1. Introduction
Mixed Rossby–gravity waves (MRGs) were the first equatorially trapped waves predicted by theory (Rattray 1965; Rosenthal 1965; Blandford 1965; Rattray and Charnell 1966; Matsuno 1966) and also the first to be discovered in the atmosphere through an analysis of their equatorial meridional wind signals (Yanai 1963; Yanai and Maruyama 1966; Maruyama 1967, 1968; Maruyama and Yanai 1967). Following these initial studies, it was quickly established that MRGs were present from the surface up to at least 30 km in the stratosphere (Yanai et al. 1968; Yanai and Hayashi 1969; Yanai and Murakami 1970a,b; Nitta 1970). Although these disturbances were first studied as free (i.e., dry) modes, the latter papers included speculation that tropospheric MRGs might also be coupled to convection. This was confirmed as satellite irradiance data became more available (Chang 1970; Wallace 1971; Wallace and Chang 1972; Zangvil 1975). A historical review of these early MRG studies is provided in Takayabu et al. (2016), with summaries also given by Liebmann and Hendon (1990), Hendon and Liebmann (1991), Dunkerton (1993), and Dunkerton and Baldwin (1995).
Extensive work on MRGs since the 1970s has documented their role in modulating convection over the Pacific (Zangvil and Yanai 1980, 1981; Yanai and Lu 1983; Liebmann and Hendon 1990; Hendon and Liebmann 1991; Magaña and Yanai 1995; Dunkerton 1993; Dunkerton and Baldwin 1995; Yang et al. 2003, 2007a,b; Yokoyama and Takayabu 2012). There is ample evidence that MRGs can transform into easterly [also known as “tropical depression (TD) type”] waves over the western Pacific (Takayabu and Nitta 1993), some of which develop into tropical storms (Dickinson and Molinari 2002; Aiyyer and Molinari 2003; Frank and Roundy 2006; Molinari et al. 2007; Chen and Huang 2009; Schreck et al. 2012). The role of vertically propagating MRG energy in forcing the quasi-biennial oscillation (QBO) is also well established (Lindzen and Holton 1968; Dunkerton 1997; Baldwin et al. 2001; Alexander et al. 2008b). Although they are fundamental components of tropical synoptic-scale variability, especially within the intertropical convergence zone (ITCZ) and the South Pacific convergence zone (SPCZ) in the west-central Pacific, we aim to show in this and a companion study (Dias and Kiladis 2016, hereafter Part II) that many aspects of MRGs have yet to be fully explored in both the troposphere and stratosphere.
The signature of convectively coupled MRGs is well defined in space–time spectra of satellite brightness temperature Tb and outgoing longwave radiation (Takayabu 1994; Wheeler and Kiladis 1999; Kiladis et al. 2009). The antisymmetric spectrum of tropical Tb about the equator is characterized by a continuum of power above the background with its peak closely following the dispersion curve of the theoretical n = 0 meridional mode from Matsuno’s shallow water theory at an equivalent depth of around 25 m (Fig. 1a). Unlike other observed convectively coupled equatorial waves (CCEWs), whose spectral signatures are well separated into eastward and westward modes, there is a continuum of n = 0 power standing above the background that peaks near the lowest wavenumbers. Following Matsuno, the eastward-propagating signal is generally referred to as the n = 0 eastward inertio-gravity wave (EIG), whose dispersion relation asymptotically approaches that of a pure gravity wave at higher wavenumbers, with the westward branch called MRGs. Note that it is important to distinguish these modes from the broad spectrum of “internal” gravity waves that are not equatorially trapped (e.g., Bergman and Salby 1994; Dunkerton 1997; Baldwin et al. 2001; Alexander et al. 2008a; Kawatani et al. 2010), often referred to as simply “inertia–gravity” waves [see review by Fritts and Alexander (2003)].
Wavenumber–frequency power spectrum of the (left) antisymmetric component of CLAUS Tb for July 1983–June 2009, summed from 15°S to 15°N, plotted as the ratio between raw Tb power and the power in a smoothed red-noise background spectrum [see Wheeler and Kiladis (1999) for details]. Contour interval is 0.1, and contours and shading begin at 1.1. Dispersion curves for the n = 2 westward inertio-gravity waves (WIG), n = 0 EIGs, and MRGs are plotted for equivalent depths of 8, 12, 25, 50, and 90 m. (right) As in (left), but for the antisymmetric component of the zonal wind at 50 hPa summed from 15°S to 15°N, with equivalent depth curves of 25, 50, 120, 300, and 2000 m, Doppler shifted for a zonal wind of +3 m s−1.
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
Signals of the free-stratospheric MRGs originally observed by Yanai and Maruyama are also readily isolated by space–time spectral analysis (Ern et al. 2008; Lott et al. 2009; Alexander et al. 2008b; Alexander and Ortland 2010). For example, Fig. 1b is similar to Fig. 1a except it shows the spectrum of antisymmetric zonal wind at 50 hPa between 15°S and 15°N from ERA-Interim (ERAI) data. The spectral peak of these waves is centered on a much larger equivalent depth than for the Tb data, at around 120 m. Because of the impact of Doppler shifting by the zonal wind, the best fit to the dispersion curve is found by an adjustment to a westerly zonal wind of 3 m s−1—an indication of the fact that MRG activity at this level is favored during the westerly phase of the QBO (Maruyama 1991; Dunkerton 1991; Alexander et al. 2008b; Alexander and Ortland 2010; Yang et al. 2011, 2012).
