WATER RESOURCES RESEARCH, VOL. 26, NO. 7, PAGES 1583-1594, JULY 1990

Saltation of Snow

National Hydrology Research Institute, Environment Canada, Saskatoon, Saskatchewan, Canada

Division of Hydrology, University of Saskatchewan, Saskatoon, Canada

Saltation of snow, the transport of snow in periodic contact with and directly above the snow surface, is governed by the atmospheric shear forces applied to the erodible snow surface, the nonerodible surface, and the moving snow particles. Empirical data measured over a snow-covered plain suggest functions for parameters important to the apportionment of atmospheric shear forces; the aerodynamic roughness height during saltation, the mean horizontal velocity of saltating particles, and the efficiency of the saltation process. The resulting mass transport expression shows an approxi- mately linear increase in snow saltation transport rate with friction velocity, in agreement with the measurements presented. The expression is sensitive to the cohesion of the snow surface, as indexed by the threshold wind speed, that wind speed at which transport ceases; for wind speeds well above the threshold condition, higher threshold wind speeds correspond to higher transport rates. An adaptation of the expression allows calculation of the mass concentration of saltating snow from measured data and the transport rate of saltating snow from the mean wind speed at 10 m height. Application of the transport rate expression using measured wind speeds, directions, and weather observations demonstrates that the directional component of annual saltating snow transport does not always correspond with wind direction frequency.

Redistribution of surface snow by wind transport signifi- cantly affects the winter microclimate, snow cover accumu- lation, and hence snowmelt runoff patterns in cold, wind- swept regions. Several authors suggest that a large component of the mass flux of wind-transported snow trav- els in saltation, which is the horizontal movement of parti- cles in curved trajectories, which are near to and periodically impact the surface [Dyunin, 1967; Male, 1980; Schmidt, 19861. Saltating snow particles are ice particles changing in form from both sublimation and abrasion. Schmidt's [I9811 observations show particles approximating spheroids with mean diameters of 200 pm, and a range of diameters from approximately 10 pm to several hundred micrometers. They are assumed to have the density of ice.

Saltating snow differs significantly from saltation of other materials in that the "bed" is not a layer of rounded particles but is a cohesive matrix of bonded crystals which are metamorphosed by the impact of saltating particles and the vapor transfer during ventilation. Saltating snow particles are derived from shattered surface snow crystals, and Schmidt [I9801 has shown that the intercrystal bond strength, "cohesion," is a much more important parameter than crystal size in calculating the force required to eject a crystal. Particle impact is expected to result in rebound, with a shattering and reestablishment of bonds in the matrix and a "splashing" of crystals from the matrix. Several investi- gators [Kotlyakov, 1961; Oura et al . , 1967; Kind, 1981;

' ~ o r m e r l ~ at Rocky Mountain Forest and Range Experiment Station, Lararnie, Wyoming, and School of Environmental Sci- ences, University of East Anglia, Nonvich, Norfolk, England.

Copyright 1990 by the American Geophysical Union.

Paper number 90WR00006. 0043- 1397/90/90 WR-00006$05.00

Schmidt, 19861 link greater cohesion to higher threshold wind speeds for transport. The threshold wind speed indexes the atmospheric shear stress not available for sustaining saltation transport [Bagnold, 19411. Numerical simulations of particle impact, such as that proposed by Anderson and H a f [I9881 for a bed of discrete particles, could be made more applicable to the case of saltating snow by the inclusion of a crystal bond strength distribution such as that measured by Martinelli and Ozment [ I 9851.

Snow saltation trajectories are not uniform; however, they rarely exceed heights of more than few centimeters [Kikuchi, 1981; Kobayashi, 1972; Maeno et a l . , 19851. The trajectories commonly have the form of a nearly vertical ascent with a very horizontal descent. The angle of particle ascent and descent does vary [Kikuchi, 19811, and this variation is important to the apportionment of shear stress within the saltating system. Anderson [I9871 discusses the theoretical deviation of particle trajectories from common saltation trajectories when subject to turbulence. Inertia severely limits the acceleration of most saltating snow particles in response to drag imposed by the short fluctuations in atmo- spheric velocity that occur very near the surface [Pomeroy, 19881. Deviations from saltation trajectories due to turbu- lence may be considered a manifestation of "modified salta- tion" [Hunt and Nalpanis, 19851; these deviations are most significant for particles of small size at high wind speeds. Pomeroy [I9881 calculated the variance of turbulent veloci- ties for 100-pm-radius snow particles and found it to be less than 30% of the variance for the turbulent velocities of an atmospheric fluid point within the lowest 40 mm of the atmosphere. The ratio of the variance of particle velocity to the variance fluid point velocity declines rapidly with parti- cle size, proximity to the surface, and wind speed. This suggests that most "saltation-sized" snow particles within a few centimeters of the surface are not greatly affected by turbulence.

Many investigators, including Anderson and Hallet [1986], Radok [1968], and Pomeroy [1989], suggest that saltation provides a source for turbulent diffusion of particles above the saltating layer and hence knowledge of saltation trans- port is critical in estimating the blowing snow transport rate that includes the suspended component. Modified saltation is a phase between saltating and suspended snow, yet because of its highly turbulent motion it is best considered part of the suspended component; it provides the connection between the two transport modes. For instance, Anderson [I9871 discusses and theoretically models the transition from saltation to suspension as height and wind speed are in- creased and particle size is decreased. Measurements of the mass flux of saltating snow at heights where the modified saltation component is not large are necessary to provide the boundary conditions for and to evaluate theoretical models. Unfortunately, such measurements, specific to the purely saltating layer of flow, have not been reported for natural snow cover and atmospheric conditions.

The following development uses averaged measurements of the mass flux of snow and wind speeds to derive a simple yet physically based model of snow transport in saltation; saltation in this case is "pure" and does not include the turbulence-modified component. 'The model is suitable for estimating the mean mass and flow characreristics of saltat- ing snow from measurements taken at synoptic meteorolog- ical stations (mean wind speed, mean wind speed at the termination of transport). The purpose of such a model is to provide tractable expressions that (1) may be used to define the importance of the saltating snow flux to the total mass flux and to examine the influence of snow surface conditions on the saltating snow flux and (2) may be used to suggest practices for the management of wind-transported snow.

