The drifting snow is one of the most important factors that affect the global ice mass balance and hydrological balance. Current models of drifting snow are usually one- or two-dimensional, focusing on the macroscopic quantities of drifting snow under temporal average flow. In this paper, we take the coupling effects between wind and snow particles into account and present a 3-D model of drifting snow with mixed grain size in the turbulent boundary layer. The Large Eddy Simulation (LES) method is used for simulating the turbulent boundary layer of the wind field and the 3-D trajectory of every motion snow particle is calculated through Lagrangian Particle Tracking method. The results indicated that the drifting snow in the turbulent boundary layer has apparent 3-D structure and snow streamers, which lead to an intermittent transport of the snow particles and spatial inhomogeneity, and the motion trajectories of snow particles, especially the small snow particles, are obviously affected by the turbulent fluctuation. The macro statistics of drifting snow indicates that the spanwise velocity of snow particles increases with height and is one order smaller than that of streamwise velocity. Furthermore, the diameter distribution of snow particles in the air along the height shows a stratification structure.
The phenomenon of the loose snow particles traveling near the land surface under the action of wind is known as drifting snow. As a typical two-phase flow, drifting snow is widely distributed in the globe and has significant impacts on the natural environment and the social economy. On one hand, drifting snow is one of the main causes of the temporal and spatial variation of snow distribution, contributes greatly to the mass balance of the Antarctic ice sheets, and further affects global climate system. The seasonal snow cover also deeply affects the hydrological balance in cold regions, thus is of glaciological and hydrological importance. On the other hand, drifting snow causes snow accumulation on the road and reduces visibility, which may seriously affect the traffic and human activities, and its resultant non-uniform distribution of snow layer may induce and aggravate various natural disasters, such as flood, avalanche, mudslides and landslide (Michaux et al., 2002). These disasters may result in not only huge direct and indirect economic losses, but also human casualties. Thus, in-depth study on the drifting snow is considered to be essential to comprehensively understanding the ice mass balance and hydrological balance.
The transport processes of snow grains have been extensively investigated (Pomeroy et al., 1993; Lehning et al., 2002; Bavay et al., 2009). Many models were proposed by taking the snow particles as continuous phase (Uematsu et al., 1993; Mann, 2000; Taylor, 1998; Déry and Yau, 1999; Fukushima et al., 1999, 2001; Xiao et al., 2000; Bintanja, 2000a, b). Obviously, the above assumption is not in agreement with the real situation. In addition, these models could reveal neither the movement mechanisms of snow particles nor the factors affecting the behaviors of snow particles. Recently, Nemoto and Nishimura (2004) studied the snow drifting process based on particle tracking in a turbulent boundary layer and their 1-D model included four sub-processes: the aerodynamic entrainment of snow grains, grain-bed collision, grain trajectories and wind modification. Later, Zhang and Huang (2008) presented a steady state snow drift model combined with the initial velocity distribution function and analyzed the structure of drifting snow at steady state. However, neither the details of the spatial variation of snow drifting nor the whole turbulent structure of wind field can be described due to limitation of their models. Furthermore, snow particles were uniform size in most previous models, which is different from the natural situation. To date, a comprehensive study on drifting snow in the turbulent field is indispensable for a thorough understanding of the complex drifting snow.
In this paper, we present a physical 3-D numerical model for drifting snow in the turbulent boundary layer based on the LES of Advanced Regional Prediction System (ARPS, version 5.3.3) by taking the 3-D motion trajectory of snow particles with mixed grain size, the grain-bed interaction, and the coupling effect between snow particles and wind field into consideration and used it to directly calculate the velocity and position of every single snow particle in turbulent atmosphere boundary layer, the transport rate and velocity distribution characteristics of drifting snow, and the mean particles size at different heights. The paper is structured as follows: Sect. 2 briefly introduces the model and methods; Sect. 3 presents the simulation results and discussions, and Sect. 4 is the conclusion.
The ARPS developed by University of Oklahoma is a three-dimensional, non-hydrostatic, compressible LES model and has been used for simulating wind soil erosion (Vinkovic et al., 2006; Dupont et al., 2013). In this paper, it is used for modeling the drifting snow.
Snow saltation movement in the air is a typical two-phase movement, in which the coupling of
particles and the wind field is a key issue. The conservation equations of momentum and subgrid
scale (SGS) turbulent kinetic energy (TKE) after filtering with considering the impact of the
presence of particles on the flow field can be expressed as (Vinkovic et al., 2006; Dupont et al.,
2013):
Because snow particles have much higher density
It is worth noting that the inertia effect of snow particles is considered by evaluating the maximum particle response time, so the particle motion is the dynamical calculation of time step, which is guaranteed to be less than the maximum particle response time.
The collision of saltating particles with the bed surface is a key physical process in saltation, as it will rebound with a certain probability and may splash new saltating particles into the air (Shao and Lu, 2000). Kok and Renno (2009) have proposed a physical splashing function based on the conservation of energy and momentum. Thus, the saltation process under various physical environments can be accurately simulated and applied to the mixed soils and drifting snow.
