Introduction
Blowing snow is a very important parameter for modelling the snow stratification and for avalanche forecasting. Several attempts have been made to measure the mass flux, visibility, and particle-size distributions as a function of wind velocity, height over the snow surface, and weather conditions. Precision measurements have been made by Reference Budd and RubinBudd (1966), Reference MellorMellor ([1960]), Reference FöhnFöhn (1980), and personal communication from W. Pope, with rocket drift gauges. Beside these purely mechanical measurements various sensors were developed to measure blowing snow by photoelectric metering (Reference Landon-Smith and WoodberryLandon-Smith and Woodberry, 1965; Reference WishartWishart, 1965; Reference RogersRogers, 1968; Reference SchmidtSchmidt, 1977). Many of these systems are not designed for remote operation with low power consumptions or do not have outputs proportional to mass flux. For these reasons we decided to build and test a new, simple, inexpensive flux meter.
System Requirements
Measured parameter: point measurement of the mean horizontal mass flux over a ridge.
Accuracy: ±50%, independent of particle size distribution, wind speed, and temperature.
Dynamic range: 0.1-100 g m-2 s-1.
Power requirements: suitable for remote, battery-operated or solar-powered weather stations.
Minimal sampling period: ≤30 min depending on available power.
Output signal: analogue or digital signal proportional to the mean flux during the sampling period.
The system described herein meets all the requirements listed above except during conditions of strong precipitation-induced blowing snow.
Theoretical Considerations For The Design
One of the most important design parameters is determined by the distribution of the mass contribution as a function of the particle size for typical particle-size distributions. Preliminary results for Alpine blowing snow (personal communication from M. Pope) show that significant mass contribution starts for particle diameters larger than 100 μm and decreases again to insignificant values for particle diameters larger than about 800 μm. This size range corresponds to a volume or mass range of 1 to 512. It is important to notice that small changes of the percentage of large particles in the particle-size distribution cause significant changes of the mass flux. The shapes of the particle-size distributions depend on snow-surface, wind, and precipitation conditions. For these reasons we decided to design a flux meter which at least roughly classifies the particle masses to get results independent of size distribution. Unfortunately photoelectric metering allows only a classification according to the projected area of the particles instead to their volumes or masses. The mean masses have to be calculated from the measured projected areas. Figure 1 shows that for identical projected mean areas of a sphere of radius r0 and a cylinder of height h = nr0 , the corresponding volumes may differ by about two orders of magnitude. Considering spheres as the mean particle shape causes an overestimation of the real flux. The evaluation of particle shapes by Pope (in press) shows that for blowing snow in the absence of precipitation the particles are almost spherical (ratio of length of axis of inertia less than about three). Blowing snow during precipitation contains plates, fractured dendrites, needles, or even aggregates of crystals. To get some idea on the contribution of the horizontal mass flux from precipitation alone we may start with a vertical precipitation flux of 1 g m–2 s–1 (vertical velocity of the crystals 0.5 m s–1, accumulation 0.05 m of new snow per hour). For a mean wind velocity of 15 m s–1 the precipitation-induced horizontal mass flux reaches 30 g m–2 s–1. This flux corresponds to the highest measured mean fluxes (0.5 m above ground) at similar wind velocities without precipitation. This estimation shows that the photoelectrically measured total fluxes (during precipitation), assuming spherical particles, may significantly exceed the real mass fluxes.
Fig. 1. Ratio of the volumes of a cylinder and a sphere (radius r0) with identical mean projected areas as a function of the height of the cylinder h = nr0.
The drift flux decreases rapidly with increasing height above the snow surface. To get consistent drift data at constant height above the snow surface (independently of the total snow depth) the photoelectric sensor is mounted under a wind-roof (0.5 m above ground), located on a ridge. The wind-roof is designed to prevent significant snow accumulation below the gauge during conditions of blowing snow. The concentration factor for the wind-roof is chosen as small as possible to decrease the disturbance of the natural wind profile.
