Snow densification in alpine and polar regions transforms snowflakes into ice crystals. On the surface of glaciers and ice sheets, fresh snow is buried by new snow each winter, thereby slowly transforming into firn and then glacial ice as it compresses and descends. In cold and dry environments (e.g., melt-free accumulation areas of mountain glaciers, interior Greenland, and Antarctica), surface snow evolves due to temperature gradient metamorphism and atmospheric interactions . In the region below the surface, snow compaction is thought to occur in three stages . In the first stage near the surface, compaction occurs by grain growth and rearrangement due to grain boundary sliding . In the second stage, compaction occurs through increasing overburden pressure inducing creep deformation and sintering . In the final stage, interconnected pores have closed off, and further compaction is caused by air bubble compression . In wet environments, such as the percolation zone of mountain glaciers and Greenland as well as many ice shelves of Antarctica, compaction occurs by a combination of dry snow compaction processes and refreezing of meltwater, which can either enhance or detract from the densification processes just mentioned . Although meltwater percolation is an important part of the compaction process in many areas e.g.,, we will only consider dry snow compaction here.

An important reason for studying snow compaction on glaciers, ice sheets, and snowfields is to connect a change in surface elevation to a volume of stored water. The total water volume stored in glaciers and snowpacks is important to know for current as well as future water resources and sea level rise considerations. Additionally, compaction is important for ice core analysis . At the bottom of the firn column, the difference between the ice age and the gas age is an important input into paleoclimate reconstructions of temperature in ice cores and must be estimated using a compaction model . Compaction can be measured through tracking relative displacements of metal markers or snow features in optical stratigraphy , autonomous phase-sensitive radio-echo sounding , vertical strain fiber-optic sensors , the relative-displacement “coffee can” technique , and a continuously recording coffee can method .

In one-dimensional compaction, the body force acting on the snow is gravity, leading to an overburden pressure σo, which increases with depth z below the surface zs. Given mass and momentum conservation, all that is needed to predict the densification rate (i.e., surface ice velocity wizs and strain rate within the snow column ∂wi/∂z) is a constitutive relationship between stress σo and strain rate ∂wi/∂z in one vertical dimension. Thus, snow compaction can be thought of as a problem of rheology, where the rheology is complicated due to variation with depth, temperature, humidity, and water content, among many other physical processes. Three common approaches to characterizing the rheology of a material are empirical analysis e.g.,, microstructural analysis e.g.,, and experimental analysis. It is this last approach that we describe in this paper.

The standard compaction law applied to most glaciers and ice sheets is a one-dimensional relationship for the rate of change of snow density ρ with depth; i.e., 1∂ρ∂t+wi∂ρ∂z=-Cρ,ρi,T,a˙,σo,∂wi∂z,…, where C is the compaction function and can depend on the snow density ρ, ice density ρi, temperature T, accumulation rate a˙, overburden pressure σo, vertical strain rate ∂wi/∂z, humidity, water saturation, grain size, and potentially many other physical processes . take C=c(a˙,T)ρ, where c is an empirical function. However, including the accumulation rate in the compaction function is dubious as it really should enter the mass conservation equations as a boundary condition . Other forms of the righthand side of Eq. () are discussed in , , , and . In this paper, we experimentally and mathematically analyze the function dependence of C on overburden pressure and strain rate.

We now summarize the outline of the paper. In Sect. , we describe the laboratory compaction experiments of and the samples used. In Sect. , we construct a continuum model to describe the experiments and implement our model numerically. Lastly, in Sect. , we compare the theoretical predictions for the snow compaction with the data, showing that the theory does an excellent job explaining the snow compaction. We additionally discuss how this can be implemented into a compaction model such as Eq. () and our plans for future work, before a short conclusion section.

