## 1. Introduction

Tropical rainfall at synoptic scales is known to be organized by waves traveling parallel to the equator. These convectively coupled equatorial waves (CCEWs) control a large fraction of the variability in the tropical weather and a number of observational studies have shown that CCEWs are strongly affected by the interactions among atmospheric circulation, moisture transport, and deep convective rainfall. Convectively coupled Kelvin waves (CCKWs) correspond to a type of CCEW that propagates eastward and that is arguably one of the most important modes of variability in the tropics, along with tropical cyclones and the much larger and lower-frequency Madden–Julian oscillation (MJO). Because of enhanced convection and precipitation, fluxes of heat, moisture, and momentum vary dramatically inside and outside the intertropical convergence zone (ITCZ), which also impacts the tropical circulation. A number of observational and theoretical studies have suggested two-way feedback mechanisms—manifestations of the dynamic and thermodynamic coupling of clouds and tropical circulation—between the ITCZ and CCKWs (Straub and Kiladis 2002; Wang and Fu 2007; Yang et al. 2007; Dias and Pauluis 2009). In particular, Straub and Kiladis (2002) show evidence of convective activity propagating along the mean axis of the eastern Pacific ITCZ, with spatial structures, propagation speeds, and dispersion characteristics of equatorially trapped Kelvin waves.

CCKWs can be identified in the power spectrum (in frequency and zonal direction) of tropical cloudiness (Takayabu 1994; Wheeler and Kiladis 1999, hereafter WK99; Wheeler et al. 2000). After removing a raw background spectrum (WK99), power spectrum peaks fall along the dispersion curves from Matsuno’s (1966) shallow water modes, such as Kelvin waves, equatorial Rossby waves, and mixed Rossby–gravity waves. These peaks in the tropical cloudiness power spectrum suggest a range of equivalent depth from 10 to 100 m. In particular, the power spectra associated with convectively coupled Kelvin waves fall within speeds from about 10 to 30 m s^{−1}. In contrast, dry Kelvin waves propagate at 40–50 m s^{−1}, corresponding to a vertical wavelength of twice the depth of the troposphere.

The slowdown of CCEWs is usually thought to be the result of a compensation between latent heat released within convective clouds and adiabatic cooling (Emanuel et al. 1994; Neelin and Zeng 2000; Frierson et al. 2004). In this sense, slower propagation would be expected in regions of higher convective activity such as in the Indian Ocean and western Pacific, in comparison to relatively drier regions such as the eastern Pacific. Moreover, as a prominent feature of the tropics, the geographic extension of the ITCZ could be a key factor in the waves’ slowdown process. In Dias and Pauluis (2009) the physical mechanisms of the interaction between the ITCZ and CCKWs were analyzed utilizing an idealized model. A number of key elements of the modeled CCKWs are consistent with the observed CCKWs. For instance, in agreement with Wheeler et al. (2000) and Straub and Kiladis (2002), the modeled CCKWs are characterized by several features that are not present in the traditional dry Kelvin wave, including weak dispersion and meridional circulation. Dias and Pauluis (2009) argued that the latter is necessary in order to have a coherent structure propagating eastward along the ITCZ, which favors convergence, enhancing precipitation within the ITCZ and causing the waves to slow down. The intensity of the meridional circulation is shown to depend on the ratio between the meridional location of the ITCZ and the zonal wavelength of the Kelvin wave. The modeled CCKWs propagate faster along an ITCZ of same width, but farther from the equator, because of the weakening of this secondary meridional circulation. Moreover, an analytical relation between the speed of CCKWs and the location and width of the ITCZ was derived, showing that the phase speed of CCKWs decreases with increasing ITCZ width. Although not discussed in the present work, it is important to notice that other physical mechanisms such as extratropical forcing and meridional and vertical shear (Webster and Holton 1982; Kasahara and Silva Dias 1986; Webster and Chang 1988; Zhang and Webster 1989; Han and Khouider 2010) are also well known to affect equatorial waves, and in particular CCKWs.

In the observational analysis of Yang et al. (2007), CCKWs propagating over the longitude range of 50°E–130°W were found to propagate more slowly than over 130°W–50°E. This result is consistent with the theoretical prediction presented in Dias and Pauluis (2009) because the observed ITCZ tends to be broader and closer to the equator along the Indian Ocean, in comparison to the ITCZ along the eastern Pacific Ocean. Importantly, the role of the background flow was not included in this observational analysis, nor was it part of the modeling study of Dias and Pauluis (2009). Within this context, the main purpose of this paper is, using observations, to test the following theoretical hypothesis for CCKWs:

Do CCKWs phase speeds increase with increasing ITCZ distance from the equator?

Do CCKWs phase speeds decrease with increasing ITCZ width?

Are CCKWs Doppler shifted by the background wind?

