The FORMOSAT-3/COSMIC constellation consists of six low-Earth orbiting (LEO) microsatellites which orbit the Earth at an initial altitude of 800 km. The satellites perform RO measurement in both the neutral atmosphere and the ionosphere . During an occultation, signals of rising or setting GPS satellites are received by a LEO satellite. While the signals pass the Earth's atmosphere they are modified by the atmospheric conditions, in particular the ionospheric electron density, which causes refraction and degradation of the GPS waves. This can be utilized to obtain information about the ionosphere and the neutral atmosphere. More detailed information on the principles of the RO technique is given by and .

The method to derive Es information from RO signals was described in . In brief, for our investigations, the signal-to-noise ratio (SNR) profiles of the GPS L1 phase measurements (UCAR, 2018) are used. The SNR is very sensitive to vertical variations of the electron density, and these occur within an Es layer. These vertically localized electron density variations lead to phase fluctuations of the GPS signal, which can be observed as changes in the received signal strength . In order to avoid influences from the different basic signal power values on the further data analysis, every SNR profile is normalized first. In the case of absence of ionospheric disturbances, the SNR value is almost constant at altitudes above 35 km. The SNR standard deviation profile is considered to be disturbed when it exceeds an empirically found threshold of 0.2. If large standard deviation values are concentrated within a thin layer of less than 10 km vertical extent, we assume that the respective SNR profile includes the signature of an Es layer. The height where the SNR value deviates most from the mean of the SNR profile is considered the altitude of the Es layer. This has been validated by comparisons with ionosonde Es observations .

Figure  shows 2007–2016 mean zonal mean Es OR. Sporadic E OR were calculated as the number of Es within a 5∘ latitude and 10 km height window, divided by the number of ROs in the respective latitude window. Figure  shows seasonal mean OR for December–February (DJF), March–May (MAM), June–August (JJA), and September–November (SON). The distributions are similar to those shown by obtained from a more limited dataset. Maximum OR are found at altitudes slightly below 100 km for low latitudes, but above 100 km for mid to high latitudes except for autumn. OR maximize in summer, which is thought to be due to increased meteor influx during that season . The summer maximum is more pronounced in the Northern Hemisphere, which is due to the South Atlantic Anomaly and the weaker magnetic field there e.g.,, so that Southern Hemisphere summer zonal mean OR are smaller than Northern Hemisphere ones. Near the Equator, Es OR are small, owing to the horizontal magnetic field at the magnetic Equator, which does not allow the electrons to follow the vertically moving ions e.g.,.

Although OR maximize in summer, they are also found in the winter hemisphere at low latitudes. The winter low-latitude maximum is clearly visible in the Northern Hemisphere (NH; Fig. a). It is also seen in the Southern Hemisphere (SH) winter (Fig. c), but the OR values are lower than those in the NH winter, again because of the South Atlantic Anomaly, where the Es OR are low. Regarding the MAM and SON distributions (Fig. b, d), they are both not really symmetric about the Equator, but the shapes of the respective NH and SH distributions are different within one panel. On the other hand, during both seasons the distributions of the respective spring hemispheres (MAM-NH and SON-SH) reveal a similar shape, and this is also true of the respective autumn hemispheres (MAM-SH and SON-NH). The SON-SH values are lower than the MAM-NH values due to the South Atlantic Anomaly. Interestingly, the MAM-SH values are larger than their counterparts in the SON-NH, which might be due to an Es hotspot over Indonesia and especially Australia that is visible in the global distribution, e.g., presented by .

At Collm (51.3∘ N, 13.0∘ E), a SKiYMET meteor radar has been operated at 36.2 MHz since summer 2004. The radar operates in an all-sky configuration, and the main parameters observed are the MLT radial winds determined from the Doppler shift of individual meteor trails. Details of the radar system and the radial wind determination principle can be found in , , and . During 2015 the radar were upgraded by increasing the peak power and replacing the Yagi antennas with crossed dipoles. The transmit frequency is still the same . The individual meteor trail reflection heights vary between about 75 and 110 km, with a maximum meteor count rate at an altitude slightly below 90 km e.g.,. The data are binned here in six different non-overlapping height gates centered at 82, 85, 88, 91, 94, and 98 km. The hourly mean reflection height may slightly deviate from the nominal heights due to the uneven height distribution of meteors within each gate . Individual radial winds calculated from the meteors are collected to form hourly mean horizontal winds using a least-squares fit of the horizontal wind components to the raw data under the assumption that vertical winds are small . Hourly values of the zonal wind shear are calculated from adjacent height gates as in and . The reference height for shear values is attributed to the center between the nominal heights of the wind values, except for the upper height gate where the real mean height is close to 97 km, so that the uppermost wind shear level is placed at 95.5 km.

