Semidiurnal Non‐Migrating Tides in the Middle Thermosphere From Far Ultraviolet Observations

Much of the longitude/local time dependence of the thermosphere is controlled by non‐migrating tides. Observations of semidiurnal (12‐hr) tides between 120 and 200 km altitude, that is, the middle thermosphere, are rare owing to the lack of systematic measurements in this region. Since late 2018, the Global‐scale Observations of the Limb and Disk (GOLD) Mission has provided unique measurements of thermospheric disk temperature and the column density ratio of atomic oxygen to molecular nitrogen ratio (ΣO/N2) from geostationary orbit. In this paper, we present an approach to deduce the strongest semidiurnal non‐migrating tides in the middle thermosphere by adapting the method of Krier et al. (2021, https://doi.org/10.1029/2021ja029563) that deduces diurnal non‐migrating tides in simultaneous observations of temperature and ΣO/N2 made by GOLD. Testing of this approach suggests that the principal sources of uncertainties in the derived semidiurnal non‐migrating tides are the limitation on the longitudes sampled, such that uncertainties are higher for tides with longer horizontal wavelength, and contamination of the local time sums by stationary planetary waves, which causes amplitudes to be overestimated. Our approach is applied to GOLD data during solstice conditions in 2019–2021. Comparison to models yield disagreements which are likely due to uncertainties intrinsic to the method and/or misrepresentation of tidal dynamics in the models. These results are the first observations of semidiurnal non‐migrating tides in the middle thermosphere from a geostationary observational platform.


Introduction
Solar atmospheric tides are planetary-scale waves with periods that are harmonics of a solar day.They are primarily excited by the absorption of solar radiation.Semidiurnal non-migrating tides are those that have a 12-hr period and are characterized by non-Sun-synchronous zonal propagation.Non-migrating tides are particularly important to (a) describing the longitudinal/local time variability of the thermosphere and its impact on the ionosphere (see England (2012) and references therein) and (b) understanding the coupling of terrestrial weather and climate to conditions in the Earth's space environment (Hagan et al., 2009;Liu, 2016).Agreement between modeled and observed longitudinal variations of thermosphere structures can only be accomplished after accounting for non-migrating tides (Ward et al., 2010).A subset of these waves is forced by large-scale latent heat release in the tropics and propagate upward to the mesosphere/lower thermosphere (MLT) owing to sufficiently long vertical wavelengths (Hagan et al., 2007).Other non-migrating tides can be forced in situ via nonlinear wave-wave interactions (Angelats i Coll, & Forbes, 2002).
The naming convention of tidal components used in this paper is as follows.The name of a tide starts with an indication of its periodicity (D = diurnal, S = semidiurnal, T = terdiurnal).The second letter denotes the zonal propagation (E = eastward, W = westward, omitted if stationary).The third character indicates the zonal wavenumber of the tide with respect to universal time.For example, SE2 is the semidiurnal eastward propagating wavenumber 2 tide and D0 is the diurnal standing tide.
The middle thermosphere, between about 120 and 200 km altitude, is a region characterized by a steep increase in neutral temperature with altitude and the exponential decrease with altitude of neutral constituent's density correlating with their respective scale heights.The most abundant species of this region are atomic oxygen and molecular nitrogen.The parameter denoted as ΣO/N 2 , the vertical column density ratio of atomic oxygen to molecular nitrogen, is a sensitive measure of relative variations in thermosphere composition caused by geomagnetic storms (Gan, Eastes, Burns, Wang, Qian, Solomon, Codrescu, McInerney et al., 2020;Gan et al., 2024;Zhang et al., 2004), atmospheric tides (England et al., 2021;Krier et al., 2023), and mean thermospheric circulation (Gan, Qian, et al., 2023;Oberheide et al., 2020;Yamazaki & Richmond, 2013).Vertically propagating tides are expected to dissipate and deposit momentum into the background atmosphere of the middle thermosphere (Forbes et al., 2022).
Variations in the MLT neutral temperature (Li et al., 2015;Zhang et al., 2006), thermospheric wind (Lieberman et al., 2013;Oberheide et al., 2007), and neutral composition (Oberheide et al., 2013) have been shown to correlate with non-migrating tides originating from the troposphere.By now the climatology of the tidal spectrum in the MLT is understood owing to the global coverage provided by low Earth orbiting satellites (Forbes et al., 2008;Zhang et al., 2006).Non-migrating tides at the heights of the upper thermosphere have been characterized in zonal wind at 400 km as observed by the Challenging Minisatellite Payload (Häusler & Lühr, 2009) as well as near 260 km by the Gravity Field and Steady-State Ocean Circulation Explorer (Gasperini et al., 2015).Empirical modeling, namely, the Climatological Tidal Model of the Thermosphere (CTMT), has been used to extend MLT temperature tides to the middle thermosphere (Oberheide et al., 2011), albeit it has limitations such as the inclusion of only tides generated at/below the MLT region.Krier et al. (2021) used a novel approach to deduce the dominant diurnal non-migrating tides in temperature and composition derived from far ultraviolet images taken from geostationary orbit, those from the Global-scale Observations of the Limb and Disk Mission (GOLD), thereby providing the first quantitative estimates of these tides in the middle thermosphere temperature.
Like their diurnal counterparts, semidiurnal non-migrating tides can be forced in the lower atmosphere and go on to propagate upward to the thermosphere while growing exponentially in amplitude (Hagan & Forbes, 2003).A subset of them is generated in situ via nonlinear wave-wave interaction, that is, the nonlinear interaction of SW2 with SPW1 can yield SW1 and SW3.SE2 can arise from the nonlinear interaction between DE3 and DW1 in the lower thermosphere (Hagan et al., 2009).Diurnal and semidiurnal non-migrating tides tend to maximize at lowlatitudes and mid-latitudes, respectively (Forbes et al., 2006;Zhang et al., 2006).
As is the case for diurnal non-migrating tides, there are few observations of semidiurnal non-migrating tides between 120 and 200 km altitude, a region where both upward propagating and in situ semidiurnal tides are expected to exist.Forbes et al. (2022) examined the semidiurnal non-migrating spectra in thermospheric winds between 100 and 280 km.The vertical-latitudinal tidal structures in the observed thermospheric winds were interpreted in the context of those provided by the Climatological Tidal Model of the Thermosphere (CTMT), an implementation of the HME approach.It was found that the factors determining the propagation of semidiurnal tides through the dissipative thermosphere is still not fully understood or represented well by Hough Mode Extensions.GOLD observations of semidiurnal non-migrating tides would serve to inform and validate efforts to model the tidal impact on temperature and composition in the middle thermosphere.
In this paper, simultaneous observations of temperature and the column number density ratio of atomic oxygen to molecular nitrogen, ΣO/N 2 , are used to diagnose semidiurnal non-migrating tides by adapting the algorithm of Krier et al. (2021).Several changes are needed to implement the algorithm to retrieve semidiurnal non-migrating tides.This includes additional data processing of GOLD temperature and ΣO/N 2 as well as reformulating the combined temperature/ΣO/N 2 least squares fit to account for the local time sum method and the idiosyncrasies of semidiurnal non-migrating tides in the thermosphere.Retrievals are focused during solstice conditions at summer mid-latitudes because of the near optimal 12-hr local time sums afforded by those times and regions.These results are the first observations of semidiurnal non-migrating tides in the middle thermosphere from a geostationary observational platform.We report the first quantitative estimates of semidiurnal non-migrating tides in middle thermospheric temperature.
The organization of this paper is the following.Section 2 overviews the GOLD daytime observations and models used in this work.Section 3 describes the analysis method used to derive semidiurnal non-migrating tides.Section 4 presents results pertaining to testing of the analysis method and application to GOLD data.Section 5 discusses the results in the context of model output.Section 6 concludes this paper with a summary and conclusions.

