Introduction

The characteristics and structures of snow change continually and influence albedo and thermal conductivity, which are important for climate. Snow-cover characteristics are strongly influenced by meteorological conditions. Therefore, we propose a one-dimensional model using meteorological data to simulate the snow-cover characteristics for the purpose of clarifying the heat balance and water cycle at the Earth's surface.

There were few all-round snow metamorphism models after a pioneer model developed by Reference AndersonAnderson (1976). Recently, Reference Brun, Martin, Simon, Gendre and ColeouBrun and others (1989) proposed an energy and mass model for operational avalanche forecasting, and the model was advanced to take into account grain-size and type of snow (Reference Brun, David, Sudul and BrunotBrun and others, 1992). However, there are few models which can directly predict snow albedo including the effects of solid impurities and liquid water.

Basic Equations

Figure 1 shows the schematic of the physical processes in the model in this paper. All of the physical variables in this model are described by snow temperature T _s, the amount of liquid water ρ1_w, the dry snow density ρdry. and the solid impurities density ρ _D for albedo.

• Snow temperature T _s

For a snow surface

In a snow cover

C _s: specific heat of snow; λ _s: thermal conductivity of snow; l _f: latent heat of fusion of ice; F: amount of snowmelt per unit time and volume; I = (1 − A _s)S↓; A _s: albedo; S↓: solar radiation; μ: extinction coefficient of solar radiation; z: depth from the snow surface; L↓: downward atmospheric radiation; T _sfc; snow surface temperature; ∈: emissivity of snow (= 0.97); σ: Stefan-Boltzmann constant; H: sensible heat flux; lE: latent heat flux.

• Amount of liquid water ρ1_w ρ1_w: mass of liquid water per unit volume; Q: downward liquid water flux (Reference ColbeckColbeck, 1978; Reference ShimizuShimizu, 1970).

• Dry snow density ρdry

W _s: load; η compactive viscosity coefficient of snow.

• Solid impurities density ρD

ρD mass of solid impurities per unit volume of snow; fD: rate of impurities flow; ρ _wet: wet snow density

(= ρdiy + ρ1_w).

Fig. 1.

Sckematic of the relation between the physical processes and snow-cover characteristics in this model.

In practice, Equations (1)-(5) are written in differential forms to calculate the profiles of the variables. (A simple version was described in the appendix inReference Kondo and Yamazaki Kondo and Yamazaki (1990).) The input data are solar and downward radiation, air temperature, specific humidity, wind speed and precipitation. Snowfall and rainfall are distinguished by air temperature (2°C) in the model. The density of new snow is assumed to be 70 kg m⁻³.

Albedo

The albedo of pure dry snow is obtained from profiles of dry snow density and the optical absorption coefficient of ice (Reference Kondo and YamazakiKondo and others, 1988). The multiple reflection is considered with an assumption that snow is constructed by ice plates and air layers(Fig. 2).

Fig. 2.

Schematic of snow albedo submodel.

The albedo, A _s, is obtained by two-stream model as

where

and

Here, μ _i is the extinction coefficient of solar radiation in i-th snow layer, r1 the reflectivity of ice (= 0.018), ρI the density of ice and kI the absorption coefficient of ice (assumed to be 10 m⁻¹). The thickness of an ice layer l _Ii is obtained as

where

is the specific surface area (the area of the surface of the ice particles in unit volume of snow) in i-th snow layer. The units of ρ _dry,i and are gem⁻³ and m²kg⁻¹, respectively. Equation (8) is an experimental equation based on the data in Reference Kondo, Numata and YamazakiNarita (1971).

In the model, it is considered that albedo is decreased by the impurities and liquid water content in snow. As the density of the impurities increases, optical absorption coefficient increases. The increase of effective absorption coefficient due to impurities, k _D, is assumed to be equal to the cross-section of the impurities per unit volume of ice. Based on a few assumptions for characteristics of impurities, the following equation can be obtained:

The units of k _D and ρ _D are m⁻¹ and kgm⁻³, respectively. The amount of aerosol fallout is given by (3.14 × 10⁻⁸)β (kgm⁻²s⁻¹), which is obtained from an assumption of the aerosol-size distribution, where β is the atmospheric turbidity defined byReference Yamamoto, Tanaka and Arao Yamamoto and others (1968).

The effect of liquid water on albedo is described through a decrease in the specific surface area. That is

where

ω is water content (= ρ1_w/ρ _wet) and γ = 1.9 (Yamazaki and others, 1991). The value of S*(ρ _wet) is obtained from Equation (8) using p _wet instead of ρ _dry.

Sensible and Latent Heat Flux

Sensible heat H and latent heat lE are written as

and

Here, Cp and ρ are specific heat at constant pressure and density of air, respectively, and U T _a and q _a are wind speed, air temperature and specific humidity, respectively. The value q _sat(T _sfc) is the saturated specific humidity at surface temperature T _sfc. Bulk coefficients C _H and C _E are set at 0.002 and 0.0021 at a reference height of 1 m, respectively (Reference Kondo and YamazakiKondo and Yamazawa, 1986).

