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BRANDES ET AL.
On the Influence of Assumed Drop Size Distribution Form on Radar-Retrieved
Thunderstorm Microphysics
EDWARD A. BRANDES, GUIFU ZHANG,
AND
JUANZHEN SUN
National Center for Atmospheric Research,* Boulder, Colorado
(Manuscript received 27 January 2005, in final form 14 June 2005)
ABSTRACT
Polarimetric radar measurements are used to retrieve drop size distributions (DSD) in subtropical thunderstorms. Retrievals are made with the single-moment exponential drop size model of Marshall and
Palmer driven by radar reflectivity measurements and with a two-parameter constrained-gamma drop size
model that utilizes reflectivity and differential reflectivity. Results are compared with disdrometer observations. Retrievals with the constrained-gamma DSD model gave better representation of total drop concentration, liquid water content, and drop median volume diameter and better described their natural
variability. The Marshall–Palmer DSD model, with a fixed intercept parameter, tended to underestimate
the total drop concentration in storm cores and to overestimate significantly the concentration in stratiform
regions. Rainwater contents in strong convection were underestimated by a factor of 2–3, and drop median
volume diameters in stratiform rain were underestimated by 0.5 mm. To determine possible DSD model
impacts on numerical forecasts, evaporation and accretion rates were computed using Kessler-type parameterizations. Rates based on the Marshall–Palmer DSD model were lower by a factor of 2–3 in strong
convection and were higher by about a factor of 2 in stratiform rain than those based on the constrainedgamma model. The study demonstrates the potential of polarimetric radar measurements for improving the
understanding of precipitation processes and microphysics parameterization in numerical forecast models.
1. Introduction
The assimilation of radar reflectivity and radial velocity information into numerical forecast models has
potential to improve short-term numerical forecasts of
convection (Sun 2004). The planned upgrade of the national network of Weather Surveillance Radar-1988
Doppler (WSR-88D) systems for polarimetric measurements should lead to further improvement. Demonstrated capabilities with research polarimetric radars include better rainfall estimation (e.g., Brandes et al.
2002) and discrimination of hydrometeor types (Vivekanandan et al. 1999; Straka et al. 2000), accurate designation of freezing levels and melting layers (Brandes
and Ikeda 2004), and a capability to retrieve drop size
* The National Center for Atmospheric Research is sponsored
by the National Science Foundation.
Corresponding author address: Dr. Edward A. Brandes, National Center for Atmospheric Research, P.O. Box 3000, Boulder,
CO 80307.
E-mail: brandes@ncar.ucar.edu
© 2006 American Meteorological Society
distributions in storms (e.g., Zhang et al. 2001; Brandes
et al. 2003, 2004a,b). Implications are enormous for
promoting the understanding of the interrelationships
between drop size distributions (DSD) and storm morphology and between solid and liquid precipitation.
The retrievals should also prove important for verifying
and refining microphysical parameterizations within
numerical forecast models. In the simulation and prediction of atmospheric moist convection, parameterization of microphysical processes is clearly a principal
source of error. Also, as more and more detailed storm
structures are simulated, the sensitivity to microphysical parameterizations becomes ever more important.
To date, polarimetric measurements have not been
used in numerical forecast models. As a first step, we
examine the dependence of radar-derived thunderstorm microphysical properties on the assumed form of
the drop size distribution. The exponential DSD model
of Marshall and Palmer (1948) with a fixed intercept
and the two-parameter constrained-gamma (C-G)
model (Zhang et al. 2001) are investigated. Although a
comparison with a two-parameter exponential DSD
model that fully exploits polarimetric radar measure-
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JOURNAL OF APPLIED METEOROLOGY AND CLIMATOLOGY
ments might be more appropriate, the Marshall–Palmer
(M–P) DSD model was chosen because of its long history in rainfall estimation and numerous applications in
numerical forecasting. Also, this study prefaces that of
Zhang et al. (2006) who compare forecasts with the
numerical model of Sun and Crook (1997) using microphysics parameterizations based on the M–P DSD
model and a simplified C-G model. The primary goal of
this study is to demonstrate benefits that can be gained
from this new observational capability.
