RESEARCH LETTER
10.1029/2024GL108233
Key Points:• Tropical temperature perturbations
near extreme deep convection rapidly
conform to convective quasi
equilibrium in a three‐hour window• Only the top 7% precipitating deep
convection can modulate hourly
tropical temperature patterns beyond a
1° radius• Top 1% precipitating deep convection
constrains nearby temperature
perturbations up to an 9° radius during
peak precipitation

When, Where and to What Extent Do Temperature
Perturbations Near Tropical Deep Convection Follow
Convective Quasi Equilibrium?
Yi‐Xian Li1,2, Hirohiko Masunaga2, Hanii Takahashi3, and Jia‐Yuh Yu1

1

Department of Atmospheric Sciences, National Central University, Taoyuan, Taiwan, 2Institute for Space‐EarthEnvironmental Research, Nagoya University, Nagoya, Japan, 3Jet Propulsion Laboratory, California Institute ofTechnology, Pasadena, CA, USA

Abstract Convective Quasi‐Equilibrium (CQE) is often adopted as a useful closure assumption to

Y.‐X. Li and J.‐Y. Yu,
Yi-Xian.Li@monash.edu;
jiayuh@atm.ncu.edu.twsummarize the effects of unresolved convection on large‐scale thermodynamics, while existing efforts toobservationally validate CQE largely rely on specific spatial domains or sites rather than the source of CQE
constraints—deep convection. This study employs a Lagrangian framework to investigate leading temperature
perturbation patterns near deep convection, of which the centers are located by use of an ensemble of satellite
measurements. Temperature perturbations near deep convection with high peak precipitation are rapidly
adjusted toward the CQE structure within the [ 2, 1] hours centered on peak precipitation. The top 1%
precipitating deep convection constrains neighboring free‐tropospheric leading perturbations up to 9°. Notable
CQE validity beyond a 1° radius is observed when peak precipitation exceeds the 93rd percentile. Thesefindings suggest that only a small fraction of deep convection with extreme precipitation shapes tropical free‐tropospheric temperature patterns dominantly.

Citation:

Plain Language Summary Convective Quasi‐Equilibrium (CQE) is a concept in atmosphericSupporting Information:
Supporting Information may be found in
the online version of this article.

Correspondence to:

Li, Y.‐X., Masunaga, H., Takahashi, H., &Yu, J.‐Y. (2024). When, where and to what
extent do temperature perturbations neartropical deep convection follow convective
quasi equilibrium? Geophysical Research
Letters, 51, e2024GL108233. https://doi.
org/10.1029/2024GL108233
Received 8 JAN 2024
Accepted 13 MAY 2024

science that explains a state where the influence of deep convection (cumulonimbus clouds) and large‐scaleatmospheric forces is balanced, causing certain thermodynamic properties to adjust toward specific reference
profiles. Previous studies have focused on how temperature changes relate to the CQE structure but in specificregions or sites while this study aims areas near deep convection—supposedly the source of CQE constraints.
Using a unique framework with data from multiple satellites, we track the evolution of temperature patterns near
deep convection and find temperatures near deep convection with extreme rainfall are adjusted toward the CQE
structure rapidly within 3 hr of maximum rainfall. However, only the deep convection with top 7% extreme
rainfall can effectively affect nearby temperature pattern beyond 1°, with the top 1% influencing up to an 9°
radius. These findings highlight the dominant impact of a small fraction of deep convection, particularly thosewith extreme rainfall, on nearby temperature perturbation patterns.

1. Introduction© 2024. The Author(s).
This is an open access article under theterms of the Creative Commons
Attribution‐NonCommercial‐NoDerivs
License, which permits use anddistribution in any medium, provided theoriginal work is properly cited, the use isnon‐commercial and no modifications or
adaptations are made.

LI ET AL.

