Downloaded 09/25/21 08:53 PM UTC 462 JOURNAL of APPLIED METEOROLOGY VOLUME 43 Shape Factor, a Normalized Number Concentration, and Work and the Disdrometer

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Drop Size Distribution Retrieval with Polarimetric Radar: Model and Application

EDWARD A. BRANDES,GUIFU ZHANG, AND J. VIVEKANANDAN National Center for Atmospheric Research,* Boulder, Colorado

(Manuscript received 24 February 2003, in ®nal form 30 September 2003)

ABSTRACT Polarimetric radar measurements are used to retrieve properties of raindrop distributions. The procedure assumes that drops are represented by a gamma distribution and retrieves the governing parameters from an empirical relation between the distribution shape and slope parameters and measurements of radar re¯ectivity and differential re¯ectivity. Retrieved physical characteristics of the drop size distribution (DSD) were generally well matched with disdrometer observations. The method is applied to select storms to demonstrate utility. Broad DSDs were determined for the core (high re¯ectivity) regions of thunderstorms. Largest drop median volume diameters were at the leading edge of the storm core and were displaced slightly downwind from updrafts. Rainy downdrafts exhibited what are believed to be equilibrium DSDs in which breakup and accretion are roughly in balance. DSDs for stratiform precipitation were dominated by relatively large drops. Median volume diameters at the ground were closely related to the intensity of an overlying bright band. The radar measurements suggest that, although DSDs in stratiform rain were also broad and nearly constant in the rain layer, they were not at equilibrium but were merely steady. DSD invariance is attributed to small total drop numbers, which result in few collisions.

1. Introduction using measurements of radar re¯ectivity and differential re¯ectivity. Subsequent research suggests that, for short Dual-polarization radars typically transmit horizon- time periods commensurate with radar measurements, tally and vertically polarized electromagnetic waves and DSDs are more typically represented by a gamma dis- receive backscattered signals. Because illuminated hy- tribution (Ulbrich 1983), drometeors are not exactly spherical and are not simi- ␮ larly oriented, wave scattering is different for the two N(D) ϭ ND0 exp(Ϫ⌳D), (1) polarizations. Waves propagating through precipitation Ϫ␮Ϫ1 Ϫ3 are subject to scattering, attenuation, phase shifts, and where N 0 (mm m ) is a number concentration pa- depolarization. Signal properties change continuously rameter, ␮ is a distribution shape parameter, ⌳ (mmϪ1) as the waves propagate, yielding information regarding is a slope term, and D (mm) is the drop equivalent particle size, shape, and orientation that can be used to volume diameter. Because the gamma DSD is described estimate the governing parameters of assumed drop size by three parameters, three measurements or relations are distribution (DSD) models (Seliga and Bringi 1976; required. The retrieval technique used here, an adap- Zhang et al. 2001; Gorgucci et al. 2002; Bringi et al. tation of that proposed by Zhang et al. (2001), is based 2002). A capability to retrieve DSD information would on measurements of radar re¯ectivity at horizontal po- be important for studying precipitation processes and larization (ZH) and differential re¯ectivity (ZDR), and an the hydrometeor properties of storms, for validating mi- empirical constraining relationship between the drop crophysical parameterizations within numerical models, size distribution shape and slope parameters. The ␮±⌳ and for improving rainfall estimates. relation was derived from drop size distribution mea- Seliga and Bringi (1976, 1978) retrieved two param- surements. Concern has been raised as to the in¯uence eters de®ning an exponential DSD, that is, a concen- of measurement errors on such relations (Chandrasekar tration parameter and the drop median volume diameter, and Bringi 1987). This issue is addressed by Zhang et al. (2003), who show that the ␮±⌳ relation is intrinsi- cally different than the linear relation associated with * The National Center for Atmospheric Research is sponsored by measurement error and that retrieved ␮ and ⌳ values the National Science Foundation. are not biased by statistical errors. Gorgucci et al. (2002) and Bringi et al. (2002) propose Corresponding author address: Dr. Edward A. Brandes, National to retrieve the DSD from re¯ectivity, differential re- Center for Atmospheric Research, P.O.Box 3000, Boulder, CO 80307. ¯ectivity, and speci®c differential phase (KDP). The pro- E-mail: [email protected] cedure yields the mean axis ratio of the drops, the DSD