Two examples of the diurnal zonal wind and wind shear variation in the MLT over Collm are shown in Fig. . In the left panels, height–time cross sections of 2007–2016 mean (a) DJF and (c) JJA mean diurnal zonal winds and wind shears S(t) over Collm are shown. In winter, maximum wind values exceed 70 (50) m s-1 in the eastward (westward) direction, while the zonal wind shear maximizes at more than -8 (+6) m s-1 km-1, similar to the values shown by . We note that during two time intervals per day, strong negative wind shear prevails, which according to the wind shear theory supports Es formation. During summer (Fig. c), the background wind shear is strongly positive, and only at the upper height gate do short time intervals of negative wind shear occur. Therefore, at altitudes below ∼93 km, Es formation is much less supported by the mean wind distribution, which is consistent with the greater mean Es heights in summer than in winter at mid latitudes (see Fig. ).

Clearly, the main contribution to zonal wind and wind shear variability, in particular during winter, is due to the SDT and partly DT. To visualize the higher-frequency wind shear variations, in the right panels of Fig. , the diurnal zonal wind shear values S(t) at 92.5 km are shown for (b) DJF and (d) JJA, together with a modeled least-squares fit SMod(t) including diurnal mean S0, as well as the 8, 12, and 24 h components: 2SMod(t)=S0+∑i=13aisin⁡2πPit+bicos⁡2πPit, with t as the time and Pi as the above-mentioned periods, and the coefficients ai and bi being determined by minimizing ∑(S(t)-SMod(t))2. The amplitudes Ai and phases Ti of the wind shear are calculated as 3Ai=ai2+bi2,Ti=Pi2πarctan⁡aibi+Pi2. Note that the phases are defined here as the time of maximum negative wind shear, so that a Pi/2 term is added to the right-hand side of the second part of Eq. (). The residuals S(t)-SMod(t), multiplied by a factor of 5 for better visibility, are added as blue line in the right panels of Fig. . Obviously, there is a quarterdiurnal signature during winter (Fig. b) and at this altitude level. Amplitudes and phases of this variation are calculated via a least-squares fit similar to Eqs. () and (), but in addition including the QDT period P4=6 h in the analysis. In contrast to winter, there is no significant 6 h component of the wind shear visible during summer (Fig. d), which is consistent with small QDT amplitudes in summer, as was shown by .

Figure  shows the 2007–2016 mean seasonal mean Collm residual zonal wind shear after removing the mean shear and the 8, 12, and 24 h components for (a) DJF, (b) MAM, (c) JJA, and (d) SON. The 6 h phases, defined as the time of maximum negative wind shear according to Eq. (), are added. The QDT at Collm is relatively strong in winter, but very weak and insignificant in summer, and consequently as in Fig. d the 6 h signal is not visible for JJA. For the other seasons the QDT shear tends to increase with altitude, although in MAM the QDT signal is not the major one at the upper height gate. In SON, the QDT seasonal means consist of a superposition of summer, transition, and winter oscillations seetheir Figs. 3 and 6. As a consequence, at the upper height gates a clear QDT signature is visible, but these are not connected with the QDT at the lower heights, also leading to a break in the vertical phase change.

We use the nonlinear Middle and Upper Atmosphere Model (MUAM) to investigate the QDT with wavenumber 4. MUAM is a three-dimensional mechanistic primitive equation model based on the COMMA-LIM model. The more recent version of the model, MUAM, is documented by , , and . MUAM extends from the surface (1000 hPa) to the lower thermosphere, while the lower 30 km zonal mean temperatures are nudged with monthly mean 2000–2010 mean ERA-Interim reanalyses of zonal mean temperature. This ensures that the zonal mean dynamics of the lower atmosphere is close to the reanalyses, while waves are allowed to form anyway. The model has a horizontal resolution of 5×5.625∘ and a vertical resolution of 2842 km in logarithmic pressure coordinates with a constant scale height of H=7 km. Parameterizations of gravity waves, of solar and infrared radiation, as well as of several ionospheric effects are included. The latter, however, are only represented based on simple empirical distributions.

