APRIL 2024

783

MASUNAGA AND TAKAHASHIThe Energetics of the Lagrangian Evolution of Tropical Convective Systems
HIROHIKO MASUNAGAaab

AND HANII TAKAHASHI

Institute for Space-Earth Environmental Research, Nagoya University, Nagoya, Japan
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Californiab(Manuscript received 22 August 2023, in ﬁnal form 17 January 2024, accepted 26 February 2024)
ABSTRACT: The convective life cycle is often conceptualized to progress from congestus to deep convection and develop
further to stratiform anvil clouds, accompanied by a systematic change in the vertical structure of vertical motion. This archetype scenario has been developed largely from the Eulerian viewpoint, and it has yet to be explored whether the same
life cycle emerges itself in a moving system tracked in the Lagrangian manner. To address this question, Lagrangian tracking is applied to tropical convective systems in combination with a thermodynamic budget analysis forced by satelliteretrieved precipitation and radiation. A new method is devised to characterize the vertical motion proﬁles in terms of thecolumn import or export of moisture and moist static energy (MSE). The bottom-heavy, midheavy, and top-heavy regimes
are identiﬁed for every 18 3 18 grid pixel accompanying tracked precipitation systems, making use of the diagnosed column
export/import of moisture and MSE. The major ﬁndings are as follows. The Lagrangian evolution of convective systems isdominated by a state of dynamic equilibrium among different convective regimes rather than a monotonic progress fromone regime to the next. The transition from the bottom-heavy to midheavy regimes is fed with intensifying precipitation
presumably owing to a negative gross moist stability (GMS) of the bottom-heavy regime, whereas the transition from themidheavy to top-heavy regimes dissipates the system. The bottom-heavy to midheavy transition takes a relaxation time ofabout 5 h in the equilibrating processes, whereas the relaxation time is estimated as roughly 20 h concerning the midheavy
to top-heavy transition.
KEYWORDS: Tropics; Convection; Convective-scale processes; Satellite observations

1. Introduction
In the general context of ﬂuid dynamics, convection is often
treated as a bulk ensemble of unresolved turbulent eddies
while their net effect on the heat and/or momentum budget
on resolved scales is explicitly analyzed. Atmospheric moist
convection serves as an example in that it is often argued to
be an implicit ingredient in driving the tropical atmosphere ina quasi-equilibrium thermodynamic state (e.g., Arakawa andSchubert 1974; Emanuel et al. 1994). Nonetheless, moist convection is more than unresolved turbulence. Precipitating
clouds are occasionally developed into a long-lived, organized
convective system, with its internal structure changing systematically over time during the life cycle. The structural evolution of mesoscale convective systems (MCSs) typically begins
with an intensifying stage characterized by cumulus congestus
clouds growing into deeper convective towers, followed by amature stage consisting of deep convective cores and extensive stratiform anvils (Leary and Houze 1979).
This well-established picture of convective evolution is
known to prevail over a broad range of spatial and temporalscales across the hierarchy of tropical convective disturbances(Mapes et al. 2006). Such examples include the quasi-2-day
wave (Takayabu et al. 1996), the convectively coupled Kelvin

Supplemental information related to this paper is available
at the Journals Online website: https://doi.org/10.1175/JAS-D-230141.s1.
Corresponding author: Hirohiko Masunaga, masunaga@nagoyau.jp

wave (Straub and Kiladis 2003), and the Madden–Julian oscillation (Lin and Johnson 1996). Congestus clouds, deep convection, and stratiform anvils each have distinct heating
proﬁles, giving rise to a bottom-heavy or top-heavy thermal
forcing to the atmosphere depending on the proportion atwhich convective/stratiform rain contributes to the total precipitation (Schumacher et al. 2004; Back et al. 2017). These
impacts of heating proﬁles on the environment and the resulting feedback on convective dynamics have been incorporated into a school of theory on convectively coupledequatorial waves (Mapes 2000; Khouider and Majda 2006;
Kuang 2008).
It should be noted that the archetype of the convective life
cycle as illustrated above has been conceptualized largely
from the Eulerian viewpoint, although not without exception(e.g., Hannah et al. 2016). In reality, MCSs move around at a
speed of around 10 m s21 at tropical latitudes (Feng et al.
2021), implying that they migrate over a distance of a few
hundred kilometers during a time span of 10 h. Given that alifetime of 10 h and a size of O(100) km fall within a typicalparameter range of organized convective systems (e.g.,
Chen and Houze 1997), an MCS is well capable of traveling
beyond its own horizontal extent in the course of its life cycle. It follows that the Lagrangian frame is no less appropriate than the Eulerian for tracking the evolving structure of
MCSs.
Automated methods to track convective systems in a sequence of satellite images have been developed over decades.
A prototype algorithm dates back to the early years of geostationary Earth-orbiting (GEO) satellites (Woodley et al. 1980),
which was later reﬁned by subsequent studies (Williams andHouze 1987; Mapes and Houze 1993; Machado et al. 1998). A

DOI: 10.1175/JAS-D-23-0141.1
Ó 2024 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding
reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).
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number of novel approaches have emerged in recent years(Futyan and Del Genio 2007; Feng et al. 2012; Fiolleau andRoca 2013a,b; Heikenfeld et al. 2019; Takahashi et al. 2021,
among others), opening pathways to a spectrum of new applications of the Lagrangian cloud tracking with a focus on radiation (Feng et al. 2011; Bouniol et al. 2016, 2021), precipitation(Feng et al. 2021; Takahashi et al. 2021), high clouds (Luo andRossow 2004), and convective dynamics (Takahashi and Luo
2014; Elsaesser et al. 2022).
Our central interest in this study lies not in the cloud-scaleevolution of an individual MCS but in the dynamic processes
governing the interactions between the atmosphere and an
ensemble of clouds that reside in it. More speciﬁcally, the focus is put on how a convecting atmospheric column gains orloses moisture and moist static energy (MSE) and how it
evolves over the Lagrangian life cycle of convective systems.
To this end, the composite life cycle of convection constructedfrom a convection tracking dataset is combined with a moisture and MSE budget analysis forced by satellite-retrieved
precipitation and radiation. A precipitation-based tracking
method is employed in order to capture congestus rainfallthat would be difﬁcult to detect with infrared-based tracking.
On the other hand, the dissipating stage of MCS accompanied
only by nonprecipitating anvil clouds would be left out of theanalysis.
Of particular interest is whether the well-known Eulerian
scenario emerges itself in the Lagrangian framework as well.
As will turn out, the Lagrangian evolution of tropical organized convection is quite different from what one might expect from conventional knowledge. Our ﬁndings are detailed
below, following an outline of the Lagrangian tracking database and the methodology of thermodynamic budget analysis.

