BMPs commonly represent hydrometeor PSD by applying a three-parameter gamma distribution in the form (Ulbrich, 1983): 1nD=N0Dαe-λD, where N0, α, and λ denote the intercept, spectral shape, and slope parameters, respectively. The kth moment of the PSD defined in Eq. (1) is given by 2Mk=∫DknDdD=N0∫Dk+αexp-λDdD=N0Γ(k+α+1)λk+α+1, where k is the order of the moment, and Γ is the Euler gamma function. The spectral shape α controls the spectral width of the PSD and greatly influences microphysical conversion rates. In most double-moment schemes, α is either fixed at a constant value (e.g., Morrison et al., 2009; Lim and Hong, 2010) or modeled as an empirical function of particle mean size related to a single growth process (e.g., Milbrandt and Yau, 2005a), implying a uniform spectral response across different microphysical regimes. However, Morrison et al. (2025) showed that PSDs exhibit distinct evolutionary behaviors when driven by different microphysical processes, as revealed by a triple-moment closure framework. Similarly, the physically based triple-moment NTU scheme predicts the zeroth- (k=0), second- (k=2), and third-order (k=3) moments, allowing α to change freely over time. The temporal evolution of these prognostic moments is driven by three key microphysical processes: initiation, vapor diffusion, and collision. According to TC20 simulations (their Fig. 8), large α values generally correspond to small ice particles with high number concentrations at higher levels, whereas small α values are linked to large aggregated or rimed ice above the melting layer or within convective cores. As a result, α is modulated not only by variations in particle size (volume) but also by other moments, specifically number concentration and a third moment (e.g., M2 or M6). This multi-moment dependency is reflected in the sensitivity of α to the intermediate parameter q, which involves all three moments: 3aα=6-3q+q(q+8)2(q-1), where q≡M0M32M23 for triple moment schemes tracking M0, M2, and M3 (TC20). In general, vapor diffusion increases α, leading to a narrower PSD (cf. CT16), whereas collision growth decreases α and broadens the PSD. These contrasting behaviors reflect the fundamental differences in how growth processes affect different moments. For example, the condensation process increases M2 linearly, increases M3 sub-linearly, but does not alter M0. On the other hand, the collision-coalescence process is mass conservative but reduces M0 and M2. Thus, α should respond differently to different processes according to Eq. (3a).

Diagnostic formulas of α in TCWA2 were derived by analyzing NTU scheme simulations, utilizing calculated moments or their proxies such as mean particle size as predictors for both the accretional and diffusional regimes. 3bαx=a+bln⁡Nx+cln⁡Nx2+dln⁡Rx1+eln⁡Nx+fln⁡Rx where x∈r,i,s,g denotes the hydrometeor category, and the empirical coefficients a through f are listed in Table 1. The mean-volume radius Rx=3Qx4πNxρx1/3 in meters is derived from the mass mixing ratio (Qx), number concentration (Nx=M0,x), and bulk density (ρx). The accretional mode is activated when cloud water coexists with a hydrometeor x, representing conditions under which collision or riming dominates the PSD evolution. In the absence of cloud water, diffusional growth or sublimation governs particle growth, and the diffusional mode is therefore applied. Figure 1a and b illustrates the dependence of α (ranging from 0 to 30) on number concentration and mean-volume radius for accretional and diffusional growth, respectively. In the accretional regime, α decreases with increasing size and decreasing Nx, indicating PSD broadening due to collisions. In contrast, diffusional growth results in larger α values for larger sizes and lower Nx, reflecting narrower spectra.

For wet SA and RI in the melting regime, the energy budget is governed by the combined effects of conductive heating and latent heat from freezing and evaporation. The concurrent influence of accretional and diffusional processes creates a mixed and spatially variable distribution of α within the melting layer in the TC20 simulations. Small particles that melt quickly and shrink show sudden spectral narrowing (large α), while larger partially melted ice particles keep smaller α values as they continue to gather liquid drops, contributing to the formation of the radar bright band. To capture this rapid transition between these two contrasting behaviors, TCWA2 employs an exponential diagnostic relation: 3cαx≥0°C=exp⁡a+bln⁡Nx+cln⁡Rx where x∈s,g. As shown in Fig. 1c, α strongly depends on particle size and quickly decreases from large values toward zero as particles approach the millimeter scale. The melting of wet SA and RI is a major source of large raindrops and is closely linked to strong radar reflectivity. However, this spectral feature, with the rapid α transition, cannot be directly mapped onto the raindrop PSD in a double-moment BMP because there is no third moment (e.g., M2) memory during hydrometeor conversion. To partially account for this unresolved PSD transition during intense precipitation, TCWA2 introduces an empirical adjustment activated when Rr6Nr≥10-16 m⁶ kg⁻¹, corresponding to about 40 dBZ radar reflectivity (in mm⁶ m⁻³). When the raindrop number concentration, Nr, exceeds a tuning threshold of 3.8 × 10³ kg⁻¹ (roughly a mean radius of 0.55 mm), αr is set to its maximum (30), consistent with the high-α regime for small sizes shown in Fig. 1c. Otherwise, αr is halved to enhance raindrop deformation effects associated with broader spectra at larger sizes. This adjustment activates only during intense rainfall and does not alter the prognostic evolution of hydrometeor mass or number. Instead, it enhances PSD flexibility, enabling the model to reproduce large ZDR and KDP signals simultaneously under high-reflectivity conditions.

