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Planetaryscale MLT waves diagnosed through multistation methods: a review
Earth, Planets and Space volume 75, Article number: 63 (2023)
Abstract
Most experimental investigations on planetaryscale waves in the mesosphere and lower thermosphere (MLT) region are based on singlestation or satellite spectral analysis methods, which suffer from intrinsic spectral aliasing/ambiguity. To overcome the aliasing, the author has developed and utilized dual and multistation spectral methods in a series of recent works. These methods were implemented on meteor radar observations and surface magnetometer observations. In the implements, a variety of waves were discovered or investigated in terms of seasonal variations and responses to sudden stratospheric warming events, such as lunar and solar tides (migrating and nonmigrating), Rossby wave normal modes, ultrafast Kelvin waves, and secondary waves of wave–wave nonlinear interactions between the previous waves. The current paper illustrates these methods using synthetic data, comparatively reviews the methods and results in plain language, and proposes a new representation, termed the adjusted Feynman diagram, to summarize the nonlinear interactions and explain their implications.
Graphical Abstract
Introduction
The atmosphere is constantly exposed to periodic disturbances. The disturbances range broadly in time and space, at least temporally from microseconds to Milankovitch orbital scales, and spatially from centimeters to the planetary scale. At the planetary scale, most disturbances are driven by planetary waves (i.e., Rossby waves), tides, and their nonlinear interaction waves. These waves’ amplitudes are often weak in the low atmosphere, increase with altitude, and maximize in the mesosphere–lowerthermosphere (MLT) region (e.g., Hirooka and Sciences 2000). The wave intensity and variety make the MLT an ideal nature lab for studying atmospheric waves. However, MLT observations are relatively sparse compared to other atmospheric regions since both balloons and spacecraft cannot permanently operate in the MLT. Continuous observations could only be collected remotely through either optical instruments onboard satellites or radio approaches on the ground. Both observational techniques have been used to investigate MLT waves, figuring out the salient wave behaviors in both case and statistical studies (e.g., Oberheide et al. 2011). However, most observational studies used singlestation or satellite methods and were, therefore, potentially affected by inherent spatiotemporal aliasing.
Singlestation analyses cannot diagnose the horizontal wavelength of planetaryscale waves (e.g., Azeem et al. (2000)), although they enable identifying waves at expected frequencies with a highfrequency resolution. On the other hand, although spacebased sensors collect data across all longitudes and allow us to determine the horizontal scale of waves, singlesatellite analyses suffer from intrinsic aliasing (e.g., Tunbridge et al. 2011; Salby 1982). The aliasing can be explained in terms of the Doppler shift of waves traveling in the earthfixed coordinate system recorded by a sunsynchronous observer because most relevant satellites obit quasisunsynchronously. For convenience, the current work uses [f, s] to denote a wave at the frequency f with the zonal wavenumber s in the earthfixed coordinate system. (In some literature, s is also denoted as m. \(s>0\) and \(s<0\) denote westward and eastward traveling waves, respectively. Since [f,s] and [\(f\),\(s\)] denote the identical wave, the current work defines the frequency as nonnegative \(f\ge 0\).) A wave [f, s] is Dopplershifted to frequency \(f'=fs\text {*1cpd}\) for a sunsynchronous observer, where cpd abbreviates cycles per day. Then, all waves [\(f+C\text {*1cpd}\), \(s+C\)] with an arbitrary C will be Dopplershifted to the same \(f'\) in the sunsynchronous coordinate system and, therefore, are not distinctive from each other in singlesatellite analyses. (Specially, when \(C \in {\mathbb {Z}}\) is an integer, [\(f+C\text {*1cpd}\), \(s+C\)] includes all potential secondary waves of wavewave nonlinear interactions between [f, s] and all migrating tides.) An example is that all migrating tides [\(n\text {*1cpd}\), n] (where \(n \in {\mathbb {N}}\) is a positive integer) are Dopplershifted to \(f'=0\) and are not distinctive from each other. Another example is the wellknown zonal wave4 structure (e.g., Immel et al. 2006). In singlesatellite analyses, this structure is characterized by \(f'=\text {4cpd}\), which might be Dopplershifted signatures of at least three potential waves [1cpd, –3], [2cpd, –2], and [0, ±4]. Although [1cpd, –3] is believed to be the main contributor (Forbes et al. 2003; Forbes et al. 2006; Hagan and Forbes 2002; Pedatella et al. 2008), [2cpd, –2] (He et al. 2011; Chen et al. 2019) and [0, ±4] (He et al. 2010; Pedatella et al. 2012a) were also reported to be the primary contributor under some conditions. Note that our Dopplershift interpretation is equivalent to the interpretation of “spacebased zonal wavenumber”\(s'\) in, e.g., Forbes and Moudden (2012) and Nguyen et al. (2016), because \(s' \equiv \vert \tfrac{f'}{1\text {cpd}}\vert\).
Many researchers tried to overcome the above aliasing by combining observations from multilongitudinal sectors (e.g., Baumgaertner et al. 2006; Murphy et al. 2006; Manson et al. 2009; Pancheva et al. 2002, 2004; Jiang et al. 2008). In a series of works, the author developed various multistation methods and implemented the methods to diagnose diverse planetaryscale waves. The current paper reviews these methods and results comparatively.
Methods
Experimental wave identifications typically refer to wave properties of, e.g., frequency, wavenumber, and polarization. The current section summarizes some methods identifying waves referring to zonal wavenumber s and frequency f based on observations recorded on regular time t grids from two or more irregularly separated longitudes \(\lambda\) at the same latitude. The fs identification can be realized through a leastsquares (LS) fitting to a predefined model as a function of f and s. Such a twodimensional (2D) fitting in a sliding window can yield a temporal resolution. However, the predefined model entails much prior knowledge, and the 2D sliding fitting is typically computationally expensive. Therefore, the author performs fs identification through two spectral analyses: linear transformations (wavelet or Fourier transformation) from the t domain to the f domain (denoted hereafter as the \(t\mapsto f\) analysis) and spectral analysis from the \(\lambda\) domain to the s domain (denoted as the \(\lambda \mapsto s\) analysis).
The phase difference technique (PDT) for diagnosing zonal wavenumber