The model describes flow that is steady state over time, in which saltation is in balance with snow surface conditions and the lowest few meters of the atmospheric boundary layer. This mean saltation behavior, in this application, is developed from averaged measurements for the following reasons. (1) Fast-response wind turbulence measurements such as those made by a sonic anemometer are not reliable near, or in, the snow saltation layer because of interference in the transmission of sound caused by high snow particle concentrations; hot-wire anemometers are subject to snow particle impact. (2) For wind speeds found during snow transport, the time required for a Lagrangian fluid point to travel from the surface to a height of 3.0 m varies from about 2.0 to 37.5 s [Hunt and Weber, 19791. For single phase flow, these are minimum times necessary for development of a steady state boundary layer to 3 m height. (3) Takeuchi's [I9801 measurements of the horizontal development of blow- ing snow flux in the lowest 0.3 m of the atmosphere suggest that about 300 m or roughly 60 s is required for the development of steady state flow. This 60 s added to the 2.0-37.5 s suggests about 1 .O-1.5 min for the development of a steady state two-phase boundary layer of 3 m depth. (4) Averaging periods several times longer than 1.5 min permit integration of snow dune and other transient flow effects on transport rate. Knowledge of these transient flow effects is not necessary for most management applications using the snow transport rate. (5) Averaging periods of the order of minutes rather than seconds will reduce any error from averaging nonlinear relationships in application of the model using hourly meteorological data.

A reasonable alternative to high-frequency measurements is to determine wind shear characteristics by measuring the vertical profiles of mean wind speed in layers of the atmo- sphere where the flow density is similar to that for single- phase flow, i.e., at heights from several tens of centimeters to several meters above the snow surface.

4 During January and February 1987 the saltating snow flux

and related atmospheric parameters were measured on a completely snow-covered and treeless plain, located 4 km west of the urban limits of the city of Saskatoon, Canada. The site is located at an elevation of 500 m above sea level and surrounded by an undisturbed, uniform fetch under summer fallow (soil cultivated so that all vegetation is removed). The uniform surface extends 600 m upwind of the measurement site. Mean snow depths of 105 mm in January and 180 mm in February had standard deviations from 30 to 60 mm. Hardness of the snow surface varied between 5 (concurrent snowfall) and 760 kN m-' (foot makes little imprint), with a high spatial variability. The ranges of meteorological conditions during measurements were as follows: air temperatures between +2" and -20°C; wind speeds at 10 m height between 5 and 15 m s-'; and threshold wind speeds between 5 and 9 m s-I . The site and conditions are noted in detail by Pomeroy [1988].

"Qualimetrics" cup anemometers indicated mean hori- zontal wind speeds at five levels, logarithmically spaced in a vertical transect from 0.35 to 3 m above the soow surface. (The use of trade and company names is for the benefit of the reader; such use does not constitute an official endorsement or approval of any service or product by the U.S. Depart- ment of Agriculture to the exclusion of others that may be suitable.) Anemometer cup rotation was monitored and averaged over 7.5-min periods and converted to mean wind speed on the basis of the manufacturer's and our own calibration.

The saltation flux measurements are 7.5-min summations of the number of particles counted by an optoelectronic snow particle detector, described by Pomeroy et al. [I9871 and Brown and Pomeroy [1989]. The counts are converted to mass fluxes using the particle size distributions found in saltating snow [Schmidt, 19811, the time of summation, and the gauge sampling area, resulting in 7.5-min averages of the mass flux (kilograms per square meter per second) of blow- ing snow through a differential area normal to the flow. Comparisons of the mass flux calculated from the particle , detector output to the mass of snow accumulated in a filter-fabric sediment trap show mean differences of 0.0021 1 kg m-' s- ' for fluxes from 0.04 to 0.19 kg m-2 s-' [Brown , and Pomeroy , 19891.

Precisely specifying measurement heights for the wind and particle flux measurements presents difficulties during blow- ing snow, as the depth of surface snow often changes by several centimeters over times of several minutes as longi- tudinal snow dunes migrate over a "more stable" snow surface. The more stable snow surface depth changes over times of tens of minutes or hours as long-term erosion/ deposition takes place. Anemometers were placed on a mast whose height above the ground was adjusted to compensate for changes in the more stable snow surface. Actual instru- ment heights above the snow cover were measured more

POMEROY AND GRAY: SALTATION OF SNOW 1585

frequently. An attempt was made to keep the particle Substitution of Friction Velocities detector at a height approximately 20 mm above the snow surface, within what appeared to be the saltation layer. Migration of snow dunes with respect to the fixed gauge usually oscillated the measurements between 0 and 30 or 40 mm above the immediate snow surface during the 7.5-min summation period. The height of the dunes and therefore the range of measurement heights increase with wind speed, tending to integrate the mass flux with respect to height within the saltation layer over time.

Framework

Bagnold [I9411 related the transport of saltating sand to the kinetic energy available to support the flow. Dyunin [I9541 applied Bagnold's ideas to blowing snow, and these concepts are used to develop an expression for the mass transport of saltating snow.

The transport rate of saltating snow, Qsal,, is the mean saltating mass crossing a lane of unit width at some mean velocity, up, that is,

W, is the mean weight of snow saltating over the surface, and g is the acceleration due to gravity (see notation section for units).

Kinetic Energy Balance

Owen [1964] explained Bagnold's concepts in terms of particle motion for an ideal case of uniform particles, by balancing the kinetic energy of saltating sand and dust with the excess kinetic energy of the atmospheric flow near the surface. He assumed that a constant shear stress, which just maintains the particle ejection process, is applied by the atmosphere on the snow surface during saltation. When these concepts are applied to blowing snow, this shear stress is greater when snow surface interparticle bond strength or cohesion is greater. The shear stress in excess of this constant level maintains the weight of saltating particles. Schmidt [1986], basing his model on Bagnold's and Owen's proposals, related the weight of saltating snow to the mag- nitude of the flow shear stress applied to the particles. A similar development, which includes the effect of nonerod- ible surfaces, subtracts the shear stress applied to nonerod- ible surface elements, T,, and the shear stress applied to the erodible surface, T,, from the total atmospheric shear stress, T, yielding the expression

The nondimensional coefficient, e , is the efficiency of salta- tion and is inversely related to the kinetic friction resulting from particle impact, rebound, and ejection of shattered crystals at the snow surface as the horizontal shear stress applied to the particles is transformed into a normal force supporting the weight of particles. The saltation efficiency has the range &I: e = 0 when the saltating flow completely loses a particle's momentum after surface impact, and e = 1 for no losses.