Although these empirical formulations do not explicitly address process-specific forcing on the PSD, they enable TCWA2 to capture the primary physical modes of α variability without the computational cost of predicting the second-order moment (M2). Using the diagnosed αx and bulk density ρx, the remaining PSD parameters, slope λx and intercept N0,x, are obtained as 4λx=M0,xΓαx+4M3,xΓαx+11/3=πρxNxΓαx+46QxΓαx+11/3,5N0,x=M0,xλxαx+1Γ(αx+1)=Nxλxαx+1Γ(αx+1).

Hydrometeor properties greatly influence microphysical processes, yet developing reliable parameterizations of ice-particle shape and density remains difficult. Thus, traditional BMPs usually treat ice particles as spheres with fixed, prescribed densities (e.g., Morrison et al., 2009; Lim and Hong, 2010), and several BMPs since 2008 (e.g., Thompson et al., 2008; Milbrandt et al., 2010) have incorporated specific mass-dimension relations with exponents smaller than 3 to represent nonspherical snow structures. In contrast, the NTU scheme predicts both particle shape and density (by tracking both volume and mass) to explicitly track ice growth history. For instance, the aspect ratio of PI follows the adaptive growth habit parameterization of Chen and Lamb (1994a) and CT16. The properties of dry SA are mainly determined by the aggregation geometry among solid-phase hydrometeors (PI and SA), while the apparent density of RI is controlled by riming growth involving CD and RD. For wet SA and RI, partial melting is diagnosed from the energy budget associated with phase change. The melted fraction is calculated from the latent heat needed to melt the ice to the freezing point, and this fraction is then used to interpolate linearly between crystal properties (shape and density) and those of liquid water.

In TCWA2, the parameterization of the RD shape follows Khvorostyanov and Curry (2002), and the liquid water density ρw is fixed at 1000 kg m⁻³. For PI, TC20 simulations indicate that particle shape varies strongly with temperature due to the alternating dominance of columnar and planar habits, as governed by the inherent growth ratio Γ(T) within the mixed-phase regime (-40 °C ≤T≤ 0 °C). The PI aspect ratio is expressed as ϕi≡DiDi03βi-1/βi+2, where Di is the particle diameter and βi is an ice adaptive growth ratio. In TCWA2, the bulk βi is approximated as a power-law function of Γ(T), with an exponent Rt that decreases from unity (highly aspherical) at initiation to near zero (quasi-spherical) as particles approach millimeter size, expressed as follows: 6Rt=aln⁡Di3+bln⁡Di2+cln⁡Di+d≤1, where Di denotes the bulk mean diameter in microns. This formulation captures the primary habit transitions between columnar and planar crystals imposed by Γ(T), while implicitly accounting for the observed reduction in intrinsic particle asphericity with increasing size (cf. CT16). The ice deposition density is retained from the NTU scheme, preserving its dependence on ice supersaturation and temperature to represent secondary growth habits (Chen and Lamb, 1994a).

For SA, data from TC20 (their Fig. 13a, c) show that bulk apparent density may fall below 300 kg m⁻³ and the aspect ratio approaches approximately 0.6 as particle size increases through aggregation, indicating a strong link between size and morphology. However, bulk density is required to derive the mean-volume radius and PSD slope parameter, so a direct size-density formulation is not practical. Instead, TCWA2 adopts an indirect diagnostic method where SA bulk density is obtained from mass concentration and air temperature via a mapped diameter from Tsai et al. (2026). The SA bulk density ρs is expressed as 7aρs=exp⁡a+bln⁡TK+cln⁡Qs+dln⁡TK2+eln⁡Qs2+fln⁡TKln⁡Qs, which yields a decrease in density from approximately 400 to 100 kg m⁻³ as Qs and TK increases above the freezing level, while rapidly transitioning to higher densities below the melting layer as temperature rises (Fig. 2a). For RI, the bulk density ρg is assumed to increase monotonically with mass content from 400 kg m⁻³ and incorporates melting effects through an extrapolated melting mass fraction. 7bρg=ρwfmw+(1-fmw)max⁡400,exp⁡aln⁡Qg3+bln⁡Qg2+cln⁡Qg+d where fmw is the melt mass fraction diagnosed from the energy budget mentioned previously. When either fmw or RI mass concentration increases, ρg increases, reflecting the transition toward a denser, partially melted state (Fig. 2b). Milbrandt et al. (2025) explicitly predict the liquid fraction in the P3 scheme to represent mixed-phase particles, whereas TCWA2 diagnoses the melted fraction for computational efficiency in operational applications.

Finally, TC20 simulations indicate that large SA aspect ratios approach approximately 0.65, while predicted graupel shapes are confined below 0.8 for large oblate particles (Jensen and Harrington, 2015). To ensure morphological consistency among size, density, and shape, TCWA2 employs a generalized diagnostic relation for oblate SA and RI: 8ϕx=a+bln⁡ρx+cln⁡ρx2+dln⁡Dx+eln⁡Dx2+fln⁡Dx3≤1, where x∈s,g, ρx is the bulk density (kg m⁻³), and Dx is the bulk diameter (µm). As shown in Fig. 2c, the aspect ratio approaches 0.65 for low-density SA and increases toward 0.8 for dense RI at millimeter scales. For sub-millimeter particles, bulk aspect ratios range from approximately 0.1 to 0.5, indicating stronger asphericity at lower densities. Below the freezing level, partial melting further shifts both shape and density toward those of liquid water. Although the above parameterizations are approximations of hydrometeor shape and density, they establish a linkage between microphysical processes and the radar forward operator, thereby allowing for a more objective evaluation of BMPs performance.

