PUBLICATIONSJournal of Advances in Modeling Earth Systems
RESEARCH ARTICLE
10.1002/2016MS000855
Key Points:
 A toy model of tropical convection isdeveloped to test the potential utilityof a moisture storage closure The degree to which moisture
convergence accounts for moisturestorage is prescribed for two
fundamental vertical modes Aspects of observations including the
moisture-rainfall curve are
reproduced with a proper choice ofparameters

Correspondence to:
H. Masunaga,
masunaga@nagoya-u.jp
Citation:
Masunaga, H. and Y. Sumi (2017), A toymodel of tropical convection with amoisture storage closure, J. Adv. Model.
Earth Syst., 9, doi:10.1002/
2016MS000855.
Received 6 NOV 2016
Accepted 1 MAR 2017
Accepted article online 7 MAR 2017

A toy model of tropical convection with a moisturestorage closure
Hirohiko Masunaga1

and Yukari Sumi2

1

Institute for Space-Earth Environmental Research, Nagoya University, Nagoya, Japan, 2Graduate School of Environmental
Studies, Nagoya University, Nagoya, Japan

Abstract A time-dependent, zero-dimensional toy model is constructed to study the large-scale variability in association with tropical moist convection. Case studies from sounding-array observations are analyzed
as a benchmark to test the model performance. The model predicts the vertical integral of vertical moistureadvection decomposed into the deep convective and congestus/stratiform modes. A closure representing
the consumption efﬁciency of water vapor into precipitation is introduced using the moisture storage ratio,or the degree to which the vertical moisture advection associated with each vertical mode accounts formoisture storage. The observations suggest that this moisture consumption is highly inefﬁcient for theconvective/stratiform mode while efﬁcient for the deep convective mode. The model solution is interpreted
as a delayed response to the diabatic forcing, unless the sum of the moisture storage ratio and gross moist
stability is negative, in which case the system is unstable. Baseline experiments with a ﬁxed moisturestorage closure overall reproduce the vertical moisture advection and precipitation as observed, but fail tosimulate a sharp pickup of precipitation in the well-known moisture-rainfall curve. This deﬁciency iseliminated when the moisture storage ratio is allowed to vary as convection intensiﬁes and dissipates.

1. Introduction
Raining cumulus clouds such as congesti and cumulonimbi are rooted on a relatively thin subcloud layer,with the cloud base height determined by the lifting condensation level (LCL) being often as low as 500 m
over tropical oceans. Given that the water vapor scale height, say 2 km, is markedly higher than LCL, lowlevel convergence brought by a deep overturning circulation would result in an accumulation of moisture
above cloud base more than below. This dynamic moisture accumulation is balanced out largely by precipitation in active convective disturbances. The classic parcel model of cumulus clouds, however, does not
involve any mechanism to condense out water vapor from the ambient air above cloud base, because moisture is by design supplied only through cloud base. The entrainment of environmental air into clouds would
relax this restriction on the nonentraining parcel model, but at the same time the entrainment reduces thein-cloud buoyancy and is unlikely to offer an efﬁcient process promoting convection after all.
Recent satellite data analyses [Masunaga, 2013, 2014] showed that while moisture convergence is the main
driver of precipitation for highly organized convective systems, precipitation from modest, less organizedsystems tends to rely largely on local evaporation and subcloud layer moisture convergence in accordancewith the parcel model expectation. It is thus speculated that the availability or unavailability of midtropospheric moisture convergence for rainfall may be a critical element in differentiating the nature of tropicalconvective systems, although the physical processes governing such distinction have yet to be clariﬁed.

C 2017. The Authors.
V

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made.

MASUNAGA AND SUMI

The main interests of this work lie in the degree to which moisture convergence is convertible into precipitation and the relevance of this degree to the whole large-scale dynamics involving moist convection. To
this end, a simple toy model of tropical convection is developed with a ‘‘moisture storage closure’’ as a prescribed measure of the inefﬁciency of precipitation production. This approach contrasts with, if not contradicts, conventional parameterizations employed for atmospheric models of different levels of complexity. A
large family of models have been built around versions of the convective quasi-equilibrium theory [e.g.,
Arakawa and Schubert, 1974; Betts, 1986; Emanuel et al., 1994] (a recent review is given by Raymond andHerman [2011]). A quasi-equilibrium postulate makes assumptions principally about the thermal effects ofmoist convection, so the roles of humidity are not as explicit as those of temperature in the quasi-

A MODEL WITH A MOISTURE STORAGE CLOSURE

1Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
equilibrium thinking per se. The relevance of tropospheric water vapor to rainfall, on the other hand, is ofcentral interest for a school of theory of tropical convection. Among the earliest along this line of researchis Raymond [2000], who derived a conceptual model under the hypothesis that precipitation is inversely
proportional to the saturation deﬁcit. A close link between tropospheric humidity and precipitation is a key
ingredient in different models of tropical waves (see review by Sugiyama [2009]) in the weak temperature
gradient (WTG) framework [Sobel et al., 2001], where a moisture perturbation is instrumental in controllingthe convective variability in lieu of buoyancy anomalies that are promptly smoothed out. An empirical relationship of precipitation (or convective heating) with midtropospheric humidity is explicitly speciﬁed insome models of convectively coupled equatorial waves [Khouider and Majda, 2006; Fuchs and Raymond,
2007; Kuang, 2008a].
The present study shares the interests with those studies in which a main focus is put on the link of rainfallwith water vapor, but is unique that precipitation is related to the moisture storage, not to the moisture
itself. It will be demonstrated that the major characteristics found in observations are reproduced with amoisture storage closure despite the gross simplicity of the model. The reproducible properties include theknown relation between precipitation and tropospheric humidity [e.g., Bretherton et al., 2004], despite thefact that the present model does not have such a relationship built in as an assumption.
The paper is structured as follows. Observations of tropical atmosphere during the hours of vigorous convection are ﬁrst examined in light of the large-scale water and thermal budget (section 2). Sounding arraymeasurements analyzed here serve as the test bed of the toy model constructed in section 3. The model
results are presented in section 4 and the ﬁndings are summarized and discussed in section 5.

2. Observations
2.1. Data
Variability in the moisture and thermal budget of the tropical atmosphere is ﬁrst investigated with sounding
array data. The observations are carried out during a coordination of the ﬁeld campaigns of the Cooperative
Indian Ocean Experiment on Intraseasonal Variability in the Year of 2011 (CINDY2011), the Dynamics of the
Madden-Julian Oscillation (MJO) (DYNAMO), and the Atmospheric Radiation Measurements (ARM) MJO
Investigation Experiment (AMIE) [Yoneyama et al., 2013]. The sounding-array data analyzed in this study aretaken from the DYNAMO array-averaged product (version 3b) provided by Johnson and Ciesielski [2013] andsee also Ciesielski et al. [2014a, 2014b]. This product also contains the array-averaged surface turbulent
ﬂuxes from the TropFlux product [Praveen Kumar et al., 2012] and the radiative heat proﬁles from the Clouds
and the Earth’s Radiant Energy System (CERES) product [Wielicki et al., 1996]. The present analysis focuseson the northern sounding array (NSA), a quadrilateral array located just to the north of the equator over thecentral Indian Ocean. All the array-mean parameters are 3 hourly except for the daily mean TropFlux surface
heat ﬂuxes. In the current analysis, the data are moderately smoothed over time by 66 h running mean,
with the exception of radiative heating rate in which the diurnal cycle is ﬁltered out by 612 h runningmean. Note that the diurnal radiative forcing is unlikely to have predominant effects on oceanic convection,
given the fact that precipitation over ocean away from coast depends only weakly on local time [e.g.,
Imaoka and Spencer, 2000].
Three different estimates of array-averaged precipitation are provided in the DYNAMO product, namely,
Tropical Rainfall Measuring Mission (TRMM) 3B42 version-7 rainfall [Huffman et al., 2007] and Q1 and Q2budget-derived precipitation, Ps and Pq, respectively, deﬁned as [Yanai et al., 1973],
hQ1 i  Dt hsi1hx@p si5LPs 1S1hQR i;

(1)



hQ2 i  2L Dt hqi1hx@p qi 5LðPq 2EÞ;

(2)

and

where Dt  @t 1v  r refers to the Lagrangian derivative operator with respect to horizontal ﬂow, v, angle
brackets denote the vertical integral from surface to the 100 hPa level, s is dry static energy (DSE), q is vapor
mixing ratio, x is vertical pressure velocity, S is surface sensible-heat ﬂux, E is surface evaporation ﬂux, QR is
net radiative heating rate, and L is the speciﬁc latent heat of vapor condensation. Note that all these parameters represent the large-scale mean variables (or the sounding-array averages) throughout this paper.

