JULY 2019

SUMI AND MASUNAGA

2005

Vertical Modes and Effective Stability of Quasi-2-Day Waves
YUKARI SUMI
Graduate School of Science, The University of Tokyo, Tokyo, Japan

HIROHIKO MASUNAGA
Institute for Space-Earth Environmental Research, Nagoya University, Nagoya, Japan(Manuscript received 12 April 2019, in final form 17 April 2019)
ABSTRACTA quasi-2-day wave is known as a convectively coupled westward inertia–gravity (WIG) wave with ashallower equivalent depth (or slower phase speed) than the dry counterpart. This study investigates therelationship between the phase speed of quasi-2-day waves and effective static stability in terms of a verticalmode perspective. By using WIG filters with different equivalent depths, different phases of the 2-day wave
are identified by filtering brightness temperature data obtained from geostationary satellites. The composite
time series and the vertical modes in the tropical atmosphere are calculated from reanalysis data. The largescale dynamical fields of the composite WIG waves are explained by the superposition of the first fourbaroclinic modes. Phase speed of the moist vertical mode is computed by applying the Radon transform to themode transform coefficient. Different vertical modes share a common phase speed, which is slower than its
dry counterpart, implying that the wave is not dispersive. To address the question of what slows the verticalmodes, the effective static stability is evaluated by defining the degree of cancellation between diabatic
heating and adiabatic cooling due to the ascent. This cancellation is confirmed to be almost complete for the
first baroclinic mode as expected theoretically. The effective static stability is found to be higher for a higher
vertical mode, but this change over different vertical modes is not as rapid as predicted from nondispersiveness. Possible reasons for this disagreement are discussed herein.

1. Introduction
A quasi-2-day wave is a synoptic-scale disturbance inthe tropical atmosphere that is accompanied by organized cloud clusters propagating westward with a period
of about 2 days (Takayabu 1994a,b; Takayabu et al.
1996; Chen and Houze 1997; Haertel and Johnson 1998;
Haertel and Kiladis 2004; Haertel et al. 2008). Characteristics of the 2-day waves have been well analyzed
using sounding observations in Tropical Ocean and
Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) intensive observation period (IOP). The COARE 2-day waves propagate
at a phase speed of about 10–30 m s21 and have a zonal
wavelength of about 1000–4000 km (e.g., Takayabu et al.
1996; Haertel and Johnson 1998).
Denotes content that is immediately available upon publication as open access.
Corresponding author: Yukari Sumi, sumi@eps.s.u-tokyo.ac.jp

The 2-day wave is the most fundamental element of
the hierarchy of the convectively coupled equatorial
waves (CCEWs) and is often observed in the active phaseof the Madden–Julian oscillation (MJO) (Nakazawa 1988;
Clayson et al. 2002). It is widely recognized that the
dynamics of the 2-day wave are explained by n 5 1
convectively coupled westward inertia–gravity (WIG)
wave (Takayabu 1994b; Takayabu et al. 1996; Haertel
and Johnson 1998; Wheeler and Kiladis 1999; Wheeler
et al. 2000; Haertel and Kiladis 2004; Haertel et al.
2008; Kiladis et al. 2009; Tulich and Kiladis 2012). The
dry equatorial waves, not coupled with convection, arederived from the shallow-water equation linearizedin equatorial beta plane (Matsuno 1966). Meanwhile,
because of coupling with moist convection, dynamical
characteristics of CCEWs are not fully explained by thelinear theory.
The activity of CCEWs has been extensively investigated by computing a space–time spectrum of tropicalconvection including cloudiness, precipitation, and water vapor (Takayabu 1994a; Wheeler and Kiladis 1999;

DOI: 10.1175/JAS-D-19-0092.1
Ó 2019 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright
Policy (www.ametsoc.org/PUBSReuseLicenses).

2006

JOURNAL OF THE ATMOSPHERIC SCIENCES

Wheeler et al. 2000; Roundy and Frank 2004; Kiladis et al.
2009; Yasunaga and Mapes 2012; and many others).
In a wavenumber–frequency diagram, spectral peaksof CCEWs lie along the theoretical dispersion curves withan equivalent depth of about 12–50 m. This is consistent
with the fact that CCEWs have a slower phase speed (orshallower equivalent depth) than their dry counterparts.
A variety of convection–wave coupling mechanisms
have been proposed to explain the reduction in thephase speed (for reviews, see Straub and Kiladis 2003;
Kiladis et al. 2009). One of the well-established ideas,
which emphasizes the prominent role of the first baroclinic mode (or deep convective mode), explains the
suppressed phase speed in terms of the reduced static
stability (Neelin and Held 1987; Neelin and Yu 1994;
Emanuel et al. 1994; Tian and Ramanathan 2003; Sobel
and Bretherton 2003; Frierson et al. 2004; Raymond
et al. 2009). It is hypothesized that adiabatic cooling
due to strong upward motion is partially canceled out
by diabatic heating due to latent heat release, which
reduces effective static stability in a convective atmosphere. In this approach, a relatively slow phase speedof CCEWs is explained by a slowdown of the dry firstmode (;50 m s21) due to a reduction of the effective
static stability. The reduced static stability can be linked
to the gross moist stability (GMS) that was originallyintroduced by Neelin and Held (1987), in which thesquare of the phase speed is proportional to GMS (Neelin
and Yu 1994; Emanuel et al. 1994; Tian and Ramanathan
2003; Frierson et al. 2004) or the normalized GMS (Sobel
and Bretherton 2003; Raymond et al. 2009). Anotherschool of theory proposes the stratiform instability
mechanism (Mapes 2000; Majda and Shefter 2001;
Majda et al. 2004; Khouider and Majda 2006a; Kuang
2008b), which focuses on the role of the second baroclinic mode (or the stratiform mode) as a key ingredient of instability. The lower-tropospheric cooling due
to the second mode decreases convective inhibition(CIN), and then drives new convection, causing CCEWs
to propagate at a phase speed similar to that of the dry
second mode (;25 m s21). It should be noted that the
reduced static stability mechanism does not necessarily
contradict the stratiform instability mechanism: a slowdown of the first mode may be caused by a reduction of
static stability, whereas the second mode could concurrently play an independent role. However, the roleof the second mode is not yet fully understood. One
possible explanation is that shallow circulation due
to the second mode can moisten the middle troposphere, which may trigger the subsequent deep convection (and hence, the first mode) (Kuang 2008b; Petersand Bretherton 2006; Haertel et al. 2008). Anotherpossible explanation is that the second mode is only a

VOLUME 76

response to the first mode (Frierson 2007; Raymond andFuchs 2007).
The multimode structure of CCEWs first appearedin the stratiform instability mechanism proposed by
Mapes (2000). Subsequently, many theoretical and observational studies have demonstrated that dynamical
structures of CCEWs can be captured by a superpositionof a few vertical modes and have extensively exploredthe concept of the possible coordination between thefirst and the higher vertical modes (Majda and Shefter
2001; Straub and Kiladis 2003; Haertel and Kiladis 2004;
Majda et al. 2004; Peters and Bretherton 2006; Khouider
and Majda 2006a; Tulich et al. 2007; Tulich and Mapes
2008; Haertel et al. 2008; Kuang 2008a; Kiladis et al.
2009; Tulich and Kiladis 2012). In multimode models,
phase speed is not likely to significantly disagree among
different vertical modes, because otherwise the wave
would necessarily be heavily dispersive and would notpropagate over a long distance. This is the crux of theissue with the artificial nature of vertical mode thinking.
In the real atmosphere, it is not necessarily clear why
different vertical modes can have a common slow phasespeed constituting a coherent wave packet without being
immediately dispersed. More observational analysis isneeded to better understand the multimode structure of
CCEWs and the slowdown of the vertical modes, andthis is still worthwhile studying. For this purpose, we
carry out a vertical mode decomposition of 2-day waves
by using satellite and reanalysis data, and explore theslowness and nondispersiveness of the vertical modes.
The change in phase speed of CCEWs is affected not
only by convective coupling but also by environmental
fields where the wave is embedded (Wheeler et al. 2000;
Straub and Kiladis 2002; Yang et al. 2003, 2007a; Roundy
2008; Dias and Pauluis 2009; Kiladis et al. 2009; Dias andPauluis 2011; Dias and Kiladis 2014; Yasunaga andMapes 2014). In addition, the phase speed, structure,and coupling mechanism can vary during the wave life
cycle. From this perspective, recent studies have focusedon the different phases of the wave by using a subdivision
approach (Yasunaga and Mapes 2014; Blanco et al. 2016).
Motivated by these studies, we investigate the multimodestructure of the slow and fast phases of the 2-day wave
by applying the vertical mode decomposition to theircomposite profiles separately. The slow and fast components (or phases) are detected by WIG filters withdifferent equivalent depth as described in section 2a.
This paper consists of the following sections. In
section 2, the data and methods are described. In section 3,
we perform vertical mode transform of 2-day waves.
In section 4, we discuss the potential factors impacting
the slowdown mechanism of vertical modes. Section 5summarizes the present findings.

