## 4 MILs' Climatology  
MILs were identified by following the procedure outlined by [PERSON] [5] and [PERSON] et al. [33] and is briefly presented here. It is worth noting that this procedure has been applied in many previous studies investigating the phenomenon of mesospheric inversion. A non-exhaustive list of such MIL studies includes: [6; 28; 34; 35; 36; 37; 38]. The identification procedure of [PERSON] and [PERSON] [5] considers the amplitude, thickness, gradient, and altitude of an MIL as the primary characteristics of an inversion layer. The amplitude is defined as the maximum temperature anomaly with respect to a mean 'unperturbed' vertical temperature profile. The thickness refers to the vertical extent of the layer spanning the bottomside to topside cold points. The gradient relates how rapidly the temperature of the layer changes over the altitude spanning the MIL. The altitude of the MIL is defined as the altitude of the bottomside cold point below the start of the positive temperature gradient. In this study, MILs are identified by using the following three criteria:  
* The bottom of the inversion is located at least 5 km above the stratopause, and the top is below 90 km;
* The temperature perturbation must be significant within measurement uncertainty;
* The amplitude of the MIL temperature perturbation must be 2\(\sigma\) larger than the temperature perturbations expected from gravity waves at the specific altitude  
Given that lidars are restricted to zenith viewing, we are unable to see the horizontal extent of the MIL. Likewise, we are restricted to nighttime measurements and are unable to measure a particular MIL continuously over the course of several days. In the context of a lidar measurement, we distinguish between'very large gravity waves' and MILs by utilizing the mean potential energy per unit mass, Equation (9), in order to characterize gravity wave activity, mean temperature deviations can be derived from potential energy following the methods of [PERSON] et al. [39]:  
Figure 3: Temperature profile illustrating the detection of different atmospheric layers that could be associated with a MIL on 4 January 2000. The measurement uncertainty for the temperature profile is indicated by the green lines. Potential MILs’ minimum altitudes are indicated by the red and pink triangles. The red triangle corresponds to the bottomside of the layer, which passed all tests and was classified as an MIL. \[E_{p}=\frac{1}{2}\frac{g^{2}}{N^{2}}(\frac{T^{\prime}}{T})^{2} \tag{9}\]  
In the definition of Equation (9), \(g\) is the gravitational acceleration, \(N\) is the Brunt-Vaisala frequency given in Equation (10), and \((T^{\prime}/T)\) is the atmospheric variance:  
\[N^{2}(z)=\frac{g(z)}{T(z)}\bigg{(}\frac{dT}{dz}+\frac{g(z)}{C_{p}}\bigg{)} \tag{10}\]  
The lidar temperature variance has been computed for each night of our study in the altitude range between 30 km and 85 km. We then calculate a potential energy profile from the mean lidar temperature profile, applying the median and the interquartile range to determine a robust estimator of the \(E_{p}\) variance. We can then compare our nightly average \(E_{p}\) profile to the climatology of potential energy due to the gravity waves observed above the OHP [40]. As a rough estimate for the reader, the climatological \(E_{p}\) at the OHP is around 10 J. kg\({}^{-1}\) at 50 km, increasing up to 100 J.kg\({}^{-1}\) at 75 km. In the OHP gravity wave potential energy climatology presented by [PERSON] et al. [40], the standard temperature deviations at 50 and 75 km are 1.6 and 4 K, respectively. We assume that these fluctuations represent \"average gravity wave activity\" at OHP. The phenomena we consider to be MILs are much larger in amplitude than even very large gravity waves. We have constructed an altitude dependent cut-off between GWs and MILs based on \(2\sigma\). Explicitly, lidar temperature anomalies with amplitudes below 3.2 K at 50 km and 8 K at 75 km are considered as GWs.