The PDT is a dualstation method, which firstly realizes the \(t\mapsto f\) analysis through a wavelet or Fourier transformation and then deals with the \(\lambda \mapsto s\) analysis through crossspectral analysis (e.g., He et al. (2018)).
A plane wave triggers coherent oscillations everywhere on the wave’s path. The coherence means that the phase difference between any two locations is timeindependent. The phase difference equals the spatial separation multiplied by the aligned wavenumber. Therefore, the aligned wavenumber can be calculated according to the experimental estimations of the phase difference and the spatial separation. Using observations from two zonally separated stations at the same latitude, one can estimate the phase difference through crosswavelet analyses or LS crossspectral analyses. Two assumptions used in the wavenumber estimation are the single and longwave assumptions. The first requires that one wave be dominantly stronger than the rest at any frequency and instant. To satisfy this assumption, we trade off between the frequency and time resolutions differently in different circumstances. The longwave assumption is required due to the Nyquist theorem in space that the shortest identifiable wavelength is twice longer than the station spacing. In the atmosphere, most planetaryscale waves are associated with integer zonal wavenumbers. Therefore, we often use a third assumption that the zonal wavelengths of the underlying waves are loworder harmonics of 360\(^\circ\) longitude. This integer zonal wavenumber assumption can relax the longwave assumption slightly.
In Fig. 1a, b, synthetic data are constructed for implementing the PDT, comprising two synthetic waves plus a unit of Gaussian noise. The synthetic waves are the diurnal and semidiurnal migrating tides, [1cpd, 1] and [2cpd, 2], which are among the most extensively studied atmospheric waves with nearzero integer zonal wavenumbers. Nearzero wavenumbers are chosen here to facilitate the longwave assumption, while the integer wavenumbers were used since noninteger wavenumber waves can be decomposed as linear combinations of integerwavenumber waves. Readers may overlook the frequency selection of the synthesized waves, since the methods introduced in the current work were developed to not favor any frequency.
The synthetic data are sampled at two random longitudes, labeled as a and b in Fig. 1b. The crosswavelet spectrum between the longitudes is displayed in Fig. 1c. The red and green peaks at \(T=\)1 and 2 day indicate zonal wavenumbers of s= 1 and 2, respectively. In addition, the power of the crosswavelet reflects the wave amplitudes. The isolation between the red and green peaks reflects that the singlewave assumption is valid. Otherwise, if the peaks overlap, the singlewave assumption fails and the dualstation PDT would not work. However, observations from more stations can help distinguish overlapping waves, as demonstrated in the subsequent two subsections.
Wavelet transformation plus leastsquares (WT & LS) approaches
The PDT described in the previous subsection relies on the single and longwave assumptions and the integerwavenumber assumption. These assumptions also enable the \(\lambda \mapsto s\) analysis through LS approaches using the outcomes from the \(t\mapsto f\) analysis that can be accomplished through, e.g., wavelet transformation, as introduced in the previous subsection (e.g., He et al. (2021a)). These LS approaches coupled with the \(t\mapsto f\) analysis are referred to as WT & LS throughout this study.
At each instant and frequency, the wavelet amplitudes (or Fourier or LombScargle amplitudes, e.g., He et al. 2021a; He et al. 2020) can be fitted to a predefined wavenumber model through LS approaches. The wavenumbers of underlying waves are supposed to be defined according to prior knowledge or to be determined through, e.g., the PDT or an LS optimization minimizing the error between the wavelet complex amplitudes and a singlewavenumber model (e.g., He et al. 2021a). According to the predefined wavenumbers, multiwavenumber models can further be implemented to estimate the amplitudes of underlying waves through LS fitting (e.g., He et al. 2018).
As an example, Fig. 2 presents an implementation of the WT & LS approach to the synthetic data displayed in Fig. 2a, b that are composed by superposing a third wave [1cpd, − 1] on Fig. 1a, b. The third wave is superposed to illustrate the capability of the WT & LS method in resolving waves overlapping at the same frequency. In Fig. 1, the waves are separated in the frequency domain, whereas in Fig. 2, two waves with comparable amplitudes overlap at frequency f = 1cpd around \(t=\) 5 day.
The synthetic data are sampled along three randomly selected longitudes, as indicated by the dashed lines in Fig. 2b, to estimate the zonal wavenumber s of the underlying waves through an LS procedure (see, e.g., Fig. S3 in He et al. 2021a). In principle, the s estimation requires samples from a minimum of two longitudes, since the fitting uses a singlewave model. However, the fewer longitudes used, the more sensitive the fitting is to noise and longitude configuration. Here, three longitudes are selected in the s estimation. The estimated s is displayed in Fig. 2c, which reveals the three synthetic waves [f, s] = [1cpd, − 1], [1cpd, 1], and [2cpd, 2] properly. Estimating the amplitudes of three waves through the WT &LS approach requires observations from at least three longitudes (see, e.g., Equation 1 in He and Chau 2019). Accordingly, the samples collected along the three longitudes are implemented further to estimate the wave amplitudes following the procedures used, e.g., in Fig. 3a, b in He et al. (2021a). The results are displayed in Fig. 2d–f, maximizing at 7, 5, and 8 ms\(^\), respectively, which is consistent with the corresponding amplitudes in Fig. 2a.
Compared with the dualstation PDT, the WT & LS approach can utilize as many stations as possible, but it lacks straightforward control over the Nyquist spatial aliasing due to the preassigned wavenumbers based on prior knowledge or by the LS method. An imprecise preassignment may lead to biased amplitude estimations. When observations are available at more longitudes, the preassignment can be avoided or relaxed through the approach introduced in the following subsection.
Harmonic regression plus wavelet transformation (HR & WT) approaches
Both PDT and WT &LS methods summarized in the previous two subsections realize the sf identification by carrying out first the \(t\mapsto f\) analysis and then the \(\lambda \mapsto s\) analysis. In principle, it is also possible to perform the \(\lambda \mapsto s\) analysis before the \(t\mapsto f\) analysis (e.g., Section 4.1 in Forbes et al. 2020).
