3.1 Cloud droplet size distribution broadening

For the control case, the liquid water mixing ratio increases linearly with height in the ascending branches and decreases in the descending branches as shown in Fig. 2a. Liquid water mixing ratio in the ascending branch is slightly smaller than that in the descending branch at the same height due to the kinetic effect (or hysteresis effect), which is consistent with Korolev et al. (2013). The saturation ratio has an increasing trend in the ascending branch after each cycle, but has a decreasing trend in the descending branch (indicated by red and blue arrows in Fig. 2b). Droplet size for two moving size grids is shown in Fig. 2c. Droplet size in the grid monotonically increases with the dry aerosol mass associated with the grid. The solid line is for the cloud droplet that formed on a dry aerosol of 503 nm and represents the largest droplet in our simulation. It grows in the ascending branch but it evaporates in the descending branch. Also, the droplet size for this grid increases after each cycle. The dashed line in Fig. 2c is for the cloud droplet that formed on a dry aerosol of 51 nm. For this cloud droplet, the changes in radius with height are similar for the initial few cycles, after which the cloud droplet deactivates and becomes a haze particle. Ultimately, the aerosol is reactivated again as a cloud droplet by the end of the simulation (green dashed line). Also notice that a second mode appears in the CDSD due to the reactivation of aerosols after about 2 h (see Fig. 2d). It should be mentioned that the critical radius, where the Köhler curve peaks and a droplet is activated, is 3.6 µm for a cloud droplet formed on a dry aerosol of 503 nm, and 0.44 µm when formed on a dry aerosol of 51 nm. Figure 2d shows that all droplet radii are larger than 4 µm at the end of updraft cycle, indicating that all cloud droplets are activated at that point. Because GCCN do not exist in our simulation and the oscillation frequency is low, all cloud droplets have enough time to grow and to be activated in the updraft region. In this study, we focus on the CDSD at the end of the updraft cycle so the growth and evaporation of unactivated cloud droplets (e.g., McFiggans et al., 2006) will not affect the final CDSD. The CDSD broadens after each cycle as the larger droplets become larger and the smaller droplets either remain similarly sized or become smaller. All these features are consistent with Korolev (1995) (see Fig. 5 in his paper).

Figure 2Thermodynamical and microphysical properties of an adiabatic cloud parcel with upward and downward oscillations. (a) Liquid water mixing ratio changes with height. (b) Cloud parcel saturation ratio changes with height. Arrows in (b) represent the evolution of the saturation ratio profile with time. (c) Radii changes of two selected cloud droplets with height. The solid line is for the largest cloud droplet that formed on a dry aerosol with a radius of 503 nm, and the dashed line is for a droplet that formed on an aerosol of 51 nm. The red and blue lines in (a)–(c) represent ascending and descending parcels, and the black dashed line indicates cloud base height. The green dashed line indicates the reactivation of that grid. The black and green circles are referred to in the text. (d) Cloud droplet size distribution changes with time. The black line represents the mean cloud droplet radius change with time. The yellow dashed line is the change in mean droplet size for the ascending-only cloud parcel with a constant velocity of 0.5 m s⁻¹, and the upper and lower dashed gray lines represent the largest and smallest cloud droplets in the ascending-only parcel.

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Korolev (1995) analytically investigates the narrowing and broadening of cloud droplet size distribution during condensation when solute and curvature effects are considered. He considers a cloud parcel oscillating vertically in simple harmonic motion. Results show that the CDSD evolution is irreversible if solute and curvature effects are considered. Irreversibility of the CDSD will not only promote the growth of large droplets, but it will also lead to the evaporation, or even deactivation of small cloud droplets, and thus broaden the CDSD. However, the relative roles of the solute effect, curvature effect, deactivation and reactivation on the broadening of droplet size distributions have not been investigated.

