INTERACTION BETWEEN WIND AND SNOW SURFACE
SHUN'ICHI KOBAYASHI and TAMOTU ISHIDA The Institute of Low Temperature Science, Hokkaido University, Sapporo, Japan
(Received in final form 21 September, 1978)
Abstract. The horizontal and vertical wind velocity fluctuations were measured using two soni<: ane-
mometers at a height of 135 em above a snow surface under a transverse snow wave-forming condition. A snow-wave was formed when the wind at a height of 1m blew at a speed of more than 7 m s- l after an approximate accumulation of from 10 to 20 em of new snow on a snowfield. For example, when a snow-wave had a wavelength of 10m and a wave height of 15 to 20 em, the measured horizontal and vertical velocity components showed that they had a frequency peak of 0.7 Hz in coherence and co-spectrum corresponding to this wavelength. The results suggest that wind turbulence and snow-wave formation interact with each other.
1. Introduction Kobayashi and Ishida (1970, 1972, 197 4) have studied wind turbulence during the time of drifting snow using sonic anemometers on snowfields in Hokkaido. When a surface layer of deposited snow is being redistributed by wind, surface microreliefs (for example, ripples, sastrugi, snow-waves, snow-dunes, snow-barchans, etc.) are formed (Kobayashi, 1971). In particular, a transverse snow-wave has a wavelength in the direction perpendicular to the wind. The wavelength is about the same size as the scale of turbulence. Descriptions and classifications of surface reliefs formed by a wind action have been made by many investigators, such as Cornish (1914 ), Hatakeyama (1936), Bagnold (1954 ), Doumani (1966 ), Oura (1966), and Kobayashi (1971 ). This paper has the main purpose of identifying the interaction between wind turbulence and snow-wave formation.
2. Classification of Deposition/Erosion Patterns When a surface layer of deposited snow is being redistributed by the wind, surface microreliefs, as shown in Figures l(a}-(f), are formed on a snowfield. The microreliefs formed during the time of snow drifting can be classified by depositional and/or erosional patterns into the following four types: (1) Deposition type (deposition dominant): dunes, ripples, barchans; (2) Erosion type (erosion dominant): sastrugi, pits; (3) Equilibrium type (deposition in equilibrium with erosion): ripples, small sastrugi; (4) Deposition/erosion type (deposition and erosion alternating): snow-waves. Ripples, as shown in Figure l(a), are small transverse waves with wavelengths from 5 to 10 em and wave heights (from trough to crest) from 2 to 5 mm. Formed at a time Boundary-Layer Meteorology 16 (1979) 35-47. 0006-8314/79/1601-0035$01.95 Copyright © 1979 by D. Reidel Publishing Company, Dordrecht, Holland, and Boston, U.S.A.
36
SHUN' I CHI KOBA Y ASH! AND T AMOTU ISHIDA
Fig. l(a}-(f). Varieties of wavy features. (a) Ripples, (b) Sastrugi, (c) pits (in Antarctica),
WIND--SNOW INTERACTIONS
d
Fig. l(a)-(f). Varieties of wavy features. (d) Longitudinal dune (in Antarctica), (e) Snow barchan (in Antarctica), (f) Snow-waves.
