Directory UMM :Data Elmu:jurnal:A:Atmospheric Research:Vol52.Issue3.Sept1999:
Atmospheric Research 52 Ž1999. 221–239
www.elsevier.comrlocateratmos
Raindrop size distributions in convective clouds
over Cuba
Daniel Martinez
a
a,)
, Enrico G. Gori
b
Instituto De Meteorologia, Ministerio De Ciencia, Tecnologia Y Medio Ambiente, Ap. 17032, CP 11700,
HaÕana 17, Cuba
b
Istituto Di Fisica Dell’atmosfera, Consiglio Nazionale Delle Ricerche, Italy
Received 6 May 1998; accepted 3 March 1999
Abstract
Raindrop size distributions, measured in the warm sector of convective clouds over Camaguey,
¨
Cuba, are analyzed; a wide range of rainfall rates is present in the dataset. Mean spectra are
presented and examined separately for updrafts and downdrafts, with additional information about
vertical velocities and cloud water content. Large drop sorting has been detected; its sense and
degree depend on rainfall rate and cloud dynamic characteristics. No significant small drop sorting
has been found. Mean spectra for individual clouds and for groups of clouds with the same rainfall
rate category have been calculated and fitted by different models; the best fits have been obtained
with a least square gamma function model. An alternate analytical procedure has been presented,
resulting appropriate to estimate rain water content and accurate in the sense of mean error.
Negative form factors, associated with upward concavity are obtained in most cases, because of
the presence of relatively high concentrations of small and large drops. q 1999 Published by
Elsevier Science B.V. All rights reserved.
Keywords: Raindrop size distributions; Convective clouds; Cloud physics; Cloud water content; Rain water
content
1. Introduction
The raindrop size distribution ŽRSD. is one of the most important parameters in cloud
microstructure, because it is related to most physical processes in the cloud and to most
integral parameters used for the cloud–environment interaction and for remote sensing
techniques. These are the rain water content ŽRWC., the radar reflectivity factor Ž Z ., the
rainfall rate Ž R ., the microwave attenuation Ž A., the optical extinction coefficient Ž S .
)
Corresponding author. E-mail: [email protected]
0169-8095r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 9 - 8 0 9 5 Ž 9 9 . 0 0 0 2 0 - 4
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
222
and others. The knowledge of RSD is of particular importance for modeling clouds and
larger scale systems, for the interpretation of radar and radiometric measurements, as
well as for applications in telecommunications, airborne transport and construction
industry. Many RSD measurements have been made at ground level. However, most of
the above-mentioned applications require the knowledge of the RSD above the ground
and inside the clouds. Consequently, in-cloud and sub-cloud measurements have been
made using airborne spectrometers ŽKlazura, 1971; Mazin and Shmeter, 1977; Carbone
and Nelson, 1978; Willis, 1984..
One of the main objectives of the study of RSD both at ground and aloft has been the
search of analytical expressions reproducing the expected spectral densities from known
integral rainfall parameters, especially rainfall rates. The most widely used RSD model
in the last decades has been the exponential distribution, proposed by Marshall and
Palmer Ž1948.. Its general form is:
N Ž D . s N0 exp Ž yl D .
Ž 1.
where D is the drop diameter Žor equivalent diameter, in non-spherical large drops.;
N Ž D . is the drop concentration per diameter interval, or drop spectral density; N0 and l
are constants for a particular RSD. For this analytical form, the RSD log–linear plot
should be a straight line with intercept N0 and negative slope l. According to Marshall
and Palmer’s results, the intercept is a constant for any type of rain or rainfall rate, and
the slope is a function of rainfall rate.
In general, N0 has been found to depend on R, and the functional dependence of l
with R may vary. These variations can be considerable, even within the same rainfall
event, if variations in air mass stability occur. For convective rain, the form of the RSD
generally shows deviations from the exponential, though it has been successfully used
sometimes ŽPruppacher and Klett, 1997, pp. 35–36..
Joss and Gori Ž1978. analyzed surface RSD, measured with a disdrometer in Locarno,
Switzerland. They observed systematic deviations from exponentiality in short time
averaged spectra, and proposed a shape parameter, whose deviation from unity is related
with the deviation of the RSD from the exponential. It depends on two integral
parameters P and Q, taken as the pth and qth moments of the RSD and of its
exponential fit, as follows:
DŽ P . DŽ Q .
D Ž P . q D Ž Q . observed
,
DŽ P . DŽ Q .
S Ž PQ . s
DŽ P . q DŽ Q .
Ž 2.
exponent
where
`
DŽ P . s
H0 D
`
H0
p
NŽ D. d D
D py 1 N Ž D . d D
.
Ž 3.
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
223
The Joss–Gori S parameters were shown to depend only on the form of the RSD. It
can be seen that SŽ PQ . s 1 corresponds to an exponential distribution, S - 1 to a
concave downward and S ) 1 to a concave upward distribution, respectively Žin
log–linear coordinates.. Joss and Gori Ž1978. found both types of cases, though the
SŽ PQ . - 0 case was the most commonly found.
Ulbrich Ž1983. proposed the use of the gamma distribution to fit RSD, considering
that the exponential can be obtained as its particular case, and that a series of
mathematical advantages can be obtained. The proposed function is of the form:
N Ž D . s Ng D m exp Ž L D . ; 0 F D F Dmax ,
Ž 4.
where m is a real number and the units of Ng are: my3 mmy1 y m . Here, m can be
interpreted as the slope that would have the spectrum for the small drop region if plotted
in a log–log diagram. If it is positive, it implies the existence of an internal mode for a
diameter greater than the minimum Ž Dmin . and, consequently, a concave downward
form, even in log–linear coordinates. For a negative m , no internal mode occurs in the
model curve, which is concave upward. The parameter L is interpreted as the slope of
the plot in log–linear coordinates, for the large drop part of the spectrum. Ng , in general,
has no useful physical meaning and shows a wide range of variation, in dependence to
m. Using a theoretical expression for the relation between any pair of integral parameters
of the gamma distribution, and a group of empirical Z–R relations, Ulbrich Ž1983.
estimated m for a variety of rain types. A great variability was found, with prevailing
negative m for orographic rain, and positive for thunderstorm rain. For stratiform rain
and showers both negative and positive values were obtained. The gamma function
model has also been used with success by Willis Ž1984. and Willis and Tattelman
Ž1989. to fit RSD measured by an airborne spectrometer in tropical storms and
hurricanes. It has also been applied to surface RSD data by Fujiu et al. Ž1996. in China
and by Richter and Goddard Ž1996. in Southern England.
Mazin and Shmeter Ž1977. used a power function model to fit RSD, measured in
warm cumulus clouds with Nevzorov’s large particle airborne spectrometer ŽLPS.. Their
spectra, with maximum diameters close to 3 mm, for the deepest clouds, looked linear in
a log–log plot, but they would have looked concave upward, if plotted in a log–linear
system the proposed distribution has the form:
N Ž D . s aDyb .
Ž 5.
In the present work, simultaneous measurements of RSDs, vertical velocity, and
cloud liquid water content in Cuban convective clouds are presented. In Section 2, a
general characterization is made of the dataset, including a brief description of instrumentation, data collection methodology and sample stratification criteria. Section 3 deals
with the results of the analysis of RSD properties, highlighting differences between
spectra in updrafts and downdrafts, and between clouds in different stages of development. Averaged spectra have been calculated for different rainfall rates. In Section 4,
individual and averaged spectra have been fitted by exponential, power and gamma
distributions. The gamma fit was made independently by a least squares routine, and by
an analytical procedure, based on the previous calculation of the Joss–Gori shape
parameter.
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
224
2. Instrumentation and data
The dataset used in the present work was collected during the 1986 and 1987 seasons
of the Cuban Weather Modification Program ŽPCMAT., run jointly by the Cuban
Institute of Meteorology ŽINSMET. and the Russian Central Aerological Observatory
ŽCAO.. Field experiments, including cloud physics observations, were accomplished in
the Camaguey
¨ Experimental Area ŽCEA., which is located in the central–eastern part of
Cuba, and is limited by a circumference of 80 km of radius, centered at the Meteorological Center of Camaguey
¨ City Ž21825X N and 77810X W.. The observed RSD correspond to
the interior of summer convective clouds, measured well below the freezing level by
Nevzorov’s large particle spectrometer ŽLPS., installed in the instrumented Aerocaribbean, INSMET IL-14. The LPS is an optical probe, developed in the Russian
Central Aerological Observatory ŽMazin and Shmeter, 1977; Nevzorov, 1996.. It is
intended for measuring particles in a range of diameters from 0.2 to more than 6 mm. It
has 12 channels, whose resolution decreases with diameter, as shown in Table 1. Its
working principle may be summarized as follows. A collimated light beam passes
through two narrow slits Ž0.12 mm wide. and is focused on a photodetector. The cloudy
air crosses the beam axis, forming a sample volume of 25 = 28 = 0.12 mm. Everytime
that a particle intersects the measuring volume, the light flow on the photodetector
decreases proportionally to the area covered by the particle, which is also nearly
proportional to the particle size for the range of measurement, as the minimum diameter
is nearly twice the slit width. This causes a response pulse that is subsequently amplified
and sent to a 12 channel integral pulse amplitude analyzer.
Considering that the position of the sensor favors the measurement of the horizontal
dimensions of the drops, which could have induced to overestimation of their sizes, a
correction for deviation of sphericity ŽPruppacher and Klett, 1997, p. 399. is introduced
in this work to calculate the equivalent drop diameter. The third row of Table 1 shows
the results.
