13

Bulletin of Glaciological Research ,- (,**0) +-ῌ,+ῌJapanese Society of Snow and Ice

ArticleThreshold wind velocity for snow drifting as a function of terminal fall velocity
of snow particles
Kengo SATOH, Shuhei TAKAHASHI
Kitami Institute of Technology, Kitami *3*ῌ2/*1, Japan(Received September +, ,**/; Revised manuscript accepted November +0, ,**/)

Abstract
The threshold wind velocity VT for snow drifting depends on the snow particle shape and its
surface condition. To explain the e#ect of particle shape on VT, we employ the terminal fall velocity
of snow particles, which, similar to VT, depends on the aero resistance coe$cient. For snow samples
obtained from a natural snow surface, VT was measured by a wind tunnel experiment, and theterminal wind velocity w was measured in a cylinder. In addition, the repose angle was also observed.
For estimating the friction between the particles, VT and w were considered to be dependent on
particle shape. For a dendrite crystal, VT and w were determined to be -ῌ1 m/s and *..ῌ*.3 m/s,
respectively; further, VT tended to be proportional to w. The VT/w ratio of other particles was larger
than that of the dendrite crystals; this is explained by the existence of a bonding force.

+.

Introduction

In Antarctica, snow drifting is an important factor in the estimation of the surface mass balance of anice sheet. Budd et al. (+300) estimated the snow drift
transport rate based on observations of the snow drift
ﬂux. Radok (+311) discussed a vertical proﬁle of drift
density by the di#usion theory of drifting snow
(Shiotani and Arai, +3/-), in which the terminal fallvelocity of the snow particles is important. Takeuchi
et al. (+31/) compiled the threshold wind velocity VT for
drifting snow, which is the wind speed at which snow
particles begin to move. VT varied with air temperature: 2ῌ+* m/s near *῏ and .ῌ/ m/s below ῍/῏. It
also depended on the snow particle shape or the aerodynamic characteristics of the snow particles. The
terminal fall velocity w depends on dynamic characteristics in a similar manner as VT, and w can function as an indicator of VT, although there are limited observations of w for drifting snow particles(Takahashi, +32/). In our study, VT was measured in asmall wind tunnel for snow samples obtained from anatural snow surface, and w was observed by simulating snowfall with the sampled snow particles. In addition, the adhesive force of the snow particles wasmeasured by a repose angle observation.

,.

Principles of the observations

The movement of a snow particle is initiatedwhen the drag force FV exerted by the wind exceeds
the horizontal resistance FS (Fig. +). The horizontal
resistance Fs consists of the friction force Ff and thebonding force between the particle and snow surface
Fb which is the maximum force for bond breaking(Fig. ,a). The force Fv acting on a snow particle can be
expressed as follows:+
FV῎ Cr*SV*,ῌ
,(+)

Fig. + Representation of force acting on snow particles in terms of threshold wind velocity andterminal fall velocity.

14

Bulletin of Glaciological Research+
C r* Sw,῏mg ῌ,(.)

from whichw῏    (/)

If a particle is assumed to be a sphere of diameterd with density r, Eq. / is modiﬁed as
w῏




 

(0)

where w is a function of the root of the diameter d fora constant r. Though V* in Eq. + is the wind velocity at
the height of the particle radius, VT was measured
over the surface at the center of the wind tunnel. In
this case, VT is proportional to V* at the anemometerheight;VT῏KV* ῍

Fig. , Horizontal components of forces acting on aparticle. (a) Fv is the drag force exerted by thewind, and Ff and Fb are the friction force andbonding force, respectively. (b) Horizontal components of forces excluding Fb.

where C is an aero resistance coe$cient; r*, the air
density; S, the cross sectional area of the particle; andV*, the wind velocity at the height of the snow particle
radius. The horizontal resistance FS is the sum of thefriction force Ff, which is not an actual friction force
but can be expressed as Ff῏mmg since it depends on
mg, and the bonding force Fb:
Fs῏mmg῎Fb ῌ(,)

where m is a friction coe$cient (static); m, the mass ofthe snow particle; g, the acceleration due to gravity;and Fb, the bonding force (the maximum force forbond breaking). When a snow particle begins to move,
Fv and Fs are balanced, and the following relation is
deduced:+
Cr* SV*,῏mmg῎Fb ῍,(-)

