## 1. Introduction

Moist convection plays an important role in the climate system by redistributing heat, moisture, momentum, and chemical species, and by perturbing the radiation budget. At the same time, the existence, intensity, and form of convection depend on its environment. Earlier studies have shown that deep convection responds differently to environmental perturbations in the lower versus upper troposphere. For example, Tulich and Mapes (2010) examined how heating and moistening tendencies from deep convection respond to perturbations to its environment and found that convection is more effective in removing temperature anomalies in the lower troposphere^{1} than those in the middle to upper troposphere. This is consistent with the findings in Kuang (2010), where a sufficiently complete set of linear response functions was constructed to characterize convective responses to temperature and moisture anomalies at different heights. In these studies, convective heating and moistening tendencies were used to summarize the convective responses, as they are the most important from a large-scale perspective.

The different sensitivities of deep convection to temperature perturbations in the lower versus upper troposphere have important implications. As temperature anomalies in the lower troposphere elicit strong responses from convection that quickly remove such anomalies, while responses to such anomalies in the upper troposphere are weaker, it results in a shallow convective quasi equilibrium (e.g., Kuang 2008; Raymond and Herman 2011), in contrast to a full-tropospheric convective quasi equilibrium. Furthermore, because convective responses to temperature anomalies in the upper troposphere are relatively weak and mostly local, this local response to removing such anomalies can be overwhelmed by nonlocal responses to temperature or moisture perturbations in the lower/middle troposphere, which, according to a number of conceptual models (e.g., Mapes 2000; Kuang 2008, 2010), enables the growth of convectively coupled waves, an important tropical variability (e.g., Wheeler and Kiladis 1999). Therefore, a mechanistic understanding of what contributes to the different convective sensitivities to temperature perturbations at different heights of the free troposphere is important and can help to guide the development of convective parameterization schemes.

Since a warm anomaly in the free troposphere serves as a buoyancy barrier, a reduction in convective updraft mass flux, and thus convective heating and drying, is to be expected. However, it is not obvious why this effect is substantially stronger when the warm anomaly is imposed in the lower troposphere versus in the upper troposphere. Tulich and Mapes (2010) speculated that the small buoyancy of the cloud updrafts in the lower troposphere might explain the stronger sensitivity there, because such cloud updrafts would be more sensitive to external perturbations that affect their buoyancy. There, however, are also other possible mechanisms. For example, in addition to having smaller buoyancy, convective updrafts in the lower troposphere also have smaller vertical velocity and higher liquid water content compared to those in the upper troposphere. A buoyancy barrier created by a warm anomaly will more easily eliminate parcels with smaller vertical velocity from the cloudy updrafts, resulting in more mass flux reduction in response to a lower-tropospheric warm anomaly. The liquid water mixing ratio in clouds is much higher in the lower troposphere than in the upper troposphere. Therefore, when a given warm anomaly is introduced in the lower troposphere, clouds are subjected to more evaporative cooling, which can further reduce the updraft buoyancy and lead to enhanced detrainment. These differences could all potentially explain why the convective response is stronger when the temperature field is perturbed in the lower troposphere. Throughout this paper, we use the convective mass flux change in response to the imposed perturbation as a proxy to investigate convective sensitivity.

To quantify contributions from these different factors, we will make extensive use of Lagrangian particle tracking, following the procedures of Tian and Kuang (2016). As the particles carry with them all necessary dynamic and thermodynamic information to construct their full Lagrangian histories, trajectory tracking provides more information than is available from the tracer encoding technique of Romps and Kuang (2010) and Nie and Kuang (2012). With the Lagrangian particles, we shall quantify the relative importance of cloud updraft buoyancy, vertical velocity, and liquid water mixing ratio differences in determining the differing convective responses to the same large-scale temperature perturbations at different heights of the atmosphere. Additionally, whether or not the empirical entrainment formula proposed in Tian and Kuang (2016) for shallow convection still holds for deep convection is also tested here.

The remainder of the paper is organized as follows. Section 2 briefly describes the model used and the experimental design. The analysis method, results, and their interpretations are presented in section 3, followed by a summary in section 4.

