Introduction

When practical questions concerning attenuation of visible radiation by falling snow were raised, cursory search of the literature revealed little relevant information. In view of the complex shape of snow particles a purely theoretical approach seemed undesirable, and so an immediate attempt was made to measure the attenuation of light transmitted through falling snow from a diffuse source. This proved unsatisfactory, since the photometric device available at that time was insufficiently sensitive for work over the necessarily short (~100 m.) transmission path. Interim working estimates of extinction coefficient were therefore made from visibility data for Antarctic blizzards, supplied prior to publication by Radok (see Reference Budd and RubinBudd and others, in press).

During the remainder of the winter (1964–65) simple measurements of visual range were made for a variety of conditions experienced at Hanover, N.H., and some control measurements were made with a telephotometer. These data have provided information on light attenuation by falling snow, and have also given rise to some interesting speculations regarding particle aggregation during snowfall.

It appears that useful studies in this area of interest can be made with very simple equipment, so that the type of work described here may commend itself to field glaciologists as a foul-weather occupation.

Light Scattering and Brightness Contrast

As an alternative to measuring beam attenuation directly, extinction coefficients for falling snow may be determined by considering the reduction of brightness contrast between a dark target and the adjacent horizon sky. Relevant theory, originated by Koschmieder and reviewed by Reference MiddletonMiddleton (1952), Reference JohnsonJohnson (1954) and others, is outlined below.

Consider a cone of vision directed horizontally through a uniform field of snow-filled air. The cone is illuminated uniformly, say by sunlight diffused through a dense overcast. Snow particles in a representative elementary volume of the cone scatter light (independently and incoherently), so that the element has a brightness, or luminance, B _h which is determined by the incident illumination and by the number and nature of the snow particles. B _h is the intrinsic brightness of the snow-filled element, but the apparent brightness sensed by an observer at the apex of the cone of vision is reduced by scattering in the intervening snow cloud. In the cone, luminous flux and brightness do not change with distance from the apex x if the air is perfectly clear; brightness B diminishes with x only because of scattering from snow particles, and attenuation follows the Bouger–Lambert law:

where σ is the extinction coefficient, or, since absorption is probably negligible for snow, the scattering coefficient. It is assumed independent of x. In general σ is a function of wavelength, but since snow particles are very large compared with the wave-length of visible radiation it is expected to be almost independent of wave-length in the visible range. Solution of equation (1) with the condition B = B _h when x = 0 yields:

When the line of sight is directed through an effectively infinite thickness of the snow cloud, i.e. towards the horizon sky, the total apparent brightness sensed by the observer Bah is found by integrating throughout the cone between the limits x = 0 and x = ∞:

When the line of sight is directed instead to a “black” target of zero luminance, horizontal distance X from the observer, the total apparent brightness sensed by the observer B _at is:

If the target is not black, but has an intrinsic brightness B ₀, the apparent brightness sensed by the observer B _at′ is the brightness of the limited cone B _at plus the attenuated brightness of the target itself:

The apparent contrast C between a distant target of sufficient size and the horizon sky is:

The intrinsic contrast C ₀ between the target and the adjacent horizon sky as X→0 is:

C ₀ may take values from −1 for a black body up to positive values for self-luminous targets. Equation (5) can now be re-written in terms of brightness contrast:

or

If we define a visual range V as the distance X between observer and target at which apparent contrast C diminishes to the minimum level distinguishable by eye (termed the liminal contrast ε), then equation (9) becomes a relation between extinction coefficient and visual range. It has been found that under a wide range of daylight conditions the average human eye can discern a target of adequate size (say greater than 1° visual angle) if the apparent contrast exceeds 2 per cent. From this there follows a standard definition of the meteorological visual range V _m as the maximum distance at which the average eye can distinguish a black target of suitable size, i.e. with C _o = −1 and ε = −0.02

Measurements of Visibility and Brightness Contrast

Simultaneous measurements of visual range and bulk snow density (mass of snow per unit atmospheric volume) were made on all occasions when conditions were judged to be suitable according to the following criteria: (i) sky completely overcast, (ii) little or no wind, (iii) snow density steady with time and distance for the duration of an observation (snow equalls were not sampled). Visual range was estimated by using as targets a succession of pine-covered hills and ridges, all of similar height to the observation point. The vertical mass flux of snow was measured by exposing a tray for a timed interval and weighing the catch. The mean vertical component of snow particle velocity was measured by timing the fall of particles through a height of approximately 3 m. at several different locations. With some care and practice this measurement could be made consistently simply by following particles, or groups of particles, by eye and timing their fall with a stop-watch. Snow density was obtained by dividing the vertical mass flux by the fall velocity.

