4.1. Waves With Vertical Wavelengths ≤ 7 km
 PE due to short vertical wavelengths is often used as an indicator of the global distribution of gravity waves (i.e., EQWs and/or 3D-gravity waves) in studies based on GPS RO because PE can be calculated from observed temperature alone. Alexander et al. [2008b] reported the global distribution of PE with vertical wavelengths ≤ 7 km (PE ≤ 7 km). For comparison with their recently obtained results, we also calculated PE ≤ 7 km, with PE given by
Here we briefly explain the procedure for extracting short vertical wavelengths of temperature. First, the background temperature profile of was defined by the 2-month average at each grid point. The background temperature was then subtracted from the temperature T of an individual profile, and the linear trend was removed. Next, a cosine-type window was applied to both ends of the data to reduce spectral leakage, and a high-pass filter using fast Fourier transform (FFT) with a cut-off vertical wavelength of approximately 7 km was applied to obtain T′ with vertical wavelengths ≤ 7 km. The analyzed height ranges in this study do not contain the altitude where the window was applied.
 Alexander et al. [2008b] binned COSMIC data into horizontal grid cells of 20° × 5° to secure a sufficient number of data points before calculating PE, which allows for the study of waves with zonal wave number ≤ 9. In the AGCM, PE is calculated every 3 h using 2-month averaged background temperature at each grid point; these values are then averaged for 2 months in each of the four terms illustrated in Figure 1. Therefore the simulated PE consists of waves with periods from approximately 6 h to 2 months, vertical wavelengths ≤ 7 km, and horizontal wavelengths of ∼380 to 40,000 km at the equator. Therefore the simulated waves have much wider spectral ranges than those shown by COSMIC data, and the value of simulated PE ≤ 7 km should be larger than that observed by COSMIC GPS RO.
 Figure 4 shows longitude-height cross-sections of PE ≤ 7 km averaged from 10°N to 10°S for the four terms examined (2-month averages), during which the QBO was in westerly and easterly shear phases in Jan–Feb and Jul–Aug. Contour lines show the zonal wind. The zonal distribution of zonal wind was well simulated despite the slightly larger amplitude of the Walker circulation as compared with that in the real atmosphere. The longitudinal variation in outgoing long-wave radiation (OLR) averaged from 10°N to 10°S is shown below each figure. Low values of OLR (note that the vertical axis of the OLR is inverted) correspond to active cumulus convection. The convective activity is large over the Indian Ocean to mid-Pacific and at approximately 60–80°W. In the UTLS region, much larger PE elongates from east to west over the eastern hemisphere than in the western hemisphere, which is consistent with the COSMIC results. Larger PE is also distributed in the eastern hemisphere around the altitude where the zonal wind changes from easterly to westerly at approximately 30 hPa (Figures 4a and 4c), which was also observed in the COSMIC GPS RO data. Simulated PE ranging from 8 to 10 J kg−1 was observed at approximately 60–80°W in the UTLS region for all four terms, where a secondary maximum of convective activity occurs. The values of PE ≤ 7 km in this simulation were larger than those calculated using the COSMIC data, resulting from the much wider spectral ranges covered in the AGCM. A quantitative comparison is presented in section 4.3.
Figure 4. Longitude-height cross-section of PE (shaded) due to waves with vertical wavelengths ≤ 7 km and zonal wind (contour lines). (a, c) Westerly and (b, d) easterly shear phases of the QBO in (a, b) Jan–Feb and (c, d) Jul–Aug. The shading and contour intervals are 2 J kg−1 and 4 m s−1, respectively. The line graph indicates the zonal variation in 2-monthly mean OLR (W m−2). Note that the vertical axis of the OLR is inverted. These values are averaged from 10°N to 10°S.
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 To clarify which phase velocities of waves are dominant, PE as a function of zonal phase velocity relative to the ground (Cx) was calculated. Figure 5 shows the height distribution of PE ≤ 7 km as a function of Cx averaged from 10°N to 10°S for the four terms, which are shown separately as averages in the eastern hemisphere (0–180°E) and western hemisphere (0–180°W). The sign of zonal wind associated with the Walker circulation changes at approximately ∼180°E (Figure 4). The 2-month mean vertical profiles of zonal wind with vertical wavelengths > 7 km (i.e., background zonal wind for waves with vertical wavelengths ≤ 7 km) averaged in the eastern hemisphere/western hemisphere and 10°S–10°N are shown by thick solid lines. The background zonal wind changes in longitude, latitude, and time direction must influence the local propagation of waves. The two dashed lines show the averaged zonal winds plus/minus their standard deviations (i.e., root mean square of longitude, latitude, and time variance of the zonal wind). Note that some stratospheric PE in the western hemisphere (eastern hemisphere) originates from waves generated in the troposphere in the eastern hemisphere (western hemisphere) and then enters the western hemisphere (eastern hemisphere). However, these wave propagations do not critically alter the features mentioned below (zonal propagations of EQWs are shown in Figures 10 and 11).
Figure 5. Height distributions of PE ≤ 7 km as a function of zonal phase velocity (Cx) in the (left) eastern (0–180°E) and (right) western (0–180°W) hemispheres (10°S–10°N average). (a, b, e, f) Westerly and (c, d, g, h) easterly shear phases of the QBO in (a–d) Jan–Feb and (e–h) Jul–Aug. The shading interval is 0.1 J Kg−1 m−1 s. The 2-month mean vertical profiles of zonal wind with vertical wavelengths > 7 km averaged zonally in the (left) eastern hemisphere and (right) western hemisphere and meridionally from 10°S–10°N are indicated by thick solid lines; the two dashed lines indicate averaged zonal wind plus/minus the standard deviation. See text for details.
