Symmetric and Antisymmetric Solar Migrating Semidiurnal Tides in the Mesosphere and Lower Thermosphere

Upward‐propagating solar tides are responsible for a large part of atmospheric variability in the mesosphere and lower thermosphere (MLT) region, and they are also an important source of ionospheric variability. Tides can be divided into the parts that are symmetric and antisymmetric about the equator. Their distinction is important, as the electrodynamic responses of the ionosphere to symmetric and antisymmetric tides are different. This study examines symmetric and antisymmetric tides using 21 years of temperature measurements by the Thermosphere Ionosphere Mesosphere Energetics and Dynamics/Sounding of the Atmosphere using Broadband Emission Radiometry. The main focus is on the solar migrating semidiurnal tide (SW2), which is one of the dominant tides in the MLT region. It is shown that symmetric and antisymmetric parts of SW2 are comparable in amplitude. However, their spatiotemporal characteristics are different. That is, the symmetric part is strongest during March–June at 30–35° latitude, while the antisymmetric part is most prominent during May–September with the largest amplitude at 15–20° latitude. The symmetric and antisymmetric parts can be well described by the first two symmetric and antisymmetric Hough modes, respectively. Amplification is observed in the antisymmetric part during the major sudden stratospheric warmings (SSWs) in January 2006, 2009, 2013 and 2019. Atmospheric model simulations for the 2009 and 2019 SSWs confirm the amplification in the antisymmetric part of SW2. The enhanced antisymmetric tidal forcing explains the previously‐reported asymmetric response of the ionospheric solar‐quiet current system to SSWs.