Solar heating in the middle atmosphere is parameterized following . This considers heating due to the most important gases such as H2O, CO2, ozone, O2, and N2. Note that these gases are taken as zonal means, different from other versions of MUAM e.g.,, so that mainly migrating tides are forced through the solar heating. Monthly mean zonal mean ozone fields up to 50 km altitude are taken from the Stratosphere-troposphere Processes And their Role in Climate project SPARC;, and an exponential decrease in ozone is applied above 50 km. Monthly volume mixing ratios for CO2 have been chosen according to measurements from Mauna Loa Observatory for the year 2005 e.g., 378 ppm for January;, and the mixing ratio is assumed constant across latitudes and longitudes. CO2 mixing ratios are taken as constant with height until 87.5 km, and then decrease exponentially. Chemical heating due to recombination of O2 and O , and heating due to extreme ultraviolet radiation, are added. This is described in more detail by .

Gravity waves in the middle atmosphere are calculated by a linear Lindzen-type parameterization based on and updated as described by and . Due to the fact that this parameterization does not account for gravity waves in the thermosphere, it is coupled with a modified parameterization following , and both parameterizations are connected via the eddy diffusion coefficient. To avoid large interactions between both parameterizations, we limit the phase speeds of the Lindzen-type scheme to 5–30 m s-1, while the Yiǧit scheme covers larger phase speeds between 35 and 105 m s-1. This way, the Yiǧit parameterization mainly affects the thermosphere, while the Lindzen-type parameterization affects the stratosphere and mesosphere. Overlaps between both parameterizations are small and the contributions of both routines to the tendency terms can be simply summed up .

In the configuration used here, the model incorporates a spin-up of 120 model days. Within that time, the heating rates are zonally averaged and therefore build up a background climatology without any tidal forcing. Note that we do not explicitly excite any kind of waves at the lower boundary during the whole simulation. In the following 90 model days, the heating rates are allowed to be zonally variable, and through their diurnal variability tides start to propagate. In this model version, the sun's zenith angle is kept fixed to the middle of the respective month. The last 30 model days are analyzed and the results of tidal analysis are presented here. Since there is no change in any forcing, the day-to-day variability is negligibly small.

Figure  shows the annual cycle of the simulated monthly 6 h zonal wind amplitudes at ∼101 km altitude, which is close to the height of maximum Es OR. QDT amplitudes for other model heights are shown in the Appendix. The QDT zonal wind amplitudes are larger in winter than in summer, which have also been observed . Winter maximums are seen at higher and mid latitudes, and are also predicted by . Winter QDT amplitudes in both hemispheres tend to maximize in early and late winter, with a relative minimum during solstice. There is also a tendency for the spring maximum at mid latitudes that was reported by . We also note maximums at 30–40∘, as reported by . Note that, since the solar forcing is parameterized using zonal mean climatologies of minor species, the MUAM amplitudes only show migrating components, while the radar observations deliver the total amplitude. However, from amplitude and phase comparisons between two stations, concluded that the major contribution to the QDT observed by radar is mainly due to the migrating tide, and comparison between model results and radar is justified.

Here, we compare the diurnal cycle of Es OR at the latitude of Collm with the wind shear observed by the MR with respect to the QDT signature, similar to the approach of and for the 12 and 8 h components, respectively. Similarly to Fig. , OR have been calculated as the number of Es divided by the number of RO, but here a latitude window of 10∘ centered at 51∘ N was chosen, and the data were sampled in hourly bins according to local time. The phases of nonmigrating tides at different longitudes are different, so that the nonmigrating components average out if we sample irrespective of longitude and therefore effectively average over longitudes. Therefore, only the migrating tidal components contribute to the diurnal cycle in our analyses. As in Fig. , the data refer to 10 km height gates. The diurnal cycles for four seasons are shown in Fig. . We note the downward propagation of Es OR signatures, which are dominated by the SDT and, to a lesser degree, by a diurnal variation see, e.g.,. The strongest amplitudes are seen in summer, while minimum amplitudes are found in winter, which is due to the overall seasonal cycle of OR at higher mid latitudes: see Fig. .

