The case study is based on scattered, isolated convective clouds that developed over Houston, Texas, on 19–20 June 2013, following the ACPC MIP simulations . During the daytime, the heating over the land develops a pressure gradient between the land and ocean. The associated afternoon sea breeze front triggers scattered convection by disturbing conditionally unstable layers. This study uses the NASA-Unified Weather Research and Forecasting (NU-WRF) model configuration that was also used in the ACPC MIP study as a basis ; however, the domains, grid spacing, and aerosol concentrations are revised in order to investigate cumulus thermals.

This case study utilizes a single large domain (998×998 horizontal grid cells) with 250 m horizontal grid spacing and without nesting (Fig. ). This type of domain setting exceeds the traditional downscaling ratio (1:3 1:5), resulting in reduced precipitation forecasting skill compared to multi-nested domains. However, it successfully generates thermal bubbles of isolated convection driven by sea breeze circulation for a given computational resource. The analysis is focused on the scattered convection that occurs due to mesoscale circulations within the domain. Vertical grid spacing stretches from approximately 50 m near the surface to 300 m near the 4 km level, with 96 vertical levels. The model top is approximately 22 km (50 hPa). The planetary boundary layer (PBL) parameterization was turned off, and only the turbulent kinetic energy (TKE) scheme is used; we found that the TKE scheme with the PBL scheme at this resolution unphysically suppresses the number of cumulus thermals within the middle of boundary layer (not shown). Other physics options include the new Goddard radiation scheme , Noah-MP land surface model, and Predicted Particle Properties (P3) scheme with a single ice species.

The P3 scheme predicts the mass and number concentrations of cloud droplets, raindrops, and ice particles, and additional tracers (rime mass and volume) are also predicted to better characterize ice properties . Aerosol activation follows , using the minimum supersaturation from (their Eq. A10). Based on regional observations , aerosol profiles spanning the boundary layer (up to 2500 m a.g.l. – above ground level) are stratified in the eight sensitivity experiments from relatively clean continental (500 cm-3) up to polluted conditions (4000 cm-3), increasing by 500 cm-3 for each sensitivity experiment. Aerosol is specified as a single-mode lognormal distribution with a fixed mean diameter (100 nm), lognormal distribution width (1.8), and hygroscopicity parameter (0.2). As in , aerosol transport (resolved and subgrid), activation, removal by droplet coalescence, and regeneration from droplet evaporation follows the method in , while the aerosol impact on ice nuclei is not considered. The polluted and clean aerosol size distributions and vertical profiles were based on the data from Deriving Information on Surface conditions from Column and Vertically Resolved Observations Relevant to Air Quality (DISCOVER-AQ) in September 2013 and satellite-based estimates near Houston on 19 June 2013. The timing of the satellite cloud condensation nuclei (CCN) observations is identical to the simulation dates. The profiles feature constant values in the boundary layer up to 2.5 km and in the free troposphere over 5 km, with a linear transition between these heights. Aerosol removal/replenishment processes are based on the semi-diagnostic methods in . This method activates cloud droplets for a given supersaturation rate and aerosol characteristics and tracks the sum of activated and unactivated aerosol through advection and mixing. Additional cloud droplets can be activated when the newly activated cloud droplets number exceeds the present number of cloud droplets. Aerosol number concentrations will be reduced only when cloud droplets are reduced by a coalescence process (i.e., autoconversion to precipitation class). The advantage of this approach is to account for activation and regeneration of aerosols without explicitly accounting for aerosols within cloud droplets see details in.

NCEP Final Analysis (FNL) was used to initialize NU-WRF on 19 June 2013 at 12:00 UTC, and it continued updating lateral boundary conditions until 20 June 2013 at 15:00 UTC. The 6 h lateral boundary conditions from GFS are spatially and temporally interpolated to update the model lateral boundary conditions at every model time step, while sea breeze dynamics are explicitly simulated by model physics and dynamics within the domain. Since thermal tracking requires 1 min temporal resolution of NU-WRF output, we focused on the 3 h time window from 19 June 21:00 UTC for thermal and cloudy updraft grid point analysis during the active convection period. Figure  shows the actual sampling domain used (a 100×100 km area), where most active convection occurs during this time window.

