Equatorial Waves Including the Madden–Julian Oscillation in TRMM Rainfall and OLR Data

Hye-Kyung Cho Department of Atmospheric Sciences, Texas A&M University, College Station, Texas

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Kenneth P. Bowman Department of Atmospheric Sciences, Texas A&M University, College Station, Texas

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Gerald R. North Department of Atmospheric Sciences, Texas A&M University, College Station, Texas

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Abstract

Four years of outgoing longwave radiation (OLR) and rainfall data from the Tropical Rainfall Measuring Mission (TRMM) are investigated to find the dominant large-scale wave modes in the Tropics. By using space– time cross-section analysis and spectral analysis, the longitudinal and latitudinal behaviors of the overall waves and the dominant waves are observed. Despite the noisy nature of precipitation data and the limited sampling by the TRMM satellite, pronounced peaks are found for Kelvin waves, n = 1 equatorial Rossby waves (ER), and mixed Rossby–gravity waves (MRG). Madden–Julian oscillation (MJO) and tropical depression (TD)-type disturbances are also detected. The seasonal evolution of these waves is investigated.

An appendix includes a study of sampling and aliasing errors due to the peculiar space–time sampling pattern of TRMM. It is shown that the waves detected in this study are not artifacts of these sampling features.

The results presented here are compared with previous studies. Consistency with their results gives confidence in the TRMM data for wave studies. The results from this study can be utilized for modeling and testing theories. Also, it may be useful for the future users of the TRMM data to understand the nature of the TRMM satellite sampling.

Corresponding author address: Hye-Kyung Cho, Department of Atmospheric Sciences, Texas A&M University, College Station, TX 77845-3150. Email: hkc@ariel.met.tamu.edu

Abstract

Four years of outgoing longwave radiation (OLR) and rainfall data from the Tropical Rainfall Measuring Mission (TRMM) are investigated to find the dominant large-scale wave modes in the Tropics. By using space– time cross-section analysis and spectral analysis, the longitudinal and latitudinal behaviors of the overall waves and the dominant waves are observed. Despite the noisy nature of precipitation data and the limited sampling by the TRMM satellite, pronounced peaks are found for Kelvin waves, n = 1 equatorial Rossby waves (ER), and mixed Rossby–gravity waves (MRG). Madden–Julian oscillation (MJO) and tropical depression (TD)-type disturbances are also detected. The seasonal evolution of these waves is investigated.

An appendix includes a study of sampling and aliasing errors due to the peculiar space–time sampling pattern of TRMM. It is shown that the waves detected in this study are not artifacts of these sampling features.

The results presented here are compared with previous studies. Consistency with their results gives confidence in the TRMM data for wave studies. The results from this study can be utilized for modeling and testing theories. Also, it may be useful for the future users of the TRMM data to understand the nature of the TRMM satellite sampling.

Corresponding author address: Hye-Kyung Cho, Department of Atmospheric Sciences, Texas A&M University, College Station, TX 77845-3150. Email: hkc@ariel.met.tamu.edu

1. Introduction

a. Background

Accurate observations of precipitation are necessary for monitoring the variability of weather and climate and are crucial to understanding the hydrological cycle. Precipitation has been measured by many conventional methods, such as rain gauges and ground-based radar, but the spatial coverage of these methods is limited to land surfaces and coastal regions (Hsu et al. 1997). More than two-thirds of global precipitation falls in the Tropics and subtropics, but the study of tropical rainfall has suffered from a lack of reliable data, especially over oceanic areas, and is complicated by large variability of rainfall in space and time. The only way to collect global-scale rainfall data is by means of spaceborne satellite sensors (North 1987). The Tropical Rainfall Measuring Mission satellite (TRMM) is specifically devoted to studying rainfall in the Tropics and subtropics of the earth by using passive microwave, visible and infrared sensors, and the first spaceborne rain radar. TRMM's orbital inclination of 35° allows it to observe the earth's surface between about 35°N and 35°S. TRMM has provided instantaneous rain-rate measurements throughout the Tropics since its launch on 27 November 1997; and it will contribute to a better understanding the release of latent heat, which is one of the largest sources of energy in the global atmospheric circulation (Kummerow et al. 2000).

An understanding of tropical circulations requires consideration of equatorial wave dynamics, but tropical disturbances cover a wide range of scales; and there is no simple theoretical framework, analogous to quasigeostrophic theory, that can be used to provide an overall understanding of large-scale tropical motions (Holton 1992). The present theoretical models of equatorial waves do not explain the physical mechanisms responsible for observed waves in the tropical atmosphere. In addition, previous observational studies for the equatorial waves are based on data that often are limited in space and time (Hartmann and Gross 1988).

Among these incompletely understood phenomena is the Madden–Julian oscillation (MJO), which is also referred to as the 30–60-day or 40–50-day oscillation. In 1971, Madden and Julian (1971) identified a 40–50-day oscillation in the tropical atmosphere by computing spectra and cross spectra of the zonal wind anomalies in the tropical Pacific. The MJO turns out to be the main intraseasonal fluctuation in tropical wind, outgoing longwave radiation (OLR), and rainfall. The MJO receives much attention because of its importance for tropical weather related to the Indian monsoon and its potential role in the El Niño–Southern Oscillation (ENSO). However, a complete theory of the MJO from initiation through dissipation has not been devised, and discrepancies remain between the observations and theory.

Given the present research status of tropical waves, including the MJO, continued observational study of many meteorological fields is crucial to better understanding of the tropical atmosphere. In this research, we seek to find the MJO and other tropical waves in the TRMM precipitation and OLR data to get an improved understanding of the equatorial precipitation dynamics. To facilitate this study, space–time spectral analysis, as well as space–time cross-section analyses is employed. Comparisons with the results from the previous research are included. Because of its high spatial resolution and unique sampling characteristics, TRMM provides a unique and valuable dataset with which to study tropical waves.

To our knowledge, the results presented here are the first attempt to search for tropical waves using global-scale precipitation records. Thus, the results from this study can be helpful to investigate the characteristics of the tropical waves and to test present theories and for the modeling of tropical waves and tropical convection.

b. Previous studies of tropical waves

The basic linear theory of equatorial waves is developed by Matsuno (1966). Matsuno shows the wave solutions that are theoretically possible in the Tropics using a simplified set of the hydrodynamical equations on an equatorial β plane. Subsequently, numerous studies progressed to identify these waves in the observational data (Wikle et al. 1997).