The Convective Quasi‐Equilibrium (CQE) theory, first introduced by Arakawa and Schubert (1974), posits thatconvective energy within cumulus ensemble remains in statistical equilibrium, balanced between large‐scale
replenishment and cloud‐scale consumption. Intrinsic to the equilibrium, moist convection actively steers vertical temperature perturbations toward specific reference profiles, a principle embedded in various moist
convective adjustments (Ahmed et al., 2020; Betts, 1973; Betts & Miller, 1986; Kuo, 1974; Manabe et al., 1965)
and parameterizations (Chikira & Sugiyama, 2010; Frierson, 2007; Moorthi & Suarez, 1992; Randall &Pan, 1993; T. Wu, 2012; G. J. Zhang & McFarlane, 1995; Zhao et al., 2018). Such adjustment of vertical temperature structures is facilitated by analytic solutions (Emanuel et al., 1994; Yu & Neelin, 1997) to develop
tropical intermediate complexity models (Neelin & Zeng, 2000; Sobel & Neelin, 2006; Zeng et al., 2000) and has
shown to have a profound effect within deep convective areas by observations (Holloway & Neelin, 2007; Liet al., 2022; W. Wu et al., 2006; Xu & Emanuel, 1989).
Deep convection, often characterized by its robust updraft core and expansive cirrus anvil canopy, has predominantly been studied using satellite observations to discern its thermodynamic characteristics across temporaland spatial scales (Del Genio & Kovari, 2002; Feng et al., 2011; Houze et al., 2015). Collocating polar‐orbiting
and geostationary satellites enables the monitoring of three‐dimensional thermodynamic structures within deep
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convection (Chakraborty et al., 2016; Chung et al., 2008; Takahashi & Luo, 2014), where the Mesoscale
Convective System (MCS) plays a crucial role — contributing over half of tropical precipitation (Fenget al., 2021; Nesbitt et al., 2006; Roca et al., 2014; Schumacher & Rasmussen, 2020; Yuan & Houze, 2010). The
presence of a convective core and its characteristics, including depth, width, intensity, and shape, influences the
behavior of MCSs (D. Wang et al., 2020; Zheng et al., 2018). Algorithms utilizing geostationary satellites havebeen employed to track the convective core and its associated MCS generating comprehensive global data sets forresearch communities (Feng et al., 2021; Fiolleau & Roca, 2013; Huang et al., 2018).
Despite extensive validations showing the proximity of tropical temperature perturbation profiles to those constrained by the CQE theory, the spatial domains were confined to specific sites or regions across observations, that
is, a Eulerian framework (Holloway & Neelin, 2007; Li et al., 2022; Nie et al., 2010) and models (Lin et al., 2015;
X. Wang et al., 2022). This leaves an intriguing gap unexplored: the immediate vicinity of tropical deep convection, presumably to be the primary force shaping temperature structures. This gap prompts us to ask: Do
temperature perturbations near tropical deep convection follow Convective Quasi Equilibrium? If so, when,
where, and to what extent do they exhibit this behavior? Addressing these questions is crucial for advancing ourunderstanding of the fundamental mechanisms driving convective processes and their influence on the broader
atmospheric dynamics. To bridge this gap, a Lagrangian framework integrating an MCS‐tracking database andCloudSat retrieval to pinpoint the center of deep convective systems is used to quantitatively assess when thetemperature perturbations, within a certain radius relative to the center, adhere to the CQE structure.

2. Data and Methodology
2.1. Data
The CloudSat satellite is equipped with a 94 GHz Cloud Profiling Radar (CPR) that detects cloud and precipitation particles. The CPR has a high‐resolution footprint of approximately 1.7 km along track and 1.3 km acrosstrack with a vertical resolution of 480 m. Its active sensing capabilities enable the radar data to provide detailed
vertical cloud structures. The Tracking Of Organized Convection Algorithm through a 3‐D Segmentation(TOOCAN) is a specialized tool developed for detecting and tracking MCS using infrared imagery from geostationary satellites (Fiolleau & Roca, 2013). The clustering method within TOOCAN utilizes an iterative processacross horizontal and temporal dimensions to decompose brightness temperature regions under 235 K into several
MCSs by repeating growing regions starting at 190 K with a 5‐K increment. To identify deep convection, radarreflectivity and cloud mask data from the CloudSat satellite's 2B‐GEOPROF product (Marchand et al., 2008), andmorphological parameters of MCS mass center locations (latitude, longitude, time) along the life cycles from the
TOOCAN database, are employed.
In this study, we examine the hourly temperature field using the European Center for Medium‐Range Weather
Forecasts' fifth global reanalysis (ERA‐5; Hersbach et al., 2020). In addition to temperature data across all
available pressure levels, we also extract total precipitation data from the ERA5 to identify the peak precipitation,
which further aligns the analyzed time series with hour 0 indicating the peak precipitation and conditions onstrong convection events. Both TOOCAN and ERA‐5 data are harmonized to a temporospatial resolution of
0.25° × 0.25° and hourly increments to ensure their congruence.
Within the scope of this study, only MCS objects with track of the mass center confined within 30° north andsouth in latitude over lands and oceans are examined. To align with the data availability across the CloudSat,
TOOCAN, and ERA‐5, we analyze data only for the year 2013. Note that only the ascending (daytime) observations of CloudSat at around local time 13:30 is used here due to its battery anomaly since 2011.
2.2. Locating Centers of Deep Convection
In this manuscript, we focus on well‐developed MCSs that contain at least one deep convective core (DCC),
detected by CloudSat, within its coverage at any given time during its lifespan recognized by TOOCAN. The
DCC criteria encompass continuous radar echo from cloud top to within 2 km of the surface, an echo of at least 10
dBZ above 10 km, and an attaching anvil horizontally spanning over 20 km with its base above 5 km, similar toprevious works (Takahashi & Luo, 2012; Takahashi et al., 2017, 2021, 2023). This integration of continuous
monitoring from geostationary satellites and vertical‐penetration ability from polar‐orbiting satellite prevents
misclassification based solely on cold brightness temperature (Liu et al., 2007) and facilitates accurate tracking of
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Geophysical Research Letters10.1029/2024GL108233