GOLD Measurements of Temperature and Composition
The Global-scale Observations of the Limb and Disk (GOLD) instrument comprises a dual-channel far ultraviolet spectrograph which resolves the airglow spectrum from ∼132 to 162 nm (Eastes et al., 2017).It is a hosted payload on a telecommunications satellite in geostationary orbit at 47.5°West and has been returning images of the Earth since October 2018 (Eastes et al., 2020).Up until 6 September 2021, regularly scheduled dayside disk scans were performed over the same region of the Earth at a 30-min cadence for 18 hr each day (McClintock et al., 2020).After 6 September 2021, the cadence was decreased to one dayside disk observation sequence per 1-2 hr to extend the lifetime of the GOLD detector.An advantage of the synoptic sampling of the GOLD instrument is the possibility of studying tides at periods much shorter than the precession period of a low Earth orbiting spacecraft which are on monthly/climatological timescales, for example, ∼60 days in the case of SABER onboard the TIMED satellite (Forbes et al., 2006) and ∼35 days in the case of ICON (Cullens et al., 2020).
GOLD infers effective temperature from the observed rotational structure of the N 2 LBH band system emissions using an algorithm adapted from that used for limb-viewing measurements made by the High-resolution Ionospheric and Thermospheric Spectrograph (Aksnes et al., 2006;Aryal et al., 2022;Lumpe et al., 2002).Mean temperature uncertainties are estimated to vary from ∼20 to 100 K depending on signal-to-noise ratio in the observed radiances which tend to be higher for lower solar zenith angles (SZA) than that at higher SZA (Eastes et al., 2020).From the ratio of observed column emission intensities at OI 135.6 nm and the N 2 LBH band system, GOLD retrieves ΣO/N 2 following the approach first proposed by Strickland et al. (1995).Typical random uncertainties in the retrieved ΣO/N 2 are about 5% (Correira et al., 2021).The O/N 2 retrieval algorithm assumes that these emissions are produced by photoelectron impact and the subsequent de-excitation.There can be an appreciable contribution to emissions at OI 135.6 nm originating from O + radiative recombination especially in the afternoon around the equatorial ionization anomaly, but Correira et al. (2021) estimates that this contribution is mostly small (<2%) for the early GOLD science operations (2018)(2019)(2020)(2021).
Effective temperature and ΣO/N 2 are both quantities derived from vertically integrated emissions and thus do not correspond to a particular altitude.These parameters are not retrieved at locations where the SZA is greater than 80°or the view angle from local nadir, that is, the emission angel (EMA) is greater than 75°.This work uses the newest versions of the GOLD Level 2 data products at the time of writing, Version 5 for temperature and Version 4 for ΣO/N 2 , which have 250 × 250 km 2 spatial resolution at nadir.