Using long-term OLR data, Wheeler and Kiladis (1999, hereafter WK) find statistically significant wave modes that are strongly coupled with the tropical deep convection. Figure 1 is reproduced from WK. On this plot, the dispersion curves of the even and odd meridional mode-numbered equatorial waves for the three equivalent depths of h = 12, 25, and 50 m are superimposed for matching with the theoretical waves in Matsuno's paper. They successfully identify equatorial Kelvin waves, n = 1 equatorial Rossby waves (ER), mixed Rossby–gravity waves (MRG), and inertio–gravity waves, as well as the MJO by removing a smooth empirical background noise spectrum. The signal for tropical depression (TD)-type waves is also detected at a wide range of the synoptic scales (Takayabu and Nitta 1993; Takayabu 1994a,b; Kiladis and Wheeler 1995). They also found that an equivalent depth on the order of 25 m is the most optimal for an association of equatorially trapped wave with convection. This provides new insight into the nature of the interaction between convection and large-scale moist dynamics which is of crucial importance to the theory and modeling of the tropical atmosphere.

Observational studies portray the MJO as an eastward-propagating, equatorially trapped, wavenumber-1, baroclinic oscillation in the Tropics (Madden and Julian 1994; Waliser et al. 1999). It originates somewhere in the Indian Ocean and moves through the Eastern Hemisphere at around 5 m s−1, accompanying deep convective activity. It then decays or propagates through the Western Hemisphere at a higher speed, but at this time does not interact much with convection. (Madden and Julian 1972; Bantzer and Wallace 1996). The appearance of westward- and northward-propagating waves with a similar period west of the date line indicates that the MJO should be considered as a multimode system in which modes have interactions with one another (Murakami et al. 1984). A number of theories have been proposed to explain the origin and mechanism of tropical waves and the MJO, but none are yet fully corroborated by the observational evidence (Waliser et al. 1999). Most difficulties are caused by the complexity of the structure, seasonality, and sporadic occurrences of the waves (Slingo et al. 1999).

2. Data and methods

a. TRMM 3G68 rainfall and 3G01 OLR product

The primary rainfall data used in this study is from the TRMM 3G68 combined rain-rate product. The TRMM “combined” algorithm merges information from two TRMM sensors, the precipitation radar (PR) and the TRMM microwave imager (TMI) 10-Ghz channel into a single retrieval to produce a “best” rain estimate utilizing the strengths of each sensor (Haddad et al. 1997).

The 3G68 data is an hourly gridded product. It includes 24 hourly grids in a single daily file. The PR and TMI fields of view are spatially averaged within each 0.5° × 0.5° latitude–longitude box that had a TRMM overpass, recording the time of the first field of view in that grid box, the average rainfall, and ancillary statistics from the instantaneous observations within the grid box during an hour. If neither the TMI nor the PR swath covered an hourly grid box, then no information is written to the files. This means that values in the file represent grid boxes that had data from at least one of the TRMM instruments (TSDIS/TRMM 1999). Rain rates r are given in mm h−1.

In addition to rainfall data, we used OLR data obtained from the TRMM satellite (3G01 data product). Because values of OLR in the Tropics respond more strongly to variations in cloudiness than any other factor, it has been used in many studies as a index of convective activity in those latitudes and to estimate precipitation (Arkin and Ardanuy 1989). Moreover, OLR data are more continuous than rainfall data and most observational studies of the equatorial waves have been done using the OLR data. Thus, it is valuable to compare the OLR data from TRMM with those from other satellites.

The TRMM 3G01 product is converted from the TRMM Visible Infrared Scanner (VIRS) channel 4 (10.8 μm) radiance to brightness temperature. The gridding and data averaging schemes for the TRMM 3G01 product are the same as for the TRMM 3G68 data. The 3G01 data provide mean, maximum, and minimum brightness temperature for each hourly 0.5° × 0.5° grid box. Among these variables, only mean brightness temperature values are used here. The area of interest in this study covers from 15°S to 15°N, where the dominant tropical waves are observed.

b. Method for averaging data

Because of the asynoptic nature in the TRMM data, TRMM 3G68 and 3G01 products are not directly amenable to standard space–time spectral analysis methods. With a 0.5° × 0.5° grid resolution, approximately 14% of grid boxes have no data on a given day depending on latitude.

To get a “synoptic” dataset that has uniform increments in time with a minimum number of missing values, we compute daily mean, area-averaged rain rates for 2.5° × 2.5° and 5° × 5° latitude–longitude grids from the 0.5° × 0.5° high-resolution dataset. For this, we assume that the data within one large grid box are spatially homogeneous and all measurements, taken during one day's cycle, are made simultaneously. Under these two presumptions, we collect all data included in one large grid box from TRMM 3G products during 1 day and daily area-averaged rainfall and OLR values are calculated from
i1520-0442-17-22-4387-e1
where N is the number of 0.5° × 0.5° boxes included in one large grid box of the new resolution system for 1 day, Ri is ith 3G68 or 3G01 datum among the N observational data, Wi is the weighting function for Ri, and Ai is the area of the ith 0.5° × 0.5° grid box. Daily values of the rain rate and OLR averaged over 2.5° × 2.5° and 5° × 5° grid boxes have ≤2.8% and ≤0.5% missing values per day, respectively. Linear interpolation in time is used to fill in the missing data because the error resulting from interpolating over time at a given point is typically much smaller than the error resulting from interpolating across space at a given time (Lucas et al. 2001). The results shown in section 3 are tested using the data at spatially different resolutions. Because the results are similar for different resolutions, we concluded that the spatial averaging does not significantly influence our results. The data used here extend from 8 December 1997 to 30 June 2002 (1666 days).

c. Method of spectral analysis

The primary identification of equatorial waves, including the MJO in both rainfall and OLR, is made by analyzing longitude–time cross sections. Longitude– time cross-section analysis provides a detailed display of the regional differences in wave propagation and provides a check on the results of the space–time spectral analysis (Zangvil 1975). To investigate the latitudinal behavior of the tropical waves, latitude–time cross sections are also analyzed. An average phase speed is crudely estimated from the cross sections by measuring the slope of lines of maximum convection.

Space–time spectral analysis is used to find the dominant waves in the rainfall and OLR data fields. The space–time spectrum is useful to look for dispersion relationships to support dynamical theories and to diagnose possible relationships between spatial and temporal scales, which may ultimately lead to the detection of wavelike dynamics (von Storch and Navarra 1999).

The results shown in section 3c are motivated by Wheeler and Kiladis's recent works (WK; Wheeler et al. 2000). They performed a space–time spectral analysis of a long (∼18 yr) twice-daily record of satellite-observed OLR over the same domain as this study. Comparisons between the results from WK and the result using the TRMM data are presented to validate the quality of TRMM data for studying wave variations. We follow the same methods as WK for spectral analysis. The details are well described in WK. For the best comparison, the same spatial resolution as WK, namely 2.5° × 2.5°, is used for this spectral analysis. The spectral methods are applied separately on symmetric and antisymmetric components about the equator. This decomposition is helpful to compare the peaks in the spectra to the waves in dynamical theory.

