The 10DWs shown in the above section are the combination of multiple modes of 10DWs with different zonal wavenumbers. Some observational studies also confirmed these 10DWs in the troposphere of NH (Huang et al., 2009; Wang et al., 2012; Huang et al., 2017) but could not provide zonal wavenumber information because the applied observation data were from a single site. To reveal the dominant modes of those 10DWs, we use the MERRA-2 data along the same lines of latitude to calculate the frequency-wavenumber spectra. The results in the above section demonstrate that the 10DWs are the strongest at ~11 km in the troposphere and ~31 km in the lower stratosphere, so we plot the frequency-wavenumber spectra at these two heights. To identify the prevailing modes of the 10DWs in 2020, a sliding 30-day window is adopted, this time with an increment of 10 days. Within each window, the time and zonal averages are removed from the raw data as the background component. Then, the frequency-wavenumber spectra are calculated for the residuals, and 34 amplitude spectra are acquired throughout the whole year. Finally, we calculate and normalize their arithmetic average.

The normalized frequency-wavenumber spectra of the zonal wind disturbance at ~11 km in 2020 are provided in the top row of Figure 4. The left panels show the results at 32°N, while the right panels show the results at 32°S. It should be noted that the horizontal axis, where positive zonal wavenumbers represent eastward propagation and negative zonal wavenumbers signify westward propagation. In the following, the westward-propagating and eastward-propagating modes with zonal wavenumber n are referred to as Wn and En, respectively, for simplicity. In the troposphere, the eastward-propagating modes are stronger than the westward-propagating modes. Among the eastward-propagating modes, those with larger wavenumbers are stronger than those with smaller wavenumbers. Furthermore, the eastward-propagating modes in the NH are weaker than those in the SH. In the NH, E3 and E6 are almost as strong. In the SH, E5 is obviously stronger than the other modes. Likewise, the normalized frequency-wavenumber spectra of the zonal wind disturbance at ~31 km in 2020 are provided in the bottom row of Figure 4. The left panels show the results at 32°N, while the right panels show the results at 32°S. Only the modes with small wavenumbers, e.g., 1 and 2, exist in the lower stratosphere. The strongest mode in the NH is W1, while the strongest mode in the SH is E2.

The westward-propagating and eastward-propagating modes in the stratosphere and above with small wavenumbers, e.g., 1 and 2 were widely revealed by previous studies (Hirooka & Hirota, 1985; Forbes & Zhang, 2015; John & Kumar, 2016; Huang et al., 2021). The dominant modes in the troposphere are the focus of our research. Figure 4 only provides average intensity of each mode at ~11 km in 2020; as such to reveal the temporal and spatial evolution characteristics of the eastward-propagating modes with large wavenumbers (E3-E6), the MERRA-2 data are harmonically fitted along the same lines of latitude using a sliding 30-day window with an increment of 1 day. Within each window, harmonic fits are performed on the residuals in the period range from 8.0 to 12.0 days in increments of 0.5 days. The formula used for the fitting is as follows:

$$f'(\lambda ,t)=A\cos \left( {\frac{{2\pi }}{T}t - s\lambda - \theta } \right)$$

2

where t, A, T, and θ are the same quantities as in Eq. (1). In addition, λ is the longitude in radians, and s is the zonal wavenumber, where positive zonal wavenumbers represent eastward propagation and negative zonal wavenumbers signify westward propagation. For a certain mode of a 10-day wave, its amplitude at any given height in each window corresponds to the one fit among these periods that has the largest amplitude.

To ensure that the fitting result represents the actual signal rather than noise, we replace the zonal wind from the MERRA-2 data with values from a randomly generated Gaussian distribution with zero mean and a standard deviation associated with the values deduced from the MERRA-2 data. Then, the same method is utilized to fit the random data, and the largest amplitude values of the fitted waves are used to estimate the worst-case noise levels (McDonald et al., 2011). The largest worst-case noise level among E3, E4, E5 and E6 10DW at 32°N and 32°S is ~1 ms−1.

Classic PW studies indicate that the atmospheric background state affects planetary-scale wave propagation by altering the refractive index (Charney & Drazin, 1961); hence, the planetary-scale wave amplitude and phase strongly depend on the background wind conditions. Therefore, we whether these eastward-propagating modes with large wavenumbers propagate freely in the atmospheric background by calculating the refractive index. The formula is as follows (Andrews et al., 1987):

$${n^2}=\frac{{{{\bar {q}}_y}}}{{\bar {u} - c}} - \frac{{{f^2}}}{{4{H^2}{N^2}}} - \frac{{{s^2}}}{{{a^2}{{\cos }^2}\varphi }}$$