JULY 2019

SUMI AND MASUNAGA

2. Data and methods
a. DataThis study analyzes 10 years (2000–09) of satellite andreanalysis data. The analysis area is the whole tropicalregion between 108N and 108S. All satellite data areprojected onto a 0.258 grid.
Infrared (IR) brightness temperature (TBB) data areused to identify convective peaks of quasi-2-day waves(see section 2b). TBB data are obtained from the globalmerged IR brightness temperature data (mergedIR),version 6, by Janowiak et al. (2001). MergedIR is a highresolution gridded global dataset derived from the merging
of the Geostationary Operational Environmental Satellite(GOES)-8/10, Meteorological Satellite (Meteosat)-7/5,
and Geostationary Meteorological Satellite data. The
spatial coverage is between 608N and 608S. The horizontal resolution is about 0.03648, and the temporal
resolution is 30 min. Owing to the high temporal resolution of mergedIR, we were able to construct a composite time series with a time increment of 30 min (seesection 2c for more details).
The precipitation data in the tropical ocean area
are obtained from the Tropical Rainfall Measuring
Mission (TRMM) data product. The TRMM satellite
was launched in November 1997 and ended in April
2015. It is a sun-asynchronous satellite orbiting withan inclination angle of about 358 and carrying a Precipitation Radar (PR), which estimates the verticalprofile of precipitation. The horizontal and verticalresolutions of TRMM PR are about 4 km and 250 m,
respectively. The near-surface precipitation data in
PR2A25 level 2, version 6 (Iguchi et al. 2000), areused to analyze the variance in precipitation of 2-day
waves.
To analyze the environmental fields associated withthe 2-day waves, we use the European Centre for MediumRange Weather Forecasts (ECMWF) interim reanalysis(ERA-Interim) data (Simmons et al. 2007). The horizontal resolution is 1.58, and the temporal resolution is
6 h. The total number of pressure levels is 38, in whichpressure intervals are not uniform. The ERA-Interim
variables utilized in this work are the geopotential
height f, temperature T, specific humidity q, zonal
wind velocity u, meridional wind velocity y, and pressure
velocity v. The analysis area covers the whole oceanic
tropical region between 108N and 108S as selected using
the ERA-Interim sea–land flag.

b. Filtering analysis
A space–time filtering of TBB is performed to identify aconvective peak of 2-day wave. Here we assume that a
convectively coupled WIG wave is a practical indicator

2007of the 2-day wave. The procedure used in the present study
is similar to that of Sumi and Masunaga (2016) and isbriefly summarized below.
The filtering is performed for every 2-yr period from2000 to 2009 (i.e., 2000–01, 2002–03, . . . , 2008–09)
because a large memory space is required for fast
Fourier transform (FFT). First, TBB is averaged overthe meridional band between 108N and 108S, and thena longitude–time cross section of the averaged TBB isconstructed. Missing data are filled in by linear interpolations over time. The mean and the linear components in time, identified by least squares fit at eachlongitude, are then removed (e.g., Masunaga et al.
2006). Second, the FFT is applied to the time series ofanomalous TBB by using a split cosine bell taperingfunction to avoid spectral leakage. Third, a WIG filter (the filter design is described below) is applied to awavenumber–frequency domain to isolate individual
WIG waves. Fourth, the filtered TBB is transformed
back to the longitude–time domain. Finally, the peaksof the WIG waves are defined as local minima of thefiltered TBB lower than 22 standard deviations (SDs).
This threshold is used to prevent the composite wave
structures from being contaminated by irrelevant background convective signals.
Figure 1 shows the power spectrum of TBB normalized by red noise background spectrum. The background spectrum is computed by repeated smoothingof the raw spectrum with a 1–2–1 filter (Wheeler andKiladis 1999; Kiladis et al. 2009; and many others).
As reported in previous studies, the spectral peak of
WIG wave is located between the dispersion curves
with an equivalent depth of about 15–45 m. It is more
linear than predicted from theoretical dispersion curves
of WIG wave, but is still dispersive, unlike the Kelvinwave.
The WIG filter is defined as a wavenumber–frequency
domain including the zonal wavenumber k 5 1–14 andthe equivalent depth he 5 12–50 m. The filter is rather
larger than WIG filters with he 5 17–50 m used in previous studies (e.g., Wheeler and Kiladis 1999). However,this difference does not seriously affect our conclusions.
To isolate different phases of WIG wave, the WIG
filtering is performed for each subfilter with different
equivalent depths (e.g., Yasunaga and Mapes 2014). The
faster and slower components of the wave are identified
by the subfilters with the higher and shallower equivalent depth, respectively. The subfilters are denoted by
h15/5, h25/15, h35/25, h45/35, and h55/45 in the order ofincreasing equivalent depths (black boxes in Fig. 1). For
instance, the h15/5 filter includes the equivalent depths
from 15 to 5 m. The wavenumber in the range k 5 1–14
is applied equally to all the subfilters.

2008

JOURNAL OF THE ATMOSPHERIC SCIENCES

FIG. 1. The raw power spectrum normalized by the background
spectrum in wavenumber–frequency (cpd) domain. A negativeand positive wavenumber mean westward and eastward mode,
respectively. The shading begins at a value of 1.2, which is significant at the 95% level with 182 degree of freedom in a chisquared test. Dotted lines represent dispersion curves of n 5 1
WIG wave with equivalent depth of 5, 15, 25, 35, 45, and 55 m.
Five subfilters are indicated by black boxes.

The spectral peak of WIG wave spreads across thesubfilter boxes, which may lead to a cross contamination
between the slower and faster components. The slower
components are accompanied by more convection, because the raw spectrum has more power at shallower
equivalent depth, and hence the contamination of slower
to faster components can be more serious. The influenceof the cross contamination on our results will be discussed
in section 4b and appendix A.
It is noted that much of the filtered signal is filterednoise. In fact, a nonpropagating convective signal issometimes identified as a propagating wavelike disturbance (see also Fig. A1 in appendix A). Instead of
the filtering method, an object-based approach has
been used to avoid a contamination of nonpropagating
convective signals (e.g., Dias et al. 2012). However, in
this study, the filtered noise can be averaged out in thecomposite fields by using a large number of the composite base points with 22 SD threshold (see section 2c).
We will show below that dynamical features of the composite WIG waves are qualitatively consistent with thosein previous studies.

c. Composite analysis
The composite base points are defined as the convective peaks of the filtered WIG waves, which are samples

VOLUME 76

in the whole equatorial ocean region between 108N
and 108S (the total sample number is 331 748). Various
satellite and reanalysis data are composited aroundthe base points. Individual base points only provide
discrete snapshots because of a coarse temporal sampling of the composited data: daily observational frequencies are four times (every 6 h) and less than twotimes in reanalysis data and low-Earth-orbiting (LEO)
satellite data, respectively. However, a statisticallycontinuous time series is constructed by compositing ahuge number of reanalysis and satellite snapshots withrespect to the observational time difference between
mergedIR and the composited data. The time intervalof the composite time series is set to 0.5 h, because thecomposite base points are detected from 0.5-hourly
mergedIR data. For a more detailed description, referto section 2e in Sumi and Masunaga (2016).
Most of the composite base points are found in thewarm pool region (not shown), which is consistent withthe fact that the convectively coupled WIG waves (or2-day waves) are most prominently observed in the IndoPacific Ocean region (Wheeler and Kiladis 1999; Kiladis
et al. 2009; and many others). Therefore, the composite
results may be affected by background conditions of thewarm pool region where the WIG waves are most active.
Regional differences will be briefly described in section 3b.

d. Radon transform
The phase speed of CCEW is usually computed as aregression coefficient of the peaks of convective anomalies in a longitude–time domain (e.g., Wheeler et al.
2000). The Radon transform (RT) method is also used in
an objective manner to calculate propagation speeds oftropical synoptic-scale disturbances (Yang et al. 2007b;
Dias and Pauluis 2011; Dias et al. 2012). In the present
study, the RT method is adopted to compute the phasespeed of the WIG waves (section 3a) and that of the
moist vertical modes (section 3c). It is noted that theestimated phase speed is insensitive to the difference
between the evaluation methods (not shown).
The procedure is briefly summarized here [see theappendix in Yang et al. (2007b) for more details].
The anomalous field of longitude–time domain is projected along a line at angle u with respect to the longitude axis. The energy of the projected data would havemaximum amplitude when the projection axis is perpendicular to the ridges and troughs of the wave in thecourse of varying u from 08 to 1808. With this angle
denoted by u? , the angle perpendicular to the projection angle (u? 1 908) gives the wave propagation direction in the longitude–time domain. Then the phasespeed is calculated as the slope of the propagation
direction.