The total atmospheric shear stress T is equal to u*'p, where u* is the friction velocity and pis the flow density. For the atmospheric boundary layer the friction velocity is determined from measurements of the vertical profile of wind speed by

where u, is the wind speed at height z; k is von KBrmBn's constant (0.4); and zo is the aerodynamic roughness height, the expected height at which the wind speed vanishes. This equation describes the horizontal wind speed in blowing snow at heights well above the saltation layer if zo is permitted to vary with the friction velocity. The shear stress applied to nonerodible elements, r,, is equal to uZ2p, where u*, is the nonerodible friction velocity and is estimated using empirical techniques outlined by Lyles and Allison [1976], Tabler and Schmidt [1986], and Pomeroy [1988]. For com- plete snow covers with no exposed vegetation, u; = 0. Following Owen, the shear stress applied to the erodible surface, T,, is equivalent to that required to maintain particle ejection and does not contribute to supporting the weight of saltating snow. When r , = 0, the total atmospheric shear stress at which particle ejection ceases is assumed to approx- imate r t . The cessation rather than initiation of particle ejection is used to partition the shear stress because the processes of particle impact, rebound, and ejection are fully operational as saltation ceases but not as it begins [Bagnold, 1941; Anderson and Huff, 19881. For this case, T, = ur2p, where u; is the threshold friction velocity, that is, the friction velocity at the cessation of saltation over a continuous snow cover. The threshold friction velocity is considered to be lower (u: = 0.07-0.25 m sC1) for fresh. loose, dry snow and during snowfall and higher (u: = 0.25-1.0 m s-') for older, wind-hardened, dense, and/or wet snow where interparticle bonds and cohesion forces are strong [Kind, 19811. The term "threshold" always refers to speeds or conditions found at the cessation of saltation.

Saltation Velocity

The saltation velocity up is the mean horizontal velocity of snow moving in the saltation layer. Saltation photographs reproduced by Maeno et al. [1979] show that the horizontal velocities of ascending snow particles increase to match the ambient horizontal wind velocity, while descending particles decelerate very slightly; hence the saltation velocity is approximately proportional to a wind speed within the saltation layer. The measurements of Abbott and Francis 119771 for saltating sand in water suggest a center of fluid drag at 0.8h, where h is the mean saltation trajectory height. Following Schmidt [1986], the wind speed at 0.8h is propor- tional to the saltation velocity.

Bagnold [1941] and Chepil [I9451 show that the vertical profiles of wind speed above layers of saltating sediment in the atmosphere meet at a constant wind speed focus within the saltation layer, and wind speeds below the height of the focus are relatively uniform, a prediction confirmed for saltating snow by the measurements of Maeno et al. [1979]. Furthermore, during saltation the wind speed at the height of this focus remains constant at the threshold value. When applied to saltating snow as shown below, the focus height is

Focua Height 26 Jan

0 1 2 3 4 5 ROUGHNESS HEIGHT 20 ht-~rn]

Fig. 1. Wind speed profile focus heights and Owen's saltation trajectory heights plotted against the aerodynamic roughness height. Wind speed profiles measured above the saltating layer of blowing snow determine the focus heights and aerodynamic roughness heights, and Owen's [I9801 equation using measured friction veloc- ities determines the saltation height. Values are 7.5-min averages.

of the order of, or greater than, 80% of the mean height of saltating particle trajectories, and therefore the threshold wind speed is proportional to the wind speed within the saltation layer and thus to the saltation velocity.

A mathematical solution and measured wind speed data are required to find focus heights for saltating snow condi- tions. Following Bagnold [1941], at and above the focus height the wind speed during drifting at some height z is found as

where zf is the height of the wind speed focus and u,(,~) is the threshold wind speed at height zf. The height of the focus is found using (3) and (4) set to threshold conditions, as

u* In (zO) - uTln (4) zf = exp

u* - u: I (5)

in which the aerodynamic roughness height zo is found as

zo = exp [In (z) - ku,lu*] (6)

The intrinsic aerodynamic roughness height zb is found using (6), with the wind speeds and friction velocities set to threshold conditions.

Figure 1 shows values of the focus height given by (5) and of the aerodynamic roughness height given by (6), using 199 vertical profiles of wind speed measured during saltating snow and at the threshold for each drifting event. Measure- ments during January 18 and 26 show a similar and rapid increase in focus height with aerodynamic roughness height, while those on January 20 show a less rapid and more linear increase. The difference in focus height-aerodynamic rough- ness height regimes is attributed to differences in surface roughness, snowpack structure, and the presence of precip- itation. The intrinsic aerodynamic roughnesses of the sur- faces on January 18 and 26 of 0.20 and 0.28 mm, respec- tively, represent conditions with very light or no snowfall; hence the snow surface is old and wind hardened. Heavy

snowfall and a fresh snow surface on January 20 are associ- ated with an intrinsic aerodynamic roughness of 0.38 mm. For all conditions the focus height is of the order of the mean saltation trajectory heights observed by Kobayashi [I9721 and Kikuchi [1981]. As referenced by Greeley and Zversen [1985], Owen [1980], using a theoretical analysis, proposed mean saltation heights calculated from the friction velocity as in (9), which are plotted for purposes of comparison in Figure 1. For old, wind-hardened snow (January 18 and 26) the focus height approaches 80% of Owen's saltation ''

heights, while for fresh snow and snowfall (January 20) the focus height is approximately 50% of the saltation height. The focus heights are an order of magnitude higher than those found by Bagnold [I9411 for sand and Chepil [I9451 for soil, demonstrating that the snowpack structure not only influences the atmospheric boundary, layer during drifting but influences it in a manner different from sediment beds.

Figure 1 suggests that the wind speed focus is frequently at a height near the center of fluid drag on mean saltating snow trajectories; hence the focus wind speed is proportional to the saltation velocity, the coefficient of proportionality being analogous to a mean particle aerodynamic drag coefficient. Because the focus wind speed is proportional to the thresh- old wind speed and hence the threshold friction velocity, the saltation velocity up is proportional to the threshold friction velocity u; as

where c is the saltation velocity proportionality constant. Equation (7) implies that for saltation with constant thresh- old conditions the saltation velocity is constant and indepen- dent of wind speeds above the saltation layer. This implica- tion is in agreement with the results .of Ungar and Haj's [I9871 numerical analysis. The threshold friction velocity varies with snowpack interparticle cohesion and the pres- ence of fresh falling snow; hence these factors also control the saltation velocity.

Form of the Saltation Transport Expression

Assembling (I), (2), and (7) and substituting friction ve- locities for shear stresses provides the saltation transport equation,

Note that the transport rate increases with the square of the friction velocity rather than the cube as proposed in many other expressions [Greeley and Zversen, 19851 for saltating I

sediment. This difference is due to the conclusion that the saltation velocity for snow is a function not of the friction velocity but rather of its threshold value.

Determination of Transport Coeficients From Measurements

Values of the product of the saltation velocity coefficient and the saltation efficiency (c . e) may be found from the measured mass flux (kilograms per square meter per second) and measured friction velocities, u*, u*,, and u;, if a rela- tionship between the mean saltation transport rate Qsalt and mean saltation mass flux qsalt is specified.

