Figure 1 shows k–ω power spectra for the eastward-propagating symmetric component of temperature (TS) at 100, 70, 50, and 20 hPa from the SL datasets of the six reanalyses, averaged over the 30-year period (1981–2010). The major portion of the spectral power appears along the Kelvin wave dispersion curves (black diagonal lines) at all altitudes. The spectral characteristics are broadly similar among the reanalyses with peaks at k=2–3. The reanalyses commonly exhibit a gradual shift of the Kelvin wave spectrum to larger equivalent depths (h) with respect to the altitude: the majority of the spectral power appears at h<60 m at 100 hPa, at h∼60 m at 70 hPa, and at h=60–240 m above 70 hPa. There exist variations in the spectral width in h: for example, compared to the other reanalyses, CFSR shows broader spectra at small equivalent depths (h<60 m) consistently at 70–20 hPa, with peaks at slightly smaller equivalent depths (70–50 hPa). In addition to the Kelvin wave signal, at 100 hPa, another peak exists at k=2 in the low-frequency range (ω<0.04 cyc d-1). As mentioned in Sect. 2, this is likely related to the intraseasonal MJO .

The relative magnitude of Kelvin wave spectral power between ERA-I, MERRA, MERRA-2, and CFSR varies with height (Fig. 1), while JRA-55 and JRA-55C show generally less power below 20 hPa compared to the other four datasets. Note that relatively small variances in JRA-55 and JRA-55C are found in the temperature field but not in the wind fields (see Fig. S1 in the Supplement). In Fig. 1, the ratio of the spectral power to the background spectrum is also shown (thin purple contours), which indicates statistical significance of the spectral signals. A ratio of 1.2 was deemed to be statistically significant at the 95 % level by , and for the majority portions of the Kelvin wave spectra in Fig. 1, the ratios are generally larger than 1.5 (i.e., 50 % larger than the background spectral power), conservatively implying that the spectral peaks are all statistically significant.

Figure 2 shows k–ω spectra for the westward-propagating symmetric component of v (vS) at 100, 70, 50, and 20 hPa. Note that the lower bound of the y axis at 100 hPa is different from that at the other levels, as the vS spectrum is much broader in frequency at 100 hPa. The 100 hPa spectrum has periods from around 2.5 d (0.4 cyc d-1) to longer than 30 d (0.033 cyc d-1), with a peak at k=-5, ω=0.1–0.2 cyc d-1 common to all of the reanalyses. In the upper troposphere, the background zonal wind near the Equator is westerly (easterly) in the western (eastern) hemisphere. MRG waves that are generated in the region of westerly (easterly) background flow can have relatively low (high) ground-based frequencies, which might result in the broad spectrum in frequency of vS at 100 hPa. The low-frequency portion of the wave spectrum at 100 hPa seems to roughly follow the MRG wave dispersion curves for a background wind (U) of +10 m s-1 (Fig. 2, long-dashed lines).

At 70 hPa, the low-frequency portion (<0.1 cyc d-1) of the spectrum is mostly filtered out, and commonly in all reanalyses, the peaks appear at k=-6, ω∼0.20 cyc d-1 and at k=-5, ω∼0.22–0.25 cyc d-1 (h∼60 m for both, assuming U=0) (Fig. 2). However, the lower bound in frequency of the spectral power is different between the reanalyses: JRA-55C, JRA-55, and MERRA-2 exhibit broader frequency spectra than the others. Above 70 hPa, another two local peaks appear in a higher-frequency region (k=-4, ω∼0.3 cyc d-1; k=-3, ω∼0.4 cyc d-1), in addition to the aforementioned peaks, and these peaks become primary at 20 hPa. At 50 hPa, the spectral shapes are generally similar among the reanalyses, although the spectral region where the majority of the power exists in CFSR is slightly shifted toward lower zonal wavenumbers when compared with the other reanalyses.

Figure 2 also exhibits a distinct feature in the vS spectra at 20 hPa, compared to the spectra at the lower altitudes: statistically significant power appears along the narrow spectral region that includes k=-5, ω∼0.5 cyc d-1 and extends into higher wavenumbers and frequencies in all of the reanalyses. Toward lower wavenumbers, it seems to merge into the aforementioned peak at k=-3, ω∼0.4 cyc d-1. The spectral power along this region is larger in JRA-55C, JRA-55, and CFSR than in the others. This portion of the spectrum is further examined in Sect. 4. As will be seen therein, the waves with this spectrum do not originate from below, and they appear with different timing and different characteristics from the lower-frequency upward-propagating MRG waves which dominate the vS spectra in the lower altitudes. In Sect. 3.2–3.4, we focus on the lower-frequency MRG waves filtered with a cut-off frequency of 0.33 cyc d-1 (period of 3 d).