JULY 2019

SUMI AND MASUNAGA

2009

e. Vertical mode
Vertical mode decomposition is a useful tool to analyze dynamical structure of tropical disturbances and is
widely used in observational and theoretical studies ofmesoscale convective systems (Mapes and Houze 1995;
Mapes 1998) and CCEWs (Haertel and Johnson 1998;
Haertel and Kiladis 2004; Khouider and Majda 2006b;
Tulich et al. 2007; Tulich and Mapes 2008; Haertel et al.
2008; Kuang 2008a; Tulich and Mapes 2010; Tulich andKiladis 2012). There are a number of ways to calculatevertical normal modes in the atmosphere (e.g., Kasahara
and Puri 1981; Silva Dias et al. 1983; Fulton and Schubert
1985). We adopt the method proposed by Fulton andSchubert (1985), in which the vertical modes are derived
from the primitive-equation system in the pressure
coordinate. The vertical mode is obtained by solving a
discrete eigenvalue equation:2›pp› c (p) 5 c2n cn (p) ,SR p n(1)where R is the gas constant of dry air, S 5 RT/pcp 2 ›p T
is the static stability of the basic state (the overbar denotes
temporal and area averages), and cp is the dry air heat
capacity at constant pressure), cn (p) is a normalized
structure function of the nth dry vertical mode, and cn
is the phase speed. Note that cn (p) consists of a normalized orthogonal system and only depends on pressure.
Dynamical profile A(x, y, p, t) is projected onto eachvertical mode component An (x, y, p, t):^ (x, y, t)c (p) .
An (x, y, p, t) 5 A
n
n

(2)

The A(x, y, p, t) represents u, y, f, and D, whereD 5 ›p (Q/S) is forced divergence and Q is heating rate.
^n (x, y, t) is obtained by
The mode coefficient A^ 5 hc jAi 5
A
n
n

ð pb
1
c A dp,pb 2 pt pt n(3)

where the bracket hAjBi represents the vertical innerproduct for functions A and B of pressure, and pt and pb
are the pressure values at the top and bottom boundaries, respectively. Here, cn (p) satisfies hcn jcm i 5 dmn ,where dmn is the Kronecker delta. The mode components Tn , vn , and Qn are calculated by substituting cn inthe thermodynamic equation, continuity equation, anddefinition of D, respectively [see Eq. (12) or Eq. (13)].
Equation (1) is solved by assuming a rigid lid at pt ,where pt 5 100 hPa and pb 5 1000 hPa. The pressure coordinate between pt and pb is divided into 36 layers with auniform thickness of 25 hPa. Then the total number ofvertical modes is 36 (n 5 0–35). ERA-Interim data in thepressure coordinate (between pt and pb ) are also vertically

FIG. 2. Longitude–time diagram of anomalous TBB (K) for a
WIG wave identified with the h25/15 filter. The slope of theblack line represents a phase speed of about 21 m s 21.

interpolated, with uniform spacing (25 hPa), by cubic
spline interpolation. It is noted that the vertical wavelength of the vertical mode varies with S, especially near
the tropopause layer where S increases rapidly. The T is
calculated as a temporal-domain-averaged temperature
between 108N and 108S over a 10-yr period (2000–09)
because regional/seasonal differences in temperature
profile are small between pb and pt (not shown).
The phase speeds of vertical modes, obtained as eigenvalues of Eq. (1), are faster in the lower mode. The
phase speeds (c1 5 52, c2 5 27, c3 5 20, c4 5 15 m s21)
and the equivalent depths (Hn 5 c2n /g: H1 5 274, H2 5 75,
H3 5 39, H4 5 23 m) of the first four vertical modes
are overall consistent with those calculated in previous
studies. For example, Tulich and Kiladis (2012), using
ERA-Interim data, showed that the phase speeds of thefirst four vertical modes are 51, 22, 18, and 14 m s21.
Haertel et al. (2008), using TOGA COARE data, showedthat the phase speeds of the first three vertical modes are
52, 26, and 19 m s21. In addition, the pressure levels of thepeaks and nodes of each vertical mode are roughly consistent with those in the previous studies (not shown).

3. Resultsa. Phase speed of composite WIG waves
Figure 2 shows a longitude–time diagram of anomalous TBB composited around the WIG peaks identified by the h25/15 filter. A convectively active areawith a negative TBB anomaly propagates westward
over a period of about two days. The phase speed of the

2010

JOURNAL OF THE ATMOSPHERIC SCIENCES

VOLUME 76

of the estimated phase speed (15–28 m s21) is consistent with that in previous studies (10–30 m s21). The
phase speeds of TBB agree well with that of precipitation, indicating that the estimated phase speeds areinsensitive to parameter choice (cloud or precipitation). These results support the robustness of the RT
method for use in the present application.

b. Vertical mode transform

FIG. 3. Phase speed of a composite WIG wave. Black and blue
circles represent the phase speeds estimated by RT of anomalous
TBB and anomalous precipitation, respectively.

composite WIG wave is computed by applying the RT
method to the composite TBB and precipitation anomalies (see section 2d for more detail). In Fig. 2, a slope of theblack line represents the phase speed of 21 m s21. The
composite precipitation is rather noisy because of a coarse
temporal sampling of TRMM satellite (not shown), but
this has little effect on the estimation of the phase speed.
Figure 3 shows the phase speed for slower and fastercomponents of WIG waves. As expected, a subfilter
with a deeper (shallower) equivalent depth results in a
faster (slower) component. It is shown that the range

To understand the gross features of WIG waves,the dynamical profiles are projected onto each verticalmode. The vertical mode decomposition has been performed for the simulated and observed 2-day waves. It
is demonstrated that the tilted dynamical structure of
the 2-day wave can be captured by the first two verticalmodes (Haertel and Kiladis 2004; Kuang 2008a). Additional higher modes (i.e., the third and fourth modes)
are also considered in other studies (Haertel et al. 2008;
Tulich and Mapes 2008; Tulich and Kiladis 2012). In thepresent study, a similar vertical mode decomposition isapplied to the composite WIG waves.
Figure 4 shows a time–pressure cross section of themode transform component vn for the first four baroclinic modes, where vn is calculated by substituting
cn in the continuity equation [see Eq. (12)]. A strong

FIG. 4. Time–pressure cross section of pressure velocity (Pa s21) of WIG wave identified with the h25/15 filter: (a) v1 , (b) v2 , (c) v3 ,(d) v4 , (e) a superposition of vn from n5 1 to n 5 4, and (f) composite v. Negative and positive values denote upward and downward
motion, respectively.

JULY 2019

SUMI AND MASUNAGA

2011

FIG. 5. Longitude–time diagram of mode transform coefficient of zonal wind u^n (the first four modes) for a WIG
wave identified with the h25/15 filter. Shades of red and blue represent westerly and easterly zonal wind anomalies
at the surface, respectively. The slope of the black line corresponds to the phase speed of vertical mode, which isestimated by RT of u^n . The red line is the same as black line in Fig. 2.

updraft with a single-sign structure due to the firstmode develops around 0 h (Fig. 4a). It reaches amaximum of about 500 hPa in the middle troposphereat 23 h. Although the first mode has the largest amplitude among the first four modes, the higher modes
are not negligible to explain the wave dynamics. The
second and third modes exhibit a dipole-like structurewith opposite signs in the lower and upper half of thetroposphere (Figs. 4b,c). These two modes are almostin phase with each other and have peaks around ;16 h,which is roughly quadrature with the first mode. The
time lag between the first mode and the second andthird modes is estimated to be about 10 h. The negative peak of the third mode in the lower troposphere
represents a shallow updraft preceding deep convection. Such dipole-like structures preceding andlagging the first mode are consistent with the congestus mode and the stratiform mode, respectively.

