## 1. Introduction

In a companion paper (Kiladis et al. 2016, hereafter K16), evidence was presented of a zonally standing mode in antisymmetric convection about the equator related to the

Observational studies based on spectral filtering show that MRG and EIG cloudiness perturbation patterns are consistent with Matsuno’s theory in that they are antisymmetric with respect to the equator and they propagate coherently with the tropospheric flow toward the west and east, respectively. The period is, on average, close to 4.5 days for convectively coupled MRGs is and is around 3 days for EIGs (Wheeler and Kiladis 1999; Wheeler et al. 2000; Kiladis et al. 2009). Their combined signals produce very strong peaks in the 3–6-day range in raw spectra (Yanai et al. 1968; Wallace and Chang 1969; K16). The results in K16 are based on a coherence analysis of circulation and Cloud Archive User System (CLAUS) tropical brightness temperature (*T*_{b}) data, along with an empirical orthogonal function (EOF) decomposition of the same data using a 2–6-day bandpass-filtered

Theories for MRG initiation in the troposphere range from lateral forcing (Mak 1969; Lamb 1973; Hayashi 1976; Wilson and Mak 1984), to nonlinear wave–convective instability of the second kind (Itoh and Ghil 1988), nonlinear resonance (Raupp and Silva Dias 2005), evaporation–wind feedback (Goswami and Goswami 1991), or meridional and vertical shear instabilities (Zhang and Webster 1989; Wang and Xie 1996; Xie and Wang 1996; Han and Khouider 2010; Zhou and Kang 2013). One particular issue that is unresolved concerns the reason that convectively coupled MRGs and EIGs are constrained to a relatively narrow longitudinal sector. Observational studies such as Zangvil and Yanai (1980), Yanai and Lu (1983), Magaña and Yanai (1995), and K16 find strong links between lateral forcing and MRGs, although such forcing is not confined to the western/central Pacific sector. Similarly, Goswami and Goswami (1991) show that both MRG and EIG instabilities only occur in their shallow-water framework when surface basic-state winds are easterlies, which is also not a feature exclusive to the western/central Pacific sector. Both Itoh and Ghil (1988) and Raupp and Silva Dias (2005) offer some compelling arguments for the geographic selection of MRG and EIG modes; however, to the authors’ knowledge, there has been no observational support for their theories. In summary, it appears that theories concerning the dynamics of MRGs and EIGs would benefit from a more complete observational assessment of the behavior of these types of disturbances.

There is ample evidence of local and remote impacts of organized subseasonal tropical rainfall (Frank and Roundy 2006; Gloeckler and Roundy 2013; Schreck et al. 2011); however, most of the current climate models poorly resolve this type of variability, and convectively coupled MRGs and EIGs are no exception. For instance, Hung et al. (2013) show that most models from phase 5 of the Coupled Model Intercomparison Project (CMIP5) do not reproduce the observed strong MRG concentration over the western/central Pacific. In addition, CMIP5 precipitation space–time power spectra along the MRG and EIG dispersion curve vary widely in comparison with observations, from a complete lack of MRG and EIG power, to having too strong of an MRG signal, or to showing only a large wavenumber-0 spectral peak. Model physical parameterizations are likely to be a major source of these differences. In particular, both the type and tuning of cumulus parameterizations, which are key to the physical processes involved in the coupling between convection and the larger-scale flow, have been shown to strongly affect a model’s ability to resolve convectively coupled equatorial waves (CCEWs) and other tropical disturbances (Lin et al. 2008; Straub et al. 2010; Frierson et al. 2011). The difficulties in modeling convectively coupled MRGs and EIGs are perhaps not surprising given the large number of disparate ideas put forth to account for their initiation and maintenance. Observational understanding of the nature of the coupling between MRG and EIG dynamics and rainfall is, therefore, an important step toward improving the ability of models to predict and simulate synoptic-scale variability in the tropics.