## 2. Models and experimental design

The cloud-resolving model (CRM) simulations were performed using the anelastic System for Atmospheric Modeling (SAM), version 6.8.2 (Khairoutdinov and Randall 2003), for a radiative–convective equilibrium (RCE) state over an ocean surface with a sea surface temperature of 301.15 K. A sponge layer is placed in the upper third of the domain. The subgrid-scale effects were parameterized using a prognostic turbulent kinetic energy 1.5-order closure scheme. We used the SAM one-moment bulk microphysics scheme and the surface fluxes were calculated using the Monin–Obukhov similarity theory. Radiation was a simple Newtonian relaxation scheme, as in the study by Pauluis and Garner (2006). There was a uniform cooling of 1.5 K day^{−1} within most of the troposphere while temperatures lower than 200 K were nudged to 200 K with a time scale of 5 days. This idealized radiative scheme was used so that mean convective heating in the upper and lower troposphere is the same. A simulation with 500-m horizontal spacing on a doubly periodic domain (128 km × 128 km) and stretched vertical grid with spacing ranging from 80 m near the surface to 1 km in the stratosphere was started from an RCE initial sounding and run for 40 days to reach a fully convecting equilibrium state. Starting from the end of day 40, restart files were output every 12 h for another 50 days. A set of 2-h-long simulations were then initialized from these restart files but with a temperature perturbation added to the initial conditions. The temperature perturbation was horizontally uniform, Gaussian shaped in pressure, with a half-width of 75 hPa, and a peak value of +0.25 K. The size of the temperature perturbation was chosen by balancing the need for a good signal-to-noise ratio, which favors large perturbations, and the need for the response to remain mostly linear, which favors small perturbations. A mostly linear response, as explained in Tian and Kuang (2016), reduces the level of confounding in the interpretation of the results. This set of simulations with perturbations, combined with the initial 50-day simulation without perturbations, provided 100 pairs of 2-h-long control and perturbed runs. The averaged differences between these pairs of runs were taken as the convective responses to the imposed temperature perturbation.

As convection responds to remove the initial temperature anomaly, the amplitude of the initially added warm anomaly gradually decreases over time, and some moisture anomalies start to develop. Because we want to examine the response of the convective ensemble to the initially imposed temperature perturbation, such drifts in the thermodynamic profiles, while not a dominant factor, complicate the interpretation. To reduce this complication, we prescribed time-invariant temperature and moisture tendencies to substantially reduce the drift in the large-scale basic state (see appendix A). We take the view that the prescribed tendencies affect convection indirectly through their effects on the (horizontal mean) temperature and humidity profiles, and these profiles uniquely determine the convective statistics.

To sample convective responses to temperature perturbations at different heights of the atmosphere, we introduced the same aforementioned temperature perturbations centered at two different pressures (Fig. 1a): 700 and 300 hPa (hereafter referred to as the 700-hPa/lower-troposphere case and the 300-hPa/upper-troposphere case). Once a warm anomaly is imposed in the atmosphere, convection reacts to remove this anomaly, which leads to a cooling tendency. It is clear from Fig. 1b that the convective response is much stronger in the lower-troposphere case than that in the upper-troposphere case, consistent with prior studies (e.g., Tulich and Mapes 2010; Kuang 2010).

We shall examine convective responses within the framework of an ensemble of entraining plumes, following Lin and Arakawa (1997) and Tian and Kuang (2016). This is achieved by tracking a total of approximately 30 million Lagrangian particles/parcels (or eight per grid box on average) embedded in the CRM, the details of which are in appendix B. Particles that are detrained over the same range of heights are grouped as one “entraining plume,” or one “parcel group.” Therefore, the term entraining plume used here does not describe physical structures such as the similarity plumes in the water tank experiments of Morton et al. (1956). Instead, a “plume” comprises parcels from different clouds, and parcels from a single cloud contribute to multiple plumes. For this study, we define cloudy updrafts to be grid boxes with vertical velocities greater than 1 m s^{−1} and nonprecipitating liquid water mixing ratio greater than 0.01 g kg^{−1}. A cloudy parcel is considered to have been detrained into the environment if it was no longer part of a cloudy updraft for more than 5 min.

