The lidar instruments are powerful tools for the study of atmospheric perturbations. They produce accurate observations with high temporal and spatial resolution, well adapted for studying atmospheric gravity waves (Chanin and Hauchecorne, 1981). Gravity wave activity has been extensively analysed using lidar throughout the middle atmosphere in several studies (Fritts and Alexander, 2003, and references therein).

Rayleigh lidar provides vertical profiles of molecular density and temperature when the atmosphere is free of aerosols (Rayleigh scattering above 30 km) from about 30 to 90 km depending on the signal-to-noise ratio (Hauchecorne and Chanin, 1980). The OHP lidar is composed of a frequency-doubled Nd:YAG laser emitting at 532 nm with a repetition rate of 50 Hz and a collector surface area composed by a mosaic of four mirrors with a diameter of 50 cm corresponding to a surface of 0.8 m2. Lidar measurements have been performed continuously at OHP since late 1978. In early years, the vertical resolution was 0.3 km, and it has been improved to 0.075 km since the mid-1990s. The temporal resolution is about 2 min 40 s. We only used night-time profiles during clear-sky conditions above OHP. During night-time the background noise decreases considerably (for more details about the lidar instrument and technique, see Keckhut et al., 1993). The number of observations used in this work is 14 profiles for August and 13 profiles for October 2012.

In order to have access to perturbations with short time and vertical scales, at least in a statistical sense, we analyse raw lidar signals with a variance method (Hauchecorne et al., 1994; Mzé et al., 2014). This method is based on the computation of the signal perturbations over short time and vertical intervals Δt,Δz and on the summation of the square of these perturbations over a large number of elementary intervals ΔT=ΔtNt,ΔZ=ΔzNz, which give an estimation of their variance. It allows extracting root-mean-square mean amplitude of small-scale perturbations that are not detectable on single profiles. The observed variance of the signal is defined as the sum of instrumental and atmospheric variances: Vobs=Vatm+Vinst. The instrumental variance is estimated assuming a Poisson noise distribution for the photon counting signal. Then, the atmospheric variance will provide an estimation of the GW activity in the middle atmosphere.

The estimation of the variance with a given thickness ΔZ=ΔzNz of the layers is equivalent to estimating the power spectral density of atmospheric fluctuations in band-pass filter with characteristics related to ΔZ. This method is equivalent to an estimation of the variance using a broad band-pass filter centred at λv∼2.4ΔZ. Between 30 and 50 km ΔZ=1.5 km and Nz=20, while ΔZ=3 km and Nz=40 above. Lidar vertical resolution (Δz) is equal to 0.075 km. ΔT=ΔtNt∼ 26.7 min, with Δt=160 s and Nt=10.

The variance method is computed for each night between 30 and 85 km of altitude and expressed in potential energy per unit mass (J kg-1) using the Brunt–Väisälä frequency from the mean lidar temperature profile, in order to characterize gravity wave activity.

The variance method is relatively simple but has the advantage of being robust, fast and using raw data. It is independent of data processing errors. More details about the variance method are presented in a recent study by Mzé et al. (2014).

A number of sensitivity studies have shown that the sponge layer has an evident strong damping effect on the zonal flow forcing. For that reason, in the following analysis we consider only the region below 68 km of altitude (indicated by a dashed line in figures).

Under the effect of the initial instability generated by the WB at t=0, the first cloud forms after about 4 min integration time, and precipitation after 8 min. In good agreement with the results of Costantino and Heinrich (2014), the cloud grows up quickly and reaches the tropopause (12.5 km of altitude) after about 20 min. At this point, a large spectrum of gravity waves is generated close to the tropopause and propagates downward (in the troposphere) and upward (in the stratosphere) direction. The initial wind is a simplified profile close to that observed near the OHP in the real simulation, on 21 October 2012, at 00:12 UTC. It is zero up to 17.5 km (in the real simulation it is slightly negative) and then it increases linearly up to 80 m s-1 at 65 km (in the real simulation the wind shear is not linear, but the intensity is similar).

Figure 1 shows the horizontal cross section of wind vertical velocity, w, which is a good proxy to observe gravity waves. Green contour lines represent absolute values of the ratio between potential temperature anomaly and its horizontal mean value, by steps of 0.02. With the term anomaly we define the difference between the local value of a physical quantity and the horizontal average. This parameter is an indicator of atmospheric buoyancy, which is a source of instability and turbulence.