deep convection centers. Note that the TOOCAN data over the western South Pacific is not available because theroutine scanning schedule of the MTSAT‐2 satellite, being operated during the study period, did not allow as
frequent observations as optimal for cloud tracking in the southern hemisphere. Although the centers are foundusing integrating satellite products, subsequent Lagrangian analysis exclusively relies on ERA‐5 reanalysis data
due to its capacity to capture temporal evolution across a vast three‐dimensional domain equally inside andoutside clouds unlike infrared satellite sounding.
2.3. Characterizing Temperature Perturbations Near Deep Convection
All following calculations and illustrations in this section are conducted within a specified radius from theconvection centers. The radius is chosen from 1 to 10° (or roughly from 100 to 1,000 km) to study the horizontal‐
scale dependence. For each convection object, defined as an MCS recognized by TOOCAN with an embedded
DCC detected by CloudSat, we identify the peak precipitation hour at every grid throughout its life duration andextract the hourly temperature profiles within 24 hr before and after accordingly. The temperature perturbations
are obtained by subtracting a mean temperature profile averaged over the relative [ 24,24] hours within theradius. To investigate temperature behaviors influenced by convection intensity, a threshold for peak precipitation
exceeding a specific percentile, ranging from the 80th to 99th percentile, is calculated and applied across the radial
distance, relative hours, and convection objects. For each hour, the temperature perturbations conditioned on thepeak precipitation over the 368 observed deep convective objects at each level are regressed against those in thefree troposphere, defined between 850 and 200 hPa, resulting in a single regression coefficient. The verticalprofile of regression coefficients, same as that presented in Holloway and Neelin (2007), depicts the leading
hourly pattern of temperature perturbations observed within the radius, reflective of a specific convection
intensity.
2.4. Quantifying Similarity of Temperature Perturbations to the Theoretical CQE Structure
The theoretical temperature perturbation profile constrained by CQE, to be compared with the leading observational profile, is referred to as the A‐profile afterward for simplicity. The A‐profile is a function of temperature
profile under assumptions of hydrostatic approximation, ideal gas law, and Clausius–Clapeyron relation (seedetailed derivations in Li et al., 2022, modified from Yu & Neelin, 1997):
⎞
⎛
pLCL
⎟
⎜
1 + γ T pLCL ))T ′ ( p)p
1⎟⎜
A( p,T( p)) ≡exp ⎜ κ ∫= ( LCL )d ln p′ ⎟, p > pLCL ,⎠⎝1 + γ(T( p))p0
T ′ p0 )1 + γ(T(p′ ))
κ

