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A Moist Static Energy Budget Analysis of Quasi-2-Day Waves
Using Satellite and Reanalysis Data
YUKARI SUMI
Graduate School of Environmental Studies, Nagoya University, Nagoya, Japan

HIROHIKO MASUNAGA
Hydrospheric Atmospheric Research Center, Nagoya University, Nagoya, Japan(Manuscript received 29 March 2015, in final form 23 October 2015)
ABSTRACT
A moist static energy (MSE) budget analysis is applied to quasi-2-day waves to examine the effects ofthermodynamic processes on the wave propagation mechanism. The 2-day waves are defined as westward
inertia–gravity (WIG) modes identified with filtered geostationary infrared measurements, and the thermodynamic parameters and MSE budget variables computed from reanalysis data are composited with respect to the WIG peaks. The composite horizontal and vertical MSE structures are overall as theoretically
expected from WIG wave dynamics. A prominent horizontal MSE advection is found to exist, although thewave dynamics is mainly regulated by vertical advection. The vertical advection decreases MSE around thetimes of the convective peak, plausibly resulting from the first baroclinic mode associated with deep convection. Normalized gross moist stability (NGMS) is used to examine the thermodynamic processes involving
the large-scale dynamics and convective heating. NGMS gradually decreases to zero before deep convection
and reaches a maximum after the convection peak, where low (high) NGMS leads (lags) deep convection.
The decrease in NGMS toward zero before the occurrence of active convection suggests an increasingly
efficient conversion from convective heating to large-scale dynamics as the wave comes in, while the increase
afterward signifies that this linkage swiftly dies out after the peak.

1. Introduction
Quasi-2-day waves are tropical atmospheric disturbances with a period of about 2 days. Their dynamics are
explained by a convectively coupled n 5 1 westward
inertia–gravity wave (WIG1) (Matsuno 1966; Takayabu
1994a). A quasi-2-day wave typically has a phase speedof about 10–30 m s21, a zonal wavelength of about
2000–4000 km, and an equivalent depth of about
12–50 m (Takayabu 1994b; Takayabu et al. 1996; Chenet al. 1996; Haertel and Johnson 1998; Wheeler andKiladis 1999; Wheeler et al. 2000; Kiladis et al. 2009;
Haertel and Kiladis 2004; Haertel et al. 2008). A 2-day
wave has distinct developing stages associated with
progress in the representative cloud types from shallow
cumuli to deep convective and stratiform clouds

Corresponding author address: Yukari Sumi, Graduate School ofEnvironmental Studies, Nagoya University, Furo-cho Chikusa-ku,
Nagoya 464-8601, Japan.
E-mail: sumi@satellite.hyarc.nagoya-u.ac.jp
DOI: 10.1175/JAS-D-15-0098.1
Ó 2016 American Meteorological Society(Takayabu et al. 1996; Chen and Houze 1997; Mapes
et al. 2006). These stages are common to convectively
coupled equatorial waves (CCEWs), Madden–Julian
oscillation (MJO), and mesoscale convective systems(Houze 2004; Kikuchi and Takayabu 2004; Mapes
et al. 2006).
In past 15 yr, theoretical models of CCEWs havebeen developed with a focus on multiple vertical modes
having different heating profiles. Mapes (2000)
proposed a ‘‘stratiform instability’’ mechanism where alower-tropospheric cooling by the second baroclinic
mode could give rise to an unstable growth working
together with upper-tropospheric heating and lowertropospheric cooling in stratiform precipitation. Haertel
and Kiladis (2004) and Haertel et al. (2008) examinedthe variations of environmental fields in 2-day waves
and MJO by using the two-mode model whose basic
states are given by Tropical Ocean and Global Atmosphere Coupled Ocean–Atmosphere Response Experiment (TOGA COARE) intensive operative period(IOP) data. Kuang (2008) discussed the ‘‘moisture

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stratiform instability’’ mechanism that relies on a moisture increase in the middle troposphere preceding deep
convection as a driver of CCEWs.
In fact, observational investigations have indicatedthat a moisture increase in the middle troposphereplays a key role in the development of deep convection
accompanied with CCEWs and MJO (Kiladis et al. 2009;
Benedict and Randall 2007). Tropical deep convection
and tropospheric moisture are known to be closely
linked to each other from a variety of observational
studies (Sherwood 1999; Bretherton et al. 2004; Petersand Neelin 2006; Holloway and Neelin 2009). In linewith these observational implications, the relationshipbetween tropospheric moisture and convective development has been built in some recent theoretical models(Neelin and Yu 1994; Raymond 2000; Sobel et al. 2001;
Fuchs and Raymond 2007; Kuang 2008). Equatorial
wave modes theorized along this line are often called a‘‘moisture mode.’’ A series of recent studies discussed
the MJO dynamics in terms of moisture modes (Maloney
2009; Kiranmayi and Maloney 2011; Andersen and Kuang
2012; Benedict et al. 2014; Kim et al. 2014; Sobel et al.
2014; Maloney et al. 2010). Meanwhile, quasi-2-day waves
have drawn far less attention than MJO with afew exceptions (Haertel and Kiladis 2004; Haertel et al.
2008) and have yet to be studied particularly in light of thepotential importance of tropospheric moisture. In thisstudy, the thermodynamic processes in quasi-2-day waves
are analyzed for clarifying their relevance to the wave
propagation mechanism. To this end, a moist static energy(MSE) budget analysis is applied to quasi-2-day waves.
This paper consists of the following sections. In
section 2, the data and composite methods are described. In section 3, features of spatiotemporal variationsof convection and environmental fields are shown in detail. In section 4, the thermodynamic in quasi-2-day waves
are examined from MSE budget analysis. In section 5,
discussion and summary are presented.