2. Lagrangian tracking of convective systems
a. IMERG-CT
The present work adopts the Integrated Multisatellite
Retrieval for GPM (IMERG)-based Convection Tracking(IMERG-CT) dataset (Takahashi et al. 2021). IMERG, ahalf-hourly, 0.18 3 0.18 global precipitation product, provides
surface precipitation estimates derived from a constellation ofsatellite passive microwave observations that are compiled
with GEO and LEO infrared measurements and are adjusted
to rain gauge network data over land (Huffman et al. 2020).
Lagrangian tracking is applied to the IMERG V06 precipitation, which is a unique signature of IMERG-CT because mostof the existing cloud-tracking methods are built upon infrared
imagery alone. The caveat is that IMERG precipitation is notentirely independent of satellite infrared measurements,
which the IMERG algorithm relies on to ﬁll in gaps in sporadic passive-microwave retrievals. Nevertheless, IMERG-CT
offers an advantage to this work since precipitation is a key
variable in the MSE budget analysis outlined later. In theoriginal IMERG-CT algorithm, an additional constraint wasapplied that infrared brightness temperature should decreasebeyond 245 K at least one time during the lifetime. This

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TABLE 1. Number of IMERG-CT samples sorted by lifetime.
Each PC family (or life cycle) is counted as a single sample.
Lifetime (h)
0.5–3
3–6
6–9
9–12
12–15
15–18
18–21
21–24

Number of samples
205 51464 37318 164
7313
3811
2010
1148
712

condition does not signiﬁcantly affect the results and is no longer applied to this study.
The algorithm to produce the IMERG-CT dataset works asfollows. First, a precipitation core (PC) is identiﬁed as a contiguous area of considerable precipitation ($5 mm h21). PCs
are tracked in a manner based on 3D (i.e., 2D in space and 1Din time) 10-connectivity segmentation as devised by Fiolleau
and Roca (2013a): If the eight pixels surrounding a reference
PC pixel and the two temporally adjacent pixels (one before
and another after) all meet the PC criterion, these 10 pixels
are considered to be part of the same PC family as the reference pixels. Repeating this procedure to all qualifying pixels
until unable to continue any further, one eventually ﬁnds a 3D
structure of pixels connecting together in space and time,
which as a whole is recorded to constitute a single family ofPCs. This PC family deﬁnes the life cycle of a convective system, which potentially contains the merging and/or splitting ofclouds at some point of the evolution.
Precipitation systems (PSs) are then deﬁned on a snapshotby-snapshot basis by extending the search for contiguous precipitating pixels beyond each PC until a lowered threshold of
0.5 mm h21 is met. By construction, any PS must enclose atleast one PC in itself, with the possibility that multiple PCs
may be embedded when PSs sought around different PCs in
the neighborhood end up in the same system. Each PS isthought of as a member of the same family as its “seeder PC”
belongs to. A more thorough description of the algorithm isfound in Takahashi et al. (2021).

b. Composite life cycle of PSs
A large population of PC/PS families are collected from the
IMERG-CT database for ﬁnding the general properties of
convective systems and their evolution over life cycle. The period analyzed is 2 years from 1 January 2007 to 31 December
2008, with the study area being tropical latitudes between
208S and 208N across all longitudes over both ocean and land.
The size of IMERG-CT samples is listed in Table 1, where
the population of PC families is broken down by lifetime
every 3 h. It is noted that a lifetime shorter than 30 min and a
PC area smaller than 400 km2, hardly resolved given the native resolution of IMERG (half-hourly and 0.18), are discarded from the analysis. It is evident from Table 1 that the
samples become increasingly fewer as the lifetime increases.
Lifetimes longer than 24 h are not included in the analysis

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Precipitation System Area
3.5

PS area [x105 km2]

15h
18h
21h
24h

3h
6h
9h
12h

3
2.5
2
1.5
1
0.5
0

6

12

18

24

Lifestage [h]FIG. 1. Composite life cycle of PS area size sorted by lifetime as
indicated in the inset legend, where 3–6 h is labeled as “6h” as an
example.

because statistical robustness is difﬁcult to ensure for exceptionally long-lived systems.
A set of composite life cycles are constructed by averaging all
the detected PC/PS families at each life stage. This is done separately for different lifetimes as classiﬁed in Table 1. Figure 1
shows the temporal evolution of the PS area throughout thecourse of life cycle. The PS area for long-lived systems grows

until it reaches a maximum and then shrinks over time, with the
maximum being larger and later as lifetime is elongated. This isa signature known in the literature (e.g., Machado et al. 1998;
Inoue et al. 2009). Short-lived systems are not precisely in line withthis rule of thumb for unknown reasons. PSs are found to be typically on the order of 105 km2 in area or a few hundred kilometersin length scale.
The composite life cycle of precipitation is presented inFig. 2. The precipitation averaged over PS (Fig. 2a) and the
maximum precipitation within PS (Fig. 2b) each follow a quasisymmetric bell-shaped curve over life cycle resembling the areal evolution shown above (Fig. 1). The bell-like symmetry,however, breaks down especially for longer-lived PSs when the
maximum and average precipitations are combined into theirratio, which steeply rises during the ﬁrst few hours regardlessof lifetime (Fig. 2c). A notable feature salient for long-lived
systems is that the maximum-to-average precipitation ratio exhibits an extended period of plateau during the hours of enhanced precipitation. It is noted that Fig. 2 would remain
qualitatively unchanged if plotted separately for the PSs overocean and those over land, although the land composite issomewhat noisy due to a small sample size (see Fig. S1 in the
online supplemental material).
The plateau came as a surprise at ﬁrst to the authors, as
there seemed no physical reason. What gives rise to this plateau, then? An explanation does exist, however, to qualitatively interpret this behavior. Let us consider a simple
conceptual model of MCS comprising an intensely precipitating convective core and a lightly raining stratiform anvil,
P 5 fc Pc 1 (1 2 fc )Ps ,

50

4

3

(c) Max-to-Average Ratio
14
12
Pmax /Pave

5

40

30

2
12
18
Lifestage [h]

24

10
8
6

20
6

(1)

where P is the system-averaged precipitation, Pc and Ps areconvective and stratiform precipitation, respectively, and fc isthe fractional cover of the convective core. Since the systemmaximum precipitation occurs in a convective area and hence
Pc may be considered to vary roughly in proportional to the
maximum precipitation in this simplistic model, the maximumto-average precipitation ratio rP is(b) Maximum Precipitation
60

Pmax [mm/h]

Pave [mm/h](a) Average Precipitation
6

0

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4
0

6

12
18
Lifestage [h]

24

0

6

12
18
Lifestage [h]24FIG. 2. Lifetime-sorted composite life cycle of (a) PS-averaged precipitation, (b) PS-maximum precipitation, and (c) the ratio of the maximum
precipitation to the average precipitation.

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rP ~

Pc
P


5 fc 1 (1 2 fc )

Ps
Pc

21
:

(2)

For a well-developed MCS where Pc well exceeds Ps, Eq. (2)can be simpliﬁed further into
rP ~

1
:
fc

(3)

Figure 2c and Eq. (3) together lead to a working hypothesisthat some mechanism might be at work to keep fc nearly constant, as necessitated to account for the plateau of rP, throughout the mature stage of long-lived convective systems.
The initial rise and subsequent plateau of rP are consistent
with Elsaesser et al. (2022), who showed that fc from the
Global Precipitation Mission (GPM) Dual-frequency Precipitation Radar (DPR) product decreases to a minimum toward
the mature stage and stays nearly constant in the rest of evolution. This unexpected behavior suggests a mechanism for
the Lagrangian convective evolution that has yet to be explored in depth. To this end, we next examine the dynamic
processes governing the Lagrangian life cycle of convective
systems. The above hypothesis will be revisited in light of theanalysis results in section 5.