In the latter part of the study period attempts were made to measure number density as well as mass density. When falls were predominantly of simple single particles the number flux could be obtained by exposing a tray for a brief timed period (about l0 sec.) and then counting the particles seen through 10 windows of a template laid over the tray. Snow-flakes and other fragile aggregations of particles usually shattered into their components on hitting the tray, but late in the study it was found that the required flux estimates could be made by exposing a warmed tray for about 5 sec. and counting the water marks. The simultaneous measurements of mass flux and number flux for single particle falls yielded estimates of mean particle mass for various types of snow crystals.

Photometric control readings were made with a Spectra brightness meter, model UB

, with stabilized power source and external galvanometer. The meter accepts light from a 1.5° solid angle. Target brightness was compared with the brightness of adjacent horizon sky for several targets lying at distances of 1.5 m. to 3.1 km. Readings were made using a filter which simulates the spectral response of the human eye, and also with blue (≈0.445 μ) and red (≈0.585 μ) filters. The extinction coefficient was obtained graphically in accordance with equation (9). Results of observations are tabulated in the Appendix.

Attenuation as a Function of Density

This work was originally intended to determine extinction coefficient for falling snow and to relate it to some descriptive index such as snow density or precipitation rate. Before proceeding to this end result, however, it seems desirable to enquire into the justification for such relationships.

Results from standard electromagnetic theory provide a relation which expresses the scattering coefficient for spherical aerosols as the total effective scattering area per unit volume:

where N is the number of particles per unit volume, a is the cross-sectional area of a particle normal to the beam, and K is the scattering area ratio, i.e. the area of wave front affected by a particle divided by the actual particle area a (strictly speaking, particle size distribution should be taken into account by writing the summation

, but here we seek simplicity).

The numerical concentration of particles N in falling or blowing snow is not easy to measure, but the mass density γ can be obtained without too much trouble. It is therefore convenient to express N as a function of γ, at least for simple particle shapes:

where l is a characteristic linear dimension of the particle, γ _i is ice particle density, and A ₁ is a dimensionless constant. The particle cross-section a is proportional to l ², so that equation (11) can be re-written as:

where the geometric coefficient A ₂ is constant for a given shape of particle and the scattering parameter K_l is a function of particle size and shape, though perhaps a weak function in view of the large particle size for snow.

As a preliminary to examination of the data for gently falling snow, we may quickly review existing data on visual range during blizzard conditions.

When cold snow is blown by strong winds the grains are usually equant and often rounded; the particle size and the size distribution at a height of 2 m. or so apparently vary rather little from time to time and place to place. Under these circumstances A ₂, K _l and l are constants, and a linear relation between σ and γ is to be expected. Visibility data for Antarctic blizzards by Budd and others (in press) show that this is indeed the case (Fig. 1), and therefore it is entirely logical to express σ as a function of γ for blowing snow. Since Budd and others give particle sizes it is possible to calculate the approximate value of K_l . If it is assumed that the visibility estimates were in accordance with the meteorological standard,Footnote * i.e. C ₀ = −1 and ε = 0.02, values of K _l between 1.8 and 2.7 are obtained, depending on how l is defined for the non-spherical particles.Footnote † These values may be compared with the limiting value of 2 given by Mie theory (Reference MiddletonMiddleton, 1952; Reference JohnsonJohnson, 1954).

Fig. 1. Extinction coefficient and visual range related to the density of suspended snow. Note that a direct comparison of σ_a for the Antarctic blizzard with σ_m for falling snow brings the data into coincidence for the mid-range of densities

When snow is falling in relatively calm weather, the size and shape of crystals and flakes vary widely with atmospheric conditions, and it would be surprising if σ remained simply related to γ. However, while A ₂ K _l /l is not likely to be constant, there could still be linear correlation between σ and γ if A ₂ K _l /l does not vary systematically with γ. The data of Figure 1 dispel this possibility; any relation between σ and γ is clearly non-linear. A first guess from the logarithmic plot is that σ might be proportional to γ ⁿ , where

.