2. Data and methods
a. DataGlobal-merged IR brightness temperature (TBB)
data (mergedIR) is used for identifying quasi-2-day
waves. The mergedIR is gridded global data in which
Geostationary Operational Environmental Satellite(GOES)-9/-10, Meteorological Satellite (Meteosat)-7/-5,
and Geostationary Meteorological Satellite (GMS) are
combined (Janowiak et al. 2001). The temporal resolution is 30 min and the horizontal resolution is 0.03648.
The surface precipitation data is obtained from
Tropical Rainfall Measuring Mission (TRMM) data

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product. TRMM satellite is a sun-asynchronous satellite orbiting with an inclination angle of about 358 andcarrying the Precipitation Rader (PR). The horizontal
and vertical resolutions of PR are about 4 km and
250 m, respectively. The near-surface precipitation
data in PR2A25 level 2, version 6, is used (Iguchi et al.
2000). Convective and stratiform precipitation types
are separated.
ERA-Interim (ERAI) data is one of European
Centre for Medium-Range Weather Forecasts (ECMWF)’sreanalysis datasets (Simmons et al. 2007). The horizontal resolution is 1.58 and the total number of pressurelevels is 38. The temporal resolution is 6 and 12 h forthree- and two-dimensional data, respectively. To calculate MSE budget terms, various ERAI data areutilized in this study (see section 4): 1) the threedimensional variables: geopotential height f, temperature T, specific humidity q, horizontal wind velocity
vector V ¼ (u, y), and pressure velocity v and 2) the
two-dimensional variables: surface sensible heat flux
(SH), surface latent heat flux (LH), surface short-wave
flux SWs , short-wave flux at the top of the atmosphere(TOA) SWt , surface long-wave flux LWs , long-wave flux
at TOA LWt , and sea–land flag. Note that conclusions inthis study hardly depend on the choice of reanalysis
datasets (not shown). National Centers for Environmental Prediction (NCEP) Reanalysis-1 (Kalnay et al.
1996) and Japanese 25-year Reanalysis (JRA25) data
(Onogi et al. 2007) have been tested and found to show
no qualitative difference from ERAI data.
In all datasets, the analysis period is the 10 yr between
2000 and 2009. The analysis area is the whole tropicalregion between 108N and 108S over oceans as selectedwith the sea–land flag. Also, all satellite data are projected onto a 0.258 grid.

b. Spectrum
Space–time spectrum analysis is performed on thehigh-resolution TBB data (mergedIR) produced fromgeostationary Earth-orbiting (GEO) satellites. TBB
spectrum is calculated in a manner similar to that usedby Wheeler and Kiladis (1999). First, raw TBB data areaveraged over the latitudinal band between 108N and
108S. A longitude–time cross section of the areaaveraged TBB is constructed for every 2 yr between
2000 and 2009. Missing data are filled in by linear interpolation over time. Second, anomalous values are
obtained by removing the mean and the linear components at each longitude. Third, the time series of theanomaly are divided into 80-day segments that consecutively overlap each other by 40 days. Then, a split cosine bell tapering function is applied to each segment
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FIG. 1. The raw power spectrum normalized by the background
spectrum (shaded) in wavenumber–frequency (cpd) domain. A
negative (positive) wavenumber means westward (eastward)
mode. Shading begins at a value of 1.3, which is significant at the
95% level with 91 DOF. The colored lines represent the dispersion
relationships of various equatorial waves for different equivalent
depths of 5 (blue) and 100 m (red): inertia–gravity wave (n 5 1),
Rossby wave (n 5 1), mixed Rossby wave (n 5 0), and Kelvin wave
(n 5 21). The large black frame represents the WIG filter used inthis study, including k 5 1–24 and he 5 5–100 m.

(FFT) is performed to obtain the power spectrum of
anomalous TBB for each segment. The power spectrum,with a frequency resolution f of 0.0125 (1/80) cycles perday (cpd) are averaged over 10 yr from 2000 to 2009.
Figure 1 shows the raw power spectrum normalized by
the background spectrum in wavenumber–frequency
domain. The background spectrum is computed bysmoothing many times the raw spectrum with a 1–2–1 filter
in frequency only. Here, the degree of freedom (DOF) isestimated about 91 (’2 3 365 3 10/80). Shading begins
at a value of 1.3, which is significant at the 95% level, basedon a chi-squared test with 91 DOF. The normalized spectrum in Fig. 1 corresponds well to the equatorial wave
signals in the previous studies (Wheeler and Kiladis 1999;
Roundy and Frank 2004; Masunaga et al. 2006; Masunaga
2007; Kiladis et al. 2009; Tulich and Kiladis 2012; Yasunaga
and Mapes 2012a,b). The westward-propagation domain
includes the signals of WIG wave, equatorial Rossby wave,
and tropical depression (TD)-type disturbances. Also, diurnal and semidiurnal disturbances are prominent. The
eastward-propagation domain includes the signals of
MJO and Kelvin wave.

c. Filtering
To isolate individual CCEW modes, Wheeler andKiladis (1999) presented a wavenumber–frequency

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filtering technique. They calculated power spectra fromoutgoing longwave radiation (OLR) data and defined thefilters based on the dispersion curves of equatorial waves
derived by Matsuno (1966). The similar technique has
been widely used in previous studies based on OLR data,
infrared brightness temperature data, precipitation data,and precipitable water data (Wheeler et al. 2000; Roundy
and Frank 2004; Masunaga et al. 2006; Masunaga 2007;
Kiladis et al. 2009; Yasunaga and Mapes 2012a,b).
This approach is applied to the present study withsome minor modifications as described below. It is assumed that convectively coupled WIG wave is a practicalindicator of a quasi-2-day wave. WIG signals are identified from space–time filtering analysis of TBB data. To
isolate WIG waves, the n 5 1 WIG filter is defined between k 5 1 and 24 and two dispersion relationships withthe equivalent depth he of 5 and 100 m (large black framein Fig. 1). This filter is somewhat broader than WIG filters
used in previous studies, although not including highest
frequencies as examined by some recent works (Tulich
and Kiladis 2012; Kikuchi 2014). The filter includes thestrong WIG signals (k ; 10, f ; 0.5–1) and stays outside
the TD signals (e.g., Kiladis et al. 2006) that appear below
the lower end of the filter near f 5 0.2–0.4 cpd. Therefore,
TD-type disturbances are unlikely to severely affect inthis analysis. To verify the justification for the WIG filter,
we tested the same filter as defined in Wheeler andKiladis (1999), including k 5 1–14 and he 5 17–50 m.
The difference of the filter design does not change ourresults seriously (not shown).
Figure 2 is the longitude–time cross section during theperiod between December 2004 and January 2005 for an
example to show how the filtering works. Unfiltered
TBB (Fig. 2a) shows that a synoptic-scale envelope ofactive convection is prominent between the Indian
Ocean (IO) and western Pacific Ocean (WP), while
2-day disturbances are only barely discernible. Westwardpropagating 2-day waves are clearly captured in the WIGfiltered TBB (Fig. 2b), particularly striking in the western
Pacific between 1408E and 1808.