Here we investigate whether the observed MRG and EIG convective variability in the western/central Pacific is mainly due to a (i) zonally standing oscillation, (ii) interference between westward and eastward wavelike disturbances, or (iii) westward and eastward wavelike disturbances that alternate in time. By interference in (ii), we include both possibilities of preexisting westward and eastward disturbances propagating toward one another and continuing to propagate toward the west and east, respectively, and also cases where westward and eastward disturbances initiate simultaneously because of external forcing. Using the same datasets as in K16, namely CLAUS

The paper outline is as follows. In section 2, we review MRG and EIG theory, and in section 3 we present an EOF analysis of MRG and EIG theoretical shallow-water modes. This analysis serves two purposes. First, by using an input dataset containing a known signal, we can test the ability of EOFs to distinguish between the types of wave patterns described above in points (i), (ii), and (iii). Second, it gives us insight into what we should expect from the

## 2. Review of MRG and EIG dynamics

*u*and

*υ*and the geopotential

*ϕ*. The parameters are the gravity wave speed

*R*are Earth’s angular velocity and radius, respectively. In the next section, we use

*H*

_{eq}= 25 m for the convectively coupled modes since that appears to best fit the data (Wheeler and Kiladis 1999; Kiladis et al. 2009; K16, their Fig. 1a). The normal modes are obtained by the method of separation of variables, which yields the dispersion relation:

*ω*is the frequency,

*k*is the zonal wavenumber, and the parameter

*n*is related to the meridional structure of each mode (Matsuno 1966). The MRG and EIG modes correspond to the case where

*β*comes from the meridional gradient. In the case of MRGs, zonal convergence is out of phase with meridional convergence. In contrast, zonal and meridional convergences are in phase in the case of EIG. Based on the dispersion relation, MRGs and EIGs also satisfy

*H*

_{eq}≈ 25 m), Fig. 2 shows that the MRG velocity field has to be 4 times larger than that of the EIG in order for both disturbances to have the same divergence amplitude (Fig. 3 in K16). In the stratosphere where

Ratio of EIG to MRG divergence [from (6)] as a function of zonal wavenumber for *H*_{eq} = 8, 12, 25, 50, 90, and 120 m.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Ratio of EIG to MRG divergence [from (6)] as a function of zonal wavenumber for *H*_{eq} = 8, 12, 25, 50, 90, and 120 m.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Ratio of EIG to MRG divergence [from (6)] as a function of zonal wavenumber for *H*_{eq} = 8, 12, 25, 50, 90, and 120 m.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

## 3. EOF analysis

### a. Idealized data

In this section, we use two idealized cases to test the ability of EOFs to distinguish between various types of interference signals discussed in the introduction. First, we superimpose the theoretical dynamical fields associated with a single MRG and EIG mode, where all fields are normalized to the same maximum divergence amplitude, and they are in phase at the initial time (as in Fig. 3 of K16). We choose the divergence field because horizontal low-level convergence is proportional to ascent in a baroclinic model; thus, it is used here as a proxy for moist convection. The second idealized case is similar, except that for half the time the dataset is made up of only MRG modes, followed by only EIG modes. Time–longitude samples of these divergence fields are shown in Fig. 3. The first case shown on the left panel is denoted “MRG+EIG” and the second shown on the right is denoted “MRG/EIG.” In both cases, the first two EOFs are in quadrature and represent propagating patterns similar to the MRG and EIG divergence seen in Fig. 2 in K16, together explaining about 99% of the total variance. As an aside, it is shown in appendix A that the amount of variance explained by the first two EOFs is substantially decreased by adding small-amplitude noise to either MRG+EIG or MRG/EIG data, suggesting that, despite the relatively small variance explained, the first two EOFs of observed

Time–longitude diagram of (a) MRG+EIG and (b) MRG/EIG nondimensional divergence fields at 7.5°N (see details in the text). The wave parameters used are *H*_{eq} = 25 m. Dark (light) gray shading is for positive (negative) anomalies.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Time–longitude diagram of (a) MRG+EIG and (b) MRG/EIG nondimensional divergence fields at 7.5°N (see details in the text). The wave parameters used are *H*_{eq} = 25 m. Dark (light) gray shading is for positive (negative) anomalies.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Time–longitude diagram of (a) MRG+EIG and (b) MRG/EIG nondimensional divergence fields at 7.5°N (see details in the text). The wave parameters used are *H*_{eq} = 25 m. Dark (light) gray shading is for positive (negative) anomalies.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Since the normalization implies that

