Mass–dimension (m–D) relationships determining bulk
microphysical properties such as total water content (TWC) and radar
reflectivity factor (Z) from particle size distributions are used in both
numerical models and remote sensing retrievals. The a and b coefficients
representing m=aDb relationships, however, can vary significantly
depending on meteorological conditions, particle habits, the definition of
particle maximum dimension, the probes used to obtain the data, techniques
used to process the cloud probe data, and other unknown reasons. Thus,
considering a range of a,b coefficients may be more applicable for use in
numerical models and remote sensing retrievals. Microphysical data collected
by two-dimensional optical array probes (OAPs) installed on the University of
North Dakota (UND) Citation aircraft during the Mid-latitude Continental Convective
Clouds Experiment (MC3E) were used in conjunction with TWC data from a
Nevzorov probe and ground-based S-band radar data to determine a and b
using a technique that minimizes the chi-square difference between the TWC and
Z derived from the OAPs and those directly measured by a TWC probe and
radar. All a and b values within a specified tolerance were regarded as equally
plausible solutions. Of the 16 near-constant-temperature flight legs analyzed
during the 25 April, 20 May, and 23 May 2011 events, the derived surfaces of
solutions on the first 2 days where the aircraft-sampled stratiform cloud
had a larger range in a and b for lower temperature environments that
correspond to less variability in N(D), TWC, and Z for a flight leg.
Because different regions of the storm were sampled on 23 May, differences in
the variability in N(D), TWC, and Z influenced the distribution of
chi-square values in the (a,b) phase space and the specified tolerance in a
way that yielded 2.8 times fewer plausible solutions compared to the flight
legs on the other dates. These findings show the importance of representing
the variability in a,b coefficients for numerical modeling and remote
sensing studies, rather than assuming fixed values, as well as the need to
further explore how these surfaces depend on environmental conditions in
clouds containing ice hydrometeors.
Introduction
Mass–dimension (m–D) relations are
required to link bulk microphysical properties, such as total water content
(TWC) and the forward model radar reflectivity factor (Z), to
ice crystal particle size distributions (PSDs). These relations are
extensively assumed in both numerical models and remote sensing retrievals
and relate a particle's mass (m) to its size, typically defined by
its maximum dimension projected onto a 2-D plane (D), by means of a
power law in the form m=aDb. Past
studies have suggested that the exponent b is related to the exponent in
surface area–dimension relationships or to a
particle's fractal dimension . The prefactor a
has some dependence on b and on the particle density.
Distribution of a and b coefficients used
for
characterizing m=aDb relationship from
past studies. Points colored by the (a) environment in which measurements
were taken and (b) technique used to derive the relations.
Prior m–D relationships have been determined using cloud probe data
obtained in a variety of environmental conditions.
Figure a shows how m–D coefficients derived from
previous studies vary depending on the types of clouds sampled. A full list
of these m–D coefficients and their corresponding references is
available as a supplement. Coefficients derived using data over mountainous
terrain e.g.,, cirrus clouds
e.g.,, convective clouds
e.g.,, regions of large-scale ascent e.g.,, and computer-generated shapes
e.g., are shown. A total of 119
relations are shown in Fig. . The range of a in
Fig. a spans 5 orders of magnitude, with variations in
a spanning 3 orders of magnitude or more, even for measurements
obtained in the same cloud type. The exponent b ranges between 1
and 3 within the same environments. The relations in
Fig. were derived using data collected by different
types and versions of cloud probes, using different algorithms to process the
data. showed that it can be difficult to
disentangle the dependence of derived microphysical parameters on
environmental conditions from the dependence on the probes used to collect
and the methods to process the data.
Figure b shows that m–D coefficients also vary
depending on the technique used to derive the m–D relations. In some
studies the maximum dimension of frozen hydrometeors was recorded before the
crystal was melted and the single particle mass was subsequently measured
, whereas other studies used
measurements of either bulk mass measured by an evaporation probe
or bulk Z values observed by a
collocated radar measurement in
combination with in situ measured PSDs. Furthermore, showed
that inconsistencies in how D is defined
can also impact m–D relations. For example, they noted that ice water content
(IWC) values derived using various definitions of D ranged between 60 %
and 160 % of the IWC derived using a smallest enclosing circle to define
D.
Remote sensing retrieval schemes and model microphysical parameterization
schemes are sensitive to the choice of m–D relationship. For example,
showed that differences in the mean extinction, IWC,
and effective radius retrieved from spaceborne remote sensors were 28 %,
9 %, and 30 %, respectively, depending on whether m–D relations
of spherical aggregates hereafter BF95 or bullet rosettes
were used. showed time series
of modeled precipitation rate with differences of 20 % to 50 %
depending on assumptions about particle density, which are affected by the
m–D relation. Later studies e.g.,
attributed differences in model output to the influence of particle mass on
terminal fall velocities.
Although many studies have established m–D relations for specific
cases, a universal m–D relationship has not been found, and a
single relation cannot be expected to represent the wide range of crystal habits and
sizes within clouds occurring at different temperatures and locations or those formed
by different mechanisms. Moreover, a single relationship cannot account for
the natural variability in cloud properties such as particle size, shape, and
density that occurs even in similar environmental conditions. Thus, an
alternate approach is more appropriate for modeling and remote sensing
studies that considers multiple m–D relations over many retrievals
or model simulations to evaluate the variability in the ensemble results.
While previous studies
e.g., have
considered how m–D relations vary with environmental conditions,
such as temperature, the derived relations were fixed regardless of potential
fluctuations for that environment. Further uncertainties were associated with
measurement errors induced by shattering of large ice crystals on probe tips
and subsequent detection within the probe's sample volume
, from the processing techniques used
, and from the statistical counting of particles
e.g.,. The approach by
evaluated the variability in the prefactor
a for an assumed exponent b for two field projects but
ultimately still derived a single m–D relationship for each dataset
based on the mean conditions.
Extending the approach of , which derived a volume
of equally realizable solutions within the phase space of the three gamma fit
parameters (concentration N0, shape μ,
and slope λ) characterizing PSDs, a novel approach is used here to
determine equally valid m–D relations for a given environment. Data
from a variety of environments sampled during the Mid-latitude Continental
Convective Clouds Experiment (MC3E) are used to establish a surface of
equally plausible a and b coefficients in the (a,b)
phase space using a technique that minimizes the chi-square difference
between the TWC and Z derived from the PSDs measured by
optical array probes (OAPs) and those directly measured by a TWC
probe and radar.
The remainder of this paper is organized as follows. Section
outlines the datasets used and the methodology to process the radar and
microphysics data, while Sect. describes the technique
employed to determine the surfaces of m–D coefficients. A brief
description of the MC3E cases used in this study is provided in
Sect. , and the surfaces of coefficients are derived and
discussed in Sect. . A summary of the technique and its
implications for numerical modeling and remote sensing retrieval schemes are
given in Sect. .
Data and methodology
The data in this study were collected within mesoscale convective systems
(MCSs) during the 2011 MC3E
. The study presented here uses data from
cloud microphysical instruments aboard the University of North Dakota (UND)
Cessna Citation II aircraft and from the Vance Air Force Base, OK (KVNX),
Weather Surveillance Radar 1988 Doppler (WSR-88D) radar.
