INTERACTION

BETWEEN

WIND AND

SNOW SURFACE

S H U N ' I C H I K O B A Y A S H I and T A M O T U I S H I D A 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 sonic anemometers 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 1 m blew at a speed of more than 7 m s-1 after an approximate accumulation of from 10 to 20 cm of new snow on a snowfield. For example, when a snow-wave had a wavelength of 10 m and a wave height of 15 to 20 cm, 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 K o b a y a s h i a n d I s h i d a (1970, 1972, 1974) h a v e s t u d i e d w i n d t u r b u l e n c e d u r i n g t h e t i m e of drifting s n o w using s o n i c a n e m o m e t e r s o n snowfields in H o k k a i d o . W h e n a surface l a y e r of d e p o s i t e d s n o w is b e i n g r e d i s t r i b u t e d b y wind, surface m i c r o r e l i e f s (for e x a m p l e , ripples, sastrugi, s n o w - w a v e s , s n o w - d u n e s , s n o w - b a r c h a n s , etc.) are f o r m e d ( K o b a y a s h i , 1971). In p a r t i c u l a r , a t r a n s v e r s e s n o w - w a v e has a w a v e l e n g t h in t h e d i r e c t i o n p e r p e n d i c u l a r to t h e wind. T h e w a v e l e n g t h is a b o u t t h e s a m e size as the scale of t u r b u l e n c e . D e s c r i p t i o n s a n d classifications of s u r f a c e reliefs f o r m e d by a w i n d a c t i o n h a v e b e e n m a d e b y m a n y i n v e s t i g a t o r s , such as C o r n i s h (1914), H a t a k e y a m a (1936), B a g n o t d (1954), D o u m a n i (1966), O u r a (1966), a n d K o b a y a s h i (1971). This p a p e r has t h e m a i n p u r p o s e of i d e n t i f y i n g t h e i n t e r a c t i o n b e t w e e n w i n d turbulence and snow-wave formation.

2. Classification of Deposition/Erosion Patterns W h e n a surface l a y e r of d e p o s i t e d snow is b e i n g r e d i s t r i b u t e d by the wind, surface m i c r o r e l i e f s , as s h o w n in F i g u r e s l ( a ) - ( f ) , a r e f o r m e d on a snowfield. T h e m i c r o reliefs f o r m e d d u r i n g the t i m e of s n o w drifting can b e classified by d e p o s i t i o n a l a n d / o r e r o s i o n a l p a t t e r n s into t h e f o l l o w i n g f o u r types: (1) D e p o s i t i o n t y p e ( d e p o s i t i o n d o m i n a n t ) : d u n e s , ripples, b a r c h a n s ; (2) E r o s i o n t y p e ( e r o s i o n d o m i n a n t ) : sastrugi, pits; (3) E q u i l i b r i u m t y p e ( d e p o s i t i o n in e q u i l i b r i u m with e r o s i o n ) : ripples, small sastrugi; (4) D e p o s i t i o n / e r o s i o n t y p e ( d e p o s i t i o n a n d e r o s i o n a l t e r n a t i n g ) : s n o w - w a v e s . R i p p l e s , as s h o w n in F i g u r e l ( a ) , a r e small t r a n s v e r s e w a v e s with w a v e l e n g t h s f r o m 5 to 10 c m a n d w a v e h e i g h t s (from t r o u g h to crest) f r o m 2 to 5 mm. F o r m e d at a t i m e Boundary-Layer Meteorology 16 (1979) 35-47. 0006-8314/79/1601-0035501.95 Copyright © 1979 by D. Reidel Publishing Company, Dordrecht, Holland, and Boston, U.S.A.

36

SHUN'ICHI KOBAYASHI AND TAMOTU ISHIDA

Fig. l(a)-(f). Varieties of wavy features. (a) Ripples, (b) Sastrugi, (c) pits (in Antarctica),

WIND-SNOW INTERACTIONS

Fig. l(a)-(f). Varieties of wavy features. (d) Longitudinal dune (in Antarctica), (e) Snow barchan (in Antarctica), (f) Snow-waves.

37

38

St-IUN'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 0.5 cm min -1 Sastrugi and pits, as shown respectively in Figures l(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 cm rain -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 l(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 l(d). They prove the presence of two wind systems in Antarctica: katabatic and cyclonic winds (Ageta 1971; Watanabe t977). 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 cm min -1 when wind speed at a height of 1 m 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 l(f), were formed when a wind blew at a speed of more than 7 m s -1 at a height of 1 m after snow had accumulated to more than 10 cm 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 slow 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 cm min -1. The range of travel speed was from 2 to 10 cm min -1, as shown in Figure 4. These values are larger than those of sastrugi which move at a speed of less than 1.0 cm min -~ as shown in the same figure. The snow-waves had wavelengths from 3 to 15 m and wave heights from

WIND-SNOW

INTERACTIONS

39

E t4 Jan,14

~b'12

1971

Ld t/3

r'~

z

8 6 PERIOD

E20

(1)

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........................