To explore the relative roles of different factors in this CDSD broadening mechanism, three more cases are tested here. For the first case, we turn off both the solute and curvature effects for all cloud droplets after 700 s; this is the time when the cloud parcel first reaches 50 m above cloud base and is just below the oscillation layer. Specifically, we set S_sat=1 for all droplets. The result is shown in Fig. 3a. The CDSD repeats for each cycle in this particular case, consistent with Korolev et al. (2013), and the total cloud droplet number concentration (n) is constant (red solid line in Fig. 3d). For the second case, we only turn off the curvature effect but retain the solute effect. Specifically, we ignore the exponential term in Eq. (2) such that S_sat=a_s. The result in Fig. 3b shows that the largest droplet (with the most solute) can grow after each cycle while the smallest droplet size (with the least solute amount) associated with a moving size grid does not change much after each cycle. However the largest droplet size that a grid can reach is much smaller than that in the control case. Because the saturated water vapor pressure over a droplet formed on larger aerosol is lower than that formed on smaller aerosol due to the solute effect, the larger droplet grows faster than the smaller droplet in the updraft region, and it evaporates slower in the downdraft region. For this case, the solute effect alone cannot explain the larger cloud droplets in the control case. In addition, n is also a constant and droplet deactivation does not occur (green dashed line in Fig. 3d). In the third case, we consider both curvature and solute effects, but we do not allow droplet reactivation. This means that once the droplet deactivates it cannot be activated again. The result in Fig. 3c shows that the growth of the largest cloud droplet is similar to the control case, but the size of the smallest cloud droplet associated with a grid also increases after each cycle. The reason for this CDSD broadening is the Ostwald ripening effect, where large droplets grow at the expense of small ones. Past studies have concluded that the ripening effect is typically slow and inefficient for droplet growth (Wood et al., 2002). The vertical oscillations near cloud base that are considered here, however, allow for droplet deactivation and result in the decrease of n with time (see Fig. 3d), as in the control case. Thus, the typically inefficient Ostwald ripening is amplified through the resulting deactivation of the smallest droplets. An early suggestion of this behavior is shown in Fig. 8 of Hagen (1979). The only difference between the control and this simulation is that n for the control case increases near the end of the simulation because of droplet reactivation (see Fig. 3d). It should be mentioned that the step changes in n in Fig. 3d are a result of using a discretized grid method to represent the continuous spectrum. A downward step in n means droplet deactivation, and an upwards step in n means droplet reactivation. Deactivation and reactivation can also be seen from the CDSD qualitatively: droplet deactivation occurs when the peak value of CDSD decreases (from red to blue as shown in Fig. 2d), while droplet reactivation occurs when a subset of smaller cloud droplets appears.

Figure 3(a) Cloud droplet size distribution (CDSD) changes with time without solute or curvature effects. (b) CDSC changes with time with the solute effect but without the curvature effect. (c) CDSD changes with time including both solute and curvature effects but where droplet reactivation is not considered. (d) Total cloud droplet number concentration (n) changes with time for the different cases. The gray region in (a)–(c) represents the range of the droplet size spectrum for the control case, and the black lines represent the mean cloud droplet radius change with time.

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From Fig. 3a and b, we can see that the solute effect contributes part of the CDSD broadening compared with the control case. But the solute effect alone is not enough to explain the growth of the largest cloud droplet. Droplet deactivation, which is related to the curvature effect, plays a crucial role here (see Fig. 3c). Because the oscillations occur within the cloud region, 50 m above cloud base, droplet deactivation is surprising to us. There are two related questions: (1) why do some cloud droplets deactivate in the cloud region while others do not? (2) Why is droplet deactivation related to the CDSD broadening?

The reason for the droplet deactivation is mainly because the cloud parcel experiences upwards and downwards oscillations. In the downdraft region, the air is subsaturated, which supports droplet evaporation. In addition, the saturated water vapor pressures over polydisperse droplets are different via both the solute and curvature effects. Smaller droplets with less solute and larger radii of curvature have higher saturated water vapor pressures, and thus evaporate faster than larger droplets in the downdraft region. Therefore, smaller droplets will evaporate first in the downdraft region.

