## Snow metamorphism

The great variability in snow microstructure is due in a small part to the initial diversity of precipitating particles but mostly to the various transformations the ice matrix undergoes because of thermodynamic relationships among the water phases. These transformations are called snow metamorphism. Since the temperature of natural snow is generally close to the triple point, mass exchanges between vapor and ice and possibly between ice, vapor, and liquid water are very active. This activity is enhanced by the large specific area of the ice/air interface, which derives from the snow microstructure. Consequently, ice crystals are continuously evolving in snow. The effects and rate of metamorphism differ according to the prevailing thermal and meteorological conditions, which explains why succeeding fresh snow layers evolve differently and why the snowpack becomes stratified. The main cause of differentiation in metamorphism is the absence or presence of the liquid phase. Thus, we distinguish between dry and wet snow metamorphism. From a qualitative point of view, the mechanisms of dry and wet snow metamorphism have been well understood for a long time. The main transformations of snow due to metamorphism are summarized in Fig. 2.7.

### Dry snow metamorphism

Dry snow is defined as snow that contains no liquid water. By this definition, the pores in dry snow are filled only by air that is usually saturated with vapor. This is typically the case when the temperature of a snow layer with few impurities is below -0.01 °C, but pure dry snow may exist at a bulk temperature even closer to 0 °C. Vapor saturation occurs because of the large specific area of the ice/air interface, which facilitates mass exchanges between ice and vapor and thus ensures a macroscopic equilibrium between the two phases when there is no interstitial air movement. At the microscale, equilibrium is not possible because of differences in temperature and in ice crystal geometry.

According to Equations (2.7) and (2.2), microscopic differences in vapor pressure at saturation due to variations in curvature and temperature have the following important consequences for snow metamorphism.

• Equilibrium growth form. Because of the crystal's shape, local variations in curvature generate local variations in the equilibrium vapor pressure. If temperature is quasi-uniform, the air around convex crystal surfaces (at the points of crystal branches, for example) tends to be at a higher vapor pressure than the air around concave surfaces (at the bond between two adjacent crystals, for example). Thus, the induced local pressure gradients

generate vapor diffusion from convex towards concave regions. To maintain equilibrium (as much as possible), diffusion is partially balanced by sublimation of the convex crystal surfaces and a corresponding deposition over concave or less convex surfaces. If this process continues, highly convex surfaces therefore shrink while concave or less convex surfaces grow. Because the equilibrium form of an individual ice crystal is a sphere (the shape with the smallest specific area or surface-to-volume ratio), grains become rounded over time. If the process continues for several weeks, the average size of snow grains increases, but at a much slower growth rate.

Kinetic growth form. If a temperature gradient is maintained through a snow sample, variations in temperature generate variations in vapor pressure at saturation and thus induce vapor diffusion from the warmest crystal surfaces toward the coldest ones. This diffusion is partially balanced by sublimation of the warmest crystal surfaces and the corresponding deposition over the coldest ones. If the temperature gradient is high enough, the growth of the coldest surfaces is rapid enough to form facets and even strias, which are the respective shapes of faceted crystals (class 4) and depth hoar (class 5). The maintenance of a strong temperature gradient through a snow layer keeps snow far from the equilibrium form and the specific area weakly decreases and even increases if initial snow belongs to classes 3 or 6.

Under natural conditions, curvature and temperature gradient effects work together and compete. Newly deposited precipitation particles (class 1) generally include dendrites or planar crystals, which have numerous sharply convex surfaces. In most situations, these convex features become rapidly rounded and snow metamorphoses into decomposing and fragmented precipitation particles (class 2). At this stage, if the temperature gradient exceeds about 5 °C m-1, snow metamorphoses into faceted crystals (class 4) and further into depth hoar (class 5) if it exceeds about 15 °C m-1. If the temperature gradient stays below 5 °C m-1, rounding continues and snow metamorphoses into rounded grains (class 3).

Wet snow metamorphism When snow holds a significant amount of liquid water (typically more than 0.1% of the volume), mass exchanges between the three water phases must be considered to explain snow metamorphism. In most cases, wet snow is not saturated and mass exchanges are not limited to the solid and liquid phases. If we neglect the effects of impurities (mainly concentrated in the liquid phase) and of air dissolution into water, the Gibbs-Duhem and Laplace equations dictate that the melting point temperature of snow at low liquid water content varies as the sum of negative linear functions of curvature and capillary pressure (Equation 2.9). In snow at higher liquid water content, the melting point temperature varies primarily as a negative linear function of curvature (Equation 2.8). In both cases, the more convex the crystal surface, the lower the melting point temperature around this surface.

