1. Introduction

Drifting and blowing snow occur when loose snow particles are available at the surface and when wind speeds increase above a certain threshold value. Then, turbulent motions are able to lift the snow particles upwards against the action of gravity. For moderately strong winds, the particles remain confined very close to the rigid surface and periodically rebound at the surface; this is generally referred to as saltation. For stronger winds, snow particles are able to leave the saltation layer and become suspended. In fully developed blowing snow, saltation and suspension occur simultaneously. In particular, the functional dependence of total snowdrift-transport rates on the wind velocity has been studied widely (e.g. Reference Budd, Dingle, Radok and RubinBudd and others, 1966; Reference RadokRadok. 1970; Reference TakeuchiTakeuchi, 1980; Reference SchmidtSchmidt, 1986; Reference Pomeroy and MalePomeroy and Male, 1992).

Snowdrifting is one of the many manifestations of two-phase flow systems. Therefore, the physical mechanisms involved in the drifting of snow are similar to those investigated ill the broad and complex field of two-phase (solid-fluid/gas) flows. Analogous processes are, for instance, the drifting of wind-blown desert sand (Reference BagnoldBagnold, 1941, Reference Bagnold1966), the transport of sediment over the ocean floor and river beds (e.g. Reference Bridge and DominicBridge and Dominic, 1984), blood flow, quicksand, sea spray and dense fog.

Snowdrifting occurs regularly over snow-covered regions such as the ice sheets of Antaretica and Greenland. The transport of snow by the wind can be a major mass-balance component over snowfields and ice sheets by spatially redistributing the precipitated snow (e.g. Reference Takahashi, Naruse, Nakawo and MaeTakahashi and others, 1988). Blowing snow over the edge of Antaretica even contributes in an absolute sense to the integrated mass balance. Sublimation of blowing snow also negatively contributes to the surface mass balance of snow-covered regions (e.g. Reference SchmidtSchmidt, 1982; Reference King, Anderson, Smith and MobbsKing and others, 1996). Furthermore, the occurrence of snowdrifting causes acceleration of the downslope katabatic winds Over sloping surfaces as a result of the increase in near-surface density of the gravity flow (Reference Kodama, Wendler and GosinkKodama and others, 1985). Drifting and blowing snow even affect the longwave-radiation balance of the lowest atmospheric layers, as pointed out by Reference Yamanouchi and KawaguchiYamanouchi and Kawaguchi (1984). Obviously, the occurrence of drifting snow influences and modifies many physical processes in the near-surface atmospheric layers.

This study focuses on the influence of drifting and blowing snow on ihe vertical turbulent mixing of momentum and scalars in the atmospheric surface layer. To keep a snow particle in suspension at a certain average height above the surface, the residual upward turbulent stress exerted on the particle must balance the downward force of gravitation. This requires expenditure of the surface-layer turbulence, which is manifested by decreases in turbulence-intensity and turbulent-length scales (Reference Adams and WeatherlyAdams and Weatherly, 1981). As the snowdrifting content decreases sharply with height, the surface layer is stably stratified and turbulence is reduced (Reference Wamser and Lykossov.Wamser and Lykossov, 1995). This mechanism is conceptually similar to a thermally stable turbulent surface layer in which negative buoyancy destroys turbulence (Reference Lumley and Bradshaw.Lumley, 1978). The effect of snowdrifting on turbulence may therefore be described by introducing an appropriate “snowdrift” Richardson number which depends on the vertical turbulent suspended snow flux. Besides showing model results in which this effect is quantified, we will present a simple, first-order parameterization of the stability effects of snowdrifting in terms of known surface-layer variables. As this study is one of the first to discuss the effects of snowdrifting on atmospheric turbulence and, since the model that is used includes various assumptions and empirical parameterizations, the results presented here are far from definitive; future work will focus on refitting the model.