᭧ 2004 American Meteorological Society

Unauthenticated | Downloaded 09/25/21 08:53 PM UTC 462 JOURNAL OF APPLIED METEOROLOGY VOLUME 43 shape factor, a normalized number concentration, and work and the disdrometer. [For instrument locations see the DSD median volume diameter. There is some ques- Brandes et al. (2002, their Fig. 1).] tion with regard to the use of KDP for DSD parameter retrieval; K is computed from measurements of dif- DP 3. DSD parameter retrieval ferential propagation phase, which can be noisy, particularly at lower rain rates. To reduce the uncertainty in The method for retrieving the DSD governing parameters follows that of Zhang et al. (2001) as modi®ed by KDP, the differential phase measurements are ®ltered in range, often over several kilometers (e.g., Ryzhkov and Brandes et al. (2003). The procedure uses the de®nitions ZrnicÂ 1996; Hubbert et al. 1993). Retrievals of drop of radar re¯ectivity and differential re¯ectivity ex- pressed in terms of the DSD parameters (N , ␮, and ⌳) mean shape with KDP can produce large variations in 0 space and time, which have not been independently ver- and drop backscattering amplitudes and an empirical i®ed. Further, Illingworth and Blackman (2002) argue relation between ␮ and ⌳ determined from drop observations: that the redundancy among ZH, ZDR, and KDP precludes the retrieval of the three DSD parameters in Eq. (1) with ⌳ϭ1.935 ϩ 0.735␮ ϩ 0.0365␮2. (2) this parameter set. Rather, KDP is used here for verifying the radar calibration (Vivekanandan et al. 2003). Bringi This relation is based upon a representative cross section et al. (2002) apply the method only to situations in of storms that included light stratiform to heavy con- Ϫ1 vective precipitation. Drop axis ratios are calculated which KDP Ն 0.3Њ km (rain rates of greater than 20 mm hϪ1). Hence, the technique has limited application with for general DSD retrieval. r ϭ 0.9951 ϩ 0.025 10D Ϫ 0.036 44D2 This paper gives an overview of the available data 0.005 030D340.000 249 2D , (3) and the DSD retrieval procedure. The method is then ϩ Ϫ applied to moderate thunderstorms, a stratiform rain where r is the axis ratio (vertical axis divided by the event, and a strong multicellular thunderstorm. Re- horizontal axis). This expression was used by Brandes trieved total drop concentrations, mass-distribution et al. (2002) for rainfall estimation. Bias factors for spectral widths, drop median volume diameters, rain- polarimetric rainfall estimators using ZH, KDP, KDP and water content, and rain rates are compared with com- ZDR, and ZH and ZDR converged to a similar value, in- putations with disdrometer observations and are used to dicating the validity of the relation. The DSD can be examine the spatial and temporal characteristics of found by using the de®nition of ZDR and the empirical DSDs within the selected storms. relation [Eq. (2)] to retrieve ␮ and ⌳ by iteration and then using the de®nition of radar re¯ectivity at hori-

zontal polarization to ®nd N 0. 2. Data Once the governing parameters of the DSD are known, other attributes can be computed. The total drop Radar measurements used in this study were collected concentration (N ,mϪ3) is given by with the National Center for Atmospheric Research's S- T band, dual-polarization Doppler radar (S-Pol) in east- Dmax N ϭ N(D) dD, (4) central Florida during a special ®eld experiment (``PRE- T ͵ CIP98'') to evaluate the potential of polarimetric radar 0 to estimate rainfall (Brandes et al. 2002). Measurements where Dmax, the diameter of the largest drop (mm), is were obtained at high temporal and spatial resolution estimated from radar re¯ectivity (Brandes et al. 2003). The over a special rain gauge network installed by the Na- drop median volume diameter (D0, mm) is de®ned as tional Aeronautics and Space Administration as part of DD0 max its Tropical Rainfall Measuring Mission. The network ͵͵DN33(D) dD ϭ DN(D) dD, (5) included a video disdrometer (Kruger and Krajewski 0 D0 2002) located 38 km from the radar. One-minute observations were available. Re¯ectivity and differential and one-half of the rainwater content is contained in drops smaller and one-half in drops larger than D .A re¯ectivity measurements from the radar were averaged 0 closely related parameter, the mass-weighted average (linear units) over ®ve range gates. Each gate was 0.15 diameter (D , mm), is computed from km in length. For comparison with the disdrometer (sec- m tions 3 and 4), the smoothed radar measurements for Dmax range bins containing the disdrometer and for 0.5Њ an- ͵ DN4 (D) dD tenna elevation were used. Gridded values of DSD pa- 0 Dm ϭ . (6) rameters (section 4) were found by averaging retrieved Dmax 3 DSD parameters over a volume having a radius of 0.75 ͵ DN(D) dD km. Proximity to the Weather Surveillance Radar-1988 0 2 2 Doppler (WSR-88D) at Melbourne, Florida, (KMLB) The variance of the mass spectrum (␴ m , mm ) is given gave dual-Doppler radar coverage over the gauge net- either by (Ulbrich 1983)