Cross spectra are also obtained between OLR and rainfall data to investigate coherent wave variations within these data. To examine the seasonal dependency, separate spectral analyses are applied to the 6-month northern summer data from March to October, as well as the 6-month winter data from November to April. The definition of the seasonal cycle is same as WK and it is based on the times of the year when the latitude of maximum mean tropical convection is either south or north of the equator.

In this study, we also carry out a space–time spectral analysis using the method introduced by Hayashi (1971, 1982). One of the advantages of the Hayashi's method is that it allows a partition between standing and traveling wave variance and is suitable to make quantitative variance comparisons (Hayashi 1977; Céron and Guérémy 1999). The standing waves defined in this method consist of coherent eastward- and westward-moving components of equal amplitude, while the traveling parts are incoherent with each other, including irregular and noise components. Using the coherence between the eastward- and westward-moving waves, the space–time spectra are partitioned into standing and traveling waves and the characteristics of both components are analyzed.

The sampling error and aliasing study in interpreting the asynoptic TRMM data are performed with the synthetic data sampled using the same method of the TRMM satellite orbit and included to strengthen the results from this study in the appendix.

3. Results

a. Longitude–time cross section

Figure 2 is a longitude–time cross section of the daily mean OLR and rainfall averaged between 15°S and 15°N from the daily 0.5° × 0.5° latitude–longitude TRMM 3G01 OLR and 3G68 rainfall data for the period from 1 January 1998 to 31 December 1999. Although the individual organization and intensity of the convective anomalies in the OLR and precipitation fields are highly variable in time and space, the overall features of the wave activity shown in Figs. 2a and 2b are very coherent. Zonally propagating waves are visible as narrow diagonally oriented features in both the OLR and precipitation. From maps below each cross section, one can clearly see that climatologically, stationary precipitation maxima occur over Africa, and the Maritime Continent in the western Pacific. There are two main regions of wave activity associated with deep convection. One is from the eastern Indian to the western Pacific Oceans between 60° and 150°E. The other is in the Atlantic Ocean around the equator.

A distinct eastward-propagating feature is most pronounced over the Indian and western Pacific Ocean. Enhanced convection appears over the Indian Ocean, intensifies, and propagates into the western Pacific at a speed of about 5 ∼ 8 m s−1. These anomalies decay usually around the date line, which is normally a region of colder surface water. The estimated period of these variations is about 30–60 days, indicating that this is the MJO. The intensity and period of the MJO varies seasonally. During the northern winter (November to April), the eastward-propagating MJO is stronger than in summer (May to October). Estimating the seasonal period of the MJO is difficult because the position and intensity of the individual events are variable and intermittent.

In addition to the eastward-moving MJO, westward-propagating features are also evident over this region. The westward-propagating features often originate in the western Pacific and propagate in the Indian Ocean with a speed of −5 to −10 m s−1. The time interval between these westward-moving waves usually ranges from 10 to 20 days. The features of these waves are similar to a predominant mode in the off-equatorial regions presented in previous observational studies (Murakami et al. 1984; Ghil and Mo 1991; Wang and Xie 1997).

In the Atlantic region, there are two areas that feature strong convection. One is over equatorial South America between 80° and 50°W; the other is the east of the Greenwich meridian over tropical Africa. The intensities of these convections in the two areas are comparable, and both of them generally have weaker propagating features than in the Indian and western Pacific. From March to May, which is the equatorial warm SST period, the convections from two areas interact with each other and intensify.

The period immediately after the launch of TRMM was affected by the extraordinarily strong El Niño event that continued until early May 1998. At that time a rapid transition from El Niño to La Niña conditions occurred. During the 1997/98 El Niño, strong convection occurred over the central Pacific instead of the Indian Ocean and the eastward-propagating waves with a period of 30– 40 days can be seen in the central and eastern Pacific. During May 1998, a strong eastward-propagating wave traveled completely around the equator. After this unusual wave passage, the region of maximum convection shifted back to the Maritime Continent. La Niña conditions prevailed until January 2002. Another strong wavelike feature circled the globe during April 2002 (not shown) and a significant increase of the rainfall over the Indian Ocean during 2002 is seen. No significant changes, however, are seen in the central Pacific Ocean.

b. Latitude–time cross section

The possible latitudinal propagation of equatorial convective waves is investigated using latitude–time cross sections, and Fig. 3 shows the results from the first 2-yr TRMM data. In order to compare the disturbances occurring within different longitudinal bands, Fig. 3a shows the rainfall values averaged over longitudes from 60° through 150°E. This region has the maximum wave activity and precipitation around the equator. Figure 3b is averaged over the eastern Pacific Ocean between the date line and 80°W. The most prominent feature seen in Figs. 3a,b is, not surprisingly, the seasonal movement of the ITCZ, which is the major area of cloud formation over the tropical oceans. The average position of the ITCZ is at about 10°N, but during the northern winter, the ITCZ moves southward and is likely to be around 5°S.

Figure 3a reveals the existence of significant northward-propagating intraseasonal variation with a 40–60-day periodicity over the Indian Ocean and the western Pacific. While no systematic propagation is apparent over the central Pacific (Fig. 3b), there is slight evidence of occasional northward-propagating features.

Over the Indian and western Pacific Ocean, there are prominent northward-propagating disturbances during the northern summer. Although the wave variations in the northern summer are stronger than in winter, some northward-propagating waves are also observed south of the equator during winter. This northward propagation is one of the well-known characteristics of the MJO (Murakami et al. 1984; Wang and Xie 1997) and has received much attention in previous research because of the relationship with active and break periods of the monsoons (Maloney and Hartmann 1998).

Over the Pacific (Fig. 3b), there are more distinct seasonal differences in the location of the ITCZ between summer and winter. During northern summer, the well-defined ITCZ is located about 10°N, while double ITCZs are often found during March through May, when the central Pacific Ocean has warmer SSTs than during the rest of year. As mentioned before, no significant propagating feature is shown over this region and there is, at best, slight evidence for northward-propagating features. Strong anomalies are detected, however, during the recent El Niño event (early 1998).

c. Space–time spectral analysis

1) Raw and background power spectra

In this section, the wave activity near the equator is described statistically using space–time spectral analysis. As a simple first step, a 1D fast Fourier transform (FFT) analysis was performed at each box. These temporal power spectra fail to detect any dynamically meaningful spectral peaks.

