• Barthazy, E., , S. Göke, , R. Schefold, , and D. Högl, 2004: An optical array instrument for shape and fall velocity measurements of hydrometeors. J. Atmos. Oceanic Technol., 21 , 14001416.

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  • Böhm, H., 1989: A general equation for the terminal fall speed of solid hydrometeors. J. Atmos. Sci., 46 , 24192427.

  • Brandes, E. A., , K. Ikeda, , G. Thompson, , and M. Schönhuber, 2008: Aggregate terminal velocity/temperature relations. J. Appl. Meteor. Climatol., 47 , 27292736.

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  • Heymsfield, A., , and M. Kajikawa, 1987: An improved approach to calculating terminal velocities of platelike crystals and graupel. J. Atmos. Sci., 44 , 10881099.

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  • Hudak, D., , H. Barker, , P. Rodriguez, , and D. Donovan, 2006: The Canadian CloudSat Validation Project. Proc. Fourth European Conf. on Radar in Hydrology and Meteorology, Barcelona, Spain, ERAD, 609–612.

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  • Jiusto, J. E., , and G. E. Bosworth, 1971: Fall velocity of snowflakes. J. Appl. Meteor., 10 , 13521354.

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  • Langleben, M. P., 1954: The terminal velocity of snowflakes. Quart. J. Roy. Meteor. Soc., 80 , 174181.

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  • Zikmunda, J., , and G. Vali, 1972: Fall patterns and fall velocities of rimed ice crystals. J. Atmos. Sci., 29 , 13341347.

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    (top) An example of snow terminal fall velocity measured by the HVSD. Each black point is a measurement of a single particle; the red dots are averages for each 0.3-mm size interval; the vertical red lines indicate the standard deviation. (bottom) The standard deviation and the relative standard deviation as a function of size.

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    (a) Geometry of the measuring planes (modified Fig. 5 of Barthazy et al. 2004) and (b) regression of beam distance across beam. (c) The position of the measured particles in (b) (colors).

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    Relative error in fall velocity measurement caused by the instrument for the left, center, and right positions in Fig. 2. Here the sensor pixel corresponds to the X0 position in Fig. 2c.

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    Sample of particle images detected at the upper and lower beams on 7 Dec 2005.

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    Occurrence of size difference measured in the upper and lower beams for a given size interval. The mean and standard deviation of the size difference and the mean size of snowflake in each size interval are also indicated.

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    Summary of instrumental errors in fall velocity measurements for deep and shallow snow events. The observed average relative standard deviations for the deep and shallow systems are also shown.

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    Area ratio Ar defined as the ratio of the area of the vertical cross section to the area of circle of maximum dimension of the particle. The average is indicated by the red dots and the solid brown line shows the best-fit power law.

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    Frequency of occurrence of different fall velocities for particles within a size interval.

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    Time–height profiles of reflectivity and Doppler velocity. Examples of Doppler spectra and the VDmax relationships with the standard deviations around the average curves are shown for the two periods indicated by the blue and red arrows.

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    An example of “homogeneous” snow. At the top is shown the time–height of VertiX reflectivity and Doppler velocity with some Doppler spectra. Below there is a series of six panels showing the mean VD relationship for the indicated time intervals and the standard deviation around the mean values. The average VD equation, its determination coefficient, the standard deviation (dashed lines), the relative standard deviation (solid curves), and the size interval of the particles for which the VD fit is made are indicated for each time interval. Figure 1 shows the average for the entire event.

  • View in gallery

    (top) Average velocity size relationship for 27 periods of homogeneous snow. (bottom) The standard deviation and the relative standard deviation for the average curve.

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    Distribution of H, Tt, and Ts in the sample population. The meaning of the different symbols is described in the text.

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    Values of the coefficient a determined by the power-law fit to measurements vs values given by (6). The case identified with the arrow is one of the outliers that is close to fulfilling our strict criteria of homogeneity. The regression (6) did not take this case into account. The dotted line indicates a 1:1 correspondence.

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    Scattergram of values. The arrows indicate the location of the two outlier cases.

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Snow Studies. Part I: A Study of Natural Variability of Snow Terminal Velocity

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  • 1 Department of Atmospheric and Oceanic Sciences, McGill University, Montreal, Quebec, Canada
  • 2 Kyungpook National University, Daegu, South Korea
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Abstract

The variability and the uncertainties in snowfall velocity measurements are addressed in this study. The authors consider (i) the instrumental uncertainty in the fall velocity measurement, (ii) the effect of unstable falling motion on the accuracy of velocity measurement, and (iii) the natural variability of homogeneous snow terminal fall velocity. It is shown that, when periods of homogeneous characteristics of snow are selected to minimize the mixture of particles of different origin, the standard deviation of snowfall velocity within each period tends to stabilize at a value between 0.1 and 0.2 m s−1.