Introduction
Atmospheric tides are global-scale oscillations with a period of a solar or lunar day and its harmonics (Forbes & Garrett, 1979;Lindzen & Chapman, 1969).Tidal variations can be observed in various atmospheric parameters such as pressure, density, temperature, and winds, from the surface to the exobase (e.g., Dai & Wang, 1999;Forbes et al., 2011;Oberheide et al., 2009;Sakazaki et al., 2012).Solar tides (period = 24 hr, 12 hr, and so on) are excited by radiative heating of the atmosphere due to periodic absorption of solar radiation by water vapor (H 2 O) in the troposphere and ozone (O 3 ) in the stratosphere, as well as by latent heating associated with tropical deep convective activity (e.g., Hagan & Forbes, 2002;Hagan & Forbes, 2003).Lunar tides (predominantly 12.42 hr) are driven mainly by gravitational force of the moon acting upon the Earth's lower atmosphere (e.g., Pedatella, Liu, & Richmond, 2012;Vial & Forbes, 1994).For the most part, tidal forcing migrates westward around the Earth with the apparent motion of the sun or moon, and thus westward-propagating "migrating" tides dominate the atmospheric response.Some tides possess characteristics of vertically propagating waves.Upward-propagating tides grow in amplitude with height as the air density decreases, until they eventually get dissipated in the mesosphere and lower thermosphere (MLT) region (ca.80-150 km).Amplitude of upward-propagating tides, therefore, attains its maximum in the MLT region.Solar tides often dominate the MLT meteorology (e.g., Forbes et al., 2008), while lunar tides are smaller than their solar counter parts (Forbes et al., 2013; J. T. Zhang & Forbes, 2013;Conte et al., 2017).In the middle and upper thermosphere (>150 km), tides (predominantly 24 hr) are locally generated by solar radiation absorption by thermospheric constituents such as O, O 2 and N 2 (Forbes & Garrett, 1976;Hagan et al., 2001).Some tides at these altitudes are also generated in situ by diurnally-varying ion drag (Jones Jr et al., 2013).The transition from predominantly upward-propagating tides to locally-generated thermospheric tides occurs around 130 km (e.g., Yamazaki et al., 2014).
Classical tidal theory expresses tides as global-scale linear waves (e.g., Forbes, 1995).The theory was developed for horizontally uniform atmosphere without any background winds or dissipation.In this case, linearized equations of the atmosphere are separable in latitude and height.The solutions to the latitude equation (known as Laplace's tidal equation) are called Hough functions, which are often denoted by a set of two numbers such as (1,1) and (2,3).The first number represents the zonal wavenumber k and the second number represents the meridional index m, which identifies the latitudinal structure of the mode.Tides with m > 0 are called gravity modes, and they are symmetric and antisymmetric about the equator if k + m is even and odd, respectively.Tides with m < 0 are called Rossby modes, and they are antisymmetric and symmetric about the equator if k + m is even and odd, respectively.For instance, the 24-hr (1, 2) mode is the first symmetric Rossby-mode diurnal tide, and the 12-hr (2,3) mode is the first antisymmetric gravity-mode semidiurnal tide.For each mode, the vertical equation provides information on whether the wave is vertically propagating or vertically trapped (i.e., evanescent).For instance, the diurnal (1, 2) mode is vertically trapped, while the semidiurnal (2,3) mode is vertically propagating with a predicted vertical wavelength of ∼80 km.As the energy propagates upward, the phase of vertically propagating modes decreases with height, which is called downward phase propagation.MLT tides commonly show downward phase propagation (e.g., Jacobi et al., 1999;Manson et al., 2002), providing evidence that they are upward-propagating from below.Thermospheric tides above ∼150 km are mostly vertically trapped modes (e.g., Forbes, 1982), so that they do not show phase variation with height (e.g., Yamazaki et al., 2023).
A solar tide at any given latitude and height can be mathematically expressed by the following formula: where A n,s and P n,s are amplitude (in units of respective parameter) and phase (in degrees), respectively.t is UT (in hours), and λ is longitude (in degrees).n is a positive integer related to the frequency of the wave; that is, n = 1, 2, 3 corresponds to the diurnal, semidiurnal and terdiurnal tides, respectively.s is an integer which is related to the zonal wavenumber k by |s| = k.Tides with s > 0 and s < 0 propagate eastward and westward, respectively.Using local solar time t*, which is defined as the tidal wave can be expressed as follows: Journal of Geophysical Research: Atmospheres ). (3) The tides that satisfy s = n depend only on local solar time and are independent of longitude.They correspond to migrating tides.
The standard tidal nomenclature is used in this paper, such as SW2 and DE3.The first letter represents the period.That is, "D" is for diurnal tides and "S" is for semidiurnal tides.The second letter indicates the direction of zonal propagation.That is, "E" is for eastward-propagating tides (or s > 0) and "W" is for westward-propagating tides (or s < 0).The last number is the zonal wavenumber (k or |s|).For instance, SW2 is the westward-propagating semidiurnal tide with zonal wavenumber 2 (n = 2, s = 2), or the migrating semidiurnal tide, and DE3 is the eastward-propagating diurnal tide with zonal wavenumber 3 (n = 1, s = 3).
MLT tides are subject to large changes during sudden stratospheric warmings (SSWs).SSWs are large-scale meteorological disturbances in the winter polar stratosphere (e.g., Baldwin et al., 2021;Labitzke & Van Loon, 2012).During SSWs, the polar stratospheric temperature at 10 hPa typically undergoes a rapid increase by more than 25 degrees Kelvin within a week, and the eastward zonal mean flow at 60°latitude at 10 hPa decreases or even turns westward.The SSWs that involve the reversal of the zonal mean flow are classified as major warmings, while those which do not involve the zonal wind reversal are called minor warmings (Butler et al., 2015).SSWs can disturb the whole atmosphere; their influence extends from the surface to the topside ionosphere (Goncharenko et al., 2021;Pedatella et al., 2018).Changes in MLT tides during SSWs can result from changes in tidal propagation (Fuller-Rowell et al., 2010;Jin et al., 2012;Sassi et al., 2013;Siddiqui et al., 2022), tidal sources (Goncharenko et al., 2012;Siddiqui et al., 2019), and tidal interactions with other waves (H.-L.Liu et al., 2010;Miyoshi & Yamazaki, 2020;He, Chau, et al., 2020).Upward-propagating tides are believed to be the main driver of SSW effects in the ionosphere (e.g., Chau et al., 2009;Goncharenko et al., 2010).The most prominent ionospheric response to SSWs has been observed in the low-latitude region, where the ionospheric plasma density and electric field show enhanced semidiurnal perturbations during SSWs (e.g., Goncharenko et al., 2021).
The ionospheric dynamo region (ca.95-150 km) is where electrical conductivity of the air is highest due to collisional interactions between E-region plasmas and neutral atmosphere (e.g., Heelis, 2004;Richmond, 1995).Electric currents are generated due to the systematic difference between electron-neutral and ion-neutral interactions.Neutral winds act as a driver, as they produce a U × B electromotive force, where U is the neutral wind velocity and B is the Earth's main magnetic field.The wind-driven currents are, in general, not divergence-free, which leads to charge accumulation.Electric fields are generated in such a manner that the total currents (winddriven plus electric-field-driven currents) are divergence-free.The dynamo process leads to the formation of a global-scale electric current system, which is known as solar quiet (Sq) current system (Yamazaki & Maute, 2017).The Sq current system usually consists of two dayside vortices: a counterclockwise vortex in the northern hemisphere and a clockwise vortex in the southern hemisphere.Early numerical studies examined characteristics of current systems driven by different Hough modes of diurnal and semidiurnal tides (e.g., Forbes & Lindzen, 1976;Hanuise et al., 1983;Richmond et al., 1976;Stening, 1989;Tarpley, 1970).It is generally found that the two-vortex structure of the Sq current system can be reproduced with the vertically-trapped diurnal (1, 2) mode and some additional upward-propagating semidiurnal modes such as (2,2), (2,3) and (2,4) modes.The intensity and shape of the Sq current system can change from day-to-day, as the Hough-mode composition of tides in the dynamo region changes.Yamazaki, Richmond, Liu, et al. (2012) reported changes in the Sq current system during the major SSWs of January 2006 and 2009.They noted that during these SSWs, Sq currents were unusually weak and strong in the northern and southern hemispheres, respectively.Yamazaki, Richmond, Liu, et al. (2012) argued that the observed north-south asymmetry of Sq could be attributed to enhanced wave forcing by antisymmetric tidal modes.They pointed out that the semidiurnal (2,3) mode is effective in driving ionospheric currents that would increase the Sq current intensity in one hemisphere and reduce it in the other hemisphere.Symmetric modes such as (2,2) and (2,4) modes are expected to drive ionospheric currents that would simultaneously increase or decrease the northern and southern Sq current intensities, and thus are not effective in inducing the north-south asymmetry.
The existence of antisymmetric tides during these SSWs are, however, yet to be observationally confirmed.
Since the electrodynamic responses of the ionosphere to symmetric and antisymmetric tidal forcings are different, we examine symmetric and antisymmetric parts of tides separately.The main objectives of this study are (a) to determine the seasonal climatologies of symmetric and antisymmetric tides based on 21 years of temperature measurements by the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument on the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) spacecraft, and (b) to examine their interannual variations, addressing their possible connections with major SSWs.The study mainly focuses on SW2, which is known to be an important driver of ionospheric electrodynamics (e.g., Chang et al., 2013;Millward et al., 2001;Richmond & Roble, 1987).SW2 is one of the major tides at dynamo-region altitudes, along with DW1 and DE3 (e.g., Oberheide et al., 2011).