In contrast to decades of study and hundreds of published papers on dry and convectively coupled MRGs, the literature on n = 0 EIG waves is quite sparse. Free n = 0 and especially n = 1 EIG waves have been reported in the stratosphere and mesosphere (Mayr et al. 2003, 2004; Tindall et al. 2006a,b). Wheeler et al. (2000) and Kiladis et al. (2009) isolated the structure of n = 0 EIG waves and showed that their dynamical fields match their expected structures to some extent, although these differ to a greater degree from their theoretical structures than other CCEWs. Yang et al. (2003, 2007a) used a similar regression approach to isolate the structures of CCEWs, except that they fit the meridional wind structures to their expected scales using the parabolic cylinder functions associated with Matsuno’s solutions as a basis [see also Gehne and Kleeman (2012)]. The n = 0 EIG structures in the Yang et al. (2007a) study (called EMRG waves in their Fig. 10a) do correspond well with that expected from theory; however, this is partly expected owing to the projection method used, which constrains the meridional wind structure to be symmetric about the equator and of a specified latitudinal scale owing to an assumed equatorial Rossby radius of 6°.
Despite strong spectral evidence of convectively coupled n = 0 EIG waves (herein “EIGs”), as opposed to the many studies of other CCEWs, apart from a case during July 1992 mentioned by Yang et al. (2003), to the authors’ knowledge there are no observational case studies of these modes in the literature. Based on the evidence presented below and in Part II, we speculate that this may be due to the fact that EIGs often occur with accompanying MRG or other CCEW equatorial wave activity, and that while the spectrum in Fig. 1a in part represents propagating disturbances, there is also a tendency for nearly standing antisymmetric cloudiness about the equator associated with the lower-wavenumber spectral signals. In this paper we will discuss observed evidence for such propagating and standing components corresponding to MRGs and EIGs and confirm that while their stratospheric counterparts exist globally, convectively coupled MRG–EIG modes strongly favor the Pacific sector—the only region where a double ITCZ exists in the mean state (Fig. 2). Section 2 discusses the structure and propagation of n = 0 modes, section 3 outlines the data and methodology, and sections 4 and 5 present results of EOF and cross-spectral analyses. Variability of the modes is discussed in section 6, their association with extratropical forcing in section 7, and a summary and discussion is presented in section 8.
(a) Annual-mean CLAUS Tb for the period July 1983–June 2009. Shading interval is 5 K. Standard deviation of four-times-daily (b) 2–96- and (c) 2–6-day-filtered CLAUS Tb for the entire 1983–2009 record. Shading interval is 2 K.
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
2. Structure of n = 0 MRG and EIG modes
Theoretical structure of the n = 0 MRG and EIG modes. (top) The geopotential (contours; dashed negative), divergence (shading; blue negative), and vector winds are shown for the MRG and EIG, respectively, with (bottom) the streamfunction displayed as contours. The divergence has been scaled to the same dimensionless value in all panels, with the same arbitrary geopotential contour interval used in (top). The streamfunction interval in (bottom right) is one-half that used in (bottom left).
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
In Fig. 3 it is obvious that MRGs are accompanied by a much stronger total circulation field than EIGs for a given divergence field. This difference will vary according to zonal wavenumber and equivalent depth for dimensional waves (Part II). MRGs are dominated by rotational flow with gyres centered on the equator (Fig. 3, left) while EIGs are primarily divergent with weak off-equatorial streamfunction signals (Fig. 3, bottom right). Both modes have cross-equatorial flow into regions of convergence, but in the case of EIGs the convergence is supplemented by zonal divergent flow, while in MRGs the zonal flow opposes the meridional convergence (Part II). The zonal wind, pressure, and divergence fields peak at Re = 1 in both waves.
Looking back at Fig. 1a, the equivalent depth of the dispersion curve that best matches the n = 0 Tb spectral signal is around 25 m, similar to most of the other CCEWs identified using satellite data (Takayabu 1994; Wheeler and Kiladis 1999; Kiladis et al. 2009). In that case Re would be around 7.5°, a value perhaps not coincidentally near the mean latitude of the ITCZ in the Northern Hemisphere (e.g., Holton et al. 1971; Chang 1973). Likewise, in Fig. 1b the implied 120-m equivalent depth yields an equatorial Rossby radius of around 11°, so we expect that the stratospheric MRG–EIG modes will have a larger meridional scale than those coupled to convection in the troposphere, as confirmed below.
3. Data and methodology
a. Datasets
We use the Cloud Archive User System (CLAUS) Tb data (Hodges et al. 2000), which has eight-times-daily global fields of Tb from July 1983 through June 2009, along with twice-daily interpolated outgoing longwave radiation (OLR) data (Liebmann and Smith 1996) and the TRMM 3B42 precipitation dataset (Huffman et al. 2007). MRG–EIG modes have relatively fast propagation speeds; therefore, the statistics for most of this study are obtained using the period that overlaps with the higher-temporal-resolution CLAUS data. OLR has also been used to extend these results and document wave activity back to 1974. The annual mean value of CLAUS Tb between 20°N and 20°S for the entire record is shown in Fig. 2a, which shows the familiar convective zones centered near the equator over land, the ITCZ between 5° and 10°N over the Pacific and Atlantic, and the SPCZ extending southeastward from the region of New Guinea.
For dynamical fields the ERAI dataset is used. Since the structures we are concerned with are relatively large scale, for computational convenience the ERAI, CLAUS Tb, and TRMM data have been interpolated to a regular 2.5° grid for our calculations, matching the spatial resolution of the OLR dataset. Virtually identical results have been obtained in tests using the higher-spatial-resolution products.
b. Filtering and cross spectra
Cross-spectral approaches have been used extensively as an objective means to document the existence, structure, and scale of equatorial waves (Yanai et al. 1968; Wallace and Chang 1969; Wallace 1971; Madden and Julian 1972). To obtain the cross spectra shown in this study, the data were first detrended and the first three harmonics of the seasonal cycle were removed, with tapering applied to 5% of the points at each end of the series. Spectral estimates were smoothed with a 301-point running average in frequency, and statistical significance was determined by a comparison of the coherence value distributions with those obtained through Monte Carlo simulations of cross spectra by generating a background obtained from running one time series backward, preserving its autocorrelation, calculating its coherence, then adding the 1000 estimates of the coherence generated by random-Gaussian-noise time series. Several parametric methods were also tested (e.g., Blackman and Tukey 1958), all yielding similar results. In the cross-spectra plots below, all of the regions of shaded coherence are significant at better than the 99% level.
