a statistical and physical description of hydrometeor ...u0758091/hydrometeors/brndes...a...

A Statistical and Physical Description of Hydrometeor Distributions in ColoradoSnowstorms Using a Video Disdrometer

EDWARD A. BRANDES AND KYOKO IKEDA

National Center for Atmospheric Research,* Boulder, Colorado

GUIFU ZHANG

University of Oklahoma, Norman, Oklahoma

MICHAEL SCHÖNHUBER

Joanneum Research, Graz, Austria

ROY M. RASMUSSEN

National Center for Atmospheric Research,* Boulder, Colorado

(Manuscript received 1 February 2006, in final form 17 July 2006)

ABSTRACT

Winter-storm hydrometeor distributions along the Front Range in eastern Colorado are studied with aground-based two-dimensional video disdrometer. The instrument provides shape, size, and terminal ve-locity information for particles that are larger than about 0.4 mm. The dataset is used to determine the formof particle size distributions (PSDs) and to search for useful interrelationships among the governing pa-rameters of assumed distribution forms and environmental factors. Snowfalls are dominated by almostspherical aggregates having near-exponential or superexponential size distributions. Raindrop size distri-butions are more peaked than those for snow. A relation between bulk snow density and particle medianvolume diameter is derived. The data suggest that some adjustment may be needed in relationships foundpreviously between temperature and the concentration and slope parameters of assumed exponential PSDs.A potentially useful relationship is found between the slope and shape terms of the gamma PSD model.

1. Introduction

Observations of particle size distributions in winterstorms are needed to verify and improve microphysicalparameterizations in numerical forecast models and toquantify winter precipitation accurately, discriminateamong hydrometeor types, and develop algorithms fordetermining particle size distributions with remote sen-sors such as polarimetric radar. This study examinesbulk characteristics of observed particle distributions at

the ground using a two-dimensional video disdrometer.The instrument is manufactured by Joanneum Re-search at the Institute of Applied Systems Technologyin Graz, Austria. It has been used previously to studythe distribution of raindrops (Williams et al. 2000;Tokay et al. 2001; Kruger and Krajewski 2002), dropaxis ratios (Thurai and Bringi 2005), and fall velocities(Thurai and Bringi 2005). To our knowledge this is thefirst application to document particle distributions inwinter storms.

We begin with a description of the disdrometer andanalysis procedures and demonstrate instrument capa-bilities. Particle observations are fit with exponentialand gamma distribution models; and the governingparameters of the distributions are determined. Physi-cal properties of winter precipitation, such as particlebulk density, shape, terminal velocity, maximum andmedian volume diameter, and snowfall rate, are inves-

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

Corresponding author address: Dr. Edward A. Brandes, Na-tional Center for Atmospheric Research, P.O. Box 3000, Boulder,CO 80307.E-mail: [email protected]

634 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 46

DOI: 10.1175/JAM2489.1

© 2007 American Meteorological Society

JAM2489

tigated. Interrelationships among particle size distribu-tion (PSD) parameters and relationships with tempera-ture and relative humidity are examined. Findings arecompared with published studies of frozen precipita-tion, and implications for microphysical parameteriza-tion in numerical forecast models are discussed.

2. Instrumentation

Kruger and Krajewski (2002) give a detailed techni-cal description of the disdrometer calibration andcomputational procedures. The instrument consists oftwo horizontally oriented line-scan cameras, separatedin the vertical by about 6 mm, which provide orthogo-nal views of hydrometeors falling through a common10 cm � 10 cm area. Blocked photo detectors for eachcamera are recorded at a line-scan frequency of 51.3kHz. Horizontal resolution is approximately 0.15 mm.Vertical resolution depends on particle terminal veloc-ity and is roughly 0.1�0.2 mm for raindrops and 0.03mm for snowflakes. The instrument is calibrated everyfew months by dropping graduated spheres with diam-eters of 0.5–10 mm into the device.

Particles as small as a single bin are designated if thelight beams are sufficiently attenuated. Particles seenby only one camera are discarded. Mismatches arecommon for small hydrometeors. The mismatches arebelieved to be associated with particles outside the vir-tual viewing area and instances of more than one par-ticle in the viewing area at the same time (Kruger andKrajewski 2002). Mismatched particles, identified byodd shapes and unrealistic terminal velocities, are re-

moved from the dataset by imposing thresholds. Esti-mates of hydrometeor properties improve as particlesize increases. Using the calibration spheres, we esti-mate that the relative standard error in the height andwidth measurement varies from 14% for a particle witha mean diameter of 0.5 mm to less than 1.5% for aparticle with a diameter of 10 mm. The error in axisratios varies from 30% to 2% over this size range. Ob-served particle terminal velocities �obs are determinedfrom the time difference a particle takes to break eachcamera plane. Estimated fall speeds can be verified bycomparing computed values for raindrops with labora-tory experiments. From the dispersion in velocities ondays with calm winds we estimate the standard error tobe 0.4 m s�1 for drops with a diameter of 0.5 mm andless than 0.2 m s�1 for drops with diameters larger than2 mm.

Recorded information for each hydrometeor in-cludes front and side silhouette images (Fig. 1), equiva-lent volume diameter, maximum width and height, anestimate of oblateness (valid for raindrops), and termi-nal velocity. A wealth of information regarding precipi-tation-sized particles from numerous storm types and avariety of temperature and humidity conditions isreadily obtained.