(1)

p

and
p κ
A( p,T( p)) ≡ ( ) , p > pLCL ,p0(2)

where p is pressure, T temperature, T′ the temperature perturbation from climatology, p0 the reference level, pLCL2s lv
the lifting condensation level, γ ≡ c εepRwith ε ≡ RRdv = 0.622 the ratio of gas constant for dry air Rd to that for
T2
pd

v

water vapor Rv, es the saturation vapor pressure with respect to liquid, lv = 2.5 × 106 J/kg the latent heat ofvapourization, cpd = 1004 J/kg/K the specific heat of dry air at constant pressure, and κ ≡ cRpdd .
For simplicity, the individual A‐profile is calculated with p0 = 1,000 hPa and pLCL = 950 hPa by input oftemperature profile interpolated to a 5‐hPa interval at each grid and hour, without considering entrainment. To
compare with the regression coefficient profiles, the A‐profile is averaged within the [ 24, 24] hours and within
the given radius, then normalized to have unity root mean square over the free troposphere. Unless specifically
noted, the term A‐profile refers to the normalized A‐profile hereafter. These settings are considered practicalgiven the robust statistics of A‐profiles in the tropics and the nature of A‐profile illustrating proportions between
vertical levels (Li et al., 2022). Finally, to quantify similarity between the A‐profile and regression coefficientprofile, vertical spatial correlation and root‐mean‐square deviation (RMSD) are calculated over the freetroposphere.
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Geophysical Research Letters10.1029/2024GL108233

Figure 1. Number of extracted temperature profiles at each 0.25° grid within 9 degrees of tropical deep convection centers for(a) all instances regardless of peak precipitation and (b) instances where the peak precipitation surpasses the 99th percentile(c.f., method) in the year of 2013.

3. Results and Discussions
3.1. Spatial Distribution of Temperature Perturbations Extracted Near Deep Convection
Prior to comparing the A‐profile and regression coefficient profile for their similarity, we explore thegeographical distribution of collocated deep convection distribution to comprehend where the temperature perturbations are analyzed.
Figure 1a shows the count of extracted temperature profiles spanning [ 24, 24] hours near tropical deep convection within a specific radius of 9° from the deep convection centers, irrespective of precipitation intensity. The
9° radius is selected because it represents the maximum distance where the CQE constraint on temperature appears valid, as later detailed in Section 3.3. The pattern generally corresponds to the ITCZ climatology, with moredeep convection over the continents, especially the Amazon and west Africa, compared to the oceans. To further
examine the temperature structure in conjunction with extreme precipitation, Figure 1b manifests the number ofextracted samples over grids where peak precipitation exceeds the 99th percentile of all instances. Similarly, it
captures a greater prevalence of extreme convective columns over continents than over oceans. The sensitivity
tests with different radii demonstrate no significant changes on the geographic patterns in both cases (not shown),
where the land‐sea contrast of deep convection occurrences has been observed by previous studies (Liu &Zipser, 2005; Liu et al., 2007; Takahashi & Luo, 2014; Takahashi et al., 2017; Wang et al., 2019).
Note that Figure 1b does not mark where mass centers of deep convection or intense DCCs locate but directly
pinpoints the grids collocated with intense precipitation within the specified 9° radius. Compared to an examination at the MCS scale, which analyzes every grid within the radius of top 1%‐precipitating MCSs (not shown),
this analysis at the individual grid level notably reduces noise among temperature perturbations. Such a difference
likely arises due to inhomogeneous precipitation pattern within MCSs and the spatial discrepancy between thesatellite‐identified mass center and the ERA‐5 precipitation center. The missing data over the Southwest Pacific in
the TOOCAN data set (see Section 2.2), roughly between 115°E 175°E, may appear concerning because of
frequent identification of DCCs (Takahashi & Luo, 2014; Takahashi et al., 2017). However, this region

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Figure 2. (a) The A‐profile (dashed) and vertical profile of regression coefficients (colored) of the temperature perturbations
within an 9° radius conditioned on the peak precipitation exceeding the 99th percentile at each level against the verticallyaveraged free troposphere. The regression coefficient profile is calculated at 21 hr relative to the peak precipitation hour.
(b)–(g) As in panel (a), but for 14, 7, 0, 7, 14, and 21 hr, respectively. Note that lighter colors indicate hours farther awayfrom the 0 hr. (h) Collection of all the profiles shown in panels (a)–(g) for comparison. (i) The hourly time series of root‐
mean‐square deviation (black) and vertical spatial correlation (red) between the A‐profile and the regression coefficientprofile over the troposphere.