Sufficiently high-resolution simulations can generally reproduce the expected thermal-like structures that are characteristic of cumulus clouds e.g.,. This provides a numerical tool to investigate the dynamics of these thermals, which, in turn, leads to a better understanding of many aspects of convection . Here we identify, track, and analyze cumulus thermals in the NU-WRF simulations described in the previous section, using the methodology of . In the following, we describe the main features of this method; for further details, please refer to their study.

To identify thermals, an automated algorithm identifies peaks in vertical velocity throughout a particular volume of the simulation at each output time step and assumes that these indicate the instantaneous locations of the thermals' centers. By comparing these locations in consecutive output time steps, the algorithm can estimate each thermal's trajectory, which also yields an estimate of their ascent rates at each time step. Assuming spherical shapes, a thermal's size can be estimated by choosing the radius that makes the average vertical velocity of the enclosed volume match the corresponding ascent rate. Notice that each thermal's ascent rate can vary between time steps, and hence, the estimated size of a thermal may also vary in time. The smallest radius permitted for a thermal is twice the model grid spacing, which is 500 m in this case. Smaller thermals are discarded. This ensures that each identified thermal corresponds to a coherent rising volume of air. showed that, indeed, thermal shapes do not deviate much from sphericity, making this a good approximation. Finally, it is worth noting that the algorithm only takes into account thermals with average ascent rates of at least 1 ms-1 and with centers that have at least 0.01 gkg-1 of cloud condensate. Furthermore, it computes each thermal's vertical momentum budget and discards any cases in which the tracked trajectory is inconsistent with it. From the sample of tracked thermals, different statistical measures can be obtained for both microphysical and dynamical properties. These can be then compared to results based on the cloudy updraft sampling framework. For consistency, our threshold criteria for selecting cloudy updraft grid points is a vertical velocity of 1 ms-1 and a cloud condensate of 0.01 gkg-1.

The mass flux captured by the tracked thermals is typically 15 %–20 % of the estimated total mass flux, as will be shown below. Despite this being a relatively small fraction, showed that the convective evolution is well represented by the thermals, suggesting that their dynamics are representative of the entire convective activity (discussed later). Untracked updrafts are typically too small or too slow to be tracked with this algorithm. Furthermore, the total mass flux is not uniquely defined and may contain spurious non-convective contributions e.g.,. Finally, it is worth noting that we find very similar properties of thermals in this study compared to what found with their higher-resolution simulations (65 m horizontal grid spacing). The only prominent difference is that our thermals are larger (R∼1.2 km, compared to R∼0.3 km), which may be expected given our coarser spatial resolution setting, but this could also be partially attributable to differences in the case study conditions. Owing to the similarity of results to those of , we expect that finer-resolution results would be more converged but similar in nature.

Figure  shows statistical composites of microphysics properties within tracked thermals from the selected background aerosol cases of 500, 1000, 2000, and 4000 cm-3 (i.e., for each subsequent doubling of aerosol concentrations). For these composites, only the time step of the maximum ascent rate of each thermal is considered. The results demonstrate that an increase in background aerosol concentrations tends to (a) increase cloud droplet nucleation rates, (b) reduce supersaturation values, (c) increase cloud droplet number concentrations, and (d) decrease rain number concentrations (Fig. a–d). Plots of average values of these quantities within thermals as a function of aerosol number concentration (not shown here) reveal that nucleation rates, supersaturation values, and cloud drop number concentration behave roughly linearly with aerosol number concentration, whereas rain number concentration decreases exponentially, which is consistent with raindrop generation by coalescence of cloud droplets. On the other hand, although number concentrations of both cloud droplets and raindrops are strongly affected by aerosol number concentration, their mixing ratios respond less strongly (Fig. e and f) and in such a way that the total liquid water mixing ratio remains more weakly impacted (not shown here).

Microphysical quantities are found to peak at thermal centers nearly universally, which reinforces the important role of thermals as the building blocks of convection from both a dynamical and microphysical point of view. For example, supersaturation values are only reached inside thermals, generating numerous cloud droplets around their cores. Streamlines of the averaged flow also indicate a more turbulent mixing around the thermal frame, whereas upstream currents are present in the core of thermals.

