Theor Appl Climatol (2010) 102:471–481 DOI 10.1007/s00704-010-0332-5

ORIGINAL PAPER

Dependence of accumulated precipitation on cloud drop size distribution Mladjen Ćurić & Dejan Janc & Katarina Veljović

Received: 6 July 2009 / Accepted: 19 August 2010 / Published online: 16 September 2010 # Springer-Verlag 2010

Abstract Convective precipitation is the main cause of extreme rainfall events in small areas. Its primary characteristics are both large spatial and temporal variability. For this reason, the monitoring of accumulated precipitation fields (liquid and solid components) at the surface is difficult to carry out through the use of rain gauge networks or remotesensing observations. Alternatively, numerical models seem to be the most powerful tool in simulating convective precipitation for various analyses and predictions. Due to a lack of comparisons between modelled and observed precipitation characteristics over a long period of time, we focus our research on comparisons between observations and three model samples of accumulated convective precipitation over a particular study area. We use a numerical cloud model with two model schemes involving the unified Khrgian–Mazin size distribution of cloud drops and a model scheme involving a monodisperse cloud droplet spectrum and the Marshall–Palmer size distribution for raindrops, respectively. For comparison, we have selected a study area with a sounding site. Our analysis shows that the model version with the Khrgian–Mazin size distribution exhibits a better agreement with the observed mean, median and range of extreme values of accumulated convective precipitation. Model simulations with the Khrgian–Mazin size distribution most closely match observations, with a correlation coefficient of 0.91. Use of the Marshall–Palmer size distribution, on the other hand, systemically underestimates the observed precipitation and has the lowest correlation coefficient among the methods, M. Ćurić (*) : D. Janc : K. Veljović Institute of Meteorology, University of Belgrade, 16 Dobracina str., 11000 Belgrade, Serbia e-mail: [email protected]

0.83. Such an investigation is crucial to improve predictions of accumulated convective precipitation for various climatological and hydrological analyses and predictions.

1 Introduction Convection plays an important role in short-term severe weather prediction and the global climate system. One of the most interesting convective phenomena is a rain shower. Knowledge of the spatial and temporal distribution of a rain shower in small catchments is critical in praxis, especially for forecasting flash floods that can lead to enormous material damage and loss of life. In many areas of the world, this is an important climatological factor. For this reason, monitoring the quantity of precipitation at an appropriate frequency and for a suitable period of time is of great importance. Precipitation data are usually taken from rain gauge networks. Heavy precipitation has been shown to be poorly depicted due to limited resolution (e.g. Lovejoy and Schertzer 2006), especially in the case of convective storms that lead to a sparse and spot-like data distribution. Meteorological radars are an improvement since they provide widespread spatial coverage at high spatial and temporal resolution and produce spatially continuous values. However, meteorological radars have important disadvantages due to the calibration of the Z–R relationship and their positioning in a complex topography (Hunter 1996; Young et al. 1999). Unfortunately, precipitation estimation by satellite is reliable over sea but not over land. Thus, remote-sensing observations still give only a very rough estimation of rain showers occurring at the ground level. In addition, some characteristics of precipitation are difficult to measure or separate with observations (for example, the hail content of the total precipitation). In

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this context, the best approach is to use various numerical models that are capable of simulating rain showers with a higher spatial and time resolution (Spiridonov and Ćurić 2005; Ćurić et al. 2008). In this work, we chose to use a cloud-resolving mesoscale model (Ćurić et al. 2003) in order to simulate rain showers. Bulk microphysical schemes are frequently used in cloudresolving mesoscale models due to their lower computational cost. This scheme assumes a distribution function for the cloud and precipitation particle sizes. Variations in the accumulated convective precipitation due to uncertainties inherent in the selection of distribution functions and their parameters must be assessed (Ćurić et al. 1998; Gilmore et al. 2004; van den Heever and Cotton 2004). Until now, cloudresolving mesoscale models have been used in studies to quantify the considerable sensitivity of the amount of precipitation accumulated from a convective storm to variations in exponential hail size distribution parameters (van den Heever and Cotton 2004; Gilmore et al. 2004). Various forms of ice crystals collide with supercooled drops or transfer to raindrops via melting in a convective cloud. The selection of the cloud drop size distribution is therefore critical in an adequate treatment of convective precipitation in mixed-phase clouds. The Marshall–Palmer distribution and different forms of gamma size distributions are the most widely used approximations of raindrop spectra. The Marshall–Palmer exponential size distribution (hereafter called MP) has been used in numerous models due to its simplicity. However, it tends to overestimate the number of both the smallest and the largest drops (Sempere-Torres et al. 1994). An alternate analytical function of the gamma type for fitting observed cloud drop datasets is the Khrgian–Mazin distribution proposed by Khrgian and Mazin (1963), Mazin et al. (1989) and Pruppacher and Klett (1997). It can be used to describe both cloud droplet and raindrop spectra. In the present study, we use the unified Khrgian–Mazin size distribution (hereafter called KM) for all of the water forms (both cloud droplets and raindrops). Such a type of distribution generates only minor concentrations of raindrops over 0.5 cm in diameter, which is found in reality. In this paper, we compare two model schemes, one that uses the KM size distribution rather than the widely used distribution with a monodisperse cloud droplet spectrum and the Marshall–Palmer size distribution for raindrops (hereafter called MMP). Most convective cloud models with bulk microphysics use the MMP scheme (Lin et al. 1983; Cotton et al. 1986; Murakami 1990; Xue et al. 2001; Ćurić et al. 2003; Gilmore et al. 2004). The Khrgian–Mazin size distribution was mainly used to approximate the cloud droplet spectrum (Hu and He 1988) despite the fact that it also includes a high concentration of raindrops. To determine whether the KM or the MMP model scheme better simulates the accumulated convective pre-

