In this study we mainly use Rayleigh–Mie–Raman (RMR) lidar systems which are located at the ALOMAR Observatory near Andenes in the Norwegian Arctic (69∘ N, 16∘ E) and at Kühlungsborn in Germany (54∘ N, 11∘ E). The climatology of each station comprises 10 years of measurements between 2002 and 2012 in winter between December and February. For the calculation of the temperature from the RMR lidar raw data we use the hydrostatic density integration as described, for example, by , based on the backscatter at 532 nm wavelength.

The ALOMAR RMR lidar is able to obtain temperatures from above the stratospheric aerosol layer (i.e. above ∼22–34 km depending on actual conditions) up to ∼90 km during both darkness and daylight conditions. The lidar is actually a twin lidar system with two independent lasers (∼20W average power at 532nm) and two tiltable telescopes of 1.8m diameter. Typical integration times for this study are 1h at 2.5km vertical resolution. Further details about the ALOMAR RMR lidar are described by and .

The RMR lidar at Kühlungsborn is similar to the system at ALOMAR. An additional detection channel for N2 vibrational Raman backscatter at 608nm allows for the correction of stratospheric aerosol effects. By this, the lowest altitude bins for Rayleigh temperature retrieval can be set to 22km. The ∼20W average power at 532nm and the four receiver telescopes of 60cm diameter each allow for 1km vertical resolution after 1h integration. Typically, the upper altitude limit is above 85km altitude. In this study the data are limited to nighttime conditions. The RMR lidar data at Kühlungsborn are complemented by temperature profiles obtained by a collocated potassium resonance lidar. This technique provides absolute temperatures between ∼85 and 105km by probing the Doppler broadening of the D1 resonance line of potassium atoms that exist in this altitude range. These data are also used as a start value for the hydrostatic temperature retrieval for the Kühlungsborn RMR lidar, while the calculation for the ALOMAR RMR lidar is initialized with climatological data. Further details of the temperature lidars at Kühlungsborn are described by and .

Note that in this study only nighttime measurements (Andenes: 134, Kühlungsborn: 71) with at least 3 h of measurement are taken into account from both lidar systems. A nighttime mean of the lidar temperature profiles is calculated to minimize the impact of small-scale effects.

To set local stratopause variabilities into the global context we use global reanalysis data from the Modern-Era Retrospective analysis for Research and Applications Version 2 (MERRA-2; ). Here we use daily means of the 3-hourly instantaneous output on 42 constant pressure levels ranging from 1000 to 0.1hPa, i.e. from the surface up to 68km. The horizontal resolution is 0.625∘ in longitude and 0.5∘ in latitude. For our analysis we use MERRA-2 reanalysis data from winter 1980/81 to 2018/19, i.e. 39 winters in total, where the term winter includes the months December, January and February. Note that MERRA-2 assimilates temperature and ozone profiles from MLS satellite in the upper stratosphere and mesosphere starting in August 2004 . Therefore the stratopause temperatures are more reliable after that , which makes MERRA-2 particularly suited for our analysis.

Here we use the regional variability of STEs to investigate the agreement between the local lidar measurements and the global MERRA-2 reanalysis data. Similar to and we identify a STE if the temperature profile is at least 15K above the climatological mean at 2hPa (near the stratopause). To calculate the deviation from the climatological mean for the lidar nighttime temperatures, the NRLMSISE-00 empirical model of the atmosphere is used as a reference atmosphere. This model comprises an all-embracing day-by-day temperature climatology. Thus the deviation from the climatological mean for the lidar temperature data is calculated by subtracting the NRLMSISE-00 model climatology from the nighttime lidar measurement for each altitude, day and location separately. Consequently we can divide the lidar nighttime temperatures into STEs and non-STEs. Note that for lidar data the STE criterion is applied exactly at the stratopause altitude and thus varying with the height of the stratopause, while for MERRA-2 data a fixed pressure level of 2hPa is used as in and . This is done due to the relatively low number of lidar measurements with strongly varying stratopause heights and for the comparability of our MERRA-2 results with other studies.

The deviation from the climatological mean in MERRA-2 data is calculated by removing the long-term mean computed from the 39 winters of MERRA-2 data, for each day, altitude and longitude separately at the latitudes where Kühlungsborn and Andenes are located. Note that daily mean values are used for MERRA-2 data, and nighttime means are used for lidar data. Thus an impact of tides and long period gravity waves especially on the lidar data can not be excluded.