Figures 4a and 4b represent the zonal wavenumber– frequency power spectra of the antisymmetric and symmetric components of OLR, while Figs. 4c and 4d are from the rainfall data. The dominant feature is the “redness,” which makes it difficult to distinguish wave signals from the broad noise background. The power is greatest at periods longer than 20 days, especially in the symmetric components. The results are similar to WK.

To extract the signals from the raw spectra, a red background spectrum is calculated by averaging the power of the antisymmetric and symmetric components and then smoothing strongly in both frequency and wavenumber using the same methods in WK. The shapes of the background spectra are quite smooth and distinctly “red” (Figs. 4e,f ). Dividing the individual power spectra in Figs. 4a–d by the background spectra from each variable, the distinct spectral peaks are separated from the raw spectrum.

2) Distinct signals in power spectra

Figure 5 shows the power spectra of the TRMM OLR and precipitation after removal of the smooth background. The major features in Fig. 1 can be seen in both figures. Both OLR and rainfall have power in the region of the MJO, the Kelvin waves, and n = 1 ER waves in the symmetric components, as well as MRG waves in the antisymmetric components. The power at periods longer than 40 days dominates each panel, but the symmetric components usually have more MJO signal in both datasets. Comparison reveals that the rainfall data have more power at very low frequencies for all wavenumbers and that the spectral peaks are broader and less distinct than OLR, especially for the MRG wave in the antisymmetric components. In the symmetric components, one notices that the distinct power peaks consist mostly of the eastward-propagating components. On the other hand, the westward components exhibit a broader power distribution, especially at high frequencies. This is the first direct evidence for equatorial wave modes in precipitation.

3) Cross spectra between OLR and rainfall

A cross-spectral analysis between the OLR and rainfall was performed in order to learn the degree of association between the two time series in the spectral domain. Results are shown in Fig. 6 for the antisymmetric and symmetric components separately. Generally, the coherence between OLR and rainfall data increases with decreasing frequency at low wavenumbers. In the coherence for the antisymmetric components, no outstanding spectral peak is observed, except in the region of MJO waves. The greatest coherence in the symmetric components is found at the regions of the MJO and n = 1 ER waves.

4) Seasonal variation in the spectra

To examine the seasonal variations in the wave activity, separate space–time spectral analyses were applied to the northern winter and summer data. Figures 7 and 8 represent the seasonal variations of the OLR and rainfall, respectively. To find the dominant wave modes, the spectrum from each season is divided by the same background defined in Figs. 4e and 4f. Decomposition into the northern winter and summer leads to the increasing power, especially for TD-type disturbances compared with Fig. 5.

During the northern summer, MRG waves in the antisymmetric components and TD-type disturbances in both symmetric and antisymmetric components are enhanced. Takayabu (1994a,b) studied the seasonal preference for these waves and found similar results. In the northern winter, the Kelvin waves, n = 1 ER waves in symmetric components, and MJO in both symmetric and antisymmetric components have more power than in the northern summer. Among these waves, the seasonal variation of n = 1 ER waves is greatest. The seasonal variation of the MJO is in accord with the previous observational results and also similar to WK.

5) Partition of power spectra into standing and traveling parts

Figures 9 and 10 show the results after the partition between the standing and traveling part in the OLR and rainfall data, respectively, using the method of Hayashi (1977). The standing parts of the antisymmetric and symmetric components are markedly red and have a similar variance pattern to the raw spectra in Fig. 4. The traveling parts contain most of the wave signals defined in Figs. 1 and 5 without the necessity of removing the ad hoc smoothed background spectrum. Compared with Fig. 5, the spectrum for the traveling part shows more distinct peaks, especially for high-frequency variance, such as the TD-type disturbances and IG waves. There is also a region of large variance around westward-moving wavenumber 6 and a period of about 7 days in both components. The signal for the MJO can be found in the traveling parts, but the power is less than that in Fig. 5. In addition, the spectral peak for n = 1 ER waves is not well separated from the standing components and this is more notable in the precipitation field. The traveling parts in the rainfall have similar features with the OLR data, but the higher-frequency waves are not well detected as in the OLR data. The results of this section suggest that the partition into standing and traveling part proposed in Hayashi (1977) is an efficient way to separate some waves from the basic state.

4. Summary and discussion

TRMM is specifically devoted to measuring rainfall in the Tropics and subtropics, and the wealth of data from the TRMM instruments gives new insights into the tropical atmosphere and the global atmospheric circulation. In this study, we investigated the characteristics of those tropical waves that are associated with the convection by using the TRMM OLR and rainfall data for the period from 8 December 1997 to 5 June 2002. Wavelike variations during this period, including the seasonal cycle, are examined in terms of both their zonal and meridional propagation using space–time cross sections.

Eastward-propagating waves with periods of 30–50 days are dominant near the equator, especially over the Maritime Continent and adjacent oceans. The characteristics of this wave agree well with the results from previous studies of the MJO. Westward-propagating waves with periods from 20–30 days are also observed consistently in this region. During the northern winter, the amplitudes of the eastward-propagating waves are large and systematic meridional propagation is not observed. In the northern summer, on the other hand, the amplitude of the westward-propagating waves is comparable with that of the eastward-moving waves, and significant northward-propagating waves appear. Thus, the northern summer has more active transient waves and more complex wave patterns. These seasonal features in the Indian and western Pacific Ocean regions are well-known characteristics of tropical intraseasonal variability (Wang and Xie 1997).

Many theoretical and modeling studies have focused on the relationships between these wave modes (Murakami et al. 1984; Wang and Xie 1997; Kemball-Cook and Wang 2001). Murakami et al. (1984) explained these waves in “group velocity” terms, namely, the individual perturbations move northwestward, but the wave packet appears to propagate eastward. Conventional wave– CISK (conditional instability of the second kind) theory explains the features of the MJO as coupled Kelvin– Rossby waves. An internal heating mechanism, such as CISK, may maintain or even amplify both the eastward Kelvin waves and westward Rossby waves away from the source regions. Because the Kelvin wave is composed solely of divergent wind and has a more favorable association with low-level convergence than the rotational Rossby waves, over warmer parts of the ocean the Kelvin waves are selectively amplified rather than the Rossby waves (Waliser et al. 1999). During the northern summer, however, the Indian monsoon is intimately tied to the phase of the MJO. Thus, the Rossby waves, which are associated with the main center of the equatorial convection, move northward and westward accompanying the “break” and “active” phase of the monsoon (Lau and Peng 1987; Maloney and Hartmann 1998). The northward-propagating waves are observed only over the Indian Ocean, and no systematic latitudinal propagation is found in other longitude bands. Nitta (1987) pointed out that there are suitable longitudes for this wave generation. Since the zonal winds in the western Pacific region vary substantially in the east–west direction in response to the summer monsoon, the atmospheric responses may depend on the longitudinal positions of the tropical heat sources. A predominance of upper-tropospheric easterlies, however, may prohibit northward-propagating features west of about 110°E (Nitta 1987).