Finally, the diagnosed PSD parameters and associated hydrometeor properties must be combined with the particle's terminal velocity to accurately model the sedimentation of bulk moments. TCWA2 employs the theoretical bulk fall speed parameterizations developed by Chen et al. (2022), which explicitly incorporate particle morphology within a multi-moment gamma-type PSD framework. Alternative formulations for each category, based on PSD and morphology, are derived by fitting offline integrations of bin-based calculations. Detailed expressions and fitted coefficients are provided in Appendix A. Note that the ice shape effect has been integrated into the parameterizations of deposition and collision retained from TC20. The parameterization of the number sink due to evaporation loss is revised in Appendix B.

Radar forward operators have been continuously enhanced to connect NWP models and radar observations, often utilizing precomputed electromagnetic scattering lookup tables or parameterizations based on assumed particle shapes and exponential PSDs. While these methods work for traditional BMPs with fixed or slightly varying spectral parameters, they are not compatible with the physical assumptions underlying TCWA2. Specifically, employing existing radar operators would require externally imposed, internally inconsistent assumptions about PSD shape, particle density, and melting effects. To address this limitation, a new bulk dual-polarization radar operator is developed to explicitly link the gamma-type PSD to the diagnosed α, melting fraction of large ice particles, and category-dependent hydrometeor properties in TCWA2.

Under the Rayleigh approximation, the backward scattering amplitudes fa,b(π) and the forward scattering amplitudes fa,b(0) are expressed as follows (Van de Hulst, 1981): 9fa,bπ=fa,b0=π2D36λa1La,b+1ε-1, where λa is the radar wavelength, D is the equivolume particle diameter, and ε is the dielectric constant determined by the ice-liquid fraction. The depolarization factors La,b correspond to oblate or prolate spheroids and depend on particle aspect ratio (Ryzhkov et al., 2011). Although the Rayleigh approximation is formally valid only for particles small compared to the radar wavelength, Ryzhkov et al. (2011) estimated that the associated errors are typically within ∼ 1 dBZ for reflectivity and 0.1–0.2 dB for ZDR at S- and C-band frequencies, given cutoff equivolume diameters ranging from 3.6 to 8 mm. These uncertainties are substantially smaller than those arising from PSD variability and microphysical assumptions in BMPs. Moreover, TCWA2 employs flexible gamma-type PSDs with diagnostically varying α, which suppress the large-diameter tail relative to the Marshall–Palmer spectrum (α=0), thereby reducing resonance effects associated with large particles.

In traditional bulk radar operators, particle-level scattering amplitudes fa,b are often approximated as power-law functions of particle diameter, then integrated over an assumed PSD (e.g., Jung et al., 2008). While this method can achieve reasonable accuracy for radar reflectivity, its effectiveness would decline for polarimetric variables such as ZDR and KDP, which are expressed as ratios or differences between two bulk scattering integrals. In such cases, small fitting errors in the particle-level fa,b may be nonlinearly amplified during subtraction or division, leading to a loss of numerical significance (i.e., large-term cancellation). To avoid this issue, TCWA2 adopts a different strategy. Instead of fitting particle-level fa,b, bulk dual-polarimetric radar variables are obtained directly from bin-based scattering calculations integrated over the PSD following Ryzhkov et al. (2011, Eq. 29). Specifically, bin calculations are performed over hydrometeor-dependent, model-resolved ranges of PSD parameters (100 combinations of αx and λx), and the resulting bulk radar signatures of DBZ,x, ZDR,x, KDP,x for each category x∈r,i,s,g are then fitted as functions of the spectral parameters (αx, λx,Nx), hydrometeor properties (φx, ρx), and radar wavelength (λa=0.11 m for S band and 0.053 m for C band). By directly fitting radar signatures constrained to the PSD variability range in TCWA2, the accuracy of the simulated radar outputs mainly relies on the fitted bulk parameterizations, thus avoiding numerical errors from intermediate calculations.

Matsui et al. (2019) demonstrated that simulated dual-polarimetric observables are highly sensitive to assumptions about particle shape, orientation, and tumbling. In TCWA2, hydrometeors are assumed to have a fixed equilibrium orientation, with no explicit canting-angle variability, tumbling, or electric-field-induced alignment. Nonspherical particles fall with their maximum dimensions aligned horizontally, corresponding to an orientation angle of 90° measured from the vertical to the particle's major axis. This broadside-falling assumption is consistent with microphysical calculations of collision cross section and fall speed, but it may enhance the simulated dual-polarimetric response because orientation randomization is not represented. The radar operator also uses a fixed horizontal-viewing geometry at a zero-elevation angle. Therefore, elevation-angle-dependent projection effects are not represented and may introduce uncertainty when comparing with radar observations at nonzero elevation angles. Lastly, the bulk apparent density and aspect ratios of SA and RI are treated as modal-mean properties independent of particle size, while the shapes of RD and PI explicitly vary with particle diameter, allowing the aspect ratio to change across the PSD for a given bulk mean size. This distinction is especially important for polarimetric variables, as spectral variability in particle shape strongly impacts ZDR and KDP. Although size–shape dependence has received limited attention in previous bulk radar operators, it is explicitly incorporated in the TCWA2 operator to ensure physically consistent polarimetric signals.

