In this study, we use the vector vorticity equation cloud-resolving model (VVM) to simulate the development of diurnal precipitation over complex topography. VVM was initially developed by Jung and Arakawa (2008), based on the three-dimensional anelastic vorticity equations. In VVM, the horizontal vorticity is predicted, and the vertical velocity is diagnosed using a three-dimensional elliptic equation. The pressure gradient force is eliminated in the equations, and the horizontal buoyancy gradient that drives the vorticity field responds to the surface fluxes directly. Thus, compared with the other models using the traditional terrain-following coordinate approach, VVM can better represent local circulation induced by heating differences. The steeper the topography is, the more significant this advantage becomes (Wu et al., 2019). The immersed boundary method is implemented (Chien and Wu, 2016; Wu and Arakawa, 2011) to represent the steep topography in Taiwan. With this representation, mountain waves, orographic precipitation, upslope wind, and other atmospheric phenomena related to topography can be reasonably simulated without having computational problems. The Noah land surface model (Noah LSM; Chen and Dudhia, 2001; Chen et al., 1996) version 3.4.1 is also coupled to VVM (Wu et al., 2019) and has been applied to evaluate the influence of land–atmosphere interactions on the afternoon thunderstorms on idealized tropical islands (Wu and Chen, 2021).

To investigate the atmospheric processes specifically over the island of Taiwan, Wu et al. (2019) developed a framework of VVM with high-resolution Taiwan topography and land use types, named TaiwanVVM. They carried out idealized simulations of summertime afternoon thunderstorms with realistic Taiwan topography. Hsieh (2019) utilized TaiwanVVM to discuss the effect of local circulation associated with fog formation at Xitou, Nantou County, Taiwan. In contrast to previous studies using TaiwanVVM, this study uses the Predicted Particle Properties (P3; Morrison and Milbrandt, 2015) microphysics scheme, implemented by Huang and Wu (2020) in VVM, to enable capturing the influences of aerosols on cloud microphysics while the aerosols are not scavenged by precipitation. The other physical parameterizations used in TaiwanVVM are the Rapid Radiative Transfer Model for GCMs (RRTMG; Iacono et al., 2008), the flux–profile relationship of Deardorff (1972) to estimate the surface fluxes, and the eddy viscosity and diffusivity coefficients depending on deformation and stability (Shutts and Gray, 1994) as the first-order turbulence closure.

For TaiwanVVM, the horizontal resolution is 500 m. The total number of vertical layers is 70, and the vertical resolution is 100 m from the sea level up to 3900 m and a stretched grid above 3900 m up to about 19 260 m (Krueger, 1988). The domain is 512×512km in size (Fig. 1). To avoid the domain boundary being cut at the edge of complex topography, which might potentially induce problems from the inflow outside the domain, the island of Taiwan is placed in the center of the domain with a sufficient area of surrounding seas. To focus on the phenomena solely related to the island of Taiwan, the topography of adjacent land around the island of Taiwan, including several islands, islets, and a part of southeast China, is not implemented in the model. Although the lateral boundary of TaiwanVVM is doubly periodic, the diurnal convection stays in the domain under a weak synoptic environment. Other detailed settings of the simulations are provided in Table 1.

A semi-realistic approach is adopted in designing TaiwanVVM simulations. That is, an observed sounding is idealized as the uniform initial condition over the entire domain, similarly to in Wu et al. (2019). Such an approach is commonly used in LESs (e.g., Grabowski et al., 2006). The direct comparison to the observations of specific cases or events is not the goal of this study. Instead, the idealization emphasizes the decisive environmental factors that modulate the development of particular convection types. By this semi-realistic approach, interactions among physical processes dominate the evolution of local circulation and convection, which can also interact with the simplified background states in the initial condition. The variability in the background environment is represented by the ensemble approach (mentioned later in Sect. 2.3), and the statistics of the semi-realistic ensemble can be compared with the observed climatological statistics from cases with similar environments.