We divided all Lagrangian parcels in cloudy updrafts at the cloud base into 100 groups based on their detrainment heights. The larger the parcel group number, the higher the parcels detrain. The relatively large number of parcel groups was adopted to enhance the homogeneity within each group while exposing differences between groups. For the same parcel group, the height range over which the parcels detrain can be different between the control and the perturbed experiments. A parcel that does not originate from the cloud base but becomes part of a cloud later is considered entrained into a particular parcel group based on this parcel’s detrainment height. Given the chaotic nature of cumulus convection, it is not meaningful to track the same individual parcels in both the control and perturbed runs and analyze changes in their patterns of behavior due to the perturbation. With our parcel grouping method, parcels belonging to the same group in the control and the perturbed runs may be viewed as the “same” parcels in a statistical sense. Therefore, we should interpret the differences between the control and perturbed runs of the same parcel group number as the differences in behavior for the same statistical parcel. The above procedure is the same as in Tian and Kuang (2016) and is summarized here, almost verbatim, for completeness. For more detailed descriptions about parcel grouping, see Tian and Kuang (2016).

## 3. Analysis and results

### a. Convective responses to temperature perturbations

The basic features of deep convection in the control run (Fig. 2) are broadly similar to those of shallow convection, and the latter was extensively examined in Nie and Kuang (2012) and Tian and Kuang (2016). Total water mixing ratio (Fig. 2a) and buoyancy acceleration (Fig. 2b) at the cloud base (~762.5 m) are almost uniform across the different parcel groups. On the other hand, a parcel that detrains higher tends to start with slightly higher vertical velocity (Fig. 2c) and farther away from the cloud edge (Fig. 2d), indicating some role of the initial conditions in determining the fate of cloudy updrafts, similar to what was seen in Tian and Kuang (2016). Here, the distance to the cloud edge is calculated as the minimum distance from the parcel location to the boundary between the cloudy updrafts and their environment, and parcel location is assumed to be the center of the grid box that it resides in. The fractional entrainment rate (fractional change in mass flux per unit height) for deep convection is much smaller compared to that of shallow convection, ranging from 0.05 to 1 km^{−1} in the bulk of the cloud layer (Figs. 2e,f), in contrast to 2–3 km^{−1} that is typical of shallow convection (e.g., Tian and Kuang 2016). This smaller entrainment rate in deep convection can be attributed to higher vertical velocities and bigger cloud radii, which will be discussed later in this section. In this study, an entrainment event only occurs if a parcel becomes part of a cloud updraft and stays as such for at least 5 min in order to remove the rapid cycling across the cloud edge.

As mentioned earlier, convective responses are recorded as the averaged differences between the perturbed and control pairs. We zoomed onto those parcel groups that are most affected by the added temperature perturbations, namely those that can detrain above the layer with the imposed temperature perturbation. When perturbed at 700 hPa, the convective responses to the temperature perturbation are qualitatively similar to those of the shallow convection case investigated in Tian and Kuang (2016) (Figs. 3a–d). The imposed warm anomaly creates a buoyancy barrier, shown as a belt of negative buoyancy anomalies in the perturbed layer (Fig. 3b). More buoyant parcels are more likely to go across the perturbation layer, and will be preferentially sampled aloft. This is observed as a slightly positive buoyancy anomaly, especially for the parcel groups that detrain at higher altitudes, because these parcels traverse the barrier more quickly and experience less entrainment-induced buoyancy reduction. Accordingly, updraft vertical velocity also increases above the perturbed layer for these parcel groups (Fig. 3c). The slight increase in the total water content in the perturbed region and its decrease above are affected by changes in both the entrainment rate and precipitation, as the total water is no longer a conserved variable in deep convection. We also observe that the cloud radius, represented by the minimum distance to cloud edge, increases below the level where the imposed temperature perturbation peaks, and decreases above it. This may be due to changes in stratification, as a stronger vertical buoyancy gradient causes the clouds to spread more horizontally, and the buoyancy barrier removes less buoyant cloudy-updraft parcels from the outer rims of the clouds, leading to a radius decrease above the perturbation layer. Overall, the convective responses here are comparable to those in Tian and Kuang (2016). The convective responses in the 300-hPa case (Figs. 3e–h) share some similarities to those in the 700-hPa case, except with generally smaller magnitudes, but also exhibit some differences. The response in total water in the 300-hPa case is less positive. This could be due to enhanced precipitation efficiency as the updrafts are slowed with the prescribed warm anomaly. There is also a belt of reduced buoyancy in response to the warm anomaly, but the positive buoyancy region in the 300-hPa case is limited the most high-reaching parcel groups. This is likely because the imposed warm anomaly is already close to or higher than the detrainment level of most of the parcel groups. Likely for the same reason, the updraft vertical velocity responses are also all negative in the 300-hPa case, similar to what is seen in the parcel groups that detrain at low altitudes. Finally, the cloud radius increases slightly below the perturbation level and decreases above, but this increase occurs farther below the peak perturbation level than that in the 700-hPa case, and the magnitude of the change is smaller.