37
38
SHUN'ICHI KOBAYASHI AND TAMOTU ISHIDA
and place where deposition was dominant, they were transformed into sastrugi (longitudinal features) as wind speed increased. Ripples move at a speed of less than O.Scmmin- 1 . Sastrugi and pits, as shown respectively in Figures 1(b) and (c), were formed when the snow surface was eroded, their alignment being in the direction of the wind. Sastrugi move at a speed of less than 1.0 em min - 1 during their formation period. They are caused by wind erosion. Sastrugi cease to move after they undergo age-hardening and the snow surface becomes stable. Dunes, as shown in Figure 1(d), which are also called whaleback dunes, are associated with strong winds blowing during the passage of a cyclonic snow-storm (blizzard) in Antarctica, at which time much new snow is deposited above the ice sheet. Dunes are aligned in the longitudinal direction. In Antarctica, sastrugi and pits have been etched on dunes in differential angles to the direction of dunes, as seen in Figure 1(d). They prove the presence of two wind systems in Antarctica: katabatic and cyclonic winds (Ageta 1971; Watanabe 1977). A barchan, as shown in Figure l(e), is commonly formed when previously deposited snow is being redistributed by a 'dry' wind like a katabatic wind in Antarctica. Measurements conducted in Antarctica disclosed that a barchan moved at a speed of 3 em min - 1 when wind speed at a height of 1m was 11 m s - 1 . Movement of a barchan is the same as the movement of a transverse snow-wave, i.e., as the movement of a drift. Snow-waves, as shown in Figure 1 (f), will be described in the next section. 3. General Characteristic of Transverse Snow-Waves
The term 'wave' has been applied to a transverse undulation which has the appearance of a sea wave. The snow-waves, as shown in Figure 1(f), were formed when a wind blew at a speed of more than 7 m s- 1 at a height of 1m after snow had accumulated to more than 10 em in thickness on a snowfield. Snow-waves are classified as a deposition/erosion type, i.e., deposition and erosion occur in an alternating sequence. Whether the snow surface had been eroded or deposited was determined experimentally by measuring a change in the snow surface level by a scaled stake. An example of the measured results is shown in Figure 2, in which the snow surface level fluctuated. In. Figure 2, period (1) shows a time sequence during active formation of waves, whereas period (2) shows a time sequence when wave formation was weakened despite the fact that snow was blowing. Period (2) is marked by stabilization of the snow surface, because the income of snow particles is balanced by the export in this snowfield. Movement of snow-waves is observed by ~low motion pictures taken with a 16-mm movie camera. An example of wave travel is shown in Figure 3, a case in which the travel speed is about 4.3 em min - 1 . The range of travel speed was from 2 to 10 em min-\ as shown in Figure 4. These values are larger than those of sastrugi which move at a speed of less than 1.0 em min - 1 as shown in the same figure. The snow-waves had wavelengths from 3 to 15 m and wave heights from
39
WIND-SNOW INTERACTIONS
~
5 14.-.--.--.-.--.--,-,--,--,-,--,--,-,--.--.-.--.--.--.~ :::5
6'
12
(j)
8
Jan.l4
1971
w ~ 10
0
z 6 ~ 4
PERIOD ( I ) - -
E 20
u
-----.:<-----
P E R I 0 D ( 2 ) ______.
__.i(i~it~a~l~l~(>vel_o_f_sno_w_s_ur_face_l_:_. _ __
_J
w 10 > w _J
o~~-L~--L-~-L--~J--L~--L-~~--L_J__L~--~_L__J
10
11
12
13
LOCAL
14
15
TIME
16
17
18
19
20
(hr)
Fig. 2. Time change in the snow surface level. (1) period of active wave formation; (2) period of stable surface. Stake (A) is located 1 m windward from stake (B). This wavy surface has a wavelength of Sm.
Jan.14 1971
(WAVE TRAVEL) ~WIND
5 m Fig. 3.
~4.5m---+
An example of wave travel. Travel speed: 4.3 em min-t.
5 to 20 em. In wave troughs, eroded patterns like sastrugi were formed. These may be related to the phenomenon known as separated flows.
4. Wind Turbulence During Snow-Wave Formation Fluctuations in the horizontal and vertical wind velocity components were measured using two sonic anemometers at the same height of 135 em for each component under a snow wave-forming condition. A snow-wave was formed when a wind blew
40
SHUN'ICHI KOBAYASHI AND TAMOTU ISHIDA
Jan.14,1971 (SAPPORO) 10
(em/min.) 0
9
8 0
w 7 w a.. 6
0
_I
w
5
oo
> a:: 3
<(
0
f-
w
>
<(
3: 0
0 0 __
j
:
• • •
·-·· ·-· 6
WIND Fig. 4.
(em/min.) 0 0.5
0
4
2
,---
0
7
8
0.3
f-
z
0.2 -W ~~
0.1
::JW
0
f-0
-0.1 9 1 o(m/ S)
SPEED
w w a..