The ratio of the saltation transport rate to mean saltating mass flux (see notation section for units), QSalt/qsdt, equals

+ 20 Jan X 26 Jan

c*e=0.68/um - 1.5 -

0 -

XX .Y ,,..,,,,

uLI

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 FRICTION VELOCITY u* [m/s)

Fig. 2. Saltation velocity and efficiency coefficient product plot- ted against the friction velocity. The product (c . e) is determined using (10) and measurements of the saltating snow mass flux and the atmospheric friction velocity. Values are 7.5-min averages.

h, the mean saltation trajectory height. Measurements of snow saltation trajectories are very limited for outdoor conditions and suggest considerable variation about mean values; however, Owen [I9641 suggests that the mean initial vertical velocity of ascending saltating particles is propor- tional to the friction velocity and hence the mean saltation height is proportional to ~ * ' / ( 2 ~ ) . As referenced by Greeley and Iversen [1985], P. R. Owen (unpublished manuscript, 1980) proposes the relationship

h = 1 . 6 ~ * ' / 2 ~ (9)

where all units are in meters and seconds and h represents an ideal, the mean trajectory height. While the proportionality constant is expected to vary with the mean angle of particle ejection from the surface and hence somewhat with surface snow conditions, the value of 1.6 in (9) provides saltation heights within the range reported by Kikuchi [I9811 for snow, saltating in a wind tunnel, and hence is an acceptable approximat ion.

Given (8) and (9), the product (c e) may be found from mass flux and wind speed profile measurements as

due to the shattering of snow surface crystals which are not ejected or are ejected at slower speeds, as is described by Anderson and Haff [1988]. If the efficiency was constant, the measured proportionality between (c . e) and u* at near- threshold conditions implies proportionality between salta- tion velocity and friction velocity. Constant monitoring of wind speeds and threshold conditions at the measurement site from early January to April of 1987 showed that wind speeds rarely fluctuated about the threshold condition as occurred on January 26 [Pomeroy, 19881. The condition under which (c . e) increased with u*, that of wind speeds at near-threshold conditions, was infrequent for that winter at Saskatoon.

The second case is more general, involves higher wind speeds and less wind-hardened snowpacks, and corresponds to friction velocities for which the significant mass fluxes occur on all days of measurement. The efficiency declines with friction velocity in this case. Because of higher wind speeds and a somewhat less hard snow surface, the more energetic saltating particles spend more energy shattering surface crystal bonds which either are not ejected or are ejected at low velocities. Anderson and Haff [I9881 calculate an increase in the number of these low-velocity ejections with increasing impact speed. For snow an increase in low-velocity ejections should occur as intercrystal cohesion decreases. The loss of energy resulting from less efficient particle to surface impact as the wind speed increases is manifest as a decreasing saltation efficiency.

For simplicity in the saltation transport equation this more general relationship between (c . e) and u* is used. Assuming that e attains its maximum value of 1.0 within the range of measurements, the maximum (c - e), equal to 2.8 at u* equal to 0.23 m s-I , provides c = 2.8. The assumption that e = 1.0 for some measurement does not affect the value of (c - e), which is determined from measurements. An inverse rela- tionship between e and u* is satisfying conceptually. In analogy to a mass moving along a static surface, the effi- ciency e is inversely proportional to the kinetic friction of transport of the mass. This kinetic friction is proportional to the velocity of the mass; hence e is inversely proportional to velocity, and the condition e = 0 cannot exist. Using the greatest value of (c . e) and noting c = 2.8, we find that such a relationship specifies e = 1/(4.2u*). A line representing the modeled values of (c e) is drawn through the measured

and is plotted against friction velocities in Figure 2. The points in Figure 2 fall into two elongated groups character- ized by two straight-line segments: one, specific to near- threshold friction velocities on January 26 and four measure- ments on January 20, indicates an increase in (c - e) with u*; the other, corresponding to the full range of friction veloci- ties and all days of measurement, indicates a decrease in (C e) with u*. The two groupings of data do not correspond to the two differing focus height-roughness height regimes.

Drifting on January 26 was highly intermittent; hardening of drift-packed surface crystal bonds, during pauses in drifting, permitted severe wind hardening of surface snow, as indicated by threshold wind speeds which increased over the day. The low particle impact velocities (due to low wind speeds) and hard snow surface suggest high and relatively unchanging efficiency particle rebound with little energy loss

points in Figure 2.

Saltation Transport Equation

Combining the results of measurements and theory leads to the expression for the transport rate of saltating snow:

The dimensionless coefficient 0.68 combines saltation veloc- ity and efficiency coefficients: c . e = 0.68/u*, where the value of the coefficient does not depend upon the assumed value of c. Saltation transport rate increases in a roughly linear manner with friction velocity, in contrast to the cubic increase proposed by Bagnold [I9411 for transport of sand and the squared increase proposed by Ungar and Haff [I9871 for saltation transport of sediment. The linear increase results from a mean horizontal velocity which does not increase the wind speed and an efficiency which increases

FRICTION VELOCITY u* Cm/sl

Fig. 3. Measured and modeled (equation (11)) snow saltation transport rates plotted against the friction velocity. Measured values are determined from the measured saltating snow mass fluxes and an assumed saltation height function for a variet of transport threshold friction velocities from 0.2 to 0.33 m s-): Modeled values are calculated for transport threshold friction velocities of 0.2,0.25,0.3, and 0.35 m s- ' . Values are 7.5-min averages.

approximately with the inverse of wind speed; Bagnold's model incorporates neither of these results, while Ungar and Haffs model incorporates only the first. Saltation models derived for sand and soil are not directly applicable to snow, a consequence of the importance differences between the cohesive, yet breakable snow surface crystal structure and a bed of discrete, noncohesive sediment grains.

Figure 3 shows values of saltation transport rate deter- mined from measured mass fluxes and friction velocities for threshold friction velocities u: from 0.20 to 0.33 m s-I with results of the model for threshold friction velocities equal to 0.20, 0.25, 0.30, and 0.35 m s- ' . For given conditions of nonerodible roughness and threshold friction velocities, QSal, is approximately proportional to the friction velocity. For near-threshold conditions, relatively higher transport rates correspond to lower threshold friction velocities; for wind speeds well above the threshold, higher transport rates correspond to higher threshold friction velocities. The cross- over between these two conditions occurs at friction veloc- ities from 0.40 to 0.55 m s-' . The crossover is a consequence of (1) the diminishing proportion of shear stress exerted on the erodible snow surface as the wind speed increases and (2) the proportionality between saltation velocity and thresh- old friction velocity. At low wind speeds the shear stress is small, and a significant proportion is exerted in shattering snow cover crystals to effect particle ejection. Soft snow covers require less stress than hard snow covers to shatter intercrystal bonds; hence for soft, low-threshold snow cov- ers the excess shear stress at low wind speeds is sufficient to result in relatively higher transport rates. At high wind speeds the shear stress is large in comparison to its threshold level, and the difference between the shear stresses ex- tracted by shattering the bonds of soft and hard snow covers is less important. In this case the lower saltation velocity

lower transport rate than that of the hard, high-threshold snow covers. With regard to the saltation transport rate, the greater efficiency and velocity of particle transport over hard snow surfaces overcome the greater difficulty in shattering particles from these surface as the wind speed exceeds near-threshold conditions. This result concurs with Schmidt's 119861 conclusion that the highest saltation trans- port rates are associated with hard, compacted (high trans- port threshold) snow covers.

