NOTES AND CORRESPONDENCE Aggregate Terminal Velocity ...

OCTOBER 2008

NOTES AND CORRESPONDENCE

2729

NOTES AND CORRESPONDENCE Aggregate Terminal Velocity/Temperature Relations EDWARD A. BRANDES, KYOKO IKEDA,

AND

GREGORY THOMPSON

National Center for Atmospheric Research,* Boulder, Colorado

MICHAEL SCHÖNHUBER Joanneum Research, Graz, Austria (Manuscript received 6 September 2007, in final form 12 May 2008) ABSTRACT Terminal velocities of snow aggregates in storms along the Front Range in eastern Colorado are examined with a ground-based two-dimensional video disdrometer. Power-law relationships for particles having equivalent volume diameters of 0.5–20 mm are computed for the temperatures ⫺1°, ⫺5°, and ⫺10°C. Fall speeds increase with temperature. Comparison with relationships found in the literature suggests that temperature-dependent relations may be surrogates for relations based on aggregate composition (e.g., plates, columns, or dendrites) and the degree of riming.

1. Introduction An investigation of aggregate terminal velocities at ground was conducted with observations from a video disdrometer manufactured by Joanneum Research at the Institute of Applied Systems Technology in Graz, Austria. Originally designed for raindrops, the instrument has demonstrated capabilities for monitoring frozen hydrometeors. Detailed information regarding particle shape, size, and terminal velocity is obtained. Although pristine hexagonal forms, capped columns, and needles are detectable, the component forms of aggregates are not easily identified from disdrometer images. Hence, no attempt was made to sort the dataset by composition. As is customary, aggregate terminal velocities ␷ t (m s⫺1) are assumed to have the form

␷t ⫽ ADb,

共1兲

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Dr. Edward A. Brandes, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307. E-mail: [email protected] DOI: 10.1175/2008JAMC1869.1 © 2008 American Meteorological Society

where D (mm) is a characteristic diameter (e.g., Langleben 1954; Lumb 1961; Jiusto and Bosworth 1971; Locatelli and Hobbs 1974; Heymsfield and Kajikawa 1987). The coefficient responds to the specific component forms that make up the aggregate (e.g., plates, columns, and dendrites) and the amount of riming. (In this study the exponent is virtually constant.) Most weather forecast models assume spherically shaped snow particles with constant density and do not explicitly predict snow crystal habit or degree of riming. Instead, a single ␷t(D) relationship similar to those in the aforementioned studies is applied. An alternate approach may be to derive fall speeds as a function of temperature with the thought that temperature dictates a range of likely particle size, composition, and degree of riming, which largely determine fall speeds. Previous applications of disdrometers to winter storms include that of Hanesch (1999) who investigated relationships between snowflake shape and terminal velocity. Yuter et al. (2006) present fall speeds for melting snowflakes. Brandes et al. (2007) found that the mean mass-weighted terminal velocity of snowflakes increased from about 0.9 to 1.3 m s⫺1 as temperatures warmed from ⫺5° to 0°C. The increase was attributed primarily to particle growth, but increased riming is also a factor. In what follows, the relation between aggregate fall

2730

JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY

speed and temperature is investigated more closely. Disdrometer capabilities for estimating properties of frozen particles are briefly summarized. Terminalvelocity relations found in the literature are reviewed. Fall speeds for disdrometer-observed aggregates are examined, and power-law relations are derived for specific temperatures. The range in velocity/size relations when stratified according to temperature is shown to roughly match that found in stratifications by the degree of riming or aggregate composition. Last, differences between observed aggregate fall speeds and calculated fall speeds described in recent studies are discussed.