The microphysical response to aerosol number concentration could cause a prominent dynamical response in thermals via changes in the rate at which latent heat is released due to condensation. For example, following the reasoning by , a reduction in supersaturation rates could result from the larger number of smaller droplets (and, hence, more available surface area for condensation) as aerosol concentrations increase. All else being equal, this could imply a faster latent heat release due to condensation. However, Fig. a indicates no prominent mean response in latent heating rates within the tracked thermals (summed over all source terms), while cloud nucleation rates increase and supersaturation rates decrease with increasing aerosol concentrations (Fig. a and b). This implies that there is no prominent change in latent heating per unit of time available for the dynamics of the thermals, which indicates similar total condensation rates despite changes in driving supersaturations. A possible explanation is that supersaturation differences are sustained within the context of negligibly different total condensate production rates within the thermal core, but that hypothesis cannot be definitively supported without additional diagnostics that separate the sources of latent heat in future work. Figure b and c also show no notable changes in their composite buoyancy (B) or vertical velocity (w). We do not find any prominent trends in terms of the thermals' composite lifetime, vertical distance traveled (DZ), or radius (R). For R and DZ, this can be inferred from the vertical profiles shown in Fig. b and d. Furthermore, the histograms of these quantities are negligibly changed (not shown).

Since many of these variables have strong vertical dependencies, we next investigate these responses in terms of vertical profiles of microphysical quantities, latent heating rates, vertical velocity, and mass flux, as estimated from cloudy updraft grid points (Fig. a–h) and from the tracked thermals (Fig. i–p). To begin with, notice that the vertical profiles in both frameworks show qualitatively consistent features at most elevations. Perhaps the most prominent difference between these two frameworks is that thermals indicate a larger contribution than cloudy updraft grid points to several quantities at levels above 6–7 kma.g.l. This is very clear in terms of vertical velocity (Fig. g and o), where both frameworks yield very similar profiles up to ∼6–7 kma.g.l. but significantly different values aloft. According to cloudy updraft grid points, vertical velocity reaches its maximum near 7 kma.g.l., whereas, according to thermals, it continues to increase, reaching its maximum near 10 kma.g.l. This suggests that the thermal sampling criteria is more selective of vigorous updrafts aloft. This also results in a slightly more top-heaviness of the profiles of other quantities, which reflects how strongly coupled microphysical processes are with updraft dynamics. In terms of mass flux, both frameworks yield a maximum near 3 kma.g.l., but thermals indicate a secondary maximum between 7 and 9 kma.g.l. Notice that this corresponds to the contribution of relatively few thermals (Fig. a), suggesting that, unlike near the cloud base where convection results from small contributions of many updrafts, convection at mid and high levels near the cloud top is more tightly linked to the contribution of relatively few but vigorous updrafts, a feature that may be better captured by the cumulus thermal framework.

It is important to point out that, throughout the 3 h period analyzed here, convection evolves and may behave differently at different stages. To assess this, Figs. S1–S3 in the Supplement show profiles, as in Fig. , where the 3 h period has been divided into three stages. These profiles reflect the fact that convection deepens with time but, otherwise, show consistency with Fig. . Furthermore, considering the entire 3 h period provides a larger sample of updrafts, which, in turn, aids in reducing the noise.

Regarding the responses to increases in aerosol concentrations, both frameworks show overall agreement. To visualize these responses more clearly, Fig.  shows the differences between profiles for each successive doubling of aerosol concentrations, their average change, and the difference between the most and least polluted cases. This figure also helps to identify in which quantities there is a consistent response to increases in aerosol concentrations. Notice how linear the response is for cloud nucleation rate and cloud droplet number concentration (Fig. a, b and i, j) and how the decrease in rain number concentration behaves exponentially, with the largest changes for lower aerosol number concentrations (Fig. c and k). On the other hand, the increase in the cloud water mixing ratio is slightly offset by the decrease in the rain water mixing ratio (Fig. d–e and l–m), so that there is a slight net decrease in total liquid water mixing ratio (not shown here). However, the variability in the decrease in rain water mixing ratio between pairs of experiments is significantly higher than in the increase in the cloud water mixing ratio, which also makes the net decrease in total water mass highly variable between simulations.