cipitation (hereafter called ACP), we compared the corresponding models and the observed ACP values for 27 convective precipitation events over a 10-year period in the study area. This study is focused on improving the forecast of accumulated cumulative precipitation that might lead to flash floods in small catchments.

2 Description of the cloud model The cloud-resolving mesoscale model (Ćurić et al. 2003, 2007) was used to achieve our stated goal. The model numerically integrates the time-dependent, non-hydrostatic and fully compressible equations. The dependent variables of the model are: the Cartesian wind components, perturbation potential temperature and pressure, the turbulent kinetic energy and mixing ratios for water vapour, cloud water and ice, rain, snow, and hail. The model uses the generalised terrain-following coordinate in the vertical direction, whilst the horizontal coordinates are the same as in the Cartesian system. The turbulence is treated by a 1.5order turbulent kinetic energy formulation. The Coriolis force is neglected. The model domain was 112×112×17 km with 600-m grid spacing in the horizontal direction and 300-m spacing in the vertical direction. The simulations were terminated at t=120 min. The sound wave terms were integrated in time with a 0.5-s time step. The other terms were computed with a large time step of 3 s. The wave-radiating condition was applied for lateral boundaries. An upper boundary with a Rayleigh spongy layer was used, whilst the lower boundary was a free slip boundary. The reference state was homogeneous in the horizontal direction, with constant values of temperature, humidity, pressure and wind velocity and direction. The cloud simulation was performed over a flat terrain. For the purpose of this study, a single-moment bulk microphysical scheme was used (Lin et al. 1983; Murakami 1990; Ćurić et al. 2003). Within such a scheme, one moment of the size distribution would be held fixed, as in other sensitivity studies, with respect to variations in arbitrarily specified size distribution parameters (van den Heever and Cotton 2004; Cohen and McCaul 2006). Two categories of non-precipitating forms (cloud water and cloud ice) and three categories of hydrometeors (rain, hail and snow) were treated. The shapes of all hydrometeors were assumed to be spherical, except for snow, which is in a hexagonal form with its maximum diameter (Lin et al. 1983). Exponential size distributions appear to provide a good fit to snow and hail size distributions (Cheng et al. 1985; Braham 1990; Gilmore et al. 2004). We therefore approximated hail and snow spectra with the corresponding exponential size spectra. Cloud ice is monodisperse (with a

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single ice crystal diameter), in agreement with Murakami (1990) and Gilmore et al. (2004). Cloud droplets were assumed to be monodisperse (a single cloud diameter of 20 μm) or KM-distributed, whilst raindrops were approximated with the Marshall–Palmer or KM size distribution (MMP and KM model schemes). The KM approach treats cloud droplets and raindrops with the unified KM size distribution as follows (Pruppacher and Klett 1997; Ćurić et al. 1998): N ðDÞ ¼

  AD2 BD exp  2 4

ð1Þ

where: ; B¼ A ¼ 1:452 r rQ R6 w M

3 RM

:

ð2Þ

In the above expression, Q is the total liquid water mixing ratio; RM is the mean radius of the drop spectrum, which may take arbitrary values; D is the drop diameter; and ρw and ρ are the liquid water and the cloud–air densities, respectively. In a single-moment scheme, we suppose that RM takes constant values of 10 and 50 μm. The KM drop spectrum splits into cloud droplets and raindrops at D=100 μm. For RM =10 μm, this spectrum provides high number concentrations of small cloud droplets and raindrops, as shown in Fig. 1. A larger RM value implies more large raindrops in the spectrum, with the removal of the mode diameters to the right. The cloud water spectrum has a number concentration calculated as Nc =ρQc/Mc, where Qc and Mc are the cloud– water mixing ratio and the mass of a single droplet, respectively (not presented in Fig. 1). The MP size

QC=QR=1 g/kg

1E+12

KM10 KM50 MP10 MP50

F(D) (m -4 )

1E+10

1E+8

1E+6

1E+4

1E+2

1E+0

distribution of raindrops (diameters larger or equal to 100 μm) generates a greater number of large raindrops as compared to the KM scheme. Hydrometeors within a mixed-phase cloud are generated or grow by the help of various mechanisms, which are presented in more detail in Appendices 1 and 2. Hereafter, the mechanisms will be presented briefly. In the model, the cloud water is initiated by condensation using a saturation adjustment scheme derived by Tao et al. (1989) or by the melting of cloud ice. Cloud ice is produced through depositional nucleation, heterogeneous freezing of cloud droplets and homogeneous freezing of cloud droplets below −40°C, in agreement with Murakami (1990). There are two mechanisms by which a cloud consisting of small droplets can produce significant numbers of large particles or precipitation. The mutual collection of water drops through collision and subsequent coalescence is commonly referred to as the coalescence process or warm rain. The growth of ice particles in a predominantly supercooled water cloud due to the Bergeron–Findeisen processes is important in the beginning of cloud evolution. Thereafter, collisions between supercooled water and ice particles are much more important. These processes are commonly referred to as cold rain or the ice crystal process. Both of these processes may act concurrently within the same cloud. In this paper, we treat hail and graupel in one spectrum (Ćurić et al. 2007). According to this scheme, hail (graupel) can be initiated with probabilistic freezing of raindrops, collisions between rain and cloud ice, collisions between rain and snow, and aggregation of snow crystals. Hail grows by the accretion of cloud water, rain, cloud ice and snow, or it can be melted or sublimated. Snow can be initiated by the autoconversion of cloud ice to snow or by collisions between raindrops and cloud ice crystals. Production terms for snow include various accretion terms (the Bergeron–Findeisen process, collisions of snow crystals with cloud ice, cloud water, raindrops, hail), snow melting and sublimation. Rainwater can be generated by the autoconversion of cloud droplets to raindrops, melting of precipitating ice or collisions with ice crystals at temperatures higher than 0°C. A raindrop grows by the accretion of cloud droplets. As noted, many important production mechanisms include cloud droplets or raindrops. This implies that the shape of the drop size distribution may critically determine the magnitudes of such interactions, as was shown by Ćurić et al. (1998, 2009). In the next sections, we will prove our statement.

3 Sensitivity tests 0.00

0.20

0.40

0.60

D(cm)

Fig. 1 Drop size distribution, F(D) (m−4) vs. diameter for model schemes used in the paper

In this section, we perform sensitivity tests to investigate the impact of cloud drop size distribution (the KM size distribution versus the MMP approach) on rain showers

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from a model cloud. The mean radius of the cloud drop spectra (RM) is taken to be 10 and 50 μm. Sensitivity tests with the KM model scheme are denoted by KM10 and KM50. We shall first analyse the impact of different size distributions on some important production rates. The KM transfer rate of cloud droplets to raindrops is larger than its MMP counterpart. This may be attributed to the larger number concentration of cloud droplets in the monodisperse spectrum (Nc =2.38×108 m−3if Qc =1 g/kg and ρ=1 kg/m3) than in the KM distribution. This implies a greater effectiveness of large cloud droplets in forming raindrops after random collisions. Once formed, a raindrop accretes cloud droplets and different kinds of crystals. These accretion rates vary over a wide range of magnitudes (~10−5–10−10 kg kg−1 s−1). Due to these large differences, comparisons of the KM and MMP production rates are presented as:   KM k ¼ log10 : ð3Þ MMP A positive k-ratio indicates that the KM production rate is larger than the MMP one and vice versa. Hereafter, we select rates at which raindrops collect cloud droplets and at which hail collects raindrops in the dry growth regime, as well as the probabilistic freezing and rain evaporation rates. Corresponding k-ratios are presented in Fig. 2. As noted, the KM10 rate at which a raindrop collects cloud droplets is lower by a factor of 6.3 (k∼−0.8) than the MMP rate for all