Tidal Specification by TIEGCM-ICON Simulations
In this work, the Thermosphere-Ionosphere-Electrodynamics General Circulation Model for the Ionospheric Connection Explorer (TIEGCM-ICON; Maute, 2017) is used to simulate GOLD effective temperature and ΣO/N 2 observations.These simulations (a) allow for the specification of the full tidal spectrum which informs which tides are strongest, (b) provide phase offsets in temperature and ΣO/N 2 used in our retrieval algorithm of semidiurnal non-migrating tides (discussed in Section 3), and (c) serve as a test bed for our retrieval algorithm.TIEGCM is a physics-based model of the upper atmosphere that can be forced at the lower boundary.TIEGCM, developed by the National Center for Atmospheric Research, is a three-dimensional nonlinear physics-based model of the of the coupled thermosphere-ionosphere system for which forcing from the lower and upper boundaries can be prescribed.
In the TIEGCM-ICON version of the model, atmospheric tides derived from the ICON observations of horizontal winds (Harding et al., 2017) and temperature (Stevens et al., 2018), both made by MIGHTI (Michelson Interferometer for Global High-resolution Thermospheric Imaging), are used to perturb the lower boundary at about 97 km altitude.The spectrum of upward propagating tides at the lower boundary is specified by Hough Mode Extensions (HMEs; Cullens et al., 2020;Forbes et al., 2017) applied to temperature and horizontal winds (∼90-103 km altitude range).There are known issues with the ICON HMEs primarily related to the limited latitudinal coverage of temperature and winds.Owing to ICON's orbit and MIGHTI's viewing geometry, only latitudes between about 10°S-40°N are sampled.The gap in coverage in the southern hemisphere leads to higher uncertainties in certain Hough modes which propagate into the modeled tides (Cullens et al., 2020;Forbes et al., 2017).Forbes et al. (2017) noted that because of the latitude sampling, SW1 and especially S0 are retrieved less accurately by the HME fitting technique applied to ICON observations.Nevertheless, TIEGCM-ICON provides realistic upward propagating tides at the boundary of the Mesosphere/Lower Thermosphere.Daily model output files are readily available starting at the beginning of the ICON Mission in late 2019.
To simulate temperature observations made by GOLD, we calculate effective temperature by operating a contribution function on the modeled temperature vertical profiles (Krier, 2024).In short, contribution functions model the altitude dependence of N 2 LBH emissions on the vertically integrated brightness.The contribution function corresponding to SZA = 70°and EMA = 0°is used, following Krier et al. (2021), because it is suitable for our tidal analysis which uses data at about the same SZA.Simulated values of ΣO/N 2 are calculated from TIEGCM output by vertically integrating down to the standard N 2 column depth of 10 17 molecule cm 2 , first specified by Strickland et al. (1995).The full tidal spectrum is extracted from the simulated temperature and ΣO/N 2 via twodimensional Fast Fourier Transforms.
Figure 1 presents amplitudes and phases in effective neutral temperature and ΣO/N 2 for select semidiurnal nonmigrating tides (SE2, S0, SW1, SW3, and SW4) during June conditions.These tides reflect the 5-day average between 26 June-30 June 2020 as simulated by TIEGCM-ICON.Notably, at northern middle latitudes, SE2 and SW1 are the dominant tides in temperature with S0 being tertiary.The hierarchy of semidiurnal non-migrating tides in ΣO/N 2 is mixed as SE2 and SW1 have high amplitudes at middle latitudes while SW3 and SW4 spike in amplitudes toward the higher latitudes despite having small amplitudes in temperature.
Figure 2 is the same as Figure 1 except for December conditions, that is, 5-day average of tides from 25 December to 29 December in 2020.At the southern middle latitudes during December 2020, SE2 is the dominant tide in temperature while the situation for ΣO/N 2 is like that during June 2020 in that several tides are prevalent.Since HMEs specify the lower boundary of TIEGCM-ICON, it is worthwhile to assess the strength of semidiurnal nonmigrating tides as calculated by HMEs.
Figure 3 presents mean HME semidiurnal non-migrating tidal amplitudes in temperature, zonal wind, and meridional wind at 150 km, shown as a function of latitude.SW4 is excluded from Figure 3 because this tide is not reported in the ICON Level 4.1 data used to prepare this paper.Average amplitudes are shown during two periods: days 172-193 and days 336-366 in 2020.Days in December 2020 are used in lieu of January due to apparently erroneous HME calculations during January 2021 at southern latitudes likely due to the lack of latitudinal coverage there by ICON/MIGHTI (Cullens et al., 2020).Note that HMEs include only upward propagating tides and thus do not include in situ generated tides, that is, those generated by absorption or child waves from nonlinear wave-wave interaction.Figure 3 clearly shows that, in the perspective of HMEs, SE2 is the dominant semidiurnal non-migrating tide at this altitude at northern middle latitudes during June 2020 and southern middle latitudes during December 2020.This also suggests that SW1 is an in situ generated tide in the output of TIEGCM-ICON.SW1 can be generated by the nonlinear interaction between SW2 and SPW1.SPW1 can be prominent owing to the offset of the geomagnetic equator from that in geographic coordinates (Jones et al., 2013).Krier et al. (2021) presented a method to extract diurnal non-migrating tides from GOLD measurements of temperature and ΣO/N 2 ; the procedure is covered in detail therein but will be summarized here.Because GOLD only retrieves temperature and ΣO/N 2 at daytime local times and the ∼140°of longitude visible from its point-of- view in geostationary orbit, typical two-dimensional Fourier methods to extract tides do not apply.The method of Krier et al. (2021) deduces two specified non-migrating diurnal tides in both temperature and ΣO/N 2 by leveraging the known phase relationships between temperature and composition.The optimal tidal parameters are determined by a direct search method (Lewis et al., 2000) in a least squares approach, see Krier et al. (2021) for details on the specific implementation for diurnal tides.A proxy for the diurnal non-migrating tides is obtained by taking 12-hr local time differences, which involve data points near dusk and dawn.Krier et al. (2021) focused on DE3 and DE2 at equatorial latitudes.The changes and adaptations necessary to modify the method of Krier et al. (2021) to extract semidiurnal non-migrating tides are listed and explained below.