Observations from an arbitrary number of stations at any instant can be decomposed into a linear combination of zonal subharmonics \(\vert s\vert =0, 1, 2,...\), through an LS harmonic regression. Each subharmonic coefficient is a complex number that denotes the subharmonic’s amplitude and zonal phase. The wavelet transformation of the time series of each complex coefficient represents the subharmonic tf spectrum. The tf spectra, using the Gabor function (Torrence and Compo 1998) and its conjugate as the mother wavelet, denote the eastward and westward traveling structures, respectively. The current paper refers this method as HR & WT.
The computation cost of the HR & WT methods is proportional to the number of selected zonal harmonics, whereas the cost of the WT & LS method is proportional to the number of stations. Therefore, analyzing observations from a large number of stations, the HR & WT methods would be computationally cheaper than the WT & LS methods.
Figure 3 presents an implementation of the HR &WT approach using the synthetic data displayed in Fig. 2b. The implementation aims at the wave amplitudes for seven wavenumbers \(s=\)1, 2, 3, 0, − 1, − 2, and − 3, for which the synthetic data are sampled at seven longitudes, comprising the three longitudes displayed in Fig. 2b and four more randomly selected longitudes as indicated by the dashed lines in Fig. 3a. The amplitudes for the seven wavenumbers are estimated following the procedures detailed in section 4 in Forbes et al. (2020), which are presented in Fig. 3b–h, respectively. The estimation captures the three waves properly. However, in addition to the three waves, some spectral signals appear in Fig. 3b–h which do not exist in the synthetic data, such as the peak of s = 0 at T = 1, t = 8 in Fig. 3e and the peak of s = − 1 at T = 0.5, t = 10 in Fig. 3f. These power leakages are associated with the finite number and uneven distribution of longitudinal samplings (see section 4 in Forbes et al. (2020) for discussions). This leakage will reduce with data sampled from more longitudes or evenly distributed longitudes. Therefore, the HR & WT approaches are more applicable to datasets evenly covering plenty of longitudes, such as observations from slowprocessing polarorbiting satellites and outputs from models. When longitudinal coverage is sparse or uneven, the WT & LS method will be more practical.
Evaluations of the above methods
The methods introduced in previous subsections have been evaluated crossly.
Wave amplitudes estimated through the WT & LS methods were compared in a statistical study with a climatological tidal model of the thermosphere (CTMT, Oberheide et al. 2011) and in a case study with results derived from observations of the Michelson Interferometer for Global Highresolution Thermospheric Imaging (MIGHTI Immel et al. 2017) instrument on the ICON satellite. The comparisons, displayed in Figs. 7 and 8 in He and Chau (2019) and Fig. 3 in He et al. (2021a), respectively, exhibit reasonable consistency. The comparison with MIGHTI results is adjusted and displayed here in Fig. 4. The method was also evaluated using virtual data generated with the Whole Atmosphere Community Climate Model with thermosphere–ionosphere eXtension (SDWACCMX) by Macotela et al. (2022). The author estimated amplitudes of selected waves through WT &LS approaches using virtual data from only three selected longitudinal sectors and compared the results with amplitudes estimated using data from all longitude sectors. The comparison exhibits reasonable consistency. Observational error propagation in the WT & LS method was analyzed through a Monte Carlo simulation (see Fig. 4 in He et al. 2018a; He et al. 2018, respectively). In addition, He and Chau (2019) quantified the susceptibility of the WT & LS amplitude estimations to neglected waves through analytical analysis and implemented the analytical solution with the empirical model CTMT. The results, displayed in Fig. 9 in He and Chau (2019), demonstrated the method’s feasibility when the neglected waves are weaker than the estimated waves.
To evaluate the PDT approach, He et al. (2020) and He et al. (2021a) collected meteor radars from multiple longitudinal sectors and used them to estimate s through the PDT using different combinations of radar pairs. The dualstation configurations result in consistent estimations (see Fig. 3 in He et al. 2020a), which are compared excellently with the WT &LS estimations (see Figures S2 and S3 in the supporting information in He et al. (2021a)).
In implementing the HR &WT approach with four longitudinally separated stations, Forbes et al. (2020) evaluated the approach using output from Thermosphere–Ionosphere–Mesosphere–Electrodynamics General Circulation Model (TIMEGCM). In Fig. 9, the author compared the fourstation estimation with alllongitude estimation, and investigated the susceptibility of the fourstation estimation on the longitudinal polarization of the fourstation configuration. The author also quantified the error propagation and its dependence on the stations’ longitudinal separation through a Monte Carlo simulation in their Figure A1 in Forbes et al. (2020).
Spectral periodic table (SPT)
Secondary wave–wave nonlinear interactions between two waves [\(f_1\),\(s_1\)] and [\(f_2\),\(s_2\)] might generate two secondary waves (SWs), denoted hereafter as [\(f_1\),\(s_1\)]–[\(f_2\),\(s_2\)]=[\(f_1f_2\),\(s_1s_2\)] and [\(f_1\),\(s_1\)]+[\(f_2\),\(s_2\)]=[\(f_1+f_2\),\(s_1+s_2\)] and termed lower and upper sidebands (LSB and USB), respectively. These relations of s and f are ensured by the phasematching among involved waves (He and Forbes 2022), and are, therefore, referred to as phasematching relations in the current work. The phasematching relation of f is equivalent to energy conservation according to the Manley–Rowe relation (He et al. 2017) which specifies that the energy of each wave involved in an interaction is proportional to the wave’s absolute frequency. If one planetary wave [\(f_{PW}\),\(s_{PW}\)] interacts with multiple migrating tides [n cpd,n] (\(n\in {\mathbb {N}}\)), the resultant secondary waves (SWs) [\(n\pm \frac{f_{PW}}{1\text {cpd}}\) cpd,\(n\pm s_{PW}\)] will populate the spectrum periodically. Using this periodicity, He et al. (2021b) developed the spectral periodic table (SPT) to extract the SW signatures in batches.