The reason why droplet deactivation is related to the CDSD broadening can be explained in two ways. From the thermodynamic point of view, the liquid water mixing ratio is roughly a constant at a given height for each cycle (see Fig. 2a). As the n decreases due to the droplet deactivation, we can expect that on average droplet size will be larger because the same amount of water will be redistributed on fewer cloud droplets. From the kinetic point of view, quasi-steady state supersaturation (s_qs) will become larger after each cycle due to droplet deactivation, as shown in Fig. 2b. s_qs, the environmental supersaturation in quasi-steady state, is inversely proportional to the integral of the mean droplet size $\stackrel{\mathrm{‾}}{r}$ and the droplet number concentration (n), $s_{\mathrm{qs}}\propto (\stackrel{\mathrm{‾}}{r}n{)}^{-\mathrm{1}}$ (e.g., Squires, 1952; Politovich and Cooper, 1988; Korolev and Mazin, 2003; Lamb and Verlinde, 2011). Here the decrease in n due to droplet deactivation is much greater than the change of $\stackrel{\mathrm{‾}}{r}$; therefore, s_qs will increase with decreasing n. This means that larger droplets grow even faster in the updraft region, and smaller droplets evaporate even faster in the downdraft region – beyond the solute effect alone. Conversely, an increase in s_qs will enhance droplet deactivation for smaller droplets, and it will also reinforce the growth of larger droplets in a positive feedback.

Table 2Microphysical properties at cloud top for different cases: r_max is the largest cloud droplet radius in a moving size grid, r_min is the smallest cloud droplet radius in a grid, $\stackrel{\mathrm{‾}}{r}$ is the mean cloud droplet size, σ is the standard deviation of the droplet radius, $\mathit{\sigma }/\stackrel{\mathrm{‾}}{r}$ is the relative dispersion and n is the cloud droplet number concentration. Case 0 is when the cloud parcel reaches the cloud top for the first time with the same setup as the control case (shown as black circle in Fig. 3). For other cases, results represent the parcel at cloud top for the last time after 3 h simulation; an example of the control case is shown as the green circle in Fig. 3. Bold values represent cases of broader cloud droplet size distribution with relative dispersions larger than 0.15.

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One question relevant to precipitation initiation is how fast can the largest cloud droplet grow in an oscillating parcel compared with droplets in an ascending-only parcel? For the latter case, the cloud parcel ascends at a vertical velocity of 0.5 m s⁻¹ for 3 h with the same initial condition as the control case. At the end of the simulation, the cloud parcel reaches about 6000 m and cloud droplets are supercooled (around 248 K), but we ignore ice nucleation in this study. The mean (yellow dashed line) and largest/smallest (upper/lower gray dashed lines) cloud droplets in an ascending-only cloud parcel are also shown in Fig. 2d. It can be seen that the size of the largest cloud droplet in a moving size grid at the cloud top in each cycle of the oscillating parcel (blue color bar) is similar to that in the ascending-only parcel (upper gray line). This is quite surprising because when the parcel reaches 1200 m for the first time (i.e., the top of the oscillation cycle), the largest cloud droplet radius is 9.07 µm (see Table 2 and Fig. 2c); however after several cycles, the largest cloud droplet radius at 1200 m is 17.3 µm. The size is similar to the largest droplet size associated with a moving size grid in an ascending-only parcel at a height of about 6000 m. This means that the largest cloud droplet size for a grid in an oscillating parcel at 1200 m is much larger than calculated from a traditional cloud parcel model (ascent only), and hence shows “superadiabatic” growth. In addition, the size of the smallest cloud droplet for a grid and the mean droplet size are larger in an ascending-only parcel. Differences between the mean droplet sizes increases after each cycle, especially at the end of the simulation due to the reactivation of numerous small droplets. Therefore, the relative dispersion, which is the ratio of the standard deviation to the mean of a droplet size distribution, also increases after each cycle, and is much larger than in an ascending-only cloud parcel.