Method to Derive Semidiurnal Non-Migrating Tides
The algorithm used in this paper to deduce semidiurnal non-migrating tides follows closely to that for retrieving the diurnal tides with a few exceptions.Three semidiurnal non-migrating tides are fitted: SE2, SW1, S0.This is due to consideration of Figures 1 and 2 which show TIE-GCM simulations suggesting that more than two semidiurnal non-migrating tides are prevalent.This contrasts with the case of diurnal non-migrating tides for which only two tides are fitted, that is, DE2 and DE3.The latitudes at which semidiurnal non-migrating tides are extracted in the following analyses are between 50°S and 20°S during January solstice and between 20°N and 50°N during June solstice.This is also intended to minimize terdiurnal leakage.A general circulation model (Du & Ward, 2010) showed that terdiurnal non-migrating tides in zonal winds, between 75 and 130 km, maximize poleward of 50°N and 50°S.Satellite data analysis (Moudden & Forbes, 2013) of neutral temperature, between 80 and 110 km, quantified the strength of terdiurnal non-migrating tides, showing that most of the largest tides in temperature are confined to equatorial regions.Although these results do not pertain directly to terdiurnal tides in middle thermospheric temperature and ΣO/N 2 , they suggest that terdiurnal tides impact these parameters less at middle latitudes than at equatorial and polar latitudes.Confirmation requires further modeling studies of terdiurnal tides in effective temperature and ΣO/N 2 which is beyond the scope of this paper.Dusk-dawn differences were used by Krier et al. (2021) to obtain a proxy for diurnal tides.In this paper, dusk + dawn sums are used as a proxy for the semidiurnal perturbations.Data points roughly 12 hr apart are added together and halved, this removes the diurnal and terdiurnal perturbations and leaves behind double amplified semidiurnal perturbations.The non-migrating semidiurnal tides are taken to be the residuals from the zonal mean of the dusk + dawn sums.It is worthwhile to mention that stationary planetary waves are removed by 12-hr differences, but stationary planetary waves are not removed by 12-hr sums.This leads to increased uncertainty in the inferred non-migrating semidiurnal tides which requires further analysis to quantify.It is well known that stationary planetary waves are generated by the interaction between migrating tides and non-migrating tides (Hagan et al., 2009).
The least squares approach used to fit for three semidiurnal tidal components entails fitting tidal perturbation equations to the dusk + dawn sum residuals.At a fixed latitude and altitude, tides are waves propagating through local time and longitude.Following Zhang et al. (2006), Equation 1gives a mathematical expression for tidal perturbations as a function of local time t LT and longitude λ.
where n is the period in cycles per day, s is the zonal wavenumber with a negative value indicating eastward propagation, Ω is Earth's rotation rate, A n,s is the amplitude of the tide, ϕ n,s is the phase of the tide.The dusk + dawn sum residuals are assumed to be composed of tidal perturbations generated by SE2, SW1, and S0.All other tides and waves are assumed to have sufficiently small amplitude.Mathematically, the dusk + dawn sum residuals, ΔT(λ), is the sum of the total tidal perturbations (SE2 + SW1 + S0) at the dusk and dawn local times, T 2 and T 1 , respectively (Equation 2).
Equation 3 shows the basis functions used to fit three semidiurnal non-migrating tides, that is, SE2, SW1, and S0, to the local time sum residuals in temperature.
where t 2 is the dusk local time and t 1 is the dawn local time.The unknown parameters in Equation 3 are {T SE2 , T SW1 , T S0 , ϕ SE2 , ϕ SW1 , ϕ SO } which are respectively the amplitudes and phases of the tides.Note that Δt = t 2 t 1 12.The ΩΔt terms are placed to correct for local time sums that involve data points not quite 12 hr apart.Equation 4 is the corresponding expression for ΣO/N 2 dusk + dawn sum residuals.
where {R SE2 , R SW1 , R S0 , Φ SE2 , Φ SW1 , Φ SW1 } denotes the set of ΣO/N 2 tidal amplitudes and phases.The ΣO/N 2 phases are constrained by the prescribed phase differences at the latitude of interest (Equations 5-7).The dusk + dawn residuals are converted to dimensionless units and normalized.The normalized ΣO/N 2 amplitudes are assumed to equal those in temperature for each tide (Krier et al., 2021).
where Θ x is the temperature-ΣO/N 2 phase differences for tide x (Figure 4).To best fit the data, the prescribed phase differences are allowed to vary ±10°of longitude.Figure 4 shows phase differences between tides in temperature and ΣO/N 2 for SE2, S0, and SW1 as a function of latitude during June and January 2020.The phase difference for SE2 varies smoothly at northern middle latitudes during June and at southern middle latitudes during January.While the phase difference for SW1 and S0 tend to jump due to their relatively low amplitudes in temperature.
In order to deduce the tides, the normalized non-migrating semidiurnal proxies for temperature and ΣO/N 2 , T obs and R obs , are simultaneously fitted to Equations 3 and 4. A least-squares scheme determines the combination of {T SE2 , T SW1 , T S0 , ϕ SE2 , ϕ SW1 , ϕ SO } that yields the lowest total squared residual T 2 res + R 2 res (Equation 8).
The set of ΣO/N 2 tidal parameters follow from the temperature-composition phase relationships and the normalization factors.
For the diurnal approach, it was necessary to remove a linear trend from the dusk-dawn differences for both temperature and ΣO/N 2 .This was done to amplify the wave-3 and wave-4 variations salient to DE2 and DE3.In the case of extracting semidiurnal non-migrating tides, this linear detrending must be carefully considered since SW1 may appear as a linear trend in the data sampled in a 140°longitude window.Consequently, the linear detrending is not performed for the ΣO/N 2 perturbations.The linear detrending is performed for the temperature perturbations because a consistent linear trend (higher LT sum on the eastern side of the disk than on the western side) is still seen in the temperature dusk + dawn sums regardless of season, suggesting that this is an instrumental or algorithm effect on the retrieval of disk neutral temperature by the GOLD instrument.
The top half of Figure 5 presents temperature semidiurnal non-migrating tidal amplitudes as a function of day of year in 2020 at 36°S and 36°N separately.SE2 is dominant at both latitudes during most of 2020, maximizing at around 10 K.At 36°N latitude, SW1 peaks at around June solstice exceeding the strength of SE2.SW1 is secondary to SE2 at 36°S during around December solstice.S0 is the third highest component of the semidiurnal nonmigrating tide at 36°N during the June solstice period.It is reasonable to fit for SE2, SW1, and S0 at these middle latitudes around solstice times.Additionally, analyzing data at middle latitudes during their respective summer solstice times allows this approach to take local solar times whose difference is approximately equal to 12 hr.This is not possible at middle latitudes during other seasons because the GOLD mission does not retrieve temperature and composition for SZA higher than 80°.This SZA restriction severely limits the local solar times that can be sampled at middle latitudes during certain seasons.By achieving a local time sum as close to perfect as possible (involving data points 12 hr local time part), the amount of leakage from terdiurnal and diurnal tides is minimized.
Stationary planetary waves are non-propagating waves which are constant as a function of time.Therefore, while the difference of wave forms at different local times removes the SPWs variations, the sum of wave forms at different local times serves to amplify variations associated with SPWs.The bottom half of Figure 5 shows the amplitudes of stationary planetary waves (SPWs), which are not removed by local time sums.SPWs of zonal wavenumber 1-4 are shown.SPW1 can reach large amplitudes comparable to that of SE2 and SW1.There appears to be some degree of correlation between the strength of SPW1 and SW1 during 2020 at 36°N.SPW2, SPW3, and SPW4 are mostly small in amplitude around June and January solstice times.GOLD temperature and ΣO/N 2 images are processed in the following fashion before applying the algorithm for retrieving non-migrating semidiurnal tides.Our aim is to derive tides as a function of latitude at middle latitudes.
In the "image" space of the GOLD images, a given row of pixels has some latitudinal variation which becomes larger going away from the equator.Therefore, it is necessary to interpolate the GOLD data from the irregular longitude/latitude grid in which they are reported to a regular grid.This is performed using Kriging (nearest neighbor) interpolation.In this method, data points and their spatial variance are used to determine trends which are applied to the grid points.Figure 6 presents an example of this interpolation for the case of a smoothly varying parameter, namely, the solar zenith angle in a northern hemisphere scan at 13.25 hr universal time (UT).The regular longitude grid used is from 110°W to 15°E in intervals of 5°.Two different regular latitude grids are used, each in intervals of 4°.For scans of the northern hemisphere, the regular grid runs from 2°N to 50°N.For scans of the southern hemisphere, the regular grid runs from 50°S to 2°S.Data from a period of days (where background conditions are roughly constant) are averaged to maximize the tidal signal and smooth geophysical noise.
Testing of the diurnal method (Krier et al., 2021) shows robustness to random noise and that ionospheric contamination is likely not an issue during the solar minimum conditions in the early years of the GOLD Mission (2018-2020).Scenario testing (not shown here) assessing the impact of using incorrect a prior phase differences showed that doing so leads to extremely low correlation between the algorithm input and reconstructed fitted tides.Therefore, if application to real data yields good correlation, then it is reasonable to assume that the fitted tides in temperature and composition are self-consistent due to the constraint on the phase difference.