The fourth mode is almost in phase with the first mode
(Fig. 4d), which may provide a modest modulation to
the first mode.
A vertically tilted and top-heavy v profile is captured by a superposition of the first four verticalmodes (Fig. 4e), which is qualitatively consistent withthe observed one (Fig. 4f). Also, other dynamical
profiles of the wave (u, y, D, f, and Q) are sufficiently
explained by the first four modes (not shown). Additional higher vertical modes (n . 4) are needed toreproduce a boomerang-like temperature structure,
as reported in previous studies (Haertel and Kiladis
2004; Haertel et al. 2008; Kuang 2008a; Tulich andKiladis 2012). Results for the other subfilters are
similar (not shown).
The tilting v profile (Fig. 4f) is common in different
ocean regions but is modified by background conditions (not shown). For example, the maximum level of2012

JOURNAL OF THE ATMOSPHERIC SCIENCES

FIG. 6. Phase speed of the moist vertical mode estimated by RTof u^n for n 5 1 (red), n 5 2 (blue), n 53 (green), and n 54 (yellow).
The phase speeds estimated form TBB (black) and precipitation(gray) are as in Fig. 3.

convection is slightly lower in the eastern Pacific thanin the warm pool region, which is explained by the regional difference of the large-scale atmospheric circulation (e.g., Back and Bretherton 2006). It is worth
performing vertical mode transform of the composite
WIG waves in different regions, and this will be addressed in future work.

c. Phase speed of the vertical mode
The phase speed of the moist vertical mode is computed by applying the RT method to the mode transform^n (x, y, t). Figure 5 shows a longitude–time
coefficient A
diagram of the coefficient u^n . In the first and fourth modes(the second and third modes), the surface wind converges(diverges) around the composite center, which corresponds to the upward (downward) motion in Fig. 4.
Figure 6 shows the phase speeds of the moist verticalmode estimated from u^n . Phase speed of the moist vertical mode is slower than the dry contour part and isroughly equal to that of the composite WIG wave estimated from TBB and precipitation anomalies. It should
be noted that un itself is not filtered but is simply composited against the filtered TBB peaks, so the agreementin the phase speed is not by design. The phase speed ofthe dry first mode is, in theory, a few times faster thanthat of the higher modes (see section 2e), whereas different moist modes have a similar phase speed between
themselves in Fig. 6. This nondispersive nature of the
moist vertical modes is also verified for other dynamicalparameters, such as geopotential, horizontal wind divergence, and heating rate (not shown).

d. Definition of an
In previous studies, a relatively slow phase speed of
the 2-day wave has been examined in terms of the effective static stability. Haertel and Kiladis (2004), using a

VOLUME 76

FIG. 7. Plot of He vs a
~ n (51 2 He /Hn ) for the first four modes, where
He is changed from 3 to 36 m with an interval of 3 m.

linear primitive-equation model, showed that a degree of
reduction of the equivalent depth of the first two verticalmodes is equal to that of static stability. Haertel et al.
(2008) showed that for COARE 2-day wave, a degree of
reduction in static stability is larger in the lower verticalmode. Following these studies, we derive a diagnostic
equation to examine the slowdown of the vertical modes.
First, the relationship between static stability S and effective static stability Se is considered. For a fixed verticalmode, the thermodynamic equation is written as follows:›t Tn 5 Svn 1 Qn ,(4)where Svn and Qn represent, respectively, adiabatic
heating (cooling) due to descent (ascent) and diabatic
source including latent and radiative heating terms.
Assume that Qn is expressed as
Qn ; 2an Svn ,(5)where an is a positive constant representing the degreeof cancellation between adiabatic heating (cooling) anddiabatic cooling (heating) (Neelin and Held 1987; Haertel
and Kiladis 2004; Haertel et al. 2008; Kiladis et al. 2009).
Substituting Eq. (5) into Eq. (4) yields›t Tn ; Se vn ,(6)

where Se is defined as
Se 5 (1 2 an )S .

(7)

It is assumed that 0 , an , 1 since Se is considered to bepositive. When an is large, Se is small, and thus, a degreeof reduction of static stability is large.
Second, the relationship between static stability andequivalent depth of the vertical mode is considered.

JULY 20192013

SUMI AND MASUNAGA^ n between 224 and 124 h for the first four modes.
FIG. 8. Scatterplot of the mode transform coefficients =  V^n and D
The coefficients are in units of 1026 s21. Different colors represent different phase speeds of WIG wave.

Here, the equivalent depth of the dry vertical mode andthat of the moist vertical mode are denoted by Hn and
Hnmoist , respectively. In a shallow-water system, Hn iswritten as2N,Hn 5 
1
g m2z,n 1 2
4h(8)

where N is a constant Brunt–Väisälä frequency, g is
the gravitational acceleration, h is the scale height,
and mz,n is the vertical wavenumber (e.g., Lindzen
1967; Wheeler et al. 2000). As Hn is a function of mz,nand S (}N2), it is proportional to S for a fixed verticalmode with constant mz,n . Assuming that Hnmoist is also
proportional to Se , we obtain

Hnmoist Se
; .
Hn
S

(9)

Equation (9) indicates that a degree of reduction ofequivalent depth (i.e., Hnmoist /Hn ) is related to that ofstatic stability (i.e., Se /S). It should be noted that Hn
and Hnmoist cannot be exactly expressed in a simple
form as shown in Eq. (8), because static stability is afunction of height in the real atmosphere. This issue,
however, is neglected here because the changes in thestatic stability profile are small except near the surface
and tropopause.
Figure 6 suggested that moist vertical modes havea common equivalent depth He (;H1moist ; H2moist ;H3moist ; H4moist ) because they have a similar phase speed

2014

JOURNAL OF THE ATMOSPHERIC SCIENCES

VOLUME 76

among themselves. Then replacing Hnmoist with He in
Eq. (9) yields
He Se
; .
Hn S

(10)

Combining Eqs. (7) and (10), we finally obtain a diagnostic equation for an :an ; a~n 5 1 2

He
,
Hn

(11)

where a~ n 5 1 2 He /Hn is a monotonically decreasing
function of He with an increment of 21/Hn . The value ofa~ n changes more steeply with respect to He in the higher
mode (Fig. 7).
Equation (11) suggests that a slowdown of the verticalmode (He /Hn , 1) can be explained by a reduction ofeffective static stability that each mode feels (1 2 an ).
There are two implications of Eq. (11). First, for a givenn, an decreases with He with an increment of 21/Hn .
Second, the higher mode has a smaller an when different
moist vertical modes have a common He . Therefore, itfollows that for nondispersive waves, adiabatic coolingcancels diabatic heating more weakly in the higher
mode. The possible application of Eq. (11) will be
discussed in section 4.

e. Estimation of an
Now an is computed according to Eq. (5). The modetransform components vn and Qn are written as
vn 5 2=  V^nðppt

cn dp,

ðp^ S c dp,
Qn 5 D
n
n

(12)

(13)

pt

^ n are the mode transform coefficientswhere =  V^n and Dof horizontal wind divergence =  Vn and forced divergence D, respectively. Following Yanai et al. (1973), theterm Q is computed as the apparent heat source by using
ERA-Interim data. Substituting Eqs. (12) and (13) into
Eq. (5) givesQ
an 5 2 n
Svn
5

^
D
n
.
=  V^(14)n

Therefore, an is calculated as the least squares coeffi^ n and =  V^n . In Haertel and Kiladis
cient between D(2004), an is computed as a ratio of Qn to Svn at a

FIG. 9. Plot of an calculated as a least squares coefficient between^ n (see Fig. 8). The vertical axis starts at 0.65. Error bars
=  V^n and D
indicate the 95% confidence intervals. Different color denotes
different modes (n 5 1–4).

pressure level of the maximum variances. Theirmethod is different from ours, as mentioned above,
but an is insensitive to the difference in the estimation methods. They showed a1 5 0:95 and a2 5 0:75
for the simulated 2-day wave with a phase speed of
16 m s21, which is roughly consistent with our result
shown below: a1 5 0:96, a2 5 0:85, and a3 5 0:75 for
the WIG wave component with a phase speed of
15 m s21 (see Fig. 9).
^ n for the
Figure 8 shows a scatterplot of =  V^n versus D^n^first four vertical modes. Amplitudes of =  V n and D
are larger (smaller) in shallower (higher) equivalent
depth, because the slower (faster) components are accompanied with more (less) convection. The first andfourth modes are tightly distributed around a linear line
(Figs. 8a,d), while the second and third modes are more
scattered (Figs. 8b,c). Then it is implied that an (andtherefore, Se ) is time independent for the first and fourthmodes, but is more time dependent for the second andthird modes. This seems to be consistent with the concept of GMS: the vertical component of GMS is separated into a time-independent part due to the first mode
and a time-dependent part due to the second mode (e.g.,
Inoue and Back 2015).
In Fig. 9, the estimated values of an are positive andsmaller than 1 as predicted by Eq. (7), except in the
fourth mode. The following results are noted: 1) Valuesof an are smaller in the higher mode (a1 . a2 . a3 ),supporting that different moist vertical modes have acommon He (or phase speed). 2) Slower (faster) components of WIG wave, accompanied by more (less)convection, have a larger (smaller) an . This result would
be consistent with previous studies suggesting that thecancellation between diabatic heating and adiabatic
cooling is large (small) in convective active (inactive)
regions (e.g., Haertel et al. 2015; Powell 2017). 3) For a

