Bulk-type cloud microphysics schemes adopt a specific functional form of the Drop Size Distribution (DSD) for each hydrometeor type (e.g., cloud water, rain, cloud ice, snow, and graupel/hail), rather than calculating the evolution of the DSD at each diameter bin (Lin et al., 1983; Rutledge and Hobbs, 1983; Schoenberg Ferrier, 1994; Meyers et al., 1997; Reisner et al., 1998; Morrison et al., 2005; Morrison and Milbrandt 2015; Thompson et al., 2008; Milbrandt and Yau, 2005). This approach to representing DSD evolution is computationally efficient, meaning that bulk-type cloud microphysics schemes can be applied to operational numerical weather or climate models. Among several microphysics schemes, the Weather Research and Forecasting (WRF) Double-moment 6-class (WDM6) bulk microphysics scheme (Lim and Hong, 2010) has been widely used across regions for research and operational purposes (Byun et al., 2011; Gao et al., 2011; Morrison et al., 2015; Lim et al., 2020).

Previous studies have shown that the WDM6 scheme tends to overestimate reflectivity, as demonstrated by contoured frequency by altitude diagram (CFAD) analyses (Min et al., 2015; Chakraborty et al., 2021). Min et al. (2015) evaluated the performance of the WRF Single-Moment 6-class (WSM6) and WDM6 cloud microphysics schemes using radar observations during summer season. They reported that although the WDM6 agrees better with radar observations than WSM6, both schemes overestimate the height of the melting level and the bright band relative to observations, as revealed by CFAD analysis. Additionally, WDM6 tends to overestimate reflectivity below the melting layer during summer monsoon cases. Chakraborty et al. (2021) compared WDM6 with other microphysics schemes, including the WSM6, Milbrandt, and Thompson Aerosol-aware schemes. They found that WDM6 tends to retain precipitation hydrometeors rather than precipitating them immediately, leading to greater accumulation of frozen hydrometeors in the upper layers. In the same study, CFAD analysis revealed that WDM6 produces the strongest reflectivity in the upper and middle levels among the microphysics schemes. This suggests unrealistic precipitation growth in the ice-phase region above the melting layer.

The WDM6 scheme adopts a rain DSD in the form of a generalized gamma distribution with a fixed shape parameter (μ), following the study by Cohard and Pinty (2000). Although most bulk microphysics schemes employ a fixed shape parameter, several observational studies have reported that the rain DSD shape parameter varies considerably (Yang et al., 2019; Uijlenhoet et al., 2003; Dolan et al., 2018; Cha et al., 2023; Lee et al., 2023). Yang et al. (2019) showed that the rain DSD shape parameter proposed by Testud et al. (2001) ranges from -2.3 to 60 over southern England during 2013–2017. Furthermore, they found that increasing the shape parameter initially leads to an underestimation of rainfall, followed by an overestimation. Uijlenhoet et al. (2003) noted that the rain DSD shape parameter varies depending on the stage of precipitation development and the type of cloud (e.g., stratiform or convective). Dolan et al. (2018) analysed 12 disdrometer datasets spanning three latitude bands – high (≥45° N), middle (23–45° N), and low (≤23° N) – and various precipitation types, including light rain, orographic precipitation, deep convection, organized midlatitude systems, and tropical oceanic rainfall. They found that the mean shape parameter values of rain observed using a two-dimensional video disdrometer (2DVD), Joss-Waldvogel disdrometer (JWD), and automated particle size velocity (PARSIVEL) units (APU) vary by region, ranging from -1.20 to 7.26. Similarly, Cha et al. (2023) compared DSD parameters observed with JWD and PARSIVEL across nine regions in Korea and East Asia and reported mean shape parameter values ranging from -0.158 to 8.37. These findings indicate that using a fixed shape parameter in the gamma DSD model is not optimal and that more representative values should be applied.

Scaling normalization is commonly employed to represent the hydrometeor DSD and has been extensively studied since the 1990s (Sempere-Torres et al., 1994; Testud et al., 2001; Lee et al., 2004; Szyrmer et al., 2005; Berne et al., 2012; Morrison et al., 2019; Lee et al., 2023). Sempere-Torres et al. (1994) performed the first scaling normalization of DSDs based on a single reference variable, the rainfall rate (R). Later, Testud et al. (2001) extended this single normalization to a double-moment normalization using the third and fourth moments of the DSD. They showed that the mean raindrop volume diameter (Dm) exhibits significant variability, while the intercept parameter (N0*) shows no correlation with R. Lee et al. (2004) demonstrated that the normalization approach of Testud et al. (2001) was for the particular cases of the normalization in Lee et al. (2004) using the third and fourth moments. To address this limitation, Lee et al. (2004) generalized the normalization concept and proposed a general form of double-moment scaling normalization, N(D)/N0′=h(D/Dm′), where the generalized characteristic number density (N0′) and generalized characteristic diameter (Dm′) can be defined using any combination of two moments. The Generalized Double-Moment scaling Normalization (GDMN) significantly reduces the scatter in the normalized DSD compared to single-moment normalization and decreases the standard deviation of the fractional error in estimations of the other moments not used in the normalization, leading to a more accurate representation of the DSD.