In a study of a shallow cumulus case (Tian and Kuang 2016), an empirical formula was proposed to relate the fractional entrainment rate to the product of the vertical velocity and cloud radius, and a nearly constant coefficient *α* was identified within the bulk of the cloud layer (*w* is the vertical velocity, *d* is the distance to cloud edge, and *α* ≈ 0.23 m s^{−1}). Here, we carried out similar analyses to see if that formula also holds for deep convection. Following Tian and Kuang (2016), we calculated the percentage changes in the updraft vertical velocity, distance to cloud edge, and fractional entrainment rate in both the 700- and 300-hPa perturbed cases relative to the control case. Figures 4 and 5 show that the relative changes in the fractional entrainment rates of parcel groups can be reproduced to a good extent by adding up the negatives of the relative changes in vertical velocity and distance to cloud edge. This indicates that the previously proposed entrainment formula for shallow convection can also be extended to deep convection. The *α* coefficient in the latter is considerably larger, around 2.2 m s^{−1} within the bulk of the cloud layer (~4000–9000 m) and becomes even larger near the cloud base (Fig. 6). The higher value within the bulk of the cloud layer is consistent with Tian and Kuang’s (2016) hypothesis that coefficient *α* may scale with the square root of the turbulent kinetic energy: as deep convection has greater turbulent kinetic energy, *α* also becomes larger. The large *α* close to the cloud base, however, needs further investigation. We suspect this may be related to the large population of shallow clouds that collapse in the lower 1000 m, which increases the turbulent mixing as indicated in, for example, Bretherton et al. (2004).

### b. Methods for examining three hypotheses

Figures 1b and 3 demonstrate that deep convection responds more strongly to temperature perturbations imposed in the lower troposphere. Note that with the idealized radiation scheme used in this study, convective heating in the upper and lower troposphere is the same in the control run and, thus, cannot explain the differing convective sensitivities. In the following, we will examine the relative importance of the cloud updraft buoyancy, vertical velocity, and liquid water mixing ratio differences in determining the different sensitivities to temperature perturbations in the lower versus upper troposphere.

Let us first consider the effects of cloud liquid water content. Once a horizontally uniform warm anomaly is introduced into the free troposphere, saturation vapor pressure increases inside the clouds within the perturbed layer, and liquid water evaporates to maintain saturation, which reduces updraft buoyancy through evaporative cooling. Evaporation can also happen when cloudy updrafts rise into the perturbed layer and entrain the now-warmer environmental air. We can infer the importance of liquid water by contrasting the actual buoyancy reduction at the two peak perturbation levels (corresponding to 700 and 300 hPa approximately). It turns out that the peak buoyancy reduction in the lower troposphere (0.0022 m s^{−2}) is only around 10% larger than that in the upper troposphere (0.0020 m s^{−2}), and this difference is negligible compared to the average buoyancy difference between the lower and upper troposphere (0.02 m s^{−2}). Thus, the difference in the amount of liquid water should not be a determining factor of the substantial (a factor of 3) difference in the convective sensitivities between the lower- and upper-troposphere cases.