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 In the eastern hemisphere (left panels), the easterly (westward wind) associated with the Walker circulation allows most of the eastward waves to propagate from the upper troposphere to the stratosphere, whereas most of the westward-propagating waves with Cx of approximately −10 m s−1 are prevented from entering the stratosphere (some westward waves with Cx of ∼−10 m s−1 could propagate upward because of local changes in the zonal wind; see the width of the standard deviation). The situation is reversed in the western hemisphere. Most westward-propagating waves are not influenced by the westerly (eastward wind) in the upper troposphere.
 In addition to differences in the zonal wind profiles in the troposphere, the wave sources (cumulus convection) are generally larger in the eastern hemisphere than in the western hemisphere, which could result in larger PE in the eastern hemisphere than in the western hemisphere in the UTLS region (Figure 4). PE generally shows large values around the tropopause, which might partially result from a sharp cold point tropopause in the tropical regions [Alexander et al., 2008b]. In addition, the wave energy accumulation might occur around the tropopause because the vertical group velocity is inversely proportional to buoyancy frequency if the zonal wind does not change vertically [Andrews et al., 1987].
 In the stratosphere, waves are markedly influenced by the phase of the QBO. In the westerly shear phase of the QBO in the eastern hemisphere (Figures 5a and 5e), the dominant Cx of waves is ∼10 m s−1 from 200 to 50 hPa, but changes to ∼20 m s−1 at approximately 30 hPa. Eastward-propagating waves generated in the eastern hemisphere do not encounter the zonal wind until ∼30 hPa. Therefore most of the eastward waves can exist until 30 hPa. On the other hand, in the westerly shear phase of the QBO in the western hemisphere (Figures 5b and 5f), isolated large PE occurs at approximately 20–30 hPa with Cx of ∼20 m s−1. These Cx values are larger than those of the mean westerly associated with the Walker circulation in the upper troposphere (note also that some PE at this altitude originates from waves generated in the eastern hemisphere, as mentioned above). In the easterly shear phase of the QBO (Figures 5c, 5d, 5g, and 5h), PE with Cx of approximately −10 m s−1 is generally larger than that with Cx of approximately 10 m s−1 over 20–30 hPa. The PE distribution in the stratosphere generally agrees with the fact that temperature variances associated with zonally and vertically propagating gravity waves in background shear flow have a functional dependence of ∣ − Cx∣−1, where is background zonal wind. Nonlinear effects and dissipation would limit the growth close to the critical level, so that temperature variances maximize below [Randel and Wu, 2005 and references therein]. The results obtained from Figures 4 and 5 indicate that the distribution of the PE results from three factors: the source strengths, the Walker circulation, and the phase of the QBO.
 Alexander et al. [2008b] showed the global distribution of PE ≤ 7 km at 15, 22, 26, and 32 km in January 2007 and 2008, when the QBO was in its easterly and westerly shear phases, respectively. For both 2007 and 2008, they found similar PE distributions at 15 km, as well as similar OLR distributions. Large PE is visible from the Indian Ocean through the Western Pacific and over South America. However, the stratospheric PE is different between the different phases of the QBO. Around the altitude where the zonal wind is ∼0 m s−1, relatively large PE elongates more zonally in the westerly than in the easterly shear phase of the QBO. Another interesting point is that large PE is found over South America in both the easterly and westerly shear phases of the QBO.
 Figure 6 shows the global distribution of PE in Jan–Feb at 100 hPa (Figures 6a and 6b) and the altitude at which zonal wind is approximately 0 m s−1 at the equator in the westerly (approximately 32–35 hPa; Figure 6c) and easterly (approximately 30–32 hPa; Figure 6d) shear phases of the QBO. The PE distributions at 100 hPa do not differ much between the two phases; large PE is observed mainly from the Indian Ocean to the western Pacific, and PE of approximately 8–16 J kg−1 is located over South America, the eastern Pacific, and the Congo Basin. These results are qualitatively similar to those observed by COSMIC. The PE distribution is quite different between the westerly and easterly shear phases of the QBO for which zonal wind is ∼0 m s−1 (Figures 6c and 6d). Zonally elongating PE with large values occurs from 10–330°E (30°W) above the equator in the westerly shear phase. During the easterly shear phase, relatively small values of PE elongate around the equator and are more scattered than those in the westerly shear phase. Large PE is observed over South America in both the westerly and easterly shear phases. The illustrated PE consists of both EQWs and 3D-gravity waves. The PE with a meridional symmetric mode may be due to EQWs, whereas that scattered in the latitudinal direction is probably due to 3D-gravity waves, as discussed in section 4.3.
Figure 6. Global distribution of PE ≤ 7 km in Jan–Feb at (a, b) 100 hPa, and (c) 32–35 hPa and (d) 30–32 hPa where zonal wind (contour lines) is approximately 0 m s−1 at the equator. (a, c) Westerly and (b, d) easterly shear phases of the QBO. The shading interval is 2 J kg−1 for (a, b) and 1 J kg−1 for (c, d). The solid and dashed lines illustrate westerly and easterly, and thick solid lines show 0 m s−1. The contour interval is 10 m s−1.