Testing the Impact of Algorithm Assumptions
In this section, results from various aliasing and sampling experiments are presented.The sampling experiments assess the impact of incomplete longitudinal coverage and using imperfect local times sums which are caused by involving data points that are not quite 12 hr local time apart in the local time sums.This approach for deducing non-migrating semidiurnal tides assumes that only three tides generate the longitudinal variation in the local time sums.The purpose of the aliasing experiments is to explore how much tides and waves which are assumed to be absent leak into the deduced SE2, SW1, and S0 tides.These tests use the tides derived from TIEGCM-ICON output to generate pure tidal data sampled in the GOLD observational geometry.The following results look at application of our algorithm at northern hemisphere latitudes during June Solstice.Four scenarios are explored, each of which are characteristic of different combinations of the two assumptions being true or false.The two assumptions are the following: (a) perfect 12-hr local time sums are achieved, (b) only SE2, SW1, and S0 are present in the data.Figures S1-S4 in Supporting Information S1 file depicts the agreement between the derived and true tidal parameters for Scenarios 1, 2, 3, and 4. The results for each of the scenarios are summarized below.
Scenario 1 entails application of the approach to data with perfect local solar times sums and in which only SE2 and SW1 are present.There is excellent agreement for all parameters except for the SW1 temperature amplitude which is underestimated due to its horizontal wavelength which extends well beyond the GOLD field-of-regard.Scenario 1 was also tested in the idealistic case of complete longitudinal coverage and yielded tidal parameters that are all within 1% of the truth, thereby demonstrating the fundamental limitation introduced by having incomplete longitudinal coverage.
Scenario 2 data permits perfect local solar times sums, but all tides are present in the data.Stationary planetary waves of zonal wavenumber 1-4 are also included since local time sums do not remove their variations.Each of the SPWs included were artificially defined to have amplitude of 5 K in temperature and a similar relative amplitude in ΣO/N 2 .In this case, phases are retrieved accurately, but all amplitudes are retrieved with varying degrees of inaccuracy due to the presence of tides/waves assumed to be absent and the restriction in longitude.A version of Scenario 2 was also performed in the case of full longitudinal coverage and yielded retrieved amplitudes that are about 5 degrees Kelvin higher than the truth, equal to the magnitude of SPWs added to the data (not shown).This clearly shows that the presence of SPWs leads to the overestimation of amplitudes.
Scenario 3 data uses the actual local solar time/SZA sampling afforded by the GOLD mission while only SE2, SW1, and S0 are present.Results for this scenario resemble those from Scenario 1.This suggests that our algorithm is somewhat robust to the imperfect local solar time sums that GOLD affords at these latitudes during this season.Scenario 4 data uses the actual GOLD sampling and all tides and SPWs are present in the data.This scenario most closely reflects the conditions expected in actual GOLD observations.Phases are still retrieved accurately for this scenario, but the amplitudes have considerable inaccuracies, but the true amplitudes still lie within the error bars.
Table 1 summarizes the median amplitude errors in percent and provide a qualitative assessment of the phase accuracy in each of the four scenarios discussed above for SE2, SW1, and S0 respectively.Evidently, the breaking of assumption of perfect 12-hr local time sums has less than an impact than the breaking of the assumption that only SE2, SW1, and S0 are present in the data.Significant error in amplitudes is seen for SW1, whose horizontal wavelength is larger than that of SE2 and S0, and S0, whose amplitude is relatively small.