The IL-14 was also equipped with an aircraft load complex ŽALC., including an
analog integrator and the corrections to produce wind vertical velocity with an error of
15% for vertical velocities Ž w . less than 10 mrs. For greater w, the error is estimated as
20%. Liquid water content from cloud droplets was measured with Nevzorov’s hot wire
probe, with sensitivity from 0.003 to 0.01 g my3 , depending on flight conditions. Work
principles and general characteristics of this equipment can be found in the works of
Table 1
Threshold values of diameter measurement for the 12 LPS channels
Channel
1
2
3
4
5
6
7
8
9
10
11
12
d min Žmm.
Dmin Žmm.
0.2
0.2
0.3
0.3
0.4
0.4
0.6
0.6
1.0
1.0
1.4
1.4
2.0
1.9
2.8
2.7
3.6
3.4
4.4
4.1
5.2
4.7
6.0
5.3
d min represents minimum measured drop dimensions for each channel.
Dmin represents minimum equivalent diameter for each channel.
The 12th channel has no well-defined maximum limit, so that its reading includes all raindrops greater than 5.3
mm. For statistical calculations, the same width of channel 11 has been assumed for channel 12.
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
225
Mazin and Shmeter Ž1977. and Nevzorov Ž1996.. Intercomparisons between Nevzorov’s
hot wire probe and the well-known PMS King probe have shown close agreement
ŽKorolev et al., 1996..
The dataset consists of 72 samples. Of them, 65 had altitudes near to 3 km and
temperatures from 7 to 108C. It also includes six cloud base passes at altitudes close to
1.5 km and temperatures of 19–238C. According to the flight methodology, clouds
should be penetrated in their central part so that the length of the in-cloud flight track
corresponds roughly to the horizontal extension of the cloud. The cases for which this
condition could not be fulfilled were excluded from the analysis. In the case that the
measured cloud was part of a mesoscale system, and was merged at the base with other
clouds, it was identified by its corresponding tower. Consequently, the volume of cloudy
air in each individual sample depends of the horizontal dimensions of the cloud, which
ranged from 0.8 to 7 km. As the LPS sampling area is of 7 cm2 , this corresponds to a
range of sampling volumes of 0.6–5 m3.
Cloud top heights ranged from 4 to 12.5 km, with average of 7.7 km and standard
deviation of 1.9 km. Cloud base heights ranged from 1.1 to 1.5 km, with an average of
1.2 km, and standard deviation of 0.1 km. Considering their dimensions and form, the
measured clouds can be classified as cumuli congesti or cumulonimbi. Previous studies
have shown that Camaguey
¨ convective clouds may be classified as nearer to continental
that to maritime, considering their cloud base droplet spectra in early stages of
development, and their characteristic maximum vertical velocities ŽPerez
´ et al., 1992;
Martınez,
1996..
´
Every second of in-cloud flight, corresponding to nearly 80 m, a set of measurements
was recorded. This was later used to calculate the averaged spectrum for each of the 72
cases, which will be called ‘cloud spectrum’. For a subset of 55 clouds, with simultaneous measurements of vertical drafts and RSD, averaged spectra for updrafts Ž w ) 1. and
downdrafts Ž w - y1. were calculated. These will be referred to as ‘draft spectra’.
Rainfall rates, calculated from cloud spectra were used for data stratification, applying
the criterion of Willis and Tattelman Ž1989. for high rainfall rate categories ŽRRC.. Two
more categories were included to account for the lower rates. The definition of the
applied RRC is shown in Table 2. No stratification was done according to flight height,
as the cloud base spectra were only six cases, and they showed no outstanding
differences with those measured higher in the clouds. An index was introduced to
Table 2
Ranges of rainfall rate ŽR., number of cases, and frequency distributions of top heights Ž H T . for the clouds of
the dataset, stratified by RRC
RRC
Range of
R Žmm hy1 .
Number
of cases
Cloud top height range Žkm.
4–5 5–6 6–7 7–8 8–9
9–10
10–11
11–12
12–13
1
2
3
4
5
- 2.5
2.5–25
25–62.5
62.5–125
)125
34
13
9
8
8
2
0
0
0
0
3
0
1
3
1
0
2
0
0
2
1
1
0
2
0
0
0
2
0
0
6
0
0
0
0
8
2
1
0
3
9
7
2
2
1
5
1
3
1
1
226
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
Fig. 1. General characterization of the dynamics of the clouds in each of the RRC. ŽA. Averaged mean Ž w .,
maximum Ž wma x ., and minimum Ž wmin . vertical velocities and absolute maximum Ž wq . and minimum Ž wy .
vertical velocities against RRC. ŽB. Number of cases with DESs 0 and DESs1 against RRC.
account for the stage of development of the clouds, to consider it in the analysis; it is
called DES, for development stage index. It is simply defined as DES s 1 when the
averaged measured cloud air vertical velocity was positive, i.e., w ) 0, showing
predominant upward motion, and DES s 0 in the cases when w F 0.
According to cloud top heights, whose frequency distributions are shown in Table 2,
most of the measured clouds are mixed phase, as the freezing level ranged from 4 to 5
km, and the y58C level ranged from 5 to 6 km. As can also be seen, all high rainfall
rate cases Ž R ) 25 mm hy1 . correspond to clouds with H T G 6 km, all of them with
cloud top temperatures below y58C.
Most clouds of the dataset had well-defined updrafts and downdrafts, with considerable vertical velocities This can be seen from Fig. 1, showing ŽA. the RRC average of
the mean, maximum and minimum values of the 1 s values of w for each case, and the
corresponding absolute maximum and minimum. Number of DES s 0 and DES s 1
cases for each RRC are also shown ŽB..
3. RSDs
3.1. RSDs in clouds
Typical cloud spectra for individual cases corresponding to each RRC are shown in
Fig. 2. As can be seen from these examples, maximum diameter is larger for higher
RRC spectra. The same holds for the large drop concentrations.
This can be better analyzed from Table 3. As can be seen, the range of variation of
Dmax is well differentiated between the lowest ŽRRC s 1,2. and highest ŽRRC s 3–5.
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
227
Fig. 2. Raindrop spectra for four typical cases, corresponding to different RRC. Filled squares represent data,
and curves represent functional fits.
rainfall rate spectra. This suggests that the presence of giant drops Ž D ) 5 mm. is
related to the development of intense rainfall in convective clouds in Camaguey.
¨
Another interesting characteristic of cloud spectra, apparent from the examples is that
they generally exhibit no absolute maximum for small drop diameters for D ) Dmin , as
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
228
Table 3
Frequency distributions of maximum diameters Ž Dma x . for the clouds of the dataset, stratified by RRC
RRC
1
2
3
4
5
Dma x Žmm.
0.4
0.6
1.0
1.4
1.9
2.7
3.4
4.1
4.7
5.3
5.9
1
0
0
0
0
4
0
0
0
0
10
0
0
0
0
8
0
0
0
0
3
2
0
0
0
8
3
0
0
0
0
5
0
0
0
0
1
0
1
1
0
2
1
1
0
0
0
3
1
1
0
0
5
5
6
those reported by Willis Ž1984. for tropical storm clouds. For the high rainfall rate cases
ŽD and E., the upward concavity of the spectra is apparent, especially for small drop
diameters.
3.2. RSDs in updrafts and downdrafts
For the lowest rainfall rates, updraft and downdraft spectra were usually not very
different ŽFig. 3.. The situation changes for the high rainfall rates, where drop sorting
Fig. 3. Updraft and downdraft spectra for two typical low rainfall rate cases Žleft. and two typical high rainfall
rate cases Žright..
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
229
takes place, mainly for the larger drops. This is the cause of marked differences between
updraft and downdraft spectra in their right zones, while the left parts of the two spectra
are very close to each other.
In relation with this, the dependence of CWC on vertical velocity for updrafts and
downdrafts, was investigated. For updrafts, a positive correlation was observed between
vertical velocity and CWC mainly for developing clouds Ž r s 0.70., as expected from
the formation and growth of droplets by condensation. No correlation exists for
downdrafts, but considerable, or even high CWC is present in most cases. The existence
of significant CWC in downdrafts reveals the presence of cloud droplets, which is
consistent with a low evaporation rate, and with the existence of high concentrations of
small raindrops.
The sign of the difference in large drop concentration between updrafts and downdrafts depends on the cloud stage of development, as can also be seen from the typical
cases shown in Fig. 3. Updraft spectrum for case 53 Žcorresponding to a developing
cloud., for example, is greater than downdraft spectrum for all drop sizes. The opposite
occurs for case 277, which correspond to a mature cloud containing, at flight level, a
powerful rainshaft and a weak updraft. This can be better understood from the analysis
of a particular case. Fig. 4 shows the evolution of the spectrum and other characteristics
for the cloud corresponding to cases 325–329. It was measured five times, on August
31, 1987, from 1713 to 1744 h. More penetrations could not be made for flight security
reasons. The gradual development of the cloud spectrum ŽA. is due to the parallel
evolution of RSD in updrafts ŽB. and downdrafts ŽC.. The maximum measurable
diameter is first attained in the updrafts, but the large drops reach higher concentrations
in the downdraft, in the last penetration. The main rainfall related integral parameters
and cloud top height kept growing all the time ŽD, E.. The constant behavior of Dmax
after the third penetration is probably only an instrumental truncation Žsee the note in
Table 1.. Mean vertical velocity remained positive in all passes. At the second
penetration, a decrease of mean and maximum velocity was registered, but later they
increased again. In the third pass, the ALC system got unbalanced and the register
saturated, so that the vertical velocity values could not be measured, though the
predominantly upward motion is certain.