2. Model Description

The drifting of snow particles in the air is considered as a two-phase turbulent flow of air mixed with rigid, non-cohesive, inertia-free spherical snow particles, separated sufficiently to be considered isolated so that particle interaction can be safely neglected (Reference Lumley and Bradshaw.Lumley, 1978). Furthermore, it is assumed that volume concentration is high enough so that the air-snow mixture may be regarded as a continuum and that the horizontal velocity of the particles equals that of the air but is low enough to ignore interaction. For this, it is necessary that the particles are small relative to the smallest length scale in turbulence, the Kolmogorov length scale, which is about 1 mm in the lower atmosphere. This is generally the case for snowdrifting with a particle diameter varying roughly between 40 and 400 μ (e.g. Reference Budd and RubinBudd, 1966).

3. Results

We will show the results of experiments with varying initial wind-speed profiles, the steepness of which are determined by the value of u_*H . Increasing values of u_*H > can be associated with stronger externally forced boundary-layer winds. Figures 1–4 show the vertical profiles of horizontal wind velocity, friction velocity, Richardson number and suspended-mass concentration, respectively, for various values of u_*H such that u_*H > u_*1 so that snowdrifting occurs.

Fig. 1.

Steady-state wind-speed profiles (thin lines) for five selected experiments represented by their initial friction velocity (u_*H). >Bold lines represent the usual logarithmic wind-speed profile in the absence of snowdrift.

Fig. 2.

Vertical profile of simulated steady-state friction velocity (thin lines) for the five selected experiments. Bold lines represent the friction-velocity profiles which would be found if the stabitity effects of suspended snow are neglected.

Fig. 3.

Vertical profile of the “snowdrift” Richardson number (Equation (4)) for the five selected experiments.

Fig. 4.

Vertical profile of the suspended snow mass for the five selected experiments.

In a stable surface layer, the vertical transfer of momentum is initially reduced by the snowdrift-induced diminution of turbulent intensity. This implies that the presence of suspended snow induces a general increase in wind speeds (although near the surface a slight reduction in wind speed occurs), the largest increases occurring at higher levels (Fig. 1). In other words, the mean velocity gradient is larger for the same shear stress when snowdrifting occurs. Obviously, this behaviour is simitar to what is commonly observed in thermally stable surface layers (e.g. Reference GarrattGarratt, 1992). Figure 2 shows that u_* increases upward, which results from the fact that the density of the air—snow mixture decreases strongly with height and that the shear stress must be constant with height in steady-state by virtue of Equation (5). The increase in u_* with height becomes stronger when the wind speed is larger and hence more snow can be kept in suspension. Also shown in Figure 2 is u_* ′, which is defined as the friction velocity which would be found by an observer unaware of the stability effects induced by snowdrifting. Because of the steeper wind-speed profile in the case of snowdrifting, the observer who assumes a logarithmic wind profile finds a value of u_* ′ that is larger than the true friction velocity throughout the surface layer. Obviously, if one neglects the stabitity effects caused by snowdrifting, the friction velocity would be significantly overestimated.

The vertical profiles in Richardson number are given in Figure 3. Interestingly, the suspended-snow Richardson number takes maximum values close to the surface. This results from the fact that the turbulent snow-particle flux (which is proportional to the snowdrift mass concentration) and the associated buoyancy destruction is maximum near the surface and decreases rapidly upward. The increase in at higher levels can be attributed to the usual decrease in turbulent shear production with height (e.g. Reference GarrattGarratt, 1992). Interestingly, the Richardson number reaches its maximum value at nearly all heights for intermediate values of the wind-speed gradient (u_*H = 0.5−0.7 m s⁻¹). tends to decrease slightly throughout the surface layer for stronger winds (or, equivalently, for larger values of u_*H ). Apparently, the increase in shear production is larger than the increase in snowdrift-induced buoyancy destruction when wind speeds become stronger. This can at least partly be attributed to the fact that the increase in suspended mass with wind speed is largest at low winds by virtye of the dependence of saltation-mass concentration on u_* * (Equation (4)). The simulated values of obtained in these experiments compare favourably with those resulting from the suspended-sediment simulations of Reference Adams and WeatherlyAdams and Weatherly (1981). Note, however, that the increase in at higher levels depends strongly on the amount of suspended mass. If the simulated drift density at higher levels was smaller, the increase in at higher levels would be les pronounced or even absent.