JULY 2019

SUMI AND MASUNAGA

2015^ n (blue) for a WIG wave identified with
FIG. 10. Time series of the mode transform coefficients =  V^n (red) and D
the filter h25/15. The coefficients are in units of 1026 s21.

given vertical mode, an appears not to decrease with He
as sharply as expected theoretically (cf. Fig. 7). The
decreasing trend of an is statistically significant in theslower components, because the overlap of the error
bars is small enough, especially for the first mode. A
relatively large overlap of error bars in the faster component indicates that the cross contamination between
the slower and faster components would occur. The
slight decrease trend of an will be further discussed in
section 4.
^ n and =  V^n for the
Figure 10 shows a time series of D
first four vertical modes. Temporal variation of the
fourth (third) mode is quite similar to that of the first(second) mode (see also Fig. 8). It is suggested that asuperposition of the first and fourth modes (the second
and third modes) may correspond to a so-called deep

convective mode (stratiform and congestus modes). For
the first and fourth modes, =  V^n is almost in phase with
Dn . This in-phase relationship seems to be consistent
with the reduced stability model, in which diabatic
heating is rapidly canceled by adiabatic cooling becauseof the ascending motion in order to maintain convective
quasi equilibrium (e.g., Emanuel et al. 1994). On the^ n for the second
other hand, =  V^n lags slightly behind D
and third modes, leading to a partial cancellation of the
diabatic heating. Results for the other subfilters are
similar (not shown).
^ 4 has a slightly larger amplitude than
In Fig. 10, D=  V^4 , which results in a4 . 1 (Fig. 9). It is predictedfrom Eq. (7) that an . 1 can lead to Se , 0 (or He , 0).
However, we consider a situation in which Se cannot
be negative because 0 , an , 1 (and thus, Se . 0) is2016

JOURNAL OF THE ATMOSPHERIC SCIENCES

VOLUME 76

TABLE 1. Phase speed and mean zonal flow for each WIG wave
identified with different WIG filters. Standard deviations of themean zonal flows calculated from the sample number of 192 areshown in parentheses. Units are m s21.

FIG. 11. Vertical profile of mean zonal wind u (m s21), which isdefined as a time mean (648 h) of the composite zonal wind at acomposite center. Different colors represent different phase speedsof the WIG wave. Error bars denote the standard deviation of u.

assumed in deriving Eq. (11). Therefore, the fourthmode with a negative Se is excluded in the following
discussion of Eq. (11). We do not necessarily deny thepossibility that effective static stability can be negative(e.g., Raymond 2001; Fuchs and Raymond 2007), but
this is beyond the scope of the present study.

4. Discussion
A diagnostic equation for an [Eq. (11)] is derived
following the reduced stability mechanism, indicatingthat a slowdown of the vertical mode is approximately
described by a reduction of static stability. However,Fig. 9 shows that an does not decrease as sharply as
predicted by Eq. (11). To validate the equation, theequivalent depth of WIG waves is computed in section~ n in section 4b.
4a, and then an is compared with aa. Estimation of He
The equivalent depth of CCEWs is estimated fromthe Doppler-shifted dispersion curve on a space–timediagram. Observational and theoretical studies showthat equatorial waves are affected not only by the
Doppler shifting due to an ambient zonal flow but also
by non-Doppler effects (e.g., Zhang and Webster 1989;
Yang et al. 2003, 2007a; Stechmann and Majda 2009). In
Takayabu (1994b), the equivalent depth of WIG waves,observed during December–February in the Southern
Hemisphere, is estimated to be about 20 m, when considering the Doppler shift of the mean westerly flow

Filter

c

U850

Uave

U200

h15/5
h25/15h35/25h45/35h55/45

15
21
23
26
28

2.0 (0.14)
1.7 (0.12)
1.7 (0.11)
1.7 (0.16)
2.0 (0.13)

3.2 (0.10)
2.8 (0.10)
2.6 (0.10)
2.9 (0.17)
3.2 (0.14)

4.5 (0.40)
4.2 (0.26)
4.4 (0.22)
4.9 (0.17)
4.8 (0.13)

at 850 hPa. Dias and Kiladis (2014) shows that theequivalent depth of CCEWs is similar across seasonsand basins. In addition, they estimate the equivalent
depth of WIG waves to be about 25 m by considering
the Doppler shift of the mean barotropic flow. In thepresent study, a similar estimation of equivalent depth
is carried out as described below.
To assess the Doppler shift, we choose three mean
zonal flows at different pressure levels: u at 850 hPa
U850 , u at 200 hPa U200 , and the pressure-averaged ufrom 850 to 200 hPa Uave . The mean zonal wind u isdefined as the time mean (648 h) of composite zonal
wind at the composite center. It is easterly in the whole
troposphere and is insensitive to the difference in the
wave speed (Fig. 11). This result suggests that the waves
are sampled from a relatively uniform backgroundcondition, regardless of their propagation characteristics. There is no significant correlation between themean zonal flow and the phase speed of WIG waves(Table 1), which is consistent with Yasunaga and Mapes(2014). It is considered that the amount of Doppler
shifting is small because the mean flow is sufficiently
slower than the intrinsic wave speed.
The values of the wavenumber k, the frequency f,
and He (U) are summarized in Table 2; He (U0 ) represents He estimated by considering no Doppler effect(u 5 0). The values in Table 2 are computed as follows:
In the case of the filter h25/15, a zonal extent of negative TBB anomaly is about 188 (see Fig. 2), leading
to l ; 2 3 188(;36 3 105 m) and k ; 10. Considering
Doppler shift due to U850 5 1.7 m s21, intrinsic frequency is calculated as f (U850 ) 5 (c 2 U850 )/l ; 0:46 cpd.
Substituting the values of k and f (U850 ) into dispersion
curve of n 5 1 WIG wave, He (U850 ) is estimated to be
17.2 m. As predicted theoretically, He (U) and f (U) aresmaller (larger) in the strong easterly (westerly) wind.

b. an versus a~nHere, we test the application of the diagnostic equation~ n (U) for thefor an [Eq. (11)]. Figure 12 shows an and afirst three vertical modes, where a~ n (U) 5 1 2 He (U)/Hn
is computed from He (U) in Table 2. It is found that an

JULY 20192017

SUMI AND MASUNAGATABLE 2. Phase speed c (m s21), zonal wavenumber k, frequency f (cpd), and equivalent depth He (m).

FiltercKf (U0 )f (U850 )f (Uave )f (U200 )He (U0 )He (U850 )He (Uave )He (U200 )h15/5
h25/15h35/25h45/35h55/45