rainwater mixing ratios. This may be attributed to the higher number concentration of small KM-distributed raindrops, which weakens the effectiveness of collisions with cloud droplets as compared to the MMP scheme. In contrast, the KM50 accretion rate is always larger than its MMP counterpart by a factor of approximately 3.8 (k∼0.5). This is entirely due to the higher number of medium-range raindrops in the KM50 spectrum as compared to the MMP distribution, resulting in more frequent collisions between a raindrop and cloud droplets. Hail is more efficient in collecting the KM-distributed raindrops than the MPdistributed ones due to the smaller cross-sectional area and mean terminal velocity of the KM-distributed raindrops (the k-ratio varies from 0.5 to 2). Raindrop evaporation is predominantly mass-dependent, whilst the heat transfer needed for evaporation is a surface process. The evaporation rate therefore depends on the mean surface area/ volume ratio. For the same rain water content, this ratio is smaller for the KM-distributed raindrops than for the MPdistributed ones. This results in reduced KM evaporation rates. The probabilistic freezing rate is highly dependent on the raindrop volume. The mean raindrop volume is larger for the MP size distribution due to the higher number of large raindrops in the spectrum (Fig. 1). The k-ratios are

100 15.5

CAPE = 2770 J/kg CIN = 5 J/kg T Td

2.0

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11.8 11.3

er 50

Pressure (hPa)

Dgacr50 0.0

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Pgfr10

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Q r (g/kg)

Fig. 2 The k-ratios versus rainwater mixing ratios (Qr, in g/kg) for accretion rate between rain and cloud water (Pracw), accretion rate between hail and rainwater in the dry growth regime (Dgacr), probabilistic freezing term (Pgfr)and rain evaporation term (er) if cloud water and hail mixing ratios are 1 g/kg

2.2

0.2 km

-20

0

10 -10 Temperature ( C)

20

30

40

Fig. 3 Sounding used in the sensitivity tests, which is represented by temperature (solid line) and dew point (dotted line) profiles plotted on a skew T/log P diagram. The dashed line represents the moist adiabate through the LCL. The numbers to the right of the thermodynamic diagram represent the hydrostatic height of the pressure levels. Symbols on the right-hand side of the same figure denote wind barbs with short and long barbs representing 5 and 10 kt, respectively

Accumulated precipitation and cloud drop size distribution 10

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a

KM10 KM50 MMP

KM10

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y(km)

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14

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KM10 KM50 MMP

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Fig. 4 Height vs. time of maximum rain water mixing ratio (a); hail mixing ratio (b); and vertical velocity (c) for all model schemes. Mixing ratio values (g/kg) and vertical velocity values (m/s) are given at t=20, 60 and 100 min of simulated time

therefore always negative for this process. This implies that the hail sizes would be smaller than in the MMP scheme, and more hailstones remain aloft. Our calculations show that accretion rates involving cloud water do not differ in magnitude for the two treated model schemes. Fig. 5 Distribution of accumulated rainfall (in mm) at the surface for„ the KM10, KM50 and MMP tests at t=120 min. Isohyets start at 5 mm with a contour interval of 5 mm

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t (min)

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ƒFig. 6

Distribution of accumulated hailfall (in mm) at the surface for the KM10, KM50 and MMP tests at t=120 min. Isohyets start at 0.05 mm for the KM10 test or at 1 mm for the KM50 and MMP tests. Contour intervals are 0.05 mm (KM10 test) and 1 mm (KM50 and MMP tests), respectively

64

KM10

y(km)

48

For the sensitivity tests, the initial sounding is presented in Fig. 3. The sounding is typical of a severe storm day over the mountainous region of Western Serbia (Ćurić 1982). The winds are veering from southeast to northwest above 2 km. The wind speed varies from 5 m/s near the ground to about 17.5 m/s at a height of 9 km. The lifting condensation level (LCL) occurs at approximately 820 mb. There is significant moisture until 580 mb. The convection is initiated using an ellipsoidal warm bubble (1.5 K temperature perturbation) with an axis of 10 km in the horizontal and 1.5 km in the vertical direction. The thermal bubble centre is positioned at (x, y)=(16, 96) km in the horizontal and 1.5 km in the vertical direction. The maximum rainwater and hail mixing ratios as well as that of the vertical velocity are the parameters that characterise the model cloud’s potential to produce both rain and hail, as well the cloud intensity. The time and height of the corresponding maximum are shown in Fig. 4a–c, respectively. The rainwater maximum takes the largest values and is located at the highest altitudes (up to 9 km) in the KM10 scheme (Fig. 4a). This may be attributed to the numerous small raindrops in this spectrum and thus the highly reduced probabilistic freezing rates. In contrast, the corresponding maximum and altitude (~1 km, below the cloud base) are considerably lower in the KM50 and MMP model schemes. This is due to the larger mean terminal velocities of raindrops