Both model and measurements show a similar increase ' with friction velocity, though some measurements on Janu- ary 20 were subject to partial gauge burial by snow and may , underestimate the transport rate. The model is tested against '

measured mass fluxes; hence modeled QSal,lh values are used as modeled mass fluxes. The mean difference (mea- sured mass flux - modeled mass flux) is -0.01024 kg rn-' s-', with a standard deviation of difference of 0.07295 kg m-2 s - I . The scatter about the mean, in part, illustrates the difficulty in representing with a steady state model a phe- nomenon that fluctuates rapidly and triggers an atmospheric boundary layer response that may not fit tidily within a measurement period. Typical mass fluxes are 0.42 kg m-2 s-' for friction velocities greater than 0.4 m s- ' . The model overestimates these fluxes by only 2.5% and hence is quite a promising predictor of net transport over a snowstorm.

Comparison of the saltation transport rates to other blow- ing snow transport rate measurements demonstrates the relative contribution to the total flux made by saltation. Schmidt [I9861 reported threshold conditions and measured the blowing snow transport rate with a fabric trap which stretched from the surface to 0.5 m above the snow surface [Schmidt et al., 19821. The model he derived from these measurements is set for typical roughness conditions present at Saskatoon with zb = 0.2 mm and Schmidt's transport to shear stress ratio, C = 5.0. The ratios of the saltating snow transport rate to Schmidt's blowing snow transport rate are listed in Table 1 for two threshold conditions and three friction velocities. For low friction velocities, saltation com- prises 5&100% of transport, while at a friction velocity of 0.7 m s-' (roughly ulo = 15 m s-I), saltation comprises as little as 8% of blowing snow transport, the rest moving in suspension (which includes modified saltation).

Drift Density of Saltating Snow The mean mass concentration of snow in "pure" salta-

tion, referred to as the saltation drift density, is a parameter 4

TABLE 1. Ratios of the Saltation Model Transport Rate to Schrnidr's [I9861 Blowing Snow Transport Rate

Threshold Friction Velocity

u; = 0.2 u; = 0.3 m s-I m s-'

Friction velocity rr* = 0.35 m s-' 0.53 1 .OR Friction velocity u* = 0.50 m s-' 0.18 0.41 Friction velocity u* = 0.70 m s-' 0.08 0.15

Ratios are calculated using z6 = 0.2 mm and Schmidt's C = 5.0 for two threshold friction velocities and three friction velocities.

'Saltation model transport rate value exceeded Schmidt's blow- possessed by soft, low-threshold snow covers results in a ing snow transport rate value.

that is necessary to define a lower boundary condition for turbulent diffusion of blowing snow [Radok, 1968; Anderson and Hallet, 1986; Pomeroy and Male, 1987; Pomeroy, 19891. Even though the drift density declines with height in the saltation layer, a computed mean value such as the "salta- tion drift density" provides a reference for the quantity of saltating snow available for modification by turbulence. Such a reference value is necessary to perform the diffusion calculations which model the suspended component of blow-

V n g snow. The saltation drift density, qsallr is defined by qsall = qsaltlup.

Values of the saltation drift density, calculated from ' measured mass fluxes and modeled saltation velocities, are plotted against friction velocity in Figure 4. The drift densi- ties reflect a range of threshold and hence snow cover and snowfall conditions. For near-threshold wind speeds (within 0.4 m s- ' of u> the saltation drift density increases in proportion to the friction velocity. The increase in drift density with friction velocity is attributed to the rapidly diminishing rate of increase of the ratio of shear stress - available for transport to mean saltation height as the friction velocity increases from its threshold level and a possibly steady saltation efficiency near the threshold condition. More commonly, well above the threshold condition, drift densities vary independently of friction velocity and cluster within a range from 0.4 to 0.9 kg mW3. Location within the cluster varies with threshold condition, the highest drift densities being associated with the lowest threshold wind speeds. Threshold wind speed is associated with both the surface snow condition and the saltation velocity; hence high drift densities are associated with less cohesive, fresh snow covers and snowfall and with low saltation velocities, the inverse being true for low drift densities. In demonstration, for winds well above the threshold level a mean drift density of 0.70 kg m-3 is associated with a threshold friction velocity of 0.2 m s-I , and, similarly, 0.51 kg m-3 with 0.23-0.27 m s-I and 0.48 kg m-3 with 0.27-0.33 m s-I. This variation of drift density with threshold condition suggests a mechanism where the snow surface condition might affect diffusion of blowing snow. The range of values within the cluster corre-

2 1 Cluster u w e r bound

7 . 7

x X x~ XY x E ~~7 cluster lower b i n d -

c.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 FRICTION VELOCITY u* [m/s]

Fig. 4. Saltation drift density (mean concentration of saltating snow in the atmosphere) plotted against the friction velocity. Drift density is calculated as the measured mass flux divided by the modeled saltation velocity. Values are 7.5-min averages.

0.00 0.20 0.40 0.60 0.80 FRICTION VELOCITY u* Cm/s]

Fig. 5. Aerodynamic roughness height measured during salta- tion and that modeled, plotted against the friction velocity. Values are 7.5-min averages.

sponds to reference drift densities that are necessary for calculation of the suspended component of blowing snow, as suggested by data shown by Mellor and Fellers [I9861 and Pomeroy [1988].

Calculation of Saltation Transport From Meteorological Observations

For conditions with complete snow covers and no exposed vegetation the transport rate of saltating snow may be calculated from standard meteorological wind speed mea- surements if an aerodynamic roughness height can be spec- ified. Owen [I9641 suggests that the roughness height above a saltating flow is proportional to the square of the friction velocity, and Tabler [I9801 confirmed such a proportionality for blowing snow over lake ice. Measurements from com- plete snow covers over fallow land (Figure 5) show a relationship between roughness height and friction velocity during blowing snow; however, the association between the parameters differs from day to day at the same site. In concurrence with Schmidt's [I9861 observations the greatest roughness values are associated with the fresh snow cover and snowfall on January 20. Fitting Owen's proposed rela- tionship to the data provides the expression

with a coefficient of determination R~ = 0.68 and a standard error equal to 0.57 mm. The dimensionless coefficient 0.1203, which is based upon measurements over complete land-based snow covers, is an order of magnitude greater than Tabler's [I9801 value of 0.02648, measured over a mixture of >75% snow and smooth lake ice. Roughness heights calculated from (12) are also an order of magnitude greater than Schmidt's [I9861 measurements over shallow, complete, lake ice-based snow surfaces, except for near- threshold conditions, where values are similar. This suggests differences between land-based and lake ice-based snow covers in the generation of aerodynamic roughness by saltating snow, possibly due to greater momentum loss during particle rebound and ejection on the land-based snow covers, which are less dense and less subject to temperature gradient metamorphism. Adams [I9811 provides a full dis-

0 0

0 0

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 10-rn WIND SPEED ulo [rn/sl

Fig. 6. Measured and modeled (equation (14)) snow saltation transport rates plotted against the 10-m wind speed. Values are 7.5-min averages.

cussion of snow cover development and characteristics on frozen lakes.