After evaluating the sensitivity to individual variables, the horizontal reflectivity factor ZH,x for category x is parameterized as 10ZH,x≅Nxfρx100.1DBZ,x-18, with a coefficient of determination (R2) reaching 0.9998. The fitted expressions for DBZ,x with the density correction function f(ρx) are provided in Appendix C1. Since the ZDR is defined as the ratio of horizontal to vertical reflectivity, the vertical reflectivity factor ZV,x for category x is then derived from 11ZV,x=ZH,x100.1ZDR,x, where ZDR,x depends on both PSD parameters and hydrometeor properties. The specific differential phase KDP,x for category x is expressed as a function of number concentration, PSD parameters, hydrometeor properties, and radar wavelength (λa, with a reference of 0.11 m): 12KDP,x≅Nxfαx,λxfρxfϕx0.11λa, with the fitted formulations of ZDR,x and KDP,x for each category, as given in Appendices C2 and C3 (R2 > 0.9919). For wet SA and RI, direct fitting to mixed-phase PSDs is difficult because particles in both groups contain both ice and liquid components. TCWA2, therefore, adopts a mean-size approximation in which particle shape and density are linearly weighted by the melting mass fraction fmw. The dielectric constant of wet ice particles, εwx, is computed using a two-step Maxwell–Garnett mixing formulation (Maxwell Garnett, 1904) and fitted as a function of bulk density and volume water fraction, fvw. 13εwx=a+bfvw-1+cρx+dfvw-2+eρx2+fρxfvw-1, where x∈s,g, and the coefficients are listed in Table 1. The volume water fraction fvw follows Ryzhkov et al. (2011, Eq. 10), incorporating ice density effects and modified with a water-skin exponent from Jung et al. (2008, Eq. 2): 14fvw=ρxfmwρw-ρwfmw+ρxfmw0.3, with DBZ,x, ZDR,x, and KDP,x computed for each hydrometeor category, the total bulk radar variables are obtained by summation: 15DBZ=10log⁡101018ZH,r+ZH,i+ZH,s+ZH,g16ZDR=10log⁡10ZH,r+ZH,i+ZH,s+ZH,gZV,r+ZV,i+ZV,s+ZV,g17KDP=KDP,r+KDP,i+KDP,s+KDP,g Although TCWA2 is a double-moment scheme with several diagnostic parameterizations, it provides an internally consistent representation of PSD variability, particle morphology, and sedimentation within an integrated framework. Additionally, the dedicated bulk dual-polarization radar operator is derived from bin-based scattering integrations and fitted over the model-resolved PSD space, offering numerical robustness, consistency with the microphysical assumptions, and computational efficiency suitable for online NWP applications.

The diagnostic variables parameterized in TCWA2 include the spectral shape and aspect ratio, both of which depend on the bulk size and hydrometeor number concentration. Figure 3 shows the instant spatial distributions of number concentration for all categories during the mature stage of the simulated squall-line system. For CD, the number concentration Nc is primarily distributed around the melting layer (below the 0 °C isotherm) and within the convective region (Fig. 3a), with magnitudes on the order of 10⁸ kg⁻¹. The RD number concentration Nr reaches about 10³ kg⁻¹ within the convective core and decreases to 10² kg⁻¹ over the stratiform region (Fig. 3b), displaying a spatial pattern similar to that of CD but extending to lower altitudes. For PI, the number concentration Ni is highest in the upper cloud region around 10⁴ kg⁻¹ and gradually decreases to 10² kg⁻¹ toward the mixed-phase region (Fig. 3c). A local enhancement of Ni appears near the mixed-phase zone, due to secondary ice production. The number concentrations of SA and RI exhibit spatial distributions similar to that of PI but with higher magnitudes, reaching approximately 10⁵ kg⁻¹ for Ns (Fig. 3d) and 10⁴ kg⁻¹ for Ng (Fig. 3e).

Figure 4a shows that raindrops are suspended within the convective core up to about 6 km in altitude, with peak diameters of roughly 1 mm located below the melting layer. In contrast, smaller bulk diameters are found over the stratiform region, reflecting weaker collision processes. For PI, the mean diameters near the cloud top are generally smaller than 0.15 mm and increase to around 0.2 mm as particles fall toward the mixed-phase region (Fig. 4b). Overall, the diameter distributions of SA and RI exhibit growing variability from the upper anvil region down to the melting layer (Fig. 4c, d), consistent with gravitational size sorting. Note that the wet SA and RI appear below the melting layer to represent partial melting, and the RI's diameter continuously increases to 1 mm. Larger particle diameters are seen along the cloud edges, likely due to sublimation effects aloft. In addition, active collision processes are present within the convective region above the melting layer, where the SA and PI, which have relatively low fall speeds, are more often collected by the RD.

For RD, the αr in Fig. 5a approaches 1 near the melting layer and within the convective core, indicating relatively broad PSDs in the accretion-dominated region. In the stratiform region without the presence of the CD, the αr slightly increases to 2 due to evaporation through the switch to diffusion mode. For PI, the αi remains below 1 within the upper anvil, reflecting small ice particles in the diffusion-dominated region (Fig. 5b). As particles descend into the mixed-phase region, αi increases beyond 1, indicating depositional growth and reduced Ni. The αs and αg show similar vertical patterns. Both are less than 1 near the cloud top, pointing to broad PSDs in the anvil region, and gradually rise to about 1.3 above the melting layer through size sorting (Fig. 5c, d). Higher values of αs and αg are also observed at cloud edges, due to sublimation removing smaller particles and narrowing the PSD. Finally, αs and αg rise sharply, exceeding 20 just below the melting layer, reflecting melting-induced narrowing of the PSD.