3. Diagnosis method of convective regimes
A useful approach to look into the dynamics of moist convection is to examine the column processes driven by verticalmotion (Chikira 2014) and the associated variability in the
moisture and MSE budget. A bulk portion of moisture resides
in the lower troposphere, so that an upward motion at low
levels, associated with a lower-tropospheric convergence, always imports moisture into the atmospheric column regardless of the depth of ascent. In contrast, MSE is exported outof the column by a top-heavy ascent, while MSE is exported
only to a lesser extent or can be even imported when the ascent is bottom-heavy because MSE increases toward both thesurface and tropopause from a midtropospheric minimum inthe typical tropical sounding. The fact that the column MSE
variability is sensitive to the vertical proﬁle of vertical motion
has been exploited in investigating the large-scale dynamics
associated with the convective life cycle (e.g., Masunaga and
L’Ecuyer 2014; Inoue and Back 2015, 2017).
A new analysis strategy is devised in this study, loosely built
upon the moisture and MSE budget diagnosis of Masunaga
and L’Ecuyer (2014), but readjusted, so it ﬁts the Lagrangian
tracking. The physical basis of the algorithm is ﬁrst outlined.

a. Formulation of diagnostic equations
The methodological basis is founded on the vertically integrated moisture and MSE budget equations,
Dt hqi 1 hv­p qi 5 E 2 P,

(4)

Dt hhi 1 hv­p hi 5 S 1 LE 1 hQR i,(5)respectively, where

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Dt ; ­t 1 vH ? =H
is the Lagrangian derivative operator with the subscript H referring to the horizontal component,dph···i ; ···gis the vertical integral to the whole thickness of the atmosphere, q is the vapor mixing ratio, h is the MSE per unit
mass, v is the vertical p velocity, E is the surface evaporation
ﬂux, P is the precipitation, S is the sensible heat ﬂux, QR is
the radiative heating rate per unit mass, and L is the speciﬁc
latent heat of vaporization. The Lagrangian framework is
adopted because it is better suited for the convection-tracking
analysis than the Eulerian approach. It is worth noting thatthe horizontal advection term is combined into the Lagrangian derivative and does not explicitly appear in the current
formulation.
For the sake of putting focus on the convection-induced
variability, the clear-sky component is subtracted from the allsky counterpart for all atmospheric and surface variables andis denoted by prime as, for instance,hQR i ; hQR i 2 hQR iclear ,(6)

where hQRi deﬁnes the column-integrated atmospheric cloudradiative effect (ACRE). The clear sky is assumed to be in its
own balanced state,
Dt hqiclear 1 hv­p qiclear 5 Eclear ,Dt hhiclear 1 hv­p hiclear 5 Sclear 1 LEclear 1 hQR iclear :Convective dynamics is deﬁned as an anomaly against this hypothetical clear sky representing the local background statewith a weak subsidence. Equations (4) and (5) are rewritten
by subtracting the clear-sky component as
Dt hqi 1 hv­p qi ’ 2P,(7)

Dt hhi 1 hv­p hi ’ hQR i :(8)Here, E and S have been omitted because evaporation ﬂux
and sensible heat ﬂux are much less sensitive to cloudinessthan precipitation and radiation. Precipitation is absent in clear
skies, so P 5 P at all times. Equations (7) and (8) claim thatthe moisture and MSE budget equations for the primed variables are forced by precipitation and the column-integrated
ACRE, respectively. As evident from its deﬁnition, ACRE refers to the effect on the atmosphere alone in this paper andshould not be confused with the cloud forcing to the whole
Earth’s system including the surface.
The vertical motion v is decomposed into the ﬁrst two baroclinic modes with all higher-order modes ignored,v 5 v1 1 v2 ,(9)where half a wavelength (a full wavelength) is equivalent to
the depth of the troposphere for v1 (v2). The sign and relative

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(a) Normalized GMS
1(b) Precipitation inefficiency

J1
J2

1
0.8
K1 and K2

0.5
J 1 and J 2

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0

-0.5

K1
K2 (q import)
K2 (q export)

0.6
0.4
0.2
0

-1
40

45

50

55

60

65

70

0

500 1000 1500 2000 2500 3000
LP [W/m2]

CWV [mm]

FIG. 3. (a) Normalized GMS as a function of CWV (mm) and (b) precipitation inefﬁciency as a function of LP (W m22).

magnitude of the two vertical modes govern the top- or bottomheaviness in the v proﬁle as will be detailed later.
A few closure assumptions are applied to the budget equations to make them solvable. First, the horizontal temperature
variability is neglected in the MSE storage term,
Dt hhi ’ LDt hqi :(10)This approximation is well justiﬁable at tropical latitudes(Charney 1963). Second, the vertical MSE advection is related
to the vertical moisture advection by invoking a version of thenormalized gross moist stability (GMS) (Raymond et al.
2007),
hvi­p higi ; 2,
Lhvi­p qi

(11)where i (51, 2) refers to the vertical modes. Using Eqs. (10)and (11), MSE is replaced by the vapor mixing ratio in thebudget Eqs. (7) and (8) as
Dt hqi 1 hv1­p qi 1 hv2­p qi 5 2P,
LDt hqi 2 g1 Lhv1­p qi 2 g2 Lhv2­p qi 5 hQR i :

(12)
(13)

Finally, the Lagrangian tendency of moisture is parameterized bya prescribed parameter h deﬁned for each of the two modes as
Dt hqi 5 2h1 hv1­p qi 2 h2 hv2­p qi ,(14)or, equivalently,
P 5 2(1 2 h1 )hv1­p qi 2 (1 2 h2 )hv2­p qi :By design, hi refers to the proportion of the dynamic moistening by vertical motion that is left behind in the atmospherewithout precipitating out (Masunaga and Sumi 2017). In thispaper, hi is called the precipitation inefﬁciency.
Eliminating the time-derivative terms in Eqs. (12) and (13)with (14), one obtains diagnostic equations for the ﬁrst- andsecond-mode components of the column-integrated vertical
moisture advection,(g2 1 h2 )LP 2 (1 2 h2 )hQR i
,
Lhv1­p qi 5 2(g2 1 h2 )(1 2 h1 ) 2 (g1 1 h1 )(1 2 h2 )(15)

(g1 1 h1 )LP 2 (1 2 h1 )hQR i
:(g2 1 h2 )(1 2 h1 ) 2 (g1 1 h1 )(1 2 h2 )(16)

Lhv2­p qi 5

Equations (15) and (16), driven by precipitation and ACRE on
the rhs, have the practical utility that the column moistening
owing to vertical motion is driven by satellite-based retrievalsof precipitation and ACRE without explicit knowledge of q
and v.
In calculating Eqs. (15) and (16), gi and hi are prescribedby a simple formula. The normalized GMS is given as a function of column water vapor (CWV) or hqi (mm),g1 5 7:55 exp(20:061hqi),(17)

g2 5 20:75,(18)

derived from a ﬁt to satellite observations of CWV from
AMSR-E measurements and radiative heating from the
CloudSat/CALIPSO product (for details, see Masunaga andMapes 2020). It is noted that the clear-sky component was not
explicitly considered in Masunaga and Mapes (2020), but instead, a background state has been set aside in their formulation, in which the ﬁrst and second baroclinic modes are
considered as a perturbation to the background state. The
clear-sky and background states are deemed to be practicallyequivalent to each other in that both represent a quasi-staticreference state against which convective dynamics (i.e., theﬁrst and second modes) is deﬁned.
The sharp decline of g1 with increasing CWV, as visuallyevident in Fig. 3a, arises from the fact that the low-level import of MSE increases with CWV and hence offsets to agreater degree the upper-level MSE export in the presence ofa deep ascent penetrating the troposphere. On the other
hand, the vertical advection of MSE is roughly proportional