We now focus on the relative importance of cloudy-updraft vertical velocity and buoyancy (ignoring evaporative cooling due to liquid water), namely how differences in these two quantities between the lower and the upper troposphere affect the response of the cloudy-updraft mass flux. The mean cloudy-updraft vertical velocity in the lower troposphere is ~3.5 m s^{−1}, which is about half of the value in the upper troposphere. As mentioned earlier, cloudy parcels with small initial vertical velocity may be preferentially eliminated from the updrafts and lead to more mass flux reduction and, hence, a stronger response. Additionally, cloudy updrafts only have marginally positive buoyancy throughout the lower troposphere, so that their rise can be inhibited more easily by the imposed buoyancy barrier (Tulich and Mapes 2010). A straightforward way to tackle this problem is to look at how cloudy-updraft mass flux changes as a function of those two variables. In a Lagrangian framework, the updraft mass flux is calculated by counting the number of cloudy parcels that cross a particular interface within a certain time interval. Here, we care more about the total mass flux change over the entire perturbed layer rather than change at any single layer within the perturbed region. Therefore, we selected two levels instead of a single interface: one below the perturbed layer (~1400 m for the 700-hPa case and ~7300 m for the 300-hPa case), where the parcel properties are relatively unaffected by the imposed perturbation, and the other above the peak of the imposed temperature perturbation, where the convective responses are the strongest (~3000 m for the 700-hPa case and ~9400 m for the 300-hPa case), and we refer to these two levels as the bottom and top levels for simplicity. We binned the parcels based on their buoyancy acceleration and vertical velocity at the bottom level right before they ascend to the next level. Within each bin, we then calculated the percentage of cloudy parcels that can go across from the bottom level to the top level. Any cloudy parcel that succeeds in ascending from the bottom to the top level will be counted as one crossing, and those that fail to do so will be counted as zero crossing. The total number of crossings is then normalized by the total number of parcels at the bottom level to obtain the percentage of cloudy parcels that cross the layer, which will be referred to as the crossing percentage. This calculation is easily done with Lagrangian particles because we keep track of each parcel, and know exactly if a parcel crosses the perturbation layer or not based on its trajectory. Crossing percentage can show how the preferential elimination of cloudy parcels depends on the parcels’ properties.

It is apparent that the parcel crossing percentage increases with the initial vertical velocity and buoyancy at the bottom level for both the perturbed and the control cases (Fig. 7): more buoyant parcels with higher vertical velocity can travel longer distances and are, thus, more likely to traverse the perturbation layer. However, the crossing percentage changes much less with the buoyancy than with the vertical velocity in the lower troposphere (Fig. 7b). The difference in the percentage reduction between the most buoyant and least buoyant parcels is only ~15%, compared to a difference of over 60% between the cloudy updrafts with the highest and the lowest vertical velocities. This preliminary analysis suggests that the difference in the vertical velocity between the lower and upper troposphere might play a more important role in affecting convective sensitivity than the differences in buoyancy throughout the perturbed layer. However, the percentage reduction seems to depend on both the initial vertical velocity and buoyancy acceleration in the 300-hPa case (Figs. 7c,d). The joint probability density function (PDF) of the cloudy parcel’s initial vertical velocity and buoyancy acceleration (not shown) indicates that they are highly correlated in the upper troposphere.

*z*and

*F*consists of both Archimedean buoyancy and perturbation pressure gradient acceleration [Eq. (3)]. In addition,

To perform the swapping experiment, we first obtain the PDFs of the cloudy parcel’s initial vertical velocity and buoyancy in the 300- and 700-hPa cases (see Fig. C1 in appendix C). Here, buoyancy refers to the total or effective buoyancy, which is the sum of the Archimedean buoyancy and the buoyancy-induced pressure gradient acceleration, because this is the actual buoyancy acceleration that a parcel experiences during its ascent (e.g., Davies-Jones 2003), and we need to change the buoyancy-induced pressure gradient accordingly when we change a parcel’s buoyancy.

The initial vertical velocity PDF is obtained at the time the parcel passes the bottom level. To quantify the effects of the initial vertical velocity, we can, for example, solve Eqs. (1)–(3) for the 700-hPa case, but with the initial vertical velocity from the 300-hPa case. To do so, we divide each PDF into 200 quantiles with the same number of parcels in each quantile (the result is insensitive to the number of quantiles used). For each parcel in the 700-hPa case, we first determine the quantile of its initial vertical velocity and then replace it with the initial vertical velocity from the 300-hPa PDF for the same quantile (the median value within the quantile is used). The (hypothetical) parcel with the modified initial vertical velocity is assumed to follow the same trajectory as the original parcel and experiences the same acceleration (and has the same liquid water content) as the original parcel at a given height (even though it might move along the trajectory faster or more slowly than the original parcel). One can similarly integrate Eqs. (1)–(3) for the 300-hPa case using the initial vertical velocity from the 700-hPa case.