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4.2. Equatorial Trapped Waves
 To investigate what kind of EQWs appear in the model, Figure 7 shows the zonal wave number versus the frequency spectra of the symmetric and antisymmetric components of temperature for the westerly shear phase in Jan–Feb. Spectra of symmetric and antisymmetric components were computed separately for each latitude before averaging from the equator to 10° (i.e., equivalent to the average of 10°S–10°N). Note that temperature data were nonfiltered in both space and time before calculation of the spectra (i.e., the vertical wavelength filter used in section 4.1 was not applied). In the stratospheric temperatures, the spectrum is not the red noise-like background spectrum dominating in OLR data or precipitation data [e.g., Wheeler and Kiladis, 1999]. Therefore symmetric and antisymmetric spectra were not divided by the background spectra.
Figure 7. Zonal wave number versus the frequency spectra of the (a, c) symmetric and (b, d) antisymmetric components of temperature for the westerly shear phase in Jan–Feb at (a, b) 83–35 hPa and (c, d) 35–15 hPa. Spectral units are log10(K2 wave number−1 cpd−1). Dispersion curves for symmetric components include Kelvin waves, n = 1 EIGWs and WIGWs, and n = 1 equatorial Rossby waves, whereas antisymmetric components include MRG waves, n = 0 EIGWs, n = 2 EIGWs and WIGWs, and n = 2 equatorial Rossby waves. Equivalent depths are selected as 8, 12, 25, 50, and 90 m, which are the same as in Figure 3. The effect of background wind is neglected to draw the dispersion curves of the equatorial waves. Note that the ranges of shading are different between the symmetric and antisymmetric components. Smoothing was conducted with a 1-2-1 moving mean in frequency.
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 The altitude ranges were selected from 83–35 hPa and 35–15 hPa. The boundary of 35 hPa corresponds roughly to the altitude at which the zonal wind changes from easterly to westerly. Dispersion curves for symmetric components include Kelvin waves, n = 1 EIGWs and WIGWs, and n = 1 equatorial Rossby waves, whereas antisymmetric components include MRG waves, n = 0 EIGWs, n = 2 EIGWs and WIGWs, and n = 2 equatorial Rossby waves. Equivalent depths were selected as 8, 12, 25, 50, and 90 m, as in Figure 3 (note that the frequency range was extended to ∼1.1 cpd because 3-hourly data were used to calculate the spectra). The effect of background wind was neglected when drawing the dispersion curves of the equatorial waves.
 For zonally propagating waves, the intrinsic frequency is written as follows:
where ω, k, and are the ground-based frequency, zonal wave number, and background zonal wind, respectively. The intrinsic frequencies and vertical wavelengths change because changes with altitude. On the other hand, the ground-based frequency ω of a wave is defined at the wave source level. Note that the ground-based frequency ω and zonal wave number k do not change with altitude, assuming a slowly varying background wind field, even through the background wind changes with altitude; the distribution of the zonal wave number versus the frequency spectra is independent of altitude. The distribution of the spectra would be changed only if a wave were to undergo critical level filtering and/or wave dissipation. Therefore in Figure 7 we selected the same dispersion curves of EQWs as those of the sources (Figure 3) to compare spectral features in the stratosphere with those of the sources. A more detailed discussion was presented by Ern et al. [2008, and references therein]. We did not use a vertical wavelength filter to preserve this wave property.
 Clear signals of Kelvin waves, MRG waves, n = 0 EIGWs, and n = 1 equatorial Rossby waves can be seen in Figure 7. The peaks corresponding to n = 1 EIGWs became much clearer when the same spectral figures were drawn using meridional wind data in which Kelvin waves do not appear (not shown). Note that the spectral distributions of the temperature at 83–35 hPa (Figures 7a and 7b) were relatively similar to those of precipitation (Figure 3) in the range of 8 ≤ he ≤ 90 m. These results imply a connection between stratospheric EQWs and tropospheric wave sources of convectively coupled EQWs. On the other hand, the wave spectrum also shows distributions outside the he of 8–90 m, especially at 35–15 hPa (Figures 7c and 7d), which are likely independent of convectively coupled EQWs. Waves with large equivalent depth could be an indication of processes involving longer vertical scales in the troposphere, which become visible at higher altitudes due to a decrease of atmospheric density and an increase in statistic stability [Ern et al., 2008]. The powers of Kelvin waves and n = 0 EIGWs with small equivalent depths (he ∼ 8 m) decreased with height, implying that these waves interact with the mean zonal wind. There are other spectral peaks with periods of approximately 1 day and a wide zonal wave number range in both symmetric and antisymmetric components. These wave numbers may correspond to the tide and 3D-gravity waves generated by the diurnal cycle of convection [Kawatani et al., 2003].
 To investigate the global distribution, sources, and propagation of EQWs, an equatorial wave filter was constructed. Kelvin waves, MRG waves, n = 0 EIGWs, n = 1 and n = 2 EIGWs and WIGWs, and n = 1 and n = 2 equatorial Rossby waves were extracted separately for further examination according to the methods of Wheeler and Kiladis . To avoid degradation of the actual amplitude of EQWs during the analyzed periods, we included data from both before and after each analyzed period. For example, when the EQWs in January–February were extracted, the last few days of December and the first few days of March were included. Then, a cosine-type window was applied only to the data from December and March. After applying FFT to the combined data, only the data from 1 January to the end of February were extracted.
 Figure 8 shows the spectral ranges between the lines indicating equivalent depths of 8 and 90 m under the assumption of zero background wind in the zonal wave number-frequency domain for odd and even modes. The equivalent depths of 8 to 90 m correspond to vertical wavelengths of 2.3 and 7.6 km with N2 = 6 × 10−4 s−2, which is the representative value in the lower stratosphere (∼80 hPa). The vertical wavelength range is similar to that used in the PE calculation in Figures 4–6. The zonal wave numbers of the EQWs were selected from 1 to 9 according to the method of Alexander et al. [2008b]; this range corresponds to zonal wavelengths from 4444 to 40,000 km over the equator. The minimum period was set at 1.1 day (∼0.9 cpd) to avoid including waves with a period of 1 day.