## 2. Data and methodology

Cloud Archive User Services (CLAUS) data from 1 July 1983 through 30 September 2004 are used to identify the ITCZ and CCKWs. The CLAUS dataset utilizes 8-times-daily geostationary and polar-orbiting images producing a global grid of brightness temperature *T _{b}* with a resolution of 0.5° (Hodges et al. 2000). A similar dataset, the National Oceanic and Atmospheric Administration (NOAA) outgoing longwave radiation (OLR) product (Liebmann and Smith 1996), has been successfully used as a statistical proxy for tropical convection (WK99; Wheeler et al. 2000); however, despite the fact that it has a nearly continuous record since 1974, its resolution is lower than the CLAUS dataset. In addition, we use National Centers for Environmental Prediction (NCEP)–National Center for Atmospheric Research (NCAR) reanalysis data (Kalnay et al. 1966) to represent the global atmospheric circulation. The data are available on a 2.5° grid and we use the monthly averaged zonal wind from the same time period as the CLAUS data.

In the remainder of this section, we present our methodology to define the mean location and width of the ITCZ and the monthly mean phase speed of CCKWs. Since ITCZ characteristics greatly vary around the globe, the analysis is focused on the five tropical ocean basins defined in Table 1 and illustrated in Fig. 1. We exclude continents from the analysis because the ITCZ is not as well defined over land. In section 2a, the method used to determine the location and width of the ITCZ as a function of longitude and time is presented. Based on this methodology, the overall features of the ITCZ’s spatial structure are discussed. In section 2b, the *T _{b}* filtering technique, analogous to that of WK99, is reviewed and the method to determine the CCKWs phase speed is described. Section 2b ends with a broad description of temporal and spatial variabilities of CCKWs phase speed.

Location of the ocean regions.

### a. The location and width of the ITCZ

A number of observational studies (Gruber 1972; Mitchell and Wallace 1992; Waliser and Gautier 1993) have focused on the annual cycle of the ITCZ location using brightness temperature satellite data. Waliser and Gautier (1993) identify the ITCZ using the highly reflective cloud (HRC) dataset (Garcia 1985), which was built to detect large-scale convective cloud systems. Other studies have used low-level convergence, low-level vorticity, and rainfall products; hence, the methodology presented in this section can certainly be refined, either by use of a different dataset or by improving the ITCZ identification technique. In the present study, we use a method similar to that of Waliser and Gautier (1993) to obtain the monthly mean location and width of the ITCZ.

*T*and is typically near the equator, with the main convective axis located from 10°S to 10°N; therefore, a natural definition for the location of the ITCZ is the meridional distance from the equator to the closest minimum

_{b}*T*. Since this definition does not take into account the zonal coherence of convective regions, the resulting axis is highly oscillatory across longitude lines. To minimize this effect, instead—similarly to the definition used in Waliser and Gautier (1993)—we use a smoother approximation where we compute the average over low

_{b}*T*locations along each longitude line; that is,

_{b}*y*is the meridional distance from the equator,

*x*is the zonal distance,

*t*represents time, and

*y*= −15° and

_{S}*y*= −15°, and

_{N}*T*

_{b0}is a threshold limit defined by the chosen percentile (e.g., 15th or 25th percentile) of the global distribution of

*T*over a fixed time interval. Following this method, there are two parameters that we have to choose: the percentile used to define

_{b}*T*

_{b0}and the frequency to update

*T*

_{b0}. First, by testing different thresholds (e.g., 15th, 20th, 25th, and 30th percentile), we found that the ITCZ location is not very sensitive to the choice of the specific percentile used in the definition of

*T*

_{b0}; hence, for simplicity, we show the results using only the first quartile of

*T*. Second, to minimize effects of seasonal variability, we have chosen to compute

_{b}*T*

_{b0}for each output of

*T*data; that is,

_{b}*T*

_{b0}is updated every 3 h. Because using the first quartile of

*T*as the threshold yields a narrow range of

_{b}*T*

_{b0}(198 <

*T*

_{b0}< 204 K), our results are also robust to the choice of the frequency to update

*T*

_{b0}. Note that, over the entire time series, the minimum

*T*is 170 K, the median is 288 K, and the maximum is 340 K.

_{b}*T*is below a threshold value:

_{b}*T*<

_{b}*T*

_{b0}at

*n*latitude grid points, the width is defined as

*n*/2 (because of the 0.5° resolution of the data). The method is tested for different choices of the threshold limit

*T*

_{b0}, similarly to the ITCZ axis. As expected (and unlike the ITCZ location), the ITCZ width is considerably wider for larger values of

*T*

_{b0}. For instance, comparing the width using the 15th percentile to the first quartile, the latter is on average 4° wider. Figure 2 shows a comparison between the ITCZ width during May 1985 and at a fixed location (180°). In this particular case, the mean width increases from 3.5° using the 15th percentile to 6.4° using the 25th percentile. In contrast to the width itself, the variations of the width for low values of

*T*

_{b0}are in good agreement. To illustrate that, Fig. 2b displays the same time series as Fig. 2a, except that the monthly mean of each of the curves from the top panel is removed and then each time series is divided by its maximum amplitude within that month. In contrast to Fig. 2a, the two curves shown in Fig. 2b are in close agreement. Although the width definition used here does not take into account the presence of a coherent structure parallel to the equator, it yields results that are consistent with well-known features of the ITCZ (e.g., a broader ITCZ over the Indian Ocean in comparison to the eastern Pacific). Another relatively simple alternative to compute the width is to use the standard deviation of the distance to the equator when the brightness temperature is below the threshold; however, this method has similar limitations to the one we use because it also cannot distinguish between the extension of contiguous or scattered regions of

*T*below the threshold.