The difficulty at this point is that there is no prima facie evidence for a simple relation between γ and the total scattering area Na. It becomes necessary to consider how particle size and concentration vary for snow-storms generally.

Concentration and Aggregation of Falling Snow Particles

As a starting point for this discussion, we explore the implications of the power relation between σ and γ mentioned above. The most clearly defined limit of the data in Figure 1 gives a maximum value of the exponent

, and if equation (13) is appropriately rearranged we see that this value implies that , and hence from equation (12) , is constant, or at least does not vary appreciably with γ. In Figure 2, is plotted against γ, and it can be seen that overall there is no significant correlation. With the exception of one set of observations the values of vary rather little over the observed range of conditions. Since is a relatively weak function there is greater variation in the values of N, but nevertheless it seems legitimate to consider whether N is kept within certain limits by some mechanism.

Fig. 2.

plotted against snow density

First of all, it may be recalled that nucleation of ice crystals from supercooled cloud droplets usually requires freezing nuclei, which are typically kaolinite particles from terrestrial dust. In some geographical locations the concentration of freezing nuclei may be fairly constant over the winter period, although the efficiency of nucleation apparently depends to some extent on temperature. If N were held between fairly narrow limits by availability of freezing nuclei, then the density of falling snow would be determined by the mass of the individual particles. Crystal form and size depend on temperature and degree of super-saturation of the atmosphere, and in general there is a correlation between air temperature at an observation station and the size of snow crystals encountered there. Intensity of snowfall also tends to be related to air temperature, but it is not known whether there is a correlation between particle size and intensity of snowfall. Using the very rough data from the present observations, and checking estimated crystal masses against Reference MasonMason’s (1962) data for generally larger crystals, particle mass has been plotted against snow density in Figure 3. There is little apparent correlation, although a systematic trend might still be found with more precise data and wider sampling. Group averages of the data suggest a slight increase of particle mass as density increases.

Fig. 3. Mean mass of single snow particles plotted against snow density

So far no mention has been made of particle aggregation, although light scattering depends on the concentration of discrete particles rather than the absolute number of crystals present. Winters at Hanover are cold, and heavy snowfall with large flakes is quite rare, but on one occasion when large snow-flakes fell with temperatures near the freezing point the concentration of flakes was relatively low—about two orders of magnitude smaller than typical concentrations for snowfall in the cold weather. Visual range on this occasion was abnormally great for snow of that density. We should therefore give some thought to the collision process by which flakes and other aggregations are formed.

Consider a large unit volume which encloses many falling particles and which moves downward at the mean fall velocity of the enclosed particles. In general, neighbouring particles move relative to each other; size, shape, and hence fall velocity, of individual particles vary, and aerodynamic instability of planar crystals causes “fluttering” oscillations. Free-air turbulence is probably unimportant, as it tends to produce in-phase displacement over relatively large volumes. A mean free path for the particles L may be defined as:

where ā _c is a mean “collision cross-section” for the particles. This will probably correspond closely to the geometric cross-section, although it could be defined to accommodate the effects of any net electrostatic attractions.

We introduce a “wandering velocity” v , which is the deviation from the mean fall velocity of all the particles, determined largely by the shape and size distribution of the particles. The frequency of collision for a particle f is therefore:

When two particles collide they may simply rebound, or they may adhere to form a single larger particle. The probability of adhesion will be determined by the geometry of the colliding grains and by the physical state of their surfaces; intricate dendrites will readily interlock, and warm crystals with high specific surface or extreme local curvature will adhere tenaciously. To express the proportion of collisions which result in adhesion we introduce an “aggregation efficiency” E.

It is now possible to write an expression for the rate at which particles are removed in effect from the cloud by collision and adhesion:

L is actually a function of N, but we now assume that the collision cross-section of two adhering particles is the sum of their individual cross-sections and so, since N a _c is constant, L becomes constant for a given snowfall.

If particles fall from their parent cloud with initial concentration N ₀ at time t = 0, the solution of equation (16) is:

Hence the concentration of discrete particles N diminishes exponentially with time during fall and the rate constant for the decay depends on the size of individual particles, on the “wandering velocity”, and on the aggregation efficiency.