Data and Method of Analysis
The data employed in this study are temperature observations (version 2.0) from the SABER instrument onboard the TIMED satellite (Esplin et al., 2023;Russell III et al., 1999) during July 2002-June 2023.Only the measurements equatorward of ±50°latitude are used, as they are not affected by yaw maneuvers.The orbital period of the TIMED satellite is approximately 90 min, and thus the space craft completes ∼15 orbits per day.Due to the slow precession rate, it takes approximately 60 days for TIMED/SABER to achieve a full 24-hr local time coverage.The highest temporal resolution that can be achieved for the solar tidal analysis is ∼60 days.In some studies, assumptions are made about the vertical and/or meridional structures of tides, which enables to infer short-term tidal variability (e.g., Lieberman et al., 2015;Ortland, 2017).The application of such a technique is not considered in this study to avoid introducing the assumptions about the spatial structure of tides.
Tidal amplitudes and phases are determined based on least squares fits of Equation 1 to the TIMED/SABER measurements at each latitude and height, including n from 0 to 3 and s from 0 to 4. This includes not only tides but also the mean field (n = 0, s = 0) and stationary planetary waves (n = 0, s ≠ 0), although they are not investigated in this study.Before fitting, the TIMED/SABER data were binned in hourly universal time (UT) bins, in 2-km altitude bins from 70 to 110 km, in 5°latitude bins, and in 15°longitude bins.This mitigates the impact of the uneven distribution of the data in the UT-longitude space on the fitting.
Uncertainties in the tidal amplitude and phase were estimated using the method described by Yamazaki and Matthias (2019).In brief, the technique is based on iterations of the fitting procedure with bootstrap samples of the data.Measurement (random) errors are added to each bootstrap sample, so that not only fitting errors but also measurement errors will be taken into account in the estimated uncertainties in the tidal amplitude and phase.Remsberg et al. (2008) conducted an uncertainty analysis of the TIMED/SABER data and reported that random errors in individual temperature measurements are typically 1.0 K (70 km), 1.8 K (80 km), 3.6 K (90 km), 6.7 K (100 km) and 15.0 K (110 km).We used these uncertainty values in our error analysis.The estimated 1σ uncertainties in the amplitude are typically 0.3-0.9K (70 km), 0.5-1.5 K (80 km), 0.6-1.4K (90 km), 1.0-1.8K (100 km) and 2.0-4.1 K (110 km) for SW2 derived using a 60-day window, while the 1σ uncertainties in the phase are 60-80°(70 km), 44-68°(80 km), 25-46°(90 km), 17-45°(100 km) and 12-44°(110 km).As can be noted, the 1σ uncertainties in the amplitude and phase increase and decrease with height, respectively.This is as expected from the growth of SW2 from 70 to 110 km.The 1σ uncertainties are on the same of order of magnitude for other tides with comparable amplitude such as DW1 and DE3.In general, the uncertainty decreases with the number of independent measurements (N) by 1 ̅̅ ̅ N √ .Thus, for climatological results involving 21 years of the data, for example, the uncertainties are expected to be roughly 1 ̅̅̅ ̅ 21 √ ∼ 0.22 times the said values.
The temperature field can be divided into symmetric and antisymmetric parts as follows: where ϕ denotes latitude (in degrees).T s and T a are symmetric and antisymmetric parts of the temperature field T.
The decomposition of T into T s and T a causes no loss of information, as the sum of T s and T a is always T. Tidal analysis was performed on either T, T s or T a .