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
This work adopts Ps as a primary reference for precipitation because Pq is signiﬁcantly noisier than Ps on subdaily time scales. It is shown in MJO-oriented studies using DYNAMO sounding-array data [Johnson et al.,2015; Ruppert and Johnson, 2015] that Pq is overall in good agreement with the TRMM 3B42 rainfall, but thesame cannot necessarily be said if a heavy temporal smoothing is not applied to Pq. High-frequency ﬂuctuations in Pq are likely attributed to the noisiness of the moisture tendency term computed directly from theobserved time series, while this difﬁculty is virtually absent in the Q1 budget owing to the intrinsic homogeneity of free-tropospheric temperature across tropical latitudes. For the same reason, the Lagrangian moisture storage analyzed here is not derived from the humidity time series as observed, but instead isrecalculated with Ps substituted in the moisture budget equation,
Dt hqi5E2Ps 2hx@p qi:(3)Equation (3) has another advantage that moisture storage is derived independently of horizontal moistureadvection. The DYNAMO sounding-array estimates of horizontal moisture advection are known to be biased
when the intrusion of a localized dry air mass is not properly captured by the sparsely deployed sounding
stations [Hannah et al., 2016].
As may have been noticed, the Lagrangian tendency is preferentially used rather than the Eulerian in the current formulation. In fact, the choice between the Eulerian and Lagrangian derivatives would not make any
practical difference in light of our research interests, since horizontal advection stays small in the moist static
energy (MSE) budget for short-term tropical variations [Inoue and Back, 2015a]. Given its minor nature, the horizontal advection term is chosen to be left hidden in the Lagrangian derivative rather than dealt with separately as an independent term. The remaining source/sink terms on the right-hand side are reduced tosurface evaporation, precipitation, and vertical moisture advection or equivalently moisture divergence,hqr  vi, when integrated over height to the depth of the troposphere. Hannah et al. [2016] demonstrated
the utility of Lagrangian moisture tendency for studying aspects of the atmospheric moisture budget.
Three 72 h sequences captured by the NSA are selected as benchmark cases. Two of the three cases containvigorous convective events having a precipitation maximum at 0000 UTC 22 October (event 1) and 0300
UTC 27 November (event 2), corresponding to the third and fourth highest peaks, respectively, in the NSAaveraged precipitation record. The top two convective events are not chosen for these cases because Ps
largely underestimates the TRMM 3B42 rainfall (not shown) and the ﬁdelity of budget parameters is not
conﬁdently assured. The third example sequence (event 3), centered around 1200 UTC 4 December, isdevoid of any notable rainfall event in between the second and third MJO episodes identiﬁed during theCYNDY2011/DYNAMO/AMIE observational period [Yoneyama et al., 2013].
2.2. Vertical Mode Decomposition
The toy model developed later is zero-dimensional in space, dealing with vertically integrated atmosphericvariables along with surface ﬂuxes. The detailed vertical proﬁles of atmospheric parameters are therefore
outside the scope, while some fundamental aspects of the vertical structure are retained by means of atruncated series of vertical modes. To transformthe observations into a comparable format to themodel framework, vertical mode decomposition isapplied to the temporal-mean NSA sounding using
a conventional methodology [Fulton and Schubert,
1985; Haertel and Johnson, 1998]. The ﬁrst four
eigenmodes of x with a rigid-rid boundary condition are shown in Figure 1. The evolution of eachmode coefﬁcient is computed by ﬁrst projecting
the horizontal divergence proﬁle onto a series of
its eigenmodes at every time step and then converting those proﬁles into x through the mass conservation equation.

Figure 1. The ﬁrst four eigenmodes of x as labeled in the panel
with the ﬁrst and second modes highlighted.

MASUNAGA AND SUMI

Vertical mode decomposition is known to be ofgreat utility in tropical meteorology in which twoor three vertical modes sufﬁce in general to explain

A MODEL WITH A MOISTURE STORAGE CLOSURE

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855

1st mode (color) for event 2

1st-2nd modes (color) for event 1

1st-2nd modes (color) for event 2

1st-4th modes (color) for event 1

1st-4th modes (color) for event 2

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

ω [hPa/h]

1st mode (color) for event 1

Figure 2. Time series of x with vertical mode decomposition applied (color shade): the ﬁrst mode only (left), the ﬁrst and second modes added together (middle), and the ﬁrst fourmodes all combined (right) for event 1 (top) and event 2 (bottom) (see section 2.3 for details of the individual events). The undecomposed x is contoured at the interval of 3 hPa h21 inall panels.

the atmospheric dynamics in association with convective heating [Mapes and Houze, 1995; Mapes, 2000; Wu
et al., 2000; Majda and Shefter, 2001; Haertel and Kiladis, 2004; Khouider and Majda, 2006; Back andBretherton, 2009]. Figure 2 presented the decomposed x proﬁles for two convective events during
DYNAMO that will be examined later in section 2.3. The ﬁrst eigenmode (color shade in left panels) is responsible for the overall temporal variability of x (contoured) but by design fails to explain its vertical tilt. The addition of the second mode (middle panels) improves the reproducibility of the detailed structure of x including
the tilt. When the third and fourth modes are included as well (right panels), further improvements are visible(e.g., the altitude of the updraft peak for event 1) but the changes are mostly minor. For the sake of convenience, only the ﬁrst two modes are considered in the analysis to follow. The ﬁrst eigenmode is hereaftertermed as the deep convective mode and the second mode as the congestus/stratiform mode.
The latter may be simply referred to as the congestus mode when the mode coefﬁcient is positive (i.e., themode has a low-level updraft) and as the stratiform mode otherwise, following conventional terminology.
2.3. Case Studies
Analysis results for event 1 are presented in Figure 3. A sharp precipitation peak is evident (top left panel) in
Ps and TRMM 3B42 rainfall. The two precipitation products are in excellent agreement during the hours ofactive convection in this particular case. The surface evaporation stays consistently small, far short of precipitation especially around its peak intensity. It conﬁrms that moisture convergence should serve as a dominant moisture source of precipitation for intensive convective systems.
The deep convective mode develops in phase with rainfall (top middle panel), with its peak straddled intime by a pair of the congestus/stratiform dipole (bottom middle panel). This is a familiar pattern known toprevail in the tropics over a broad range of time scales [Mapes et al., 2006, and references therein], although
it is somewhat surprising to ﬁnd it at this level of clarity in a single convective event. As seen earlier in Figure 2, the deep convective mode dominates the vertical motion in amplitude (note that the color scales aredifferent between the two panels) as also evident in the vertically integrated vertical moisture advection, 2

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
ω
deep conv (1st) mode
cong/strat (2nd) mode

LP from Q1 budget

Total

LP from TRMM 3B42
LE
q storage

ω

<QR>

S+LE

S+LE+<QR>

Figure 3. Time series of observations for event 1. (top left) Ps (solid), TRMM 3B42 rainfall (dotted), and E (dot-dashed), all in the energy-ﬂux unit (W m22). (Bottom left) Surface heat ﬂuxes
or S 1 LE (dot-dashed), total diabatic forcing or S1LE1hQR i (solid) with the contribution of hQR i shaded in red where heating and in blue where cooling (W m22). (top middle) The
vertical proﬁle of x for the deep convective mode (hPa h21). (bottom middle) The vertical proﬁle of x for the congestus/stratiform mode (hPa h21). (top right) 2hx@p qi (shade) with its
decomposition into the deep convective mode (solid) and the congestus/stratiform mode (dashed) in the energy-ﬂux unit (W m22). Also plotted in dotted hatch is hDt qi (red where
positive and blue where negative). (bottom right) 2hx@p hi (shade), where h denotes moist static energy, with its decomposition into the deep convective mode (solid) and thecongestus/stratiform mode (dashed).

hx@p qi (top right panel). The total 2hx@p qi closely resembles precipitation as expected and is explained tothe ﬁrst approximation by the deep convective mode. The remaining small difference is mostly accounted
for by the congestus/stratiform mode, lending conﬁdence in the assumption that these two modes sufﬁce
in practice to reproduce the total variability. Interestingly, moisture storage (derived by equation (3))
accords well with the congestus-mode vertical advection, whereas it nearly vanishes after the congestus
mode switches to the stratiform mode. The enhancement of the congestus mode accompanying a positive
moisture storage results in an import of MSE prior to the peak convection, followed by a rapid buildup of
the MSE export associated with the deep convective and stratiform modes as convection dissipates (bottom
right panel). The change of MSE in sign from import to export is characteristic of a so-called dischargerecharge model of tropical disturbances [Peters and Bretherton, 2006], evidenced also by satellite observations [Masunaga and L’Ecuyer, 2014].
All these aspects of the thermodynamic evolution are found also in another case of intensive convection(event 2 in Figure 4). Precipitation, represented in unison by Ps and TRMM 3B42 rainfall, intensiﬁes toward
the peak while evaporation remains minor in magnitude (top left panel). The deep convective mode here
again evolves in parallel with precipitation (top middle) and is accompanied by the congestus/stratiform
mode switching its sign (bottom middle), although the congestus mode is virtually absent and dipole symmetry is not as evident as in event 1. Vertical moisture advection is broken down into the deep convective
and congestus/stratiform mode in a similar manner as seen in event 1 (top right), and moisture storage
varies coherently with the congestus-mode component of 2hx@p qi. Event 2 is likely inﬂuenced by a preceding disturbance visible at 1200 UTC 25 November, which might have somewhat distorted the dynamics
intrinsic to the disturbance of present interest centered at 0300 UTC 27 November.

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
ω
deep conv (1st) mode
cong/strat (2nd) mode

LP from Q1 budget

Total

LP from TRMM 3B42
LE
q storageω

S+LE

S+LE+<QR><QR>

Figure 4. Time series of observations for event 2. (top left) Ps (solid), TRMM 3B42 rainfall (dotted), and E (dot-dashed), all in the energy-ﬂux unit (W m22). (bottom left) Surface heat ﬂuxes
or S 1 LE (dot-dashed), total diabatic forcing or S1LE1hQR i (solid) with the contribution of hQR i shaded in red where heating and in blue where cooling (W m22). (top middle) The
vertical proﬁle of x for the deep convective mode (hPa h21). (bottom middle) The vertical proﬁle of x for the congestus/stratiform mode (hPa h21). (top right) 2hx@p qi (shade) with its
decomposition into the deep convective mode (solid) and the congestus/stratiform mode (dashed) in the energy-ﬂux unit (W m22). Also plotted in dotted hatch is hDt qi (red where
positive and blue where negative). (bottom right) 2hx@p hi (shade) with its decomposition into the deep convective mode (solid) and the congestus/stratiform mode (dashed).