The difference between updraft and downdraft spectra appears not only in the values
of maximum diameters, but also in the distribution of the different moments of the RSD
between the smaller and larger drops. To analyze this effect, different weighted median
diameters are used. The most extended in RSD formulations is the volume median
diameter D 0 , dividing the contribution to the RWC from left and right part of the RSD
in two equal parts. The median reflectivity diameter D 0 Z has also been used, because of
the greatest weight given to larger drops. In general, a median diameter of order p can
be defined as:
`
H0 D
p
D0p
H0
NŽ D. d D s2
D p N Ž D . d D.
Ž 6.
In Fig. 5, median diameters of order p for updraft spectra have been plotted against
the same parameters for downdrafts, for p s 1–4 and 6 ŽA–E.. Maximum diameters
have also been plotted ŽF.. The fact that the main difference between updraft and
230
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
231
downdraft spectra is given by the larger drops and depends on the cloud development
stage, is well expressed in the graphs. The distribution of points at both sides of the
y s x line is quite symmetric. Increasing the order, the separation between filled and
hollow circles Žrespectively corresponding to DES s 0 and DES s 1. increases. The
value of the diameter for which this separation is noticeable also increases with order.
This means that higher order moments, being the most affected by large drop concentrations, show the greatest dependence on vertical velocity and cloud development stage.
Note that the strictly symmetric disposition of the points corresponding to the greater
values of maximum diameter, either in updrafts or downdrafts ŽFig. 5F. may be an
artifact, due to the open maximum limit of the LPS 12th channel.
4. Functional fits to RSDs in clouds
4.1. Functional fits to indiÕidual cases
The observation of individual cloud spectra ŽFig. 2. shows that, with some exceptions, they do not look linear in log–linear coordinates, and they are generally concave
upward. Log–log plots, Žnot shown. resulted linear in some cases. This suggests a
gamma distribution, but does not allow discarding the possibility of exponential or
power fits in certain cases. Thus, the three models were tried and compared.
For different purposes, such as parameterization in cloud models, or remote sensing
applications, the estimation of RSD functional form from calculated or measured
integral parameters is desirable. This approach has been called ‘analytical’ ŽWaldvogel,
1974; Willis, 1984.. In the present section, a variant of this method will be followed to
obtain additional gamma fits for individual cloud spectra, and the results will be
compared with the least squares estimations. The method consists in obtaining the form
factor m of the gamma distribution, by applying the following relation, developed by
Ulbrich Ž1983., after calculating the Joss–Gori form parameter SP Q from the experimental DSD.
pqq
ms
Ž 7.
pqqq2m
As the choice of P and Q depends in general on the particular task, five combinations were tested: w p,q x s w6,2x; w6,3x; w6,4x; w4,3x and w3,2x. The relations Ž8. and Ž9.,
used to calculate L and Ng have also been taken from Ulbrich Ž1983.:
Ž 8.
L D 0 s 3.67 q m .
P s ap
G Ž p q m q 1.
Ž 3.67 q m .
pq m q1
Ng D 0pq m q1
Ž 9.
Fig. 4. Evolution of spectra in a developing cloud measured on August 8, 1987. The five passes correspond to
17:13 Ž1., 17:21 Ž2., 17:28 Ž3., 17:37 Ž4. and 17:44 Ž5.. ŽA. Cloud spectra; ŽB. updraft spectra; ŽC. downdraft
spectra; ŽD. integral parameters; ŽE. cloud top height and vertical velocity.
232
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
Fig. 5. Median diameters of order p for ps1,4,6 ŽA–E. and maximum diameters ŽF. for updrafts against the
same magnitude for downdrafts. For ps 2 and 6, the usual symbols have been used. The rest of the median
diameters have been denoted by D 0 p . Subindexes ‘up’ and ‘dw’ refer, respectively, to updraft and downdraft
spectra. Lines represent y s x locus.
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
233
D 0 is available from data, but it was calculated from Ž10., as a function of Dm , the so
called mass-weighted average diameter ŽUlbrich, 1983., a quantity close to D 0 both in
value and physical meaning, but statistically more representative and depending only on
the moments of the distribution.
`
Dm s
H0 D N Ž D . d D
4
`
H0 D N Ž D . d D
s
3
ž
4qm
3.67 q m
/
D0 .
Ž 10 .
Three independent criteria were applied for goodness of fit evaluation: the logarithmic mean squared error ŽLMSE., given by:
LMSEs
n
1
Ý
n
2
log Ni Ž obs . y log Ni Ž fit . ,
Ž 11 .
1
the relative mean error ŽRME., given by:
RME s
1
n
Ý
n
1
Nobs y Nfit
Ž 12 .
Nobs
and the relative error of estimation of the RWC, given by:
d RW C s
ABS Ž RWC obs y RWC fit .
RWC obs
,
Ž 13 .
In the above equations, ‘obs’ refers to the observed RSD, ‘fit’ refers to its functional
fit, Ni is the drop count of each channel, and ‘n’ is the number of LPS channels with
non zero drop counts.
The advantage of LMSE as a criterion, used previously by Willis Ž1984. is that it
gives equal weight to all parts of the spectrum, according to the approximately
exponential dependence; so, it is a good indicator for the reproduction of the form of the
original spectrum by the fit. The RME, used previously by Fujiu et al. Ž1996., removes
the dependence on the diameter by dividing the absolute error by the spectral density of
each channel, and renders the error estimation independent from the integral characteristics of the spectrum.
Only those RSD with Dmax G 1 mm were fitted, for having at least five spectral
density estimations, implying more than two degrees of freedom in regression. Thus, a
total of eight fits were tried for each of the 67 RSD fulfilling this condition. The
exponential, power and gamma least squares fits will be denoted, respectively as EXls,
POls and GAls. The analytical gamma fits corresponding to the SP Q parameters will be
denoted as GApq Ž p s 6,4,3; q s 2,3,4.. The curves in Fig. 2 shows the graphical
performance of POls, EXls, GAls, GA62 and GA64 for four characteristic cases. As can
be seen from these examples in a qualitative way, the best fit in most cases is GAls, but
the exponential and potential fits provide acceptable approximations for the lower
rainfall rates, for which the maximum diameter is small. The power fit overestimates
large drop concentration for the high rainfall rates. The exponential fit seems to suit the
right tales of the high RRC, but it neglects curvature and underestimates small drop
234
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
concentration. A preliminary analysis of the errors suggested searching differences in the
performance of the models depending on RRC.
4.2. Functional fits to mean spectra for each rainfall rate category
Cloud RSD for individual cases, corresponding to the same RRC, were bin-averaged
for each of the measured diameter intervals. For RRCs 1, five clouds had Dmax - 1
mm, and hence, less than five non-zero bins. They were excluded from the averaging.
The largest drop bins in the averaged spectra were considered for curve fit only if they
were represented in more than a fourth of the total number of clouds in the corresponding RRC.
Fig. 6 shows GAls fits for mean spectra for the five RRC. The slope of the right wing
of the spectrum decreases with rainfall rate.
The numerical values of the parameters and the errors for the three least squares fits
and the best performing analytical gamma fit are shown in Table 4. The analysis of the
errors shows that the GAls fit has always the least LMSE, and RME. All five analytic
fits give good predictions of RWC, but GA64 has very high LMSE, which is relatively
high in GA63 too Žnot shown.. The power fit performs better than the exponential from
the point of view of mean errors, particularly for the lower RRC, but the exponential
proved very efficient in predicting LWC for mean spectra. In general, EXls performs
better for mean spectra than for individual cases, as had been obtained by Joss and Gori
Ž1978. for surface rain.
The analysis of Table 4 shows that, although the different gamma fits have different
parameters, this does not have so much influence in the fit error. The parameters can
adjust themselves to a certain extent to provide a good fit. For example, the GAls and
Fig. 6. Bin averaged observed cloud spectra for the five RRC and the corresponding least squares gamma fit
ŽGAls..
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
235
Table 4
Parameters and estimation errors of averaged spectrum fits for different RRC for four functional fit models
Parameters and error
estimations
RRC
1
2
3
4
5
Exls
N0 Žmy3 mmy1 .
l Žmmy1 .
LMSE
RME
d RW C
5926
4.09
0.22
0.74
0.05
8780
2.78
0.1
0.5
0.02
4827
1.53
0.13
0.61
0.39
7904
1.51
0.12
0.58
0.11
11452
1.41
0.12
0.58
0.01
31.7
4.08
0.06
0.37
0.37
186.1
3.23
0.05
0.33
0.41
411.1
3.03
0.07
0.43
0.4
658.1
2.92
0.1
0.38
0.51
1160
2.8
0.1
0.44
0.54
GAls
Ng Žmy3 mmy1y m .
m
L Žmmy1 .
LMSE
RME
d RW C
24.1
y4.47
y0.627
0.01
0.19
0.03
779.8
y2.34
0.789
0.01
0.23
0.08
1400
y1.95
0.616
0.03
0.32
0.11
2892
y1.73
0.682
0.03
0.31
0.15
4505
y1.64
0.599
0.03
0.36
0.13
GA62
Ng Žmy3 mmy1y m .
m
L Žmmy1 .
LMSE
RME
d RW C
699
y1.55
2.5
0.14
0.52
0.26
1910
y1.01
1.03
0.07
0.35
0.13
5237
y0.73
1.2
0.06
0.46
0.11
7813
y0.72
1.1
0.06
0.48
0.13
POls
a Žmy3 mmy1yb .
b
LMSE
RME
d RW C
2490
y1.01
1.9
0.06
0.38
0.21
LMSEs logarithmic mean squared error.