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Fig. 7 Study area with the locations of the 26 rain gauge stations

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Table 1 Geographic coordinates, altitudes and the mean convective precipitation at rain gauge stations in the study area

Table 2 Mean convective precipitation per event (mm) at rain gauge stations

Rain gauge stations

Latitude

Longitude

Rain gauge stations

Jabuka Pančevo Starčevo Omoljica Banatsko Novo Selo Banatski Brestovac

44°57′ 44°53′ 44°48′ 44°45′ 44°59′ 44°44′

N N N N N N

20°36′ 20°40′ 20°43′ 20°45′ 20°47′ 20°49′

E E E E E E

70 80 80 80 105 80

Dolovo Skorenovac Kovin Beograd Beograd-Surčin Ralja Vrčin Sopot Mali Požarevac Grocka Umčari

44°54′ 44°46′ 44°45′ 44°48′ 44°49′ 44°34′ 44°39′ 44°31′ 44°32′ 44°41′ 44°35′

N N N N N N N N N N N

20°53′ 20°55′ 20°59′ 20°28′ 20°18′ 20°34′ 20°35′ 20°35′ 20°40′ 20°43′ 20°44′

E E E E E E E E E E E

95 75 75 132 96 230 250 170 270 150 140

Udovice Selevac Kolari Smederevo Batajnica Besni Fok Vranić

44°38′ 44°30′ 44°35′ 44°39′ 44°54′ 44°58′ 44°36′

N N N N N N N

20°51′ 20°53′ 20°54′ 20°55′ 20°17′ 20°25′ 20°22′

E E E E E E E

180 180 100 120 80 75 230

Stepojevac Umka

44°31′ N 44°40′ N

20°18′ E 20°18′ E

125 85

Altitude (m)

in their spectra as well as the larger probabilistic freezing and rain evaporation rates (Fig. 2). The altitudes of the maximum values are well correlated with the precipitation zone. In contrast, the MMP test produces the largest hail mixing ratio maximum (e.g. 13.5 g/kg at t=20 min) due to the larger probabilistic freezing rates (Fig. 4b). As noted, the cloud dynamics are predominant during the first 20 min of simulated time for each model scheme (Fig. 4c). Afterwards, the MMP vertical velocity maximum that provides the largest values (up to 45 m/s) oscillates around a height of 11 km, which is in contrast with observations of convective storms. Both KM maxima are below 40 m/s, and they are more oscillatory in character, especially during the first 80 min of simulated time. Precipitation is the final product of complex microphysical and dynamical interactions. Figure 5 represents the accumulated rain (in mm) at the surface for the KM10, KM50 and MMP tests at t=120 min. As noted, the rain precipitation area has a V shape due to the cloud splitting to cyclonic (to the right) and anticyclonic (to the left) cells in a

Jabuka Pančevo Starčevo Omoljica Banatsko Novo Selo Banatski Brestovac Dolovo Skorenovac Kovin Beograd Beograd-Surčin Ralja Vrčin

Mean convective precipitation per episode (mm)

Rain gauge stations

Mean convective precipitation per event (mm)

21.7 22.0 23.1 21.9 22.0

Sopot Mali Požarevac Grocka Umčari Udovice

23.7 21.0 21.1 23.4 23.8

22.3

Selevac

22.0

21.7 21.4 21.9 23.6 21.3 25.4 23.6

Kolari Smederevo Batajnica Besni Fok Vranić Stepojevac Umka

20.3 22.5 21.7 20.1 23.8 23.7 22.2

strong-sheared environment (Weisman and Klemp 1982; Ćurić et al. 2003; van den Heever and Cotton 2004). The tracks of the split cells diverge more for the KM model scheme than for the MMP one. The contributions to rainfall of both cells are nearly equal for the MMP test. In most cases, this does not occur. The cyclonic cells, thus, prevail (Grasso and Hilgendorf 2001). The cyclonic cell dominates for the KM50 test (less for the KM10 test) and is the most common feature. Accumulated hail at the surface (Fig. 6) has the form of hailstreaks, in agreement with hail observations (Changnon Jr and Kunkel 2006). Another important behaviour of the hail precipitation is that the contribution of the cyclonic cell to hailfall is considerably greater than that of the anticyclonic one for the KM model scheme. Unfortunately, a possible explanation of such cell behaviour has not yet been put forth (van den Heever and Cotton 2004). In general, the KM10 model scheme produces a larger amount of accumulated total precipitation (3.6×1010 ℓ ) than the MMP one (2.6×1010 ℓ )