3.2 Sensitivity studies

In this subsection, we investigate the effects of several factors on the CDSD in the adiabatic parcel model with vertical oscillations. Previous studies show that aerosol number concentration and vertical velocity are the two most important factors controlling cloud properties in an adiabatic cloud parcel model (e.g., McFiggans et al., 2006; Reutter et al., 2009; Chen et al., 2018). Two regimes are frequently considered: an aerosol-limited regime exists when there is an ample supply of water, and the cloud droplet number concentration is limited by the aerosol number concentration; and an updraft-limited regime exists when supersaturation is starved, and the cloud droplet number concentration is limited by the updraft velocity. In the updraft-limited region, cloud droplets will compete with each other for the limited available water, and the larger aerosols will suppress the activation of smaller aerosols (Ghan et al., 1998; Feingold and Kreidenweis, 2000; Feingold et al., 2001). Based on Reutter et al. (2009), the aerosol-limited regime exists when the ratio of the vertical velocity to droplet number concentration, w∕n, is larger than 10⁻³ m s⁻¹ cm³ and the updraft-limited region occurs when the w∕n ratio is smaller than 10⁻⁴ m s⁻¹ cm³. For the control case, the w∕n ratio is $\mathrm{7}\times {\mathrm{10}}^{-\mathrm{4}}$ m s⁻¹ cm³, which is in the transitional regime. In this subsection, we choose several values of aerosol number concentration and vertical velocity to investigate the CDSD in the aerosol-limited and updraft-limited regimes. In addition, we also test the effect of the recirculation layer thickness on the CDSD broadening.

3.2.1 Effect of total aerosol number concentration

We test two other aerosol number concentrations, 10² and 10⁴ cm⁻³, and keep the median radius and geometric standard deviation the same as the control case (see Fig. 4a and c). These values are chosen to represent the conditions for clean clouds (10² cm⁻³) and polluted clouds (10⁴ cm⁻³), which are consistent with previous studies (e.g., Xue and Feingold, 2004; Chen et al., 2018). Considering a vertical velocity of 0.5 m s⁻¹, they also represent the aerosol-limited regime (the 10² cm⁻³ case leads to a w∕n ratio of $\mathrm{5}\times {\mathrm{10}}^{-\mathrm{3}}$ m s⁻¹ cm³) and the transition regime (the 10⁴ cm⁻³ case leads to a w∕n ratio of $\mathrm{4}\times {\mathrm{10}}^{-\mathrm{4}}$ m s⁻¹ cm³). The results show that the CDSD for the relatively clean case (10² cm⁻³) behaves similarly to the solute effect alone (compare Figs. 3b and 4b) – there is neither droplet deactivation nor reactivation. The CDSD broadening is due to the ripening effect alone, which is not as efficient as when it is accompanied by deactivation as in the control case. For the relatively polluted case (10⁴ cm⁻³), both droplet deactivation and reactivation occur (see Fig. 4d). The largest cloud droplet acts similarly to that in the control case, while the smallest cloud droplet is larger 1.5 h into the simulation but then begins to become smaller compared with the control case. We interpret these observations as follows. For the clean case, all aerosols are activated, and all droplets are able to grow to a relatively large size, making them unlikely to deactivate. However for the polluted case, not all CCN are activated; there are consequently some smaller droplets that cannot grow very large and will evaporate first in the downdraft region. Another explanation from Korolev (1995) is that the CDSD broadening occurs when air supersaturation (S_e) is smaller than the critical supersaturation for the smallest cloud droplets (S_sat(r_small)). For this condition, the smallest cloud droplets evaporate and the largest cloud droplets might grow slightly if S_e>S_sat(r_large) or evaporate slightly if S_e<S_sat(r_large), thus leading to broadening. If the water vapor mixing ratio in air is much larger on average than the saturated water vapor mixing ratio over droplet, only narrowing of the CDSD occurs. Because in-cloud supersaturation decreases with increased aerosol concentration, it is expected that the Ostwald ripening is more efficient in polluted cloud, which is also consistent with (Srivastava, 1991).