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to that of moisture when vertical motion is conﬁned to lowlevels, leading to a practically constant value of g2. Of particular importance is that g2 is negative, implying that the second
baroclinic mode has the potential of driving an unstable
growth of moist-convective circulation (Raymond et al. 2009)
unless, since the GMS as currently deﬁned is based solely on
the vertical component, the vertical MSE advection is morethan offset by the horizontal advection.
Precipitation inefﬁciency is also known to have contrasting
behaviors between the ﬁrst and second modes. It was shownby Masunaga and Sumi (2017) using the DYNAMO sounding
array data (Johnson and Ciesielski 2013) that the dynamic
moistening by a deep, ﬁrst-mode ascent is nearly entirely consumed as precipitation, whereas hardly so by a shallow ascent.
To reﬂect this, the present work employs their empirical parameterization with simpliﬁcation,

h1 5 0,and(19)

h2 5 0:8 1 0:2 exp(21023 LP),

for hv2­p qi , 0,h2 5 1,for hv2­p qi $ 0,(20)

for LP (W m22). As plotted in Fig. 3b, h2 slightly decreases
from unity as precipitation picks up, accounting for a nonlinearly increasing moisture with intensifying rainfall as
found in the literature (e.g., Bretherton et al. 2004; Holloway
and Neelin 2009; Masunaga 2012). This formula is implemented where the second-mode ascent moistens the column(i.e., hv2­pqi , 0), but otherwise no precipitation is expected
to occur and h2 is ﬁxed at one. The present results are insensitive to details in the characterization of h1 and h2 as far
as h1 stays close to zero and h2 does not deviate too far
from unity.
Note that h2 is conditioned on the sign of hv2­pqi , which
formally requires the prior knowledge of h2 in Eq. (16). In
practice, g2 1 h2 and 1 2 h2 in Eq. (16) are always positive
for all possible values of g2 and h2 as given by Eqs. (18) and(20). It follows that the sign of hv2­pqi may be determineduniquely regardless of h2.
Equations (15) and (16) constitute a simple model to derive
conventionally unobservable variables (vertical advection of q
and h) from satellite-retrieved parameters (P and hQRi). Key
assumptions behind this model and their validity against in
situ sounding array data were discussed in depth by Masunaga
and Sumi (2017). They found that the observations overall support the mode-speciﬁc precipitation inefﬁciency (h1 and h2)
and GMS (g1 and g2) as currently assumed. The same ideas
were later applied with updates to the model of Masunaga andMapes (2020), which was demonstrated to capture fundamental behaviors in observations.

b. Data
The diagnostic equations [Eqs. (15) and (16)] require surface
precipitation both inside and outside the PSs (see section 2 forthe deﬁnition of PS). Surface precipitation is taken from the
IMERG v6 dataset (Huffman et al. 2020). CRE integrated
over height hQRi is obtained from the Clouds and the Earth’s

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Radiant Energy System (CERES) Synoptic 18 (SYN1deg) data
product (Doelling et al. 2016). The CERES SYN1deg productis an hourly, 18 3 18 gridded dataset of the radiative ﬂux ﬁelds
in the all- and clear-sky atmospheres, derived with the cloud
and aerosol observations from GEO and LEO satellite imagers and calibrated to the TOA ﬂuxes from the CERES instruments. In the CERES SYN1deg algorithm, the clear-sky
radiance is obtained from radiative transfer calculations withall clouds removed under the otherwise identical atmosphericcondition to the all-sky radiance. CWV taken from the hourly
ﬁfth major global reanalysis produced by ECMWF (ERA5)
data (Hersbach et al. 2020) is used for evaluating g1 in
Eq. (17). All the variables are averaged to a common
18 3 18 grid across the global tropics from 208S to 208N andinterpolated over time to half-hourly steps during thewhole 2 years of 2007 and 2008.

c. Adaptive domain
The aim of the present budget analysis is to assess the net
thermodynamic effects of moist convection on the atmospheric environment of O(100–1000) km in horizontal scale.
This line of research has been an area of active interest in
tropical meteorology, originating from early work based onﬁeld campaign soundings (e.g., Yanai et al. 1973). A geographically ﬁxed array of observations, however, is impractical for applications to the Lagrangian tracking of migrating
clouds. This study instead introduces an adaptive domain or astudy domain varying in size and location as it chases after
moving convective systems.
An adaptive domain is deﬁned as the minimum longitude 3
latitude rectangle enclosing each PS from the IMERG-CT
database. As such, grid pixels in the vicinity of a PS are
included together with the PS itself as part of the large-scale
environment constituted of both raining and rain-free components. If multiple PSs from the same family (see section 2afor deﬁnition) emerge in the same snapshot, the minimum
rectangle is adjusted so that it encompasses all those PSs at atime. This operation prevents any adaptive domain from being
divided into multiple subdomains nor absorbed into anotherdomain for a given PS family throughout its lifetime. The
adaptive domains are updated every half hour to adjust themselves to deforming PSs as they move. The four sides of a domain are required to be a multiple of 18 in longitude andlatitude, so the domain precisely ﬁts in a block of 18 3 18 gridboxes. Figure 4 schematically illustrates an adaptive domain
with a PS embedded inside.
The adaptive domain area can greatly exceed the enclosed
PS area when the PS is heavily elongated and/or split into separate systems. To assess the impact of such cases on the composite statistics, a test analysis was run with the “outlier”
cases excluded in which the areal ratio of PS to the surrounding adaptive domain is lower than a certain threshold, say,
30%. By exclusion of these events in which precipitation tends
to be heavily diluted out, the domain-averaged precipitation
composited from the remaining samples is found to be somewhat higher than in the control case. The key results otherwise
turn out to be practically unchanged (not shown).

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Bottom-heavy (BH)
Mid-heavy (MH)Convective regimes

Top-heavy (TH)Precipitation System (PS)1o1oAdaptive Domain

FIG. 4. An example schematic of the adaptive domain (see section 3c) and convective regimes embedded in thedomain (section 3d).

d. Bottom-heavy, midheavy, and top-heavy regimes
The net export (or import when the sign is reversed) of moisture and MSE is a useful measure of the thermodynamic consequences of vertical motion, founding the basis of three
convective regimes summarized in Fig. 5. The column-integrated
moisture export and MSE export are obtained by combining
the ﬁrst two baroclinic modes,
moisture export : hv­p qi 5 hv1­p qi 1 hv2­p qi ,(21)

MSE export : hv­p hi 5 2g1 Lhv1­p qi 2 g2 Lhv2­p qi :(22)
Three convective regimes1 are diagnosed depending on thesign of hv­pqi and hv­phi . Considered ﬁrst is the “positive”
second baroclinic mode, where the mode is assigned a positive
sign when comprising a low-level ascent and an upper-levelsubsidence. The positive second baroclinic mode, in tandemwith a coexisting ﬁrst baroclinic ascent, gives rise to a verybottom-heavy v proﬁle with a layer of updraft conﬁned mostly
to the lower troposphere. A low-level convergence and a midlevel divergence accompanying this shallow circulation would
import both moisture and MSE into the atmospheric column,that is, hv­pqi , 0 and hv­phi , 0. This case is classiﬁed into
the bottom-heavy regime. Although deep, vigorous convectionis unlikely to dominate the bottom-heavy regime, a combined
effect of the moisture and MSE imports, implying a negative
GMS, potentially triggers a rapid growth of moist convection(Raymond et al. 2009).
The midheavy regime arises when the layer of updraft develops so deep that it brings about a divergence in the upper
troposphere. This is associated with a net export of the column MSE while importing moisture into the column, that is,1