In addition, the variability of snow terminal fall velocity is examined with three control variables: surface temperature Ts, echo-top temperature Tt, and the depth of precipitation system H. The results show that the exponent b in the power-law relationship V = aDb has little effect on the variability of snowfall velocity: the coefficient a correlates much better with the control variables (Ts, Tt, H) than the exponent b. Hence, snowfall velocity can be modeled with a varying coefficient a and a fixed exponent b = 0.18 (V = aD0.18) with good accuracy.

* Current affiliation: University of Miami, Miami, Florida

+ Previous affiliation: McGill University, Montreal, Quebec, Canada

Corresponding author address: Isztar Zawadzki, Department of Atmospheric and Oceanic Studies, McGill University, 805 Sherbrooke St. West, Montreal, Quebec H3A 2K6, Canada. Email: isztar.zawadzki@mcgill.ca

Abstract

The variability and the uncertainties in snowfall velocity measurements are addressed in this study. The authors consider (i) the instrumental uncertainty in the fall velocity measurement, (ii) the effect of unstable falling motion on the accuracy of velocity measurement, and (iii) the natural variability of homogeneous snow terminal fall velocity. It is shown that, when periods of homogeneous characteristics of snow are selected to minimize the mixture of particles of different origin, the standard deviation of snowfall velocity within each period tends to stabilize at a value between 0.1 and 0.2 m s−1.

In addition, the variability of snow terminal fall velocity is examined with three control variables: surface temperature Ts, echo-top temperature Tt, and the depth of precipitation system H. The results show that the exponent b in the power-law relationship V = aDb has little effect on the variability of snowfall velocity: the coefficient a correlates much better with the control variables (Ts, Tt, H) than the exponent b. Hence, snowfall velocity can be modeled with a varying coefficient a and a fixed exponent b = 0.18 (V = aD0.18) with good accuracy.

* Current affiliation: University of Miami, Miami, Florida

+ Previous affiliation: McGill University, Montreal, Quebec, Canada

Corresponding author address: Isztar Zawadzki, Department of Atmospheric and Oceanic Studies, McGill University, 805 Sherbrooke St. West, Montreal, Quebec H3A 2K6, Canada. Email: isztar.zawadzki@mcgill.ca

1. Introduction

Deposition, riming, and aggregation are the processes by which ice particles grow. Their fall velocity is one of the important input parameters in all descriptions of ice particle growth (Heymsfield and Kajikawa 1987), and understanding and modeling it is essential in studying the development of precipitation in cold clouds. Thus, numerous observational and theoretical studies of this subject have been performed, from the early work of Langleben (1954), through Jiusto and Bosworth (1971) and Böhm (1989), to the very recent efforts by Brandes et al. (2008). The latter establishes a relationship between surface temperature and fall velocity of snow, and in their paper the reader will find numerous references describing the steady progress in the field.

The balance between gravity and drag forces determines the terminal fall speed of snow. The larger particles may be further slowed by self-induced turbulence. These forces are determined by shape and density that in turn are determined by microphysical processes acting during growth: deposition and its habit, aggregation, and riming. These processes are governed by very interrelated parameters: particle concentration, temperature, humidity (controlled by w, the vertical air motion), and time/length of growth (height of snow system, H). At the top of a widespread precipitation system at a quasi-steady state, air is saturated with respect to water; thus, temperature alone determines the initial habit of growth and the concentration of active freezing nuclei (initial number of particles). Aggregation is influenced by habit (thus by temperature and humidity), number concentration, depth of the snow system, and perhaps temperature.

It is common to describe the relationship between the diameter D of the hydrometeor and its terminal fall velocity V by a power-law relationship: V = aDb, where a, b are constants determined from observations or by theoretical considerations.

Our hypothesis is that the description of the characteristics of snow in terms of ambient parameters must include the depth H of precipitation systems, the surface temperature Ts, and the temperature at the top of the system, Tt, where particles originate. Other relevant variables, such as the strength of the vertical motions that determine the available humidity for growth and the degree of riming, are more difficult to determine and will not be considered here. Thus, the main purpose of the present work, with results described in section 5, is to explore whether a clear relationship between (a, b) and the environmental parameters—H, Tt, and Ts—can be established.

In section 3 we will determine the capability limits of our instrument to measure the variability of fall velocity for a given size. Section 2 describes our database and the instruments used for this study.

2. Data

Data on snow velocity and size used in this study were obtained at the Centre for Atmospheric Research Experiments (CARE) site (80 km north of Toronto) during the winter of 2005/06 as part of the Canadian Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO)/CloudSat Validation Project (C3VP), described by Hudak et al. (2006). On 36 days solid precipitation, originating in single-layered systems, reached the ground. Within those days we identified 27 periods of snowfall during which H, Tt, and Ts remained nearly constant throughout the period.