In Fig. 6a, the bulk ice density ρi remains nearly constant at 910 kg m⁻³ as solid ice. For SA, the ρs starts around 400 kg m⁻³ near the cloud top and gradually decreases to 100 kg m⁻³ as particle mass increases (Fig. 6b). This feature reflects the formation of low-density, porous particles through aggregation. In contrast, the ρg also begins near 400 kg m⁻³ in the upper cloud region and increases to 700 kg m⁻³ with increasing particle mass (Fig. 6c), indicating that enhanced riming produces more compact particle structures. Below the melting layer, both ρs and ρg exhibit locally elevated values due to partial melting, which increases particle compactness and effective density as liquid water accumulates on ice particles (Fig. 6b, c). The relatively low ρg near the convective core can be partly attributed to the diagnostic treatment of RI density in TCWA2. The SLL parameterization provides a practical approach, but it may require further refinement as more observational constraints become available. Overall, the bulk density distributions in TCWA2 illustrate distinct growth pathways of deposition, aggregation, and riming for three groups of ice particles, respectively.

Figure 7a presents that the RD aspect ratio φr is about 0.9 within the convective core and increases toward unity in the stratiform region. For PI, the aspect ratio φi is parameterized using the temperature-dependent habit function Γ(T), leading to distinct shape transitions with height. Columnar ice crystals dominate at temperatures colder than -20 °C, while plate-like habits are prevalent between -10 and -20 °C. Above -10 °C, the ice habit shifts back toward columnar types (Fig. 7b). Consequently, the φi varies widely from 7 to 0.1. For oblate SA, the aspect ratio φs indicates highly aspherical particles (∼ 0.4) in the upper cloud region, and gradually increases to 0.6 near the melting layer through ice aggregation (Fig. 7c). The RI φg shows systematically larger aspect ratios, increasing to about 0.8 below the melting layer due to liquid water partitioning during melting (Fig. 7d).

To demonstrate the relationship between hydrometeor properties and bulk fall speed, Fig. 8 displays the mass-weighted fall speed as a function of bulk diameter for four precipitation categories. The RD bulk fall speed rises quickly for diameters smaller than 2 mm and gradually levels off at about 8 m s⁻¹. This damping behavior results from increased drag caused by large drop deformation when φr falls below 0.8. For PI, the bulk fall speed varies significantly, mainly influenced by crystal shape. Particles with more aspherical habits (i.e., larger deviations of φi from one) fall more slowly, as shown by the orange and purple points, while more spherical crystals (φi close to one), represented by the blue points, attain higher fall speeds. This emphasizes the high sensitivity of PI sedimentation to habit-dependent effects. The bulk fall speed of SA shows a consistent dependence on particle density, increasing steadily from low- to medium-density aggregates. Additionally, RI displays a similar density-dependent relationship but with extra scatter influenced by particle shape. Overall, RD has the highest average bulk fall speed (3.71 m s⁻¹), followed by RI (0.85 m s⁻¹), SA (0.46 m s⁻¹), and PI (0.16 m s⁻¹), which aligns more closely with the traditional method using fixed parameters prescribed for specific hydrometeors (i.e., Locatelli and Hobbs, 1974).

Although TCWA2 captures the main features of variable PSDs, regime-dependent microphysical variables, and property-dependent fall speeds, the diagnostic approach used here is inevitably less complex than the NTU scheme, which explicitly tracks particle growth memory through prognostic moments.

The dual-polarimetric radar variables of DBZ, ZDR, and KDP are parameterized in TCWA2 as functions of the PSD, hydrometeor properties, and number concentration. This subsection investigates the spatial distribution of these radar products in the simulated convective system and explains how individual hydrometeors contribute to polarimetric signatures.

Figure 9 presents the spatial distributions of simulated horizontal radar reflectivity (DBZ) along with its breakdown into contributions from RD, PI, SA, and RI. The total radar reflectivity (Fig. 9a) depicts a deep convective structure marked by strong echoes exceeding 40 dBZ that extend from near the surface up to about 7 km altitude, followed by a downstream shift to a stratiform region around 260 km in horizontal distance. Below the melting layer, RD dominates the reflectivity signal, highlighting the role of melting ice particles in generating large liquid drops associated with the radar bright band (Fig. 9b). Above the freezing level, SA mainly contributes to the outflow cloud deck reflectivity (0–25 dBZ), reflecting efficient aggregation within the main ice cloud region (Fig. 9d). In contrast, RI accounts for the enhanced reflectivity (> 20 dBZ) in the mixed-phase region, indicative of active riming (Fig. 9e). The PI contribution is relatively weak (< 0 dBZ) and mostly confined to upper levels, due to its small particle size (Fig. 9c). This hydrometeor-based breakdown reveals a clear vertical partitioning of reflectivity linked to deposition, aggregation, riming, and melting processes in TCWA2.