The disdrometer was installed at the National Centerfor Atmospheric Research Snowfall Test Site at Mar-shall, Colorado (Rasmussen et al. 2001). Other instru-mentation included thermometers, a hygrometer, an-emometers, snow gauges, and a visibility sensor. Dis-drometer measurements are influenced by the wind(Nešpor et al. 2000). Problems are exacerbated for

FIG. 1. Sample video disdrometer images. Front (gray) and side (black) profiles are shown. Size increments are (a), (b) 2 and (c),(d) 1 mm.

MAY 2007 B R A N D E S E T A L . 635

snow particles because of their small terminal veloci-ties. Wind-affected observations can be identified bythe distribution of hydrometers within the viewing re-gion and increased scatter among terminal velocity es-timates. To minimize wind effects, the disdrometer wasplaced within a double-fence intercomparison refer-ence wind shield. A Geonor Model T-200 snow gaugehaving a resolution of 0.0254 mm was also placed withinthe wind shield. Gauge performance for measuringsnow accumulations has been evaluated by Rasmussenet al. (2001). In general, the gauge agrees within �5%of that from manual measurements. Our analyses arerestricted to events with ambient wind speeds of lessthan 4 m s�1. Visual inspection of particle spatial dis-tributions disclosed that this threshold eliminated ob-servations with obvious wind effects. Nevertheless,some undersampling, particularly of small particles, islikely (Nešpor et al. 2000). Undersampling of small par-ticles as a result of wind losses and aforementionedmeasurement issues dictates that hydrometeor proper-ties and concentrations for particles smaller than about0.4 mm are regarded as suspect.

3. Data and analysis

Measurements were made during October–April for2003 to 2005. The dataset consists of 113 h of observa-tions from 52 storm days. Only snow was observed for23 events, and only rain was observed for 7 events. Theremaining storms typically began as rain that laterchanged to snow. Surface temperatures were as low as�17°C, but approximately 80% of the observationswere obtained at temperatures above �5°C. For a smallnumber of storms, an observer was on site to record thedegree of riming and hydrometeor habits. Riming wasusually light—that is, dendrites were readily identified;on occasion, however, graupel was observed.

Winter precipitation along the Front Range primarilyoccurs under upslope conditions (Mahoney et al. 1995).About one-half of the events were postfrontal. The oth-ers were split almost equally between leeside cyclo-genesis and traveling surface low pressure systems.Observed particle distributions were dominated by ag-gregates. Storms dominated by graupel, ice pellets,dendritic crystals, and ice needles were not observed.Therefore, no attempt was made to discriminate amonghydrometeor habits.

Particle size distributions were fit with the exponen-tial model (e.g., Marshall and Palmer 1948; Gunn andMarshall 1958)

N�D� � N0 exp��D�, �1�

where N0 (mm�1 m�3) is a concentration intercept pa-

rameter, � (mm�1) is a slope term, and D (mm) is the

particle equivalent volume diameter. The observationswere also fit with the gamma model (e.g., Ulbrich 1983)

N�D� � N0D� exp��D�, �2�

where N0 (mm��1 m�3) is now a number concentra-

tion parameter, is a distribution shape or curvatureparameter, and � (mm�1) is a slope term sensitive tothe larger particles. The governing parameters in (1)and (2) were estimated from the third and sixth mo-ments and the second, fourth, and sixth moments of theobserved particle distributions, respectively. The pro-cedure is described by Vivekanandan et al. (2004). Themoments, using the gamma model as an example, arecalculated from

Dn� � �Dmin

Dmax

Dnn�D� dD � N0��n�1�

� �� n � 1, �Dmax� � �� n � 1, �Dmin��,

�3�

where n is the moment number, �() is the incompletegamma function, Dmin is the diameter of the smallestparticle in the distribution, and Dmax is the largest par-ticle. The procedure yields three equations with threeunknowns that are solved by an iterative method. Thegoverning parameters of the PSD were computed for5-min samples. In a typical case, each spectrum con-tained hundreds to thousands of hydrometeors.

Once the PSD is known, other attributes can be com-puted. The median volume diameter D0 of the particlesis defined as

�Dmin

D0

D3N�D� dD � �D0

Dmax

D3N�D� dD, �4�

where one-half of the precipitation volume is containedin particles smaller than D0 and one-half is contained inparticles larger than D0. The total number concentra-tion NT is

NT � �Dmin

Dmax

N�D� dD, �5�

and the mean terminal velocity � t is

� t �

�Dmin

Dmax

N�D��obs�D� dD

�Dmin

Dmax

N�D� dD

, �6�

where �obs is the observed particle velocity. Other char-acteristic velocities can be computed, for example, by


weighing each observation according to its volume ormass.

Random sampling from a population of hydromete-ors causes a bias in estimated PSD attributes (Smith etal. 1993; Smith and Kliche 2005). The bias, an under-estimate, decreases as the sample size increases. Withour sample sizes the bias in most PSD attributes shouldbe small. For example, based on the results of Smith etal. we estimate that the bias in D0 is less than 5%. Anexception is Dmax, for which the bias could be as largeas 30%.

Precipitation characteristics for a long-lived snowevent on 20 November 2004 are plotted in Fig. 2. Thesurface air temperature was 0.6°C initially, cooled to0°C at 0645 UTC, and varied between 0° and �3°Cafterward. The top panel shows snowfall rate (waterequivalent in millimeters per hour) as measured bysnow gauge and computed from disdrometer measure-ments using a density–size relation described in section4a. Other panels show D0 and Dmax, NT, and � t . Ingeneral, displayed parameters show fair stability fromsample to sample. Increases in precipitation rate at0915 and 1520 UTC coincide with increases in D0 andDmax as well as an increase in total number concentra-tion. The rate increase at 0915 UTC was marked by adecrease in particle terminal velocity, whereas the rateincrease at 1520 UTC shows an increase. Relativelyheavy snowfall rates after 2030 UTC do not show asignificant increase in particle size but show an order-of-magnitude increase in number concentration.