In contrast to the microphysical quantities, latent heating rates, vertical velocity and mass flux do not reveal such prominent and consistent responses to aerosol concentrations (Fig. f–h and n–p), and this is true in both frameworks. As expected, changes in vertical velocity closely follow changes in latent heating rates, but both are small on average, with a high level of noise between different pairs of experiments and more so in the cumulus thermal framework. For example, in the comparison between 4000 and 500 cm-3, we find an increase of ∼10 % in the vertical velocity near heights of 6 and 11 kma.g.l. consistent with findings by. The average response for a doubling of aerosol concentrations at these altitudes also suggests an increase, but it is much weaker (∼2 %); however, not all individual pairs of cases show such an increase, and the amplitude of the individual responses is usually larger than the average one.

Regarding mass flux, notice that its estimate based on tracked thermals is ∼15 % of the cloudy updraft estimate (Fig. h and p). As shown by , the relatively low fraction captured by thermals results from mainly small and slow thermals that are harder to identify and track with our method; however, it is representative of the entire convective activity. In fact, notice that the changes in mass flux for each doubling are consistent between the thermal estimate and the cloudy updraft grid point estimate (Fig. h and p). Here, too, the average response for doubling aerosol concentrations is weaker than the individual responses. In fact, it is nearly zero everywhere, except for a slight increase around 4 kma.g.l. in the thermals' framework, which can be linked to an increase in the number of tracked thermals (Fig. e).

Similar results are seen for other quantities relevant for cumulus thermals (Fig. ). A certain degree of correspondence can be seen between buoyancy changes (Fig. g) and vertical velocity changes (Fig. o), with hardly any average response when doubling the aerosol concentrations, despite significant (but not consistent) changes between individual pairs of simulations. Changes in the average vertical distance traveled by thermals (DZ; Fig. h) is also similar to changes in the vertical velocity of thermals, especially its average response for a doubling of aerosol concentrations (Fig. o). This indicates that the average thermal lifetime (not shown here) is also invariant to aerosol number concentrations.

All these quantities related to the thermals' dynamics seem to respond only very weakly to changes in aerosol number concentrations, compared to the natural variability between each pair of simulations. This is a known limitation when investigating the aerosol invigoration of convection. Several studies have emphasized the difficulty of rising above the noise level when trying to identify aerosol indirect effects e.g.,. For a given microphysics and dynamics framework, our results support this view from both the cloudy updraft and the thermal frameworks regarding fundamental dynamical properties, since results vary widely, depending on which pair of experiments is taken into account. However, we also see some indication of a change in the sign of the trend across the full dynamic range of aerosol variability. For instance, the doubling aerosol initially increases buoyancy near 6 kma.g.l. but ultimately decreases buoyancy at that elevation by a similar amount when reaching the highest aerosol concentration. Similar responses can be seen in terms of w, DZ, and mass flux, which is consistent with an aerosol-limited regime e.g.,.

Average thermal size, which we estimate here with its radius R, shows no systematic change related to aerosol number concentrations (Fig. b and f). However, we do find a response in the number of tracked thermals, particularly between 2–4 kma.g.l., where most thermals develop. This response also seems to depend on the particular range of aerosol variability, with more thermals being tracked as aerosol concentrations increase in the cleaner regime (500–2000 cm-3) and fewer thermals being tracked when aerosol concentrations are doubled in the more polluted regime (2000–4000 cm-3).

We have shown how our two sampling criteria provide a general agreement in terms of the microphysical and dynamical responses to increases in aerosol number concentrations. However, we have also noted differences which reveal important features of thermal and grid point analyses. The scatterplots in Fig.  show how relevant quantities averaged within thermals compare to the same quantities averaged over cloudy updraft grid points, both for different vertical layers (circle dots) and for the entire columns (crosses) in the different aerosol number concentration experiments. In general, these plots confirm that both thermal and cloudy grid points analyses are close to each other, but interesting features emerge from their comparison.

