RESEARCH LETTER Key Points: CCEW phase speeds tend to shift as one would expect based on Doppler effects Observed CCEW amplitudes vary strongly by season and basin Observed CCEWs equivalent depths are similar across seasons and basins Supporting Information: Readme Figures S1 S27 Correspondence to: J. Dias, juliana.dias@noaa.gov Citation: Dias, J., and G. N. Kiladis (2014), Influence of the basic state zonal flow on convectively coupled equatorial waves, Geophys. Res. Lett., 41, 6904 6913, doi:. Received 08 AUG 2014 Accepted 12 SEP 2014 Accepted article online 17 SEP 2014 Published online 6 OCT 2014 Influence of the basic state zonal flow on convectively coupled equatorial waves Juliana Dias 1,2 and George N. Kiladis 2 1 Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado, USA, 2 NOAA Earth System Research Laboratory, Physical Sciences Division, Boulder, Colorado, USA Abstract Observational data are used to test the hypothesis that the basic state modulates the dispersion properties of convectively coupled equatorial waves (CCEWs). This hypothesis is based on shallow water theory, which predicts that the zonal speed of propagation of such tropospheric equatorial modes is altered by the equivalent depth and the basic zonal flow. Localized space-time spectra are calculated to investigate how CCEW spectral peaks vary across the tropics and how they are affected by the variations in zonal wind observed geographically and by season. Doppler shifting by the basic state barotropic zonal flow is readily identified. Once this Doppler shifting is taken into account, the equivalent depths of CCEWs inferred from global power spectra are surprisingly uniform, both geographically and temporally. However, there are also detectable modulations that appear consistent with changes in vertical shear of the zonal flow, along with other shifts that are not as easily explained. 1. Introduction Convectively coupled equatorial waves (CCEWs) represent the leading modes of synoptic scale-organized convection in the tropical troposphere. By means of wave number-frequency spectral analysis of satellite cloudiness data, Takayabu [1994], Wheeler and Kiladis [1999], and Wheeler et al. [2000] identified the mean relationship between the scale and structure of CCEWs and tropical rainfall. In these studies, the dynamical structures of CCEWs were shown to largely correspond with that expected from the theory of equatorially trapped shallow water modes developed by Matsuno [1966], and it was established that the spectral peaks of tropical cloudiness line up with the dispersion curves associated with the predicted normal modes, such as equatorial Rossby waves (ER), Kelvin waves (KW), mixed Rossby-gravity waves (MRG), n = 0 eastward inertio-gravity waves (EIG0), and westward inertio-gravity waves (WIG). Matsuno s theory is based on a vertical decomposition of the primitive equations linearized about a basic state of rest, which reduces the problem to a set of independent linear shallow water equations associated with each vertical mode. The equivalent depth (H eq ) is the key parameter that links the vertical mode to the horizontal structure, that is, the zonal gravity wave speed and meridional structure of the shallow water solution. Observed CCEW spectral peaks are consistent with H eq in the range 12 50 m, which is much shallower than the peak projection response to deep convective heating that implies an H eq of about 200 m [e.g., Fulton and Schubert, 1985]. Despite decades of study, there is still no generally accepted theory for the reduction of H eq due to convective coupling (see review by Kiladis et al. [2009]). One set of ideas suggests that the observed H eq is more consistent with the second baroclinic mode than the first baroclinic mode implied by Matsuno s theory. Cloud-resolving model studies [Tulich et al., 2007; Tulich and Mapes, 2008; Kuang, 2010] and theoretical work [Mapes, 2000;Majda and Shefter, 2001;Raymond and Fuchs, 2007; Kuang, 2008] suggest that the shallower H eq is dominant because CCEWs are triggered by low-level moist convective processes (e.g., shallow convection) that typically leads deep convection. Another theory for the observed H eq states that the effect of convection on large-scale waves is to reduce the effective static stability of the atmosphere [Neelin and Held, 1987; Neelin and Yu, 1994; Emanuel et al., 1994; Frierson et al., 2004; Raymond et al., 2009], which, in turn, reduces the phase speed of the waves, along with altering their scale and horizontal structure. In this view, deep convective heating is coupled to the first baroclinic structure of Matsuno s modes within CCEWs. Aside from moist convective processes, equatorial waves can also be affected by the basic flow in which they are embedded. The simplest effect is due to Doppler shifting, and there is observational evidence that this is important regionally [Yang et al., 2003, 2007]. Theoretical studies have shown that the inclusion of a zonal DIAS AND KILADIS 2014. American Geophysical Union. All Rights Reserved. 6904