contributes relatively less to the global occurrence of tropical deep convection observed with coexisting highradar echo top height and low cloud top brightness temperature (Liu et al., 2007). Therefore, while the absence ofdata poses a constraint, its impact on the study's outcomes and conclusions could be regarded as minor.
3.2. Leading Observations and Their Similarity to the CQE Theoretical Structure
To derive the representative profile of observational temperature perturbations for comparison with the A‐profile,or the CQE theoretical structure, we adopt the regression method from Holloway and Neelin (2007). The
regression coefficients represent the slope of the regression and can be interpreted as expected temperature change
at the given level with a 1‐K change in the vertically averaged free tropospheric temperature. Figures 2a–2g
display the A‐profile (dashed line) and leading observational patterns (colored lines) obtained through theregression at specific hours relative to the peak precipitation with a 7‐hr interval, and Figure 2h collects all these
profiles for a comprehensive comparison. Note that the profiles are calculated using temperature profiles
conditioned on the 99th percentile peak precipitation (8.537 mm/hr) within an 9° radius of the deep convection
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centers, corresponding to the 14,915 occurrences distributed in Figure 1b. To quantify the similarity between the
regression coefficient profile and A‐profile, Figure 2i showcases the time series of RMSD and vertical spatial
correlation, both computed over the free troposphere between 850 and 200 hPa at each hour. Of special note is thatthe vertical spatial correlation is identical to the cosine similarity between profiles, and hence a positive correlation suggests a leading temperature perturbation profile that increases with height, mirroring the A‐profile
pattern within the free troposphere. All correlations mentioned in the text refer to the vertical spatial correlation.
Throughout most hours, the leading temperature perturbations tend to decrease with altitude in the free troposphere, opposing that seen in the A‐profile, as depicted in Figures 2a–2c and 2e–2g. The prevailing shallow
cumulus with substantial entrainment might be a cause of the bottom‐heavy structure (Lin et al., 2015; Singh &O’Gorman, 2013), which leads to negative correlations and elevated RMSD in Figure 2i. This suggests that at anhourly scale or within a day, the CQE principle is mostly limited (Donner & Phillips, 2003; Lin et al., 2015; G. J.
Zhang, 2003). In contrast, the highest correlation and lowest RMSD occur at the peak hour, flanked by abrupt
increases within the [ 4, 2] hour range. The correlations at the [ 2, 1] hours are found significantly different fromall hours beyond the [ 2, 1] hour window with a 95% confidence level using Fisher's Z‐test. This is consistent
with Figure 2d, which captures similar increasing perturbations with height between both profiles over the freetroposphere, remarkably closely aligned between 700 and 300 hPa. The convective cold top, marked by a negative
minimal perturbation around 100 hPa (Holloway & Neelin, 2007), remains consistently robust across all thehours. All the observational characteristics mentioned above hold true when assessed across different radii andpeak‐precipitation thresholds within the [89th, 99th] percentile range except for the higher correlations found fortighter radii and stricter thresholds (not shown). The rest of the manuscript will exclusively utilize the correlation
to assess the CQE validity on temperature, as RMSD exhibits a similar response with opposite trend.
3.3. CQE Validity as a Function of Relative Hour, Relative Distance and Peak Precipitation Percentile
We have demonstrated how top‐1%‐precipitating temperature perturbations align with the A‐profile near deep
convection, focusing on evolution of the vertical leading patterns. To provide a more comprehensive scrutiny, we
validate the proximity of temperature perturbations to the CQE structure using spatial vertical correlation as afunction of the hour relative to peak precipitation, distance relative to the deep convection centers, and thethreshold percentile of the peak precipitation.
Figure 3a suggests that the CQE robustly constrains the leading temperature perturbations within the [ 2, 1] hour.
For each radius, Fisher's Z‐test at a 95% confidence level is performed at each hour to identify significantly
different hours by comparing the correlation at each hour with those at all the hours 3 hours before and after. The
positive correlation reaches the farthest distance of 9° during the peak hour with significant differences observed
within [2, 9] degrees, along with the maximum correlation of ∼0.86 among all hours. The robustness of the CQE
constraints on temperature within the 3 hr aligns with the timescale of convective adjustment commonly
considered (see Section 5b in a review of Arakawa, 2004). This supports the CQE principle that convective
adjustment is relatively fast compared to large‐scale forcing. Interestingly, the influence of the CQE on temperature appears more profound one hour before the peak precipitation, where the positive correlation reaches 9°
with significant differences within [3, 10] degrees, compared to one hour after, where it reaches 7° with significant difference limited within [4, 8] degrees. In addition, the correlations at the 2 hr are significantly positive
within [ 3,7] degrees while those at the 2 hr are all negative without significance. Such asymmetric horizontal
extent of validity can be observed by conditioning on percentiles equal to or higher than 95% (not shown). The
CQE constraints on temperature quickly deteriorate after 1 hour following the peak precipitation, causing temperature perturbations to deviate from the CQE more rapidly than the build‐up of positive correlations before thepeak hour. This aligns with the observations that the peak of the first baroclinic mode, or deep convective mode
favoring the CQE structure, is followed by the peak of second baroclinic mode, which on the contrary disfavors
the CQE (Masunaga & L’Ecuyer, 2014). Note that, while the second baroclinic mode could act against CQE, thesmoothing effect of horizontal temperature perturbations is still efficient by means of fast dry gravity waves.
Notably, other positive correlations appear around [ 22, 15] and [12, 17] hours, primarily confined within a 1°
radius. The hourly time series of maximum precipitation among the top 1% precipitating grids suggests that thisphenomenon is likely due to a few extreme precipitation events which happen to peak around these hours (notshown) while these minor peak correlations are more pronounced when considering a smaller radius and higher