To investigate the influence of CCN on diurnal precipitation over complex topography, we perform experiments with two scenarios of aerosol concentration. In the clean scenario, the aerosol number mixing ratio is fixed at 3×108kg-1 in the entire domain, which is within the range of the clean conditions in the marine environment (Andreae, 2009). Under the normal scenario, on the other hand, the aerosol number mixing ratio increases to 3×1010kg-1, which lies in the range of the urban environment of Taipei City, Taiwan (Lin, 2012). In P3, the number of activated CCN (Nc) is determined by 1Nc=Na2[1-erfln⁡s0s2(1+β)ln⁡σd, where s is supersaturation, s0 is mean geometric supersaturation, β is the soluble fraction of an aerosol particle, and σd is the dispersion of the dry spectrum. s0 depends on the chemical properties of the soluble part of the dry aerosol, including density, surface tension, the van 't Hoff factor, osmotic potential, and molecular weight. When s=s0, only half of the total aerosols would be activated as CCN (Khvorostyanov and Curry, 2006; Morrison and Grabowski, 2007, 2008). Thus, the initial atmospheric conditions are identical, but the aerosol concentration scenarios are different. We can expect that the difference in convection development and convective properties results from the impact of aerosol concentration.

To find appropriate representations for the environment of diurnal precipitation under weak synoptic-scale weather forcing in summer, the selection procedure was carefully designed (Fig. 2). First, using the Taiwan Atmospheric Events Database (TAD; Su et al., 2018), we selected the days with the weak southwesterly flow or weak synoptic weather conditions during the summers (May to September) between 2005 and 2014. Then, using the Central Weather Bureau surface rain gauge observations, we calculated the average diurnal precipitation cycle of 115 well-functioning weather stations for each day. To find the days with a prominent diurnal precipitation cycle, only the days with precipitation in the afternoon greater than that in the morning, as well as with the diurnal precipitation cycle within 2 standard deviations, were selected. Next, using 3-hourly Tropical Rainfall Measuring Mission Multi-Satellite Precipitation Analysis (3B42) version 7, we chose the days when precipitation occurred on the island of Taiwan but the coverage of precipitation in the surrounding areas (20.625–26.375∘ N and 118.125–123.875∘ E) was less than 20 %, making sure that the precipitation occurred locally on the island of Taiwan. There are 218 dates from the summers between 2005 and 2014 that pass the criteria mentioned above. The observed composite precipitation of these 218 dates is displayed in Fig. 4a. The precipitation over the mountains is much more intense than that on the plains. The most significant precipitation hotspot is located around the Alishan Range, which is the green box in Fig. 1 (area S). Another precipitation hotspot is situated in the Snow Mountain Range and the northern tip of Central Mountain Range, the blue box in Fig. 1 (area N), although the observation sites are relatively scarce there. For these two precipitation hotspots, the mountain ridges next to the plains have notably more rainfall than the mountain ridges behind them. Finally, we selected 30 dates for which to perform semi-realistic simulations. The selection of these 30 cases generally covers the rainfall variability in the 218 dates and serves as the ensemble members representing favorable environments for orographic locking diurnal precipitation in summer.

The simulations were driven by the simplified Banqiao Station soundings at 08:00 Taiwan standard time of these 30 cases. The thermodynamic and dynamic parameters of the initial soundings are shown in Table A1. Although variability appears in the initial convective available potential energy (CAPE), convective inhibition (CIN), precipitable water (PW), K index, and mean low-level southwesterly of the 30 simulated cases, high CAPE (mostly higher than 1100Jkg-1 with the maximum value surpassing 3300Jkg-1), low CIN (mostly less than 70Jkg-1), and high PW (generally greater than 45 mm with the maximum value almost reaching 60 mm) indicate that these soundings are conditionally unstable and moist, which is considered favorable for convection to develop. Low-level southwesterlies exist in 27 soundings, and 26 of them are southwesterly below 1500 m on average.

Aside from atmospheric conditions, the initial settings of physical parameterizations are listed below. The chemical properties of aerosols are set as ammonium sulfate, and the size distribution of aerosols follows a lognormal size distribution, with a mean size of 0.05 µm (Morrison and Milbrandt, 2015). The initial condition of the ocean and the land is relatively simple. The surface temperature of the sea and land is prescribed as the temperature of the lowest level of the initial sounding. To drive Noah LSM, land properties are necessary for model inputs. The daily averaged soil moisture over the island of Taiwan from the Global Land Data Assimilation System (GLDAS; Rodell et al., 2004) version 2.0 is assigned to the topsoil layers for all land grids in the model. The initial settings of the terrain elevation, slope type, land use, green vegetation fraction, and soil texture are the same as in Wu et al. (2019).