d. Definition of the peak of WIG wave
A convective center of WIG wave is detected byutilizing a longitude–time cross section of filtered TBB
data. First, zonal local minima of filtered TBB are
searched at each time. Only a minimum locating at thecenter of a search area of 6108 is picked up, ignoring
noisy fluctuations. All local minima are searched bymoving the search area in zonal direction. Next, theminima colder than minus two standard deviations s aredefined as the longitudinal location of convective centers of WIG waves (WIG peaks). This threshold is used
to prevent the composite WIG wave structures from

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FIG. 2. A longitude–time cross section of (a) raw TBB, (b) WIG-filtered TBB, and (c) WIG-filtered TBB and WIG peaks (black dot).
The area surrounded by a black box in (b) is displayed in a larger size in (c). The longitudinal band is between 608E and 1208W. The time
period is between 31 Dec 2004 and 31 Jan 2005.

being contaminated by irrelevant background convective
signals. The standard deviation is computed fromlongitude–time cross section with the sample number ofabout 50 457 600 (;2 3 365 3 48 3 360/0.25) for each 2 yr.
Finally, the latitude of the convective center is defined as a
meridional local minimum of raw TBB within the bandfrom 108N to 108S. The total number of WIG peaks is355 491 in 10 yr between 2000 and 2009. Figure 2c shows
the WIG convective centers distributing along WIG
ridges in Fig. 2b. It is seen that robust wave passages arecaptured while leaving weak, noisy signals are excluded.
The geographic distribution of WIG peaks is shownin Fig. 3. The WIG waves are most active in the WP and
Africa, as revealed by the previous analysis (Wheeler
and Kiladis 1999; Kiladis et al. 2009; Tulich and Kiladis2012). It is noted that only peaks in the ocean are usedin the composite analysis.

e. Composite analysis method
A quasi-2-day wave accompanies with an individual
convective event with a period from a few hours to 1 day.
Such a fast temporal variation is unable to be tracked
continuously by satellite and reanalysis data owing totheir coarse temporal sampling. For example, daily observation frequencies are four times (6 hourly) and lessthan two times in reanalysis and low-Earth-orbiting(LEO) satellite data, respectively. To circumvent thisproblem, Masunaga (2012) developed a new composite
technique to statistically reconstruct fast temporal variations in the atmosphere, where two LEO satellite datasets are combined. In his technique, TRMM data areused to detect composite base points and the other LEO
data are composited around them (his Fig. 2), where
the time axis is defined as the observational time difference between the two LEO satellites. A huge numberof the base points are required to obtain a statisticallycontinuous time series because individual composite
samples only provide discrete snapshots.
A similar approach is applied to this study withmajor modifications. First, peaks in the filtered GEO
satellite data (mergedIR) are used to detect the compositing base points as described in sections 2c and 2d.
Second, various other satellites and reanalysis data arecomposited around the base points to construct the statistical continuous time series. Noise that happened to fallin the WIG filter (Fig. 2b) is probably included when the
WIG filter is applied but would have been averaged out inthe composite time series once a composite analysis isperformed with a large number of the base points of thecomposite with a 2s threshold to filter out weak signals.
The composite time 0 h defines the moment when WIG
wave is most convectively active as implied by TBB minima. The time resolution of the composite time series is
0.5 h because the composite base points are time stamped
by 0.5-hourly mergedIR data. As such, the time difference
between a base point and 6-hourly ERAI is technicallysorted by 0.5-hourly bins, although the composited parameters may be heavily smoothed over time.

3. Features of the composite 2-day waves
In this section, the horizontal structure and propagation character of the composite WIG waves are described. Figure 4 shows the horizontal structure ofanomalous TBB, precipitation, geopotential height, andhorizontal wind during a sequence of WIG phases.
Anomalies represent the deviation from the temporalmean over 648 h in the composite time series, which arecomputed by subtracting the mean from the value ateach grid point in Fig. 4. The computing method of
anomaly is similar to that used in Takayabu et al. (1996)
and Benedict and Randall (2007). Shading and arrows in

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FIG. 3. Geographic distribution of WIG peaks detected by the method in section 2d. The total
peak number is 355 491.

Fig. 4 are plotted only where statistically significant at
the 95% level as tested by the Student’s t test.
The area with a negative TBB anomaly (convective
active area) has a zonal scale of about 1000 km (Fig. 4a).
Therefore, the zonal wavelength of the composite
WIG wave is estimated about 2000 km, corresponding
to that of COARE 2-day wave (Takayabu 1994b;
Takayabu et al. 1996; Haertel and Johnson 1998; Haertel
and Kiladis 2004). The convective envelope moves
westward with a period of about 2 days and has a minimum TBB around 0 h by definition. The TBB minimum
lags the precipitation maximum by a few hours, which isconsistent with the previous studies (Takayabu et al.
1996; Haertel and Johnson 1998; Haertel and Kiladis
2004). The somewhat noisy distribution of precipitation
anomaly (Fig. 4b) is likely to result from the small
sample size of PR2A25 data, because TRMM satellite
has less than twice-daily observations at the same location. The propagation speed is estimated to be about
17 m s21 given that the convective active area moves
over about 308 in 48 h. It has been reported that thephase speed of quasi-2-day waves spans over a wide
range from 10 to 30 m s21(Takayabu 1994b; Takayabu
et al. 1996; Haertel and Johnson 1998; Haertel andKiladis 2004). The synoptic-scale structure of geopotential height and wind at 200 hPa is reminiscent of
the theoretical n 5 1 WIG mode (Matsuno 1966) in thatdivergence and convergence are on the left and right,
respectively, of the low pressure field [see Fig. 3 inKiladis et al. (2009)]. The negative TBB anomaly is almost in quadrature with the geopotential height and is inphase with the wind divergence field (Wheeler et al.
2000; Haertel and Kiladis 2004; Kiladis et al. 2009).
The convective center of the composite WIG waves isshifted away from the equator to around 58N (Fig. 4a).
This could result from the effect of intertropical convergence zone, which is located near the equator (Indian
Ocean; IO) and off the equator around 78N (Pacific
Ocean). In fact, the WIG convective center appears
around 78N in WP and around 08 in IO (not shown). The
horizontal structure of geopotential height and wind isoffset to the north of the equator (Fig. 4c), while that oftheoretical n 5 1 WIG is equatorial symmetric, probably
owing to the same reason as mentioned above.