To construct the SPT, the authors first calculated a frequency crossspectrum between the two stations, \({\tilde{c}}(f)\), through the LombScargle analysis. Then, the authors chopped the \(\tilde{c}(f)\) spectrum into 0.5cpdwidth pieces, in analogy to the concept of a period in the periodic table of elements. Each piece is characterized by \(\delta f: = \left\frac{f_{PW}}{1\text {cpd}}  \left \lfloor {\frac{f_{PW}}{1\text {cpd}}} \right\rceil \right\) either increasing monotonically from 0 to 0.5 or decreasing from 0.5 to 0. The pieces were wrapped following the magenta arrow in Fig. 5. In the resultant spectral table, each row from left to right is associated with increasing \(\delta f\) from 0 to 0.5. In principle, all the potential SWs associated with the same PW at \(f_{PW}\) are located in the same column, which are characterized by \(\delta f \equiv \min n\pm \frac{f_{PW}}{1\text {cpd}}\) and termed a family, in analogy to the concept of a group in the periodic table of elements. Their product \(P(\delta f):=\prod \limits {\tilde{c}}(\delta f)\) is expected to be significantly beyond the noise level. Therefore, maxima of \(P(\delta f)\) were used to identify family candidates. For each candidate, the phasematching relation of the zonal wavenumber \(n\pm s_{PW}\) can be used as a constraint of \({\tilde{c}}(\delta f)\) for estimating the zonal wavenumbers through an LS regression. Finally, the LS estimations were compared with their PDT estimations for evaluations.
Similar to the periodic table of elements which can potentially be extended with more elements and periods, the SPT in Fig. 5 can potentially be extended to include interactions of higherorder tidal harmonics.
Adjusted Feynman diagram (AFD): a representation of wave–wave interactions
Inspired by Hasselmann (1966), the author adjusts the Feynman diagram to represent MLT wave–wave nonlinear interactions. As an example, Fig. 6a displays an adjusted Feynman diagram (AFD) of the LSB generation between a planetary wave [\(\frac{1}{10\text {d}}\), 1] and a migrating tide [2cpd, 2]. The AFD comprises three arrows in the fs plane, intersecting at a vertex. Each arrow denotes one wave. The arrows pointing to the vertex denote the parent waves which existed before the interaction, whereas the arrow pointing away from the vertex denotes the secondary wave (SW), which is generated in the interaction. The projections of each arrow onto the x and yaxes denote the wave’s frequency and zonal wavenumber, respectively. Since the author defines \(f\ge 0\), the parent waves locate either at the left side of the yaxis (\(f>0\)) or on the yaxis (\(f=0\)), and the SW locates either at the right side (\(f>0\)) or on the yaxis (\(f=0\)). The red and blue colored arrows denote energy sources and sinks, respectively, according to the Manley–Rowe relation (He et al. 2017). In the interaction represented in Fig. 6a, the planetary wave is an energy sink, indicating that the wave is amplified in the interaction, although it is a parent wave. Such an amplified parent wave was called an antiwave (e.g., Hasselmann 1966). This amplification was termed planetary wave amplification by stimulated tidal decay (PASTIDE, He et al. 2017). The Manley–Rowe relation is known as the Planck relation in Quantum mechanics. One quantum mechanic counterpart of PASTIDE is the LASER (light amplification by stimulated emission of radiation). The AFD for the USB generation between the planetary wave and tide is displayed in Fig. 6b, comprising two energy sources and one sink. In each of the AFDs, either LSB or USB, the blue arrows’ vector sum equals the red arrows’ sum, required by the phasematching relations.
Detected planetaryscale waves
The above methods were implemented on networks of meteor radars at high, mid, and lowlatitude and surface magnetometers at the geomagnetic equator, as summarized in Fig. 7. The implementations revealed diverse planetaryscale waves as specified in Table 1 and Fig. 8. The current Section summarizes these results in three categories as follows.
Multiday oscillations
In a series of case or statistical studies at different latitudes, various waves were diagnosed at periods longer than 1 day, as summarized in the fs depiction in Fig. 8b. These waves can be categorized into Rossby wave normal modes (RWNMs, e.g., Sassi et al. 2012), quasi2day waves (Q2DWs, e.g., Salby 1981; Rojas and Norton 2007), ultrafast Kelvin waves (UFKWs, e.g., Forbes 2000), and secondary waves (SWs, e.g., Forbes and Moudden 2012) of the previous waves’ nonlinear interactions.
Diagnosed RWNMs include the 16, 10, and 6day normal modes, namely, [\(\frac{1}{16\text {d}}\), 1], [\(\frac{1}{10\text {d}}\), 1], and [\(\frac{1}{56\text {d}}\), 1] at mid and highlatitude in the northern hemisphere, through the PDT using meteor wind observations, mostly during arctic SSWs (e.g., He et al. (2020c)). Quasi6 and 10day RWNMs were also diagnosed in the antarctic SSW 2019 in both the northern (He et al. 2020a) and southern hemispheres (Wang et al. 2021), through the PDT. In addition, a wave [\(\frac{1}{16\text {d}}\), 2] was detected around an SSW and was explained as the USB of the interaction between the 16day RWNM [\(\frac{1}{16\text {d}}\), 1] and stationary planetary wave [0, 1] (He et al. 2020c), as denoted in the AFD in Fig. 9.
Through the PDT and multiyear composite analyses, He et al. (2021b) diagnosed midlatitude Q2DWs [\(\frac{1}{41\text {h}}\), 4] and [\(\frac{1}{50\text {h}}\), 3] that maximize annually in July.
Through both the PDT and WT &LS methods, enhancements of lowlatitude Q2DWs [\(\frac{1}{50\text {h}}\), 3] and [\(\frac{1}{46\text {h}}\), 2] are observed in early 2020 (He et al. 2021a). Their amplitudes compare consistently with those derived from Michelson Interferometer for Global Highresolution Thermospheric Imaging (MIGHTI) at 95–100 km altitude where the two data sets overlap (He et al. 2021a). The authors attributed these Q2DW enhancements to their seasonality.
In a case study, Forbes et al. (2020) implemented the HR &WT and detected UFKWs [\(\frac{1}{24\text {d}}\), –1] from surface magnetic field perturbations collected by four equatorial magnetometers. The results revealed the capabilities of the surface observations being used to infer the MLT dynamics.
Near12h waves
The enhancement of the semidiurnal lunar tide M2 [\(\frac{1}{12.4\text {h}}\), 2] during SSWs was broadly reported in the atmosphere and ionosphere (e.g., Yamazaki 2013; Forbes and Zhang 2012; Forbes et al. 2013; Pedatella et al. 2012b; Zhang and Forbes 2013; Liu et al. 2021). However, the M2 estimation in singlestation analyses might be contaminated by another 12.4h wave, namely, the LSB of the interaction between the semidiurnal solar migrating tide (SW2) and the 16day RWNM: [\(\frac{1}{12.4\text {h}}\), 1]=[2cpd, 2]–[\(\frac{1}{16\text {d}}\), 1] (e.g., Kamalabadi et al. 1997) as represented in Fig. 6c. The PDT was developed originally for distinguishing these two 12.4h waves: He et al. (2018) confirmed the M2 at boreal midlatitude during SSW 2013 whereas He et al. (2018a) reported the LSB at boreal highlatitude during SSW 2009 which confirmed the potential contamination.