Seasonal Variations
The seasonal dependence of the solar migrating semidiurnal tide, SW2, is investigated using 21 years of temperature measurements by TIMED/SABER.For this purpose, the data from different years were combined, and tidal analysis was performed for different months.Figure 1 shows the amplitude (upper 12 panels) and phase (lower 12 panels) of SW2 for each month.It is noted that the evaluation of the tide for a given month involves the measurements made during a 60-day period centered around the 15th day of the month, and thus there is some overlap in the data used for the tidal analysis for the adjacent months.
The results in Figure 1 show that the amplitude is generally greater at higher altitudes, and the phase tends to decrease with height.As mentioned before, these are fundamental characteristics of upward-propagating tides.
The amplitude is largest during June-August.The northern peak is seen at 25-30°N, while the southern peak is at 10-15°S, indicating north-south asymmetry.There is also difference in the phase between the northern and southern peaks.The seasonal dependence of SW2 depicted in Figure 1 is consistent with earlier reports (e.g., Pancheva & Mukhtarov, 2011).
The symmetric part of SW2 is presented in Figure 2 in the same format as Figure 1.Amplitude maxima are seen at ±30-35°latitude throughout the year.The peak amplitude is largest during April-May.The phase does not strongly depend on the season, especially at midlatitudes in the ionospheric dynamo region (>95 km), where the amplitude is large.The antisymmetric part of SW2 is presented in Figure 3.According to the latitude-height structure, the antisymmetric part of SW2 can be classified into two seasonal types.That is, the northernhemisphere (N.H.) summer type occurs during May-September with amplitude maxima at ±15-20°latitude, while the N.H. winter type occurs during November-March with amplitude maxima at ± 40°latitude and ±10-15°latitude.The transitions between the two types occur around April and October.The phase structure is also different between the N.H. summer and winter types.
The three major tides in the ionospheric dynamo region are DW1, SW2 and DE3.While the symmetric and antisymmetric parts of SW2 are comparable in amplitude (Figures 2 and 3), DW1 and DE3 are predominantly symmetric about the equator regardless of season, as seen in Figures 4 and 5.The results suggest that SW2 is the most important source of antisymmetric tidal forcing to the ionosphere.The remainder of the paper focuses on further characterization of the symmetric and antisymmetric parts of SW2 at dynamo-region heights.
The Hough-mode composition of SW2 is examined.Figures 6a and 6b show the latitudinal structures of the first two symmetric and antisymmetric Hough modes of the semidiurnal tide, respectively, in the temperature field, as derived using the computation software provided by H. Wang et al. (2016).These Hough modes are all upwardpropagating gravity modes.The (2,2) mode has a single peak at the equator, while the (2,4) mode has midlatitude crests around ±30-35°latitude besides an equatorial trough.The temperature perturbations of the (2,4) mode are out of phase between the equatorial and midlatitude peaks.For antisymmetric modes, the (2,3) mode has peaks about ±25°latitude.The perturbations are out of phase at the northern and southern peaks.The (2,5) mode has amplitude maxima at ±10°and ±40°.
Observed SW2 can be represented by the sum of Hough functions as follows: Θ 2,m (ϕ) is the Hough function for the semidiurnal (2,m) mode.Θ 2,2 (ϕ), Θ 2,3 (ϕ), Θ 2,4 (ϕ) and Θ 2,5 (ϕ) can be found in Figures 6a and 6b.Once the constants a 2,m and p 2,m are determined, the entire structure of temperature perturbations in latitude and longitude can be reproduced by Equation 6.Although in some literature, a 2,m and p 2,m were simply called "amplitude" and "phase" (e.g., Forbes et al., 2013;Jin et al., 2012), in this paper, they are referred to as "Hough-function amplitude" and "Hough-function phase," respectively.This is to distinguish them from the amplitude and phase of a tide (i.e., A n,s and P n,s in Equation 3).It is noted that A n,s and P n,s vary with latitude, while a 2,m and p 2,m are single values independent of latitude.For SW2 (n = 2, s = 2), Equation 3 can be rewritten as:      x(ϕ) cos (2π where x(ϕ) = A 2,2 cos 2π ) and y(ϕ) = A 2,2 sin 2π ) .Meanwhile, for given m, Equation 6 can be rewritten as: where X = a 2,m cos 2π ) and Y = a 2,m sin 2π ) .X and Y (and hence a 2,m and p 2,m ) can be determined in such a way to minimize the difference between Equations 7 and 8. Symmetric (m = 2, 4, …) and antisymmetric (m = 3, 5, …) Hough functions were fitted to the symmetric and antisymmetric parts of SW2 (Figures 2 and 3), respectively.Since we are primarily concerned with symmetric and antisymmetric tidal forcing in the ionospheric dynamo region, Hough function decomposition of SW2 was performed at 109 km.
Figures 6c and 6e show that the symmetric part of SW2 at 109 km during May can be well represented by the first two symmetric modes, namely the (2,2) and (2,4) modes.It is noted that the symmetric part of SW2 is most prominent during May (Figure 2).The (2,4) mode is mainly responsible for the observed amplitude maxima at ±30-35°latitude.Figures 6d and 6f show the results for the antisymmetric part of SW2 during August, when the antisymmetric part of SW2 is most evident (Figure 3).The meridional structure of the antisymmetric part of SW2 is mainly explained by the (2,3) mode.