The residuals of Es OR after removing daily mean and 8, 12, and 24 h components, i.e., calculated in the same manner as the wind shear residuals in Fig. , are shown in Fig. . The QDT phases are calculated according to the right-hand side of the second part of Eq. (), but without the P4/2 term, since we are interested in the times of maximum Es. The zonal wind shear phases shown in Fig.  are also added in the lower parts of the respective panels of Fig. . Again, solid symbols indicate that the amplitudes are significant at the 5 % level according to a t-test. The QDT in Es is strongest in summer, as are the overall OR (see Fig. ). Vertical phase gradients in summer are large, but smaller in winter and autumn, which is also the case with the QDT in neutral winds . In autumn, significant QDT amplitudes are only found in a small height range at 95–100 km. Generally, the QDT in Es disappears below 90–95 km. In all seasons, negative wind shear phases and Es phases agree within their standard deviations in the upper two radar height gates, i.e., above 90 km. This is also the case in summer, although both wind shear and Es amplitudes are not significant then. This indicates that the QDT in Es actually forms at the nodes of the negative QDT wind shear component. Therefore the Es formation process, which is responsible for the strong 12 h component in Es, also acts for the much weaker QDT.

The Es QDT amplitudes are not necessarily directly related to the respective wind shear amplitudes, because Es intensities and OR depend on wind shear, but also on the ionization of metallic ions, and the latter are thought to exhibit a seasonal cycle owing to the variability of meteor influx . Therefore, e.g., at mid latitudes, the largest Es OR amplitudes are found in summer (see Fig. ), while the largest QDT in the neutral wind is seen during winter (Figs.  and ). Further, wind shear theory includes the influence of the horizontal magnetic field leading to a latitudinal dependence of Es OR, as well as to anomalies like the South Atlantic Anomaly. To take this into account when showing the global distribution of Es QDT amplitudes, following we calculated relative amplitudes A4,r=A4/ OR. We show their global distribution at an altitude of 101 km in Fig. a. The zonal wind shear QDT at the nearest model level at 100.9 km height, calculated from MUAM monthly simulations, is shown in Fig. b. The respective distributions at other altitudes are shown in the Appendix. Both parameters in Fig.  show a corresponding maximum in the winter higher mid latitudes in both hemispheres. In the Southern Hemisphere, the QDT in both Es and wind shear is small during boreal winter. In the Northern Hemisphere summer, a further QDT maximum in Es is found, which is, however, only weakly represented in the wind shear. During the equinoxes, corresponding maximums are again found at lower and mid latitudes, especially in the Northern Hemisphere. Overall, there is a striking similarity between Es and wind shear QDT, in particular taking into account that the wind shear here is not taken from observations, but from numerical modeling with tides being forced self-consistently and not based on observed distributions.

Figure  shows in the left column seasonal mean relative Es amplitudes as latitude–height plots, while the corresponding modeled QDT shear amplitudes are presented in the right column. We show values only between 90 and 110 km, because below and above the OR become small (Figs.  and ), and therefore the A4,r distribution tends to become more irregular. There is an overall close correspondence between Es and wind shear during the solstices (Fig. a, b, e, f). Wind shear amplitudes then are large in the mid- to high-latitude winter hemisphere, with large values down to the upper mesosphere, while corresponding Es amplitudes are large there, too. A secondary maximum in the lower thermosphere winter near 30∘ is also seen in both Es and wind shear. In JJA, there is also a qualitatively corresponding maximum in the summer hemisphere. This is also visible in the DJF wind shear, however, only very weakly expressed in DJF Es. During boreal spring (Fig. b, c), there are corresponding maximums in Es and wind shear QDT at higher mid latitudes, peaking at 60∘ S and at 45–60∘ N. There is also an indication of a joint maximum close to the Equator. The latter, however, is probably coincidental, since the wind shear theory according to Eq. () does not hold for magnetic latitudes below about 20∘. There is a modeled QDT wind shear maximum near 30∘ S during MAM, which is not visible in Es, similar to the situation during austral spring (Fig. g, f). Generally, the correspondence between wind shear and Es QDT is weakest during SON. This may partly be due to the seasonal cycle and the fact that seasonal averages are taken from monthly data that may be very different (e.g., a clear minimum/maximum in September/November at higher northern mid latitudes; see Fig. ). Also, the overall correspondence between Es and wind shear QDT mainly refers to the position of maximums and minimums, while the absolute values may differ. Still, however, there is a correspondence visible between the global distribution of the QDT in Es and the one seen in wind shear, indicating the presence of the wind shear mechanism also for the QDT.