The fourth regime, that is, the moisture export combined with
the MSE import, rarely occurs in the currently analyzed data andhence is not considered.

hv­pqi , 0 and hv­phi , 0. A caveat is that the sign of hv­phi
depends on a delicate balance between the lower-tropospheric
convergence and upper-tropospheric divergence of MSE, which
can be individually large but nearly cancel each other out in avigorous deep ascent. The resulting smallness of hv­phi would
make horizontal MSE advection dominant over the verticalcomponent in the Eulerian column import of MSE. It is reminded that the horizontal advection does not explicitly appear
in the Lagrangian formulation of the MSE budget [Eq. (5)], formally allowing that the column import and export of MSE are
differentiated depending solely on the sign of hv­phi .
The vertical motion structure turns into a more top-heavy
proﬁle as the negative second mode, made up of an updraft
aloft and a downdraft below, comes into play. A midlevel
convergence sandwiched by high- and low-level divergences
associated with such a top-heavy structure results in a net
export of both moisture and MSE from the column. These
conditions, hv­pqi , 0 and hv­phi , 0, deﬁne the top-heavy
regime.
Vertical motion is tied with moist convection to the extent
that large-scale adiabatic cooling is offset by cloud diabatic
heating. Shallow and congestus clouds are presumed to dominate the bottom-heavy regime (Takayabu et al. 2010), whilethe v proﬁle in the midheavy regime is accounted for almost
exclusively by the ﬁrst baroclinic mode, ascribable dynamically to deep convective heating (Schumacher et al. 2004).
Top-heavy v proﬁles, on the other hand, are typical of a mature MCS comprising deep convective cores and extensive
stratiform anvils (Mapes and Houze 1995). The characteristic
cloud types representative of each regime are included inFig. 5.
The three convective regimes are identiﬁed with Eqs. (21)and (22) on each 18 3 18 pixel (see Fig. 4) making use of the
IMERG precipitation, the CERES ACRE, and the ERA5CWV via Eqs. (15)–(18). This procedure is repeated to all
pixels that fall within each adaptive domain. The results arethen projected onto the composite life cycle illustrated in
section 2b.

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JOURNAL OF THE ATMOSPHERIC SCIENCES

Bottom heavy (BH)

Mid heavy (MH)p

VOLUME 81

Top heavy (TH)

p

p

Z

Z

Moisture import&
MSE import

Z
Moisture export&
MSE export

Moisture import&
MSE export

FIG. 5. Schematic illustration of convective regimes, namely, bottom-heavy, midheavy, and top-heavy, based on the sign of moisture/
MSE import. Thick curves are the v proﬁles typical of each regime, with the sign ﬂipped so that a positive value implies an updraft. Grayarrows indicate the horizontal winds inferred from the v proﬁle under mass conservation. Cloud cartoons show the cloud type characteristic of each regime (see text for details).

4. Energetics of the Lagrangian life cycle of convection
The main results from the analysis are presented in this section.

a. Phase diagram of convective regimes
It follows from Eqs. (15)–(22) that the column-integrated
moisture and MSE exports are determined once precipitation(LP) and ACRE (hQRi ) are given. Figure 6a shows thecolumn moisture import mapped onto the LP–hQRi plane asderived from Eqs. (15)–(22). CWV must be known to estimate g1 by Eq. (17), so the 2-yr mean CWV projected onto2

the LP–hQRi plane is ﬁrst constructed using the same datasets as employed in the analysis (section 2b). In general, moisture is imported into an atmospheric column where ACRE islarge while exported for smaller ACREs, with the boundary
in between shifting to a higher ACRE as precipitation increases. It is expected from Eqs. (15) and (16) that the contours would be linear if gi and hi were constant, but in reality,
they are heavily curved. This arises primarily from the CWV
dependence of g1 in Eq. (17). CWV systematically increases
with precipitation (Bretherton et al. 2004) and with ACRE
(Masunaga and Bony 2018) in the tropical atmosphere, leading
2(a) q import [kW/m ](c) Phase diagram(b) h import [kW/m ]

150

1

150

3

150

Bottom-heavy

0
50
-0.5

0

-1
0

1000

2000

3000

100

1
0

50

-1
-2

0

-3
0

1000

2

2000
2

LP [W/m ]LP [W/m ]3000

Net ACRE [W/m2]

100

Net ACRE [W/m2]

Net ACRE [W/m2]2
0.5

Mid-heavy100Top-heavy

50

0
0

1000

2000

3000

2

LP [W/m ]FIG. 6. (a) The column-integrated moisture import hv­pqi derived from Eqs. (15)–(22) as a function of precipitation (LP; W m22) and
ACRE (hQRi ; W m22). The moisture import (export) (in kW m22) is shaded in red (blue), overlaid by solid (dashed) contours at an interval of 0.25 kW m22. The thick curve depicts the zero contour. (b) As in (a), but for the MSE import and export. (c) Phase diagram as
derived from (a) and (b).
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(b) Net ACRE

800

80

750

75

700

70

ACRE [W/m2]

LP [W/m 2](a) Precipitation

650
600
550

65
60
55

500

50
0

6

12
18
Lifestage [h]240(c) Precipitation
10

12
18
Lifestage [h]

24

200

10

150

1500
1
1000
500

ACRE [W/m2]

2000
2

6

(d) Net ACRE

2500

LP [W/m ]791

MASUNAGA AND TAKAHASHIAPRIL 2024

100
1
50
0

0

0.1
0

6

12
18
Lifestage [h]

24

-50

0.1
0

6

12
18
Lifestage [h]24FIG. 7. The composite life cycle of domain-averaged (a) precipitation (W m22) and (b) ACRE (W m22), where different curves represent different lifetimes (cf. Table 1 and Fig. 1). The composite life cycle of 18-scale (c) precipitationand (d) ACRE in PDF (shaded) and cumulative PDF (contoured every 10 percentiles from the 10th to the 90th) withthe domain average overlaid (dashed) for reference. Only the lifetime of 24 h is shown for visual clarity.

to a gradient of g1 in the precipitation–CRE plane. The P
dependence of h2 also has an effect of distorting the contours,
although only to a lesser extent, as the effect is conﬁned to near
the upper-left corner of the plot.
A net import of MSE occurs when precipitation is relatively
weak and ACRE is large, but otherwise MSE is exported out of
the column (Fig. 6b). Combining Figs. 6a and 6b, one obtains
the “phase diagram” of the bottom-, mid-, and top-heavy regimes mapped on the LP–hQRi plane (Fig. 6c). The bottomand midheavy regimes are partitioned by the curve on which the
MSE import vanishes, while the mid- and top-heavy regimes are
separated where the moisture import gives way to the export.
This phase diagram serves as a quick reference (in a statisticalsense) for ﬁnding out which of the three regimes a given observation ﬁnds itself in and for predicting when one regime transitions to another as precipitation and ACRE change over time.

