Compared to solar thermal tides, the amplitudes of atmospheric lunar tides are small. In spite of this, emphasized that atmospheric lunar tides are attractive for theoretical and observational studies because their frequencies and forcing are better determined than for any other atmospheric waves. The main lunar tide is the semidiurnal tide M2 which has a period of 12.42 h and a zonal wavenumber 2. The lunar tide is maximal in the tropics where its surface air pressure variation is between 5 and 10 Pa . This pressure variation could be transformed to an oscillation of the geopotential height of a fixed pressure level close to the ground. Assuming a pressure gradient of 9 Pam-1, the geopotential height amplitude would be about 0.5 to 1.1 m at the ground level. The simulation of has values of about 0.4 m as oscillation of geopotential height of the pressure level at ground.

Similar to atmospheric gravity waves, the lunar tide strongly amplifies during its way upward to the lower thermosphere. Recently, it has been recognized that the lunar tide is amplified in addition by a factor 2 or 3 during sudden stratospheric warmings (SSWs), so that the lunar tide should be considered for a better modeling and understanding of space weather . emphasized that the lunar tide is only amplified during SSWs that are accompanied by strong polar night jet oscillations (PJO).

showed that the amplification of the lunar tide by the perigee transit of the Moon can be utilized as a signal to retrieve the average travel time of the lunar tide from the surface to the ionosphere. With a delay of about 3 d the lunar tide arrives in the dynamo region where it induces ionospheric electric currents. These currents are associated with magnetic field variations which can be observed by ground-based magnetometers. reported about big L days where the lunar tidal variation in horizontal intensity of the geomagnetic field is stronger than usual. found a close relationship between the big L days in ionospheric total electron content (TEC) and the occurrence of SSWs. The spatio-temporal variations of TEC are monitored by ground stations of the Global Navigation Satellite System (GNSS). The lunar tide-induced TEC variations are mainly due to electrodynamic lifting of the equatorial ionospheric plasma as a consequence of the electric field variations in the dynamo region . A relationship between geomagnetic lunar tides and SSWs was previously detected in magnetometer data by who found that the M2 amplitude during SSW events is approximately 3 times as large as that for non-SSW winters.

Observations of the lunar tide in the middle atmosphere have been reported for the parameters temperature and horizontal wind, and articles about numerical simulations of lunar tides in the middle atmosphere are usually focusing on these parameters . An exception is the study of who performed a numerical simulation of the lunar tide from the ground to 100 km height and who presented all the simulations results for the lunar semidiurnal tide in geopotential height. Thus, the results of the present observational study of Aura/MLS can be compared to the simulation by .

Geller showed that the lunar tidal amplitude in geopotential height at the equator is about 220 m at 90 km height in January and about 99 m in July. This amplitude variation is due to a change of the vertical thermal structure which favors a more propagating wave (smaller vertical wavelength) in January and a more standing wave (larger vertical wavelength) in July. The phase progression with height is a bit larger in January than in July . However, the phase profiles are relatively steep in both months favouring a (2,2) wave mode of the lunar tide. (2,2) refers to the tidal wave mode (m,n) where m is the zonal wave number and n is the meridional mode number. The latitude structure is described by the Hough function with n–m internal nodes where the tidal amplitude is zero. The (2,2) mode is symmetric with a maximum of the tidal amplitude at the equator and a decrease of the amplitude towards the poles. While the (2,2) mode propagates vertically with a long wavelength of about 300 km, the (2,4) mode has a shorter vertical wavelength. found that the (2,2) mode of the lunar tide is dominant below 70 km height. Above, the (2,4) wave mode with a short vertical wavelength of 40–60 km increases in amplitude. Observational studies from ground-based radars often identify a (2,4) mode of the lunar tide with vertical wavelengths less than 60 km in the mesosphere . However, observations by show both: in some months vertical wavelengths larger than 100 km and in other months vertical wavelengths less than 60 km.