To understand the microstructural origin of macroscopic snow material properties, the role of snow microstructure in avalanche initiation, and snow metamorphosis, performed continuous compression tests on snow samples collected near Dartmouth College in Hanover, NH (USA). In their experiments, focused on nine samples, ranging from (i) relatively warm, freshly fallen, low-density snow to (ii) snow that was collected during cold air temperatures (ca. -7 to -9 ∘C) and placed in a -30 ∘C cold room for 1 year, allowing for the snow crystals to sinter. In the sintering process, bonds form between snow crystals and the snow evolved into round and well-connected snow grains during the year in the cold room . In this paper, we focus on the four sintered samples from , as they are most applicable to alpine snowpacks and firn on glaciers. The naming convention for these sintered low-temperature (SLT) samples follows , where the highest density sample is SLT-1 and the lowest density sample is SLT-4 (see Table ). Before compaction, the samples were vibrated to construct samples with different densities, undoubtedly breaking some of the sintered bounds. These sintered and vibrated samples represent a range of densities with relatively small differences in specific surface area (SSA; surface-to-volume ratio, viz. Table ). The snow samples were then extruded as cylinders, 15.7 mm in diameter and 18 mm tall, and placed on a Skyscan material testing compression stage (i.e., a French-press or cafetière-à-piston-style coffee maker) inside of a microscopic computed tomography (micro CT) machine. A schematic of the compression stage is shown in Fig. . Both the top and bottom plates are impermeable but the air contained within the snow is able to flow out the side of the open sample. The samples were compressed at a constant rate of 12.7 mm h-1 (i.e., 111 m yr-1 or a pore-scale Reynolds number of Re=10-4 with 1 mm as a characteristic pore size) for the full 5.7 mm dynamic range of the Skyscan compression stage. To avoid termination effects, we only consider the first 5 mm of the range . The compression stage measured the load required to maintain a constant displacement rate for each snow sample. then plot the loading stress as a function of the displacement, from which we can derive a stress-strain curve. In the next section, we develop a model for constant-displacement-rate compaction experiments, and in Sect.  we compare the model predictions for stress versus displacement against the experimental data.

Compaction occurs in a variety of natural and industrial processes, such as sedimentary basin formation, paper pulp dewatering, and particle flocculation, and has motivated numerous experiments and mathematical models e.g.,. In this section, we describe the theory outlined by , although we write the theory in terms of porosity ϕ rather than the solid fraction.

For dry snow that is composed of solely air and ice particles, snow density ρ is given as 2ρ=ϕρa+(1-ϕ)ρi, where ϕ is the porosity, i.e., the void space . The density of air is ρa and the density of pure ice is ρi, both of which we treat as constant. Although air density increases when pressurized in a confined space, in an open system where the air is able to flow, the air density is approximately constant. In the firn column of a glacier or ice sheet, the transition between freely flowing air and confined air is known as the pore-close-off depth. Below the pore-close-off depth the air density will increase with pressure, and above the pore-close-off depth, the air is approximately constant density.

Now from the expression in Eq. (), it is clear that density variation in snow is equivalent to variation in porosity: near the surface of a glacier or snowpack, the snow air content is high and then density is closer to ρa, whereas at depth, the snow air content is low, and the density approaches ρi. Treating snow as a mixture of air and ice, mass conservation of each component may be written as 3∂ρaϕ∂t+∇⋅ρaϕua=ms,4∂ρi(1-ϕ)∂t+∇⋅ρi(1-ϕ)ui=-ms, where ua is the air velocity, ui is the ice velocity, and ms is the mass exchange between each phase due to sublimation. Again, we treat the densities of each substance as constant, meaning that it is so easy for air to escape from the snow that the air density does not change. We also neglect phase change (i.e., ms=0) and restrict our attention to one vertical dimension. Thus, we can write this model as 5∂ϕ∂t+∂∂zϕwa=0,6-∂ϕ∂t+∂∂z(1-ϕ)wi=0. Now adding Eqs. () and () gives the insight that decreasing the porosity requires evacuating the incompressible air out of the pore space or, alternatively, air motion allows for snow compaction; i.e., 7∂∂zϕwa+(1-ϕ)wi=0. In other words, there is an exact volumetric trade-off where the pore space occupied by air is filled by ice during compaction.