Our information is obtained primarily from (i) a particle imager, the Hydrometeor Velocity and Shape Detector (HVSD), (ii) a vertically pointing radar, and (iii) aircraft temperature soundings.

The particle imager HVSD was originally developed at the Eidgenössische Technische Hochschule (ETH) in Zürich, Switzerland, and is described by Barthazy et al. (2004). This instrument is a two-beam optical disdrometer that provides fall velocities of snowflakes for 0.15-mm size intervals, enabling the derivation of particle size distribution.

Furthermore, to guide us in the data selection we use a secondary, but important, source of information: a vertically pointing X-band radar (hereafter VertiX). This radar gives time–height records of reflectivity, Doppler velocity, and Doppler spectra with a resolution of 2 s in time and 37.5 m in range. The instrument is described in Zawadzki et al. (2001). VertiX indicates whether a particular snow system was uniform in time and whether the falling snow originated from a single layer during the entire event and whether it had a clearly and persistently defined echo-top height. The time–height profile of Doppler velocity from VertiX indicates whether the vertical velocity (fall velocity plus air motion) is uniform in time, while the Doppler spectra give information about the presence of turbulence and strong vertical motions during the event. Vertical echo structures, in particular echo-top height H, were given by VertiX data. In addition to the HVSD and VertiX we have a continuous record of surface temperature Ts, as well as frequent temperature soundings taken by aircraft during takeoff or landing at the Toronto International Airport located about 60 km south of the site. These soundings are used to estimate the echo-top temperature Tt.

3. Variability and uncertainty in measurements of snow terminal fall velocity

To set the stage, let us start with an example of velocity measurements taken with the HVSD, shown in Fig. 1. Each of the black dots of the upper panel in Fig. 1 represents one of the over 16 000 observed particles for which terminal velocity is given as a function of maximum horizontal size for each size interval. The red dots and the red bars are the average values and standard deviation,
i1520-0469-67-5-1591-e1
respectively. The lower panel shows σ and the relative value, σ/V, as functions of size (here, unless indicated otherwise, snowflake size is given by the maximum value measured from the two beams, Dmax). The terminal velocities in Fig. 1, as well as similar observations made with modern devices found in the literature, have a common striking feature: great variability. For a fixed value of terminal velocity all particle sizes are possible. This is accepted in general as self-evident given the great variability in particle morphology. Thus, it appears that snow properties are stochastic in nature and only the mean values (first-order statistic) are of practical interest. However, if the stochastic component is as important as it appears, it may have implications for the physical processes involving snow. For example, it may affect the rate of aggregation.

On the other hand, the difficulties of measuring snow properties are also recognized, and it is not clear what portion of the observed complexity is instrument induced and what fraction is due to the natural variability of snow properties. We will attempt here to give a first-order assessment of this issue by an evaluation of our observational uncertainties.

a. Errors induced by the geometry of the HVSD

Knowledge of the exact physical dimensions of the two measuring planes and the vertical offset between beams is necessary for an accurate determination of the size and fall velocity of a hydrometeor, as well as the counts of number concentrations. The geometry of two measuring planes and beam distance regression are illustrated in Fig. 2.

The length of the measuring area is given by the dimension of the gap in front of the rectangular tube, measured by the line passing through the center (from CL to CT) of the measuring area, determined to be 108.8 mm (z direction in Fig. 2a). The width of the measuring area decreases slightly toward the line scan camera (tube side), due to its trapezoid shape, from 81.0 to 72.5 mm (x direction in Fig. 2a). The two horizontal measuring planes have a slightly trapezoidal shape, resulting in different beam distance depending whether snowflakes fall near the lamp side (LL, CL, RL) or tube side (LT, CT, RT). At the same time there is a closing between beams in the cross-beam direction, as seen in Fig. 2c. The distance between beams was determined at nine different (edge) locations (LL, CL, RL, LC, CC, RC, LT, CT, and RT in Fig. 2a), where the first letters (R, C, L) represent right, center, and left, respectively, and the second letters (L, C, T) designate the lamp, center, and tube side, respectively. For instance, RL indicates the right of the lamp side. Since the position of particles along the cross-beam direction is known, a regression equation, y = Ax0 + B, is used to obtain the beam distance, where x0 is the position of snowflakes along the width (x direction in Fig. 2a). This regression was introduced in the data analysis software so as to compensate for the lack of parallelism across the beam.