_{b}(a) ITCZ width at longitude 180° for the month of May 1985. The solid line corresponds to computation using the 15th percentile of *T _{b}* and the dashed line corresponds to the 25th percentile. (b) As in (a), but the lines correspond to the normalized variation from the mean ITCZ width.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

(a) ITCZ width at longitude 180° for the month of May 1985. The solid line corresponds to computation using the 15th percentile of *T _{b}* and the dashed line corresponds to the 25th percentile. (b) As in (a), but the lines correspond to the normalized variation from the mean ITCZ width.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

(a) ITCZ width at longitude 180° for the month of May 1985. The solid line corresponds to computation using the 15th percentile of *T _{b}* and the dashed line corresponds to the 25th percentile. (b) As in (a), but the lines correspond to the normalized variation from the mean ITCZ width.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

The annual cycle of the ITCZ computed using data from 1984 to 2002 is shown in Fig. 3, which includes the continents. The location of the ITCZ (left panel) over the Indian Ocean (IO; region I in Fig. 1) oscillates between the Northern Hemisphere (NH; near 5°N) during the boreal summer and the Southern Hemisphere (SH; near 5°S) during the boreal winter. In the western and central Pacific (WP and CP; regions II and III), the ITCZ tends to be closer to the equator, whereas in the eastern Pacific (EP; region IV), the ITCZ is located around 10°N during the boreal summer shifting toward the equator during the boreal winter. The behavior of the ITCZ over the Atlantic Ocean (AO; region V) is similar to the EP, except that in average it stays closer to the equator (near 5°N). Figure 3 also indicates that the ITCZ location favors the Northern Hemisphere, which is a result of interactions between the ocean and atmosphere, combined with the geometry of the continents (Philander et al. 1996). Notice that over the IO and WP the monsoons and ocean warm pools spread out the ITCZ (Webster 1987). In contrast, the ITCZ is clearly narrower over the AO and EP, and the CP is a transition zone. Overall, the ITCZ is broader during May–October. This pattern repeats in most years of the time series, except during El Niño events such as 1987 and 1997, which are characterized by an above average ITCZ width and southward displacement of the ITCZ over the central Pacific.

Annual cycle computed from 20 yr of daily mean (left) ITCZ location and (right) ITCZ width, as defined by Eqs. (1) and (4), respectively. We define *T*_{b0} as the 25% percentile of daily *T _{b}*. The vertical lines correspond to the boundaries of the tropical ocean basins shown in Fig. 1.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

Annual cycle computed from 20 yr of daily mean (left) ITCZ location and (right) ITCZ width, as defined by Eqs. (1) and (4), respectively. We define *T*_{b0} as the 25% percentile of daily *T _{b}*. The vertical lines correspond to the boundaries of the tropical ocean basins shown in Fig. 1.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

Annual cycle computed from 20 yr of daily mean (left) ITCZ location and (right) ITCZ width, as defined by Eqs. (1) and (4), respectively. We define *T*_{b0} as the 25% percentile of daily *T _{b}*. The vertical lines correspond to the boundaries of the tropical ocean basins shown in Fig. 1.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

### b. The filtered convectively coupled Kelvin waves

The technique used to isolate the variability associated with Kelvin waves consists of spectral filtering; it is analogous to the technique used in WK99. For each latitude between 15°S and 15°N, the data are split into 20 segments of 18 months with the period from May to April centered (segments overlap by 3 months). Before computing the Fourier decomposition in time and zonal direction, the mean and linear trend of each segment are removed, and the ends of the time series are tapered to zero. The filtering corresponds to inversion of the two-dimensional Fourier transform where only the wavenumber and frequency corresponding to CCKWs are retained. The same spectral region from Straub and Kiladis (2002) is used—zonal wavenumber between 1 and 15, frequency between 0.05 and 0.4 cpd, and phase speeds between 9 and 35 m s^{−1}.

Figure 4 shows the averaged spectrum from 15°S to 15°N of the symmetric and antisymmetric components of *T _{b}*, as well as the spectrum of the ITCZ width and location. Notice the similarities between the symmetric

*T*and the ITCZ width, and also the similarity between the antisymmetric

_{b}*T*and the ITCZ location. The spectrum of the ITCZ width has a peak along the region associated with CCKWs and also a slower westward-propagating component corresponding to convectively coupled equatorial Rossby waves. The spectrum of the ITCZ location reveals a peak similar to the mixed Rossby–gravity dispersion curve. Although the large correlation is likely due to the fact that all the spectra shown are red, the correlation between the symmetric

_{b}*T*and the ITCZ width spectra is 0.98, and the correlation between the antisymmetric

_{b}*T*and the ITCZ location spectra is 0.99. These results suggest that variations of the symmetric component of the

_{b}*T*spectrum can be interpreted as a widening of the ITCZ by tropical disturbances, whereas the antisymmetric component corresponds to a shift in the ITCZ axis. Interestingly, the fact that the power spectrum of the ITCZ width peaks along the CCKW spectral region is consistent with the ITCZ expansion and contraction being associated with the propagation of CCKWs. In fact, Dias and Pauluis (2009) show that the ITCZ expands during the ascent phase of the modeled wave and contracts during the subsidence phase.