Wave occurrence over the central Pacific is very important to global climate variations associated with ENSO. Specifically, equatorial Rossby waves, as well as Kelvin waves, generated over this region have received much attention because of their possible roles in air–sea interaction and impacts on the midlatitudes. The results presented in this study show that during the warm SST period from March to July, a double ITCZ and more active wave variations over the Pacific Ocean are typically found. Without any other information, it is difficult to categorize these waves theoretically, but undoubtedly it is indicative of a strong interaction between the tropical waves and the upper ocean.

Two recent studies give some insights into the possible relationship between the MJO and these features observed in the Pacific Ocean. Nieto and Schubert (1997) pointed out the contribution of barotropic instability in the lower troposphere for the double ITCZs, and Maloney and Hartmann (2001) found that the MJO is important to generate this barotropic instability. The MJO is characterized by altering periods of easterly and westerly zonal wind anomalies across the tropical Pacific Ocean during the northern summer. It provides a significant source of eddy kinetic energy and supports tropical cyclogenesis (Maloney and Hartmann 2000, 2001).

It is interesting to investigate the effects of the recent ENSO on tropical wave characteristics. Due to the short record, a clear statistical relationship cannot be found between ENSO and the MJO, but strong Kelvin wave forcing related to the MJO tends to occur prior to peaks of ENSO warming. The MJO is much weaker immediately after the warming peak of ENSO (Zhang and Gottschalck 2002). The strong MJO forcing contributes to a substantial weakening of the low-level easterly winds through the equatorial Pacific Ocean. As a consequence, the weaker-than-average easterly winds contribute to a deepening of the oceanic thermocline and an increase in SST anomalies in the central equatorial Pacific (Kousky et al. 2002). This is particularly true during the 1997/98 El Niño event. Unfortunately, TRMM did not observe the onset of this warm ENSO episode, but Fig. 2 shows an extraordinarily strong wave that travels completely around the equator without any interruption during May 1998. The MJO is weaker than normal after this. In addition, convection is mainly found over the central Pacific Ocean during the El Niño event. However, after the passage of this strong wave the major convection shifts back to the monsoon region, and the convection over the Pacific Ocean is weak. Takayabu et al. (2001) suggested that the strong MJO forcing during May 1998 may have intensified the easterly trade winds over the eastern equatorial Pacific Ocean, finally terminating the 1997/98 El Niño event abruptly. Another strong Kelvin wave is observed during April 2002, and weaker-than-average low-level easterly winds are reported throughout the equatorial Pacific from May 2002 (Kousky et al. 2002), but the significance of this strong wave is not clear in the OLR and precipitation data.

Space–time spectral analysis proves to be a useful method to detect wave modes and to compare with theoretical dispersion relationships. The spectra from “raw” OLR and rainfall data have basically red patterns, except for broad indications of propagating waves with phase speeds ranging from 3 to 8 m s−1 in each direction. To isolate distinct waves, the raw spectra are divided by a smooth red background spectrum. In the symmetric components, pronounced spectral peaks for the Kelvin waves and n = 1 ER waves are found. In the antisymmetric components, MRG waves are clearly detected. Unlike linear equatorial waves, the MJO is found at low frequencies, with a broad band of low zonal wavenumbers. TD-type disturbances are also detected in the synoptic-scale westward-propagating wave regions. The spectral results presented in this study not only give information on tropical waves, but also confirm the quality of the TRMM data for wave studies. Wheeler and Kiladis (1999) performed the same space– time spectral analysis on 18 yr of twice-daily OLR data. Their major findings can also be seen in 4 yr of TRMM OLR and precipitation data.

The seasonal evolution of TRMM data presented in this study is also consistent with the previous observational studies (Salby and Hendon 1994; Madden and Julian 1994), as well as WK. During the northern summer, the synoptic-scale disturbances, including TD-type disturbances, are more active; whereas during the northern winter, n = 1 ER waves emanate with moderate intensity and the MJO and Kelvin waves slightly increase in intensity. The results from Takayabu and Nitta (1993) indicate that the TD-type wave dominates when under the influence of monsoon westerlies with larger vertical wind shear.

The coherence between the OLR and rainfall data was also investigated through cross-spectrum analysis. As expected, the data are quite consistent, especially for the low-frequency waves.

Some previous studies reported the characteristics of intraseasonal standing waves over the Indian Ocean and the western Pacific Ocean (Murakami et al. 1986; Hsu et al. 1990; Yanai et al. 2000). To check their observations, Hayashi's method was used to separate standing and propagating waves. The spectra of the standing waves have strong signals for the intraseasonal period standing waves in both the symmetric and antisymmetric components. The spectra of the traveling waves contain spectral peaks for the dominant waves found in Fig. 5, without the necessity of defining an ad hoc smoothed background spectrum. However, the n = 1 ER wave was not detected more in the precipitation field rather than in the OLR, and the MJO was much weaker in its traveling parts. It can be inferred that intraseasonal standing waves are connected with the n = 1 ER and MJO, especially in the rainfall field. However, the possibility cannot be eliminated that they are not really standing waves, but simply interference of two unrelated propagating waves (Straus and Lindzen 2000). Another impression of a standing oscillation in convection may come forth because of the modulation of the eastward-propagating convective disturbance by an amplitude envelope with maxima in the eastern Indian and western Pacific Oceans and a minimum over the Maritime Continent (Zhang and Hendon 1997). Thus, future study will compare Hayashi's method with the method in the WK by investigating the differences between the Figs. 5, 9, and 10 to develop better skill of extracting the dominant wave characteristics from the basic states. We also leave the study of the statistical methods for asynoptic satellite data and their practical application into the TRMM data as a crucial future work to increase the utilization of the TRMM data in the research of the tropical waves (Salby 1982a,b).

Results of this study are from the first 4 yr of TRMM data. Further study using more data from TRMM is crucial to improving the present results and achieving a better understanding of tropical waves and their relationships with recent climate changes. Future study will also investigate the relationship between TRMM precipitation and OLR data to improve the current rainfall retrieval algorithm.