Identification of coincident aircraft and radar data
The use of airborne microphysical measurements and radar data collected from
the ground allowed sampling of the same region of the cloud from
microphysical and remote sensing perspectives. Use of the Airborne Weather
Observation Toolkit radar matching algorithm and the Python ARM Radar
Toolkit Py-ART; permitted calculation of radar Z in
the vicinity of the aircraft for each second of in situ cloud distributions
measured during flight. The algorithm organizes all radar gates in a 3-D
space for efficient acquisition of radar
parameters at nearby radar range gates. The interpolation
technique is then applied to data at the eight nearest gates within 500 m of
the aircraft's location, ignoring vertically adjacent gates beyond a range of
65 km as the beamwidth exceeds the 500 m threshold to obtain an averaged
Z at the aircraft location.
List of constant-temperature flight legs used in the analysis for
which coincident data between the ground-based radar and UND Citation exist.
Start and end times, mean altitude, and temperature are displayed.
Mean temp.Mean alt.Start timeEnd time(∘C)(km)(UTC)(UTC)25 April 2011 -22.06.811:42:5011:49:00-26.57.411:05:2011:14:45-26.57.411:21:2011:34:05-35.58.310:03:0510:08:45-35.58.310:11:1010:20:15-35.58.310:28:3010:35:45-35.58.310:51:1510:59:1020 May 2011 -5.55.013:41:2513:52:00-10.55.913:54:0514:00:05-16.06.914:35:3014:40:35-23.07.914:16:3014:32:1523 May 2011 -25.07.921:49:5521:55:15-25.07.922:06:4522:11:00-34.59.122:32:5022:37:15-34.59.122:41:3522:48:20-34.59.122:58:4023:03:40
To compare microphysical properties with radar-measured Z for constant
altitude flight legs at a similar environmental temperature, only those times
when the radar and microphysical datasets are coincident and the temperature
varies by less than 1 ∘C were considered. To reduce uncertainty due
to counting statistics in the measured PSDs, microphysical data were averaged
over a 10 s period. Each 10 s period determined required radar echo and
microphysical data for all 1 s samples to ensure that the aircraft and
matched radar Z were completely in cloud during the 10 s period. The TWC
measurements and matched radar Z were then averaged over the same 10 s
period, with each 10 s interval assigned as a coincident point.
Table lists the start and end times, mean altitude, and
temperature for each of the 16 constant-temperature flight legs flown when
the UND Citation was in cloud. Observations where the mean TWC for a 10 s
interval <0.05 g m-3 were ignored as the values were considered
either below the noise threshold of the Nevzorov probe or optically thin
cloud. To further constrain the study to periods when clouds were dominated
by ice-phase hydrometeors such that TWC ≈ IWC and to reduce the
impact of liquid-phase hydrometeors on the derived TWC and Z,
observations were excluded from the analysis if the concentration from the
cloud droplet probe exceeded 10 cm-3 at any point during the 10 s
interval, which usually corresponds to the presence of water
. Of the coincident observations considered,
13 % were excluded from the analysis based on these criteria. A total of
489 coincident observations were retained for this analysis.
Radar measurements
Data from the KVNX S-band (10 cm wavelength) radar were used in this
study. Although the NASA dual-polarization (N-Pol) S-band Doppler radar
was deployed during MC3E, mechanical issues prevented reliable collection of
data for two of the three events examined here. Radars at other wavelengths
collected data during MC3E. However, attenuation through liquid portions of
the cloud e.g., and
non-Rayleigh scattering by larger particles
e.g., could not be accounted for and
prompted exclusive use of the S-band radar.
Radar reflectivity factor values for gates near the UND Citation
(Sect. .1) were used to obtain the average value of Z, using
the radar matching algorithm only if the following criteria were met:
the correlation coefficient ρHV≥0.75, sigma differential phase
SDP ≤12 deg2 km-2, differential reflectivity is represented by -2≤ZDR≤3 dB, and reflectivity texture (defined as the standard
deviation in Z of the nearest five gates) <7 dBZ. These ranges represent
acceptable values for echoes based on previous studies .
Radar gates not meeting these criteria were masked, reducing the likelihood
of including gates with excessive signal noise due to clutter or a weak signal,
contamination by the aircraft, or other factors. For instances where the
matched Z changed by more than 2 dBZ for subsequent 1 s points (fewer
than 1 % of the observations), all radar gates factored into the
radar matching algorithm were inspected by eye to ensure that no outlier
values were responsible for the jump in the matched Z. Of the observations
that were manually inspected, all appeared spatially consistent with no
outliers present and as such remained in the averaging routine of the
matching algorithm discussed in Sect. .1.
Microphysical measurements
During MC3E the Citation aircraft sampled clouds in situ, with most data
collected in ice-phase clouds between the melting layer and cloud top
. A suite of microphysical instruments was installed on
the aircraft, including OAPs, which were used to image particles and derive
PSDs, and a TWC probe. Specifics on the instrumentation and steps used to
process the data are described below.
OAP data
A cloud imaging probe (CIP), a 2D cloud (2D-C)
probe, and a High Volume Precipitation Spectrometer (HVPS), version 3 (HVPS-3), sized
particles by shadowing photodiode arrays attached to fast response
electronics. Data from the 2D-C and HVPS-3 were combined to create a
composite PSD, permitting particles between 150 µm and 19.2 mm to
be considered in the analysis. The 2D-C was used instead of the CIP in the
analysis, even though the CIP has a larger sample volume, because the inclusion
of anti-shattering tips on the 2D-C reduced the impact of shattered artifacts
e.g.,. Previous studies
showed that use
of algorithms to identify shattered artifacts is sometimes needed even when
the OAP is equipped with anti-shattering tips. Artifacts are identified by
examining the frequency distribution of the times between which particles
enter the sample volume inter-arrival time;. When
artifacts are present, this distribution follows a bimodal distribution, with
naturally occurring particles having a mode with longer inter-arrival times
and shattered artifacts having a mode with shorter inter-arrival times
e.g.,. During MC3E there was only one mode in the
inter-arrival time distribution corresponding to the naturally occurring
particles at all times, suggesting that there were few shattered
artifacts. Therefore, no shattering removal algorithm was used for the 2D-C
and HVPS. Following , the number distribution function N(D)
was determined using the 2D-C for particles with D<1 mm and the HVPS-3 for
D>1 mm. The 1 mm cutoff was chosen since N(D) for the two OAPs agreed
within 5 % on average for 0.8≤D≤1.2 mm and was used for all PSDs irrespective of periods when the
difference between N(D) for the OAPs exceeded 5 % in the overlap
region. Given uncertainties in the probe's sample area and limitations of its
depth of field for smaller particle sizes , particles
with D<150µm were not included in the analysis.
The OAP data were processed using the University of Illinois/Oklahoma OAP
Processing Software UIOOPS;. Numerous morphological
properties were calculated (e.g., particle maximum dimension, projected area,
perimeter, area ratio, and habit) for individual particles, and PSDs were
determined for each second of flight. Following and
, only particles imaged with their center within the OAP's
field of view were considered, as otherwise there is too much uncertainty in
particle size. Particles were identified as having their center within the
field of view if their maximum dimension along the time direction exceeded
the largest length where the particle potentially touched the edge of the
photodiode array.
TWC data
The TWC was determined from the Nevzorov probe using the power
required to melt or evaporate ice particles impinging on the inside of a cone
e.g.,. The probe used had a deeper
cone than previous designs, with a 60∘ vertex angle (as opposed to
a 120∘ angle) that prevented many particles from bouncing out of
the cone. Because previous studies suggested that particles with D>4 mm can bounce out of even the deeper cone ,
TWC may be underestimated when such particles are present. However,
showed that the ratio of the Nevzorov IWC to
that derived from the measured PSDs using the BF95 relation did not
significantly vary with particle maximum dimension. Of the coincident points
belonging to constant altitude flight legs in this study, 79.2 % of
the observations had cumulative mass estimates using the BF95 relation from
particles with D≤4 mm contributing at least
80 % to the total mass. Therefore, measurements of TWC were
included irrespective of whether Dmax>4 mm.
