The simulations performed here employs variations of the Tel Aviv University (TAU) bin microphysics parameterization (Feingold et al., 1988; Tzivion et al., 1987, 1989) coupled to a single-column Eulerian framework. The 1-D model is based on the Kinematic Driver (KiD) model (Shipway and Hill, 2012), but instead of prescribing w for each time t and height z, it is calculated from the simplified vertical momentum equation, considering the buoyancy of the parcel and the weight of the liquid water, as well as the reaction force on the parcel resulting from the acceleration of the air in the neighborhood (Pruppacher and Klett, 2012):

where $\mathit{\gamma }\equiv m^{\prime }/\mathrm{2}m\approx \mathrm{0.5}$, with m and m^′ being the mass of the parcel and the mass of the air displaced by the parcel, respectively; g is the gravity acceleration; θ and θ^′ are the potential temperature of the ascending parcel and the environment, respectively; and q_l is the liquid water mixing ratio. The entrainment rate $\mathit{\mu }\equiv \frac{\mathrm{1}}{m}\frac{\mathrm{d}m}{\mathrm{d}z}$ considers the lateral mass flux along the axis of a vertical plume of radius R(t,z). It is assumed to follow the inverse radius dependence: $\mathit{\mu }=\frac{C}{R}$, where C≈0.2 is the entrainment parameter. The equation for the radius of the plume is

where ρ represents the density of the air.

In our simulations, a 1 s time step was used for both dynamics and microphysics algorithms during an integration time of 1800 s (30 min). For the vertical domain, a 120-level grid was defined with a 50 m grid spacing from an altitude of 0 to 6000 m.

As initial conditions, vertical profiles of potential temperature and water vapor mixing ratio (q_v) from an in situ atmospheric sounding corresponding to 17:30 Z on 11 September 2014, from Manacapuru, Brazil (Fig. 1), were provided. A constant temperature perturbation of 2.5 K was introduced at the surface to force the convection.

Figure 1Vertical profiles employed as initial conditions in the simulations.

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The contribution of the entrainment in the equations for the evolution of θ, q_v and N_a is expressed as $\mathit{\mu }(X-X^{\prime })w$, where X and X^′ represent the in-cloud and environmental values for each of the magnitudes mentioned, respectively.

We employ a phase space defined by two bulk properties of the DSD (hereafter “bulk phase space”): N_d (cm⁻³), which coincides with the zeroth moment of the DSD, and D_eff, the droplet effective diameter (µm), which is the ratio between the third and second moments.

Sensitivity tests in the bulk phase space provide a very efficient means of evaluating how a specific parameter variability can affect the evolution of cloud-top DSDs. Here, we test the sensitivity of N_d and D_eff at the cloud top to variations in N_a, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$, σ_a and κ, using ranges normally found in the Amazonian atmosphere (Gunthe et al., 2009; Martin et al., 2010; Pöhlker et al., 2016) (Table 1). Two sets of parameters are tested: set 1 applies to the tests employing bins for the aerosol, whereas set 2 is used for the simulations with a bulk treatment of the aerosol.

The choice of the intervals of values for the aerosol properties was made in a way that allowed for the exploration of the largest subset of realizable values of the parameters, while maintaining a reasonable computation time. For certain combinations of the size distributions parameters, the PSD can be very narrow, with a very small concentration of aerosols larger than the activation threshold. This configuration, along with a small N_a, generates clouds with a very low water content and unrealistically high supersaturations, when a bin treatment of the aerosol is used. To prevent this kind of situation, ranges were chosen as to produce averaged N_d at cloud top >10 cm⁻³, while maintaining the largest possible variety for each parameter. In order to test the response to ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}=\mathrm{0.05}$ µm, for instance, we needed to use values of N_a larger than 800 cm⁻³, and σ_a larger than 1.6. We did obtain realistic outputs from simulations with lower N_a, such as 200 cm⁻³, but only when using PSDs with larger ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a. As we needed a fully applicable parameter-space, this explains the choice of the intervals described.