Berne et al. (2012) compared the temporal and spatial variability of DSDs using the single-moment normalization method of Sempere-Torres et al. (1994) and the GDMN method of Lee et al. (2004) over the École Polytechnique Fédérale de Lausanne (EPFL) during March 2009 and July 2010. They showed that GDMN better represents the temporal and spatial variability of DSDs. The GDMN method can be also successfully applied to the retrieval of spatiotemporally varying DSDs from dual-polarimetric radar measurement (Kwon et al., 2020; Lee et al., 2023; Shin et al., 2024). Meanwhile, Morrison et al. (2019) extended the GDMN analysis by developing a generalized nth moment normalization using three or more moments with single- and double-moment normalization using disdrometer data obtained from the U.S. Department of Energy (DOE) Atmospheric Radiation (ARM) program sites worldwide. They noted that increasing the number of reference moments can reduce variability in the DSD parameters.

To evaluate the applicability of the GDMN method for representing the DSD in a bulk-type cloud microphysics scheme, this study develops a new version of WDM6, termed WDM6-GDMN, by incorporating the GDMN approach to represent the rain DSD, following Lee et al. (2004). The performance of WDM6-GDMN is examined through simulations of an isolated summer convection event using observed normalized DSD functions. Section 2 outlines the proposed method, Sect. 3 describes the experimental setup, case study, and observational data, and Sects. 4 and 5 present the results and the summary and conclusions, respectively.

Figure 5 shows the observed radar reflectivity at 06:00, 07:00, and 08:00 UTC on 6 August 2013, along with the simulated radar reflectivity from the WDM6 and GDMN experiments. The Column Maximum (CMAX) observations at 06:00 UTC reveal two distinct regions of strong reflectivity (Fig. 5a): one over the northwestern Korean Peninsula, which moves south-eastward over time, and another, consisting of a smaller convection cell, in the south of the peninsula, moving eastward (Fig. 5a–c). By 08:00 UTC, these two convection cells converge, producing strong reflectivity over the southeastern Korean Peninsula (Fig. 5c). The WDM6 scheme does not reproduce this movement accurately, especially for the northwestern system. Although the initial stages of the two precipitation cells in WDM6 are similar to the observations (Fig. 5a and d), the stronger northwestern cell fails to move south-eastward, shifting mostly eastward instead (Fig. 5d–f). As a result, the two precipitation cells move eastward independently and do not merge, as observed. Conversely, GDMN reproduces the movement of the two precipitation cells more realistically (Fig. 5g–i). Although weak reflectivity signals remain over the western Korean Peninsula, the two cells merge in GDMN, producing intense reflectivity over the southeastern peninsula at 08:00 UTC, consistent with observations (Fig. 5g–i).

The spatial distribution of cumulative surface precipitation on 6 August 2013 is shown in Fig. 6a and b. Compared with AWS observations, WDM6 overestimates precipitation over the northeastern Korean Peninsula and underestimates it in the west and south (Fig. 6c). In WDM6, precipitation cells approaching from the northwest move eastward, leading to excessive surface precipitation in the northeast. In contrast, GDMN produces more precipitation in the southeast due to the merging of two precipitation cells in that area, thereby reducing the bias of WDM6 relative to AWS (Fig. 6d). This reduction in bias is attributed to GDMN's more accurate simulation of the movement of precipitation cells initially located in the northwest. To quantify the spatial distribution of accumulated surface precipitation, statistical skill scores – including Root Mean Square Error (RMSE), Bias, Probability of Detection (POD), False Alarm Ratio (FAR), Equivalent Threat Score (ETS) and Fractions Skill Score (FSS) – are calculated for WDM6 and GDMN (Table 2). Although GDMN underestimates total surface precipitation relative to observations, all statistical skill scores show improvement compared with WDM6. The calculated FSS also indicates that GDMN outperforms WDM6.