4 Study area, data and comparison with observations Further insight in the characteristics of the KM model scheme may be obtained by a comparison of the modelled accumulated convective precipitation and actual observed values. To do this, a study area with northern latitudes between 44°20′ and 45°02′ N and eastern longitudes between 20°15′ and 21°03′ E was considered (Fig. 7). The geographical coordinates of 26 rain gauges with their

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478 Table 3 General sample statistics for areal accumulated convective precipitation at the surface (>5 mm) expressed in 109 ℓ after t=120 min for the observations and three model schemes (KM10, KM50 and MMP) for the study area

Sample characteristics Sample size Mean Median Variance SD Standard error Minimum Maximum Range Lower quartile Upper quartile Quartile range Skewness Kurtosis

altitudes are presented in Table 1. It should be noted that the area is mainly flat in the region north of the Danube, with the altitudes of the rain gauge stations below 100 m. The rest of the area south of the Danube is covered with small forest hills, with the altitudes of the rain gauge stations in the interval between 100 and 300 m. The convective precipitation per event for all rain gauges is presented in Table 2. The amounts are rather uniform north of the Danube, but more heterogeneous with larger amounts in the southern part of the area. Belgrade, with the sounding site, lies in the central part of the area. We selected samples of 27 convective precipitation events over a 10-year period (1976–1985). The analysis was performed with the following criteria: (a) The intensity of the convective precipitation is higher than or equal to 2.5 mm/h; (b) the duration is <2 h; c) the event is observed by at least 13 rain gauges in the study area; and (d) the event only contributes to the daily accumulated precipitation. Model convective storms were simulated at different locations outside the study area but within the model domain (Section 2), in agreement with observations made using the same characteristics of a thermal bubble. For comparison, we have made an effort to reproduce the observed storms in real time. In this subsection, we attempt to determine whether the model versions with the KM size distribution better reproduce the observed ACP values in comparison to their MMP counterparts for events over a 10-year-long period. Model simulations with three model schemes were performed for the 27 precipitation events. We used the Belgrade midday routine soundings as the initial condition of the model. We compared three samples of model ACP values with the observed ones using statistical analysis.

Observed values

KM10

KM50

MMP

27 20.90 20.65 21.17 4.60 0.9 11.0 29.1 18.14 17.30 24.42 7.12 −0.19 −0.64

27 24.30 23.98 28. 5.30 1.04 12.1 34.04 21.94 21.56 28.0 6.43 −0.28 −0.19

27 20.14 20.12 26.55 5.15 1.01 10.76 30.06 19.30 16.75 23.89 7.14 −0.65 −0.18

27 16.14 15.76 20.73 4.55 0.89 9.76 24.75 14.99 12.08 19.26 7.19 −0.88 0.30

General sample statistics for the observed and three model values of ACP are presented in Table 3. The mean, median, minimum and maximum values of ACP are the lowest for the MMP scheme and differ most from the observed values. In contrast, the analysed statistical characteristics are most successfully reproduced by the KM50 scheme, whilst the KM10 scheme overestimates them. The variance and standard deviation of ACP obtained using the MMP scheme are closer to the observed values. The standard error differs only slightly among the analysed precipitation events. We can conclude that the MMP scheme systematically underestimates the observed ACP values. For further analysis, we next applied linear regression between the observed and simulated ACP values. Figure 8 represents the regression lines obtained with ACP data points using the KM10, KM50 and MMP model schemes versus the observed values, respectively. The best linear correlation was found between the observations and the KM50 scheme (r=0.91), and the weakest one was obtained with the MMP scheme (r=0.83). The slope parameters of the regression lines show a marginal difference. In contrast, the differences in the intercept parameters are important. This is due to a systematic underestimation or overestimation of the observed ACP values as compared to their model counterparts. Disagreements between the modelled and observed ACP may be predominantly attributed to the different cloud drop size distributions used in the model. Other important factors are: the spot-like precipitation distribution derived from the rain gauge data; a storm initialisation time and space that is not fully coherent with the corresponding sounding; different propagation speeds of the model and the observed cloud; the bulk microphysical scheme used and so on. Due to the great number of simulations and thus large computer resources

ACPKM10 (10 9 )

Accumulated precipitation and cloud drop size distribution 36 34 r2 = 0.8188; r = 0.9049; y = 2.506+ 1.043x 32 30 28 26 24 22 20 18 16 14 12 10 16 22 10 12 14 18 20 24

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forecast of accumulated convective precipitation over the treated study area. This is of great importance for waterrelated sectors as well as for climatological analyses.