RMEs relative mean error.
d RW C s relative error of estimation of the RWC.
GA62 fits do not seem to differ drastically in the graphs, but the second one produces
systematically lower absolute values of m.
5. Discussion of results
The characteristics observed in cloud and draft spectra cannot be totally explained
with available data, and will be the subject for future research work. The first regularity
to be explained is the lack of clear differences between concentrations of small raindrops
in updrafts and downdrafts. According to Willis Ž1984. Žp. 1651., it would have been
expected that condensation and coalescence between droplets had caused a surplus of
236
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
small raindrops in updrafts and evaporation effects had caused a deficit of small
raindrops in downdrafts. The first effect is clearly observed, while the second is not. The
explanation may be related with small evaporation rate in downdrafts, which seem to be
mostly precipitation-driven. The turbulent transport of water vapor, droplets and small
raindrops at the boundaries between updrafts and downdrafts are also factors to be
considered.
Essentially the same phenomenon of a high concentration of small raindrops has been
observed in surface rain, for Mexico City by Garcıa
´ and Montanez
˜ Ž1992., in Shenyan,
China, by Fujiu et al. Ž1996., and by Richter and Goddard Ž1996. for the case of warm
air advection in Southern England. These latter underlined the decisive role of environmental air in drop size distribution characteristics. For in-cloud measurements in small
and medium-sized cumuli, Klazura Ž1971. and Mazin and Shmeter Ž1977. found
relatively high concentrations of small drops. Carbone and Nelson Ž1978. measured
RSD in more developed convective clouds, with rainfall rates up to 40 mm hy1 . Their
spectra are upwardly concave in their small drop part. They found small drop deficit,
relative to the corresponding Marshall–Palmer distribution, for the value of R calculated
from data, but not relative to an exponential least squares fit. Considering their form, the
spectra could have been fitted to a gamma function distribution, with negative m.
However, Willis Ž1984. and Willis and Tattelman Ž1989. obtained gamma fits with
positive m , showing a deficit of small drops. Their sample included in-cloud air in
tropical storms and hurricanes at different altitudes.
Another important regularity to be explained is the way in which large drop sorting
takes place. In Section 3.1, the difference in Dmax between the lowest two RRC and the
higher ones was emphasized. The fact that it is accompanied by a difference in cloud top
height suggests the predominance of different mechanisms of rain formation for both
groups of RRC.
As shown above, the first giant drops generally form in updrafts. They may be the
result of an active collision–coalescence process, favored by the high humidity of the
feeding air, and high vertical velocities. Recycling and continuous growth of large drops,
previously present in the downdrafts and entrained to the updrafts, may also influence
giant drop growth, as discussed in the work of Pruppacher and Klett Ž1997. Žp. 33..
However, as evidenced from Fig. 1A, vertical velocity does not vary drastically for the
different RRC, but giant drops and high rainfall rates only occur for clouds, whose tops
are higher than 6 km. The importance of the coalescence–riming–graupel mechanism of
rain formation in deep tropical clouds with cloud base temperature above 108C is well
known ŽSilverman, 1986.. Consequently, formation and re-circulation of graupel, and
possibly hail development may occur, favoring the formation of giant drops, after the
melting of hail. This hypothesis cannot be conclusive because of the limited number of
these small clouds in the dataset. Unfortunately, the LPS was unable to distinguish
between large drops and hailstones. The only way to detect the presence of hail was the
different sound of drops and hailstones hitting the aircraft fuselage. It was reported in
some cases in the on-board diary, following this qualitative method.
On the other hand, observations of giant drops in tropical maritime warm clouds have
been made by Beard et al. Ž1986. and Black and Willis Ž1996., showing that, at least for
this type of clouds, the collision–coalescence process may produce these drops without
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
237
the participation of the ice phase. However, previous research has shown that cloud base
droplet spectra for Camaguey
¨ clouds in their earliest stages of development are much
narrower than those typical of maritime clouds ŽPerez
´ et al., 1992..
6. Summary and conclusions
RSDs and other measured cloud physical parameters have been presented for a
sample of 73 penetrations of convective clouds, most of them at an altitude close to 3
km, way below the freezing level. The measurements have been made over Camaguey,
¨
Cuba, in the summers of 1986 and 1987. A wide rainfall rate range, from almost 0 to
297 mm hy1 was studied. Five RRC were established to stratify the sample. Giant drops
are generally present in the high rainfall rate spectra, which never occurs in clouds with
tops lower than 6 km, corresponding to the y8ry108C isotherm. High concentrations
of small drops are present in almost all cases, which causes upward concavity in the
spectra. To consider the action of vertical drafts in drop sorting, separated mean RSD
were calculated for the updraft and downdraft part of the clouds. The analysis showed
that the high concentration of small drops is common to updrafts and downdrafts.
Relatively high cloud water contents were also measured in both kinds of drafts.
Large drop sorting was detected for the high rainfall rates, in the sense of higher
concentrations of large and giant drops in the updrafts of developing clouds and in the
downdrafts of clouds with prevailing downward motions. This is consistent with the
existence of a well-developed collision–coalescence process, possibly combined in some
cases with graupel and hail formation, hailstone melting or partial melting and large
drop recycling. Additional data analysis and modeling work would be needed to arrive
to definitive conclusions in this sense.
Exponential, power and gamma least squares functional fits were applied to the
observed spectra. Five additional analytical gamma function fits were obtained by
deriving the gamma form factor from the previous calculation of the moments of the
distribution and the Joss–Gori form parameters from observed data. The results for the
bin averaged mean spectra were compared, using as criteria the logarithmic mean square
error, the relative mean error and the error in RWC estimation. The traditionally used
exponential model generally underestimates small drop concentrations, and neglects the
concavity of the spectrum, and so, does not describe properly the form of the observed
distributions. However, it predicts well the liquid water content for averaged spectra.
Power function fits were also tried, but, although they perform properly for the low
rainfall rate spectra and the small drop wing of higher rainfall rate spectra, they
overestimate large drop concentrations. The least squares gamma fit ŽGAls. performs the
best in most cases, according to two mean error criteria, and has also a relatively low
error of RWC estimation. Raindrop spectra in Camaguey
¨ clouds are radically different
from those in tropical cyclone clouds, reported by Willis Ž1984. and Willis and
Tattelman Ž1989.. This is not surprising, considering the differences in dynamics,
pointed out by Jorgensen and LeMone Ž1989., between oceanic and continental clouds,
and the fact that Camaguey
¨ clouds are nearer to continental than to maritime, according
to their dynamical and microphysical characteristics, found by Perez
´ et al. Ž1992. and
Ž
.
Martınez
1996 .
´
238
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
Acknowledgements
We are grateful to V. Petrov, V. Beliaev, C. Perez
and M. Valdes,
´
´ for their
participation in the experiments and preliminary data processing, and also to the rest of
the colleagues of the Russian Central Aerological Observatory and the Cuban Institute
of Meteorology and the piloting crews of Aerocaribbean, who participated in data
collection. G. Angulo and C. Fernandez
also participated in preliminary data processing.
´
We are especially grateful to A. Nevzorov, P. Willis, J. Joss, G. McFarquhar O. Nunnez,
´ ˜
and three anonymous reviewers for their valuable suggestions. The Governments of
Cuba and the former USSR supported the experiments. One of us ŽD. Martınez
´ .
acknowledges the International Center for Theoretical Physics Programme for Training
and Research in Italian Laboratories, Trieste, for providing a grant for a research
fellowship and the Instituto di Fisica dell’Atmosfera, Rome, for providing its installations and general support.
References
Beard, K.V., Johnson, D.V., Baumgardner, D., 1986. Aircraft observations of large raindrops in warm,
shallow, convective clouds. Geophys. Res. Lett. 13, 991–994.
Black, R.A., Willis, P.T., 1996. Giant drops observed from large aircraft. 12th International Conference on
Clouds and Precipitation. Aug. 19–23, 1996. Proceedings, Vol. 1., Zurich,
Switzerland, pp. 27–30.
¨
Carbone, R.E., Nelson, L.D., 1978. The evolution of raindrop spectra in warm-based convetive storms as
observed and numerically modeled. J. Atmos. Sci. 22, 2302–2314.
Fujiu, G., Jicheng, L., Zihua, L., Taoshi, L., 1996. Studies on the raindrop spectrum characteristics in three
types of precipitus clouds. 12th International Conference on Clouds and Precipitation. Aug. 19–23, 1996.
Proceedings. Vol. 1., Zurich,
Switzerland, pp. 15–18.
¨
Garcıa,
´ F., Montanez,
˜ R.A., 1992. Ground-based raindrop spectra observations for the analysis of summer
convective showers in Mexico City. 11th International Conference on Clouds and Precipitation. Aug.
17–21, 1992. Proceedings. Vol. 1., Montreal, Canada, pp. 151–153.
Jorgensen, D.P., LeMone, M.A., 1989. Vertical velocity characteristics of oceanic convection. J. Atmos. Sci.
46, 621–640.
Joss, J., Gori, E.G., 1978. Shapes of raindrop size distributions. J. Appl. Meteor. 17, 1054–1061.
Klazura, G.E., 1971. Measurements of precipitation particles in warm cumuli over Southeast Texas. J. Appl.
Meteor. 10, 739–750.
Korolev, A.V., Strapp, W.J., Nevzorov, A.N., 1996. On the accuracy of Nevzorov airborne hot wire
LWCrTWC probe. 12th International Conference on Clouds and Precipitation. Aug. 19–23, 1996.