Non-Seasonal Variations
Non-seasonal variations in the symmetric and antisymmetric parts of SW2 in the dynamo region are investigated.Tidal analysis was performed on the TIMED/SABER measurements obtained during a 60-day period centered around the 15th day of each month of each year.Amplitude variations in the symmetric and antisymmetric parts of SW2 are depicted in Figures 7a and 7b, respectively.The overall meridional structure remains the same throughout the period of analysis for both the symmetric and antisymmetric parts.That is, an amplitude maximum is visible around 32.5°latitude in the symmetric part and around 17.5°latitude in the antisymmetric part throughout the period 2002-2023.For both the symmetric and antisymmetric parts, the seasonal change dominates the amplitude variability.In order to highlight non-seasonal variations, climatological annual cycles are removed from the amplitude time series at each latitude, and the residuals are plotted in Figures 7c and 7d.The amplitude anomalies show complex temporal variability.The amplitude anomalies at 32.5°latitude are depicted in Figure 7d.It is noted that the four largest amplitude anomalies (>8 K) in the antisymmetric part (red) coincide with the major SSWs in January 2006January , 2009January , 2013January and 2019.These SSWs are particularly strong and long-lasting events, which are sometimes referred to as polar-night jet oscillation (PJO; e.g., Hitchcock et al., 2013;Conte et al., 2019).The amplitude anomaly in the symmetric part of SW2 does not seem to be affected by these PJO events.The symmetric and antisymmetric parts of SW2 during the SSW-PJO events in January 2006January , 2009January , 2013January and 2019 are further investigated.The Northern Annular Mode (NAM) index is used to define the interval of each event.Pedatella and Harvey (2022) demonstrated that there is a good correlation between changes in the MLT SW2 amplitude and the NAM index at 10 hPa based on model simulations.Figure 8 shows the NAM index during the N.H. winters of 2005/2006, 2008/2009, 2012/2013 and 2018/2019.A reduction and slow recovery of the NAM index indicate the occurrence of SSW-PJO events.For each event, a 60-day interval starting from the day when the NAM index turns negative was selected (highlighted in Figure 8), and the data during these intervals from different years were combined to determine the symmetric and antisymmetric parts of SW2. Figure 9 shows the amplitude and phase for the symmetric and antisymmetric parts of SW2 derived using the temperature data during the four SSW events.The results for the symmetric part are similar to those of the January climatology presented in Figure 2.However, the amplitude and phase patterns for the antisymmetric part are quite different from those of the January climatology shown in Figure 3.For instance, during the SSW events, amplitude maxima are found around ±30°latitude, in contrast to the climatological amplitude maxima at ± 40°latitude and ±10-15°latitude.
Figure 10 compares Hough-function representations of SW2 at 109 km between the SSW events and January climatology for the symmetric part (panels a-d) and antisymmetric part (panels e-h).The first two Hough modes can capture the symmetric and antisymmetric parts of SW2 not only for the January climatology but also for the SSW events.As mentioned, the meridional structures of the amplitude and phase during the SSW events are similar to those of the January climatology for the symmetric part, but they are considerably different for the antisymmetric part.The difference is primarily found in the (2,3) mode, the contribution of which is enhanced by a factor of 2-3 during the SSW events.This is demonstrated in Table 1, which shows the Hough-function amplitude and phase of the semidiurnal (2,2), (2,3), (2,4) and (2,5) modes during the SSW events (for the individual events as well as the composite of the four events) and for the January climatology.It is noted that during the SSW events, not only the Hough-function amplitude but also the Hough-function phase of the (2,3) mode deviates from the climatological value.