Thermal Conductivity

The thermal conductivity is parameterized using porosity P (= 1 − ρ _dry/ρI) as follows.

where λ_a is the thermal conductivity of the transverse structure piled by ice plates and air layers same as Figure 1 (the smallest theoretical conductivity), and λ_b is that of the longitudinal structure (the largest theoretical conductivity). They are written as

and

where

Here, λ_I is the thermal conductivity of ice (2.2 W m⁻¹K⁻¹), λ_A the thermal conductivity of air (2.14 × 10⁻²W m⁻¹K⁻¹), λ′ _A the effective thermal conductivity of air corrected for vapor diffusion, L the latent heat of sublimation of ice (2.83 MJ kg⁻¹), D _v the diffusion coefficient of water vapor (6.5 × 10⁻⁵m²s⁻¹) and C the saturated vapor density.

Compactive Viscosity

The compactive viscosity coefficient of wet snow is obtained from the equation for dry snow (Reference KojimaKojima, 1957; Reference ShinojimaShinojima, 1967) using a multiplicative factor which describes the decrease of the compactive viscosity coefficient due to liquid water. That is

where A(w) is assumed as

Here, η ₀ (6.9 × 10⁵ kg s m⁻²), K (2.1 × 10⁻³ m³ kg⁻¹), α _s(9.58 × 10⁻² °C⁻¹) and β _s (= 18) are constants.

Short-Period Simulation

Diurnal variation of profiles of temperature, water content and snow density were simulated for the following observation. The detail of the observation has been described in Yamazaki and others (1991). Period: 23-28 February and 9-15 March 1989. Place: Mount Zao Bodaira, Yamagata Prefecture, Japan (a flat soil tennis court).

Observations: air temperature, humidity, wind speed, solar radiation, albedo, snow depth, snow type, snow temperature, water content, density and solid impurities density.

Solid impurities density was measured using a method of light absorption (Reference Kondo and YamazakiKondo and others, 1988). The impurity particles are almost mineral. The range of measured absorption coefficients, k _D, in the top 5 cm of the snow cover is from 0.1 m⁻¹ (26 February) to 9.8 m⁻¹(15 March). Figure 3 displays the values of observed and calculated albedo.

Fig. 3.

Comparison of calculated and observed snow albedo at Zao Bodaira. Δ: effcts of impurities or liquid water are not considered; ×: taking only impurities into account; O: effects of impurities and liquid water are considered.

Figure 4 shows the simulated and observed snow temperature profiles for 23 February. The snow temperature was measured using a thermistor thermometer. The solar radiation was shut off with a board at the moment of measurement. The initial profiles of this simulation are given at 0700 h. Diurnal variation patterns of water content are also in agreement with the observations (figure not shown)

Fig. 4.

An example of calculated and observed snow temperature profiles.

Long-Period Simulation

Period: 1986–1988 (3 winters).

Place: Sapporo, Hokkaido, Japan.

Data: Meteorological data and snow-pit observations at the Institute of Low Temperature Science, Hokkaido University (e.g. Reference Endo, Aki and MizunoEndo and others, 1986; Reference Ishikawa and MotoyamaIshikawa and Motoyama, 1986) and meteorological data at Sapporo District Meteorological Observatory.

Figure 5 shows the time series of albedo in 1986. The values of important parameters are listed in Table 1. In this calculation it is assumed that the impurities do not flow out because the value of fD is set to zero for the snowmelt season. However, the estimated albedo is smaller than the observations (Reference Ishikawa and MotoyamaIshikawa and Motoyama, 1986) in the later period. The actual amount of aerosol fallout may be larger than the value which is used in this simulation. At the start of the season, the albedo is slightly underestimated. The reason for this is unclear, however, the observed values in this year are higher than usual.

Fig. 5.

Comparison of calculated and observed snow albedo for 1986. The observed values are averaged from 1100 to 1200 h (wave length: 0.29-3.0 μm).

Table 1

The list of the main parameters

In Figures 6 and 7 the calculated snow depth and snow water equivalent are compared with the observations (Reference Ishikawa and MotoyamaIshikawa and Motoyama, 1986; Endo and others, 1986). The calculated values are overestimated in snowmelt season, because the albedo is large.

Fig. 6.

Comparison of calculated and observed snow depth for 1986.

Fig. 7.

Comparison of calculated and observed snow water equivalent for 1986.

Concluding Remarks

The evolution of snow-cover characteristics was simulated with a one-dimensional energy balance model. This model includes new parameterizations of the albedo, thermal conductivity and compactive viscosity. In particular, snow albedo was predicted taking into account the effects due to solid impurities and liquid water. The agreement between observed and calculated components was obtained for short- and long-period simulations. However, the handling of impurities requires much more study.