A Morlet wavelet analysis is applied to the zonal wind at a given altitude for each SMR, resulting in spectra W̃(f,t)n, where f, t, and n=1,2,…,5 represent the frequency, time, and an index of SMRs. W̃(f,t)n corresponds to the phasor representation used e.g.,. We attribute the coherence among W̃(f,t)n to waves traveling in the longitudinal direction with zonal wave number mk, (k=1,2,…,K) and complex amplitude ãk. At given f and t, we fit ãk from W̃(f,t)n following Eq. (5) in . 1W̃1,W̃2,…,W̃5′=Ẽ5×KãK×1 Here, the kth entry of ãk is defined as ãk, and the entry of Ẽ in the kth row and nth column is defined as Ẽn,k:=ei2πmkλn, representing the phase of kth wave detected by the nth SMR at longitude λn. When K<=5, Eq. () allows ã to be estimated with preassigned mk, as demonstrated in Fig. 4 in . Since our five SMRs are mainly from three distinct longitudinal sectors, our implementation entails K≤3. Although two of the five radars provide redundant information as they are in the same longitude sector as other SMRs, we use all five for a broader temporal coverage and higher statistical significance. We assign m following Fig.  for reasons detailed in Sect. .

Note that in estimating ã, we assume that the meridional variation of all waves is negligible among the SMRs. To test this assumption, we ran the climatological tidal model of the thermosphere CTMT, derived from TIDI and SABER. Semidiurnal components in the zonal wind at 50∘ N are highly correlated with those at 40∘ N: the correlation coefficients associated with SW2 and SW1 are 0.94 and 0.99 (not shown here). For the latitude dependence of the semidiurnal tide and its seasonal variation, readers are referred to and , for example.

In principle, ã could be estimated through the LSR or a short-time Fourier transform (STFT) within a sliding window e.g.,. Using a Gaussian window with the proper width, the LSR or STFT might even yield results identical to ours. The width of the window, ΔT, proportionally determines the time resolution σt∝ΔT, which is coupled with the frequency resolution σf according to the Fabor's uncertainty principle σtσf≥14π. The resolution in our wavelet analysis is determined by the Morlet factor as specified in Sect. .

Tides are characterized by oscillations at periods which are integral fractions of a solar or lunar day. In the atmosphere, the solar tides are primarily forced by daily variation in the absorption of sunlight . At a period of 12 h, the migrating component SW2 is known to be the dominant tide e.g.,, while the nonmigrating components, SW1 and SW3, are also frequently reported e.g.,. At the latitude for our study (49∘ N), SW1 and SW3 are expected to be more intensive than other semidiurnal nonmigrating tides on climatological averages (not shown here) according to the tidal model cf.. These solar tides, according to the classic tidal theory , have amplitudes ∼20 times larger than those of lunar gravitationally forced tides. Despite these theoretical predictions, oscillations at 12.4 h have been clearly detected in the upper atmosphere and ionosphere and explained as the lunar tide M2, particularly around SSWs e.g.,. The occurrence of M2 was also confirmed by a wave number identification using a dual-SMR network m=2 at 12.4 h during SSW 2013, cf.. The significant M2 tide was attributed to the lunar forcing resonance due to a shift of a local maximum (namely the Pekeris peak) in the atmospheric frequency response, which is supported by a comparison in a numerical experiment using GSWM driven by two specifications of a climatological-mean background atmosphere and that during SSWs .

In addition to the M2, also oscillating at the period 12.4 h is a westward-traveling structure with zonal wave number m=1, namely the lower sideband (LSB) of the modulation of the 16 d planetary wave (PW) on SW2 tide as explicitly detected and explained in. LSB's m and f are determined by their parent waves according to the nonlinear interaction resonance conditions Ψ̃LSB=Ψ̃SW2Ψ̃PW*. Here, Ψ̃•:=ei2π(f•t+m•λ) represents the phase of a wave • e.g.,. The 16 d PW is a normal wave, and its intrinsic period of 12.5 d is determined by the resonant properties of the atmosphere e.g.,. Having been Doppler shifted by the prevailing eastward wind during winter, the PW is observed at a period up to 20 d, with an average of 16 d for the climatology of the 16 d PW, cf.. The corresponding LSB occurs in the frequency range of fLSB=(2-1/12.5,2-1/20)d-, associated with LSB at TLSB=12.4±0.1 h. Similarly to LSB, an upper sideband (USB), at TUSB=11.6±0.1 h and m=3, might also be excited by the modulation, following the resonance conditions Ψ̃USB=Ψ̃SW2Ψ̃PW as explicitly detected in. To include the periods of most potential LSB and USB, in our wavelet analysis cf., we set the Morlet factor to 128 so that the passed frequency band corresponds to 12.4±0.1, 11.6±0.1 h, and 12.0±0.1 h. These period bands are narrow enough to prevent power leakage or aliasing between each other.