b. Composite life cycle
The composite life cycle of precipitation and ACRE is plotted in Fig. 7. The evolution of precipitation (Fig. 7a) is byconstruction parallel to the PS-averaged precipitation shown
earlier in Fig. 2a, except that spatial averaging is applied to theadaptive domain containing the environment as well as the PSs
it surrounds. A bell-shaped curve of precipitation is absent in
ACRE (Fig. 7b), indicative of relatively thin anvil clouds that
extend beyond the raining cores toward the end of the life cycle(Protopapadaki et al. 2017), masking the variability intrinsic toprecipitation. Precipitation and ACRE span the ranges of
550–750 and 60–70 W m22, respectively, during the composite
life cycle, which may appear to be conﬁned in a somewhat limited parameter space when projected on the phase diagram(Fig. 6c). It is noted, however, that the pixel-by-pixel PDFs ofprecipitation and ACRE are each spread over a much wider
range, skewed toward small values for precipitation (Fig. 7c)
while distributed more or less symmetrically about the mean for
ACRE (Fig. 7d). The evolutionary tracks of individual eventswould thus have a broad swath sweeping through the phase diagram when remapped on the LP–ACRE plane.
Among the factors responsible for the statistical spread in
the PS properties is diurnal cycle. The composite evolution of

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JOURNAL OF THE ATMOSPHERIC SCIENCES(b) Mid-heavy(c) Top-heavy

0.8

0.8

0.6

0.6

0.6

0.4
0.2
0
-0.2

(1/P)(dP/dt) [1/h]0.8(1/P)(dP/dt) [1/h](1/P)(dP/dt) [1/h](a) Bottom-heavy

VOLUME 81

0.4
0.2
0
-0.2

-0.4
6

12
18
Lifestage [h]

24

0.2
0
-0.2

-0.4
0

0.4

-0.4
0

6

12
18
Lifestage [h]

24

0

6

12
18
Lifestage [h]24FIG. 8. The composite life cycle of the precipitation growth rate (h21) for the (a) bottom-heavy regime, (b) midheavy regime, and(c) top-heavy regime. Different curves represent different lifetimes (cf. Table 1 and Fig. 1). See text for a detailed deﬁnition of the precipitation growth rate.

ACRE broken down into different local times experiences salient diurnal variability, especially in the shortwave component
as expected (Fig. S2). The diurnal variation, however, is largely
averaged out when all local times are combined together.
To glimpse the dynamics unique to each convective regime,
the domain-averaged precipitation growth rate for a regime R
is examined by
R
1 dP 
1
1 Pj (t 1 Dt) 2 Pj (t)(t) ’
cosyj ,∑
P dt
PDt
∑ cosy j2R j (t)
j2R

(23)

j

where the subscript j refers to a block of 18 3 18 grid pixels
within the adaptive domain, yj is the latitude of the jth grid
pixel, and t is the life stage in hours with Dt being the temporalinterval ﬁxed at half an hour. The regime R is diagnosed at
the life stage of t in Eq. (23) regardless of whether the grid
pixel evolves into another regime at t 1 Dt. If the grid pixel
falls outside the updated adaptive domain at t 1 Dt, however,the precipitation growth rate for that grid pixel is removed
from the statistics.
Figure 8 shows the composite life cycle of the precipitation
growth rate. The growth rate consistently stays positive in thebottom-heavy regime (Fig. 8a), as predicted from the negativeness of GMS intrinsic to a very bottom-heavy v proﬁle. In
contrast, the precipitation growth rate remains positive onlyin the ﬁrst half of the lifetime before turning to negative during the rest of life cycle in the midheavy regime, with its magnitude remaining small throughout (Fig. 8b). The decaying
phase begins even earlier and is more prominently negative inthe precipitation growth rate for the top-heavy regime (Fig. 8c).
This is physically plausible in that moist convection is expected
to be no longer dynamically sustainable as the atmosphere exports both moisture and MSE in the top-heavy regime.
Less expectedly, however, the life cycle progresses withoutaccompanying any appreciable shift in occurrence from regime to regime. The relative occurrence of each regime iscomputed at every life stage as∑ cosyjf R (t) 5

j2R∑ cosyj

,

(24)

j

where the summation in the denominator is applied to all grid
pixels within the adaptive domain. The composite life cycle of
the regime-by-regime occurrence is shown in Fig. 9. The regime occurrence is found to be surprisingly ﬂat over time regardless of lifetime, setting aside a barely discernible increase
for the bottom-heavy regime and a subtle decrease for the
top-heavy regime. It is indicated that the Lagrangian evolution of precipitation as we have seen does not involve any
noticeable “regime shift” as one might naively anticipate fromthe conventional scenario of the convective life cycle.
Figure 10 shows the PDF of vertically integrated vertical
moisture advection (2Lhv­pqi ) and vertical MSE advection(2hv­phi ). The probability distribution is nearly invariant
over time except where moisture import exceeds 90% in cumulative PDF (the uppermost contour in Fig. 10a) and where
MSE export is exceptionally large (below 10% or the lowermost contour in Fig. 10b). Recalling that the sign of these variables is diagnosed to deﬁne the convective regimes, one ﬁnds
that the ﬂatness of the convective regime occurrence (Fig. 9)
originates from the quasi-steadiness in the statistical distribution of the column processes.
The results above do not necessarily imply that each regime
remains “frozen” in itself throughout the life cycle. Figure 11
illustrates what is actually taking place behind the seemingly
quasi-static evolution in light of the transition rate from one
regime to another,lR"R (t) 5∑

j2R"R

cosyj∑ cosyj

,

(25)

j

where R " R stands for the pixels that belong to the regime
R at t but transition to another regime R at t 1 Dt. By

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(a) Bottom-heavy(b) Mid-heavy(c) Top-heavy

60

60

50

50

50

40
30
20
10

Occurrence [%]60

Occurrence [%]Occurrence [%]793

MASUNAGA AND TAKAHASHIAPRIL 2024

40
30
20
10

0
6

12
18
Lifestage [h]

24

30
20
10

0
0

40

0
0

6

12
18
Lifestage [h]

24

0

6

12
18
Lifestage [h]24FIG. 9. As in Fig. 8, but for the relative occurrence (%) of each regime.

construction, the rate of temporal change in the regime occurrence is related to the transition ratios as
df BH
5 (lMH"BH 2 lBH"MH )dt
1 (lTH"BH 2 lBH"TH ) 1 s BH ,

(26)

df MH
5 (lTH"MH 2 lMH"TH )dt
1 (lBH"MH 2 lMH"BH ) 1 s MH ,

(27)

df TH
5 (lBH"TH 2 lTH"BH )dt
1 (lMH"TH 2 lTH"MH ) 1 s TH ,(28)

where BH, MH, and TH denote bottom-heavy, midheavy,
and top-heavy, respectively. The last term sR summarizes thesource and sink arising from the pixels emerging into or disappearing from the adaptive domain as the PS migrates and expands/shrinks as time proceeds from t to t 1 Dt. In practice,
the adaptive domain only barely moves over a time step of
30 min given a typical migration speed of 10 m s21. For easeof discussion, the sR terms are not considered from now on.
Figure 11a shows a pair of transition rates inside the ﬁrst
parenthesis in Eq. (26). The transition rate from midheavy tobottom-heavy stays around 8% until dropping down toward
the end of the life cycle. This is, however, counteracted almost
precisely by the inverse transition. The transition rates from
top-heavy to midheavy (Fig. 11b) and from bottom-heavy to
top-heavy (Fig. 11c) share the common feature that the transitions in opposite directions are virtually a mirror image to theother. It follows that the regime-to-regime transitions largely
cancel out when combined into a pair within each parenthesis
in Eqs. (26)–(28). These cancellations mathematically explain
why the occurrence of each regime is devoid of any notable(a) vertical q advection(b) vertical h advection
100

1000
10
500
1
0

-500

0.1
0

6

12
18
Lifestage [h]

24

500

-<wdh/dp> [W/m 2]-<wdq/dp> [W/m 2]

1500

100

0
10
-500
1
-1000

-1500

0.1
0

6

12
18
Lifestage [h]24FIG. 10. (a) The composite life cycle of 18-scale vertically integrated vertical moisture advection 2Lhv­pqi in PDF(shaded) and cumulative PDF (contoured every 10 percentiles from the 10th to the 90th) with the domain averageoverlaid (dashed) for reference. Positive (negative) values mean a moisture import into (export out of) the column.
Only the lifetime of 24 h is shown for visual clarity. (b) As in (a), but for vertical MSE advection 2hv­phi .