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Figure 3. (a) Vertical spatial correlation between the regression coefficient profiles using temperature perturbations
conditioned on peak precipitation exceeding the 99th percentile and the A‐profile over the free troposphere. Each box
indicates the correlation at a specific hour relative to the peak precipitation (x‐axis, spanning from 24 to 24 hr in 1 hrincrements) within a certain radius with respect to the deep convection center (y‐axis, extending from 1 to 10° in 1°
increments). The black box indicates the [ 2, 1] hours interval and the black dots represent the hours that are significantly
different. (b) As in panel (a), but the correlations are calculated at the peak hour using temperature perturbations exceeding
different percentile thresholds of the peak precipitation (x‐axis, ranging from 80% to 99% in 1% increments).

peak precipitation (not shown). Note that these secondary peaks are not statistically significant (i.e., not hatched in
Figure 3a).
Figure 3b demonstrates that during the hour of peak precipitation, only the leading temperature perturbations withthe top 10% peak precipitation exhibit a comparable pattern of increasing perturbations with height, akin to the
CQE structure within a 1° radius, while only those in the top 7% extend beyond 1°. This suggests that not all
convective objects can effectively adjust the neighboring temperature through the CQE constraints, but only theextreme ones among deep convection. This is consistent with the “circus tent” concept, which suggests that deep
convection with the highest free‐tropospheric saturation moist static energy (or the sub‐cloud moist static energy,
theoretically equivalent as expected from the CQE), play a dominant role in convective adjustment processes inthe tropics (Bao et al., 2021; Williams et al., 2023; Zhou & Xie, 2019). The high threshold of peak precipitations
and the minor peaks of positive correlations in Figure 3 both reinforce our understanding that most of the CQE
constraints arise from a very small fraction of deep convection, consistent with previous literature (Bao &Stevens, 2021).

4. Concluding Remarks
Although previous studies have extensively examined the validity of CQE on temperature, many of them havefocused on specific spatial domains or sites, that is, a Eulerian framework, rather than directly addressing thesource of CQE constraints—deep convection. This study aims to investigate the evolution of leading temperature
perturbation patterns near deep convection, consisting of MCSs identified by stationary satellites and deep
convective cores observed by the CloudSat at a time. By employing a Lagrangian framework following the deep
convection centers, this approach enables the quantification of when, where, and to what extent these perturbations resemble the CQE structure. Our key findings for the broad tropics (30°N–30°S) can be succinctly
summarized as follows:

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•
•

•

10.1029/2024GL108233(When) Conditioned on the top 1% peak precipitation and within the relative [ 2, 1] hours,(Where) temperature perturbations obeying the CQE structure, defined as a positive vertical spatial correlation
between the free‐tropospheric leading observational and analytic theoretical profiles, reaches a distance up
to 9°,(To what extent) accompanied by higher correlations before the peak precipitation than after, with a maximum
correlation of ∼0.86 during the peak hour.