Discussion
DW1, SW2 and DE3 are known to be important sources of wave forcing to the ionosphere (e.g., England, 2012;Fang et al., 2013;Jones Jr et al., 2014;Yamazaki & Richmond, 2013).The symmetric and antisymmetric parts  of SW2 are comparable in amplitude, while DW1 and DE3 are largely symmetric about the equator (Figures 2-5).This leaves SW2 as the primary source of antisymmetric tidal forcing to the ionosphere from below.The lack of asymmetry in DW1 and DE3 is probably due to the fact that antisymmetric modes of DW1 and DE3 have a small vertical wavelength (see, e.g., Truskowski et al., 2014), which makes the wave susceptible to dissipation and thus makes its vertical propagation difficult.It is noted that spatiotemporal features of both symmetric and antisymmetric parts of DE3 presented in Figure 5 are in agreement with those in Oberheide and Forbes (2008).
The symmetric and antisymmetric parts of SW2 can be well represented by the first two symmetric and antisymmetric Hough functions, respectively (Figure 6).The antisymmetric (2,3) mode is the dominant mode of SW2 in the dynamo region during June-September.The early modeling study by Forbes and Vial (1989) predicted that the amplitude of the (2,3) mode would be relatively small in the MLT region throughout the year.However, the later study by Truskowski et al. (2014) acknowledged the importance of the (2,3) mode, and this study also confirms its major contribution to SW2, especially during the N.H. summer months from May to September.
In this study, the Hough function decomposition of SW2 was performed using the TIMED/SABER measurements at a fixed altitude, specifically at 109 km.This is because our primary interest is in symmetric and antisymmetric tidal forcing in the ionospheric dynamo region.In some other studies, both meridional and vertical structures of tides are assumed, and two-dimensional (latitude vs. height) functions are fitted to the data obtained from the entire MLT region (e.g., Dhadly et al., 2018;Forbes et al., 1994).Such a technique could under-or overestimate the amplitude and phase of Hough modes at a fixed height (e.g., 109 km) depending on the accuracy of the assumed height structure of Hough modes, and thus the results are not necessarily comparable with those in this study.
The antisymmetric part of SW2 has two seasonal types; the N.H. summer type during May-September and the N.H. winter type during November-March, with the transitions between the two types around April and October (Figure 3).The N.H. summer type is mainly contributed by the (2,3) Hough mode, while the N.H. winter type consists of weak contributions from both (2,3) and (2,5) modes.Antisymmetric tidal forcing can lead to north-south asymmetry of the ionospheric Sq current system (Stening, 1989;Yamazaki, Richmond, Liu, et al., 2012).Asymmetric ionospheric dynamo between the northern and southern hemispheres drives interhemispheric field-aligned currents (IHFACs; i.e., electric currents that flow from the ionospheric dynamo region of one hemisphere to the other hemisphere along magnetic field lines), which cause an apparent longitudinal separation in the central positions of the northern and southern Sq current vortices (Fukushima, 1979;Takeda, 1982Takeda, , 1990)).Takeda (2002), using ground-based magnetometer data from the Asian sector, showed that there are two types in the shape of the Sq current system: the N.H. summer type during April-September and the N.H. winter type during October-March.The N.H. summer type is characterized by a large longitudinal separation between the northern and southern Sq vortices, which is much reduced in the N.H. winter type.Similar results are also reported for other longitude sectors (Chulliat et al., 2016).We suspect that IHFACs driven by the antisymmetric part of SW2, in particular the (2,3) mode, are responsible for the north-south asymmetry in the shape of the N.H. summer type of the Sq current system.Indeed, satellite observations of IHFACs have revealed that the global pattern of IHFACs has also two types; one during the N.H. summer and the other during N.H. winter (Park et al., 2011(Park et al., , 2020;;F. Wang et al., 2022).More studies are required to clarify the relationship among the semidiurnal (2,3) mode, IHFAC and north-south asymmetry of the Sq current system.
After removal of the climatological annual cycle presented in Figures 2 and 3, considerable variability still remains in the residual amplitude of both symmetric and antisymmetric SW2 (Figures 7c and 7d).We have pointed out that enhancements in the amplitude of the antisymmetric part at mid latitudes coincide with the long-lasting Arctic SSW-PJO events in January 2006, 2009, 2013and 2019 (Figure 7e).It is noted that an SSW-PJO event also took place in January 2004 (e.g., Conte et al., 2019), but the change in the antisymmetric part of SW2 in January 2004 is much smaller than those during the other SSW-PJO events (Figure 7e).Also, Figure 7e does not depict any enhancement in symmetric or antisymmetric part of SW2 during the recent major SSW event in January 2021, although its ionospheric effects, which are believed to be due to enhanced SW2, have been reported (Oberheide, 2022; Pedatella, 2022; R. Zhang et al., 2022;Jones Jr et al., 2023).Figure 7e shows the largest amplitude anomaly in the symmetric part of SW2 in September 2019.Interestingly, this coincides with the rare SSW event in the Southern Hemisphere (Yamazaki et al., 2020).Since Antarctic SSWs are rare, the correspondence between their occurrence and the amplification in the symmetric part of SW2 remains unclear.More studies are required to identify the necessary conditions for the amplification in the symmetric and antisymmetric parts of SW2.Yamazaki, Richmond, Liu, et al. (2012) examined the response of the ionospheric Sq current system to the SSW events in January 2006 and 2009 based on the analysis of ground-based magnetometer data from the American sector.They observed unusually weak and strong Sq currents in the northern and southern hemispheres, respectively.They also performed numerical experiments using the Thermosphere Ionosphere Electrodynamics General Circulation Model (TIE-GCM; Richmond et al., 1992;Qian et al., 2014).The TIE-GCM is a physicsbased model of the coupled thermosphere-ionosphere system.The model has its lower boundary at an altitude of ∼97 km, where tidal forcing from below can be specified.Yamazaki, Richmond, Liu, et al. (2012) used Hough functions to specify tidal forcing at the lower-boundary of the TIE-GCM.They demonstrated that the north-south Sq asymmetry observed during the January 2006 and 2009 SSWs can be reproduced if the model is forced by the semidiurnal (2,3) mode with modified amplitude and phase.They noted that the amplitude of the (2,3) mode needs to be increased by a factor of 6 from the seasonal climatology predicted by the Global Scale Wave Model (Hagan & Forbes, 2002, 2003) in order to achieve reasonable agreement with the observations.The present study, based on a composite analysis of TIMED/SABER temperature data during the four SSW events in January 2006January , 2009January , 2013January and 2019, has , has shown that the amplitude of the semidiurnal (2,3) mode is enhanced by a factor of 2-3 during these events compared to the January climatology.Our results support the hypothesis by Yamazaki, Richmond, Liu, et al. (2012); that is, the north-south asymmetry of the Sq current system during the January 2006 and 2009 SSWs is due to the enhancement of the semidiurnal (2,3) mode.There is a discrepancy in the amplification factor of the (2,3) mode, which requires more studies.Also, the mechanism for the enhancement of the (2,3) mode remains to be identified.During SSWs, the meridional distribution of ozone changes significantly (Goncharenko et al., 2012;Siddiqui et al., 2019).Ozone is a primary source for the generation of the semidiurnal tide (Forbes & Garrett, 1979), and thus the change in its meridional structure might affect the Hough-mode composition of SW2.