Proceedings, Vol. 1., Zurich,
Switzerland, pp. 346–349.
¨
Marshall, J.S., Palmer, W.M.K., 1948. The distribution of raindrops with size. J. Meteor. 5, 165–166.
Martınez,
D., 1996. Turbulence parameters and vertical drafts in tropical convective clouds over Camaguey,
´
¨
Cuba. 12th International Conference on Clouds and Precipitation. Aug. 19–23, 1996. Proceedings, Vol. 1.,
Zurich,
Switzerland, pp. 566–569.
¨
Mazin, I.P., Shmeter, S.M. ŽEds.., 1977. Cumulus clouds and the deformation of the fields of meteorological
elements related with them ŽIn Russian.. Trudy TsAO, 134, 280 pp.
Nevzorov, A.N., 1996. CAO aircraft instrumentation for cloud physics. 12th International Conference on
Clouds and Precipitation. Aug. 19–23, 1996. Proceedings, Vol. 1., Zurich,
Switzerland, pp. 371–374.
¨
Perez,
C., Martınez,
D., Petrov, V.V., 1992. Microstructure, mixing and turbulence in cumulus clouds over
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Cuba and the Caribbean Sea. Proc. WMO Workshop on Cloud Microphysics and Applications to Global
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Pruppacher, H.R., Klett, J.D., 1997. Microphysics of clouds and Precipitation. Kluwer Academic Publishers,
954 pp.
Richter, C., Goddard, J.W., 1996. The dependence of dropsize distribution shape factor on precipitation events.
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¨
Silverman, B.A., 1986. Static mode seeding of summer cumuli. A review. In: Braham, R.R. ŽEd.., Precipitation Enhancement. A Scientific Challenge. Meteorological Monograph No. 43. American Meteorological
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Ulbrich, C.W., 1983. Natural variations in the analytical form of the raindrop size distribution. J. Appl.
Meteor. 22, 1764–1775.
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3–15.
www.elsevier.comrlocateratmos
Raindrop size distributions in convective clouds
over Cuba
Daniel Martinez
a
a,)
, Enrico G. Gori
b
Instituto De Meteorologia, Ministerio De Ciencia, Tecnologia Y Medio Ambiente, Ap. 17032, CP 11700,
HaÕana 17, Cuba
b
Istituto Di Fisica Dell’atmosfera, Consiglio Nazionale Delle Ricerche, Italy
Received 6 May 1998; accepted 3 March 1999
Abstract
Raindrop size distributions, measured in the warm sector of convective clouds over Camaguey,
¨
Cuba, are analyzed; a wide range of rainfall rates is present in the dataset. Mean spectra are
presented and examined separately for updrafts and downdrafts, with additional information about
vertical velocities and cloud water content. Large drop sorting has been detected; its sense and
degree depend on rainfall rate and cloud dynamic characteristics. No significant small drop sorting
has been found. Mean spectra for individual clouds and for groups of clouds with the same rainfall
rate category have been calculated and fitted by different models; the best fits have been obtained
with a least square gamma function model. An alternate analytical procedure has been presented,
resulting appropriate to estimate rain water content and accurate in the sense of mean error.
Negative form factors, associated with upward concavity are obtained in most cases, because of
the presence of relatively high concentrations of small and large drops. q 1999 Published by
Elsevier Science B.V. All rights reserved.
Keywords: Raindrop size distributions; Convective clouds; Cloud physics; Cloud water content; Rain water
content
1. Introduction
The raindrop size distribution ŽRSD. is one of the most important parameters in cloud
microstructure, because it is related to most physical processes in the cloud and to most
integral parameters used for the cloud–environment interaction and for remote sensing
techniques. These are the rain water content ŽRWC., the radar reflectivity factor Ž Z ., the
rainfall rate Ž R ., the microwave attenuation Ž A., the optical extinction coefficient Ž S .
)
Corresponding author. E-mail: [email protected]
0169-8095r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 9 - 8 0 9 5 Ž 9 9 . 0 0 0 2 0 - 4
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
222
and others. The knowledge of RSD is of particular importance for modeling clouds and
larger scale systems, for the interpretation of radar and radiometric measurements, as
well as for applications in telecommunications, airborne transport and construction
industry. Many RSD measurements have been made at ground level. However, most of
the above-mentioned applications require the knowledge of the RSD above the ground
and inside the clouds. Consequently, in-cloud and sub-cloud measurements have been
made using airborne spectrometers ŽKlazura, 1971; Mazin and Shmeter, 1977; Carbone
and Nelson, 1978; Willis, 1984..
One of the main objectives of the study of RSD both at ground and aloft has been the
search of analytical expressions reproducing the expected spectral densities from known
integral rainfall parameters, especially rainfall rates. The most widely used RSD model
in the last decades has been the exponential distribution, proposed by Marshall and
Palmer Ž1948.. Its general form is:
N Ž D . s N0 exp Ž yl D .
Ž 1.
where D is the drop diameter Žor equivalent diameter, in non-spherical large drops.;
N Ž D . is the drop concentration per diameter interval, or drop spectral density; N0 and l
are constants for a particular RSD. For this analytical form, the RSD log–linear plot
should be a straight line with intercept N0 and negative slope l. According to Marshall
and Palmer’s results, the intercept is a constant for any type of rain or rainfall rate, and
the slope is a function of rainfall rate.
In general, N0 has been found to depend on R, and the functional dependence of l
with R may vary. These variations can be considerable, even within the same rainfall
event, if variations in air mass stability occur. For convective rain, the form of the RSD
generally shows deviations from the exponential, though it has been successfully used
sometimes ŽPruppacher and Klett, 1997, pp. 35–36..
Joss and Gori Ž1978. analyzed surface RSD, measured with a disdrometer in Locarno,
Switzerland. They observed systematic deviations from exponentiality in short time
averaged spectra, and proposed a shape parameter, whose deviation from unity is related
with the deviation of the RSD from the exponential. It depends on two integral
parameters P and Q, taken as the pth and qth moments of the RSD and of its
exponential fit, as follows:
DŽ P . DŽ Q .
D Ž P . q D Ž Q . observed
,
DŽ P . DŽ Q .
S Ž PQ . s
DŽ P . q DŽ Q .
Ž 2.
exponent
where
`
DŽ P . s
H0 D
`
H0
p
NŽ D. d D
D py 1 N Ž D . d D
.
Ž 3.
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
223
The Joss–Gori S parameters were shown to depend only on the form of the RSD. It
can be seen that SŽ PQ . s 1 corresponds to an exponential distribution, S - 1 to a
concave downward and S ) 1 to a concave upward distribution, respectively Žin
log–linear coordinates.. Joss and Gori Ž1978. found both types of cases, though the
SŽ PQ . - 0 case was the most commonly found.
Ulbrich Ž1983. proposed the use of the gamma distribution to fit RSD, considering
that the exponential can be obtained as its particular case, and that a series of
mathematical advantages can be obtained. The proposed function is of the form:
N Ž D . s Ng D m exp Ž L D . ; 0 F D F Dmax ,
Ž 4.
where m is a real number and the units of Ng are: my3 mmy1 y m . Here, m can be
interpreted as the slope that would have the spectrum for the small drop region if plotted
in a log–log diagram. If it is positive, it implies the existence of an internal mode for a
diameter greater than the minimum Ž Dmin . and, consequently, a concave downward
form, even in log–linear coordinates. For a negative m , no internal mode occurs in the
model curve, which is concave upward. The parameter L is interpreted as the slope of
the plot in log–linear coordinates, for the large drop part of the spectrum. Ng , in general,
has no useful physical meaning and shows a wide range of variation, in dependence to
m. Using a theoretical expression for the relation between any pair of integral parameters
of the gamma distribution, and a group of empirical Z–R relations, Ulbrich Ž1983.
estimated m for a variety of rain types. A great variability was found, with prevailing
negative m for orographic rain, and positive for thunderstorm rain. For stratiform rain
and showers both negative and positive values were obtained. The gamma function
model has also been used with success by Willis Ž1984. and Willis and Tattelman
Ž1989. to fit RSD measured by an airborne spectrometer in tropical storms and
hurricanes. It has also been applied to surface RSD data by Fujiu et al. Ž1996. in China
and by Richter and Goddard Ž1996. in Southern England.
Mazin and Shmeter Ž1977. used a power function model to fit RSD, measured in
warm cumulus clouds with Nevzorov’s large particle airborne spectrometer ŽLPS.. Their
spectra, with maximum diameters close to 3 mm, for the deepest clouds, looked linear in
a log–log plot, but they would have looked concave upward, if plotted in a log–linear
system the proposed distribution has the form:
N Ž D . s aDyb .
Ž 5.
In the present work, simultaneous measurements of RSDs, vertical velocity, and
cloud liquid water content in Cuban convective clouds are presented. In Section 2, a
general characterization is made of the dataset, including a brief description of instrumentation, data collection methodology and sample stratification criteria. Section 3 deals
with the results of the analysis of RSD properties, highlighting differences between
spectra in updrafts and downdrafts, and between clouds in different stages of development. Averaged spectra have been calculated for different rainfall rates. In Section 4,
individual and averaged spectra have been fitted by exponential, power and gamma
distributions. The gamma fit was made independently by a least squares routine, and by
an analytical procedure, based on the previous calculation of the Joss–Gori shape
parameter.