The lunar tide in temperature is about 0.008 K at ground in Batavia and increases to about 0.6 K in 90 km height above the equator (TIMED/SABER observation in ). reported a lunar tidal amplitude of 0.4 K in 80 km height in NIMBUS satellite data. The simulation of showed an amplitude of 0.45 K in July above the equator and 1.7 K in January. The small value of the lunar temperature tidal amplitude at ground observed by was confirmed in a new study by who found a lunar tidal amplitude of 0.007 K observed by 38 buoys across the tropical Pacific and Atlantic. The phase of the lunar temperature tide is quite close to the phase of the lunar tide in geopotential height . The lunar temperature tide has in first order an adiabatic relationship to the lunar tide in pressure (or geopotential height) .

The present study will focus on the lunar tide in geopotential height in the mesosphere observed by Aura/MLS from 2004 to 2021. The reason is that the lunar tidal signal is significant at mesospheric heights and we will see that the lunar tidal signal in geopotential height is clearer than in temperature. The reason is possibly that the pressure perturbation by the gravitational forcing of the Moon is the primary perturbation and the temperature perturbation is a consequence of the pressure perturbation. It seems that the present study is the first observational investigation of the lunar tide in geopotential height (or pressure) in the middle atmosphere. The study describes the Aura/MLS mission and the data analysis in Sect. 2. The results are presented in Sect. 3. A discussion of the results is in Sect. 4, and the conclusions are given in Sect. 5.

The FFT amplitude spectrum of the oscillations of geopotential height z of a fixed pressure level (0.0046 hPa) in about 82 km height is computed for the Aura/MLS measurements in the equatorial latitude belt (10° S to 10° N) from 2004 to 2021. Figure  shows a significant spectral peak at the expected Moon period (1/half lunar month indicated by the red line). The z amplitude of the lunar tidal variation reaches a value of about 38 m which is high above the spectrum continuum of 1 m or less. In addition, there are significant spectral peaks at the frequency of the annual oscillation and its harmonics (green lines in Fig. ).

The FFT amplitude spectrum of the oscillations of temperature T of a fixed pressure level (0.0046 hPa) in about 82 km height is depicted in Fig. . The spectral line at the Moon period is still significant and reaches an amplitude of about 0.35 K while the spectrum continuum is at 0.15 K or less. Compared to the lunar z peak in Fig. , one can say that the lunar tidal peak in temperature in Fig.  is not so prevailing as that in geopotential height. Figure  shows further significant spectral peaks at the periods 2.3, 2.7, 3.2, 4.0, 5.3, and 8.0 d. These planetary wave-like oscillations also occur in the geopotential height fluctuations in Fig. . It is not clear why these oscillations of the equatorial zonal mean series appear at these discrete frequencies. In the literature, there are only a few reports about zonal mean oscillations in the middle atmosphere or planetary waves with zonal wave number zero . A further investigation of these oscillations in Figs.  and  is beyond the scope of the present study which is focused on the lunar tide.

Using the FFT spectra at each pressure level, the vertical profile of the lunar tidal amplitude in geopotential height z is derived and shown in Fig. .

The z amplitude of the lunar tide in the Aura/MLS observations goes from 0.2 m at 16.6 km height to 55 m at 90.4 km height. The simulation of showed z amplitudes of the lunar tide from 0.55 m at 15 km height to 99 m at 90 km height in July while these values increase to 0.71 m at 15 km and 220 m at 90 km height in January. Thus, the mean values of the lunar tide in geopotential height observed by Aura/MLS are roughly smaller by a factor of about 2–3 than the lunar tide in geopotential height simulated by . I assumed here that the mean of the January and July value of the z amplitude from can be roughly compared to the annual mean of the z amplitude from Aura/MLS.

The vertical profile of the lunar tidal amplitude in temperature T is shown in Fig. . The T amplitude of the lunar tide in the Aura/MLS observations goes from 0.02 K at 16.6 km height to 0.68 K at 90.4 km height. The simulation of showed T amplitudes of the lunar tide from 0.01 K at 15 km height to 1.1 K at 90 km height in July while these values are 0.01 K at 15 km and 3.0 K at 90 km height in January. Again, the simulated lunar tidal amplitudes in the upper mesosphere at 90 km height are roughly stronger by a factor 2–3 than the Aura observations of the mean lunar tide. The mean lunar tidal amplitude of about 0.68 K at 90.4 km height observed by Aura/MLS is in a good agreement with the temperature lunar tide in TIMED/SABER observations which are about 0.6 K at 90 km height . Also, the small value of 0.02 K in the lower stratosphere roughly fits to the 0.007 K amplitude of the lunar tide at the surface assuming an increase of the lunar tidal amplitude with height.