The fall velocity V observed by the HVSD as a function of the distance Bd between the upper and lower beams and the times t1 and t2 that the particle is detected at the upper and lower beams respectively is calculated by
i1520-0469-67-5-1591-e2
We first consider the uncertainty in fall velocity measurement due to the geometry of the instrument. An approximation for the variance of V, σV2, can be expressed as
i1520-0469-67-5-1591-e3
where
i1520-0469-67-5-1591-eq1
and 9470 is the scan rate of the HVSD. With
i1520-0469-67-5-1591-e4
we can express (3) as
i1520-0469-67-5-1591-e5

Given the high precision in measuring time, the contribution of the first term to the variance of V is at least two orders of magnitude smaller than of the second term for all expected fall velocities of snow. Figure 3 illustrates σV of the fall velocity (and σV/V) given by the second term. The figure shows that the instrumental uncertainty of fall velocity measurements is quite dependent on the location where the particle is detected within the sensitive area. The measuring planes tilts slightly toward tube side, inducing a different uncertainty in fall velocity measurements depending on where the hydrometeor falls within the measuring area. On average, the instrumental uncertainty of the fall velocity measurement is around 12% for fall speeds below 2 m s−1.

b. Wobbling of snowflakes

Fall velocity is determined from the time it takes for the center of mass (CM) of a particle to travel the distance between the upper and lower beams of the HVSD. The CM position is estimated from the midpoint of the vertical extent of the particle, which is not the same as the CM, and the wobbling of particles further introduces an uncertainty in velocity measurements. Snowflakes rarely fall in a straight line; rather, they exhibit an unstable falling motion. Zikmunda and Vali (1972) suggested that mass distribution and flow disturbance caused by surface features resulted in complicated fall patterns. In addition, Kajikawa (1992) classified unstable fall patterns into three types: nonrotation, swing, and rotation or spiral (based on the observations of unrimed crystals such as dendritic crystals). We will now quantify the effect of wobbling using our observations. In addition to the effect of wobbling on the accuracy with which we can measure the variability of fall velocity, there is also the more fundamental interest in studying the way snow wobbles as it falls.

We start with an example of sample images of snowflakes detected in the upper and lower beams of the HVSD separated by 9-mm distance, as illustrated in Fig. 4.

Even within such a short fall path (9 mm on the average) the change in the images is apparent. Our subjective impression suggests that these snowflakes swing clockwise (1227:06) or counterclockwise (1138:19, 1138:35, 1227:57, 1228:56) or rotate (122923). In any case, we see different cross sections in the upper and lower beams. These differences may be due in part to differences in sensitivity between the beams. Since wobbling is a random event and sensitivity differences are systematic, the separation of the two effects should be clear in the statistics of a large number of observations. Observations periods of snow listed in Table 1 were used to determine the probability of occurrence of the difference in vertical extent between the two beams. These observations correspond to four shallow with a cold surface (H < 4 km, −14°C < Ts < −9°C) and four deep and warm (H > 4 km, −1 > Ts > −6°C) snow systems.

As an example of the frequency of occurrence of differences in size, two different size intervals of snowflakes are illustrated in Fig. 5.

In Fig. 5, the mean value, for example, 0.05 mm for the 4–5 mm size interval, indicates the systematic difference in vertical size as measured in upper and lower beams. We attribute this small bias to the difference in sensitivity between the two beams. On the other hand, the standard deviation σ of the size differences is interpreted as due to wobbling. Size differences due to the sensitivity difference in the upper and lower beams are much smaller as compared to the effect of wobbling. Figure 5 shows that size differences due to the beam sensitivity difference are small, less that 3% of the particle size, and the difference decreases, as snowflakes grow, to negligible levels. Relative size differences between the upper and lower beams due to wobbling also decrease with size. Detailed analysis of the wobbling shows that this effect introduces a difference in estimated size (relative to size) of 40% for the smallest particles, decreasing to ∼10% of particle size for D ∼ 5 mm—around 5% of the size of the larger particles. Wobbling is less noticeable in particles falling from deeper systems. This could be due to their faster fall speed, which allows less time for wobbling between the two beams.

The relative error in fall velocity measurement due to wobbling is the ratio between the time that the particle travels a distance equal to the uncertainty in the position of center of mass (half the σ in vertical size) and the time that the center of mass travels the 9 mm between the two beams. This is equal to of the standard deviation. The differential sensitivity introduces errors ( of the mean difference between upper and lower beams) below 1% while the effect of wobbling leads to uncertainties in fall velocity measurements of 1.5% for the smaller particles, increasing to 4.5% and 2.5% for the deep and shallow systems, respectively. The difference between deep and shallow systems is attributable to the faster fall velocities in the deep systems, probably associated with higher particle densities. The results of this data analysis are summarized in Fig. 6 for both deep and shallow systems. These values correspond to averages, and there is considerable case-to-case variability.