basic flow alters the dynamics of equatorially trapped waves in fairly complex ways beyond simple Doppler shifting [Kasahara and da Silva Dias, 1986; Webster and Chang, 1988; Zhang and Webster, 1989, 1992; Zhang, 1993; Wang and Xie, 1996; Stechmann and Majda, 2009; Han and Khouider, 2010]. An interesting aspect of these studies is that the degree and nature of the impact of zonal basic flows on the structure of equatorial waves varies mode by mode. For instance, in the case of a constant basic flow, Zhang and Webster [1989] show that non-doppler effects are negligible for the higher frequency modes, while substantially affecting the low-frequency modes such as ERs. ER dynamics are also more sensitive to vertical shear than higher frequency modes such as KWs [Wang and Xie, 1996], and meridional shear can lead to equatorial wave instabilities [Han and Khouider, 2010]. In addition, the effects of meridional shear in altering shallow water modes tend to increase with increasing zonal wave number [Zhang and Webster, 1989; Ferguson et al., 2009]. Despite the potential for large impacts on the dispersion relationships and structures of CCEWs by the zonal flow, it is an observational fact that observed tropical precipitation power spectra are in close agreement with Matsuno s dispersion curves for a resting basic state when using global data. Thus, the impacts of basic flow variations must be subtle, and here we attempt to isolate the effects of regional and seasonal changes in the zonal basic state. 2. Data and Methods 2.1. Regional Space-Time Power Spectra We use 1983 2009 Cloud Archive User Services (CLAUS) brightness temperature (T b )data[hodges et al., 2000] and ERA-Interim reanalysis (ERAI) [Deeetal., 2011] to represent deep convection and the large-scale circulation. Our results are completely reproducible using outgoing longwave radiation, except that the higher-frequency WIGs are better resolved by CLAUS. Tropical Rainfall Measuring Mission data also yield consistent results. Regional space-time power spectra (RPS) of T b localized by longitudinal sectors are calculated following the tapering method from Dias et al. [2013a], where data are tapered in time and space using a Hann window. The longitudinal sectors are referred to by their middle longitude, such that S j [lon j lonw 2, lon j + lonw 2] (1) where the index j refers to a longitude between 0 and 360. In the next section we use intervals of 90 with lonw = 90 starting at 0, and latitudinal averages are calculated from 15 Sto15 N. Different sectors were tested, and altering these choices does not affect our main conclusions. A Fourier transform in time and longitude is calculated using the T b windowed data, yielding RPS i j (k,ω).asin Dias et al. [2013a], for a given location (lon j ), date (i), planetary wave number (k), and frequency in cycles per day (ω), we derive a time series of space-time spectral power by sliding the temporal time-window throughout the entire record. The background RPS are calculated in the same way as in the global power spectrum (GPS) from Wheeler and Kiladis [1999], except that we calculate the background for each S j, and in the case of composites, only the RPS in the given sample are used to calculate the background. This is necessary because of the large differences in the raw spectra across seasons and sectors. We use the same tapering parameters as in Dias et al. [2013a]: the total longitude window is 120, and the time-window is 44 days. This approach is analogous to a spatial-temporal wavelet analysis [e.g., Kikuchi and Wang, 2010], and tests show that our results are not very sensitive to the choice of windowing function. The 120 width is wider than lonw, representing a compromise between spectral resolution of synoptic scales and data localization. The time-window is chosen to be substantially larger than the period of the lowest wave frequencies of CCEWs. To both summarize the results and facilitate comparisons, in addition to displaying the ratio between the total RPS and the corresponding background as is typically done, we display the mean frequency weighted by the square of this ratio. That is, for every wave number on the abscissa, the ordinate is the frequency given by ω(k) = r 2 (k,ω)ω(k) r2 (k,ω) (2) where r(k,ω) is the ratio of power above the background if that exceeds the background by 10%; this is set to zero otherwise. We calculate this mean for the symmetric and antisymmetric RPS and separately for the westward and eastward branches. The westward branch averages are also split into low and high frequencies, with a cutoff set at 3 days. The symmetric eastward estimate is calculated for periods shorter than DIAS AND KILADIS 2014. American Geophysical Union. All Rights Reserved. 6905