The spectral shapes of Kelvin and MRG waves obtained from the ML datasets (not shown) are very similar to those from the SL datasets (Figs. 1 and 2) for each reanalysis, whereas their spectral power is larger by up to about 35 %, depending on the altitude and reanalysis. This will be further discussed in Sect. 3.3.

In the following sections, we define Kelvin waves as the symmetric mode with h=8–240 m, k=1–10, and periods of 2–20 d following , unless otherwise stated. These spectral components include a major portion of the Kelvin wave variances (Fig. 1), while excluding contributions of the other disturbances at low phase-speed ranges at 100 hPa (see also Fig. 9). The MRG waves are defined as the antisymmetric mode with h>8 m, -10≤k<0, and periods of longer than 3 d, as previously mentioned, where h is for U=0. The perturbations filtered for these spectral components are denoted as, for example, TKelvin for the Kelvin wave temperature or vMRG for the MRG wave meridional wind.

To investigate the spatial distributions, the Kelvin and MRG signals are reconstructed in physical space by filtering their spectral components. Figure 3a shows the distributions of TKelvin and vMRG variances at 100, 70, 50, and 20 hPa during the period of 1981–2010, averaged for ERA-I, MERRA-2, CFSR, and JRA-55 (the reanalyses that assimilate satellite data are included as the ensemble members, but MERRA is excluded since it exhibits very similar spatial distributions to MERRA-2, as will be seen in Fig. 3b). The ensemble-mean variances are normalized by their maximum values on each horizontal plane. Both the Kelvin and MRG wave variances are confined near the Equator, and the locations of their maxima slant eastward in the vertical, consistent with the equatorial wave theory. Note that, while the phase of MRG waves propagates westward, their wave packet travels eastward when the background flow is westerly or weak easterly which is a preferred condition for MRG wave propagation in the stratosphere. The maxima of variances for the Kelvin (MRG) waves are located in the eastern (western) hemisphere in the lower stratosphere: 70 and 50 hPa, consistent with previous observational studies . At 20 hPa, the Kelvin and MRG waves show rather broad distributions in longitude. The distributions of the wave variances at 100 hPa are closely related to those of background zonal wind in the upper troposphere . The easterly (westerly) upper tropospheric wind in the eastern (western) hemisphere allows Kelvin (MRG) waves to more readily propagate vertically, resulting in the hemispheric difference in the wave variances. In addition, the MRG wave variances at 100 hPa are distributed quite broadly in longitude (Fig. 3a). showed that during the northern winter the MRG wave variances in the upper troposphere are much larger in the western hemisphere, with peaks at about 140 and 30∘ W, than in the eastern hemisphere, whereas during the northern summer, the variance distribution becomes broad, stretching to the western Pacific (see their Fig. 6).

Figure 3b shows the wave variances at the Equator in each of the six reanalyses. The longitudinal variations of the Kelvin wave variances seem broadly similar among the reanalyses, with peaks at approximately 45∘ E at 100 hPa and at 75–110∘ E at 50 hPa, although the Kelvin waves in CFSR show much smaller longitudinal variations than those in the other reanalyses at 100 and 50 hPa. The MRG waves have maximum variances in the eastern Pacific and South America at 70 and 50 hPa, and these maxima are roughly twice the minima over the Maritime Continent and western Pacific, common to all of the reanalyses. However, at 50 hPa, detailed distributions in the eastern hemisphere show some differences among the reanalyses: the peak in ERA-I is eastward of that in the others (e.g., 35–45∘ eastward of CFSR, JRA-55, and JRA-55C), and the variance in CFSR seems to extend further to the west (∼150∘ W). Differences between JRA-55 and JRA-55C appear over the Indian Ocean for Kelvin waves and the eastern Pacific for both waves, whereas the differences are very small on the Maritime Continent and western Pacific at all altitudes. These could be explained in part by relatively large numbers of radiosonde observations near the Maritime Continent compared to the other regions around the Equator in these two reanalyses (Fig. 5 in ; Fig. 2.8c and 2.9c in ).

The locations of the wave variances vary with respect to the QBO winds in the stratosphere (not shown). For example, MRG wave variances averaged for the 50 hPa easterly QBO are located to the west of those for the westerly QBO. However, the MRG wave variances are much smaller in the easterly QBO phases, and their contribution to the climatological-mean variances shown in Fig. 3 is relatively small. Therefore, the results in Fig. 3 are more representative of the westerly QBO phases for MRG waves and, similarly, easterly phases for Kelvin waves.