Among characteristic velocities, mean values are thesmallest because more numerous and slower-fallingsmall particles have the same weight as less plentifuland faster-falling large particles. Mass-weighted termi-nal velocities are slightly less than volume-weighted ve-locities because mass increases more slowly than vol-ume as the particle diameter increases.

The gamma distribution has been widely accepted bythe meteorological radar community for raindrops(e.g., Jameson 1991; Schuur et al. 2001; Bringi et al.2002; Illingworth and Blackman 2002) because itreadily describes a variety of observed distributionswhile maintaining a simple and efficient functionalform. Application to snowflakes needs some justifica-tion. Modelers and observationalists often assume thatparticles in winter storms are exponentially distributed.Figure 3 presents PSD examples from a snow event on18 March 2003. The sharp downturn at the smallest sizein the top panel is believed to be a manifestation ofsmall particle detection issues (section 2). Fitted rela-tions for truncated-exponential and truncated-gammadistributions are overlaid. Computed properties andgoverning parameters for the two PSD models are sum-

marized in Table 1. Cursory inspection of Fig. 3 sug-gests that the gamma distribution model gives a betterrepresentation of distributions that are nonlinear in thesemilogarithmic plot—that is, the upward-curving dis-tribution of 2125–2130 UTC and the downward-turningdistribution of 2230–2235 UTC.

The frequency of the gamma PSD shape factor differs for winter snowstorms and rainstorms (Fig. 4).The distribution for snow is skewed with a mode of �1or close to exponential. Twenty-two percent of the sare negative, an indication that small particle con-centrations often exceed that of an exponential dis-tribution. Negative s are common with ice particledistributions derived from aircraft observations (e.g.,Heymsfield et al. 2002; Heymsfield 2003). The mode value for winter rain is 6. Only, 4% of the values areless than zero. While the mode values could be used todefine a special gamma distribution with a constant ,that simplification could lead to significant error if thetotal particle concentration, coalescent, or evaporativeproperties of the distribution are desired.

Figure 5 presents a time series of D0 and Dmax, esti-mates of for a truncated-gamma PSD, and � and NTfor both truncated-gamma and truncated-exponentialPSDs. The data are for an event on 1 November 2004during which precipitation began as rain, became mixedphase, and finally changed to snow. Snowfall rates var-ied between 0.5 and 6 mm h�1. Temperatures fell from4°C at 0000 UTC to �1°C at 0500 UTC. The averagegamma PSD shape parameter for the rain stage is ap-proximately 3. The shape parameter decreases to nega-tive values as the precipitation begins the change tosnow (0045 UTC). Negatives at this stage are associatedwith bimodal spectra composed of a few relatively largewetted snowflakes and large numbers of small rain-drops and ice particles that are narrowly distributed.The gamma model can be inappropriate in these situ-ations. As the precipitation turns to all snow (�0145UTC), the shape factor increases to about 0, indicatingthat the distribution is near exponential. After 0315UTC, averages between �1 and �2. Examination ofthe particle spectra at this stage reveals superexponen-tial distributions much like the middle panel in Fig. 3.Whenever is negative, the slope of the fitted trun-cated-exponential PSD is larger than that of the trun-cated-gamma PSD.

The truncated-exponential PSD overestimates par-ticle total number concentration for rain and underes-timates the concentration for mixed-phase and snowportions of the storm (Fig. 5, bottom panel). The expo-nential model is a poor fit during the mixed-phasestage, and for some spectra the iterative procedure usedto compute the PSD governing parameters does not

MAY 2007 B R A N D E S E T A L . 637

converge to a solution. The truncated-gamma distribu-tion overestimates the concentration for negative .The difference arises from the handling of small par-ticles. The disdrometer-observed number concentra-

tion turns downward as the diameter approaches 0, asin the top panel of Fig. 3; while the fitted distributionturns upward. [The fitted distributions are truncated atDmin � 0.1 mm to prevent an infinite number of par-

FIG. 2. Time series of observed PSD attributes computed for a storm on 20 Nov 2004: (top) snow rate S as computed from disdrometerobservations and measured by snow gauge, (2d from top) median volume diameter D0 and maximum particle size Dmax, (3d from top)total number concentration NT , and (bottom) characteristic terminal velocities � t .


Fig 2 live 4/C

ticles with the gamma PSD model.] For the most part,disdrometer observations of NT lie between estimatesfor the truncated-exponential and truncated-gammaPSDs. However, the gamma model provides a muchbetter fit to the observed hydrometeor distributionsduring the rain and mixed-phase portions of the event.During the rain stage, the fitted NT estimates with the

gamma distribution model are higher than the disdrom-eter by a factor of 1.2, whereas the estimate with theexponential model is larger by a factor of 4.9.

4. PSD physical attributes

a. Snow bulk density

A critical issue for estimating liquid equivalents andfor quantifying PSD attributes with polarimetric radarmeasurements is the relationship between particle sizeand density. To determine density, snowflake volumeswere computed from the disdrometer observations.Each silhouette image (e.g., Fig. 1) is composed of nu-merous two-dimensional sections whose dimensions aredetermined by the spatial and temporal resolution ofthe cameras. Each areal subsection was assumed to be“coin” shaped. The total volume estimate was found bysumming the component volumes. The final particlevolume estimate was taken as the geometric mean ofthe individual estimates from both cameras. Bulk snowdensity �s was determined from the 5-min disdrometer-derived precipitation volume and the correspondinggauge-measured precipitation mass. Sensitivity to thenumber of particles, calculated volumes for often highlyirregular particle shapes, and gauge quantization at lowprecipitation rates dictates that density estimates areapproximate. However, density calculations for intensesummer rain events average close to 1 g cm�3, indicat-ing that the method has merit.