To further evaluate the performance of GDMN method, CFADs from the KWK and GDK sites are analysed for two simulations (Fig. 7). The location of the two sites used in the CFAD analysis are shown in Fig. 3b. The highest precipitation during the analysis period was recorded at KWK, whereas only a small amount was observed at GDK, despite the model simulations indicating substantial precipitation at this site. In Fig. 7, the x axis represents reflectivity, the y axis represents altitude, and the colour scale indicates the frequency ratios (%) at specific altitudes and reflectivity values. The reflectivity values range from 0 to 60 dBZ. Black solid lines denote the cumulative reflectivity frequencies at the 25th, 50th, and 75th percentiles. The simulated radar reflectivity is calculated following the methodologies of Koch et al. (2005), Stoelinga (2005), and Park et al. (2025). The model incorporates a simple water coating method for melting solid-phase particles to represent the bright band.

At KWK, between 10 and 12 km in altitude, weak reflectivity values of approximately 6–7 dBZ are frequently observed, gradually increasing toward 4 km, where precipitation particles grow. The median reflectivity below 4 km is about 20 dBZ. In WDM6, the reflectivity is generally overestimated relative to radar observations. No distinct melting layer is simulated, and the increase in reflectivity with decreasing altitude down to the 4 km is similar to observations. However, between 6 and 10 km, the increase occurs more rapidly than in the observations. WDM6 produces a median reflectivity of 30 dBZ below 4 km, about 10 dBZ higher than radar observations, and frequently reproduces strong reflectivity exceeding 40 dBZ. In contrast, GDMN exhibits a slower increase in reflectivity with decreasing altitude compared with WDM6 above 6 km, making it closer to observations. Although GDMN does not reproduce the narrow, distinct bright band observed near the melting layer likely due to the forward operator within WRF, it well reproduces reflectivity intensity below 4 km, bringing it into better agreement with the observations. GDMN also shows a decrease in reflectivity toward the surface, indicating possible substantial evaporation in the lower layers.

For the GDK site, peak reflectivity is observed at 4 km, indicating the presence of a distinct melting layer at this altitude. Radar observations also show gradually increasing reflectivity from 12 to 4 km, similar to the KWK site, followed by a decrease from 4 km to the surface. In WDM6, reflectivity increases rapidly from the top to 4 km, and the bright band is not distinctly simulated. This rapid increase in reflectivity, inconsistent with observations, was also noted in Min et al. (2015), indicating improper growth of solid particles in the WDM6 scheme. Additionally, WDM6 overestimates reflectivity below 4 km by about 10 dBZ compared with radar observations. Particle growth in GDMN is slower than in WDM6 over the upper layers, consistent with observations. GDMN also reproduces a clear trend of decreasing reflectivity below the melting layer at GDK. Although discrepancies with the observations still remain after applying the GDMN approach, particularly above approximately 6 km, GDMN overall reduces the overestimation of reflectivity compared with WDM6.

Figure 8 shows the 1 h accumulated surface precipitation along the cross-section shown in Fig. 3b from 04:00 to 08:00 UTC on 6 August. The cross-section spans the path of the precipitation cell, which AWS observations show was initially located over the northwestern Korean Peninsula and moved south-eastward over time (Fig. 3b). This movement is well simulated in GDMN but not captured in WDM6 (Fig. 5d–i). In both WDM6 and GDMN, precipitation begins to develop in the northwest (Location “A” in Fig. 8) after 04:00 UTC (Fig. 8a) and gradually shifts south-eastward (Location “B” in Fig. 8) with time (Fig. 8b–d). WDM6 simulates a persistent precipitation peak near “A” from 04:00 to 06:00 UTC failing to capture the south-eastward movement of precipitation cells (Fig. 5). By contrast, in GDMN, the precipitation peak shifts from Area A to Area B between 04:00 to 08:00 UTC (Fig. 8a–d). Between 05:00 and 06:00 UTC, the maximum precipitation location in GDMN is similar to that in WDM6; however, it subsequently shifts markedly south-eastward. The difference between the two schemes arises from GDMN's more realistic representation of precipitation cell movement.

To identify the main causes of the differences in simulated precipitation and reflectivity between GDMN and WDM6, thermodynamic fields and mixing ratios are further analysed in Figs. 9 and 10, focusing on 07:00 and 08:00 UTC, when two precipitation cells merged into a single cell. Figure 9 presents the vertical structures of temperature and wind along the cross-sectional area shown in Fig. 3b. In WDM6, a strong updraft develops near 37.3° N at 07:00 UTC and dissipates by 08:00 UTC, producing intense precipitation at the same location during this hour (Figs. 9a and 8d). As the precipitation cells originating in the northwest move eastward, the updraft driving the convection at 07:00 UTC is no longer visible at 08:00 UTC. In contrast, GDMN simulates an updraft farther southeast, near 36.5° N at 07:00 UTC (Fig. 9a and c). The south-eastward-moving precipitation cell merges with the eastward-moving cells and continues moving south-eastward, maintaining an updraft near 35.5° N at 08:00 UTC (Fig. 9d). Additionally, GDMN produces a colder environment below 1 km compared with WDM6 (Fig. 9). The stronger and more extensive near-surface cooling in GDMN drives the precipitation cells south-eastward, whereas the relatively weak near-surface cold environment in WDM6 limits the south-eastward propagation of the system.