Acknowledgments

The authors wish to thank to Thomas T. Wilheit and Benjamin Giese for their comments on an early version of the manuscript. The authors also thank referees for their insightful comments on the first draft of this paper. The TRMM 3G68 data were obtained from the TRMM Science Data and Information System (TSDIS) at Goddard Space Flight Center, NASA. This research was funded by NASA Grant NAG 5-9649.

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APPENDIX

Sampling Error and Aliasing Study

There are extensive reports of the diurnal cycle in surface temperature, cloud cover, precipitation, and OLR. The existence of diurnal variability poses problems for the analysis of satellite data due to the possibility of an undersampled diurnal cycle being aliased to other frequencies (Salby and Callaghan 1997). The non-sun-synchronous TRMM orbit allows observation of the diurnal cycle of the rainfall by constructing a “mean diurnal cycle,” based on the averages at individual local times. This is not possible from conventional sun-synchronous satellites. The unique sampling pattern of TRMM, however, can cause problems in interpreting the data because the satellite visits the same area at intervals either longer or shorter than 24 h, depending on the latitude (Shin and North 1988). Specifically, in performing space–time spectral analysis, the asynoptic sampling might lead to significant misinterpretations of individual spectral peaks (Wu et al. 1995).

In this study, the aliasing of irregularly sampled data is tested by using synthetic data with diurnally modulated wave signals sampled using the TRMM orbit. The daily averaged values are calculated using the same method in section 2b to get a “synoptic” dataset. The results are compared with a benchmark calculation that uses true synoptic sampling of the synthetic data at 3-, 6-, 12-, or 24-h intervals. Figure A1 shows a time–longitude cross section from the synthetic data that is a combination of a 48-day eastward-propagating wave and a 1-day westward-propagating wave, which acts as a diurnal modulation of the slow eastward-moving wave. Both waves have wavenumber-1 structure and equal amplitudes. Figure A1 shows the synthetic data sampled every 3, 12, and 24 h (Figs. A1a–c, respectively) and TRMM sampling at three different latitudes (Figs. A1d–f). The 3-h time resolution (Fig. A1a) is fine enough to resolve the diurnal cycle, so it is given as a reference of the correct sampling. Compared with Fig. A1a, the 12-h sampling (Fig. A1b) represents the low-frequency wave well, but the diurnal wave is hardly detected. With 24-h sampling (Fig. A1c), the westward-moving diurnal wave appears stationary, and the eastward-moving wave moves in and out of phase with the stationary diurnal wave. The TRMM satellite sampling at the equator (Fig. A1d) exhibits a pattern similar to the 3-h synoptic sampling case for the slow wave, but off-equatorial TRMM sampling does not represent the real pattern successfully.

To test the effect of the errors shown in Fig. A1 on the spectral estimates, the spectral method used in section 3c is performed on these synthetic data with the same 4-yr record length. Increasing the length of a data sequence improves the spectral resolution, which is the ability to distinguish two signals that are close in frequency (Wu et al. 1995). The 4-yr data are good enough to distinguish waves with the 60-day or shorter period. Figure A2 shows the spectra of the various regularly sampled data. Figures A2a,c,e (2 waves) show the spectra from the data containing a 48-day eastward-moving wavenumber-1 wave and a westward-moving diurnal cycle. Figs. A2b,d,f (3 waves) add a 24-day westward wavenumber-1 wave, to the waves in the left column. The comparison between the “2 waves” and “3 waves” is useful to investigate the effects of additional wave signal on the spectral estimates and provides an insight about the aliasing caused by the unresolved waves in the atmosphere. All three waves have the same amplitude. Six-hour regular sampling resolves all of the true signals without any aliasing and power loss (Fig. A2a). To compare the resolved power in each spectrum, the individual peaks are labeled with the relative power, which is a fraction of true power of that peak in Fig. A2a. The 12-h regular sampling (Fig. A2b) has all true signals with same powers, but it has a spurious eastward-moving wavenumber-1 signal with a 1-day period. This shows that 12-h sampling cannot resolve a diurnal traveling wave. The spectra computed with 24-h sampling cannot represent any diurnal signal because the Nyquist frequency of the 24-h sampling is 2 days. The diurnal wave in the 24-h sampling is aliased to zero frequency, but the spectra of low-frequency waves are not affected much. The additional wave does not make any difference on the spectra in the case of the regular sampling.

Figure A3 shows the spectra using TRMM sampling of the synthetic data. Daily averaged TRMM data cannot resolve the diurnal wave, but the amplitudes of the low-frequency waves are not affected much. As expected, the spectra computed using TRMM sampling depend on latitude. Spurious spectral peaks introduced by aliasing of the diurnal cycle are more serious as one moves to higher latitudes. In Fig. A3a, a weak aliased peak near 22 days can be caused by the fact that the TRMM satellite orbit precesses through 1 day over this period. The required time for the sampling of the diurnal cycle, namely the period that the satellite observations revisit the same location at the same local time, increases at higher latitudes. Thus, false peaks found in Fig. A3c– f can be also interpreted in terms of the orbital geometry, but the broadening in the aliased frequency around the 42-day period makes more serious errors than at the equator. Moreover, TRMM sampling of additional waves causes additional false peaks, which make it more difficult to interpret the spectra correctly.

To attempt to remedy the inherent weakness of the TRMM sampling pattern, the synthetic data are sampled with both the TRMM sampling pattern and either a 12- or 24-h regular synoptic sampling pattern; daily averaged values are then calculated. This simulates combining data from TRMM and multiple sun-synchronous satellites. The goal here is not to resolve the diurnal cycle, but to check for aliasing of the diurnal wave to low frequencies. All of the synthetic datasets analyzed in Fig. A4 contain a 48-day eastward-moving wave, a 24-day westward-moving wave, and a diurnal cycle; and all waves have the same amplitude. Although the spurious peak around 22 days in Fig. A3a,b disappears, adding 24-h regular sampling to TRMM sampling at the equator produces small spurious low-frequency peaks (Fig. A4a). These peaks are not apparent when TRMM sampling is combined with 12-h sampling (Fig. A4b). The same test is also performed on the TRMM sampling at higher latitudes (Figs. A4c–f). Although the aliasing of the 42-day wave shown in Figs. A3c–f still exists, the powers of these aliases are significantly decreased. These results show that the combination between the TRMM sampling and a regular sampling data with at least a 12-h time interval is efficient to remove the spectral noise, especially at higher latitudes. From these results, we conclude that TRMM sampling can resolve low-frequency waves at the equator without serious contamination by the diurnal cycle. Off the equator, however, care should be taken to avoid spurious spectral peaks.

Fig. 1.
Fig. 1.