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
224
2. Instrumentation and data
The dataset used in the present work was collected during the 1986 and 1987 seasons
of the Cuban Weather Modification Program ŽPCMAT., run jointly by the Cuban
Institute of Meteorology ŽINSMET. and the Russian Central Aerological Observatory
ŽCAO.. Field experiments, including cloud physics observations, were accomplished in
the Camaguey
¨ Experimental Area ŽCEA., which is located in the central–eastern part of
Cuba, and is limited by a circumference of 80 km of radius, centered at the Meteorological Center of Camaguey
¨ City Ž21825X N and 77810X W.. The observed RSD correspond to
the interior of summer convective clouds, measured well below the freezing level by
Nevzorov’s large particle spectrometer ŽLPS., installed in the instrumented Aerocaribbean, INSMET IL-14. The LPS is an optical probe, developed in the Russian
Central Aerological Observatory ŽMazin and Shmeter, 1977; Nevzorov, 1996.. It is
intended for measuring particles in a range of diameters from 0.2 to more than 6 mm. It
has 12 channels, whose resolution decreases with diameter, as shown in Table 1. Its
working principle may be summarized as follows. A collimated light beam passes
through two narrow slits Ž0.12 mm wide. and is focused on a photodetector. The cloudy
air crosses the beam axis, forming a sample volume of 25 = 28 = 0.12 mm. Everytime
that a particle intersects the measuring volume, the light flow on the photodetector
decreases proportionally to the area covered by the particle, which is also nearly
proportional to the particle size for the range of measurement, as the minimum diameter
is nearly twice the slit width. This causes a response pulse that is subsequently amplified
and sent to a 12 channel integral pulse amplitude analyzer.
Considering that the position of the sensor favors the measurement of the horizontal
dimensions of the drops, which could have induced to overestimation of their sizes, a
correction for deviation of sphericity ŽPruppacher and Klett, 1997, p. 399. is introduced
in this work to calculate the equivalent drop diameter. The third row of Table 1 shows
the results.
The IL-14 was also equipped with an aircraft load complex ŽALC., including an
analog integrator and the corrections to produce wind vertical velocity with an error of
15% for vertical velocities Ž w . less than 10 mrs. For greater w, the error is estimated as
20%. Liquid water content from cloud droplets was measured with Nevzorov’s hot wire
probe, with sensitivity from 0.003 to 0.01 g my3 , depending on flight conditions. Work
principles and general characteristics of this equipment can be found in the works of
Table 1
Threshold values of diameter measurement for the 12 LPS channels
Channel
1
2
3
4
5
6
7
8
9
10
11
12
d min Žmm.
Dmin Žmm.
0.2
0.2
0.3
0.3
0.4
0.4
0.6
0.6
1.0
1.0
1.4
1.4
2.0
1.9
2.8
2.7
3.6
3.4
4.4
4.1
5.2
4.7
6.0
5.3
d min represents minimum measured drop dimensions for each channel.
Dmin represents minimum equivalent diameter for each channel.
The 12th channel has no well-defined maximum limit, so that its reading includes all raindrops greater than 5.3
mm. For statistical calculations, the same width of channel 11 has been assumed for channel 12.
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
225
Mazin and Shmeter Ž1977. and Nevzorov Ž1996.. Intercomparisons between Nevzorov’s
hot wire probe and the well-known PMS King probe have shown close agreement
ŽKorolev et al., 1996..
The dataset consists of 72 samples. Of them, 65 had altitudes near to 3 km and
temperatures from 7 to 108C. It also includes six cloud base passes at altitudes close to
1.5 km and temperatures of 19–238C. According to the flight methodology, clouds
should be penetrated in their central part so that the length of the in-cloud flight track
corresponds roughly to the horizontal extension of the cloud. The cases for which this
condition could not be fulfilled were excluded from the analysis. In the case that the
measured cloud was part of a mesoscale system, and was merged at the base with other
clouds, it was identified by its corresponding tower. Consequently, the volume of cloudy
air in each individual sample depends of the horizontal dimensions of the cloud, which
ranged from 0.8 to 7 km. As the LPS sampling area is of 7 cm2 , this corresponds to a
range of sampling volumes of 0.6–5 m3.
Cloud top heights ranged from 4 to 12.5 km, with average of 7.7 km and standard
deviation of 1.9 km. Cloud base heights ranged from 1.1 to 1.5 km, with an average of
1.2 km, and standard deviation of 0.1 km. Considering their dimensions and form, the
measured clouds can be classified as cumuli congesti or cumulonimbi. Previous studies
have shown that Camaguey
¨ convective clouds may be classified as nearer to continental
that to maritime, considering their cloud base droplet spectra in early stages of
development, and their characteristic maximum vertical velocities ŽPerez
´ et al., 1992;
Martınez,
1996..
´
Every second of in-cloud flight, corresponding to nearly 80 m, a set of measurements
was recorded. This was later used to calculate the averaged spectrum for each of the 72
cases, which will be called ‘cloud spectrum’. For a subset of 55 clouds, with simultaneous measurements of vertical drafts and RSD, averaged spectra for updrafts Ž w ) 1. and
downdrafts Ž w - y1. were calculated. These will be referred to as ‘draft spectra’.
Rainfall rates, calculated from cloud spectra were used for data stratification, applying
the criterion of Willis and Tattelman Ž1989. for high rainfall rate categories ŽRRC.. Two
more categories were included to account for the lower rates. The definition of the
applied RRC is shown in Table 2. No stratification was done according to flight height,
as the cloud base spectra were only six cases, and they showed no outstanding
differences with those measured higher in the clouds. An index was introduced to
Table 2
Ranges of rainfall rate ŽR., number of cases, and frequency distributions of top heights Ž H T . for the clouds of
the dataset, stratified by RRC
RRC
Range of
R Žmm hy1 .
Number
of cases
Cloud top height range Žkm.
4–5 5–6 6–7 7–8 8–9
9–10
10–11
11–12
12–13
1
2
3
4
5
- 2.5
2.5–25
25–62.5
62.5–125
)125
34
13
9
8
8
2
0
0
0
0
3
0
1
3
1
0
2
0
0
2
1
1
0
2
0
0
0
2
0
0
6
0
0
0
0
8
2
1
0
3
9
7
2
2
1
5
1
3
1
1
226
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
Fig. 1. General characterization of the dynamics of the clouds in each of the RRC. ŽA. Averaged mean Ž w .,
maximum Ž wma x ., and minimum Ž wmin . vertical velocities and absolute maximum Ž wq . and minimum Ž wy .
vertical velocities against RRC. ŽB. Number of cases with DESs 0 and DESs1 against RRC.
account for the stage of development of the clouds, to consider it in the analysis; it is
called DES, for development stage index. It is simply defined as DES s 1 when the
averaged measured cloud air vertical velocity was positive, i.e., w ) 0, showing
predominant upward motion, and DES s 0 in the cases when w F 0.
According to cloud top heights, whose frequency distributions are shown in Table 2,
most of the measured clouds are mixed phase, as the freezing level ranged from 4 to 5
km, and the y58C level ranged from 5 to 6 km. As can also be seen, all high rainfall
rate cases Ž R ) 25 mm hy1 . correspond to clouds with H T G 6 km, all of them with
cloud top temperatures below y58C.
Most clouds of the dataset had well-defined updrafts and downdrafts, with considerable vertical velocities This can be seen from Fig. 1, showing ŽA. the RRC average of
the mean, maximum and minimum values of the 1 s values of w for each case, and the
corresponding absolute maximum and minimum. Number of DES s 0 and DES s 1
cases for each RRC are also shown ŽB..
3. RSDs
3.1. RSDs in clouds
Typical cloud spectra for individual cases corresponding to each RRC are shown in
Fig. 2. As can be seen from these examples, maximum diameter is larger for higher
RRC spectra. The same holds for the large drop concentrations.
This can be better analyzed from Table 3. As can be seen, the range of variation of
Dmax is well differentiated between the lowest ŽRRC s 1,2. and highest ŽRRC s 3–5.
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
227
Fig. 2. Raindrop spectra for four typical cases, corresponding to different RRC. Filled squares represent data,
and curves represent functional fits.
rainfall rate spectra. This suggests that the presence of giant drops Ž D ) 5 mm. is
related to the development of intense rainfall in convective clouds in Camaguey.
¨
Another interesting characteristic of cloud spectra, apparent from the examples is that
they generally exhibit no absolute maximum for small drop diameters for D ) Dmin , as
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
228
Table 3
Frequency distributions of maximum diameters Ž Dma x . for the clouds of the dataset, stratified by RRC
RRC
1
2
3
4
5
Dma x Žmm.
0.4
0.6
1.0
1.4
1.9
2.7
3.4
4.1
4.7
5.3
5.9
1
0
0
0
0
4
0
0
0
0
10
0
0
0
0
8
0
0
0
0
3
2
0
0
0
8
3
0
0
0
0
5
0
0
0
0
1
0
1
1
0
2
1
1
0
0
0
3
1
1
0
0
5
5
6
those reported by Willis Ž1984. for tropical storm clouds. For the high rainfall rate cases
ŽD and E., the upward concavity of the spectra is apparent, especially for small drop
diameters.
3.2. RSDs in updrafts and downdrafts
For the lowest rainfall rates, updraft and downdraft spectra were usually not very
different ŽFig. 3.. The situation changes for the high rainfall rates, where drop sorting
Fig. 3. Updraft and downdraft spectra for two typical low rainfall rate cases Žleft. and two typical high rainfall
rate cases Žright..
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
229
takes place, mainly for the larger drops. This is the cause of marked differences between
updraft and downdraft spectra in their right zones, while the left parts of the two spectra
are very close to each other.