Figure 10 illustrates the spatial distribution of simulated differential reflectivity (ZDR) and highlights three common polarimetric features seen in convective systems. First, a prominent ZDR column (Illingworth et al., 1987; Loney et al., 2002) appears just below and above the freezing level, mainly caused by RD (Fig. 10b). This feature results from large, oblate liquid drops lifted by strong updrafts and from reduced particle terminal velocity. It is well represented by the diameter-dependent RD shape, the fall speed formulation, and the flexible PSD modeling in TCWA2. Second, a midlevel ZDR band (Kumjian and Ryzhkov, 2008) about 1 dB appears in the melting layer, caused by partially melted SA and RI with increased bulk density during melting (Fig. 10d, e). Third, a clear cloud-top ZDR enhancement (> 1 dB) is simulated at higher levels, mainly due to PI (Fig. 10c). Elevated ZDR in this zone indicates the presence of columnar solid ice crystals with relatively low fall speeds. Throughout the broader cloud deck, ZDR remains relatively low (0.1–0.4 dB), mostly due to SA, given the dominant reflectivity partition. Although PI shows high ZDR values locally within the mixed-phase region, its contribution to the total ZDR is much less than that of RI.

Figure 11 presents vertical cross sections of simulated specific differential phase (KDP) and reveals a distinct KDP column (Hubbert et al., 1998; Loney et al., 2002) stretching from near the surface through the mixed-phase region into upper levels at 7 km, collocated with the strongest updrafts. This feature is mainly caused by RI within the mixed-phase region and by RD below the freezing level (Fig. 11b, e). The continuous enhancement of KDP along the vertical indicates the presence of many aspherical, dense hydrometeors lifted by strong upward motion. In addition, these particles contribute to localized KDP (> 0.4 ° km⁻¹) near the melting layer. Above the freezing layer, KDP remains relatively weak across various cloud decks and is mostly associated with dry SA (Fig. 11d). Contribution from PI at temperatures warmer than -40 °C is generally minor due to its low number concentration (Fig. 11c).

Figure 12 further demonstrates how ZDR depends on spectral shape, aspect ratio, and apparent density for each type of hydrometeor. For RD, ZDR steadily rises with bulk diameter and reaches higher (lower) values for smaller (larger) αr at the same diameter, indicating more (less) deformation in broader (narrower) PSDs. For PI, ZDR varies greatly at small diameters, mainly influenced by particle shape, with highly aspherical crystals producing higher ZDR despite weaker reflectivity. Dry SA typically results in small ZDR values that decrease as diameter increases, while wet SA shows moderate ZDR that quickly declines as particles become denser and more spherical during melting. Similarly, dry RI particles have low ZDR, and wet RI falls within the intermediate ZDR range.

In summary, these results show that TCWA2 consistently performs across PSD variability, microphysical diagnostics, sedimentation behavior, and polarimetric radar products, and provides a physical foundation for applying TCWA2 to proof-of-concept real-case simulations and radar-based model assessments discussed in later sections.

The MPAS model is configured with a variable-resolution mesh comprising 97 031 horizontal cells, with the model top at 30 km. The mesh includes a circular refinement zone centered at (24.0° N, 121.0° E) with a radius of 850 km. Inside this area, a high horizontal resolution of 2 km is maintained within about a 150 km radius, gradually expanding to 6 km toward the outer edges. This setup covers roughly 18.7–29.2° N in latitude and 115.3–126.8° E in longitude (Fig. 13), covering the operational radar network over Taiwan. The 2 km grid spacing is coarser than the 1 km idealized WRF simulation and may underresolve convective-core updrafts and hydrometeor lofting. Thus, this real-case convection-permitting simulation is suitable for evaluating overall precipitation organization and qualitative polarimetric structures, but it should not be interpreted as a complete assessment of TCWA2 behavior within fully resolved convective cores. For time integration, the model uses a 36 s time step with 4 acoustic sub-steps (the default is 2) to ensure numerical stability and high-resolution efficiency. For physical parameterizations, the scale-aware New Tiedtke cumulus scheme (Wang, 2022) is used to ensure consistent performance across different mesh resolutions. Boundary layer and surface processes are represented by the Yonsei University PBL scheme (Hong et al., 2006) and Monin–Obukhov similarity theory (Monin and Obukhov, 1954), respectively. Land-surface interactions and radiative transfer are handled through the Noah Land Surface Model (Chen and Dudhia, 2001) and the RRTMG scheme (Iacono et al., 2008). To provide reference context rather than a definitive scheme intercomparison, additional simulations are performed using the single-moment WSM6 (Hong and Lim, 2006) and semi-double-moment Thompson (Thompson et al., 2008) schemes, alongside the TCWA2 simulation.

For the case study, an afternoon thunderstorm observed during the TAHOPE field campaign on 24 June 2022, was selected for evaluation. This event involved a typical convective system that caused precipitation over western central Taiwan, with a maximum of over 190 mm accumulated within 12 h. The heavy rainfall and well-developed leading convective structure make this case ideal for assessing microphysical performance using ground-based radar data. Each experiment started at 00:00 UTC on 24 June 2022, and lasted for 12 h, with initial and boundary conditions provided by the National Centers for Environmental Prediction (NCEP) Global Forecast System (GFS). Model outputs are compared with two observational datasets. First, precipitation forecasts are verified against ground-based observations from the CWA's Quantitative Precipitation Estimation and Segregation Using Multiple Sensors (QPESUMS) system (Chang et al., 2021). Second, simulated polarimetric radar variables are contrasted with Plan Position Indicator (PPI) observations from four operational CWA radar sites: RCWF (25.07° N, 121.77° E; S band), RCLY (22.53° N, 120.38° E; C band), RCNT (24.14° N, 120.58° E; C band), and RCSL (25.0° N, 121.4° E; C band). These radar stations collectively cover the leading convective areas of the event. During this thunderstorm, a leading squall line formed around 06:00 UTC on 24 June and dissipated by approximately 11:00 UTC. Accordingly, six-hourly radar data were sampled to match the model output frequency. The results are analyzed with particular focus on the characteristics of dual-polarimetric radar variables.

