Weather conditions that determine snow transport distances at a site in Wyoming

Ronald D. Tabler and R.A. Schmidt

Rocky Mountain Forest and Range Experiment Station, Laramie, Wyoming, U.S.A.

and Rocky Mountain Forest and Range Experiment Station,

Fort Collins, Colorado, U.S.A.

ABSTRACT: A mathematical model for the sublimation of wind-blown snow has recently been published. In a simplified form, the model predicts the distance that a particle of given size will travel before completely sublimating; critical variables are particle speed, relative humidity and temperature of the air, and total insolation. Measurements of these conditions, at a site in southeastern Wyoming (elevation 2500 m) during all drifting events over the 1970-71 winter, indicate average transport distances of 460 and 900 m, for particle diameters of 0.010 and 0.015 em, respectively.

RES~~: On a publie recemment un modele mathematique de la sublimation de la neige poussee par le vent. Apres simplification, le modele donne la distance qu'une particule, de dimensions donnees, parcourera avant sublimation totale; les variables critiques sont: la vitesse de la particule, l'humidite relative et la temperature de l'air et !'insolation totale. Des mesures de ces conditions, faites en un point sud-est du Wyoming (altitude 2500 m), pendant les periodes de grand vent de l'hiver 1970-71, indiquent des distances moyennes de 460 et de 900 m pour des particules de diametre de 0.010 et de 0.015 em respectivement.

INTRODUCTION

The design of snow fence systems, either to control blowing and drifting snow on highways or to increase water yield from wind-swept areas, requires an estimate of the amount of blowing snow at the fence site. One_proposed method [1] requires the average distance of snow transport, Rm, defined as the distance the average sized snow particle will be transported before it completely sublimates. Schmidt [2] has recently published a mathematical model for sublima­tion rate which can be used to calculate this transport distance if particle size and speed, air temperature and relative humidity, and total insolation are known.

To apply this model at a snmdence site in southeastern Wyoming, air temperature, relative humidity, windspeed, and total solar radia­tion were recorded throughout the 1970-71 winter. A recording snow flux gauge indicated the time and duration of snowdrifting events for which weather conditions were analyzed. Particle sizes were not measured, but comparative values of Rm were calculated for the most probable range of particle sizes.

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SUBLIMATION MODEL

Sublimation of a snow particle suspended in a turbulent air stream results from a vapour pressure gradient across the boundary layer surrounding the particle. Rate of sublimation depends on how rapidly air is exchanged in this boundary layer. Energy, in the form of heat, is transferred to or from the particle surface by conduc­tion, convection, and radiation. Water vapour is transferred by diffusion and convection. Schmidt [2] demonstrates that the express­ion used by Thorpe and Mason [3] to quantify sublimation from an individual ice sphere applies to particles within a cloud of blowing snow as well, since the interparticle spacing exceeds the depth of particle boundary layers under natural concentrations of snow suspend­ed in turbulent air. For a spherical particle with diameter, x, sub­limating in an environment with relative humidity RH at temperature, T, the rate is

(dm/ dt)x [

RH ) 1 [1

sM ) 1lx(Nu) 1 - 100 + KT RT - 1 Q

1 [1 M ) 1 K~ R~ - 1 + -D-­

PsT

( 1)

The latent heat of sublimation (Ls), thermal conductivity of air (K), diffusivity of water vapour in air (D), and saturation water vapour density at temperature T, denoted by PsT• are all temperature and pressure dependent. The universal gas constant R, and M, the molecu­lar weight of water, are constants. Particle ventilation is account­ed for by Nusselt's number (Nu) = xK/D. Heat transfer by radiation is the term Q. The value of RH, in this case, is the percentage of water vapour with respect to saturation over ice.

Several approximations can be made to simplify the expression for (dm/dt) . To begin, let the pressure be considered constant at some elevatfon, so that the denominator of dm/dt can be expressed as some function of temperature

f (T) K1rs [L SM - 1] + _1_

RT Dp5T

(2)

At 750mb (mean atmospheric pressure for the study site), this function is approximated by f(T) ~ 6. 6• 103 (T2 - 8.2T + 254) where T is the temperature in °C. In a similar manner

(3)

is well approximated by g(T) ~ 15.82(85.6- T), from -20°C to zero. Thorpe and Mason gave the~rimental equation for ventilation

factor as (Nu) ~ 1.88 + 0.58 lxV/v where V is the ventilation veloc­ity and v is the kinematic viscosity of air. If we assume that the ventilation is simply the particle fall velocity, and further that this fall velocity is a linear function of particle diameter [4], then

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for V ~ 2400x, the Nusselt number is approximated by (Nu) ~ 1.88 (1 + 37x) in the temperature range O>T>-20°C.