Figure 8. Equatorial wave filter for (a) odd and (b) even modes. Superposed are the dispersion curves of each EQW for two equivalent depths of 8 and 90 m. Hatched areas between the two lines are the filtering range of (a) Kelvin, n = 1 EIGWs and WIGWs, n = 1 equatorial Rossby waves, (b) MRG waves, n = 0 EIGWs, n = 2 EIGWs and WIGWs, and n = 2 equatorial Rossby waves. The minimum period is set at 1.1 day (∼0.9 cpd) to avoid including waves with a period of 1 day. Some overlapping areas occur between the Kelvin waves and n = 1 EIGWs and between n = 0 and n = 2 EIGWs.
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 Ern et al.  reported that EQW activity in a slow phase speed wave band between equivalent depths of 8 and 90 m was mainly modulated by the QBO and that higher equivalent depths (90–2000 m) showed less pronounced variation due to the QBO but more variation due to the stratopause SAO. As we could analyze data up to approximately 32 km in this model, we focused on wave activity related to the QBO. Therefore the selected wave band was set between 8 and 90 m.
 In this section, we describe application of the equatorial wave filter to unfiltered data (i.e., original data with no temporal or spatial filtering). The propagation properties of EQWs in relation to background winds from the troposphere to the stratosphere can be investigated using this filter since the ground-based frequency ω and zonal wave number k of a wave do not change unless a wave has dissipated, as mentioned above. The spectral range of n = 1 EIGWs may include some fast Kelvin waves (he > 90 m) with periods of less than ∼3 days (see Figures 7a, 7c, and 8a). However, most fast Kelvin waves do not exist in the filtering range of n = 1 EIGWs. In fact, fast Kelvin waves have periods of approximately 6–10 days [Hirota, 1978; Hitchman and Leovy, 1988]. Ultra-fast Kelvin waves with periods of 3–4 days [Salby et al., 1984] are dominant at much higher altitudes than those examined here. There are some areas of overlap between the Kelvin waves and n = 1 EIGWs, and between n = 0 and n = 2 EIGWs.
 The amplitudes of the extracted wave component in temperature associated with Kelvin waves and MRG waves are ∼2–3 K and ∼1 K, respectively. These results were consistent with those from COSMIC GPS RO data [Alexander et al., 2008b] and SABER data [Ern et al., 2008]. We confirmed that the period, zonal phase speed relative to the ground, and vertical wavelength were also simulated realistically in the model. Not only temperature, but also the zonal and meridional wind and geopotential height of EQW components, were extracted using the equatorial wave filter. In addition, we confirmed that the spatial structures of extracted EQWs in the stratosphere agreed with those derived theoretically by Matsuno .
 To investigate source distributions of these EQWs, the equatorial wave filter was also applied for OLR data. The spectral characteristics of the zonal wave number versus the frequency spectra of OLR (not shown) are similar to those of precipitation shown in Figure 3. Signals of convectively coupled EQWs that appeared in OLR data were extracted using the same equatorial wave filter shown in Figure 8. Therefore we can investigate the wave source distributions, propagations, and global distributions of EQWs.
 Figure 9 illustrates the global distribution of OLR variance due to each convectively coupled EQW component and the PE associated with each EQW at 32–35 hPa in Jan–Feb during the westerly shear phase of the QBO. The variances of equatorial Rossby waves with equivalent depths of 8–90 m are much larger in the upper troposphere than in the stratosphere. The connection of the equatorial Rossby waves between the upper troposphere and stratosphere is not very clear (not shown). The vertically propagating responses of convectively coupled n = 1 equatorial Rossby waves are confined to within a few kilometers of the wave forcing (see discussions by Wheeler et al., 2000 and references therein). Therefore EQWs without equatorial Rossby waves are discussed in this section. Wheeler et al.  reported the global distributions of OLR variance due to each EQW for northern/southern summer using data for a period of ∼17 years (see their Figure 3). They also presented the temporal variation of OLR variance for each EQW. Our results shown in Figure 9 are from 2-month averaged fields, and the range of the equatorial wave filter is slightly different from that used by Wheeler et al. . Therefore it is not necessary for the model results to be identical to the observational climatological results. Nevertheless, the distributions of simulated OLR variances are similar to those observed, and the simulated variance of EQWs is in the range of the time variation of observed OLR variance.
Figure 9. Global distribution of (left) OLR variance due to each convectively coupled EQW component and (right) PE due to each EQW at 32–35 hPa in Jan–Feb during the westerly shear phase of the QBO. (a, b) Kelvin waves, (c, d) MRG waves, (e, f) n = 0 EIGWs, (g, h) n = 1 EIGWs, (i, j) n = 1 WIGWs, (k, l) n = 2 EIGWs, and (m, n) n = 2 WIGWs. The shading intervals are 15 W2 m−4 for (a), 5 W2 m−4 for (c), 3 W2 m−4 for (e, g, i, k, m), 0.5 J kg−1 for (b), and 0.05 J kg−1 for (d, f, h, j, l, n).