Figure 1 shows a continuous generation and propagation of gravity wave packets from 40 to 90 min of integration time, by steps of 10 min. In the stratosphere, energy is transported upward and eastward on the right side of the domain, westward on the left side. This can be seen following the temporal displacement of vertical velocity intensity peaks, which is reliably representative of GW group velocity.

There is a strong asymmetry in wave propagation between eastern and western part, which is driven by the eastward wind shear. According to Eq. (1), a change in the horizontal wind speed has a direct effect on α (the angle between the vertical and the isophases), which decreases (increases) with altitude on the left (right) side of the storm.

On the left side (upwind) α remains small but positive ,and waves can propagate vertically in the stratosphere (until α is zero and waves become evanescent). Their amplitude, increasing with altitude, may reach the threshold value beyond which wave saturation can occur. On the right side (downwind), α increases with altitude (mostly visible in the lower stratosphere) and wave intrinsic frequency decreases. According to Eq. (1), when U0 is strong enough so that ω→0, waves break and can no longer penetrate into the upper stratosphere. This process is generally referred to as wind filtering.

Horizontal phase velocity of wave packets at a given altitude can be analysed in more detail looking at Fig. 2, where the vertical wind speed is shown as a function of time (y axis) and space (x axis). Slopes of isophases represent the horizontal velocity of wave packet phases. As a reference, dark green lines overplotted on figures show horizontal velocity slopes relative to 16.7 (dotted), 33.3 (solid) and 66.7 (dashed) m s-1 (i.e. 60, 120, 240 km h-1, respectively). On the left side of the domain, horizontal phase speed of GW generated between 40 and 60 min of integration time, and within a radius of 100 km from storm centre, is close to 16.7 m s-1. Horizontal wavelengths are equal to about 2–3 km, at 40 km of altitude, and 4 km at 65 km. This may give us a rough estimate of wave periods of a few minutes. Oldest waves, generated in the first 20 min of GW activity (20–40 min of integration time), have much stronger phase velocities (up to approximately 66.7 m s-1) and longer horizontal wavelengths, up to 10 km. This is in good agreement with the results of Lane and Sharman (2006) and the order of magnitude of convective GW wavelengths they found, between 2 and 10 km.

Top image of Fig. 2 (40 km) shows how phase velocity of wave packets may vary with time. For instance, at t=60 (or t=80) min and 900 < x < 950 km, isophase slopes rapidly increase, turning almost vertically (phase speed deceleration), and decrease again little afterwards (acceleration). As expected, phase velocity appears higher downwind than upwind, and also more constant with time. Comparing top and bottom images of Fig. 2, its seems that phase velocity is likely to be higher, on average, in the upper levels of the stratosphere.

Figure 3 shows the horizontal momentum flux, calculated averaging the u′w′ product over an area of 300 km on the left (blue line) and right side (red line) of the storm.

The averaged value over the whole 600 km region (left and right side together) is reported in green. Before the generation of convective GW, HMF is almost zero in the stratosphere, while after t=20 min, GW dynamics induces a vertical transport of horizontal momentum. At t=30 min, HMF is negative upwind, where u′w′‾ is generally negative (air parcels oscillate along a slant path in a upward–westward direction), and positive downwind. Stratospheric peak values of HMF are equal to -73 × 10-3 and 76 × 10-3 N m-2 at 14.6 km of altitude. With increasing altitude HMF tends rapidly to zero, with absolute values smaller than 2 × 10-3 N m-2 beyond 40 km on the left side and 30 km on the right side. Upwind gravity waves, no subjected to wind filtering, penetrate deeper in the stratosphere. Considering the whole region, the balance between negative and positive values is slightly negative, with a peak value in the stratosphere of -6 × 10-3 N m-2 (at 23.5 km).

With increasing time, the maximum altitude of absolute HMF values larger than 2 × 10-3 N m-2 increases up to 49 (left side) and 38 km (right side). At t =50 min, it reaches 60 (left side) and 37 km (right side). Considering the whole region, the overall HMF is positive between 13.5 and 22 km of altitude (with a maximum of 12 × 10-3 N m-2, at z=14 km) and negative above (with a peak of -12 × 10-3 N m-2, at z=29 km). At t= 70 min, HMF is positive between 17.5 and 28.5 km of altitude (with a maximum of 17 × 10-3 N m-2, at z=23 km) and negative above (with a peak of -8 × 10-3 N m-2, at z=43 km).

