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 approximately 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 significantly affects the winter microclimate, snow cover accumulation, and hence snowmelt runoff patterns in cold, windswept regions. Several authors suggest that a large
component of the mass flux of wind-transported snow travels in saltation, which is the horizontal movement of particles 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 investigators [Kotlyakov, 1961; Oura et al., 1967; Kind, 1981;
' ~ o r m e r lat~ Rocky Mountain Forest and Range Experiment
Station, Lararnie, Wyoming, and School of Environmental Sciences, University of East Anglia, Nonvich, Norfolk, England.
Copyright 1990 by the American Geophysical Union.
Paper number 90WR00006.
0043- 1397/90/90WR-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 atmospheric velocity that occur very near the surface [Pomeroy,
19881. Deviations from saltation trajectories due to turbulence may be considered a manifestation of "modified saltation" [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 velocities for 100-pm-radius snow particles and found it t o 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 particle 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 transport 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 increased 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 saltating snow from measurements taken at synoptic meteorological 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 blowing snow flux in the lowest 0.3 m of the atmosphere suggest
that about 300 m o r 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 atmosphere where the flow density is similar to that for singlephase flow, i.e., at heights from several tens of centimeters
to several meters above the snow surface.
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 horizontal 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 o r product by the U.S. Department 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 blowing 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 blowing snow, as the depth of surface snow often changes by
several centimeters over times of several minutes as longitudinal 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 instrument heights above the snow cover were measured more
4
,
,
POMEROY
AND GRAY:SALTATION
OF SNOW
frequently. An attempt was made to keep the particle
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 magnitude of the flow shear stress applied to the particles. A
similar development, which includes the effect of nonerodible surfaces, subtracts the shear stress applied to nonerodible 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 saltation 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.
1585
Substitution of Friction Velocities
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 complete 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 approximate 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 proportional 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
1
0
2
3
ROUGHNESS HEIGHT 20 ht-~rn]
5
4
Fig. 1. Wind speed profile focus heights and Owen's saltation
trajectory heights plotted against the aerodynamicroughness 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 velocities 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 conditions. 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
zf
= exp
I
u* In (zO)- uTln (4)
u* - u:
(5)
in which the aerodynamic roughness height zo is found as
zo = exp [In (z) - ku,lu*]
snowfall and a fresh snow surface on January 20 are associated 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 threshold 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 threshold conditions the saltation velocity is constant and independent of wind speeds above the saltation layer. This implication 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 presence 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 velocities for shear stresses provides the saltation transport
equation,
(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. Measurements 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 roughness height regimes is attributed to differences in surface
roughness, snowpack structure, and the presence of precipitation. The intrinsic aerodynamic roughnesses of the surfaces on January 18 and 26 of 0.20 and 0.28 mm, respectively, represent conditions with very light or no snowfall;
hence the snow surface is old and wind hardened. Heavy
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
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 relationship between the mean saltation transport rate Qsaltand
mean saltation mass flux qsaltis specified.
The ratio of the saltation transport rate to mean saltating
mass flux (see notation section for units), QSalt/qsdt,
equals
.
I
+X
--
20 Jan
26 Jan
c*e=0.68/um
1.5 -
0
XX
0.00
0.10
0.20
.Y
0.30
,,..,,,,
0.40
uLI
0.50
0.60
0.70
FRICTION VELOCITY u* [m/s)
Fig. 2. Saltation velocity and efficiency coefficient product plotted 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 proportional 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
and is plotted against friction velocities in Figure 2. The
points in Figure 2 fall into two elongated groups characterized by two straight-line segments: one, specific to nearthreshold friction velocities on January 26 and four measurements on January 20, indicates an increase in (c e) with u*;
the other, corresponding to the full range of friction velocities 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
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 nearthreshold conditions implies proportionality between saltation 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 relationship between e and u* is satisfying conceptually. In
analogy to a mass moving along a static surface, the efficiency 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
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 velocity 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
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 transport rates are associated with hard, compacted (high transport threshold) snow covers.