Due to a deficient treatment of the activation scavenging, when a bulk treatment of the aerosol is used, the lower values of the aerosol parameters at which a reasonably dense cloud can be generated are much smaller. By not allowing the PSD to freely evolve, there is a continuous, spurious source of large aerosols that induces unrealistically high values of N_d and can destabilize the model if some thresholds for N_a, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a (900 cm⁻³, 0.08 µm and 1.9, respectively) are exceeded. Therefore, in this case, the upper and lower limits for each parameter had to be decreased.

Table 1Aerosol parameters used for the sensitivity tests using bin and bulk approaches for the aerosol: intervals for values and steps between them. For additional details, the reader is referred to the text.

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The sensitivities were calculated as the slope of the linear fit between Y and X_i in logarithmic scale for normalization:

where Y represents either N_d or D_eff, and X_i is the aerosol property affecting Y. S_Y(X_i) represents the relative change in Y for a relative change in X_i and places less reliance on the absolute measures of parameters (Feingold, 2003; Reutter et al., 2009; Ward et al., 2010). The subscript X_k indicates that when calculating the sensitivity to X_i, the other aerosol parameters are held constant. For each value at which X_k is fixed, we will obtain a new value of S_Y(X_i), i.e., we can also calculate S_Y(X_i) as a function of X_k, S_Y(X_i,X_k).

The latter differentiates our approach from previous studies. Feingold (2003) included the variability of all X≠X_i when calculating the linear regression between ln Y and ln X_i, only distinguishing the results for two subsets of N_a. Similarly, Reutter et al. (2009) analyzed the sensitivities to ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$, σ_a and κ for three combinations of N_a and w, but all values of Y calculated at a given value of X_i were averaged prior to fitting. This analysis was then expanded by Ward et al. (2010), who calculated $S_{N_{\mathrm{d}}}\left(\mathit{\kappa }\right)$ for different values of ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a used to initialize the parcel model. Now, we use a more general approach that allows us to study the responses of both N_d and D_eff to changes in each aerosol characteristic, as a function of the other aerosol parameters used to initialize the model.

The control run of the model produced a shallow cumulus that grew to a depth of 4000 m in about 30 min. Figure 2 shows the evolution of w, N_d and D_eff, characterized by the following aerosol initial parameters: N_a=800 cm⁻³, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}=\mathrm{0.08}$ µm, σ_a=1.9 and κ=0.1. The cloud top is defined as the last model level, from the surface to the top, where the N_d was larger than 1 cm⁻³. Note that there is a maximum of N_d at cloud top for all times. As droplets ascend and mix with new droplets, they grow by diffusion of vapor and collision–coalescence. As a consequence, D_eff is larger in upper levels.

Figure 2Evolution of w (m s⁻¹), N_d (cm⁻³) and D_eff (µm) in the simulation.

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The bulk phase space view is introduced in Fig. 3 to discuss the isolated effect of each parameter, when keeping the other aerosol PSD properties constant. Overall, following the cloud top in the phase space, two local maximums of N_d are found. The first one corresponds to the smallest D_eff (<5 µm) and is related to the maximum in the nucleation rate. This represents the first steps in cloud formation, where the droplets are very small and there is no significant vertical cloud development. The second one, which is also the global maximum, is reached when the cloud is deeper, as a consequence of the accumulation of droplets advected by the updraft. Regardless of the N_d fluctuations, the cloud-top D_eff shows an overall monotonic increase with altitude, except at the end of the simulation where the updraft decelerates.