In relation with this, the dependence of CWC on vertical velocity for updrafts and
downdrafts, was investigated. For updrafts, a positive correlation was observed between
vertical velocity and CWC mainly for developing clouds Ž r s 0.70., as expected from
the formation and growth of droplets by condensation. No correlation exists for
downdrafts, but considerable, or even high CWC is present in most cases. The existence
of significant CWC in downdrafts reveals the presence of cloud droplets, which is
consistent with a low evaporation rate, and with the existence of high concentrations of
small raindrops.
The sign of the difference in large drop concentration between updrafts and downdrafts depends on the cloud stage of development, as can also be seen from the typical
cases shown in Fig. 3. Updraft spectrum for case 53 Žcorresponding to a developing
cloud., for example, is greater than downdraft spectrum for all drop sizes. The opposite
occurs for case 277, which correspond to a mature cloud containing, at flight level, a
powerful rainshaft and a weak updraft. This can be better understood from the analysis
of a particular case. Fig. 4 shows the evolution of the spectrum and other characteristics
for the cloud corresponding to cases 325–329. It was measured five times, on August
31, 1987, from 1713 to 1744 h. More penetrations could not be made for flight security
reasons. The gradual development of the cloud spectrum ŽA. is due to the parallel
evolution of RSD in updrafts ŽB. and downdrafts ŽC.. The maximum measurable
diameter is first attained in the updrafts, but the large drops reach higher concentrations
in the downdraft, in the last penetration. The main rainfall related integral parameters
and cloud top height kept growing all the time ŽD, E.. The constant behavior of Dmax
after the third penetration is probably only an instrumental truncation Žsee the note in
Table 1.. Mean vertical velocity remained positive in all passes. At the second
penetration, a decrease of mean and maximum velocity was registered, but later they
increased again. In the third pass, the ALC system got unbalanced and the register
saturated, so that the vertical velocity values could not be measured, though the
predominantly upward motion is certain.
The difference between updraft and downdraft spectra appears not only in the values
of maximum diameters, but also in the distribution of the different moments of the RSD
between the smaller and larger drops. To analyze this effect, different weighted median
diameters are used. The most extended in RSD formulations is the volume median
diameter D 0 , dividing the contribution to the RWC from left and right part of the RSD
in two equal parts. The median reflectivity diameter D 0 Z has also been used, because of
the greatest weight given to larger drops. In general, a median diameter of order p can
be defined as:
`
H0 D
p
D0p
H0
NŽ D. d D s2
D p N Ž D . d D.
Ž 6.
In Fig. 5, median diameters of order p for updraft spectra have been plotted against
the same parameters for downdrafts, for p s 1–4 and 6 ŽA–E.. Maximum diameters
have also been plotted ŽF.. The fact that the main difference between updraft and
230
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
231
downdraft spectra is given by the larger drops and depends on the cloud development
stage, is well expressed in the graphs. The distribution of points at both sides of the
y s x line is quite symmetric. Increasing the order, the separation between filled and
hollow circles Žrespectively corresponding to DES s 0 and DES s 1. increases. The
value of the diameter for which this separation is noticeable also increases with order.
This means that higher order moments, being the most affected by large drop concentrations, show the greatest dependence on vertical velocity and cloud development stage.
Note that the strictly symmetric disposition of the points corresponding to the greater
values of maximum diameter, either in updrafts or downdrafts ŽFig. 5F. may be an
artifact, due to the open maximum limit of the LPS 12th channel.
4. Functional fits to RSDs in clouds
4.1. Functional fits to indiÕidual cases
The observation of individual cloud spectra ŽFig. 2. shows that, with some exceptions, they do not look linear in log–linear coordinates, and they are generally concave
upward. Log–log plots, Žnot shown. resulted linear in some cases. This suggests a
gamma distribution, but does not allow discarding the possibility of exponential or
power fits in certain cases. Thus, the three models were tried and compared.
For different purposes, such as parameterization in cloud models, or remote sensing
applications, the estimation of RSD functional form from calculated or measured
integral parameters is desirable. This approach has been called ‘analytical’ ŽWaldvogel,
1974; Willis, 1984.. In the present section, a variant of this method will be followed to
obtain additional gamma fits for individual cloud spectra, and the results will be
compared with the least squares estimations. The method consists in obtaining the form
factor m of the gamma distribution, by applying the following relation, developed by
Ulbrich Ž1983., after calculating the Joss–Gori form parameter SP Q from the experimental DSD.
pqq
ms
Ž 7.
pqqq2m
As the choice of P and Q depends in general on the particular task, five combinations were tested: w p,q x s w6,2x; w6,3x; w6,4x; w4,3x and w3,2x. The relations Ž8. and Ž9.,
used to calculate L and Ng have also been taken from Ulbrich Ž1983.:
Ž 8.
L D 0 s 3.67 q m .
P s ap
G Ž p q m q 1.
Ž 3.67 q m .
pq m q1
Ng D 0pq m q1
Ž 9.
Fig. 4. Evolution of spectra in a developing cloud measured on August 8, 1987. The five passes correspond to
17:13 Ž1., 17:21 Ž2., 17:28 Ž3., 17:37 Ž4. and 17:44 Ž5.. ŽA. Cloud spectra; ŽB. updraft spectra; ŽC. downdraft
spectra; ŽD. integral parameters; ŽE. cloud top height and vertical velocity.
232
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
Fig. 5. Median diameters of order p for ps1,4,6 ŽA–E. and maximum diameters ŽF. for updrafts against the
same magnitude for downdrafts. For ps 2 and 6, the usual symbols have been used. The rest of the median
diameters have been denoted by D 0 p . Subindexes ‘up’ and ‘dw’ refer, respectively, to updraft and downdraft
spectra. Lines represent y s x locus.
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
233
D 0 is available from data, but it was calculated from Ž10., as a function of Dm , the so
called mass-weighted average diameter ŽUlbrich, 1983., a quantity close to D 0 both in
value and physical meaning, but statistically more representative and depending only on
the moments of the distribution.
`
Dm s
H0 D N Ž D . d D
4
`
H0 D N Ž D . d D
s
3
ž
4qm
3.67 q m
/
D0 .
Ž 10 .
Three independent criteria were applied for goodness of fit evaluation: the logarithmic mean squared error ŽLMSE., given by:
LMSEs
n
1
Ý
n
2
log Ni Ž obs . y log Ni Ž fit . ,
Ž 11 .
1
the relative mean error ŽRME., given by:
RME s
1
n
Ý
n
1
Nobs y Nfit
Ž 12 .
Nobs
and the relative error of estimation of the RWC, given by:
d RW C s
ABS Ž RWC obs y RWC fit .
RWC obs
,
Ž 13 .
In the above equations, ‘obs’ refers to the observed RSD, ‘fit’ refers to its functional
fit, Ni is the drop count of each channel, and ‘n’ is the number of LPS channels with
non zero drop counts.
The advantage of LMSE as a criterion, used previously by Willis Ž1984. is that it
gives equal weight to all parts of the spectrum, according to the approximately
exponential dependence; so, it is a good indicator for the reproduction of the form of the
original spectrum by the fit. The RME, used previously by Fujiu et al. Ž1996., removes
the dependence on the diameter by dividing the absolute error by the spectral density of
each channel, and renders the error estimation independent from the integral characteristics of the spectrum.
Only those RSD with Dmax G 1 mm were fitted, for having at least five spectral
density estimations, implying more than two degrees of freedom in regression. Thus, a
total of eight fits were tried for each of the 67 RSD fulfilling this condition. The
exponential, power and gamma least squares fits will be denoted, respectively as EXls,
POls and GAls. The analytical gamma fits corresponding to the SP Q parameters will be
denoted as GApq Ž p s 6,4,3; q s 2,3,4.. The curves in Fig. 2 shows the graphical
performance of POls, EXls, GAls, GA62 and GA64 for four characteristic cases. As can
be seen from these examples in a qualitative way, the best fit in most cases is GAls, but
the exponential and potential fits provide acceptable approximations for the lower
rainfall rates, for which the maximum diameter is small. The power fit overestimates
large drop concentration for the high rainfall rates. The exponential fit seems to suit the
right tales of the high RRC, but it neglects curvature and underestimates small drop
234
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
concentration. A preliminary analysis of the errors suggested searching differences in the
performance of the models depending on RRC.
4.2. Functional fits to mean spectra for each rainfall rate category
Cloud RSD for individual cases, corresponding to the same RRC, were bin-averaged
for each of the measured diameter intervals. For RRCs 1, five clouds had Dmax - 1
mm, and hence, less than five non-zero bins. They were excluded from the averaging.
The largest drop bins in the averaged spectra were considered for curve fit only if they
were represented in more than a fourth of the total number of clouds in the corresponding RRC.
Fig. 6 shows GAls fits for mean spectra for the five RRC. The slope of the right wing
of the spectrum decreases with rainfall rate.
The numerical values of the parameters and the errors for the three least squares fits
and the best performing analytical gamma fit are shown in Table 4. The analysis of the
errors shows that the GAls fit has always the least LMSE, and RME. All five analytic
fits give good predictions of RWC, but GA64 has very high LMSE, which is relatively
high in GA63 too Žnot shown.. The power fit performs better than the exponential from
the point of view of mean errors, particularly for the lower RRC, but the exponential
proved very efficient in predicting LWC for mean spectra. In general, EXls performs
better for mean spectra than for individual cases, as had been obtained by Joss and Gori
Ž1978. for surface rain.