We begin with a review of the M–P and C-G DSD
models. Polarimetric radar measurements are then
used to retrieve DSD attributes using the two models.
Retrieved parameters and estimates of microphysical
processes are compared with disdrometer observations.
The spatial distribution of retrieved DSD characteristics and estimated evaporation and accretion rates are
also examined.
2. Examined drop size distribution models
Drop distribution forms selected for study are the
exponential model of Marshall and Palmer (1948) and
the constrained-gamma model of Zhang et al. (2001).
Marshall and Palmer determined that, except for small
diameters, raindrop size distributions can be approximated with
N共D兲 ⫽ N0 exp共⫺⌳D兲,
共1兲
where N0, a concentration parameter, is 8000 m⫺3
mm⫺1, ⌳ (mm⫺1) is a distribution slope term, and D
(mm) is the drop equivalent volume diameter. There
has been much discussion about the appropriateness of
this restrictive DSD model and about whether the intercept N0 is constant. Sauvageot and Lacaux (1995)
show that N0 varies and increases significantly as rain
rate R increases. Sauvageot and Lacaux also determined that ⌳ decreases with R for R ⬍ 20 mm h⫺1 and
has a near-constant value in the range of 2.2⫺2.3 mm⫺1
for heavier rain rates. A constant intercept or slope
parameter in (1) reduces the DSD to a single-parameter distribution. If true, rainfall could be estimated
with a single radar measurement (radar reflectivity factor), and microphysical parameterizations in numerical
forecast models could be reduced to a single moment
(e.g., Lin et al. 1983; Walko et al. 1995; Sun and Crook
1997). A constant N0 dictates that only the slope term
can respond to a change in precipitation intensity. Assuming a constant slope and allowing N0 to vary dictates that the mass-weighted terminal velocity is constant. In either case a severe constraint is placed on the
VOLUME 45
DSD. A more general approach would be to let N0 and
⌳ vary, but currently there is little basis for specifying
both parameters in numerical models. In this study the
M–P DSD model is used for illustration purposes. To
be consistent with other applications of the exponential
DSD model, for example, the microphysical parameterization schemes of Kessler (1969), we assume a maximum drop size of infinity.
The constrained-gamma model was developed by
Zhang et al. (2001) for retrieving DSD parameters from
polarimetric radar measurements. Raindrops are assumed to be represented by the gamma size distribution
(Ulbrich 1983)
N共D兲 ⫽ N0D␮ exp共⫺⌳D兲.
共2兲
With three parameters [the concentration parameter
N0 (m⫺3 mm⫺1⫺␮), a shape term ␮, and ⌳], (2) is capable of describing a variety of drop size distributions
and has been widely accepted by the radar-meteorology
community. Note that (2) reduces to (1) when ␮ ⫽ 0.
Retrieval of the governing parameters in (2) requires
three measurements or relationships. Among polarimetric variables radar reflectivity and differential reflectivity are the most suitable for drop size distribution
retrieval (Brandes et al. 2004b). Because independent
information from remote radar measurements is limited, physical constraints are required for retrieval.
Zhang et al. (2001) closed the system with an empirical
constraint between ␮ and ⌳ that was derived from disdrometer measurements. Brandes et al. (2003) redefined the constraining relation as
⌳ ⫽ 0.0365␮2 ⫹ 0.735␮ ⫹ 1.935
共3兲
to accommodate heavy rainfalls better. With (3) and
radar measurements of radar reflectivity at horizontal
polarization ZH and differential reflectivity ZDR, the
governing parameters of (2) can be obtained with an
iterative procedure and the retrieved DSD can then be
integrated to obtain physical properties of the distribution. This approach is computationally intensive. An
alternate method (Brandes et al. 2004a) is to use the
C-G model [(2) and (3)] to derive a set of polynomial
relations for retrieving DSD attributes. In brief, the
procedure is to assume a range of ␮ and to compute ⌳
with (3). A value for N0 is specified and calculations are
made of reflectivity, differential reflectivity, total drop
concentration NT (m⫺3), liquid water content W (g
m⫺3), and drop median volume diameter D0 (mm) using (2), the selected N0, and the tabulated values of ⌳
and ␮. Ratios of NT and W with ZH are independent of
N0 and are functions of ␮ (or ⌳) determined by ZDR
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BRANDES ET AL.