Figure 4(a) Aerosol size distribution for a low number concentration of 10² cm⁻³. (b) Cloud droplet size distribution changes with time for the low aerosol number concentration case. (c) Aerosol size distribution for the high number concentration of 10⁴ cm⁻³. (d) Cloud droplet size distribution changes with time for the high aerosol number concentration case. Gray lines in (a) and (c) represent the control case with a total aerosol number concentration of 10³ cm³, and gray regions in (b) and (d) are the range of the cloud droplet size spectrum for the control case.

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3.2.2 Effect of vertical velocity

Two vertical velocities (0.1 and 1.0 m s⁻¹) are used to test their influence on CDSD broadening. These values are chosen based on observations that updrafts in stratocumulus clouds are in the order of 0.1 m s⁻¹ and in cumulus clouds are in the order of 1.0 m s⁻¹ (Ditas et al., 2012; Katzwinkel et al., 2014). Results also show that they correspond to the aerosol-limited regime (the 1.0 m s⁻¹ case leads to a w∕n ratio of 10⁻³ m s⁻¹ cm³) and the transitional regime (the 0.1 m s⁻¹ case leads to a w∕n ratio of $\mathrm{5}\times {\mathrm{10}}^{-\mathrm{4}}$ m s⁻¹ cm³). For a relative low velocity of ±0.1 m s⁻¹, the cloud parcel only experiences one and a half cycles within 3 h (see Fig. 5a). The parcel reaches the cloud base in around 1 h, which is significantly later than the control case due to the small velocity (see Fig. 5a). However, the largest cloud droplet size ultimately becomes similar to that in the control case, and we also see the cloud droplet number concentration decrease due to droplet deactivation. No droplet reactivation occurs because the small velocity generates a low supersaturation in the updraft region, which is unfavorable for droplet reactivation. For a relative high velocity of ±1.0 m s⁻¹, the cloud parcel can cycle more times within 3 h (see Fig. 5c). The parcel reaches cloud base faster than the control case (see Fig. 5c). Here we keep the thickness of the recirculation layer constant. Therefore, larger vertical velocity results in a higher oscillation frequency. Both droplet deactivation and reactivation occur in this case, and the largest and smallest cloud droplets behave similarly to the control case.

Figure 5(a) The height of cloud parcel changes with time for the low velocity case of ± 0.1 m s⁻¹. (b) Cloud droplet size distribution changes with time for the low velocity case. (c) The height of the cloud parcel changes with time for the velocity of ±1.0 m s⁻¹. (d) Cloud droplet size distribution changes with time for the high velocity case. Red and blue lines in  (a) and (c)  represent ascending and descending parcels, and gray lines represent the control case with velocity of ±0.5 m s⁻¹. The gray regions in (b) and (d) are the range of cloud droplet spectrum for the control case.

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3.3 Discussion

We have studied the effects of total aerosol number concentration, updraft velocity and the thickness of the recirculation layer on CDSD broadening. However we note that there are other parameters used in this study that can lead to the uncertainties in the results. For example, Takeda and Kuba (1982) found that using an insufficient number of model grids will lead to the narrow CDSD reported by Mordy (1959). Kreidenweis et al. (2003) found that both the spectral discretization and the uncertainty in the value of the mass accommodation coefficient can lead to uncertainty in the results. To test the effects of the mass accommodation coefficient and spectrum discretization on the CDSD, two more sensitivity studies are conducted. One case is to set the mass accommodation coefficient (α_m) to 0.06 based on Shaw and Lamb (1999). It is expected that a smaller value of α_m might suppress the growth of cloud droplets. The other case is to change the number of grids from 100 to 200, while keeping other parameters the same as in the control case.

Figure 6(a) The height of cloud parcel changes with time for the thin recirculation layer of 150 m. (b) Cloud droplet size distribution changes with time for the thin recirculation layer case. (c) Aerosol size distribution for the thick recirculation layer of 350 m. (d) Cloud droplet size distribution changes with time for the thick recirculation layer case. Red and blue lines in  (a) and (c)  represent ascending and descending parcels, and gray lines in represent the control case with recirculation layer of 250 m. The gray regions in (b) and (d) are the range of cloud droplet size spectrum for the control case.