Development of equally plausible (a,b) surfaces
In this section, a method for determining a surface of equally realizable
solutions for m–D coefficients in the phase space of (a,b)
coefficients is described. The surface of these coefficients is determined
through a procedure that minimizes the χ2 differences between the
TWC and Z derived from N(D) and those directly
measured by the Nevzorov and ground-based radar, respectively. The
minimization procedure is carried out for each constant-temperature flight
leg (defined by temperature varying by less than 1 ∘C) for the
MC3E cases studied. This approach follows that of ,
who developed volumes of equally realizable N0, μ, and λ
characterizing the observed N(D) as gamma distributions for observations
obtained during the Indirect and Semi-Direct Aerosol Campaign (ISDAC) and the
NASA African Monsoon Multidisciplinary Analyses project (NAMMA).
List of constant-temperature flight legs and the ratio between
Zdiff and TWCdiff valid at
the (a,b) values that minimize χ2.
25 April 2011 20 May 2011 23 May 2011 Times (UTC)ZdiffTWCdiffTimes (UTC)ZdiffTWCdiffTimes (UTC)ZdiffTWCdiff11:42:50–11:49:002.0213:41:25–13:52:004.9221:49:55–21:55:151.5211:05:20–11:14:450.8113:54:05–14:00:056.3122:06:45–22:11:001.8211:21:20–11:34:051.6214:35:30–14:40:353.222:32:50–22:37:150.9910:03:05–10:08:450.814:16:30–14:32:153.9922:41:35–22:48:201.8210:11:10–10:20:151.522:58:40–23:03:400.3210:28:30–10:35:458.5810:51:15–10:59:101.76
For an individual 10 s sample, the TWC and Z derived
from the PSD for a specific a and b are given by
TWCSD and ZSD,
respectively, as
TWCSD=∑j=1N(aDb)N(Dj)dDj,
and
ZSD=(6πρice)|Kice|2|Kw|2∑j=1N(aDb)2N(Dj)dDj,
following the method of and accounting for the
different dielectric constants for water (|Kw|2=0.93) and ice
(|Kice|2=0.17). Uncertainties in TWCSD and
ZSD are discussed later in this section. The metric defining the
difference between the TWC and Z derived from N(D) for a specific a
and b and those
directly measured by the Nevzorov and ground-based radar,
respectively, is given by TWCdiff and Zdiff as
follows:
TWCdiff=TWC-TWCSD(a,b)TWC×TWCSD(a,b)2,
and
Zdiff=Z-ZSD(a,b)Z×ZSD(a,b)2.
In this study, TWCdiff and Zdiff are computed for
all points in the domain of values encompassing 5×10-4<a<0.35 g cm-b and 0.20<b<5.00 at
increments of 5×10-4 g cm-b and 0.01, respectively.
Given a priori assumptions of Z being proportional to the square of
a particle's mass, the square root of reflectivity was used in Eq. () so that TWCdiff would be similar
to Zdiff on average and so that each would have
approximately equal weight in determining a and b. Although
radar Z measurements involve a significantly greater sample volume
than that of OAPs and a bulk content probe, TWCdiff
and Zdiff were not weighted proportionally to the
sample volume in order to ensure that both bulk moments had some impact on
the derived a and b. Given that larger ice crystals are
fractionally more important than small crystals in determining
ZSD compared to TWCSD and given
varying contributions of larger crystals to ZSD and
TWCSD, TWCdiff has a
greater impact on the χ2 minimization procedure some of the time, while
Zdiff has a greater impact at other times. The ratios between
Zdiff and TWCdiff for each
flight leg are given in Table and range between 0.32 and
8.58, with a mean of 2.62 for the 16 flight legs. No attempt is made to force
equal weight for Zdiff and
TWCdiff for each coincident point because there are
periods when cloud properties influence TWC differently than
Z.
TWCdiff+Zdiff
in (a,b) phase space for (a) a 10 s coincident point
beginning at 13:56:15 UTC on 20 May 2011, (b) integrated over the
encompassing flight leg between 13:54:14 and 13:59:35 UTC, and normalized by
the number of observations N. The black dot in (b) denotes the a and
b minimizing χ2.
At first, the sum of TWCdiff+Zdiff is used to
identify (a,b) values that characterize an individual 10 s data point. An
example of TWCdiff+Zdiff computed in the (a,b) phase
space for a 10 s averaged PSD measured beginning at 13:56:45 UTC on
20 May 2011 is shown in Fig. a. The color representing
TWCdiff+Zdiff is shaded on a logarithmic scale to
more easily show the range of values. The smallest swath of values,
arbitrarily chosen as being TWCdiff+Zdiff≤1
within the region outlined in black, spans b values from 1.13 to 4.72. The
curvature in the outlined region highlights the correlation of a and b,
showing that a similar m can be obtained using very different b values by
adjusting a accordingly. Considering both TWCdiff and
Zdiff allows the shape and placement of the smallest swath of
values to be adjusted according to two different moments of the PSD, since
conditions impact TWC differently than Z. Using two constraints on the
χ2 minimization technique therefore provides additional insight into
the microphysical properties as discussed in Sect. .
The chi-square statistic for a flight leg, defined as
χ2(a,b)=1N∑i=1N[TWCdiff(i)+Zdiff(i)],
involves a summation over all N 10 s coincident observations
represented by the index i and normalized by N. When χ2 is
computed by summing over all N points in the flight leg, the region with the
smallest χ2 (χ2≤1; outlined region in
Fig. b) is smaller than the region in
Fig. a, which shows χ2 for a single point, because
different (a,b) values minimize χ2 for each of the individual PSDs in
the 5 min period depicted. Therefore, overall the χ2 values are
higher than the TWCdiff+Zdiff computed for each (a,b). The point in
Fig. b corresponds to the a and b point
that minimizes χ2, represented hereafter as χmin2,
which represents the most likely a and b value.
To represent the uncertainty in the derived coefficients for each flight leg,
all a and b values fulfilling χ2≤χmin2+Δχ2 are assumed to be equally plausible solutions. Analogous to
, the confidence region is defined as Δχ2=max(χmin2,Δχ12,Δχ22). The
χmin2 characterizes the robustness of the minimization
procedure affected by the natural parameter variability over a flight leg,
Δχ12 represents uncertainties in the PSD due to statistical
sampling uncertainties, and Δχ22 represents measurement
uncertainties. Similar to their study, Δχ12 is determined here as
Δχ12=1N∑i=1N12TWCSD,min(i)-TWCSD(i)TWCSD,min(i)×TWCSD(i)2+ZSD,min(i)-ZSD(i)ZSD,min(i)×ZSD(i)2+12TWCSD,max(i)-TWCSD(i)TWCSD,max(i)×TWCSD(i)2+ZSD,max(i)-ZSD(i)ZSD,max(i)×ZSD(i)2.
The different terms in Eq. () represent the difference in
the minimum and maximum TWC or Z derived from the minimum
and maximum N(D) using the most likely (a,b) values minimizing
χ2 (TWCSD,min and TWCSD,max
or ZSD,min and ZSD,max) and those derived from the measured N(D) (TWCSD or
ZSD). Following , the
minimum and maximum N(D) values are determined by subtracting or adding the
square root of the number of particles counted in each size bin to the number
of particles counted in the bin when computing N(D). This technique
represents uncertainty in the actual particle counts for each size bin as
given by Poisson statistics .