Lag–longitude diagram of MRG+EIG divergence at 7.5°N (shading) and meridional wind on the equator regressed onto the first PC of MRG+EIG. Both variables are nondimensional, and contour intervals are 1 std dev starting at 1 std dev. Dark gray shading (solid contours) is for positive anomalies and light gray shading (dashed contours) is for negative anomalies.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Lag–longitude diagram of MRG+EIG divergence at 7.5°N (shading) and meridional wind on the equator regressed onto the first PC of MRG+EIG. Both variables are nondimensional, and contour intervals are 1 std dev starting at 1 std dev. Dark gray shading (solid contours) is for positive anomalies and light gray shading (dashed contours) is for negative anomalies.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Lag–longitude diagram of MRG+EIG divergence at 7.5°N (shading) and meridional wind on the equator regressed onto the first PC of MRG+EIG. Both variables are nondimensional, and contour intervals are 1 std dev starting at 1 std dev. Dark gray shading (solid contours) is for positive anomalies and light gray shading (dashed contours) is for negative anomalies.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

While the EOF patterns are very similar in both MRG+EIG and MRG/EIG cases, their differences can be readily captured by lag correlating the first two PCs. Specifically, the lag correlation between the first two PCs in both MRG+EIG and MRG/EIG cases are in very close agreement, including the overall weak lag correlation (solid black and dashed black lines in Fig. 5); however, computing the lag correlation during the MRG period only (solid gray line in Fig. 5) or EIG only (dashed gray line in Fig. 5) reveals much stronger correlations at the expected lags, given the prescribed MRG and EIG periods. To summarize, the EOF analysis applied to idealized cases suggests that weak lag correlation between the first two PCs of

Lag correlation between the first two PCs of MRG+EIG (solid black), MRG/EIG (dashed black), MRG only (solid gray), and EIG only (dashed gray).

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Lag correlation between the first two PCs of MRG+EIG (solid black), MRG/EIG (dashed black), MRG only (solid gray), and EIG only (dashed gray).

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Lag correlation between the first two PCs of MRG+EIG (solid black), MRG/EIG (dashed black), MRG only (solid gray), and EIG only (dashed gray).

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

### b. Observations

*τ*,

*n*and

*τ*units are days. The set of independent time intervals

As in Fig. 5, but for lag correlation between the first two PCs of

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

As in Fig. 5, but for lag correlation between the first two PCs of

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

As in Fig. 5, but for lag correlation between the first two PCs of

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

The choice of TW is motivated by the decorrelation time of PC1 and PC2 using the entire record, which is about 5 days. A short time window of 5 days can be useful in detecting a specific event, whereas a longer time window can capture a period where a particular type of disturbance is dominant; thus, for the following analysis, a range of TWs are used and interpreted accordingly. To physically interpret the time-windowed analysis, recall that based on the EOF patterns shown in K16, positive (negative) correlation at negative lags represents westward (eastward) propagation (see their Fig. 5). The correlation statistical significance is tested based on the *p* value calculated by transforming the correlation at each lag to create a *t* statistic with

(a) PDF of

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

(a) PDF of

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

(a) PDF of

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

It is important to note that, based on our correlation criteria, the propagating segments for all choices of TW represent only about 30% of the total time intervals (Fig. 7b). A similar composite to the ones shown in Fig. 7c, including only the remaining 70% of the cases (i.e., noncorrelated intervals) reveals only weak correlations at all lags, with a very similar structure as the one shown in Fig. 6 (not shown). This result suggests that, while periods of MRG/EIG are present for about 30% of the record in observations, for the remaining 70% we cannot conclude whether the weak correlation between PC1 and PC2 is due to independent standing oscillations or a simultaneous interference between westward and eastward disturbances (MRG+EIG). To resolve this issue, many variations of this method have been attempted, but these produce similarly inconclusive results, which motivated the completely independent analysis presented next.