2. Disdrometer measurements Kruger and Krajewski (2002) give a detailed description of the disdrometer calibration and computational procedures. The instrument consists of two horizontally oriented line-scan cameras, separated in the vertical direction by about 6 mm, which provide orthogonal views of hydrometeors falling through a common 10 cm ⫻ 10 cm area. Shadowed photodiodes for each camera are recorded at a line-scan frequency of 51.3 kHz. Vertical resolution depends on particle terminal velocity and is about 0.03 mm for snowflakes. Horizontal resolution is approximately 0.15 mm. The instrument is calibrated by dropping graduated spheres of known size into the device. Recorded information for each hydrometeor includes orthogonal silhouette images, equivalent volume diameter, maximum width and height, and terminal velocity. Particles as small as a single photodiode are designated if both light beams are sufficiently attenuated. Particles seen by only one camera are discarded. Mismatches, common for small hydrometeors at high precipitation rates, are believed to be associated with particles outside the primary viewing area and instances with more than one particle in the viewing area at the same time (Kruger and Krajewski 2002). Mismatched particles, identified by unrealistic terminal velocities and axis ratios and significant differences in particle vertical dimensions for the two channels, are also removed. The procedure to compute equivalent volume diameters follows that of Brandes et al. (2007). Each silhouette image is composed of numerous two-dimensional sections whose size is determined by the spatial and temporal resolution of the cameras. Each areal subsection was assumed to be “coin”-shaped. A total volume estimate was made for each camera by summing the component volumes. The equivalent volume diameter was then computed as the geometric mean of the two estimates.

VOLUME 47

Derived properties of particles with diameters ⬍0.5 mm are uncertain. Estimates of hydrometeor properties improve as size increases. Calibration datasets disclose that the relative standard error in height and width measurements varies from 14% for a spherical particle with a mean diameter of 0.5 mm to ⬍1.5% for a particle with a diameter of 10 mm (Brandes et al. 2007). Particle terminal velocities are computed from the time difference a particle takes to break each camera plane. It is difficult to estimate the likely error in snowflake terminal velocities because of their dependence on density, shape, size, and orientation. From the dispersion in terminal velocities for raindrops, Brandes et al. estimate that standard error varies from 0.4 m s⫺1 for drops with diameters of 0.5 mm to ⬍0.2 m s⫺1 for drops larger than 2 mm. Errors for snowflakes should be less because of the finer vertical resolution. Disdrometer measurements have been collected in snowstorms since October of 2003. There currently are observations from more than 75 storm days. For most storms the disdrometer was operated at the National Center for Atmospheric Research Snowfall Test Site at Marshall, Colorado. Other available measurements are temperature, relative humidity, wind speed and direction, surface pressure, and visibility. Station height is 1.742 km MSL; observed pressures average about 820 hPa. All terminal velocities in this report have been adjusted to a pressure P of 1000 hPa with

␷t,1000 ⫽ ␷t,obs共Pobs ⲐP1000兲0.5, which yields a reduction of ⬃10%. Wind flow about the disdrometer can adversely impact particle measurements (Nešpor et al. 2000). To minimize wind effects, the instrument was placed within a double wind fence. Analyses were restricted to events with ambient wind speeds of ⬍4 m s⫺1. Experiments with multiple wind anemometers placed within the wind shields reveal that wind speeds within the inner shield are reduced by about two-thirds. Residual wind effects and measurement resolution issues cause terminal velocities of particles with diameters of less than 0.5 mm to be noisy. For a select number of storms, the disdrometer was placed within a refrigerated room with a 1 m ⫻ 1 m hatch in the roof. This arrangement eliminated issues regarding wind flow about the disdrometer, but particle distributions may be influenced by the breakup of snowflakes striking the sides of the hatch and drifting snow on the building roof. At times, precipitation consisted of more than one frozen hydrometeor type. The most common situation occurred during transition periods between aggregates

OCTOBER 2008

2731

NOTES AND CORRESPONDENCE

TABLE 1. Representative aggregate terminal-velocity relations. The units are meters per second for ␷t, millimeters for D, and degrees Celsius for T. Diameter definitions are maximum dimension for Lumb, Locatelli and Hobbs, and Heymsfield et al. and equivalent sphere for Jiusto and Bosworth. Study Lumb (1961)