Estimates of the measurement uncertainty from the OAPs, Nevzorov probe, and
ground-based radar also influence the uncertainty in the derived
coefficients. The uncertainty due to measurement error Δχ22 is
defined as
Δχ22=1N∑i=1N12TWCSD,measmin(i)-TWCSD(i)TWCSD,measmin(i)×TWCSD(i)2+ZSD,measmin(i)-ZSD(i)ZSD,measmin(i)×ZSD(i)2+TWCmeasmin(i)-TWC(i)TWCmeasmin(i)×TWC(i)2+Zmeasmin(i)-Z(i)Zmeasmin(i)×Z(i)2+12TWCSD,measmax(i)-TWCSD(i)TWCSD,measmax(i)×TWCSD(i)2+ZSD,measmax(i)-ZSD(i)ZSD,measmax(i)×ZSD(i)2+TWCmeasmax(i)-TWC(i)TWCmeasmax(i)×TWC(i)2+Zmeasmax(i)-Z(i)Zmeasmax(i)×Z(i)2.
The terms TWCSD,measmin,
TWCSD,measmax, ZSD,measmin,
and ZSD,measmax represent the minimum and maximum
TWC or Z derived using a 50 % uncertainty in the measured N(D).
This uncertainty follows , where up to a 50 %
difference in the number concentration for particles with D>0.1 mm was
determined. Uncertainties in the bulk measurements of TWC and Z must also
be considered in the generation of the uncertainty surfaces, with the minimum
and maximum possible bulk values represented as
TWCmeasmin, TWCmeasmax,
Zmeasmin, and Zmeasmax. Following
, it was assumed that there was a 2 % uncertainty
when Dmax≤4 mm and an 8 % uncertainty for other periods
to address the possibility of particles bouncing out of the cone of the
Nevzorov probe. A radar reflectivity uncertainty of 1 dB
is subtracted from or added to the measured Z to
determine Zmeasmin and Zmeasmax.
Frequency of χmin2/Δχ12 (blue shading) and
χmin2/Δχ22 (red shading), where χmin2,
Δχ12, and Δχ22 are derived for each flight leg
used in analysis.
Figure illustrates the frequency distribution of the
ratio between χmin2 and Δχ12 (blue shading) and
between χmin2 and Δχ22 (red shading) for all 16
flight legs. Of all 16 legs considered, 15 have a ratio between
χmin2 and Δχ12 greater than 1, meaning that
χmin2>Δχ12, and 50 % of the observations
have ratios greater than 10. For 5 of the 16 legs, the ratio between
χmin2 and Δχ22 is greater than 1, indicating that
the χ2 obtained from the (a,b) minimization procedure is
greater than the difference between moments derived from the minimum and
maximum N(D) and from the minimum and maximum TWC and
Z due to measurement errors for nearly one-third of the periods in
this study. This means that the natural parameter variability over a flight
leg is sometimes more important for the derived uncertainty of m–D
coefficients, whereas at other times measurement errors are more important.
This is further discussed in Sect. .
Zc(D) as a function of D derived using modified
m–D coefficients from BF95 (black) and from the 5th (blue), 25th
(green), 50th (orange), 75th (red), and 95th (magenta) percentiles from
the set of equally plausible m–D coefficients in the order of increasing b
and a values for the 14:16:30–14:32:15 UTC flight leg on 20 May 2011.
Mean radar reflectivity matched at the aircraft's position for the same
period is listed in top left.
At first, the b coefficients greater than 3 shown in
Fig. may seem counterintuitive as the mass of a
particle cannot be greater than that of an ice sphere. Furthermore, a particle's
density would increase with increasing D for b>3. But,
due to the covariability in a and b, b>3 does
not necessarily imply that the particle has a mass greater than a sphere.
Nevertheless, equally plausible b values greater than 3 were closely
inspected, as past studies e.g., have disregarded
b>3 as a possible exponent in an m–D relation. To
investigate the impact of b>3, a linear sequence of b
values in the plausible surface was generated for each flight leg, and the
5th, 25th, 50th, 75th, and 95th percentiles of b
were determined. The corresponding a from each of these b values
was identified, and the cumulative reflectivity distribution functions,
defined as
Zc(D)=6π×ρice2|Kice|2|Kw|2∫0D(aD′b)2N(D′)dD′,
were computed using the mean N(D) for the period and the particle mass
derived with these a and b values. Figure shows an example of the
Zc(D) over the range of particle sizes observed from the
-23∘C flight leg on 20 May 2011 using these a and b
coefficients. The Zc(D) derived using BF95 coefficients, with the
prefactor a (=0.002 g cm-1.9) modified following the correction
factor of applicable for the definition of D used
here, is also shown for reference. It is worth noting that the modified BF95
coefficients may reasonably resolve the particle mass for some particle sizes
for the PSD depicted in Fig. . While the lower values of a and
b yield larger Zc(D) for smaller D than the larger values
of a and b, the derived total reflectivity Zt=∫DminDmaxZ(D)dD for the 5th and
95th percentiles of b are within 11.38 mm6 m-3 of the mean
matched radar Z of 18.36 mm6 m-3 (12.64 dBZ), a difference of
62 % in the mean. In contrast, the difference
in the mean from the Zt computed with modified BF95 coefficients
is much higher, 88.6 %, suggesting values of b>3 are indeed giving
plausible results for the range of particle sizes observed.
When the seven flight legs that have some values of b>3 in the
surface of equally plausible solutions are considered, Z values for
the 5th and 95th percentiles of
b are within 82.4 % of the mean matched radar Z.
While this value is greater than the 50.5 % difference for the other
flight legs and for the period illustrated in Fig. , Z
values for the 5th and 95th percentiles
are more consistent with the mean matched radar Z compared to that
computed with the modified BF95 relationship.
Thus, the bulk variables such as Z derived using b>3
are physically plausible for the distributions examined here, given the
covariability in a and b. However, this conclusion may only
apply when the coefficients are applied over the range of particle sizes
observed during MC3E and assuming PSDs with similar shapes. For example, for
the 95th percentile of b (b=3.61) and
the corresponding value of a used to construct Fig. ,
ice particles with D<3.83 cm have particle masses less
than those of spherical particles with a density of solid ice for the same
maximum dimension. In contrast, if the covariability in a and
b was not taken into account when choosing the corresponding
a value, then a particle could have a mass greater than that of a
spherical particle for a much smaller D. While the technique
highlights the possibility of a wide range of m–D coefficients for a
given environment, equally plausible solutions containing b>3
are still not considered in the remainder of this study to remain consistent
with previous studies and to avoid any chance of unphysical behavior should
the equally plausible coefficients be extrapolated to PSDs from remote
sensing retrievals or microphysics parameterization schemes that extend to
particle sizes larger than in the original dataset.
0.5∘ PPI scan of corrected radar reflectivity from the KVNX
radar for (a) 11:26:51 UTC on 25 April 2011, (b) 14:04:34 UTC on 20 May 2011, and (c) 23:02:54 UTC
on 23 May 2011. Black lines denote the Citation flight track for
the constant-temperature leg corresponding to the radar image shown.