## 4. An object analysis of MRGs and EIGs

An alternative approach to investigating synoptic convective organization is to objectively identify contiguous cloud regions (CCRs) in

### a. Method

While we refer to Dias et al. (2012) for details on the algorithm implementation, it is worthwhile to summarize some of its relevant features. The first step is to search for contiguous regions in the three-dimensional space of latitude, longitude, and time where

Example of a coherent CCR: gray contours correspond to (a) longitude–latitude–time view, (b) longitude–time view, and (c) latitude–time view. In (b) and (c) the blue lines show

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Example of a coherent CCR: gray contours correspond to (a) longitude–latitude–time view, (b) longitude–time view, and (c) latitude–time view. In (b) and (c) the blue lines show

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Example of a coherent CCR: gray contours correspond to (a) longitude–latitude–time view, (b) longitude–time view, and (c) latitude–time view. In (b) and (c) the blue lines show

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

In the present analysis, we retain only CCRs that last for at least 1 day, span more than 500 km in latitude, and with centroids located between 120°E and 120°W and between 20°S and 20°N [we refer to this region as the tropical Pacific (TP)], which is the same area used for the EOF analysis in K16. The threshold is chosen to maximize the number of CCRs that last for more than 2 days and that span at least 1000 km meridionally. To calculate this optimized threshold, we detect CCRs at thresholds ranging from the 1st to the 25th percentiles of the distribution of

### b. CCR statistics

Figure 9a shows a two-dimensional histogram of CCR centroids in the space of latitude and longitude

(a) Longitude–latitude count of CCRs based on their centroids

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

(a) Longitude–latitude count of CCRs based on their centroids

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

(a) Longitude–latitude count of CCRs based on their centroids

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Figures 9d and 9e show the partition of the zonally coherent CCRs ^{−1} value is chosen based on the mean error in our estimate of

Aside from the similar propagation characteristics between CCRs in the STP and NTP, these two sets are also similar in lifespan and size (not shown). In addition, for about 80% of NTP CCRs, there is at least one STP CCR that occurs less than 3 days apart from that NTP CCR, where the timing is based on the difference between the two

### c. CCR composites

Composites are calculated by shifting *t* test at each grid point. This calculation was done using 2–20-day bandpass-filtered

(a) Lag–longitude NTP westward CCR composites of 2–20-day ^{−1}, starting at +0.1 (solid) and −0.1 m s^{−1} (dashed).

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

(a) Lag–longitude NTP westward CCR composites of 2–20-day ^{−1}, starting at +0.1 (solid) and −0.1 m s^{−1} (dashed).

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

(a) Lag–longitude NTP westward CCR composites of 2–20-day ^{−1}, starting at +0.1 (solid) and −0.1 m s^{−1} (dashed).

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

As in Fig. 10, but for lag–latitude NTP CCR composites. In all panels, the zonal average is calculated from 160°E to 160°W. Shading and contours also as in Fig. 10.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

As in Fig. 10, but for lag–latitude NTP CCR composites. In all panels, the zonal average is calculated from 160°E to 160°W. Shading and contours also as in Fig. 10.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

As in Fig. 10, but for lag–latitude NTP CCR composites. In all panels, the zonal average is calculated from 160°E to 160°W. Shading and contours also as in Fig. 10.

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Figure 10 compares the lag–longitude composites of NTP CCRs, based on the partitioning shown in Fig. 9d. By construction, the anomalous ^{−1} for westward-moving CCRs and 25 m s^{−1} for the eastward ones.