Object-based tracking analyses, which combine cloud object connecting and rain cell tracking algorithms, are developed to obtain the statistics related to convective structures and the intensity of precipitating systems. Figure 3 is a conceptual example of the algorithm. The x–z cross-section of two three-dimensional cloud objects is shown at the top of Fig. 3, with their projection to the surface presented beneath. The cloud object connecting is performed by the six-connected segmentation method (Tsai and Wu, 2017). It connects horizontally and vertically adjacent cloudy (cloud liquid water plus cloud ice mixing ratio greater than 10-4) grid boxes as the same cloud object. In this study, only the convective cloud objects, defined by a cloud base lower than 0.5 km, a cloud depth thicker than 1.0 km, and the center of cloud mass higher than 0.5 km, are analyzed. These criteria are chosen to include the shallow cumulus clouds during the developing stage of convection. For the vertically overlapped cloud objects, the cloud projection on the surface is determined by the lowest cloud object detected bottom-up from the surface.

The bottom of Fig. 3 shows the x dimension of a two-dimensional rain cell, which is formed by a four-way connection of rainy grids with a rain rate greater than 5 mmh-1. By co-locating the rain cell with the cloud object above, we could establish the relationship between the precipitation and the convective structure. We simplify the condition of cloud object overlapping by assuming that the precipitation on the surface is completely contributed by the lowest cloud object. Still, a rain cell could be covered by multiple cloud objects. For instance, both cloud objects in Fig. 3 partially cover the rain cell. For this rain cell, the accumulated rain rate covered by cloud object no. 1 is greater than that of cloud object no. 2, and we would co-locate the rain cell fully with cloud object no. 1. In other words, the rain cell would be co-located with the cloud object that contributes most precipitation to it.

To further evaluate the evolution of precipitating systems, we perform iterative rain cell tracking (IRT; Moseley et al., 2013, 2019). This links the rain cells at each time step and forms the rain tracks, providing a Lagrangian framework that focuses on the life cycle of the diurnal precipitating systems. By the time connection of the rain cells and the co-location between rain cells and cloud objects, the life cycle of precipitating systems is established. We can assess the progression of convective organization and the CCN effect on it.

Figure 4b demonstrates the composite simulated precipitation in Taiwan of our 30 cases. The simulated precipitation pattern captures the key features in the observed climatology of diurnal precipitation under weak synoptic weather in summertime (Fig. 4a and Lin et al., 2011), particularly the characteristics of more precipitation over the mountains than on the plains and the location of the two major precipitation hotspots (i.e., areas S and N in Fig. 1).

The timing of sufficient solar heating and surface fluxes and the establishment of local circulation determine the initiation time of diurnal precipitation, which is highly influenced by the topography (Kuo and Wu, 2019). As increasing CCN suppress the warm rain processes and delay the rain initiation, the changes in the initiation time of precipitation reflect one of the crucial effects of increasing CCN on diurnal precipitation over complex topography.

To visualize the precipitation timing associated with the topography, a three-dimensional perspective is adopted using VAPOR (Clyne et al., 2007). It is clear to see that, under the clean scenario (Fig. 5a), the development of the initiation time of precipitation is earlier over the mountain ridges and later in the river valleys. The strong buoyancy gradient induced by the heating difference between the mountain ridges and their ambient atmosphere produces convergent valley breezes that cause early precipitation over the mountain ridges. As a result, diurnal precipitation can be initiated at noon or even earlier over the mountain ridges. In the river valleys, on the other hand, diurnal precipitation can be postponed until 15:30 or even later (not shown in Fig. 5a).

Figure 5b illustrates the delayed timing of precipitation initiation when CCN concentration increases (i.e., the normal scenario minus the clean scenario). For the highlighted areas with the significant postponement, the precipitation is usually initiated before 13:00 under the clean scenario. This phenomenon is especially evident in area S, where the western slopes and ridges (pointed out by the yellow arrow in Fig. 5) have a precipitation initiation at around noon and a significant rain postponement for about 1.5 h. Thus, we conclude that increasing CCN delay the initiation time of precipitation. This significant delay in precipitation initiation could prevent local circulation from being disrupted by rainfall, which provides the convective clouds with a longer time to develop. If this hypothesis stands, the convection supported by the persisting local circulation could lead to a stronger intensity and higher degree of organization. Therefore, we next compare the statistics associated with the convective structures diagnosed by object-based tracking analyses on diurnal precipitating systems to examine the relationship between the delay in precipitation initiation and the convective intensity.