15
21
23
26
28

13.89
9.92
10.35
10.12
10.20

0.50
0.50
0.57
0.63
0.69

0.43
0.46
0.53
0.59
0.64

0.39
0.43
0.51
0.56
0.61

0.54
0.40
0.46
0.51
0.57

14.8
22.0
31.0
42.2
52.2

10.0
17.2
24.8
34.8
42.6

7.60
14.2
22.0
30.2
27.4

5.40
11.0
16.6
23.2
31.0

best matches with a~ n (U850 ) in the slower components,
especially for the first mode. The values of a1 anda~ 1 (U850 ) are almost the same between the h15/5 andh35/25 filters. A decreasing rate of a1 is nearly equal
to that of a
~ 1 (U850 )(521/H1 ). The values of a2 anda~ 2 (U850 ) are similar between the h15/5 and h25/15 filters, while a decreasing rate of a2 is more modest thanthat of a~ 2 (U850 )(521/H2 ). The a3 is roughly equal toa~ 3 (U850 ) only for the h15/5 filter. A decreasing rate
~ 3 (521/H3 ). There areof a3 is different from that of asome possible explanations for the disagreement between~n:an and a1) The value of an would depend on the accuracy of thedata used in the analysis. A slight decrease trend ofan (Figs. 9 and 12) and a4 . 1 (Fig. 9) would partiallyresult from the uncertainties in ERA-Interim data
because of the cumulus parameterization in a data
assimilation system. In fact, Kim and Alexander (2013)shows that a rapid variation associated with WIG
wave cannot be resolved well in reanalysis data.
This issue may lead to a relatively large overlap oferror bars of an in the higher modes (Figs. 9 and
12). However, we verified that the ERA-Interim ismore suitable for a study of WIG waves than otherreanalysis data (see appendix B for more detail). In
future work, our results will be compared with ancomputed from satellite or in situ data.
2) The estimation of an has its own uncertainties. We
assume that an is a time-independent parameter andcompute an as the least squares coefficient between^ 1 changes almost linearly with^ n and =  V^n . As DD
=  V^1 (Fig. 8), it is indicated that a1 is less dependent
on time as expected theoretically. On the other hand,^ n does not vary
for the second and third modes, D
linearly with =  V^n , indicating that an would be
more time dependent. Therefore, the estimation
error of an can be large on the second and thirdmodes. While a time dependency of an is not considered in the present study, we will address this issue
in future work.
3) Difference of an between the slower and fastercomponents (or the shallower and higher equivalent
depths) may be smaller than expected from Eq. (11)because of cross contamination, which may lead to aslight decrease trend of an .
4) In the literature, effective static stability has been
introduced to explain a large-scale motion associatedwith the first mode that is tied to deep convection(e.g., Neelin and Held 1987; Yano and Emanuel1991; Neelin and Yu 1994; Emanuel et al. 1994): theatmosphere is assumed to be in a state of precipitation to ignore the descending area with no precipitation, and then convective heating is rapidly
canceled by adiabatic cooling because of ascending
motion. As expected theoretically, a1 is roughly
identical to a
~ 1 (U850 ). Then it is implied that the
reduced stability mechanism may be more suitablefor explaining the slowdown of the first mode.
However, it is not intuitively obvious how the
reduced stability mechanism explains a slowdown of~ n for thethe higher modes. In fact, the result of an 6¼ asecond and third modes implies that the slowdownof the higher modes cannot be fully explained by the
reduced stability mechanism.
5) A slight decrease trend of an can be interpreted inanother way: an is constrained to be approximately
constant for particular vertical modes and does
not vary as predicted by Eq. (11). If this is true,
the range of slowdown of the vertical modes thatcan be explained by the reduced static stability islimited.
In future work, similar analysis will be applied toother CCEWs and mesoscale disturbances in the tropics.
This will provide insight into the multimode structure of
CCEWs and is still worth studying.

5. Summary
The present study examines the slowdown of verticalmodes associated with quasi-2-day waves by using
satellite and reanalysis data. The 2-day waves are
identified by filtered TBB (mergedIR), where the fasterand slower components of the 2-day wave are separated
by adopting WIG filters with different equivalent depths
within a range from 5 to 55 m. ERA-Interim data arecomposited around the convective peaks of WIG waves
to analyze the large-scale fields associated with the

2018

JOURNAL OF THE ATMOSPHERIC SCIENCES

VOLUME 76

FIG. A1. Longitude–time diagram of WIG waves identified witheach subfilter as well as the nonfiltered TBB anomalies from 10 to
20 January 2005. The color shading illustrates the nonfiltered TBB
anomaly. The wave envelops are indicated by contours drawn at
the value of 22 standard deviations. Different colors denote different waves: h55/45 (black), h45/35 (blue), h35/25 (green), h25/15(purple), and h15/5 (red).

FIG. 12. Plots of an (black) vs a
~ n (U) (blue) for the first threemodes: (top) n 5 1, (middle) n 5 2, and (bottom) n 5 3. The verticalaxis starts at 0.7 in the top panel. The value of a
~ n (U) 5 1 2 He (U)/Hn
is computed by using He (U) in Table 2. Blue squares, circles, triangles,~ n (u850 ), a
~ n (u200 ), and a~ n (uave ),and diamonds indicate a~ n (u0 ), a
respectively.

waves. The composite fields are further decomposed by
the vertical modes calculated with the mean temperature
profile (ERA-Interim) in the tropical region between
108N and 108S.
It is shown that the dynamical structure of WIG waves
are explained largely by a superposition of the first fourbaroclinic modes. The resulting vertically tilted, topheavy structure is consistent with the multimode structure, including the deep, congestus, and stratiform
modes (e.g., Mapes 2000; Haertel et al. 2008). The phasespeed of moist vertical modes is computed by Radontransform of the mode coefficient in the longitude–timedomain. The moist vertical modes are slowed down
compared to its dry counterpart, while different verticalmodes have similar phase speeds, so the wave is only
weakly dispersive.
In the effective stability mechanism, the slower phasespeed of the first mode is attributed to the reduction ofeffective stability. It is not obvious, however, if the slowdown of the higher modes is also explained by a similar framework. The effective stability increases with theequivalent depth as expected from the nondispersive nature, but the rate of increase is not as steep as theoretically
predicted. This disagreement is partially reconciled particularly for the first mode when the Doppler effect is taken
into account, but fails to account for higher modes. The
reasons include the limitation in the estimation method
and physical robustness of the effective stability for thehigher vertical modes. Further investigations of theseoutstanding issues should be addressed in future work.
Acknowledgments. The authors thank four anonymous reviewers for their helpful and constructive
comments and criticisms, which greatly improved thismanuscript. This work was supported by Japan Society for the Promotion of Science Grant-in-Aid for
Scientific Research B (19H01966). The global-merged IR

JULY 20192019

SUMI AND MASUNAGA

FIG. A2. Time series of the filtered TBB (K) composited around the peaks of the wave identified with the (a) h25/15
and (b) h45/35 filters.

brightness temperature data were provided by National
Centers for Environmental Prediction (NCEP) Climate
Prediction Center (CPC). The TRMM PR (2A25)
datasets were provided by the Japan Aerospace
Exploration Agency (JAXA). ERA-Interim data were
provided by ECMWF. The NCEP Reanalysis data were
provided by the NOAA-CIRES Climate Diagnostics
Center, Boulder, Colorado. JRA-25 data were provided
from the Japanese 25-year Reanalysis (JRA-25), thecooperative research project carried out by the Japan
Meteorological Agency (JMA) and the Central ResearchInstitute of Electric Power Industry (CRIEPI).

Therefore, it is considered that the cross-contamination
effect is not so serious in our results.

APPENDIX B
Comparison with Other Reanalysis Datasets
To test that ERA-Interim is suitable for our analysis,
we compare apparent heat source Q calculated from

APPENDIX A
Cross Contamination
Figure A1 shows WIG wave components identified
with each subfilter, as well as the nonfiltered TBB
anomalies, from 10 to 20 January 2005. The slower
and faster components overlap or penetrate eachother in several places, implying that cross contamination may have occurred in our analysis. To assess
the effect of the contamination, the filtered TBB is
composited around the peak of the faster (the h25/15
filter) and slower (the h45/35 filter) components. The
slower (faster) component’s contamination of the faster(slower) component can be significant, if the amplitudeof the slower (faster) component is large around the peakof the faster (slower) component. In Fig. A2, the amplitudes of the faster (slower) component are significantly
small around the peak of the slower (faster) component.

FIG. B1. The composite Q profile at t 5 0 of the composite timeseries for WIG waves identified with the h25/15 filter. The values of
Q is calculated from JRA-25 (blue), NCEP (green), and ERAInterim (black) reanalysis and TRMM (red).