In addition to the LSB and M2, there are at least four other near12h waves that are reported to be active during SSW (see Fig. 8c), including the USB [\(\frac{1}{11.6\text {h}}\), 3]=[2cpd, 2]+[\(\frac{1}{16\text {d}}\), 1] (Fig. 6d), the SW2 [2cpd, 2], and nonmigrating tides [2cpd, 1] and [2cpd, 3] (e.g., Pedatella and Forbes 2010; Pedatella and Liu 2013). Implementing the WT &LS method on multiyear observations of five boreal midlatitude meteor radars, He and Chau (2019) revealed that the three tides do not enhance around the SSW center day, but the LSB, USB, and M2 do. The results suggested that the reported enhancements of the nonmigrating tides are misinterpreted from the LSB and USB due to the nonorthogonality of spectral basis functions used for extracting the waves. Due to similar nonorthogonality, the experimental M2 and SW2 estimations might contaminate each other. Equation 20 in He et al. (2018) quantified the nonorthogonality between the 12.0 and 12.4h sinusoid functions in a rectangle window as a function of the window width, which is displayed here in Fig. 11. The equation specifies that the nonorthogonality minimizes at zero when \(\ {e}^{i2\pi \Delta f\Delta T}=1\) or \(\Delta f\Delta T \in {\mathbb {N}}\), where \(\Delta f\approx \frac{1}{12.0\text {h}}\frac{1}{12.4\text {h}}\) denotes the frequency difference between the two waves and \(\Delta T\) denotes the window width. Therefore, the nonorthogonality minimizes at \(\Delta T=\frac{1}{\Delta f}, \frac{2}{\Delta f},...\), namely, \(\Delta T\approx 14.8\text {d}, 29.5\text {d},...\). These windows should be prioritized in relevant studies to minimize the contamination between the 12.0 and 12.4h tides. Otherwise, as a counterexample and as revealed by Fig. 11, using a 21day window would result in approximately 20% of the SW2 amplitude being estimated as an M2 amplitude (Chau et al. 2015). It is advisable to avoid using a 21day rectangular window for estimating M2, as it maximizes the contamination from SW2 and leads to an overestimation of M2 amplitude. When there are potentially multiple or unknown contaminating waves, the author recommends using a broad window and enveloping the data with, e.g., a Gaussian function (see the green dashed line in Fig. 11.).
Similar to the 16day RWNM, the 10day RWNM can also interact with SW2 (He et al. 2021a), generating near12h LSB ([\(\frac{1}{12.6\text {h}}\), 1]=[2cpd, 2]–[\(\frac{1}{1\text {d}}\), 1], Fig. 6a) and USB ([\(\frac{1}{11.4\text {h}}\), 1]=[2cpd, 2]+[\(\frac{1}{10\text {d}}\), 1], Fig. 6b). These LSB and USB might also be interpreted as nonmigrating tides due to the nonorthogonality.
Other tides and SWs
The PDT was also used to diagnose s of other solar tidal subharmonics in two case studies using two boreal midlatitude meteor radars. He et al. (2020) identified six subharmonics during the 2017–2018 winter and found that all subharmonics are migrating components. The 12, 6 and 4h components weaken around the central day of SSW 2018. Among the six migrating subharmonics, the lowest frequency four were broadly studied but the 4.8 and 4h subharmonics have been overlooked. He et al. (2022) attributed the overlook to inappropriate noise models used in the existing literature.
The lowest frequency four migrating components were also identified in the summer 2019 (He et al. 2021b). The SPT analysis revealed that all four migrating components interacted with two Q2DWs, [\(\frac{1}{40\text {h}}\), 4] and [\(\frac{1}{50\text {h}}\), 3], generating a variety of SWs as sketched in the AFDs in Fig. 10. Among these interactions, those sketched in Fig. 10e, h, and i involving the 8 and 6h migrating tides were observed interacting with PWs for the first time.
Summary
Observations of mesosphere and lower thermosphere (MLT) region are sparse compared to those below and above MLT. Therefore, most experimental studies on planetaryscale waves in the MLT region have been carried out using singlesatellite or station methods, which are subject to intrinsic temporal–spatial aliasing. To overcome this issue, several multistation methods have been developed recently that use groundbased observations recorded on a regular time grid in multiple longitudinal sectors. These methods transform the observations from the time and longitude domain to the frequency and zonal wavenumber domain using various techniques such as Fourier and wavelet transforms, Lomb–Scargle spectral analyses, and leastsquares regression. By implementing these methods to meteor winds and surface magnetic field observations from low, mid and highlatitudes, researchers have identified various waves, including Rossby wave normal modes (RWNMs), quasi2day waves (Q2DWs), ultrafast Kelvin waves (UFKWs), and tides and their upper and lower sidebands (USB and LSB) arising from nonlinear interactions with RWNMs or Q2DWs. The activities of these waves during sudden stratospheric warming events (SSWs) have also been investigated in detail, such as the amplification of 16, 10, and 6day RWNMs, semidiurnal lunar (M2) tide, and the 11.4 and 11.6h USBs (zonal wavenumber 3) and 12.4 and 12.6h LSBs (zonal wavenumber 1) of tideRWNM interactions, and the weakening of 6 and 4h migrating tides. These works revealed that the estimation of M2 might be contaminated by the 12.4h LSB in singlestation analyses in existing literature due to zonal wavenumber ambiguity. The M2 estimation might also be contaminated by the 12h solar migrating tide (SW2) due to nonorthogonal spectral basis functions used for extracting the waves. On the other way around, the existence of M2 can also contaminate the SW2 estimation due to the nonorthogonality. Similarly, the contamination associated with the nonorthogonality also occurs potentially between the USBs and the 12h nonmigrating tide with zonal wavenumber 3 (SW3), and between the LSBs and the 12h zonalwavenumber1 tide (SW1). A multiyear composite analysis revealed that the LSB and USB amplifications might have been misinterpreted as SW1 and SW3 amplifications, respectively, in some existing literature. In addition, LSBs and USBs generated by interactions between four solar migrating tides and two quasi2day waves were observed 3 months before the Antarctic SSW 2019.
The results summarized above were obtained by applying the methods at specific latitudes, primarily at midlatitudes in the Northern Hemisphere. To broaden the scope of these findings and overcome aliasing issues, future studies should expand to different latitudes and the Southern Hemisphere. To further improve the methodology and remove the assumptions made in multistation approaches, future research should incorporate data from both satellite and groundbased observations across various latitudes, as demonstrated in pioneering work by Zhou et al. (2018).
Availability of data and materials
As a review, the current work is based on data and materials that have been published in the original papers. No new dataset has been used in the current work.
Abbreviations
 MLT region:

Mesosphere and lower thermosphere region
 SSW:

Stratosphere sudden warming event
 PDT:

Phase difference technique
 LS:

Leastsquares
 HR&WT:

Harmonic regression plus wavelet transformation
 WT&LS:

Wavelet transformation plus leastsquares method
 SPT:

Spectral periodic table
 Q2DW:

Quasi2day wave
 RWNM:

Rossby wave normal mode
 UFKW:

Ultrafast Kelvin wave
 SW:

Secondary wave
 USB:

Upper sideband
 LSB:

Lower sideband
 M2:

Semidiurnal lunar tide
 SW1, SW2, and SW3:

Semidiurnal tide traveling westward with zonal wavenumber 1, 2, and 3
 AFD:

Adjusted Feynman diagram
 PASTIDE:

Planetary wave amplification by stimulated tidal decay
 LASER:

Light amplification by stimulated emission of radiation
 SDWACCMX:

The Specified Dynamics version of the Whole Atmosphere Community Climate Model with thermosphereionosphere eXtension
 CTMT:

The climatological tidal model of the thermosphere
 MIGHTI:

Global Highresolution Thermospheric Imaging
 2D:

2Dimensional
 TIMEGCM:

Thermosphere–IonosphereMesosphere–Electrodynamics General Circulation Model
References
Azeem SM, Killeen TL, Johnson RM, Wu Q, Gell DA (2000) Spacetime analysis of TIMED Doppler Interferometer (TIDI) measurements. Geophys Res Lett 27(20):3297–3300
Baumgaertner AJG, Jarvis MJ, McDonald AJ, Fraser GJ (2006) Observations of the wavenumber 1 and 2 components of the semidiurnal tide over Antarctica. J Atmos SolarTerrestrial Phys 68(11):1195–1214
Chau JL, Hoffmann P, Pedatella NM, Matthias V, Stober G (2015) Upper mesospheric lunar tides over middle and high latitudes during sudden stratospheric warming events. J Geophys Res Space Phys 120(4):3084–3096
Chen T, Wan W, Xiong J, Yu Y, Ren Z, Yue X (2019) A Statistical Approach to Quantify Atmospheric Contributions to the ITEC WN4 Structure Over Low Latitudes. J Geophys Res Space Phys 124(3):2178–2197
Forbes JM (2000) Wave coupling between the lower and upper atmosphere: Case study of an ultrafast Kelvin wave. J Atmos SolarTerrestrial Phys 62(17–18):1603–1621
Forbes JM, He M, Maute A, Zhang X (2020) Ultrafast Kelvin Wave Variations in the Surface Magnetic Field. J Geophys Res Space Phys 125(9):e2020
Forbes JM, Heelis R, Zhang X, Englert CR, Harding BJ, He M, Chau JL, Stoneback R, Harlander JM, Marr KD, Makela JJ, Immel TJ (2021) Q2DWTide and Ionosphere Interactions as Observed From ICON and GroundBased Radars. J Geophys Res Space Phys 126(11):e2021
Forbes JM, Moudden Y (2012) Quasitwoday wavetide interactions as revealed in satellite observations. J Geophys Res Atmos 117:12
Forbes JM, Russell J, Miyahara S, Zhang X, Palo S, Mlynczak M, Mertens CJ, Hagan ME (2006) Tropospherethermosphere tidal coupling as measured by the SABER instrument on TIMED during JulySeptember 2002. J Geophys Res Space Phys 111:10
Forbes JM, Zhang X (2012) Lunar tide amplification during the January 2009 stratosphere warming event: Observations and theory. J Geophys Res Space Phys 117(12):1–13
Forbes JM, Zhang X, Bruinsma S, Oberheide J (2013) Lunar semidiurnal tide in the thermosphere under solar minimum conditions. J Geophys Res Space Phys 118(4):1788–1801
Forbes JM, Zhang X, Talaat ER, Ward W (2003) Nonmigrating diurnal tides in the thermosphere. J Geophys Res Space Phys 108(A1):1–10
Hagan ME, Forbes JM (2002) Migrating and nonmigrating diurnal tides in the middle and upper atmosphere excited by tropospheric latent heat release. J Geophys Res Atmos 107(24):1–15
Hasselmann K (1966) Feynman diagrams and interaction rules of wavewave scattering processes. Rev Geophys 4(1):1–32
He M, Chau JL (2019) Mesospheric semidiurnal tides and near12  h waves through jointly analyzing observations of five specular meteor radars from three longitudinal sectors at boreal midlatitudes. Atmos Chem Phys 19(9):5993–6006
He M, Chau JL, Forbes JM, Thorsen D, Li G, Siddiqui TA, Yamazaki Y, Hocking WK (2020) Quasi10Day Wave and Semidiurnal Tide Nonlinear Interactions During the Southern Hemispheric SSW 2019 Observed in the Northern Hemispheric Mesosphere. Geophys Res Lett 47(23):e2020
He M, Chau JL, Forbes JM, Zhang X, Englert CR, Harding BJ, Immel TJ, Lima LM, Bhaskar Rao SV, Ratnam MV, Li G, Harlander JM, Marr KD, Makela JJ (2021) Quasi2Day Wave in LowLatitude Atmospheric Winds as Viewed From the Ground and Space During JanuaryMarch, 2020. Geophys Res Lett 48:13
