The least squares fitting method used in previous studies to derive the amplitude and phase of Q5DOs from satellite observations is based on Eq. (1) but without fitting the first term on the right-hand side (e.g., Huang et al., 2017; Qin et al., 2021). Generally, a 20 d sliding window with a step of 1 d is used to simultaneously extract the amplitudes of TPWs with zonal wave numbers from 3 to -3 (westward to eastward). The daily amplitudes of the Q5DOs are obtained with the largest value in the wave periods between 4 and 7 d. The fitting result is marked at the end day of each 20 d window. To better understand the original least squares fitting method, the synthetic data are used to first simulate the effect of a rapid and large change in SPWs when calculating the amplitudes of Q5DOs. As shown in Fig. 1a and b, three components of waves with a zonal wave number of 1 are given in the synthetic data: an SPW with an amplitude of 100 m and eastward- and westward-propagating Q5DOs with respective amplitudes of 100 and 60 m. The phases are set as 0, -π/4, and π/5 for the SPW and the westward and eastward-propagating Q5DOs, respectively. To simulate the effect of SPWs on TPWs, rapid large changes in the amplitudes of SPW are given on day 100 with magnitudes from 100 to 500 m and on day 150 with magnitudes from 500 to 100 m (see Fig. 1a).

Figure 1c presents the amplitudes of the westward- and eastward-propagating Q5DOs fitted by the least squares fitting method. As shown in Fig. 1c, abnormal fluctuations after days 100 and 150 are captured, which correspond to the occurrence of rapid large changes in the amplitudes of SPWs. However, Fig. 1c suggests that the fitted Q5DOs are not largely influenced by the SPWs when rapid large changes are not given in the amplitudes of SPWs (before day 100 or from days 120 to 150). Additionally, Fig. 1c indicates that abnormal fluctuations in Q5DOs induced by SPWs are not equivalent to two oppositely propagating directions. An enhancement and a decrease in the amplitudes of westward- and eastward-propagating Q5DOs can be simultaneously observed. Results shown in Fig. 1d are the same as those in Fig. 1c but are derived based on different phases of the westward and eastward Q5DOs in the synthetic data, where π/4 and -π/5 are given in the westward and eastward Q5DOs. Comparing the results between Fig. 1c and d, it is interesting to note that the effect of a rapid large change in SPWs on the derived Q5DOs also depends on the phase relationships. Yamazaki and Matthias (2019) suggested that the effect of SPWs could be ignored when the activities of Q10DWs in the oppositely propagating direction were not simultaneously enhanced. However, according to our simulations, this criterion is not suitable for the analysis of Q5DOs with different phases. Our simulation indicates that the influence of a quick and large change in SPWs should not be ignored when extracting Q5DOs during SSWs from satellite observations based on the least squares fitting method. Thus, in this study, we develop a new fitting method to derive the Q5DOs by suppressing the effect of a rapid and large change in SPWs.

As the daily amplitude of SPWs (Akt) cannot be directly derived when Q5DOs exist, the primary goal of the new method is to eliminate the rapid and large changes in Akt. The following steps are performed, in which SPWs and Q5DOs are considered within the same wave numbers.

Based on the new fitting method, we present the fitting result in Fig. 2. As shown in Fig. 2b, the fitted amplitudes of the Q5DOs are generally consistent with the amplitudes given in the original synthetic data. The apparent fluctuations in Q5DOs induced by SPWs have been removed. Note that, based on the new fitting method, the fitted amplitudes are not dependent on the phases of Q5DOs. The new fitting method will provide the same results as those shown in Fig. 2b when Q5DOs have different phases (not shown). Thus, the fitted amplitudes from the new method do not rely on the phase relationships of those waves. Figure 2 demonstrates that the new method is effective to suppress the effect of rapid large change in SPWs; however, a further experiment with synthetic data containing the enhancement of both SPWs and Q5DOs is needed to demonstrate that the new method can properly capture the changes in Q5DOs during SSWs. Moreover, we also add signals of SPWs and Q5DOs with wave number 2 in the synthetic data to approach the real situation in satellite observations. Figure 3 shows the results of the further experiment. The synthetic data used in Fig. 3 consist of six components: SPWs with wave number 1 and 2 (SPW1 and SPW2, respectively), westward-propagating Q5DOs with wave number 1 and 2 (W1 and W2, respectively), and eastward-propagating Q5DOs with wave number 1 and 2 (E1 and E2, respectively). The daily variation in the amplitudes of SPWs and Q5DOs are shown in Fig. 3a and b, respectively. The phase of SPW1 and SPW2 as well as the W1, E1, W2, and E2 Q5DOs are set as 0, π/6, -π/4, π/5, -π/4, and π/3, respectively. Figure 3c and d present the fitting results for the least squares fitting method and the new fitting method. As shown in Fig. 3d, the result manifests that the variations in Q5DOs can be captured based on the new method and that the effect of rapid large change in SPWs can be limited.