Event 3, shown in Figure 5, is a convectively quiescent period. As expected, no notable variation is observedin any parameters studied. The deep convective mode in particular is weaker in amplitude by an order ofmagnitude, that is, 100 W m22 at most in 2hx@p qi for event 3 as opposed to 900 W m22 or larger at theconvective peak for events 1 and 2. The difference between the convective and nonconvective events is far
less striking in the congestus/stratiform mode and moisture storage, given that their magnitude is as small
as 100 W m22 regardless of the presence or absence of convection. The close relevance between thecongestus-mode component of 2hx@p qi and moisture storage is visible also in event 3 (top right panel),
not as evidently as in event 1 but with better clarity than in event 2.
The observations inspire us with a working hypothesis that moisture storage may be ascribed primarily tothe congestus-mode component of vertical moisture advection, while its deep-mode counterpart largely
precipitates out rather than being accumulated in moisture storage. This idea is mathematically translated
as follows in terms of a nondimensional parameter named the moisture storage ratio, g,hi
Dt hqi  2 gdc hx@p qidc 1gc=s hx@p qic=s ;(4)where the subscripts ‘‘dc’’ and ‘‘c/s’’ designate the deep convective mode and the congestus/stratiform
mode, respectively. By deﬁnition, gdc and gc=s (or g collectively where the general properties of the moisturestorage ratio are discussed) are a prescribed measure of the degree to which vertical moisture advection
associated with each vertical mode is responsible for moisture storage. The aforementioned hypothesis
claims that gc=s  1 and gdc  0. The high moisture storage ratio for the congestus mode suggests that the
cumulus-cloud types typical of this mode, namely raining shallow cumuli or cumuli congesti, should behighly inefﬁcient in converting the dynamically accumulated moisture into precipitation. The possible

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
ω
deep conv (1st) mode
cong/strat (2nd) mode

Total

LP from Q1 budget

q storageLE
LP from TRMM 3B42

ω

<QR>

S+LE

S+LE+<QR>

Figure 5. Time series of observations for event 3. (top left) Ps (solid), TRMM 3B42 rainfall (dotted), and E (dot-dashed), all in the energy-ﬂux unit (W m22). (bottom left) Surface heat ﬂuxes
or S 1 LE (dot-dashed), total diabatic forcing or S1LE1hQR i (solid) with the contribution of hQR i shaded in red where heating and in blue where cooling (W m22). (top middle) The
vertical proﬁle of x for the deep convective mode (hPa h21). (bottom middle) The vertical proﬁle of x for the congestus/stratiform mode (hPa h21). (top right) 2hx@p qi (shade) with its
decomposition into the deep convective mode (solid) and the congestus/stratiform mode (dashed) in the energy-ﬂux unit (W m22). Also plotted in dotted hatch is hDt qi (red where
positive and blue where negative). (bottom right) 2hx@p hi (shade) with its decomposition into the deep convective mode (solid) and the congestus/stratiform mode (dashed).

reasons behind this include the lack of mechanism for cumulus clouds to effectively condense out midtropospheric moisture as argued in section 1. In contrast, the deep convective mode is considered to involve
efﬁcient dynamics for conversion from moisture into precipitation so gdc  0. The underlying physical processes will be discussed in section 4.4.
The moisture storage term in equation (4) would vanish as integrated over a life cycle of convective disturbances, so that þ
þ052 gdc hx@p qidc dt1gc=s hx@p qic=s dt ;(5)
where gc=s and gdc are assumed to be constant. Figures 3 and 4 suggest that 2hx@p qidc yields a ﬁnite positive value when integrated over time during a convective event, while 2hx@p qic=s is quasi-sinusoidal andthus would be largely integrated out, that is,þþ
2 hx@p qidc dt > 0; 2 hx@p qic=s dt  0:
(6)
Equations (5) and (6) together offer additional evidence for gdc  0. Unfortunately, no useful constraint on
the possible range of gc=s is derived by this line of argument.
It is noted that the formulation itself does not pose any mathematical constraint on the range of g. From aphysical point of view, on the other hand, gdc and gc=s are unlikely to fall far outside the range between 0
and 1, because g  0 implies a signiﬁcant drying that proceeds in spite of dynamic vapor accumulationand g  1 requires an unusually large evaporation overcompensating the moisture loss due to

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
precipitation. Such situations, although may temporarily occur in exceptional cases, are in general not typical of tropical convective disturbances and will not be considered in the experiments below.

3. Model Design
A toy model is now developed to test the working hypothesis proposed in the previous section. The model
is formulated on the basis of the moisture and MSE budget equations,hi
Dt hqi52 hx@p qidc 1hx@p qic=s 1hx@p qibg 1E2P;(7)hi
Dt hhi52 hx@p hidc 1hx@p hic=s 1hx@p hibg 1DF;(8)

where h is MSE, P is precipitation, DF  S1LE1hQR i is the diabatic forcing, and the subscript ‘‘bg’’ denotes
the background component that formally represents the unperturbed state unaccounted for by neither the
deep convective nor congestus/stratiform mode. Evaporation, precipitation, and the diabatic forcing are also
separated into the steady background state and a time-dependent perturbation indicated by prime symbol,
E5Ebg 1E 0 ;P5Pbg 1P0 ;DF5DFbg 1DF 0 :The background terms are assumed to satisfy the steady state budget equations on their own in theabsence of the two convective modes, that is,
Ebg 2Pbg 2hx@p qibg 50;(9)DFbg 2hx@p hibg 50:(10)

The background state is factored out by inserting equations (9) and (10) into the budget equations (7) and(8), respectively, leading tohi
Dt hqi52 hx@p qidc 1hx@p qic=s 1E 0 2P0 ;(11)hi
Dt hhi52 hx@p hidc 1hx@p hic=s 1DF 0 :(12)For later reference, an alternative form of the moisture budget equation is derived by substituting equation(4) for the left-hand side of equation (11)hi002 ð12gdc Þhx@p qidc 1ð12gc=s Þhx@p qic=s 5Pdc
1Pc=s
5P0 ;
(13)
0
where P0dc and Pc=s
are the breakdown of precipitation into each vertical mode,
0
Pdc
52ð12gdc Þhx@p qidc ;052ð12gc=s Þhx@p qic=s :
Pc=s

(14)

In equation (13), E 0 has been omitted because the variability of evaporation is practically unimportant compared to P0 as shown in Figures 3–5 (top left). It is implied by equations (4) and (14) that vertical moistureadvection is partitioned into moisture storage and precipitation by the proportions of gi and 12gi , respectively, for each vertical mode represented by the subscript i, so precipitation increases with decreasing gi fora given magnitude of 2hx@p qii .
The MSE budget equation is conveniently rewritten in terms of q rather than h. Following a common practice in tropical meteorology, the contribution of DSE is neglected in the MSE tendency term,

MASUNAGA AND SUMI

Dt hhi  LDt hqi:

(15)

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deep conv mode
cong/strat mode

Event 1

Event 2The temperature perturbation, while may not becompletely smoothed out on a subdaily timescale, is known to appear in a pair of low-level
cooling and upper-level heating during the timeof deep convection [e.g., Mapes and Houze,
1995] and therefore practically vanishes in thevertical integral [Masunaga, 2013].
Vertical MSE advection is replaced by vertical
moisture advection in terms of a version of thegross moist stability (GMS) [Neelin and Held,1987],
cdc 52cc=s 52

Elapsed time [h]hx@p hidc;
Lhx@p qidc
hx@p hic=s
Lhx@p qic=s

;

(16)

where cdc and cdc (or collectively c) is deﬁned
here individually for each vertical mode in aslightly modiﬁed form of the normalized GMS[Raymond et al., 2007, 2009]. GMS for a given vertical eigenmode may be deemed as an externallyspeciﬁed constant when the moisture and DSE proﬁles are ﬁxed. Figure 6 conﬁrms that GMS for the deep convective mode is almost invariant at 0.2–0.3 in the two convective events studied in section 2.3. GMS remains
stable over time when computed for an individual vertical mode owing to the coherence between the numerator and denominator in equation (16), apart from a modest variability due to slight ﬂuctuations in the verticalhumidity structure. This stability also holds for the congestus/stratiform mode, whereas the congestus/stratiform mode consistently has a negative GMS, marking a sharp contrast with the deep convective mode.

Figure 6. Time series of GMS for different vertical modes for event 1(blue) and event 2 (red). The deep convective mode and thecongestus/stratiform mode are indicated by closed and open circles,
respectively.

The fact that the two representative vertical modes have a GMS of opposite signs constitutes an important
aspect of the present model as will become clear in section 4.
Equations (12), (15), and (16) are combined into,hi
LDt hqi5L cdc hx@p qidc 1cc=s hx@p qic=s 1DF 0 :Implementing equation (17) with the moisture storage closure (4), one ﬁnds a diagnostic equation forvertical moisture advection,hi
2L ðgdc 1cdc Þhx@p qidc 1ðgc=s 1cc=s Þhx@p qic=s 5DF 0 :100

2-d<ωdq/dp> dc/dt [W/m h]

Event #1
Event #250

(18)

Equation (18) is not closed without an additional
assumption relating 2hx@p qidc and 2hx@p qic=s . An
empirical closure is introduced here as,

0

-50

2

-100
-300

-200

-100

0

100

200

300

2

-<ωdq/dp> c/s [W/m ]Figure 7. Scatterplot of 2hx@p qic=s (abscissa) versus the time
derivative of 2hx@p qidc (ordinate) from event 1 (closed circles)
and event 2 (open circles). A straight line with the inclination of
0.3 h21 is shown as a reference.