Figure 6a shows the relation between �s and particlemedian volume diameter. Data points from specificdays tend to cluster. This is illustrated by observationsfrom a roughly 5-h segment on 28 November 2004.Clustering attests to the importance of meteorologicalconditions, revealing the prevalence of aggregates orsnow pellets and whether riming is light or heavy.

FIG. 3. Observed 5-min PSDs (number concentration plotted vsequivalent diameter) for a long-lived snow event on 18 Mar 2003.Computed PSD properties are given in Table 1. Surface tempera-tures varied between 0° and 0.3°C.

TABLE 1. Computed PSD attributes for the distributions in Fig.3. Fitted values are presented for truncated particle size distribu-tions. Here CT is the number of particles observed in the 5-mininterval. Gauge-observed snowfall rates S (liquid equivalents) arealso shown. Units used: NT (m

�3), N0 (m�3 mm��1 or

m�3 mm�1), D0 (mm), � (mm�1), and S (mm h�1). Times are in

UTC.

Time interval 2050–2055 2125–2130 2230–2235

NT 1.16 � 103 1.89 � 102 3.68 � 103

CT 3010 564 11 914D0 1.79 6.37 2.09N0 (gamma) 1.39 � 10

3 7.18 � 101 2.33 � 104

(gamma) �0.90 �0.78 1.23� (gamma) 1.40 0.35 2.25N0 (exponential) 1.91 � 10

3 1.82 � 101 1.85 � 104

� (exponential) 1.80 0.35 1.77S 2.70 2.59 3.03

MAY 2007 B R A N D E S E T A L . 639

The overlaid red curve, a least squares fit applied inan attempt to determine a climatological relation, isgiven by

�s�D� � 0.178D0�0.922, �7�

where D0 is in millimeters and �s is in grams per cen-timeter cubed. This relation excludes four outliers. Thecorrelation coefficient is 0.82. A relation for snow par-ticle mass m (in grams) corresponding to (7) is

m�D� � 8.90 � 10�5D02.1. �8�

Note that (7) does not match the observations at me-dian volume diameters of less than 1 mm. Also, thereare no observations with D0 of less than 0.6 mm. Hence,the relation does not apply to this portion of the sizespectrum.

Table 2 presents several density–snowflake size rela-tions found by others. All relations are plotted in Fig.6a. Particle diameter definitions vary. Magono and Na-kamura (1965) and Holroyd (1971) use the geometricmean of the particle major and minor axes as seen fromabove. The Muramoto et al. (1995) relation is based onthe maximum horizontal dimension, and the Fabry andSzyrmer (1999) relation is based on an equivalentvolume diameter. Heymsfield et al. (2004) define thediameter to be that of the minimum circumscribedcircle that encloses the projected area of the particle.The Magono and Nakamura relation is for dry and wetsnows. Holroyd used the dry snow data of Magonoand Nakamura. The Heymsfield et al. (2004) data arealso for dry snow. The relation of Fabry and Szyrmer isan average relation that summarizes several studies.The particulars of the Muramoto et al. study are notknown.

Differences in the definition of particle diameter, in-strumentation, and precipitation climatological charac-teristics are all likely contributors to the scatter amongrelationships and make direct comparison difficult.Equation (7) is intermediate among the selected rela-tions and closely agrees with those of Holroyd (1971)and Fabry and Szyrmer (1999). Higher densities withthe Magono and Nakamura relation probably followfrom their inclusion of wet snowflakes. Lower densitiesfound by Muramoto et al. likely results from their useof the maximum particle dimension.

In the top panel of Fig. 2, snowfall rates computed bymultiplying the volume of individual particles by den-sities from (7) are compared with the snow rate mea-sured with a gauge. Overall, the comparison is good,but differences that are as large as 0.5 mm h�1 occurfor the heavier snow rates after 2000 UTC. Snow par-ticles during this storm stage were more dense than isgiven by (7). Median volume diameters are relativelysmall (�2 mm) for this stage, suggesting that an impor-tant proportion of particles may not have been de-tected.

Particle bulk density is plotted against surface tem-perature in Fig. 6b. A fitted curve has been added toshow the mean trend. The correlation coefficient forthe entire temperature range is low (�0.14). There is,however, a tendency for low densities to become morefrequent as temperatures warm above �5°C. This isbelieved to be a consequence of increased aggregation(discussed further in section 4c). Figure 6c examines therelationship between bulk density and relative humid-ity. A mean tendency is also evident for the frequencyof low-density aggregates to increase as humidity in-creases above about 95%, but again the correlation islow (�0.12). A negative correlation might be expected

FIG. 4. Relative frequency of the truncated-gamma PSD shape factor for winter (left) snowstorms and (right) rainstorms. Thetotal number of snow and rain spectra is 916 and 308 and the number of storm days is 30 and 11, respectively.


because high humidity fosters particle growth by aggre-gation (Hosler et al. 1957). It is undoubted that morethan surface meteorological conditions determine bulkdensity.

b. Aggregate aspect ratios

To discriminate among particle habits with polari-metric radar, mean particle dimensions and orienta-tions must be known. The shapes of raindrops and pris-

tine ice crystals are well known. Less is known aboutthe mean shape of aggregates. Aspect ratios r, definedhere as the ratio of the maximum vertical dimensiondivided by the maximum horizontal dimension, are il-lustrated in Fig. 7. Although this ratio differs from thatobtained by fitting the images with ellipses and dividingthe minor axis by the major axis, the current definitionallows comparison with radar measurements of differ-ential reflectivity in a statistical sense. Aspect ratio scat-

FIG. 5. Time series of PSD attributes for 1 Nov 2004 showing (top) D0 and Dmax, (2d fromtop) the shape parameter for a truncated-gamma PSD, (3d from top) the slope parameter� for truncated-gamma and truncated-exponential PSDs, and (bottom) estimates of totalconcentration NT .

