On the Influence of Assumed Drop Size Distribution Form on Radar-Retrieved Thunderstorm Microphysics

Transcription

1 FEBRUARY 2006 B R A N D E S E T A L. 259 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 brandes@ncar.ucar.edu 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 American Meteorological Society

2 260 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 45 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 D N 0 exp D, where N 0, 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 N 0 is constant. Sauvageot and Lacaux (1995) show that N 0 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 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 N 0 dictates that only the slope term can respond to a change in precipitation intensity. Assuming a constant slope and allowing N 0 to vary dictates that the mass-weighted terminal velocity is constant. In either case a severe constraint is placed on the 1 DSD. A more general approach would be to let N 0 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 D N 0 D exp D. With three parameters [the concentration parameter N 0 (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 to accommodate heavy rainfalls better. With (3) and radar measurements of radar reflectivity at horizontal polarization Z H and differential reflectivity Z DR, 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 N 0 is specified and calculations are made of reflectivity, differential reflectivity, total drop concentration N T (m 3 ), liquid water content W (g m 3 ), and drop median volume diameter D 0 (mm) using (2), the selected N 0, and the tabulated values of and. Ratios of N T and W with Z H are independent of N 0 and are functions of (or ) determined by Z DR 2 3

3 FEBRUARY 2006 B R A N D E S E T A L. 261 FIG. 1. Radar (left) reflectivity and (right) differential reflectivity measurements from a thunderstorm observed in Florida at 1926 UTC 17 Sep alone for the C-G model. Logarithms of the ratios are then fit with polynomial relations. Diameter D 0 is assumed to be a function of Z DR alone. The resulting relations are W Z H Z 2 DR 1.124Z DR, 4 N T 2.085Z H Z 2 DR 2.066Z DR, and 5 3 D Z DR Z DR 1.479Z DR 0.717, with Z H in millimeters to the sixth power divided bycubic meters and Z DR in decibels. [A corresponding relation for rainfall rate is given by Brandes et al. (2004a).] Expressions (3) (6) are for truncated DSDs for which D min (mm), the size of the smallest drop, was assumed to be 0.1 mm and D max, the largest drop, was estimated from radar reflectivity (Brandes et al. 2003). As a physical parameter, the total number concentration is examined rather than N 0. Equations (4) (6) assume that drop axis ratios r the vertical dimension divided by the horizontal dimension are given by r D D D D 4 [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 D 0 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 6 7 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 The improvement comes from drop size information contained in the Z DR 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 N m 3 mm 1 ], retrieval relations for W, D 0, and N T are W Z H 0.571, 8 D W 0.25, and 9 N T 8000D Equations (8) (10) assume 0 D mm and use the integral definitions of W and Z H and the approximation D / (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, D 0, and N T 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

4 262 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 45 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 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 ( 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 ( 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 ( and 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 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

5 FEBRUARY 2006 B R A N D E S E T A L. 263 FIG. 3. Observed 1-min DSD for strong convection (1917 UTC), weak convection (2100 UTC), and stratiform rainfall (2230 UTC) on 17 Sep 1998; C T 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 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 N T N 0 /. Because N 0 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 D 0 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 N T and D 0 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 D 0 are located in the reflectivity cores (Fig. 1). With the C-G model, D 0 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 R e (g m 3 s 1 ) and the accretion rate of cloud drops R c (g m 3 s 1 ) for the M P DSD are single-parameter relations given by R e W and 11 R c W For simplicity we have assumed a saturation deficit of 1.0gm 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-

6 264 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 45 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 m e is (Kessler 1969) The total evaporation rate is D max dm e D R e N D dd. D min dt 14 dm e m dt e D From the integral expression for rainwater content and (2) we obtain

7 FEBRUARY 2006 B R A N D E S E T A L N 0 w 10 3 D max, 4 D min, 4 W, 15 where denotes an incomplete gamma function and w is the density of water. After substituting (13), (15), w 1.0gcm 3, and m e 1.0gm 3, the total evaporation rate becomes R e D max, 13 5 D min, 13 5 W. D max, 4 D min, 4 16 Again following Kessler, the accretion of cloud water for a collection efficiency of 1.0 is D max D 2 R c 10 6 D min 4 t D m c N D dd, 17 where t (D) (ms 1 ) is the drop terminal velocity and m c is the cloud water content. Substituting (2), the expression for N 0 [(15)], m c 1.0gm 3, w 1.0gcm 3, and the drop terminal velocity relation of Brandes et al. (2002), yields t D D D D 4, R c 2 c l l 1 l 0 18 D max, l 3 D min, l 3 W, D max, 4 D min, 4 19 where c l represents the constants in the terminal velocity relation. For convenience, alternate forms for R e and R c consistent with (4) (6) are R e Z H Z 2 DR 1.448Z DR R c Z H Z 2 DR 1.087Z DR. and 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 (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,

8 266 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 45 FIG. 5. Time series comparison for 17 Sep showing estimated (top) evaporation rate R e and (bottom) accretion rate R c 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 N 0 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-

9 FEBRUARY 2006 B R A N D E S E T A L. 267 FIG. 6. As in Fig. 4, but for estimated (top) R e and (bottom) R c 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. REFERENCES Brandes, E. A., and K. Ikeda, 2004: Freezing-level estimation with polarimetric radar. J. Appl. Meteor., 43, , G. Zhang, and J. Vivekanandan, 2002: Experiments in rainfall estimation with a polarimetric radar in a subtropical environment. J. Appl. Meteor., 41, ,, and, 2003: An evaluation of a drop distribution based polarimetric radar rainfall estimator. J. Appl. Meteor., 42,

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