The currently used filters for the MRG waves exclude the low-frequency perturbations (<0.1 cyc d-1) at 100 hPa revealed in Fig. 2, of which the spectrum follows the dispersion curves for U∼10 m s-1. An additional calculation, the same as with Fig. 3, is performed but for the 100 hPa low-frequency components of vS using the filters for 0.033≤ω<0.1 cyc d-1 and -10≤k<0 (Fig. S2). It is observed in all the reanalyses that the low-frequency perturbations are located mostly in the western hemisphere where westerlies exist (over the eastern Pacific and Atlantic), consistent with the Doppler-shifted dispersion curves discussed in Fig. 2.

To further investigate the spatial structures of the Kelvin and MRG waves, including the circulation patterns and horizontal scales of representative wave modes in each reanalysis, an empirical orthogonal function (EOF) analysis is used following the technique outlined in . In that study, EOFs were calculated from the covariance matrix of a 2–6 d filtered meridional wind in 20∘ N–20∘ S from ERA-I data to isolate MRG waves. The filter band was based on the well-documented strong spectral peak in the equatorial meridional wind centered at around the 4.5 d period (Fig. 2). Dynamical fields associated with each EOF were obtained by projecting unfiltered ERA-I data at each grid point onto the associated principal component (PC) time series.

Here, we use data interpolated to 2.5∘ resolution for the six reanalyses considered. A similar technique as in is used to obtain the statistical structures of the equatorial Kelvin waves, except that a 2–25 d eastward-only filter band is applied to the equatorial zonal wind based on the spectral peaks in Fig. 1. The results are very robust to changes in the filtering, as long as the filter band contains the spectral peaks shown in Figs. 1 and 2, and to changes in latitudinal extent of the EOF basis.

In all cases, EOF “pairs” are obtained with respective PC time series that correlate at better than 0.96 (with time lag), which together represent the propagating pattern of Kelvin or MRG waves. Thus, each mode can be represented by either EOF pattern and its associated PC. Figure 4 shows the projected structures of the leading modes (EOF 1) of 50 hPa Kelvin waves from the six reanalyses for the period of 1981–2010. Wind vectors are shown at the locations where they are statistically significant, taking into account temporal and spatial autocorrelation. In all cases, zonal wavenumber 1 structures are obtained as the leading modes in the tropics, with zonal wind perturbations in phase with geopotential, as expected from theory. Especially in the tropics, remarkably similar patterns are found in all of the reanalyses. The wind and geopotential perturbations exhibit much larger amplitudes in the eastern hemisphere than in the western hemisphere, consistent with the result in Fig. 3. Higher-order Kelvin EOF pairs represent integer zonal wavenumber structures, with k=2 Kelvin structures comprising the second EOF pair (EOFs 3 and 4). The second EOF pairs also show reasonable agreement among the reanalyses (not shown). The k=1 patterns shown in Fig. 4 account for the largest portion of total variance in the equatorial 2–25 d eastward zonal wind in each reanalysis, which amounts to a maximum of 42 % in the case of ERA-I and somewhat smaller amounts in the other reanalyses (Table 2).

Lag regressions based on the PC time series show the eastward propagation of the Kelvin wave signals at a mean phase speed of around 35 m s-1 (not shown), which is rather faster than the 20–30 m s-1 found in previous studies, although these phase speeds are highly dependent on the state of the QBO e.g.,. While these features will be explored in further detail in a future study, the main point here is that the various reanalyses appear to be quite suitable for studying the mean statistical structure of these waves, as well as their variability.

The situation for the MRG wave signals is rather different (Fig. 5). The leading MRG EOFs represent localized wave packets that are confined over the eastern Pacific to Atlantic sector, as expected from the location of the variance peaks in Fig. 3. The EOF patterns have classical structures of equatorial gyres and off-equatorial antisymmetric geopotential perturbations as predicted by . The reanalyses each represent structurally very similar MRG wave patterns, except for CFSR which has a zonally broader signal (i.e., longer wavelengths), consistent with the spectra shown in Fig. 2. In addition, the MRG wave patterns in JRA-55, JRA-55C, and CFSR are displaced somewhat to the west of the other reanalyses, again reflecting their variance distributions shown in Fig. 3b. The higher-mode EOF pairs of MRG waves (not shown) have wave structures over the Indian and Pacific oceans, with good agreement among the reanalyses for up to the fourth mode pairs.

Kelvin and MRG signals at other levels from 70 to 10 hPa are broadly similar to those shown here, although the structures change considerably at 100 hPa (not shown). In summary, there is reasonable agreement between the reanalyses in their statistical representation of Kelvin and MRG wave structures, at least for the leading modes.