Various methodologies have been used in the past to obtain the statistical horizontal and vertical structure of MRGs–EIGs from gridded analyses, typically using filtered data as a basis to isolate scales of interest. Here we utilize filters based on the Fourier space–time decomposition described in Wheeler and Kiladis (1999), where we include all wavenumbers for temporal filtering. We also utilize westward- or eastward-only filters that retain either negative or positive wavenumbers, respectively, excluding wavenumber 0. For comparison purposes we display the four-times-daily standard deviation of 2–96-day-filtered Tb in Fig. 2b. This filter effectively removes the diurnal and seasonal cycles, along with interannual variability, and shows that the regions of largest intraseasonal and synoptic-scale variability generally coincide with regions of deepest mean convection implied in Fig. 2a.
As seen in Fig. 1, most of the power above the background of the tropospheric and stratospheric n = 0 modes lies within 2–6 days and this period brackets the filter bands we use for the EOF analyses in this study. Figure 2c shows that the 2–6-day-filtered standard deviation in fact accounts for the bulk of the subseasonal wave activity over many regions and for more than 70% of the signal seen in Fig. 2b over the Pacific. In particular, such synoptic-scale activity accounts for most of the variability within the ITCZ and SPCZ, far exceeding that of the MJO, for instance. Also included within the 2–6-day band are easterly waves or TD-type disturbances (Takayabu and Nitta 1993; Wheeler and Kiladis 1999) but it will be shown that these are well separated from MRG–EIG disturbances by the EOF approach used here.
c. EOF approach
As pointed out by Dunkerton and Baldwin (1995), the use of point correlations as an analysis tool is compromised when representing the statistical structures of waves from gridded analyses because of the relatively small “coherence length” of fluctuations when all scales of motion are included. When linear dynamics is dominant, EOFs can be a useful alternative if used judiciously and with an eye toward the underlying dynamics (e.g., Monahan et al. 2009).
EOFs are calculated using the covariance matrix of filtered Tb within 20°S–20°N for both global domains and domains localized in longitude. The dynamical fields associated with each EOF are obtained by projecting unfiltered ERAI data at each grid point onto the associated principal component (PC) time series. In the following analysis, streamfunction and statistically significant Tb fields will be shown, with associated wind perturbations displayed only if either the zonal or meridional wind is significant at the 95% level, after taking into account the autocorrelation in each field as described by Livezey and Chen (1983).
4. EOF results
We first demonstrate the ability of spatial EOFs to isolate the stratospheric MRG modes first detected by Yanai and Maruyama (1966). Figure 4 shows the first four EOFs of 2–6-day meridional wind at 50 hPa for the entire tropical belt between 20°S and 20°N. As with the rest of the plots in this paper, these patterns have been scaled to a +2 standard deviation perturbation in their respective PCs—a typical amplitude for strong events. The EOFs isolate gyres centered on the equator having the structure of MRG waves in Fig. 1. The first two EOFs form a quadrature pair and are correlated at 0.76 with PC1 leading, which from Figs. 4a and 4b implies westward propagation. Similarly, the third- and fourth-EOF PCs are correlated at 0.67, with the fourth leading, also implying westward-propagating MRGs. Together the first two EOFs explain nearly 15% and the third and fourth explain 13% of the filtered variance. Results using the 2–6-day vorticity or the antisymmetric components of the zonal wind, geopotential, or temperature are quite similar (not shown).
The 50-hPa streamfunction and wind regressed onto four-times-daily 1979–2012 PCs corresponding to (a) EOF1, (b) EOF2, (c) EOF3, and (d) EOF 4 of the 2–6-day-filtered 50-hPa meridional wind between 20°S and 20°N. All fields are scaled to a +2 standard deviation PC perturbation. Streamfunction contour interval is 4 × 105 m2 s−1, with solid contours and yellow shading corresponding to positive perturbations. Vector winds are shown if either the zonal or meridional wind is statistically significant at the 95% level. The largest wind vectors correspond to speeds of around 3 m s−1.
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
The EOF pairs in Figs. 4a–d represent two distinct scales of stratospheric MRGs. The gyres in Figs. 4a and 4b are zonal wavenumber-4 disturbances that propagate westward at around 30 m s−1, giving a mean period of around 4 days, although this varies depending on season and the phase of the QBO. In Figs. 4a and 4b, this leading pair is localized to the sector extending from the Americas, across the Atlantic into Africa. The next EOF pair represents a zonal wavenumber-3 pattern with its loading more zonally distributed but also with a minimum over the Pacific. These waves propagate at a much faster phase speed of around 45 m s−1, yielding a period close to 3.5 days. Higher-mode pairs out to at least EOFs 7 and 8 account for MRG activity over other sectors. One point of critical relevance to our arguments below is that westward-propagating MRGs can be isolated in the stratosphere by EOF analysis of 2–6-day-filtered data, without having to filter explicitly for westward propagation.
Although stratospheric waves are not the primary focus of this study, some points about these modes are relevant here. While there are statistically significant correlations between the circulations shown in Fig. 4 and Tb, their amplitudes are small (<2°K). This agrees with results of Hendon and Wheeler (2008), who also showed weak but significant coherence between antisymmetric zonal winds at 50 hPa and OLR. Interestingly, despite the weak convective coupling, circulations that are significantly correlated with these EOFs do extend down through the troposphere to the surface, although these are also relatively small amplitude (not shown). Consistent with past results such as Maruyama (1991), Dunkerton (1991), Alexander et al. (2008b), and Yang et al. (2011, 2012), there is a large modulation of the activity of these waves by the QBO—a topic that will be pursued in a future study.