The grain-bed interaction is a stochastic process, in which the impact particles may rebound with
a certain probability. If a particle rebounds into the air, it can be described using three
variables: the velocity
The rebound probability can be expressed as:
For the velocity after rebound
The newly ejected particles and the “dead particles” (not rebounded) will reach equilibrium when the saltation process becomes stable.
The number of newly ejected particles is usually proportional to the impact velocity and can be
written as (Kok and Renno, 2009):
Once a new particle is splashed into the air, it can also be characterized by its velocity
The speed of the ejected particles is exponentially distributed. Kok and Renno (2009) developed
a physical expression of the average dimensionless speed of the ejected particle as follow:
Kok and Renno (2009) indicated that the angle
The blowing snow process in the turbulent boundary layer is simulated and the simulation results are
compared with the existent experiment results. Computational region is set as
In this model, the grid has a uniform size of
A turbulent boundary layer over a snow bed is generated using the dynamic Smagorinsky–Germano
subgrid-scale (SGS) model by setting the soil type, reasonable roughness and an initial field with
turbulent fluctuations. For the flow field, we applied the rigid ground boundary condition at the
bottom, the open radiation boundary in the top, the periodic boundary condition in the spanwise
direction, the open radiation boundary condition at the end of the domain along the streamwise
direction and the periodic boundary condition in the inflow with the cycle location at
The size distribution of snow particles in this paper is fitted to the experiment results obtained
from field observations of SPC (Schmidt, 1984), that is
The density of snow particles and air are 912 and
The processes of snow blowing with the wind velocity of
Almost all the flows at atmospheric boundary layer are turbulent. Therefore, the simulation of turbulent boundary layer is the key and basis for accurately simulating the drifting snow. Enough time is supplied for forming a stable turbulent boundary layer before particles taking off. The computational region is relative small and the inflow contains the real turbulent fluctuation.
Figure 3 shows the cloud map of velocity along the streamwise direction (
It can be observed from Fig. 3 that homogeneous turbulent fluctuations are distributed in the fully developed boundary layer. When the stable drifting snow is formed, the wind velocity will significantly decrease in the drifting snow region due to the reaction force of the snow particle and the turbulent fluctuations gradually become non-uniform in the drifting snow region. This is mainly due to the presence of the snow streamers-resulted great difference in the number concentration of snow particles at different positions (details are shown in Sect. 3.2).
In addition to the turbulent fluctuation, the wind profile can also be obtained by the time
averaging and spatial averaging of a time-series of wind velocities (
When the turbulent boundary layer is fully developed, the snow particles will be released and the motion feature of snow particles could be further obtained.
The saltation process, either in the field or in the wind tunnel, exhibits a temporospatial discontinuity. This discontinuity is affected by many factors such as turbulent fluctuation, topography, surface moisture, roughness elements, etc. (Stout and Zobeck, 1997; Durán et al., 2011). Different from previous models which are unable to clearly describe the drifting snow structure, our 3-D model could be used to directly calculate the 3-D trajectory of every snow particle and further intuitively demonstrate the overall structure of snow saltation layers because it describes the macroscopic performance of a large amount of drifting snow particles.
Figure 5a and b shows the typical trajectories of snow particles with diameter of 100 and 300
A large amount of snow particles move complexly in the turbulent boundary layer, constituting the
overall structure of the drifting snow. Figure 6 shows the top view of the snow-driving particles
concentration and the horizontal cloud map of streamwise wind velocity at corresponding moment, in
which, (a) and (b) represent the moment of
It can be observed from Fig. 6 that snow streamers with high saltating particle concentration obviously swing forward along the downwind direction, merging or bifurcating during the movement. It can also be found that the snow streamers with elongated shape differ greatly in length, but only 0.1–0.2 m in width. From the corresponding slices of wind velocity cloud map, it can be seen that many low-speed streaks exist in the near-wall region of the turbulent boundary layer. By comparing the concentration and corresponding velocity cloud map, it can be found that the particle concentration shows a direct proportional relationship with the local wind velocity, that is, only few snow particles present in the low-speed streaks.
The in-homogeneous take off and splash of the snow particles in the turbulent wind field are the main reasons for the formation of snow streamers. The shape and size of streamers largely depend on the flow structure of the turbulent boundary layer. In addition, during the full development of drifting snow, the saltating particles and wind field are in the state of dynamic balance due to the feedback effect of each other. When the number concentration of snow particles at a certain position is high enough, the local wind velocity will be significantly reduced, resulting in a lower splash level. Thus the streamer will gradually weaken or even disappear. In contrast, the local wind speed in the low concentration region will increase, which enhances the splash process, so the snow particles will grow rapidly and form a streamer. Furthermore, the fluctuating velocity may also change the movement direction of snow particles. All the above reasons together cause the serpentine forward of the snow streamers.