For buoyancy, the PDFs are constructed for all heights, each also divided into 200 quantiles as before (see Fig. C1 in appendix C for an example). To quantify the effects of buoyancy, when solving Eqs. (1)–(3), say, for the 700-hPa case, we can replace the parcel buoyancy with that of the 300-hPa case. To do so, we again assume that a hypothetical parcel moves along the same trajectory as does the original parcel, albeit at a faster or slower rate, and experiences the same acceleration as the original parcel at a given height (before its buoyancy is replaced by a corresponding value from the 300-hPa case). When evaluating Eq. (2), at each height of the 700-hPa case, we find the corresponding height in the 300-hPa case so that these two heights have the same pressure difference from their respective bottom levels. For each parcel in the 700-hPa case at each height, we then determine the quantile of its buoyancy (taken as the buoyancy of the original parcel when it is at this height) and replace this buoyancy with the buoyancy for the same quantile from the PDF at the corresponding height of the 300-hPa case (vertical interpolation is used to obtain the PDF at the corresponding height when needed). This replacement is the same as that for the initial vertical velocity, except that it is now done for all heights that the parcel travels through. More details on the replacement or swapping procedure can be found in appendix C.

The above procedure allows us to isolate the effects of the initial vertical velocity and buoyancy. For example, after we replace the initial vertical velocity of a cloudy-updraft parcel that originates from the lower troposphere with that from the upper troposphere, the parcel still has the nearly neutral buoyancy characteristic of the lower troposphere but with a much higher vertical velocity typical of the upper troposphere. If we subject this new parcel to the same temperature perturbation in the lower troposphere, it is expected that it will respond differently compared to the original parcel. This sampled difference in response is caused by the change in vertical velocity alone, thus reducing the confounding effect from buoyancy to a great extent. The same experiment can be repeated over buoyancy acceleration as well, and therefore this set of swapping experiments would allow us to assess the relative importance of buoyancy and initial vertical velocity.

Based on this idea, we analyze the cloudy-updraft mass flux change in each of the following configurations: 1) keeping a parcel’s initial vertical velocity at 700 hPa the same but replacing its buoyancy with that of 300 hPa, 2) keeping a parcel’s buoyancy at 700 hPa the same but replacing its initial vertical velocity with that of 300 hPa, and 3) replacing both parcels’ initial vertical velocity and buoyancy at 700 hPa with those of 300 hPa. After replacement, we recalculate the cloudy-updraft mass flux change. As we are interested in the relative not the absolute change in the updraft mass flux, we simultaneously change the parcel properties in the control and perturbed cases so that we can obtain the relative mass flux change before and after changing each variable, and determine their relative contributions.

### c. Results and discussion

The cloudy-updraft mass flux before the swapping experiment is shown in Fig. 8. The computed updraft mass flux in all cases is binned to parcel’s initial vertical velocity for more direct comparison. The peak mass flux value occurs at about 3.5 m s^{−1} in the lower troposphere and 6 m s^{−1} in the upper troposphere, corresponding to the mean cloudy-updraft vertical velocities at these two heights. The convective response signal is represented by the mass flux reduction between the control and the perturbed cases. In the lower-troposphere case, updraft mass flux gets reduced by around 16% due to the imposed warm anomaly compared to the approximately 7% mass flux reduction in the upper-troposphere case, which is broadly consistent with the relative magnitude of different convective sensitivities between the lower- and the upper-troposphere cases observed in Fig. 1b and, for example, Tulich and Mapes (2010).

Figure 9 shows the new cloudy-updraft mass flux distribution at 700 hPa after replacing the cloudy parcel’s buoyancy acceleration with that of 300 hPa but keeping the initial vertical velocity unchanged (Fig. 9a), and replacing the parcel’s initial vertical velocity with that of the 300-hPa case but keeping the buoyancy acceleration unchanged (Fig. 9b). After replacing the parcel’s buoyancy acceleration with values from the 300-hPa case, the peak mass flux reduction in the perturbed run relative to the control run is now 10% compared to the original 16% reduction. This is because even though a parcel still has the same initial vertical velocity, its buoyancy generally increases after being replaced by its corresponding 300-hPa-case buoyancy, which helps the parcel gain more vertical momentum to overcome the barrier and leads to less mass flux reduction. The mass flux reduction is even smaller after we replace the parcel’s initial vertical velocity in the 700-hPa case with that from the 300-hPa case, in which the reduction goes down to 6.7%. Once we replace both the cloudy parcel’s initial vertical velocity and buoyancy acceleration in the lower troposphere with values from the upper troposphere, the mass flux reduction becomes almost negligible.