0.4
n:>
<( (/)
( u.)
Relation between travel speed of a snow-wave and wind speed at a height of 1 m. In the figure, open and solid circles show the speed for snow waves and for sastrugi, respectively.
at a speed of more than 7 m s- l at a height of 1 m after an accumulation of 20 em of new snow on a snowfield. The snow-wave had a wavelength of 10m and a wave height of 15 to 20 em. Calculation of coherence and co-spectrum of the horizontal and vertical components of wind velocity were calculated using procedures outlined by Blackman and Tukey (1958). Long-period fluctuations were eliminated by a high-pass filter as follows:
1 m
Y; =X; - -2 (xi-m+!
+ 2xi-m+2 + · · · + (m -l)X;- 1 + mx;
+ (m -1)x;+l + · · · + Xi+m-1), where X; is the fluctuation of either the horizontal velocity u; or the vertical velocity w;, at the ith sampling time, i.e., i times .dt (.dt = 0.25 s), y; is the output of the filter, m is the number of lags, and i = 1, 2, · · · N. In practice, the following
41
WIND-SNOW INTERACTIONS
recurrence formulas are used instead of the above equation to reduce computing time: Yi+I =
y;
+x;+! -x;
V;+I = V;- X;-m
1
+-2 V;+I, rn
+ 2x;- Xi+m,
where rn = 10. If rn = 10, the output power falls to 0.35 times the input power at 0.2 Hz, and 0.036 times at 0.1 Hz; for greater input power, at more than 0.4 Hz, there is no change in the ratio of output to input power. In order to obtain the smooth distribution of the spectra, the averaging procedure known as Hamming (Blackman and Tukey, 1958) has been used. The smoothed quantities Fu(r/), Fw(-ry), COuw(71), and Wuw(71) are called, respectively, the power spectrum of U;, the power spectrum of w;, the cospectrum and the quadrature spectrum. Using these spectra, the coherence CH(71) and the phase lag Y(77) are defined in the following equations: 2
2
CH( ) = COuw(71 ) + Ouw(71 ) 71 ' Fu (77) · F w ( 71) Y(77)=tan-l( Ouw(71)). COuw(71)
5. Wind Speed and Air Temperature Profiles A temperature profile under neutral conditions is shown in Figure 5(a). The corresponding wind speed profile over a wavy snow surface plotted on a semilog scale is given by the solid line in Figure 5(b ), i.e., the vertical profile of wind speed Uz at height z above the snow surface is expressed by the following equation:
3
3 where U* = JTj p, (the friction velocity), p is the air density (1.4 X 10- gjcm ), k ( =0.4) is von Karman's constant, Z 0 is the roughness parameter, Tis the shear stress and d is the zero-plane displacement, so that Vz = 0 on Z = (Z0 - d). When the zero-plane displacement is assumed to be zero, the vertical wind profile is no longer a straight line in Figure 5(b) (dashed curve). The best value of d was 20 em, lowering to 15 em during the observation period.
6. Change of Coherence and Co-Spectrum During Snow-Wave Formation 1
Fluctuations in wind velo~ity were recorded on a chart running at a speed of 1 ems - . Values of wind speed were read at intervals of 0.25 s, each run having a duration of
42
SHUN'ICHI KOBAYASHI AND TAMOTU ISHIDA
#P-012
250
2 00
# P-012 1000
(a) '0
+
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500
(b)
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10 0
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- -
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l'l
snow surface
I
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0
I
+
d=20cm
I
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d
-50
-8 -7 -6 -5 -4 -3 -2 -1 TEMPERATURE("C )
Fig. 5(a), (b).
0
10
0
1
2
3
4
WIND
5
6
7
SPEED
8
9 10 11 1 2 ( m/s)
u.
Profiles of temperature (a) and wind speed (b).
