The coupled process between the snowpack model (UEB) and the soil water model (STEMMUS-FT) was illustrated in Fig. 1. The sequential coupling is employed to couple the soil model with the current snowpack model. The role of the snowpack is explicitly considered by altering the water and heat flow of the underlying soil. The snowpack model takes the atmospheric forcing as the input (precipitation, air temperature, wind speed and direction, relative humidity, shortwave and longwave radiation) and solves the snowpack energy and mass balance (Eqs. A8 and A9; subroutines: ALBEDO, PARTSNOW, PREDICORR), which provides the melt water flux and heat flux as the surface boundary conditions for the soil model STEMMUS-FT (subroutines: h_sub and Enrgy_sub for the advanced coupled models and Diff_Moisture_Heat for the basic coupled model). The soil–snow coupling variables are the snowmelt water flux Mr, the convective heat flux due to snowmelt water Qm, and the heat conduction flux Qg. STEMMUS-FT then solves the energy and mass balance equations of soil layers in one time step. To highlight the effect of the snowpack on the soil water and vapor transfer process, we constrained the soil surface energy boundary as the Dirichlet-type condition (take the specific soil temperature as the surface boundary condition). Surface soil temperature was derived from the soil profile measurements and was not permitted to be higher than zero when there is snowpack. In such way, the reliability of the soil surface energy boundary condition is maintained and the snow thermal effect is implicitly considered. The snowmelt water flux, in addition to the rainfall, was added to the topsoil boundary for solving soil water transfer. To ensure numerical convergence, the adapted time step strategy was used. Half-hourly meteorological forcing measurements were linearly interpolated to the running time steps (Subroutine Forcing_PARM). The precipitation rate (validated at 3 h time intervals) was regarded uniformly within the 3 h duration (refer to Table S6.1 in the Supplement for details). The general description of the primary subroutines in STEMMUS-UEB was presented in Table 2. It includes the main functions, input and output, and their connections with other subroutines (linked with Tables S6.1 and S6.2 in the Supplement for the description of model input parameters and outputs for this study; see the detailed description in Tarboton and Luce, 1996; Zeng and Su, 2013; Yu et al., 2018a).

The detailed physically based two-phase flow soil model (STEMMUS) was first developed to investigate the underlying physics of soil water, vapor, and dry air transfer mechanisms and their interaction with the atmosphere (Zeng et al., 2011a, b; Zeng and Su, 2013). It is achieved by simultaneously solving the balance equations of soil mass, energy, and dry air in a fully coupled way. The mediation effect of vegetation on such interactions was recently incorporated via the root water uptake sub-module (Yu et al., 2016) and by coupling with the detailed soil and vegetation biogeochemical process (Wang et al., 2021; Yu et al., 2020a). It facilitates our understanding of the hydrothermal dynamics of respective components in the frozen soil medium (i.e., soil liquid water, water vapor, dry air, and ice) by implementing the freeze–thaw process (hereafter STEMMUS-FT, for applications in cold regions, Yu et al., 2018a, 2020c).

The frozen soil physics considered in STEMMUS-FT includes three parts: (i) the ice blocking effect on soil hydraulic conductivities (see Sect. S2.2.2 in the Supplement), (ii) the inclusion of ice effect in the calculation of soil thermal capacity and conductivity (see Sect. S2.2.8), and (iii) the exchange of latent heat flux during phase change periods. With the aid of Clausius–Clapeyron relation, which characterizes the phase transition between liquid and solid phase in the thermal equilibrium system, the soil water characteristic curve (e.g., van Genuchten, 1980) is then extended to consider the freezing temperature dependence, i.e., soil freezing characteristic curve (Hansson et al., 2004; Dall'Amico et al., 2011). The fraction of soil liquid–solid water at a given temperature was then calculated prognostically with the soil freezing characteristic curve. Soil hydraulic parameters were further used in the Mualem (1976) model to compute the soil hydraulic conductivity. The ice effect is considered by reducing the soil saturated hydraulic conductivity as a function of ice content (Yu et al., 2018a).