The horizontal and vertical structures are nearly
identical between the Indian and Pacific Oceans (notshown). A secondary difference is that a maximum
height of convection is lower in the eastern Pacific thanin the Indo–western Pacific, which could be explained by
the regional difference of the tropical large-scale circulation (e.g., Back and Bretherton 2006).

4. MSE budget analysis
a. Column-integrated MSE budget terms
MSE (m) is nearly conserved in moist adiabatic processes and defined ass 5 f 1 cp Tandm 5 s 1 Lq ,

(1)
(2)

where s is dry static energy, L is the latent heat of condensation (2:5 3 106 J kg21 ), and cp is dry air heat capacity at constant pressure (1004 J K21 kg21). The budgetof column-integrated MSE is expected as follows:h›t mi 5 2hV  =mi 2 hv›p mi 1 hQR i 1 LH 1 SH .

(3)

Angle brackets represents mass-weighted vertical integration from theÐ top of the atmosphere pt to the surp
face ps ; hAi 5 g21 pst dp A, where g is the gravitational
acceleration, ps is 1000 hPa, and pt is 100 hPa. The lefthand side (lhs) is the MSE tendency. The first andsecond terms on the right-hand side (rhs) represent
horizontal MSE advection (HADV) and vertical MSE
advection (VADV), respectively. The other terms on
rhs are MSE source terms: the radiative heating QR ,surface latent heat flux (LH), and surface sensible heat
flux (SH). The term hQR i is calculated as the differencein net radiative flux between the surface and top of theatmosphere:
hQR i 5 SWt Y2 SWt [2 SWs Y 1 SWs [2 LWt [2 LWs Y 1 LWs [ .

(4)Upward and downward arrows represent upward anddownward flux, respectively. The budget terms arecomputed in a square grid box with the size of 1:58. The

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FIG. 4. 12-hourly snapshots of the horizontal structure of the composite WIG waves: (a) anomalous TBB, (b) anomalous precipitation,and (c) anomalous geopotential height (shaded) and horizontal wind (vectors) at 200 hPa. Negative anomaly of TBB is represented bycontours in (a) and (b). Anomalies are computed by subtracting the time-mean value during 648 h in the composite time series from thevalues at each grid point. Shades and arrows indicate regions significant at the 95% level. The horizontal axis is a relative longitude
between 2208 and 1208. The vertical axis is latitude between 108N and 108S. Time increases downward from 224 to 124 h in incrementsof 12 h.

box is centered on the longitude of composite center(WIG peak). Our results are insensitive to the differencein the grid size being varied between 1:58 and 7:58.
Figure 5 shows the composite time series of the budget
terms. As 6-hourly ERAI data is composited around thebase points, the composite time series has wiggles every

6 h as the result of sampling inhomogeneity. The tendency term has a maximum of 100 W m22 before theWIG peak (about 215 h) and a minimum of 2100 W m22
after the peak. This suggests that MSE increases before
deep convection (MSE recharge) and decreases afterconvective development (MSE discharge). Total MSE

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FIG. 5. Composite time series of integrated MSE budget terms:
the MSE tendency (black), total MSE advection (red), surface flux
(green), radiative heating (blue), and the budget residual (purple).
The residual is computed by subtracting the right-hand side fromthe left-hand side of the budget equation.

advection term, or sum of the horizontal and verticaladvections, fluctuates between 2160 and 0 W m22 inphase with the tendency term. The advection increases
MSE import before deep convection and approaches
its maximum 0 around 215 h. Then, it drops when deep
convection develops and reaches its minimumof 2160 W m22 after the WIG peak. Although the MSE
advection shows a striking variability, the MSE source
terms scarcely change with time even during active
convection. Radiative and surface fluxes stay nearly
constant at 260 W m 22 (LW 5 2180 W m 22 andSW 5 1120 W m22) and 1130 W m22 (SH 5 20 W m22
and LH 5 1110 W m22), respectively. The budget residual is calculated by subtracting the rhs from the lhs in
Eq. (3), which fluctuates almost in phase with the tendency term. It might result from uncertainties of reanalysis data, including 1) the convection and cloudparameterizations used in the data assimilation and 2) thecoarse spatiotemporal resolution. Although the budget
is not closed in this analysis, the MSE recharge–discharge
process is mainly regulated by the MSE advection.
Haertel et al. (2008) performed a budget analysis ofintegrated MSE in 2-day waves based on a two-mode
model, where TOGA COARE sounding data are used
as a basic state of the model. MSE source term is in
opposite phase to the tendency and the amplitude islarger than our result (see their Fig. 10). Inoue and Back(2015) also showed that the surface heat flux and radiative
heating are in opposite phase to the tendency in COARE
2-day wave (see their Fig. 2). Although the radiative
heating in Fig. 5 is likely to be out of phase with thetendency (a maximum of the radiative heating lags theconvective peak), the surface flux nearly stays constant. It
is considered that MSE source terms might be underestimated in reanalysis data than in situ and satellite data,
because the flux data in ERAI are provided as a time
cumulative value of 6 or 12 h. In fact, column-integrated
radiative heating, computed from CloudSat radiative flux

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FIG. 6. As in Fig. 5, but for zonal (red), meridional (green), andvertical (blue) MSE advection.

data 2B-FLXHR-lidar (L’Ecuyer et al. 2008), has a larger
amplitude of about 620 W m22 than ERAI data (notshown). However, such difference would not have serious
effects on the present study because the amplitudes of
MSE source terms are negligibly smaller than that of
MSE advection during the active phase of WIG waves.
Total MSE advection is separated into the horizontal
and vertical components. Figure 6 shows the composite
time series of the advection terms, smoothed by moving
average with temporal window of 66 h. The result isinsensitive to changing a width of the window.
The notable change of VADV occurs mostly within
612 h about the time of deep convection. An increaseof negative VADV around the WIG peak corresponds
to an enhancement of MSE export due to the first baroclinic mode (Peters and Bretherton 2006; Haertel et al.
2008). The previous studies also suggested that the second baroclinic mode increases positive VADV (MSE
import) leading deep convection, while it is not apparent inthis analysis. HADV is further separated into the zonal andmeridional components: 2hV  =mi 5 2hu›x mi 2 hy›y mi.
Zonal advection shows a striking variability, leading to
an import of MSE before the convective peak, followed
by a remarkable MSE export shortly after the peak.
Meridional advection with a small amplitude stays
negative throughout the evolution.
Previous studies have reported that MSE recharge–
discharge processes are mainly regulated by VADV in
convectively coupled gravity waves. Inoue and Back(2015) showed that the MSE tendency is dominated by
VADV in COARE 2-day waves and that HADV andMSE source terms are canceled out by each other. Haertel
et al. (2008), based on two-mode model, showed that therecharge–discharge process in 2-day wave is mainly controlled by VADV, although they did not compute HADV
directly. Using a diabatically accelerated cloud-resolving
model, Peters and Bretherton (2006) showed that VADV
dominate the MSE tendency both in a mesoscale convective event and a convectively coupled Kelvin wave.
They also indicated that VADV is more dominant over
HADV in the Kelvin wave. In this study, however, zonal