Figure 1. Space-time spectra of CLAUS T b where each panel shows the total power above the background (shading). Rows correspond to the (a, b) global spectra and the regional spectra for sectors centered at (c, d) 0,(e,f)90 E, (g, h) 180,and(i,j)90 W. Figures 1a, 1c, 1e, 1g, and 1i display the symmetric (antisymmetric) components. Symbols correspond to the weighted frequency calculated using (2) where the size of the symbol is proportional to the amount of power above the background (the smallest circle represents 10% of power above the background). Shading interval is 0.1, starting at 1.1. The dispersion curves are shown for H eq = 25 m with U = 0 ms 1 (black), U = 5 ms 1 (solid gray), and U = 5 ms 1 (dashed gray). DIAS AND KILADIS 2014. American Geophysical Union. All Rights Reserved. 6906
Table 1. Mean Zonal Barotropic Wind and Mean Shear by Seasons, Annual Mean, and 10th and 90th Percentiles of the Entire Record a Sector Zonal Wind S 0 S 90 E S 180 S 90 W December February (DJF) barotropic 0.1 2.4 2.0 0.3 U 200 hpa U 850 hpa 13.6 2.6 7.8 16.3 March May (MAM) barotropic 0.6 1.4 0.9 0.8 U 200 hpa U 850 hpa 15.3 0.3 11.7 12.5 June August (JJA) barotropic 0.8 3.5 3.3 2.3 U 200 hpa U 850 hpa 10.1 13.6 6.6 7.8 September November (SON) barotropic 0.0 2.9 2.0 1.3 U 200 hpa U 850 hpa 15.2 4.6 7.6 7.3 Annual barotropic 0.1 2.6 2.1 0.9 U 200 hpa U 850 hpa 13.6 5.2 8.3 10.9 90% barotropic 0.6 1.1 0.4 0.9 U 200 hpa U 850 hpa 17.1 2.7 14.6 17.5 10% barotropic 0.9 3.9 3.7 2.7 U 200 hpa U 850 hpa 9.8 14.0 2.5 4.1 a Each column corresponds to the mean over the sector indicated in the top row. 20 days and k > 3 to separate KWs from the Madden-Julian oscillation (MJO). The purpose of this calculation is to smoothly follow r(k,ω) maxima in order to compare how closely they follow Matsuno s dispersion curves. Other methods, such as tagging the actual maximum power above the background, yield nearly identical but noisier results. To illustrate the approach, Figure 1 shows the GPS as well as the RPS for sectors S 0, S 90 E, S 180, and S 90 W, where RPS is the mean over 44 day segments overlapping by 10 days, using the entire record. Matsuno s dispersion curves are overlaid for H eq = 25 m (gravity wave speed of about 15 m s 1 ) assuming a basic state of rest, along with the dispersion curves Doppler shifted by U =±5ms 1. These curves indicate that basic state advection alone should result in substantial shifts in the spectral signals, as discussed in detail by Zhang and Webster [1989] and Yang et al. [2003]. Figure 1 is discussed further in section 3. 2.2. The Basic Zonal Flow To assess the relationship between the background zonal flow and the local spectral power, we calculate the mean ERAI zonal velocity between 15 S and 15 N within each S j and then calculate a 21 day running mean. We have also tested a zonal wind latitudinally weighted by T b variance, but that did not impact our main conclusions. The running average was chosen to be larger than observed periods of CCEWs in order to represent the background mean state. Once again, the main conclusions are not sensitive to either the length of the running averaged choice, or an alternative definition using a high-pass filter (not shown). Table 1 summarizes the annual and seasonal mean zonal barotropic wind, mass weighted from 1000 hpa to 100 hpa, and the vertical shear defined as zonal wind at 200 hpa minus that at 850 hpa, along with their 10th and 90th percentiles for the entire record. We note that the barotropic flow is nearly always weak or easterly. 3. Results 3.1. Global Versus Regional Power Spectra Figure 1 clearly illustrates that CCEW amplitude varies geographically in a substantial way, as has been shown in previous work [Kiladis et al., 2009, and references therein]. In contrast to these studies, one advantage of our technique is that it allows for a varying background in space and time. Figure 1 shows that while KWs and ERs are seen in all sectors, MRGs and EIG0s are much more predominant at S 90 E and especially at S 180. Only very weak EIG0 and MRG signals stand above the background at S 0 and S 90 W, respectively. Similarly, the global n = 1 WIG signal is primarily seen at S 90 E, whereas the n = 2 WIG signal is only visible at S 180. In addition, ERs are stronger at S 90 E and S 180, and even though our frequency cutoff is close to the MJO period of around 45 days, its spectral peak is most prominent at S 90 E, as expected. In contrast to the amplitude variations, the scaling of the waves is remarkably stable across the globe despite the large differences in the basic state. This is evident even in the implied deviations from linear theory seen for individual waves. For example, KWs are seen to be weakly dispersive in all sectors, with their spectral signals closest to the resting state 25 m dispersion relation at low wave numbers, curving toward DIAS AND KILADIS 2014. American Geophysical Union. All Rights Reserved. 6907