As sketched in Fig. , the abovementioned six waves occupy three near-12 h periods associated with three zonal wave numbers. When implementing Eq. () to quantify the waves, we assume that in comparison with the mentioned waves, other potential waves are negligible at each of the periods. Specifically, we assume that at T=12.0±0.1 h the most important waves are tides SW1, SW2, and SW3 (m1=1, m2=2 and m3=3); at T=12.4±0.1 h M2 and LSB are dominant (m1=1 and m2=2); and at T=11.6±0.1 h only the USB (m1=3) exists. With these assignments of m and according to Eq. (), we repeat the estimation of ã on the grids of date t and altitude h at each of the three periods, resulting in the amplitudes for all six waves, ã•(h,t), where • represents LSB, M2, the USB, SW1, SW2, or SW3. The corresponding amplitude |ã•(h,t)| is displayed in Fig. .

The SSW CA results in Fig. a–f suggest that, among the six components, only three, namely LSB, M2, and USB, exhibit a sharp maximum immediately following PVW, whereas the others, namely SW1, SW2, and SW3, do not: their intensities largely decrease from 40 d before PVW to 50 d after. The enhancements of LSB, USB, and M2 around SSWs are consistent with existing studies, both statistical studies with single-radar approaches e.g., and case studies e.g.,. However, our finding that SW1 and SW3 do not show enhanced intensity during SSWs are at variance with most existing studies e.g.,. LSB and USB enhancements associated with nonenhancing SW1 and SW3 support the hypothesis that LSB and USB were detected at low-frequency resolution and misinterpreted as SW1 and SW3, respectively . In a case study on SSW 2009, evidence for the SW1 misinterpretation was extracted with an intercontinental-scale dual-SMR network extending along 80∘ N , while in another case study on SSW 2013, similar evidence was identified for the SW3 misinterpretation with a similar network at 54∘ N . Here, we report the first multi-year statistical evidence. Besides the responses of LSB and USB, supporting the hypothesis is the decreasing SW2 at PVW (note that the color is scaled for SW2 amplitude in a range broader than those of others). The declining SW2 feeds the LSB and USB enhancements: SW2 provides 100 % and 97 % of the energy of the LSB and USB, respectively, according to the Manley–Rowe relations detailed in .

Although LSB, USB, and M2 composite behaviors look similar to each other in Fig. a–c, their patterns show remarkably different year-to-year variability as shown in Fig. a–c. To investigate the year-to-year variability, we conduct a CA similar to |ã| displayed in Fig. a–f but for the complex amplitude ã. In contrast to the |ã| in Fig. a–c, where all three components maximize during SSW, in |ã| (not shown here) only M2 maximizes, whereas LSB and USB do not. Determined by the phases of both SW2 and the PW at SSWs, ã of LSB and USB exhibit more randomness than that of M2, the phase of which is determined only by the M2 phase at SSW. The consistency between |ã| and |ã| of M2 might be attributed either to a potential association between SSW and a particular lunar phase as suggested by, e.g., or simply to the limited sampling number of M2 enhancement events during SSWs (see Fig. b).

To explore possible relationships between the enhancements of different waves, we search, in Fig. a–c, the maximum amplitude in a 30 d wide window following each PVW, as a measure of the intensity of the corresponding enhancements. The maxima are marked as magenta plus symbols in Fig. . A clear association is found between the LSB and M2 enhancements. As illustrated in Fig. , the seven events are clustered mainly into three groups. In the case of other combinations, i.e., USB vs. M2, or LSB vs. USB, we have not found any noticeable relationship. This result suggests that LSB and USB are independent of each other during SSWs. The lack of coupling between the sidebands has been discussed in detail in Sect. 4.5 in .

We further investigate three clusters in Fig.  according to a classification of associated major SSWs cf.: vortex-split or displacement marked by solid and unfilled black circles, respectively. Clearly, three clusters circled in the magenta lines in Fig.  are associated with the SSW classification: (a) the strongest LSB and intermediate M2 occur in vortex-displacement events, (b) the intermediate LSB and strongest M2 occur in vortex-split events, and (c) the weakest M2 and weakest LSB occur mostly in nonmajor SSW events. Here, nonmajor SSW events refer to the minor and final SSWs cf.. The only exception in this classification is the 2015 event.

