-90 and 1991-92. Air temperature and snow depth are shown for reference. Large flux events (high rates of densification) are marked. The fractional height (dotted line) of the flux-gradient minimum appears to rise from near the base to near the top of the snow as a function of time. A smoothed line (solid) has been drawn through the data to highlight the trend.

In summary, our temperature model predicts a vapor-transport system that moves mass most vigorously from low (but not the base) to high in the pack. Vapor-flux gradients vary more than an order of magnitude, and are in the same range as those determined by other means (Table 6). The location of the minimum densification rate coincides with a notably weak layer.

Estimates of Grain-to-Grain Vapor Transport

In 1986-87, about one out of ten grains grew, while the rest shrank (Fig.7). If one assumes spherical grains and no layer-to-layer vapor fluxes, this would yield an increase in size by a factor of 2-3,which is close to what was observed (Fig.5). We can use these grain-growth measurements to estimate the net inter-particle flux.

First, we estimate the net inter-particle flux from photo-grammetric measurements of the width, height and thickness of 70 grains photographed in March 1986. From these measurements, we calculate that the end-of-winter mass of individual grains varied from 1 to 8 x 10 ⁻⁵ kg. The grains began to grow in November 1985, at which time their mass was nearly an order of magnitude smaller. Subtracling the initial mass from the final mass and dividing by the elapsed time, the growth rates ranged from 1 to 11 x 10⁻¹² kg s⁻¹. Dividing by the average surface area of an individual grain, estimated from the photographs, we calculate that the net inter-part vapor flux averaged 0.24 to 0.90 x 10⁻⁷ kg m⁻² s⁻¹ during the winter.

Calculation of Grain-to-Grain Vapor Transport

We can also use the changes in the average grain-size shown in Figure 5 to estimate the net inter-particle flux. The growth curves have been replotted as a function of the elapsed time since each layer was rleposited (Fig.17), with grain diameter replacing grain radius for convenience. Two distinct trajectories (grain radius vs time) are suggested by the data, one for the lower snow layers (4 and 10 cm), and one for the tippet snow layers (20 and 30 cm). The lower layers were less dense than the upper layers and on average 5 ° C higher in temperature, two conditions conducive to more rapid growth

The trajectories shown in Figure 17 are exponential curves:

(12)

where r is the grain radius (mm), r ₀ is the initial radius, equal to 0.52 mm for both upper and lower layers, and r_∞ is the radius a grain converges to with time. For the lower layers, r _∞ = 2.13 mm and α = 0.0209: for upper layers, r _∞ = 1.88 mm and α= 0.0102.

Fig. 17. Growth-rate curves for two upper and two lower layers of snow, 1986-87, with the net inter-particle vapor flux implied by this growth rate (see text for details) shown on the right axis.

When differentiated, Equaiion (12) gives the growth rate:

(13)

The exponential forms in Equations (12) and (13) indicate a growth rate (dr/dt), proportional to the number of grains present in a layer at a given time (i.e. the number of vapor sources), and arc consistent with Equation (3), which indicates that this number decays exponentially with time.

Equation (13) can be used with a simple model of spherical grains to estimate the net inter-particle vapor flux. Approximating snow grains as ice spheres is not that extreme an assumption given the similarity of the curve for spheres and the other curves in Fig. 3. The model also requires the assumption that the rate of water-vapor condensation on a spherical grain is proportional to its surface area. Under these assumptions, the rate of change of mass of an average grain with time (dm/di) is:

(14)

where ρ_i is the density of ice. and we have made use of the relationship dV/dt = (∂V/∂r)(∂r/∂t) between the volume (V) and the radius to derive Equation (14). The rate of condensation of water vapor per unit area (j(t)) on a growing grain multiplied by lhe surface area of the grain (S = 4πr²) must equal the time rate of change of mass, thus:

(15)

Based on the data in Fig. 17, the net condensation rate or net inter-particle flux rate, j(t), was initially high, but then diminished, asymptotically approaching zero as the winter progressed. The warmer, more rapidly growing grains of the lower layers initially experienced net fluxes several times greater than those in the colder, Upper layers of snow. The average values calculated with the model (0.52 and 0.68 x 10⁻⁷ kg m⁻² s^{− 1} for upper and lower layers, respectively) are consistent with the average values estimated from photogrammetric measurements. These rates are for “average” grains. Actual fluxes must have varied widely.