MASUNAGA AND SUMI

(17)

d
0
52vð12gc=s Þhx@p qic=s ; (19)hx@p qidc 5vPc=s
dt

where v is a positive constant. Equation (19) is intuitively inspired by the observations in the top right panel of Figures 3 and 4 that 2hx@p qidc increases withtime when 2hx@p qic=s is positive in sign (the congestus regime) and decreases once it turns to negative(the stratiform regime). This relationship is interpreted
implicitly as the convective heating by congestus or

A MODEL WITH A MOISTURE STORAGE CLOSURE

9Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
0stratiform clouds, Pc=s(see equation (14)), is involved in the growth and dissipation of the deep convective mode.
Theoretical arguments relevant to this empirical insight are found in the literature [Khouider and Majda, 2006,
2008; Haertel et al., 2008]. Figure 7 shows that the observations clearly verify the positive correlation between 2
hx@p qic=s and the time derivative of 2hx@p qidc in support of the proportionality claimed by equation (19). Equations (18) and (19) together yield a prognostic equation describing the evolution of 2hx@p qidc ,

2s L

d
DF 0
1Lhx@p qidc ;hx@p qidc 5
dt
gdc 1cdc(20)

gc=s 1cc=s:
vð12gc=s Þðgdc 1cdc Þ

(21)

where
s 

It is beneﬁcial to ﬁrst illustrate fundamental aspects of the solution under the simplest condition of a ﬁxed
DF 0 . The analytic solution of equation (20) in this special case is





t
DF 0
t
1
:
(22)
2Lhx@p qidc 52Lhx@p qit50
exp
2
12exp
2
dc
gdc 1cdcssEquation (22) claims that 2Lhx@p qidc converges toward DF 0 =ðgdc 1cdc Þ as time proceeds regardless of theinitial condition, 2Lhx@p qit50
dc , for s > 0. The e-folding time scale, s , dictates how quickly this processcompletes. The solution is analogous to the radiance as a function of optical depth when radiation penetrates through a homogeneously absorbing media, given the mathematical identity between equation (20)and the nonscattering radiative-transfer equation.
The behavior of the solution is entirely different when s is negative, in which case 2hx@p qidc would growexponentially to inﬁnity. Recalling that 0ⱗgⱗ1 and cdc 5 0.2–0.3, one can safely expect the denominator in thedeﬁnition of s (equation (21)) to be positive deﬁnite, with rare exceptions aside. On the other hand, gc=s 1cc=s
in the numerator is only marginally positive or could be even negative because cc=s < 0 (Figure 6), implying thepossibility that s could fall to below zero as well. Figure 8 schematically illustrates the evolution of 2hx@p qidc
and 2hx@p qic=s for two contrasting cases with opposite signs of s . The solution is constrained on a straightline described by equation (18), which is rightward decreasing when gc=s 1cc=s > 0 (or s > 0) and increasingotherwise. Any point laid on the right-half of this plane (i.e., 2hx@p qic=s > 0) proceeds upward along this line
while it moves downward if on the left-half of the plane, obeying equation (19). As a consequence, the solution
evolves toward the y axis until eventually resting on it in cases of s > 0 but moves away from the y axis without limit when s < 0. It turns out that the stability of the solution depends on the sign of s .
A brief physical interpretation to this stability criterion may be given as follows. An individual term on theleft-hand side of equation (18) is rewritten using h5s1Lq and equations (14) and (16) as2Lðgi 1ci Þhx@p qii 5hx@p sii 2LPi0 ;(23)

where the subscript i again represents each vertical mode. In the discussion to follow, hx@p qii is assumed to be
negative (i.e., moistening) as is typically the case when convection is developing. It is inferred from equation (23)that diabatic heating LPi0 is short of the work required-L<ωdq/dp>dc
to drive a large-scale ascent against adiabatic cooling,
unstable (τ* <0)stable (τ* >0)
hx@p sii , when gi 1ci > 0. This shortage needs to becompensated by the diabatic forcing DF 0 so that thelarge-scale ascent is sustained. It follows that the deep
ΔF’
ηdc+γdc
convective mode and the congestus/stratiform mode,
if both have a positive gi 1ci , must compete with eachother for a ﬁxed supply of DF 0 , so the system remains-L<ωdq/dp>c/s
stable. The vertical mode with a negative gi 1ci , on the0contrary, would produce a diabatic heating more thanenough to fuel its own vertical motion.
Figure 8. Schematic to illustrate the model solution for aconstant DF0 in terms of 2Lhx@p qidc (ordinate) and 2Lhx@p qic=s(abscissa). See test for details.

MASUNAGA AND SUMI

If gc=s 1cc=s is negative, the congestus/stratiform
mode gives rise to a residual heating capable of

A MODEL WITH A MOISTURE STORAGE CLOSURE

10Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
feeding the deep convective mode as well as itself, enabling a collaborative growth of the two verticalmodes. The system therefore could be destabilized once assisted by a positive feedback between themodes as prescribed by equation (19).
A negative s has the potential to provide a partial explanation for rapidly intensifying convective disturbances in nature, whereas unrealistic without any mechanism to eventually restabilize the system after the
instability criterion is met. Such a mechanism is not offered in the current simplistic modeling framework.
Throughout the rest of the paper, therefore, gc=s 1cc=s is restricted to above zero for all model runs.

4. Model Results
4.1. Experimental Setup
Equation (20) together with the ancillary equation (21) constitutes the core of the current model, delineating the
progress in 2hx@p qidc over time. Its congestus/stratiform counterpart, 2hx@p qic=s , is obtained from the solution using equation (19). Precipitation, P0 , is then evaluated so the moisture budget equation (13) is satisﬁed.
Table 1 summarizes the parameter settings for different model runs. The key prescribed parameters characterizing the model are the moisture storage ratios, gc=s and gdc , listed in the ﬁrst two columns. The default
assumptions are gc=s 50:99 and gdc 50 as discussed in section 2. Note that gc=s is set to be slightly smaller
than unity because the right-hand side of equation (19) collapses to zero when gc=s is exactly 1. For all
experiments except c/svar and fullvar, v (third column) is chosen to satisfy vð12gc=s Þ50:3 h21 so that equation (19) best reproduces the observations plotted in Figure 7. Once gc=s ; gdc , and v are given, s is evaluated with equation (21) as shown in the ﬁfth column. The GMS is ﬁxed for each vertical mode, namely
cdc 50:2 and cc=s 520:5, for all experiments. Any change to cc=s within a plausible range as expected from
Figure 6 does not alter the overall evolution as long as gc=s 1cc=s stays above zero.
The last column of Table 1 indicates a parameter controlling the variability in diabatic forcing, mF. The diabatic forcing in equation (20) is speciﬁed to vary with time as
DF 0 5DFb0½12cos ð2pmF tÞ ;2(24)

where DFb0 is ﬁxed at 150 W m22. Figure 9 (left) depicts equation (24) with three different values of mF as listed in
the top three rows of Table 1. This ‘‘cosine-bell’’ model is aimed at roughly mimicking the observations of diabatic
forcing (Figures 3 and 4, bottom left), where the two convective events each exhibit a modest enhancement of
DF 0 , driven chieﬂy by radiative heating for event 1 (Figure 3) and by evaporation for event 2 (Figure 4). Such asystematic variation in the diabatic forcing is not observed for the nonconvective event (Figure 5). Note that E 0 ,while ignored in the moisture budget equation, is potentially important for the MSE budget as a part of DF 0 .
4.2. Baseline Experiments
First examined are baseline experiments with constant gdc and gc=s . When these parameters are ﬁxed, ananalytic solution of equation (20) is derived with the cosine-bell diabatic forcing (24) under the initial condition of 2hx@p qidc 50 as
DFb0
h
i3
2ðgdc 1cdc Þ 11ð2pmF s Þ2




t
2
11ð2pmF s Þ 12exp 22cos ð2pmF tÞ22pmF s sin ð2pmF tÞ ;s

(25)

pmF DFb0hi3
vð12gc=s Þðgdc 1cdc Þ 11ð2pmF s Þ2t
2pmF s exp 2
1sin ð2pmF tÞ22pmF s cos ð2pmF tÞ :s

(26)

2Lhx@p qidc 5

2Lhx@p qic=s 5

Exponential terms in equations (25) and (26) are ill-deﬁned when s equals zero. In this special case, thesolution may be directly derived from equations (18) and (19) with gc=s 1cc=s 50,

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11

Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
2Lhx@p qidc 5

Table 1. Experimental Setups. For All Runs, cdc 50:2 and cc=s 520:5
Experiments
m24
m48
cntrl
c/s0.5dc0.5
dc0.05c/svarfullvargc=s

gdc

v (h21)

s (h)

m21
F (h)

0.99

0

30

8.17

24
48
72

0.50
0.99
0.99
Equation (30)Equation (30)

0
0.5
0.05
0
Equation (31)

0.60
30
30
3
3

0
2.33
6.53
Variable
Variable

2Lhx@p qic=s 5DFb0½12cos ð2pmF tÞ ;2ðgdc 1cdc Þ(27)

pmF DFb0
sin ð2pmF tÞ:
vð12gc=s Þðgdc 1cdc Þ
(28)