In this section, we analyze long-term variations in the Kelvin and MRG wave activity and discuss the impact of satellite data upon the assimilated wave activity based on the comparison of JRA-55 and JRA-55C. Figure 6 shows annual-mean time series of 100 hPa TKelvin and vMRG variances, as defined in Sect. 3.1. The results from the ML datasets are presented (solid lines), except for MERRA for which the SL datasets are used (dashed lines) due to the lack of ML data. The variance of TKelvin in ERA-I fluctuates between about 0.5 and 0.85 K2 with an increasing trend, whereas that in JRA-55C fluctuates between ∼0.35 and 0.58 K2 without such an obvious trend. Despite this difference, much of the interannual variability in the Kelvin wave activity is reflected in both time series, with a Pearson linear correlation of 0.85 (0.88 after trend removal) between the two. Similar correlations are obtained between the other series, with the TKelvin variances ranging between those in ERA-I and JRA-55C. By comparing JRA-55 and JRA-55C, it appears that assimilation of satellite data leads to an increase in the Kelvin wave temperature variance, as also found in the previous section (Fig. 3b).

The difference in the variance between JRA-55 and JRA-55C is further investigated in Fig. 7, which presents the time series of the annually averaged difference at altitudes from 100 to 5 hPa. Because the wave variance in the stratosphere is strongly dependent on the QBO e.g.,, 1-2-1 smoothing is applied to the annual time series to filter out the quasi-biennial fluctuations and focus on the longer-term variations. It is evident that the difference (JRA-55 minus JRA-55C) in the TKelvin variance is generally positive from 100 to 20 hPa during the 30-year period, indicating the enhancement of the Kelvin wave amplitude by assimilation of satellite data in JRA-55. In addition, it is noteworthy that at 100 hPa, the difference is systematically larger from around 2000, compared to that before late 1990s: it is roughly 6 % (up to ∼8 %) in the years before 1998, whereas it increases to ∼10 % in 1998–2000 and becomes ∼20 % afterward. A similar systematic change of the difference in the TKelvin variance is even more evident in the upper stratosphere: at 10 and 5 hPa (Fig. 7) and above (not shown), where the difference is mostly negative until 1998 and becomes positive after 1999. Such changes are not found at 50 hPa.

The systematic change in the impact of satellite data assimilation on the Kelvin wave amplitude in JRA-55 around 1998 might be due to the TOVS–ATOVS transition, given the timing of the change see Fig. 8 infor the timelines of satellite data used in JRA-55 and other reanalyses. The Advanced Microwave Sounding Unit A (AMSU-A) instruments in the ATOVS suite were introduced in 1998. Compared to the Stratospheric Sounding Unit (SSU) instruments in the TOVS suite, the AMSU-A observations have better vertical coverage with a higher vertical resolution see Fig. 7 infor the vertical weighting functions of the SSU and AMSU-A measurements. The vertical weighting functions of the two suggest that the AMSU-A instruments could be advantageous over the SSU, particularly at an altitude of ∼100 hPa which the latter does not cover. This may explain the systematic change found at 100 hPa around 1998 (Fig. 7). The higher vertical resolution of the AMSU-A instruments is expected to mostly benefit the upper stratosphere where radiosonde sounding observations do not reach (i.e., above ∼10 hPa). This is consistent with the observed systematic change being most prominent in the upper stratosphere. In addition to the TOVS–ATOVS transition, there is also a possibility that the Global Navigation Satellite System Radio Occultation (GNSS-RO) has been influencing the assimilated waves in JRA-55 since 2006 Fig. 10 in, although this cannot be identified by the current analysis.

Recalling that JRA-55C does not exhibit a long-term trend in Kelvin wave activity at 100 hPa (Fig. 6), the trend shown in JRA-55 is probably not an actual change in the true atmosphere but instead an artifact arising due to the satellite transition. The change in the 100 hPa TKelvin variance around 1998 is 17 % in JRA-55 when it is measured by the difference between the variances averaged for the two periods before and after 1998, i.e., 1981–1997 (P1 hereafter) and 1999–2010 (P2), relative to the 30-year mean variance. The changes in the other reanalyses are comparable to or smaller than that in JRA-55 (e.g., 19 % in ERA-I, which is the largest value). This suggests that the long-term changes in the 100 hPa Kelvin wave activity shown in those reanalyses also could result largely from the satellite transition.