He M, Chau JL, Hall CM, Tsutsumi M, Meek C, Hoffmann P (2018) The 16Day Planetary Wave Triggers the SW1TidalLike Signatures During 2009 Sudden Stratospheric Warming. Geophys Res Lett 45(22):12631–12638
He M, Chau JL, Stober G, Hall CM, Tsutsumi M, Hoffmann P (2017) Application of ManleyRowe Relation in Analyzing Nonlinear Interactions Between Planetary Waves and the Solar Semidiurnal Tide During 2009 Sudden Stratospheric Warming Event. J Geophys Res Space Phys 122(10):10783–10795
He M, Chau JL, Stober G, Li G, Ning B, Hoffmann P (2018) Relations between semidiurnal tidal variants through diagnosing the zonal wavenumber using a phase differencing technique based on two groundbased detectors. J Geophys Res Atmos 123(8):4015–4026
He M, Forbes JM (2022) Rossby wave second harmonic generation observed in the middle atmosphere. Nat Commun 13(1):7544
He M, Forbes JM, Chau JL, Li G, Wan W, Korotyshkin DV (2020) Highorder solar migrating tides Quench at SSW onsets. Geophys Res Lett 47(6):1–8
He M, Forbes JM, Jacobi C, Li G (2022) Six solar migrating tidal harmonics observed in midlatitude MLT region. Geophys Res Lett. 34:67
He M, Forbes JM, Li G, Jacobi C, Hoffmann P (2021) Mesospheric Q2DW Interactions With Four Migrating Tides at 53\(^\circ\)N Latitude: Zonal Wavenumber Identification Through DualStation Approaches. Geophys Res Lett 48(8):e2020
He M, Liu L, Wan W, Lei J, Zhao B (2010) Longitudinal modulation of the O/N2 column density retrieved from TIMED/GUVI measurement. Geophys Res Lett 37:20
He M, Liu L, Wan W, Wei Y (2011) Strong evidence for couplings between the ionospheric wave4 structure and atmospheric tides. Geophys Res Lett 38(14):2–7
He M, Yamazaki Y, Hoffmann P, Hall CM, Tsutsumi M, Li G, Chau JL (2020) Zonal wave number diagnosis of Rossby wavelike oscillations using paired groundbased radars. J Geophys Res Atmos 125:12
Hirooka T, Sciences P (2000) Normal mode Rossby waves as revealed by UARS / ISAMS observations. J Atmos Sci 57(9):1277–1285
Immel TJ, England SL, Mende SB, Heelis RA, Englert CR, Edelstein J, Frey HU, Korpela EJ, Taylor ER, Craig WW, Harris SE, Bester M, Bust GS, Crowley G, Forbes JM, Gérard JC, Harlander JM, Huba JD, Hubert B, Kamalabadi F, Makela JJ, Maute AI, Meier RR, Raftery C, Rochus P, Siegmund OHW, Stephan AW, Swenson GR, Frey S, Hysell DL, Saito A, Rider KA, Sirk MM (2017) The ionospheric connection explorer mission: mission goals and design. Space Sci Rev 214(1):13
Immel TJ, Sagawa E, England SL, Henderson SB, Hagan ME, Mende SB, Frey HU, Swenson CM, Paxton LJ (2006) Control of equatorial ionospheric morphology by atmospheric tides. Geophys Res Lett 33:15
Jiang G, Xu J, Xiong J, Ma R, Ning B, Murayama Y, Thorsen D, Gurubaran S, Vincent RA, Reid I, Franke SJ (2008) A case study of the mesospheric 65day wave observed by radar systems. J Geophys Res Atmos 113(16):1–12
Kamalabadi F, Forbes JM, Makarov NM, Portnyagin YI (1997) Evidence for nonlinear coupling of planetary waves and tides in the Antarctic mesopause. J Geophys Res Atmos 102(D4):4437–4446
Liu J, Zhang D, Goncharenko LP, Zhang S, He M, Hao Y, Xiao Z (2021) The latitudinal variation and hemispheric asymmetry of the ionospheric lunitidal signatures in the American sector during major sudden stratospheric warming events. J Geophys Res Space Phys 126(5):1–11
Macotela EL, He M, Pedatella N, Chau JL (2022) Seasonal amplitude variation of mesospheric near12hour waves at midlatitudes: radar observations 3 and whole atmosphere model results. J Geophys Res Space Phys 67:8
Manson AH, Meek CE, Chshyolkova T, Xu X, Aso T, Drummond JR, Hall CM, Hocking WK, Jacobi C, Tsutsumi M, Ward WE (2009) Arctic tidal characteristics at Eureka (80 N, 86 W) and Svalbard (78 N, 16E) for 2006/07: Seasonal and longitudinal variations, migrating and nonmigrating tides. Annales De Geophysique 27(3):1153–1173
Murphy DJ, Forbes JM, Walterscheid RL, Hagan ME, Avery SK, Aso T, Fraser GJ, Fritts DC, Jarvis MJ, McDonald AJ, Riggin DM, Tsutsumi M, Vincent RA (2006) A climatology of tides in the antarctic mesosphere and lower thermosphere. J Geophys Res Atmos 111(23):1–17