_{b}Mean of 20 yr of the log_{10} of the power spectrum of the (a) symmetric component of *T _{b}*, (b) antisymmetric component of

*T*, (c) ITCZ width, and (d) ITCZ axis, with planetary wavenumber on the

_{b}*x*axis and frequency (cpd) on the

*y*axis. The solid lines correspond to phase speeds of 9 and 35 m s

^{−1}. The vertical dotted line is located at

*k*= 1 and the two horizontal dotted lines bound the region between 0.05 and 0.4 cpd.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

Mean of 20 yr of the log_{10} of the power spectrum of the (a) symmetric component of *T _{b}*, (b) antisymmetric component of

*T*, (c) ITCZ width, and (d) ITCZ axis, with planetary wavenumber on the

_{b}*x*axis and frequency (cpd) on the

*y*axis. The solid lines correspond to phase speeds of 9 and 35 m s

^{−1}. The vertical dotted line is located at

*k*= 1 and the two horizontal dotted lines bound the region between 0.05 and 0.4 cpd.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

Mean of 20 yr of the log_{10} of the power spectrum of the (a) symmetric component of *T _{b}*, (b) antisymmetric component of

*T*, (c) ITCZ width, and (d) ITCZ axis, with planetary wavenumber on the

_{b}*x*axis and frequency (cpd) on the

*y*axis. The solid lines correspond to phase speeds of 9 and 35 m s

^{−1}. The vertical dotted line is located at

*k*= 1 and the two horizontal dotted lines bound the region between 0.05 and 0.4 cpd.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

CCKW episodes have been associated with peaks in the variance of the filtered *T _{b}* (WK99; Straub and Kiladis 2002). The phase speed of single episodes of CCKWs can be estimated using linear regression; however, because our primary interest is to quantify the mean phase speed of the CCKWs activity over limited space and time regions, the Radon transform (RT) method is used instead. This method has been described in detail and applied in Yang et al. (2007). Here, the filtered data are first averaged in the meridional direction from 15°S to 15°N, then split in longitude subregions (according to the five ocean regions defined in Table 1), and finally arranged in monthly time series. That is, each data subset corresponds to a month from May 1984 to May 2002 and an ocean basin. The RT yields the projected data along lines at each angle from 0° to 180° in the (

*x*,

*t*) plane. When the line is perpendicular to the crests and troughs of the wave, the energy of the projected data along this line has a maximum, and hence the angle perpendicular to the projection gives the wave propagation direction in the longitude–time domain and, thus, the phase speed. The spread around the peak in energy points to the question of the error in the RT estimate, particularly because we would like to compare phase speed magnitudes. There are two main sources for the spread. First, during a given month and along an ocean basin, there may be more than one Kelvin wave–like event propagating at different speeds (or the same wave packet can be dispersive). The second error is the finite size domain (both in time and space). In fact, even for a pure nondispersive wave packet, the spread in energy can be shown to depend on the relationship between phase speed and domain dimensions. Despite the limitations of the method, the phase speed of CCKWs for each ocean basin and each month is defined as the speed corresponding to the peak in energy in each data subset, and this approach gives reasonable results.

Apart from the statistical significance of the peak in energy, another issue with the use of the RT method is that CCKWs have been observed to be weakly dispersive (Wheeler et al. 2000; Roundy 2008) and the Radon transform method cannot account for wave dispersion. Actually, by filtering the data in the same spectral region, but including only wavenumbers *k* > 4, we obtain a phase speed estimate that is always smaller than when the smaller wavenumbers are included (not shown). The reduced phase speed suggests that CCKWs are dispersive. Because the phase speed difference is larger over the EP and AO in comparison to WP and IO, it is possible that long waves are more common in the EP and AO. However, whether the wavenumber depends on the geographic location of the wave requires a more thorough analysis of individual CCKWs episodes that is beyond the scope of the present work. In spite of these issues, in the following analyses all wavenumbers from *k* = 1 to 15 are included.

The main statistics of the phase speed in each ocean basin are summarized in Table 2 and the annual cycle in each basin is shown in Fig. 5. Notice that a comparison of panels in Fig. 5 indicates that CCKWs tend to propagate faster along the EP and AO than along the WP and IO. Note too that the phase speed peaks in November in all ocean basins and the slowest waves are found in the boreal winter in all basins, except IO. Comparing the ITCZ axis and width (Fig. 3) to the phase speed (Fig. 5), there is no clear evidence of relationship among their annual cycle; however, some features are noticeable. The ITCZ is distinctly broader and CCKWs propagate more slowly over the WP and IO in comparison to the EP and AO for most of the year (except for February, March, and April). The phase speed and the ITCZ vary the least in the WP, except that the ITCZ migrates from the Northern Hemisphere during the boreal summer to the Southern Hemisphere during the boreal winter. The fastest waves along the WP, CP, and IO occur when the ITCZ crosses the equator. The modulation of the phase speed by the ITCZ is further investigated in the following sections.

Statistics of phase speed (m s^{−1}) split by ocean basin.