Jiusto and Bosworth (1971) Locatelli and Hobbs (1974) Heymsfield et al. (2007b)

Relation

␷t ␷t ␷t ␷t ␷t ␷t ␷t ␷t ␷t ␷t

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

0.523D0.25 0.692D0.25 0.742D0.25 0.832D0.25 0.676D0.2 0.978D0.2 0.79D0.27 0.81D0.16 1.31e⫺0.0138T(0.1D)0.185⫺0.0084T 2.00e0.0004T(0.1D)0.244⫺0.0049T

Composition, aggregate type, or data source Unrimed, horizontal type Unrimed, vertical type Rimed, horizontal type Rimed, vertical type Dendrites Plates and columns Densely rimed Unrimed Synoptic CRYSTAL-FACE

FIG. 1. Aggregate terminal velocity plotted vs hydrometeor size for select relations in Table 1. Diameter definitions vary (Table 1).

2732

JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY

VOLUME 47

TABLE 2. Number of storm days and data points for the temperature categories in Fig. 3.

FIG. 2. The distribution of maximum hydrometeor equivalentvolume diameter plotted against wet-bulb temperature (6257 fiveminute spectra).

and snow grains. Storms were selected that were dominated by large aggregates. The measurements were edited lightly to reduce the impact of mismatched particles, outliers, and mixtures of hydrometeor types. The dataset was arbitrarily divided into periods of homogeneous precipitation as short as 5 min. The distribution of terminal velocities was determined as a function of size, and modal values were computed for 1-mm size intervals. It was assumed that the modal values were representative of the dominant particle type. Hydrometeors whose terminal velocity differed from the modal value by more than 0.5 m s⫺1 were discarded. This procedure eliminated 1%–10% of the particles.

3. Survey of terminal-velocity relations A number of aggregate terminal-velocity relations found by others are listed in Table 1 and are plotted in Fig. 1. Note that definitions of particle diameter vary. The observed size range in the cited studies matches or exceeds that shown. Relationships of Lumb (1961) are based on observations made by Magono (1953, 1954). The curves represent different aggregate compositions and whether or not riming was detected. Horizontal type refers to flat snowflakes whose shapes approximate that of plates; vertical type refers to irregular aggregations of snow crystals. Also shown are relations found by Jiusto and Bosworth (1971) and Locatelli and Hobbs (1974). Jiusto and Bosworth’s relations are stratified for compositions of columns and plates and for dendrites. Relations of Locatelli and Hobbs are for rimed and unrimed particles. Particle counts in the latter study were small, 27

Temperature category

No. of storms

No. of particles

⫺10°C ⫺5°C ⫺1°C

6 6 6

8700 10 094 24 302

and 28, respectively, which may account for a limited size range. There is general agreement among the relations. More rimed and dense forms are characterized by higher velocities, and velocities increase with particle size. However, fall speeds for a specific size vary by 25%–40%. Computed precipitation and aggregation rates would vary accordingly. Heymsfield et al. (2007b) present terminal-velocity relations that are a function of temperature (Table 1). They are believed to be valid for aggregates as large as 20 mm (A. Heymsfield 2008, personal communication). The relations are based on calculations with aircraft images obtained from several synoptic events and the Cirrus Regional Study of Tropical Anvils and Cirrus Layers–Florida-Area Cirrus Experiment (CRYSTALFACE) field programs. Fall speeds were computed using the procedure described by Mitchell and Heymsfield (2005), which does not specifically address riming effects. The relations predict higher particle fall speeds as temperatures decrease (e.g., Fig. 1). This apparently arises from a growth in particle cross-sectional area and a reduction in particle density as temperatures rise. The terminal-velocity relation for CRYSTAL-FACE shows reduced temperature dependence (not plotted) but higher terminal velocities. Falls speeds for temperatures ⱖ⫺10°C range from ⬃1.3 m s⫺1 for particles with a diameter of 2 mm to 2.4 m s⫺1 for particles with a diameter of 20 mm. Higher terminal velocities in CRYSTAL-FACE are thought to result from greater particle densities (Heymsfield et al. 2007a).