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 Wheeler et al.  reported that waves of relatively short vertical wavelength (≤10 km) are indeed important for the coupling of large-scale dynamics and convection. Therefore the global distributions of PE due to EQWs shown in Figure 9 may be related to the activity of convectively coupled OLR. Convectively coupled EQWs would be a sufficient condition for the generation of EQWs propagating into the stratosphere. Concerning MRG waves, Magaña and Yanai  discussed that extratropical wave propagating into the tropical region is another possible source of MRG waves. In general, large PE associated with EQWs is observed from the equator to 10°N–10°S. For Kelvin waves, large PE in the stratosphere (Figure 9b) is generally located to the east of source distributions (Figure 9a), suggesting that Kelvin waves generated in the upper troposphere propagate eastward. In contrast, larger PE of n = 0 EIGW (Figure 9f) occurs over the Indian Ocean despite larger variance in OLR (Figure 9e) in the Pacific than in the Indian Ocean. It was also interesting that the distributions of large PE with MRG waves and other EQWs do not correspond directly with those of the large variance in OLR.
 EQWs propagate eastward or westward from the source region and should be influenced by the background zonal wind. To investigate the propagation of EQWs, 2-month averages of zonal and vertical energy fluxes () (overbars denote the 2-month average) were used. The ratio of energy flux to energy density is equivalent to the ratio of wave action flux to wave action density when Wentzel-Kramers-Brillouin (WKB) theory holds in space and time. A number of previous studies have provided detailed discussions of wave action flux and wave action density [Dunkerton, 1983; Andrews et al., 1987, and reference therein]. The direction of wave group propagation is expressed by energy flux F to energy density E as follows [cf. Gill, 1982]:
where E is wave energy (kinetic energy plus potential energy) per unit mass, and gx and gz represent zonal and vertical intrinsic group velocity, respectively. Because the energy fluxes are parallel to the intrinsic group velocity of gravity waves in the context of WKB theory, the sign of energy flux shows the direction of gravity wave propagation relative to the mean wind.
 Longitude-height cross-sections of PE and zonal/vertical energy flux () in Jan–Feb during the westerly and easterly shear phases of the QBO are shown in Figures 10 and 11, respectively (10°S–10°N average). Note that energy fluxes are 2-monthly averaged pictures that should look different at any given time. The line graph indicates the zonal variation in OLR variance due to each wave component averaged from 10°N to 10°S (note the different ranges of the ordinate axes for Kelvin waves, MRG waves, and other EQWs). The top right panel shows the 2-monthly mean and 10°S–10°N average zonal wind.
Figure 10. Longitude-height cross-sections of PE and zonal and vertical energy fluxes () due to each EQW in the westerly shear phase in Jan–Feb (10°N–10°S average). PE due to (a) Kelvin waves, (c) n = 0 EIGWs, (d) MRG waves, (e) n = 1 EIGWs, (f) n = 1 WIGWs, (g) n = 2 EIGWs, and (h) n = 2 WIGWs averaged between 10°N and 10°S. (b) Zonal wind averaged from 10°N to 10°N. The line graph indicates zonal variation in OLR variance (W2 m−4) due to each EWQ component. Note the different ranges of the ordinate axes for Kelvin waves, MRG waves, and other EQWs. The arrow unit is 250 J kg−1 m s−1 for (a) and 30 J kg−1 m s−1 for (c–h). The vertical component of energy flux is multiplied by a factor of 1000 to coincide with their direction parallel to the propagation of wave path. The shading interval is 0.5 J kg−1 for (a) and 0.1 J kg−1 for (c–h). The shading interval is 4 m s−1 for zonal wind for (b).
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 PE associated with Kelvin waves is most dominant among EQWs (note that the shading intervals in Figures 10 and 11 are different between PE associated with Kelvin waves and those associated with other EQWs). In the troposphere, there are large sources of Kelvin waves in the eastern hemisphere, and most Kelvin waves generated in the eastern hemisphere propagate into the stratosphere (Figures 10a and 11a). Kelvin waves are also excited in the western hemisphere, but energy fluxes suddenly become weak below 100 hPa. These results indicate that the vertical shear of the Walker circulation in the western hemisphere prevents most Kelvin waves from entering the stratosphere by the same mechanism (critical-level filtering) presented in Figure 5. Interestingly, some Kelvin waves generated at approximately 180°E propagate eastward and encounter the westerly in the troposphere at approximately 170–150°W. The value of the energy flux decreases rapidly in the zonal direction. These results indicate that Kelvin waves dissipate not only via vertical propagation, but also via zonal propagation [Fujiwara and Takahashi, 2001; Suzuki and Shiotani, 2008].
 In the westerly shear phase of the QBO, most Kelvin waves in the eastern hemisphere propagate from the troposphere into the middle stratosphere, and their propagation becomes much weaker around 20–30 hPa (Figure 10a), when the QBO is in the westerly phase. On the other hand, in the easterly shear phase, most of the Kelvin waves stop propagating vertically within the range of the westerly phase of the QBO around 50 hPa (Figure 11a). These different propagation characteristics result in the different global distributions of PE around the altitude at which the zonal wind approaches zero (see Figures 6c and 6d).
 Even in the westerly shear phase of the QBO (Figure 10a), some Kelvin waves in the eastern hemisphere seem to dissipate around 70 hPa, where zonal wind is nearly 0 m s−1 (2-month mean zonal wind averaged from 10°S–10°N is shown in Figure 10b, but there are some areas where zonal wind is positive over the equator; see the vertical profiles of zonal wind with standard deviations shown in Figure 5a). Spectral analysis indicates that very slow Kelvin waves with Cx < ∼5 m s−1 dissipate around 70–100 hPa (not shown).