The vertical profiles of lunar tidal phase in geopotential height (red line) and temperature (blue line) are shown in Fig. . Similar to , both phase profiles are close together and slowly increase with height (as for a gravity wave with downward phase progression). The vertical wavelength is roughly around 500 km in the mesosphere but in case of the temperature phase profile a shorter vertical wavelength of about 80 km is present from 77 to 95 km. The simulation of shows vertical phase gradients at mesospheric heights which indicate a vertical wavelength of about 500 km.

For the derivation of the seasonal and interannual variations of the lunar tidal amplitude in geopotential height, digital filtering was applied. Figure  shows the time series of the lunar z amplitude in about 82 km height. In agreement with the simulation by , the lunar tide is about 2–3 times stronger in January than in July. The amplitude is around 80 m in January and around 30 m in July. A maximum of 130 m is achieved in January 2013. Larger amplitudes could be reached if the number of filter coefficients would be reduced so that a faster response of the digital filter to the seasonal variation would be obtained. However, the penalty of a faster response time of the filter would be that the separation of the lunar tide from other planetary wave-like oscillations would be less reliable.

The Aura/MLS observations also permit the study of the latitudinal dependence of the lunar tidal amplitude. Here, Fig.  only shows the mean lunar tidal amplitude in geopotential height z in about 82 km height as derived from the FFT spectra at latitudinal belts from 80° S to 80° N where the belts are 10° wide in latitude with a spacing of 10°. Figure  shows that the lunar tidal amplitude z is maximal at 0–10° N. This equatorial maximum decreases to the half at about 30° N (or 30° S). This shape (full width at half maximum) well agrees with the lunar tide simulation of (Fig. 10 in Geller's article). The only differences are the slight increase of the amplitude at high latitudes in the Aura/MLS observations of Fig.  and the slight interhemispheric asymmetry of the curve (maximum is shifted by 5° to the Northern Hemisphere. Both effects could be due to the amplification of lunar tides by SSWs (with strong PJO) which are more common in the northern winter hemisphere.

Finally, the climatology of the lunar tidal amplitude is derived by digital filtering as function of latitude (Fig. ). The seasonal variation is obtained by bandpass-filtering of the time series at the period of a half lunar month and by sorting the 17 years-long amplitude envelope as function of Day of Year. The lunar tide is strong at high latitudes in polar winter. It is not certain if this effect is due to the lunar tide or to stronger planetary wave-like oscillations of the winter hemisphere at high latitudes. From 0 to 20° N, a maximal tidal amplitude of about 80 m is present in January and February (in agreement with the time series in Fig. ). This result is in agreement with numerous reports on the lunar tide and the geomagnetic lunar tide which is stronger in January than in July . A new result might be the maxima of the lunar tide at March equinox at mid-latitudes. In the past, there was a study about the geomagnetic lunar tide which emphasized that the lunar tide was amplified around the December solstice and the equinoxes .

This study evaluated for the first time the signal of the lunar tide in Aura/MLS data of the middle atmosphere. In addition, it was the first time that the lunar tide in geopotential height was analysed in observations of the middle atmosphere. The results showed that the Aura/MLS data are appropriate and convenient for the study of the lunar tide. At mesospheric heights, significant spectral peaks are observed at a period of a half lunar month, particularly in the spectra of geopotential height but also in temperature. This lunar semimonthly variation of the lunar tide is due to the near-polar, sun-synchronous orbit of the Aura satellite (Fig. ).

Generally, it seems that the geopotential height measurements are more valuable for investigations of the lunar tide than the temperature observations because the lunar tidal signal is stronger in geopotential height than in temperature. It also can be argued that the geopotential height perturbation is the primary effect of the lunar tide and the temperature lunar tide arises from the adiabatic relationship to the pressure lunar tide .

The vertical profiles of the lunar tides in geopotential height and temperature of the Aura/MLS observations are smaller by a factor 2–3 compared to the simulation of . The vertical gradient of the observed lunar tidal amplitudes are similar to the simulated gradient in . The amplitude of the temperature lunar tide at mesospheric heights derived from Aura/MLS well agrees with the temperature lunar tide of TIMED/SABER observations .