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Table 2 summarizes the microphysical properties at cloud top for different cases. When the cloud parcel first reaches about 1200 m, the largest cloud droplet radius associated with a moving size grid (r_max) is 9.1 µm (case 0). If the cloud parcel continues rising for 3 h as for the ascending-only case, r_max=17 µm at 6000 m. However if the parcel experiences recirculation within cloud region, r_max can also be around 17 µm as long as deactivation occurs, except for the low N_a case (see Table 2). If reactivation also occurs, the smallest cloud droplet radius associated with a moving size grid r_min is around 5 µm and the relative dispersion is larger than 0.1. It is interesting to note that low mass accommodation has a negligible effect on r_max, but it has a stronger impact on r_min. This will result in a broader CDSD compared with the control case. In addition, a low mass accommodation coefficient inhibits the growth of cloud droplets and leads to more activated cloud droplets (Xue and Feingold, 2004). Results for 200 grids are similar to those from the control case, which means that the 100 grids used in this study are enough to limit the uncertainty due to spectrum discretization.

From the above, we see that droplet deactivation and droplet reactivation play crucially important roles in CDSD broadening in this study. Deactivation of smaller droplets is important for the growth of larger cloud droplets (e.g., see Figs. 2d, 3c, 4d, 5b, d and 6b). Droplet deactivation occurs in the descending branch for smaller droplets due to both the curvature and solute effects (Ostwald ripening). The evaporation of smaller cloud droplets with less solute makes water vapor available for the growth of other larger cloud droplets. On average, the largest cloud droplet size for a moving size grid increases with time after each cycle.

Results from the sensitivity studies show that the relative dispersion is larger than 1.5 for relatively polluted conditions when both deactivation and reactivation occur (see Table 2), which is consistent with the values from observations and simulations (e.g., Miles et al., 2000; Liu and Daum, 2002; Chandrakar et al., 2016). However the relative dispersion has also been found to be larger than 1.5 for relatively clean conditions (e.g., Miles et al., 2000; Lu and Seinfeld, 2006; Chandrakar et al., 2016). This might be due to other mechanisms, such as supersaturation fluctuations in a turbulent environment or the collision coalescence process. It should be mentioned that the CDSD observed in previous studies might have the problem of instrumental broadening due to low instrument resolution or long-distance averaging of the sampling volume (Brenguier et al., 2011; Devenish et al., 2012). A broad CDSD is also observed by recent holographic measurements, which limit the effect of instrument broadening and have much higher temporal and spatial resolution than other instruments, such as particle-counting probes (Beals et al., 2015; Glienke et al., 2017; Desai et al., 2018).

We note that deactivation is suppressed for a thin recirculation layer ΔH=150 m as shown in Fig. 6b, and therefore the CDSD broadening is not as efficient as the control case. However, the vertical oscillations of an air parcel due to turbulence might be much smaller than 150 m. Wood et al. (2002) did not observe the enhanced CDSD broadening by deactivation and reactivation with a shallower recirculation layer. One interesting question is whether deactivation or reactivation can be inhibited for a very thin recirculation layer. To answer this question, three more cases are carried out with recirculation layers of 50, 5 and 1 m. All these cases have the same setup as the control case except for the thickness of recirculation layer. The CDSD and total cloud droplet number concentration for each case are shown in Fig. 7. It can be seen that reactivation is inhibited for all cases, but deactivation always occurs. More interestingly, the CDSD for all these three cases are similar, and the decrease of total cloud droplet number concentration due to deactivation is also similar. The evolution of the CDSD for a thin recirculation layer is independent of air motion and degrades to a steady state where the CDSD broadening is due to Ostwald ripening in a still environment.