We also performed a similar experiment by replacing the parcel’s initial vertical velocity and buoyancy acceleration in the 300-hPa case with corresponding 700-hPa case values. Since parcels take on smaller vertical velocity and buoyancy values after the replacement, the cloudy mass flux is reduced more compared to the case without replacement, and the replacement of the initial vertical velocity causes more reduction (12%) than the replacement of buoyancy acceleration (8%). The above results are self-consistent and indicate that both buoyancy and initial vertical velocity play a role in determining the different convective responses to temperature perturbations at different heights of the free troposphere, but the initial updraft vertical velocity is a more important factor than buoyancy. These results indicate the importance of including cloudy-updraft vertical velocity in convective parameterization schemes in order to properly account for the differing convective sensitivities to temperature perturbations at different heights of the free troposphere, and can be used to train/validate different convection schemes in large-scale models.

## 4. Conclusions

To better understand why deep convection responds differently to large-scale temperature perturbations imposed at different heights of the atmosphere, we used Lagrangian particles to track the full history of cloudy parcels and a swapping technique to disentangle the relative contributions from highly correlated variables. It was found that a warm anomaly creates a buoyancy barrier that inhibits convective activities. In the lower troposphere, this anomaly not only affects local convective activities but also has extended effects higher up in the troposphere, which is in contrast to the much smaller and mostly local effects when the same anomaly is added in the upper troposphere.

Differing convective sensitivities to temperature perturbations in the lower versus upper troposphere have key consequences on the instability of convectively coupled waves. The nonlocal responses to temperature perturbations in the lower troposphere can overwhelm the more local and weaker responses to temperature anomalies in the upper troposphere. This allows for a positive upper-tropospheric convective heating anomaly in an upper-tropospheric region that is anomalously warm, thus producing potential energy, which could lead to unstable waves (e.g., Kuang 2010). In this paper, we have examined the roles of three factors in determining the different convective sensitivities to temperature perturbations in the lower versus upper troposphere: cloud liquid water, vertical velocity, and buoyancy. An idealized radiative scheme was used so that mean convective heating in the upper and lower troposphere was the same and thus could not explain the differing sensitivities.

We find that greater cloud liquid water in the lower troposphere is not a controlling factor because cloud buoyancy reduction peaks at similar values in response to the same imposed warm anomalies in the upper and lower troposphere. We find that differences in cloud updraft buoyancy and vertical velocity between the upper and lower troposphere both play a significant role, and vertical velocity turns out to be more important than buoyancy. Therefore, cloud updraft vertical velocities need to be correctly modeled in convective parameterization schemes. There have been a number of schemes that explicitly include cloud-scale velocities (Bretherton et al. 2004; Neggers et al. 2009; Chikira and Sugiyama 2010). However, a comprehensive understanding of the dominant balance of forces that regulates the vertical velocity distribution is still lacking, and how to appropriately account for the perturbation pressure terms in the vertical momentum budget still needs further investigation. These issues will be discussed in our forthcoming paper (Tian et al. 2018, manuscript submitted to *J. Adv. Model. Earth Syst.*).

We also apply the same analysis framework in Tian and Kuang (2016) to observe how deep convection responds to the imposed temperature perturbations, and we see that both shallow and deep convection share quite similar features in their responses to the imposed temperature anomalies. Furthermore, the empirical relationship for fractional entrainment rate proposed in Tian and Kuang (2016) is found to be valid for both shallow and deep convection, but differs in the empirical coefficient *α*, and this coefficient might be inferred from turbulent kinetic energy.

This study is focused on the responses of convection to small environmental perturbations. We have chosen 700 and 300 hPa to represent the lower and upper troposphere, respectively, because we can obtain good signal-to-noise ratios when imposing warm anomalies at these two heights. Selecting different heights in the lower and upper troposphere might change the results quantitatively, but we believe the qualitative picture of the relative importance among the different factors should remain the same.

## Acknowledgments

This research was supported by the Office of Biological and Environmental Research of the U.S. DOE under Grant DE-SC0018120 as part of the ASR Program and NSF Grant AGS-1649819. The authors thank Marat Khairoutdinov for making the SAM model available; Martin Singh, Kaighin McColl, and Giuseppe Torri for helpful comments; and two anonymous reviewers and the editor, Olivier Pauluis, for their detailed and careful reviews. The Harvard Odyssey cluster provided much of the computing resources for this study.