2020

JOURNAL OF THE ATMOSPHERIC SCIENCES

ERA-Interim, NCEP–NCAR (Kalnay et al. 1996), andJRA-25 (Onogi et al. 2007) reanalysis data with TRMM
latent heat product (Shige et al. 2004). Figure B1 shows
Q profile at composite center. A maximum of TRMM Q
profile is at 450 hPa, which matches best with ERAInterim: a maximum of Q profile is between 400 and
500 hPa for ERA-Interim, while it is at 300 hPa for JRA-25
and NCEP. ERA-Interim is considered to be more appropriate for a study of 2-day waves than other reanalysis data.
REFERENCES
Back, L. E., and C. S. Bretherton, 2006: Geographic variability inthe export of moist static energy and vertical motion profiles inthe tropical Pacific. Geophys. Res. Lett., 33, L17810, https://doi.org/10.1029/2006GL026672.
Blanco, J. E., D. S. Nolan, and B. E. Mapes, 2016: Convectively
coupled Kelvin waves in aquachannel simulations: 2. Life cycleand dynamical-convective coupling. J. Geophys. Res. Atmos.,
121, 11 319–11 347, https://doi.org/10.1002/2016JD025022.
Chen, S. S., and R. A. Houze Jr., 1997: Diurnal variation and lifecycle of deep convective systems over the tropical Pacificwarm pool. Quart. J. Roy. Meteor. Soc., 123, 357–388, https://doi.org/10.1002/qj.49712353806.
Clayson, C. A., B. Strahl, and J. Schrage, 2002: 2–3-day convective
variability in the tropical western Pacific. Mon. Wea. Rev., 130,
529–548, https://doi.org/10.1175/1520-0493(2002)130,0529:
DCVITT.2.0.CO;2.
Dias, J., and O. Pauluis, 2009: Convectively coupled waves propagating along an equatorial ITCZ. J. Atmos. Sci., 66, 2237–
2255, https://doi.org/10.1175/2009JAS3020.1.
——, and ——, 2011: Modulations of the phase speed of convectively coupled Kelvin waves by the ITCZ. J. Atmos. Sci.,
68, 1446–1459, https://doi.org/10.1175/2011JAS3630.1.
——, and G. N. Kiladis, 2014: Influence of the basic state zonal
flow on convectively coupled equatorial waves. Geophys. Res.
Lett., 41, 6904–6913, https://doi.org/10.1002/2014GL061476.
——, S. N. Tulich, and G. N. Kiladis, 2012: An object-based
approach to assessing the organization of tropical convection. J. Atmos. Sci., 69, 2488–2504, https://doi.org/10.1175/
JAS-D-11-0293.1.
Emanuel, K. A., J. D. Neelin, and C. S. Bretherton, 1994: On largescale circulations in convecting atmospheres. Quart. J. Roy.
Meteor. Soc., 120, 1111–1143, https://doi.org/10.1002/qj.49712051902.
Frierson, D. M., 2007: Convectively coupled Kelvin waves in an
idealized moist general circulation model. J. Atmos. Sci., 64,
2076–2090, https://doi.org/10.1175/JAS3945.1.
——, A. J. Majda, and O. M. Pauluis, 2004: Large scale dynamics ofprecipitation fronts in the tropical atmosphere: A novel relaxation limit. Commun. Math. Sci., 2, 591–626, https://doi.org/
10.4310/CMS.2004.v2.n4.a3.
 and D. J. Raymond, 2007: A simple, vertically resolved
Fuchs, Z.,model of tropical disturbances with a humidity closure. Tellus,
59, 344–354, https://doi.org/10.1111/j.1600-0870.2007.00230.x.
Fulton, S. R., and W. H. Schubert, 1985: Vertical normal modetransforms: Theory and application. Mon. Wea. Rev., 113,
647–658, https://doi.org/10.1175/1520-0493(1985)113,0647:
VNMTTA.2.0.CO;2.
Haertel, P. T., and R. H. Johnson, 1998: Two-day disturbances inthe equatorial western Pacific. Quart. J. Roy. Meteor. Soc., 124,
615–636, https://doi.org/10.1002/qj.49712454611.

VOLUME 76——, and G. N. Kiladis, 2004: Dynamics of 2-day equatorial
waves. J. Atmos. Sci., 61, 2707–2721, https://doi.org/10.1175/
JAS3352.1.
——, ——, 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, https://doi.org/10.1175/
2007JAS2314.1.
——, K. Straub, and A. Budsock, 2015: Transforming circumnavigating Kelvin waves that initiate and dissipate theMadden–Julian oscillation. Quart. J. Roy. Meteor. Soc., 141,
1586–1602, https://doi.org/10.1002/qj.2461.
Iguchi, T., T. Kozu, R. Meneghini, J. Awaka, and K. Okamoto,2000: Rain-profiling algorithm for the TRMM Precipitation
Radar. J. Appl. Meteor., 39, 2038–2052, https://doi.org/10.1175/
1520-0450(2001)040,2038:RPAFTT.2.0.CO;2.
Inoue, K., and L. Back, 2015: Column-integrated moist static
energy budget analysis on various time scales during TOGACOARE. J. Atmos. Sci., 72, 1856–1871, https://doi.org/10.1175/
JAS-D-14-0249.1.
Janowiak, J. E., R. J. Joyce, and Y. Yarosh, 2001: A real-time
global half-hourly pixel-resolution infrared dataset andits applications. Bull. Amer. Meteor. Soc., 82, 205–217,
https://doi.org/10.1175/1520-0477(2001)082,0205:ARTGHH.
2.3.CO;2.
Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year
Reanalysis Project. Bull. Amer. Meteor. Soc., 77, 437–471,
https://doi.org/10.1175/1520-0477(1996)077,0437:TNYRP.
2.0.CO;2.
Kasahara, A., and K. Puri, 1981: Spectral representation of threedimensional global data by expansion in normal mode functions. Mon. Wea. Rev., 109, 37–51, https://doi.org/10.1175/
1520-0493(1981)109,0037:SROTDG.2.0.CO;2.
Khouider, B., and A. J. Majda, 2006a: A simple multicloud parameterization for convectively coupled tropical waves. Part I:
Linear analysis. J. Atmos. Sci., 63, 1308–1323, https://doi.org/
10.1175/JAS3677.1.
——, and ——, 2006b: Multicloud convective parametrizations
with crude vertical structure. Theor. Comput. Fluid Dyn., 20,
351–375, https://doi.org/10.1007/s00162-006-0013-2.
Kiladis, G. N., M. C. Wheeler, P. T. Haertel, K. H. Straub, andP. E. Roundy, 2009: Convectively coupled equatorial
waves. Rev. Geophys., 47, RG2003, https://doi.org/10.1029/
2008RG000266.
Kim, J.-E., and M. J. Alexander, 2013: Tropical precipitation variability and convectively coupled equatorial waves on submonthly time scales in reanalyses and TRMM. J. Climate, 26,3013–3030, https://doi.org/10.1175/JCLI-D-12-00353.1.
Kuang, Z., 2008a: Modeling the interaction between cumulus
convection and linear gravity waves using a limited-domain
cloud system-resolving model. J. Atmos. Sci., 65, 576–591,
https://doi.org/10.1175/2007JAS2399.1.
——, 2008b: A moisture-stratiform instability for convectively
coupled waves. J. Atmos. Sci., 65, 834–854, https://doi.org/
10.1175/2007JAS2444.1.
Lindzen, R. S., 1967: Planetary waves on beta-planes. Mon.
Wea. Rev., 95, 441–451, https://doi.org/10.1175/1520-0493(1967)
095,0441:PWOBP.2.3.CO;2.
Majda, A. J., and M. G. Shefter, 2001: Models for stratiform instability and convectively coupled waves. J. Atmos. Sci., 58,
1567–1584, https://doi.org/10.1175/1520-0469(2001)058,1567:
MFSIAC.2.0.CO;2.
——, B. Khouider, G. N. Kiladis, K. H. Straub, and M. G. Shefter,
2004: A model for convectively coupled tropical waves:

JULY 2019

SUMI AND MASUNAGA

Nonlinearity, rotation, and comparison with observations.
J. Atmos. Sci., 61, 2188–2205, https://doi.org/10.1175/15200469(2004)061,2188:AMFCCT.2.0.CO;2.
Mapes, B. E., 1998: The large-scale part of tropical mesoscaleconvective system circulations: A linear vertical spectral
band model. J. Meteor. Soc. Japan, 76, 29–55, https://doi.org/
10.2151/jmsj1965.76.1_29.
——, 2000: Convective inhibition, subgrid-scale triggering energy,
and stratiform instability in a toy tropical wave model.
J. Atmos. Sci., 57, 1515–1535, https://doi.org/10.1175/15200469(2000)057,1515:CISSTE.2.0.CO;2.
——, and R. A. Houze Jr., 1995: Diabatic divergence profiles inwestern Pacific mesoscale convective systems. J. Atmos. Sci.,
52, 1807–1828, https://doi.org/10.1175/1520-0469(1995)052,1807:
DDPIWP.2.0.CO;2.
Masunaga, H., T. S. L’Ecuyer, and C. D. Kummerow, 2006: The
Madden–Julian oscillation recorded in early observations fromthe Tropical Rainfall Measuring Mission (TRMM). J. Atmos.
Sci., 63, 2777–2794, https://doi.org/10.1175/JAS3783.1.
Matsuno, T., 1966: Quasi-geostrophic motions in the equatorial
area. J. Meteor. Soc. Japan, 44, 25–43, https://doi.org/10.2151/
jmsj1965.44.1_25.
Nakazawa, T., 1988: Tropical super clusters within intraseasonalvariations over the western Pacific. J. Meteor. Soc. Japan, 66,
823–839, https://doi.org/10.2151/jmsj1965.66.6_823.
Neelin, J. D., and I. M. Held, 1987: Modeling tropical convergencebased on the moist static energy budget. Mon. Wea. Rev.,
115, 3–12, https://doi.org/10.1175/1520-0493(1987)115,0003:
MTCBOT.2.0.CO;2.
——, and J.-Y. Yu, 1994: Modes of tropical variability underconvective adjustment and the Madden–Julian oscillation.
Part I: Analytical theory. J. Atmos. Sci., 51, 1876–1894,
https://doi.org/10.1175/1520-0469(1994)051,1876:MOTVUC.
2.0.CO;2.
Onogi, K., and Coauthors, 2007: The JRA-25 Reanalysis. J. Meteor.
Soc. Japan, 85, 369–432, https://doi.org/10.2151/jmsj.85.369.
Peters, M. E., and C. S. Bretherton, 2006: Structure of tropicalvariability from a vertical mode perspective. Theor. Comput.
Fluid Dyn., 20, 501–524, https://doi.org/10.1007/s00162-0060034-x.
Powell, S. W., 2017: Successive MJO propagation in MERRA-2
reanalysis. Geophys. Res. Lett., 44, 5178–5186, https://doi.org/
10.1002/2017GL073399.
Raymond, D. J., 2001: A new model of the Madden–Julian oscillation. J. Atmos. Sci., 58, 2807–2819, https://doi.org/10.1175/
1520-0469(2001)058,2807:ANMOTM.2.0.CO;2.
 Fuchs, 2007: Convectively coupled gravity and mois——, and Z.
ture modes in a simple atmospheric model. Tellus, 59A, 627–
640, https://doi.org/10.1111/j.1600-0870.2007.00268.x.
 Fuchs, 2009: The me——, S. L. Sessions, A. H. Sobel, and Z.
chanics of gross moist stability. J. Adv. Model. Earth Syst.,
1 (3), https://doi.org/10.3894/JAMES.2009.1.9.
Roundy, P. E., 2008: Analysis of convectively coupled Kelvinwaves in the Indian Ocean MJO. J. Atmos. Sci., 65, 1342–1359,
https://doi.org/10.1175/2007JAS2345.1.
——, and W. M. Frank, 2004: A climatology of waves in the equatorial region. J. Atmos. Sci., 61, 2105–2132, https://doi.org/
10.1175/1520-0469(2004)061,2105:ACOWIT.2.0.CO;2.
Shige, S., Y. N. Takayabu, W.-K. Tao, and D. E. Johnson, 2004:
Spectral retrieval of latent heating profiles from TRMM PR
data. Part I: Development of a model-based algorithm.
J. Appl. Meteor., 43, 1095–1113, https://doi.org/10.1175/15200450(2004)043,1095:SROLHP.2.0.CO;2.