The variance of vMRG exhibits generally similar interannual variations at 100 hPa between all of the reanalyses (Fig. 6), while the magnitudes of the variance are different. The vMRG variance has an increasing long-term trend from the early 1990s, even in JRA-55C. The vMRG variance at 100 hPa in JRA-55 is always larger than that in JRA-55C by roughly 6 %–12 % (Figs. 6 and 7), reflecting the impact of satellite data on the assimilated MRG waves. In addition, the assimilation of satellite data increases the analyzed MRG wave activity at 5 hPa in JRA-55 (Fig. 7). This increase is up to ∼55 %, and it becomes even larger at higher altitudes (see Fig. 8). Notable differences between P1 and P2 in the satellite impact on the assimilated MRG wave activity are not found below 5 hPa (Fig. 7). At 5 hPa, the impact is significantly larger in P2 than in P1 (note that two different scales are used in the y axis in the bottom panel of Fig. 7, below and above 20 %, separated by the dashed horizontal line).

Figure 8 presents the vertical profiles of TKelvin and vMRG variances averaged for P1 and P2 along with the differences between the two periods. In the ML results, the TKelvin variance is maximized at approximately 80 hPa in ERA-I and at 70 hPa in the others for both periods. These peak altitudes are generally consistent with that obtained using GNSS-RO data in 2007–2014 by Fig. 6: 18 km (75–80 hPa), considering vertical grid spacings of the reanalyses. The TKelvin variance decreases with height from these levels to ∼30 hPa and gradually increases from ∼20 hPa in both periods in all the reanalyses, except in MERRA-2 during P1. Further examination indicates that sometimes the temperature seems not to be properly assimilated in the upper stratosphere during P1 in MERRA-2 (Fig. S3; note the lack of Kelvin wave peaks in 1984, 1989, and 1991), which may result in the decrease above ∼20 hPa in P1 in MERRA-2. It has previously been reported that the monthly mean zonal wind at 10 hPa in MERRA-2 shows significant differences from observations, in particular with larger annual/semiannual oscillations, until the mid-1990s , which is perhaps partly related to the underrepresentation of the Kelvin wave temperature shown in Fig. S3. Errors in a mean state of assimilated fields can cause degradation of any variables describing waves during the first-guess model integration, which in turn degrade the assimilated fields 3 or 6 h later. In P2, the TKelvin variance at 70–30 hPa has similar magnitudes between ERA-I, MERRA-2, and CFSR, while JRA-55 exhibits a relatively smaller variance than these reanalyses, as is also evident in Fig. 1. In the upper stratosphere, the variance is notably larger in ERA-I in P2 than in the others (also see Fig. S3).

The difference in the TKelvin variance between P1 and P2 is positive in the whole stratosphere in all of the reanalyses (Fig. 8). In JRA-55C, the difference is small at 100 hPa (∼6 %), as already discussed (Fig. 6), and above 10 hPa (4 %–8 %), although it reaches ∼20 % at 50–20 hPa. Given the small difference in Kelvin wave activity at ∼100 hPa, the increase in the activity at 50–20 hPa could result from interaction with the QBO which itself has variability in its morphology. However, since only five (seven) QBO cycles are included in P2 (P1), further investigation of the long-term change associated with the QBO might be less meaningful in a statistical sense. Another possible reason could be the increased number of radiosonde observations around 30 hPa with time see Fig. 15 in. In JRA-55, the difference is large in the upper stratosphere (∼25 % or larger), and it is even larger in ERA-I, MERRA-2, and CFSR. Similarly to the case of JRA-55, the large increase in Kelvin wave activity in the middle and upper stratosphere in the other reanalyses could also be in large part due to the transition of satellite instruments. From Fig. S3, it is evident that the difference in the Kelvin wave variances at 10 hPa between ERA-I and the others increases abruptly around 2000, which may support such an impact of the satellite instrument change in ERA-I.

In all the reanalyses, the vMRG variance decreases with height below ∼20 hPa and increases above ∼10 hPa (Fig. 8). In both periods, MERRA-2 and CFSR exhibit larger variances than the others, particularly in the upper stratosphere. In P1, it is seen in the upper stratosphere that the vMRG variance in the ML result of CFSR varies less smoothly in the vertical and that the variance in MERRA-2 is about 2–3 times larger than that in the others. These features are still present to a lesser extent in P2. Examination of time series of the vMRG variance (Fig. S3) shows that some unexpected peaks appear in P1 in MERRA-2 with exceptionally large magnitudes even during the easterly QBO phase, which are primarily responsible for the large value of the vMRG variance in P1 shown in Fig. 8. The unexpected peaks disappear suddenly starting in 1999 (Fig. S3), likely emphasizing the impact of the AMSU-A observations on the assimilated wave activity in MERRA-2. Regarding the vertical fluctuations exhibited in CFSR, the reason for this is unknown. The profiles of the difference between P1 and P2 may imply that above 10 hPa, the large increase in the vMRG variance in JRA-55 (10 %–25 %) and ERA-I (∼40 %), compared to the small increase in JRA-55C (∼5 %), is probably in large part due to the satellite transition.