2 min, because the drifting snow phenomena occur very quickly. An example of coherence during a snow-blowing period, the case of period (1) in Figure 2, is shov· _ in Figure 6. A remarkable peak is seen at a frequency of 0.7 Hz marked with an asterisk on the figure. The snow-wave had a computed length of 10m, which agreed with the observed wavelength of the snow-wave. On the other hand, when the process of snow-wave formation weakened in the case of period (2), as shown in Figure 2, a frequency peak could not be found though snow was drifting (see Figure 7). The co-spectrum, which is related to momentum transfer, also had a peak at a frequency of 0.7 Hz (marked with an asterisk in Figure 6), corresponding to the wavelength of the snow-wave. In this case, the upward transfer of momentum was dominant. This suggests that wind turbulence and snow-wave formation interact with each other. Here it should be noted that there is poor statistical reliability of spectra from a data set only two minutes long, because of the slow movement of the wave. If a single observed peak, with 10 deg of freedom, is observed to be 8 (cm 2 sec - 2 )/Hz as shown in Figure 6, then we have 80% confidence that the true long-run value lies between 5.0 and 16.3 (cm 2 sec - 2 )/Hz. Nevertheless, there is a physical explanation for the peak frequency which corresponded to the wavelength of the snow-wave observed. Asai (1970) made an investigation of three-dimensional features of a perturbation superimposed in a plane Couette flow with unstable stratification. He showed that
WIND~-SNOW
'u
_.E"
43
INTERACTIONS
10
v
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0
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=>
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c
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"0
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l
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l _____L_t_l-----'-..J
0.5
F RE Q UE NC Y
~;s
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( Hz )
12~1 -------,----,-,----,E
10
05
L-ll ___1________1_____._.1
c=-S-0
~
-----jl_____
~0 --~~~ 001
005
01
FREQUENCY
05
1
3
(Hz)
f'~'·r=~-,---=~---- ~ I
w 30
<(
I
I
L,---"--__L_-'---'~'-LLL--l_._.L.._.l_~-____L
0.01
005
01
FREQUENCY
Fig. 6.
05
1
(Hz)
Co-spectrum, coherence and phase angle for covariances uw under an active wave-forming condition.
vertical momentum transfer tends to move upward against a shear for a transverse perturbation, while it tends to move downward for a longitudinal perturbation. This indicates that a transverse perturbation transforms the kinetic energy of the perturbation to that of the mean flow through an upward transfer of the horizontal momentum against the shear. However, Asai's work would seem irrelevant to the present study because it pertains to unstable conditions, whereas the data over the snow were for neutral conditions. Another possibility may be the 'fluidization of snow' caused by wind action (Kuroiwa, 1975). A snow cloud associated with an avalanche or heavy blowing is an example of fluidized snow, and recently its dynamic behaviour is being investigated very actively (Maeno and Nishimura, 1978).
44
SHUN"ICHI KOBAYASHI AND TAMOTU ISHIDA
upward
#5·072
0
I~ u.
----------~---·--·--~-if=--! ~
-5
0
::r
~
c ~
_, 0
0
"0
u w n. cf)
-15
0
u
FREQUENCY
I Hz I
li~~'~ 001
005
01
FREQUENCY
:~(-; ~~0~7-2
_' .''
05
1
3
(Hz)
~ 1- --~--- -
~ 30~
""
r
n.
o''
I I
0.01
005
01
F R E Q UE N C Y
Fig. 7.
05
I I H• I
Co-spectrum, coherence and phase angle for covariances uw under a stable surface condition 40 minutes after an occurrence in Fig. 6.
7. Spacings of Snow-Waves and Scale of Wind Turbulence
After a snow drift has formed, observations can be made of areal variations. The stratigraphic structure of the pit wall of a long trench about 8 m in length along a prevailing wind direction has been studied. The analysis showed that the stratification was caused by alternating erosion and deposition in each portion of the long pit wall. A traverse observatioR of a snowfield disclosed both eroded and non-eroded regions on the snow surface, whereupon measurements were made of the mean distance between two adjacent eroded areas. Each eroded region consisted of groups of small sastrugi. The wavelengths of snow-waves and the spacings of adjacent eroded regions were nearly of the same order of magnitude as the scale of
45
WIND-SNOW INTERACTIONS
turbulence obtained by the use of a sonic anemometer. According to Inoue (1952), the scale of the 'largest turbulon' of the longitudinal wind speed is derived from the auto-correlation function R (.1t) of the fluctuating wind speed at time t0 and t0 + .1t, in which R(.1t) for the small time Jag may be approximated by the following equation:
(Jt)m ,
R(Llt)-1- To
where T 0 denotes the time taken for the 'largest turbulon' to pass through a measuring point, and m is constant. The value of m has between* and~ in the present paper. Thus, the scale of the 'largest turbulon' Lis defined by the following equation:
L=To·ii, where ii is the mean wind speed at a given height. Here it is noted that the value of L is always much larger than that of the Taylor microscale A (e.g., Tennekes and Lumley, 1972) but that it agrees approximately with the integral scale defined by
/ '=
I
Cuu d(.1t).