The maximum possible particle flux is also an important design parameter. The maximum horizontal mass flux under conditions of strong drift and precipitation (accumulation 0.1 m new snow per hour corresponding to 3 g m–2 s–1, mean vertical particle speed 0.5 m s–1, horizontal 25 m s–1, gust factor 1.6) amounts to about 600 g m–2 s–1. For a mean particle diameter of 200 µm the corresponding maximum particle flux reaches 5 × 1o8 m–2 s–1.
Fundamentals of the Design
The drifting snow particles cross a 0.02 m long infrared light beam with a rectangular cross-section 1 mm high and 0.5 mm wide (Fig. 2). A highly sensitive photodiode detects the shadows of the particles. The corresponding voltage pulse height is proportional to the projected area of a particle crossing the light beam assuming that the particle dimensions are smaller or equal to the width of the light beam. All pulses higher than a certain lowest level are resolved into five classes. To get a good approximation for the real flux and a simple electronic design, the classes are defined according to the following rules: (1) Each class corresponds to a certain length of pulse train. The number of pulses in the train is proportional to the volume of a mean size particle of the class. (2) The discriminator levels separating the five classes have to be set proportional to the corresponding particle areas. The relative classification chosen is given in Table I.
Fig. 2. The flux meter in position.
The electronic design is described in the Appendix.
Table I. Particle classification rules
Power requirements
The sensor is designed for battery operation. The power requirements depend on the sampling rate. If n is the number of samples per minute, the mean supply currents are: +12 V, (2+3n) mA; —12 V, 1n mA. For a six months operating period, the following battery specifications are required: +12 V, 5(2+3n) Ah; —12 V, 5n Ah. In our case, the flux meter is integrated in a remote station, powered by solar cells. The data are transmitted by radio to the Institut.
Limits of accuracy determined by the design
Figure 3 shows approximate fits to typical particle-size distributions f(d) (where d is the particle diameter) measured by M. Pope (personal communication in 1979). The data may be very roughly fitted by the sum of two In-normal distributions (dashed curves). The distributions show long tails toward large particle sizes. In the plots the mass contribution curves d 3 f(d) are calculated from the size distributions f(d). The histogram overlay corresponds to the classification. The bars near the tops of the columns show the representative particle volumes of the classes. These values have to be compared with the real means of the classes given by the histogram. The maximum deviation amounts to 13%, the weighted mean deviations for the two distributions are 5% and 1%.
Fig. 3. Approximate fits to typical particle-size distribution f(d). The histogram corresponds to the chosen classification. The bars near the tops of the columns show the representative particle volumes of the classes. The shape of the tail of the particle-size distribution towards high particle diameters depends strongly on snow-surface and wind conditions.
If the particle-size distributions are cut at a particle diameter of 1 mm, the off-range mass contributions are: example 1: 4%; example 2: 20%. Because the sensor registers all particles with diameters larger than 500 µm as 500 µm particles, the real deviations are considerably smaller.
The slit height of only 1 mm causes an additional restriction to the precision of the flux meter. Particles crossing the boundary of the light beam are measured with too small a mass. On the average, the recorded volumes of the particle section in the beam amounts only to 0.735 of the effective section volume (Fig. 4). This number corresponds to the highest possible error. For the particle-size distribution given in Figure 3, the erroneous mean systematic mass reduction amounts to 10%-15%.
Fig. 4. Diagram to show that the mass contribution of particles crossing the boundary of the light beam is too small.
The maximum slit height is limited by the maximum possible particle counting frequency of the system and by the light-beam optics. The light intensity has to be homogenous to within ±5% over the beam cross-section. The beam shape is defined by one limiting slit mounted in Front of the light source. The light from the high-directivity light-emitting diode is collimated and refocussed to the photodiode by lenses (Fig. 5). The maximum possible detectable particle frequency (particles with volumes corresponding to class 2) is limited by the electronics to 6.5 kHz. The sensitive counting area amounts to 2 × 10–5 m 2. Particle fluxes up to about 3 × 108 m–2 s–1, corresponding almost to the maximum expectable particle flux during strong precipitation, are treated correctly.
Fig. 5. Light source.