lower H eq (slower gravity wave speeds) at higher frequencies and wave numbers. A similar deviation with respect to the 25 m dispersion curve is seen for EIG0s in Figure 1h, while the opposite tendency of higher equivalent depths at higher frequencies is observed for MRGs, n = 2 WIGs, and especially for n = 1 WIGs in the sectors where their signals are detectable. Theory, as shown by a comparison with the Doppler-shifted dispersion curves, dictates that zonal advection cannot explain these wave number-dependent deviations from the dispersion relationships, which must be due to vertical and/or horizontal shear or other nonlinear effects. A possible exception is the ER signal, which closely follows the 25 m resting curve at S 90 E and more uniformly deviates toward higher H eq in the other sectors, especially at S 0. Even there, however, there is a hint of a departure from the linear dispersion at the higher wave numbers, this time toward lower H eq. 3.2. Seasonal Regional Power Spectra To investigate RPS seasonal dependence, Figure 2 displays the weighted frequencies (ω) calculated using equation (2), except here we split the data by seasons (DJF for December February, etc.). Also shown in the Figures 2c, 2f, 2i, and 2l are the vertical profiles of the zonal wind for each season and sector. Figure 2 shows that many of the departures from the linear dispersion curves noted in Figure 1 persist throughout the year, but there are also substantial deviations in scale and amplitude tied to the seasonal cycle. For instance, the n = 1 WIG is detectable in all sectors but primarily in MAM (green stars), whereas the n = 2 WIG is only strong at S 180 but is seen in all seasons, along with a weak signal at S 90 E during all seasons except JJA. KWs again appear to be weakly dispersive across seasons as well as across regions, and EIG0s and MRGs, which are absent in the S 90 W annual climatology shown in Figure 1, appear in the MAM RPS. We also note that there is a tendency for EIG0s and KWs to propagate more slowly during JJA in comparison to MAM and DJF, particularly at S 180, with an opposite tendency for n = 2 WIGs. This is consistent with the stronger barotropic easterlies seen then in the zonal wind vertical profile and in Table 1. Significantly, the observed H eq for each mode deviates only slightly from 25 m across seasons when the zonal flow is taken into account, especially for lower wave numbers. Changes in amplitude are not so easily accounted for, however. For example, the dominance of WIGs during MAM at 0 cannot be simply explained by the zonal wind profile, since this varies little over the annual cycle at this location. 3.3. Doppler Shifting by the Barotropic Flow Observed doppler shifting of CCEWs by the basic flow have been noted by Yang et al. [2003, 2007]; however, their analysis was based on a very short period. Here we apply the RPS technique to test this hypothesis more thoroughly in space and time. Figure 3 is similar to Figure 2, except that it shows ω for two subsets of dates: strong easterlies (blue symbols) and weak basic flow (red symbols). Strong easterlies (SE) are defined as when the mean zonal barotropic wind over all dates is less than its 10th percentile. Similarly, weak basic flow (WE) is defined as all dates where the barotropic wind exceeds the 90th percentile, which tends to be close to zero (Table 1). Figures 3c, 3f, 3i, and 3l display the zonal wind vertical profile averaged over SE and WE dates. One conclusion that can be immediately drawn from Figure 3 is that there is evidence for the expected Doppler shift by the barotropic flow for KWs and ERs in all sectors and for n = 2 WIGs, EIG0s, and MRGs at S 180. For example, the difference in the barotropic flow between WE and SE is about 3 m s 1 in all sectors except S 0 (Table 1), and this is manifested as a general shift in the signal toward lower frequencies (downward) for eastward waves and toward higher frequencies for westward waves. As might be expected, the difference between ω is minimized at S 0, which is consistent with this sector having the smallest difference between SE and WE. Interestingly, the dispersion characteristics of n = 1 WIGs at S 90 E do not seem to be affected by the barotropic flow, in that the observed frequencies cross through the U = 0 dispersion curve even in the WE case. Note also that the implied amount of phase speed shifts inferred from the RPS are more consistent with the differences in the barotropic flow than with just the much larger upper level flow differences shown in Figures 3c, 3f, 3i, and 3l. These composites are calculated using data from all seasons, but the phase speed differences can be seen within seasons as well (not shown). There is also some evidence of Doppler shifting when comparing KW ω across sectors. For instance, during SE periods, KWs propagate substantially more slowly in all sectors except for S 0, which also has the smallest difference in the barotropic flow. It is notable that this signal is more noticeable for higher-frequency KWs, again supporting the notion that higher wave numbers are more affected by non-doppler effects. The large separation between the blue and red ER symbols at S 90 W in Figure 3j cannot be entirely explained based on Doppler shifting by the barotropic flow. In this case, while the SE zonal wind profiles at S 180 (Figure 3i) and S 90 W (Figure 3l) are both nearly barotropic and very similar to one another, ER peaks match DIAS AND KILADIS 2014. American Geophysical Union. All Rights Reserved. 6908