For spherical particles, radiation transfer is Q = (~X2 /4)(1- Q) (1 + \)S 0 where S0 is the solar radiation flux in cal/cm2sec. The albedo of the horizontal snow surface(\) may range from 0.7 to 1.0, and if the particle surface albedo (s) is near 0.5 and varies in the same way as surface albedo, then (1- s)(l + \) ~ 1. If we convert from S0 in cal/cm2sec to Sin cal/cm2hr then Q ~ 7·lo-S~x2 s.

With these approximations, an average sublimation rate may be estimated from (4)

(dm) = 8.95·lo-4x

8(1 + 37X

8)(1- RH/100) + 5.22·Io-7(85.6- T)X!s

dt Xs T - 8.2T + 254

for the temperature range 0 to -20°C, at 750 mb. The factors required to evaluate this expression are (1) relative humidity (RH) in per cent, (2) temperature (T) in °C, (3) solar radiation (S) in cal/cm2hr, and (4) the average sublimation particle diameter (Xs) in centimetres.

"Particle life" may be estimated from the sublimation rate (4) and the average particle mass (m). If m = Pp~X6/6 where Xm is the nominal diameter of a particle with average mass, for a steady-state, average sublimation rate, the time in which a particle of average mass would completely sublimate is m/(dm/dt)ys· During this time,

d:ifting snow moves.downwind at ~-average rate (UP)' so that a d1stance may be def1ned by Rm = Upm/(dm/dt)Xs' Th1s corresponds to

the maximum contributing distance defined by Tabler [1], and is here set equal to the distance a particle of average mass would travel at the average drifting velocity before it completely sublimated.

The average velocity of the solid phase (Up) may be defined as the ratio of drift transport to drift content, and some estimate of the relation between this velocity and mean windspeed is available from the Antarctic data [4]. Drift content is the mass of blowing snow contained in a vertical column of the atmosphere. Drift trans­port is the rate at which this mass moves downwind through a vertical plane. If the horizontal speed of the snow particles in a given layer is taken to be the same as the windspeed (thus ignoring any "slip", or particle resting periods during sal tat ion), mean drifting speed can be expressed as

u p

/"n U dz 0 z z

f n dz 0 z

(5)

where Uz and nz are the windspeed and mass concentration at level z. Data published by Budd, Dingle, and Radok [4] for the Byrd Snowdrift Project were used to derive an estimate of mean drift speed. Drift transport (from 1 mm to 300 m) divided by drift content for the same layer was plotted against u10 , the windspeed at the 10-m height (Fig. 1). The fitted equation, expressed in terms of the 2-m wind­speed (assuming U2 ~ 0.84Ulo) is

uP = 1.12(U2 - 5.87) (6)

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With this approximation for Up, the relationship for maximum contributing distance becomes

R m

o.ss7p (u2 - s.s7)x3 m

(dm/ dt)x 5

(7)

This relation raises several questions. First, what is the density of a drifting snow particle? Then, what is the relation between the diameter of a particle with average mass (Xm) and the diameter (Xs) of a particle exhibiting the average sublimation rate? Further, how are both of these related to the average particle diameter (X)? Answers to these questions require detailed examination of the tem­perature and humidity profiles during drifting, and are the subject of further studies. For the present, three assumptions are made:

a) the particle density is that of ice (Pp = 0.916 gm/cm3) b) average particle diameter is equal to average sublimation

diameter (X = Xs) c) the diameter of a particle with avera~ mass i~ equal to 1.1

times the average particle diameter(~~ 1.33X3). This approximation has been revised from that given by Mellor [5], based on unpublished data by Schmidt.

Substituting these assumptions in (7) provides the relationship between weather factors and snow transport distance

R m

o.7ls(u2 - s.s7)x3

(dm/dt)x (8)

where Rm is in metres, and (dm/dt)x is calculated from (4) with average particle size (X). The following portion of the paper leads to an evaluation of this relation for an entire winter season at a specific site.

STUDY AREA AND INSTRUMENTATION

The study site is located on Pole Mountain in southeastern Wyoming at latitude 41°15'N, longitude 105°2l'W, elevation about 2500 m. Mean precipitation from November 1 to April 1 is about 150 mm, most of which is relocated by the wind. Shortgrass vegetation predominates on the gently rolling topography, with trees limited to stream bottoms and some of the higher hills with more favourable moisture conditions. The study area is a 111-acre watershed upon which a 3.8-m snowfence, 396m long, has been erected to increase snow accumulation [1].