Note that some sawtooth-shaped points can be seen in the fitting results in Figs. 1, 2, and 3. The sawtooth-shaped points are caused by removing the linear declination on the time series. This process is required in both the original and new methods to eliminate the effect of seasonal trends in the observational data on the fitting of Q5DOs. The sawtooth-shaped points can be eliminated in the simulation by not removing the seasonal trends, but we keep them in both the original and new methods in the simulations in order to be consistent with the processes of dealing with the observational data.

The SPWs and TPWs can be both captured in the mesosphere region, and their origins have been reported in some previous studies. The mesospheric SPWs are usually believed to be related to the upward wave signals from the troposphere and the lower stratosphere which rely on the structure of the polar vortex (e.g., Harvey et al., 2018). In addition, wave–wave interactions, gravity wave forcing, and auroral heating can also generate mesospheric SPWs (e.g., Lu et al., 2018; Xu et al., 2013; Smith, 2003). The mesospheric TPWs are generally considered to be the result of atmospheric instabilities, and many recent studies have noticed the relationship between extremely strong TPWs and SSW events (Liu et al., 2004; Ma et al., 2020; Yamazaki et al., 2021). The mesospheric TPWs during SSWs can also be secondarily generated in situ by wave–wave interactions (e.g., Xiong et al., 2018; Wang et al., 2021). Nevertheless, the trigger mechanisms of mesospheric TPWs are still not fully understood due to a lack of long-term and high-resolution observational data in this region. Thus, satellite observations are widely used to reveal the feature of mesospheric TPWs. However, as indicated by our simulations, previous studies have ignored the effect of rapid and large changes in SPWs when calculating the variations in TPWs during SSWs. Using the geopotential height data provided by the Aura MLS measurement, we extract the variations in the traveling Q5DOs at 60∘ N during Arctic SSWs. The effectiveness of the new fitting method is discussed by comparing the results from the original least squares fitting method with those from the new method. The daily amplitudes of the Q5DOs are obtained with the largest value in the wave periods between 4 and 7 d. The fitting result is marked at the end day of each 20 d window. The traveling Q5DOs with a wave number of 3 and amplitudes below 10 hPa are not shown due to their weak amplitudes. In the present study, the pressure regions from 10 to 1 hPa, from 1 to 0.01 hPa, and from 0.01 to 0.001 hPa are discussed as the stratosphere, mesosphere, and lower thermosphere, respectively.

As observations from the Aura satellite are available after August 2004, the variations in traveling Q5DOs are investigated during eight midwinter major SSWs from 2005 to 2021 in the present study. Table 1 presents the eight midwinter major SSWs with their onset dates. The date with the maximum positive temperature gradient between 90 and 60∘ N at 10 hPa is defined as the SSW onset date, which is obtained around the date of the first wind reversal during each major event (e.g., Andrews et al., 1987). Note that the onset date used in the present study is only to roughly determine the commencement of SSWs, and our discussions are not sensitive to the nonuniform definitions of SSW onsets (e.g., Butler et al., 2015). In the present work, the SSW in the winter of 2009–2010 is classified as a minor event, as the wind reversal occurred 18 d after the onset date. To distinguished it from the SSW in February 2018, the SSW with the onset date of 28 December 2018 is referred to as the “2019 SSW” in this study. The SSWs before 2013 have been widely studied in previous publications (e.g., Choi et al., 2019; Charlton and Polvani, 2007; Butler et al., 2017; Liu et al., 2019; Rao et al., 2019), and the details of the three major SSWs from 2018 to 2021 are also available in many recent reports (e.g., Rao et al., 2018, 2020, 2021; Wang et al., 2019; Davis et al., 2022; Okui et al., 2021; Wright et al., 2021).