Event overview
The Citation aircraft sampled different ice-phase environments during the
25 April, 20 May, and 23 May 2011 flights. provide an
overview of all MC3E cases, while give a synoptic
scale overview of the MCSs examined in this study. These particular events
were chosen because of variations in how the storms
evolved and
the location of in situ measurements relative to the convective system.
Figure shows a 0.5∘ plan-position indicator (PPI)
scan of corrected radar reflectivity from the KVNX radar for each event. The
PPI was obtained during the middle of the UND Citation flight leg, depicted by
the black line in Fig. .
The first event involved an upper-level trough that produced ascent aloft and
generated thunderstorms across northern Oklahoma around 06:00 UTC on 25 April 2011.
As these storms traversed northward along an elevated frontal boundary
overnight, their bases decoupled from the boundary layer as daytime solar
radiation ceased. The discrete cells evolved into an MCS and moved into
southern Kansas by 11:00 UTC (Fig. a), when the Citation sampled
weaker embedded convection and broader stratiform precipitation. The second
MCS, with a north-to-south-oriented squall line which was part of a larger
system, developed from a line of convective cells originating in western Texas
along a dry line around 10:00 UTC on 20 May 2011 and propagated into the
deployment region in northern central Oklahoma. The Citation aircraft primarily
flew within the trailing stratiform region of the MCS
(Fig. b). The third MCS originated as a series of discrete
supercell thunderstorms along a surface dry line in western Oklahoma and
moved eastward into the MC3E domain by 21:00 UTC on 23 May 2011 before
transitioning to a more linear MCS feature. Microphysical measurements were
made in the anvil region of these strong thunderstorms
(Fig. c).
Distribution of matched Z(a, c, e) and TWC from the
Nevzorov probe (b, d, f) for each constant-temperature leg on 25 April (a, b),
20 May (c, d), and 23 May 2011 (e, f). Whiskers represent the
5th and 95th percentiles, box edges are
the 25th and 75th percentiles, and the
line in the middle is the median. Cases where multiple legs of the same
temperature exist are shown in chronological order.
To provide context for the bulk characteristics sampled during each event,
box plots of Z matched at the aircraft's location, and TWC values from the
Nevzorov probe for each constant-temperature flight leg are given in
Fig. . The whiskers represent the 5th and 95th
percentiles from coincident observations, the box edges denote the 25th and
75th percentiles, and the red line in the middle is the median.
Distributions are listed in the order of decreasing temperature, with instances
of multiple legs having the same average temperature shown in chronological
order. While the bulk TWC and Z may differ for flight legs of similar
average temperature on a given day, as in the -26.5 and -35∘C
environments on 25 April (Fig. a–b), a greater or smaller
TWC correlates with a greater or smaller Z for most cases. The variability
in the TWC and Z as it relates to the construction of surfaces of equally
plausible m–D coefficients is discussed in the next section.
Results
This section discusses how the (a,b) surfaces vary between different
cases, as a function of temperature, depending on the determination of radar
reflectivity and depending on whether PSDs had large mass contributions from
particles with D>4 mm.
Radar absolute Z calibration
While S-band radars within the Next Generation Weather Radar (NEXRAD) WSR-88D network are calibrated
individually and among one another upon initial installation, biases in Z
can develop over time . described a
technique that uses self-similarity in the Z, ZDR, and specific
differential phase (KDP) fields to estimate the absolute Z bias
for events in rain. This method was employed for the cases in this study, and
biases in Z of -1.08 (25 April), -0.65 (20 May), and 1.43 dBZ
(23 May 2011) were found. These corrections were applied to the value of Z
calculated as explained in Sect. . The surfaces of m–D
coefficients derived using the matched radar Z and those with the bias
corrections applied were similar, with the range of equally plausible b
values differing, on average, by 6.4 % after the corrections were made.
Surfaces of equally plausible a and b values from
the m=aDb relation from each
near-constant-temperature leg on 23 May 2011 for all coincident observations
(red) and only those where cumulative mass for D>4 mm is
≤20 % (blue). Flight legs of the same temperature are shown in
chronological order.
Accounting for mass contributions from larger particles
As discussed in Sect. .3.2, the Nevzorov probe is prone to
larger particles (D>4 mm) bouncing out of the collection
cone, resulting in potential TWC underestimations. Mass contents were
derived from the PSDs using the modified BF95 coefficients to identify time
periods in which the contribution of mass from particles with D>4 mm was likely greater than 20 %. Of all 10 s PSDs used
in this study, 20.9 % had mass contributions from the larger
particles exceeding 20 % of the total mass.
Figure illustrates the similarity in the
(a,b) surfaces generated using all coincident observations (red
shading) and only those using observations with mass from larger particles
contributing ≤20 % of the total mass (blue shading) for the 23 May 2011 event. Regions of overlap between the two approaches only appear
as purple shading. The sensitivity test shows that omitting observations where
larger particles contribute fractionally more to the total mass yields an area
of equally plausible (a,b) surfaces for the 23 May event, differing,
on average, by 1.4 %. As such, all coincident observations are used
for this study irrespective of the fractional contributions of particles with
D>4 mm to the mass.
Surfaces of equally plausible a and b values for
near-constant-temperature flight legs for (a–c) 25 April, (d–g) 20 May,
and (h–i) 23 May 2011 events. Multiple legs occupying the same temperature
are assigned a different color within a panel.
Environmental impact on m–D coefficients
Surfaces of equally plausible m–D coefficients in the (a,b)
phase space from all flight legs outlined in Table are shown
in Fig. . For each event, flight legs are grouped by the
same environmental temperature, with the different colors corresponding to the
time periods given in each panel. These surfaces are influenced by how
TWC and Z derived from the PSDs relate to observed
TWC and Z and by the variability in each within a flight
leg. The observed trends in the (a,b) surfaces and how they are
affected by N(D), TWC, and Z are discussed further
below.
To compare surfaces of equally plausible solutions between different
environments and also between periods with the same temperature, the
percentage of overlap between any two flight legs is computed and shown as a matrix in
Fig. . The percentage of overlap is determined by counting
the number of (a,b) pairs contained in both equally plausible surfaces for
the conditions listed in the row and column in the matrix and dividing by the
number of (a,b) pairs in the surface for the condition listed in the row
multiplied by 100 %. There are two values in the matrix corresponding to
each comparison between two flight legs, with differences between the two
values resulting from dividing the area of the equally plausible surface from
the corresponding column by that in the corresponding row in the matrix.
Thus, it is possible for the percentage of overlap between two flight legs to
be greater when normalized by an equally plausible surface that is smaller in
area and to be smaller when normalized by a larger equally plausible
surface. It is worth noting that the percentage of overlap does not always
follow an organized trend with respect to moving away from the gray diagonal
line in the matrix, as depicted in the top right corner of
Fig. a. The lack of organized overlap values in some regions
of the matrix could be influenced by the sensitivity in computing the overlap
region over a fine resolution of (a,b) values within the domain described
in Sect. or perhaps could change in a more organized
manner if there were a more statistically representative sample for these
calculations to be made. Using the (a,b) surfaces from the
-26.5∘C flight legs on 25 April (Fig. b) as an
example, 62 % of the (a,b) surface for the 11:05:20–11:14:45 UTC
period (labeled -26.5∘C, I; Fig. a) overlaps
with the later -26.5∘C flight leg, while 65 % of the (a,b)
surface for the 11:21:20–11:34:05 UTC period (labeled -26.5∘C,
II) overlaps with the earlier -26.5∘C flight leg. The difference
occurs because there are 1132 (a,b) pairs in the surface for the
11:05:20–11:14:45 UTC period and 1077 (a,b) pairs in the surface for the
11:21:20–11:34:05 UTC period. Flight legs having the same temperature are
ordered chronologically as in Fig. and are differentiated with
a Roman numeral. Differences of the (a,b) surfaces between flight legs are
further discussed below.