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FIG. 7. Time–pressure cross sections of anomalies of the MSE budget terms (m2 s23 ): (a) MSE tendency,(b) horizontal MSE advection, (c) vertical MSE advection, and (d) Q1 minus Q2. Anomalies are computed bysubtracting the time-mean value during 648 h in the composite time series from the values at each pressure level.
Shading represents regions significant at the 95% level.

advection is as large as VADV. This result is discussed
for validity in sections 4c and 4d.

b. Vertical structures of the budget terms
To analyze the vertical structure of the budget terms,
we use the budget equation of MSE:›t m 5 2V  =m 2 v›p m 1 Q1 2 Q2 ,(5)where Q1 is the apparent heating source, Q2 is moisturesink, andQ1 2 Q2 5 QR 2 Fs,ed 2 LFq,ed ,(6)

where MSE eddy term (Q1 2 Q2 ) is the sum of radiative
heating QR and MSE eddy transport (2Fs,ed 2 LFq,ed ).
Eddy transport of sensible heat flux (2Fs,ed ) and latent
heat flux (2LFq,ed ) are not provided by reanalysis data.

Also, vertical information of QR is not provided in
ERAI data (see section 4a). Instead of Eq. (6), Q1 2 Q2
is calculated from Eq. (5) by subtracting the advection
terms from the tendency term (Yanai et al. 1973). Notethat the MSE eddy transport usually exceeds radiative
heating in the region where deep vertical motion exists(Yanai et al. 1973), although their vertical profiles are
missing in this study. In fact, Masunaga and L’Ecuyer(2014) estimated the MSE eddy transport and radiative
heating from satellite data, showing that the former ismuch larger than the later during tropical convection
with the time scale of about 1–2 days.
Figure 7 shows the time–pressure cross section of thebudget terms in Eq. (5). A darker red (blue) color
means a stronger MSE import (export). The tendency
term vertically tilts with time and has a maximum
around 750 hPa before deep convection. HADV (mainlywith zonal advection) is almost in phase with the tendency

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at each pressure level. Meridional advection is negligibly small as one might be expect from Fig. 6 (notshown). The amplitude of VADV is comparable to thatof HADV, although the variability is more confined intime to the period of intense convection between 212
and 112 h. Before deep convection, a positive anomaly
emerges in the lower troposphere accompanied by anegative anomaly from the middle to upper troposphere (Fig. 7c). A strong updraft in the whole troposphere causes MSE increase (decrease) in the lower(upper) half of the troposphere. After WIG peak, a decrease of upper-tropospheric updraft weakens the negative VADV (positive anomaly), while an increase oflower-tropospheric downdraft enhances a negative VADV
(negative anomaly). The MSE eddy term shows a prominent contribution to the budget, which is absent by definition in the column integrated terms in Fig. 5.
The vertical dipole structure of VADV could suggestthat the thermodynamic process might be explained by a
two-mode theory (Haertel and Kiladis 2004; Peters andBretherton 2006; Haertel et al. 2008). During an enhancement of deep convection, the first mode causes
MSE export and import in the upper and lower troposphere, respectively, leading to an increase of negative
VADV (MSE discharge process). The second modecauses MSE import in the lower troposphere before
deep convection, leading to an increase of positive
VADV (MSE recharge process). The superposition ofthese modes results to a vertical tilting structure of
VADV. In this study, an increase of positive VADV(Fig. 6) is much smaller than previous studies (Haertel
et al. 2008; Inoue and Back 2015). A slight vertical tilt in
VADV (Fig. 7) may imply the presence of a shallow
updraft likely associated with the second mode leading the development of the first mode, although
overwhelmed in amplitude by the first mode. This indicates that the second mode has a more minor contribution to the thermodynamic process than the firstmode. The above discussion will be dealt with again in
section 5c.

c. Decomposition of advection terms
Variables within advection terms are separated into
the background and perturbation fields to further investigate MSE recharge–discharge process. The background is defined as the 10-yr-averaged 6-hourly
climatology of ERAI data that consists of a 365-daylong time series of global maps. The perturbation represents the local deviation from the climatology. Here,zonal velocity u can be written as u 5 u 1 u0 , where thebar and the prime represent the background and perturbation, respectively. The zonal advection term
(2u›x m) is separated into the four components:

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FIG. 8. As in Fig. 5, but for the MSE advection separated by the
background and perturbation fields: (a) zonal, (b) meridional, and(c) vertical MSE advection. The colored lines correspond to thelabeled components in the keys.

2u›x m 5 2u0 ›x m 2 u0 ›x m0 2 u›x m0 2 u›x m .

(7)

The meridional and vertical advection terms aredecomposed in the same way. The results are insensitive
to the different definition of the background.
Figure 8 shows the composite time series of the decomposed advection terms. The background components (2u›x m 2 y›y m and 2v›p m) stay small andnegative (steady MSE exports). The zonal advection islargely modulated by 2u›x m0 , while 2y›y m0 and
2v›p m0 are nearly 0. Meridional advection depends on
2y 0 ›y m0 . Small amplitudes of 2u0 ›x m0 and 2y 0 ›y m0 indicate that perturbation u0 and y 0 weakly acts against m0 .
Vertical advection is mainly dominated by 2v0 ›p m involving perturbation v0 . This agrees with the previous
studies indicating that VADV is associated with thelarge-scale vertical motion (Peters and Bretherton 2006;
Haertel et al. 2008; Inoue and Back 2015).
We examine the vertical structure of the background andperturbation components included in zonal MSE advection. Figure 9 shows snapshots of the longitude–pressure

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FIG. 9. Snapshots of the longitude–pressure cross sections of MSE zonal gradient (m s22) and zonal wind (m s21): ›x m, ›x m0 , u and u0 . The
horizontal and vertical axes represent relative longitude and pressure, respectively. Time increases to the right (224, 215, 25, 15, 110 h).