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PERIOD(2)

i i

,

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u

~initial

lever of snow s u r f a c e )

,

_J

w l0 > ILl __1

0

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13 LOCAL

~

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14 TIME

1

15

,I

1

i

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I

~

17

I

11~

,

I

19

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 5 m.

Jcm.14

1971

(WAVE

•

TRAVEL

)

>WIND

14:30 10cm

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f

-'- -..

16:15 .

5 t

-"

I

rn I

:"

.--. (

t

I

4.5m

•

I

Fig. 3. An example of wave travel. Travel speed: 4.3 cm min 1.

5 to 20 cm. In wave troughs, e r o d e d p a t t e r n s like sastrugi were f o r m e d . T h e s e may be related to the p h e n o m e n o n k n o w n as s e p a r a t e d flows.

4. Wind Turbulence During Snow-Wave Formation F l u c t u a t i o n s in the h o r i z o n t a l a n d vertical w i n d velocity c o m p o n e n t s were m e a s u r e d using two sonic a n e m o m e t e r s at the same height of 135 cm for each c o m p o n e n t u n d e r a snow w a v e - f o r m i n g c o n d i t i o n . A s n o w - w a v e was f o r m e d w h e n a wind blew

40

SHUN'[CHI

KOBAYASIqI

AND

TAMOTU

ISHIDA

J a n . 1 4 , 1971 ( SAPPORO )

crn/min.)

10

9 8 c~ hi Ld

7

n

6

u3

.

5

iii

>

o•

4

> <

0.4

o

3 O O

hi

(cm/min. ) t~ 0.5 Ld

O

<

n-

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__J

2

0.2

1 -- :r. . . .

uJ 13_ u')

•

0

Z --m

0.1

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0

~o

rY>

<

5 ,

6 l

WIND

7 i

6 i

9 t

SPEED

10 (m/s)" I ,

-0.1

(U,)

Fig. 4. 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 s p e e d of m o r e t h a n 7 m s -1 at a height of 1 m after an a c c u m u l a t i o n of 20 cm of new snow on a snowfield. T h e s n o w - w a v e had a w a v e l e n g t h of 10 m a n d a wave height of 15 to 20 cm. C a l c u l a t i o n of c o h e r e n c e a n d c o - s p e c t r u m of the h o r i z o n t a l a n d vertical c o m p o n e n t s of w i n d velocity were calculated using p r o c e d u r e s o u t l i n e d by B l a c k m a n a n d T u k e y (1958). L o n g - p e r i o d fluctuations were e l i m i n a t e d by a high-pass filter as follows: 1 yi = x ~ - - - ~ ( x ~ _ m + ~ + 2x~ ~+2 +" • • + ( m rrl

1)xi

l + mx~

+ ( m - 1)xi+l +" • • "+Xi+m--1), w h e r e xi is the fluctuation of either the h o r i z o n t a l velocity ui or the vertical velocity wi, at the ith s a m p l i n g time, i.e., i times A t (At = 0.25 s), Yi is the o u t p u t of the filter, m is the n u m b e r of lags, a n d i = 1 , 2 , . . - N .

In practice, the following

WIND-SNOW INTERACTIONS

41

recurrence formulas are used instead of the above equation to reduce computing time: Y i + I = Yl

1 +Xi+l --Xi - 1 - ~ m

ldi+ 1 ,

Ui+l=Vi--Xi m+2Xi--Xi+m,

where m = 10. If m = 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 H a m m i n g (Blackman and Tukey, 1958) has been used. The smoothed quantities F,(r/), F~(r/), CO,w07), and W~w(r/) are called, respectively, the power spectrum of ui, the power spectrum of wi, the cospectrum and the quadrature spectrum. Using these spectra, the coherence CHO?) and the phase lag Y(r/) are defined in the following equations:

CH O?) =

CO.w(~)2+O~w(n) 2 F,(rt)" F~(rl)

y ( ~ ) : tan-l(Q~w(~)

5. Wind Speed and Air Temperature Profiles A t e m p e r a t u r e 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 O~ at height z above the snow surface is expressed by the following equation: uz : u,/k

. In (z + d ) / Z o ,

where U , = ~ , (the friction velocity), p is the air density (1.4 x 10 -3 g/cm3), k (=0.4) is von K a r m a n ' s constant, Z0 is the roughness parameter, T is the shear stress and d is the zero-plane displacement, so that Uz = 0 on Z = (Zo-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 cm, lowering to 15 cm during the observation period.