2021

Silva Dias, P. L., W. H. Schubert, and M. DeMaria, 1983: Largescale response of the tropical atmosphere to transient convection. J. Atmos. Sci., 40, 2689–2707, https://doi.org/10.1175/
1520-0469(1983)040,2689:LSROTT.2.0.CO;2.
Simmons, A., S. Uppala, D. Dee, and S. Kobayashi, 2007: ERAInterim: New ECMWF reanalysis products from 1989 onwards. ECMWF Newsletter, No. 110, ECMWF, Reading,United Kingdom, 25–35.
Sobel, A. H., and C. S. Bretherton, 2003: Large-scale waves interactingwith deep convection in idealized mesoscale model simulations.
Tellus, 55A, 45–60, https://doi.org/10.3402/tellusa.v55i1.12084.
Stechmann, S. N., and A. J. Majda, 2009: Gravity waves in shear
and implications for organized convection. J. Atmos. Sci., 66,
2579–2599, https://doi.org/10.1175/2009JAS2976.1.
Straub, K. H., and G. N. Kiladis, 2002: Observations of a convectively coupled Kelvin wave in the eastern Pacific ITCZ.
J. Atmos. Sci., 59, 30–53, https://doi.org/10.1175/1520-0469(2002)
059,0030:OOACCK.2.0.CO;2.
——, and ——, 2003: The observed structure of convectively coupled Kelvin waves: Comparison with simple models of coupledwave instability. J. Atmos. Sci., 60, 1655–1668, https://doi.org/
10.1175/1520-0469(2003)060,1655:TOSOCC.2.0.CO;2.
Sumi, Y., and H. Masunaga, 2016: A moist static energy budget analysis
of quasi-2-day waves using satellite and reanalysis data. J. Atmos.
Sci., 73, 743–759, https://doi.org/10.1175/JAS-D-15-0098.1.
Takayabu, Y. N., 1994a: Large-scale cloud disturbances associated with equatorial waves. Part I: Spectral features of thecloud disturbances. J. Meteor. Soc. Japan, 72, 433–449, https://doi.org/10.2151/jmsj1965.72.3_433.
——, 1994b: Large-scale cloud disturbances associated withequatorial waves. Part II: Westward-propagating inertiogravity waves. J. Meteor. Soc. Japan, 72, 451–465, https://doi.org/10.2151/jmsj1965.72.3_451.
——, K.-M. Lau, and C.-H. Sui, 1996: Observation of a quasi-2-day
wave during TOGA COARE. Mon. Wea. Rev., 124, 1892–
1913, https://doi.org/10.1175/1520-0493(1996)124,1892:
OOAQDW.2.0.CO;2.
Tian, B., and V. Ramanathan, 2003: A simple moist tropicalatmosphere model: The role of cloud radiative forcing.
J. Climate, 16, 2086–2092, https://doi.org/10.1175/15200442(2003)016,2086:ASMTAM.2.0.CO;2.
Tulich, S. N., and B. E. Mapes, 2008: Multiscale convective wave
disturbances in the tropics: Insights from a two-dimensional
cloud-resolving model. J. Atmos. Sci., 65, 140–155, https://doi.org/10.1175/2007JAS2353.1.
——, and ——, 2010: Transient environmental sensitivities of explicitly simulated tropical convection. J. Atmos. Sci., 67, 923–
940, https://doi.org/10.1175/2009JAS3277.1.
——, and G. N. Kiladis, 2012: Squall lines and convectively coupledgravity waves in the tropics: Why do most cloud systems
propagate westward? J. Atmos. Sci., 69, 2995–3012, https://doi.org/10.1175/JAS-D-11-0297.1.
——, D. A. Randall, and B. E. Mapes, 2007: Vertical-mode andcloud decomposition of large-scale convectively coupled gravity
waves in a two-dimensional cloud-resolving model. J. Atmos.
Sci., 64, 1210–1229, https://doi.org/10.1175/JAS3884.1.
Wheeler, M., and G. N. Kiladis, 1999: Convectively coupledequatorial waves: Analysis of clouds and temperature in thewavenumber–frequency domain. J. Atmos. Sci., 56, 374–399,
https://doi.org/10.1175/1520-0469(1999)056,0374:CCEWAO.
2.0.CO;2.
——, ——, and P. J. Webster, 2000: Large-scale dynamical fields
associated with convectively coupled equatorial waves.

2022

JOURNAL OF THE ATMOSPHERIC SCIENCES

J. Atmos. Sci., 57, 613–640, https://doi.org/10.1175/15200469(2000)057,0613:LSDFAW.2.0.CO;2.
Yanai, M., S. Esbensen, and J.-H. Chu, 1973: Determination of bulk
properties of tropical cloud clusters from large-scale heat andmoisture budgets. J. Atmos. Sci., 30, 611–627, https://doi.org/
10.1175/1520-0469(1973)030,0611:DOBPOT.2.0.CO;2.
Yang, G.-Y., B. Hoskins, and J. Slingo, 2003: Convectively coupledequatorial waves: A new methodology for identifying wave structures in observational data. J. Atmos. Sci., 60, 1637–1654, https://doi.org/10.1175/1520-0469(2003)060,1637:CCEWAN.2.0.CO;2.
——, ——, and ——, 2007a: Convectively coupled equatorial
waves. Part I: Horizontal and vertical structures. J. Atmos. Sci.,
64, 3406–3423, https://doi.org/10.1175/JAS4017.1.
——, ——, and ——, 2007b: Convectively coupled equatorial
waves. Part II: Propagation characteristics. J. Atmos. Sci., 64,
3424–3437, https://doi.org/10.1175/JAS4018.1.

VOLUME 76

Yano, J.-I., and K. Emanuel, 1991: An improved model of theequatorial troposphere and its coupling with the stratosphere.
J. Atmos. Sci., 48, 377–389, https://doi.org/10.1175/15200469(1991)048,0377:AIMOTE.2.0.CO;2.
Yasunaga, K., and B. Mapes, 2012: Differences between moredivergent and more rotational types of convectively coupled equatorial waves. Part I: Space–time spectral analyses.
J. Atmos. Sci., 69, 3–16, https://doi.org/10.1175/JAS-D11-033.1.
——, and ——, 2014: Differences between faster versus slower
components of convectively coupled equatorial waves. J. Atmos.
Sci., 71, 98–111, https://doi.org/10.1175/JAS-D-13-03.1.
Zhang, C., and P. J. Webster, 1989: Effects of zonal flows onequatorially trapped waves. J. Atmos. Sci., 46, 3632–3652,
https://doi.org/10.1175/1520-0469(1989)046,3632:EOZFOE.
2.0.CO;2.