In response to minimize the potential model-comparison uncertainties from various model structures and to figure out which process matters, three levels of complexity of mass and heat transfer physics are made available in the current STEMMUS-FT modeling framework (Yu et al., 2020c). First, the 1D Richards equation and heat conduction were deployed in STEMMUS-FT to describe the isothermal water flow and heat flow (termed BCD). The BCD model considers the interaction of soil water and heat transfer implicitly via the parameterization of heat capacity, thermal conductivity, and the water phase change effect. The water flow is fully affected by soil temperature regimes in the advanced coupled water and heat transfer model (termed ACD model). The movement of water vapor, as the primary linkage between soil water and heat flow, is explicitly characterized. STEMMUS-FT further enables the simulation of temporal dynamics of three water phases (liquid, vapor, and ice), together with the soil dry air component (termed ACD-air model). The governing equations of liquid water flow, vapor flow, airflow, and heat flow were listed in Appendix A1 (see the more detailed model description in Zeng et al., 2011a, b; Zeng and Su, 2013; Yu et al., 2018a, 2020c).

The Utah energy balance snowpack model (UEB; Tarboton and Luce, 1996) is a single-layer physically based snow accumulation and melt model. Two precipitation types, i.e., rainfall and snowfall, are discriminated by their dependence on air temperature. The snowpack is characterized using two primary state variables, snow water equivalent (SWE), and the internal energy U. Snowpack temperature is expressed diagnostically as the function of SWE and U together with the states of the snowpack (i.e., solid, solid and liquid mixture, and liquid). Given the insulation effect of the snowpack, snow surface temperature differs from the snowpack bulk temperature, which is mathematically considered using the equilibrium method (i.e., balances energy fluxes at the snow surface). The age of the snow surface, as the auxiliary state variable, is utilized to calculate the snow albedo (see Appendix A3). When the snowpack is shallow, the albedo is the weighting function of the snow albedo and the bare-ground albedo. The solar radiation penetration in the shallow snowpack is exponentially attenuated and expressed in the weighting factor. The melt outflow is calculated using Darcy's law with the liquid fraction as inputs. The conservation of mass and energy forms the physical basis of UEB (Tarboton and Luce, 1996, as presented in Appendix A2).

UEB is recognized as one simple yet physically based snowmelt model. It captures the snow process well (e.g., diurnal variation of meltwater outflow rate, snow accumulation, and ablation; see the general overview of UEB model development and applications in Table S6.3). It requires little effort in parameter calibration and can be easily transferable and applicable to various locations (e.g., Gardiner et al., 1998; Schulz and de Jong, 2004; Watson et al., 2006; Sultana et al., 2014; Pimentel et al., 2015; Gichamo and Tarboton, 2019), especially for data-scarce regions like the Tibetan Plateau. We thus selected the original parsimonious UEB (Tarboton and Luce, 1996) as the snow module to be coupled with the soil module (STEMMUS-FT).

On the basis of the aforementioned STEMMUS-UEB coupling framework, the various complexities of vadose zone physics were further implemented as three alternative model versions. First, the soil ice effect on soil hydraulic and thermal properties and the heat flow due to the water phase change were taken into account, while the water and heat transfer is not coupled in STEMMUS-FT and is termed the BCD model. Second, the STEMMUS-FT with the fully coupled water and heat transfer physics (i.e., water vapor flow and thermal effect on water flow) was applied and termed the ACD model. Lastly, on top of the ACD model, the air pressure was independently considered as a state variable (therefore, the airflow) and termed the ACD-air model. With the abovementioned model versions (STEMMUS-FT_Snow), taking into account the no-snow scenarios (STEMMUS-FT_No-Snow), Table 3 lists the configurations of all six designed numerical experiments. The model parameters used for all simulations for the tested experimental site are listed in Table S6.2.

Maqu station, equipped with a catchment-scale soil moisture and soil temperature (SMST) monitoring network and micro-meteorological observing system, is situated on the northeastern edge of the Tibetan Plateau (Su et al., 2011; Dente et al., 2012; Zeng et al., 2016). According to the updated Köppen–Geiger climate classification system, it can be characterized as a cold climate with dry winter and warm summer. The annual mean precipitation is about 620 mm, and the annual average potential evaporation is about 1353.4 mm. Precipitation in Maqu is uneven over the year, with most of the precipitation events occurring from May to October and little precipitation or snowfall during the wintertime. The average annual air temperature is 1.2 ∘C, and the mean air temperatures of the coldest month (January) and the warmest month (July) are about -10.0 and 11.7 ∘C, respectively. Alpine meadows (e.g., Cyperaceae and Gramineae), with a height varying from 5 to 15 cm throughout the growing season, are the dominant land cover in this region. This site is seasonally snow covered, with temporal snow in the non-growing season, which is due to the intermittent snowfall and the rapid snow melting and sublimation caused by the high air temperature and strong solar radiation in the daytime. The general soil types are sandy loam, silt loam, and organic soil for the upper soil layers (Dente et al., 2012; Zheng et al., 2015; Zhao et al., 2018a). The soil texture and hydraulic properties were listed in Table S6.2, and how these were used in STEMMUS-UEB is illustrated in Fig. 1 and Table 2.