()
The results from measurements are shown in Figure 8. The scale of turbulence is in agreement with the scale of patterns marked by erosion on the snow surface, such as
E
'
25 :< w'
>
''
<{
'
...
~
w 20 c..n Q:: w > c..n
z<{
''
15
0
lL..
I
''
•
Q:: 1-
0
0
•
•
0
:::J
:::J
'
0
1-
"'
•
>
'
z
'"
10 ~
"" 0
0
2
3
4
WIN 0 Fig. 8.
0
0
• 0 0
5
0
5
w z w __J
20 u
CD 0::
1(.!)
z w __J w
__J
L
''
•• " '0 • 0" '
10
A.
6
7
SPEED
8 AT
LL
0
""
,5 w __J
"" I
9
' 'I
10
(/)
10 11 12 13 14 15 16 17 1m HEIGHT; U 1 (m/s)
Relation between wavelength of transverse wave or scale of wind turbulence and wind speed at a height of 1 m.
46
SHUN'ICHI KOBAYASHI AND TAMOTU ISHIDA
wavelengths of snow-waves or the distances between groups of sastrugi. In addition the scale of turbulence and the spacings of snow-waves vary inversely with mean wind speed. In particular, the spacings of snow-waves approach zero when the mean 1 wind speed at a height of 1 m above the snow surface rises above 15 m s - • This means that a transverse feature will disappear in a strong wind, because of the predominance of longitudinal characteristics of the wind. Thus, transverse features (ripples, waves, barchans) occur with winds of under 15m s- 1 , whereas longitudinal 1 features (dunes, sastrugi) result when winds rise above 15m s- . Observations of drifting snow by Dalrymple (1966) and Yamada (1974) in Antarctica support this; according to their observations, the threshold wind velocity differentiating between drifting snow (with snow particles moving at low levels) or blowing snow (with snow particles moving at high levels) was 13 to 14 m s- 1 • Similar features have been reported in relation to clouds, i.e., in the presence of weak shear, the clouds are aligned perpendicular to the flow; as the shear grows stronger, the cloud pattern becomes oriented along the flow direction (Mal, 1930). 8. Concluding Remarks The results of a study of some interactions between wind turbulence and transverse snow-wave formation are summarized as follows: (1) Movement of a transverse snow-wave is the same as movement of drift, i.e., deposition and erosion occur in an alternating sequence. The irregularity of deposition is influenced by the scale of the wind turbulence. (2) When a transverse snow-wave is forming, the wind shear near the snow surface is weakened, as shown in Figure 5(b). (3) The momentum transfer has a spectral peak corresponding to the wavelength of the transverse snow-wave. In this case, the upward direction is dominant. (4) The wavelengths of the transverse snow-waves are nearly of the same order of magnitude as the scale of wind turbulence. One of the most interesting aspects of this phenomenon is the similarity with wavelike clouds in the atmosphere and with wind waves in the oceans. Although there is a large literature on wind-waves, as well as theoretical work on flows over sand waves (e.g., Taylor and Dyer, 1977; Kendall, 1970), a theoretical treatment for snow-waves has not been made to date. Acknowledgements Part of this work was supported by funds from Cooperative Program (No. 75157) provided by the Ocean Research Institute, University of Tokyo. The authors wish to express their thanks to Drs Tomio Asai and Ryuji Kimura of this Institute, for their helpful discussions and encouragement throughout the study. They are indebted to Dr Eiichi Inoue of the National Institute of Agricultural Science for his continuing stimulation. The authors also are grateful to the referees for their valuable comments.