MAY 2007 B R A N D E S E T A L . 641

ter is large for small particles. Some of the scatter, inparticular at the smallest sizes, probably stems frominstrument sensitivity. Large ratios associate with ag-gregates whose axis of elongation is closer to vertical

than horizontal. Many small particles with small ratiosare branched crystals much like that seen in Fig. 1d.The scatter in aspect ratios decreases as size increases.At a diameter of 2 mm, ratios range from approxi-mately 0.4 to 5. At a diameter of 8 mm, the range is onlyfrom 0.5 to 1.5.

The curve in Fig. 7 is a fit applied to modal values ofaspect ratios for 0.2-mm size bins. Ratios increaseslightly with size from 0.9 to 1.0. Although an increasein aspect ratios is the usual case, distributions in whichthe aspect ratio decreases slowly with size can also befound. The scatter is large; hence the fitted relation isprobably not significant. The usual case for large aggre-gates seems to be an aspect ratio between 0.9 and 1.0 inthe mean. This finding is consistent with that of Ma-gono and Nakamura (1965) who show aggregates to belargely spherical (their Fig. 2).

c. Terminal velocity

Examples of observed particle terminal velocities fora storm on 5 March 2004 are given in Fig. 8. The toppanel shows mixed-phase precipitation detected be-tween 0100 and 0115 UTC. The temperature fell from5.5° to 0.5°C during the period. Raindrops, ice pellets,and wetted aggregates were observed. From 0145 to0200 UTC the temperature was approximately 0.1°C(middle panel). Hydrometeor habits were dendrites,plates, stellars, and aggregates of these forms. Duringthis stage, terminal velocities were weakly dependenton size, varying from about 0.8 m s�1 for a particle withan equivalent volume diameter of 1 mm to 1.1 m s�1

for a particle with a diameter of 11 mm. In the mean,the observed velocities for larger particles are within0.1 m s�1 of that reported by Locatelli and Hobbs(1974) for unrimed aggregates (their Fig. 20). Between0220 and 0235 UTC the temperature lowered to�0.5°C. Hydrometeor habits were classified by an ob-server as irregular snow particles and lump graupel.Comparison with relations of Locatelli and Hobbsshows the observed terminal velocities to be slightlyhigher than their densely rimed aggregates but not ashigh as their low-density graupel. The temporal varia-

FIG. 6. Relationships between bulk density and (a) particle me-dian volume diameter, (b) ambient temperature, and (c) relativehumidity. The red curve in (a) is (7); expressions for the remainingcurves are given in Table 2. The dataset consists of 768 spectrafrom 28 storm days. Yellow data points in (a) are for 0345–0855UTC 28 Nov 2004.

TABLE 2. Snowflake density–particle size relation comparison.

Study Relation

Magono and Nakamura (1965) �s � 2D�2

Holroyd (1971) �s(D) � 0.17D�1

Muramoto et al. (1995) �s(D) � 0.048D�0.406

Fabry and Szyrmer (1999) �s(D) � 0.15D�1

Heymsfield et al. (2004) �s(D) � 0.104D�0.95

Eq. (7) �s(D) � 0.178D�0.9220


Fig 6 live 4/C

tion in the size–terminal velocity relation seen in Fig. 8is typical of winter storms along the Front Range.

Interrelationships among mass-weighted terminal ve-locity, size (D0), and ambient temperature are illus-trated in Fig. 9. Individual data points are color codedfor density. Small, high-density particles with terminalvelocities of greater than 1.2 m s�1 (Fig. 9a) are indica-tive of small lump graupel or snow pellets (Zikmundaand Vali 1972; Locatelli and Hobbs 1974). Large, lessdense particles are aggregates. Terminal velocities ofthese particles closely match those found by Locatelliand Hobbs for aggregates. Fall speeds for aggregatesincrease slowly with size despite the decrease in particledensity. This result is similar to that of Locatelli andHobbs who found that aggregates fall faster than theirconstituents. The terminal velocity for spherical snow-flakes can be computed (Pruppacher and Klett 1997) as

� t � � 4g�sD3CD�a�0.5

, �9�

where g is the acceleration of gravity, CD is the dragcoefficient, and �a is the density of air. Fall velocitiesincrease as particle density and size increase and de-crease as the drag coefficient increases. Particle bulkdensity and size are inversely related [e.g., (7)]. Becausebulk density varies according to D�0.922, the terminalvelocity is a little more sensitive to D than �s is. Theirproduct increases slowly as D increases. [That the ex-ponent in the density–size relation may be greater than�1 is supported by the Muramoto et al. (1995) andHeymsfield et al. (2004) studies.] This could partly ex-plain the increase in terminal velocity seen for largeaggregates. Drag coefficient impacts have not been in-vestigated. Magono and Nakamura (1965) determinedthat the drag coefficient for dry snow particles was near

constant. Fall speeds could increase if the drag coeffi-cient decreased with particle size. We intuitively expectdrag to increase for large fluffy aggregates.