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Dmax el [Eq. (1)] associate with high concentrations of small (D Ϫ D )23DN(D) dD ͵ m dropsÐnot usually observed at the leading edge of 0 2 strong convection. The constrained-gamma model [Eq. ␴ m ϭ or Dmax (2)] is designed to match disdrometer observations and ͵ DN3 (D) dD will, in general, reproduce the negative ␮s seen by the 0 disdrometer. However, the retrieval model with integra- 22 ␴ m ϭ (M54Ϫ 2MDm ϩ MD3 m)/M3, (7) tion of drop diameters from 0 mm to Dmax may not be applicable in this region of some storms. In this study, with M3, M4, and M 5 being the third, fourth, and ®fth ZDR values of greater than 3 dB are ignored. Imposition moments of the drop size distribution. The rainwater of an upper limit for Z dictates that retrievals in large Ϫ3 DR content (W, gm ) is computed from drop regions, particularly at the leading edge of strong Ϫ3 Dmax convection, should be viewed skeptically. ␳␲w ϫ 10 W ϭ N(D)DdD,3 (8) For radar measurements with small Z (Ͻ0.3 dB), a 6 ͵ DR 0 simple set of estimators was derived using the con- and the rainfall rate (R, mm hϪ1) is computed from strained-gamma model. Rain and radar parameters were calculated for ⌳ in the range of 0.5±13 mmϪ1 and a Dmax Ϫ43 ®xed value of N 0. Ratios of the parameters NT, W, and R ϭ 6␲ ϫ 10 D ␷ t(D)N(D) dD, (9) ͵ R with ZH are independent of N0 and are functions of 0 ␮ (or ⌳), determined by ZDR alone for the constrained- Ϫ3 where ␳w is the density of water (g cm ) and ␷t(D)(m gamma DSD. We took logarithms of the ratios and ®t sϪ1), the drop terminal velocity, is computed as in Bran- them with polynomial functions to obtain des et al. (2002). 2 (0.728Z DRϪ2.066Z DR) The retrieval method has been previously veri®ed by NTHϭ 2.085Z ϫ 10 , (10)

Zhang et al. (2001) and Brandes et al. (2003). Trends 2 Ϫ4 (0.223Z DRϪ1.124Z DR) W ϭ 5.589 ϫ 10 ZH ϫ 10 , and (11) in the integral parameters NT, D 0, Dm, W, and R are 2 readily retrieved, and biases are generally small. Be- (0.165Z DRϪ0.897Z DR) R ϭ 0.007 60ZH ϫ 10 . (12) cause NT (a low moment of the DSD) is derived from 6 Ϫ3 radar re¯ectivity and differential re¯ectivity (high mo- The units of ZH and ZDR are linear (mm m ) and deci- ments of the DSD), a precise model is required to cap- bels, respectively. Expressions for D0 and ␴m are func- ture the variance in NT. Drop concentrations tend to be tions of ZDR alone and are given by underestimated with video disdrometers (Tokay et al. 32 2001). Hence, the estimated drop concentrations from D0DRDRDRϭ 0.171Z Ϫ 0.725Z ϩ 1.479Z radar and the disdrometer are regarded as approximate. DSD parameters retrieved with the constrained-gamma ϩ 0.717 and (13) 2 method are fairly insensitive to variations in Dmax and ␴m ϭϪ0.0247Z DRϩ 0.519Z DR ϩ 0.163, (14) have less bias than retrievals with assumed ®xed values of ␮. Estimated rain rates have distinct advantages over with ZDR in decibels. The relations in Eqs. (10)±(14) are estimates derived from power-law relations using radar unique for the axis ratio relation used and are funda- re¯ectivity or speci®c differential phase measurements mentally different from that derived from DSD simu- (Brandes et al. 2003). The accuracy of ␮ and ⌳ retrievals lations or measurements (e.g., Gorgucci et al. 2002). In depends on rain rate. Retrievals and observations are general, the integral values [Eqs. (4)±(9)] gave a better well matched for re¯ectivity of greater than 30 dBZ (see match with disdrometer observations. The polynomial Brandes et al. 2003, their Fig. 3). Correspondence de- relations [Eqs. (10)±(14)] are computationally ef®cient creases at light rain rates as the relative error in the and provide good estimates of NT, W, R, D0, and ␴m radar measurements and statistical variability associated for ZDR Յ 3 dB. For large ZDR, as occasionally seen at 4 with small disdrometer drop counts increases. the leading edge of strong convection, NT can be 10 Because of measurement and model error, the above mϪ3 (or more). The DSDs for such measurements are integrals become unreliable for retrieving DSD param- dominated by large drops, possibly supported by ice eters when ZDR Ͻ 0.3 dB or ZDR Ͼ 3 dB. Radar-measured cores, and are probably not well represented by the con- and disdrometer-calculated values of ZDR of greater than strained-gamma model. 3 dB are occasionally observed at the leading edge of A retrieval for the standard deviation (width) of the strong convection. Related DSDs typically are charac- mass spectrum ␴m, not shown previously, is presented terized by small numbers of very large drops, and cal- in Fig. 1. Spectrum widths for the disdrometer obser- culations with disdrometer observations using the mo- vations were computed from the DSD moments; radar- ments method (e.g., Kozu and Nakamura 1991) often based widths were computed from the polynomial re- yield negative ␮s. Some of these negative values may lation in Eq. (14). The radar-retrieved values tend to be arise simply from inadequate sampling of large drop a little smaller on average than that computed for the concentrations. Negative ␮s with the gamma DSD mod- disdrometer measurements (Ͻ0.1 mm). However, it is