Matrix of overlap area between the equally plausible (a,b)
surfaces corresponding to each row and column for (a) 25 April, (b) 20 May,
and (c) 23 May 2011. The overlap area for each square is normalized by the
area of the (a,b) surface corresponding to the flight leg listed in
each row.
25 April case
While differences exist between the (a,b) surfaces for the near-constant-temperature legs on 25 April (Fig. a), these surfaces have
considerable overlap with each other for a<0.01 g cm-b and b<2.5
(Fig. a–c). The -22 and -26.5∘C legs have
similar sets of equally plausible solutions, with (a,b) surfaces
overlapping between 46 % and 91 % (Fig. a). Less
agreement in the (a,b) surfaces is observed among the -35∘C
flight legs, with the surfaces overlapping by an average of 27.8 % among the
different combinations. The differences in the size of the surfaces is
primarily influenced by the natural variability within cloud (Δχ2=χmin2) for five of the seven legs and by the uncertainty due to
measurement errors (Δχ2=Δχ22) for the remaining legs.
The areas of the (a,b) surfaces for the -22 and -26.5∘C legs
were, on average, 31.2 % smaller than the surfaces associated with the
-35∘C environment (Fig. a–c). Three of the
four -35∘C legs have surfaces larger than the -22 and
-26.5∘C environments, as the surface of equally plausible m–D
coefficients extends beyond the maximum value a of 0.017 g cm-b and
b of 3.00 found for the -22 and -26.5∘C legs. To explain the
variation in these (a,b) surfaces for the different temperatures, the
distributions of microphysical quantities for the times corresponding to
these surfaces were examined.
Representative particle images from the HVPS-3 for each
near-constant-temperature flight leg on 25 April 2011.
Mean N(D)(a, b, c) and cumulative M(D)(d, e, f) for each constant-temperature leg on
25 April (a, d), 20 May (b, e), and
23 May 2011 (c, f). Cases where multiple legs of the same
temperature exist are shown in chronological order.
To examine the variability in hydrometeors, particle images and distributions
of bulk microphysical properties were analyzed for each flight leg. Example
particle images from the HVPS-3, which provide information on the size and
habit of ice-phase particles with D>1 mm, are plotted in
Fig. . The pictured particles represent a subset of those
imaged for the time period given and were chosen at random in an attempt to
obtain a representative sample of hydrometeors. Figure shows
the mean N(D) and cumulative mass distribution function M(D) using the
modified BF95 relationship for each flight leg analyzed in this study.
Figure details the distribution of number concentration
Nt, median mass diameter Dmm, and a metric for
particle sphericity obtained from the PSDs derived from the 2D-C and HVPS-3
data at each 10 s coincident observation. The Dmm is derived
using the modified BF95 coefficients for comparison among the different flight
legs. The whisker and box edges are the same as in Fig. .
Particle sphericity ζ is
defined by
ζ=A1/2/P,
where a is the cross-sectional area directly measured by the probe
and P is the perimeter determined from the sum of all pixels within
a width of one diode of the edge of the particle and the diode resolution.
described how a higher ζ denotes more-circular
particles. Sphericity values shown in Fig. represent a
mass-weighted mean of ζ for all particles using mass estimated from the
modified BF95 relation within each 10 s observation.
Figures , , and are
ordered in the same manner as in Fig. , with instances of
multiple legs having the same average temperature shown in chronological
order.
As evidenced by the particle images and mean N(D) at T=-22 and
-26.5∘C (Figs. a–c, a), the
presence of aggregates exceeding 5 mm is more common compared to lower
temperatures (Fig. d–g), where the ice crystals and
aggregates appear to be skewed towards smaller sizes. Distributions of
Dmm (Fig. b) and TWC
(Fig. b) also indicate this trend, with a median
Dmm for the 11:05:20–11:14:45 UTC (T=-26.5∘C)
flight leg of 2.2 mm, while the -35∘C periods have a median
Dmm ranging between 1.1 and 1.7 mm.
As in Fig. , but for number concentration
Nt, median mass diameter Dmm, and mass-weighted mean
sphericity.
To illustrate that the range of equally plausible (a,b) coefficients is
sometimes explained more by the variability in cloud parameters than the
uncertainty in measurement errors, the distributions of bulk microphysical
variables, TWC, and Z are compared between the 11:05:20–11:14:45 UTC
(T=-26.5∘C) and 10:03:05–10:08:45 UTC (T=-35∘C)
periods. The -26.5∘C flight leg had ranges in Nt,
Dmm, sphericity, Z, and TWC between the 25th and 75th
percentiles (interquartile range hereafter) of 21.5 L-1, 1.3 mm, 0.04,
5.2 dBZ, and 0.73 g m-3, respectively, while the same variables for
the -35∘C period had smaller interquartile ranges of
7.4 L-1, 0.1 mm, 0.02, 4.0 dBZ, and 0.17 g m-3
(Figs. a, b; a–c). The distribution of
χ2 in the (a,b) phase space is expected to differ when the variability
in N(D) throughout a flight leg is different between two periods since
different a and b values are likely to yield TWCSD and
ZSD similar to the observed TWC and Z.
Figure illustrates the distribution of χ2 for the
two periods, with the outlined region representing χ2 values that are
≤2 for comparison. The region containing χ2≤2 is 90.8 %
smaller for the -26.5∘C flight leg compared to the
-35∘C period and indicates that the TWCSD and
ZSD derived from all possible a and b values remain fairly
consistent over the course of the -26.5∘C flight leg due to the
smaller interquartile ranges in the TWC, Z, and bulk microphysical
properties. As such, low χ2 values are present over a larger range of
m–D coefficients for the -35∘C leg.
χ2 statistic in (a,b) phase space for the (a) 11:05:20–11:14:45 UTC
and (b) 10:03:05–10:08:45 UTC flight legs on 25 April 2011. Outlined regions represent χ2≤2, and the dots
represent χmin2.
Although the distribution of χ2 is an important factor in determining
the area of an equally plausible surface, the Δχ2 confidence
region, which is equal to χmin2 for four
of the flight legs on this day and equal to Δχ22 for three of the flight legs on this day, can also influence the area of (a,b)
surfaces. While the allowable tolerance is greater by factor of 2 for the
-26.5∘C leg, the equally plausible (a,b) surface is 3.4 times
smaller compared to the -35∘C flight leg
(Fig. b, c) because of the magnitude and distribution of
χ2 values in the (a,b) phase space. Put another way, more χ2
values considered within the (a,b) phase space are greater than the
χmin2+Δχ2 criteria to be considered equally
plausible solutions compared to the -35∘C leg.
Same as in Fig. , but for the 20 May 2011 case.
20 May case
The wide range of temperatures sampled during the 20 May event was associated
with a large variation in Z (Fig. c), with median values
ranging between 12.5 dBZ (T=-23∘C) and 27.1 dBZ (T=-5.5∘C). Representative particle images (Fig. )
highlight differences in particle size and habit between the higher
temperature flight legs (T=-5.5 and -10.5∘C) and the lower
temperature periods (T=-16 and -23∘C), with images and mean
N(D) (Fig. b) from the -5.5 and -10.5∘C legs
indicating a greater frequency of larger ice crystals and aggregates with
D≥2 mm. A Mann–Whitney U test confirms that Dmm
(Fig. e) and sphericity (Fig. f)
between the higher and lower temperature environments are statistically
different at the 99 % confidence level, with notably larger and fewer
spherical particles observed during the -5.5 and -10.5∘C flight
legs. Furthermore, median Z values for the -5.5 and -10.5∘C periods
(22.3–27.1 dBZ) are up to 30.7 times greater than for the -16 and
-23∘C legs (12.2–12.5 dBZ), while the median TWC is up to 1.9
times (0.3 g m-3) greater for the -5.5 and -10.5∘C legs.