In conclusion, the upward transport of HMF is very efficient, with an average vertical speed of about 30 km h-1. In 40 min (from t=30 to 70 min) the peak of negative HMF moves by 20 km (from z=23 to 43 km), as well as the highest altitude with HMF smaller than -2 × 10-3 N m-2 (from z=40 to 60 km).

Note that we are using here the hypothesis that storm anvil, and hence GW source, is punctual and centred in the middle of the domain. This is not completely exact, as we can see in Fig. 1, showing that storm edges move outward up to 30 km away from domain centre (t=70 min). Cloud edges are the most convective parts of a storm and GW are mostly generated at storm borders. Then, in a bi-dimensional simulation, we have actually two GW sources, very close and symmetric. In the region located just above cloud top (e.g. at t=70 min, Fig. 1) a complex interaction between the downwind wave (propagating rightward from the left edge of the storm) and the upwind wave (propagating leftward from the right edge of the storm) occurs. Looking closely at the time development of this interaction field, it really seems that it does not propagate beyond the outer edges of the cloud. In particular the downwind wave is not advected eastward beyond the right edge, remaining bounded in the region within ±30 km, with a zero phase speed (this is maybe due to the interference of the two waves that are symmetric, with very similar characteristics and intensities). To test the consistency of our approximation (i.e. to consider the storm as punctual), we calculated HMF eliminating the area within ±30 km. Above 20 km of altitude, results are very close to those shown in Fig. 3, where the area within ±30 km is considered. We believe that our hypothesis, always remaining an approximation, does not alter considerably the results of our analysis. For real case simulations, this approximation is supposed to work even better as convective cloud anvils are smaller.

Note also that another factor (other than wave breaking) that may contribute to the decrease of eastward momentum in Fig. 3 is the fact that eastward-propagating waves leave the ±300 km subdomain (where momentum fluxes are diagnosed) by its lateral side. To verify this point, we have calculated the HMF in almost the entire domain (±900 km). Above 20 km of altitude, these results are very close to those obtained averaging over a subdomain of 300 km. In particular, qualitative features of HMF vertical profiles (e.g. such altitudes where HMF decreases or increases) are very similar for the two cases, indicating that a ±300 km averaging window is large enough not to lose essential information.

Mean HMFs, averaged over the whole 600 km subregion, have the same order of magnitude of those calculated by a highly resolved global model in the lower stratosphere of midlatitude NH, of a few 10-3 N m-2 (see Sect. 1.2). In our case, values are stronger, probably because measurements are only performed during storm occurrence. In addition, the smaller grid spacing of our simulation may resolve small GW scales associated with important HMFs that are neglected in global models.

Figure 4 shows the horizontal drag force calculated at t=70 min, on the left (blue line, left image) and right side (red line, middle image) of the domain, and the overall average value of left and right side together (green). As expected, downwind wave breaking leads to a strong negative peak of instantaneous drag force, down to -140 m s-1 day-1 at 52.5 km of altitude. The thinner black line shows the mean horizontal wind anomaly with respect to the left half of the domain. U-wind anomaly has a peak of -0.75 m s-1 at 52 km, coincident with that of drag force, and shows a similar vertical profile to GW drag distribution. Upwind wave breaking is significantly weaker and the resulting drag force (red line) is positive but very small, with two peaks in the lower and two in the upper stratosphere, at 18.5 km (13.6 m s-1 day-1), 25 km (11.9 m s-1 day-1), 54.3 km (6.8 m s-1 day-1) and 61.4 km (8.5 m s-1 day-1). The resulting acceleration of the mean wind is almost irrelevant, oscillating between positive and negative values. Near the tropopause, approximately below 15 km, the large drag force values and wind anomalies on both left and right sides are probably due to outward air flux above cloud top, driven by storm convective dynamics.

Considering the whole area, the mean stratospheric drag force is slightly positive between 20 and 40 km (vertical average of 4.37 m s-1 day-1) and strongly negative above (vertical average of -40.1 m s-1 day-1), with a strong deceleration up to -68 m s-1 day-1at 52 km. Wind anomaly is somewhat proportional to drag force, with a peak at 54.8 km equal to -0.37 m s-1.