Next, we adopt the same procedure to isolate the structure of convectively coupled MRG–EIG modes, except that 2–6-day-filtered brightness temperature 
Figure 5 shows the first two EOFs (shading) and also the associated 850-hPa circulation pattern obtained by regression onto the PCs. Individual seasonal EOFs using the same domain are nearly identical to these annual results and with the corresponding EOFs of year-round global data for 20°S–20°N as well as to EOFs calculated from combined MRG–EIG filtered data using the filtering boxes shown in Fig. 1a, with correlations between respective PCs in excess of 0.75 in all cases. The seasonal dependence of the modes will be discussed more fully below and in Part II.
As in Fig. 4, but for (a) EOF1 and (b) EOF2 of 
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
Earlier investigations such as Hendon and Liebmann (1991) and Dunkerton (1993) used antisymmetric OLR to isolate MRGs, and although we did not use this decomposition as a basis in our EOF calculations, the two leading modes still show an out-of-phase fluctuation in Tb between the ITCZ and SPCZ in Fig. 5 [the overwhelming tendency for such out-of-phase relationships in Tb at these frequencies in the central Pacific was first shown by Wallace (1971)]. MRG-like circulations accompany these convective signals, with negative Tb or enhanced convection consistent with the theoretical patterns of low-level convergence in Fig. 3a. EOF1 and EOF2 explain 3.4% and 3.0% of the variance, respectively, which is marginal in terms of their separation as independent modes by the criteria of North et al. (1982). These EOFs do represent a propagating pair but only at a correlation of 0.19 with the first mode leading (Part II). While this would result in a westward-propagating cloud field, the zonal scales of the Tb signals also differ significantly. Thus, we will initially analyze these EOFs separately.
In Fig. 6 we lag regress the total Tb, 850-hPa streamfunction and wind on the PC1 time series of EOF1 in Fig. 5a. A striking feature of the behavior of Tb in Fig. 6 is that its propagation is primarily in the meridional, rather than zonal, direction. In contrast, MRG-like circulations in the 850-hPa streamfunction fields are propagating westward through the region of zonally standing convective signals. The zonally standing versus propagating nature of these signals is brought out by Fig. 7, which shows a time–longitude plot of Tb at 7.5°N along with the rotational component of the meridional wind on the equator for 25 days surrounding the peak in PC1. Fluctuations in the rotational meridional wind propagate westward at 25 m s−1—a nearly identical phase speed as the leading-mode stratospheric MRGs described above and the tropospheric MRGs reviewed in section 1. If instead the divergent meridional wind is plotted, this shows a dominant zonally standing component, and other fields related to the mass circulation such as the horizontal divergence and vertical motion also display similar behavior consistent with Tb in Fig. 6 (not shown). Also tested was whether the Tb signals truly represent convection rather than simply high-cloudiness perturbations that might be advected by the upper-tropospheric flow. This was verified in two ways: by calculating EOFs using four-times-daily TRMM 3B42 precipitation data and also by projecting the TRMM data onto the PCs obtained above from Tb data for the overlapping period (1998–2009). These results retain all of the same antisymmetric, zonally standing, but poleward propagating characteristics as the Tb data show (not shown).
As in Fig. 5, but for 
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
Time–longitude diagram of Tb at 7.5°N and the rotational component of the meridional wind on the equator regressed onto the four-times-daily 1983–2009 PC of 
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
Standing antisymmetric signals of cloudiness between the ITCZ and SPCZ on the time scale of n = 0 modes have been reported in the literature, but this observation has not received any recent attention. In his comprehensive spectral study of radiosonde and Tb data over the Pacific, Wallace (1971) remarked on the fact that “…the n = 0 mode is not evident in the brightness data as a zonally propagating wave. It may be more appropriate to view it as a standing oscillation confined to the central Pacific.” Following up on this observation, Holton (1972) showed that westward-propagating MRG circulations with upward and eastward group velocity could be produced in a linearized primitive equation model forced by standing antisymmetric heating with a period of 5 days. Hendon and Liebmann (1991) speculated that standing 5-day convective patterns might have been peculiar to the single season that Wallace analyzed, motivated by the fact that they isolated a strong westward-propagating signal using bandpass-filtered antisymmetric OLR for 3.5–6.5-day periods during October–November. Takayabu and Nitta (1993) also produced a zonally standing Tb pattern by lag correlating 2.5–10-day-filtered Tb with itself at the base point 5.5°N, 179.5°E. In fact, their Tb pattern does also display distinct poleward propagation in both hemispheres (their Fig. 5a), although this point was not emphasized in their study.
We can reproduce the westward-propagating point correlation result of Hendon and Liebmann (1991) as in their time–longitude diagrams using identically filtered OLR or Tb, but when 2–6- or 2–10-day-filtered data are used (either symmetric or antisymmetric), the signal is dominated by a zonally standing pattern analogous to that in Fig. 7. The difference between these results holds during all times of the year, so it is not just a seasonal dependence. As shown by Part II, one source of this discrepancy lies in the fact that the 3.5-day cutoff used by Hendon and Liebmann (1991) excludes the bulk of EIG power seen in Fig. 1a. Even point correlations derived from 3.5–6.5-day antisymmetric OLR or Tb data still show much stronger meridional than zonal propagation as in Fig. 6 when mapped at lag, while the associated circulations still propagate westward (not shown). Hints of this meridional propagation are actually seen in plots of Liebmann and Hendon (1990) and Hendon and Liebmann (1991) (e.g., their Figs. 11 and 8, respectively); however, a focus on time–longitude diagrams at one latitude in those studies and in Dunkerton and Baldwin (1995) seems to have precluded a more thorough documentation of this tendency.
