Quantification of the Radiative Effect of Aerosol–Cloud Interactions in Shallow Continental Cumulus Clouds

a. Cloud averaging and inhomogeneity

To quantify the radiative effect of covariation between ACI and other background meteorological conditions in our simulations, we need to link the cloud microphysics on the smallest simulated scales to the total cloud radiative effect on the larger cloud-field scale. Many improvements to our quantitative understanding of the interactions between clouds and radiation have been made through use of the two-stream approximation to radiative transfer (Schuster 1905). An early example is Neiburger (1949), who showed that a stratus cloud’s albedo should approach unity at a particular rate as the thickness of the cloud becomes large. The cloud albedo is defined as the fraction of incoming radiation that is scattered by clouds, integrated over the upward and downward hemispheres. Meador and Weaver (1980) gave a general expression for the two-stream approximation solution for the relationship between cloud albedo α and cloud optical thickness τ₀ in a broadband solar shortwave sense:

where γ depends on the degree of forward scattering, and it is assumed that radiation is only scattered and not absorbed, clouds are horizontally homogenous and in a single layer, the underlying surface is black, and the sun is at the zenith.

Since the sun is typically some angle θ from the zenith, we replace τ₀ in Eq. (2) with τ ≈ τ₀/cosθ, as in Bohren (1987). Sagan and Pollack (1967) showed that the amount of scattered light over a full hemisphere can be approximated simply by the amount of scattering at a particular angle near 55° and −55° from the normal that defines the forward and backward hemispheres, respectively, giving γ = 2/(1 − g), with the scattering asymmetry parameter approximated as a constant g = 0.85 for liquid cloud drops at visible wavelengths (Hu and Stamnes 1993).

In essence, our analysis uses Eq. (2) to quantitatively link a radiative descriptor (albedo) to a descriptor of cloud liquid microphysics, for which we use optical depth τ₀. This framework was developed in Liu et al. (2011), and includes the concept of the relative cloud radiative effect (rCRE) after Betts (2007). The rCRE is intended to quantify the radiative effect of clouds in Earth’s atmosphere with minimal sensitivity to radiative variability caused by noncloud effects. The general form of this rCRE-based framework has been used by Xie and Liu (2013), Seifert et al. (2015), Sena et al. (2016), and Feingold et al. (2017) to give quantitative estimates of the magnitude of radiative-microphysical relationships. But this framework, even while idealized, is subject to the large horizontal inhomogeneity albedo bias described and corrected for in Cahalan et al. (1994), but neglected in these previous works. We apply this correction to the rCRE-based analysis framework here as follows.

As in Liu et al. (2011), the shortwave rCRE for a domain of interest is defined as the fraction of incoming shortwave radiation reflected by clouds (the average cloud albedo ${\overline{\alpha }}^c$) multiplied by the fraction of the domain occupied by those clouds:

where we note the overline with a “c” indicates that an average is performed over all points defined as cloudy (not the full domain); that is, for any variable x, ${\overline{x}}^c=\overline{x\left(\tau >1\right)}$. We take the area of this same spatial region in our LES where τ = τ₀/cosθ > 1, and divide by the domain area to define the cloud fraction f. The threshold used for this definition is equivalent to a constant threshold of cloud radiative effect, similar to defining cloud shadow edges using a constant cloud shadow darkness threshold. This is most consistent with the LES radiation calculations, and therefore the calculated rCRE.

To evaluate Eq. (3), we must address the assumptions used in Eq. (2) to calculate the average cloud albedo. Liu et al. (2011) gave a slightly more complex expression than Eq. (3) that relaxed the assumption of nonzero surface albedo and nonzero cloud absorption, and this yielded better comparison between surface measurements of the shortwave rCRE and satellite and model output. Liu et al. (2011) also outlined a method to treat multilayer clouds as an “effective” single-layer cloud, as in Ramanathan (1987). This effective cloud layer can account for vertical heterogeneity, that is, multilayer stratiform clouds, but does not account for horizontal inhomogeneity in a single layer.