In the experiments, the sample has an initial height of z=h0=18 mm (see Fig. ), and, therefore, the overburden pressure is negligible when compared to the applied stress. We take the vertical coordinate system to increase downward. At the top of the sample z=0, the wall is impermeable; i.e., wa=wi=0. Thus, we can integrate Eq. () to find that 8wi+ϕwa-wi=0. We model the airflow through porous snow using Darcy's law, which is given as 9ϕwa-wi=-k(ϕ)μ∂p∂z for permeability k(ϕ), air viscosity μ, and air pressure p. As described by , Eq. () relates the solid ice velocity wi to the Darcy velocity of the airflow through the pores. Thus, combining Eqs. () and (), we find that 10wi=k(ϕ)μ∂p∂z; that is, the ice particle velocity is determined by the pressure gradient driving air motion.

Force balance in the vertical direction is given as 11∂Σ∂z=0, where Σ is the vertical bulk compressive stress . We define effective pressure N as the difference between the compressive stress and air pressure, i.e., N=Σ-p, and write Eq. () as 12wi=-k(ϕ)μ∂N∂z. As is typical in soil mechanics and other applications of compaction , we can relate the effective pressure N to the porosity ϕ through an empirical constitutive relationship, so that N is identically equal to N(ϕ). This relationship is a form of plasticity as there is a one-to-one correspondence between load and density without reference to displacement, strain rate, or stress history. Inserting this into Eq. () and putting that into Eq. () gives 13∂ϕ∂t=∂∂z(1-ϕ)-N′(ϕ)k(ϕ)μ∂ϕ∂z, which is a nonlinear diffusion equation for the porosity, and N′(ϕ)=dN/dϕ is the derivative of the effective pressure with respect to the porosity.

Here we compare the predictions of the theory outlined in the previous section with the experimental data of , which we described in the Sect. . A key output of these experiments was the applied load required to compact the snow as a function of displacement, during the constant-displacement-rate experiments. These data are shown in Fig. . The lowest density snow SLT-1 withstands the least stress as a function of displacement. As the density of the snow samples increases, so does the required stress for a given displacement. A similar observation can be made for the theory: the applied load is given by Eq. () and can be written dimensionally as 30Σ=N01-ϕnϕmatz=h, which states that the applied load Σ increases as the porosity ϕ decreases (see Fig. ). In other words, for a given displacement, the stress in the theory will also increase as the initial density increases, which is also true for the experimental data, meaning that we have chosen a sensible constitutive law. The theory for the load as a function of the displacement makes sense qualitatively, but we can compare the theory to the experiments in the following way: we run the full model, Eqs. ()–(), and evaluate the stress in Eq. () using the value of the porosity ϕ at the bottom of the sample z=h and plot it against the displacement δ=h0-h. The results for reasonable parameters are shown as black lines in Fig. .

The model, therefore, contains three inputs and seven parameters. The most important and most uncertain parameter is γ. Due to the fact that the permeability prefactor k0 and the plastic effective pressure prefactor N0 are unconstrained, we choose representative values for each sample. The permeability prefactor k0 only appears within γ, whereas the plasticity prefactor N0 appears in γ and is required to scale the theory output to compare with experiments. The next four parameters are the exponents a, b, m, and n, which have expected values, but again have not been measured for these samples. Therefore, we only examine small departures from typical values. Additionally, as we will see, there is a small amount of friction in the compression device, and so a small constant stress σf is added to the stress output from the theory; i.e., Σ=N(ϕ)+σf at z=h. This approach was also used by in their experiments. The inputs are the initial porosity of the sample, which is set by the initial density measured by . Another input to the problem is the initial height of the sample, h0. The total experiment run time tf is also an input and set by the final displacement divided by the displacement rate; i.e., tf=(h0-hf)/W, which is about 24 min.

The values of the fixed parameters used for the theory lines in Fig.  are the typical exponents (i.e., a=3, b=2, m=2, n=2) and the load information σf=3 kPa and N0=30 kPa. The value of γ for each curve is shown on the figure and is γ=0.46 (SLT-1), γ=0.25 (SLT-2), γ=0.29 (SLT-3), and γ=0.18 (SLT-4). In selecting these values, we successfully (1) found fixed load parameters that worked for all of the experiments and (2) kept the constitutive exponents as the typical values. Thus, γ is the only parameter that varies between theory predictions for each experiment, which is to be expected due to potential variation of k0 with grain size. It is worth emphasizing that this is not a systemic nonlinear best-fit analysis for seven parameters; rather, the agreement between theory and experiment demonstrates a physically motivated model with a single underconstrained parameter.