CCKW phase speed annual cycle. The dashed lines correspond to the average value in each ocean basin and the distance between the dotted lines shows the standard deviation month by month.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

CCKW phase speed annual cycle. The dashed lines correspond to the average value in each ocean basin and the distance between the dotted lines shows the standard deviation month by month.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

CCKW phase speed annual cycle. The dashed lines correspond to the average value in each ocean basin and the distance between the dotted lines shows the standard deviation month by month.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

## 3. Linear regression analysis

In the previous section we computed the monthly mean time series of ITCZ width, location, and CCKW phase speed for each ocean basin, and their annual cycle (shown in Figs. 3 and 5). In this section, we aggregate the data in order to investigate the linear relationship between each pair of variables. The variables are the ITCZ width *w*, the ITCZ distance from the equator *d*, the CCKWs phase speed *c _{p}*, and the zonal barotropic wind

*U*. We define

_{B}*U*as the pressure-averaged zonal flow from 850 to 250 hPa, followed by the latitudinal average from 15°S to 15°N and the longitudinal average within each basin.

_{B}An event is defined as *ε*(*n*) = [*w*(*n*), *d*(*n*), *c _{p}*(

*n*),

*U*(

_{B}*n*)], noticing that the index

*n*is not time, since data was aggregated from the five ocean basins. That is, an event actually corresponds to the monthly average value on a single ocean basin. Hence, the total sample size here is

*n*= 12 × 18 × 5 (months × years × ocean basins) = 1080. The histograms in Fig. 6 display the distribution of each variable in

*ε*. Figure 6a shows that the ITCZ width peaks at 5°, along with a weaker but distinct peak at 10°. Figure 6b shows that the ITCZ is more frequently observed close (

*d*< 3°) to the equator. The CCKWs’ phase speed histogram displayed in Fig. 6c peaks at about 15 m s

^{−1}; their support is consistent with the filtering window (9–35 m s

^{−1}), and the tail of the distribution is longer to the right of the peak. The barotropic flow (shown in Fig. 6d) is most frequently westward, peaking at about −2.5 m s

^{−1}, which is typical of the equatorial region.

Histograms of (a) ITCZ width, (b) ITCZ axis, (c) CCKW phase speed, and (d) background wind.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

Histograms of (a) ITCZ width, (b) ITCZ axis, (c) CCKW phase speed, and (d) background wind.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

Histograms of (a) ITCZ width, (b) ITCZ axis, (c) CCKW phase speed, and (d) background wind.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

To investigate the linear dependence among CCKWs phase speed, ITCZ width, location, and barotropic flow, correlation coefficients are computed, as well as the linear regression line of each pair of elements of *ε*. The summary of the analysis can be seen in Figs. 7 and 8, where each panel corresponds to the scatterplot of each pair of elements of *ε*. At the top of each panel, both the correlation coefficient and the 95% confidence interval of the slope *β* of the linear regression line are displayed. Figure 7a shows the scatterplot of *c _{p}* and

*d*. The linear regression analysis in this case indicates that the phase speed increases with increasing distance from the equator with correlation

*w*×

*c*of the aggregated dataset, the regression line, and the theoretical curve from Dias and Pauluis (2009) for a

_{p}*k*= 6 CCKW along an ITCZ centered at the equator. Notice that most of the data lie between 400 and 1400 km; within this region, the slope of the regression line (dashed) is in reasonable agreement with the theoretical curve (solid). The offset is due to the model parameter that sets the moist gravity wave speed to 15 m s

^{−1}, which is faster than some of the slower observed Kelvin waves.

The gray markers represent the scatterplot for each pair of variables: (a) phase speed and ITCZ distance, (b) phase speed and ITCZ width, and (c) phase speed and zonal barotropic wind. The correlation coefficient and the 95% confidence interval of the slope of the regression line are displayed on top of each panel. The solid line corresponds to the linear regression line; the dashed lines correspond to the upper and lower bound for its slope.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

The gray markers represent the scatterplot for each pair of variables: (a) phase speed and ITCZ distance, (b) phase speed and ITCZ width, and (c) phase speed and zonal barotropic wind. The correlation coefficient and the 95% confidence interval of the slope of the regression line are displayed on top of each panel. The solid line corresponds to the linear regression line; the dashed lines correspond to the upper and lower bound for its slope.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

The gray markers represent the scatterplot for each pair of variables: (a) phase speed and ITCZ distance, (b) phase speed and ITCZ width, and (c) phase speed and zonal barotropic wind. The correlation coefficient and the 95% confidence interval of the slope of the regression line are displayed on top of each panel. The solid line corresponds to the linear regression line; the dashed lines correspond to the upper and lower bound for its slope.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

As in Fig. 7, but for (a) ITCZ distance and width, (b) zonal barotropic wind and ITCZ distance, and (c) zonal barotropic wind and ITCZ width.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

As in Fig. 7, but for (a) ITCZ distance and width, (b) zonal barotropic wind and ITCZ distance, and (c) zonal barotropic wind and ITCZ width.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

As in Fig. 7, but for (a) ITCZ distance and width, (b) zonal barotropic wind and ITCZ distance, and (c) zonal barotropic wind and ITCZ width.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

CCKW phase speed and ITCZ width relationship. Comparison among model (solid curve), data (gray markers), and linear regression fit (dashed line).

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

CCKW phase speed and ITCZ width relationship. Comparison among model (solid curve), data (gray markers), and linear regression fit (dashed line).