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and drop counts, are plotted in Fig. 2. The region between the curves is thought to represent ``good'' data points. Note that the upper curve excludes a small number of data points for re¯ectivities of less than 35 dBZ. These points are all characterized by small sample sizes Ϫ3 (NT Ͻ 200 m ) and are dominated by large drops that, in general, are poorly sampled with disdrometers because of the small sample volume. Radar measurements

of ZH and ZDR closely agreed with disdrometer observations (Brandes et al. 2002); hence, the two curves were used to ``edit'' the radar measurements. Only measurements between the curves were accepted for anal- FIG. 1. Drop distribution spectral width (␴m) as retrieved by radar and computed from disdrometer measurements for a series of storms ysis. Hail increases radar re¯ectivity, and a tendency to on 17 Sep 1998. tumble while falling reduces the differential re¯ectivity relative to the rain-only case. Measurements contaminated by large hail lie to the right of the lower curve clear that the constrained-gamma method provides use- in Fig. 2Ðgenerally in the region with re¯ectivity Ͼ40 ful detailed information about DSDs. dBZ. Displacement from the curve increases as hail size The method for retrieving DSD parameters makes use increases (Aydin et al. 1986; Brandes and Vivekanandan of known raindrop characteristics and shapes and is not 1998). At low re¯ectivity, statistical errors in the ZDR applicable to solid (frozen) precipitation. Hence, retriev- measurement and ground targets can result in data points als are reproduced only for storm levels well below the below the lower curve. When applied to radar data, the melting level. Hail and nonmeteorological targets, such upper boundary eliminates some measurements that are as insects and ground clutter, may contaminate the radar dominated by unusual distributions of large drops. The measurements and adversely in¯uence the retrievals. To boundary also eliminates measurements strongly in¯u- minimize contamination, we adapted the procedure of enced by nonmeteorological factors such as ground re- Aydin et al. (1986). Radar re¯ectivity and differential turns and insects. re¯ectivity, computed for video disdrometer observa- The sharp edge of data points at the lower boundary tions collected during PRECIP98, regardless of rain rate in Fig. 2 is a common feature of calculations with ob-

FIG. 2. Radar re¯ectivity plotted against differential re¯ectivity as computed from disdrometer observations. The curved lines delineate observations with drop counts Ն200 and are used to discriminate radar measurements contaminated by nonmeteorological targets and hail from ``rain only'' measurements. DSD shape parameter values are indicated (▫: ␮ Ͻ 0; ϩ:0Յ ␮ Ͻ 4; ϫ: 4 Յ ␮ Ͻ 8; ⅙: ␮ Ͼ 8).

Unauthenticated | Downloaded 09/25/21 08:53 PM UTC MARCH 2004 BRANDES ET AL. 465 served drop size distributions and radar measurements. as measured by radar and computed from disdrometer There is a tendency, in particular at moderate re¯ectivity observations, is shown in Figs. 4a,b. Although there was values, for the shape parameters of the DSDs in Fig. 2 a separation of 38 km between the instruments, the radar to increase from left to right. Thus, for a particular ZDR sampling volume is many orders of magnitude larger value, the DSD narrows (becomes less exponential) as than that of the disdrometer, and the radar beam was the re¯ectivity (along with rain intensity) increases. For 400 m above the disdrometer, the agreement is good. a speci®ed ZH value, there is also a tendency for ␮, ⌳, There are signi®cant differences. Some disdrometer- and R to increase toward the lower boundary. The latter based radar re¯ectivities associated with the leading data points are predominantly convective. DSDs at the convection are 3±8 dB higher than that measured by lower boundary are somewhat narrower and have great- radar. At 1515 UTC, a transition period between con- er slopes for the large drop portion of the distribution vective and stratiform rainfall, the disdrometer re¯ec- than those at the upper boundary. Moreover, they have tivities fall below 10 dBZ while the radar-measured val- the minimum drop median volume diameters, the high- ues remain larger than 17 dBZ. For the more stratiform est total drop concentrations, and highest rain rates for component of the precipitation (1520±1600 UTC), the a particular re¯ectivity value. In converse, DSDs near range of extreme values in the wavelike re¯ectivity trace the upper boundary have the largest median volume are slightly wider with the disdrometer. A similar re- diameters for a speci®ed re¯ectivity. This boundary, as lationship can be seen with differential re¯ectivity. The the plotted data suggest, is less de®ned because of the suppressed maxima and raised minima in the radar mea- sampling issues associated with small concentrations of surements are attributed to greater smoothing with the large drops. larger radar measurement volume. A mean (calibration) Hu and Srivastava (1995) studied drop breakup and bias is not evident. accretion effects on simulated DSDs. They found that, Figures 4c,d show the comparison for total drop con- regardless of the initial distribution, essentially parallel centration and for median volume diameter using the equilibrium DSDs in which drop breakup and accretion procedure described in section 3. Trends are well were balanced evolved over periods of 10 min and falls matched. On average, radar-derived drop concentrations of 2 km. Hence, equilibrium DSDs are believed to be are smaller. The difference is most noticeable for the manifest by constant values of ␮ and ⌳ (Sauvageot and period of stratiform rainfall. It is possible that the dis- Lacaux 1995; Atlas and Ulbrich 2000). With our re- crepancy lies with retrieving the zero-order moment of trieval model, constant ZDR is an indication of constant the DSD from higher-order measurements, that the rain- ␮ and ⌳. Hu and Srivastava determined that the time observed DSDs may not be described well by the con- required for an equilibrium distribution to form was strained-gamma distribution, or that there is signi®cant inversely proportional to the rain rate. DSDs at the lower advection of precipitation below the elevated radar boundary in Fig. 2 are believed to be near equilibrium beam. Differences in NT are not necessarily transferred and not quite exponential. They may bear little resem- to W, R, D0, Dm, and ␴m, because these parameters have blance to their initial distribution upon melting. In con- higher correlation with ZH and ZDR. Note that a signif- trast, data points near the upper boundary represent icant mean bias is not evident in the retrieved D 0s. Re- DSDs dominated by large drops and relatively low rain ¯ectivity and differential re¯ectivity measurement pairs rates and drop numbers. Hence, these DSDs are believed during the period of stratiform rain vary between 20 to evolve more slowly and to retain more of their initial dBZ and 0.4 dB and 32 dBZ and approximately 1 dB. characteristics. If these measurements were plotted on Fig. 2, they would be near the upper boundary, suggesting that the DSDs were dominated by relatively large drops. 4. DSD retrieval examples A dual-Doppler wind ®eld analysis (storm relative) In this section, the DSD retrieval method is applied is presented in Fig. 5. At 0.5 km (Fig. 5a) the prevailing to several storm types. Results are compared with dis- ¯ow is from the storm's left rear (east-southeast). Note drometer observations to verify the utility of the retriev- that the disdrometer site is in a re¯ectivity gradient re- als. Also, the spatial and temporal distributions of the gion on the north side of the southernmost convective retrieved ®elds are examined for consistency with storm band. Advection of precipitation, particularly small kinematics. drops, below the radar beam could have caused the higher drop concentrations detected by the disdrometer. Low-level vertical velocities in the convective bands a. Moderate thunderstorms are predominantly downdrafts as suggested by the gen- During the late morning hours on 21 August 1998 eral acceleration of the horizontal wind from east to two short east±west lines or bands of moderate thun- west. With height the updrafts became widespread. derstorms developed in east-central Florida (Fig. 3a). Winds throughout much of the southern band veer (be- The convection drifted toward the west-southwest. The come more southerly) with height between 0.5 and 2 southernmost line passed over the disdrometer. A com- km (Fig. 5b). In contrast, the northern convective band parison of radar re¯ectivity and differential re¯ectivity, exhibits strong northeasterly ¯ow at 2 km. This ¯ow