The Maqu SMST monitoring network spans an area of approximately 40 km × 80 km with an elevation ranging from 3200 m to 4200 m a.s.l. (33∘30′–34∘15′ N, 101∘38′–102∘45′ E). SMST profiles are automatically measured by 5TM ECH2O probes (METER Group, Inc., USA) installed at different soil depths, i.e., 5, 10, 20, 40, and 80 cm. The micro-meteorological observing system consists of a 20 m planetary boundary layer (PBL) tower providing the meteorological measurements at five heights above ground (i.e., wind speed and direction, air temperature and relative humidity), and an eddy covariance system (EC150, Campbell Scientific, Inc., USA) equipped for measuring the turbulent sensible and latent heat fluxes and carbon fluxes. The equipment for four-component downwelling and upwelling solar and thermal radiation (NR01-L, Campbell Scientific, Inc., USA) and liquid precipitation (T200B, Geonor, Inc., USA) are also deployed. A dataset from 1 December 2015 to 15 March 2016 was utilized in this study. Independent precipitation data (3 h time interval) during the same testing period from an adjacent meteorological station were used as the mutual validation data.

Yakou super snow station (38∘00′36 N, 100∘14′24 E, 4145 m) is located in the upstream Heihe basin in the northeastern Tibetan Plateau. It is a high-elevation snow-covered site with the wet summers and dry winters. The dominant land type is tundra with frozen ground below. There is a unique seasonal variation of snow depth with the maximum snow depth usually being in the springtime (32 cm during the period 2014–2017). Loam is the main soil type with the silt loam near the surface and sandy soil for the deeper soil layers.

The integrated hydrometeorological, snow cover, and frozen ground data were published and available from the Cold and Arid Regions Science Data Center at Lanzhou (Che et al., 2019; Li et al., 2019; Li, 2019). The meteorological data (air temperature, wind speed, precipitation, downward shortwave and longwave radiation, and relative humidity) were recorded by the automatic meteorological station (AMS). In situ measurements of snow cover properties (snow depth and snow water equivalent) were obtained using the state-of-the-art instruments (SR50A and GammaMONitor, Campbell Scientific, USA). Soil moisture profiled at 4, 10, 20, 40, 80, 120, and 160 cm soil depth was measured using ECH2O-5 probes (METER Group, Inc., USA). In addition to the seven soil depths, the surface soil temperature (0 cm) was also recorded using the Avalon AV-10T sensors (Avalon Scientific, Inc., USA). The eddy covariance system was equipped at the Yakou site for measuring land surface turbulent fluxes. The dataset from 1 September to 31 December 2016 was used to validate the model performance in mimicking the dynamics of snow water equivalent, soil hydrothermal regimes, and land surface evaporation. The calibrated soil hydraulic and snow cover properties were listed in the Supplement in Table S6.2.

The time series of surface albedo, calculated as the ratio of upwelling shortwave radiation to the downwelling shortwave radiation and estimated using BCD, ACD, and ACD-air models, is shown in Fig. 2 together with precipitation. As the snowpack has a higher albedo than the underlying surface (e.g., soil, vegetation) compared to the observations, models without snow module presented a relatively flat variation of daily average surface albedo and lacked the response to the winter precipitation events (Fig. 2, Table 4). With the snow module, STEMMUS-UEB models can mostly capture the abrupt increase of surface albedo after winter precipitation events. The mismatches in terms of the magnitude or absence of increased albedo after precipitation events indicated that the model tended to underestimate the albedo dynamics. The shallow snowfall events might be not well captured by the model (see Sect. 4.1). Three model versions (BCD-Snow, ACD-Snow, and ACD-air-Snow) produced similar fluctuations regarding the presence of snow cover with slight differences in terms of the magnitude of albedo.