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

CCKW phase speed and ITCZ width relationship. Comparison among model (solid curve), data (gray markers), and linear regression fit (dashed line).

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

Figure 7c does not show any clear evidence of Doppler shift effect of the mean phase speed of CCKWs by the mean equatorial barotropic flow since the correlation between *U _{B}* and

*c*is close to zero. However, the joint distribution of

_{p}*c*and

_{p}*U*shows hints of this type of behavior since most of the very fast CCKW events (

_{B}*c*≫ 25 m s

_{p}^{−1}) are associated with either weak or eastward barotropic zonal flow (

*U*> 0). Further analyses of these fast CCKW events show that most of them occur over the eastern Pacific and Atlantic Oceans during the Northern Hemisphere fall and winter, when the ITCZ is weaker. We discuss these fast cases in more detail in the next section.

_{B}While the correlation analyses indicate the sensitivity of the propagation speed to both the width and location of the ITCZ, the contributions of these individual factors cannot be completely decoupled from each other. Indeed, ITCZ width and ITCZ distance are negatively correlated, as shown in Fig. 8a. Note that the joint distribution of these two variables indicates that the ITCZ is rarely simultaneously broad and away from the equator. Moreover, Figs. 8b and 8c show that both the ITCZ distance and width are negatively correlated with *U _{B}*, but the correlation is considerably stronger between

*w*and

*w*and

*U*suggests that events of strong eastward flow (

_{B}*U*≫ 0) are associated with narrow ITCZ events. The correlation between phase speed and the zonal flow at 850 (

_{B}*u*) and 250 hPa (

_{l}*u*) is, respectively,

_{t}*u*=

_{s}*u*−

_{t}*u*yields a correlation of

_{l}Although none of the correlation coefficients between the elements *ε* is particularly large, the pairwise joint distribution described above highlights a potential modulation of CCKW phase speed by variations of the ITCZ spatial distribution that is consistent with the theoretical prediction. The low value of the correlation coefficients here should be viewed as an indication that factors other than the mean width and location of the ITCZ are affecting the propagation speed of CCKW. In addition, the statistical analysis does not find a significant correlation between propagation speed and barotropic wind. This means that our analysis of the CLAUS data does not provide strong evidence for a Doppler shift of the CCKW, although we did find some circumstantial evidence for it. Interestingly, there seems to be a stronger statistical relationship between vertical wind shear and propagation speed.

## 4. Conditional probability analysis

To further test the coupling between the ITCZ and CCKWs proposed by Dias and Pauluis (2009), in this section the global phase speed variations, beyond linear regression techniques, are investigated. More specifically, in section 4a, we analyze the CCKW phase speed in relation to the ITCZ, isolating the effects of the ITCZ north–south displacement, from its expansion/contraction. In section 4b, we focus on the main characteristics of fast eastward-propagating events (*c _{p}* > 25 m s

^{−1}).

### a. Relationship among the ITCZ width, location, and phase speed of CCKWs

To distinguish the modulation of the phase speed by the ITCZ width from the ITCZ location, the aggregated dataset (from section 3) is split into subsets conditional to the location and width of the ITCZ. The mean ITCZ width over the entire time series and ocean basins is 7° and the mean distance from the equator is 3.5°N (the mean location is at 2.1°N). The first subset contains all the cases where the ITCZ location and width are below the mean (“close/narrow” subset). The second subset contains all the cases where the ITCZ location and width are above the mean (“far/wide” subset). The third subset has the cases where the ITCZ location is below the mean and the width is above the mean (“close/wide” subset). The last subset contains the remaining cases (“far/narrow” subset). The main statistics of the phase speed in each of the subsets are shown in Table 3. The subsets represent respectively 18.2%, 7.3%, 16.3%, and 58.2% of the total dataset. That is, most frequently the ITCZ is farther from its mean location with width below the mean; wide ITCZs with an axis above the mean are the rarest case, and ITCZs that are closer than the mean distance are approximately equally distributed between narrow and wide cases.

Statistics of phase speed (m s^{−1}) split by ITCZ location close to or far from the equator (i.e., below or above mean) and narrow or wide ITCZ (i.e., below or above mean width).

Here we apply a normal kernel smoothing technique (Bowman and Azzalini 1997) to estimate probability density functions (PDFs) and the estimated PDF in each subset is shown in Fig. 10a. Notice that the shift among the peaks of each curve is consistent with the theoretical prediction from Dias and Pauluis (2009). For instance, the PDF shift between close/narrow (gray dashed) and far/narrow (gray solid), as well as between close/wide (black solid) and far/wide (black dashed), indicates that the phase speed increases when the ITCZ moves away from the equator both when the ITCZ is narrow and when it is wide. Similarly, the PDF shift between close/wide (black solid) and close/narrow (gray dashed), and between far/wide (black dashed) and far/narrow (gray solid), suggests that the phase speed decreases when the ITCZ is broader, both when the ITCZ is near the equator and when it is farther away.

PDF of CCKW phase speed (m s^{−1}) split into subsets conditional to the ITCZ width and distance from the equator, for (a) all data and (b) only the “weak *U _{B}*” cases.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

PDF of CCKW phase speed (m s^{−1}) split into subsets conditional to the ITCZ width and distance from the equator, for (a) all data and (b) only the “weak *U _{B}*” cases.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

PDF of CCKW phase speed (m s^{−1}) split into subsets conditional to the ITCZ width and distance from the equator, for (a) all data and (b) only the “weak *U _{B}*” cases.