To bring the local values at Andenes and Kühlungsborn into the global context regarding PWs, the three predominant zonal wave numbers 1, 2 and 3 are fitted to the MERRA-2 data at both latitudes separately. The following description is illustrated in Fig. : for each day the original daily mean data (blue line) at each latitude at 2hPa are decomposed into the three dominating wave numbers 1, 2 and 3 (green line for wave 1, orange line for wave 123, i.e. all three waves together) using a least-squares procedure. Thus we somewhat reconstructed the original data based on the different PW components. Here we call the maximum absolute value of a wave the amplitude of that wave and the value between zero and an arbitrary location on the wave the local displacement. In this study the local displacement is at 11∘ E for Kühlungsborn and 16∘ E for Andenes respectively. With the help of these two concepts (amplitude and local displacement) not only the change in the amplitude of a PW but also the change in phase of the wave is taken into account. For example, the amplitude of a PW could remain the same, but changes in phase still can change the value of the local displacement at a given location.

Following this approach we reconstruct the whole time series for Andenes (69∘ N, 16∘ E) and Kühlungsborn (54∘ N, 11∘ E) based on PWs with wave numbers 1, 2 and 3. To visualize the differences and similarities of the original time series and the reconstructed time series, we correlate them for each location separately (i.e. Andenes and Kühlungsborn) as will be shown in the next section.

Figure  shows all available winter nighttime mean profiles between 2002 and 2012 for Andenes (Fig. 2a) and Kühlungsborn (Fig. 2b) and those identified as STEs (Fig. 2c, d). Profiles in Fig. 2c and d are a subset of Fig. 2a and b, respectively. There is a measurement rate of 13 % for Andenes and 7 % for Kühlungsborn of all winter days for this period. If one compares the long-term mean profiles (blue line) of Andenes and Kühlungsborn in Fig. a and b, it is noticeable that the altitude of the stratopause is 4km lower at Kühlungsborn (47km) than at Andenes (51km). The stratopause altitude during a STE is about 3km lower at Kühlungsborn and 7km lower at Andenes than the corresponding mean stratopause altitude. Possible causes of the difference in height of the stratopause between mid-latitudes and high latitudes as well as between STEs and the long-term mean are discussed in Sect. . For Andenes 41 out of the 134 measured profiles are classified as STEs (30 %), while for Kühlungsborn 20 out of 71 measured profiles are classified as STEs (28 %). Note that we will call the period 2002–2012 the “lidar period” in the following.

Figure a and b show vertical profiles of STEs (grey lines) and their average (red line) from MERRA-2 reanalysis data between winter 1980/81 and 2018/19 as well as the temperature climatology for all available 39 winters (blue line). The climatological stratopause altitude (cf. blue line) is 57km at Andenes and about 49 km at Kühlungsborn and thus higher compared to the lidar data at both locations (50/47km respectively). However similar to the lidar data the stratopause is in general higher at Andenes than at Kühlungsborn. As already mentioned in Sect. , MLS observations are assimilated in MERRA-2 starting in summer 2004 improving the upper stratosphere and lower mesosphere . We will call the period 20005–2019 the “MLS period” in the following. Restricting the analysed period to the MLS period results in a climatological stratopause altitude of 53km at Andenes and 48km at Kühlungsborn (see Supplement). Thus the difference between lidar observations and MERRA-2 reanalysis decreases for Andenes and even vanishes for Kühlungsborn.

The reason why we define STEs at 2 hPa can be seen in Fig. c and d, showing again vertical profiles of STEs but with subtracted climatology. In general a STE ranges between 35 and 55km with its maximum around 44km, which is in good agreement to findings of and and to the lidar data (see Fig. c and d). The maximum mean STE temperature differs between lidar and MERRA-2 data by about 2K at Andenes and 4K at Kühlungsborn. The STE occurrence rate is 14% at Andenes and 17% at Kühlungsborn. Differences between MERRA-2 and lidar data are discussed in the following.

Figure  compares the absolute mean STE profiles at Andenes and Kühlungsborn derived from lidar observations (blue) and MERRA-2 data (red). At Andenes the difference of the absolute peak temperature between lidar and MERRA-2 is 3K at 2hPa, while it is 6K at Kühlungsborn. In general, the MERRA-2 profiles are colder than the lidar profiles indicating that small-scale disturbances like gravity waves are not well represented in MERRA-2.

There are quite large differences in the STE occurrence rates between lidar (30%/28%) and MERRA-2 (14%/17%). However, when we apply the STE criterion (15K above climatological mean) not to the fixed pressure level of 2hPa but similar to the lidar data to the varying stratopause height then the STE occurrence rates derived from MERRA-2 are 27%/23% and thus much closer to the lidar STE occurrence rates.

Note that the differences further decrease when we additionally restrict MERRA-2 to the lidar period (30%/23%) and that we get slightly lower rates when we restrict to the MLS period (25%/18%). Thus differences in the STE occurrence rates between lidar and MERRA-2 vanish at Andenes and decrease at Kühlungsborn when restricting to the lidar period. The remaining differences between lidar and MERRA-2 can be attributed for one thing to the different statistical basis, which is much better for the MERRA-2 data than for the lidar data since there are 296/230 STE profiles in the MERRA-2 data restricted to the lidar period but only 41/20 in the lidar data at Andenes/Kühlungsborn. On the other hand small-scale dynamics like gravity waves are not perfectly represented in MERRA-2 in the upper stratosphere, which could explain some part of the differences between lidar and MERRA-2. A further discussion on the influence of small-scale dynamics can be found in Sect. . Differences of the STE occurrence rates between the lidar period and MLS period can also occur due a year-to-year variability of the STE occurrence.