FIG. 1. Radar (left) reflectivity and (right) differential reflectivity measurements from a thunderstorm observed
in Florida at 1926 UTC 17 Sep 1998.
alone for the C-G model. Logarithms of the ratios are
then fit with polynomial relations. Diameter D0 is assumed to be a function of ZDR alone. The resulting
relations are
2
W ⫽ 5.589 ⫻ 10⫺4ZH ⫻ 10共0.223ZDR⫺1.124ZDR兲,
2
NT ⫽ 2.085ZH ⫻ 10共0.728ZDR⫺2.066ZDR兲,
and
共4兲
共5兲
3
2
D0 ⫽ 0.171ZDR
⫺ 0.725ZDR
⫹ 1.479ZDR ⫹ 0.717,
共6兲
with ZH in millimeters to the sixth power divided bycubic meters and ZDR in decibels. [A corresponding
relation for rainfall rate is given by Brandes et al.
(2004a).] Expressions (3)–(6) are for truncated DSDs
for which Dmin (mm), the size of the smallest drop, was
assumed to be 0.1 mm and Dmax, the largest drop, was
estimated from radar reflectivity (Brandes et al. 2003).
As a physical parameter, the total number concentration is examined rather than N0. Equations (4)–(6) assume that drop axis ratios r—the vertical dimension
divided by the horizontal dimension—are given by
r ⫽ 0.9951 ⫹ 0.025 10D ⫺ 0.036 44D2 ⫹ 0.005 303D3
⫺ 0.000 249 2D4
共7兲
[Brandes et al. 2002; the relation as given in Brandes et
al. (2002) contains a typographical error that was repeated in subsequent publications but is corrected
here—see Brandes et al. (2005)].
Previous studies (Zhang et al. 2001; Brandes et al.
2003, 2004a,b) suggest that D0 can be retrieved with an
accuracy of about 0.2 mm. Errors in polarimetric rainfall estimates based on DSD retrieval are reduced from
that associated with radar reflectivity alone (Brandes et
al. 2003, their Table 3). Root-mean-square errors for
rainfall accumulations, after removing systematic bias,
were lowered from 7.8 to 6.7 mm, and the correlation
between radar-estimated and gauge-observed rainfalls
increased by 0.05 to a value of 0.92. The improvement
comes from drop size information contained in the ZDR
measurement. Retrievals of rainwater content have not
been verified quantitatively, but improvements over reflectivity-based algorithms are likely comparable to
that for rainfall estimates.
For the M–P DSD [(1) with N0 ⫽ 8000 m⫺3 mm⫺1],
retrieval relations for W, D0, and NT are
0.571
,
W ⫽ 3.45 ⫻ 10⫺3ZH
D0 ⫽ 1.64W0.25,
NT ⫽
8000D0
.
3.67
共8兲
共9兲
and
共10兲
Equations (8)–(10) assume 0 ⱕ D ⱕ ⬁ mm and use the
integral definitions of W and ZH and the approximation
D0 ⫽ 3.67/⌳ (Ulbrich 1983). An upper drop size limit of
infinity yields rainfall accumulations that are ⬃7%
larger than DSDs that are truncated to match observations (Brandes et al. 2003). With the M–P DSD model,
W, D0, and NT are all derived from the reflectivity measurement alone.