We conclude from Fig. 6 that the uncertainty in velocity measurements by the HVSD is on average ∼13%. The HVSD has a very low vertical profile and, furthermore, it is shielded from the wind; here, the errors due to turbulence near the ground are considered negligible (a collocated sonic anemometer would have been useful to quantify this problem). The errors assessed in this section must be kept in mind when we discuss velocity measurement of snow later.

c. Uncertainty due to the limited view of the particles

Different particle imagers have different capabilities for providing information on the shape and size of snow particles. The HVSD, like all 2D optical particle imagers, cannot see the true maximum size Dmax unless the maximum dimension is aligned perpendicularly to the light beam. Neither can it see the horizontal area swept by the particle (the same is true for a “3D” disdrometer). Thus, the question that must be addressed is the influence of these limitations on the derived parameters of V = aDb and on the observed scatter around the mean.

There is no general agreement on what is the most significant measure of particle size from the point of view of fall velocity. Perhaps the most convenient way of elucidating this question is by choosing the size measure that minimizes σ/V in the velocity–size relationship. We have explored this option with our data and found that the equivalent diameter De (the diameter of a circle with the same area as the particle) systematically gives the lowest σ/V. However, with the HVSD we can only determine the side-view equivalent diameter (i.e., the equivalent diameter of the vertical cross section), not the one perpendicular to the fall direction. Thus, an instrument with the capability of a more complete view of particles is needed to confirm our tentative finding. In what follows we will use the maximum detected size since this is the most widely used measure—it facilitates intercomparison of results and is the one needed for most microphysical computations. Figure 7 provides the average transformation between the two.

In summary, after subtracting the instrumental variances in measurement from the observed values we see that the natural variability in particle fall velocity is below 18% for the small particles and rapidly decreases with size to below 10% for the shallow systems or a few percent for the deep systems. This is a rather small variability. We performed some computations (not shown here) using geometrical sweep-out with unit aggregation efficiency that indicate these fluctuations in fall velocity have a negligible effect on aggregation rate.

Finally, we look at the distribution functions of velocities for a given size (Fig. 8). The distributions are skewed toward larger velocities. To understand this imbalance, we first point out that the true Dmax is underestimated on average; therefore, the velocity of a larger particle is assigned to a smaller one (shifted from right to left in Fig. 1). Moreover, the larger complex particles are more likely to depart from a spherical shape; consequently the underestimation of Dmax will increase with size, leading to a net shift of particles from left to right. Hence, the distribution of velocity around the average becomes asymmetrical with more particles falling faster than slower, and the effect will be more pronounced when the change in fall velocity with size is greater. This is precisely what is observed in Fig. 8. There is no reason to suspect that this could be an instrumental effect. If the observed asymmetry in the distribution is due to the systematic underestimation of Dmax by the HVSD, this could lead to an overestimation of the exponent b and coefficient a. Also, this effect can add to the scatter, increasing with decreasing size, around the mean velocity. This would suggest that the natural variability in fall velocity is even smaller than the values given above. On the other hand, the distribution of fall velocity can be skewed owing to physical reasons that are presently unknown.

4. Radar observations

Before continuing with the study of fall speed variability let us introduce our other source of information, namely the vertically pointing X-band radar, VertiX. From these observations we will be guided in the classification of situations according to the characteristics of the time–height profile of reflectivity, of Doppler velocity, and of the observed Doppler spectra.

Figure 9 shows a particular event that occurred on 7 December 2005. Note the color scale for the Doppler height–time profiles: it was devised to maximize the visual appreciation of small variations in Doppler velocity. During the period between 1110 and 1200 the system height is close to 2.5 km. Convective activity is present at all times, as shown by great time variability of Doppler velocity, with periods of positive (upward) Doppler velocity (warm colors) indicating updrafts stronger than the fall velocity of particles. The Doppler spectra are very broad—in excess of what could be expected from the variability in the terminal fall velocity of snow—and “wiggly,” indicating changes in vertical air velocity with height. In contrast, from 1210 to 1240 the top of precipitation descended to 2 km; we see less time variability of the Doppler velocity and the Doppler spectra are narrower, all indicating a quiet, more “stratiform,” period of precipitation. In this period the change of velocity with height just compensates for the slowing fall speed due to the increase in air density.

Snow may travel long horizontal distances but, in a weak wind shear, so will the entire profile—thus maintaining the correspondence between the vertical profiles of VertiX observations and the HVSD ground measurements. However, strong horizontal variability in the precipitation pattern combined with wind shear may decouple observations at the ground from aloft. To reestablish the correspondence between observations at the ground and aloft from radar observations, a good horizontal uniformity in space is required. The vertical profiles of Doppler spectra are particularly indicative of the spatial variability of snow characteristics. Thus, the relationship of these time–height profiles with observations at the ground must be done with caution and in reference to the situation. VertiX observations give an indication (but not a proof) of the homogeneity of a particular time period of snowfall.