The simulation results are assessed using a hierarchy of precipitation, reflectivity, and dual-polarimetric diagnostics. Figure 14 displays three simulations that generally replicate the elongated north–south precipitation band over western and central Taiwan, with spatial pattern correlations between 0.53 and 0.55. The mean of the highest 10 precipitation values is 198 mm for WSM6, 188 mm for Thompson, and 173 mm for TCWA2, compared with 181 mm for QPESUMS. The WSM6 run produces a larger area of intense rainfall exceeding 90 mm over central Taiwan, indicating it overestimates the most intense precipitation amounts. Thompson and TCWA2 generate a narrower distribution of heavy rainfall, closer to the observed value, for this metric.

As the location of the leading convective system is identified, a comprehensive evaluation is performed using grid-to-grid comparison against hourly radar observations from 06:00 to 11:00 UTC at four radar stations. Figure 15 shows joint height–DBZ probability density distributions comparing simulated and observed DBZ. The observed joint distribution displays a frequent occurrence of reflectivity values between 2–10 km altitude, with reflectivity values mainly ranging from 10 to 30 dBZ. Above the melting layer near 5 km altitude, reflectivity steadily decreases with height, indicating ice growth through riming and aggregation. Enhanced reflectivity exceeding 30 dBZ around 5 km corresponds to the radar bright band caused by the melting of large ice particles. The WSM6 simulation captures the overall vertical structure but has a broader distribution of high reflectivity, suggesting an overproduction of large hydrometeors. The Thompson scheme shows better vertical coherence compared to WSM6 but still has an overly extended high reflectivity region below 5 km, especially near the bright band peak. The pronounced bright band signatures may partly result from the combined effects of the fixed ice-particle density assumptions in the microphysics schemes and the simplified treatment of dielectric properties during melting in the radar operator. In contrast, the TCWA2 run shows the closest agreement with the observations, achieving a pattern correlation of 0.99. The vertical placement and intensity of the reflectivity core depict a realistic transition from moderate reflectivity at lower levels, to peak echoes within the melting layer, and to weaker values aloft. Notably, the reflectivity value around 25 dBZ and the enhanced bright band signals at 4–5 km altitude are captured in both narrow and broad PSD representations. The lower occurrence of moderate reflectivity at upper levels may be related to the current RI initiation treatment or the lack of accumulated riming-history memory, and warrants further examination in future work. Nevertheless, this improved depiction suggests that TCWA2 more effectively constrains the hydrometeor PSD through microphysical growth and sedimentation processes, offering a robust baseline for subsequent assessment of dual-polarimetric radar variables.

To further compare the dual-polarimetric radar products simulated by TCWA2, Fig. 16 shows joint distribution comparisons of ZDR and KDP between the TCWA2 simulation and observations. The observed joint height–ZDR features a concentrated core between 0.1 and 1.5 dB below 4 km altitude, with a near-surface tail extending toward 3 dB, indicating the presence of oblate RD in a broad spectrum, along with size-sorting processes. Above the melting layer, the ZDR core diminishes below 1 dB and gradually weakens with height up to about 15 km. A slightly enhanced ZDR signal appears above 12 km in the TCWA2 simulation from the aspherical PI on the cloud top. Overall, the TCWA2 simulation generally reproduces both the vertical confinement and magnitude of the ZDR core, with a pattern correlation of 0.81. Since the KDP value depends on the radar wavelength, comparisons are performed separately for S and C band observations, corresponding to different sampling points. Both joint height–KDP comparisons show similar levels of agreement. Observations reveal enhanced KDP below 5 km, with maxima extending toward the surface, reflecting high liquid water content and active collisional breakup processes. TCWA2 captures this vertical distribution with pattern correlations of 0.90 and 0.93 for the two KDP configurations, respectively. However, TCWA2 occasionally overestimates KDP near 10 km altitude, which may partly reflect limitations in the microphysical parameterizations of RI, the fixed horizontal-viewing geometry, and the broadside-falling assumption for particle orientation without explicit tumbling or electric-field-induced alignment. These uncertainties indicate the need for further refinement.