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FIG. 3. Radar re¯ectivity depictions of (a) short lines of moderate thunderstorms on 21 Aug 1998, (b) the trailing stratiform precipitation region of the storm system occurring on 17 Sep, and (c) a strong thunderstorm complex on 22 Sep. The disdrometer site is indicated by a white solid circle. Range marks are at 10-km intervals; rays are at 30Њ intervals. North is toward the top of the ®gure. Measurements from the Melbourne WSR-88D are in (a) and S-Pol measurements are in (b) and (c).

overtakes the southern band at 4 km (not shown) and in the core of the southern convective band (the region helps to sustain the elevated updrafts. Re¯ectivity max- with re¯ectivity Ͼ40 dBZ) are fairly uniform between ima in the southern band slope southward with height; 1.4 and 1.8 mm. The region with D 0 Ͼ 1.6 mm is those in the northern band slope eastward. displaced slightly from the region with re¯ectivity Ͼ40 Physical DSD parameters at 0.5 km are presented in dBZ. This result could be a size-sorting effect in which Fig. 6. Inspection reveals that peak drop concentrations large drops are the ®rst to fall from the elevated storm in the southern convective line are 3000±5000 mϪ3. core. Maximum concentrations in the northern line are about Rainwater contents with the southern convective band 3000 mϪ3. Concentrations at the edges of the bands are are mostly 0.5±3 g mϪ3, and rainfall rates are all Ͻ60 Ͻ300 mϪ3. Estimated drop median volume diameters mm hϪ1. Somewhat smaller values characterize the

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FIG. 4. Radar±disdrometer time plots of (a) radar re¯ectivity (ZH), (b) differential re¯ectivity (ZDR), (c) total drop concentration (NT), and

(d) drop median volume diameter (D0). The radar antenna elevation is 0.5Њ. northern band. An interesting feature of the southern tivity have long been exploited through the use of pow- band is that the 1 g mϪ3 rainwater content contour nearly er-law relations. coincides with the 40-dBZ contour. This relation is also true for the 20 mm hϪ1 rain-rate contour. This relation- b. Stratiform rain ship is altered slightly in the northern band. The correspondence is an indication of reduced dependence on On 17 September, a line of thundershowers moved differential re¯ectivity in this case. Correlations be- from west to east across central Florida. As the line tween rainwater content and rain rate with radar re¯ec- approached the east coast, a large region of stratiform

FIG. 5. Storm-relative, dual-Doppler wind ®eld analyses at (a) 0.5- and (b) 2-km height for the thunderstorms of 21 Aug (1448 UTC). Melbourne WSR-88D re¯ectivity contours for 30 and 40 dBZ are shown (heavy solid contours). Thin solid (dashed) contours show updrafts (downdrafts) Ն0.5 m sϪ1 at 0.5 km. The velocity contour interval is 1 m sϪ1 at 2 km. North is toward the top of the ®gure.