In order to analyze the aero resistance coe$cientC, we employ w of a snow particle, which is dependent
on C and is measurable. Since the air resistance balances the force of gravity in still air, often decreasing
with su$cient distance, the air drag force also balances the force of gravity, and(1)

Assuming a logarithmic vertical distribution ofwind velocity, K is represented by the surface roughness Z*, the particle radius Z+ and the anemometerheight Z,; K῏ln(Z,/Z*)/ln(Z+/Z*). When Z* is *.*/ mm, Z+῏*./ mm and Z,῏,/ mm (centre of wind tunnel section), K῏,.1.
If we apply this relation to natural wind for thenormal anemometer height of +./ m to /.* m, K acquires a value between ../ and /.*. Using Eqs. -, . and
1, VT is derived asVT῏K  

(2)

Since it is di$cult to measure the aero resistance
coe$cient C in Eq. 2, VT is represented as a function of
w by using Eq. .;VT῏K 


(3)

If Fb in Eq. 3 is considered to be negligible, as
shown in Fig. ,b, VT is reduced toVT῏K  (+*)

Thus, VT is principally proportional to w, asshown in Eq. 3. If we can measure the friction coe$cient m, the bonding force Fb per mass of the particle can be obtained from Eq. 3.

-.

Observation method

The observations of VT and w were performed
daily between 1 and 3 a.m. from + January to ,2
February ,**- at the athletics ﬁeld of the Kitami
Institute of Technology, Kitami, Hokkaido. The observation methods were almost identical to those of
Satoh et al. (,**-). In order to measure VT, a small

Satoh et al.

Fig. - Small wind tunnel used in the measurement ofthreshold wind velocity. Working section; width:/* mm, height: /* mm and length: ,** mm. A particle was procured from a natural snow surface without disturbing it and was placed at the bottom of
the wind tunnel.

wind tunnel (working section; length: ,** mm, width:/* mm and height: /* mm) was used outdoors (Fig. -).
The snow sample (length: ,** mm, width: /* mm, anddepth: ,* mm) was transferred from a natural snow
surface using a case, so as not to disturb it and wasplaced at the bottom of the wind tunnel. The windspeed in the wind tunnel was gradually increased, andVT was measured as the wind speed at which the snow
particles began to move. Wind speed measurements
were carried out at ,/ mm high, leeward end of thewind tunnel, with a thermal thermistor anemometer.
The experiment was repeated ﬁve times and the average of VT was used for the calculations.
The velocity of falling snow and snow particle
size were measured in a cylinder having a diameter of
2 cm and height of +2* cm by using two infrared ray
sensors (Fig. .).
The two sensors were positioned ,* mm apartinside a box below the cylinder, and they measured
the time taken by a particle to traverse the two infrared rays (lapse time). The terminal velocity was obtained from the lapse time, and the snow particle size
was obtained from the reduction in infrared ray intensity. The upper sensor was employed for data logging,while the lower sensor was used to measure particle
size. Because the lower sensor was wider than theupper one, the ‘edge e#ect’, an error in particle size
measurement, which occurs when a particle passes atthe edge of an infrared ray, was avoided. In our system, the cross section of the infrared ray beam used in
the upper sensor was . mm and + mm in width andthickness, respectively, while that of the beam in thelower sensor was +* mm and + mm in width and thickness, respectively.
In every observation, the photomicrograph of the15

Fig. . System for terminal fall velocity measurement.
Particles were allowed to fall in a cylinder. Their
speed and size were measured when they passed
through the two infrared rays of the sensors.

snow particle was taken in a cold room and the particle shape was distinguished.

..

Results

Threshold wind velocity
Observations for VT, w and repose angle (q) wereconducted every morning between + January and ,2February ,**-. In Fig. /, VT and metrological conditions are shown. The time of the observations areindicated by snow particle-shaped symbols on the airtemperature trend. Temperatures during the observations were ῍*.2῎ to ῍,+.0῎, while the air temperature was between ῍,...῎ and -.*῎.
In Fig. /, we suppose that VT is dependent on
particle shape and elapsed time after snowfall. VT was-ῌ0 m/s for the dendrite crystals indicated as * in Fig.
0 and /ῌ+, m/s for the broad branches indicated as ῐ.
In the case of dendrite crystals with rime, indicated as
῏, VT was 0ῌ+- m/s. These valuesῌlarger than thosefor dendriteῌindicate that the snow particles becomeheavier by riming, and the value of VT depends on theamount of rime.
After snowfall, VT increased to +*ῌ+/ m/s in ashort period of + to - days. When the snow particles
experienced melting and refrozen (melting metamorphism), VT reached +/ m/s, which is the wind speed
limit for wind tunnel system. A similar variation wasobserved in w; however, variation was not so large as
VT.
In order to apply the value of VT obtained fromwind tunnel experiments to the usual anemometer
..+