## APPENDIX A

### Reducing Drift in the Large-Scale Basic State

We prescribed time-invariant temperature and moisture tendencies to reduce drift in the large-scale basic state. The added tendencies were obtained iteratively by subtracting the rate of temperature and humidity drifts over the 2-h period from the initial large-scale forcing, until the drift becomes substantially smaller than the original drift (Fig. A1), compared to what was shown in Fig. 1. For example, the moisture drift over the 2-h period in the 700-hPa case is now maximized at 0.01 g kg^{−1}, compared to about 0.04 g kg^{−1} without prescribing the time-invariant temperature and moisture tendencies.

## APPENDIX B

### Description and Validation of the Lagrangian Model and Validation of Offline Calculations of the Crossing Percentage

The Lagrangian particles are embedded in the CRM and their trajectories are computed at every time step by first linearly interpolating the three-dimensional CRM velocity to the particle position using 1D interpolation (*w* is interpolated in the *z* direction, *u* is interpolated in the *x* direction, etc.) and then integrating using the second-order Adams–Bashforth scheme. Random-walk-associated subgrid-scale (SGS) turbulence is not included here as SGS diffusion is small compared to other terms in the height range that we consider. Each Lagrangian particle is treated as a point entity, and there is no interaction among different particles. The particle positions are then output every minute, along with snapshots of other 3D variables such as vertical velocity, buoyancy acceleration, and pressure gradient force.

Before using the Lagrangian parcel model to diagnose convective processes, which is done extensively in the paper, we verify that it can faithfully reproduce the Eulerian statistics.

Here, the subscript *i* refers to each individual particle; *p*′ is the perturbation pressure; *F* is the total acceleration term, which is the sum of Archimedean buoyancy and total perturbation pressure accelerations; and

Now we validate offline calculations of the crossing percentage based on Eqs. (1)–(3) against the crossing percentage computed directly from CRM outputs of the trajectories. With model output of particle trajectories, we know the particle location at each time step and can tell exactly whether or not a cloudy parcel can cross a barrier. In the offline calculation based on Eqs. (1)–(3), we assume that we only know the particle initial vertical velocity and associated accelerations along the trajectory without knowing whether or not it can transcend the barrier upfront. Therefore, we integrate the vertical momentum equation for each parcel from the time when it is at the bottom level forward. If this parcel succeeds in reaching the chosen top level with vertical velocity greater than 1 m s^{−1}, we would count this as one crossing. Otherwise, it would be counted as zero crossing, and the result is presented in Fig. B2. Except for parcels with very small initial velocities (smaller than 2 m s^{−1}), results from the offline calculation agree well with the crossing percentages calculated using CRM-output trajectories directly.

## APPENDIX C

### More Details on the Swapping Procedure

The swapping of properties is based on the probability distributions of parcel properties for the two cases. Figure C1 shows the probability density function of a cloudy parcel’s initial vertical velocity and buoyancy when it enters our chosen bottom layers in the lower- (700 hPa) and upper-troposphere (300 hPa) cases, respectively. The initial vertical velocity peaks around 3.5 m s^{−1} in the lower troposphere and 6 m s^{−1} in the upper troposphere. The distribution of the buoyancy acceleration peaks around 0 in the lower troposphere, which is consistent with the near-neutral buoyancy observation in the literature, and the cloudy parcel’s buoyancy acceleration is more positive in the upper troposphere.

Figure C2 shows an example of the nonlinear mapping used in the swapping experiment to map the buoyancy acceleration of cloudy parcels at 700 hPa to that of 300 hPa at the layer right below perturbation. Cloudy parcels in both the lower and upper troposphere are divided into 200 quantiles based on their buoyancy values in ascending order. The median buoyancy values within the same quantile in 700 and 300 hPa are paired up and represented as a dot in Figure C2. A third-order polynomial is used to fit this relationship, which provides the mapping for swapping the buoyancy. All cloudy parcels that enter the bottom layer at 700 hPa are rendered in the same fashion so that the resulting buoyancy PDF resembles that of 300 hPa. This mapping is done in a layer-by-layer manner. The mapping of the initial vertical velocity follows the same procedure, but it is only done for the bottom layer rather than for all layers across the imposed temperature perturbation.

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^{1}

Throughout the paper, the lower troposphere will refer to the part of the lower troposphere above the cloud base.