Comparisons of fitted amplitudes of traveling Q5DOs are shown in Figs. 4 and 5 for wave number 1 during the 2008 SSW and for wave number 2 during the 2013 SSW, respectively. Results for each case are given for 81 d, covering the period from 40 d before to 40 d after the SSW onset date (day 0). Figure 4 presents the amplitudes of W1 and E1 Q5DOs obtained from both the original (Fig. 4a, d) and new (Fig. 4b, e) methods during the 2008 SSW. The differences are calculated by subtracting the fitting result of the original method from the new method (Fig. 4c, f). Amplitudes are not fitted in the white area (where the data availability is less than 60 %) in each window. As shown in Fig. 4a, the W1 Q5DOs fitted by the original least squares fitting method reveal a significant response to the onset of the 2008 SSW. The amplitudes of the W1 Q5DOs in the mesosphere are larger than 500 m from days 0 to 20 with a maximum amplitude of 628 m on day 5. Figure 4b suggests that the amplitudes obtained from the new method are lower than 500 m during the 2008 SSW. The maximum amplitude obtained from the new method is 466 m on day 5, which is about 75 % of the amplitude obtained from the original least squares fitting method. The negative differences shown in Fig. 4c are generally larger than 200 m from days 0 to 20 in the mesosphere, indicating that the amplitudes of W1 Q5DOs after the onset of the 2008 SSW might be overestimated by the original least squares fitting method. Nevertheless, positive differences larger than 100 m are also captured before the SSW onset (day -15) around 1 hPa, as shown in Fig. 4c, revealing that the amplitudes of W1 Q5DOs obtained from the original method can also be underestimated during the 2008 SSW. For the amplitudes of E1 Q5DOs during the 2008 SSW, the original least squares fitting method may present an overestimation before the onset date and an underestimation after the onset date. As shown in Fig. 4f, the positive and negative differences both have maximum amplitudes over 200 m in the mesosphere around the onset date.

Figure 5 presents the same results as Fig. 4 but for the amplitudes of the W2 and E2 Q5DOs during the 2013 SSW. As shown in Fig. 5, strong enhancements of W2 Q5DOs and weak amplitudes of E2 Q5DOs after the 2013 SSW are captured by the original least squares fitting method. However, results from the new method after the onset of the 2013 SSW suggest that, based on the original least squares fitting method, the amplitudes of W2 Q5DOs might be overestimated and the amplitudes of E2 Q5DOs may be underestimated. The maximum positive and negative differences are both over 100 m. In order to understand the common differences between the two methods, we calculate the differences during the eight SSWs and present the results in Figs. 6, 7, 8, and 9 for the W1, W2, E1, and E2 components, respectively.

As shown in Figs. 6 and 7, the difference in the fitted westward-propagating Q5DO amplitudes between the new and original methods are usually negative after the SSW onsets, suggesting that the amplitudes of the westward-propagating Q5DOs might be overestimated by the original least squares fitting method after the SSW onsets. However, the difference in the fitted eastward-propagating Q5DO amplitudes between the new and original methods (as shown in Figs. 8 and 9) are usually positive after the SSW onsets, indicating that the amplitudes of the eastward-propagating Q5DOs might be underestimated by the original least squares fitting method after the SSW onsets. Additionally, the E1 Q5DOs before the SSW onsets might also be overestimated by the original least squares fitting method, as seen in Fig. 8. The enhancements of traveling Q5DOs during SSWs reported in previous studies are usually westward propagating after the SSW onsets and eastward propagating before the SSW onsets (e.g., Gong et al., 2018; Yu et al., 2022). Thus, our analyses indicate that the previously reported Q5DOs obtained by satellite measurements during SSWs might be contaminated by SPWs. The amplitudes of the enhancement of Q5DOs during SSWs might be overestimated. Additionally, the westward-propagating Q5DOs before the SSW onsets and the eastward-propagating Q5DOs after the SSW onsets might be underestimated by the original least squares fitting method. Therefore, in future studies of the activities of Q5DOs during SSWs based on satellite observations and reanalysis data, the variations in different wave components in Q5DOs have to be carefully derived by eliminating the effects of SPWs.

Generally, the TPWs, including the Q5DOs, dominate in the mesosphere and lower thermosphere, which are enhanced seasonally during winter and spring and largely control the winds and temperatures in the middle atmosphere (e.g., Gong et al., 2018, 2019; Pancheva et al., 2018; Yamazaki et al., 2020, 2021). The vertical and latitudinal propagation of the TPWs can also transport energy and lead to coupling on a global scale (e.g., Koushik et al., 2020; Ma et al., 2022). Thus, extracting the real amplitudes of the traveling waves is also important to reveal the characteristics in the mesosphere and the vertical couplings in the middle atmosphere. Some extremely strong TPWs are found to be related to the occurrence of SSWs, but their trigger mechanisms have not been fully understood (e.g., Ma et al., 2020; Yamazaki et al., 2021). However, the rapid and large change in the SPWs during SSWs can lead to contamination when deriving the real amplitudes of TPWs based on satellite observations or reanalysis data. The new method proposed in the present study can capture a more accurate variation in the amplitudes of TPWs than the old one. The new method is based on examinations during SSW events due to the assumption that a rapid and large change in SPWs is usually observed during SSWs. Nevertheless, the new method can also be used to extract the amplitudes of TPWs in the mesosphere during other seasons and cases, such as the spring final warmings and other disturbances in stratospheric vortices. Based on the new method, the common feature of the TPWs revealed by satellite observations in the mesosphere and lower thermosphere can be reevaluated, and the trigger mechanism of the mesospheric TPWs during SSWs can be further understood.