72

As plotted in Figure 9 (right), the solutions
are qualitatively reminiscent of the observations shown in Figures 3 and 4 (top right).
Recalling the earlier discussion in the case of a constant DF 0 (Figure 8), one would expect that 2Lhx@p qidc
approaches DF 0 =ðgdc 1cdc Þ over a ﬁnite time of s as implied by equation (22). As such, the behavior of 2
hx@p qidc may be interpreted as a delayed response to the slowly varying diabatic forcing. The delay
would be minimal when s  m21
F , whereas it would become increasingly difﬁcult to keep up with achange of diabatic forcing as m21
F decreases toward s . This gives an explanation to the differences among
the three runs shown in Figure 9. The run with the fastest varying DF 0 (m24) has a small amplitude and an
incomplete cycle of 2hx@p qi compared to the others, because the system is not allowed a sufﬁcient time
to respond to the forcing. The evolution is otherwise self-similar across a range of mF. The observed undulation of the diabatic forcing has a cycle of roughly one wavelength (Figure 3) or 1.5 wavelengths (Figure4) over a 72 h period, suggesting that 48–72 h is a realistic choice for m21
F . All the experiments to follow
will be run with m21
F 572 h.
The sensitivity of model results to gc=s and gdc is next investigated. Figure 10 compares the control run(gc=s 5 0.99) (left column) and another run with a reduced gc=s (c/s0.5, right column). The two experiments
are nearly identical to each other except for a minor difference in the delay time in the atmospheric
response to diabatic forcing. The peak of 2hx@p qidc occurs at 42 h or slightly later for the control, while
located at 36 h in experiment c/s0.5. Experiment c/s0.5 corresponds to the case with s 5 0, where the delay
is absent as indicated by equation (27). Precipitation evolves closely following vertical moisture advection,
but with a further delay due to a ﬁnite moisture storage (equation (13)). Moisture storage for experiment c/
s0.5 is by design as half in amplitude as 2hx@p qic=s .
Additional runs are carried out where gdc is modiﬁed instead of gc=s . Figure 11 (left) shows that an increase of
gdc from 0 to 0.5 leads to a signiﬁcant decrease of amplitude. This is explained by the term DFb0 =ðgdc 1cdc Þ inequations (25) and (26). The sum of gdc and cdc scales the strength of convective disturbances for a given diabatic forcing (see Figure 8), as GMS does in its original deﬁnition. A positive gdc effectively enhances the GMS
and hence reduces the amplitude of vertical moisture advection and precipitation. Moisture storage now
remains positive over an extended period of time because the ﬁrst term dominates the second in the righthand side of equation (4). The evolution more resembles the control experiment when gdc is chosen to be amore modest value of 0.05 (Figure 11, right).

Diabatic forcing

2

ΔF’ [W/m ]

100
50

800

deep conv mode
cong/strat mode

2

150

ν24
ν48
cntrl

Simulated vertical q advection
1000-<ωdq/dp> [W/m ]

200

0

600
400
200
0
-200

0 6 12 18 24 30 36 42 48 54 60 66 72
t [h]

0 6 12 18 24 30 36 42 48 54 60 66 72t [h]Figure 9. (left) DF0 as a function of time for m21
F 5 24 h (experiments m24), 48 h (m48), and 72 h (cntrl). (right) The solutions of 2hx@p qidc
(solid) and 2hx@p qic=s (dotted) for the three runs.

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
c/s0.5 η( c/s=0.5)

1000
800

2

-<ωdq/dp> and Dq/Dt [W/m ]2-<ωdq/dp> and Dq/Dt [W/m ]cntrl (ηc/s=0.99)
deep conv mode
cong/strat mode

600
400
200
0
-200

1000
800

q storage (-)
q storage (+)

600
400
200
0
-200

0 6 12 18 24 30 36 42 48 54 60 66 72
t [h]

0 6 12 18 24 30 36 42 48 54 60 66 72
t [h]c/s0.5 η( c/s=0.5)

cntrl (ηc/s=0.99)-<ωdq/dp>
P’

2

800

1000
-<ωdq/dp> and P’ [W/m ]2-<ωdq/dp> and P’ [W/m ]

1000

600
400
200
0
-200

800
600
400
200
0
-200

0 6 12 18 24 30 36 42 48 54 60 66 72

0 6 12 18 24 30 36 42 48 54 60 66 72

t [h]

t [h]

Figure 10. (top row) 2hx@p qidc (solid), 2hx@p qic=s (dotted), and Dt hqi (shaded in red where positive and in blue where negative). (bottom row) Total 2hx@p qi (shaded) and P0 (solid).
(left column) Experiment cntrl. (right column) Experiment c/s0.5.

4.3. Moisture-Precipitation Relation
The current model provides column water vapor (CWV) by integrating equation (17) over time,ðð hi DF 0dt;
hq0 i5 Dt hqidt5
cdc hx@p qidc 1cc=s hx@p qic=s 1
L(29)

where hq0 i is the perturbation of CWV to its background state. This quantity, when compared to P0 fromequation (13), yields the model-predicted relation of CWV with precipitation. Precipitation is known to growincreasingly as the ambient air moistens in the satellite-based climatology [Bretherton et al., 2004; Peters andNeelin, 2006] as well as from in situ observations of short-term variability [Holloway and Neelin, 2010]. Similarly, an abrupt intensiﬁcation of precipitation is conﬁrmed once CWV exceeds 60 mm in the sounding arraydata analyzed here (Figure 12). The CWV-precipitation relationship is assessed as an additional benchmark
to test the model against observations.
The results are plotted in Figure 13 for different baseline experiments examined above. Simulated tracksof the CWV-precipitation relationship are found to vary sensitively depending on the choice of gc=s andgdc . Unfortunately, however, all the runs fail to reproduce the rapid pickup of precipitation as expectedfrom the observations. Precipitation is almost linearly correlated with CWV in some runs (cntrl and c/s0.5),
and the rate of rainfall pickup saturates rather than grows further as CWV becomes larger for other runs
(dc0.5 and dc0.05). All the trajectories make a clockwise turn, as opposed to the satellite observations
composited in the context of the convective life cycle [Masunaga, 2012]. A thorough parameter search(not shown) has proven that this apparent deﬁciency is not mitigated by merely tuning gc=s or gdc toanother ﬁxed value.

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
dc0.05 η( dc=0.05)

1000
800

2

-<ωdq/dp> and Dq/Dt [W/m ]2-<ωdq/dp> and Dq/Dt [W/m ]dc0.5 (ηdc=0.5)
deep conv mode
cong/strat mode

600
400
200
0
-200

1000
q storage (-)
q storage (+)

800
600
400
200
0
-200

0 6 12 18 24 30 36 42 48 54 60 66 72
t [h]

0 6 12 18 24 30 36 42 48 54 60 66 72
t [h]dc0.5 (ηdc=0.5)

dc0.05 η( dc=0.05)-<ωdq/dp>
P’

2

800

1000
-<ωdq/dp> and P’ [W/m ]2-<ωdq/dp> and P’ [W/m ]

1000

600
400
200
0
-200

800
600
400
200
0
-200

0 6 12 18 24 30 36 42 48 54 60 66 72

0 6 12 18 24 30 36 42 48 54 60 66 72

t [h]

t [h]

Figure 11. (top row) 2hx@p qidc (solid), 2hx@p qic=s (dotted), and Dt hqi (shaded in red where positive and in blue where negative). (bottom row) Total 2hx@p qi (shaded) and P0 (solid).
(left column) Experiment dc0.5. (right column) Experiment dc0.05.

4.4. Experiments With Variable g
The moisture storage ratio, g, is assumed to be invariant in the model runs shown so far. This restriction limits the ﬂexibility in the conversion efﬁciency from vertical moisture advection to precipitation and is presumably one of the reasons why the model is incapable of reproducing the increasingly intensifying rainfall
as CWV increases (Figure 13). The assumption of constant g is relaxed in two experiments to follow so that
gc=s and gdc are each variable according to a prescribed function (deﬁned below).
Event #1
Event #2

50

P [mm/d]

40
30
20
10
0
56

58

60
<q> [mm]62

64

Figure 12. The evolutionary track of the observations in the
CWV-precipitation plane for events 1 (ﬁlled circles) and 2 (opencircles) (see section 2 for the deﬁnition). Adjacent points on eachline are 3 h apart.