Application to June Solstice in 2020
In this subsection, extraction of semidiurnal non-migrating tides from GOLD observations between 20°N and 50°N latitude is shown.GOLD data from 20 June-11 July 2020 are averaged in GOLD image space (row, column, scan number).Outliers are not involved in the average, neither are data on days with sufficiently high geomagnetic activity (K p of at least 4) or solar activity (daily F10.7 cm sfu greater than 2.5 standard deviations over the mean during the period).The averaging is done to amplify the tidal signal relative to geophysical day-today variations not related to tides.During this time around northern summer solstice, SE2, SW1, and S0 are extracted from the local time sum residuals.Figure 7 shows the latitude versus longitude distribution of the local  time sum residuals and the reconstructed tides (SE2 + SW1 + S0) in both temperature and ΣO/N 2 .ΣO/N 2 perturbation amplitudes exceed 10% peak-to-peak and temperature perturbations exceed 30 K from the zonal mean.Figures 7b and 7d show the superposition of SE2, SW1, and S0 tides that best fits the dusk + dawn sums.
Figure 7e shows the correlation coefficients between the left-and right-hand sides of panels (a)-(d) of Figure 7.
Most retrievals have a correlation coefficient greater than 0.5 suggesting that the total tidal variation due to SE2, SW1, and S0 reasonably reproduces the dusk + dawn sums.
Figures 8 and 9 respectively show the estimates of tidal amplitudes and phases in both temperature and ΣO/N 2 as a function of latitude.The error bars in Figure 8 serve as an estimate of the uncertainty in the derived amplitudes.
The error bars reflect the root mean square deviation between the dusk + dawn sums and the reconstructed fitted tides at each latitude.The error bars likely overestimate the uncertainty of the derived amplitude but comparing the size of the error bars gives a sense as to which latitudes the fit does not perform well.Tables S1 and S2 in Supporting Information S1 file tabulate the amplitudes and phases presented in Figures 8 and 9 as well as those corresponding to latitudes where the correlation coefficient threshold is violated.
Unlike the spectra provided by HMEs (Figure 3) and those simulated by TIEGCM-ICON (Figures 1 and 4), S0 is comparable in strength to both SE2 and SW1.In fact, the amplitudes of the three tides are equal at most latitudes, suggesting that the three tides together generate the perturbations in the local time sums.While SE2 is known to be an upward propagating tide, S0 and SW1 are likely in situ generated considering their absence in HME upward propagating tide spectra.This is consistent with the findings of Forbes et al. (2022) who interpreted SW1 and S0 tides in ICON/MIGHTI horizontal winds as in situ generated waves.The relatively strong S0 retrieved from GOLD data suggests that the TIEGCM-ICON underestimates the in situ generated S0.This could arise from misrepresented nonlinear wave-wave interactions.Although, additional longitudinal coverage in the measured effective temperature and ΣO/N 2 would increase the veracity of the retrieved SW1 and S0.It is possible that the restriction in longitude negatively impacts the estimation of SW1 and S0 to the extent that the constraint on the temperature-composition phase differences is not enough.Additionally, the presence of SPWs in the local time sums can cause tides such as S0 to be overestimated (see the results of Section 4.1).
To show the consistency of the retrieval across years, Figures 10 and 11 offer a comparison of the retrieved amplitudes and phases during 2019, 2020, and 2021.The retrieved phases are consistent across years, except for SE2 between about 20°N and 30°N latitude where the LST sums use data around 10 hr LST apart so that some terdiurnal leakage may occur.In all 3 years, the amplitudes of the three tides mostly track each other.This shows that, regardless of a specific fitting window, the three tides retrieved (SE2, SW1, and S0) are required to generate the observed variations in the LST sums.The retrieved amplitudes are estimates of amplitudes of semidiurnal non-migrating tides in the middle thermosphere temperature and composition.A caveat to these results is that the retrieved amplitudes are most likely overestimated due to the contamination of unknown in amplitude stationary planetary waves (SPWs) which are not removed by the operation of LST sums.The

Application to December Solstice in 2020
Figure 12 shows the latitude versus longitude distribution of the local time sum residuals and the reconstructed tides (SE2 + SW1 + S0) in both temperature and ΣO/N 2 , but now for mean GOLD data from 1 to 20 January 2020 at latitudes spanning 50°S to 20°S.It is notable that the local sum patterns at northern middle latitudes examined during the June solstice time and those at southern middle latitudes during January solstice time are inphase for temperature and out-of-phase for ΣO/N 2 .That is, the longitudes of maxima and minima for the two times analyzed are the same in temperature, but different for ΣO/N 2 .Figure 12e shows high correlation between the dusk + dawn sums and the reconstructed fitted tides at latitudes between about 40°S and 20°S.Figures 13 and 14 show the retrieved amplitudes and phases during this time.Tables S3 and S4 in Supporting Information S1 file tabulate the amplitudes and phases presented in Figures 13 and 14 as well as those corresponding to latitudes where the correlation coefficient threshold is violated.Like those retrieved for the June solstice time, the SE2, SW1, and S0 amplitudes seem to track with one another with no apparent dominant tide.In this case, the strength of SE2 equals that of S0 at most latitudes.

Case Study of January 2019 Sudden Stratospheric Warming
Sudden stratospheric warming (SSW) events are of great interest to the study of the coupling of the lower atmosphere to the near-Earth space environment via upward propagating atmospheric waves.SSWs are largescale events that mostly occur during January-February in the northern polar region, characterized by a reversal in the polar vortex and rapid rises in polar stratospheric temperatures (Butler et al., 2015).It has been proposed that the mechanism for SSW generation is driven by the intensification of upward propagating planetary wave activity, with either zonal wavenumber 1 or 2 (Matsuno, 1971).Enhanced S0 and SW1 tides have been reported in ionospheric perturbations at times of SSW (Pedatella & Forbes, 2010) and it has been attributed to the nonlinear interaction between the migrating semidiurnal tide and planetary waves (Angelats i Coll & Forbes, 2002;Teitelbaum & Vial, 1991).Modeling has reproduced this to some extent (Chang et al., 2009;Liu et al., 2021;Pedatella & Liu, 2013).Others have proposed that interaction between the semidiurnal migrating tide and quasi-16-day waves can play a role in enhancing semidiurnal non-migrating tides during SSWs (He et al., 2018).
Due to position its station in geostationary orbit, GOLD has made possible a new perspective to the impacts of PW-tidal interactions on the EIA during SSWs (Gan, Eastes, Burns, Wang, Qian, Solomon, Codrescu, and McClintock, 2020;Gan, Oberheide, et al., 2023).Because of the unique sampling cadence of GOLD, if GOLD can reveal a response in the semidiurnal non-migrating tides in middle thermospheric temperature and ΣO/N 2 to   an SSW, it would prove valuable to exploring the origin of these amplitude enhancements.This subsection discusses the January 2019 SSW event.A depletion of more than 10% in the GOLD ΣO/N 2 was reported, demonstrating the connection between the polar stratospheric changes with those in the global thermospheric composition (Oberheide et al., 2020).The event peaked around 2 January 2019.In what follows, proxies for the semidiurnal non-migrating tides and their retrieved amplitudes and phases are compared for times before and during the 2019 January SSW.The changes are discussed in context with another northern wintertime when there was not a major SSW.
Figure 15 offers a direct comparison of the dusk + dawn sums and reconstructed fitted tides (SE2 + SW1 + S0) before the SSW, days 300-320 in 2018, and during the SSW, days 1-20 in 2019.Not much difference is seen in the temperature perturbations, but there is an apparent amplification of the trough in the ΣO/N 2 perturbations on the extreme western side of the longitudinal span sampled.Making a similar comparison to the same date ranges in the 2019-2020 winter when there was not an SSW event, there is no such notable amplification in the ΣO/N 2 perturbations (not shown).Thus, it is not clear whether the amplification in the ΣO/N 2 perturbations are from the SSW or intraseasonal variability.The changes in the ΣO/N 2 pattern could be caused by changes in the semidiurnal non-migrating tides or SPWs causing the perturbations.
Figure 16 shows the correlation coefficient of the fits during the two times shown in Figure 15.In general, the fit is better during the SSW than before the SSW.Figures 17 and 18 show the retrieved amplitudes and phases respectively before and during the SSW.Besides a difference in SE2 phases between about 45°S and 30°S, there is no notable difference between the tidal parameters during the two times.Found in Supporting Information S1 file, Figures 5-7 are the same as Figures 16-18, except for the same date ranges in the 2019-2020 winter when there was not an SSW.As for the times before and during the January 2019 SSW, there is little difference between the retrieved tidal parameters during these times, although the corresponding time "during" the SSW in the non-SSW year sees lower amplitudes in ΣO/N 2 across the board.Therefore, no conclusions can be made that GOLD sees a response in the semidiurnal non-migrating tides to the January 2019 SSW.