Additional relevant information provided by the Doppler spectra of VertiX is the occurrence of a mixed phase. As shown in Zawadzki et al. (2001), rapid changes in Doppler velocity with fall distance, often followed by bimodal spectra, are indicative of the presence of supercooled cloud and supercooled drizzle coexisting with snow. These situations produce heavy riming, sod we therefore avoided such periods.

5. Case-to-case variability in VD relationships

We now proceed with the study of the fall velocity variability between different periods of snow and its relationship with environmental conditions. The objective here is the description of snow characteristics, both to provide guidance for the analysis of radar data and as our first step toward developing physically meaningful parameterizations of snow in numerical models.

From our database we select snow periods for which the following criteria are satisfied:

  • (i) The characteristics of the VertiX’s time–height record are uniform during the period, as illustrated in the example of Fig. 9;
  • (ii) There is no evidence of heavy riming in the Doppler spectra of VertiX;
  • (iii) The echo-top height is nearly constant;
  • (iv) There is nearly constant surface and echo-top temperature;
  • (v) A sample size of particles in each period is greater than 1000 for particles for which only one particle is in the field of view of both beams of the HVSD and therefore an unambiguous match between the upper and lower beams of the HVSD can be established;
  • (vi) The detected sizes extend over a minimal range of diameter with a maximum size of at least 4.5 mm (the condition of a broad size spectrum is necessary for a robust definition of the velocity-size relationship); and
  • (vii) The VD relationship is robust in the sense that the power-law, V(D) = aDb, fit to the average values for each size is good that the determination coefficient is at least R2 = 0.85.

In this manner 10 time periods, lasting from one-half to six hours, were identified. These periods are homogenous in that they satisfy these criteria. In addition, for each of the 10 cases the VD relationship is closely maintained during an event. Figure 10 shows an event that occurred on 18 February 2006. The average fall velocity measurements for this event were shown in Fig. 1. It is an example of such homogeneity in a snow event lasting 2.5 h. The six subperiods show very limited fluctuations in the VD relationship and a high robustness in the VD fit. VertiX also shows very uniform characteristics of the data throughout the entire period.

The concept of homogeneity used here may be useful in understanding the physical characteristic of snow, but from the point of view of providing guidance for the description of snow in analysis of radar data and in numerical modeling it is a limitation rather than a useful criterion because homogeneity is difficult to find in snow systems. Thus, by relaxing conditions vi and vii we have increased the sample size to 27 events, including all periods for which we had good quality HVSD measurements, and at the same time the environmental parameters (echo-top height H, surface temperature Ts, and temperature at the echo top Tt) were nearly constant during each period.

Figure 11 shows the average VD relationship and its variability for all these data as well as its level of uncertainty. As we have seen, instrumental limitations introduce an error close to 13%. In homogeneous periods of snow the value of σ/V is size dependent and is larger for the smaller particles. For the larger particles it is usually below 20% and is sometimes close to the instrumental uncertainties (i.e., the natural variability of fall velocity is not detectable with the HVSD). When case-to-case variability is added, σ/V exceeds 30%. That is, after removing the instrumental uncertainty, the case-to-case variability of the fall velocity of the larger particles is σ/V ≈ 0.2 whereas within homogeneous periods of snow it is below 0.15. The small particles have greater uncertainty in fall velocity but the case-to-case variability contributes very little to the total scatter around the mean velocity.

We now apply a statistical analysis of our data so as to establish the dependence of the VD relationship on the variable environmental parameters—H, Tt, and Ts—of snow growth. All statistical computations are done using the Statistical Package for the Social Sciences (SPSS). Figure 12 shows the distributions of these variables for the cases selected.

Each point in Fig. 12 represents one of the 27 separate cases of snow. The solid circles in Fig. 12 are the 10 cases of “homogeneous” snow periods. The open circles are two periods of a very unusual situation of a quasi-isothermal atmosphere over the entire depth of snow. The most common temperature profile is a near-isothermal layer over a limited depth followed by a near-adiabatic profile. This characteristic establishes the correlation between H and (Tt, Ts) seen in all the panels of Fig. 12. Thus, the two periods represented by the open circles are outliers that offer the opportunity of testing the robustness of statistical correlation analysis between variables.

We first consider the 10 homogeneous snow periods. Table 2 shows the correlation coefficient between a and the parameters H, Tt, and Ts.

