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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]
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DOI: 10.1175/JAM2489.1
© 2007 American Meteorological Society
JAM2489
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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.
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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
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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
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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 .
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Fig 2 live 4/C
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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
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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.
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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
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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
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Fig 6 live 4/C
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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.
644 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
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.
646 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
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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
648 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
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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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