The problem of assuming horizontal homogeneity is the same as the “plane-parallel albedo bias” problem, which was explained using a simple model of the fractal nature of stratocumulus by Cahalan et al. (1994). The horizontally averaged albedo of a cloud field cannot be uniformly related to the horizontally averaged optical depth without knowing the subaveraging-scale distribution (horizontal inhomogeneity) of the cloud layer. Cahalan et al. (1994) showed that fields of clouds are not randomly inhomogeneous, but have characteristic spatial distributions of properties that are well described by fractals.

Our simulations approximately resolve the scale at which cloud optical depth homogeneity can be assumed (LES columns of 100 m × 100 m horizontal) over a domain large enough to include many clouds with varying optical depths where cloud heterogeneity cannot be ignored (24 km × 24 km). We take the average of Eq. (2) and define an inhomogeneity correction factor c₁ after Cahalan et al. (1994):

This measure of the horizontal inhomogeneity can be calculated at any particular time in the simulations as ${\log }_{10}\left(c_1\right)={\log }_{10}\left({\overline{{\tau }_0}}^c\right)-{\overline{{\log }_{10}\left({\tau }_0\right)}}^c$. Note the cosine of the solar zenith angle is a constant over the whole domain and is factored out of the average without affecting c₁. In general, c₁ will only equal 1 if the distribution of optical depth values τ₀ being averaged is monodisperse.

A similar “inhomogeneity” problem is encountered when we attempt to use the idealized relationship between optical depth τ₀, cloud liquid water path L, and cloud drop number concentration N expressed by Boers and Mitchell (1994), Zhang et al. (2005), and others, which is valid at some idealized subcloud scale. Once again, to apply this relationship to the large scale would assume cloud horizontal homogeneity, so we approximate the cloud average as

where L is in grams per meter squared, N is cloud drop number per volume of the cloudy part of the column (cm⁻³), and $c_{T,p}$ is a function of temperature and pressure in an adiabatic case. Following Marshak et al. (2006), a reduced constant factor c₀ is likely to be the best correction for this type of averaging bias, as opposed to changing the magnitude of the powers on L or N. We obtain τ₀ for the lhs of Eq. (4) and the ${\overline{L}}^c$ and ${\overline{N}}^c$ for the rhs from the gridscale microphysics in each LES, averaging over all grid columns with τ > 1 at each minute from the 12 selected simulations when cloud exists (over 100 h). We find that setting c₀ = 0.07 for all times in our dataset minimizes the mean absolute error between the lhs and rhs of Eq. (5) at 1.35 in units of optical depth. Given values of optical depth typically larger than 10, we used Eq. (5) to evaluate the rCRE in Eq. (4), substituting ${\overline{{\tau }_0}}^c$ with ${\overline{L}}^c$ and ${\overline{N}}^c$, giving a typical mean absolute error in rCRE of a few percent due to this assumption at any particular time in our analysis. The total error in approximating rCRE using Eqs. (3)–(5) is reduced further by setting c₁ to a best-fit value at each particular time in our analysis (see section 4c).

b. Mutual information

In parallel with using the equations defined above, we also quantify covariability between meteorological processes affecting cloud amount and cloud brightness using a statistical technique. Based on the understanding of information as entropy (Shannon 1948), one can measure the mutual information (MI) between two variables. Following Dawe and Austin (2013),

where P(A), P(B), and P(A, B) are the marginal and joint probability density functions (PDFs) for the variables A and B. For example, if P(A) and P(B) are perfectly independent, the mutual information equals zero, and if perfectly correlated or anticorrelated, the MI is one.