Both model and measurements show a similar increase
with friction velocity, though some measurements on January 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 (measured mass flux - modeled mass flux) is -0.01024 kg rn-'
FRICTION VELOCITY u* Cm/sl
s-', with a standard deviation of difference of 0.07295 kg
m-2 s - I
. The scatter about the mean, in part, illustrates the
Fig. 3. Measured and modeled (equation (11)) snow saltation
transport rates plotted against the friction velocity. Measured values difficulty in representing with a steady state model a pheare determined from the measured saltating snow mass fluxes and an nomenon that fluctuates rapidly and triggers an atmospheric
assumed saltation height function for a variet of transport threshold
friction velocities from 0.2 to 0.33 m s-): Modeled values are boundary layer response that may not fit tidily within a
calculated for transport threshold friction velocities of 0.2,0.25,0.3, measurement period. Typical mass fluxes are 0.42 kg m-2
and 0.35 m s-'. Values are 7.5-min averages.
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.
approximately with the inverse of wind speed; Bagnold's
Comparison of the saltation transport rates to other blowmodel incorporates neither of these results, while Ungar and ing snow transport rate measurements demonstrates the
Haffs model incorporates only the first. Saltation models relative contribution to the total flux made by saltation.
derived for sand and soil are not directly applicable to snow, Schmidt [I9861 reported threshold conditions and measured
a consequence of the importance differences between the the blowing snow transport rate with a fabric trap which
cohesive, yet breakable snow surface crystal structure and a stretched from the surface to 0.5 m above the snow surface
bed of discrete, noncohesive sediment grains.
[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
Figure 3 shows values of saltation transport rate deter- transport rate to Schmidt's blowing snow transport rate are
mined from measured mass fluxes and friction velocities for listed in Table 1 for two threshold conditions and three
threshold friction velocities u: from 0.20 to 0.33 m s-I with friction velocities. For low friction velocities, saltation comresults of the model for threshold friction velocities equal to prises 5&100% of transport, while at a friction velocity of
0.20, 0.25, 0.30, and 0.35 m s-'. For given conditions of 0.7 m s-' (roughly u l o = 15 m s-I), saltation comprises as
nonerodible roughness and threshold friction velocities, QSal, little as 8% of blowing snow transport, the rest moving in
is approximately proportional to the friction velocity. For suspension (which includes modified saltation).
near-threshold conditions, relatively higher transport rates
correspond to lower threshold friction velocities; for wind
speeds well above the threshold, higher transport rates Drift Density of Saltating Snow
correspond to higher threshold friction velocities. The crossThe mean mass concentration of snow in "pure" saltaover between these two conditions occurs at friction veloc- tion, referred to as the saltation drift density, is a parameter
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
TABLE 1. Ratios of the Saltation Model Transport Rate to
Schrnidr's [I9861 Blowing Snow Transport Rate
(2) the proportionality between saltation velocity and threshold friction velocity. At low wind speeds the shear stress is
Threshold Friction
small, and a significant proportion is exerted in shattering
Velocity
snow cover crystals to effect particle ejection. Soft snow
u; = 0.2
u; = 0.3
covers require less stress than hard snow covers to shatter
m s-I
m s-'
intercrystal bonds; hence for soft, low-threshold snow covers the excess shear stress at low wind speeds is sufficient to Friction velocity rr* = 0.35 m s-'
0.53
1 .OR
result in relatively higher transport rates. At high wind Friction velocity u* = 0.50 m s-'
0.18
0.41
0.08
0.15
speeds the shear stress is large in comparison to its threshold Friction velocity u* = 0.70 m s-'
level, and the difference between the shear stresses exRatios are calculated using z6 = 0.2 mm and Schmidt's C = 5.0 for
tracted by shattering the bonds of soft and hard snow covers two threshold friction velocities and three friction velocities.
is less important. In this case the lower saltation velocity
'Saltation model transport rate value exceeded Schmidt's blowpossessed by soft, low-threshold snow covers results in a ing snow transport rate value.