Citation: Journal of the Atmospheric Sciences 68, 7; 10.1175/2011JAS3630.1

To complement the PDF analyses, a two-sample *t* test is applied in order to address the statistical significance of the difference among the means shown in Table 3. The null hypothesis is that data of two subsets are independent random samples from normal distributions with equal means and unknown variances, in contrast to the alternative that the mean of the first sample is less than the mean of the second sample. The test also yields a 95% lower bound on the difference of population means. The result is returned in *h*, where *h* = 1 indicates a rejection of the null hypothesis at the 5% significance level. This is the Behrens–Fisher problem (Weerahandi 1995) and the hypothesis testing results are shown in Table 4. In agreement with the theoretical prediction, the null hypothesis is rejected at the 5% significance level in all cases, except in the comparison between close/wide and far/wide that is rejected only at the 10% significance level. The largest difference in phase speed is found between the far/wide and far/narrow cases; the least phase speed change is found between the close/wide and far/wide subsets.

Two-sample *t* test of the null hypothesis of equal means and unknown variances, against the alternative that the mean of the first sample is less than the mean of the second sample (details in the text). The variable *h* = 1 indicates rejection of the null hypothesis and *δc _{p}* is the lower bound in the difference between means.

Consistent with the weak correlation found between *U _{B}* and

*c*, when retaining only cases where the magnitude

_{p}*U*is less than 1 m s

_{B}^{−1}, or a half standard deviation (“weak

*U*” subset), the overall features of the phase speed PDF are not affected (see Fig. 10b). The results of the two-sample test are also very similar. Hence, fast waves are more likely to be found when the ITCZ is wide and close to the equator, and slow waves are more likely to be found when the ITCZ is narrow and away from the equator even when the barotropic wind is weak.

_{B}### b. The fast eastward-propagating events

In this section we turn to the analysis of the fast eastward-propagating events corresponding to the right tail of the histogram shown in Fig. 6c. Here, we define fast events as cases where the phase speed is above 25 m s^{−1}. Over the 18 yr of monthly data shown in the previous sections, there are 27 such cases, and their geographic and temporal distribution are summarized in Table 5. Notice that most events occur during the Northern Hemisphere fall. Because most of these fast cases occur over the EP and AO, we focus the following analyses on these two ocean basins. Apart from the majority of the fast cases being associated with an ITCZ located farther from the equator and narrower than the mean width, among the 22 fast events along the EP and AO, 73% of them are associated with stronger eastward upper (200 mb) tropospheric flows in comparison to the average within the corresponding ocean basin. However, the CCKW mean phase speed conditional on cases of strong vertical shear is not significantly above the overall mean phase speed.

Frequency of fast phase speed events split by ocean basin and 3-month period [June–August (JJA), September–November (SON), December–February (DJF), and March–May (MAM)].

There are two possible mechanisms that explain these more rapid propagation speeds over the EP and AO. First, upper tropospheric equatorial westerlies over these two ocean basins, from October to May, are a well-known climatological feature. During this period, intrusions of high potential vorticity into the tropics are often observed (Waugh and Polvani 2000) and have been linked to deep convection (Kiladis and Weickmann 1992; Kiladis 1998); hence, these faster events may correspond to disturbances that are triggered by extratropical flow intrusions and they may not have Kelvin-like characteristics. However, it should be noted that these extratropical waves and CCKWs propagate at similar speeds and are not necessarily independent, as the study by Straub and Kiladis (2003) demonstrates. The second mechanism for these waves could be a Gill-type (Gill 1980) equatorial response to heating (the heating source being the warm pool in the WP and the Amazon basin over South America). In this case, the faster waves are less coupled to deep convection and behave more like a dry Kelvin wave, leading to the faster propagation speed (they show up in the data because they may be linked to stratiform clouds). A more detailed analysis of these waves requires the identification of each particular fast disturbance in the *T _{b}* data and the analysis of their structure and life cycle, which is beyond the scope of this paper.

In any case, the frequency of these fast events is likely higher than what our study suggests because the phase speed is an average over a 1-month period. That is, within the same time window, the phase speed is an average that may include fast and slow events. Importantly, although the linear regression results (shown in section 3) are not affected by excluding these fast cases from the analyses, the fast cases discussed in this section highlight that the spectral filtering may include equatorial disturbances that are not CCKWs.

## 5. Summary and discussion

To test the theoretical predictions of Dias and Pauluis (2009), CLAUS brightness temperatures and NCEP–NCAR reanalysis data were used to access the relationship among geographic variations of the ITCZ, changes in CCKW phase speed, and the barotropic flow. More specifically, an 18-yr-long monthly time series, over the tropical ocean basins, was obtained for each of these variables. Several features of this study are consistent with previous observations. For instance, the regional and seasonal fluctuations of the location of the ITCZ are consistent with Waliser and Gautier (1993), as well as the broadening of the ITCZ over the Indian Ocean and western Pacific (Webster 1987). Moreover, the propagation speed of CCKWs is in agreement with the composite analyses of Yang et al. (2007), which suggests that CCKWs propagate faster over the Western Hemisphere in comparison to the Eastern Hemisphere.