Figure 3a shows the sensitivity of cloud-top DSDs to the initial concentration of aerosols. Note that an increase of N_a increases N_d for the most part, as expected. The nucleation enhancement induces a smaller D_eff due to water vapor competition, for the same liquid water content (not shown). Thus, if the water vapor amount is kept constant, the diffusional growth for each droplet is slowed. The latter manifests as a trend to the horizontal orientation in the lower portion of the trajectories in the bulk phase space, corresponding to the smallest sizes (<10 µm), where the diffusion of water vapor is the predominant droplet growth mechanism. It is interesting to note that all profiles evolve towards similar values in their maximum N_d and D_eff. This is related to a buffering effect of the entrainment. Note that the entrainment term, in the temperature, water vapor and aerosol tendency equations, is proportional to the difference of the values of those variables between the cloud and the environment. The larger aerosol content will induce the strongest modifications in the fields, thus increasing the contribution of the entrainment term. This feedback effect decreases the sensitivity of the maximum N_d and D_eff attained in the cloud to the aerosol loading. Also notable are the N_d>N_a values in the control run, which result from the vertical gradient of w shown in Fig. 2. Because the updrafts are stronger below cloud top, there is a tendency to accumulate droplets in the layers analyzed here.

Figure 3Illustration of the sensitivity of cloud top bulk properties to (a) the aerosol number concentration (cm⁻³), (b) the median radius of the PSD (μm), (c) the geometric standard deviation of the PSD (–) and (d) the aerosol hygroscopicity (–). The markers represent the averaged DSDs for the time steps when the cloud top remains at the same model level during its growth. The colors distinguish between simulations using different values of the parameter specified at the top of the graphs. The control simulation is represented by black markers in the figures.

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Note that the fraction of activated droplets in the first level is similar among all simulations in Fig. 3a (close to one-third of N_a), which is a consequence of all of the other aerosol PSD parameters being kept constant. In reality, increased pollution in the Amazon is usually followed by changes in the aerosol PSD shape, given the different properties of background and biomass burning or urban particles. Therefore, it is important to analyze the effects of every aerosol PSD parameter separately to fully understand the pollution effect in Amazonian clouds.

Figure 3b and c show the sensitivity of cloud-top DSDs to ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a while keeping the other parameters at their control standards. The effects of increasing aerosol size and PSD width are similar to the consequences of increasing N_a. By increasing ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ or σ_a, more droplets are activated due to the larger availability of aerosols with sizes above the activation threshold. Thus, nucleation increases, whereas diffusional growth decreases. The latter is visible throughout the trajectories in Fig. 3b and c.

The tests in Fig. 3 evidence a type of saturation effect for the larger values of N_a, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a tested, i.e., the sensitivity decreases as these parameters increase. This behavior is mainly explained by the consumption of water vapor supersaturation. Even if continuous water vapor supply from the surface occurs, the supersaturation can be completely consumed, depending on the aerosol availability and the diffusional growth rate. If the number of activated aerosols is able to consume all of the supersaturation, given certain z and t, an increase of its quantity will not introduce differences in the DSD.

Finally, Fig. 3d shows that the effects of varying κ are very small. Nevertheless, this is a result for one single combination of N_a, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a, i.e., the control values of the parameters; according to Ward et al. (2010), the sensitivity to κ can vary as a function of N_a and ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$. Additionally, it is known that the sensitivity to κ increases substantially as κ decreases (Petters and Kreidenweis, 2007). However, this effect is more or less evident depending on the values of the other parameters. Hence, to characterize the sensitivity of DSDs to aerosol properties, we should explore the multiparameter space composed by all combinations of discrete values of the parameters from its interval of realizable values.

To illustrate this sensitivity variation, we calculated $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left(X_i\right)$ and $S_{{\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}}\left(X_i\right)$, with X_i being N_a, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$, σ_a or κ. ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ and ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$ are the time averages of N_d and D_eff at cloud top for each simulation, respectively. From Eq. (3), $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left(N_{\mathrm{a}}\right)$, for example, is the slope of the linear fit between the values of ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ and N_a in logarithmic scale, for a given combination of ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$, σ_a and κ. The sensitivity to one aerosol parameter can then be calculated a number of times equivalent to all possible combinations of the values of the other parameters in Table 1.