MASUNAGA AND SUMI

It was argued in sections 1 and 2.3 that cumulus
convection lacks an efﬁcient mechanism to condense out moisture in the ambient air above cloudbase, offering a possible explanation for gc=s  1 assuggested from the observations. There is, on theother hand, a potential pathway for midtropospheric moisture to precipitate out through cumulus dynamics. Since the large-scale mean cloudmass ﬂux at cloud base is balanced out to a large
degree by subsidence mass ﬂux from above at thesame height [e.g., Yanai et al., 1973, Figure 13], anappreciable portion of the subcloud-layer moisture
fed into convective clouds can be in theory traced
back to the downward moisture ﬂux from the midtroposphere. While a dry subsidence into the subcloud layer in general would work against the

A MODEL WITH A MOISTURE STORAGE CLOSURE

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
c/s0.5 (ηc/s=0.5)

35

35

30

30

25

25

<P’> [mm/d]<P’> [mm/d]

cntrl (ηc/s=0.99)

20
15
10

15
10

5

5

0

0

-5

-5
-1

0

1

2

3

4

5

6

7

-1

0

1

2

3

4

<q’> [mm]<q’> [mm]dc0.5 (ηdc=0.5)

dc0.05 (ηdc=0.05)

35

35

30

30

25

25

<P’> [mm/d]<P’> [mm/d]

20

20
15
10

6

7

5

6

7

20
15
10

5

5

0

0

-5

5

-5
-1

0

1

2
3
4
<q’> [mm]

5

6

7

-1

0

1

2
3
4
<q’> [mm]Figure 13. Simulated tracks of the CWV-precipitation relationship in terms of hq0i (abscissa) and P0 (ordinate) for four baseline experiments. Arrows indicate the evolutionary direction
along the track. (left top) cntrl, (right top) c/s0.5, (left bottom) dc0.5, and (right bottom) dc0.05.

maintenance of moist convection, a strong convective downdraft could mechanically force further convection once cold pools develop. Risi et al. [2008] found by use of stable water isotopes that the proportion of
unsaturated downdraft in the subcloud-layer moisture budget amounts to 34% for light precipitation and
45% for strong precipitation, the latter of which in particular is comparable to the contribution of environmental subsidence of 43–45%. It is therefore hypothesized that convective downdrafts, once developed toa certain degree, could intensify themselves without being diluted anymore by environmental subsidence,pulling down more and more midtropospheric moisture that is then fed into subsequent cumuli. This process, if occurring fast enough at a subhourly time scale typical of cumulus processes compared to the largescale variability taking place over daily time scales as seen in Figures 3 and 4, would be regarded virtually as
an instantaneous condensation of midtropospheric water vapor from the moisture budget perspectives.
A modiﬁcation to gc=s to mimic this effect is made in a rather heuristic manner,P0;
gc=s 5g0;c=s 12
Pc=s

(30)

where g0;c=s 50:99 and LPc=s 54000 W m22. The choice of Pc=s here is arbitrary but must be large enoughto satisfy P0 =Pc=s  1, because otherwise gc=s could decrease across the stability criterion of gc=s 1cc=s 50.
Equation (30) embodies a primitive expectation that an increase in P0 would reinforce convective downdrafts with cold pools and hence feedback on precipitation through equation (14). Given the lack of any
theory that analytically formulates g in terms of precipitation, gc=s is related linearly with P0 with all higherorder terms left out of consideration for the sake of brevity.
Equation (20) combined with equation (30) is integrated numerically with other parameters being equal tothe control setup in experiment c/svar (Figure 14, left). Although gc=s changes only mildly from 0.99 to about

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
c/svarηdc and ηc/s

0.8

1

ηc/s
ηdc

0.8
ηdc and ηc/s

1

fullvar

0.6
0.4

0.6
0.4

0.2

0.2

0

0
0 6 12 18 24 30 36 42 48 54 60 66 72
t [h]

0 6 12 18 24 30 36 42 48 54 60 66 72
t [h]fullvar
35

30

30

25

25

<P’> [mm/d]<P’> [mm/d]

c/svar
35

20
15
10

20
15
10

5

5

0

0

-5

-5
-1

0

1

2
3
4
<q’> [mm]

5

6

7

-1

0

1

2
3
4
<q’> [mm]567Figure 14. (top row) gc=s and gdc as a function of time. (bottom row) Simulated tracks of the CWV-precipitation relationship in terms of hq0i (abscissa) and P0 (ordinate). Arrows indicate
the evolutionary direction along the track. (left column) Experiment c/svar. (right column) Experiment fullvar.

0.8 then back toward 1, the evolutionary track in the CWV-precipitation plane has improved signiﬁcantly
compared to the baseline experiments (Figure 13). Precipitation now grows increasingly with CWV toward
its higher end as often found in the literature [Bretherton et al., 2004; Raymond et al., 2007; Holloway and
Neelin, 2010].
The hypothetical process of feeding cumuli with midtropospheric moisture by way of the subcloud layer
would be further facilitated in the presence of the deep convective mode, as deep convection likely generates considerably stronger downdrafts than congesti. This may explain why gdc is closer to 0 rather than 1
in the ﬁrst place, but yet it is somewhat extreme that gdc is ﬁxed at zero all the time. To relax this assumption, gdc is also perturbed in a similar manner to gc=s (but within a range close to zero) as


P0
;
(31)
gdc 5g0;dc 12
Pdc
where g0;dc 50:1 and LPdc 5600 W m22 (Experiment fullvar). The shape of the trajectory remains similar toexperiment c/svar, but the rising and returning branches of the trajectory are now split apart (Figure 14,
right). Such a counter-clockwise turn in the CWV-precipitation curve is as found in the observations by
Masunaga [2012]. The magnitude of the variability is about 6 mm for hq0 i and 30 mm/d for P0 , which compares well with the observations plotted in Figure 12.
Vertical moisture advection and precipitation simulated in experiments c/svar and fullvar are shown in Figure 15. Unlike the striking differences in the CWV-precipitation curve, the evolution of vertical moistureadvection and precipitation is overall reminiscent of the previous experiments (Figures 10 and 11). It is evident, however, that the new runs with variable g, especially the fullvar run, exhibit a sharper rise of precipitation toward the peak than the baseline experiments. The new runs turn out to be better in line with theobservations in which the precipitation peak width is narrow compared to the modestly undulating diabatic

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Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
fullvar

1000
800

2

-<ωdq/dp> and Dq/Dt [W/m ]2-<ωdq/dp> and Dq/Dt [W/m ]c/svardeep conv mode
cong/strat mode

600
400
200
0
-200

1000
q storage (-)
q storage (+)

800
600
400
200
0
-200

0 6 12 18 24 30 36 42 48 54 60 66 72
t [h]

0 6 12 18 24 30 36 42 48 54 60 66 72
t [h]c/svarfullvar-<ωdq/dp>
P’

2

800

1000
-<ωdq/dp> and P’ [W/m ]2-<ωdq/dp> and P’ [W/m ]

1000

600
400
200
0
-200

800
600
400
200
0
-200

0 6 12 18 24 30 36 42 48 54 60 66 72

0 6 12 18 24 30 36 42 48 54 60 66 72

t [h]

t [h]

Figure 15. (top row) 2hx@p qidc (solid), 2hx@p qic=s (dotted), and Dt hqi (shaded in red where positive and in blue where negative). (bottom row) Total 2hx@p qi (shaded) and P0 (solid).
(left column) Experiment c/svar. (right column) Experiment fullvar.

forcing (Figures 3 and 4). An additional sanity test for those runs is performed with the simulated tracks projected onto Figure 7, where the second closure assumption (19) is checked for validity. Figure 16 conﬁrms
that the experiments c/svar and fullvar stay within a range consistent with the observations.

2-d<ωdq/dp> dc/dt [W/m h]

100

5. Conclusion and Discussion

Event #1
Event #2
fullvarc/svar

50

0

-50

-100
-300

-200

-100

0

100

200

300

2

-<ωdq/dp> c/s [W/m ]Figure 16. Scatterplot of 2hx@p qic=s (abscissa) versus the time
derivative of 2hx@p qidc (ordinate) from event 1 (closed circles)
and event 2 (open circles). A straight line with the inclination of
0.3 h21 is shown as a reference. Simulated tracks from the c/svarand fullvar experiments are overlaid.

MASUNAGA AND SUMI

The large-scale dynamics associated with tropicalmoist convection was studied using sounding arrayobservations and a time-dependent, zerodimensional model. The model predicts the vertical
integral of vertical moisture advection (or equivalently of moisture divergence) with a verticalmode decomposition applied into the deep convective and congestus/stratiform modes, denoted by
2hx@p qidc and 2hx@p qic=s , respectively. The variability in vertical moisture advection is a proxy of
the intensity of large-scale updraft since the temperature and humidity proﬁles do not change drastically over a time scale of interest (subdaily to afew days). Each vertical mode is assigned the moisture storage ratio, gdc and gc=s deﬁned in equation(4), as a measure of the fraction of vertical moistureadvection is left unconsumed without being