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FIG. 6. Retrieved total drop concentration, median volume diameter, rainwater content, and rain rate at 0.5 km for the 21 Aug thunderstorms (1448 UTC). Light shading shows radar re¯ectivity Ն30 dBZ, as measured with the S-Pol radar; dark shading shows re¯ectivity Ն40 dBZ. The disdrometer site is shown by a dot. precipitation with embedded moderate convection from radar measurements. Although the radar-derived formed in its wake (Fig. 3b). Stratiform precipitation, D0s are a little smaller than their disdrometer counter- which drifted northward over the disdrometer site, was parts (ϳ0.1 mm), the overall agreement is excellent. particularly uniform. A radar-re¯ectivity bright band as- The re¯ectivity bright band associates with large, sociated with the melting layer ®rst appeared over the partly melted particles whose precise structure and com- disdrometer site at 2135 UTC. Maximum brightband position are not known. However, the large increase in re¯ectivity in the column above the disdrometer is re¯ectivity of 12±13 dB between the break in the re- shown in Fig. 7a. Brightband intensity increased to a ¯ectivity lapse (ϳ5 km) and the brightband maximum peak value of 46 dBZ at 2212 UTC. Re¯ectivity in the is suggestive of large aggregates rather than convective column then waned until a second wave of less-intense debris or graupel (see Klaassen 1988). Regardless, Fig. precipitation moved over the site. Peak brightband re- 7 reveals a close relationship between brightband in- ¯ectivity with this wave was 37 dBZÐmeasured at 2258 tensity and the size of drops deposited at the surface. UTC. Pro®les of radar re¯ectivity and differential re- This relationship was examined by Huggel et al. (1996) ¯ectivity at the time of the two maxima are presented who determined that well-de®ned bright bands were an in Fig. 8. (The 0ЊC level as determined from a 2200 indicator of broad drop distributions with comparatively UTC sounding released at the Kennedy Space Center high concentrations of large drops. A similar relation was 4.99 km.) holds here. Figure 7b shows drop median volume diameters as Examination of the radar pro®les (Fig. 8) discloses computed from disdrometer observations and estimated little change in either radar re¯ectivity or differential

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and a differential re¯ectivity of 1 dB at 2212 UTC and a re¯ectivity of 29 dBZ and a differential re¯ectivity of 0.6 dB at 2258 UTC. These measurements are close to the upper boundary in Fig. 2. Associated median volume diameters computed for the rain layer are 1.65 and 1.46 mm. DSD shape factors, as calculated from disdrometer measurements and retrieved from radar measurements, are roughly 0±2 and 2±4 for the heavier rainfall in the two waves. Hence, the distributions are broad, particularly for the earlier period when the bright band is strongest. The constancy of the radar measurements suggests that DSDs resulting from melted snow¯akes are little altered over the lowest 3 km. Rain rates calculated from the disdrometer measurements are at most 11 mm hϪ1 and are typically much less. The Hu and Srivastava (1995) study suggests that at these rates an equilibrium DSD, in which drop breakup and accretion are in balance, would evolve slowly. Hence, the observed distributions, although broad and steady, are likely dominated by the large drops created by aggregation and have not FIG. 7. (a) Maximum brightband re¯ectivity at the disdrometer site achieved the equilibrium conditions of DSDs at the low- for the stratiform rainfall event of 17 Sep. (b) The median drop er boundary in Fig. 2. diameter as estimated from disdrometer observations and radar measurements. Retrieved spatial distributions of NT, D 0, W, and R at 2212 UTC are presented in Fig. 9 for a height of 0.5 km. Except for the western edge of the rainfall shield re¯ectivity in the rain layer below 3 km. If drop growth and for a moderate embedded convective cell near the by accretion were the dominant process in the layer, northern edge of the domain (x ϭϪ30, y ϭ 43 km), re¯ectivity and differential re¯ectivity would increase parameter gradients are small. Where re¯ectivity ex- toward the ground. If large drop breakup was the dom- ceeded 35 dBZ, retrieved drop concentrations are gen- inant process, re¯ectivity and differential re¯ectivity erally 300±600 mϪ3. Median volume diameters vary would decrease toward ground. from 1.6 to 1.9 mm. Retrieved rainwater contents are Rain-layer radar measurements at the disdrometer site mostly less than 0.3 g mϪ3, and rain rates roughly vary (Fig. 8) show an average re¯ectivity of about 35 dBZ from 4 to 8 mm hϪ1. Minute-to-minute drop counts at

FIG. 8. Pro®les of radar re¯ectivity and differential re¯ectivity, as measured by radar at 2212 and 2258 UTC for

the stratiform event of 17 Sep. Median volume drop diameters (D0) for the rain layer (surface±3 km), computed from the integral relations, are also shown.