Use of (3) and (12) gives u* in terms of the 10-m wind speed, ulo, as

For a complete snow cover without exposed vegetation, u*, = 0, and a central measured value for u: is 0.25 m s-I. Substitutingg = 9 . 8 m ~ - ~ , ~ = 1 .2kgm-~,u:= 0.25ms-I, u*, = 0.0 m s- ' , and u* as defined in (13) into (1 1) results in

which provides an approximation to snow saltation transport using 10-m wind speeds over nonvegetated plains during drifting or blowing snow events.

The results of (14), and saltation transport rates from measured mass fluxes and friction velocities, are plotted against the 10-m wind speed, calculated from an extrapola- tion of measured vertical profiles of wind speed, in Figure 6. Both measured and modeled transport rates show approxi- mately linear increases with 10-m wind speed. Defining "measurement" as the measured mass flux multiplied by the saltation height (9) determined from measured friction ve- locities and "model" as the result of (14) calculated using the corresponding 10-m wind speeds, we find that the mean difference (measurement - model) is 0.00045 kg m-I s- ' , with a standard deviation of 0.00191 kg m-' s-I. For wind speeds greater than 10 m s- ' the mean difference is less than 6% of the transport rate, and the standard deviation is less than 25%. The roughly linear relation means that this equa- tion may be applied for a variety of averaging times, given that steady state boundary layer conditions are present.

Orientation of Snow Control and Management Works

Takeuchi [I9891 has shown that the size of drifts upwind of snow fences is sensitive to the saltation rather than total blowing snow transport to these fences; hence the direction and magnitude of saltation transport over a winter have a

bearing on the orientation, location, and size of drifts around barriers. A practical application of the model in determining the orientation of snow control and management works with regard to the "trapping" of saltating snow is demonstrated below. Figures 7 and 8 show "roses" of the wind directional frequency and the average annual saltation transport on a 1000-m fetch in eight principal directions (N, NE, E, SE, S, SW, W, NW) for Regina and Prince Albert, Saskatchewan, Canada. Wind directional frequency is a parameter easily abstracted from summaries of meteorological station records such as those published by the Canadian Climate Centre and other agencies. The saltation transport is calculated on an hourly basis by (1 1) and (13) using synoptic meteorological observations of wind speed and the presence of "drifting" or "blowing" snow and is corrected for the depletion of snow cover over the 1000-m fetch calculated by a blowing snow transport, sublimation, and erosion model [Pomeroy, 1988, 19891. Hourly values are summed over the winter; annual values presented are averages of the winters 1970-1976.

The city of Regina (latitude 50"26'N, longitude 104"401W) is situated in the central North American grasslands on a flat, highly exposed lacustrine plain, which for the most part is cultivated to spring wheat and other cereal grains. The primary land surfaces surrounding Regina during the winter are stubble (harvested stalks of grain), which comprises -70% of the area, with the remainder as summer fallow (bare ground), native prairie grasses, roads, and farmsteads. In contrast, the city of Prince Albert is located 265 km north of Regina in an oilseed and cereal grain growing region on the border between the mixed grassland and deciduous forest "Parklands" and the southern fringe of the boreal forest. The forests near Prince Albert result in a less exposed environment than around Regina. The mean 10-m wind speeds during the months of the year with snow cover at Regina and Prince Albert are 6.0 and 3.8 m s-' , respectively; the mean annual snowfalls, expressed as snow water equiv- alent, are 115 and 110 mrn, respectively; and the mean annual saltation transports, the average over 1000-m fetches of stubble and fallow, expressed as a percent of the annual snowfall, are 16 and 7%, respectively.

Figure 7a shows that winds at Regina occur from the NW, W, SE, and E directions 75.4% of the time and these winds produce, on average, 87.8% of the annual saltation transport (Figure 7b). The shapes of roses of wind and saltation transport are similar, with the exception of the S and SW directions, where winds occur with a frequency of 14.1% but produce only about 2.7% of the saltation transport. The S and SW winds at Regina often bring the "chinook" and a midwinter warming, resulting in the low transport values for those directions. At Regina the most effective orientation of a snow control measure which traps the saltating load would be perpendicular to the direction of the dominant prevailing winds, namely, SW to NE.

One should not assume matching symmetry in the direc- tional distributions of saltation transDort and wind. Wind speeds may have a directional bias, as also may snowfall, snow depth, snow surface cohesion, and other factors which affect saltation transport. The roses of wind frequency and saltation transport at Prince Albert (Figure 8) exhibit major departures in their distribution patterns. For example, winds from the NE occur with a frequency of 13.1% and produce 18.5% of the annual saltation transport, whereas those from the SW, occurring with approximately the same directional

R E G I N A W I N D D I R E C T I O N A L FREQUENCY <NOV. TO APR. )

NORTH <a. 6%)

NU c i a . 8 % )

WEST (17. 2 % ) E A S T < l a . 1%)

S E <23.3%)

SOUTH <7. 4%)

Fig. 7a

R E G I N A S A L T A T I N G SNOW F L U X ONLY

NORTH (5. 0%) I

N U <25. 8%) N E (3. 8 % )

WEST (18 . 1%) - E A S T ( 2 1 . 7 % )

sw < l . O X ) S E <24. 7%)

SOUTH < 1 . 7 % )

Fig. 76 Fig. 7. Directional roses of annual (a) winter month wind direction frequency and (b) directional percentage of

saltating snow transport over a 1000-m fetch for Regina, Canada. Annual values are determined from hourly synoptic meteorological measurements from November to April and the saltation transport model (equations (11) and (13)), corrected for snow depletion over the fetch. The values are averaged over 1970-1976.

frequency (13.3%), produce only 0.7% of the annual trans- port, for which warm S-SW winds are again to blame. The wind rose suggests that barriers placed on any alignment within the segment bounded by NE-SW and NW-SE direc- tions would provide an equal measure of control of blowing snow. However, the transport rose rules out a NW-SE orientation because of the low amounts of saltating snow originating from the S and SE directions. A control placed with a N-S alignment would provide protection and "trap- ping" action against 85.4% of the winds and 89.6% of the annual transport of saltating snow.