Figures 4, 5, 6 and 7 show S_Y(X_i) as a function of all values of N_a, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$, σ_a and κ considered. Generally, ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ can be almost 3 times more sensitive to changes in the aerosol parameters than ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$, which stems from the mathematical definition of these physical magnitudes. For each value on the x axes of Figs. 4, 5, 6 and 7, there are several combinations of the other two parameters; as a result, there are several points for each value on the x axes in the figures.

Figure 4Sensitivities of the droplet number concentration and effective diameter to the aerosol number concentration, S_Y(N_a), as a function of (a) the median radius of the PSD (µm), (b) the geometric standard deviation of the PSD (–) and (c) the aerosol hygroscopicity (–).

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Figure 5Sensitivities of the droplet number concentration and effective diameter to the median radius of the PSD, $S_Y\left({\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}\right)$, as a function of (a) the aerosol number concentration (cm⁻³), (b) the geometric standard deviation of the PSD (–) and (c) the aerosol hygroscopicity (–).

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The impact of N_a on cloud droplets depends on the values of ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a, but does not vary with κ, as can be seen in Fig. 4. For smaller values of ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a, S_Y(N_a) reaches its maximum and presents a large dispersion. Conversely, it tends to be concentrated around a minimum sensitivity value as these parameters increase. Hence, for smaller aerosols, the relative importance of the aerosol properties can be very different from that at larger sizes.

Figure 5a shows that the sensitivity to ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ decreases for higher values of N_a and σ_a. Similar to the behavior of S_Y(N_a), the lower variability in $S_Y\left({\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}\right)$ corresponds to the values of N_a and σ_a where the absolute value of the mean sensitivity is the lowest. Conversely, the effects of κ on the sensitivity to ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ are negligible (Fig. 5c).

The same applies to the sensitivity to σ_a (Fig. 6), substituting σ_a with ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ as the independent variable in Fig. 6b. It is remarkable that the maximum absolute values of S_Y(σ_a) are higher than S_Y(N_a), $S_Y\left({\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}\right)$ and S_Y(κ) obtained here. Nevertheless, even when the values of S_Y(σ_a) indicate that σ_a has a high relative impact on ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ and ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$ for certain circumstances, we should keep in mind that the effect of varying a parameter is determined by its range of realizable values. For example, assuming that the maximum and minimum values specified in Table 1 determine the entire variation of the parameters in a given situation, it follows that a 0.6 change in σ_a (an increase ratio of 1.38) could induce a 10.6-fold increase in ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$, whereas a variation of 0.06 µm in ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ (a 2.2 increase ratio) could increase ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ 21.6 times, if we consider the maximum values of $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left({\mathit{\sigma }}_{\mathrm{a}}\right)$ and $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left({\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}\right)$, respectively. In turn, a 2800 cm⁻³ change in N_a (corresponding to a 4.5 increase ratio), would only increase ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ by a factor of 6.1 at most.

Figure 6Sensitivities of the droplet number concentration and effective diameter to the geometric standard deviation of the PSD, S_Y(σ_a), as a function of (a) the aerosol number concentration (cm⁻³), (b) the median radius of the PSD (µm) and (c) the aerosol hygroscopicity (–).

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Figure 7Sensitivities of the droplet number concentration and effective diameter to the aerosol hygroscopicity, S_Y(κ), as a function of (a) the aerosol number concentration (cm⁻³), (b) the median radius of the PSD (µm) and (c) the geometric standard deviation of the PSD (–).