The observed spatial and temporal dynamics of soil temperature from five soil layers were used to verify the performance of different models (Fig. 3). The initial soil temperature state can be characterized as the warm bottom and cool surface soil layers (based on in situ observations). The freezing front (indicated by the zero-degree isothermal line, ZDIL) developed downwards rapidly until the 70th day after 1 December 2015, when it reached its maximum depth. Following this, the freezing front stabilized as an offset effect of latent heat release (termed the zero-curtain effect). Such influence can be sustained until all the available water to that layer is frozen, at which point the latent heat effect is negligible compared to the heat conduction. At shallower layers, the atmospheric forcing dominates the fluctuation of thermal states. The isothermal lines (e.g., -2 ∘C) had a larger variation than that of ZDIL. At deeper soil layers, the temporal dynamics of isothermal lines were smoother than that of ZDIL, indicating that the effect of fluctuated atmospheric force on soil temperature was damped with the increase of soil depth. Compared to the observations, BCD-Snow model presented an earlier development of the freezing front and arrival of the maximum freezing depth (60th day after 1 December 2015). The deeper and more fluctuated freezing front indicates that a stronger control of atmospheric forcing on soil thermal states was produced by BCD-Snow model. The ACD models can capture the propagation characteristic of the freezing front well in terms of the variation magnitude and maximum freezing depth. There is no significant difference in soil thermal dynamics between the model with and without the snow module, except at the surface soil layers (Table 4).

Figure 4 shows the spatial and temporal dynamics of observed and simulated soil water content in the liquid phase (SWCL). The SWCL of active layers depends to a large extent on the soil freezing and thawing status. Soil is relatively wet at soil layers of 10–60 cm for the starting period. Its temporal development was disrupted by the presence of soil ice and tended to increase wetness during the thawing period. A relatively dry zone (θL<0.06 m3 m-3) above the freezing front was found, indicating the nearly completely frozen soil during the stabilization stage. The initial wet zone of soil moisture was narrowed down and the rewetting zone tended to enlarge from BCD-Snow simulation due to its early freezing and thawing of soil (Fig. 4b). The position of the dry zone occurred earlier due to the early reaching of the stabilization period in the BCD-Snow model (Fig. 3b). For the ACD models, the position and development of initial wet zone, rewetting zone, and dry zone are similar to those from the observations, indicating the soil moisture dynamics can be captured well by the ACD models. Compared to the STEMMUS-FT_Snow model, there was no observable difference in the SWCL dynamics at deeper soil layers from STEMMUS-FT_No-Snow simulations. The surface SWCL was found affected from STEMMUS-FT_Snow simulations (Table 4).

Figure 5 shows the comparison of time series of observed and model-simulated surface cumulative latent heat flux using three models with and without consideration of the snow module. Considerable overestimation of latent heat flux was produced by the BCD-Snow model: 121.79 % more than was observed. Such overestimations were largely reduced by ACD and ACD-air models. There is a slight underestimation of cumulative latent heat flux in the ACD-Snow and ACD-air-Snow models, with values of -8.33 % and -7.05 %, respectively. Compared with STEMMUS-FT_Snow simulations, there is less latent heat flux produced by the STEMMUS-FT_No-snow simulations. This is mainly due to the sublimation of snow cover, which cannot be simulated by the STEMMUS-FT_No-snow models. The difference in cumulative latent heat flux between STEMMUS-FT with and without snow module increases from BCD to ACD-air schemes, with the values of 2.02 %, 7.69 %, and 8.97 % for BCD, ACD, and ACD-air schemes, respectively.

To further elaborate the effect of snowpack on LE, we presented the diurnal variations of LE and its components at two typical episodes with precipitation events (freezing and thawing period, respectively). The relative contribution of liquid and vapor flow to the total mass transfer after precipitation events was separately presented in Figs. 8 and 9, i.e., the liquid water flux driven by temperature qLT, matric potential qLh and air pressure qLa, water vapor flux driven by temperature qVT, matric potential qVh, and air pressure qVa.

After a winter precipitation event, land surface albedo increases considerably (Fig. 2), indicating the presence of the snowpack. However, such snowfall events were episodic with small magnitudes (similar to those in Li et al., 2017), which means that they are difficult to capture well. Such difficulties can be partially attributed to the inherent uncertainties in precipitation measurements (both the precipitation amount and types). Due to the spatial variability of precipitation, the accurate observation of winter precipitation has proven to be a challenge, especially during windy winters (Barrere et al., 2017; Pan et al., 2017). It is necessary to have more snowpack-relevant measurements (e.g., the high-resolution measurements of the spatiotemporal field of wind speed, precipitation, and snowpack variations) to understand the dynamics of snowpack and its effect on energy and water fluxes. Furthermore, the temporal resolution of precipitation measurements adopted in this study is relatively coarse (3 h). In the current precipitation partition parameterization, the amount of snowfall was determined as a function of precipitation and air temperature thresholds. Given the coarse temporal resolution of precipitation measurements, the model may produce a time shift of snowfall events or even the misidentification of snowfall. The simple relation between the air temperature and precipitation types may be not suitable for this region because air temperature is not the best indicator of precipitation types, as argued by Ding et al. (2014). Other factors, i.e., relative humidity, surface elevation, and wet-bulb temperature, are also very relevant and should be taken into account for the discrimination of precipitation types. The other uncertainty lies in the representation of the snow process. For example, the wind blow effect and canopy snow interception, which have been recognized as important to the accurate simulation of snowpack dynamics (Mahat and Tarboton, 2014), are not taken into account in detail. Last but not least, the interpretation of surface albedo dynamics needs to be adapted to the specific site, especially regarding the shallow snow situations (Ueno et al., 2007, 2012; Ding et al., 2017; Wang et al., 2017). The albedo of the underlying surface should also be properly accommodated to this Tibetan meadow system. Regardless of the aforementioned uncertainties, our proposed model was capable of capturing the surface albedo variations with precipitation (Fig. 2) and can be seen as acceptable for analyzing snow cover effects in such a harsh environment.