'
',
4
V
'
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 "saltation 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 blown g snow. The saltation drift density, qsallris 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 densities reflect a range of threshold and hence snow cover and
snowfall conditions. For near-threshold wind speeds (within
0.00
0.20
0.40
0.60
0.80
FRICTION VELOCITY u* Cm/s]
0.4 m s-' of u> the saltation drift density increases in
proportion to the friction velocity. The increase in drift
Fig. 5. Aerodynamic roughness height measured during saltadensity with friction velocity is attributed to the rapidly tion and that modeled, plotted against the friction velocity. Values
diminishing- rate of increase of the ratio of shear stress are 7.5-min averages.
available for transport to mean saltation height as the friction
velocity increases from its threshold level and a possibly
sponds to reference drift densities that are necessary for
steady saltation efficiency near the threshold condition.
calculation of the suspended component of blowing snow, as
More commonly, well above the threshold condition, drift
suggested by data shown by Mellor and Fellers [I9861 and
densities vary independently of friction velocity and cluster
Pomeroy [1988].
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 Calculation of Saltation Transport From
speeds. Threshold wind speed is associated with both the Meteorological Observations
For conditions with complete snow covers and no exposed
surface snow condition and the saltation velocity; hence high
drift densities are associated with less cohesive, fresh snow vegetation the transport rate of saltating snow may be
covers and snowfall and with low saltation velocities, the calculated from standard meteorological wind speed meainverse being true for low drift densities. In demonstration, surements if an aerodynamic roughness height can be specfor winds well above the threshold level a mean drift density ified. Owen [I9641 suggests that the roughness height above
of 0.70 kg m-3 is associated with a threshold friction velocity a saltating flow is proportional to the square of the friction
of 0.2 m s-I, and, similarly, 0.51 kg m-3 with 0.23-0.27 m velocity, and Tabler [I9801 confirmed such a proportionality
s-I and 0.48 kg m-3 with 0.27-0.33 m s-I. This variation of for blowing snow over lake ice. Measurements from comdrift density with threshold condition suggests a mechanism plete snow covers over fallow land (Figure 5) show a
where the snow surface condition might affect diffusion of relationship between roughness height and friction velocity
blowing snow. The range of values within the cluster corre- 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 relationship to the data provides the expression
2
1
Cluster u w e r bound
~~7
7 .
x
E
c.00
0.10
0.20
0.30
7
xX
x~
XY
cluster lower b i n d
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.
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 nearthreshold 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-
bearing on the orientation, location, and size of drifts around
barriers. A practical application of the model in determining
0
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
0.00
2.00
4.00
6.00
8.00
10.00 12.00 14.00 16.00
cover over the 1000-m fetch calculated by a blowing snow
10-rn WIND SPEED ulo [rn/sl
transport, sublimation, and erosion model [Pomeroy, 1988,
Fig. 6. Measured and modeled (equation (14)) snow saltation 19891. Hourly values are summed over the winter; annual
transport rates plotted against the 10-m wind speed. Values are values presented are averages of the winters 1970-1976.
7.5-min averages.
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
cussion of snow cover development and characteristics on cultivated to spring wheat and other cereal grains. The
frozen lakes.
primary land surfaces surrounding Regina during the winter
Use of (3) and (12) gives u* in terms of the 10-m wind are stubble (harvested stalks of grain), which comprises
speed, ulo, as
-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
For a complete snow cover without exposed vegetation, u*, of Regina in an oilseed and cereal grain growing region on
= 0, and a central measured value for u: is 0.25 m s-I.
the border between the mixed grassland and deciduous
Substitutingg = 9 . 8 m ~ - ~ ,1 .~2 k=g m - ~ , u : = 0.25ms-I, forest "Parklands" and the southern fringe of the boreal
u*, = 0.0 m s-', and u* as defined in (13) into (1 1) results in 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 equivwhich provides an approximation to snow saltation transport alent, are 115 and 110 mrn, respectively; and the mean
using 10-m wind speeds over nonvegetated plains during annual saltation transports, the average over 1000-m fetches
drifting o r blowing snow events.
of stubble and fallow, expressed as a percent of the annual
The results of (14), and saltation transport rates from snowfall, are 16 and 7%, respectively.