cross sections of ›x m, ›x m0 , u, and u0 . Each snapshot showsdifferent timings from the WIG developing stages: an earlystage of WIG convection (224 h), maximum of the MSE
tendency (215 h), near-precipitation maximum (25 h),minimum of the MSE tendency (15 h), and a decaying
stage of WIG convection (110 h). The zonal gradientof the background MSE ›x m has a small negative and alarge positive value to the east and the west of the origin,respectively. The perturbation MSE gradient ›x m0 islarger than ›x m and is positive and negative on the west
and the east of m0 , respectively. An eastward tilting of›x m0 represents that the perturbation MSE m0 tilts eastward (not shown). The background zonal wind u iseasterly (;5 m s21) everywhere from the surface to thetropopause, which is slower than the propagation speedof the composite WIG wave (;17 m s21). This implies
that a synoptic-scale convective system accompaniedwith WIG waves could not be driven westward by the
background easterly wind. The perturbation wind u0 islarge in amplitudes near the tropopause, which is a well-known character of 2-day waves (Takayabu et al. 1996;
Haertel and Johnson 1998; Haertel and Kiladis 2004).
Overall, u constantly overwhelms u0 in magnitude and›x m0 shows a more distinct variability than ›x m, whichleads to a striking value of 2u›x m0 as will be seen below.
Figure 10 shows the longitude–pressure cross sectionsof decomposed zonal MSE advection. The zonal advection 2u›x m is dominated by 2u›x m0 in the free troposphere, while the other components are less prominent as
expected from Fig. 8. The term 2u›x m0 has the same
dipole structure as ›x m0 . Because u is predominantly
easterly in the free troposphere (Fig. 9), 2u›x m0 generates MSE increase to the west of a positive m0 and MSE
decrease to the east of it. In the next section, we compare
MSE recharge–discharge processes in different background zonal wind to verify the above scenario.

d. Zonal wind effect
To examine the possible roles of the background zonal
wind on the MSE budget, we define a barotropic flow

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FIG. 10. As in Fig. 9, but for the four components of zonal MSE advection decomposed by the background and perturbation fields: 2u0 ›x m,
2u›x m, 2u0 ›x m0 , 2u›x m0 , and 2u›x m (m2 s23 ).

as a vertically averaged u between 700 and 800 hPa,
where the tendency and MSE advection have striking
variabilities (see Fig. 7). The base points of composite(WIG peaks) are categorized into five groups by thebarotropic flow at location of the points as shown in
Table 1. Categories C1 and C5 correspond to the strongest easterly and westerly flow, respectively. Each of C1–
C4 includes about 20%–25% and C5 includes about 8%.
Figure 11 shows the composite time series of anomalous MSE tendency, HADV, and VADV for different
barotropic flow. The anomaly is the deviation from thetemporal mean over 648 h during the composite timeseries. Whenever the background barotropic flow iseasterly (C1–C3) or westerly (C4–C5), the tendency and
HADV are in phase with each other. As the barotropic
easterly flow becomes stronger, their amplitudes becomelarger owing to 2u›x m0 . Meanwhile, VADV is insensitive
to the direction and magnitude of the background windspeed. In the easterly flow, 2u›x m0 and 2u0 ›x m0 are inphase with each other (see Fig. 8), leading to a large
amplitude of HADV (e.g., C1 in Fig. 11). Although
2u0 ›x m0 is identical to that in the easterly flow, it is out ofphase with 2u›x m0 in the westerly flow (not shown),
canceling each other out in HADV (e.g., C5 in Fig. 11).
The coherence of 2u›x m0 with the MSE tendency
implies that this term is largely left unconsumed during

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TABLE 1. Category of barotropic flow between 700 and 800 hPa.
Category

Barotropic flow (m s–1)

Percentile

C1
C2
C3
C4
C5

#25.7
25.4 to 23.0
22.7 to 0.0
0.3 to 3.9$4.2

0 to 26
26 to 52
52 to 72
72 to 92
92 to 100a passage of the WIG wave and thus is not critical of
2-day wave dynamics. As u is slower than the wave
propagation speed of about 17 m s21, the term 2u›x m0
brings an MSE anomaly from the near side of the WIG
peak itself. Under the easterly background, this results
in a positive MSE advection because the near side lies
ahead of the convective peak where the lower troposphere is moist. When the background is westerly, incontrast, the horizontal MSE advection feels a dry
anomaly behind the wave peak and counteracts the
moistening due to the wave itself, which is primarily
regulated by the vertical advection.

e. Gross moist stability
The MSE recharge–discharge process in WIG waves
is dominated by the vertical advection associated with alarge-scale circulation. Here, GMS is adopted to investigate the convective–wave coupling mechanism.
Neelin and Held (1987) first introduced GMS as a tool tosimplify the relationship between moist convection andlarge atmospheric circulations. In their two-dimensionalmodel, GMS was a time invariant parameter and wasmeaningful only when GMS is positive because otherwise the system would be unstable and time dependent.
Recently, GMS has been tested in much broader contexts including the so-called normalized GMS (NGMS)(Raymond and Fuchs 2007; Raymond et al. 2009; Sessions
et al. 2010; Kuang 2011; Hannah and Maloney 2011;
Andersen and Kuang 2012; Benedict et al. 2014; Sobel
et al. 2014). For example, Raymond et al. (2009) defined
NGMS as ‘‘the ratio of net lateral outflow of MSE from an
atmospheric column to some measure of convective intensity within the column.’’ The sign and magnitude of
NGMS are a measure to characterize how convective
heating and large-scale dynamics work together. Utilizing
the definition of NGMS in Raymond et al. (2009), we
examine the roles of convection in the large-scale dynamics associated with WIG waves. NGMS is defined as
2hv›p mi,NGMS 5 2
2Lhv›p qi(8)where the denominator is moisture vertical advection andnumerator is MSE vertical advection. Unlike Raymond
et al. (2009), only VADV is included in the numerator,

FIG. 11. As in Fig. 5, but for (a) MSE tendency, (b) horizontal
MSE advection, and (c) vertical MSE advection. The WIG peaks
are categorized by the background barotropic flow between 700
and 800 hPa (see Table 1). Categories C1 (blue), C2 (purple), andC3 (green) represent easterly flow, and C4 (red) and C5 (yellow)
represent westerly flow.