One of the analytical predictions of Dias and Pauluis (2009) states that because of the increasing amount of compensation between latent heat and adiabatic cooling, the propagation speed of CCKWs along an equatorial ITCZ decreases with increasing ITCZ width. More specifically, the theoretical decrease in phase speed is fast when the ITCZ width is below 4° and slows down for wider ITCZs. Since in this analysis both spatial and time averages were computed, narrow ITCZs are not well resolved. For instance, aggregating all data (from different basins), the ITCZ distance from the equator is below 1° in 38% of the dataset, of which only 2% are associated with ITCZs narrower than 4°. Despite that, linear regression analyses show a negative correlation between ITCZ width and CCKW phase speed. Furthermore, a two-sample *t* test shows that the mean phase speed when the ITCZ is close to the equator and wide is less than when the ITCZ is close to the equator and narrow at the 95% confidence level. Figure 9 illustrates that these results are in reasonable agreement with the theoretical prediction.

Another model prediction from Dias and Pauluis (2009) is that CCKWs propagate faster as the ITCZ moves away from the equator. In agreement with the theory, linear regression analyses show positive correlation between CCKW phase speed and ITCZ distance. That is, the observed CCKWs propagate faster when the ITCZ is farther from the equator. Moreover, within the subset of the data where the ITCZ width is below the mean (“narrow” ITCZs), the mean phase speed increases if one compares cases where the ITCZ location is below and above the mean position. A similar relationship was found within the subset of ITCZ width above the mean (“wide” ITCZs). This increase in phase speed is significant at the 95% level, using a two-sample *t* test.

Interestingly, we found little evidence of Doppler shift when analyzing the correlation between the mean barotropic wind *U _{B}* and the CCKW propagation speed

*c*. In fact, stronger correlation was found between phase speed and zonal vertical shear than between phase speed and zonal barotropic wind. This result indicates that the phase speed of CCKWs likely depends more strongly on the vertical structure of the background flow, which has been suggested in a number of modeling studies (Zhang and Webster 1989; Mapes 2000; Majda and Shefter 2001; Han and Khouider 2010). The role of the vertical wind structure in coupling the ITCZ and CCKWs could potentially be clarified using an idealized model that includes the second baroclinic mode such as the one used by Han and Khouider (2010).

_{p}The data analysis suggests regional and temporal variations of CCKWs phase speed. Overall CCKWs propagate faster during November along all ocean basins, and they propagate more slowly during February/March, except along the IO where the slowest waves occur during August. During the boreal summer, the fastest waves are observed in the EP and AO, where the ITCZ is narrow and far from the equator (above 5°N). During the boreal winter, the ITCZ remains narrow over the eastern Pacific, but moves closer to the equator and waves slow down. From the boreal summer to winter and over the western Pacific and the Indian Ocean, the ITCZ tends to shift from the Northern to Southern Hemisphere, but the ITCZ width and CCKW mean phase speed do not show any significant change. We also found that the fastest events (*c _{p}* > 25 m s

^{−1}) occur mainly in November over the EP and AO. Because extratropical flow intrusions into the tropics are often observed in this period, these faster events may not correspond to CCKWs.

The methods for identifying the ITCZ and computing the CCKWs phase speed have obvious limitations. For instance, averaging low brightness temperature only yields the actual axis and width of the ITCZ when there are no other localized areas of low temperature within the equatorial band. In addition, the monthly mean phase speed of CCKWs is obtained assuming that at least one Kelvin wave has occurred every month and in each ocean basin. In reality, some of the filtered data may not correspond to any coherent structure similar to a Kelvin wave. Other than a case by case analysis, these are difficult aspects to handle in a systematic way. One alternative is to disregard cases where the variance of the filtered data is too small (i.e., small variance may indicate that the filtered data does not correspond to any coherent propagating signal). Including only cases where the variance is larger than one standard deviation does not qualitatively affect the linear regression analysis or the two-sample *t* test. Despite these limitations, the fact that the data still suggest a relationship between the geographic distribution of tropical cloudiness and the phase speed of CCKWs assures the robustness of the coupling between CCKWs and the ITCZ through compensation between adiabatic cooling and latent heat release.

There are a number of other potential coupling mechanisms between the ITCZ and CCKWs that were not addressed here. For instance, the analysis in Dias and Pauluis (2009) suggests that the ITCZ expands (contracts) during the ascent (subsidence) phase of the wave. It would be interesting to find observational evidence of this modulation. Another related question is of whether Kelvin waves can broaden the ITCZ enough to set their own propagation speed. Furthermore, the differences between the horizontal and vertical structures of the slower and faster CCKWs and, more generally, the relationship between CCKWs and the background flow are other relevant topics that merit further observational and theoretical investigation.

## Acknowledgments

We thank Paul Roundy and two anonymous reviewers for constructive feedback that helped to clarify the manuscript. The authors also thank George Kiladis for his comments and suggestions for the present paper. The work presented here is part of the first author’s Ph.D. thesis. Juliana Dias was partially supported by the New York University Dean’s Dissertation Fellowship. This work was also supported by NSF under Grant ATM-0545047.

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