There is a relationship between mass-weighted ter-minal velocity and temperature (Fig. 9b). As tempera-tures warm above about �5°C, fall speeds increase no-ticeably on average from about 0.9 to 1.3 m s�1. Theincrease, seen for all density categories, is most likelyrelated to aggregation and corresponding increases inparticle size. The relation between particle median vol-ume diameter and temperature is presented in Fig. 9c.Our dataset is limited in that not all temperatures arerepresented, but, as temperature increases above �7°Cor so, aggregation, as suggested by the mean increase inparticle size, becomes ever more active and the spreadin particle median volume diameters increases. Hosler

FIG. 7. Particle aspect ratios: maximum vertical dimension di-vided by the horizontal dimension. The line shows the modalshape of particles �10 mm. The data are from 0200 to 0220 UTC1 Nov 2004. Precipitation was dominated by irregular ice crystalsand aggregates.

FIG. 8. Observed hydrometeor terminal velocities for three timeperiods in the storm of 5 Mar 2004. Fitted relations are overlaid.The raindrop relation is from Brandes et al. (2002).

MAY 2007 B R A N D E S E T A L . 643

et al. (1957) found a temperature of �4°C and Hobbs etal. (1974) found a temperature of �5°C as the point atwhich particle stickiness increases and aggregation isenhanced. [The increase in stickiness is attributed tothe growth of a quasi-liquid layer that forms on icesurfaces (Furukawa et al. 1987; Rosenberg 2005).]Largest D0s in our dataset were at temperatures greaterthan �1°C. This agrees with the findings of Hobbs et al.They also found a secondary dendritic growth region inthe temperature range from �12° to �17°C. It is un-

fortunate that the dataset collected to date has too fewobservations in this range to verify this finding.

d. Snowfall rate

Snowfall rate (S, liquid equivalent) and bulk densityin winter precipitation are inversely related (Fig. 10a).1

This fact is not surprising given that heavy snowfallrates are often characterized by aggregates and warmertemperatures. Heavy rates with dense pristine ice crys-tals or snow pellets simply were not observed. Snow-falls with very light rates and very low bulk densitieswere seldom observed. Perhaps there are too few par-ticles at low precipitation rates to grow large aggre-gates.

Snow particle terminal velocity and snowfall rate areweakly related (Fig. 10b). Most of the data points atlight snowfall rates are aggregates and have a terminalvelocity of approximately 1 m s�1. On-site particle ob-servations support the notion that data points with lightsnowfall rates and relatively high �t are snow pellets orlump graupel. The increase in mass-weighted terminalvelocity at high snow rates for aggregates is believed toarise primarily from an increase in particle size (Fig.10c). For the Colorado Front Range, the most commonsituation appears to be a snowfall rate of about 1 mmh�1 and a D0 on the order of 1.5 mm.

For low snow rates there is no obvious relation withthe shape parameter of the gamma PSD (Fig. 10d). Thecurvature term varies considerably from small negativevalues to more than 5. Although the sample size issmall, heavy snow rates tend to be slightly superexpo-nential.

5. Numerical model microphysics parameterization

Some numerical forecast models incorporating sec-ond- or higher-moment particle size distributions suchas (1) and (2) close the system of unknowns by fore-casting the precipitation mass and using empirical rela-tionships between the governing parameters of the dis-tribution and temperature (e.g., Reisner et al. 1998;Hong et al. 2004; Thompson et al. 2004). Disdrometer-derived values of N0 for storms in Colorado, assuminga truncated-exponential PSD, are plotted against tem-perature in Fig. 11 (top panel). A fit to the data is

N0 � 7 � 103�T0 � T �

0.6, �10�

1 All data points with snowfall rates that exceed 4 mm h�1 arefrom two storms.

FIG. 9. Attributes of winter-storm PSDs: (a) mass-weighted ter-minal velocity and median volume diameter, (b) mass-weightedterminal velocity and temperature, and (c) median volume diam-eter and temperature. Data points are color coded according tothe estimated-density key in (a). The dataset is the same as inFig. 6.


Fig 9 live 4/C

where T0 � 273.15 K and T is the observed temperature(K). The solid thin line is a relationship used by Honget al. and Thompson et al.,

N0 � 2 � 103 exp 0.12�T0 � T ��, �11�

that was derived from observations described by Houzeet al. (1979). The dataset of Houze et al. consists of 37spectra obtained by aircraft from four winter storms inthe state of Washington at temperatures of �42° to6°C. The particle sensor had a measurement resolutionof 70 m and a data window width of 1050 m. Ingeneral, particle sizes would have been estimated fromthe shape of partial images. Houze et al. truncated thesize distribution on the small end when the data de-parted from an exponential distribution.

Also shown is the relationship

N0 � 7.63 � 103 exp0.107�T0 � T �� 12�

derived by Field et al. (2005). The data were obtainedduring 16 aircraft flights around the British Isles. Par-ticles as large as 6400 m were sampled with an array ofinstruments over a temperature range of �55° to 10°C.