Figure 11 is similar to Fig. 10, but for lag–latitude composites. In the three cases,

The relationship between STP and NTP CCRs is quantified by calculating the pattern correlation between the composites at each lag using only NTP versus only STP CCRs. The pattern correlation for all variables is calculated after flipping north and south grid points and switching the sign of the meridional wind of the STP composites, which means that, when NTP and STP composites are similar, the pattern correlation should be positive. Focusing once more only on zonally coherent CCRs and the associated circulation at 850 hPa, Fig. 12 shows the pattern correlation of NTP and STP

Lag–pattern correlations between NTP and STP CCR composites for

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Lag–pattern correlations between NTP and STP CCR composites for

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

Lag–pattern correlations between NTP and STP CCR composites for

Citation: Journal of the Atmospheric Sciences 73, 5; 10.1175/JAS-D-15-0231.1

## 5. Summary and conclusions

Results from two completely independent analyses, one based on EOFs and the other one based on object tracking, are used to characterize the behavior of synoptic-scale convective activity in the tropical central Pacific. The EOF analysis is based on the results from K16, and the object-based algorithm involves detecting contiguous regions of enhanced cloudiness in the space of longitude, latitude, and time, referred to as CCRs. To motivate the latter approach, we first calculate EOFs of idealized data consisting of only MRG and EIG theoretical modes. This preliminary analysis reveals commonalities with the observational results from the EOF decomposition of K16. In particular, it indicates that alternating or overlapping patterns of westward and eastward disturbances are hardly distinguishable from one another based on the EOF approach. This result motivates the hypothesis that the standing pattern of oscillation in the tropical central Pacific discussed in K16 is due to an interference between MRG- and EIG-like disturbances. To further investigate this hypothesis, we then apply an independent analysis based on the object-based algorithm mentioned above. In agreement with the antisymmetric component of the space–time power spectrum of

The CCR composites revealed that these synoptic disturbances follow theoretical linear dynamics to a remarkable degree. For instance, consistent with K16 and with theoretical MRGs, the westward-moving CCRs are associated with ^{−1} toward the west with a period of about 4.25 days. Also, as predicted by MRG theory,

The structure of eastward-moving CCRs is also in agreement with EIGs from shallow-water theory in that their ^{−1}, and their period is about 3.5 days. The zonally standing composites are particularly interesting as they show low-level zonal winds moving toward the east, which is likely characterizing the EIG component of an “MRG+EIG”-like interference signal, along with meridional winds propagating to the west consistent with the MRG component of this interference. Our results, therefore, imply that 25%–30% of standing CCRs are cases of a simultaneous interference signal that is also consistent with shallow-water theory in that they behave similarly to our idealized MRG+EIG test case.

While the analyses presented here in combination with K16 provide substantial observational evidence of the expected contrast in behavior between MRGs and EIGs from shallow-water theory, there are also notable differences. For example, the observed disturbances often deviate from Matsuno’s symmetric or antisymmetric amplitudes with respect to the equator, and this could be due to a variety of processes. For instance, at times there is likely to be interference with other types of equatorial waves, such as Kelvin, equatorial Rossby, or easterly waves. Another potential impact to the behavior of the waves involves horizontal and vertical wind shear within the basic state, which is known to affect the eigenfunctions of the shallow-water system (Zhang and Webster 1989, 1992; Zhang 1993; Han and Khouider 2010; Monteiro et al. 2014). In addition, nonlinear effects coming from moisture–wave coupling may play a substantial role in altering the disturbance structures. Another important distinction from linear theory is related to the poleward propagation of

## Acknowledgments

We thank Brant Liebmann, Takeshi Horinouchi, and two anonymous reviewers for thoughtful and insightful comments on an original draft of this paper and Maria Gehne for guidance on several aspects of the analysis techniques used here.

## APPENDIX A

### MRG and EIG Interference Signal with Noise

*r*is modeled using a univariate lag-1 autoregressive process:

*n*is the discrete time step,

*α*is the assumed lag-1 autocorrelation, and

*r*so that its amplitude is at most 10% of the wave amplitude, the leading EOFs and PCs of

While a complete analysis of the impact of various types of noise in the EOF decomposition is beyond the scope of this paper, the examples shown here suggest that the relative low amount of variance explained may be due to noise and/or multiscale convective processes not necessarily related to MRGs and EIGs.

## APPENDIX B

### The Propagation Coherence Criteria

*t*is time. The mean CCR in the meridional and zonal directions are defined as

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