1. Introduction
A quasi-2-day wave is a synoptic-scale disturbance in the 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 s−1 and have a zonal wavelength of about 1000–4000 km (e.g., Takayabu et al. 1996; Haertel and Johnson 1998).
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 phase of 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 = 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, are derived from the shallow-water equation linearized in equatorial beta plane (Matsuno 1966). Meanwhile, because of coupling with moist convection, dynamical characteristics of CCEWs are not fully explained by the linear theory.
The activity of CCEWs has been extensively investigated by computing a space–time spectrum of tropical convection including cloudiness, precipitation, and water vapor (Takayabu 1994a; Wheeler and Kiladis 1999; 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 peaks of CCEWs lie along the theoretical dispersion curves with an equivalent depth of about 12–50 m. This is consistent with the fact that CCEWs have a slower phase speed (or shallower equivalent depth) than their dry counterparts.
A variety of convection–wave coupling mechanisms have been proposed to explain the reduction in the phase 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 speed of CCEWs is explained by a slowdown of the dry first mode (~50 m s−1) due to a reduction of the effective static stability. The reduced static stability can be linked to the gross moist stability (GMS) that was originally introduced by Neelin and Held (1987), in which the square 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). Another school 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 s−1). 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 role of 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; Peters and Bretherton 2006; Haertel et al. 2008). Another possible explanation is that the second mode is only a response to the first mode (Frierson 2007; Raymond and Fuchs 2007).
The multimode structure of CCEWs first appeared in 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 superposition of a few vertical modes and have extensively explored the concept of the possible coordination between the first 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 not propagate over a long distance. This is the crux of the issue 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 phase speed constituting a coherent wave packet without being immediately dispersed. More observational analysis is needed to better understand the multimode structure of CCEWs and the slowdown of the vertical modes, and this 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 the slowness 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 and Pauluis 2011; Dias and Kiladis 2014; Yasunaga and Mapes 2014). In addition, the phase speed, structure, and coupling mechanism can vary during the wave life cycle. From this perspective, recent studies have focused on 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 multimode structure of the slow and fast phases of the 2-day wave by applying the vertical mode decomposition to their composite profiles separately. The slow and fast components (or phases) are detected by WIG filters with different 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 5 summarizes the present findings.
2. Data and methods
a. Data
This study analyzes 10 years (2000–09) of satellite and reanalysis data. The analysis area is the whole tropical region between 10°N and 10°S. All satellite data are projected onto a 0.25° grid.
Infrared (IR) brightness temperature (TBB) data are used to identify convective peaks of quasi-2-day waves (see section 2b). TBB data are obtained from the global-merged IR brightness temperature data (mergedIR), version 6, by Janowiak et al. (2001). MergedIR is a high-resolution 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 60°N and 60°S. The horizontal resolution is about 0.0364°, 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 (see section 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 with an inclination angle of about 35° and carrying a Precipitation Radar (PR), which estimates the vertical profile of precipitation. The horizontal and vertical resolutions 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), are used to analyze the variance in precipitation of 2-day waves.
To analyze the environmental fields associated with the 2-day waves, we use the European Centre for Medium-Range Weather Forecasts (ECMWF) interim reanalysis (ERA-Interim) data (Simmons et al. 2007). The horizontal resolution is 1.5°, and the temporal resolution is 6 h. The total number of pressure levels is 38, in which pressure intervals are not uniform. The ERA-Interim variables utilized in this work are the geopotential height ϕ, temperature T, specific humidity q, zonal wind velocity u, meridional wind velocity υ, and pressure velocity ω. The analysis area covers the whole oceanic tropical region between 10°N and 10°S as selected using the ERA-Interim sea–land flag.
b. Filtering analysis
A space–time filtering of TBB is performed to identify a convective peak of 2-day wave. Here we assume that a convectively coupled WIG wave is a practical indicator of the 2-day wave. The procedure used in the present study is similar to that of Sumi and Masunaga (2016) and is briefly summarized below.
The filtering is performed for every 2-yr period from 2000 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 over the meridional band between 10°N and 10°S, and then a longitude–time cross section of the averaged TBB is constructed. Missing data are filled in by linear interpolations over time. The mean and the linear components in time, identified by least squares fit at each longitude, are then removed (e.g., Masunaga et al. 2006). Second, the FFT is applied to the time series of anomalous TBB by using a split cosine bell tapering function to avoid spectral leakage. Third, a WIG filter (the filter design is described below) is applied to a wavenumber–frequency domain to isolate individual WIG waves. Fourth, the filtered TBB is transformed back to the longitude–time domain. Finally, the peaks of the WIG waves are defined as local minima of the filtered TBB lower than −2 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 smoothing of the raw spectrum with a 1–2–1 filter (Wheeler and Kiladis 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 Kelvin wave.