For the most part, our N0s are larger by a factor of3–5 than that found by Houze et al. (1979). Althoughdifferences in instrumentation and data processing maycontribute to this result, particle distributions in Colo-rado may simply be narrower, having higher concentra-tions of small particles and fewer large particles, thanthose in the Pacific Northwest. Our N0s are within afactor of 1.4 of that found by Field et al., except fortemperatures near 0°C. The Colorado data, obtained atthe ground, show a factor-of-4 decrease in the interceptparameter on average as temperatures warm above�5°C and aggregation broadens the distribution. PSDbroadening is supported by a corresponding decrease in� (Fig. 11, bottom panel). The fitted �–T relation is

� � 2.27�T0 � T �0.18. �13�

A corresponding fit to the data of Houze et al. is

� � 1.0 � 0.1�T0 � T �. �14�

Our �s are about 1 mm�1 larger. Houze et al. deter-mined correlation coefficients of �0.66 between tem-perature and N0 and �0.90 between the slope of the

FIG. 10. (a) Bulk snow density �s, (b) mass-weighted terminal velocity � t, (c) median volume diameter D0, and (d) the gamma PSDshape parameter plotted against snowfall rate expressed as liquid equivalent (S � 0.2 mm h�1). The dataset is the same as in Fig. 6.

MAY 2007 B R A N D E S E T A L . 645

exponential PSD and temperature. Correlation coeffi-cients for the disdrometer observations are �0.63 be-tween logN0 and T and �0.41 between � and T. Al-though correlation could be improved somewhat by av-eraging over periods longer than 5 min or averagingover small temperature intervals, the coherence seen inour data from sample to sample (e.g., Figs. 2 and 5) andthe fact that the estimated snowfall rates closely matchthe gauge observations are evidence that the fluctua-tions are largely meteorological.

The behavior of � and N0 depends on which micro-physical processes (nucleation, depositional growth, ag-gregation, or ice multiplication) dominate (Lo and Pas-sarelli 1982; Mitchell 1988, 1991) and which ice growthregimes are active (Gordon and Marwitz 1984). Hence,differences between in-cloud PSD attributes deter-mined by aircraft and disdrometer-derived attributes atground are likely. Observations suggest that precipita-tion processes drive the slope of the distribution to alimiting value of approximately 1 mm�1 (Lo and Pas-sarelli 1982; Ryan 1996). The observations in Fig. 11agree with that finding. Mitchell and Heymsfield (2005)

attribute the limiting value to the growth of particlesand a reduction in the dispersion of fall speeds thateventually shuts down the aggregation process.

Figure 12 shows concentration parameters for trun-cated-exponential and truncated-gamma PSDs plottedagainst snowfall rate. The rates are computed from thedisdrometer observations. A fit to the data for the trun-cated-exponential distribution yields

N0 � 5 � 103S�1.2. �15�

For snowfall rates of less than 3 mm h�1, the concen-tration parameter varies by more than two orders ofmagnitude. Although the dataset size for heavy snowrates is limited, there is close agreement between theobservations and (15) for S � 3 mm h�1. Sekhon andSrivastava (1970) determined that N0 and S were re-lated by

N0 � 2.50 � 103S�0.94. �16�

This relation, used by Reisner et al. (1998) in their nu-merical model microphysical parameterization scheme,

FIG. 11. Concentration and slope parameters for truncated-exponential PSDs plotted against surface air temperature. Rela-tions fitted to the observations are shown by thick solid lines; fitsto the data of Houze et al. (1979) are given by thin solid lines. AnN0–T relation found by Field et al. (2005) is shown by a dashedline. The dataset is the same as in Fig. 6.

FIG. 12. The relationship between N0 for truncated-exponentialand truncated-gamma PSDs with snowfall rate. Relations fitted tothe observations are shown by thick solid lines. Equation (10)from Sekhon and Srivastava (1970), for an exponential PSD, isshown by a dashed line. The dataset is the same as in Fig. 6.


is also plotted in Fig. 12. The concentration given by(15) is larger by a factor of 2.4 than that of (16) for asnow rate of 0.5 mm h�1. This ratio reduces to 1.2 at arate of 8 mm h�1. The differences would seem to beinsignificant in view of the data scatter at low snowrates and the few disdrometer observations at highsnow rates.

The N0s for the truncated-gamma PSD are plottedversus S in the bottom panel of Fig. 12. A fit to thedata is

N0 � 1.3 � 104S�1.45. �17�

Increased scatter, attributed to greater freedom whenfitting the observed particle distributions with a three-parameter model, would seem to preclude the utility of(17).

Relationships between the shape and slope param-eters for precipitation near ground, assuming trun-

cated-gamma PSDs, are shown in Fig. 13. Fitted rela-tions for snow, rain, and mixed-phase precipitation are

� � �0.004 99�2 � 0.798� � 0.666, �18�

� � �0.003 25�2 � 0.698� � 1.71, and �19�

� � �0.000 120�2 � 0.602� � 2.06. �20�

Equations (18)–(20) are applicable if the true PSD is agamma distribution—that is, if the distributions areconcave upward or downward. Caution should be ex-ercised when using such relationships because, as notedby Chandrasekar and Bringi (1987), derived PSD at-tributes can be correlated because of errors in the com-putation of particle moments. The issue is discussedfurther by Zhang et al. (2003) who argue that, althoughobservational error does contribute to correlation be-tween computed PSD properties, the derived relationscontain useful meteorological information. Seifert

FIG. 13. The distribution of and � for (a) snow, (b)rain, and (c) mixed-phase precipitation. A truncated-gamma PSD is assumed.

MAY 2007 B R A N D E S E T A L . 647

(2005) conducted a study with a stochastic dropbreakup/coalescence model that suggests relations simi-lar to (18)–(20) represent fundamental properties ofdrop distributions in convective storms. Hence, a physi-cal relationship is believed to exist between these twoparameters. Such a relationship may be useful when atwo-parameter PSD model is needed.