because it is an inherent driver of the recharge–discharge
process in WIG waves as presented in sections 4c and 4d.
Figure 12a shows the composite time series of the denominator and numerator in Eq. (8). The noise issmoothed out by applying a 66-h moving average.
The result is insensitive to the smoothing. The sign ofNGMS reflects that of VADV because the vertical moisture advection is always positive (moisture import).
Figure 12b shows that NGMS decreases gradually before
deep convection and approaches 0 around 220 h when
VADV is nearly 0. NGMS increases as deep convection
passes its peak and arrives at its maximum 0.2 around 16 h.
Masunaga and L’Ecuyer (2014) calculated the similar
composite time series of NGMS from satellite observations of tropical convection, varying with time on anhourly to daily time scale for the horizontal scale ofabout 100 km. Although their analysis was not targetedon any particular wave mode, NGMS decreases toward
0 before deep convection and increases following

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755

FIG. 12. As in Fig. 5, but for (a) vertical MSE advection (blue) andvertical moisture advection (black) and (b) normalized GMS.

convection development, closely resembling Fig. 12b.
Moisture and MSE convergence are largely controlled byvertical advection in Masunaga and L’Ecuyer (2014). The
large contribution of VADV might be because the horizontal scale of the convection in their analysis is muchsmaller than that in the composite WIG waves (Fig. 4).
Also, it is possible that the contribution of the second
baroclinic mode is underestimated in ERAI data. In
section 5b, we will discuss again why the contribution of
the second mode is smaller in this study than in previous
studies (Haertel et al. 2008; Inoue and Back 2015).
Figure 13 shows the zonal distribution of precipitation
and NGMS. The area of active precipitation with thezonal scale of about 1000 km moves westward as expected from Fig. 4. Convective rainfall peaks a few hours
earlier than stratiform rainfall in 6 5 h, which is a wellknown property of tropospheric mesoscale convective
systems (Houze 2004). NGMS has a significantly highvalue to the east of the precipitation peak. NGMS has alower value to the west of the peak, in which VADV
closes to 0. This might suggest that an area favorable(unfavorable) for the development of convection
emerges to the west (east) of the convective envelope.

5. Discussion
a. Comparison with the other CCEWs and MJO
In convectively coupled gravity waves, HADV is
known to be less important than VADV to controlling

FIG. 13. Snapshots of zonal distribution of precipitation and
GMS. Solid lines represent total (black), convective (red), andstratiform (blue) rainfall rate. Dotted line represents normalized
GMS. The horizontal axis is the relative longitude between 2208
and 1208. (top)–(bottom) The time increases from 224, 215, 25,
15, 110 h.

a MSE recharge–discharge process. Peters andBretherton (2006), based on a cloud-resolving model,
showed that VADV is dominant over HADV in the
Kelvin wave. Haertel et al. (2008) and Inoue and Back(2015) showed that a column MSE is mainly controlled
by VADV in COARE 2-day waves (WIG waves). The
results in this study also support the previous studies,indicating that VADV plays a primary role in maintaining the thermodynamic process. However, HADVin Fig. 5 is larger than that in the COARE 2-day wave
(Inoue and Back 2015). This discrepancy could be756

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explained by the difference of the background zonal
wind. In the present study, 2u›x m0 and 2u0 ›x m0 are inphase because of background easterly wind (see Fig. 9),
leading to an emphasis of HADV. Meanwhile, thebackground zonal wind is strong westerly in the lower
troposphere during TOGA COARE IOP (Chen et al.
1996; Ciesielski et al. 1997; Yanai et al. 2000). HADV in
COARE 2-day wave could be weakened because
2u›x m0 and 2u0 ›x m0 are opposite in phase under the
background westerly (see also section 4d).
In MJO, both VADV and HADV are responsible for
the MSE recharge–discharge process. HADV could
play an important role in the eastward-propagation
mechanism of the MJO (Maloney 2009; Maloney et al.
2010; Kiranmayi and Maloney 2011; Benedict et al. 2014;
Kim et al. 2014; Sobel et al. 2014). As noted in the previous paragraph, an importance of HADV is small in the
WIG wave and Kelvin wave. This corresponds to theresult in Inoue and Back (2015) showing that a relative
contribution of HADV increases as the time scale of the
disturbance grows to intraseasonal scales.
There are some differences in the thermodynamic
process of the MJO compared to the WIG wave. First,
MSE source terms have the same amplitude as the MSE
advection and play an important role in regulating acolumn MSE. Also, radiative and surface fluxes fluctuate larger as the time scale of the disturbance becomes
longer (e.g., Inoue and Back 2015). Second, the meridional
advection has a striking variability during the MJO propagation. Third, the perturbation wind (u0 and y 0 ), associatedwith the wave kinetics, causes equal or larger MSE advection than the background wind (u and y). Meanwhile, eddy MSE advection (2u0 ›x m0 and 2y 0 ›y m0 ) stays
small in WIG waves. This difference may be because thevariation of environmental fields changes with the spatiotemporal scale of CCEWs (Kiladis et al. 2009). For
example, the amplitude of u0 in WIG is about 10% ofthat in MJO (Benedict and Randall 2007; Haertel et al.
2008; Kiladis et al. 2009).

b. Comparison with two-mode model
The two-mode model well captured MSE recharge–
discharge processes in the convectively coupled gravity
waves (Peters and Bretherton 2006; Haertel et al. 2008).
The shallow vertical motion with the second baroclinic
mode causes a positive VADV preceding deep convection (MSE recharge process). The strong vertical motion
associated with the first baroclinic mode causes a negative
VADV (MSE discharge process). After the convective
peak, the descending motion in the lower troposphereand the ascending motion in the upper tropospherewith the second mode cause a negative VADV (MSE discharge process). Also, previous studies have emphasized

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a prominent contribution of the second mode in the 2-day
wave dynamics. Haertel and Kiladis (2004), based on a
two-dimensional inviscid primitive equation linearized
about a basic state, showed that the wave dynamics andstructure are well described by a superposition of the firstand second modes. Yasunaga and Mapes (2012a) andYasunaga and Mapes (2014) showed a significant coherence between rain and midtroposphere divergence in
WIG waves, suggesting an importance of the second
mode to the wave dynamics.
The framework of the two-mode model could be applied to this study. In Fig. 6, a negative VADV around a
WIG peak might be explained by the first mode. A
positive VADV with the second mode is not visuallyevident, although a slight vertical tilt provides a hint of
the second mode (Fig. 7). The lack of a clear signature ofshallow moistening by the congestus mode (Peters andBretherton 2006; Haertel et al. 2008) might be a uniquefeature of WIG waves or possibly due to the technicallimitation in reproducing a shallow updraft in reanalysis
data. It could be difficult for reanalysis data to resolve a
rapid variation associated with shallow circulation owing to the coarse resolution. Despite an obscure contribution of the second mode, this study hardly denies a
primary importance of the second baroclinic mode in
the wave dynamics. It would be beneficial elsewhere to
test carefully to what extent a premoistening of thesecond mode is important in the WIG wave dynamics incomparison with the vertical mode decomposition
technique used in recent investigations (Handlos andBack 2014; Masunaga and L’Ecuyer 2014).