3. Comparison of DSD attributes retrieved with
the Marshall–Palmer and constrained-gamma
models
Radar measurements used in this study were obtained with the National Center for Atmospheric Research S-band dual-polarization radar in east-central
Florida. In situ DSD measurements were collected with
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FIG. 2. DSD physical parameters retrieved from polarimetric radar measurements obtained
on 17 Sep 1998 using the C-G and M–P DSD models. Disdrometer observations show validation.
a video disdrometer (Kruger and Krajewski 2002) located 38 km from the radar. Figure 1 shows reflectivity
and differential reflectivity measurements from one of
several thunderstorms that passed over the disdrometer
site (x ⫽ ⫺27 km, y ⫽ 27 km) on 17 September 1998.
The beam center (0.5° antenna elevation) was roughly
350 m above the disdrometer.
Figure 2 shows a time series of radar-based retrievals
of total drop concentration, rainwater content, and
drop median volume diameter using the M–P and C-G
DSD models. Attributes computed from disdrometer
observations provide verification. It is important to
note that the disdrometer results are computed directly
from the observations and do not assume a particular
DSD model. The disdrometer observations are 1-min
samples that roughly correspond in scale to the radar
retrievals. The latter are based on averages of reflectivity and differential reflectivity taken over five range
bins (750 m). Although the radar–disdrometer comparison is influenced by the disparity in sampling volumes, the minute-to-minute scatter in instrument differences is relatively small. Systematic differences may
arise from DSD variations not captured by the simple
␮–⌳ relation [(3)], vertical gradients of precipitation
(after 2200 UTC), or the horizontal advection of precipitation in regions of reflectivity or differential reflectivity gradients (2045–2115 UTC). Comparisons between radar-derived and disdrometer-observed DSD
properties are also influenced by radar beam smoothing, which tends to reduce maxima and raise minima.
It is apparent that for strong convection (1915–1935
and ⬃2030 UTC) the M–P DSD model [(10)] underestimates the total drop concentration. Drop concentrations are overestimated for stratiform precipitation that
formed from convective debris (1940–2020 and 2130–
2300 UTC). During strong convection (e.g., 1917 UTC)
observed drop distributions were broad and roughly
exponential (Fig. 3). A fit to the data using the third
and sixth moments of the DSD yielded an intercept of
20 900 m⫺3 mm⫺1. This large intercept agrees with the
study of Marecal et al. (1997) who found that in convection the number of small drops exceeded that of the
M–P model. Observed drop populations for weak convection (2100 UTC) and for stratiform precipitation
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BRANDES ET AL.
FIG. 3. Observed 1-min DSD for strong convection (1917 UTC),
weak convection (2100 UTC), and stratiform rainfall (2230 UTC)
on 17 Sep 1998; CT is the number of drops detected, R is the rain
rate (mm h⫺1), and Z is the radar reflectivity at horizontal polarization (dBZ ).
(2230 UTC) are more narrowly distributed and have a
relatively low number concentration. Computed intercepts for these DSDs are 22 800 and 2200 m⫺3 mm⫺1,
respectively. It is evident from Fig. 3 that the M–P DSD
model is a poor fit for the 1-min drop distributions
observed during these storm stages and overestimates
the number of small drops. Overestimated small drop
populations with an exponential DSD can cause a significant rainfall overestimate bias even when the intercept is permitted to vary (Brandes et al. 2003, their
Table 3).
Total drop concentrations retrieved with the M–P
DSD model remain fairly uniform throughout the observational period. For an exponential DSD and an upper drop size limit of infinity, the total drop concentration and the intercept are related by NT ⫽ N0/⌳. Because N0 is fixed, a decrease in reflectivity associates
with an increase in the DSD slope. The number concentration of large drops decreases disproportionately
while small drop concentrations decrease minimally. A
fixed intercept also causes the underestimate of rainwater content in intense convection. The most notable
feature of the retrieved D0 is that drop sizes in the
stratiform rain shield, which forms in the declining
stages of the storm complex, are underestimated by
about 0.5 mm. This result is consistent with the overestimate in number concentration.