Figure 2. (a, b, d, e, g, h, i, and k) Similar to Figure 1, except that only the weighted frequencies are displayed by season. (c, f, i, and l) The mean vertical wind profiles by sectors and seasons. In Figures 2a 2l, rows correspond to sectors 0, 90 E, 180,and90 W where blue correspond to DJF, green to MAM, red to JJA, and brown to SON. DIAS AND KILADIS 2014. American Geophysical Union. All Rights Reserved. 6909
Figure 3. Similar to Figure 2, except that blue (red) correspond to a composite over strong easterlies (weak basic flow) as defined in the text. The mean barotropic flows in each subset are shown in Table 1. DIAS AND KILADIS 2014. American Geophysical Union. All Rights Reserved. 6910
more closely with the U = 5ms 1 Doppler-shifted dispersion curve at S 180 when compared to those at S 90 W. However, as mentioned earlier, it is likely that we are seeing substantial non-doppler effects in the case of ERs, which are more significant for those waves [Zhang and Webster, 1989;Wang and Xie, 1996]. Note that the SE profile at S 90 W is similar to the JJA mean in Figure 2l, whereas the WE profile is closest to the DJF mean. Similar shifts in the spectra are seen so that part of the signal in Figure 3j is likely due to the seasonal cycle. It is important to emphasize that not just the zonal flow but also the thermodynamic basic state in each subset of dates is expected to be very different, because the average JJA precipitation in the eastern Pacific is much higher than in DJF. The contrast between zonal flow effects on EIG0s and MRGs is also suggestive. For example, in the SE case at 180 (Figure 3h), MRG and EIG0 signals closely follow the U = 5.0 ms 1 dispersion curve at all wave numbers, consistent with the relatively large 3.9 m s 1 barotropic component for that case. However, for WE when the barotropic flow is 0.4 m s 1, ω is close to the U = 0 ms 1 dispersion curve between k = 4 and k = 8, and beyond that, the EIG0 (MRG) ω shifts toward lower (higher) values. The pronounced smaller-scale MRG departure from the dispersion curve persists for WE at S 180, as well as in the case of seasonal means (Figure 2h). We note the possibility that these higher wave number westward spectral peaks may be due to easterly waves, in which case, our results suggest that these are also Doppler shifted as would be expected due to the basic flow difference. 3.4. Is There a Steering Level for Equatorial Waves? Evidence in the previous section supports the notion of Doppler shifting of the dispersion curves by the barotropic flow in which CCEWs are embedded. While Figures 1 3 show that an H eq of around 25 m is fairly robust across seasons and basins, at least for low wave numbers, the zonal wind vertical profiles show substantial variability in space and time, especially at upper levels. That raises the question of whether CCEWs are preferentially advected by the flow at any one level, as is the case of the midtropospheric steering level for extratropical baroclinic waves. For example, KWs appear to be more influenced by the upper level flow at S 90 E in Figures 2d and 3d since in both these cases their spectral signals shift as expected toward lower H eq when upper level easterlies are stronger, even though the low-level flow changes are in the opposite direction. To further investigate the effects of the flow at various levels, two subsets of dates in each sector are analyzed analogously to the barotropic flow case. One contains all dates where the zonal basic flow at a given level is weaker than the 10th percentile, and the second one where the zonal basic flow at that level exceeds the 90th percentile. In the supporting material, we provide composite spectra for the extremes in zonal flow at all individual levels between 100 and 1000 hpa. Here it can be seen that the higher wave number MRG shifts in Figure 3h are more consistent with changes in the low-level wind, a result in agreement with that obtained by Liebmann and Hendon [1990]. At the same time, the opposite appears to be true for the KWs, that is, stronger upper level westerlies above about 600 hpa are associated with faster eastward propagation. Perhaps significantly, overall changes in propagation speeds appear to be more easily detected in the case of the weakly dispersive KW and for the EIG0 at higher wave numbers where their dispersion relation is more linear and they behave more like gravity