On the basis of conservation of energy (Xu and Li, 2003), the net heat flux of snow cover is equal to the intersystem heat increase over the same interval. The snow exchanges energy with the atmosphere above and ground below. The net energy flux of the snowpack can be expressed as Qnet=Qr+Qs+Ql+Qp+Qg where Qnet is net input energy for snowpack (J m-2); Qr is the net radiation flux (W m-2); Qs is flux of sensible heat (W m-2); Ql is flux of latent heat (W m-2); Qp is heat flux supplied by precipitation (W m-2); Qg is ground heat flux (W m-2) always being neglected.

Based on the water balance, daily snow melt was estimated using the net energy of the snow cover.

The snow melt can be divided into three phases: warming, ripening and output (Dingman, 1994). Wliq=Wliq+(PrE+ΔW)Ws=Ws+(PsSΔW) where Wliq is liquid water content (m); Ws is snow water equivalent (m); ΔW is the energy for water transfer; Ps is snowfall (m); S is sublimation (m); E is evaporation (m); Ci is heat capacity of snow (J kg-1 ∘-1); λf is latent heat of fusion (kJ kg-1).

When Wliq is larger than the holding capacity of the snow cover, the outflow from the snow cover is described as Wout=Wliq-Wliq_max where Wliq_max is the holding capacity of the snow cover (m) and Wout is the outflow from the snow cover (m).

We hypothesized the following assumptions could be applied to our model, including the influence of soil water vapor migration and heat flow transfer can be ignored; the soil is considered vertical and non–plastic soil; vertical heat and moisture flux; and Darcy's law applies to moisture movement in both saturated and unsaturated conditions (Shoop and Bigl, 1996). Based on these assumptions, water and heat movement in seasonally frozen soil can be treated as a one-dimensional problem.

Water movement in freezing and thawing soil can be expressed as Richard's equation and a one-dimensional heat transport equation for a freezing or thawing column of soil. Cs∂T∂t=∂∂Zλ∂T∂Z+Lfρi∂θi∂t ∂θl∂t=∂∂ZDθl∂θl∂Z-Kθl-ρiρl∂θi∂t where D(θl) is the diffusivity of soil (m2 s-1); K(θl) is unsaturated hydraulic conductivity of soil (m s-1); ρs is density of ice (kg m-3); ρl is density of liquid water (kg m-3); θl is volumetric liquid water content (%); θs volumetric ice content (%); Cs is volumetric heat capacity of soil (J m-3∘-1); λ is thermal conductivity of soil (W m-1∘-1); Lf is latent heat of fusion (J kg-1); T is soil temperature (∘); Z is depth (m); t is time (s).

Based on the water balance, the water content θl_top in the top layer of soil relied on the outflow from snow pack Wout, and evaporation from soil surface Esoil, the liquid content in top soil layer was given by θl_top=β(Wout-Esoil) where β is the constant coefficient.

Observation of air temperature, snow surface temperature, humidity, rainfall, wind velocity, atmospheric pressure and hours of daylight provided a good dataset for driving the snow cover model. Comparisons between simulated and measured results for snow depth in 2005–2006 are shown in Fig. 2.

According to Fig. 2, the snow depth trends were the same for both the simulated results and the measured values. At Qumalai, snow accumulation was measured from late October through the middle of May the following year, when the melting process finished. Between mid-March and early November, the snow depth reached a maximum.

The daily mean freezing depth was used in the following analysis. Figure 3 shows the depth of frozen soil from October 2005 to June 2006. The simulation of the frozen soil trend for the whole period was consistent with measurements.

The daily mean soil temperature grew progressively colder in October, and the surface soil began to freeze at the same time. Over time, the soil froze deeper, and after 141 days (11 March), it was frozen to 95 cm.

The majority of the site began to thaw in April, when temperatures grew warmer. The shallow layer of soil began thawing on 15 March. By early April, the layer below 140 cm also began to thaw, though the layers between 70 and 140 cm depth remained frozen. The remaining frozen soil thawed from both the top and bottom depths.

Figure 4 shows the simulated and measured daily average temperatures for every day of 2005–2006. According to Fig. 4, both the simulated and measured soil temperatures show the same trend. The soil temperature warmed from late March into late October, which corresponds to when the soil thawed (Fig. 3).

To detect the numerical model efficiency, efficiency coefficients (EC) and root mean squared errors (RMSE) were used. RMSE for simulated daily soil temperatures of 2005–2006 ranged from 1.82 to 1.9∘, and EC was 0.94. There was no distinct difference for the surface soil. This means that there was no system error for the model used at these sites, and the simulated temperature for surface soil was coincident with the measured value.

Soil liquid water content was measured at 50 cm from April to October 2005 at Qumalai. The comparisons between simulated and measured results are shown in Fig. 5.

The simulated liquid water content decreased in early May, when there was heavy evaporation and no precipitation. As the evaporation decreased, the liquid water content increased, while precipitation created an increase in soil moisture.

From the simulation results, we found that during the thawing period, the soil liquid water content was significantly affected by the surface conditions, including precipitation and evaporation.