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JOURNAL OF THE ATMOSPHERIC SCIENCES(b) MH <-> TH

MH->BH
BH->MH0

6

12
18
Lifestage [h]

24

5
4
3
2
1
0
-1
-2
-3
-4
-5

.

TH-> MH

MH->TH06(c) TH <-> BH

Transition rate [%/h]Transition rate [%/h]Transition rate [%/h](a) BH <-> MH
10
8
6
4
2
0
-2
-4
-6
-8
-10

12
18
Lifestage [h]VOLUME 81

5
4
3
2
1
0
-1
-2
-3
-4
-5

24

BH->TH

TH->BH0

6

12
18
Lifestage [h]24FIG. 11. The composite life cycle of the transition rate (% h21) between two selected regimes: (a) bottom-heavy and midheavy, (b) midheavy
and top-heavy, and (c) top-heavy and bottom-heavy. Different curves represent different lifetimes (cf. Table 1 and Fig. 1). See text for a detailed
deﬁnition of the transition rates.

It was discussed near the end of section 2 that a plateau in
the maximum-to-average precipitation ratio (Fig. 2c) might
imply that the fractional cover of vigorous convective cores in
an MCS should be nearly invariant over a span of time. This
speculation agrees with the current result in which the relative
occurrence of each convective regime including the midheavy,
relevant to deep convective heating, stays practically constant(Fig. 9).
Figure 12 shows the precipitation breakdown into the three
convective regimes,∑ Pj cosyj

PR (t) 5

j2R∑ cosyj

:

(29)

j2R

The midheavy regime consistently produces more intense
rainfall than the other regimes, making a major contribution
to the total precipitation. The predominance of the midheavy
regime, backed up by its bell-shaped evolutionary patternreminiscent of the PS-averaged precipitation (Fig. 2a), supports the presumption for Eq. (3), namely, Pc .. Ps.
The population of different cloud types is known in the literature to evolve systematically during the Lagrangian life cycle of convection. Regardless of spectral windows in use suchas infrared brightness temperature (Machado et al. 1998), infrared and visible radiances (Luo and Rossow 2004), and infrared split-window channels (Inoue et al. 2009), these studies

Regime-sorted Precipitation
2000
1750

MH

1500
2

5. Discussion

all agree that deep convective cores initially dominating thecloud system gradually give way to cirrus clouds that eventually thin out toward the end of the life cycle. It is not entirely
clear how to reconcile the apparent contradiction between themonotonic progression of cloud population and the dynamic
equilibrium among convective regimes. Among possible explanations is that every 18 3 18 grid box would contain multiple types of clouds at a time, hindering a speciﬁc cloud typefrom mapping uniquely onto a certain convective regime.

LP [W/m ]temporal evolution (Fig. 9), despite the fact that transitions
are active at all times. This result leads to a somewhat surprising conclusion that the three convective regimes are in dynamic equilibrium throughout the Lagrangian life cycle ofmoving convective systems rather than evolving monotonically from one stage to the next.

1250
1000
750
TH

500
250

BH

0
0

6

12
18
Lifestage [h]24FIG. 12. The composite life cycle of precipitation (W m22) separated by different convective regimes of bottom-heavy (dotted),
midheavy (solid), and top-heavy (dashed). Different curves represent different lifetimes (cf. Table 1 and Fig. 1).

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(a) Precipitation(b) Net ACRE

6000

140

5000

120

4000

2

ACRE [W/m ]2

LP [W/m ]795

MASUNAGA AND TAKAHASHIAPRIL 2024

3000
2000

100
80
60

1000

Domain Ave.
Precip Max.

0

40
0

6

12
18
Lifestage [h]

24

0

6

12
18
Lifestage [h]24FIG. 13. The composite life cycle of (a) the domain-maximum precipitation (solid; W m22) and (b) the ACRE
cosampled with the domain-maximum precipitation (solid; W m22). The domain-averaged precipitation and ACRE
are reproduced from Fig. 7 (dashed) in each panel for comparison. Different curves represent different lifetimes(cf. Table 1 and Fig. 1).

More importantly, the convective regimes deﬁned in termsof the v top-heaviness are different in nature from the cloud
regimes or weather states identiﬁed from the cloud properties(Jakob and Tselioudis 2003; Rossow et al. 2005). Nonprecipitating clouds such as cirrus clouds, for instance, are not amajor explanatory factor to differentiate the v proﬁles as
illustrated in Fig. 5. Precipitating clouds are more likely linked
to large-scale dynamics, but this linkage is largely left unexplored in observations of the convective life cycle. A notable
exception is the latent heating proﬁles from the GPM DPR
product analyzed by Elsaesser et al. (2022), which were found
not to change drastically in top-heaviness during the lifetime(see their Fig. 2). The stability of heating structure is physically consistent with the invariance in the fractional occurrence of each convective regime (Fig. 9).
A fundamental question yet to be addressed is how the midheavy precipitation intensiﬁes and dissipates (Fig. 12). The midheavy regime by deﬁnition has a positive GMS (section 3d), in
which case precipitation is tightly coupled with the diabatic forcing to the atmosphere (Neelin and Held 1987). This is proven by
combining Eqs. (7) and (8) under the steady-state approximation (Dt ’ 0) into
LP ’ 2Lhv­p qi ’

hQR i
,g(30)

where g is the nondecomposed normalized GMS [cf. Eq. (11)],hv­p hig ;2:
Lhv­p qiIt is inferred from Eq. (30) that precipitation varies in proportion
to the diabatic forcing, which is reduced to ACRE in the current

formulation of the MSE budget Eq. (8), to the extent that g isdeemed as invariant. The domain-mean ACRE, however, is notquite in parallel with the bell-shaped curve of precipitation as observed (Fig. 7b). This qualitative discrepancy may not be entirely
unexpected, given that the bell shape is attributed mainly to a relatively small number of intense events as evidenced by the cumulative PDF of precipitation shown by Fig. 7c.
Figure 13 presents the composite pattern of the domainmaximum precipitation and the cosampled ACRE. ACRE isfound to exhibit a salient bell-shaped feature when sampled exclusively with heavy precipitation, now matching the theoretical expectation of Eq. (30). For LP ; 5000 W m22 (Fig. 13a)and hQRi ; 120 W m22 (Fig. 13b), g is estimated from
Eq. (30) to be ;0.024. This value is even smaller than g1 couldbe (see Fig. 3a), necessitating a more bottom-heavy v proﬁlethan the ﬁrst-baroclinic mode typical of the midheavy regime.
Heavy precipitation associated with such a bottom-heavy ascent is interpreted as a burst of convection invigorated as thebottom-heavy regime, which by deﬁnition has a negative GMS
and potentially nurtures a self-sustaining growth of convection,transitions to the midheavy regime. This might be among theprocesses lying behind the fact that the heaviest rains tend to
be produced not by the deepest convection but by relatively
bottom-heavy clouds (Hamada et al. 2015).
To examine the time scale of this process, let us consider ahypothetical system that initially consists only of the bottomheavy regime developing into the midheavy regime. The rate ofchange in the midheavy occurrence in Eq. (27) is simpliﬁed as
df MH
5 lBH"MH ,
dt