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FIG. 9. Retrieved drop size distribution parameters and disdrometer site as in Fig. 6 but for the stratiform region of the 17 Sep event (2209 UTC). Light shading shows radar re¯ectivity Ն30 dBZ, as measured with the S-Pol radar; dark shading shows re¯ectivity Ն35 dBZ. the disdrometer site varied from 399 to 517 mϪ3 between itation core (x ϭϪ32, y ϭ 26 km) and the ambient 2209 and 2215 UTC. During this period, median volume wind. The latter ¯ow can be seen entering the storm at diameters were 1.40±1.88 mm, rainwater contents were 2 km (Fig. 10b). That the south-to-southeasterly ¯ow is 0.21±0.28 g mϪ3, and rain rates were 4.4±6.4 mm hϪ1. the primary updraft source is con®rmed by the conser- vation of momentum at higher levels in the updraft (Figs. 10c,d). c. Strong thunderstorm complex At mid- and upper storm levels (e.g., 6 km) hydro- Figure 10 shows a storm-relative, dual-Doppler wind meteors growing and lifted in updrafts were carried to- ®eld analysis at 1956 UTC for a strong thunderstorm ward the storm's rear (northwest) and western portions complex that developed on 22 September. The complex of the precipitation core. Size sorting undoubtedly oc- was part of a larger area of showers that propagated curred as particles passed through the updraft. The larg- toward the east (Fig. 3c). Individual convective cells est particles would have fallen ®rst, remaining near the formed in the updraft and re¯ectivity gradient region at updraft core, and lighter particles would have been car- the storm's leading edge [between x ϭϪ28, y ϭ 20 ried farther to the storm's rear. Frozen particles lifted km and x ϭϪ15, y ϭ 30 km (0.5 km, Fig. 10a)] and in updrafts would eventually have fallen within down- moved northwestward through the complex. New cells drafts and weaker updrafts in the storm's northern and formed in response to convergence between low-level western quadrants. Hydrometeors with suf®cient ter- out¯ow that originated in weak downdrafts along the minal velocities and size to prevent their evaporation storm's northern edge and in a somewhat stronger rainy would have fallen to lower levels and been borne by downdraft within the westward extension of the precip- the wind back toward the front of the storm complex.

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FIG. 10. Wind ®eld analyses as in Fig. 5 but at 1956 UTC for the thunderstorm complex of 22 Sep. Re¯ectivity contours beginning at 30 dBZ and at 10 dB intervals, as measured with the Melbourne WSR-88D, are shown. Thin solid (dashed) contours show updrafts (downdrafts). Contour intervals are 1 m sϪ1 at 0.5 km and 2 m sϪ1 at and above 2 km. The location of measurement pro®les for downdraft (D) and updraft (U) regions, as in Fig. 13, are indicated.

Figure 11 presents retrieved DSD physical parameters ZDR measurements at the leading edge of the storm core for 1956 UTC. Inspection reveals that high total drop were nearly 4 dB. The implication is that huge drops, concentrations coincide with the storm core, that is, the perhaps supported by ice cores, were present. The ex- region with radar re¯ectivity roughly Ն50 dBZ. Con- treme ZDRs exceed the limits of the data used to develop centrations greater than 10 000 mϪ3 are indicated. A the constrained-gamma model (Fig. 2). Moreover, as- peak value, on the order of 32 000 mϪ3, corresponds sociated DSDs may be poorly described by the con- with a newer convective cell near the leading edge of strained-gamma model. Note also that exclusion of ZDR the storm core (x ϭϪ25, y ϭ 27 km). In truth, the measurements Ͼ3 dB from the analysis will cause un- concentrations for this cell seem too high. (Video dis- derestimates of D0, W, and R. Retrievals for two older drometer data are not available for this event.) Some storm cells (near x ϭϪ32, y ϭ 32 km and x ϭϪ27,

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FIG. 11. Retrieved drop size distribution parameters as in Fig. 6 but at 1956 UTC for the 22 Sep thunderstorm. Light shading shows re¯ectivity Ն40 dBZ, as measured with the S-Pol radar; dark shading shows re¯ectivity Ն50 dBZ. The locations of measurement pro®les for downdraft (D) and updraft (U) regions in Fig. 13 are indicated. y ϭ 34 km) with relatively high drop concentrations along the 50-dBZ contour vary from 3 to7gmϪ3, and and large drop median volume diameters are believed rain rates vary from 70 to 130 mm hϪ1. The variation to be plausible. in the retrieved parameters comes from the differential

The distribution of large D 0s seems dictated, for the re¯ectivity measurements and illustrates potential prob- most part, by the con®guration of updrafts and advection lems with rain-rate algorithms based on re¯ectivity by the horizontal wind. Centers of large D 0 are displaced alone. downwind relative to the updraft. Drop concentrations Examination of the retrieved DSD shape parameters and D 0s at the rear of the storm complex fall below 100 (Fig. 12) indicates that the DSD is broad within the mϪ3 and 1.4 mm, respectively. The maximum retrieved storm core. Broadening is implied by small values of rainwater content is 7.6 g mϪ3. This extreme value re- ␮. In fact, there is a large region of slightly negative sides in the trailing portions of the storm core where ␮s. This result could be due to retrieval model error, radar re¯ectivity is high (Ͼ50 dBZ), total drop con- but, as noted earlier, the model is merely reproducing centrations are large (Ͼ10 000 mϪ3), and drop median characteristics of the disdrometer observations. In gen- volume diameters are relatively small (Ͻ1.8 mm). Peak eral, the negatives are believed. However, at the leading rainwater content and rain rate (ϳ150 mm hϪ1)at0.5 edge of the storm core, aforementioned issues regarding km roughly coincide with the rainy downdraft. Unlike large drops and related concentrations of small drops the moderate convective case discussed earlier, there is may come into play. Although small ␮ values exist in much crossing of the re¯ectivity contours by the rain- spots along the northern edge of the storm, in general, water content and rain-rate contours. Rainwater contents DSDs at the edge of the storm, where rain rates are