6. Change of Coherence and Co-Spectrum During Snow-Wave Formation Fluctuations in wind velocity were recorded on a chart running at a speed of 1 cm s 1 Values of wind speed were read at intervals of 0.25 s, each run having a duration of

42

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I

I

,

KOBAYASHI

,

AND

TAMOTU

ISHIDA

r

# P-012

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(a)

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TEMPERATURE ("C ) Fig. 5(a), (b).

1

{

-1

0

i O 1

, , , l l , J , l 111 2 3 4 5 6 7 8 910 WIND SPEED (m/s) Us

Profiles of temperature (a) and wind speed (b).

2 min, because the drifting snow p h e n o m e n a occur very quickly. An example of coherence during a snow-blowing period, the case of period (1) in Figure 2, is shown 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 10 m, 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 m o m e n t u m 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 m o m e n t u m was dominant. This suggests that wind turbulence and snow-wave formation interact with each other. H e r e it should be noted that there is poor statistical reliability of spectra from a data set only two minutes tong, because of the slow m o v e m e n t of the wave. If a single observed peak, with 1 0 d e g of freedom, is observed to be 8 (cm ~ 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. H e showed that

WIND-SNOW

4

10 . . . .

,

"E

INTERACTIONS

q~-T-

, TTI-~I'-----

,

43

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-10

r

0,01

005

0.1

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FREQUENCY

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t

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9°~ - - ;

(Hz)

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~

0.05 0.1 FREQUENCY

------r

--T

~

r TM

;-;;2

1'

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05 ( Hz ~ - ~

.........

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) I i i i

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.

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0.,01

_1

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1

05

FREQUENCY

( Hz

3

)

Fig. 6. Co-spectrum, coherence and phase angle for covariances g~-under an active wave-forming condition.

vertical m o m e n t u m transfer tends to m o v e upward against a shear for a transverse perturbation, while it tends to m o v e 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 m o m e n t u m against the shear. H o w e v e r , 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. A n o t h e r 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

%

KOBAYASHt

AND

TAMOTU

ISHIDA

# 5-072

iv "E ,a v

up w a r d

f

0

I D -10

-15

6 -200--01

'

' J-~ ~ ' ,I 0.05 01

I

L~

0

_ I 001

k

J

05

I

05 (Hz)

1

( Hz

FREQUENCY

L 0,05

9o'k ............ ~l~ s-°72/

01 EREOUENCY

I

. . . . .

3

I ......

L -

,il 0.01

v 0,05

0.1 FREQUENCY

' vV'l, 1 05

1

( H~ )

Fig. 7. Co-spectrum, coherence and phase angle for covariances u~ under a stable surface condition 40 minutes after an occurrence in Fig. 6.

7. Spacings of S n o w - W a v e s and Scale of W i n d Turbulence After a s n o w drift has f o r m e d , o b s e r v a t i o n s can be m a d e of areal variations. T h e stratigraphic structure of the pit wall of a long trench about 8 m in length along a prevailing wind direction has been studied. T h e analysis s h o w e d that the stratification was caused by alternating erosion and deposition in each portion of the long pit wall. A traverse observation of a snowfield disclosed both e r o d e d and n o n - e r o d e d regions on the s n o w surface, w h e r e u p o n m e a s u r e m e n t s w e r e m a d e of the m e a n distance b e t w e e n two adjacent eroded areas. Each e r o d e d region consisted of groups of small sastrugi. T h e w a v e l e n g t h s of s n o w - w a v e s and the spacings of adjacent e r o d e d regions w e r e nearly of the s a m e order of m a g n i t u d e as the scale of

WIND-SNOW

45

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 (At) of the fluctuating wind speed at time to and to + At, in which R (At) for the small time lag may be approximated by the following equation: R (At)-1-(

A~0t)" ,

where T0 denotes the time taken for the 'largest turbulon' to pass through a measuring point, and m is constant. The value of m has between ½and z3 in the present paper. Thus, the scale of the 'largest turbulon' L is defined by the following equation:

L=--To.a, where a 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

l-= f C,, d(At). o

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

-/

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0 0

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Fig. 8.

23

x

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._J

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t

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m

LL 0

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SPEED

AT

1 10 lm

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HEIGHT;

co

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15 16

0 1

U~ ( m / s )

Relation between w a v e l e n g t h of transverse w a v e 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 wind speed at a height of 1 m above the snow surface rises above 15 m s -1. 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 15 m s -s, whereas longitudinal features (dunes, sastrugi) result when winds rise above 15 m s -1. Observations of drifting snow by Dalrymple (1966) and Y a m a d a (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) M o v e m e n t of a transverse snow-wave is the same as m o v e m e n t 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 m o m e n t u m 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. O n e of the most interesting aspects of this p h e n o m e n o n 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 T o m i o 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

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