The circulations in Fig. 6 are somewhat distorted, more zonally elongated versions of the MRG structures generally pictured in the literature (e.g., Liebmann and Hendon 1990; Hendon and Liebmann 1991; Dunkerton and Baldwin 1995; Wheeler et al. 2000; Kiladis et al. 2009), so a legitimate question arises as to whether these structures are a statistical artifact of the EOF approach. This was tested by performing the same EOF analysis on westward-propagating (excluding wave 0) 2–6-day-filtered brightness temperature 
As in Fig. 5, but for (a) EOF1 and (b) EOF3 of 
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
The first pair, as represented by EOF1 (Fig. 8a), recovers the structures discussed in previous tropospheric MRG studies, including the poleward and eastward tilts of the Tb signals that are only weakly present in the zonally standing disturbances of Fig. 5. A similar but less exaggerated anisotropy in the streamfunction contours was also seen by Liebmann and Hendon (1990), Takayabu and Nitta (1993) Dunkerton and Baldwin (1995), Kiladis et al. (2009), and Roundy and Janiga (2012). This EOF displays the westward propagation of the Tb signal in lockstep with their associated circulations at around 22 m s−1, without any evidence of meridional propagation (not shown). The second pair, as shown by EOF3 (Fig. 8b) represents the transition of MRG to TD disturbances that frequently occurs over the western Pacific (Liebmann and Hendon 1990; Takayabu and Nitta 1993; Numaguti 1995; Dickinson and Molinari 2002; Straub and Kiladis 2003b; Kiladis et al. 2009). This transition is considered to be due in part to deformation and the accumulation of wave energy brought on by the convergence of the basic-state zonal westerly flow over the region (Sobel and Bretherton 1999; Kuo et al. 2001; Aiyyer and Molinari 2003; Chen and Huang 2009).
The fact that zonally standing signals constitute the leading modes in the Tb fields in Figs. 6 and 7 means that these signals are indeed dominant, since if westward propagation was preferred, these would have shown up as the leading EOF modes of 2–6-day-filtered data as in the case of the stratospheric EOFs of Fig. 4. This point will also be demonstrated using synthetic data by Part II. While the modes represented by Fig. 8 are not completely independent of those shown in Fig. 5a, with correlations running in the 0.58 range between the respective PC pairs, we demonstrate below and in Part II that while they often do occur independently, the zonally standing EOF results in Figs. 6 and 7 are dominant.
5. Cross-spectral results
Cross-spectral analysis between various dynamical fields and OLR has been particularly effective at isolating the structure of convectively coupled MRGs (e.g., Yanai et al. 1968; Wallace 1971; Chang and Miller 1977; Zangvil and Yanai 1980, 1981; Liebmann and Hendon 1990; Hendon and Liebmann 1991; Dunkerton 1993). Based on analyses of global data, the latter three studies argued that convectively coupled MRG activity is primarily confined to the central Pacific near the date line, with dry MRG circulations still prevalent over the east Pacific and Atlantic. More recent work using spatially windowed space–time spectra also indicates that convectively coupled MRG–EIG activity peaks over the Pacific (Dias and Kiladis 2014), although completely localizing their signals is not possible using their methodology. Maps of MRG–EIG Tb variance also show maxima over the Pacific with some variability extending into other ocean basins (Wheeler and Kiladis 1999; Roundy and Frank 2004; Kiladis et al. 2009), but in those cases it is not clear how much of the signal is truly indicative of the filtered mode versus the potential for misrepresentation by the inclusion of “background” variability. In this section we employ cross-spectral approaches using dynamical fields, along with Tb, to better nail down the geographic extent of the signals due to MRG–EIG modes in the troposphere and stratosphere.
Figure 9a shows the raw spectral power of the meridional wind on the equator at 50 hPa plotted at each longitude for frequencies between 0.011 and 0.5 cycles per day (i.e., periods from 90 to 2 days). This spectrum is quite remarkable in that, even though we have only removed a trend without the “prefiltering” that has often been used in early studies (e.g., Yanai et al. 1968), it shows a very pronounced peak at 3–5-day periods present at all longitudes, without any sign of reddening at lower frequencies typically seen in most spectra of observed parameters in the atmosphere. Past spectral studies (Yanai et al. 1968; Yanai and Hayashi 1969; Dunkerton 1993) as well as the EOF results of Fig. 4 leave little doubt that this signal is dominated by MRG activity. The equivalent plot for 850-hPa meridional wind in Fig. 9b shows two strong spectral peaks representing activity between 3 and 4 days, but these are only evident over the central and east Pacific and Atlantic, giving way to a more typical red spectrum at other longitudes. The implications of this spectral power distribution will be discussed further below.
Spectral power by frequency and longitude of ERAI equatorial meridional wind at (a) 50 and (b) 850 hPa. Cross-spectral (c) coherence squared and (d) phase between 50-hPa zonal wind at 10°N and 10°S. Cross-spectral (e) coherence squared and (f) phase between westward-only-filtered 50-hPa zonal wind at 10°N and 10°S. Shading in (d) and (f) represents phase, with white showing in-phase and gray showing out-of-phase coherence and reddish shading denoting 10°N leading by the phase angle on the scale. The entire record from 1979 through 2012 was used to calculate all of the plots.
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
In Figs. 9c and 9d, the squared coherence and phase between 50-hPa zonal wind at 10°N and 10°S is plotted. Tests show that using a latitude of 10° maximizes the coherence, consistent with the 11° theoretical equatorial Rossby radius for a 120-m equivalent depth discussed in section 2. There are three main bands of statistically significant coherence in Fig. 9c, one of which is in phase at around 0.2 cycles per day centered over the Indian Ocean and western Pacific. This peak can be associated with “fast” stratospheric Kelvin waves (Ern et al. 2008; Alexander et al. 2008b; Alexander and Ortland 2010; Yang et al. 2011; Maury and Lott 2014), the structure of which is readily isolated by applying the EOF technique described above to appropriately filtered data (not shown). The other two peaks in Fig. 9c are out of phase at between 0.3 and 0.4 cycles per day and at less than 0.05 cycles per day, both of which extend across all longitudes. Regions of high in-phase coherence would be expected for any large-scale propagating features with a symmetric structure such as Kelvin waves. Out-of-phase relationships, however, are likely generated by antisymmetric wave structures.