4. Observations a. Snowflake size as a function of temperature Figure 2 shows the distribution of maximum hydrometeor size (equivalent volume diameter) for the wetbulb temperature range from ⫺17° to 5°C. Sample sizes (5 min) typically consist of hundreds to thousands of particles. Maximum particle size for individual samples is probably underestimated. Simulations by Smith et al. (1993) with exponential raindrop distributions suggest that maximum diameters could be underestimated by ⬃30% for samples with 500 particles. Doubling the number of particles would reduce the error to about

OCTOBER 2008

NOTES AND CORRESPONDENCE

2733

FIG. 3. Hydrometeor terminal velocities plotted vs equivalent-volume diameter for various temperatures. Fitted relations (2)–(4) are overlaid. Standard deviations are shown by vertical bars.

23%. Snowflakes are approximated well by exponential distributions (e.g., Brandes et al. 2007; their Fig. 4); hence, the results of Smith et al. should be applicable. Interest here is on the envelope of maximum sizes, which should be less affected by sampling issues. The wet-bulb temperature is used to better represent temperatures at the surface of the aggregates. Snowflakes increase in size as temperatures warm, presumably because of dendritic growth and aggregation. Maximum particle sizes are at temperatures that are slightly warmer than 0°C. Melting rapidly reduces the size of the hydrometeors as temperatures continue to rise. In this dataset, melting appears to be completed for wet-bulb temperatures ⬎4°C. Snowflake size/temperature distributions similar to

Fig. 2 have been presented by Magono (1953), Hobbs et al. (1974), and Rogers (1974). These studies show maximum particle sizes occurring at dry-bulb temperatures of about ⫺1°C. Maximum particle diameters ranged from 14 to 45 mm. Snowflakes with dimensions as large as 60 mm at a temperature of 0.5°C have been reported (Auer 1971). The largest aggregate measured to date with our disdrometer had an equivalent volume diameter of almost 31 mm and a maximum dimension of 51 mm. It was observed on 12 May 2004 at a wet-bulb temperature of 0.7°C and near calm wind conditions. The particle terminal velocity was 1.56 m s⫺1 (reduced to 1000 hPa), and its aspect ratio (maximum vertical dimension divided by its maximum horizontal dimension) was 0.63.

2734

JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY

VOLUME 47

FIG. 4. Hydrometeor terminal velocities for four storms in the ⫺10°C category.

b. Temperature-dependent terminal-velocity relations We now derive power-law terminal-velocity relations in the putative form given by (1). In the absence of a procedure for discriminating among possible aggregate constituents or quantifying the degree of riming, we stratify the data according to temperature T and derive relations for ⫺1°, ⫺5°, and ⫺10°C. To ensure a significant number of events with large aggregates, a window of ⫺7° ⬍ T ⬍ ⫺3°C was used at ⫺5°C. Two-degree windows were used at ⫺1°C (⫺2° ⬍ T ⬍ 0°C) and ⫺10°C (⫺11° ⬍ T ⬍ ⫺9°C). For some events only every other or every third observed snowflake was used to prevent any one storm from contributing more than

25% to the total snowflake count. The number of storms and the number of particles in each temperature window are given in Table 2. The distribution of particle terminal velocities at the three temperatures is presented in Fig. 3. Select storms for the ⫺10°C category are plotted in Fig. 4 to show storm-to-storm variation. Traditional least squares fits to all particles in each temperature category were influenced by the large numbers of poorly sampled small particles and did not agree with terminal velocities of the largest particles very well. To improve the fit for large particles, the mean terminal velocity in 1-mm size bins was computed. Plotted curves in Fig. 3 are fits to these data. The derived relations are

OCTOBER 2008

NOTES AND CORRESPONDENCE

2735

FIG. 5. (top) Hydrometeor terminal velocities plotted vs equivalent-volume diameter for two time periods on 13 Feb. (bottom) Representative snowflakes. The distance between bars is 1 mm.