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Bulletin of Glaciological Research

Fig. / Threshold wind velocity, daily snowfall depth and temperature from + January to ,2 February ,**-,
where C is +. The symbols used are as follows: * represents the dendrite crystals; ῍, melting dendrite
crystals; ῎, dendrite crystals with rime; ῏, melting dendrite crystals with rime; ῐ, broad branches; ῑ,melting broad branches; ῒ, broad branches with rime; ΐ, melting broad branches with rime῔, surface
hoar; ῌ, needles; ῕, granular particles. In the name of particle shape, “melting” (shown by solid symbols)
is for the particles, which experienced melting metamorphism, although they were usually refrozen atmeasurement time.

heights in experiments conducted in natural wind, VTshould be multiplied by +.0 and +.3 for anemometerheights of +./ m and / m, respectively. However, theturbulence encountered in natural wind tends tocause snow drifting at a small average wind speed.
These opposite factors compensate each other; therefore, VT in a wind tunnel is close to that in naturalwind. The minimum value of VT for a dendrite crystal
in the wind tunnel experiment was around - m/s andthat in natural wind is around . m/s; their ratio is
approximately +.-. Therefore, VT obtained from windtunnel measurements can be applied to natural drifting snow by using a multiplication factor of around+.-.
.., Terminal fall velocity
The relation between w and particle size areshown in Fig. 0. The data of around +** particles wereaveraged for daily experiments. The lines in Fig. 0 are
based on the relation expressed in Eq. 0, with di#erent
particle densities. C is about *./ for a sphere and +.* fordisc under +*. of Re (Reynolds number), whereas it

may be larger than +.* for a dendrite crystal due to its
complicated shape, and close to *./ for round shapeparticle. C for lines in Fig. 0 was assumed to be + for a
disc. w of the dendrite crystals was primarily *./ῌ*.2
m/s, which is located below an equidensity line ofaround ,/ kg/m-. In the case of the dendrite crystals
with rime, the range of w was *.0ῌ+.+ m/s, which were
distributed on the /*ῌ+** kg/m- lines. For the melting
particles, the range of w was *.0ῌ+.- m/s, which were
distributed between the +**ῌ,** kg/m- lines. The large
w values of the melting particles were due to theirlarge density, which resulted from their sphericalshape. In other words, in the case that a snow particle
melts with maintaining a mass constant, the area ofcross section S in Eq. / decreases with decreasing
diameter, and w increases with an increase in densityr, according to Eq. /.
The relation between VT and w is shown in Fig. 1.
The variation in VT was larger than that in w for theentire data. For the dendrite crystal, the range of VT
was -ῌ1 m/s and that of w was *..ῌ*.3 m/s, and VT
tended to be proportional to w. In this case, the ratio ofSatoh et al.

17

Fig. 0 Relation between terminal fall velocity and particle size. Curved lines of equal density are derived
from Eq.0. Symbols are the same as in Fig. /.

Fig. 1Threshold wind velocity VT and terminal fall velocity w. Symbols are the same as in Fig. /.

VT/w was around 1ῌ+*; this can be explained by using
the friction coe$cient in Eq. +*. For the dendrite
crystal with rime, VT was /ῌ+, m/s and w was *./ῌ+.*m/s. In their relation is more scatter and lager thanthe proportional relation for dendrite crystals. Since
the e#ect of mass increase by rimes is same for VT andw, one of the causes of large scattering VT is relatively

large cohesion due to high temperature when it fell.
For the broad branch, VT was 0ῌ+/ m/s and w was *.0
ῌ*.2 m/s. For surface hoar, granular particles andother melting particles, the VT values exceeded themeasurement limit of +/ m/s. The VT/w ratios of theseparticles were larger than those of the dendrite crystals. This large ratio is probably due to the existence