measured mass fluxes and friction velocities, are plotted
Figure 7a shows that winds at Regina occur from the NW,
against the 10-m wind speed, calculated from an extrapola- W, SE, and E directions 75.4% of the time and these winds
tion of measured vertical profiles of wind speed, in Figure 6. produce, on average, 87.8% of the annual saltation transport
Both measured and modeled transport rates show approxi- (Figure 7b). The shapes of roses of wind and saltation
mately linear increases with 10-m wind speed. Defining transport are similar, with the exception of the S and SW
"measurement" as the measured mass flux multiplied by the directions, where winds occur with a frequency of 14.1% but
saltation height (9) determined from measured friction ve- produce only about 2.7% of the saltation transport. The S
locities and "model" as the result of (14) calculated using the and SW winds at Regina often bring the "chinook" and a
corresponding 10-m wind speeds, we find that the mean midwinter warming, resulting in the low transport values for
difference (measurement - model) is 0.00045 kg m-I s-',
those directions. At Regina the most effective orientation of
with a standard deviation of 0.00191 kg m-' s-I. For wind a snow control measure which traps the saltating load would
speeds greater than 10 m s-' the mean difference is less than be perpendicular to the direction of the dominant prevailing
6% of the transport rate, and the standard deviation is less winds, namely, SW to NE.
than 25%. The roughly linear relation means that this equaOne should not assume matching symmetry in the direction may be applied for a variety of averaging times, given tional distributions of saltation transDort and wind. Wind
that steady state boundary layer conditions are present.
speeds may have a directional bias, as also may snowfall,
snow depth, snow surface cohesion, and other factors which
Orientation of Snow Control
affect saltation transport. The roses of wind frequency and
and Management Works
saltation transport at Prince Albert (Figure 8) exhibit major
Takeuchi [I9891 has shown that the size of drifts upwind of departures in their distribution patterns. For example, winds
snow fences is sensitive to the saltation rather than total from the N E occur with a frequency of 13.1% and produce
blowing snow transport to these fences; hence the direction 18.5% of the annual saltation transport, whereas those from
and magnitude of saltation transport over a winter have a the SW, occurring with approximately the same directional
0
0
0
R E G I N A W I N D D I R E C T I O N A L FREQUENCY
NORTH
<NOV.
TO A P R . )
<a. 6 % )
NU
cia. 8%)
WEST
(17. 2 % )
EAST
SE
< l a . 1%)
<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%)
NU
<25. 8 % )
WEST
( 1 8 . 1%)
sw
<l.OX)
I
NE
(3.8 % )
-
EAST
SE
(21.7%)
< 2 4 . 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 transport, 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 directions 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 "trapping" 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 measurements 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 friction 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 efficiency 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-
P R I N C E ALBERT WIND D I R E C T I O N A L FREOUENCY
NORTH
WEST
(7. 5 % )
CNOV TO APR)
I
EAST
(10. 1%)
SE
(lB.P%>
CP. 2 % )
SOUTH
(7. 1%)
Fig. 8a
P R I N C E ALBERT SALTATING SNOW FLUX ONLY
NORTH
( 1 0 . 1%)
I
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 (1I ) 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 approximately in proportion to the friction velocity or less approximately 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-'.
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 measured 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
*
4
8
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 because 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.
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-'.
mean
drift density (mass per atmospheric volume)
qsal,
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-'.
-'.
-'
4
.
i
-'
Acknowledgments. T. Brown of the Division of Hydrology,
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
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 Graduate Studies Scholarships and Natural Sciences and Engineering
Research Council of Canada operating grant A4363.
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AND GRAY:SALTATION
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D. M. Gray, Division of Hydrology, University of Saskatchewan,
Saskatoon, Saskatchewan, Canada S7N OWO.
J. W. Pomeroy, National Hydrology Research Institute, Environment Canada, 11 Innovation Boulevard, Saskatoon, Saskatchewan,
Canada S7N 3H5.
(Received January 13, 1989;
revised October 30, 1989;
accepted December 29, 1989.)