(31)

assuming that all other processes are absent. The relaxation
time with respect to this transition is deﬁned from Eq. (31) as

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JOURNAL OF THE ATMOSPHERIC SCIENCES

t BH"MH
;
MH

f MH

l

BH"MH

:

(32)

If the midheavy regime then transitions into the top-heavy regime, the relaxation time regarding this second transition isestimated similarly ast MH"TH
;
MH

f MH

l

MH"TH

:

(33)

It is found from Figs. 9b and 11a that f MH ’ 40% h21 andlBK"MH ’ 8% h21 at t 5 0. Substituting these estimates into
Eq. (32), one obtains

t BH"MH; 5 h:
MH(34)

Applying lMH"TH ’ 2% h21 (Fig. 11b) to Eq. (33) leads tot MH"TH; 20 h:MH(35)

Equation (34) suggests that the transition from the bottomheavy regime, supposedly a feeding agent for convective intensiﬁcation, takes the relaxation time of 5 h before substantiallyinﬂuencing the midheavy regime through the equilibratingprocesses. This estimate is roughly consistent with the time ittakes for the maximum-to-average precipitation ratio to climb
up before reaching a plateau (Fig. 2c). The relaxation time
tBH"MHis hence a measure of the time required for the midMH
heavy regime to “metabolize” itself through the transition
with the bottom-heavy regime. The second time scale of 20 h
in Eq. (35) is longer than the ﬁrst, suggesting in theory that thesystem-maximum precipitation gives way to the decaying stage
more slowly than the earlier intensifying stage. This is conﬁrmed by the gradual decline of the maximum-to-average ratio toward the end of the life cycle (Fig. 2c). Note thatequilibration is expected to be completed only when the relaxation time is shorter than the system lifetime.
The dynamic equilibrium scenario in the Lagrangian convective life cycle marks a contrast with the conventional
scenario of MCS evolution built largely on the Eulerian perspective. The two pictures may seem intuitively inconsistent
but can be reconciled analogically in light of the typical life cycle of a long-lived squall line as an example. A propagatingsquall line projected on Eulerian coordinates begins witha line of convective clouds entering the study domain andends with a trailing stratiform anvil eventually leaving out ofthe domain, depicting a life cycle lasting no longer than thetime that takes for the system to pass through the domain.
In contrast, the Lagrangian picture is more of a quasi-static
system persisting for an extended period of time, resulting
from a sequence of short-lived cells being renewed continuously as they move. This stationarity in the Lagrangianframework may be translated into a state of dynamic equilibrium when interpreted in the context of large-scale
thermodynamics.
The merging and splitting of clouds are a potential sourceof ambiguity. The dynamic equilibrium as pictured aboveis built on the assumption that regime-to-regime transitions
occur within each PS, whereas this premise is potentially

VOLUME 81

complicated by multiple concurrent PSs before merger or after splitting that are mixed together into the composite evolution. To look into this, a sensitivity test is run with theselected samples that consist exclusively of the life stages atwhich only a single PS exists. This test case is intended to see
to what extent composite statistics may be distorted by coexisting PSs. As shown in Fig. S3, the evolution remains essentially unchanged from Figs. 2 and 9, implying that the merging
and splitting unlikely impose a major effect on composite
time series.

6. Conclusions
The Lagrangian life cycle of organized moist convection
was investigated with focus on the underlying physical processes by tracking migrating precipitation systems and theirlarge-scale environment. The composite evolution analyzed
with the IMERG-CT dataset was found to depict an initial
pickup and a subsequent decline in both the system mean andmaximum precipitations as known in the literature. A lesserknown feature is that the maximum-to-average precipitation
ratio exhibits a plateau in between, especially for long-lived
systems. The plateau suggests that the fractional cover of localized intense precipitation such as deep convective cores
may stay for a certain span of time in the course of the convective life cycle.
In an attempt to explore the mechanism behind these observations, the study domain enclosing tracked precipitation
systems was broken down on a 18 3 18 grid basis into thebottom-heavy, midheavy, and top-heavy regimes. Each regime is deﬁned depending on the sign of column moisture/
MSE export diagnosed from a water/energy budget analysis
forced by the satellite retrievals of precipitation and radiation.
In the bottom-heavy regime, an atmospheric column imports
both moisture and MSE owing to a very bottom-heavy v
proﬁle, whereas the v structure is less bottom-heavy in themidheavy regime, resulting in an import of moisture and an
export of MSE. Column moisture and MSE are both exported
when v is even top-heavier, which deﬁnes the top-heavy regime. As such, these three regimes are intended to separate
the strength of moist convection from the thermodynamic
perspectives.
A key ﬁnding from the regime-sorted composite analysis isthat the transition from one regime to another is closely balanced against the inverse process, implying a state of dynamic
equilibrium in which the relative occurrence of each regime
stays nearly invariant during the whole life cycle. The growthof precipitation toward the peak is arguably activated by anegative GMS intrinsic to the bottom-heavy regime, leading
to a burst of convection in the transition from the bottomheavy to the midheavy regimes. This transition, evaluatedwith a dimensional analysis to have a relaxation time of about
5 h, takes place as part of the equilibrating processes for systems lasting longer than the relaxation time. This time scale isclose to the time taken by the initial rise of the maximum-toaverage precipitation ratio before it is settled into a plateau.
The more gradual decay of precipitation toward the end of

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MASUNAGA AND TAKAHASHIAPRIL 2024

the life cycle could be interpreted as the transition from themidheavy regime to the top-heavy regime.
Some fundamental aspects of the Lagrangian evolution of
convective systems remain to be answered. Key outstanding
questions include what determines the timing and magnitudeof the precipitation peak during the life cycle and what controls the system lifetime. The analysis method devised in thisstudy will be further reﬁned in future work in an attempt toshed new light on these long-standing problems.
Acknowledgments. The authors greatly appreciate insightful feedback from Johnny Luo on an early version of themanuscript and are grateful to three anonymous reviewers
for their thoughtful and helpful comments. HM was supported by the Japan Society for the Promotion of Science(JSPS) Grants-in-Aid for Scientiﬁc Research (KAKENHI)
19H01966 and the Japan Science and Technology Agency(JST) Moonshot R&D Program JPMJMS2389. The work of
HT was funded by NASA’s Atmosphere Observing System
under WBS/RTOP:107686/04.02.06.
Data availability statement. The IMERG data are available
from Goddard Earth Sciences Data and Information Services
Center (GES DISC) (https://disc.gsfc.nasa.gov/datasets/GPM_
3IMERGHH_06/summary) (Huffman et al. 2019). The CERESSYN1deg product is provided by NASA Langley Atmospheric
Science Data Center (https://asdc.larc.nasa.gov/project/CERES/
CER_SYN1deg-1Hour_Terra-Aqua-MODIS_Edition4A) (NASA/
LARC/SD/ASDC 2017). The ERA5 CWV data are available from
Copernicus Climate Change Service (C3S) Climate Data Store(CDS) (https://cds.climate.copernicus.eu/cdsapp#!/dataset/reanalysisera5-single-levels) (Hersbach et al. 2023).

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