## 4 Comparisons of Tidal Characteristics with Theory  
While several modes can be observed for a single latitude (Figure 2), it is not always obvious how to identify the values corresponding to migrating modes as expected from theory (as provided by the GSWM model). The closest values are selected as the migrating  
Figure 2: Some examples of histograms for the distribution of diurnal tidal temperature anomalies from ERA5 meteorological analyses over all longitude grid points for three different cases. Distributions are expressed in density probability and the number of longitude (population) steps over the 144 beams. Sampling the full longitudes: (**Top**), at 32 km in altitude on the latitudinal band around 10\({}^{\circ}\) N; (**Middle**), at 66 km in altitude on the latitudinal band around 60\({}^{\circ}\) N; (**Bottom**), at 64 in km altitude on the latitudinal band around 80\({}^{\circ}\) N.  
mode in spite of the fact that sometimes it does not correspond to the strongest mode. However, good agreements can be reported between the tides retrieved from ERA5 analyses using the methodology described in Section 3.2 and the GSMW [20] monthly mean tidal characteristics. Comparisons of the phases show good agreements within error bars for both modes and for low and higher latitudes in the altitude range up to 60 km. Above this altitude, times of maximum differ and are out of phase by half a period (Figures 3 and 4). Over the equator, tidal amplitudes are similar to the GSM with, however, different vertical behavior (Figure 3 top) around 50 km for the semi-diurnal mode and above 60 km for the diurnal mode. At mid-latitude (Figure 4 top), results similar to past analyses show amplitudes nearly half [21] the expected ones up to 60 km, showing larger observed amplitudes than expected. Phases are in good agreement between observations and simulations with some drifts with altitude probably associated with the damping of the variability in the mesosphere [32].  
Tidal amplitudes as a function of latitude and altitude correspond to the well-known butterfly-wing distribution (Figure 5). A different behavior is obtained at the equator with a maximum around the stratopause and around the equatorial zone in a \(\pm\) 15\({}^{\circ}\) band. From this figure, conclusions similar to the ones associated with the vertical profiles above the equator (Figures 3 and 4) can be drawn. Larger amplitudes than the ones predicted by the model are extracted from the ERA5 series. Above 60 km, retrieved responses appear to be more inconsistent. Differences can be also observed for phases between mid-latitude and the equator. Amplitudes for the semi-diurnal mode are similar in magnitude, but the retrieved space pattern exhibits more noise (Figure 6). Also, the large increase in amplitude simulated at the equatorial latitude above 60 km reported by GSWM (Figure 5) is not captured by ERA5.  
Figure 3: Vertical profiles of amplitudes expressed in temperature (**top**) and phases expressed in hours (**bottom**) of the semi-diurnal (**left**) and diurnal (**right**) tides extracted from ERA5 meteorological analyses at the equator (blue lines) compared with the characteristics given by the GSMM (black lines). Horizontal bars for the phase estimates correspond to the uncertainties.