The vertical distribution of suspended mass in the various experiments is shown in Figure 4. The relative increase in suspended mass with u_*H is largest at higher levels, which is due to the fact that both the saltation-mass concentration and the vertical diffusivity of snow increase with wind speed. In all experiments, the suspended-mass concentration decreases by approximately two orders of magnitude with height. This decrease is smaller than that obtained from measurements over a uniform, completely snow-covered plain in Canada in winter (Reference Pomeroy and GrayPomeroy and Male, 1992). This discrepancy may be due to the fact that terminal fall velocities of especially the smallest particles are underestimated by Equation (7) (mean fall velocities range from 1.1m s⁻¹ at the surface to 0.04–0.08 m s⁻¹ at z = H), which leads to an overestimate of the suspended mass of the lightest particles at high levels.

Figure 5 shows the relative change in the turbulent-eddy diffusivity K_m caused by snowdrifting. Clearly, it indicates that when snowdrifting occurs the eddy diffusivity of momentum decreases compared to a simitar surface layer without snowdrifting. Although the absofute values of K_m become larger for increasing values of u_*H , the largest reduction in K_m relative to the case without snowdrifting occurs for intermediate values of u_*H (by as much as 25%) The relatively strong decrease in K_m near the surface results from a diminution of turbulence due to buoyancy as well as from a decrease in friction velocity (see Fig. 2), whereas at higher levels the decrease in K_m is caused exclusively by the increase in Richardson number.

Fig. 5.

Vertical profile of the ratio of simulated “snowdrift” eddy diffusivity for (K_m) and the “no snowdrift” eddy diffusivity for the five selected experiments.

Now it is also clear why surface-layer wind measurements indicated that the surface-roughness length appeared to increase with u_* in the case of snowdrifting (e.g. Reference JoffreJoffre, 1982; Reference ChamberlainChamberlain, 1983). Not only does the wind-speed profile turn around the focus point (see Fig. 1), the stabitity-induced curvature also increases with increasing u_*. When roughness lengths are determined from extrapolating downward the wind-speed profile, assuming that the log-arithmic wind profile can be extended down to the roughness height (which is not the case since the momentum balance of the saltation layer cannot be described by Equation (5)), both effects contribute to the apparent increase in z_0m with u_* during snowdrifting.

Needless to say, the decrease in the vertical turbulent exchange in the surface layer when snowdrifting occurs cannot be neglected in atmospheric models as well as in analysing surface-layer meteorofogical data. In the next section, we will present a simple parameterization which can be used to quantify the “pseudo”-buoyancy snow effect of suspended snow on the exchange coefficients when snowdrifting physics are not expliciny taken into account or when snowdrifting measurements are not available.

4. Parameterization

Assuming that u_* is independent of height, the vertical wind-speed profile in the suspension layer can be derived analytically from the steady-state Equation (5):

Now consider Figure 6, in which the Richardson number is plotted against height for three selected experiments. Obviously, increases roughly linearly with height for z > 0.5 m. Hence, we may approximate the vertical profile in Richardson number according to the linear relation in which the coefficient β may be identified with the inverse Obukhov length . Equation (15) can then simply be rewritten as follows:

The coefficients α and β are parameterized in terms of u_* by applying linear regressions of the simulated vertical profiles of against height for the various experiments (represented by u_*H ). Figure 7 shows the resulting dependence of α and β on u_*. Clearly, α and β reach maximum values in the range of friction vetocities considered. This is a manifestation of the fact that the Richardson number takes maximum values for intermediate values of the wind-speed gradient, as explained in the previous section. The coeflicicnts α and β are accurately described by the following fourth-order polynomial fits (correlation coefficient > 0.99):