Figure 13 shows some very large values of and �.Their frequency is low (Fig. 4). Large s and �s arecharacteristic of narrow PSDs and commonly occur atthe beginning and ending of storms when small num-bers of small particles are observed. Computed valuesalso tend to be noisier during these storm stages. Issueswith small particles (section 2) would also contribute tothe narrowing of these distributions. Significant precipi-tation is characterized by broad PSDs with small valuesof and � (e.g., Fig. 5). Nevertheless, our s and �s aresomewhat larger than those found for aircraft observa-tions (e.g., Heymsfield et al. 2002; Heymsfield 2003).The latter studies include high concentrations of cloudparticles that are not detected by the disdrometer. Also,relations found by Heymsfield and collaborators arerepresentative of cloud particle distributions through-out the storm depth, whereas relations found here areapplicable for precipitation-sized particles at theground.

For a particular , � for snow is smaller in the meanthan for rain. As a consequence, the fitted relationslope for snow is larger than that for rain, especially forheavier-precipitation events. The dataset for mixed-phase precipitation (Fig. 13c) is much like that for rain.

6. Summary and discussion

The video disdrometer is a powerful observationaltool for studying the microphysical properties of winterstorms. What the instrument lacks in resolution is madeup for by the sheer volume of observations readily ob-tained for precipitation-sized particles in a variety ofstorms and under different meteorological conditions.The observations should prove to be important for veri-fying and developing microphysical parameterizationsin numerical forecast models and for the interpretationof polarimetric radar observations.

Our results show that, while PSDs near the ground inwinter storms are closer to exponential on average thanraindrops, the distributions often turn markedly up-ward or downward (e.g., Figs. 3 and 4) and hence thereare benefits for modeling the hydrometeors with agamma distribution. However, the advantage with thegamma distribution is largely the capability to handledistributions of mixed-phase particles and the raindroppopulations that stem from melting. The gamma model

adds complexity to a numerical model because anotherparameter is introduced. However, the existence ofrelationships between and �, as in Fig. 13, is impor-tant because it effectively reduces the three-parametergamma distribution to two parameters. It consequentlyshould not be necessary to impose more severe assump-tions on the PSD, such as a constant .

Using precipitation volume measurements from thedisdrometer, we derived a relationship for bulk density[(7)] that is an almost inverse linear relation (1/D) withparticle size. Although the correlations are weak(�0.15), further refinement of the relation may be pos-sible when temperatures warm above �5°C or so byconsidering the influence of temperature and humidityon snowflake density.

Heavy snow rates along the Front Range in easternColorado typically involve relatively warm tempera-tures and aggregates with median volume diametersthat are greater than 5 mm. The increase in snow vol-ume more than offsets the reduction in bulk density asparticles grow in size. Heavy snow rates are also sup-ported by increases in particle terminal velocity. In ourdata the shape parameter of the gamma distributionmodel in heavy snows is often negative (22% of thetime), indicating the presence of superexponential con-centrations of small particles. Negative s can be aproblem when using the gamma model to calculate NT.To avoid an infinite or unrealistic NT, the distributioncan be truncated at a small particle size. For remotesensing NT is not an overly important issue because itscorrelation with radar measurements is relatively low.The problem is greater for a two-moment numericalmodel that predicts NT and then uses it to derive othervariables. A solution may be to develop a parameter-ization scheme based on predictions of precipitationmixing ratio and radar reflectivity and avoiding the useof NT .

Empirical relationships between temperature and theconcentration and slope parameters of the exponentialPSD were evaluated. At a specific temperature, N0 var-ied by an order of magnitude and the range in � wasbroad. Over the temperature range of �20° to �5°C, asmall decrease in the magnitude of both parameterswas noted. The N0 decreased from approximately 10

4.6

to 104.3 m�3 mm�1 and � decreased from 4 to 3 mm�1.As temperatures warmed above �5°C, N0 and � de-creased further in the mean to about 103 m�3 mm�1 and2 mm�1, respectively, which is an indication that aggre-gation had broadened the PSD. Observed N0s and �swere larger than in the Pacific Northwest, suggestingthat PSDs in Colorado are narrower and are composedof smaller particles. Parameter N0 and snowfall rate Sare weakly correlated at snowfall rates of less than 3


mm h�1, with N0 varying by more than two orders ofmagnitude. Hence, the relation does not appear to beuseful.

This study emphasized bulk characteristics of storms.Future efforts will include detailed studies of particleevolution within winter storms. Of particular interestare the conditions that determine whether significantaggregation takes place. Also, as the dataset grows, in-terrelationships between variables described here willbe refined and habit-specific relations will be devel-oped. Hydrometer properties such as bulk density andterminal velocity are clearly determined by more thansize. A “fuzzy logic” approach may prove useful whenthe theoretical form of the relation is not known butinterrelationships among bulk attributes of particle dis-tributions and their dependence on environmental fac-tors such as temperature and humidity are desired.

Acknowledgments. This research responds in largepart to requirements of and funding from the FederalAviation Administration (FAA). The views expressedare those of the authors and do not necessarily repre-sent the official policy or position of the FAA. Thestudy was also supported by funds from the NationalScience Foundation designated for U.S. Weather Re-search Program activities at the National Center forAtmospheric Research. In addition, GZ was partly sup-ported by the National Science Foundation throughATM-0608168. The authors are indebted to Dr. Wil-liam D. Hall for his constructive and thoughtful reviewof the manuscript and to Drs. Paul R. Field and An-drew J. Heymsfield for insightful discussions regardingparticle distributions in storms.

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