The association between the SSW classification and M2 strength is consistent with the conclusion drawn from more SSW events using equatorial magnetic field observations e.g.,. Here, our multi-SMR-jointed analysis allows us to separate LSB and M2 components that share the same period. Our results imply that LSB has contaminated previous M2 estimations based on single-site observations, particularly during vortex-displacement SSWs. The association between the vortex displacement SSW and M2 implies an alternative explanation for the LSB signatures at T=12.4 h with m=1. Although it has never been proposed in existing literature, the LSB signature might be, according to the resonant condition, a secondary wave of the nonlinear interaction between stationary PW with zonal wave number 1 structure (sPW1) and M2. Although our analysis is not sufficient to exclude the possibility of the sPW1–M2 interaction, evidence from case studies was reported to only support the PW–SW2 interaction, including the triple co-occurrence and triple coherency of the three involved waves, and the accompany or occurrence of the USB e.g.,. On the contrary, against the sPW1–M2 interaction is the fact that the LSB signature was observed without co-occurrence of significant M2 e.g.,. Accordingly, throughout the current work, we explain the LSB signature as a secondary wave of the PW–SW2 interaction.

In the amplitude plots shown in Figs. j–l and d–f the vertical variations are characterized by larger amplitudes at higher altitudes. MLT waves are often excited in and propagated from the stratosphere or troposphere. The upward propagating waves amplify exponentially with increasing altitude as the air density decreases and then eventually dissipate. Such a simple vertical structure is associated with the fact that in Fig.  the temporal variations, both enhancements and weakenings, typically extend into broad altitude ranges. We vertically average the amplitudes shown in Fig. d–f for an one-dimensional representation, and display the average as a function of the DoY in Fig. a. The averaged components are shown as a scatterplot in Fig. b, c, and d, against each other, i.e., for SW1 vs. SW2, SW3 vs. SW2, and SW3 vs. SW1. The most salient feature of the scatters is that SW2 is almost always the dominant component, except in late October when SW3 is comparable to SW2. The scatters are further averaged seasonally displayed as the solid red, green, and blue lines, summarizing the main seasonal variations of SW1, SW2, and SW3. SW2 is characterized by two comparable peaks in September and in December and steep decreases in September–October and March–April (DoY 250–300 and 0–80), which is consistent with the seasonal variation of the 12.0 h harmonic amplitude (S2) observed from single-radar analyses as used in, e.g.,, although the SW1 and SW3 are not negligible in comparison with SW2. SW1 is characterized by a single peak appearing in winter and a minimum in summer, which are largely consistent with thermospheric seasonal variation of SW1 at 50∘ N according to CHAMP observations Fig. 12, in. SW3 is characterized by two peaks in earlier May and October (around DoY 130 and 280). Similar annual dual peaks of SW3 were observed from SABER measurements Fig. 2.7 in and also obtained at 50∘ N at 88 km altitude from the 3 years (2006–2008) of simulated data from the Canadian Middle Atmosphere Model Data Assimilation System Figs. 6b and 10c in. Interestingly, in Fig. a, d, the relative importance of SW1 and SW3 switches around early April and November (DoY 90 and 310): in summer SW3 is stronger than SW1, but SW1 is stronger in winter. These seasonal variations might be associated with the climatology of the background mean wind e.g.,.

In comparison with some previous studies, the amplitudes in Fig.  appear to be weaker e.g., for at least three potential reasons. First, based on single-site observations, most existing studies did not separate waves with different wave numbers but had to explain the total oscillations at 12 or 12.4 h as approximations of SW2 or M2 e.g.,. Second, most existing studies used windowing functions much narrower than ours, resulting in broader passbands and capturing more energy e.g.,. Third, some studies present the amplitude of total wind including both zonal and meridional e.g.,, while here we focus only on the zonal component. For a quantitative comparison, in the next section, we present a comparison with an independent empirical tidal model, CTMT.

Figure a–f present a composite analysis in the same manner as in Fig. j–l but for the amplitude of complex average, |ã|. The similarities between Figs. j–l and a–c indicate that the phases of solar tides are consistent from year to year.