Comparisons of DSD parameters retrieved with the
C-G model [using (4)–(6)] and disdrometer observations (Fig. 2) attest to the utility of the method. The
magnitudes of retrieved parameters closely agree with
observations. Biases in the retrievals for NT and D0 are
relatively small. A somewhat larger bias (an underestimate) is evident for the rainwater content retrieval during the intense convective stages. This result may be
due in part to radar beam smoothing.
Examination of the drop distributions for the weak
convection and stratiform precipitation (Fig. 3) reveals
that associated reflectivity values are nearly equal (37.8
and 38.4 dBZ, respectively). Relative to the respective
values for the stratiform sample, the number of detected drops for the weak convection is larger by more
than a factor of 3 and the rain rate is larger by a factor
of 2. The broader size distribution during the stratiform
rainfall associates with drop median volume diameters
that are larger than that during the weak convective
stage. The C-G retrieval is sensitive to such DSD variations through the dependence on differential reflectivity.
Spatial distributions of retrieved parameters for the
two DSD models are illustrated in Fig. 4. The more
uniform total drop concentration with the M–P DSD
model is readily apparent. Overall, the distributions of
rainwater content and drop median volume diameter
are more uniform as well. With the M–P DSD model,
the largest D0 are located in the reflectivity cores (Fig.
1). With the C-G model, D0 is determined largely by
differential reflectivity. Although drops are large in the
storm core, the largest drops are with new convection at
the leading edge of the storm complex (e.g., near x ⫽
⫺23 km, y ⫽ 33 km and x ⫽ ⫺27 km, y ⫽ 23 km).
Estimated maximum rainwater contents for both DSD
models are in the storm core. However, values are
much higher with the C-G model (6 vs 2.5 g m⫺3) because median drop diameters are smaller.
4. Microphysical processes
In this section, we develop parameterized expressions for microphysical processes to demonstrate potential impacts of the assumed DSD form on numerical
forecasts. For illustration we use the parameterization
scheme of Kessler (1969). Following his Table 4, the
evaporation rate of raindrops Re (g m⫺3 s⫺1) and the
accretion rate of cloud drops Rc (g m⫺3 s⫺1) for the
M–P DSD are single-parameter relations given by
Re ⫽ 5.03 ⫻ 10⫺4W13Ⲑ20
Rc ⫽ 5.08 ⫻ 10⫺3W7Ⲑ8.
and
共11兲
共12兲
For simplicity we have assumed a saturation deficit of
1.0 g m⫺3 in (11) and a cloud water content of 1.0 g m⫺3
in (12). Also, it is assumed in (12) that the raindrop
collection efficiency is 1.0 and that drop terminal ve-
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FIG. 4. Spatial distribution of retrieved rain DSD physical parameters for the 17 Sep storm at 1926 UTC using
the (left) C-G and (right) M–P DSD models.
locities are proportional to the square root of the drop
diameter.
Comparable evaporation and accretion rate expressions were derived for the C-G model. The evaporation
rate of a single drop at a saturation deficit me is (Kessler
1969)
dMe
⫽ 3.55 ⫻ 10⫺7meD8Ⲑ5.
dt
共13兲
The total evaporation rate is
Re ⫽
冕
Dmax
Dmin
dMe共D兲
N共D兲 dD.
dt
共14兲
From the integral expression for rainwater content and
(2) we obtain
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BRANDES ET AL.
⌳共␮⫹4兲
W,
␳w␲ ⫻ 10⫺3 关␥共⌳Dmax, ␮ ⫹ 4兲 ⫺ ␥共⌳Dmin, ␮ ⫹ 4兲兴
6
共15兲
where ␥ denotes an incomplete gamma function and ␳w
is the density of water. After substituting (13), (15),
␳w ⫽ 1.0 g cm⫺3, and me ⫽ 1.0 g m⫺3, the total evaporation rate becomes
Re ⫽ 6.78 ⫻ 10⫺4⌳7Ⲑ5
⫻
关␥共⌳Dmax, ␮ ⫹ 13Ⲑ5兲 ⫺ ␥共⌳Dmin, ␮ ⫹ 13Ⲑ5兲兴
W.