In this section, we apply the object-based tracking analyses, which diagnose both the spatial and the temporal connectivity of convective systems. The CCN effect could be different between organized and non-organized types of convection and among various stages of the convective life cycle (Rosenfeld et al., 2008). For the convective clouds that are locked by topography, the stochastic features of convection can be reduced. Thus, the following statistics concern the two major precipitation hotspot areas (areas S and N in Fig. 1), where orographic locking processes enhance the appearance of organized convective systems. We identify the organized regime by the size of the convective systems. For a simulated case under the clean scenario in a precipitation hotspot area, once the 75th percentile of the maximum cloud size during the lifetime of the diurnal precipitating systems is greater than 3×104km3, the area of the case was considered the organized regime. The classifications of the organized regime in areas S and N among the 30 simulations are listed in Table A1 in Appendix A. We then analyzed the result of the object-based tracking algorithm to identify the structural characteristics of convection modified by increasing aerosols in the mature stage of the convective life cycle (i.e., maximum intensity within the lifetime) for the organized and non-organized regimes.

Figure 6 presents the counts of the occurrence of precipitating systems with the maximum rain rate larger than 100 mmh-1. For the precipitation hotspot in area S (the green box in Fig. 1), the counts of the extreme precipitating systems increase significantly from 32 to 52 when CCN concentration rises. As for the precipitation hotspot in area N (the blue box in Fig. 1), the increment of the counts of the extreme precipitating systems due to rising CCN is only 1 (from 36 to 37), which is less than that in area S. Furthermore, the major hotspot in area S remains in about the same location, while the major hotspot shifts toward the ridges in the northeast in area N.

In addition to the spatial distribution of the extreme precipitating systems, the effect of increasing CCN can also be identified in the frequency of extreme precipitation, as shown by the probability density function (PDF) of the maximum rain rate of cloud objects (Fig. 7). For the organized regime, the probability of extreme precipitation is higher than that for the non-organized regime. Under the clean scenario, the probability of a 100 mmh-1 rain rate is 2.75×10-6 for the organized regime (blue dots in Fig. 7a), higher than that (4.84×10-7) for the non-organized regime (blue dots in Fig. 7b). Furthermore, increasing CCN result in different responses in the PDF of the two regimes. Rising CCN lead to a notable enhancement in the probability of heavy precipitation for the organized regime, which is less significant for the non-organized regime. The probability of a 100 mmh-1 rain rate for the organized regime increases by 1.50×10-6, and that for the non-organized regime decreases by 7.45×10-8. On the other hand, the reduction in the probability of light precipitation for the organized regime is lower than that for the non-organized regime when CCN concentration rises. The probability of a 1 mmh-1 rain rate for the organized regime decreases by 11.7 %, and that for the non-organized regime reduces by 12.4 %.

We further focus on the convective structure and intensity of the mature stage of the diurnal precipitating systems. Figure 8a demonstrates the box-and-whisker plot of the maximum rain rate during the lifetime of each precipitating system, representing the strength of the precipitation in the mature stage. CCN concentration is more influential on the extreme precipitation of the diurnal precipitating systems for the organized regime. The 99th percentile (P99) of the maximum rain rate increases by 10.84 mmh-1 for the organized regime when CCN concentration rises but only by 5.66 mmh-1 for the non-organized regime. The box-and-whisker plots of the maximum cloud depth and the maximum cloud size during the lifetime of each precipitating system are displayed in Fig. 8b and c, respectively, representing the characteristics of the cloud structures in the mature stage. The P99 of the maximum cloud depth increases by 0.86 km for the organized regime when CCN concentration rises, while it decreases by 0.30 km for the non-organized regime. The maximum cloud size increases by 1.43×104km3 for the organized regime when CCN concentration rises and by 1.90×104km3 for the non-organized regime. Figure 8d and e illustrate the box-and-whisker plots of the maximum in-cloud vertical velocity and the maximum core ratio during the lifetime of each precipitating system, representing the cloud dynamical features in the mature stage. The core ratio is defined as the fraction of the cloud with the vertical velocity larger than 0.5 ms-1, characterizing the updraft region. Generally, increasing CCN leads to a more intense in-cloud upward motion and a more concentrated core area for the organized regime. The P99 of the maximum in-cloud vertical velocity increases by 0.54 ms-1 for the organized regime when CCN concentration rises but decreases by 2.12 ms-1 for the non-organized regime. The P99 of the maximum core ratio decreases by 5.32 % and 4.59 % for the organized regime and the non-organized regime, respectively.

In summary, the CCN effect is more significant on the diurnal precipitating systems for the organized regime. The occurrence of the tracked extreme diurnal precipitating systems is notably enhanced. Also, the P99 of the maximum rain rate, cloud depth, and in-cloud vertical velocity during the lifetime of the diurnal precipitating systems increases by 9.4 %, 4.4 %, and 1.3 %.