Thermal properties

As in all aggregate materials, the thermal properties of snow depend upon the microstructure; for snow this reflects on crystal type, crystal organization, and connectivity. Because snow undergoes metamorphism with time and ambient conditions, its thermal properties also change. Although the measurements of Izumi and Huzioka (1975) confirm that metamorphism induces anisotropy in physical properties, there are no quantitative analyses that can predict thermal properties based on microstructure. Arons and Colbeck (1995) give a historical perspective and a useful summary of the attempts to quantitatively predict material properties in snow based on the microstructure, and conclude that until improvements are made in the ability to characterize the snow geometry, little progress is likely. Thus, there is a poor physical understanding of the geometric effects of microstructure on the aggregate properties. The most useful estimates to date of the thermal properties of snow are based on macroscopic measurements of the aggregate that are then correlated to the snow type and physical characteristics such as density and grain size.

Thermal conductivity

The thermal conductivity of a medium controls the speed at which heat will be transferred within the medium. In one dimension, the flux of heat at a point is given by the Fourier equation as where F is the flux, keff is the thermal conductivity, T is temperature, and x is the spatial coordinate along the direction of flow. The measurement of thermal conductivity in snow includes the effects of heat conduction through connected grains and through the air space, along with the "hand-to-hand" transfer of latent heat by water vapor. In this process, vapor sublimes from warmer ice grains, diffuses through the pore space, and condenses on colder grains. With this understanding, keff as defined above is actually an "effective" thermal conductivity that includes both the diffusion of heat and vapor transport processes. Singh (1999) demonstrates that for temperatures above -10 ° C and approaching the melting point, the effective thermal conductivity of depth hoar can be one-half to two-thirds due to vapor transport. For snow of higher density and larger bonds, it is likely that most of the effective conductivity is due to conduction through the ice. Singh also notes a marked increase in thermal conductivity with increasing liquid water content. Sturm and Johnson (1992) made measurements of the thermal conductivity of depth hoar and added these measurements to the compilation of Mellor (1977). Figure 2.10 is the compiled figure by Sturm and Johnson (1992), which depicts measurements of the thermal conductivity of snow as a function of snow density. Here, the earlier measurements of Sturm and Johnson are replaced with a quadratic curve from Sturm et al. (1997), based on about 500 new measurements of thermal conductivity. Although a general trend of increasing thermal conductivity with increasing density is clear, there is wide scatter in the graph as a result of variations in microstructure and temperature. For a given density, Sturm etal.'s (1997) compilation of measurements by others has a standard deviation of about 0.1 W m-1 K-1, and their own measurements have a standard deviation somewhat less than this.

The specific heat of snow reflects on the amount of energy that must be put into a unit amount of snow in order to change its temperature. Because snow is a conglomerate of ice, air, and liquid water, the specific heat is calculated as the weighted average of the parts:

Specific heat cp,s — (Pa$acp,a + [email protected],i + PAcp,£)/Ps,

Figure 2.10. Effective thermal conductivity of snow versus snow density (from Sturm and Johnson, 1992, with modifications. Published 1992 American Geophysical Union. Reproduced/modified by permission of American Geophysical Union).

where cp,s is the specific heat of snow, cp,a is the specific heat of air (= 1005 J kg-1 K-1), cp i is the specific heat of ice (= 2114 J kg-1 K-1), cp^ is the specific heat of water (= 4217 J kg-1 K-1), and 0k are the constituent volume fractions. Values given for specific heats are at 0 °C and at 1 atmosphere of pressure. Because the constituent specific heats are temperature dependent, the specific heat of snow has a mild temperature dependence.

2.3.2 Advective-diffusive heat transfer

For the prediction of heat transfer in snow as a medium, the ice-vapor-air-liquid water system is assumed to be in thermal equilibrium on scales of grain size and less. Heat transport occurs though advective and diffusive processes:

where t is time, vk is the flow velocity of fluid k (a = air, t = water), and q is the thermal source term. If there is no fluid (air or water) flow through the snow, the energy transfer is by simple heat conduction. The source term includes the latent heat effects:

When water is the fluid flowing through the snow, Lit is the latent heat of fusion and S is the melt-freeze phase change source term. When the fluid flow through the snow is air, S is the vapor source term and Liv is the latent heat of sublimation.