At 07:00 and 08:00 UTC, compared to WDM6, GDMN exhibits stronger cooling in the lower layer behind the precipitation cells (Fig. 10a and b). The differences in cloud and rain mixing ratios between GDMN and WDM6 show a similar trend. At 07:00 UTC, the cloud mixing ratio increases significantly near 36.5° N in GDMN, coinciding with a region of strong updrafts and an enhanced rain mixing ratio (Fig. 10c and e). A comparable pattern occurs around 35.5° N at 08:00 UTC (Fig. 10d and f), where vigorous upward motion generates a supersaturated environment, leading to cloud formation and, consequently, precipitation both aloft and at the surface.

Figure 11 shows the differences (GDMN minus WDM6) in the time-area averaged vertical distributions of prognostic hydrometeor mixing ratios (cloud water (qC), cloud ice (qI), rain (qR), snow (qS), and graupel (qG)) and number concentrations (cloud condensation nuclei (NN), cloud water (NC), and rain (NR)). Below 9 km, both the rain mixing ratio and the number concentration are significantly higher in GDMN compared to WDM6. In addition, GDMN exhibits slightly higher cloud water mixing ratios and number concentrations below 6 km, as well as greater graupel mixing ratios above 3 km. The enhanced graupel production in GDMN is likely a consequence of stronger convective updrafts in the model.

To identify the microphysical processes responsible for the increased generation of rain and cloud water in GDMN compared to WDM6, the vertical profiles of the time-area averaged major source and sink terms for the rain and cloud mixing ratios are analysed in Fig. 12. In GDMN (solid line), the condensation process of water vapor into cloud water (Pcond) is enhanced below 9 km relative to WDM6 (dashed line) under saturated conditions, serving as the primary source of the cloud mixing ratio (Fig. 12a). The higher cloud mixing ratio in GDMN also intensifies accretion with rain (Pracw) (Fig. 12b), which acts not only the main sink of cloud water but also the main source of rain (blue solid line in Fig. 12c). Among the major sources of rain, Pracw produces a greater rain mixing ratio in GDMN than in WDM6. Moreover, graupel melting (Pgmlt), the second-largest source of the rain mixing ratio, is higher in GDMN owing to greater graupel production. The larger rain content in GDMN further increases ascending rain, which subsequently enhances graupel formation through raindrop freezing (Pgfrz) and accretion of rain by graupel (Pgacr) (Fig. 12d). Notably, rain evaporation (Prevp) is the primary sink of rain in the lower atmosphere, particularly below 5 km. Enhanced Prevp in GDMN compared to WDM6 leads to increased evaporation of rain into water vapor, cooling the lower atmosphere. As a result, precipitation cells in GDMN tend to propagate south-eastward, consistent with the observed behaviour.

This study applies the Generalized Double-moment scaling Normalization (GDMN) method to parameterize the rain drop size distribution (DSD) in the WDM6 microphysics scheme. The WDM6 and GDMN method employ the same generalized gamma DSD, and therefore they produce identical simulation results when the same shape parameters are used. The current version of WDM6 adopts fixed shape parameters (c=1 and μ=2) following the previous study (Lim and Hong, 2010). The key advantage of GDMN method is that it constrains the shape parameters using the normalized DSD, which is considerably more stable than the raw DSD. As a result, GDMN method provides a robust and observationally constrained normalized function, h(x), whereas direct estimation of the generalized gamma shape parameters from raw DSDs, as in WDM6, is subject to substantial variability and often leads to unreliable parameter estimates. This study utilizes the fitted generalized gamma function, hGG(x), rather than the observed mean normalized function, hmean(x), because disdrometer measurements are subject to significant uncertainty for small drops. By adopting the GDMN method, the variability of the DSD is more effectively captured, thereby reducing uncertainties of DSD. The 0th and 3rd moments are used to represent the DSD within GDMN, and sixteen microphysical processes related to rain reformulated. Long-term observed rain shape parameter values (c=2.6, μ=0.29), calculated from 2DVD data collected over the Boseong area during May–October of 2018 and 2019, are applied to the DSD through the GDMN method in WDM6. Numerical experiments are conducted for a cold-type summer rainfall event characterized by strong convection.