The raw power spectrum normalized by the background spectrum in wavenumber–frequency (cpd) domain. A negative and 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 chi-squared test. Dotted lines represent dispersion curves of n = 1 WIG wave with equivalent depth of 5, 15, 25, 35, 45, and 55 m. Five subfilters are indicated by black boxes.
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
The WIG filter is defined as a wavenumber–frequency domain including the zonal wavenumber k = 1–14 and the equivalent depth he = 12–50 m. The filter is rather larger than WIG filters with he = 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 of increasing 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 = 1–14 is applied equally to all the subfilters.
The spectral peak of WIG wave spreads across the subfilter 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 influence of 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 filtered noise. In fact, a nonpropagating convective signal is sometimes 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 the composite fields by using a large number of the composite base points with −2 SD threshold (see section 2c). We will show below that dynamical features of the composite WIG waves are qualitatively consistent with those in previous studies.
c. Composite analysis
The composite base points are defined as the convective peaks of the filtered WIG waves, which are samples in the whole equatorial ocean region between 10°N and 10°S (the total sample number is 331 748). Various satellite and reanalysis data are composited around the 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 two times in reanalysis data and low-Earth-orbiting (LEO) satellite data, respectively. However, a statistically continuous time series is constructed by compositing a huge number of reanalysis and satellite snapshots with respect to the observational time difference between mergedIR and the composited data. The time interval of the composite time series is set to 0.5 h, because the composite base points are detected from 0.5-hourly mergedIR data. For a more detailed description, refer to section 2e in Sumi and Masunaga (2016).
Most of the composite base points are found in the warm pool region (not shown), which is consistent with the fact that the convectively coupled WIG waves (or 2-day waves) are most prominently observed in the Indo-Pacific Ocean region (Wheeler and Kiladis 1999; Kiladis et al. 2009; and many others). Therefore, the composite results may be affected by background conditions of the warm 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 a regression 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 of tropical 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 phase speed of the WIG waves (section 3a) and that of the moist vertical modes (section 3c). It is noted that the estimated phase speed is insensitive to the difference between the evaluation methods (not shown).
The procedure is briefly summarized here [see the appendix in Yang et al. (2007b) for more details]. The anomalous field of longitude–time domain is projected along a line at angle θ with respect to the longitude axis. The energy of the projected data would have maximum amplitude when the projection axis is perpendicular to the ridges and troughs of the wave in the course of varying θ from 0° to 180°. With this angle denoted by
e. Vertical mode
Equation (1) is solved by assuming a rigid lid at
The phase speeds of vertical modes, obtained as eigenvalues of Eq. (1), are faster in the lower mode. The phase speeds (c1 = 52, c2 = 27, c3 = 20, c4 = 15 m s−1) and the equivalent depths (
3. Results
a. 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 area with a negative TBB anomaly propagates westward over a period of about two days. The phase speed of the 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 the black line represents the phase speed of 21 m s−1. 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.
Longitude–time diagram of anomalous TBB (K) for a WIG wave identified with the h25/15 filter. The slope of the black line represents a phase speed of about 21 m s−1.
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
Figure 3 shows the phase speed for slower and faster components 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 of the estimated phase speed (15–28 m s−1) is consistent with that in previous studies (10–30 m s−1). The phase speeds of TBB agree well with that of precipitation, indicating that the estimated phase speeds are insensitive to parameter choice (cloud or precipitation). These results support the robustness of the RT method for use in the present application.
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.
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
b. Vertical mode transform
To understand the gross features of WIG waves, the dynamical profiles are projected onto each vertical mode. 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 vertical modes (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 the present study, a similar vertical mode decomposition is applied to the composite WIG waves.
Figure 4 shows a time–pressure cross section of the mode transform component
Time–pressure cross section of pressure velocity (Pa s−1) of WIG wave identified with the h25/15 filter: (a)
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
A vertically tilted and top-heavy ω profile is captured by a superposition of the first four vertical modes (Fig. 4e), which is qualitatively consistent with the observed one (Fig. 4f). Also, other dynamical profiles of the wave (u, υ, D, ϕ, and Q) are sufficiently explained by the first four modes (not shown). Additional higher vertical modes (n > 4) are needed to reproduce a boomerang-like temperature structure, as reported in previous studies (Haertel and Kiladis 2004; Haertel et al. 2008; Kuang 2008a; Tulich and Kiladis 2012). Results for the other subfilters are similar (not shown).
The tilting ω profile (Fig. 4f) is common in different ocean regions but is modified by background conditions (not shown). For example, the maximum level of convection is slightly lower in the eastern Pacific than in 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 coefficient
Longitude–time diagram of mode transform coefficient of zonal wind
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
Figure 6 shows the phase speeds of the moist vertical mode estimated from
Phase speed of the moist vertical mode estimated by RT of
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
d. Definition of
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 linear primitive-equation model, showed that a degree of reduction of the equivalent depth of the first two vertical modes 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 vertical mode. Following these studies, we derive a diagnostic equation to examine the slowdown of the vertical modes.