For heat transfer, the thermal effects of vapor transport are more pronounced in dry snow than in wet snow. For dry snow, the transport of water vapor through the snow is described by dpv dpv d d t a d x d x

where pv is water vapor density, va is the air flow (ventilation) velocity, Ds is the diffusion coefficient for vapor flow in snow, and S is the source of water vapor due to phase change, enhanced by air flow through the snow as

where h is the mass transfer coefficient, A is the specific surface of the snow, and Pv,sat is the saturation vapor density.

Albert and McGilvary (1992) demonstrated that the heat transfer associated with vapor transport is significant in the determination of the overall temperature profile of a ventilated snow sample, but that the major temperature effects are controlled by the balance between the heat carried by the dry air flow through the snow and heat conduction due to the temperatures imposed at the boundaries. The relative thermal response of the snow to air flow (advection) and imposed temperature gradients (conduction) is characterized by the Peclet number:

keff where Ax is the characteristic distance. Low Pe numbers correspond to heat conduction-controlled temperature profiles, while Pe numbers greater than one

Temperature (°C)

Figure 2.11. (a) In-snow temperature profiles at 6 h intervals for 6 February 1987 in Hanover, NH, during a mostly clear and calm day. (b) The same as for (a), except at 1900 hr on 9 February when the wind speed was 7 m s-1.A comparison of this curve with that at 1800 hr on 6 February demonstrates the effects of windpumping.

Temperature (°C)

Figure 2.11. (a) In-snow temperature profiles at 6 h intervals for 6 February 1987 in Hanover, NH, during a mostly clear and calm day. (b) The same as for (a), except at 1900 hr on 9 February when the wind speed was 7 m s-1.A comparison of this curve with that at 1800 hr on 6 February demonstrates the effects of windpumping.

correspond to advective profiles. For high porosity snows such as fresh snow or hoar, the thermal conductivity is sufficiently low that air flow through snow can cause advection-controlled temperature profiles with Pe >1. This was demonstrated in a field experiment (Albert and Hardy, 1995) where the immediate temperature effects from ventilation appeared when flow was induced in natural seasonal snow. For lower porosity, higher conductivity snow, such as wind-packed snow or firn, the thermal conductivity is sufficiently high that the Pe number is almost always low under natural conditions, making it likely that the temperature profile will be dominated by the heat conduction profile, even though there may be significant air flow through the snow (Albert and McGilvary, 1992).

As an example of observed thermal effects in a seasonal snow cover, we compare diurnal temperature profiles for calm and windy days within a 50 cm deep snowpack in Hanover, NH (Jordan et al., 1989) during the winter of 1987. Figure 2.11a shows profiles for 6 February, when wind speeds rarely exceeded 1ms-1 and clear skies prevailed for much of the day. Night-time radiational losses and daytime solar gains dominated the surface energy budget and caused large swings in temperature within the upper 20 cm of the snowpack. Such diurnal temperature patterns are typical for temperate snow covers experiencing sunny conditions and little turbulent exchange with the atmosphere.

In contrast, Fig. 2.11b shows a temperature profile from the same snowpack on 9 February 1987 at 1900 hr, when the wind speed was 7ms-1. The concave shape of the profile, compared with the convex shape for 6 February at 1800 hr, is an example of convective heat transfer through windpumping. Within this two-week data set, concave profiles consistently coincided with higher wind speeds (Jordan and Davis, 1990). Albert and Hardy (1995) present modeled and measured snow temperatures that clearly demonstrate the thermal effects of ventilation in light seasonal snow. Fresh seasonal snow has a thermal conductivity that is low compared to the ability of the snow to advect heat through the flow of interstitial air. This is in contrast to wind-packed snow and polar firn, where the temperature is usually dominated by the conduction profile even when ventilation is occurring (Albert and McGilvary, 1992). Inert tracer gas measurements in polar conditions have shown that natural ventilation can occur to depths of tens of centimeters to meters in polar snow and firn even in the presence of windpack (Albert and Shultz, 2002).

Ventilation affects both measurements of the chemistry (Albert et al., 2002) and measurements of the thermal properties in snow. For example, Sturm et al. (2002) show that thermal conductivity values inferred from measured temperature profiles in snow-covered sea ice during the SHEBA experiment (surface heat budget of the Arctic Ocean) are higher than measurements with a needle probe at the same site. Jordan etal. (2003) propose that advective heat transport through windpumping may account for this discrepancy and show that high winds (>10 m s-1) can penetrate dense, wind-packed snow on sea ice down to 10 or 20 cm depth. Further investigation is needed to parameterize the effects of windpumping, and possibly of other multidimensional heat transfer mechanisms, for one-dimensional modeling.

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