3

where the parameter *N* refers to the buoyancy frequency, \(\bar {u}\) is the zonal-mean zonal wind, *φ* is the latitude in radians, *a* is the mean Earth radius, *f* is the Coriolis parameter (*f =* 2Ωsin*φ*; Ω = 7.292\(\times\)10−5 rad−1), *H* is the scale height, *s* is the zonal wavenumber, *c* is the phase velocity, and \(\stackrel{-}{{q}_{y}}\) is the basic northward potential vorticity gradient, which can be written as follows (Andrews et al., 1987):

$${\bar {q}_y}=\beta - {\bar {u}_{yy}} - \frac{{{f^2}}}{{{\rho _0}}}{\left( {\frac{{{\rho _0}}}{{{N^2}}}{{\bar {u}}_Z}} \right)_Z}$$

4

In the above formulas, the overbars denote zonal averages, while the subscripts denote partial derivatives. The parameter *β* is the Rossby parameter (*β =*\(2\Omega \sin \varphi /a\); *a, φ* and Ω are the same quantities as those in Eq. (3)), and \({\rho _0}\) is the basic state density. We obtain all parameters from the MERRA-2 data.

Refractive index squared diagnoses the influence of the zonal mean flow on planetary-scale wave propagation in the meridional and vertical plane. The wave propagates meridionally and/or vertically in the regions where *n*2 > 0 but are evanescent or reflected in the regions where *n*2 < 0. Negative values of *n*2 (indicating regions of evanescence) are shaded in red.

Figure 5 shows time-altitude contours of zonal wind amplitudes for E3, E4, E5 and E6 10DW at 32°N in 2020. The time variations of these four waves show both similarities and differences. They are all strong in the spring and winter, and weak or even indiscernible in the summer and autumn, which is consistent with the seasonal variation of the subtropical jet (top row of Figure 1). This indicates that these wave activities and the eastward jets might be positively correlated. However, the strongest activities of these four waves occurs in different months, and their magnitudes is also somewhat different. The largest amplitudes of the E3, E4, E5, and E6 10DWs are respectively about 5.6, 5.5, 6.2, and 5.4 ms−1, which occur in November, February, March, and January respectively.

Likewise, the time-altitude contours of zonal wind amplitudes for E3, E4, E5 and E6 10DW at 32°N in 2020 are provided in Figure 6. Just like those in the NH, the time variations of these four waves display some similar and different characteristics. They are all no negligible throughout the year, and the amplitudes in spring, autumn and winter are greater than those in summer, which agrees well with the seasonal variation of the subtropical jet in the SH (bottom row of Figure1). This also implies a positive correlation between the wave activities and the eastward jet. E5 is obviously the strongest mode, and E4 is stronger than E3 and E6. Similar to the situation in the NH, the strongest activities of these four waves in the SH occurs in different months. The largest E3, E4, E5, and E6 10DW amplitudes in zonal wind are respectively about 4.9, 8.2, 10.5, and 5.6 ms−1, which appear in September, June, May, and July respectively.

We highlight the evanescence regions, where wave propagation is prohibited, with red shadow in these two figures. The wave amplitude is large in the freely propagating region, while small in the evanescence region, obviously. In both hemispheres, there are very thick evanescence regions above the freely propagating regions at ~15 km, which prevent these waves from propagating upward across the tropopause; as such these eastward-propagating modes with large wavenumbers are weak or even disappear in the stratosphere and above. Since the background zonal wind within the whole display height range (0-32 km) at 32°N from July to August is westward or weakly eastward, there is no freely propagating region.

The above result implies a positive correlation between the wave activities and the eastward jet. Then the correlation coefficients between the amplitudes of E3, E4, E5 and E6 10DWs and the zonal-mean zonal wind at ~11 km in 2020 is calculated. Figure 7 and Figure 8 show the zonal wind amplitudes of the eastward-propagating 10DWs with zonal wavenumbers from 3 to 6 at ~11 km deduced from MERRA-2 data at 32°N and 32°S, respectively. Yellow line marks the zonal-mean zonal wind. For a better representation, the zonal-mean zonal wind is scaled by 7. In both hemisphere, the seasonal variations of E3, E4, E5 and E6 10DWs are consistent with those of the zonal-mean zonal wind. The correlation coefficients between the E3/E4/E5/E6 10DW and the zonal-mean zonal wind are 0.57/0.68/0.71/0.67 in the NH, while 0.31/0.64/0.09/0.17 in the SH. All the coefficients except the one for the E5 10DW in the SH are far greater than the value corresponding to the 99% confidence level of a test of statistical significance, which implying strong and positive correlations. The coefficient for the E5 10DW also signifies a positive correlation, but not as strong as the others. We speculate that there are other reasons for the time variation of the E5 10DW in addition to the background zonal wind.