In contrast to precipitation water from rainfall, precipitation water from snowfall enters the soil considerably lagged in time due to the water storage by snow cover (You et al., 2019). With the snow module, precipitation was partitioned into rainfall and snowfall. Part of the snowfall evaporated into the atmosphere as sublimation, and the other part, together with the rainfall, infiltrated into the underlying soil. It resulted in the delay of incoming water to the soil with a lower amount compared to that without consideration of the snow module. This amount of incoming water increased the evaporation after precipitation (Figs. 6 and 7). The other source for the enhanced evaporation flux after precipitation is snow sublimation, which is absent from the model without the snow module. Sublimation occurs readily under certain weather conditions (e.g., with freezing temperatures, enough energy). It can be more active in regions with low relative humidity, low air pressure, and dry winds. Such an amount of sublimation has been reported as being important from the perspective of climate and hydrology (e.g., Strasser et al., 2008; Jambon-Puillet et al., 2018), especially in high-altitude regions with low air pressure. During the freezing period, the evaporation enhancement can be also sourced from the sublimation of surface ice. The amount of the ice sublimation appeared to decrease during the freezing period in the presence of a transient snowpack (e.g., Fig. 8c vs. Fig. 8d). This is consistent with the results of Hagedorn et al. (2007), who investigated the effect of snow cover on the mass balance of ground ice with an artificially continuous annual snow cover. According to their results, the snow cover enhanced the vapor transfer into the soil and thus reduced the long-term ice sublimation. The relative contribution of increased surface soil moisture, snow sublimation, and surface ice sublimation to the enhanced evaporation is dependent on the pre-precipitation soil moisture and temperature states, air temperature, and the time and magnitude of precipitation events. Under the conditions of the low pre-precipitation SWCL with a freezing soil temperature (e.g., Fig. 8e, 11th vs. 12th day after 1 December), the precipitation falls on the surface as snowfall and rainfall (most freezes as ice). The sublimation from surface ice can contribute to most of the total mass transfer (e.g., Fig. 8e, 11th day after 1 December). If the soil temperature rises above the freezing temperature, there will be no sublimation of surface ice, in terms of contributing to the enhanced evaporation (e.g., Fig. 9e, 102nd day after 1 December).

The model with different complexities of soil mass and energy transfer physics behaves differently in response to the winter precipitation events. During the freezing period, there is no significant difference in soil moisture simulated using the BCD models with and without the snow module. The precipitation water freezes at the soil surface, which cannot be transferred downwards with the BCD model physics. The sublimation, from either the snow or the surface ice, contributes to the precipitation-enhanced evaporation for the BCD model. As with vapor flow, the surface ice increases the soil moisture at lower layers via the downward isothermal vapor flux (Fig. 8). The surface ice sublimation and increased moisture-induced soil evaporation enhancement can be identified from the ACD model simulation. The role of airflow was negligible for the mass transfer during the freezing period.

When it comes to the thawing period, the BCD model produced a certain amount of liquid water flow, contributing considerably to the mass transfer. The obvious fluctuation of SWCL was noticed due to the thawing water and precipitation event. The main source for the increased evaporation was interpreted as isothermal liquid water flow, while for the ACD model the situation becomes more complex. Thawing surface ice and snowmelt water may coexist at the soil surface, resulting in different soil moisture response to precipitation events. The ice sublimation, snow sublimation, and increased soil moisture contribute to the evaporation enhancement after precipitation. When considering airflow, dry air interacts with soil ice and liquid and vapor water in soil pores (Yu et al., 2018a) and alters the soil moisture state. It thus considerably changes the relative contribution of each component to the mass transfer (Fig. 9).