Potential energy is calculated according to Eq. (6), averaging spatially over the whole domain (left and right side together). Figure 5 shows vertical profiles of potential (blue), kinetic (red) and total energy (black) at t=30, 50 and 70 min of integration time. Green dashed line represents conservative growth rate (Sect. 2.3). As expected, the development of the storm and the consequent excitation and propagation of GWs increase enormously the wave energy in the stratosphere. At t=30 min, mean total energy between 20 and 60 km is very small and equal on average to 1 J kg-1. Above the tropopause, its vertical profile is almost constant with height, a sign of a small GW penetration in the upper levels. At t=50 min, stratospheric total energy is 10 times larger, on average, than 20 min before (11 J kg-1) and attains a maximum value of 20 J kg-1 at 55 km. At t=70 min, the growth rate is almost exponential between 30 and 50 km. Mean total energy of the stratosphere reaches 30 J kg-1, with a strong peak of 68 J kg-1 at z=50 km. Above this altitude, the sudden decrease in Ep suggests the presence of wave breaking by saturation. This mechanism is supposed to increase atmospheric instability and may explain the further increase in kinetic energy for z > 50 km. The close profile and mean values of potential and kinetic energy confirm, to a certain extent, the hypothesis of constant energy repartition between Ek and Ep (Sect. 2.3) in particular below 40 km.

Similarly to the idealized case, in the following paragraphs we focus on the study of stratospheric dynamics up to about 58 km of altitude, where the damping layer starts. This altitude is marked in figures with a dashed line.

The case study is the 19–24 October experiment. In particular, on 21 and 22 October, a very strong rain thunderstorm occurs in the western Mediterranean Basin. From a depression over eastern Spain, cold air converges with warm and humid air that flows from northern Africa toward southern and eastern France and Germany. Over France, a cold front forms and moves eastward from Pyrenees (Spain–France border) to south-east. Figure 6 shows infrared data acquired by the IR channel (at 10.8 µm) of the geostationary meteorological satellite METEOSAT9 and provided by EUMETSAT (European Organization for the Exploitation of Meteorological Satellites). From left to right, images are relative to 21 October, at 14:00 and 21:00 UTC, and 22 October at 04:00 UTC. The IR 10.8 µm channel provides information on cloud top temperature. Colder cloud tops, generally associated with more vertically developed clouds, are whiter in the colour scale. Figure 6 shows a large mesoscale cloud system which is advected eastward over the western Mediterranean Basin. The bright and white spot in southern France at 21:00 and 04:00 UTC suggests the presence of strongly convective clouds probably associated with severe weather conditions over the OHP region (indicated in figure by an orange square).

Temporal and spatial coincidences between real and modelled meteorology are fairly good. According to WRF simulation, in the morning of 21 October (from 09:00 UTC) a heavy rainstorm appears in grid 3, from the south-east boundary. It passes twice over the OHP station from 16:00 to 18:00 UTC (21 October) and from 20:00 (21 October) to 04:00 UTC (22 October), with the strongest rain event between 22:00 and 00:00. Figure 7 shows the horizontal cross section of vertical velocity (grey colour scale) at 10 and 40 km of altitude, for 21 October, at 23:00 UTC. Contour lines represent the rain water mixing ratio (qrain), by steps of 1 g kg-1. The maximum of the total column (from the ground to top of the atmosphere) qrain value (at 3 km resolution) is reported under the image, and it is equal to 44.4 g kg-1. The position of OHP station is shown in green.

In good agreement with satellite observations, Fig. 7 shows a strong rain storm occurring over the OHP region. The severe convective dynamics (clearly visible in Fig. 8) perturbs significantly the vertical velocity field. While at low altitude levels (left image), the strongest w-speed perturbations are mostly coincident with the highest peak of rain (central part of the domain) and the highest mountain (as the Alps, in the north-east of the domain), at higher altitudes w field is more coherent and shows a large spectrum of gravity waves that propagate in the stratosphere to very long distances. The yellow square represents the grid box (referred to as virtual station, VS) where the product of vertical velocity and qrain is maximum, i.e. the place where storm convective dynamics is supposed to have the largest impact on the atmosphere. At 40 km of altitude, the concentric rings of w crests seem to indicate that the most active GW field originates from the virtual station and propagate upwind, in the westward direction. Downwind (eastward), GW propagation appears much less efficient, probably because of the wind filtering by wave breaking.