Consequently, local temperature gradients within a wet snow sample conduct heat away from the concave or less convex crystal surfaces towards the more convex surfaces. This process generates melting of the most convex surfaces, including the smallest crystals, and refreezing of liquid water (if it is available) onto the concave or less convex surfaces. These mass exchanges lead to rounding and growth of the crystals, in a process that minimizes the specific area and tends towards the equilibrium spherical form. A limiting factor for wet snow metamorphism is the capacity for liquid water to move easily from areas of melting to areas of refreezing. At very low water content, liquid water menisces are rare and disconnected, thus limiting the efficiency of this mechanism. In such cases, exchanges between vapor and solid phases may still dominate. When wet snow freezes, liquid menisces are included in the adjacent ice grains, which explains the rapid growth of grains during melt-freeze cycles.

### 2.2.4 Grain size and growth rate

Since snow texture affects most of its physical properties (among them its albedo), it is of primary importance for snow modeling to have a quantitative knowledge of metamorphism processes. A major difficulty comes from the lack of aunique parameter that may quantitatively describe snow texture. In most cases, the microstructure of a snow sample is described by the grain size and metamorphism is quantified through a growth rate. According to the International Classification, the size of a grain is its greatest extension and the grain size of a snow sample is the average size of its characteristic grains. But for a given physical process, it may be more relevant to consider other textural definitions, such as the geometrical size, the optical grain size, the pore size, or the specific area. In the simplified case where we consider snow as a packing of rounded grains, the effects of curvature on the vapor pressure and on the melting point depression dictate that the smallest grains, which have the most convex surfaces, will always shrink at the profit of larger grains. For that reason, we speak of the growth rate due to metamorphism. In actuality, however, snow shows a large set of possible grain shapes and the use of grain size or growth rate as a texture descriptor is very reductive. Precipitation particles (class 1) illustrate this difficulty: their size may be relatively large (typically one millimeter or more for plates or stellar dendrites), but they are generally very planar and along their perimeter (if we consider them as two-dimensional particles) we find very sharp surfaces that shrink very rapidly. Consequently grain size decreases while the planar crystal transforms into a granular crystal that is larger from the thermodynamic point of view.

Quantification of snow metamorphism has been investigated in three different ways, experimentally, theoretically, and numerically.

• Experimental. Different experiments on dry and wet snow metamorphism have been conducted on snow samples submitted to most temperature, temperature gradient, and liquid water content conditions that can be encountered in nature. Wakahama (1968) and Raymond and Tusima (1979) have investigated the growth rate of a population of snow grains immersed in liquid water. Giddings and Lachapelle (1962), Akitaya (1974), and Marbouty (1980) investigated the growth rate of snow submitted to a high temperature gradient. Brun (1989) investigated the growth rate of snow grains as a function of liquid water content. Brun et al. (1992) investigated the growth rate and the faceting of fresh snow submitted to a low or medium temperature gradient. These experiments have been conducted in the cold laboratory or in the field and consider the effect of grain size as well as grain shape. Brun et al. (1992) summarized these experiments in a complete set of quantitative metamorphism laws that describe the rate of change in growth and shape of snow. These empirical laws use an original formalism that makes it possible to describe snow with a continuous set of parameters {dendricity, sphericity, grain size}. Experiments also have been conducted on sintering and on the growth rate of bonds linking snow grains (Keeler, 1969a). This last process is of little interest for snow atmosphere exchanges but it is of major importance for the evolution of the mechanical properties of snow layers and the stability of the snowpack.

• Theoretical. Considering snow as a packing of idealized particles of various sizes and at various temperatures, functions for vapor diffusion between two adjacent grains have been analytically derived. The corresponding growth rate has been deduced, making it possible to compare the relative efficiencies of temperature gradient metamorphism with that due to variations in curvature (Colbeck, 1973, 1980, 1983; Gubler, 1985). Colbeck proposes the following equation to describe the average growth rate of snow particles:

where m is the mass growth rate of a particle, C a shape factor, Gg an enhancement factor, Dg the diffusion coefficient of water vapor in air, T the temperature, T' the snow temperature gradient, and pv,sat the vapor density at equilibrium.

Similarly, the growth rate of grains in water-saturated snow has been calculated. These calculations provide a theoretical background to explain the high growth rates observed at high temperature gradients and at high liquid water contents. The extension of calculated growth rates from dry and wet metamorphism to a population of grains having a distribution of sizes and shapes is difficult but possible. Several snow models use metamorphism laws deduced from these theoretical works to calculate the evolution of grain size of the different snow layers of a snowpack.

• Numerical. More recently, high-resolution numerical models have been developed to calculate vapor diffusion caused by temperature gradients inside an idealized snow matrix (Christon et al., 1987). Recent availability of high-resolution three-dimensional images of snow samples (see Fig. 2.8) and the huge increase of computer performance opens fascinating perspectives for the development and validation of snow microstructure models (Flin et al., 2003). These models will allow the establishment of more reliable metamor-phism laws in the near future.

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