5 Conclusions

a 26

28

30

ACPKM50 (10 9 )

ACPO (10 9 )

32 2 30 r = 0.8350; r = 0.9138; y = –1.241+ 1.0233x 28 26 24 22 20 18 16 14 12 10 8 10 12 14 18 20 24 16 22 26

b 28

30

ACPO (10 9 )

The research presented in this paper compares the accumulated convective precipitation from observations and three model versions for a 10-year-long period over a study area with a sounding site in its central region. The main purpose of this research was to determine what cloud drop size distribution best reproduces the observed accumulated convective precipitation. As a tool, statistical analysis was applied. We used an observed dataset of 27 accumulated convective precipitation events and three model counterparts for the same precipitation events. The microphysical models employed the unified Khrgian–Mazin gamma size distribution (KM model scheme) and a scheme involving a monodisperse cloud water spectrum and the Marshall–Palmer size distribution for raindrops (MMP model scheme). A KM model scheme with a cloud drop spectrum mean radius of 10 and 50 μm (hereafter called KM10 and KM50, respectively) and a MMP model scheme were investigated. Our comparisons using statistical analysis led to the following results: –

26 24

r2 = 0.6955; r = 0.8340; y = –1.105+ 0.8253x

–

20

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22

18 16 14 12

c

10 8 10

12

14

16

18

20

22

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9

ACPO (10 ) Fig. 8 Relationship between modelled and observed accumulated convective precipitation (ACP) using a KM10 (ACPKM10); b KM50 (ACPKM50); and c MMP (ACPMMP) model schemes. The correlation coefficient, its square and the regression line equation are given

required, we have used a single-moment scheme with carefully adjusted microphysical parameters. In the next step, the most sophisticated double-moment scheme should be introduced in operative praxis. From the analysed results, we can conclude that the KM50 scheme is the most favourable for the simulation and

–

The KM10 and KM50 size distributions are better able to produce accumulated convective precipitation than the MMP scheme; The best agreement for the mean, median and range between extreme values of accumulated convective precipitation with the observed counterparts occurred for the KM50 scheme. Simultaneously, the MMP scheme produced systematic and considerable underestimation of the cited statistical characteristics. Linear regression analysis showed the best correlation coefficient between the observed and modelled accumulated convective precipitation using the KM50 scheme (0.91). In contrast, the weakest agreement was observed using the MMP scheme, with a correlation coefficient of 0.83. Results for the treated study area give us sufficient confidence to conclude that the unified Khrgian–Mazin size distribution better reproduces the accumulated convective precipitation over a small area during convective precipitation events. This is important for different climatological and hydrological analyses and predictions as well as for more appropriate input to hydrological models.

Acknowledgements This research was supported by the Ministry of Science of Serbia under Grant 146006. Anonymous reviewers have made many suggestions in the reviewing process that increased the completeness and clarity of this paper. Their contributions are appreciated.

M. Ćurić et al.

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Appendix 1

Appendix 2

Definition of symbols used in microphysical parameterisation

Microphysical equations

Symbol Pdepi Pint Pimlt Pidw Pihom Pihet Piacr Praci Praut Pracw Prevp(er) Pracs

P(Q)sacw

Psacr Psaci Psaut Psfw Psfi

Psdep(ds) Pssub(ss) Psmlt(ms) Pwacs Pgaut Pgfr(fg) D(Q)gacw D(W)gaci D(Wgacr) Pgsub(Sg) Pgmlt(mg)