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concave upward (␮ ϭϪ1.0) for the updraft pro®le and is slightly positive (␮ ϭ 1.4) for the rainy downdraft. A simple interpretation is dif®cult because of the three- dimensional nature of the storm ¯ow and the rapidity at which the convective elements evolve. However, the

downdraft region with a higher rain rate, smaller D0, and slightly peaked DSD would seem to be closer to an equilibrium distribution than the updraft region.

5. Summary and conclusions A method for retrieving the governing parameters of gamma DSDs from remote polarimetric radar measurements was applied to subtropical rainfall events. The method utilizes radar re¯ectivity and differential re¯ectivity measurements and an empirical relation between the DSD shape and slope parameters. The constraining ␮±⌳ relation was derived previously from disdrometer observations. Overall, good agreement was found between retrieved DSD parameters and disdrometer observations for a moderate thunderstorm and a stratiform FIG. 12. Retrieved DSD shape parameter at 1956 UTC for the 22 Sep thunderstorm. Re¯ectivity and pro®le locations are as in rain event. Retrievals for a strong convective system Fig. 11. seemed reasonable except at the leading edge of the convective core where total drop concentrations may have been overestimated. The problem was ascribed to light, tend to be more peaked, with typical ␮ values of peculiar DSDs with unusual large drop concentrations 2±6. that are not well represented by the constrained-gamma Figure 13 displays radar re¯ectivity and differential model. re¯ectivity pro®les within the updraft region and the DSD retrievals for the stratiform event near its peak rainy downdraft. The updraft pro®le shows peak low- intensity revealed total drop concentrations on the order level re¯ectivity of more than 50 dBZ and differential of 300±600 mϪ3, median volume diameters of 1.6±1.9 re¯ectivities that exceed 2 dB. Re¯ectivity in the rainy mm, rainwater contents of 0.3 g mϪ3, and rain rates of downdraft is about 4 dB less, but the differential re- 4±8mmhϪ1. The retrievals were in good quantitative ¯ectivity is much lower. The retrieved DSD shape is agreement with disdrometer measurements. It was found

FIG. 13. Pro®les of radar re¯ectivity and differential re¯ectivity and drop median volume diameter as in Fig. 8 but at 1956 UTC for the 22 Sep thunderstorm. Pro®les are for updraft and downdraft regions (see Fig. 10).

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that drop D 0s at the ground were highly correlated with method may be possible. The differential propagation the strength of the overlying radar-re¯ectivity bright phase measurement is not used in the retrieval method band. As the intensity of the bright band increased from examined here. The parameter is sensitive to the total less than 35 to more than 45 dBZ, D0s increased from rain content and drop shape and is insensitive to atten- 1 to 1.8 mm. Precipitation near the ground was char- uation, and it potentially could be used as an additional acterized by relatively broad DSDs with small values constraint to reduce bias. Also, the issue of drop con- (0±4) of the shape parameter. Re¯ectivity and differ- centrations at the leading edges of convective cells ential re¯ectivity were constant in the rain layer. Ob- needs further investigation. Nevertheless, this study served rain rates were light (generally Ͻ10 mm hϪ1), demonstrates that useful DSD information can be de- drop median volume diameters were large (1.4±1.9 duced from polarimetric radar measurements. mm), and total drop concentrations were low (Ͻ600 mϪ3). Although the stratiform rain DSDs were broad Acknowledgments. The authors are grateful to Drs. and constant with height, an equilibrium distribution, in Anton Kruger and Witold Krajewski of The University which drop coalescence and breakup are roughly in bal- of Iowa for providing the video disdrometer data. Ben- ance, may not have had suf®cient time to evolve. jamin Hendrickson and Kyoko Ikeda of the National Total drop concentrations in excess of 10 000±30 000 Center for Atmospheric Research (NCAR) helped with mϪ3 were determined in the core region of a strong data analysis and the construction of the ®gures. The convective storm (de®ned as the region with re¯ectivity assistance of the NCAR Atmospheric Technology Di- Ն50 dBZ). Highest concentrations were in leading por- vision staff throughout the PRECIP98 ®eld program and tions of the core where updrafts prevailed. This region the analysis phase of this study is gratefully appreciated. was characterized by some of the largest retrieved me- This research was supported by funds from the National dian volume diameters (2.0±2.4 mm). Retrieved DSD Science Foundation designated for the U.S. Weather Re- shape factors along leading portions of the storm core search Program at NCAR. had negative values, which imply large numbers of small drops with the gamma DSD model. Some dis- REFERENCES drometer-derived negatives are undoubtedly real; others may arise simply from inadequate sampling of large Atlas, D., and C. W. Ulbrich, 2000: An observationally based con- drop populations. 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