waves. While the mean frequencies do not suggest a universal steering effect by the zonal basic flow at any particular level for all CCEWs, the studies discussed in section 1 suggest that it is likely that the vertical shear of the zonal flow influences the preferential scale of CCEWs as well as their amplitude. We find some evidence that, for instance, the spectral signal associated with n = 1 and n = 2 WIGs are associated with stronger low-level (below 700 hpa) easterlies (see the supporting information), which is in agreement with Tulich and Kiladis [2012]. Interestingly, KW, EIG0, MRG, and ER amplitudes do not seem to be systematically affected by the strength of shear, especially for k < 8. We have reproduced our calculations by seasons separately and confirmed the Doppler shift effect and the lack of a universal steering level, along with significant amplitude variations by sector. We have also tested varying window lengths and the use of a band-pass filter instead of a running mean to calculate spectra, along with alternate techniques to estimate the background spectra, and found that Doppler shifting along with the other signals described above are extremely robust. While beyond the scope of the present work, the authors plan an in depth analysis of other effects of vertical and meridional shear in comparison to theory in a future work. DIAS AND KILADIS 2014. American Geophysical Union. All Rights Reserved. 6911
4. Discussion Our analysis shows a clear Doppler shifting effect on CCEWs by the basic barotropic zonal flow which amounts to phase speed variations of the order of a few meters per second. The RPS analysis shows that despite the fact that the basic flow and the amount of moist convection vary widely around the equator, an H eq for CCEWs of around 25 m is systematically observed across seasons and basins once Doppler shifting is taken into account, with a possible exception for ERs. While the observed changes in the dispersion relations are consistent with Doppler shifting, it can only explain relatively small shifts in the dispersion properties, leading to the conclusion that this is not the primary mechanism that reduces the theoretical first baroclinic gravity wave speed ( 40 m s 1 ) to the CCEWs-observed gravity wave speed ( 15 m s 1 ). Our results also suggest that the observed H eq is not systematically altered by the changes that shear at any level would imply. It is important to note that the largest departures from the Matsuno s dispersion curves occur at relatively large wave numbers. From a theoretical standpoint, this may not be too surprising given that in the case of KWs, nonlinear interactions due to the basic zonal flow [Ferguson et al., 2009] and moist convection [Dias and Pauluis, 2009] affect smaller-scale waves more strongly, and it is likely that other more divergent modes such as EIG0s and WIGs behave similarly. While different data sets and definitions of the main parameters affect our estimated frequencies, these changes are small compared to the nearly uniform H eq and the Doppler shifted signals by the barotropic flow. On the other hand, the amplitude of the power above the background associated with the estimated dispersion curves, that is, where in frequency and wave number the spectral peak is maximized, is more sensitive to these parameters, and therefore, it points to a potential limitation of our approach regarding the scale selection of CCEWs. Moreover, our analysis does not exclude the possibility that phase speeds are also modulated by moist processes such as shown in Roundy [2008], Dias and Pauluis [2011], Yasunaga and Mapes [2013], and Dias et al. [2013b]. The robustness of the H eq is consistent with theories that state that the speed of propagation of CCEWs is determined by normal modes of the tropical atmosphere, for instance, to first-order approximation, the second baroclinic mode from Matsuno s theory [Mapes, 2000;Majda and Shefter, 2001;Raymond and Fuchs, 2007;Kuang, 2008]. In this view where CCEW initiation strongly relies on the vertical profile of the tropical heating, our study supports the idea that the difficulty in reproducing the proper frequencies and scales of these waves in models [e.g., Lin et al.,2006]lies on the triggering mechanisms, rather than on the wave dynamics, since modeled rainfall and vertical heating profiles have large biases in the tropics. The present work raises interesting questions such as why certain modes such as KWs and ERs are ubiquitous in the tropics, whereas MRGs, EIGs, and WIGs are much more geographically constrained. We have also highlighted the striking but less easily explained relationship between ERs and the zonal flow, which the authors plan on investigating further. A practical contribution of this work is the development of a technique to assess the relationship between CCEWs and any other variable of interest. 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