Calibration
The lowest possible setting of the discriminator DI corresponds to a particle diameter of about 50 µm. Knowing the approximate distribution of the mass contribution (Figs 3 and 4) we set DI to 100 µm. The sensor is calibrated by spinning a wire (diameter 50 µm) through the light beam. The battery-operated motor equipped with radial test wires simulates particle speeds between 5 m s–1 and 35 m s–1 and particle frequencies from < 10 Hz up to > 1000 Hz. The spinning wire corresponds to a particle with a diameter of 253 µm. This value corresponds to the D3 discriminator level (Table I). This fact allows for a very accurate adjustment of the total amplification (Fig. 6, absolute particle size). The linearity between the projected area and the resulting analogue pulse height was tested by using wire of different diameters, spun through the light beam parallel or perpendicular to the slit. Reference SchmidtSchmidt (1977) showed that snow particles behave like sand particles or the test wire (i.e. opaque objects). The switch SI (Fig. 7) is used to test the stability of the light source and of the preamplifier, as well as of the mechanical adjustment of the sensor, by measuring a d.c. voltage at the test output.
Fig. 6. Level discriminator circuit.
Fig. 7. Control circuit.
The mean mass flux Φ in units of kg m–2 s–1 as a function of the total count C is given by Equation (1).
where T is the time between two counter outputs in minutes, n the number of samples per minute, τ the duration of one sample (0.8 s), F the sensitive area (2 × 10–5 m2), C the total count during the period T, and d I the particle diameter corresponding to the lowest discriminator level in metres.
Simultaneous measurements with different types of snow flux gauges to get additional field calibrations have not yet been performed for different reasons; the only existing and laboratory-calibrated photoelectric system (Reference SchmidtSchmidt, 1977) was not available and has the same intrinsic inaccuracy related to the changing grain shapes. A Mellor-type snow trap is now used by Reference FöhnFöhn (1980), but those measurements were not performed at the same location. Nevertheless the correspondance of the results is satisfactory.
Summary of the systematic and statistical errors
The systematic errors depend on the particle-shape distribution, the particle-size distribution, and the boundary effects. Estimates of the errors are listed in Table II.
Table II. Estimates of the systematic errors
Source (1) of error (Table II) is inherent in the method and cannot be decreased by technical improvements.
The sources of possible statistical errors are the sampling rate and drift fluctuations.
The probability of a drift particle crossing into the sensitive area is very low. These events may be considered to be distributed according to a Poisson distribution.
To calculate the standard deviation, the total particle count instead of the recorded total mass-proportional count must be estimated.
The mass-proportional count C (Equation (I)) is equal to
where Ci is the count in class i, B the design constant defined by Equation (I), M i the pulse train length corresponding to the ith class, n i the relative portion of the ith class to the total mass contribution, p i the particle count in class i, σ i the standard deviation for class i, and σthe standard deviation for the flux.
The quantity D given by the equation
depends on the mass contribution curve. The values for the two distributions given in Figures 3 and 4 are D 1 = 0.80, D 2 = 0.86. For the relative error of the flux measurement we get
In our case √ B is equal to 9.27 × 10–3. For n = 1 min -1, T = 30 min, ϕ = 1 g m–2 s–1, we get r ϕ = 7%, for a flux of 0.1 g m–2 s–1 about 21%.
This estimate shows that even for very low sampling rates this contribution to the total error is insignificant compared with the possible systematic errors.
The last error contribution is determined by the gustiness of the wind. Reference CramerCramer (1960) gives the following relationship for the standard deviation σv of wind fluctuations when the mean wind speed is v :
If the wind speed is sampled n times per minute during time T, the resulting standard deviation of the measured mean is
The transportation of snow mass is roughly proportional to the third power of the actual wind speed. The resulting relative standard deviation of the flux is
The error contribution is again less than 10%.
In retrospect, the contributions of the statistical errors for mean drift fluxes of some g m–2 s–1 are less than 50% of the possible systematic errors. If the contribution from precipitation is not significant, the accuracy of a flux measurement is in the range of ±50%.