This yields the vertical wind profile relation in Equation (16), which includes the pseudo-buoyancy effects due to snowdrifting, in terms of known surface-layer quantities. The quality of the parameterization can be inferred from Figure 8, in which the simulated and parameterized wind-speed profile are depicted for three values of the initial friction vetocity. Clearly, the increase in wind speed due to the occurrence of snowdrifting seems to be parameterized reasonably well. However, ihe accuracy of the parameterization decreases significantly for large u_*H . This is because the decrease in friction vetocity near the surface (see Fig. 2), which is largest for high wind speeds, is not taken into account in the parameterization. Hence, the parameterized wind-speed gradient near the surface is overestimated.

Fig. 6.

Variation of Richardson number with height for three selected experiments. The solid lines represent linear fits for z > 0.5 m.

Fig. 7.

Dependence of α and β on the friction vetocity of the initial wind profile, α and β are the coefficients of the linear fits of the Richardson number vs height (see Fig. 6).

Fig. 8.

Comparison of simulated and parameterized vertical wind-speed profiles for three selected cases. The numbers represent Ihe values of u_*H (m s⁻¹ in each case.

If the eddy diffusivities of scalars (e.g. potential temperature and specific humidity) are assumed to be identical to K_m , the parameterization may also be used to quantify the snowdrift effect on the vertical scalar profiles. This parameterization provides a simple way to lake into account the buoyancy effects due toa stable snowdrift-density profile if no quantitative information about snowdrifting is available. However, one must bear in mind that the model was not tested for different values of the various model parameters (e.g. u_*t ). Therefore, the parameterization presented here should be regarded as giving only a first-order estimate of the effect of suspended snow on the turbulent exchange coefficients in the atmospheric surface layer. In future studies, a more accurate parameterization will be constructed.

5. Discussion and Conclusions

In this study, we have described a model of the atmospheric surface layer that includes snowdrifting. The model calculates the vertical profiles of wind speed and suspended-snow mass in 16 size classes using first-order closure of the turbulent fluxes of momentum and suspended snow. Turbulence decreases to keep the suspended-snow particles at a certain average height against the action of gravity; hence the turbulent energy available for the vertical turbulent transport of momentum and suspended snow is reduced. Consequently, the eddy diffusivities decrease if snowdrifting occurs, causing larger vertical gradients to be necessary to maintain the same turbulent fluxes (Reference Lumley and Bradshaw.Lumley, 1978). The “pseudo”-buoyancy destruction in the case of a stably stratified air-snow mixture is conceptually equivalent to that of a thermally stable atmosphere and can therefore be described in terms of the appropriate Richardson number.

The model results clearly demonstrate the importance of snowdrifting on the vertical wind-speed profile. As a result of the stable stratification of suspended snow, the wind speeds increase throughout most of the atmospheric surface layer, especially at the highest levels. The eddy diffusivity decreases (by 10–25% ) compared to a similar case without snowdrifting. The associated Richardson number takes large values ctose to the focus since the turbulent suspended-snow flux is largest ctose to the saltation suspension interface. decreases sharply to about 0.5 m, above which it increases roughly linearly with height. Note that increases in with height may be overestimated in this study, as the decrease in suspended-snow concentration with height is underestimated by the model. The steady-state momentum-budget equation can be solved analytically if is assumed to depend linearly on height, yielding a log-linear wind-speed profile. The coefficients of the linear relation between and height are parameterized in terms of friction velocity by calculating the model steady-state solution for various initial wind-speed profiles. This leaves an explicit log-linear vertical wind-speed profile in terms of known surface-layer quantities. It is demonstrated that the parameterized wind-speed profiles resemble fairly well the simulated ones. This simple parameterization givesa first-order estimate of the effects of snowdrifting on the vertical profiles of wind (and also scalars) and can be used when detailed information about snowdrifting quantities are lacking. One should bear in mind that the results of this study rely to some extent on a number of assumptions and empirical parameterizations. However, we are convinced that these do not influence our results in a qualitative sense.