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Note that S_Y(σ_a) changes its sign as ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ increases (Fig. 6b). This is related to variations in the effect of σ_a depending on the relation $\frac{r_{\mathrm{c}}}{{\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}}$. Considering a log-normal PSD, the number of aerosols for which r>r_c, i.e., the number of activated droplets, is positively correlated with σ_a if $\frac{r_{\mathrm{c}}}{{\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}}>\mathrm{1}$, and negatively correlated otherwise. If $\frac{r_{\mathrm{c}}}{{\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}}=\mathrm{1}$, the number of activated droplets does not depend on σ_a. The positive values obtained by Feingold (2003) for the sensitivity of droplet size on σ_a, as well as the negative values reported by Reutter et al. (2009) for the sensitivity of the N_d on σ_a should be due to the inclusion of larger aerosols, favoring the diminution of the r_c-to-${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ ratio.

Finally, the sensitivity to κ is the lowest of those analyzed here (Fig. 7). Note that an increase ratio of 5 in the value of κ modifies ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ by a factor of 1.38 at most. This is also consistent with its small influence on the sensitivities of the other parameters, as mentioned above. The symmetric distribution of the sensitivity with respect to the abscissas axis evidences a random impact of κ on the cloud-top DSDs here. This randomness results from the uncertainties involved in the determination of the cloud-top location, the calculation of ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ and ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$, and in the fitting procedure employed to obtain S_Y(κ), which predominates in the presence of such low values of S_Y(κ). However, it should be considered that the effects of the aerosol composition can be significantly increased in weak updraft conditions (Ervens et al., 2005; Anttila and Kerminen, 2007; Reutter et al., 2009).

Despite the limited dynamical capabilities of our 1-D framework, here we adopted a simplified approach to consider the mixing between the in-cloud and environmental properties. We considered that the column in the model is located in the center of a plume with radius R(t,z), which mixes homogeneously with the radially entrained air at each z. The entrainment affects w, the temperature, the humidity and the amount of aerosols in the column. Past studies in the Amazon have assumed that the entrainment mixing in Amazonian clouds is close to the extreme inhomogeneous case, given that the D_eff remains relatively constant horizontally (Freud et al., 2011). However, the recent studies of Pinsky et al. (2016) and Pinsky and Khain (2018) indicate that homogeneous and inhomogeneous mixing can be indistinguishable for polydisperse DSDs, especially for wide distributions. Additionally, these studies show the inadequacy of previous in situ techniques to identify the mixing type (the so-called mixing diagrams). Based on this finding, we will stick to the homogeneous case in the present study as a first approximation. Further studies would be needed to assess the effects of inhomogeneous mixing, and this comparison is beyond the scope of this paper.

Some cloud-top mixing is resolved in the model grid. However, it can be affected by the numerical diffusion and dispersion introduced by the scheme that solves the advective terms. The representativeness of the mixing induced by such an advection at cloud top must be analyzed carefully, and is outside the scope of this paper. For now, we limit our analysis to the results with and without the inclusion of some lateral entrainment rates, as a proxy for the effect of the dilution caused by mixing with the air in the neighborhood of the clouds.

By using bins for the aerosol, we allow the PSD to evolve freely. This way, after activation, the tail of the PSD can only be filled again if new particles are advected, entrained or replenished due to droplet evaporation. Furthermore, as the newly activated droplets fill several bins of the DSD, the development of wider DSDs is favored, accelerating collection processes. This method has been extensively employed (Yin et al., 2000a, b, 2005; Altaratz et al., 2008; Hill et al., 2008; Mechem and Kogan, 2008) to substitute the explicit calculation of the diffusional growth of the aerosol from its dry sizes, which has a much higher computational demand. Leroy et al. (2007) analyzed the influence of a similar assumption on the liquid and ice water content and the aerosol particles, drops and ice crystal spectra simulated by a 1.5-D model; he found notable consistency between both approaches, even when the bin resolution was strongly decreased, as well as a reasonable sensitivity to the initial aerosol spectra. We use this approach here to test the importance of including a more detailed treatment of the PSD in the model, when investigating the aerosol effect on cloud-top DSDs.