A MODEL WITH A MOISTURE STORAGE CLOSURE

17

Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
immediately precipitated out. Case studies from the CINDY2011/DYNAMO/AMIE sounding array data
were examined as an observational benchmark to test the model ﬁdelity. The observations suggest that
gc=s is on the order of unity, inferring that congestus clouds do not involve an efﬁcient mechanism to produce rainfall despite a column moistening by vertical moisture advection. This does not contradict thenaive expectation that the efﬁciency of cumulus precipitation would be limited because a signiﬁcant portion of moisture convergence takes place above cloud base in the humid tropical atmosphere. As such, g
may be regarded as a parameter representing the nature of cumulus convection and thus serves as a closure dealing with unresolved processes for the large-scale dynamics.
A second closure relation is introduced by equation (19) to empirically characterize the connection between
the two vertical modes, where the deep convective mode is parameterized to grow or decline dependingon the sign of the congestus/stratiform mode as evidenced by the observations. The resulting prognostic
equation (20) together with an ancillary parameter s summarizing the closure assumptions (equation (21))
founds the basis for the model. The solution indicates that 2hx@p qidc behaves as a delayed response to the
diabatic forcing DF 0 with an e-folding time given by s when s
0. The magnitude of 2hx@p qidc isapproximated well by DF 0 =ðgdc 1cdc Þ if the diabatic forcing changes slowly compared to s as is usually thecase (see Table 1). The observations suggest that gdc stays small near zero and cdc is 0.2–0.3, and therefore
the diabatic forcing is ampliﬁed by a factor of 3–5 in 2hx@p qidc . The value of cdc may be closer to zero inan environment where the midtropospheric moisture is more abundant than the present sounding overthe Indian Ocean. The updraft could be much stronger for a given diabatic forcing in such an environment.
If s is negative, the system becomes unstable as 2hx@p qidc diverges to inﬁnity. The stability criteria on s
is in practice controlled by the sign of gc=s 1cc=s , where cc=s refers to the GMS associated with the congestus/stratiform mode. Under general circumstances gc=s 1cc=s is a small positive number and the system isstable, although the possibility of a negative gc=s 1cc=s is not ruled out since cc=s < 0 by nature.
A negative GMS would in theory cause a thermodynamic instability [Raymond, 2000; Raymond et al., 2009] andcould explain a self-sustaining growth of the system in numerical simulations of tropical convection [Bretherton
et al., 2005; Sessions et al., 2010]. The condition for instability is cc=s 1gc=s < 0 instead of cc=s < 0 in the present
formulation, so that a negative GMS per se would not give rise to an instability unless cc=s further decreases tobelow 2gc=s . In this regard, gc=s acts as a moderator to GMS that relaxes, to a certain degree, the stability criterion as previously documented. This role of gc=s might be instrumental for interpreting negative values of GMS
in nature, which are observed without invoking violent instability where a bottom-heavy circulation climatologically prevails as is typical of the tropical eastern Paciﬁc [Back and Bretherton, 2006]. As discussed above,
however, the possibility remains that an instability could take place when gc=s 1cc=s reduces to a negative value, since no absolute physical constraint seems to exist against it. We do not pursue this issue in this work.
GMS for different vertical modes, by deﬁnition, does not add up to the value calculated with all the modes
combined ﬁrst into the vertical advection terms. The latter deﬁnition of GMS changes rapidly as convection
develops and dissipates [Sessions et al., 2010; Masunaga and L’Ecuyer, 2014; Inoue and Back, 2015b], while
cdc and cc=s are each less variable by design, especially in sign. The quasi-constant nature of the modespeciﬁc GMS is an important element of the present model.
The simulations for an externally speciﬁed diabatic forcing as a cosine-bell function of time qualitatively capture
the observed variation in 2hx@p qidc and 2hx@p qic=s (Figures 10 and 11). In baseline experiments with ﬁxed gdc
and gc=s , the evolution of 2hx@p qidc is, as expected, an ampliﬁed response to the cosine-bell DF 0 with a slighttemporal delay. This appears to infer that moist convection only mirrors the external forcing with little internal freedom to drive its own dynamics, which is obviously at odds with the conventional wisdom in mesoscale meteorology. This deﬁciency is mitigated to a reasonable extent by allowing gdc and gc=s to vary with time, giving a ﬂexibility
to the model so it can somehow handle the dynamics inherent in convective systems. Two of the model runs werecarried out with a heuristic assumption relating g with precipitation in attempt to take into account a positive feedback on precipitation at the intensifying stage of convective systems. The precipitation peak is sharpened in theseruns as actually observed in contrast with an overly broad peak simulated in the baseline experiment.
The CWV-precipitation relationship offers another useful test bed for the model. The steep rise in precipitation with the environmental moistening as well known in the literature is not reproducible when gdc and
gc=s are ﬁxed. Runs with variable gdc and gc=s , on the other hand, are successful in simulating major properties of the observations in the CWV-precipitation curve. It may be worth stressing that the known CWV-MASUNAGA AND SUMI

A MODEL WITH A MOISTURE STORAGE CLOSURE

18

Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
precipitation curve is able to be reproduced by a simplistic conceptual model in a prognostic way, giventhat a consensus has not been reached in the physical reasoning behind it [Raymond and Zeng, 2005; Muller
et al., 2009; Neelin et al., 2009; Raymond et al., 2009; Yano et al., 2012]. The current model predicts the CWVprecipitation curve as a trajectory tracing the life cycle of convective systems, where changes of g in thecourse of the evolution play a crucial role in controlling the trajectory morphology. The evolution begins
with a nearly horizontal path in the CWV-precipitation plane because of a positive moisture storage withminimal precipitation (i.e., g  1), while the path makes a quick turn into a vertical stretch as g reduces.
Note that the CWV-precipitation curve here is identical in shape for any initial condition (or any backgroundstate) with no reference to the absolute values of CWV and precipitation. This ﬁnding is consistent with thesatellite data analysis by Masunaga [2012], in which a family of self-similar evolutionary tracks emerges
regardless of column relative humidity (CRH) in the environment. The universality in the climatological CRHprecipitation relationship [Bretherton et al., 2004] therefore does not apply as is on an instantaneous basis,but is explicable only as the statistical ensemble of many different tracks each representing the individuallife cycle of convection [Masunaga, 2012; Ahmed and Schumacher, 2015].
The moisture storage closure (equation (4)) relates precipitation to the moisture tendency instead of moisture itself. The sharp rainfall pickup in the present model is thus not a manifestation of the moisture control
on rainfall, but results from the increasing efﬁciency of precipitation production as convection intensiﬁes.
Although a moist environment would be arguably more favorable for intense convection than a drier one,parameterizing precipitation (or convective heating) in terms of midtropospheric humidity, as done forexample in an explicit manner by Raymond [2000], may not necessarily be a fully appropriate strategy inlight of causality.
Equations (14) and (19) together lead to
0
d Pdc
0
5vPc=s;dt 12gdc(32)

00
implying that Pdc
is in quadrature with Pc=s
, with a minor deviation when gdc changes with time. In the present model, therefore, the stratiform heating delays behind the deep convective heating by a ﬁxed phaselag (90 ) rather than by a ﬁxed time lag, unlike the early work on the stratiform instability assuming a constant lag of 3 h [e.g., Mapes, 2000; Majda and Shefter, 2001]. The constant time lag was questioned by
Kuang [2008b], based on his ﬁnding that the time lag enhances linearly with an increasing wavelength ofsimulated waves. The present model is in line with Kuang [2008b] in this aspect.

The diabatic forcing is considered as strictly external to the model in the present paper, although the model
formulation readily allows to connect the diabatic forcing with the model outputs if desired. This is a possible future update to the model, as is to incorporate the potential impacts of the radiative-convective feedbacks and wind-induced surface heat exchange (WISHE).

Acknowledgments
The authors thank Johnny Luo, Kuni
Inoue, and an anonymous reviewer for
their valuable comments and helpful
suggestions. The CINDY2011/
DYNAMO/AMIE sounding array data
set was produced by Dick Johnson andPaul Ciesielsky and is distributed at
http://johnson.atmos.colostate.edu/
dynamo/products/array_averages/.
This work was supported by the Japan
Society for the Promotion of Science(JSPS) grant-in-aid (KAKENHI) for
Scientiﬁc Research (B) (26287113) and
Challenging Exploratory Research(26610150).

MASUNAGA AND SUMI

The model formulation satisﬁes the thermal and moisture budget equations while relying on multiple tunable assumptions in need of further reﬁnement. While the observations narrow down the plausible rangesof gdc and gc=s , robust theoretical reasoning for this parameterization has yet to be sought. By deﬁnition, gdc
and gc=s are related to the conversion efﬁciency of moisture convergence into precipitation throughcumulus-scale processes associated with each vertical mode. A potentially useful strategy to quantify this
efﬁciency may be analyzing cloud-resolving model (CRM) runs forced by speciﬁed large-scale dynamics.
Equations (30) and (31), or the dependence of g on precipitation for the c/svar and fullvar runs, are highly
ad hoc and should be updated as well with more sophisticated physics.
The current zero-dimensional model as a whole is obviously limited in practical utility, but a moisture storage closure as examined in this paper would be worth testing with a full-dimensional model capable of simulating realistic convective disturbances.

References
Ahmed, F., and C. Schumacher (2015), Convective and stratiform components of the precipitation-moisture relationship, Geophys. Res.
Lett., 42, 10,453–10,462, doi:10.1002/2015GL066957.
Arakawa, A., and W. H. Schubert (1974), Interaction of a cumulus cloud ensemble with the large-scale environment: Part I, J. Atmos. Sci., 31,
674–701.