Pgwet

Definition Depositional growth of cloud ice Initiation of cloud ice Melting of cloud ice to cloud water Depositional growth of cloud ice at the expense of cloud water Homogenous freezing of cloud water to cloud ice Heterogeneous freezing of cloud water to cloud ice Accretion of rain by cloud ice, producing snow or graupel depending on the amount of rain Accretion of cloud ice by rain, producing snow or graupel depending on the amount of rain Autoconversion of cloud water to rain Accretion of cloud water by rain Evaporation of rain Accretion of snow by rain, producing graupel if rain or snow exceeds threshold and T <73.16 K or rain if T > 273.16 K Accretion of cloud water by rain, producing snow (Psacw) if T < 273.16 K or rain (Qsacw) if T > 273.16 K Accretion of rain by snow, producing graupel if rain or snow exceeds threshold; if not, produces snow Accretion of cloud ice by snow Autoconversion (aggregation) of cloud ice to snow Bergeron–Findeisen process (deposition and riming), transfer of cloud water to snow Bergeron–Findeisen process embryos (cloud ice) used to calculate transfer rate of cloud water to snow (Psfw) Deposition growth of snow Sublimation of snow Melting of snow to rain, T > 273.16 K Accretion of snow by cloud water to form rain, T > 273.16 K Autoconversion (aggregation) of snow to graupel Probabilistic freezing of rain to graupel Accretion of cloud water by graupel Accretion of cloud ice by graupel Accretion of rain by graupel Sublimation of graupel Melting of graupel to form rain, T > 273.16 K (in this regime, Qgacw is assumed to be shed as rain) Wet growth of graupel, may involve Wgacs and Wgaci and must include Dgacw or Wgacr, or both. The amount of Wgacw that is not able to freeze is shed as rain

The conservation equations for the mixing ratios of water vapour (qv), cloud water (qc), rainwater (qr), cloud ice (qi), hail/graupel (qh) and snow (qs) for the single-moment scheme are written in a general form for a prognostic variable qθ as:
@ rVqq qq @ þ Dqq þ Sqq : ðrqq Þ ¼ ADV ðrqq Þ þ ð4Þ @t @z The right-hand side terms are, in order, advection, sedimentation, a diffusion term and a source (sink) for mixing ratio qθ. The parameters r and Vqq refer to the reference air density and terminal velocity. The sources (sinks), Sqq , for the corresponding hydrometeors are: Sqc ¼ rðc  ec Þ þ Pimlt  ðPihom þ Pidw þ Psacw þ Praut þPracw þ Psfw þ Dgacw þ Qsacw þ Qgacw Þ; ð5Þ Sqi ¼ rðdi  si Þ þ Pihom þ Pihet þ Pidw  ðPimlt þ Psaut þ Psaci þ Praci þ Psfi þ Dgaci þ Wgaci Þ; ð6Þ Sqr ¼ rðer þ m  f Þ þ Qsacw þ Praut þ Pracw
þ Qgacw  Piacr þ Wgacr þ Psacw þ Pgfr þ Dgacr ; ð7Þ Sqg ¼ ð1  d 2 ÞPraci þ Dgaci þ Wgaci þ Dgacw þ ð1  d2 ÞPiacr þ Pgacs þ Dgacs þ Wgacs þ Pgaut þ ð1  d 1 ÞPracs þ Dgacr þ Wgacr þ ð1  d 1 ÞPsacr þ Pgfr ;

ð8Þ Sqs ¼ Psaut þ Psaci þ Psacw þ Psfw þ Psfi þ d 2 Praci þ d 2 Piacr þ d 1 Psacr  ½Pgacs þ Dgacs þ Wgacs þ Pgaut þ ð1  d 1 ÞPracs  :

ð9Þ where Wgacr ¼ Pwet  Dgacw  Wgaci  Wgacs . For T > 273.16 K, Psaut ¼ Psaci ¼ Psacw ¼ Praci ¼ Piacr ¼ Psfi ¼ Psfw ¼ Dgacs ¼ Wgacs ¼ Dgacw ¼ Dgacr ¼ Pgwet ¼ Pracs ¼ Psacr ¼ Pgfr ¼ Pgaut ¼ Pimlt ¼ Pidw ¼ Pihom ¼ 0: ð10Þ For T < 273.16 K, Qsacw ¼ Qgacw ¼ Pgacs ¼ 0:

Accumulated precipitation and cloud drop size distribution

The symbols c, e, f, m, d and s in Eqs. 4–6 refer to the rates of condensation, evaporation of cloud droplets and raindrops, freezing of raindrops, melting of hail (graupel) and snow, deposition of ice particles, and sublimation of ice particles, respectively. In the preceding equations, δ1 =1 for a grid box in which qr and qr < 10−4 g/g and is otherwise defined as 0.δ2 =1 for a grid box in which qr < 10−4 g/g and is otherwise defined as 0. Dgaci, Dgacr and Dgacs(Wgacs, Wgacr and Wgacs) are production rates for dry (wet) growth of hail. The explicit formulation of these hydrometeor transformations can be found in Lin et al. (1983), Murakami (1990) and Ćurić et al. (2009). A definition of the symbols used in the microphysical parameterisation is given in Appendix 1.

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