Drift flux 0.5 m above the ground surface was sampled by a rec­ording gauge (Fig. 2), as described by Tabler and Jairell [6). The intake orifice of the snowtrap is 3.14 cm2 • A recording cup-type anemometer and wind vane were also located at the study site to mea­sure the 2-m windspeed and direction. A hygrothermograph and pyrheliograph 1 (also at 2m) were located about 2 km from the study

These two instruments were operated by the University of Wyoming, Water Resources Research Institute, as part of another study. The authors gratefully acknowledge the use of these data.

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site, in comparable terrain and at about the same elevation. Data were collected over the entire 1970-71 winter.

WEATHER CONDITIONS DURING DRIFTING

Mean hourly snow transport at the 0.5-m height was determined from the flux gauge record for all driftin~ events. The smallest flux discernible on the chart was 0.3 g/cm hr; 398 hours of drifting were identified, accounting for a total of 618.3 g/cm2 . Figure 3 presents frequency histograms showing how the 0.5-m flux was distrib­uted with respect to size of events, month of the year, time of day, air temperature, relative humidity, total incoming solar radiation, and windspeed.

Mean seasonal values of the contributing distance (Rm) are use­ful in the design of snow control systems [1]. To calculate these values, the mean value for any given weather variable (Y) over the season, can be obtained from

398 l: FY

y = i=1 398 (9)

1:F i=1

where Y is any given mean hourly weather variable and F is the hourly snow transport at 0.5-m height. Mean seasonal values of u2 , RH, T, and S calculated from (9) were 11.8 mps, 72.1 per cent, -7.7°C, and 16 . 0 cal/cm2hr, respectively.

CALCULATION OF SNOW TRANSPORT DISTANCE

To determine a transport distance for a particle with average mass during a drifting event, the weather variables of (4) should be those at the height of average snow transport, ordinarily in the first few centimetres above the snow surface. Measurements of rela­tive humidity this close to the snow surface are difficult to obtain. As a result, there is little information about the relative humidity profile in the first metre above the snow surface during drifting. For the purposes of this test, values of air temperature, relative humidity, and solar radiation at 2 m were used to calculate (dm/dt)X from ( 4).

Particle sizes were not measured. The majority of published values for the size of wind-blown snow particles are in the range 0.008 to 0.05 em. The most frequently reported size is 0.01 em [5]. Rather than attempting to choose a particle size that might be appro­priate for the test site, we calculated Rm for the most likely range of particle sizes (Fig. 4), using the mean weighted values for weath­er variables given by (9). Over the diameter range from 0.006 to 0.02 em, the curve of Figure 4 can be closely approximated by

R ~ 1.057·106 X 1 · 68 m s (10)

for the conditions u2 = 11.8 mps, RH = 72.1 per cent, T = -7.7°C, and S = 16.0 cal/cm2hr. For particle diameters of 0.01, 0.015, and

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0.02 em, snow transport distance would be 457, 899, and 1421 m, respectively.

These values may be compared with the transport distance calcu­lated from snow storage behind the snowfence at the study site. The following approximation used by Tabler [1] assumes that the fence is 100 per cent efficient in trapping the snow that approaches it, and that no barriers to snow movement exist up to the distance ~:

R m (11)

where qf is the total water-equivalent storage per lineal metre of fence, qP. is the total water equivalent storage of precipitation falling airectly on the fence drift, 9 is the "snow transfer coeffi­cient" (defined as the proportion of precipitation relocated by wind), and P is the precipitation received over the accumulation season. Equation (11) yielded a value of about 1150 m for the average trans­port distance. As the fence contained less than 60 per cent accumu­lation at the peak time, the assumption of 100 per cent trapping efficiency may not be too restrictive, and the assumption of no up­wind barriers is valid at this site.

CONCLUSIONS

The mathematical model for sublimation rate of snow particles (4) can be used to calculate the transport distance of drifting snow once the air temperature, relative humidity, solar radiation, parti­cle speed, and particle size are known. Before precise determina­tions of Rm will be possible, additional research must determine par­ticle speed as a function of windspeed, and must also provide infor­mation to improve estimates of the relative humidity in the first few centimetres above the snow surface.

REFERENCES

[1) TABLER, R. D. (1971). Design of a watershed snowfence system, and first-year snow accumulation. Western Snow Conference, Billings, Montana, Proc. 39: pp. 50-55.

[2) SCHMIDT, R.A. Jr. (1972). Sublimation of wind transported snow -- a model. USDA Forest Serv. Res. Paper RM-90. Rocky Mt. Forest and Range Exp. Sta., Fort Collins, Colorado, 24 pp.

[3) THORPE, A., and MASON, B. (1966). The evaporation of ice spheres and ice crystals. British J. Applied Physics 17: pp. 541-548.