Measurements and results
The sensor has been in operation as part of the “Gaudergrat” remote station for five months. After some initial electronic and calibration problems had been solved, the flux gauge produced good data for three months. The data were transmitted to the Institut on Weissfluhjoch by radio where they have been analysed on line by a Hewlett Packard desk computer 9815. A typical plot of the mean flux over the “Gaudergrat” crest recorded 0.5 m above ground under the wind-roof together with the corresponding mean wind speed (recorded 3 m above ground) is shown in the preceding paper (Reference GublerGubler, 1981, fig. II). The system allows the investigation of the dependence of the mean mass flux on the development of the snow surface conditions (erodibility of the surface) and on the mean wind speed as well as the effect of low to medium precipitation. The given example includes a south-east wind situation with no precipitation and low erodibility, resulting in low fluxes even at high mean wind speeds, followed by a north-west wind situation with low to medium precipitation increasing the erodibility and causing precipitation-induced flux, resulting in significantly higher fluxes at comparably low mean wind speeds. Integral representations of the total flux across the crest at a given height above the snow surface are given by Reference GublerGubler (1980). These types of calculations allow an estimation of the loading and unloading of potential fracture zones of avalanches under blowing-snow conditions. This is a very important mechanism in determining the danger of avalanches. Investigations allowing the computation of the total flux over a crest from single measurements of the flux and the wind speed at given heights above the snow surface are put forward by Reference FöhnFöhn (1980).
Appendix
Electronic set-up
Light source receiver (Fig. A1)
The high-directivity light source is operated by a stabilized voltage supply. Stability could be slightly improved by replacing the stabilized voltage source by a stabilized current source. To be able to modulate the light source for calibration purposes, the supply voltage may be turned on and off by an analogue switch (IC32). The light receiver consists of a low-noise photodiode followed by a three-stage voltage amplifier (IC35–IC37).
Fig. A1. Electronic circuit diagrams of the light source and receiver.
Discriminator circuit (Fig. 6)
The output signal from the receiver is either connected (IC20) to a precision full-wave rectifier (IC24, IC25) for calibration purposes or to a three-stage variable (absolute particle size) post-amplifier (IC21–IG23) followed by the five level discriminators (IC26–IC30). The band widths of the pre-amplifiers and the post-amplifiers are about 2–70 kHz. The lower limit is determined by low-speed particles (drift velocity 5 m s–1, slit width 5 × 10–4 m). The high limit is given by small particles having a high velocity (drift velocity 50 m s–1, particle diameter 10–4 m). The discriminator levels are calibrated by means of a special voltage reference (relative particle sizes).
Control unit (Fig. 7)
The timing circuit (IC3–IC6) activates the system for 0.8 s each time an external trigger pulse fires the one-shot IC3. The settling time for the analogue electronics and the light source is fixed to 1 s (IC3). The circuits are turned on by the relays R1, R2. The output frequency of the 100 kHz quartz clock (IC7) is divided down to 10 kHz and I Hz (IC1, IC2). The 100 kHz are fed to the down-counters IC14, IC15(frequency in the pulse trains), the 10 kHz source is used to modulate the light source for intensity calibrations and mechanical adjustments (IC19, S1). The 1 kHz output is fed to the down-counter IC5 which determines the time interval of one measurement. A measuring cycle is started by clearing the resets of the dividers IC1, IC2 and of the down-counters IC14, IC15. The outputs of the levsel discriminators are connected to the flip-flops IC12, IC13, IC18. Any pulse from the level discriminators arriving at the flip-flops before the reset is cleared by the timing circuit only causes a reset of the flip-flops 100 µs later (IC11b, IC11a). Pulses arriving after the reset is cleared set the flip-flop according to the original pulse height (projected particle area) and simultaneously fire the one-shot IC11b. IC11b pre-sets the down-counters IC14, IC15. The most significant flip-flop which is set determines the pre-set (IC9). 100 µs later, even the slowest particle is now inside the light beam, the analogue pulse has reached its maximum, the down-counters start to count down to zero. The counting frequency is 100 kHz. During the count-down the 100 kHz pulses are fed to a scaler IC16, IC17 by IC10, IC8, If the down-counter reaches zero, the 100 kHz source is turned off (IC10), the flip-flops are reset (IC11a) and a new particle can be analysed.
The scaler output may be connected to a counter. The total count over a certain time period is proportional to the mean mass flux during this period.