Further causes of the above described differences between lidar measurements and MERRA-2 reanalysis data are the higher temporal and vertical resolution of the lidar measurements at higher altitudes in comparison to MERRA-2 data as well as the different length of the daily averaging interval. While lidar profiles are nighttime averages, daily means are calculated for the MERRA-2 reanalysis data. Thus lidar temperature profiles are biased by solar tides and gravity waves with longer periods. When averaging nighttime measurements with a length of at least 3 h, then only a part of a tide or large period gravity wave is averaged out. This effect is much weaker in MERRA-2 data since a daily mean is calculated here. Thus especially the bias from solar tides and long period gravity waves is much weaker since the entire oscillation is used in the averaging process. Note that we checked the STE occurrence rates using only nighttime MERRA-2 data and found, depending on which part of the night is used, increased occurrence rates by up to 3% at both locations. Thus the lidar STE occurrence rates might be slightly lower when daily mean data would be available.

Even though there are differences in magnitude of STEs between lidar and MERRA-2, they are in good agreement regarding the structure of the STEs; hence MERRA-2 reanalysis data can be used to bring local observations into the global context.

To bring local variability into the global context, we compare the time series at Andenes and Kühlungsborn at 2hPa with the time series from the amplitude of wave numbers 1, 2, 3 and 123 (all together) at the same latitude and with the time series of the local displacement for this very location (also for wave numbers 1, 2, 3 and 123).

Figure  shows the temperature deviation from the climatological daily mean of the original time series at one location at 2hPa (blue line), the time series of the amplitude for the respective PWs (green line) and the reconstructed time series (orange line) based on the respective PW (local displacement) for winter 2009/10. This is done for Andenes (left column) and Kühlungsborn (right column) and for the different wave numbers (each row represents a different wave number) using MERRA-2 data. The embedding of the local variability into the global context is shown as an example for winter 2009/10. A similar analysis for all available 39 winters can be found in the Supplement.

First, focusing on wave 123 for Andenes and Kühlungsborn, it is evident that the blue (original time series) and the orange line (reconstructed time series) lie almost perfectly on top of each other. The correlation coefficient of the original and reconstructed time series for winter 2009/10 is 0.99 for Andenes and 0.97 for Kühlungsborn. Thus 99% of the day-to-day variability of the temperature at Andenes can be explained by the variability of PWs with wave number 1, 2 and 3 in MERRA-2 data. The correlation of the original time series with the amplitude (green line) results in correlation coefficients of almost zero for Andenes and only 0.28 for Kühlungsborn. The discrepancy between the correlation coefficients of the amplitude and local displacement results from the fact that the amplitude describes only the increase and decrease of the amplitude of the respective PW, while the local displacement also takes the change in phase into account. This can be observed especially well around day 0 in Kühlungsborn (upper row in Fig. ). While the amplitude increases, the value of the local displacement decreases dramatically, due to the shift in phase of the waves. Around day -10, Kühlungsborn lies under the positive part of the wave packet where the temperature maximizes. In the next 10 d, the phase shifts in such a way that Kühlungsborn is now located under the minimum half of the wave packet where the temperature minimizes. The amplitude is blind to such a phase shift event explaining its much lower correlation coefficient. During other occasions, for example between day -10 and 10 at Andenes, the amplitude and local displacement values are much more synchronous, pointing to a stationary phase. Due to the much better description of the local variability by the local displacement or reconstructed time series (orange line) we will focus on this in the following.

Considering the individual wave numbers the largest correlation coefficient between the original time series and the reconstructed time series occurs for wave 1 at both locations (Andenes: rA=0.9, Kühlungsborn: rK=0.84). Thus wave number 1 dominates the variability of the winter stratopause. In both stations wave 2 (rA=0.75,rK=0.74) is slightly less important than wave 1 (rA=0.9,rK=0.84). Wave 3 plays by far the smallest role (rA=0.5,rK=0.65).

Similar to the example winter 2009/10 described in detail above, we correlated the original time series of all available winters with the reconstructed time series of every individual wave number as well as all wave numbers together. The individual correlation coefficients for both locations can be found in Table . In general 98% of the daily stratopause variability at Andenes and Kühlungsborn can be explained by the variability of PWs with wave number 1, 2 and 3. Also the other characteristics of the individual wave numbers found in winter 2009/10 can be generalized. Note that we conducted this analysis also to temperature time series at 1hPa, the climatological stratopause altitude at Kühlungsborn, and received the same results (not shown).