 In contrast to Kelvin waves, PE due to MRG waves is large in the western hemisphere in the UTLS region. It is interesting that most MRG waves generated from the Indian Ocean to the western Pacific do not enter the stratosphere, despite the large sources (Figures 10d and 11d). The easterly associated with the Walker circulation filters most of the MRG waves in the upper troposphere. MRG waves generated at approximately 150°E–150°W propagate eastward and upward, and contribute to the PE in the stratosphere; note that the intrinsic zonal phase velocity x of MRG waves is negative (i.e., westward), while the intrinsic zonal group velocity x of MRG waves is positive (i.e., eastward). In contrast to Kelvin waves, more MRG waves propagate into the middle stratosphere (up to ∼20 hPa) during the easterly shear phase of the QBO (Figure 11d). In this phase, some MRG waves reach approximately 0–30°E in the stratosphere. Careful examination of Figure 6d indicates an off-equatorial PE distribution at approximately 30°W–30°E. MRG waves contribute to the PE around this region.
 In the westerly shear phase, most of the n = 0 EIGWs propagate until 20–30 hPa (Figure 10c), where the zonal wind is 0–8 m s−1, and generate large PE at this altitude. In the easterly shear phase, some n = 0 EIGWs are influenced by the westerly at 70–30 hPa, but more n = 0 EIGWs seem to propagate toward the upper stratosphere (Figure 11c), in contrast to Kelvin waves. This is because n = 0 EIGWs generally have faster Cx values than Kelvin waves, which can pass through weak westerly zones. These results may be dependent on the strength of the background zonal winds, which have seasonal and interannual variation.
 Around 70–30 hPa, more n = 1 WIGWs propagate into the middle stratosphere in the easterly shear phase (Figure 11f) than in the westerly shear phase of the QBO (Figure 10f), as for MRG waves. However, in general, n = 1 and n = 2 EIGWs/WIGWs are not influenced much by the background zonal wind (Figures 10e–10h and 11e–11h) to the same extent as Kelvin waves and n = 0 EQWs because n = 1 and n = 2 EIGWs/WIGWs generally have larger Cx with zonal wave numbers ranging from 1 to 9 (see Figure 8; the gradient between the zonal wave number versus the frequency is proportional to Cx). There is also a clear correspondence of wave generation in the troposphere and wave sources for n = 1 and n = 2 EQWs. For example, the maximum upward propagation of n = 1 and n = 2 WIGWs in the troposphere occurs at approximately 120–180°E, where the sources of these waves are large.
 Waves with larger Cx may interact with the SAO, whereas some n = 1 and n = 2 waves with relatively small Cx should interact with the QBO. As clearly illustrated in this section, the distributions of stratospheric PE caused by EQWs are greatly affected by the source distribution, the Walker circulation, and the QBO phase. Therefore the stratospheric variation (e.g., interannual and seasonal variation) of PE should be associated with tropospheric variation [Tsuda et al., 2008] in addition to the QBO phase. The simulation results should serve as useful information to gain a better understanding of the observed PE distribution.
4.3. Ratio Between EQWs and 3D-Gravity Waves
 The PE associated with EQWs has equatorially trapped structures (Figure 9), and most PE associated with 3D-gravity waves must show no mode to the equator. The PE ≤ 7 km shown in Figure 6 is the sum of EQWs ≤ 7 km and 3D-gravity waves ≤ 7 km. In this section, we discuss investigation of the extent to which EQWs and/or 3D-gravity waves contribute to the PE ≤ 7 km. To assess this question in accurate detail, the equatorial-wave filter discussed in section 4.2 was used for temperature data with vertical wavelengths ≤ 7 km (T′ ≤ 7 km). In calculating the temperature associated with EQWs, overlaps between Kelvin waves and n = 1 EIGWs and between n = 0 EIGWs and n = 2 EIGWs were avoided. That is, a Kelvin and n = 1 EIGW, n = 1 WIGW, and n = 1 equatorial Rossby wave filter was applied for the symmetric component of T′ ≤ 7 km and an MRG wave, n = 0 and n = 2 EIGW, n = 2 WIGW, and n = 2 equatorial Rossby wave filter was applied for antisymmetric T′ ≤ 7 km. The PE ≤ 7 km associated with EQWs was then calculated. Here PE values associated with 3D-gravity waves were regarded as the residual of total PE ≤ 7 km minus the PE ≤ 7 km associated with EQWs.
 Figure 12 shows the global distribution of PE ≤ 7 km, PE associated with EQWs ≤ 7 km, 3D-gravity waves ≤ 7 km, and PE associated with EQWs divided by total PE ≤ 7 km at 20–30 km (∼57–15 hPa) in Jan–Feb during the westerly and easterly shear phases of the QBO. PE ≤ 7 km is generally larger in the westerly than in the easterly shear of the QBO (Figures 12a and 12b). The PE with EQWs ≤ 7 km is also larger in the westerly shear phase with dominant symmetric structures. On the other hand, in the easterly shear phase, off-equatorial structures of PE with EQWs ≤ 7 km were clearly seen around 0–60°E and 0–160°W due to the small activity of Kelvin waves in the easterly shear phase, as discussed in section 4.2. The north-south extent of EQWs is described by the equatorial radius of deformation y0 as follows:
The amplitude of EQWs becomes 1/ where y = y0 (y is the meridional axis, and y = 0 at the equator; see Andrews et al. ). The value of y0 for EQWs with vertical wavelength of 7 km is approximately 1100 km. Therefore PE (as a function of T′2) associated with EQWs ≤ 7 km should be reduced to less than 1/e ≈ 37% within 1100 km. For example, PE with a value of ∼3 (J kg−1) in the westerly shear phase over the middle Pacific decreases to a value of approximately ∼1.2 (J kg−1) around 10°N (Figure 12c), which is consistent with the estimated meridional range.