An advantageous statistical property of MI is that it does not assume a functional form for the relationship between variables, so it measures linear and nonlinear relationships equally well. Another advantage is that we can calculate the MI between some variable A and B, but first condition the PDF of B on a third variable C. This is equivalent to finding the MI between A and B along surfaces of constant C, revealing the information A and B share with each other while the variability that is explained by C is effectively removed. This is called the conditional mutual information (CMI):

For example, if the MI between A and B is independent of C, the MI and the CMI will be equal to each other, indicating that C does not contain significant information about the A–B relationship. But if some variability in C matches the pattern of variability in B (either a correlation or anticorrelation), the CMI will be smaller than the MI due to C containing information about A that is effectively redundant given B. Finally, if the CMI is larger than the MI by some amount, this indicates the amount of unique information about A that has been added by C in addition to B. Similar to a traditional budget analysis, we will use this CMI analysis to find which variables are needed to reproduce the total rCRE variability, and to quantify how much information each variable contributes.

a. Overview of variability

Substituting Eqs. (4) and (5) into Eq. (3) shows that the rCRE is a function of f, ${\overline{L}}^c$, and ${\overline{N}}^c$, but also of the solar zenith angle θ and the degree of horizontal inhomogeneity c₁. The domain-scale variation in these properties (from time step to time step, and between different days) comes from the net result of variation in the domain-scale meteorological forcing that drives the simulations. The variation of ${\overline{N}}^c$ relative to the variation of $\overline{N_A}$ throughout a day, or between days, is also the net result of variation in the domain-scale meteorological forcings. We will use the term “meteorological variability” to indicate any meteorological properties indicative of variability in this specific context of ${\overline{N}}^c$ relative to $\overline{N_A}$ variation, that is, meteorological variability directly related to ACI. Examples of meteorological properties directly related to ACI could include the cloud-base updraft speed, the vertical gradient of the aerosol concentration, whether the cloud is entraining air with much lower aerosol concentration than at cloud base, etc. The goal of this work is not to determine the most important meteorological variables, but to develop a framework in which the radiative effects of such variability can be quantified, allowing for future work to rank ACI-related meteorological processes by their radiative effect.

The net result of other variability in meteorological processes realized in our simulations will be referred to as “cloud field property variation.” Examples of meteorological variations that are realized as cloud field property variations are the various input soundings and advective tendencies that drive the simulations that lead to different values of ${\overline{L}}^c$ and f. The amount that the inhomogeneity c₁ is less than 1 is proportional to the width of the distribution of τ₀ that is averaged in ${\overline{{\tau }_0}}^c$. Further investigation will be needed to determine if variation in the value of c₁ for the same ${\overline{{\tau }_0}}^c$ is more likely ACI-related meteorological variability (i.e., variation in N for the same ${\overline{N}}^c$) or cloud field property variability (i.e., variation in L for the same ${\overline{L}}^c$). Based on the form of Eq. (5), we suspect cloud field property variability contributes to most to c₁ variability. In this section, we will attempt to quantify the radiative effects of ACI-related meteorological variability relative to the concurrent radiative effects of cloud field property variability. In Fig. 5 we quantify and illustrate the range of variation exhibited by rCRE, f, ${\overline{L}}^c$, and ${\overline{N}}^c$ for the 12 selected simulations.

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Summary of simulation results in rCRE vs cloud fraction f state space: (left) probability density in grayscale, (center) the mean ${\overline{N}}^c$ at the times when simulations occupy each (rCRE, f ) space, and (right) the same, but for the mean ${\overline{L}}^c$.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0269.1

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Based on Eq. (3), the variability along a line of constant f in Fig. 5 shows the extent to which rCRE varies with ${\overline{\alpha }}^c$. The left panel of Fig. 5 shows that the simulations span an albedo range from about 30%–60%. The cloud fraction ranges from 0% to just above 50%, although noting the probability density scale, a cloud fraction below 20% is most common. The middle panel shows the same distribution, but now colored by the average of all of the ${\overline{N}}^c$ values; and the rightmost panel shows the same but colored for the average of ${\overline{L}}^c$ values. It may appear from Fig. 5 that variability in cloud albedo is more strongly related to variability in ${\overline{L}}^c$ than to ${\overline{N}}^c$. However, we cannot conclude which pattern of variation ultimately contains more information about the rCRE variation without considering covariability between ${\overline{N}}^c$, ${\overline{L}}^c$, and f.