␷t ⫽ 0.55D0.23, T ⫽ ⫺10⬚C,

共2兲

␷t ⫽ 0.67D0.25, T ⫽ ⫺5⬚C, and

共3兲

␷t ⫽ 0.87D0.23, T ⫽ ⫺1⬚C.

共4兲

Mean standard deviations (Fig. 3) vary from 0.17 m s⫺1 at ⫺10°C to 0.22 m s⫺1 at ⫺1°C. There is a fairly systematic increase in average fall speed as temperatures warm. At ⫺10°C, aggregates with an equivalent volume diameter of 10 mm have a terminal velocity of a little more than 0.9 m s⫺1. Snowflakes of that size at ⫺5°C have a fall speed that averages about 1.2 m s⫺1. At ⫺1°C, the average fall speed is 1.5 m s⫺1—more than 0.5 m s⫺1 higher than at ⫺10°C. Relations (2)–(4) bracket those in Fig. 1. Riming most likely causes the increase in fall speed. This conclusion is supported by Fig. 5, which shows particle terminal velocities for two short time periods during the 13 February 2007 event. Terminal velocities

for 1700–1705 UTC (upper-left panel) averaged 0.14 m s⫺1 higher than during 2250–2255 UTC. The lower panels show representative dendrites and aggregates of dendrites. Those for the earlier period are more heavily rimed, as indicated by their thickened branches and more whitish cast.

5. Summary and discussion The relationship between aggregate terminal velocity and temperature at ground for storms along the Front Range in eastern Colorado was investigated with measurements from a video disdrometer. Fall speeds increased with temperature. At ⫺10°C and an equivalent volume diameter of 10 mm, mean fall speeds were less than 1 m s⫺ 1. Fall speeds for aggregates of this size at temperatures of ⫺5° and ⫺1°C were 1.2 and 1.5 m s⫺1, respectively. The derived relations, believed to be applicable to orographic and stratiform snow-

2736

JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY

storms, capture the variation seen in studies that stratify the aggregates according to composition and the degree of riming. The mean rise in speed with temperature for a specific particle size is largely ascribed to riming. Higher temperatures are associated with an increased likelihood of and potentially higher concentrations of supercooled liquid. The presence of supercooled liquid may also foster particle growth and aggregation, which could lead to even more riming (Harimaya 1975). The increase in snowflake terminal velocity with size is characteristic of observational studies but differs from the calculations of Mitchell and Heymsfield (2005) and Khvorostyanov and Curry (2005), which predict a decrease in velocities for unrimed aggregates with diameters larger than about 10 mm. Mitchell and Heymsfield attribute the decline to a thickening of the boundary layer that forms about aggregates and the larger “effective” area the particles present as they fall. Our observations also differ with the calculations of Heymsfield et al. (2007b), who determined that terminal velocities for aggregates decrease as temperatures rise. The apparent discrepancy between the observed and calculated terminal velocities is believed to stem from the nontreatment of riming effects in the calculations. With the simple power-law form the distribution of terminal velocities tends to flatten for large aggregates. Mitchell and Heymsfield (2005) argue that the flattening is associated with a reduced likelihood of collisions and aggregation. Although some flattening of the terminal velocity distributions occurred in our datasets, the velocity range is relatively large at particular sizes, presumably because of factors such as snowflake composition, degree of riming, shape, and orientation. Further, the range in mean fall speeds of small and big aggregates is also large. Hence, the likelihood of collisions remains high. Acknowledgments. This study was supported by funds from the National Science Foundation designated for U.S. Weather Research Program activities at the National Center for Atmospheric Research (NCAR). Data collection was largely in response to requirements of and funding from the Federal Aviation Administration (FAA). The views expressed are those of the authors and do not necessarily represent the official policy or position of the FAA. REFERENCES Auer, A. H., 1971: Some large snowflakes. Weather, 26, 121–122. Brandes, E. A., K. Ikeda, G. Zhang, M. Schönhuber, and R. M.