18

Bulletin of Glaciological Researchof the bonding force, as expressed in Eq. 3.
Repose angle
In order to examine the friction coe$cient m in Eq.
+*, the repose angle, which is the maximum slope
angle created when snow particles pile up, was observed. In order to measure the repose angle, the snow
particles sampled from a natural snow surface wereallowed to fall and accumulate on a disk of diameter 0
cm placed in a box (Fig. 2). The accumulated snow
forms a cone, and its slope angle was measured as therepose angle. The friction coe$cient m is given as..-m῎tanqῌ

Since it is di$cult to separate Fb and the gravity
force mg, the Fb/mg ratio was obtained by substituting VT and w in Eq. +,. Fb/mg is a ratio of the bonding
force of a particle to its gravity force; and we refer toit as ‘normalized bonding force’ in this study. From
the observed VT, w and m values, this normalizedbonding force is evaluated by using Eq. +,. For thedendrite crystals in the evaluation, the m values were(++)

The relation between the repose angle q and threshold wind velocity VT is shown in Fig. 3, where them values are plotted on the upper axis. The curves inthe ﬁgure are lines of equal values of w derived from
Eq. +*. The dendrite crystals exhibited q values of 2*ῌ
3*῏, which were mostly on or below the w῎*./ m/sline. For the melting particles, q was measured as ,/ῌ/*῏ (m῎*./ῌ+.,) and VT was +/ m/s (limit of the measurement), which were largely beyond the lines of theirw values (w῎+ῌ, m/s). The large deviation from thelines of w for the melting particles was due to theexistence of the bonding force Fb shown in Eq. 3.
...

Bonding force
The bonding force Fb is deduced from Eq. 3
VT,Fb
῎ , , ῍mῌmg K w(+,)

Fig. 2 Outline of repose angle measurement. Snow
particles were allowed to fall on a disk, and theslope angle of the cone was measured.

Fig. 3 Relation of repose angle q to threshold wind velocity. Friction coe$cient is plotted on the upper axis.
Curves are lines of equal value of terminal fall velocity w. Symbols are the same as in Fig. /.

Satoh et al.

limited to +* because the accuracy of a repose angle is
not high enough when it exceeds 2/῏. For melting
particles with a VT of +/ m/s, i.e. the limit of the windtunnel experiment, the VT/w ratio was assumed to beapproximately ,*; this was estimated from melting
particles with w and VT values of *.0 m/s and +/ m/s,
respectively. The variation in Fb/mg for the year ,**is shown in Fig. +*. Fb/mg of new precipitation was
smaller than ,*; especially that of dendrite crystals
was usually almost *. After snowfall, Fb/mg increased
and reached /* (a limit of measurement) in one or two
days.
Fig. ++ shows the relation between Fb/mg and w.
For the dendrite crystals, Fb/mg was negligible, except
for two cases in which surface hoar or crystals withrime was slightly mixed. The range of Fb/mg was *ῌ/*for the dendrite crystals with rime and /ῌ+* for the19

broad branches. For the melting particles, Fb/mg waslargeῌgreater than /*.
Though it is di$cult to measure Fb and mg separately, Fb can be estimated as +./῍,./῎+*῍.(N) for a
precipitation equivalent to + mm-diameter ice sphere.

/.

Discussion

Variation of VT and w is considered as follows.
When Fb is negligible small, VT is proportional to w as
Eq. +*. In this case, when diameter and mass changed
with particle metamorphism, VT and w vary in thesame proportional relation. When Fb became large
without change of w, the proportional coe$cient will
increase large as Eq. 3. This concept is shown in Fig.
+,, in which two thick arrows show these two types ofvariation. In the natural condition, these variations

Fig. +* Variation of normalized force Fb/mg, threshold wind velocity VT and friction coe$cient m for
January and February ,**-. Symbols on the Fb/mg line are the same as in Fig. /20

Bulletin of Glaciological Research

Fig. ++

Normalized bonding force Fb and terminal fall velocity w. Symbols are the same as in Fig. /Fig. +, Concept of variation of VT and w was shown. Two thick arrows show these two types ofvariations bonding force and shape change. This arrow shows combination of these variations forparticles after snowfall.