The shear stress in excess of that exerted by the wind on the snow surface and roughness elements provides the basis for an expression describing the mean flow of saltating snow

over a variety of snow surfaces. This model and measure- ments of average mass flux and wind speed over complete nonvegetated snow covers suggest certain properties of saltating snow: (1) the mean horizontal velocity of saltating snow particles is usually proportional to the threshold fric- tion velocity; (2) the efficiency of saltation is inversely proportional to the friction velocity, except for very hard snow covers and near-threshold conditions when the effi- ciency increases with friction velocity; (3) for near-threshold conditions the mean mass concentration of saltating snow increases with friction velocity; and (4) well above the threshold the mean mass concentration of saltating snow varies widely between 0.35 and 0.9 kg mP3, the highest values associated with low threshold wind speeds.

The model provides a transport rate expression which exhibits, for near-threshold conditions, the maximum saltat-

PRINCE ALBERT WIND DIRECTIONAL FREOUENCY CNOV TO APR)

NORTH (7. 5%) I

WEST (10. 1%)

Fig. 8a

PRINCE ALBERT SALTATING SNOW FLUX ONLY

NORTH (10. 1%) I

EAST ( lB .P%>

SE CP. 2%)

SOUTH (7. 1%)

I SOUTH (0. 3%)

Fig. 8b Fig. 8. Directional roses of annual (a ) winter month wind direction frequency and (b) directional percentage of

saltating snow transport over a 1000-m fetch for Prince Albert, Canada. Annual values are determined from hourly synoptic meteorological measurements from November to April and the saltation transport model (equations (1 I ) and (13)), corrected for snow depletion over the fetch. The values are averaged over 1970-1976.

ing snow transport rates associated with the lowest threshold wind speeds. At higher wind speeds, maximum transport rates are associated with high threshold wind speeds. This reflects, for wind speeds exceeding near-threshold levels, that the high efficiency and velocity of particle transport over hard snow surfaces are more important to saltating snow transport than the dficulty in shattering and ejecting particles from these surfaces. For constant threshold and aerodynamic roughness conditions the saltating snow transport rate increases approx- imately in proportion to the friction velocity or less approxi- mately in proportion to the wind speed at 10 m height. The proportionality of the relationship between saltation transport rate and wind speed is due to the unchanging saltation velocity and drift density, the decrease in saltation efficiency, and the squared increase in mean saltation height as wind speed increases. The proportionality means that the errors created by

applying the model to friction velocities or wind speeds of averaging times greater than 7.5 min will be small. Saltation of

*

snow is shown to comprise 50-100% of blowing snow transport in the lowest 0.5 m of the atmosphere for 10-m wind speeds of 7.5 m s-', diminishing to &15% for wind speeds of I5 m s-'. 4

While important near the threshold, for wind speeds well in excess of threshold conditions it is not a major component of blowing snow transport.

A model for calculating the saltation transport rate from 10-m wind measurements is presented. Comparison of mea- sured and modeled rates shows sufficient agreement to permit use of this model in calculating the saltation transport rate on an hourly basis by using standard meteorological measurements. As a demonstration the model is used to calculate the mean annual saltation transport for various wind directions at two locations in western Canada. Snow eroded

and transported by saltation comprises 16 and 7% of the annual snowfall at these locations, the smaller value corresponding to the less exposed location. Symmetry between the distributions of wind frequency and saltation transport does not exist be- cause of the directional bias of the friction velocity and of factors which affect the threshold friction velocity. The design of snow control measures and snow accumulation models which take into account snow received as saltation transport should take this asymmetry into account.

8

NOTATION

saltation velocity proportionality constant, dimensionless. Schmidt's ratio of transport rate to shear stress, s- ' . efficiency of saltation, dimensionless. gravitational acceleration constant, m s -'. mean height of saltating particle trajectories, m. von KArmAn's constant, dimensionless. mean saltation mass flux, kg m-' s-'. saltation transport rate, kg m-' s-'. wind speed at a height of 10 m, m s-'. friction velocity, m s -' . nonerodible friction velocity, m s -' . threshold friction velocity, m s-'. mean horizontal velocity of saltating particles, m s-'. wind speed at transport threshold, m s-' . horizontal wind speed at height z , m s-'. weight of saltating snow over a unit area of snow cover, N m-'. height above snow surface, m. aerodynamic roughness height, m. intrinsic (nontransport) aerodynamic roughness height, m. height of focus of wind speeds during saltation, m s-' .

qsal, mean drift density (mass per atmospheric volume) of saltating snow, kg m-3.

p flow density, kg mP3. r atmospheric boundary layer shear stress, N m-'.

r , atmospheric shear stress applied to nonerodible surface, N m-'.

7, atmospheric shear stress applied to erodible surface, N m-'.

Acknowledgments. T. Brown of the Division of Hydrology, 4 University of Saskatchewan, developed the snow particle detectors

used in this experiment and assisted in the field program. P. L. Landine of the Division of Hydrology programmed the calculations

i of saltating snow transport for Regina and Prince Albert. R. A. . Schmidt of the Rocky Mountain Forest and Range Experiment Station, Fort Collins, Colorado, and anonymous referees provided helpful comments on the manuscript. These persons are thanked. Support was received from Saskatchewan Research Council Grad- uate Studies Scholarships and Natural Sciences and Engineering Research Council of Canada operating grant A4363.

Abbott, J. R., and J . R. D. Francis, Saltation and suspension trajectories of solid grains in a water stream, Philos. Trans. R . Soc. London, Ser. A , 284, 225-254, 1977.

Adams, W. P., Snow and ice on lakes, in Handbook of Snow, Principles, Processes, Management and Use, edited by D. M. Gray and D. H. Male, pp. 437474, Pergamon, New York, 1981.

Anderson, R. S., Aeolian sediment transport as a stochastic pro- cess: The effects of a fluctuating wind on particle trajectories, J. Geol., 1987, 497-512. 1987.

Anderson, R. S., and P. K. Haff, Simulation of aeolian saltation, Science, 241, 820-823, 1988.

Anderson, R. S., and B. Hallet, Sediment transport by wind: Toward a general model, Geol. Soc. Am. Bull., 97,523-535, 1986.

Bagnold, R. A., The Physics of Blown Sand and Desert Dunes, 265 pp., Chapman and Hall, London, 1941.

Brown, T., and J. W. Pomeroy, A blowing snow detection gauge, Cold Reg. Sci. Technol., 16, 167-174, 1989.

Chepil, W. S., Dynamics of wind erosion, 11, lnitiation of soil movement, Soil SCI . , 60(5), 39741 1, 1945.