b. Mutual information

Which variables ultimately contain the most information about the total rCRE? To quantify these relationships, we calculate the mutual information between the rCRE and the f, ${\overline{L}}^c$, and ${\overline{N}}^c$ variables from all 12 simulated days, at 1-min resolution when clouds exist. To obtain confidence intervals, we randomly (with replacement) subselected 62% of our dataset, bootstrapping in this way 1000 times. For a measure of the signal strength, we find the mutual information between a random permutation of the rCRE variable (effectively noise) and f, ${\overline{L}}^c$, and ${\overline{N}}^c$, and report these results as MI^* and CMI^*. This gives a measure of the baseline signal above which we have more confidence in the result. The results of the calculations are shown in Table 1. The first column of Table 1 quantifies what can be seen qualitatively in Fig. 5: that the current value of rCRE is strongly related to the current value of f, less so for ${\overline{L}}^c$, and even less so for ${\overline{N}}^c$. That is, over all simulations, 51% of the information content of the rCRE variation is shared with the f variability, 18% with the ${\overline{L}}^c$variability, and only 11% with the ${\overline{N}}^c$ variability. These information contents are not necessarily unique sets of information about rCRE.

Table 1.

The mutual information (MI) and conditional MI (CMI) are calculated for the rCRE, cloud fraction f, all-cloud-averaged liquid water path ${\overline{L}}^c$, and all-cloud-averaged drop number concentration ${\overline{N}}^c$; shown are mean ± 1 std dev confidence intervals generated by 1000 bootstrap subselections with replacement. The asterisk indicates that the rCRE variable is first randomly permuted; i.e., it gives the signal obtained when this method is used on noise (signal/noise threshold). The difference between CMI and CMI* gives a measure of the baseline signal above which we may have more confidence in the result.

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To understand this further, we calculate the CMI in the second column of Table 1. Conditioning the MI between rCRE and f on ${\overline{L}}^c$ results in a CMI of 70%, similar to the unconditioned MI between rCRE and f of 51% added to the unconditioned MI between rCRE and ${\overline{L}}^c$. This indicates that knowing the covariation between ${\overline{L}}^c$ and f gives no additional information about rCRE if ${\overline{L}}^c$ and f are already known. In contrast, conditioning the MI between rCRE and ${\overline{L}}^c$ on ${\overline{N}}^c$ increases the shared information content to 50% compared to the MI of either ${\overline{L}}^c$ or ${\overline{N}}^c$ alone. This is greater than the 18% + 11% that might be expected as a maximum CMI for these two variables. This indicates that not only does ${\overline{N}}^c$ contain unique information about the rCRE, but even more so if ${\overline{L}}^c$ is known; that is, the covariation between rCRE and ${\overline{L}}^c$ acts to mask the strength of the rCRE–${\overline{N}}^c$ relationship. Most notably, the maximum information about the rCRE from two variables is obtained from the pair of (f, ${\overline{N}}^c$), with a CMI of 75%. This relationship also shows that the pattern of covariation between rCRE and f acts to mask the strength of rCRE–${\overline{N}}^c$ relationship.

What does it mean for one variable to mask the dependence of another? This can be understood by considering Fig. 5 to be a 3D landscape, with a given (f, ${\overline{\alpha }}^c$) coordinate defining a particular rCRE, and movement in a given direction describing a change in the rCRE. From the differential analysis, one might have expected the ${\overline{L}}^c$ variable to give the best approximation to ${\overline{\alpha }}^c$. Indeed, they do have a strong relationship, but the pattern of variation on this landscape of cumulus clouds is such that much of what the ${\overline{L}}^c$ tells us about ${\overline{\alpha }}^c$ can be implied from knowing both f and ${\overline{N}}^c$. Put another way, instead of holding ${\overline{L}}^c$ constant, we used MI to account for ${\overline{L}}^c$ variation by its covariation with f, and the remaining variation in rCRE was even better explained by ${\overline{N}}^c$. Holding ${\overline{L}}^c$ constant is not the same as holding all cloud field property variability constant; there is some other background meteorological variability that acts to mask the strength of the relationship between ${\overline{N}}^c$ and rCRE.