Figure 12. (a, b) Global distribution of total PE with vertical wavelength ≤ 7 km at 20–30 km in Jan–Feb. The PE due to (c, d) EQW ≤ 7 km components and (e, f) 3D-gravity wave components ≤ 7 km. (g, h) The ratio between total PE ≤ 7 km and PE due to EQWs ≤ 7 km. The left and right panels show the westerly and easterly shear phases of the QBO, respectively. The shading intervals are 0.5 J kg−1 for (a, b, e, f), 0.3 J kg−1 for (c, d), and 3% for (g, h).
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 PE with 3D-gravity waves ≤ 7 km shows a scattered structure with a wide meridional range (Figures 12e and 12f). Large values are observed over areas that include the Congo basin, South America, the Indian Ocean to the mid-Pacific, and northern Australia. GPS RO data also show localized PE, such as over South America [Tsuda et al., 2000, 2008; Alexander et al., 2008b], most likely due to 3D-gravity waves. In Jul–Aug, large PE due to 3D-gravity waves ≤ 7 km is distributed around the Indian monsoon region (not shown) in association with large amounts of precipitation (Figure 2d), which is also consistent with GPS RO measurements made by the Challenging Mini-Payload (CHAMP) satellite as described by Tsuda et al. .
 Of the total PE at 20–30 km, approximately 30% from the Indian Ocean to the eastern Pacific is contributed by EQWs ≤ 7 km (Figures 12g and 12h) over the equator, indicating that 3D-gravity waves ≤ 7 km contribute ∼70% to PE ≤ 7 km at this altitude. The contribution of EQWs becomes smaller away from the equator. The contribution of EQWs to PE is smaller over the Atlantic Ocean. In the stratosphere, the ratio in the westerly shear phase is generally larger than in the easterly shear phase of the QBO due to greater propagation of Kelvin waves, which have the largest amplitude of T′ among EQWs. The latitude-height cross-section of the ratio averaged between 10°S and 10°N in both Jan–Feb and Jul–Aug reaches up to 40% at the altitude where Kelvin wave activity is large (not shown).
 The model covers a much wider spectral range than those in COSMIC GPS RO data as mentioned in section 4.1. This is one possible reason why simulated total PE ≤ 7 km was generally larger than that in COSMIC. In fact, the strength of PE ≤ 7 km associated with EQWs (Figures 12c and 12d) was comparable to PE with an equatorially trapped structure observed in COSMIC GPS RO data [Alexander et al., 2008b].
4.4. Gravity Waves With Periods ≤ 24 Hours
 As noted in section 1, Tsuda et al.  reported large PE over the tropical Atlantic Ocean, and Kawatani et al.  concluded that the source of this PE was 3D-gravity waves generated by a diurnal cycle of convection around the Bay of Guinea. Relatively large PE was also simulated over the Amazon and Indonesia. Kawatani et al.  presented results obtained in only 1 week of June. In this section, gravity waves with periods ≤ 24 h for 2-monthly means in Jan–Feb and Jul–Aug are investigated during the easterly shear phase of the QBO. Note that the 24-h cycle has a repeatable well-defined phase at each point, whereas intraday variance arises from short-lived disturbance at any time of the day. Gravity waves with periods ≤ 24 h are generated by both the diurnal cycle of convection and another intraday variability. We extracted wave components by applying a high-pass filter with a cut-off period of 24 h. The terms of the applied cosine-type window and the extracted terms are the same as those explained in section 4.2. The high-pass filter was applied to temperature, geopotential height and wind components to obtain PE ≤ 24 h and energy flux ≤ 24 h. Note that the vertical wavelength filter was not applied.
 Figure 13 shows the 2-monthly mean amplitudes of cumulus precipitation with periods ≤ 24 h for Jan–Feb and Jul–Aug during the easterly shear phase of the QBO (different from the previous figures, the starting point of longitude is 180°W). The values indicate the ratio of precipitation amplitude with periods ≤ 24 h to that with periods > 24 h. The ratio was calculated when the 2-monthly mean convective precipitation exceeded 1 mm day−1. Cumulus precipitation ≤ 24 h includes not only 24 h, but also 6- to 23-h components. Spectral analysis of cumulus convection ≤ 24 h over the land indicated a dominant period at 24 h, with a secondary maximum at 12 h. We also calculated the amplitude of cumulus precipitation with periods including only 12-h and 24-h components. In this case, the amplitude became much smaller over the ocean. Regions with strong amplitudes of cumulus convection ≤ 24 h are located over South America, the Congo Basin, and around Indonesia in Jan–Feb, and over South America, the Bay of Guinea, and the India, Indochina, and Indonesia region in Jul–Aug. Therefore gravity waves ≤ 24 h should be excited over these regions. Ricciardulli and Sardeshmukh  used satellite-observed blackbody temperature (TBB) data to show the amplitude of the diurnal cycle of convection relative to the long-term mean. The simulated distributions in the present study were similar to their results.
Figure 13. Amplitude of cumulus convection with periods ≤ 24 h for (a) Jan–Feb and (b) Jul–Aug during the easterly shear phase of the QBO. Values indicate the ratio between precipitation with periods ≤ 24 h and that with periods > 24 h where the 2-monthly mean values of convective precipitation are ≥1 mm day−1.
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 Figure 14 shows PE ≤ 24 h at 100–60 and 35–15 hPa in Jan–Feb and Jul–Aug during the easterly shear phase of the QBO, as well as the three dimensional (3-D) energy fluxes () ≤ 24 h at 100–60 and 60–35 hPa. Horizontal energy fluxes were vectored (the points of origin for the energy fluxes show the strength, position, and propagating direction of gravity waves, and the arrowheads do not show the reaching point of a gravity wave) and vertical energy fluxes with values of more than 0.2 (J Kg−1 m s−1) are shaded. For example, if an area shows southwestward vectors with shading, gravity waves propagate southwestward and upward on average. Therefore global three-dimensional propagation of gravity waves ≤ 24 h can be seen in Figure 14.