In general, there is excellent agreement between the theoretical predictions and the experimental data. For the bottom two samples, SLT-3 and SLT-4, the theory captures all of the major data trends and all falls well within any experimental scatter. For the top sample SLT-1, the theory does a reasonable job in capturing the data trend, yet for these parameters it does not follow the data points exactly. A better fit can be obtained by changing the parameters, indicating that this sample may be a different compaction regime than SLT-3 or SLT-4. However, the fit is reasonable enough that this sample is likely just on the edge of a new regime, if at all. In contrast to the other samples, the theory does not adequately capture the data trend of the middle sample, SLT-2. This sample is interesting because it has almost the same density and specific surface area as SLT-3, yet it responded very differently to compression loading. Due to the small size of the samples, i.e., 15.7 mm in diameter and 18 mm tall, the likelihood of defects or inhomogeneities dominating the results is quite high. Thus, the snow bonds in SLT-2 from prior sintering could have been particularly stubborn, requiring more load for a given displacement.

Another possibility for why the two snow samples SLT-1 and SLT-2 did not agree as well with the theory as SLT-3 and SLT-4 is that pressure sintering during the experiment allowed for greater bonding of snow crystals. Adding pressure is an efficient method of accelerating the rate of sintering and can lead to sinter rates that are orders of magnitude faster than by ambient surface energy differences alone . For this reason, attribute the increase in required load to accelerating the sintering and coarsening processes occurring within the snow samples. also analyze sintering during compaction experiments and find that the sintering rate is enhanced. The plastic compaction theory we present in this paper does not include pressure processes and, therefore, would not be able to describe the stress required to break sintered bonds, although this is a promising direction for future research.

The plastic compaction theory presented here can be related back to the general firn compaction model given in Eq. (). Rearranging Eq. () gives 31∂ϕ∂t+wi∂ϕ∂z=(1-ϕ)∂wi∂z, which is in the form of Eq. (). Inserting Darcy's law () with mass conservation () and (), the compaction function C is given by 32C=-ρa-ρi1-ϕ∂wi∂z=ρa-ρi1-ϕ∂∂zN′(ϕ)k(ϕ)μ∂ϕ∂z, which shows the connection between compaction and air evacuating pore space as well as the role of the constitutive relations N(ϕ) and k(ϕ). In this way, measuring the parameters of these constitutive relations in the laboratory allows for predictions of compaction using C from Eq. ().

For generalized viscous compaction, starts from Eq. () and connects the divergence ∂wi/∂z to an effective pressure Pe through a compaction viscosity ηc; i.e., 33∂wi∂z=Peηc, which is the compaction law used in studies of temperate ice and is equivalent to the viscous closure of a Röthlisberger channel . The compaction law, Eq. (), implies that 34C=-ρa-ρi1-ϕPeηc. If we take the effective pressure Pe to be the overburden pressure, i.e., Pe=ρgz, thereby setting the air pressure to zero, we find the pressure-compaction model described by following . Although this compaction model is attractive because it connects viscous processes to compaction, by setting the air pressure to zero (or even a constant), this compaction model fails to capture the evacuation of air from the pore space, which we have shown to be a very important process in the mechanical compaction of laboratory snow samples. In future work, we will adapt the model we present here to include viscous compaction following . Instead of neglecting the air pressure, as in the work, we will explicitly solve Darcy's law and determine the resulting compaction.

In this paper, we articulated a mathematical model to describe the snow compaction experiments of . This model consists of mass and momentum conservation as well as porosity-dependent permeability and plastic effective pressure constitutive equations. The outputs of the model are the snow density as a function of time and space as well as the stress as a function of displacement. Comparing the model outputs to the experimental data of shows excellent agreement, especially for the low-density sintered snow. As the density increased, small discrepancies between model and theory emerged, potentially due to the necessity of creep or sample defects. Nevertheless, the excellent agreement between theory and experiments suggests that measuring compaction in the lab is a promising direction forward for understanding snow compaction and that the plastic effective pressure as a function of porosity may be a key constitutive relation to quantify.