Thus, the difference in particle properties and bulk properties TWC and Z
can be used to explain differences in (a,b) coefficients observed between
the legs on this day.
Same as in Fig. , but for the 23 May 2011 case.
Microphysical properties such as the effective density ρe of
ice hydrometeors can impact TWC differently than they do Z. The
ρe, defined here as the ratio of TWC derived assuming the
modified BF95 relationship to the integrated volume of particles enclosed by
an oblate spheroid with an aspect ratio of 0.6
e.g.,, is estimated to evaluate its influence on
TWC and Z. Median ρe ranges between 0.05 and
0.08 g cm-3 for the -5.5 and -10.5∘C periods and between
0.18 and 0.21 g cm-3 for the -16 and -23∘C flight legs.
These trends along with minimal riming evident from the 2D-C particle images
suggest that particles are on average less compact for the higher temperature
legs. Furthermore, the presence of larger aggregates as suggested by greater
values of Dmm (Fig. e), lower sphericity
(Fig. f) and ρe, and the representative
particle images from the HVPS-3 (Fig. a, b) are consistent
with an increasing Z when observed by longer wavelength radars
e.g.,.
Since differences in ρe appear to affect the TWC and Z on
20 May, the variability in N(D) is not the only factor influencing the
equally plausible (a,b) surfaces depicted in Fig. d–g.
Figure b illustrates that only the -16 and
-23∘C legs have similar (a,b) surfaces, with 85 % of the
(a,b) coefficients from the -16∘C leg overlapping with the
-23∘C flight leg. Minimum values of b for the -5.5 and
-10.5∘C flight legs, where less compact particles were observed,
were 1.84 and 1.66, respectively, while minimum b values for the -16 and
-23∘C legs were 1.09 and 1.06 for similar a values
(Fig. d–g). Looking at the (a,b) surfaces another way,
values of a for the -5.5 and -10.5∘C legs were as large as
0.031 g cm-b, while a exceeds 0.05 g cm-b for b=3 during
the -16 and -23∘C flight legs. Although the Δχ2
confidence region is equal to Δχ22 for the four flight legs on this
day and has Δχ2 values that are within 1 % of each other, the
distribution of χ2 greatly influences the extent of these surfaces in
the (a,b) phase space, with an area for the -5.5 and -10.5∘C
flight legs that is on average 2.9 times smaller than the -16 and
-23∘C periods. When considering the m=aDb relation whose
size D and exponent b are held fixed, lower values of a as observed
during the -5.5 and -10.5∘C legs suggest that particles on
average have smaller m compared to the -16 and -23∘C legs and
are consistent with smaller ρe observed for the -5.5 and
-10.5∘C periods.
Same as in Fig. , but for
(a) 14:16:30–14:32:15 UTC on 20 May and (b) 21:49:55–21:55:15 UTC on 23 May 2011. Outlined regions
represent χ2≤1, and the dots represent χmin2.
23 May case
The 23 May case was unique from the other two cases in that the bulk Z
varied less between the different temperature environments
(Fig. e), with median Z ranging only between 16.9 and
18.2 dBZ. Representative particle images (Fig. ) in
addition to the mean N(D) (Fig. c) and the cumulative
M(D) (Fig. f) indicate that the sizes and shapes of ice
hydrometeors are similar for all five flight legs. Additionally,
distributions of Dmm (Fig. h) and sphericity
(Fig. i), with median values of each varying by 0.4 mm
and 0.04, respectively, further support this similarity in cloud properties
between the different environments. Equally plausible (a,b) surfaces were
also similar irrespective of temperature (Fig. h, i), with
the four flight legs after the 21:49:55–21:55:15 UTC period having surfaces
that overlap on average 62.1 % among the different combinations
(Fig. c). The 21:49:55–21:55:15 UTC leg is the only period
on this day where the Δχ2 confidence region is determined by the
natural variability in the cloud (χmin2) rather than the
uncertainty due to measurement errors (Δχ22). As such, the (a,b) surface for this period has minimal overlap with the other equally
plausible surfaces. Closer examination of the bulk TWC
(Fig. f) indicates that values at the fifth percentile for
the 21:49:55–21:55:15 UTC period are 65.2 % less than the remaining
flight legs, which impacts the distribution of χ2 values and the (a,b) values that are within the χmin2+Δχ2 threshold.
Although surfaces of equally plausible solutions trend larger in area for
lower temperature environments on 25 April and 20 May, the area of (a,b)
surfaces among the five flight legs on 23 May are on average 2.2 (3.8) times
smaller compared to the 25 April (20 May) event. To examine how the
distribution of χ2 in the (a,b) phase space is affected by differences
in the variability in TWC and Z throughout a flight leg, the
14:16:30–14:32:15 UTC period on 20 May and the 21:49:55–21:55:15 UTC
period on 23 May are compared because of their similar temperature and
the χmin2+Δχ2 threshold used to determine the (a,b)
surfaces. Figure illustrates the distribution of
χ2 for the two periods, with the outlined region representing χ2
values that are ≤1 for the purpose of comparison. The region containing
χ2≤1 is 88.2 % smaller for the 23 May flight leg compared to
the 20 May period and highlights how different a and b values can yield a
χ2 value that is within the given tolerance based on differences in the
observed TWC and Z distributions. When bulk TWC and Z values are compared
against the 25 April (20 May) events, the median Z from flight legs on
23 May is on average 34.4 % (25.9 %) lower, while the median TWC is
90.3 % (43.9 %) greater. As mentioned in Sect. , the
sampling strategy on 23 May was different from the stratiform clouds observed
with the previous two events in that measurements were primarily made in the
anvil region of supercell thunderstorms. Previous studies
e.g., noted that the prefactor a had less of
a temperature dependence within anvil cirrus clouds, consistent with trends
in a for the 23 May flight legs.
Conclusions
This paper presented a novel approach for characterizing the variability in
mass–dimension (m–D) coefficients characterizing particle size
distributions (PSDs) during the Mid-latitude Continental Convective Clouds
Experiment (MC3E). The technique outlined here extends the approach of
, who derived a volume of equally realizable
solutions in the phase space of gamma fit parameter coefficients for
characterizing PSDs. Ground-based radar measurements of reflectivity Z
from the Vance Air Force Base, OK, radar were matched to the location of the
Cessna Citation II aircraft, where total water content (TWC)
measurements from the Nevzorov probe were made and PSDs were derived from
optical array probe data. These collocated datasets permitted use of a
χ2 minimization technique where all χ2 values within a tolerance
Δχ2 of the minimum χ2 were considered equally plausible
solutions to the m=aDb relationship
for a flight leg of similar temperature. The tolerance was determined by
considering uncertainties due to natural variability in cloud conditions for
a particular environment, the statistical sampling of particles from the
PSDs, and uncertainties in the measurements themselves.