Figure 14. PE with periods ≤ 24 h at (a, b) 100–60 hPa and (g, h) 35–15 hPa, and three-dimensional energy fluxes () ≤ 24 h at (c, d) 100–60 hPa and (e, f) 60–35 hPa in (left) Jan–Feb and (right) Jul–Aug during the easterly shear phase of the QBO. Horizontal energy fluxes are indicated by vectors, and vertical energy fluxes ≥ 0.2 J kg−1 m s−1 are shaded. The arrow unit is 300 J kg−1 m s−1, and horizontal energy fluxes ≥ 30 J kg−1 m s−1 are shown. The shading intervals are 0.5 J kg−1 for PE ≤ 24 and 0.1 J kg−1 m s−1 for ≤ 24 h.
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 In Jan–Feb, there were three localized strong PE areas over South America, the Congo Basin, and around Indonesia at 100–60 hPa, while in Jul–Aug, localized PE was observed over South America. the Bay of Guinea and the Indian monsoon region. These results correspond well to the distribution of the amplitude of cumulus precipitation ≤ 24 h (Figure 13). The 3D-gravity waves generated in these areas propagate three-dimensionally and contribute large PE to the stratosphere in areas a short distance from the source region. (Although figures are not shown here we checked the properties of three-dimensional propagation both in zonal-vertical and in meridional-vertical cross-sections of energy fluxes). For example, energy fluxes indicate that large PE at 35–15 hPa around South America is due to 3D-gravity waves generated convectively over the Amazon. Gravity waves with periods of 24 h can propagate until 30°N or 30°S when background zonal winds are zero. The dominant period of the gravity waves generated over land is 24 h, as mentioned above. Therefore most gravity waves could propagate until 30°N or 30°S and then be reflected to the lower latitudes. Further details were reported previously by Kawatani et al. , who described meridional-vertical cross-sections of energy fluxes. We also calculated PE with periods including only 12-h and 24-h components. The PE is relatively large over the lands and areas a short distance from the lands, while the PE becomes much smaller far away from the land such as over the middle Pacific. These results indicated that most PE ≤ 24 h over the middle Pacific is due to gravity waves generated by intraday variability, not diurnal cycle of convection.
 Interestingly, there were some differences between Jan–Feb and Jul–Aug. In Jul–Aug, as demonstrated by Kawatani et al. , convectively generated gravity waves around the Bay of Guinea propagate southwestward across the equator with large upward propagation (Figures 14d and 14f), which results in large PE over the tropical Atlantic Ocean in the stratosphere (Figure 14h). In contrast, in Jan–Feb, a great deal of precipitation occurs over the Congo Basin, rather than the Bay of Guinea (see Figures 2c and 13a). As clearly shown, convectively generated gravity waves around the Congo Basin propagate westward and upward with altitude (Figures 14c and 14e) and then generate large PE over the tropical Atlantic Ocean in the Southern Hemisphere. Therefore the main source of PE over the tropical Atlantic Ocean in Jan–Feb is precipitation over the Congo Basin, which is different in Jul–Aug. Precipitation over the Bay of Guinea makes a secondary contribution to PE over the tropical Atlantic in Jan–Feb.
 In Jul–Aug, precipitation ≤ 24 h over the Indian monsoon region acts as a strong source of 3D-gravity waves ≤ 24 h, and large PE ≤ 24 h is distributed in this region (Figures 14b and 14h). Large PE ≤ 7 km is also located over the Indian monsoon region in Jul–Aug (not shown), indicating that Indian monsoon precipitation is a strong source of gravity waves with a wide range of periods. It is also of interest that the propagating directions of gravity waves over the Indian monsoon region are westward in the midlatitudes in Jan–Feb (Figure 14e) and eastward the midlatitudes in Jul–Aug (Figure 14f), which may be responsible for the different background zonal wind directions (i.e., westerly in Jan–Feb and easterly in Jul–Aug).
 Tsuda et al.  reported localized large PE over the Atlantic, but Alexander et al. [2008b] found no distinct large PE in this region, as in our simulated PE ≤ 7 km (Figure 12). One possible explanation is that Tsuda et al.  selected gravity waves with vertical wavelengths ≤ 10 km, whereas Alexander et al. [2008b] extracted wave components of ≤7 km. Kawatani et al.  reported that the vertical wavelengths of gravity waves generated over the Bay of Guinea are 5–10 km, and dominant vertical wavelengths are ∼8–10 km. Therefore the different results may reflect the selection of different vertical wavelength ranges. In addition, PE ≤ 7 km includes both 3D-gravity waves and EQWs, whereas PE ≤ 24 h does not consist of EQWs. This may also explain why isolated PE ≤ 24 h is clearly observed over the tropical Atlantic Ocean.
 Using an aqua planet AGCM, Sato et al.  showed that zonally averaged gravity waves ≤ 24 h generated over the equatorial region propagated poleward. The AGCM used here, which included a realistic boundary condition, also showed that gravity waves ≤ 24 h generated in the low latitudes generally propagate to higher latitude with height. Global distribution of sources and three-dimensional propagation of gravity waves ≤ 24 h obtained in the present study provide additional information regarding where gravity waves are generated and the 3-D direction in which gravity waves propagate (Figures 13 and 14).