Figure 7Cloud droplet size distribution (CDSD) changes with time for different thicknesses of recirculation layers: (a) ΔH=50 m, (b) ΔH=5 m and (c) ΔH=1 m. (d) Total cloud droplet number concentration (n) changes with time for the different cases. The gray region in (a)–(c) represents the range of the droplet size spectrum for the control case, and the black lines represent the mean cloud droplet radius change with time.

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One interesting result is that the size of the largest cloud droplet associated with a moving size grid within each cycle is similar to that in the ascending-only parcel (i.e., approximately within one micrometer), as shown in Fig. 8. The general trends approximately follow the growth rate that is independent of aerosol number concentration, vertical velocity and the thickness of the oscillation layer, as long as deactivation occurs. This suggests that the growth of the largest cloud droplets strongly depends on the amount of time such droplets remain in the cloud (residence time of cloud droplets), rather than the temporal variability of supersaturation in updrafts and downdrafts. The reason for this is that the environmental (i.e., the in-cloud) saturation ratio (S_e) is buffered by the equilibrium saturation ratio (S_sat) over smaller droplets. Figure 9 shows the changes of S_e and S_sat over two droplets (same used as in Fig. 2c) in the control case. Instead of being symmetric around one for the pure water case (ignoring solute and curvature effects), S_e in the oscillating parcel is symmetric around S_sat over the small cloud droplets. For example before 1.5 h, droplets formed on r_a=51 nm are the smallest cloud droplets in the population, and the average S_e (gray line) during one oscillation is roughly symmetric around the blue line (Fig. 9). The fact that S_e is buffered by S_sat over small cloud droplets is mainly because the number concentration of the smallest cloud droplet (36 cm⁻³ in the control case) is much larger than that of large cloud droplet ($\mathrm{1.8}\times {\mathrm{10}}^{-\mathrm{9}}$ cm⁻³). When those small droplets deactivate (between 1.5 and 2.5 h), S_sat (blue line) for those deactivated droplets is the same as S_e (gray line). During this period, S_e is symmetric around S_sat over the remaining small droplets (larger than the droplets formed on r_a=51 nm but smaller than for r_a=503 nm). When the droplets formed on r_a=51 nm are reactivated (after 2.5 h), S_e is symmetric around S_sat(r_a=51 nm) again until they are deactivated. It should be mentioned that the number concentration of those reactivated droplets increases steadily after each cycle after 2.0 h (See Fig. 3d). By the end of the simulation, the number concentration of the reactivated droplets is similar to that of the remaining large droplets (about 150 cm⁻³). Therefore, the effect of those reactivated droplets on the environmental saturation ratio becomes stronger after 2.0 h (see Fig. 9).

Figure 8The largest cloud droplet size after each cycle is plotted for the different previously discussed cases: blue dots, control case; red dots, no reactivation case; pink dots, high number concentration case; green dots, high vertical velocity case; and black dots, thin oscillation layer case. The gray line is for the ascending-only case from Fig. 4, and the red line represents the growth of a droplet.