Previous studies discussed amplification of the lunar semidiurnal tide (M 2 ) during SSW events.Fejer et al. (2010) hypothesized that enhanced semidiurnal perturbations in the equatorial electrojet (EEJ) during SSWs are due to lunar tidal forcing.This was supported by the fact that the phase of the semidiurnal EEJ perturbations during SSWs changes over time, and the rate of the change is consistent with that of M 2 (∼0.83 hr per solar day).Later studies established a robust relationship between the occurrence of SSW and the amplification of M 2 variations in the EEJ (Kumar et al., 2023;Park et al., 2012;Siddiqui et al., 2015;Yamazaki, Richmond, & Yumoto, 2012).Enhanced M 2 variations have also been observed in low-latitude ionospheric plasma densities (Goncharenko et al., 2010;Lin et al., 2019;J. Liu et al., 2019).Numerical studies have clarified that both solar and lunar semidiurnal tides in the MLT region can be affected by SSWs (Maute et al., 2016;Pedatella & Liu, 2013;Pedatella, Liu, et al., 2014;Pedatella, Liu, Richmond, Maute, & Fang, 2012;Siddiqui et al., 2021).Although the Hough-Function Amplitude a and Phase p (See Equation 6) for the Semidiurnal (2,2),(2,3),(2,4),and (2,5) Modes in TIMED/SABER Temperature at 109 km During the Major Sudden Stratospheric Warming (SSW) Events in January 2006, 2009, 2013, and 2019 (See Figure 8) distinction between the solar (12.0 hr) and lunar (12.42 hr) semidiurnal tides can be challenging in the analyses of ground-based observations due to the proximity of the two periods, observations from slowly-precessing satellites such as TIMED can largely avoid this aliasing problem.As Forbes et al. (2013) demonstrated, the apparent periods of solar and lunar semidiurnal tides are well separated in measurements from the TIMED satellite.Thus, our results for SW2 based on a 60-day window are unlikely contaminated by M 2 , even for those during SSWs.It remains to be investigated whether the amplification of the antisymmetric part of SW2 makes any significant contribution to semidiurnal perturbations in the EEJ and low-latitude plasma densities during SSWs.Also, future work should examine the symmetric and antisymmetric parts of lunar tides and clarify their roles in ionospheric M 2 variations during SSWs.
One limitation of this study in addressing the response of the symmetric and antisymmetric parts of SW2 to SSWs is in the temporal resolution.Resolving tides in the TIMED/SABER data requires at least 60 days of measurements, which determines the maximum temporal resolution that can be achieved in our analysis.Variations are not captured in Figure 7c-7e if the amplitude anomaly is short-lived relative to the 60-day window.This might be the reason why the amplification of the antisymmetric part of SW2 is observed only during long-lasting SSW-PJO events, and not during other major SSWs like those in February 2008 and January 2010, for which the duration of the event is much shorter.
To better understand the temporal variability of SW2 during SSWs, we use numerical simulations.In our previous study (Siddiqui et al., 2021), we used the Whole Atmosphere Community Climate Model, eXtended version (WACCM-X; H.-L. Liu et al., 2018) to examine the upper atmosphere response to the SSWs in January 2009 and 2019.A specified dynamics (SD) set up was adopted in the simulations, which was achieved by constraining the temperature and wind fields from 0 to 50 km with the Modern Era Retrospective Analysis for Research and Applications, version 2 (MERRA-2; Gelaro et al., 2017).As demonstrated by Siddiqui et al. (2021), the SD-WACCM-X simulations can reproduce the middle atmosphere response to the SSWs well.Here we examine the symmetric and antisymmetric parts of SW2 derived from the same SD-WACCM-X runs.The amplitude and phase of SW2 were determined using the Fourier-wavelet method introduced by Yamazaki (2023b).
Figure 11 shows the amplitude of the symmetric and antisymmetric parts of SW2 in temperature (panels a-b), zonal wind (panels c-d) and meridional wind (panels e-f) at 10 4 hPa (approximately 110 km altitude) during the SSW-PJO events in January 2009 (panels a, c and e) and 2019 (panels b, d and f), as derived from the SD-WACCM-X simulations.In each panel, the meridional structures of the first two relevant Hough functions are also indicated.The amplification of the antisymmetric part of SW2 can be clearly seen for the January 2009 event.
The amplitude enhancement lasts for approximately 20 days from the day 25-45 in both the temperature and wind fields.This roughly corresponds to the period when the NAM index is lowest (Figure 8).In the wind fields, a weaker enhancement first occurs in the symmetric part and as it decays, the antisymmetric part enhances.This is also the case for the January 2019 event.That is, an enhancement of the antisymmetric part follows that of the symmetric part.The enhancements of the symmetric and antisymmetric parts are comparable in the January 2019 event, although this does not seem to be consistent with the TIMED/SABER observations presented in Figure 7e.
For both the January 2009 and 2019 events, the enhancement of the antisymmetric part is most prominent at 20-30°latitude in temperature and at 40-50°latitude in winds, which are attributable to the amplification of the semidiurnal (2,3) mode.Figure 12 confirms that the Hough-function amplitude of the semidiurnal (2,3) mode exceeds those of the (2,2), (2,4) and (2,5) modes during these events, which is consistent with the TIMED/SABER results summarized in Table 1.The Hough-function amplitude of the (2,3) mode in SD-WACCM-X temperature at 110 km increases by a factor of 1.5-2 during the SSW events.Large discrepancies are noted between the observed and simulated Hough-function phases (Table 1 and Figure 12), indicating that further improvement of SW2 in the model is needed.Our SD-WACCM-X results do not agree with those reported earlier by Jin et al. (2012).Jin et al. (2012) examined changes in the Hough-mode composition of SW2 in the dynamo region during the January 2009 SSW using another global atmospheric model.They concluded that the main response of SW2 to the SSW is the amplification of the symmetric (2,2) and (2,4) modes.They also noted that the amplitude of the (2,3) mode decreases prior to the SSW.More modeling studies are encouraged to clarify the response of symmetric and antisymmetric tides to SSWs.