An interesting aspect of equation (17) is the way in which N at ground level varies with initial concentration N ₀ when other parameters are held constant; no matter how abundant the particles at the cloud base, in general the concentration at ground level is limited. This is illustrated in Figure 4 for a range of values of

chosen to conform with the following estimates of magnitudes: E ~ 10⁻¹ to 1, to 10⁻¹, , t ~ 10³ sec. On these assumptions there is virtually no aggregation of very small particles during a typical fall, but large stellar dendrites aggregate so fast that the concentration of resulting snowflakes at ground level can never be very high.

Fig. 4. Concentration of discrete particles at ground level (N) as attraction of particle concentration at the base of the parent cloud (N_o), according to equation (17)

The mid-winter snowfalls observed in Hanover typically had spatial dendrites of about 0 5 mm. diameter, and assuming these to have an aggregation efficiency of about 0.8 the corresponding value of

is 20. For this condition interpolation on Figure 4 shows only a slow change of N with N ₀; the actual curve (not plotted in Fig. 4) traverses the range of observed values for N (5×10⁻³ to 15 × 10⁻³ cm.⁻³). Since mass density γ is a measure of N ₀ for a given type of snow, we thus have a possible explanation for the apparent constancy of N with γ which prompted this discussion.

For the rare falls of large (5 mm.) stellar dendrites mentioned above, the value of

is about 10⁻³ cm.⁻³. Although the curve for this value is unplottable on the scale of Figure 4, it gives the maximum concentration N after a fall of 10⁻³ sec. of order 10⁻⁴ cm.⁻³, which may be compared with the observed snow-flake concentrations of 2 to 4×10⁻⁵ cm.⁻³.

Presentation of Light Attenuation Data

In view of the foregoing it seems unprofitable to fit an empirical regression line to the data of Figure 1. In fact, for practical purposes there seems to be no special merit in relating extinction coefficient to snow density if a wide range of snow types has to be accommodated. Since fall velocity does not vary greatly (Fig. 5), it is just as satisfactory, and much more convenient, to relate extinction coefficient to the more easily measureable precipitation rate. This procedure is, of course, only applicable when snow is falling directly in calm weather.

Fig. 5. Histogram giving frequency distribution for snow particle and snow-flake fall velocity (covering all observations made at Hanover, N.H., during winter of 1964/65)

Figure 6 shows the correlation between visual range and snow accumulation rate and also between extinction coefficient and snow accumulation rate. The regression line yields the relations:

Fig. 6. Extinction coefficient and visual range related to snow accumulation rate for snow falling in wind free conditions

where V _m is in km. and A, the snow accumulation rate, is in g. cm.⁻² hr.⁻¹;

where σ ₀, the extinction coefficient according to equation (10), is in km.⁻¹ and A is in g. cm.⁻² hr.⁻¹. The extinction measurements made photometrically show that σ ₀ is actually about twice the true value for extinction coefficient; for the range of conditions checked by control measurements the true extinction coefficient σ _m follows the relation:

The exponent in (19) and (20) is approximately equal to

, but this does not appear to have any obvious dimensional significance.

It should be noted here that visual range and extinction coefficient become largely dependent on other factors, notably water vapour, when snowfall is very light.

The discrepancy between σ _o and σ _n, is apparently due mainly to a gradient of sky brightness above the hill-tops used as targets. Contrast was measured between the target and the sky a few degrees of elevation above it, while the visual estimates depended on contrast between the target and the sky immediately adjacent to it. Measurements showed that sky brightness immediately above a hill-top was significantly lower than that a few degrees higher, probably because of the low albedo of the broad hill-top. This effect is roughly comparable to the “water sky” phenomenon known to polar navigators. A less important factor was the inherent contrast of the targets C ₀; instead of the assumed value −1, C ₀ was measured at values close to −0.9 for each of the three filters.

Targets of small horizontal extent should not disturb the sky brightness, and therefore the values of σ ₀ obtained from the data of Budd and others are probably close to the true values. When corrected coefficients σ _m from this study are compared with the σ ₀ values for the Antarctic blizzard, they are of the same magnitude in the range of snow densities common to both studies.

The simple photometric measurements gave no significant correlation between extinction and wave-length in the narrow range observed (≈0.4 μ−0.6 μ).

Acknowledgements

The writer is grateful to Dr. Andrew Assur for drawing attention to this problem and for encouraging the work described here. He is also indebted to Dr. Uwe Radok and Mr. William Budd for a review of the manuscript.