关␥共⌳Dmax, ␮ ⫹ 4兲 ⫺ ␥共⌳Dmin, ␮ ⫹ 4兲兴
共16兲
Again following Kessler, the accretion of cloud water
for a collection efficiency of 1.0 is
Rc ⫽ 10⫺6
冕
Dmax
Dmin
␲D2
␷ 共D兲mcN共D兲 dD,
4 t
共17兲
where ␷t(D) (m s⫺1) is the drop terminal velocity and
mc is the cloud water content. Substituting (2), the expression for N0 [(15)], mc ⫽ 1.0 g m⫺3, ␳w ⫽ 1.0 g cm⫺3,
and the drop terminal velocity relation of Brandes et al.
(2002),
␷t ⫽ ⫺0.1021 ⫹ 4.932D ⫺ 0.9551D2 ⫹ 0.079 34D3
⫺ 0.002 362D4,
共18兲
yields
3 ⫻ 10⫺3
cl⌳⫺l⫹1
Rc ⫽
2
l⫽0
4
兺
⫻
关␥共⌳Dmax, ␮ ⫹ l ⫹ 3兲 ⫺ ␥共⌳Dmin, ␮ ⫹ l ⫹ 3兲兴
W,
关␥共⌳Dmax, ␮ ⫹ 4兲 ⫺ ␥共⌳Dmin, ␮ ⫹ 4兲兴
共19兲
where cl represents the constants in the terminal velocity relation. For convenience, alternate forms for Re
and Rc consistent with (4)–(6) are
2
Re ⫽ 3.99 ⫻ 10⫺7ZH ⫻ 10共0.3147ZDR⫺1.448ZDR兲 and
共20兲
2
Rc ⫽ 2.84 ⫻ 10⫺6ZH ⫻ 10共0.1723ZDR⫺1.087ZDR兲.
共21兲
With the M–P DSD model, evaporation and accretion rates [(11) and (12)] are determined entirely by the
rainwater content estimate from radar reflectivity.
Drop size distribution information is incorporated principally through the use of (1) in the determination of
265
(8). With the C-G model, drop size information is incorporated through ⌳ and ␮ and their association with
the differential reflectivity measurement.
5. Comparison of retrieved evaporation and
accretion rates
For illustration purposes, application is made to radar measurements obtained at 0.5° antenna elevation.
Cloud water would not normally exist at this height. An
elevation above cloud base could be selected, but the
purpose here is to demonstrate the importance of the
assumed DSD model in microphysical parameterization. This height allows a comparison with estimates
based on disdrometer observations. Figure 5 compares
retrieved evaporation and accretion rates using the
M–P and C-G DSD models. Evaporation and accretion
rates computed with the C-G DSD model [(16) and
(19)] are much larger in the convective cores of storms.
Numerically simulated thunderstorms using the C-G
DSD model would have stronger cold pools and outflows than those initialized with the M–P model, and
they would propagate differently. Also, the C-G DSD
model provides more accurate estimates of massweighted terminal velocity and rainfall rate than does
the M⫺P DSD model (not shown).
Spatial distributions of estimated evaporation and accretion rates (Fig. 6) reveal patterns consistent with the
retrieved DSD parameters (Fig. 4). The range in estimated rates for both physical processes is much larger
with the C-G DSD model. Both models show maximum
evaporation and accretion in the storm core. However,
rates with the C-G model are about a factor of 2–3
higher than those with the M–P model. At the leading
edge of the storm, where drops tend to be large and the
total surface area of the drops tends to be reduced relative to the situation with small drops, evaporation and
accretion are smaller with the C-G model. Also, in trailing (left side) regions of the storm characterized by
more stratiform precipitation, rates are reduced. Total
drop concentrations with the C-G model tend to be
lower and the drops tend to be larger, resulting in more
narrow distributions with less evaporation and fewer
collisions. The potential improvement that might be
gained in an application with a numerical model is discussed by Zhang et al. (2006). However, Figs. 5 and 6
establish the importance of having the correct DSD.