VOLUME 47

Rasmussen, 2007: A statistical and physical description of hydrometeor distributions in Colorado snowstorms using a video disdrometer. J. Appl. Meteor. Climatol., 46, 634–650. Hanesch, M., 1999: Fall velocity and shape of snowflakes. Ph.D. dissertation, Swiss Federal Institute of Technology, 117 pp. Harimaya, T., 1975: The riming properties of snow flakes. J. Meteor. Soc. Japan, 53, 384–392. Heymsfield, A. J., and M. Kajikawa, 1987: An improved approach to calculating terminal velocities of plate-like crystals and graupel. J. Atmos. Sci., 44, 1088–1099. ——, A. Bansemer, and C. H. Twohy, 2007a: Refinements to ice particle mass dimensional and terminal velocity relationships for ice clouds: Part I: Temperature dependence. J. Atmos. Sci., 64, 1047–1067. ——, G.-J. van Zadelhoff, D. P. Donovan, F. Fabry, R. J. Hogan, and A. J. Illingworth, 2007b: Refinements to ice particle mass dimensional and terminal velocity relationships for ice clouds: Part II: Evaluation and parameterizations of ensemble ice particle sedimentation velocities. J. Atmos. Sci., 64, 1068–1088. Hobbs, P. V., S. Chang, and J. D. Locatelli, 1974: The dimensions and aggregation of ice crystals in natural clouds. J. Geophys. Res., 79, 2199–2206. Jiusto, J. E., and G. E. Bosworth, 1971: Fall velocity of snowflakes. J. Appl. Meteor., 10, 1352–1354. Khvorostyanov, V. I., and J. A. Curry, 2005: Fall velocities of hydrometeors in the atmosphere: Refinements to a continuous analytical power law. J. Atmos. Sci., 62, 4343–4357. Kruger, A., and W. F. Krajewski, 2002: Two-dimensional video disdrometer: A description. J. Atmos. Oceanic Technol., 19, 602–617. Langleben, M. P., 1954: The terminal velocity of snowflakes. Quart. J. Roy. Meteor. Soc., 80, 174–181. Locatelli, J. D., and P. V. Hobbs, 1974: Fall speeds and masses of solid precipitation particles. J. Geophys. Res., 79, 2185–2197. Lumb, F. E., 1961: Relation between the terminal velocity and dimensions of snowflakes. Meteor. Mag., 90, 344–347. Magono, C., 1953: On the growth of snow flake and graupel. Yokohama National University Sci. Rep., Volume 2, 18–40. ——, 1954: On the falling velocity of solid precipitation elements. Yokohama National University Sci. Rep., Volume 3, 33–40. Mitchell, D. L., and A. J. Heymsfield, 2005: Refinements in the treatment of ice particle terminal velocities, highlighting aggregates. J. Atmos. Sci., 62, 1637–1644. Nešpor, V., W. F. Krajewski, and A. Kruger, 2000: Wind-induced error of rain drop size distribution measurement using a twodimensional video disdrometer. J. Atmos. Oceanic Technol., 17, 1483–1492. Rogers, D. C., 1974: The aggregation of natural ice crystals. Department of Atmospheric Research Rep. AR110, University of Wyoming, Laramie, WY, 35 pp. Smith, P. L., Z. Liu, and J. Joss, 1993: A study of samplingvariability effects in raindrop size observations. J. Appl. Meteor., 32, 1259–1269. Yuter, S. E., D. Kingsmill, L. B. Nance, and M. Löffler-Mang, 2006: Observations of precipitation size and fall speed characteristics within coexisting rain and snow. J. Appl. Meteor. Climatol., 45, 1450–1464.