Discussion of Results
The presence of any other semidiurnal non-migrating component, terdiurnal non-migrating tide, or stationary planetary wave that may have substantial amplitude leads to uncertainty in the derived amplitudes and phases of the three pre-selected semidiurnal non-migrating tides.The local time sum approach does not remove variations associated with stationary planetary waves and allows leakage of terdiurnal non-migrating tides if the local times used are not exactly 12 hr apart.Tests examining the performance of our algorithm in various sampling and aliasing scenarios showed that the limited longitudinal sampling afforded by GOLD and the presence of waves assumed to be nonzero lead to error in the retrieved tidal parameters.The estimation of tides with long horizontal wavelength, that is, SW1, can be especially problematic because the approximately 100-140°of longitude sampled does not cover even half of its horizontal wavelength.The presence of stationary planetary waves leads to overestimation of amplitudes.A local time sum involving data points even 10 hr part can lead to substantial terdiurnal tide leakage.In all scenarios, the retrieved phases are accurate for the highest amplitude tides, that is, SE2 and SW1.The sources of uncertainties discussed above emphasize that the retrieval of semidiurnal non-migrating tides is more complicated than that for their diurnal counterparts.Application of our algorithm to periods when SPW activity is weak, which may depend on latitude, season, and/or year, would be valuable in assessing the degree of contamination of SPWs.Problematically, SPWs can be generated by nonmigrating tide interaction with other waves (Hagan et al., 2009).
Application during both January and June solstice conditions at middle latitudes showed comparable amplitudes in SE2, SW1, and S0, contrary to amplitudes calculated from Hough Mode Extensions as well as those  simulated by the Thermosphere-Ionosphere-Electrodynamics General Circulation Model which suggest that SE2 and SW1 are larger than S0.This may confirm that SW1 and S0 are in situ tides in the middle thermosphere, consistent with the interpretation by Forbes et al. (2022) of semidiurnal horizontal winds between 100 and 280 km altitude.Tidal spectra derived from TIMED/SABER cooling rates at 125 km during September 2013 indicate that SE2 and SW1 are the strongest semidiurnal non-migrating tides at around 40°N latitude (Nischal et al., 2019).Although, SW1 is prominent near the equator.Similarly, during December 2013, the strongest semidiurnal nonmigrating tides in TIMED/SABER cooling rates were SE2, SW1, and SW3 (Nischal, 2019).While these cooling rate tides are lower in altitude than the effective altitude of GOLD temperature retrieval, they suggest that SW1 and SE2 should be present in temperature higher up because cooling rates can be interpreted as changes in temperature.
Extraction of the tides around the January 2019 major sudden stratospheric warming do not suggest that there is a response in the semidiurnal non-migrating tides to the changing conditions below the middle thermosphere.This contrasts with previous studies of the tidal response to SSW events, such as that of Pedatella and Forbes (2010) who looked at the ionospheric response which mostly depends on the modulation of the E-region dynamo by MLT tides.The lack of a tidal response at the altitude (∼160 km) of GOLD data suggests that the tidal response depends on altitude.

Summary and Conclusions
This paper has presented a method for the extraction of three pre-selected semidiurnal non-migrating tides from simultaneous retrievals of temperature and ΣO/N 2 in the middle thermosphere.The approach is an adaptation to that presented by Krier et al. (2021) who discussed an algorithm for extracting diurnal non-migrating tides.Several tests were performed to assess the impact of several assumptions crucial to the implementation of this approach, including that only the three pre-selected tides cause perturbations in the data.The testing showed that retrieval of tides with a relatively long horizontal wavelength, such as SW1, is not done well owing to the incomplete longitudinal coverage.SE2 is retrieved well, although its amplitude is overestimated depending on the strength of SPWs.Systematic analyses of semidiurnal non-migrating tides should consider the SPWs or be focused on periods during which SPWs are known to be weak.Application to summer solstice periods at middle latitudes provide the first estimate of semidiurnal non-migrating tides in both middle thermospheric temperature and ΣO/N 2 .These results also serve as the first retrieval of these tides in the upper atmosphere using a geostationary observational platform.Retrieval of SE2, SW1, and S0 from GOLD data indicate that all three tides are required to generate the observed variations in the semidiurnal non-migrating proxy.This contrasts with TIEGCM-ICON modeling which show SE2 and SW1 to be dominant.Although the uncertainties in the retrieval of SW1 and S0 are especially high due to the limited longitude sampling of the GOLD instrument and the presence of SPWs in local time sums.Nonlinear tidal dynamics not represented in TIEGCM, or incomplete lower boundary tidal forcing may explain this difference between modeling and observations.To resolve these differences, future work aided by models should investigate the in situ generation, propagation, and dissipation through the middle thermosphere of semidiurnal non-migrating tides.