These results suggest that temperature perturbations near deep convection are rapidly adjusted toward the CQE
structure in a few hours, consistent with the idea that the weak temperature gradient approximation allows tropicalgravity waves to rapidly propagate strong signals from deep convection to affect the surrounding environmental
temperature (Ahmed et al., 2021; Bretherton & Smolarkiewicz, 1989; Sobel et al., 2001; Y. Zhang & Fueglistaler, 2020). Within such a short timeframe, on the order of an hour, the CQE's influence on temperature isnoticeable only when analyzing grids where peak precipitation exceeds the 89th percentile, implying that only asmall fraction of deep convection is capable of influencing the neighboring temperature over distances greaterthan 100 km. Temperature perturbations with the top 1% peak precipitation near deep convection conform to the
CQE structure up to 9° from the centers during the peak hour, as one might expect from the typical Rossby radiusof deformation, which ranges from hundreds to thousands of kilometers.
Most importantly, this study underscores the dominant role of deep convection with extreme precipitation in
shaping the leading patterns of tropical free‐tropospheric temperature. Our approach reflects the composite
temporospatial pattern of deep convective clusters under different precipitation intensities. The findings suggestthat the observed invalidity of CQE in a fine temporal resolution in previous studies, where non‐deep‐convective,
ordinary‐deep‐convective, and extreme‐deep‐convective events are not carefully separated from each other,leading to a deviated macrostate from an equilibrium dominated by extreme deep convection. On the other hand,the current method in this paper actively focuses on the prime impacts of intense deep convection that aredeliberately chosen for analysis. It is important to note that these two perspectives do not conflict; the traditionalview tends to be a passive observation over a mixed environment with occasional random deep convection. Since
deep convection occurs frequently and randomly across the tropics, our approach bridges the gap between the roleof individual convective systems and their collective consequences on the tropical regions.
It is important to note that the conclusion largely depends on the selection of domain. When narrowing down fromthe broad tropics (30N–30S) to equatorial (10N–10S) band, the vertical pattern of regression coefficients does notalter much across the [ 24, 24] hours and degrees of radii, and resembles that of Figure 2d with high correlation
with the A‐profile (see Figure S1a). The pattern with high correlation exhibits for any threshold of peak precipitation percentile larger than zero while for a direct comparison, Figure S1c maintains the same abscissa values
used in Figure 3b. As the deep tropics is more favorable for deep convection, the temperature perturbations tend to
be closer to the A‐profile in the upper troposphere. Such high CQE validity may be also associated with the highfrequency of extreme precipitation events, which tends to occur within the deep tropics (c.f., Figure 1b). With thehelp of weak temperature gradient effect, deep tropical grids are frequently and continuously covered by the large
impact radius of extreme convections, shaping the tropospheric temperature close to the A‐profile at any hour (seeFigure S1b). For the deep tropics (10°N–10°S), the conclusion of “when, where, and to what extent” changes to•••(When) peak precipitation is non‐zero, the peak‐hour temperature perturbations(Where) obeying the CQE structure is ubiquitous,(To what extent) accompanied by positive vertical spatial correlations throughout the [ 24, 24] hours.

However, to discuss and yield conclusions applicable for a broad tropical region, the manuscript focuses on thefindings using the domain of 30° north and south as Section 2.1 states.
The current study provides an interesting angle in understanding how valid the CQE constrains tropical free‐tropospheric temperature near the deep convection in a Lagrangian view. However, owing to the requirementof monitoring vertical temperature structure changes with a high spatiotemporal resolution, the ERA‐5 reanalysis
data is utilized here instead of comparable satellite observations as those used during the collocation of deep
convection. Also, although the collocation strengthens our confidence of collecting well‐developed MCSs
coincided with deep convective cores at a certain time point, one cannot assure that deep clouds always exist
around the mass centers along the evolution. Overall, we consider the methodological framework to be highly
optimized for such an analysis but future work on improving the collocating procedures and expanding the

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studying period, even toward how the CQE validity might change under a climate‐change scale, is needed tofurther understand the relationship between the CQE and deep convection.

Data Availability Statement
Information about the CloudSat can be accessed through the CloudSat Data Processing Center, including thelevel‐2 radar data used here (Marchand et al., 2008). Details regarding the Tracking Of Organized Convection
Algorithm using a 3‐dimensional segmentation are available (TOOCAN, 2019). The hourly temperature andprecipitation data from ERA‐5 are publicly accessible (Hersbach et al., 2020).
Acknowledgments
YXL was supported by the National
Science and Technology Council in
Taiwan through Grants 111‐2111‐M‐008‐
062 and 112‐2917‐I‐564‐008. JYY wassupported through Grants NSTC112‐2811‐
M008‐026. We express our gratitude to Dr.
Thomas Fiolleau for his valuable
discussions regarding the TOOCAN
database and for granting public accessto it.

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