If only one parameter is to be used, then the depth of precipitation is the choice. Both top and surface temperatures have the same skill in predicting a. The correlation between b and Tt is 0.26 and it is negligible with the other two parameters. Partial correlation analysis indicates that precipitation depth is, in effect, the dominant factor with a small contribution from the surface temperature and top temperature, once the influence of H is taken into account. A rank regression leads to the following relationship:
i1520-0469-67-5-1591-e6
The determination coefficient of this relationship is R2 = 0.94. The large margins of uncertainty in the coefficient of Ts and Tt indicate that, once the influence of storm depth is taken into account, the statistical significance of the contribution of these parameters to determine a is very low. If only H is considered, the linear regression equation becomes
i1520-0469-67-5-1591-e7
with the determination coefficient of R2 = 90. Figure 13 shows the scattergram of the measured values versus values given by (6) for these 10 cases. The exponent b does not show any significant dependence on any of the variables.

Next, we consider the larger set of 27 cases. Equation (6) does not represent well this entire population. Thus, we perform the same regression analysis as before on the extended dataset. The simple correlation between a and the environmental variables is shown in Table 3. The values are shown in brackets for all cases and for cases excluding the two outlier cases for which the temperature profile was quasi-isothermal. The drop in the correlation coefficient with Ts after the inclusion of the two outlier cases is likely due to the lack of correlation between depth of precipitation and surface temperature for these two cases.

As for the homogeneous cases, partial correlation analysis and ranked regression indicate that depth of snow is the dominant factor, although the contribution of the surface temperature is not negligible. The model regression equation is
i1520-0469-67-5-1591-e8
with a determination coefficient R2 = 0.69. If H is the only variable taken into account, the simplified model equation becomes
i1520-0469-67-5-1591-e9
with a determination coefficient R2 = 0.61. For this set of cases there is a weak correlation between b and surface temperature leading to
i1520-0469-67-5-1591-e10
with a low determination coefficient of R2 = 0.24.

The scattergram corresponding to (8) is shown in Fig. 14.

6. Discussion and conclusions

We have carefully analyzed the limitations of our HSVD observing instrument to assess the significance of the observed variability of snow terminal fall velocity (more details can be found in Jung 2008). Subtracting the variance due to instrumental uncertainty from our observed variance during selected homogeneous snow periods as well as for all observations (Fig. 11), we can give an approximate estimate of the natural standard deviation of snowfall velocity. A summary is given in Table 4. The variability within homogeneous periods is slightly greater for small particles. As we have seen, shallow systems show a narrower scatter around the mean velocity. This variability can be mainly attributed to random change in particle morphology, even if the habit of individual crystals composing the snowflake is the same. As mentioned before, these values are upper limits since they are affected by the two-dimensional limitations of our instrument. A more precise estimate could be provided by measurements with 3D imagers.1 Moreover, although we have tried to isolate homogeneous periods of snow for which we assume some uniformity in microphysics, this homogeneity is certainly not completely strict. That is, some of the variability may be due to nonhomogeneity in the formation of snow particles.

The case-to-case variability for larger particles is the largest. These particles are the ones more relevant from the point of view of detection by radars operating in the Raleigh scattering region. The second part of this work was aimed at determining the importance of some of the controlling parameters of this variability—that is, precipitation depth as well as the temperatures at the surface and at the top of the precipitation system. This is a delicate problem since the controlling parameters are more interrelated among themselves than they are with the fall velocity of snow. To sort out these dependencies, we rely on partial correlation analysis, which requires a good number of cases to ensure a good significance level. Statistically derived relationships between variables rely on the assumption of random distribution of values. We tried to approach this requirement (Fig. 12 shows the limitations of our sample). Our analysis shows that the single most important controlling parameter is the precipitation depth. In absence of this information, surface temperature may be used as a proxy; however, it appears that physically it is the depth that is the significant parameter. It is indicative, although not conclusive, that, when the two cases of low surface temperature are included, the correlation with depth increases and decreases with surface temperature (Table 3).

We made a number of other tests: the correlation between the temperature difference between top and bottom with a is slightly better than precipitation depth. Using the vertical Doppler velocity from VertiX, the time of particle residence within the precipitation depth was computed and correlated with a. This correlation is better than with precipitation depth. However, for microphysics parameterization purposes it is much simpler to use precipitation depth since no tracking of particles is necessary.

In our observations the exponent b varied by a factor of 2, roughly from 0.1 to 0.2. However, we did not find any convincing correlation between the exponent b and the controlling ambient parameters. There is a trend for b to increase with H but this trend is not very significant. The reason for the low statistical significance involving correlations with b may be that b is poorly determined when the range of sizes is limited, which is often the case (it is interesting to note that the range of sizes tends to be broader in shallow systems). Although the measured average b ≈ 0.15, we found that a higher single b leads to a better correlation between a and H. For the parameterization of snowfall speed we will take V = aD0.18 with a given by (8) where Ts and H are the temperature at the position of the snow and its distance from the top, respectively.