In each reanalysis, the variances obtained from the SL datasets are smaller than those from the ML datasets in both periods (Fig. 8) by 3 %–29 % (0 %–38 %) for Kelvin (MRG) waves depending on the altitudes and reanalyses. The differences in the variances between the ML and SL results (ML minus SL) normalized by the variances in the ML results at each altitude are reported in Table 3. The smaller amplitudes of the waves in the SL datasets compared to those in the ML datasets result from vertical interpolation of reanalysis output variables, which inevitably damps wave perturbations in the standard-level products of reanalyses. The damping by the interpolation is more significant for waves with smaller vertical wavelengths e.g.,. In addition, for a given pressure level, the damping of waves in the SL datasets also depends on the distance between the given level and its adjacent model level. That is, the interpolation effect must be less if a model level is very close to the given pressure level. For example, the distance between 100 hPa and its adjacent model level in MERRA-2 is very small (<50 m), and thus the difference between the SL and ML results of MERRA-2 is only 3 %–4 % at 100 hPa (Table 3). The same (opposite) is true for the 10 hPa (50 hPa) in ERA-I, with a small (large) difference at that level. In JRA-55 and JRA-55C, the difference in the vMRG variance between the ML and SL results is less than 5 % at all levels, implying that the ML fields of these two reanalyses contain less of the MRG wave perturbations with small vertical wavelengths (∼2Δz–4Δz, where Δz is the vertical grid spacings of their model), compared to the other reanalyses.

In this section, we compare the vertical EP flux (Fz) spectra and the wave forcing of the QBO calculated by the EP flux divergence among the reanalyses. As for the figures discussed in Sect. 3.1, we obtain k–ω spectra for each of the symmetric and antisymmetric modes but for Fz averaged over 5∘ N–5∘ S (Figs. 9 and 10, respectively; see Sect. 2 for justification of the narrowed latitude band). The spectral characteristics of Fz for the symmetric and antisymmetric modes are qualitatively similar to those of TS and vS, respectively, for each reanalysis. For the symmetric mode, the Fz spectra are aligned along the Kelvin wave dispersion curves with h values that are similar to those for TS, along with exhibiting a gradual shift of the spectra to the higher h with increasing altitudes (Figs. 1 and 9). On the other hand, the symmetric Fz and TS spectra differ in that the Fz (TS) spectra have peaks at k=3–5 (2–3) in all of the reanalysis and that the magnitudes of Fz in JRA-55 and JRA-55C are not smaller compared to those in the others as was the case for temperature (Fig. 1). As already mentioned, JRA-55 and JRA-55C have relatively weak Kelvin wave temperature amplitudes but not wind amplitudes (Fig. S1), and the major term of Fz for Kelvin waves is the vertical momentum flux (w′u′‾). The reason that the peak of Fz occurs at larger zonal wavenumbers than in the TS spectra is because the vertical wind perturbations tend to have large amplitudes at higher zonal wavenumbers: the ratio of vertical to horizontal wind amplitudes is proportional to k/m (where m is the vertical wavenumber), as from the continuity equation. In the Kelvin wave dispersion relationship, k is independent of m, so that the ratio is proportional to k for given m.

For the antisymmetric mode, Fz has similar spectral shapes and roughly the same (k,ω) peaks as those of vS for each reanalysis below 20 hPa (Figs. 2 and 10). Consistent with the vS spectra, the frequency bounds of the Fz spectra at 70 hPa are quite a bit lower in JRA-55, JRA-55C, and MERRA-2 than in the other reanalyses. At 20 hPa, the Fz spectra differ from the vS spectra in that the relatively low-frequency and high-wavenumber (and thus low phase-speed) components are largely suppressed in the Fz spectra. Indeed, these low-frequency components are more emphasized in the meridional EP flux (Fϕ) spectra (Fig. S4), implying that the low-frequency MRG waves tend to propagate laterally rather than vertically. In addition, even negative Fz values appear in the 20 hPa spectra at k=6–7 in MERRA-2, while this feature is less clear in the other reanalyses. This is suggestive of coexistence of upward- and downward-propagating waves at this level which could be another reason for the small Fz in the low-frequency range.