occur simultaneously as shown by thin arrows in Fig.
+,. In our measurement, after snowfall VT and w become large day-by-day, in which especially VT increased rapidly and reached to the measurement limit(+/ m/s) in , or - days. The data distribution of Fig. 1
shows that the dendrite crystal and broad branches
are on a same proportional relation, but dendrite crystal with rime scattered largely. This deviation would
be caused by existence of Fb adding to lager cohesion
under high temperature as described in ..,. Though
the reason of this large Fb cannot be speciﬁed, one of
speculation is that dendrite crystals with much rimes
are heavier and tend to catch snow surface like ananchor, and another one is that air temperature, when
this crystals fall, was relatively high and bonds growrapidly.
In Eq. 3, which shows the relation between VT andw, contains Fb/mg. Though Fb and mg are di$cult tomeasure individually, Fb/mg was obtained from VT, w
and m by Eq. +,. Here m was measured by q. q is sensitive to particle shape. q of dendrite crystals wereabove 2*ῌ and q for melting particles were less than /*ῌ.
In Fig. 1, the proportional coe$cient between VT and
w was about ++, which is ῍ ῌ in Fig. +* without Fb andequivalent to q of 2/ῌ. Thus m from observed repose
angle was suitable to Eq. +* for dendrite crystals andbroad branches. For other particles, Fb is not negligible and VT becomes larger and data scattered asshown in Fig.+,.
For bond-neck strength, Schmidt (+32*) examinedSatoh et al.

a relation between threshold drag forces Fc to break abond of ice spheres. Assuming Fv is same as Fc and Cῌ
+ in Eq. +, a ratio of x/R is estimated from the relationof Schmidt(+32*); when V* is / m/s, +* m/s, +/ m/s, Fc is,.- Pa, 3.* Pa, ,* Pa and x/R can be *.*-, *.*/, *.*1 respectively. Though it is di$cult to observe x/R in the ﬁeld
experiments, this ratio of x/R obtaind above seems tobe appropriate for the particle size and neck radius ofdendrite crystal.

0.

Concluding remarks

The following results were obtained from themeasurements of threshold wind velocity VT, terminal wind velocity w and repose angle q for snow
samples obtained from a natural snow surface.
+) VT values for dendrite crystals, dendrite crystals
with rime and broad branches were -ῌ0 m/s, 0ῌ+m/s and /ῌ+, m/s, respectively.
,) VT for the dendrite crystals subsequent to snowfall
was small, i.e. -ῌ0 m/s; however, it increased to +*ῌ
+/ m/s in a short period of +ῌ- days, accompanying
a change in particle shape.
-) The w values for the dendrite crystals, dendrite
crystals with rime, broad branches and melting
particles were *..ῌ*.1 m/s, *./ῌ+., m/s, *.0ῌ*.2 m/s
and *.0ῌ+.. m/s, respectively.
.) In the case of the dendrite crystals, VTtends to be proportional to w. The VT/w ratio of the other particles
was larger than that of the dendrite crystals, which

21

is explained by the existence of a bonding force.
/) The repose angle q was measured as 2*ῌ3*῍ for thedendrite crystals and ,/ῌ/*῍ for the melting particles.
0) The normalized bonding force Fb/mg between theparticles was greater than /* for the melting particles, while it was negligible for the dendrite crystals.
1) In order to apply the VT values obtained in thisstudy to natural snow drifting, they should be
multiplied by +.-.

References
Budd, W. F., Dingle, W. R. J. and Radok, U. (+300): The Byrd
Snow Drift Project Outline and Basic Results (Part II; S/.
S0), Antarctic Res. Ser. 3, 1+ῌ+-..
Radok, U. (+311): Snow Drift. Journal of Glaciology, +3, +,-ῌ+-2.
Satoh, K., Takahashi, S. and Tanifuji, T. (,**-): Threshold
wind velocity for snow particle movement and its variation with snow surface condition. Journal of The Japanese Society of Snow and Ice, 0/, +23ῌ+30. (in Japanese)
Schmidt, R. A., Threshold wind-speeds and elastic impact insnow transport. Journal of Glaciology, ,0 (3.), ./-ῌ.01,+32*.
Shiotani, M. and Arai, H. (+3/-): A short note on the snow
storm. Second National Congress for Applied Mechanics,
Science Council of Japan.
Takahashi, S. (+32/): Characteristics of Drifting snow at Mizuho Station, Antarctica. Annals of Glaciology, 0, 1+ῌ1/.
Takeuchi, M., Ishimoto, K., and Nohara, T. (+31/): A Study ofDrift Snow Transport. Journal of The Japanese Societyof Snow and Ice, -1, ++.ῌ+,+. (in Japanese)