c. Westward-propagation mechanism of 2-day waves
Previous studies based on ground and satellite measurements showed that most of the convective systems
accompanied with 2-day waves move westward (Takayabu
1994b; Takayabu et al. 1996; Chen and Houze 1997;
Haertel and Johnson 1998; Haertel and Kiladis 2004).
It is outside the scope of this study to explore the origins
of westward propagation because the 2-day wave iscurrently extracted by a WIG filter and by design moves
westward. Nevertheless, a brief discussion on the selection of propagation direction in the general context
might be beneficial.
Here, the propagation dynamics is examined in termsof thermodynamic process. As the low (high)-NGMS
area exists to the west (east) of preexisting convection,
new convection prefers to arise to the west of the preexisting convection. Hence, a synoptic-scale convective
system accompanied with the WIG wave could move
westward. This dynamics mainly depend on the genericproperties of inertia–gravity waves, and the same dynamics with opposite signs would presumably help

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convection move eastward when coupled with theeastward inertia–gravity (EIG) waves.
The present analysis presents no clear evidence thatthe background state plays roles in breaking the east–west symmetry in the propagation direction. Recently,
Tulich and Kiladis (2012), using the Weather Researchand Forecast (WRF) Model, indicated that the zonal
wind shear in the lower troposphere is important towestward propagation of 2-day wave. Key parameters
responsible for the propagation characteristics of 2-day
waves will be addressed in more depth in the future.

d. Phase speed
As shown in section 4, the thermodynamic process ismainly regulated by a large-scale vertical circulation andmight not depend on the background zonal wind speed.
Meanwhile, a further analysis is needed to discuss therelationship between the wave propagation dynamics
and the background wind (e.g., Tulich and Kiladis 2012).
Previous studies have discussed in various detail the
WIG phase speed, which is slower than the theoretically
expected one. In the vertical mode perspective, Haertel
and Kiladis (2004) showed that the shallow equivalent
depth could result from a superposition of the firstand second baroclinic modes. The first baroclinic theory
explained the phase speed of CCEWs by a constant
GMS (Emanuel et al. 1994; Neelin and Yu 1994;
Raymond et al. 2009). Based on a cross-spectral analysis,
Yasunaga and Mapes (2012a) and Yasunaga and Mapes(2014) showed that the fast and slow WIG waves havedifferent moist processes. Takayabu (1994b) and Dias
and Kiladis (2014) reported Doppler shifting by the
background zonal wind in WIG waves. Furthermore,
Dias and Kiladis (2014) showed that the equivalent
depth is uniform between the different climatologicalregions if the Doppler shifting is considered. It is still
unclear, however, how the phase speed of the WIG
wave is modified by the background zonal wind and the
moist process. To approach the above problem, in a
future study we will calculate the first and second baroclinic mode adequately in the different background fieldand estimate their modifications to the phase speed.

6. Summary
The present study explores the driving mechanism of
quasi-2-day waves with a focus on the underlying thermodynamic processes. Satellite and reanalysis data areutilized for analyzing the MSE budget of convectively
coupled WIG waves, which is considered to account for
quasi-2-day waves to the extent that the synoptic-scale
structure is approximated by linear dry dynamics. WIG
waves are detected by a filtering analysis of brightness

757

temperature (mergedIR). The budget terms calculated
from the reanalysis data (ERAI) are composited around
the WIG peaks to create statistically continuous timeseries. The horizontal dynamic structure of composite
WIG waves is confirmed to be overall as expectedfrom previous studies.
MSE advection mainly regulates MSE variability.
Meanwhile, surface fluxes and radiative heating bringabout nearly constant MSE import and export, respectively, and hardly contribute the wave variability
throughout the evolution. An enhancement of negative
vertical advection decreases column MSE around
WIG peak, plausibly resulting from the first baroclinic
mode developed in association with deep convection.
Zonal MSE advection causes a significant MSE increase
and a decrease before and after deep convection, respectively, overwhelming vertical advection in magnitude. The zonal advection is mainly dominated by2u›x m0 , while the term should be considered separately
to the MSE recharge–discharge process. It is impliedthat vertical advection rather than horizontal advection
modulates the thermodynamic process in the WIG
waves. This result is consistent with the previous studies
explaining the recharge–discharge process by a largescale vertical circulation (Peters and Bretherton 2006;
Haertel et al. 2008; Inoue and Back 2015). A slightcontribution of the second baroclinic mode in thisstudy might result from the technical limitation in
reproducing a shallow updraft in reanalysis data.
NGMS is used to examine thermodynamic processes
involving the large-scale dynamics and convective
heating. Before deep convection, NGMS gradually decreases from a positive value to a significant small positive value near 0. According to the development of deep
convection, NGMS increases and reaches its maximum
after the WIG peak. The results suggest that the low(high) NGMS leads to (lags) deep convection. In theconvective envelope associated with WIG waves, newconvection is allowed to develop to the west of the preexisting convection where NGMS decreases toward zero.
The results obtained in the present study are nearlyconsistent with the previous investigations based on thetwo-mode theory, although the second baroclinic mode
is invisible in this analysis. Ongoing work is testing
carefully to what extent a premoistening of the second
mode works in the WIG wave dynamics.
Acknowledgments. The authors thank Dr. Dave
Raymond for the meaningful discussion and his insightful comments. The authors also thank Dr. George
Kiladis and one anonymous reviewer for their helpfulcomments and suggestions, which improved this manuscript. The global-merged IR brightness temperature

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data were provided by the National Centers for Environmental Prediction (NCEP) Climate Prediction Center(CPC). The TRMM PR (2A25) datasets were provided
by the Japan Aerospace Exploration Agency (JAXA).
ECMWF ERA-Interim data were provided by ECMWF.
This work is partly supported by the Japan Society for the
Promotion of Science (JSPS) Grant-in-Aid (KAKENHI)
for Scientific Research (B) (26287113).
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