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This symmetric property of S_e can be also explained using the quasi-steady supersaturation s_qs which for pure water droplets is expressed as $s_{\mathrm{qs}}\sim \frac{w}{nr}$ (Lamb and Verlinde, 2011). This can be obtained from the analytical expression of supersaturation in an adiabatic cloud parcel: $\frac{\mathrm{d}{S}_{\mathrm{e}}}{\mathrm{d}t}=Aw-Bnr(S_{\mathrm{e}}-\mathrm{1})$, where A and B are parameters depending on thermodynamic properties (Korolev and Mazin, 2003). A symmetric distribution of w around zero will generate a symmetric distribution of s_qs around zero (i.e., S_e around one). If curvature and solute effects are considered, s_qs will be symmetric around s_k given the same condition of w, because $\frac{\mathrm{d}{S}_{\mathrm{e}}}{\mathrm{d}t}=Aw-Bnr(S_{\mathrm{e}}-S_{\mathrm{sat}})$ and thus $s_{\mathrm{qs}}\sim \frac{w}{nr}+s_k$, where $s_k=S_{\mathrm{sat}}-\mathrm{1}$ is the equilibrium supersaturation ratio over a monodisperse droplet. In the updraft region, all droplets grow and the effect of s_k is negligible. In the downdraft region and for polydisperse cloud droplets, the large number of small cloud droplets buffers the environmental conditions. Therefore S_e is symmetric around S_sat over smaller droplets before they deactivate in the oscillating parcel. $\stackrel{\mathrm{‾}}{S_{\mathrm{e}}}-S_{\mathrm{sat}}$ controls the growth of a large droplet and it is positive on average. That is why the large droplets can grow after each cycle. In addition, the influence of S_e fluctuations on droplet growth is small if S_sat over a large droplet is much lower than S_e and its fluctuations. The extreme examples of this phenomenon are when droplets form on GCCN in warm clouds (Jensen and Nugent, 2017) or ice particles form in mixed phase clouds. Therefore, the growth of the large droplet here is dominated by its in-cloud lifetime. Previous studies show that although the mean lifetime of cloud droplets is usually less than half an hour, the residence time for some lucky cloud droplets can be longer than 1 h (e.g., Feingold et al., 1996; Kogan, 2006; Andrejczuk et al., 2008). Those long-lifetime cloud droplets might contribute to large droplets in the cloud, similar to long-lifetime ice particles in mixed-phase clouds (Yang et al., 2015).

Figure 9Changes of the environmental saturation ratio (gray) and the equilibrium saturation ratios over two droplets (red and blue) with time in an oscillating parcel. The blue line is for a droplet formed on a dry aerosol with radius of 53 nm and the red line is for a droplet formed on a dry aerosol with radius of 503 nm. The smaller cloud droplet (formed on a dry aerosol with radius of 53 nm) deactivates at approximately 1.5 h and reactivates at approximately 2.5 h.

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However if all cloud droplets are deactivated, CDSD broadening does not occur (see Fig. 6d). Without droplet deactivation, the CDSD can also broaden due only to the solute effect, as is the case when the curvature effect is ignored (Fig. 3b) or when the total aerosol number concentration is low (Fig. 4b). CDSD broadening due to the ripening effect without droplet deactivation is not as significant as it is with droplet deactivation, but it might also be important after several hours as suggested by Wood et al. (2002).

Droplet reactivation usually occurs in the updraft region after several cycles, and those reactivated droplets will be deactivated again in the downdraft region. Formation of smaller cloud droplets can broaden the CDSD at smaller sizes, decrease the mean cloud droplet size, and increase the relative dispersion. Meanwhile, the generation of new cloud droplets also suppresses the growth of larger cloud droplets (see Fig. 2d).

In summary, the results of this study show that the CDSD can be broadened in a vertically oscillating cloud parcel if both solute and curvature effects are considered, consistent with the findings of previous studies (e.g., Korolev, 1995). Although our model uses an idealized setup, the sensitivity studies help explore the conditions under which this mechanism may be important in the real clouds. The results show that CDSD broadening due to Ostwald ripening can be enhanced in relatively polluted conditions when deactivation and reactivation occur, such as typically exists for continental clouds. For relatively clean conditions like marine clouds, other CDSD broadening mechanisms might be more relevant, such as the collision coalescence process or supersaturation fluctuations due to turbulence. When deactivation and reactivation occur, the simulation results show that the smallest cloud droplets do not change significantly after each oscillation cycle, while the largest cloud droplets grow on average after each cycle. The growth of the largest cloud droplet depends on its in-cloud lifetime. This is because, due to the solute effect, the saturation water vapor pressure over larger cloud droplets is smaller than the environmental water vapor pressure that is buffered by numerous smaller cloud droplets with smaller amounts of solute. It should be mentioned that the system is buffered by smaller cloud droplets formed on smaller CCN when the number concentration of those droplets is much higher than that for the largest cloud droplets formed on the largest CCN. This may not be true under relatively clean conditions, where the environmental supersaturation can be affected by droplets formed on the largest CCN.