To illustrate this point further, cross-spectral results for westward-propagating 50-hPa zonal wind between 10°N and 10°S are shown in Figs. 9e and 9f. The westward coherence squared in Fig. 9e is extremely high and out of phase, exceeding 0.65 globally and centered on waves with a period of 3 days (~0.33 cycles per day), with minimum values over the central Pacific, consistent with the leading MRG EOF patterns shown in Fig. 4. The out-of-phase westward signals centered at 0.12 cycles per day and the lowest-frequency peak at the bottom of Figs. 9e and 9c are likely due to the free external zonal wavenumber-1 and -2 Rossby modes with periods of around 7 and 28 days, respectively, which have antisymmetric zonal wind structures [e.g., the 2–2 and 1–4 modes from Madden (2007)], the spectral peaks of which can also be seen in Fig. 1b. As expected the 0.2 cycles per day “fast” Kelvin peak in Figs. 9c and 9d is absent in Figs. 9e and 9f.
We now apply the same procedure to Tb in Fig. 10. For these spectra the maximum coherence is between 7.5° on either side of the equator, in agreement with the smaller equivalent depth and equatorial Rossby radius of convectively coupled modes discussed in section 2. Unfiltered Tb coherence squared and its phase are shown in Figs. 10a and 10b. In this case coherence is much lower than for zonal wind in the stratosphere in Figs. 9c and 9e (note change in scale), with the most prominent signal out of phase and spanning a broad frequency range corresponding to periods of between 3 and 5 days. The maximum covariance lies within a 60° region centered just west of the date line and, so, likely represents at least in part the antisymmetric fluctuations represented by the 
As in Fig. 9, but for coherence squared and phase between (a),(b) CLAUS Tb at 7.5°N and 7.5°S, (c),(d) westward-only-filtered CLAUS Tb at 7.5°N and 7.5°S, and (e),(f) eastward-only-filtered CLAUS Tb at 7.5°N and 7.5°S. The entire record from 1983 through 2009 was used to calculate all of the plots.
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
To test whether convectively coupled MRG–EIG convective signals could be present in one hemisphere only, we examined cross spectra between off-equatorial Tb and wind at all 37 available levels in ERAI from the surface to 1 hPa. The most significant results were obtained using 850-hPa meridional wind at the equator and Tb at 7.5°N, as shown in Figs. 11a and 11b. The strongest signals show out-of-phase relationships, as would be expected from Fig. 3 and also Figs. 5 and 8, where equatorial southerlies at 850 hPa are accompanied by negative Tb anomalies in the ITCZ and positive anomalies in the SPCZ. Highest coherence is again seen over the central Pacific for 3–5-day periods, with secondary peaks along the coast of South America (near 90°W) and the Atlantic. Similar results were obtained using Tb at 7.5°S (not shown). Consistent with analyses by Liebmann and Hendon (1990), Hendon and Liebmann (1991), Numaguti (1995), and Dunkerton and Baldwin (1995), coherence between Tb and circulation drops off rapidly above 500 hPa, although there is some evidence for convective coupling with lower-frequency equatorial eddies in the mid- to upper troposphere over a range of longitudes in the Indian sector (not shown).
As in Fig. 9, but for coherence squared and phase between (a),(b) 850-hPa meridional wind at the equator and CLAUS Tb at 7.5°N, (c),(d) westward-only 850-hPa meridional wind at the equator and zonal wind at 7.5°N, and (e),(f) eastward-only 850-hPa meridional wind at the equator and zonal wind at 7.5°N.
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
We also attempted to isolate MRG–EIG circulations by calculating the coherence between equatorial meridional and off-equatorial zonal wind. For example, in Figs. 11c and 11d we show the coherence and phase between 850-hPa westward-propagating meridional wind on the equator and westward-propagating zonal wind at 7.5°N. At nearly all longitudes and frequencies, the phase relationship is such that the meridional wind leads the zonal wind by ¼ cycle (i.e., close to +90° in Fig. 11b), indicative of westward propagating eddies centered on the equator (Fig. 3, left panels). Once again, coherence in the 3–5-day period is largest centered on the date line; however, other regions of coherence are also seen over the easternmost Pacific (90°W) and Atlantic (30°W). Figures 11c and 11d also provide evidence for lower-tropospheric equatorial eddies over the western Pacific, Indian Ocean, and Atlantic that are not visible in Figs. 11a and 11b and so are not strongly coupled to convection. In particular, MRG-like circulations have been reported over the Indian Ocean (Yasunaga et al. 2010; Muraleedharan et al. 2015; Chen et al. 2015), although these are often identified as off-equatorial eddies (e.g., Kerns and Chen 2014). Spectra similar to those in Fig. 11 for other levels reveal equatorial eddies prevalent in the mid- to upper troposphere, especially over the Indian sector, although these have a much broader range of frequencies than those over the central Pacific, extending to periods beyond 10 days (not shown).
Figures 11e and 11f are similar plots except for 850-hPa eastward-only meridional wind on the equator and eastward zonal wind at 7.5°N. As with signals associated with eastward Tb in Figs. 10e and 10f the coherence is largest near the date line and over Africa in the higher frequency range of EIGs, with a phasing of +90° that would be consistent with eastward-propagating EIG circulations in the right panels of Fig. 3. There is also some signal over the Atlantic, along with a lower-frequency quadrature peak extending down to 0.1 cycles per day over the eastern Pacific. The nature of this eastward 5–10-day peak is not clear, although we speculate that it might be related to equatorial eddies excited by extratropical wave activity propagating into the equatorial region of the east Pacific (e.g., Kiladis 1998; Yang et al. 2007b), to be discussed further below.