Figures 8 and 9 illustrate the behavior of the sensitivity to each aerosol parameter in three different hypothetical situations. The first column shows the results from the simulations described in the previous section, the results without entrainment are shown in the second column, and the simulations using a bulk approach for the aerosol (with entrainment) are represented in the third column. For the plots shown in the first three rows in Figs. 8 and 9, the value of κ is fixed to 0.1. The response of the sensitivities to changes in κ are not shown due to its smaller influence compared with the other parameters. The panels in the last row in Figs. 8 and 9 show S_Y(κ) at σ_a=1.9 and σ_a=1.5, for the cases with a bin and bulk treatment of the aerosol, respectively. The variations of S_Y(κ) due to changes in σ_a are similar to the variations due to ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$, which is represented on the y axes of the figures.

The values of the aerosol parameters in the tests without bins for the aerosols (third column in Figs. 8 and 9) are usually lower than in the previously discussed tests. The reason for this is that, with this configuration, when the original values of the parameters are used, there is a very high nucleation rate that leads to unrealistic values of N_d and ends up destabilizing the model. This is reasonable, considering that once the aerosol is removed from activation, the remaining unactivated aerosols are spread over all sizes, perpetuating the conditions for droplet formation. At the same time, this permits clouds to develop under conditions where there would be a negligible nucleation rate if a bin treatment of the aerosol were employed.

Figure 8Sensitivity of ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ to the aerosol properties in three different configurations of the model: with entrainment and bins for the aerosols (a, d, g, j), without entrainment (b, e, h, k) and without bins for the aerosol (c, f, i, l).

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Figure 9Sensitivity of ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$ to the aerosol properties in three different configurations of the model: with entrainment and bins for the aerosols (a, d, g, j), without entrainment (b, e, h, k) and without bins for the aerosol (c, f, i, l).

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Figure 8b, e, h and k show that, without entrainment, $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left(X_i\right)$ is lower for low values of N_a, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a, due to a faster depletion of the aerosols of suitable sizes for activation. A secondary decrease in the sensitivity is found in more polluted situations, with larger aerosols and wider sizes distributions. The latter effect is caused by the supersaturation depletion related to an increase in the amount of activating aerosols. This behavior contrasts with the responses in the entrainment case, where the lower supersaturations and the supply of additional aerosols from the environment enhance the water vapor depletion and inhibit the aerosol depletion effects.

When the entrainment is not considered, $S_{{\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}}\left(N_{\mathrm{a}}\right)$ reaches very low absolute values or even positive values for an intermediate interval of the independent variables, and increases its absolute value otherwise (Fig. 9b). The same behavior is shown for $S_{{\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}}\left({\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}\right)$ and $S_{{\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}}\left({\mathit{\sigma }}_{\mathrm{a}}\right)$ (Fig. 9e, h). The positive sensitivity evidences a less intense water vapor competition. At these points, increasing the N_a, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and/or σ_a will create more droplets, given the positive values of $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left(N_{\mathrm{a}}\right)$, $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left({\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}\right)$ and $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left({\mathit{\sigma }}_{\mathrm{a}}\right)$ discussed above, increasing w by latent heat release, and in turn the supersaturation. Thus, if the increment in the number of droplets is not as intense as required to cause a significant water vapor depletion, all of the droplets will grow in the presence of such high supersaturations, thereby increasing D_eff. Conversely, for the smallest values of N_a, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a, the sensitivity decreases its absolute value again or even becomes negative. In this situation, only the largest aerosols in the right tail of the PSD are activated. Larger drops have a slower rate of growth by condensation, and the collision–coalescence rate may also be decreased due to a lower variety of fall speeds. Thus, even at high supersaturations, the growth of these droplets can be slower. In addition, when N_a is increased and the shape of the distribution is maintained, the largest increments in the amount of aerosol occur near the center of the PSD (mode values). Therefore, the leftmost sizes in the right tail will be favored, leading to a decrease in D_eff after activation. If the droplet growth rate is not intense enough to balance this trend, it will result in negative sensitivity.