Plot of
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
Equation (11) suggests that a slowdown of the vertical mode
e. Estimation of
Figure 8 shows a scatterplot of
Scatterplot of the mode transform coefficients
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
In Fig. 9, the estimated values of
Plot of
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
Figure 10 shows a time series of
Time series of the mode transform coefficients
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
In Fig. 10,
4. Discussion
A diagnostic equation for
a. Estimation of
The equivalent depth of CCEWs is estimated from the Doppler-shifted dispersion curve on a space–time diagram. Observational and theoretical studies show that 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 at 850 hPa. Dias and Kiladis (2014) shows that the equivalent depth of CCEWs is similar across seasons and 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 the present 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:
Vertical profile of mean zonal wind
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
Phase speed and mean zonal flow for each WIG wave identified with different WIG filters. Standard deviations of the mean zonal flows calculated from the sample number of 192 are shown in parentheses. Units are m s−1.
The values of the wavenumber k, the frequency f, and
Phase speed c (m s−1), zonal wavenumber k, frequency f (cpd), and equivalent depth He (m).
b.
Here, we test the application of the diagnostic equation for
The value of
would depend on the accuracy of the data used in the analysis. A slight decrease trend of (Figs. 9 and 12) and (Fig. 9) would partially result 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 of error bars of in the higher modes (Figs. 9 and 12). However, we verified that the ERA-Interim is more suitable for a study of WIG waves than other reanalysis data (see appendix B for more detail). In future work, our results will be compared with computed from satellite or in situ data. The estimation of
has its own uncertainties. We assume that is a time-independent parameter and compute as the least squares coefficient between and . As changes almost linearly with (Fig. 8), it is indicated that is less dependent on time as expected theoretically. On the other hand, for the second and third modes, does not vary linearly with , indicating that would be more time dependent. Therefore, the estimation error of can be large on the second and third modes. While a time dependency of is not considered in the present study, we will address this issue in future work. Difference of
between the slower and faster components (or the shallower and higher equivalent depths) may be smaller than expected from Eq. (11) because of cross contamination, which may lead to a slight decrease trend of . In the literature, effective static stability has been introduced to explain a large-scale motion associated with the first mode that is tied to deep convection (e.g., Neelin and Held 1987; Yano and Emanuel 1991; Neelin and Yu 1994; Emanuel et al. 1994): the atmosphere 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,
is roughly identical to . Then it is implied that the reduced stability mechanism may be more suitable for explaining the slowdown of the first mode. However, it is not intuitively obvious how the reduced stability mechanism explains a slowdown of the higher modes. In fact, the result of for the second and third modes implies that the slowdown of the higher modes cannot be fully explained by the reduced stability mechanism. A slight decrease trend of
can be interpreted in another way: 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 that can be explained by the reduced static stability is limited.
Plots of
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
In future work, similar analysis will be applied to other 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 vertical modes associated with quasi-2-day waves by using satellite and reanalysis data. The 2-day waves are identified by filtered TBB (mergedIR), where the faster and 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 are composited around the convective peaks of WIG waves to analyze the large-scale fields associated with the waves. The composite fields are further decomposed by the vertical modes calculated with the mean temperature profile (ERA-Interim) in the tropical region between 10°N and 10°S.
It is shown that the dynamical structure of WIG waves are explained largely by a superposition of the first four baroclinic modes. The resulting vertically tilted, top-heavy structure is consistent with the multimode structure, including the deep, congestus, and stratiform modes (e.g., Mapes 2000; Haertel et al. 2008). The phase speed of moist vertical modes is computed by Radon transform of the mode coefficient in the longitude–time domain. The moist vertical modes are slowed down compared to its dry counterpart, while different vertical modes have similar phase speeds, so the wave is only weakly dispersive.
In the effective stability mechanism, the slower phase speed of the first mode is attributed to the reduction of effective 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 the equivalent 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 the higher vertical modes. Further investigations of these outstanding 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 this manuscript. This work was supported by Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research B (19H01966). The global-merged IR 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), the cooperative research project carried out by the Japan Meteorological Agency (JMA) and the Central Research Institute of Electric Power Industry (CRIEPI).
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 each other 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 amplitude of the slower (faster) component is large around the peak of 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. Therefore, it is considered that the cross-contamination effect is not so serious in our results.
Longitude–time diagram of WIG waves identified with each 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 −2 standard deviations. Different colors denote different waves: h55/45 (black), h45/35 (blue), h35/25 (green), h25/15 (purple), and h15/5 (red).
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
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.
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
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 ERA-Interim, NCEP–NCAR (Kalnay et al. 1996), and JRA-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 ERA-Interim: 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.
The composite Q profile at t = 0 of the composite time series for WIG waves identified with the h25/15 filter. The values of Q is calculated from JRA-25 (blue), NCEP (green), and ERA-Interim (black) reanalysis and TRMM (red).
Citation: Journal of the Atmospheric Sciences 76, 7; 10.1175/JAS-D-19-0092.1
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