The antisymmetric OLR power divided by the background power. Contour interval is 0.1, and shading begins at a value of 1.1 for which the spectral signatures are statistically significant above the background at the 95% level (based on 500 degrees of freedom). Superimposed are the dispersion curves of the even meridional mode-numbered equatorial waves for the three equivalent depths of h = 12, 25, and 50 m. (b) Same as in (a) except for the symmetric component of OLR and the corresponding odd meridional mode-numbered equatorial waves. Frequency spectral bandwidth is 1/96 cycles per day. (Reprinted from WK)

Citation: Journal of Climate 17, 22; 10.1175/3215.1

Fig. 2.
Fig. 2.

Time–longitude cross section of (a) OLR and (b) precipitation for the daily 0.5° × 0.5° latitude–longitude grid (TRMM products 3G01 and 3G68). Each value in the diagram is averaged over 0.5° of longitude and a latitude range of 15°S–15°N for a particular day. The period of the data is from 1 Jan 1998 to 31 Dec 1999

Citation: Journal of Climate 17, 22; 10.1175/3215.1

Fig. 3.
Fig. 3.

Time–latitude cross section of precipitation for the daily 0.5° × 0.5° latitude–longitude grid (TRMM product 3G68). Each value in the diagram is averaged over 0.5° of latitude and a longitude range of (a) 60°–150°E over the Indian and western Pacific Ocean, and (b) 180°–90°W over the central Pacific Ocean. The data period is same as in Fig. 2

Citation: Journal of Climate 17, 22; 10.1175/3215.1

Fig. 4.
Fig. 4.

Zonal wavenumber–frequency power spectrum of the OLR (a) antisymmetric components, and (b) symmetric components, calculated for the period from 1 Jan 1998 to 31 Dec 2001 using the 96-day segments. (c) As in (a) but from the precipitation data. (d) As in (b) but from the precipitation data. (e) Zonal wavenumber–frequency spectrum of the “background” for the OLR. (f) As in (e) but for the precipitation data. The power has been summed over the latitude range 15°S–15°N and the magnitude of its base-10 logarithm is shown.

Citation: Journal of Climate 17, 22; 10.1175/3215.1

Fig. 5.
Fig. 5.

(a) The antisymmetric OLR spectrum shown in Fig. 4a and (b) the symmetric spectrum shown in Fig. 4b divided by the background power of Fig. 4e to find the spectral peaks for the statistically significant waves. (c) The antisymmetric rainfall spectrum shown in Fig. 4c and (d) the symmetric spectrum shown in Fig. 4d divided by the background power of Fig. 4f. Superimposed are the dispersion curves of the even and odd meridional mode-numbered equatorial waves for the three equivalent depths of h = 12, 25, and 50 m

Citation: Journal of Climate 17, 22; 10.1175/3215.1

Fig. 6.
Fig. 6.

The coherence between (a) the antisymmetric components and between (b) the symmetric components of the OLR and precipitation

Citation: Journal of Climate 17, 22; 10.1175/3215.1

Fig. 7.
Fig. 7.

(a) As in Fig. 5a but the power is calculated only from Apr to Nov. (b) As in (a) but for the power of symmetric components shown in Fig. 5b. (c) As in (a) but the power is calculated only from May to Oct. (d) As in (a) but for the power of symmetric components only from May to Oct. Note that the background spectrum used is same as in Fig. 5

Citation: Journal of Climate 17, 22; 10.1175/3215.1

Fig. 8.
Fig. 8.

(a) As in Fig. 5c but the power is calculated only from Apr to Nov. (b) As in (a) but for the power of symmetric components shown in Fig. 5d. (c) As in (a) but the power is calculated only from May to Oct. (d) As in (a) but for the power of symmetric components only from May to Oct

Citation: Journal of Climate 17, 22; 10.1175/3215.1

Fig. 9.
Fig. 9.

(a) Zonal wavenumber–frequency spectrum for the standing parts in the antisymmetric OLR components. (b) As in (a) but for the symmetric components. (c) As in (a) but for the traveling parts. (d) As in (a) but for the traveling parts in the symmetric components

Citation: Journal of Climate 17, 22; 10.1175/3215.1

Fig. 10.
Fig. 10.

As in Fig. 9 but for the precipitation data

Citation: Journal of Climate 17, 22; 10.1175/3215.1

i1520-0442-17-22-4387-fa01

Fig. A1. Time–longitude cross section of the synthetic data with a 48-day eastward-propagating wave and 1-day westward-propagating wave. Both waves have wavenumber-1 structure. The synthetic data are sampled every (a) 3, (b) 12, (such as a sun-synchronous satellite), and (c) 24 h. (d) The daily average values from the synthetic data sampled such as the TRMM satellite at the equator. (e) As in (d) but at 15°N. (f ) As in (d), but at 30°N. Panel (a) is given as a reference of the correct sampling

Citation: Journal of Climate 17, 22; 10.1175/3215.1

i1520-0442-17-22-4387-fa02

Fig. A2. (a) Wavenumber–frequency spectra of the synthetic data sampled every 6 h. Synthetic data have a 48-day eastward-propagating wave and westward diurnal wave, and both waves are wavenumber 1. (b) As in (a) but the synthetic data have another 24-day westward-propagating wavenumber-1 wave. (c) As in (a) but sampled every 12 h. (d) As in (b) but sampled every 12 h. (e) As in (a) but sampled every 24 h. (f ) As in (b) but sampled every 24 h. Labels indicate the relative power divided by the true signal in (a)

Citation: Journal of Climate 17, 22; 10.1175/3215.1

i1520-0442-17-22-4387-fa03

Fig. A3. (a) As in Fig. A2a but sampled using the TRMM satellite at the equator. (b) As in Fig. A2b, but sampled such as the TRMM satellite at the equator. (c) As in (a) but at 15°N. (d) As in (b) but at 15°N. (e) As in (a) but at 30°N. (f ) As in (b) but at 30°N

Citation: Journal of Climate 17, 22; 10.1175/3215.1

i1520-0442-17-22-4387-fa04

Fig. A4. (a) As in Fig. A3b, but the synthetic data are sampled at the equator using the combined TRMM and 24-h regular sampling. (b) As in (a) but the synthetic data are sampled using the combined TRMM and 12-h regular sampling. (c) As in (a) but at 15°N. (d) As in (b) but at 15°N. (e) As in (a) but at 30°N. (f ) As in (b) but at 30°N

Citation: Journal of Climate 17, 22; 10.1175/3215.1

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  • Bantzer, C. H., and J. M. Wallace, 1996: Intraseasonal variability in tropical mean temperature and precipitation and their relation to the tropical 40–50 day oscillation. J. Atmos. Sci, 53 , 30323045.