The EP flux spectra are reconstructed as a function of phase speed to investigate the vertical propagation and dissipation of Kelvin and MRG waves during different QBO phases and to quantify the QBO forcing by those waves as well as the phase-speed ranges responsible for the forcing. For this, we construct bins of phase speeds (cj) with a width of 2 m s-1 (Δc) and integrate the k–ω spectral densities of Fϕ and Fz across the corresponding bins (i.e., (2πa)ω/k∈[cj-Δc/2,cj+Δc/2], where ω is in cycles per second and a is the mean radius of the Earth).

Prior to this procedure, we converted the k–ω spectra to much finer resolution in frequency using linear interpolation for each k. This is required because the frequency resolution of the original spectra (Δω=1/90 cyc d-1) is coarse in terms of c for small k (e.g., for k=1, (2πa)Δω/k∼5 m s-1>Δc), causing artificial peaks and noise. For the finer spectra, we set Δω to be 500 times smaller than the original, after confirming that the results converge without displaying artificial peaks for varying Δω around such values.

Figure 11 shows vertical profiles of 5∘ N–5∘ S averaged EP flux divergence and Fz as a function of phase speed, for Kelvin and MRG waves at c>0 and c<0, respectively, composited for four different phases of the QBO. The composite is made by selecting 1 month for each QBO cycle when the zonal wind tendency is largest at a given altitude and then averaging over the 13 cycles available. The observed monthly mean near-Equator wind profiles compiled by the Freie Universität Berlin (FUB) are used to select the months. The four composites presented in Fig. 11 are for the months of maximum westerly and easterly tendencies at 20 and 50 hPa. The composite averages of the FUB wind are also plotted in Fig. 11 (thick green lines) as well as those of each reanalysis wind (thin red lines). The results are generated using the ML datasets of the four reanalyses available for EP flux calculation (ERA-I, MERRA-2, JRA-55, and JRA-55C), since the EP flux could be largely underestimated when using the SL datasets. Furthermore, the height dependence of the amplitude damping, as reported in Table 3, could also affect the estimation of the vertical divergence of EP flux see Fig. 3 infor a large difference in the EP flux divergence between ML and SL datasets of ERA-I.

The EP flux divergence of the Kelvin waves is found to be quite different between reanalyses not only in magnitude but also in its peak altitude. At the maximum westerly tendency phase at 20 hPa (Fig. 11, the first row), the vertical EP flux is notably the largest in ERA-I at ∼20 hPa. However, the EP flux divergence at this level is the largest in MERRA-2, implying that the assimilated wave fields exhibit the most severe dissipation in MERRA-2 among the reanalyses around this level. ERA-I and JRA-55 show the peaks of the EP flux divergence at ∼15 and 12 hPa, respectively. It is important to note that the easterly-to-westerly transition (i.e., zero wind) occurs at different altitudes between the reanalyses (Fig. 11, red lines; also refer to Fig. 16 in , for different time evolutions of the phase transition) and that the maximum Kelvin wave forcing is found to occur at around the zero-wind altitudes in each reanalysis. These suggest that the mean wind in the assimilated field could affect the representation of wave behaviors in reanalyses in the shear zone. The phase speed at which the forcing occurs increases with altitudes from 10–30 m s-1 at 20 hPa to 15–40 m s-1 at 10 hPa but with large spread at 10 hPa among the reanalyses. The Kelvin wave forcing integrated for the phase speed has maxima of 9.2, 10.2, and 10.0 m s-1 month-1 in ERA-I, MERRA-2, and JRA-55, respectively. When the westerly tendency is maximized at 50 hPa (Fig. 11, the second row), the peak of the Kelvin wave forcing appears slightly below (above) 40 hPa in ERA-I and MERRA-2 (JRA-55). Similar to the 20 hPa transition phase, the vertical EP flux is the largest in ERA-I, while the EP flux divergence at ∼40 hPa is larger in MERRA-2 (8.6 m s-1 month-1), which may also be related to the different mean-wind profiles in the reanalyses. Note that the mean winds in ERA-I and JRA-55 have weaker shear at 40–50 hPa with weaker westerly magnitudes above 40 hPa compared to the observed winds (Fig. 11, thick green lines), and they might affect the wave dissipation rate in these reanalyses. At the other two phases of the QBO (Fig. 11, the third and last rows), the zonal wind is westerly at 70 hPa, causing less upward propagation of Kelvin waves in the lowermost stratosphere.