A similar approach was systematically applied to raw and eastward- or westward-filtered Tb, along with various combinations of wind data for all levels from the surface to the midstratosphere. This exercise included sensitivity testing using alternate latitude combinations to account for displacement of eddy circulations off the equator as was reported by Randel (1992), Dickinson and Molinari (2002), and Kerns and Chen (2014). The outcome of this objective census of equatorial wave activity will be reported in a separate study, but of relevance here, the results reveal that the westward MRG-like and eastward EIG-like circulations detected in Fig. 11 are present from the surface to around 700 hPa. Above this level the signals weaken rapidly near the date line and strengthen somewhat over the Indian sector but in the lower-frequency range as in the westward signals of Figs. 11c and 11d. As mentioned above, these eddies display some coupling to convection but this coherence is much lower than in Fig. 11a over the central Pacific. Between 150 and 100 hPa, signals with periods centered on 3 days begin to appear over the Indian sector, and these spread to all longitudes and strengthen rapidly into the stratosphere, with the strongest evidence for stratospheric MRG–EIG circulations at 50 hPa as in Figs. 9b and 9c.
In summary, synoptic-scale eddies centered on the equator are detectable at most longitudes within the troposphere, but these are most strongly coupled to convection in the 3–5-day range for lower-tropospheric circulations over the central and eastern Pacific, Atlantic, and Africa. There is also some evidence of weaker convective coupling between mid- and upper-tropospheric MRG-like eddies over the Indian sector but at a lower-frequency corresponding to periods in the 5–10-day range. Thus, the cross-spectral results described here confirm that the global n = 0 Tb spectral signal in the 2–6-day range in Fig. 1a is dominated by the activity over the central Pacific, as was also inferred by the original studies of Liebmann and Hendon (1990) and Hendon and Liebmann (1991), as well as by Dias and Kiladis (2014).
6. Seasonality and temporal variability
Substantial variability in MRG–EIG activity is seen across a wide range of time scales (Dunkerton and Baldwin 1995). As with indices of the MJO such as RMM (Wheeler and Hendon 2004) and OMI (Kiladis et al. 2014), metrics of MRG–EIG activity can be obtained from their PC time series. To extend the temporal coverage of this wave activity we have recalculated EOFs discussed above using twice-daily OLR data from 1974 to 2014, with nearly indistinguishable results for the period of overlap between the two datasets apart from a rescaling that arises from the difference between Tb and OLR.
It is well known that MRG activity is maximized during the boreal summer and fall (Liebmann and Hendon 1990; Hendon and Liebmann 1991; Dunkerton 1993; Dunkerton and Baldwin 1995; Magaña and Yanai 1995; Roundy and Frank 2004; Huang and Huang 2011; Horinouchi 2013) along with convective signals associated with the EIG portion of the spectrum (Wheeler and Kiladis 1999; Dias and Kiladis 2014). Figure 12 shows the monthly mean standard deviation (std dev) of the PCs associated with the four EOFs of 
Monthly mean standard deviation of the 2–6-day-filtered (blue, red) and 2–6-day westward-filtered (green, yellow) OLR PCs corresponding to the 
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
A time series of the monthly average std dev of the PCs associated with the first two EOFs of 
Monthly mean standard deviation of the OLR PCs corresponding to the 
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
To further investigate the nature of this interannual variability, we calculate an anomalous monthly std dev for each PC by subtracting its respective climatological monthly mean std dev for the period 1979–2014. The monthly activity of the two PCs depicted in Fig. 13 deseasonalized in this way is correlated at 0.40, which is much higher than the 0.19 correlation between the corresponding four-times-daily raw PCs obtained in section 4. We similarly calculate monthly anomalies as above of the Niño-3.4, Niño-3, and Niño-4 indices commonly used to measure the state of the El Niño–Southern Oscillation (ENSO). The correlation is negative and statistically significant, though weak, between all of the PC std dev series and the Niño-3 and Niño-3.4 indices, peaking at −0.32 and −0.30 between Niño-3 and for activity of the pair of PCs that represent the westward 2–6-day MRG patterns in Fig. 8. Interestingly, the Niño-3 region represents SST fluctuations from 5°S to 5°N and from 150° to 90°W and is the index based on SST farthest away from the center of MRG–EIG convective activity represented by the EOFs, whereas the correlations with the Niño-4 index that is more centered near the date line are effectively zero. This indicates that the activity is affected more by large-scale changes in the basic state as opposed to being driven more by local SST.
Horinouchi (2013) also found a weak but negative correlation between MRG activity and equatorial SST using a somewhat different measure based on MRG filtered OLR as well as a strong positive (negative) correlation with anomalous seasonal equatorial precipitation (OLR). We examine this aspect by correlating the monthly OLR anomalies at each grid point with each monthly PC std dev anomaly time series as was done above for the ENSO indices. In every case, monthly MRG–EIG activity was positively correlated with monthly OLR anomalies along the equator and negatively correlated within the mean position of the ITCZ throughout the entire equatorial Pacific. While this supports the hypothesis of Horinouchi (2013) that increased MRG activity should be associated with a decrease in convective activity along the equator, the correlations were small, peaking at only around 0.40 near the date line for the westward-propagating MRG mode in Fig. 8a (not shown). It turns out that this signal is also primarily driven by ENSO, which becomes evident when looking at lagged relationships, since the correlations are very persistent and evolve slowly over monthly time scales. Taken together, the results of this section do provide some evidence for the idea that MRG–EIG activity is favored where there is a minimum in equatorial SST and a strong double ITCZ (Hendon and Liebmann 1991) and that this favorable configuration is disrupted by El Niño. Investigation of the origin of other large variations in activity such as those illustrated in Fig. 13 is ongoing and will be reported in a future study.