For an independent quantitative comparison, we present the seasonal variation of the solar tides according to CTMT , in Fig. g–l. The CTMT results exhibit some consistency with our results, especially on SW2. SW2 in Fig. h maximizes during August–September and December–January, and in between these periods there is a minimum. The vertical gradient is steeper during the August–September maximum than during December–January.

These features are similar to those in Fig. b. However, in the CTMT results, the maxima, or the minimum can hardly be observed (Fig. h). These morphological discrepancies might arise from the low temporal resolution of the model. The effective resolution is about 2 months, during which the satellite observations used for the model cover the whole local time once. In contrast, the September maximum and the minimum are narrower than 2 months; therefore they might be smeared out. In the case of phase, our result in Fig. e also exhibits similarities to the CTMT results in Fig. k.

SW3 is compared in Fig. c, f, i, and l, from which similarities in both amplitude and phase occur mainly in fall. Although SW3 in the CTMT results also exhibits a second maximum, it occurs during February–March, up to 2 months before the second annual peak in early May from our results (see Figs. c and  a). This difference might be associated with the seasonally uneven sampling of observations used for the model: the satellite takes 2 months to cover all local time sectors. In the case of SW1, major discrepancies are found in both amplitudes and phases. For instance, the December maximum in our results (Fig. a) could not be found in the CTMT (Fig. g).

These qualitative findings are supported quantitatively by Fig. a, b, and c, where the in-phase and quadrature components of our estimations vs. the model are shown as scatterplots for SW1, SW2, and SW3. The highest correlation coefficient is observed in SW2. Since the temporal and vertical resolutions of our results are higher than the CTMT, we smear our results down to the resolutions of CTMT and calculate the correlation coefficients again, yielding correlations coefficients slightly higher than those in Fig.  by up to 0.01 (not shown). The correlations are not high overall, reflecting mainly the different assumptions used by the two approaches. Our approaches assume that the meridian tidal variation among our SMRs is negligible, whereas CTMT, as well as any other tidal analyses using single-satellite approaches, assumes the tides are static in the data-binning window. Evaluating these assumptions comparatively entails an independent model with high resolutions in both time and space. Besides, our results might be contaminated by the neglected tides, which is quantified in the next section, whereas the CTMT tidal components might be contaminated by aliasing from waves with similar periods, e.g., the secondary waves and M2.

For estimating the solar tides and as explained in Sect. , in our approach we assume that our targeting components, i.e., SW1, SW2, and SW3, are the dominant components at 12.0 h. This assumption might be too strong given that other neglected semidiurnal tidal components have also been reported e.g.,. The current section quantifies the bias due to the existence of other neglected components according to CTMT.

Arrange Eq. () into two parts, namely the targeting components with amplitudes ã3×1tar and the neglected components with ã(K-3)×1neg: W̃1,W̃2,…,W̃5′=Ẽ5×KãK×1:=Ẽ5×3tarã3×1tar2+Ẽ5×(K-3)negã(K-3)×1neg. Multiply (Ẽ5×3tar)-1:=(Ẽ5×3tar)TẼ5×3tar-1(Ẽ5×3tar)T, resulting in (Ẽ5×3tar)-1W̃1,W̃2,…,W̃5′=ã3×1tar+(Ẽ5×3tar)-13Ẽ5×(K-3)negã(K-3)×1neg. Here, the term on the left is the estimated amplitude of the targeting components, while the first term on the right is the corresponding real amplitude. Therefore, their difference, i.e., the second term on the right, corresponds to the bias due to the neglected tidal components. According to CTMT, we estimate the bias and display its absolute value in Fig. . Contributing to the bias are semidiurnal components SE3, SE2, SE1, S0, and SW4. Overall, the bias is below 2 m s-1 but above 90 km in summer, which suggests our main conclusions in previous sections are not affected by our assumption that SW1, SW2, and SW3 are the dominant components. Actually, when determined by the configuration of the SMRs, the matrices of both (Ẽ5×3tar)-1 and (Ẽ5×3tar)-1Ẽ5×(K-3)neg are very well conditioned (with condition numbers of 1.6 and 2.4), suggesting our estimations are not sensitively affected by the errors in both the wavelet spectra and the neglected semidiurnal components.

Our comparison from the previous section has stressed the additional information that our results bring, specifically, those on SW1 and SW3. In future efforts, we plan to add more ground-based observations and try to combine them with satellite-based wind and temperature observations cf. to improve our understanding of mesospheric tides. Although the current work focuses on the near-12 h waves at midlatitudes, our joint data set analysis approach could be extended to other periods, e.g., diurnal or terdiurnal tides.