c. Contributions to albedo variability

To investigate which sources of cloud field property variation are affecting rCRE in this way, we normalize the rCRE variability by f. From Eq. (3), this means we only consider variability in rCRE due to variation in ${\overline{\alpha }}^c$. In the top row of panels in Fig. 6, we plot the ${\overline{\alpha }}^c$ versus ${\overline{L}}^c$, and along any particular slice of ${\overline{L}}^c$, colors are used to show the variability in ${\overline{\alpha }}^c$, ${\overline{N}}^c$, and $\overline{N_A}$. The top right panel of Fig. 6 illustrates that some of the ${\overline{\alpha }}^c$ variability is likely due to the increased slant path of radiation through clouds at different solar zenith angles.

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The average cloud albedo ${\overline{\alpha }}^c$ (y axis) vs (top) average cloud liquid water path ${\overline{L}}^c$ (x axis) and (bottom) ${\overline{L}}^c/\hspace{0.17em}\cos \theta$ (x axis). Coloring by the average ${\overline{N}}^c$ and $\overline{N_A}$, and normalizing the x axis by cosine of solar illumination angle θ (cf. top- and bottom-center panels), shows a change in the apparent variability of relationships between ${\overline{\alpha }}^c$, ${\overline{N}}^c$, and $\overline{N_A}$. The inhomogeneity c₁ could explain further variability.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0269.1

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To understand the radiative effect of ACI over the 12 simulated days, we compare the change in albedo from different simulations having different ${\overline{N}}^c$ and $\overline{N_A}$ while holding all else equal. We can approximately account for the radiative effect of illumination angle variability by binning our data by ${\overline{L}}^c/\hspace{0.17em}\cos \theta$ (bottom row of panels in Fig. 6). This represents how we expect the radiative effect of a long slant path through a small ${\overline{L}}^c$ to be equivalent to a shorter path through a larger ${\overline{L}}^c$. The bottom right panel of Fig. 6 suggests that variation in the distribution of cloud optical depths being averaged (inhomogeneity c₁) also contributes to variation in average cloud brightness. The form of Eq. (4) shows the dependence of ${\overline{\alpha }}^c$ on c₁, so we can equivalently remove the effect of this variability by binning by ${\overline{\alpha }}^c/c_1$ (Fig. 7). In this case, the signal of drop concentration variability on cloud albedo becomes even more clear. The surface aerosol signal still appears somewhat inconsistently related to the ${\overline{N}}^c$. But now this variation approximately represents variability in ACI, because Fig. 7 illustrates the variation in albedo when variations due to cloud field heterogeneity are approximately removed.

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The distribution of ${\overline{N}}^c$ and $\overline{N_A}\hspace{0.17em}$ over normalized metrics of average cloud brightness (y axis) vs average cloud amount (x axis). Normalization by heterogeneity of the cloud field and the cosine of the solar zenith angle is intended to isolate the variability in albedo due to ACI between ${\overline{N}}^c$ and $\overline{N_A}$.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0269.1

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d. Contributions to rCRE variability

We quantify the total response of the rCRE to an aerosol-related cloud drop number perturbation, similar to Platnick and Oreopoulos (2008), Quaas et al. (2009), and others, by differentiating Eq. (3) with respect to ${\overline{N}}^c$. To numerically separate variation in rCRE due to ${\overline{\alpha }}^c$ from fluctuations in f, we first take the natural log of both sides of Eq. (3), separate terms using the logarithmic product identity, and then perform differentiation:

Equation (8) says that the total change in the rCRE for a change in ${\overline{N}}^c$ can be expressed mainly by three different terms. It is not exact because the approximations in Eqs. (4) and (5) are not exact. The first term on the rhs of Eq. (8) is the partial derivative of Eq. (4) and could be approximated by the increase of albedo with ${\overline{N}}^c$ along lines of constant ${\overline{L}}^c$ in Fig. 6. But some of the variability in albedo with ${\overline{N}}^c$ at constant ${\overline{L}}^c$ is due to variability in cosθ and c₁. Thus, the partial change in ${\overline{\alpha }}^c$ due only to the radiative effect of ${\overline{N}}^c$ variability is better approximated by Fig. 7, that is, after normalization has been performed.