The key findings of the paper are as follows:
The distribution of χ2 values in the (a,b) phase space shows that the
a and b parameters are highly correlated, as expected. The
degree to which these χ2 values vary throughout a flight leg is
influenced by how the PSDs, TWC from the Nevzorov probe, and
Z from radar vary within a flight leg of similar temperature. Flight
legs that have little variability in the microphysical properties and an
allowable tolerance equal to the minimum χ2 in the (a,b) phase
space, such as the 10:03:05–10:08:45 UTC period on 25 April, occupy a
surface area in the (a,b) phase space that is up to 8.7 times larger
than flight legs where microphysical properties vary more, such as the
11:05:20–11:14:45 UTC leg on the same day.
Surfaces of equally plausible solutions appear dependent on temperature for the
25 April and 20 May events. The range of plausible a and b coefficients
is larger for flight legs of lower temperature, and 80 % of the surfaces compared
between the lowest and highest temperature for each day overlap by less than 50 %.
Cases with little dependence of the surfaces of equally plausible solutions on
temperature, like the flight legs analyzed on 23 May, can be explained in terms of the
regions of cloud sampled and the types of ice hydrometeors observed. A mean overlap of
62.1 % between four of the five (a,b) surfaces on that day is consistent
with previous studies e.g., that note little dependence
in the a coefficient on temperature in anvil cirrus clouds.
The minimum χ2 in the (a,b) phase space determines the allowable
tolerance Δχ2 for 5 of the 16 flight legs when determining the set
of equally plausible a and b coefficients, whereas the
combined uncertainty due to measurement error from the OAPs, Nevzorov
TWC probe, and radar determines the Δχ2 for the remaining
11 flight legs. This means that the uncertainty in the m–D
coefficients is driven by uncertainties in the measurements the majority of
the time, with the natural parameter variability over a flight leg being a driving
factor for 31 % of the flight legs observed. Thus, efforts to reduce
measurement errors could reduce the uncertainty in derived (a,b)
coefficients.
The covariability in a and b permits possible solutions of
b>3 for the ranges of particle sizes observed in 7 of the 16
flight legs analyzed. For these flight legs this covariability means that
the Z derived from a and b and the PSDs is still
within 82.4 % of the mean matched radar Z, which is
marginally greater than the 50.5 % difference when b is not
greater than 3.
Flight legs where the cloud particles have lower effective density
ρe, such as the -5.5 and -10.5∘C flight legs on 20 May,
yield minimum b values in the (a,b) phase space as much as 0.78 larger
than clouds with a higher ρe like the -16 and -23∘C
legs on the same day. These differences can be explained by the different impacts of
ρe on TWC compared to Z.
A key finding of this study is that a range of a and b
coefficients should be considered equally plausible for a given
environment due to the natural variability in cloud conditions and
measurement uncertainties, even within a similar temperature range. This
variability results in a large range of a and b as equally
plausible solutions (as indicated in this study) and could explain the range
in m–D coefficients determined in past studies
(Fig. ), where a coefficients can vary by 3
orders of magnitude and b coefficients can vary in value between 1 and 3 for
measurements taken in similar environmental conditions. The technique used in
this study provides insight into how equally plausible m–D
coefficients can arise because the dependence of derived microphysical
parameters on environmental conditions is sometimes more important than
measurement uncertainties based on the instruments used to collect the data,
but this is not always the case. Furthermore, it is shown that the dependence of the
(a,b) coefficients on temperature is still notable even when
considering the ranges of equally plausible solutions. Future studies should
further ascertain the extent to which the dependence of (a,b) on
other environmental parameters is robust enough to be distinguished from the
natural variability in the surface or its variability due to measurement
errors.
While representing m–D coefficients as a range of equally plausible
solutions may address shortcomings of microphysical parameterization schemes
and remote sensing retrievals that employ a single m–D relationship
for a given ice species or environment, caution should be taken if the
results presented here are applied to ranges of particle size or environments
outside of those sampled (e.g., ones with different observed habits or
various degrees of riming). The results presented here illustrate that
a similar TWC and Z can be obtained regardless of the
a and b values chosen, with coefficients randomly selected
from a surface of solutions allowing one to represent how the uncertainty in
(a,b) impacts any derived quantity. Thus, the large variability in
the derived (a,b) for an equally plausible surface does not necessarily
indicate that there is a large uncertainty in quantities derived using the
a and b coefficients. Future work should assess how the
representation of modeled processes and retrieved quantities are influenced
by the variability in a and b coefficients as well as which
environmental drivers and cloud microphysical properties influence the size
of derived surfaces of equally plausible solutions and the extent to which
measurement errors need to be reduced to better refine these surfaces. The
approach presented in this study can be applied to additional studies that
make use of collocated radar and microphysical measurements in other cloud
and meteorological environments and improve the statistical robustness of
plausible m–D parameters for given environmental conditions. Such
studies may help us to further understand how surfaces of equally plausible
(a,b) solutions are affected by different environments and the
variability in cloud conditions therein, as well as the dependence of these
solutions as a function of other cloud or environmental properties.
Code and data availability
The radar
and OAP
data used in this study are found on the NASA GHRC MC3E data
archive. The software packages used to match the radar data to the aircraft's
location
and to process the OAP data are openly available as GitHub repositories. The data containing
the matched radar and microphysical properties
used in this
study are archived and available online.
List of variables and their descriptions
aPrefactor component in mass–dimension relationshipaParticle cross-sectional areabExponent component in mass–dimension relationshipχ2Chi-square statistic for each (a,b) over a flight legχmin2Lowest χ2 value in the (a,b) phase space for a flight legΔχ12Threshold determined from uncertainty in the particle size distribution due to sampling statisticsΔχ22Threshold determined from combined uncertainty due to measurement errorsΔχ2Maximum value of χmin2, Δχ12, or Δχ22DParticle maximum dimensionDmmMedian mass diameterIWCIce water contentKDPSpecific differential phase|Kice|2Dielectric constant for ice|Kw|2Dielectric constant for waterM(D)Mass distribution functionN(D)Number distribution functionNtTotal number concentrationPParticle perimeterρeEffective densitySDPSigma differential phaseTEnvironmental temperatureTWCTotal water content measurementTWCdiffMeasure of normalized difference between the Nevzorov TWC and that derived from the N(D) for a given (a,b) defined by Eq. ()TWCSDTWC derived from the N(D) for a given (a,b)ζParticle sphericityZRadar reflectivity factorZc(D)Cumulative reflectivity distribution function up to size D′Z(D)Reflectivity distribution functionZdiffMeasure of normalized difference between the radar Z and that derived from the N(D) for a given (a,b) defined by Eq. ()ZDRDifferential reflectivityZSDZ derived from the N(D) for a given (a,b)ZtDerived total reflectivity from the mean N(D) for a given (a,b)
The supplement related to this article is available online at: https://doi.org/10.5194/acp-19-3621-2019-supplement.
Author contributions
JF prepared the paper and performed all the calculations,
with contributions from all co-authors. GM provided the idea and formulated
the framework for the study, SN provided the framework for matching radar
gates to an aircraft's position, WW processed particle data from the optical
array probes, PZ conducted the radar bias calculations used in this study,
and RR and HM provided feedback on the ideas and calculations presented.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
This research was supported by the US Department of Energy grants
DE-SC0014065 and DE-SC0016476 (through UCAR subcontract SUBAWD000397), by the
NASA Precipitation Measurement Missions grant NNX16AD80G, and by the National
Science Foundation grant AGS-1247404. We thank all participants of MC3E for
collecting the data used in this study.
Edited by: Martina Krämer
Reviewed by: Darrel Baumgardner and one anonymous referee
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