7. Upper-tropospheric structure
While the evolution of Tb and lower-tropospheric circulations associated with each of the MRG–EIG modes differ little throughout the year, this is not the case in the upper troposphere. As an example, Fig. 14 shows the evolution of 200-hPa flow associated with EOF1 at half-day intervals from day −1 through day 0, obtained by calculating June–August-only EOFs of 
As in Fig. 5, but for 
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
The eddy pattern along 30°S in Fig. 14 represents the eastern portion of a wave train propagating into lower latitudes from the southern Indian Ocean. Figure 15 shows a global view of the 200-hPa streamfunction at day−9, where a statistically significant, continuous wave train is seen originating over the South Atlantic, which then arcs into the subtropical jet over and to the east of Australia. Such a signal was not only seen by Magaña and Yanai (1995) associated with MRG activity (their Fig. 10a), but a very similar pattern was also obtained by Straub and Kiladis (2003a) prior to the development of convectively coupled Kelvin waves in the ITCZ (their Fig. 2). Similar precursor wave trains are also obtained for the second EOF in Fig. 5b, despite its higher-wavenumber structure, and the westward-propagating MRG mode in Fig. 8a, but not for the pattern in Fig. 8b. Notwithstanding some differences in the phasing of these waves, it thus appears that at least Kelvin, MRG, and the zonally standing modes studied here can be similarly forced by incoming wave activity originating in the southern extratropics during austral winter.
As in Fig. 14, but for day−9.
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
As a contrasting example, Fig. 16 shows the evolution of 200-hPa flow for EOF1 calculated from December–February 
As in Fig. 14, but for 
Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0230.1
The sequence of Fig. 16 is typical of the equatorward propagation resulting from Rossby wave propagation into the “westerly duct” region during northern winter (Webster and Holton 1982; Kiladis and Weickmann 1992a,b, 1997; Tomas and Webster 1994). Along with studies cited above, such wave activity has been implicated in exciting a variety of equatorial modes including MRGs, equatorial Rossby waves, and Kelvin waves in observations (Kiladis and Wheeler 1995; Kiladis 1998; Yang et al. 2007b; Yang and Hoskins 2013) and in simple models (e.g., Lim and Chang 1981; Itoh and Ghil 1988; Zhang and Webster 1992; Zhang 1993; Hoskins and Yang 2000; Biello and Majda 2004; Ferguson et al. 2009). The idea that lateral forcing is important for equatorial wave initiation is also in accord with observational results of Zangvil and Yanai (1980), Yanai and Lu (1983), and Randel (1992), who found significant meridional convergence of wave activity fluxes associated with upper-tropospheric MRG activity. Note that our approach does not explicitly measure wave activity associations per se but, rather, is dependent on the specific timing and phasing of individual synoptic-scale events between the tropics and higher latitudes. It is evident that the upper-tropospheric equatorial eddies depicted in Figs. 14 and 16 are not very reminiscent of MRG modes—consistent with the spectral results discussed in section 6. Also notable is the fact that during the transition seasons, even though very similar Tb EOFs are obtained, there is little evidence of extratropical forcing (not shown). This is consistent with the notion that the extratropical wave activity identified here is only one possible mechanism for the excitation of the equatorial modes, which likely also includes local forcing by convection (e.g., Hayashi 1970; Itoh and Ghil 1988; Khouider et al. 2013) or interference with other equatorial modes (e.g., Raupp and Silva Dias 2005).
8. Summary and discussion
The connection between the westward and eastward spectral peaks of antisymmetric space–time Tb (Fig. 1a) has motivated us to address the question of whether this signal represents a continuum of wave activity or whether convectively coupled n = 0 MRGs and EIGs occur independently of each other. Using EOF analysis of Tb filtered at 2–6 days 
Nevertheless, strong evidence of EIG circulations is seen in cross spectra. Our hypothesis, discussed in further detail in Part II, is that the statistical signal isolated here results from a mixture of westward and eastward propagating, along with zonally standing events that represent predominantly MRG or EIG modes that at times constructively interfere to produce a standing 
It is well known that the ITCZ and SPCZ vary coherently over a wide range of space and time scales. On interannual time scales, they tend to vary in phase, with a tendency for both to be enhanced and to shift equatorward during El Niño conditions (e.g., Chung and Power 2015), with opposite tendencies during La Niña. This in phase variability is also observed on intraseasonal time scales associated with the MJO (e.g., Matthews and Kiladis 1999). However, at synoptic time scales we have shown that out-of-phase variability is more dominant owing to the antisymmetric nature of the divergence associated with the n = 0 Matsuno modes. The localization of the MRG–EIG convective signal to the west-central Pacific is another intriguing issue. This extent of this localization is not necessarily reflected in maps of MRG–EIG Tb variance that have been published in past literature (e.g., Wheeler and Kiladis 1999; Roundy and Frank 2004; Kiladis et al. 2009; Huang and Huang 2011), which we strongly suspect are unduly influenced by unrelated background variance in regions outside the west-central Pacific. It would be of great interest to evaluate general circulation models with respect to their abilities to simulate this localization and to assess the extent to which this is related to the presence of realistic versus erroneous “double ITCZs” in the basic states of such simulations.
Several other unresolved questions remain regarding the dynamics of MRGs–EIGs. The poleward and eastward meridional tilts of the convective signals are especially pronounced for the 
While there is strong evidence for extratropical forcing of MRG–EIG activity in this and many previous studies, the precise dynamics responsible for this presumed forcing have yet to be fully uncovered. In addition, the mechanisms responsible for the meridional propagation in 
Acknowledgments
We thank Brant Liebmann, Takeshi Horinouchi, and two anonymous reviewers for their insightful comments on an initial draft of this manuscript. ERA-Interim data was provided by NCAR.
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