The analysis of Table 4 shows that, although the different gamma fits have different
parameters, this does not have so much influence in the fit error. The parameters can
adjust themselves to a certain extent to provide a good fit. For example, the GAls and
Fig. 6. Bin averaged observed cloud spectra for the five RRC and the corresponding least squares gamma fit
ŽGAls..
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
235
Table 4
Parameters and estimation errors of averaged spectrum fits for different RRC for four functional fit models
Parameters and error
estimations
RRC
1
2
3
4
5
Exls
N0 Žmy3 mmy1 .
l Žmmy1 .
LMSE
RME
d RW C
5926
4.09
0.22
0.74
0.05
8780
2.78
0.1
0.5
0.02
4827
1.53
0.13
0.61
0.39
7904
1.51
0.12
0.58
0.11
11452
1.41
0.12
0.58
0.01
31.7
4.08
0.06
0.37
0.37
186.1
3.23
0.05
0.33
0.41
411.1
3.03
0.07
0.43
0.4
658.1
2.92
0.1
0.38
0.51
1160
2.8
0.1
0.44
0.54
GAls
Ng Žmy3 mmy1y m .
m
L Žmmy1 .
LMSE
RME
d RW C
24.1
y4.47
y0.627
0.01
0.19
0.03
779.8
y2.34
0.789
0.01
0.23
0.08
1400
y1.95
0.616
0.03
0.32
0.11
2892
y1.73
0.682
0.03
0.31
0.15
4505
y1.64
0.599
0.03
0.36
0.13
GA62
Ng Žmy3 mmy1y m .
m
L Žmmy1 .
LMSE
RME
d RW C
699
y1.55
2.5
0.14
0.52
0.26
1910
y1.01
1.03
0.07
0.35
0.13
5237
y0.73
1.2
0.06
0.46
0.11
7813
y0.72
1.1
0.06
0.48
0.13
POls
a Žmy3 mmy1yb .
b
LMSE
RME
d RW C
2490
y1.01
1.9
0.06
0.38
0.21
LMSEs logarithmic mean squared error.
RMEs relative mean error.
d RW C s relative error of estimation of the RWC.
GA62 fits do not seem to differ drastically in the graphs, but the second one produces
systematically lower absolute values of m.
5. Discussion of results
The characteristics observed in cloud and draft spectra cannot be totally explained
with available data, and will be the subject for future research work. The first regularity
to be explained is the lack of clear differences between concentrations of small raindrops
in updrafts and downdrafts. According to Willis Ž1984. Žp. 1651., it would have been
expected that condensation and coalescence between droplets had caused a surplus of
236
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
small raindrops in updrafts and evaporation effects had caused a deficit of small
raindrops in downdrafts. The first effect is clearly observed, while the second is not. The
explanation may be related with small evaporation rate in downdrafts, which seem to be
mostly precipitation-driven. The turbulent transport of water vapor, droplets and small
raindrops at the boundaries between updrafts and downdrafts are also factors to be
considered.
Essentially the same phenomenon of a high concentration of small raindrops has been
observed in surface rain, for Mexico City by Garcıa
´ and Montanez
˜ Ž1992., in Shenyan,
China, by Fujiu et al. Ž1996., and by Richter and Goddard Ž1996. for the case of warm
air advection in Southern England. These latter underlined the decisive role of environmental air in drop size distribution characteristics. For in-cloud measurements in small
and medium-sized cumuli, Klazura Ž1971. and Mazin and Shmeter Ž1977. found
relatively high concentrations of small drops. Carbone and Nelson Ž1978. measured
RSD in more developed convective clouds, with rainfall rates up to 40 mm hy1 . Their
spectra are upwardly concave in their small drop part. They found small drop deficit,
relative to the corresponding Marshall–Palmer distribution, for the value of R calculated
from data, but not relative to an exponential least squares fit. Considering their form, the
spectra could have been fitted to a gamma function distribution, with negative m.
However, Willis Ž1984. and Willis and Tattelman Ž1989. obtained gamma fits with
positive m , showing a deficit of small drops. Their sample included in-cloud air in
tropical storms and hurricanes at different altitudes.
Another important regularity to be explained is the way in which large drop sorting
takes place. In Section 3.1, the difference in Dmax between the lowest two RRC and the
higher ones was emphasized. The fact that it is accompanied by a difference in cloud top
height suggests the predominance of different mechanisms of rain formation for both
groups of RRC.
As shown above, the first giant drops generally form in updrafts. They may be the
result of an active collision–coalescence process, favored by the high humidity of the
feeding air, and high vertical velocities. Recycling and continuous growth of large drops,
previously present in the downdrafts and entrained to the updrafts, may also influence
giant drop growth, as discussed in the work of Pruppacher and Klett Ž1997. Žp. 33..
However, as evidenced from Fig. 1A, vertical velocity does not vary drastically for the
different RRC, but giant drops and high rainfall rates only occur for clouds, whose tops
are higher than 6 km. The importance of the coalescence–riming–graupel mechanism of
rain formation in deep tropical clouds with cloud base temperature above 108C is well
known ŽSilverman, 1986.. Consequently, formation and re-circulation of graupel, and
possibly hail development may occur, favoring the formation of giant drops, after the
melting of hail. This hypothesis cannot be conclusive because of the limited number of
these small clouds in the dataset. Unfortunately, the LPS was unable to distinguish
between large drops and hailstones. The only way to detect the presence of hail was the
different sound of drops and hailstones hitting the aircraft fuselage. It was reported in
some cases in the on-board diary, following this qualitative method.
On the other hand, observations of giant drops in tropical maritime warm clouds have
been made by Beard et al. Ž1986. and Black and Willis Ž1996., showing that, at least for
this type of clouds, the collision–coalescence process may produce these drops without
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
237
the participation of the ice phase. However, previous research has shown that cloud base
droplet spectra for Camaguey
¨ clouds in their earliest stages of development are much
narrower than those typical of maritime clouds ŽPerez
´ et al., 1992..
6. Summary and conclusions
RSDs and other measured cloud physical parameters have been presented for a
sample of 73 penetrations of convective clouds, most of them at an altitude close to 3
km, way below the freezing level. The measurements have been made over Camaguey,
¨
Cuba, in the summers of 1986 and 1987. A wide rainfall rate range, from almost 0 to
297 mm hy1 was studied. Five RRC were established to stratify the sample. Giant drops
are generally present in the high rainfall rate spectra, which never occurs in clouds with
tops lower than 6 km, corresponding to the y8ry108C isotherm. High concentrations
of small drops are present in almost all cases, which causes upward concavity in the
spectra. To consider the action of vertical drafts in drop sorting, separated mean RSD
were calculated for the updraft and downdraft part of the clouds. The analysis showed
that the high concentration of small drops is common to updrafts and downdrafts.
Relatively high cloud water contents were also measured in both kinds of drafts.
Large drop sorting was detected for the high rainfall rates, in the sense of higher
concentrations of large and giant drops in the updrafts of developing clouds and in the
downdrafts of clouds with prevailing downward motions. This is consistent with the
existence of a well-developed collision–coalescence process, possibly combined in some
cases with graupel and hail formation, hailstone melting or partial melting and large
drop recycling. Additional data analysis and modeling work would be needed to arrive
to definitive conclusions in this sense.
Exponential, power and gamma least squares functional fits were applied to the
observed spectra. Five additional analytical gamma function fits were obtained by
deriving the gamma form factor from the previous calculation of the moments of the
distribution and the Joss–Gori form parameters from observed data. The results for the
bin averaged mean spectra were compared, using as criteria the logarithmic mean square
error, the relative mean error and the error in RWC estimation. The traditionally used
exponential model generally underestimates small drop concentrations, and neglects the
concavity of the spectrum, and so, does not describe properly the form of the observed
distributions. However, it predicts well the liquid water content for averaged spectra.
Power function fits were also tried, but, although they perform properly for the low
rainfall rate spectra and the small drop wing of higher rainfall rate spectra, they
overestimate large drop concentrations. The least squares gamma fit ŽGAls. performs the
best in most cases, according to two mean error criteria, and has also a relatively low
error of RWC estimation. Raindrop spectra in Camaguey
¨ clouds are radically different
from those in tropical cyclone clouds, reported by Willis Ž1984. and Willis and
Tattelman Ž1989.. This is not surprising, considering the differences in dynamics,
pointed out by Jorgensen and LeMone Ž1989., between oceanic and continental clouds,
and the fact that Camaguey
¨ clouds are nearer to continental than to maritime, according
to their dynamical and microphysical characteristics, found by Perez
´ et al. Ž1992. and
Ž
.
Martınez
1996 .
´
238
D. Martinez, E.G. Gori r Atmospheric Research 52 (1999) 221–239
Acknowledgements
We are grateful to V. Petrov, V. Beliaev, C. Perez
and M. Valdes,
´
´ for their
participation in the experiments and preliminary data processing, and also to the rest of
the colleagues of the Russian Central Aerological Observatory and the Cuban Institute
of Meteorology and the piloting crews of Aerocaribbean, who participated in data
collection. G. Angulo and C. Fernandez
also participated in preliminary data processing.
´
We are especially grateful to A. Nevzorov, P. Willis, J. Joss, G. McFarquhar O. Nunnez,
´ ˜
and three anonymous reviewers for their valuable suggestions. The Governments of
Cuba and the former USSR supported the experiments. One of us ŽD. Martınez
´ .
acknowledges the International Center for Theoretical Physics Programme for Training
and Research in Italian Laboratories, Trieste, for providing a grant for a research
fellowship and the Instituto di Fisica dell’Atmosfera, Rome, for providing its installations and general support.
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