6. Summary and discussion
Thunderstorm microphysical properties were retrieved using the Marshall–Palmer and constrainedgamma DSD models. It was shown that the M–P model,
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FIG. 5. Time series comparison for 17 Sep showing estimated (top) evaporation rate Re and
(bottom) accretion rate Rc using Kessler parameterizations. Results are shown for disdrometer observations and radar retrievals using the C-G and M–P DSD models.
based on a single radar moment (radar reflectivity),
oversimplifies raindrop size distributions and has a detrimental impact on retrieved DSD parameters. Total
drop concentrations are too uniform because the intercept of the distribution is fixed, effectively setting the
number concentration of small drops. DSD variability
is dictated only by changes in distribution slope; a
change in reflectivity primarily affects large drop concentrations. The net result is that total drop concentrations are significantly overestimated in stratiform precipitation. In addition, there is a tendency to underestimate total drop concentrations in the core region of
convective storms. Other retrieved DSD parameters
(e.g., the rainwater content and drop median volume
diameter) are also too uniform with the M–P DSD
model. Rainwater contents in thunderstorm cores may
be underestimated by a factor of 2–3. This result is
because drop median volume diameters are determined
entirely by reflectivity and are generally overestimated.
Overall, retrievals of total drop concentration, rainwater content, and drop median volume diameter with the
two-parameter C-G DSD model, based on radar reflectivity and differential reflectivity, exhibit considerably
less bias and readily reproduce trends in observed
DSDs.
Retrieval errors also influence derived microphysical
processes, such as evaporation and accretion. Estimated evaporation rates in thunderstorm cores with the
C-G model were much higher than with the M–P
model. This difference follows from smaller median
volume diameters and consequently a larger surface
area for the drop ensemble. Accretion rates in storm
cores for the C-G model are higher, largely because
drop total number concentrations tend to be higher.
Little can be done to improve the exponential model.
There is currently little basis for selecting a particular
intercept value, except perhaps one based on climatological data. A parameterization scheme that sets the
distribution slope to a constant value will be problematic as well. In either case all DSD parameters are
uniquely determined by rainwater content or radar reflectivity alone. It is clear that this situation is not physically correct. Accretion rate should depend on an accurate representation of drop size as well as rainwater
content. Even if a procedure not involving radar could
be found to specify N0 and ⌳, changes in DSD shape,
which can be important for microphysical processes,
would not be accounted for.
With increased computational power, numerical
models that predict two moments (typically mass mix-
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BRANDES ET AL.
FIG. 6. As in Fig. 4, but for estimated (top) Re and (bottom) Rc using Kessler parameterizations.
ing ratio and total number concentration) have become
popular. Although the sensitivity of these models to
parameterization schemes is easily established experimentally, the necessary observations on which modelers can make particular parameter selections are often
not available. Observational studies with polarimetric
radar may lead to the discovery of useful interrelationships that can help to refine parameterization schemes
(e.g., Brown and Swann 1997). Two-moment prediction
schemes are consistent with the C-G DSD retrieval
model. The biggest impact of radar-derived DSD attributes is likely to be with two-moment forecast models that can assimilate polarimetric observations.
The enabling hypothesis for the C-G DSD model is
that a relationship exists between ␮ and ⌳ and that the
relationship is not dominated by statistical error in the
disdrometer-estimated DSD moments. This issue is addressed by Zhang et al. (2003) who show that the ␮–⌳
relation contains useful information. The derived relation [(3)] appears applicable for storms in Oklahoma
(Brandes et al. 2003), but variations in different climatic
regimes are likely. Hence, an effort is being made to
acquire additional datasets.
Acknowledgments. The authors are grateful for the
assistance of Drs. Anton Kruger and Witold Krajewski
of The University of Iowa who provided the disdrometer observations. This research was supported by the
NCAR Director’s Opportunity Fund and by funds from
the National Science Foundation that have been designated for U.S. Weather Research Program activities at
NCAR.
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