Journal of Geophysical Research: Atmospheres
10.1029/2023JD040222 YAMAZAKI AND SIDDIQUI

Summary and Conclusions
Atmospheric tides are an important source of ionospheric variability.The distinction between the symmetric and antisymmetric parts of tides is important, because the ionosphere responds to them differently (e.g., Yamazaki, Richmond, Liu, et al., 2012).This study has examined characteristics of the symmetric and antisymmetric parts of solar tides in the mesosphere and lower thermosphere region using 21 years of neutral temperature observations by the SABER instrument on the TIMED satellite.
The three major components of the tide that are particularly important in atmosphere-ionosphere coupling are migrating diurnal tide (DW1), migrating semidiurnal tide (SW2) and eastward-propagating diurnal tide with  5).This leaves SW2 as the main source of antisymmetric tidal forcing to the ionosphere from below.
Spatiotemporal characteristics of the symmetric and antisymmetric parts of SW2 are different.The symmetric part of SW2 is observed throughout the year with amplitude maxima at ±30-35°latitude (Figure 2).The amplitude of the symmetric part is largest during March-June when the (2,4) mode is strong (Figure 6g).The antisymmetric part of SW2 has two seasonal types.The N.H. summer type is observed during May-September with amplitude maxima at ±15-20°latitude, when the (2,3) mode is strong (Figure 6h).The N.H. winter type is observed during November-March, characterized by weak amplitude maxima at ±40°latitude and ±10-15°l atitude.The contribution of the (2,3) mode to the N.H. winter type is much less than that to the N.H. summer type.
The transitions between the N.H. summer and winter types occur around April and October.
Amplification is observed in the antisymmetric part of SW2 during the sudden stratospheric warming (SSW) polar-night jet oscillation (PJO) events in January 2006January , 2009January , 2013January and 2019 (Figure 7e).The analysis of the four SSW-PJO events reveals the amplification of the (2,3) mode by a factor of 2-3 compared with the January climatology (Table 1).The SD-WACCM-X simulations for the January 2009 and 2019 events reveal the enhancement in the amplitude of the antisymmetric part of SW2 in dynamo-region winds (Figure 11).This explains, at least qualitatively, the previously-reported asymmetric response of the ionospheric Sq current system to SSWs (Yamazaki, Richmond, Liu, et al., 2012).A Hough function decomposition of SW2 in SD-WACCM-X temperature at 110 km reveals amplification of the (2,3) mode by a factor of 1.5-2 during the SSWs (Figure 12).

Figure 1 .
Figure 1.Monthly climatologies of the migrating semidiurnal tide (SW2) in temperature as derived from 21-year observations by TIMED/SABER.The upper and lower 12 panels show the amplitude and phase of SW2, respectively.

Figure 2 .
Figure 2. Same as Figure 1 except for the symmetric part.

Figure 3 .
Figure 3. Same as Figure 1 except for the antisymmetric part.

Figure 4 .
Figure 4. Monthly climatologies of the migrating diurnal tide (DW1) in temperature as derived from 21-year observations by TIMED/SABER.The upper and lower 12 panels show the amplitude of the symmetric and antisymmetric parts of DW1, respectively.

Figure 6 .
Figure 6.Hough function representation of the migrating semidiurnal tide (SW2) in temperature.(a) Meridional structures of the first two symmetric Hough modes.(b) Meridional structures of the first two antisymmetric Hough modes.(c) Amplitude of the symmetric part of SW2 at 109 km for May as derived from TIMED/SABER temperature observations and a least squares fit by the first two symmetric Hough functions.(d) Amplitude of the antisymmetric part of SW2 at 109 km for August as derived from TIMED/SABER temperature observations and a least squares fit by the first two antisymmetric Hough functions.(e)-(f) Same as (c)-(d) except for phase.(g) Month-to-month variation in the Hough-function amplitude for the first two symmetric semidiurnal modes at 109 km.(h) Same as (g) except for the first two antisymmetric semidiurnal modes.In (g) and (h), the error bars represent the magnitude of year-to-year variability.

Figure 7 .
Figure 7. Temporal variation in the amplitude of the migrating semidiurnal tide (SW2) in temperature at 109 km as derived from TIMED/SABER observations using a 60-day window.(a) Symmetric part.(b) Antisymmetric part.(c) Same as (a) except for the amplitude anomaly from the climatological annual cycle.(d) Same as (b) except for the amplitude anomaly from the climatological annual cycle.(e) Amplitude anomalies for the symmetric (blue) and antisymmetric (red) parts of SW2 at 32.5°latitude.The major sudden stratospheric warming (SSW) events in January 2006, 2009, 2013, and 2019 are indicated by vertical green lines.

Figure 8 .
Figure 8. Temporal development of the major sudden stratospheric warming (SSW) events in January 2006, 2009, 2013 and 2019, as represented by the Northern Annular Mode (NAM) index at 10 hPa.

Figure 9 .Figure 10 .
Figure 9. Amplitude (left two panels) and phase (right two panels) of the symmetric and antisymmetric parts of the migrating semidiurnal tide (SW2) in temperature as derived from TIMED/SABER observations during the four major sudden stratospheric warming (SSW) events in January 2006, 2009, 2013 and 2019.The period selected for each SSW event is indicated in Figure 8.

Figure 11 .
Figure 11.The amplitude of the symmetric and antisymmetric parts of the migrating semidiurnal tide (SW2) in temperature (T ), zonal wind (u) and meridional wind (v) at 10 4 hPa (∼110 km altitude) as derived from the SD-WACCM-X model simulations for the major sudden stratospheric warming (SSW) events in January 2009 (left) and January 2019 (right).The Northern Annular Mode (NAM) index at 10 hPa reached its minimum on 29 January 2009 and 4 January 2019.In each panel, meridional structures of the first two relevant Hough modes are indicated.

Table 1
, As Well As for the Composite of the Four SSW Events, and the January Climatology Note.It is noted that the January climatology does not exclude the SSW events.