To account for distinct microphysical regimes and isolate the influence of melting-layer processes, the simulated and observed radar signatures are analyzed by stratifying the results to assess whether the simulated raindrop and ice-particle characteristics align with observations at various heights. Figure 17 illustrates the joint distribution of ZDR and DBZ by dividing them into three altitude ranges. In the low-level region below 4 km, observations show multiple ZDR modes associated with liquid-phase hydrometeors. A primary ZDR core is centered around 0.2 dB within 20–30 dBZ, along with a broader distribution below 1 dB across 10–35 dBZ. A secondary mode with higher ZDR values, reaching about 2 dB, appears between 30 and 45 dBZ, likely related to increases in RD accretion and self-collection. Occasional instances exceeding 2.5 dB above 40 dBZ are also observed, with this significant deformation mainly caused by melting of large oblate ice particles in a broad PSD. The TCWA2 simulation closely reproduces both the magnitude and variability of the observed ZDR–DBZ relationship dominated by RD. Within the mixed-phase layer between 4 and 6 km, the observed ZDR cores for strong reflectivity (> 40 dBZ) are substantially reduced, while sporadic ZDR enhancements (around 0.1–0.15) are contributed by partial melting into liquid drops. TCWA2 effectively simulates this transition from rain-dominated conditions to mixed-phase wet SA and RI across the melting layer. Above 6 km, both observations and simulations exhibit low ZDR and DBZ values, generally below 1 dB and 30 dBZ, consistent with near-spherical ice particles. TCWA2 captures key aspects of this pattern, achieving a pattern correlation of 0.99. The relative magnitude of ZDR against DBZ is broadly consistent with the expected influence of ice-phase hydrometeor properties. However, this interpretation remains sensitive to radar-operator assumptions such as particle orientation and viewing geometry. Overall, the agreement between simulated and observed ZDR–DBZ distributions across all altitude ranges provides encouraging evidence that TCWA2 can represent key aspects of the vertical evolution of hydrometeor size and shape in this convective system.

Figure 18 further depicts the joint distributions of KDP at S band against DBZ by dividing the results into three altitude zones. Since KDP is directly proportional to the number concentration of nonspherical particles, this analysis offers a strict constraint on how the number density related to asphericity is represented. In the low-altitude region below 4 km, observations exhibit a main distribution centered near 30 dBZ and ranging roughly from 10 to 50 dBZ, with most KDP values staying below 1° km⁻¹. A secondary population with higher KDP reaching up to 2.5° km⁻¹ appears at strong reflectivity over 45 dBZ, mainly from large melting oblate ice particles. The TCWA2 simulation produces a narrower core between 30 and 40 dBZ and occasionally overestimates KDP at reflectivities above 40 dBZ. However, it still captures the observed relationship between KDP and DBZ. In the mixed-phase layer between 4 and 6 km, the observed KDP distribution has a similar shape, but with lower peak values and less frequent occurrences, reflecting a smaller amount of liquid water confined to partial melting. TCWA2 replicates this weaker KDP signal during phase changes. Above 6 km, both observations and simulations show low KDP and DBZ values, generally under 1° km⁻¹ and 45 dBZ, aligned with ice-dominated hydrometeors and minimal liquid water occurrence. Although TCWA2 matches the overall pattern of the observed distribution, the simulated core shifts slightly from the observed 10–25 dBZ range to 15–30 dBZ. This bias is probably caused by uncertainties in observational sampling with S-band radar at a relatively long distance from the convective system.

Figure 19 shows the joint distributions of KDP and DBZ at C band, divided by altitude, offering an independent assesment of the simulated hydrometeor signatures with a shorter radar wavelength. Compared to S-band data, C-band KDP is more sensitive to raindrop size and number concentration. In the low-level layer below 4 km, observations reveal a dominant KDP core within 30–40 dBZ, mostly below 1° km⁻¹, with noticeable KDP increases when reflectivity exceeds 30 dBZ. Peak KDP values reach about 5° km⁻¹ and occur more often at high reflectivity than in S-band data, reflecting the stronger wavelength dependence of KDP at C band. The TCWA2 simulation closely reproduces both the magnitude and reflectivity dependence of the observed distribution. In the layer between 4 and 6 km, where mixed-phase hydrometeors are present, observed KDP values decline and become more tightly clustered, reflecting diminishing liquid water content and phase changes. TCWA2 captures this vertical shift of melting and liquid water depletion across the melting layer with a pattern correlation of 0.97. Above 6 km, both observations and simulations show weak KDP values, generally below 1° km⁻¹, consistent with ice-dominated hydrometeors and limited supercooled liquid water. Although there are occasional discrepancies with KDP over 1° km⁻¹, possibly associated with RI signatures and radar-operator assumptions. Nevertheless, TCWA2 captures the overall distribution, with a high pattern correlation of 0.98.

The MPAS real-case analysis is intended as a proof-of-concept demonstration, not as a comprehensive operational validation or definitive scheme intercomparison. The simulations show that TCWA2 captures key aspects of the observed vertical reflectivity patterns and height-dependent dual-polarization signatures, with encouraging pattern correlations. However, discrepancies remain at upper levels and may partly reflect observational limitations, ice-phase uncertainties in the RI parameterizations, the fixed horizontal-viewing geometry in the radar operator, the broadside-falling particle-orientation assumption, and the absence of explicit tumbling or canting-angle variability. These issues remain topics for future process-level evaluation.

All simulations were run with MPI parallelization across 32 CPU cores. The total wall-clock times, with microphysics-component times in parentheses, were 402 (22), 511 (34), and 550 (118) s for the WSM6, Thompson, and TCWA2 simulations, respectively. Compared with Thompson, the TCWA2 microphysics component required about 3.5 times as much computational time, primarily due to revised parameterizations and additional calculations of dual-polarization radar variables. However, the total wall-clock time increased by only 39 s, or about 7.6 %, in this model configuration. Thus, TCWA2 remains computationally practical because it retains a double-moment prognostic framework while diagnosing additional microphysical and radar-related properties.