Similarly, the second term on the rhs of Eq. (8) is the partial change in ${\overline{\alpha }}^c$ due only to the radiative effect of ${\overline{L}}^c$ variability, scaled by the actual concurrent change in ${\overline{L}}^c$ with the ${\overline{N}}^c$ variation. As before, if we attempt to compare the change in ${\overline{\alpha }}^c$ for some ${\overline{L}}^c$ on one day to the change in albedo for the same ${\overline{L}}^c$ on a different day, we must normalize by the inhomogeneity c₁ and solar zenith angle cosθ, which also likely vary between days with the same ${\overline{L}}^c$. The final term on the rhs of Eq. (8) is the total response of the cloud fraction concurrent with the domain-sensible perturbation of the ${\overline{N}}^c$. Like the ${\overline{L}}^c$ contribution term, this f term represents cloud field property variation occurring at the same time as the ACI-related cloud drop number variation.

We are motivated to quantify the cloud field property variation to understand if it is large enough to overwhelm the signal of an aerosol indirect effect. To quantify the concurrent contributions of each variable to the rCRE budget, we perform temporal numerical differentiation to approximate the values of the terms in Eq. (8). The results of temporal numerical differentiation can depend sensitively on the method and time scale used, so care must be taken. We quantify the relative contributions of changes to cloud brightness and amount due to both aerosol variability and concurrent background meteorological variation on a time scale relevant for shallow cumulus. This implies a time scale slightly shorter than the characteristic time scale of mixing aerosol into the clouds, or about 5 min. We have used 3, 5, 10, 15, 20, and 30 min for the differentiation interval to examine the sensitivity of this method. For the shorter time scales, the results are qualitatively similar, but starting around 15 min, the time scale is too long to resolve the relatively fast changes in the cumulus layer, and temporal differentiation begins to describe the differences between different cloud-field states, instead of quantifying the magnitude of the changes that occur as the field evolves. Here we show the results for model output evaluated using a 5-min differentiation time scale. We used the simple form of “forward” numerical differentiation:

where i indicates a particular time and δ is the temporal differentiation interval. In practice we do this for each term on the rhs of Eq. (8) as well as the lhs, but we omit this from Eq. (9) for clarity. A statistical summary of the range of values found for the differential terms in Eq. (8) is shown in the left panel of Fig. 8. We also performed differentiation as in Seifert et al. (2015) and previous work, letting contributions from inhomogeneity c₁ contribute to calculated changes in albedo by always setting c₁ = 1 (Fig. 8, right panel).

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The change of the rCRE concurrent with changes in ${\overline{N}}^c$ (solid colors) summarized for the 12 simulated days of cumulus at SGP, accounting for inhomogeneity c₁ and letting c₁ = 1 (dashed gray). The Twomey contribution is the partial change in the albedo due only to changes in ${\overline{N}}^c$ (cyan). The contribution to the albedo change attributable to changes in ${\overline{L}}^c$ is in orange, and the partial change in rCRE due only to f variation is in green. Box and whiskers show the median, interquartile, and 95th/5th percentile.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0269.1

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The terms in Fig. 8 are unitless log-relative changes; that is, for the total response of the rCRE to a perturbation of ${\overline{N}}^c$ (gray), a value of 100% indicates that a doubling in ${\overline{N}}^c$ is accompanied by a doubling in rCRE, whereas −100% indicates that a doubling in ${\overline{N}}^c$ is accompanied by a halving of rCRE. We see that for the total response of the rCRE (gray), the median is small and positive, but the collection of all simulated rCRE responses covers a wide range and can be negative as well as positive. This is consistent with Sena et al. (2016), who found for stratocumulus over SGP that correlations between rCRE and a measure of surface aerosol concentration are just as likely to be positive as negative.