    • Search Google Scholar
    • Export Citation
  • Céron, J. P., and J. F. Guérémy, 1999: Validation of the space–time variability of African easterly waves simulated by the CNRM GCM. J. Climate, 12 , 28312855.

    • Search Google Scholar
    • Export Citation
  • Ghil, M., and K. Mo, 1991: Intraseasonal oscillations in the global atmosphere. Part I: Northern Hemisphere and tropics. J. Atmos. Sci, 48 , 752779.

    • Search Google Scholar
    • Export Citation
  • Haddad, Z. S., E. A. Smith, C. D. Kummerow, T. Iguchi, M. R. Farrar, S. L. Durden, M. Alves, and W. S. Olson, 1997: The TRMM “Day-1” radar/radiometer combined rain-profiling algorithm. J. Meteor. Soc. Japan, 75 , 799809.

    • Search Google Scholar
    • Export Citation
  • Hartmann, D. L., and J. R. Gross, 1988: Seasonal variability of the 40–50 day oscillation in wind and rainfall in the tropics. J. Atmos. Sci, 45 , 26802702.

    • Search Google Scholar
    • Export Citation
  • Hayashi, Y., 1971: A generalized method of resolving disturbances into progressive and retrogressive waves by space Fourier and time cross-spectral analysis. J. Meteor. Soc. Japan, 49 , 125128.

    • Search Google Scholar
    • Export Citation
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  • Fig. 1.

    The antisymmetric OLR power divided by the background power. Contour interval is 0.1, and shading begins at a value of 1.1 for which the spectral signatures are statistically significant above the background at the 95% level (based on 500 degrees of freedom). Superimposed are the dispersion curves of the even meridional mode-numbered equatorial waves for the three equivalent depths of h = 12, 25, and 50 m. (b) Same as in (a) except for the symmetric component of OLR and the corresponding odd meridional mode-numbered equatorial waves. Frequency spectral bandwidth is 1/96 cycles per day. (Reprinted from WK)

  • Fig. 2.

    Time–longitude cross section of (a) OLR and (b) precipitation for the daily 0.5° × 0.5° latitude–longitude grid (TRMM products 3G01 and 3G68). Each value in the diagram is averaged over 0.5° of longitude and a latitude range of 15°S–15°N for a particular day. The period of the data is from 1 Jan 1998 to 31 Dec 1999

  • Fig. 3.

    Time–latitude cross section of precipitation for the daily 0.5° × 0.5° latitude–longitude grid (TRMM product 3G68). Each value in the diagram is averaged over 0.5° of latitude and a longitude range of (a) 60°–150°E over the Indian and western Pacific Ocean, and (b) 180°–90°W over the central Pacific Ocean. The data period is same as in Fig. 2

  • Fig. 4.

    Zonal wavenumber–frequency power spectrum of the OLR (a) antisymmetric components, and (b) symmetric components, calculated for the period from 1 Jan 1998 to 31 Dec 2001 using the 96-day segments. (c) As in (a) but from the precipitation data. (d) As in (b) but from the precipitation data. (e) Zonal wavenumber–frequency spectrum of the “background” for the OLR. (f) As in (e) but for the precipitation data. The power has been summed over the latitude range 15°S–15°N and the magnitude of its base-10 logarithm is shown.

  • Fig. 5.

    (a) The antisymmetric OLR spectrum shown in Fig. 4a and (b) the symmetric spectrum shown in Fig. 4b divided by the background power of Fig. 4e to find the spectral peaks for the statistically significant waves. (c) The antisymmetric rainfall spectrum shown in Fig. 4c and (d) the symmetric spectrum shown in Fig. 4d divided by the background power of Fig. 4f. Superimposed are the dispersion curves of the even and odd meridional mode-numbered equatorial waves for the three equivalent depths of h = 12, 25, and 50 m

  • Fig. 6.

    The coherence between (a) the antisymmetric components and between (b) the symmetric components of the OLR and precipitation

  • Fig. 7.

    (a) As in Fig. 5a but the power is calculated only from Apr to Nov. (b) As in (a) but for the power of symmetric components shown in Fig. 5b. (c) As in (a) but the power is calculated only from May to Oct. (d) As in (a) but for the power of symmetric components only from May to Oct. Note that the background spectrum used is same as in Fig. 5

  • Fig. 8.

    (a) As in Fig. 5c but the power is calculated only from Apr to Nov. (b) As in (a) but for the power of symmetric components shown in Fig. 5d. (c) As in (a) but the power is calculated only from May to Oct. (d) As in (a) but for the power of symmetric components only from May to Oct

  • Fig. 9.

    (a) Zonal wavenumber–frequency spectrum for the standing parts in the antisymmetric OLR components. (b) As in (a) but for the symmetric components. (c) As in (a) but for the traveling parts. (d) As in (a) but for the traveling parts in the symmetric components

  • Fig. 10.

    As in Fig. 9 but for the precipitation data

  • Fig. A1. Time–longitude cross section of the synthetic data with a 48-day eastward-propagating wave and 1-day westward-propagating wave. Both waves have wavenumber-1 structure. The synthetic data are sampled every (a) 3, (b) 12, (such as a sun-synchronous satellite), and (c) 24 h. (d) The daily average values from the synthetic data sampled such as the TRMM satellite at the equator. (e) As in (d) but at 15°N. (f ) As in (d), but at 30°N. Panel (a) is given as a reference of the correct sampling

  • Fig. A2. (a) Wavenumber–frequency spectra of the synthetic data sampled every 6 h. Synthetic data have a 48-day eastward-propagating wave and westward diurnal wave, and both waves are wavenumber 1. (b) As in (a) but the synthetic data have another 24-day westward-propagating wavenumber-1 wave. (c) As in (a) but sampled every 12 h. (d) As in (b) but sampled every 12 h. (e) As in (a) but sampled every 24 h. (f ) As in (b) but sampled every 24 h. Labels indicate the relative power divided by the true signal in (a)

  • Fig. A3. (a) As in Fig. A2a but sampled using the TRMM satellite at the equator. (b) As in Fig. A2b, but sampled such as the TRMM satellite at the equator. (c) As in (a) but at 15°N. (d) As in (b) but at 15°N. (e) As in (a) but at 30°N. (f ) As in (b) but at 30°N

  • Fig. A4. (a) As in Fig. A3b, but the synthetic data are sampled at the equator using the combined TRMM and 24-h regular sampling. (b) As in (a) but the synthetic data are sampled using the combined TRMM and 12-h regular sampling. (c) As in (a) but at 15°N. (d) As in (b) but at 15°N. (e) As in (a) but at 30°N. (f ) As in (b) but at 30°N