Nguyen VA, Palo SE, Lieberman RS, Forbes JM, Ortland DA, Siskind DE (2016) Generation of secondary waves arising from nonlinear interaction between the quasi 2 day wave and the migrating diurnal tide. J Geophys Res Atmos 121(13):7762–7780
Oberheide J, Forbes JM, Zhang X, Bruinsma SL (2011) Climatology of upward propagating diurnal and semidiurnal tides in the thermosphere. J Geophys Res 116(A11):A11306
Pancheva D, Merzlyakov E, Mitchell N, Portnyagin Y, Manson A, Jacobi C, Meek C, Luo Y, Clark R, Hocking W, MacDougall J, Muller H, Kürschner D, Jones G, Vincent R, Reid I, Singer W, Igarashi K, Fraser G, Fahrutdinova A, Stepanov A, Poole L, Malinga S, Kashcheyev B, Oleynikov A (2002) Globalscale tidal variability during the psmos campaign of juneaugust 1999: interaction with planetary waves. J Atmos SolarTerrestrial Phys 64(17):1865–1896
Pancheva D, Mitchell NJ, Manson AH, Meek CE, Jacobi C, Portnyagin Y, Merzlyakov E, Hocking WK, MacDougall J, Singer W, Igarashi K, Clark RR, Riggin DM, Franke SJ, Kürschner D, Fahrutdinova AN, Stepanov AM, Kashcheyev BL, Oleynikov AN, Muller HG (2004) Variability of the quasi2day wave observed in the MLT region during the PSMOS campaign of JuneAugust 1999. J Atmos SolarTerrestrial Phys 66(6–9):539–565
Pedatella NM, Forbes JM (2010) Evidence for stratosphere sudden warmingionosphere coupling due to vertically propagating tides. Geophys Res Lett 37:11
Pedatella NM, Forbes JM, Oberheide J (2008) Intraannual variability of the lowlatitude ionosphere due to nonmigrating tides. Geophys Res Lett 35:18
Pedatella NM, Hagan ME, Maute A (2012) The comparative importance of DE3, SE2, and SPW4 on the generation of wavenumber4 longitude structures in the lowlatitude ionosphere during September equinox. Geophys Res Lett 39:19
Pedatella NM, Liu HL (2013) The influence of atmospheric tide and planetary wave variability during sudden stratosphere warmings on the low latitude ionosphere. J Geophys Res Space Phys 118(8):5333–5347
Pedatella NM, Liu HL, Richmond AD, Maute A, Fang TW (2012) Simulations of solar and lunar tidal variability in the mesosphere and lower thermosphere during sudden stratosphere warmings and their influence on the lowlatitude ionosphere. J Geophys Res Space Phys 117:A8
Rojas M, Norton W (2007) Amplification of the 2day wave from mutual interaction of global Rossbygravity and local modes in the summer mesosphere. J Geophys Res Atmos 112:D12
Salby ML (1981) The 2day wave in the middle atmosphere: Observations and theory. J Geophys Res Oceans 86(C10):9654–9660
Salby ML (1982) Sampling Theory for Asynoptic Satellite Observations. Part I: SpaceTime Spectra, Resolution, and Aliasing. J Atmos Sci 39(11):2577–2600
Sassi F, Garcia RR, Hoppel KW (2012) LargeScale Rossby Normal Modes during Some Recent Northern Hemisphere Winters. J Atmos Sci 69(3):820–839
Torrence C, Compo GP (1998) A practical guide to wavelet analysis. Bull Am Meteorol Soc 79(1):61–78
Tunbridge VM, Sandford DJ, Mitchell NJ (2011) Zonal wave numbers of the summertime 2 day planetary wave observed in the mesosphere by EOS Aura Microwave Limb Sounder. J Geophys Res Atmos 116(11):1–16
Wang JC, Palo SE, Forbes JM, Marino J, MoffatGriffin T, Mitchell NJ (2021) Unusual Quasi 10Day Planetary Wave Activity and the Ionospheric Response During the 2019 Southern Hemisphere Sudden Stratospheric Warming. J Geophys Res Space Phys 126(6):e2021
Yamazaki Y (2013) Large lunar tidal effects in the equatorial electrojet during northern winter and its relation to stratospheric sudden warming events. J Geophys Res Space Phys 118(11):7268–7271
Zhang JT, Forbes JM (2013) Lunar tidal winds between 80 and 110 km from UARS HRDI wind measurements. J Geophys Res Space Phys 118(8):5296–5304
Zhou X, Wan W, Yu Y, Ning B, Hu L, Yue X (2018) New approach to estimate tidal climatology from ground and spacebased observations. J Geophys Res Space Phys 123(6):5087–5101
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He, M. Planetaryscale MLT waves diagnosed through multistation methods: a review. Earth Planets Space 75, 63 (2023). https://doi.org/10.1186/s40623023018085
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DOI: https://doi.org/10.1186/s40623023018085