A MODEL WITH A MOISTURE STORAGE CLOSURE

19

Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
Back, L. E., and C. S. Bretherton (2006), Geographic variability in the export of moist static energy and vertical motion proﬁles in the tropicalPaciﬁc, Geophys. Res. Lett., 33, L17810, doi:10.1029/2006GL026672.
Back, L. E., and C. S. Bretherton (2009), A simple model of climatological rainfall and vertical motion patterns over the tropical oceans,
J. Clim., 22, 6477–6497, doi:10.1175/2009JCLI2393.1.
Betts, A. K. (1986), A new convective adjustment scheme. Part I: Observational and theoretical basis, Q. J. R. Meteorol. Soc., 112, 677–691.
Bretherton, C. S., M. E. Peters, and L. E. Back (2004), Relationships between water vapor path and precipitation over the tropical oceans,
J. Clim., 17, 1517–1528.
Bretherton, C. S., P. N. Blossey, and M. Khairoutdinov (2005), An energy-balance analysis of deep convective self-aggregation above uniform SST, J. Atmos. Sci., 62, 4273–4292.
Ciesielski, P. E., R. H. Johnson, K. Yoneyama, and R. K. Taft (2014a), Mitigation of Sri Lanka island effects in Colombo sounding data and its
impact on DYNAMO analyses, J. Meteorol. Soc. Jpn., 92, 385–405.
Ciesielski, P. E., R. H. J. H. Yu, K. Yoneyama, M. Katsumata, C. N. Long, J. Wang, S. M. Loehrer, K. Young, S. F. Williams, W. Brown, J. Braun, andT. van Hove (2014b), Quality-controlled upper-air sounding dataset for DYNAMO/CINDY/AMIE: Development and corrections, J. Atmos.
Oceanic Technol., 31, 741–764.
Emanuel, K. A., J. D. Neelin, and C. S. Bretherton (1994), On large-scale circulations in convecting atmospheres, Q. J. R. Meteorol. Soc., 120,
1111–1143.
 and D. J. Raymond (2007), A simple, vertically resolved model of tropical disturbances with a humidity closure, Tellus Ser. A, 59,
Fuchs, Z.,
344–354.
Fulton, S. R., and W. H. Schubert (1985), Vertical normal mode transforms: Theory and application, Mon. Weather Rev., 113, 647–658.
Haertel, P. T., and R. H. Johnson (1998), Two-day disturbances in the equatorial western Paciﬁc, Q. J. R. Meteorol. Soc., 124, 615–636.
Haertel, P. T., and G. N. Kiladis (2004), Dynamics of 2-day equatorial waves, J. Atmos. Sci., 61, 2707–2721.
Haertel, P. T., G. N. Kiladis, A. Denno, and T. M. Rickenbach (2008), Vertical-mode decompositions of 2-day waves and the Madden-Julian
Oscillation, J. Atmos. Sci., 65, 813–833.
Hannah, W. M., B. E. Mapes, and G. S. Elsaesser (2016), A Lagrangian view of moisture dynamics during DYNAMO, J. Atmos. Sci., 73, 1967–
1985.
Holloway, C. E., and J. D. Neelin (2010), Temporal relations of column water vapor and tropical precipitation, J. Atmos. Sci., 67, 1091–1105.
Huffman, G. J., R. F. Adler, D. T. Bolvin, G. Gu, E. J. Nelkin, K. P. Bowman, Y. Hong, E. F. Stocker, and D. B. Wolff (2007), The TRMM multisatellite precipitation analysis: Quasi-global, multi-year, combined-sensor precipitation estimates at ﬁne scale, J. Hydrometeorol., 8,
38–55.
Imaoka, K., and R. W. Spencer (2000), Diurnal variation of precipitation over the tropical oceans observed by TRMM/TMI combined with
SSM/I, J. Clim., 13, 4149–4158.
Inoue, K., and L. E. Back (2015a), Column-integrated moist static energy budget analysis on various time scales during TOGA COARE,
J. Atmos. Sci., 72, 1856–1870.
Inoue, K., and L. E. Back (2015b), Gross moist stability assessment during TOGA COARE: Various interpretations of gross moist stability,
J. Atmos. Sci., 72, 4148–4166.
Johnson, R. H., and P. E. Ciesielski (2013), Structure and properties of Madden-Julian Oscillations deduced from DYNAMO sounding arrays,
J. Atmos. Sci., 70, 3157–3179.
Johnson, R. H., P. E. Ciesielski, J. H. Ruppert Jr., and M. Katsumata (2015), Sounding-based thermodynamic budgets for DYNAMO, J. Atmos.
Sci., 72, 598–622.
Khouider, B., and A. J. Majda (2006), A simple multicloud parameterization for convectively coupled tropical waves: Part I: Linear analysis,
J. Atmos. Sci., 63, 1308–1323.
Khouider, B., and A. J. Majda (2008), Multicloud models for organized tropical convection: Enhanced congestus heating, J. Atmos. Sci., 65,
895–914.
Kuang, Z. (2008a), A moisture-stratiform instability for convectively coupled waves, J. Atmos. Sci., 65, 834–854.
Kuang, Z. (2008b), Modeling the interaction between cumulus convection and linear gravity waves using a limited-domain cloud systemresolving model, J. Atmos. Sci., 65, 576–591.
Majda, A. J., and M. G. Shefter (2001), Models for stratiform instability and convectively coupled waves, J. Atmos. Sci., 58, 1567–1584.
Mapes, B. (2000), Convective inhibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model, J. Atmos.
Sci., 57, 1515–1535.
Mapes, B., S. Tulich, J. Lin, and P. Zuidema (2006), The mesoscale convection life cycle: Building block or prototype for large-scale tropicalwaves?, Dyn. Atmos. Oceans, 42, 3–29.
Mapes, B. E., and R. A. Houze Jr. (1995), Diabatic divergence proﬁles in western Paciﬁc mesoscale convective systems, J. Atmos. Sci., 52,
1807–1828.
Masunaga, H. (2012), Short-term versus climatological relationship between precipitation and tropospheric humidity, J. Climate, 25,
7983–7990.
Masunaga, H. (2013), A satellite study of tropical moist convection and environmental variability: A moisture and thermal budget analysis,
J. Atmos. Sci., 70, 2443–2466.
Masunaga, H. (2014), Free-tropospheric moisture convergence and tropical convective regimes, Geophys. Res. Lett., 41, 8611–8618, doi:
10.1002/2014GL062301.
Masunaga, H., and T. S. L’Ecuyer (2014), A mechanism of tropical convection inferred from observed variability in the moist static energy
budget, J. Atmos. Sci., 71, 3747–3766.
Muller, C. J., L. E. Back, P. A. O’Gorman, and K. A. Emanuel (2009), A model for the relationship between tropical precipitation and column
water vapor, Geophys. Res. Lett., 36, L16804, doi:10.1029/2009GL039667.
Neelin, J. D., and I. M. Held (1987), Modeling tropical convergence based on the moist static energy budget, Mon. Weather Rev., 115,
3–12.
Neelin, J. D., O. Peters, and K. Hales (2009), The transition to strong convection, J. Atmos. Sci., 66, 2367–2384.
Peters, M. E., and C. S. Bretherton (2006), Structure of tropical variability from a vertical mode perspective, Theor. Comput. Fluid Dyn., 20,
501–524, doi:10.1007/s00162-006-0034-x.
Peters, O., and J. D. Neelin (2006), Critical phenomena in atmospheric precipitation, Nat. Phys., 2, 393–396, doi:10.1038/nphys314.
Praveen Kumar, B., J. Vialard, M. Lengaigne, V. S. N. Murt, and M. J. McPhaden (2012), Tropﬂux: Air-sea ﬂuxes for the global tropicaloceans—Description and evaluation, Clim. Dyn., 38, 1521–1543, doi:10.1007/s00382-011-1115-0.
Raymond, D. J. (2000), Thermodynamic control on tropical rainfall, Q. J. R. Meteorol. Soc., 126, 889–898.

MASUNAGA AND SUMI

A MODEL WITH A MOISTURE STORAGE CLOSURE

20

Journal of Advances in Modeling Earth Systems 10.1002/2016MS000855
Raymond, D. J., and M. J. Herman (2011), Convective quasi-equilibrium reconsidered, J. Adv. Model. Earth Syst., 3, M08003, doi:10.1029/
2011MS000079.
Raymond, D. J., and X. Zeng (2005), Modelling tropical atmospheric convection in the context of the weak temperature gradient approximation, Q. J. R. Meteorol. Soc., 131, 1301–1320.
 Fuchs (2007), A theory for the spinup of tropical depressions, Q. J. R. Meteorol. Soc., 133, 1743–1754.
Raymond, D. J., S. L. Sessions, and Z.
 Fuchs (2009), The mechanics of gross moist stability, J. Adv. Model. Earth Syst., 1, 9, doi:
Raymond, D. J., S. L. Sessions, A. H. Sobel, and Z.
10.3894/JAMES.2009.1.9.
Risi, C., S. Bony, and F. Vimeux (2008), Inﬂuence of convective processes on the isotopic composition (d18 O and dD) of precipitation andwater vapor in the tropics: 2. Physical interpretation of the amount effect, J. Geophys. Res., 113, D19306, doi:10.1029/2008JD009943.
Ruppert, J. H., Jr., and R. H. Johnson (2015), Diurnally modulated cumulus moistening in the preonset stage of the Madden-Julian Oscillation during DYNAMO, J. Atmos. Sci., 72, 1622–1646.
Sessions, S. L., S. Sugaya, D. J. Raymond, and A. H. Sobel (2010), Multiple equilibria in a cloud]resolving model using the weak temperature
gradient approximation, J. Geophys. Res., 115, D12110, doi:10.1029/2009JD013376.
Sobel, A. H., J. Nilsson, and N. L. Polvani (2001), The weak temperature gradient approximation and balanced tropical moisture waves,
J. Atmos. Sci., 58, 3650–3665.
Sugiyama, M. (2009), The moisture mode in the quasi-equilibrium tropical circulation model. part I: Analysis based on the weak temperature gradient approximation, J. Atmos. Sci., 66, 1507–1523.
Wielicki, B. A., B. R. Barkstrom, E. F. Harrison, R. B. Lee III, G. L. Smith, and J. E. Cooper (1996), Clouds and the Earth’s Radiant Energy System(CERES): An Earth Observing System experiment, Bull. Am. Meteorol. Soc., 77, 853–868.
Wu, Z., E. S. Sarachik, and D. S. Battisti (2000), Vertical structure of convective heating and the three-dimensional structure of the forced
circulation on an equatorial beta plane, J. Atmos. Sci., 57, 2169–2187.
Yanai, M., S. Esbensen, and J.-H. Chu (1973), Determination of bulk properties of tropical cloud clusters from large scale heat and moisturebudgets, J. Atmos. Sci., 30, 611–627.
Yano, J.-I., C. Liu, and M. W. Moncrieff (2012), Self-organized criticality and homeostasis in atmospheric convective organization, J. Atmos.
Sci., 69, 3449–3462.
Yoneyama, K., C. Zhang, and C. N. Long (2013), Tracking pulses of the Madden-Julian Oscillation, Bull. Am. Meteorol. Soc., 94, 1871–1891.

MASUNAGA AND SUMI

A MODEL WITH A MOISTURE STORAGE CLOSURE

21