Figure 12 shows the vertical divergence of the EP flux, (ρ0cos⁡ϕ)-1∂Fz/∂z, as a function of phase speed, along with Fz as in Fig. 11. In the main regions of the Kelvin wave forcing during the westerly-shear phases (Fig. 12, the first and second rows), the vertical divergence of the EP flux for the Kelvin waves is roughly 50 %–75 % of the total divergence shown in Fig. 11. The vertical divergence is the largest in ERA-I among the reanalyses, whereas the total divergence of the EP flux is larger in MERRA-2 (Fig. 11). This indicates that the contribution of meridional flux divergence to the Kelvin wave forcing is relatively large in MERRA-2 (40 %–50 %, not shown) compared to that in the other reanalyses (25 %–45 %). The Kelvin wave flux tends to be slanted toward the Equator in westerly-shear winds e.g.,. The larger wind shear in MERRA-2 is likely the cause for the larger contribution of meridional divergence of the flux.

MRG waves dissipate largely in the lower stratosphere when the zonal wind is easterly at 70 hPa (Fig. 11, the first and second rows). The MRG wave forcing is about 2–3 m s-1 month-1 below 70 hPa in these QBO phases in the reanalyses. When the 20 hPa easterly tendency is at its maximum (Fig. 11, the third row), MRG waves propagate through the lower stratospheric westerlies and dissipate at around 30–20 hPa. The forcing by the MRG waves at these altitudes is largest in JRA-55 (∼1.6 m s-1 month-1), where it occurs at phase speeds between -10 and -30 m s-1. It is also found in Fig. 11 (the first and last rows) that the EP flux divergence of MRG waves exists in the westerly-shear zone at 10–20 hPa in MERRA-2 when the easterly jet is located at 20–40 hPa. This signal also exists in the other reanalyses but with much weaker magnitudes. The waves responsible for this signal seem not to originate from below, as the vertical EP flux from the lower altitudes is interrupted by the easterly winds. The phase speeds of the signal (around -20 m s-1) correspond to the spectral region where Fz is suppressed in Fig. 10 (but Fϕ is emphasized in Fig. S4) mentioned previously. The EP flux divergence in this region is about 3–4 m s-1 month-1 in MERRA-2, although it is uncertain to estimate because of the large discrepancy between MERRA-2 and the others and the larger westerly shear in MERRA-2 compared to the observed (Fig. 11). Previously, also found that MRG waves with the phase speed of -19 m s-1 appear at 20 hPa when there exist easterlies (westerlies) at 50 hPa (20 hPa), using ERA-I. Recently, using a climate model, simulated MRG waves existing in a westerly-shear zone (see their Fig. 9b), which were generated locally by barotropic instability of the westerly QBO jet. These waves have a spectral peak at around k=7–8 and ω∼0.2 cyc d-1 (see their Fig. 8a) which is similar to that detected in MERRA-2 (Fig. S4). The simulated MRG waves induced westward forcing of 7.5–15 m s-1 month-1.

In Fig. 12, it is seen that for MRG waves, the contribution of vertical divergence of the EP flux to the total divergence is very small everywhere (∼10 %). In addition, the EP flux divergence and Fz for each of the composited 13 QBO cycles, as well as the individual profiles of the mean winds, are included in the Supplement (Fig. S5). While there are substantial cycle-to-cycle variations, the overall structures of the EP flux and its divergence for all cases are generally similar to those of the composite averages shown in Fig. 11.

As mentioned in the introduction, the momentum budget of the QBO is currently not fully constrained by observations, and it is believed to be essential for modeling the QBO. The different altitudes of Kelvin wave forcing around 15 hPa among the reanalyses, as well as the different magnitudes, shown in Fig. 11 may imply a limitation in quantifying wave forcing of the QBO using a reanalysis dataset. Advances in observation and assimilation of stratospheric waves might be needed to reduce the spread of the assimilated waves and to further improve global models. In addition, given that the representation of equatorial waves in reanalyses is dependent on the assimilated mean winds (Fig. 11), the reality of small-scale gravity-wave parameterization in forecast models, which can affect the details of the QBO winds, might also be important for the assessment of the large-scale wave forcing. The rather weak magnitudes of the MRG wave forcing in the easterly-shear zone in all of the reanalyses imply that other waves (especially gravity waves) might play a more important role in driving the easterly phase of the QBO than MRG waves and/or that current observational measurements, as well as models and assimilation methods, cannot fully capture the MRG waves which have relatively small vertical wavelengths. Modeling studies have demonstrated that adequate representation of the stratospheric equatorial waves requires vertical resolutions of ∼700 m or finer in the lower to middle stratosphere , while the vertical spacings of the prediction models used for the current reanalyses are about a factor of 2 larger than that (Table 1). It is noteworthy that the aforementioned simulation of locally generated MRG waves by used a 500 m vertical resolution, in which the MRG waves induced large momentum forcing (∼10 m s-1 month-1), although they did not act to drive the QBO but to neutralize the westerly jet.