Figure 1 .
Figure 1.Tidal amplitudes (a and b) and phases (c and d) in effective neutral temperature and ΣO/N 2 shown as a function of latitude for SE2, S0, SW1, SW3, and SW4.Each color corresponds to a single tide as indicated by the legend.Calculated for mean June 2020 conditions as simulated by the version of the Thermosphere-Ionosphere-Electrodynamics General Circulation Model driven by tidal perturbations derived from observations made by the Ionospheric Connection Explorer (TIEGCM-ICON).Phases are in units of universal time of maximum at 0°longitude.Temperature and ΣO/N 2 amplitudes are in units of degrees Kelvin and percent relative to the zonal mean, respectively.

Figure 3 .
Figure 3. SE2, S0, SW1, and SW3 amplitudes at 150 km in temperature, zonal wind, and meridional wind shown as a function of latitude.From Hough Mode Extensions applied to ICON data.(a)-(c) and (d)-(f) show mean tides in 2020 from days 172-193 and days 336-366, respectively.Each color corresponds to a single tide as indicated by the legend.

Figure 4 .
Figure 4. From TIEGCM-ICON simulations during 2020, temperature-ΣO/N 2 phase differences for SE2, S0, and SW1 as a function of latitude during June (a) and January (b).Phase differences are shown in units of hour.Each color corresponds to a single tide as indicated by the legend.

Figure 5 .
Figure 5. SE2, S0, SW1, SW3, and SW4 amplitudes in effective neutral temperature simulated by TIEGCM-ICON shown as a function of day of year in 2020 at 36°N (a) and 36°S (b).Each color corresponds to a single tide as indicated by the legend.Panels (c) and (d) are the same as (a) and (b), except for stationary planetary waves (SPWs), which are not removed by local time sums.SPWs of zonal wavenumber 1-4 are shown as indicated by the legend.

Figure 6 .
Figure 6.Solar zenith angle (SZA) at each pixel for an arbitrary scan of the northern hemisphere performed by the Globalscale Observations of the Limb and Disk (GOLD) imager in the irregular longitude/latitude grid in which GOLD data is reported (a) and a regular grid to which all GOLD data is interpolated as described by the text (b).Red/orange/yellow indicate SZA between about 0°and 45°.Black/purple/blue indicate SZA between about 45°and 90°.

Figure 7 .
Figure 7. Residuals from the zonal mean for the dusk + dawn sums and the reconstructed semidiurnal non-migrating tidal field from the retrieved SE2, SW1, and S0 in neutral temperature, (a) and (b), and ΣO/N 2 , (c) and (d).Based on GOLD data averaged between 20 June and 11 July 2020.Shown between 20°N and 50°N latitude and longitudes within the GOLD field-of-regard where conditions for the dusk + dawn sums are satisfied.(e) Correlation coefficients between the dusk + dawn sums and tidal reconstructions in temperature (blue) and ΣO/N 2 (red) shown as a function of latitude.

Figure 8 .
Figure 8. Retrieved amplitudes from GOLD data averaged between 20 June and 11 July 2020 as a function of latitude for SE2, (a and b), SW1, (c and d), and S0, (e and f ).Errors bars reflect the root mean square deviation of the least squares fit at each latitude.Only latitudes where the least squares fit in both temperature and ΣO/N 2 yields a correlation coefficient greater than 0.5 are shown.

Figure 9 .
Figure 9. Retrieved phases (universal time of maximum at 0°longitude) from GOLD data around June solstice as a function of latitude for (a) SE2, (b) SW1, and (c) S0.Temperature is shown in blue, ΣO/N 2 in red.Only latitudes where the least squares fit in both temperature and ΣO/N 2 yields a correlation coefficient greater than 0.5 are shown.

Figure 10 .
Figure 10.Same as Figure 8, except retrievals from data averaged between 20 June-11 July in 2019 (red) and 2021 (blue) are compared to those in 2020 (black).

Figure 11 .
Figure 11.Same as Figure 9, except retrievals from data averaged between 20 June-11 July in 2019 (unfilled triangles and dotted lines) and 2021 (asterisks and dashed lines) are compared to those in 2020 (filled lines and solid lines).

Figure 12 .
Figure 12.Same as Figure 7, except between latitudes 50°S and 20°S for GOLD data averaged from 1 to 20 January 2020.

Figure 13 .
Figure13.Same as Figure8, except between latitudes 50°S and 20°S for GOLD data averaged from 1 to 20 January 2020.

Figure 14 .
Figure 14.Same as Figure 9, except between latitudes 50°S and 20°S for GOLD data averaged from 1 to 20 January 2020.

Figure 15 .
Figure 15.Residuals from the zonal mean for the dusk + dawn sums and the reconstructed semidiurnal non-migrating tidal field from the retrieved SE2 + SW1 + S0 in neutral temperature and ΣO/N 2 .Shown between 50°S and 20°S latitude and longitudes within the GOLD field-of-regard where conditions for the dusk + dawn sums are satisfied.From GOLD data averaged between days 300-320 in 2018, (a)-(d), and days 1-20 in 2019, (a')-(d').These two periods are representative of before and during the 2019 January Sudden Stratospheric Warming, respectively.

Figure 16 .
Figure 16.Correlation coefficients between the dusk + dawn residuals and the reconstructed total semidiurnal non-migrating tide based on the retrieved SE2 + SW1 + S0 before (filled circles and solid lines) and during (unfilled triangles and dotted lines) the January 2019 Sudden Stratospheric Warming.Temperature in blue, ΣO/N 2 in red.

Figure 18 .
Figure 18.Same as Figure 11, except comparing SE2, SW1, and S0 phases before and during the January 2019 Sudden Stratospheric Warming (SSW).The solid lines and filled circles indicate before the SSW.The dotted lines and unfilled triangles indicate during the SSW.

Table 1
Summary of the Median Amplitude Errors in Percent and a Qualitative Assessment of the Phase Accuracy for the SE2, SW1, and S0 Tides in Each of the Four Scenarios Described in the Text