Dyunin, A. K., Solid flux of snow-bearing air flow, Tr. Transp. Energ. Znst. Akad. Nauk. SSSR Sib. Otd . , 4, 7148 , 1954. (Translated from Russian by G. Belkov, NRC Tech. Transl., 1102, 25 pp., 1963.)

Dyunin, A. K., Fundamentals of the mechanics of snow-storms, in Physics of Snow and Ice, vol. 1, part 2, edited by H. Oura, pp. 1065-1073, lnstitute of Low Temperature Science, Sapporo, Japan, 1967.

Greeley, R., and J. D. Iversen, Wind us a Geological Process on Earth, Mars, Venus and Titan, 333 pp., Cambridge University Press, New York, 1985.

Hunt, J. C. R., and P. Nalpanis, Saltating and suspended particles over flat and sloping surfaces, 1, Modelling concepts, in Proceed- ings of the International Workshop on the Physics of Blown Sand, vol. 2, edited by 0 . E. Barndorfl-Nielsen, pp. 9-36, University of Aarhus, Aarhus, Denmark, 1985.

Hunt, J. C. R., and A. H. Weber, A Lagrangian statistical analysis of diffusion from a ground-level source in a turbulent boundary layer, Q. J. R. Meteorol. Soc. , 105, 423443, 1979.

Kikuchi, T., A wind tunnel study of the aerodynamic roughness associated with drifting snow, Cold Reg. Sci. Technol., 5 , 107- 118, 1981.

Kind, R. J., Snow drifting, in Handbook of Snow, Principles, Processes. Management and Use , edited by D. M. Gray and D. H. Male, pp. 338-359, Pergamon, New York, 1981.

Kobayashi, D., Studies of snow transport in low-level drifting snow, Contrib. Inst. Low Temp. Sci. Hokkaido Univ., Ser. A , 24, 1-58, 1972.

Kotlyakov, V. M., Results of study of the ice sheet in Eastern Antarctica, Antarctic Glaciology, ZAHS AZSH Publ. 55, 88-99, 1%1.

Lyles, L., and B. E. Allison, Wind erosion: The protective role of simulated standing stubble, Trans. ASAE, 19, 61-64, 1976.

Maeno, N., K. Araoka, K. Nishimura, and Y. Kaneda, Physical aspects of the wind-snow interaction in blowing snow, J. Fuc. Sci. Hokkaido Univ., Ser. 7, 6(1), 127-141, 1979.

Maeno, N., R. Naruse, K. Nishimura, I . Takei, T. Ebinuma, S. Kobayashi, H. Nishimura, Y. Kaneda, and T. Ishida, Wind- tunnel experiments on blowing snow, Ann. Glaciol., 6, 63-68, 1985.

Male, D. H., The seasonal snowcover, in Dj~namics of Snow andZce Masses, edited by S. Colbeck, pp. 305-395, Academic, San Diego, Calif., 1980.

Martinelli, M., and A. Ozment. Some strength features of natural snow surfaces that affect snow drifting, Cold Reg. Sci. Technol., 11, 267-283, 1985.

Mellor, M., and G. Fellers. Concentration and flux of wind-blown snow, Spec. Rep. 86-11, 16 pp., U.S. Army, Cold Reg. Res. and Eng. Lab., Hanover, N. H., 1986.

Oura, H., T. lshida, D. Kobayashi, S. Kobayashi, and T. Yamada, Studies on blowing snow, 11, in Physlcs of Snow and Ice, vol. 1, part 2, edited by H. Oura, pp. 1099-1117, Institute for Low Temperature Science, Sapporo, Japan, 1967.

Owen, P. R., Saltation of uniform grains in air, J. Fluid Mech., 20, 225-242, 1964.

Owen, P. R., Sand movement mechanism, paper presented at Workshop on Physics of Desertscation, Int. Cent. for Theor. Phys., Trieste, Italy, 1980.

Pomeroy, J. W., Wind transport of snow, Ph.D. dissertation, 224 pp., Univ. of Saskatchewan, Saskatoon, Canada. 1988.

Pomeroy, J. W., A process-based model of snow drifting, Ann. Glaciol., 13, 237-240, 1989.

Pomeroy, J. W., and D. H. Male, Wind transport of seasonal

1594 POMEROY AND GRAY: SALTATION OF SNOW

snowcovers, Seasonal snowcovers: Physics, Chemistry, Hydrol- ogy, edited by H. G. Jones and W. Orville-Thomas, NATO Adv. Study Inst. Ser., Ser. C , 211, 119-140, 1987.

Pomeroy, J. W., T. Brown, and D. H. Male, Measurement of blowing snow properties using optical attenuation devices, Snow Property Measurement Workshop, Tech. Memo. 140, pp. 347- 389, Snow and Ice Subcomm., Assoc. Comm, on Geotech. Res., Natl. Res. Counc. of Can., Ottawa, 1987.

Radok, U., Deposition and erosion of snow by the wind, Res. Rep. 230,23 pp., U.S. Army Cold Reg. Res. and Eng. Lab., Hanover, N. H., 1968.

Schmidt, R. A., Threshold windspeeds and elastic impact in snow transport, J. Glaciol., 26(94), 4 5 3 4 7 , 1980.

Schmidt, R. A., Estimates of threshold windspeed from particle sizes in blowing snow, ColdReg. Sci. Technol., 4, 187-193, 1981.

Schmidt, R. A., Transport rate of drifting snow and the mean wind speed profile, Boundary Layer Meteorol., 34, 213-241, 1986.

Schmidt, R. A,, R. D. Tabler, and R. L. Jairell, A new device for sampling mass flux of blowing snow, Proc. West. Snow Conf., 50, 102-1 11, 1982.

Tabler, R. D., Self-similarity of wind profiles in blowing snow allows outdoor modelling, J. Glaciol., 26(94), 421-434, 1980.

Tabler, R. D., and R. A. Schmidt, Snow erosion, transport and deposition in relation to agriculture, Proceedings of the Sympo- sium, Snow Management for Agriculture, Great Plains Agric. Counc. Publ. 120, edited by H. Steppuhn and W. Nicholaichuk, pp. 11-58, Water Stud. Inst., Saskatoon, Canada, 1986.

Takeuchi, M., Vertical profiles and horizontal increase of drift snow transport, J. Glaciol., 26(94), 481-492, 1980.

Takeuchi, M., Snow-collection mechanisms and the capacities of snow fences, Ann. Glaciol., 13, 248-251, 1989.

Ungar, J. E., and P. K. HaE, Steady state saltation in air, Sedimen- tology, 34, 289-299, 1987.

D. M. Gray, Division of Hydrology, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N OWO.

J. W. Pomeroy, National Hydrology Research Institute, Environ- ment Canada, 11 Innovation Boulevard, Saskatoon, Saskatchewan, Canada S7N 3H5.

(Received January 13, 1989; revised October 30, 1989;

accepted December 29, 1989.)