By comparing the left panel with the right panel of Fig. 8, we see that ignoring the variability of inhomogeneity c₁ manifests primarily as a wider range of partial albedo changes associated with changes in ${\overline{L}}^c$. By accounting for c₁ we have increased the accuracy of quantification of the average cloud albedo, allowing us to better isolate the effect of varying aerosol from other processes. This illustrates how detectability of the radiative effect of changes in ${\overline{N}}^c$ is increased by accounting for inhomogeneity c₁.

e. ACI and meteorological variability

Now that we have quantified the radiative effect of variation in ${\overline{N}}^c$, we can investigate the effect of meteorological conditions leading to different amounts of ${\overline{N}}^c$ being realized for the same prescribed aerosol $\overline{N_A}$, and thus we gain understanding of the radiative effect of meteorological covariation with ACI. We can express the albedo effect of aerosol concentration variation mathematically as

and use the final term on the rhs as a metric of ACI. In Fig. 9 we plot the ${\overline{N}}^c$ versus $\overline{N_A}$ for all 12 simulations; each point is a snapshot every 1 min when clouds are present. In the top left panel, we include dashed lines to indicate potential constant values of $d\ln {\overline{N}}^c/d\ln \overline{N_A}=\hspace{0.17em}\text{ACI}$. We color the points by a set of meteorological fields of interest: ${\overline{L}}^c$, the vertical velocity averaged over cloud-base points, the inhomogeneity c₁, and the decoupling index. The decoupling index is the difference between the lifted condensation level (LCL) and the actual cloud base, normalized by the LCL altitude. A small value of the decoupling index indicates a cloud likely well connected to the surface, and should have higher correlation between ${\overline{N}}^c$ and $\overline{N_A}$. Because aerosol activation is a function of updraft velocity, we expect stronger average cloud-base updraft speeds to be associated with higher values of ${\overline{N}}^c$ for the same $\overline{N_A}$. Variability in the inhomogeneity c₁ and ${\overline{L}}^c$, which represents cloud field property variation, are included for comparison.

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The ${\overline{N}}^c$ vs $\overline{N_A}$ for 12 simulated days. Each point is the average from an individual LES snapshot. The top-left panel includes dashed lines indicating different slopes for the ACI relationship expressed in ${\overline{N}}^c$ vs $\overline{N_A}$ space. Color shading for variables that reflect the meteorological state in the LES are given to illustrate the potential role for meteorological variability to explain some of the ACI variability.

Citation: Journal of the Atmospheric Sciences 77, 8; 10.1175/JAS-D-19-0269.1

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Clusters of points on Fig. 9 approximately represent different simulated days, although there are some regions of overlap. Different days appear to realize very different values of the ACI metric. Some points that have higher $\overline{N_A}$ for the same ${\overline{N}}^c$ also have higher decoupling index values. This could be part of the explanation for the lack of correlation between ${\overline{N}}^c$ and $\overline{N_A}$, though we see from the pattern of variation with inhomogeneity and cloud-base vertical velocity that these factors could contribute as well.

One might expect a maximum value of the ACI metric to be a slope of 1, but only if we were considering some idealized subcloud scale, for example, d lnN/d lnN_A. The appearance of slopes steeper than one or negative on Fig. 9 is representative of the complex interactions between the domain-scale quantities ${\overline{N}}^c$ and $\overline{N_A}$. A process potentially exhibiting strong control of ACI here is the mixing of upper-tropospheric air into clouds rooted in the boundary layer. From Fig. 3, we know some clouds can entrain and mix with free tropospheric air that has an aerosol concentration much less than the value at the surface, even if the boundary layer is well mixed. This would lead to a pattern on Fig. 9 of the $\overline{N_A}$ changing slowly, while the ${\overline{N}}^c$ changes more abruptly, often decreasing sharply, leading to steep slopes and a large value of the ACI metric.