[4) BUDD, W.F., DINGLE, R.J., and RADOK, U. (1966). The Byrd Snow Drift Project: outline and basic results. In: Studies in Ant­arctic Meteorol., Antarctic Research Series:-vol. 9. Amer. Geophys. Union, pp. 71-134.

[5] MELLOR, M. (1964). Blowing snow. Cold Regions Science and En­gineering. Part III, Sect. A3c. U.S. Army Materiel Command, Cold Regions Res. and Eng. Lab., 79 pp. Hanover, New Hampshire.

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[6] TABLER, R.D., and JAIRELL, R.L. (1971). A recording gage for blowing snow. USDA Forest Serv. Res. Note RM-193, 1 pp. Rocky Mt. Forest and Range Exp. Sta., Fort Collins, Colorado.

cn20r------r------,,....-----r----.------. Q.

E --c Q) -c 0 (,) --·.:: '0

' -... 0 Q. en c

15

10

e 5 -0 -0 -II a.

l:::l 0 5

Up=0.94U10-6.57 (r=0.93)

10 15 20 u,o= wind speed at 10-metre height (mps)

25

Fig. 1. Average drift speed as a function of the 10-metre windspeed from Budd, Dingle and Radok's data [4] for the Byrd snowdrift project

124

,

Fig. 2. Recording drift gauge used to sample flux at the 0.5-metre height

......

"' Vl

N' E ~ 2

Hourly transport rote

0 10 20 30

Oct. Nov.

£Dec. ~Jan.

:::!: Feb. Mar. Apr.

0 10 20 30 40 50 60

1-

'?~ en :::!: ~~ ·.·· ... :::J 14 0

!~ . :X:

0 510

Percent of total transport

2- m wind speed

2-m air temperature

-30~ -2g -2

~ -15 0 -10

!! ~~L +10

Incoming radiation -,------,----

... JQ !----'-, 'Nightime

""" ..... .. E ~ 0 ~

0 10 20 30 40 50

Fig . 3. Percentage dist r ibut ion of se l ected factor s during a ll drift i ng events at the s tudy si t e over the 1970 - 71 winter

U)

f -CD E .. 1,000 -E

Q: -CD (,) c:: c -U)

"C -~ 0 a. 100 U) c:: c ~ ....

10 L...L--~----'--------.J 0.001 0.01 0.10

Particle diameter (X,) ,em

Fig . 4. Snow transport distance as a function of particle diameter, using the mean values for winter conditions during drifting

over the 1970- 71 winter (U2 = 11.8 mps, RH = 72.1%, T = -7.7°C, and S = 16.0 cal/cm2hr)

DISCUSSION

M.A. Bilello (U.S.A.) -My first comment is that I believe that in conducting a study of sublimation rates of drifting snow, it would be more important to know the saturation vapour pressure values just above the snow surface rather than the relative humidity th1o r.1etres above the surface

Secondly, snow drifting studies under laboratory conditions have shown that drifting snow flakes break do~~ to smaller particle sizes as they bounce along the surface. This erosional-mechanical process would play an important part in your study on sublimation r ates.

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Have you taken these points into account in your research?

R.D. Tabler (U.S.A.) - We use measurements taken at a height of two metres because the relative humidity of the air at this hei~ht is readily measurabl e using a relatively standard measurement procedure. We realize that intui tively we are really interested in the processes closer to the particles. It must be emphasized, however, that what has been presented is a simp lification of steady-state conditions. It will not apply to short-term estimates. Rather we are using av­erage values of the parameters over a period of time, which, in cer­tain cases, is 1 or 2 days and in others the total season. The long­er the period over which the average values were obtained the better the predicted values are.

R.A. Schmidt (U.S.A.) - We have a study in progress directed to the problem of measuring the humidity profile in blowing snow. One objective of this study is to locate or define a he ight that will provide more representative humidity measurements than those obtained at a height of 2 metres. We anticipate that this height will be less than SO em.

The process of saltation and the resulting mechanical breakage were recognized as important factors that had to be considered in the development of a general description of sublimation in wind-blown snow. These considerations are reflected in our assumption that the ice particles are spherical rather than being similar in shape to actual snowflakes, and in the assumption concerning snow concentra­tion profiles, which include the large flux of snow transported by saltation in the first few metres above the surface.

R.D. Tabler (U.S.A. ) - To obtain meaningful data, weather con­ditions should represent those at the centroid of the drift or the point in the drift profile with equal transport above and below. We would normally consider that measurements should be made within a few centimetres above the snow surface for this height. One can therefore see that our estimates are even more superficial in this respect. However, we feel that for steady-state conditions one can use our method to obtain usuable estimates of snow transport.

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