This study of snowfall speed is limited to winter cases with surface temperatures below freezing. We deliberately excluded cases with Doppler spectra indicating the presence of supercooled water, as discussed in Zawadzki et al. (2001). In the presence of appreciable supercooled cloud water, riming may lead to denser snow with higher velocity.

As said in the introduction, fall velocity is determined by the balance of drag and gravity forces, the latter being determined by the mass of particles. Thus, to understand the observed variability in fall velocity we need to turn our attention to snow density and its relation to fall velocity, which is the subject of Part II of this work.

Acknowledgments

This work was partially supported by a grant from the Canadian Foundation for Climate and Atmospheric Sciences (CFAS). The effort of the Canadian CALIPSO/CloudSat Validation Project (C3VP) team in setting up and maintaining the field data collection is gratefully acknowledged as well as the Canadian Space Agency for their funding of C3VP. The editing of the manuscript by Aldo Bellon is appreciated.

REFERENCES

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Fig. 1.
Fig. 1.

(top) An example of snow terminal fall velocity measured by the HVSD. Each black point is a measurement of a single particle; the red dots are averages for each 0.3-mm size interval; the vertical red lines indicate the standard deviation. (bottom) The standard deviation and the relative standard deviation as a function of size.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 2.
Fig. 2.

(a) Geometry of the measuring planes (modified Fig. 5 of Barthazy et al. 2004) and (b) regression of beam distance across beam. (c) The position of the measured particles in (b) (colors).

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 3.
Fig. 3.

Relative error in fall velocity measurement caused by the instrument for the left, center, and right positions in Fig. 2. Here the sensor pixel corresponds to the X0 position in Fig. 2c.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 4.
Fig. 4.

Sample of particle images detected at the upper and lower beams on 7 Dec 2005.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 5.
Fig. 5.

Occurrence of size difference measured in the upper and lower beams for a given size interval. The mean and standard deviation of the size difference and the mean size of snowflake in each size interval are also indicated.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 6.
Fig. 6.

Summary of instrumental errors in fall velocity measurements for deep and shallow snow events. The observed average relative standard deviations for the deep and shallow systems are also shown.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 7.
Fig. 7.

Area ratio Ar defined as the ratio of the area of the vertical cross section to the area of circle of maximum dimension of the particle. The average is indicated by the red dots and the solid brown line shows the best-fit power law.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 8.
Fig. 8.

Frequency of occurrence of different fall velocities for particles within a size interval.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 9.
Fig. 9.

Time–height profiles of reflectivity and Doppler velocity. Examples of Doppler spectra and the VDmax relationships with the standard deviations around the average curves are shown for the two periods indicated by the blue and red arrows.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 10.
Fig. 10.

An example of “homogeneous” snow. At the top is shown the time–height of VertiX reflectivity and Doppler velocity with some Doppler spectra. Below there is a series of six panels showing the mean VD relationship for the indicated time intervals and the standard deviation around the mean values. The average VD equation, its determination coefficient, the standard deviation (dashed lines), the relative standard deviation (solid curves), and the size interval of the particles for which the VD fit is made are indicated for each time interval. Figure 1 shows the average for the entire event.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 11.
Fig. 11.

(top) Average velocity size relationship for 27 periods of homogeneous snow. (bottom) The standard deviation and the relative standard deviation for the average curve.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 12.
Fig. 12.

Distribution of H, Tt, and Ts in the sample population. The meaning of the different symbols is described in the text.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 13.
Fig. 13.

Values of the coefficient a determined by the power-law fit to measurements vs values given by (6). The case identified with the arrow is one of the outliers that is close to fulfilling our strict criteria of homogeneity. The regression (6) did not take this case into account. The dotted line indicates a 1:1 correspondence.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Fig. 14.
Fig. 14.

Scattergram of values. The arrows indicate the location of the two outlier cases.

Citation: Journal of the Atmospheric Sciences 67, 5; 10.1175/2010JAS3342.1

Table 1.

Time periods used for determining the difference in the vertical size of particles between the upper and lower beam.

Table 1.
Table 2.

Correlations between a and the indicated variables.

Table 2.
Table 3.

Correlations between a and the indicated variables for 25 and 27 (in parentheses) cases.

Table 3.
Table 4.

Estimated standard deviation in snowfall velocity around the mean.

Table 4.
1

The question of determining experimentally which definition of particle size leads to the lesser σ in fall velocity is interesting and could be also elucidated with 3D imagers.

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