The results demonstrate that incorporating GDMN into the WDM6 scheme alleviates some limitations of the original WDM6 scheme, which misrepresents precipitation cell propagation and produces accumulated precipitation patterns that deviate from observations for this case. In the modified scheme, precipitation cells move south-eastward and subsequently collide with another cell generated within the Korean Peninsula, leading to increased precipitation in the southeastern regions. Overall, the modified WDM6 reproduces a more realistic propagation of precipitation cells and a more accurate spatial distribution of accumulated surface precipitation than the original scheme, achieving higher scores across all statistical skill metrics for surface precipitation.

To evaluate the simulated vertical reflectivity structure, CFAD analysis is performed using data from two radar sites, KWK and GDK. The original WDM6 scheme exhibits excessively rapid particle growth between the upper and middle layers (4–5 km) and consistently overestimates reflectivity at all altitudes. In contrast, the modified WDM6 scheme produces more gradual particle growth from the top to the middle layers, with significantly reduced reflectivity, resulting in better agreement with observed patterns. These results indicate that the modified rain DSD influences not only warm-rain microphysical processes but also mixed-phase ones, including accretion between liquid- and solid-phase particles. Consequently, the simulated reflectivity in the upper layer is more realistically represented.

Thermodynamic analysis along the path of cell movement reveals that the modified WDM6 scheme successfully triggers convective updrafts over the southeastern Korean Peninsula, which are absent in the original WDM6 scheme. In the modified WDM6, enhanced condensation of water vapor into cloud water over the updraft region increases cloud formation and promotes the production of numerous small raindrops through collision–coalescence and graupel melting. Furthermore, the modified WDM6 simulates a relatively cooler lower environment due to stronger rain evaporation, which enhances near-surface cooling. This intensified cooling drives the south-eastward propagation of precipitation cells, consistent with observations.

None of the bulk or bin microphysics schemes analysed in this study accurately reproduce the observed precipitation distribution across the Korean Peninsula on 6 August 2013 (Figs. 13 and 3b). All simulations employing bulk-type cloud microphysics schemes, which adopt a gamma DSD with constant shape parameters, generate spatial precipitation patterns similar to the original WDM6, with excessive precipitation over the central–eastern Korean Peninsula compared to AWS observations (Fig. 13a–c). The prescribed DSD function and fixed shape parameters in bulk schemes and other parameterization assumption or introduced uncertainties in microphysics schemes constrain their ability to realistically simulate precipitation. Tapiador et al. (2026) emphasized that the DSD strongly influences the physical properties of precipitation systems, including total surface area and scattering characteristics of drops, and highlighted the need for DSD modelling to be grounded in a physically consistent framework in which drop shapes follow an appropriate probability density function (PDF). Even the SBM scheme, which explicitly predicts the particle size distribution, fails to reproduce the observed spatial distribution of accumulated precipitation (Fig. 13d).

To address the limitations of the single-case analysis, we additionally conduct a single month-long simulation over East Asia for July 2023, encompassing a wide range of summertime convective systems. The model configuration, including the physical parameterizations and initial and lateral boundary conditions, is identical to that used in the single-case simulation, except that a horizontal grid spacing of 10 km and a model time step of 60 s are employed. Sea surface temperature (SST) is prescribed using the Optimum Interpolation SST (OISST) dataset (Huang et al., 2021). The performance of the original WDM6 and the modified WDM6 is evaluated against hourly precipitation from the Integrated Multi-satellitE Retrievals for GPM (IMERG) product (Huffman et al., 2015) (Fig. 14). Compared with the original WDM6, the modified WDM6 scheme reduces the precipitation over Northeastern China and increases the precipitation over the South Sea of Korea, resulting in better agreement with the observations. The statistical skill scores for surface precipitation also demonstrate the improved performance of the modified WDM6, with the pattern correlation (PC) increasing from 0.47 to 0.52 and the bias decreasing from 1.04 to 1.02. These results indicate that the performance improvements achieved by the GDMN approach are maintained in a one-month simulation encompassing a broad range of summertime convective systems, suggesting that its benefits extend beyond the single convective case examined in this study. Our findings also highlight the potential of the GDMN approach for the examined case under the specific model configuration used in this study, while recognizing that further evaluation is required before its superiority can be generalized to a broader range of convective conditions. Therefore, our study should be regarded primarily as a proof-of-concept demonstration based on limited cases simulations. Additional multi-case evaluations under diverse meteorological conditions, along with the extension of the GDMN framework to other hydrometeor types, are needed in future work.