Snow transport rate (STR) is one of the most important indicators of the strength of the drifting
snow. In this simulation, the snow particles will be collected if they pass through the section
located at
The average particle concentrations under different friction wind velocities as a function of height
are shown in Fig. 8a. It is clear from the figure that the average particle concentrations at
different friction wind velocity similarly fluctuate with height, that is, they decrease with height
increasing. And the greater the friction wind velocity, the greater the maximum height the snow
particles can achieve. Further analysis shows that the difference of average particle concentrations
under different friction wind speed is proportional to height. For example, at the height of
0.001
Figure 8b shows the relationship of STR per unit area to the saltation height. As shown in Fig. 8b,
the variations in the STR per unit area with height at different friction wind speeds are
equivalent, that is, the STR per unit area decreases with height increasing. However, the STRs per
unit area differ greatly at the same height at different friction wind speeds. In the same
condition, the STRs per unit area at height
As one of the most important aspects to evaluate the accuracy of a drifting snow model, the velocity information of snow particles in the air is worthy of attention although it is seldom given in previous models. The location and velocity of every snow particle can be obtained, and the most important of all, the spanwise velocity of snow particles can be directly obtained in our simulation. The velocity distribution of snow particles in the air and the initial take-off velocity of the ejected particles are given in Sects. 3.4.1 and 3.4.2, respectively.
The velocity distribution of snow particles in the air is shown in Fig. 9, in which (a) is the
average velocity of snow particles along the streamwise direction as a function of height and (b)
displays the corresponding velocity probability distribution of snow particles. It can be observed
from Fig. 9a that the average velocity of snow particles along the streamwise direction increases
with the height increasing and with the friction wind velocity at the same height increasing. It can
be seen from Fig. 9b that the probability distribution of snow particles' velocity along the
streamwise direction obeys the unimodal distribution. In other words, it distributes mainly at
It should be noted that the high-speed particles in our simulation are significantly more than those captured in the experiments (Fig. 9b). This is mainly because the concentration of snow particles decreases with height increasing, making it increasingly difficult to be captured the high-speed snow particles.
Figure 10 shows the spanwise velocity of snow particles in the air, where (a) is the distribution of the absolute value of spanwise velocity along the elevation and (b) is the corresponding probability distribution of snow particles' velocity. It is observed that the absolute value of spanwise velocity increases with height in an order of magnitude less than that of the streamwise velocity. The variation in the spanwise velocity with height at different friction wind velocity is not obvious.
From the above analysis it is quite evident that the velocity distribution of snow particles in the air is not sensitive to the wind velocity. The main reason for that is the wind velocity in the full development drifting snow slightly varies due to the reaction force of snow particles in the air.
The initial take-off velocities of snow particles (including rebound particles) in a fully developed drifting snow field are presented in Fig. 11, in which the (a), (b), (c) and (d) show the distributions of streamwise, spanwise, vertical and resultant velocities, respectively. It is clear that all the velocity components obey the unimodal distribution and are not affected by the friction wind velocity.
Although there is no obvious difference in take-off velocity at different wind velocity, we can see that a large amount of particles may saltate at higher saltation height and greater friction wind speed. It may be inferred that turbulent fluctuation plays an important role in the lifting of snow particles.
The snow particles with mixed size close to natural situation are applied in our simulation. In this section, the size distribution of snow particles in the air is analyzed.
Figure 12 shows the size distributions of snow particles in the air at different friction
velocities, and their comparison with the experimental results. All the data have been normalized to
the average of overall snow particles due to different characteristics of snow particles in
different experiments. To avoid the random error, those data with snow particle influx less than
In this study, we establish a 3-D drifting snow model in the turbulent
boundary layer, simulate the development process of drifting snow with mixed
particle size and draw the following main conclusions:
Turbulent fluctuation may significantly affect the trajectory of small snow particles with
equivalent diameter The drifting snow in a turbulent boundary layer is very intermittent. Fully developed
drifting snow swings forward toward the downwind in the form of snow streamers and the wind
velocity is proportional to the concentration of snow particles at different locations of the
turbulent boundary layer. The velocities of snow particles along streamwise and spanwise directions increase with the
height, and the latter is one order of magnitude less than the former in general. In addition, the
velocity distribution is not sensitive to the wind speed. The mean diameter of snow particles in the air is obviously distributed in layers. It is
constant for snow particles with height less than 0.05
This work is supported by the State Key Program of National Natural Science Foundation of China (91325203), the National Natural Science Foundation of China (11172118, 41371034), and the Innovative Research Groups of the National Natural Science Foundation of China (11121202), National Key Technologies R & D Program of China (2013BAC07B01).
Diagram of computational region.
Equivalent diameter probability distribution of snow particles.
The cloud map of flow field at
The wind profile at
The 3-D trajectories of snow particles with different diameters (
The top view of the particle concentration and the horizontal section of wind velocity
cloud map at corresponding moment (
Variation of the snow transport rate (STR) per width with
Distribution of
Distribution of the initial
The mean equivalent diameter distribution of snow particles in the air vs. height.