Figure 10Mean and standard deviation of ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ and ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$ at cloud top from the simulations with entrainment and bins for the aerosols (a, b), without entrainment (c, d) and without bins for the aerosol (e, f).

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In Figs. 8c and 9c it can be seen that, when a bulk approach is used for aerosols, the absolute value of S_Y(N_a) increases monotonically as ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a increase and is not affected by the supersaturation depletion; this is due to that fact that independently of r_c, there will always be a certain amount of aerosol such that r>r_c.

Furthermore, the absolute value of $S_Y\left({\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}\right)$ increases for higher values of N_a, which coincides with the results of Feingold (2003) and Rissman et al. (2004), and for lower values of σ_a (Figs. 8f, 9f). The same applies to S_Y(σ_a) (Figs. 8i, 9i), substituting σ_a by ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ as the independent variable. However, the maximum value of the sensitivity to the size-related parameters is significantly decreased compared with the simulations with a bin treatment of the aerosol. $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left({\mathit{\sigma }}_{\mathrm{a}}\right)$ even reaches slightly negatives values for the larger ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ in these tests (not shown), which is related to the previously mentioned variations in the effect of σ_a depending on the position of r_c with respect to the size distribution function.

Finally, it can be observed in Figs. 8l and 9l that, when the model uses a bulk approach for aerosol species, S_Y(κ) is larger for higher N_a and smaller ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$, which is in agreement with the results of Ward et al. (2010). The figures evidence that ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ and ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$ are much more sensitive to κ when considering a bulk approach for the aerosols than when its size distribution is explicitly represented in the model. Note that, in the former case, S_Y(κ) can be about 50 % of S_Y(N_a), which is a significant influence. However, perhaps the most relevant difference between these simulations and those using bins for the aerosol is the change in the sign of S_Y(κ). Although, at first, higher values of κ would determine a smaller r_c, it also contributes to a faster depletion of the larger aerosols, leading to a reduction in the nucleation rate afterward. That is the cause of the negative (positive) values of $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left(\mathit{\kappa }\right)$ ($S_{{\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}}\left(\mathit{\kappa }\right)$) obtained in the tests using bins for the aerosol (Figs. 8k, 9k). Conversely, the latter has no effect on the results when the PSD is fixed, and, therefore, positive (negative) values of $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left(\mathit{\kappa }\right)$ ($S_{{\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}}\left(\mathit{\kappa }\right)$) are obtained.

Overall, our analysis shows that increases in N_a, ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ and σ_a produce higher ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ (positive sensitivity) and smaller ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$ (negative sensitivity) when both entrainment and aerosol bins are included in the simulations. This coincides with the results of Cecchini et al. (2017), who found cloud-top averages of $S_{{\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}}\left(N_{\mathrm{a}}\right)$ and $S_{{\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}}\left(N_{\mathrm{a}}\right)$ of 0.84 and −0.25, respectively, from aircraft measurements over the Amazon forest.

The values of sensitivities reported by Feingold (2003), Reutter et al. (2009), and Ward et al. (2010) are included in the range of sensitivities obtained here, in addition to the variability added by the diverse universe of situations found over the aerosol parameter space. However, comparisons between our results and previous research are not straightforward, considering the influence of the cloud evolution here. For example, the rate of depletion of the aerosol and the water vapor will determine how much nucleation will occur above cloud base. A large supersaturation can initially cause a fast activation rate, but will decrease the intensity of this process afterwards. The response to changes in the aerosol properties in this case might be different from a case with a moderate and more spatially distributed activation rate. In the simulations with a bulk treatment of the aerosol, the aerosol depletion is slower. Thus, for a certain time interval, each cloud-top level behaves like an independent cloud base regarding the intensity of the nucleation. This explains the similarities between the sensitivities obtained from the simulations using a bulk approach for the aerosols and those from previous research.