Cloud and rain number concentration, as well as cloud mass mixing ratio (Fig. a and e), appear to be similar between thermal and cloudy grid points but have slightly higher values within thermals than for cloudy updraft grid points. This is more prominent at higher altitudes, where thermals tend to be larger and vigorous, and the same applies for rain number concentrations. In other words, at higher elevations, thermals differ more from the average cloudy conditions than at lower elevations, which emphasizes their important role in the deepening of the convective cloud. At the near-surface level (∼1 kma.g.l.), the cloud number and mass concentrations are lower than the cloudy updraft grid points, most likely due to the thermal's internal circulations that may include downdrafts and/or condensate-free volumes of air, but, nevertheless, are dynamically connected to the rising thermals and their internal microphysical processes.

Rain mass mixing ratios also appear to be higher in thermals than in cloudy updraft grid points but have, on average, similar values in both cases (Fig. f). When separated by height, thermals at higher altitudes tend to have higher rain mass mixing ratios than cloudy updraft grid points, but the opposite is true at lower altitudes. This can be explained if one thinks of thermals at upper levels as the regions where rain is starting to form and, hence, have more rain mass than the average cloudy updraft grid points, whereas rain at lower levels tends to be concentrated at downdraft regions where rising thermals are limited. An interesting feature here is that the average values per experiment cross the 1:1 line in such a way that thermals have higher rain mixing ratios than the average cloudy updraft grid points in the cleaner cases but lower rain mixing ratios in the polluted cases. This would be in line with raindrops being larger (and fewer) in the polluted cases, making them fall faster and less likely to be inside a rising thermal.

Averaged over the entire vertical column, thermals and cloudy updraft grid points respond almost equally in terms of nucleation rates to varying aerosol number concentrations (Fig. c). However, thermals tend to have slightly higher nucleation rates in the upper levels and lower nucleation rates in the lower levels compared to cloudy updraft grid points. This small difference may be because thermals in the upper levels tend to sample the larger, faster, and, hence, less diluted updrafts, while the cloudy updraft grid points may also sample weaker, shorter-lived updrafts where nucleation rates are lower. On the other hand, at lower altitudes, thermals tend to be smaller and more numerous, likely sampling similar updrafts as cloudy updraft grid points, but thermals include a larger volume of air surrounding the updrafts, slightly reducing their average nucleation rates.

In terms of overall column averages, we see that, for both thermals and cloudy updraft grid points, latent heating rates, and vertical velocity appears to be similar (Fig. d and g). Regarding the relation between thermal averages and cloudy updraft grid points, there are important differences with altitude. For example, the average vertical velocity of cloudy updraft grid points and thermals follows the 1:1 line closely up to about 6 ms-1. Average vertical velocity for thermals, in particular above an altitude of about 6 kma.g.l., does exceed this value, while the average for cloudy updraft grid points does not. To understand this, notice that the mass flux captured by thermals (Fig. e) has a first maximum just below 4 kma.g.l. and a second maximum around 8–9 kma.g.l. The first maximum coincides with the layer where most smaller and short-lived thermals are found within the boundary layer; the second maximum has about half the mass flux of the first but only about a sixth of the number of thermals (Fig. a). Thus, the thermals above 6–7 kma.g.l. are not as numerous, but larger ones individually contribute much more to the total mass flux than those in the boundary layer. Increasing the vertical velocity threshold for the cloudy updraft grid point definition, while it does not modify the aerosol sensitivities found here, yields closer values between frameworks for several quantities at upper levels but at the expense of larger differences at middle and lower levels that result in less overall consistency (Figs. S4 and S5). Further investigation of the detailed differences between the two frameworks at upper levels is left for a future study, with a focus extended to ice microphysical processes.

Finally, the fact that latent heating rates tend to be higher for thermals than for cloudy updraft grid points at a higher altitude (Fig. g) suggests that thermals are capturing the most relevant regions where condensation occurs and, thus, the most relevant convective regions of the cloud. Latent heating rates of thermals largely exceed those of cloudy updraft grid points at higher altitude but underestimate at near-surface level. These are very similar patterns of those combined from cloud and rain mass mixing ratio (Fig. e–g). Overall, these results highlight how both frameworks are generally consistent, while subtle differences between them can provide additional useful information.