WIND-SNOW INTERACTIONS
47
References Ageta, Y.: 1971, 'Some Aspects of the Weather Conditions in the Vicinity of the Mizuho Plateau, East Antarctica', Nankyoku Shiryo (Antarctic Record) 41, 42-61 (in Japanese). Asai, T.: 1970, 'Three-Dimensional Features of Thermal Convection in a Plane Couette Flow', f. Meteorol. Soc. Japan 48, 18-29. Bagnold, R. A.: 1954, The Physics of Blown Sand and Desert Dunes, Methuen, London, 265 pp. Blackman, R. B. and Tukey, J. W.: 1958, The Measurement of Power Spectra, Dover Publications, Inc., New York, 190pp. Cornish, V.: 1914, Waves of Sand and Snow, T. Fisher Unwin, London, 383pp. Dalrymple, P. C.: 1966, 'A physical climatology of the Antarctic Plateau', Antarctic Res. Ser. 9 (AGU) 195-231. Doumani, G. A.: 1966, 'Surface structures in snow', In: Physics of Snow and Ice, Part 2 (H. Oura, ed), Inst. Low Tern. Sci., Sapporo, 1119-1136. Hatakeyama, H.: 1936, 'Snow cover and wind', Oyo Buturi (Applied Physics) S, 588-592 (in Japanese). Inoue, E.: 1952, 'On the Structure of Wind Near the Ground', Bull. Natl.Inst. Agric. Sci., Ser. A 2, 1-9 (in Japanese with English Abstract). Kendall, J. M.: 1970, 'The Turbulent Boundary Layer Over a Wall With Progressive Surface Waves', J. Fluid Mech. 41, 259-281. Kobayashi, S.: 1971, 'Development and Movement of Wavy Features on the Snow Surface During Snow Drifting', Teion Kagaku (Low Temperature Science), Ser. A 29, 81-94 (in Japanese with English Summary). Kobayashi, S. and Ishida, T.: 1972, 'On the Wind Turbulence During Drifting Snow, II', Teion Kagaku (Low Temperature Science), Ser. A 30, 73-84 (in Japanese with English Summary). Kobayashi, S. and Ishida, T.: 1974, 'On. the Wind Turbulence During Drifting Snow III. Example of Observation when Snow-Wave is Forming', Teion Kagaku (Low Temperature Science), Ser. A 32, 81-87 (in Japanese with English Summary). Kuroiwa, D.: 197 5, Mechanics and Structure of Snow as a Dispersed System, proceedings of the Grindelwald Symposium, IAHS-AISH Pub!. No. 114, 3-15. Maeno, N. and Nishimura, K.: 1978 'Studies of Fluidized Snow I, Formation of Fluidized Snow and its General Property', Low Temperature Science, Ser. A 36 (in press). Mal, S.: 1930, 'Forms of Stratified Clouds', Beitr. Phys. Atmos., 17,40-68. Oura, H.: 1966, 'Studies of blowing snow 1',In: Physics of Snow and Ice, Part 2 (H. Oura, ed.), Inst. Low Tern. Sci., Sapporo, 1085-1097. Taylor, R. A. and Dyer, K. R.: 1977, 'Theoretical Models of Flow Near the Bed and Their Implications for Sediment Transport', in: The Sea, Vol. 6, 579-601. Tennekes, H. and Lumley, J. L.: 1972, A First Course in Turbulence, MIT Press, 300pp. Watanabe, 0.: 1978, 'Distribution of surface features of snow cover in Mizuho plateau', Mem. Natlinst. Polar Res. (Special Issue) 7, 44-62. Yamada, T.: 197 4, 'Surface Meteorological Condition in the Region Between Syowa Station and Mizuho Camp, Mizuho Plateau, East Antarctica', Nankyoku Shiryo (Antarctic Record) SO, 1-20 (in Japanese with English Abstract).