From our analysis, it turns out that ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$ is the most influential parameter that determines the sensitivity to aerosols at cloud top, in contrast with the importance that has been conventionally attributed to the N_a. To further illustrate this, Fig. 10 shows the mean and standard deviation of ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ and ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$ for each value of N_a tested, in each of the above referenced situations: with entrainment and bins for aerosols (a–b), without entrainment (c–d) and without bins for aerosols (e–f). The length of the standard deviation bars reflects the changes in ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}$, σ_a and κ.

For the first (and most complete) situation considered, it can be seen that the state of the system is not sufficiently determined by N_a, especially if the PSD is displaced to smaller radii (Fig. 10b). For instance, when increasing N_a by a factor of 3 in Fig. 10b, from 800 to 2400 cm⁻³, there is still some overlapping between the corresponding standard deviation bars in the phase space. However, the bars are significantly smaller if larger aerosols are considered (Fig. 10a), indicating a tendency to approach the generally accepted knowledge, i.e., increasing the importance of N_a in determining the characteristics of the DSDs. These results highlight the importance of including the PSD characteristics in aerosol–cloud interaction studies, especially when ${\stackrel{\mathrm{‾}}{r}}_{\mathrm{a}}\le \mathrm{0.08}$ µm. These parameters can produce changes in the DSD as large as those caused by changes in the N_a. These findings are also relevant given the current discussion about the importance of ultrafine aerosol particles in the development of deep convective clouds over the Amazon (Wang et al., 2016; Fan et al., 2018).

In turn, Fig. 10c–d show that, when the entrainment is not considered in the model, the variability of ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ and ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$ does not present a significant dependence on the aerosol size; it is a function of ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ and ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$ on their own. In other words, the location of the points in the phase space determines their standard deviation. Points located in the upper left corner in Fig. 10c, for instance, have approximately the same standard deviation as points in the same location in Fig. 10d. The difference between both graphics resides in the position of the points: for smaller aerosols (Fig. 10d), ${\stackrel{\mathrm{‾}}{N}}_{\mathrm{d}}$ will be lower and ${\stackrel{\mathrm{‾}}{D}}_{\mathrm{eff}}$ will be higher, than for large aerosols (Fig. 10c).

Conversely, the results indicate that the importance of N_a may be overestimated if a bulk treatment of the aerosol is employed (Fig. 10e–f). In this case, it can be seen that there is a reduction of the overlap between the standard deviation bars, especially for larger and sparser aerosols.

The simulations performed here represent an idealized cloud resulting from observed humidity and temperature profiles. However, even if we assume that it represents a realizable situation, corresponding to an average behavior, it does not include the variety of possibilities existing in real cases. Important processes such as turbulent entrainment and dynamic feedbacks can introduce a significant departure from the idealization we are considering. Full dynamical models account for dynamics feedbacks and several subgrid processes that could enhance or reduce the range of sensitivities that are demonstrated here. Nevertheless, the qualitative behavior of our main results, i.e., the dependency of the DSD sensitivity to the aerosol properties according to its position in the full parameter space, might not change. For example, Gettelman (2015) simulated several warm rain cases using the KiD model and climatological cases with a global model, utilizing a double-moment microphysics scheme, in order to analyze the sensitivity of the aerosol–cloud interaction to cloud microphysics. They found that the tests in the KiD model were consistent with the global sensitivity tests. This is an aspect we intend to study in a following work, in order to build on the present results.

The code for the modifications introduced in the KiD model, as well as in the TAU scheme, can be made available upon request from the corresponding author.

LHP performed the model simulations, the model–data analysis and prepared the paper. LATM and MAC provided guidance regarding the definition of the model initial conditions. LATM, MAC and MSG provided guidance regarding the choice of the variables and the interval of values and the model–data analysis. All authors contributed to the design of the study and the preparation of the paper.

The authors declare that they have no conflict of interest.

This research has been supported by the São Paulo Research Foundation (grant nos. 2015/14497-0, 2016/24562-6, and 2017/04654-6).

This paper was edited by Barbara Ervens and reviewed by two anonymous referees.