The Aura/MLS observations at the equator also showed that the lunar tide in January is about 2–3 times stronger than in July. This agrees with the simulation of . In addition, observations of the geomagnetic lunar tide and the lunar tide in GNSS TEC showed that the lunar tide is more amplified in January than in July. Particularly, the SSWs in northern hemispheric winter amplify the lunar tide by a factor of 2–3 at least when strong PJOs are present .

The phase profile of geopotential height is close to the phase profile in temperature. The phase profiles indicate a long vertical wavelength of about 500 km in the mesosphere, in agreement with . A long vertical wavelength may indicate a prevailing (2,2) mode of the lunar tide. Simulations of showed that the (2,2) mode is dominant at altitudes below 70 km. The phase profile of the temperature lunar tide of Aura/MLS shows more variations on short vertical scales than that in geopotential height. In the upper mesosphere, a vertical wavelength of about 80 km is present in the temperature data. The observed increase of the phase with height is in agreement with the downward phase progression of a gravity wave with upward energy propagation.

A convincing agreement between simulation and Aura/MLS is obtained for the latitudinal dependence of the lunar tide in geopotential height. The amplitude peaks a bit northward of the equator and the full width at half maximum is about 60°. Differences between the simulation and the observations are only at high latitudes in winter where relatively large amplitudes are obtained (Fig. ). According to the simulation the amplitudes of the lunar tide are very small at high latitudes. However, ground-based radar observations showed that the lunar tide at high latitudes in winter is a relevant phenomenon . On the other hand, it cannot be excluded that the climatology of the lunar tidal amplitude in Fig.  is partly contaminated by planetary wave-like oscillations which the digital filter did not separate from the lunar tidal signal. In addition, the annual mean of the lunar tide in Fig.  is just around 15 m at northern polar latitudes. This seems to be not consistent with the large tidal amplitude values (up to 148 m) at northern polar latitudes in winter in Fig. . The retrieval of the annual mean of the lunar tide is certainly more reliable than the retrieval of the monthly means of the lunar tide, since the lunar tide signal can be better separated from atmospheric noise in case of the FFT of the whole time series (Fig. ).

The climatology of the lunar tide in Fig.  at low latitudes (0–20° N) agrees with the well-known observational fact that the lunar tide is stronger in January than in July. The simulation of also showed this characteristic which is due to a difference in the vertical thermal structure of the middle atmosphere. A change in the temperature profile can shift the resonance period of the atmosphere to the frequency of the lunar tide, so that an amplification of the lunar tide occurs .

For the first time, the lunar tide in geopotential height was observed by Aura/MLS. The observations were compared to the simulation of the lunar tide by . The observed mean lunar tide is up to 55 m (geopotential height amplitude) in the mesosphere while the simulation shows amplitudes between 99 m in July and 220 m in January at 90 km height. Thus, the observed amplitude is in average about 3 times smaller than predicted by the simulation. It can be that the boundary conditions and the vertical thermal structure of the model atmosphere have to be adjusted for a better agreement.

The amplitude of the mesospheric lunar temperature tide of Aura/MLS is about 0.68 K and agrees well with those of TIMED/SABER which is about 0.6 K at 90 km height .

Generally, the vertical profiles of the lunar tidal amplitudes in Aura/MLS data are reasonable down to the tropical tropopause. That means, comparable small values are retrieved in the lower stratosphere as reported by or observed at ground . The vertical phase gradients suggest a (2,2) mode of the lunar tide with a vertical wavelength of about 500 km, in agreement with simulation results of and .

The latitudinal dependence of the mesospheric lunar tide in geopotential height agrees well with the simulation of having a maximum at the equator and a reduction to the half of the maximum at 30° N. The climatology shows that the equatorial lunar tidal amplitude is stronger in January than in July. At high latitudes in winter, the lunar tide in geopotential height was up to 148 m (Fig. ). However, a contamination of the lunar tidal signal by planetary wave-like oscillations cannot be excluded.

Generally, the present study showed that the lunar tide might be best studied in the parameter geopotential height. Unfortunately, there was only one published article which showed a simulated lunar tide in geopotential height . The Aura/MLS observations of the lunar tide are valuable for future simulations of the lunar tide.