GW dynamics is shown in deeper detail in Fig. 8, representing a vertical cross section of zonal (top image) and meridional (bottom image) vertical velocity. The blue-red colour scale is for the rain water mixing ratio, while the grey scale is for the cloud mixing ratio. The vertical black line indicates the location of VS. Wind vectors are overplotted and arrow's length is proportional to the horizontal wind intensity (0.1∘ of latitude or longitude is equal to 10 m s-1). Deep convection occurs when cloud top reaches the tropopause. This is almost the case above the VS, where the tropopause is at 12.5 km and cloud top attains 10 km of altitude. Above cloud top, the ascending flow has a strong positive vertical speed and interacts with the tropopause, from where a large GW field originates. This mechanism of wave generation is very close to that observed in the idealized case. In the top image (zonal cross section), GWs propagate upward and outward, with respect to the storm. According to linear theory, wave propagation is much more efficient upwind, where the strong zonal wind (up to 80 m s-1 at 70 km) turns the isophases. The wave packets reach the left border of the domain, 300 km westward (the distance in kilometres from the left boundary is indicated on the upper x axis). Downwind, isophases turn horizontal with increasing altitude up to 45 km (critical level), where vertical wavelengths go to zero and GWs disappear. Similarly, meridional wave propagation (bottom image) is much more efficient upwind (northward) than downwind. On the other hand, meridional wave activity is weaker than a zonal one as wind intensity is lower and close to zero, with punctual reversals at 30, 40 and 55 km. On the right side of the image, above 44∘ N (northern part of the domain), the presence of high mountains (Alps) together with a strong northward low-tropospheric wind generates intense orographic waves. The wind reversal above the tropopause, however, prevents them from propagating further southward and upward into the lower stratosphere.

We quantify the drag force exerted on the mean flow by the thunderstorm that occurred during the night of 21 October, from 22:00 to 00:00 UTC. During this time period, the storm attains its maximum intensity and it is approximately located above the OHP. For each WRF 5 min output, HMF is calculated averaging horizontally the u′ and w′ product over a square of ±40 grid points (i.e. ±120 km in the S–N and E–W directions) from OHP position. From the vertical derivative of HMF we obtain the instantaneous drag force, which is then averaged temporally over the 2 h time period. Error bars represent the confidence level of mean values. They are calculated as σ/(n-2)1/2, where n is the number of instantaneous measurements within each altitude bin and σ is their standard deviation. Spatially and temporally averaged drag force vertical profiles for the left side (blue), right side (red) and the whole square region (green) are shown in Fig. 9.

Above 16 km, stratospheric zonal wind (solid black line) is directed eastward and increases with altitude up to 65 m s-1 at 60 km. Meridional wind (dashed black line) is particularly weak and changes direction several times with increasing altitude. This is a typical meteorological condition in the Northern Hemisphere during autumn and winter. In good agreement with the idealized case, upwind drag force is small in the lower stratosphere, where no wave saturation effect is expected. A double negative peak equal to -2.31 and -3.60 m s-1 day-1occurs at 41.2 and 45.1 km of altitude. Then the deceleration approaches to zero at 50 km of altitude, increases beyond this altitude and attains -6.5 m s-1 day-1 at 56.1 km, which is the highest altitude level not directly affected by the damping. Sensitivity studies with different sponge layer depths show that the strongly negative drag force values in the last 10 km layer (down to -51.7 m s-1 day-1at 65.3 km) should be considered as an artefact due to the wind deceleration imposed by the sponge layer.

Downwind forcing shows two peaks in the lower stratosphere, at 25.5 and 32.3 km, of 1.84 and 1.74 m s-1 day-1, and a stronger peak in the stratosphere of 4.3 m s-1 day-1, at 42.0 km. Also in this case, there is a good qualitative agreement with the idealized case, with positive peaks both in the lower part and the higher part of the stratosphere.

Averaged over the whole region (green line), temporal mean of drag force vertical profile has two positive peaks of 1.38 and 1.39 m s-1 day-1(at 31.7 and 43 km) and two negative peaks of -1.87 and -4.53 m s-1 day-1(at 45.6 and 56 km). Averaged vertically, drag force is positive in the lower stratosphere between 20 and 40 km (0.23 m s-1 day-1) and negative in the upper layers between 40 and 58 km (-1.00 m s-1 day-1).

Real case drag force values, even if consistent with those of the idealized case, are however much weaker, by at least 1 order of magnitude. This is true also in case of HMF estimates. For instance, Fig. 10 (left image) shows the instantaneous HMF vertical profile at 23:00. On the left side, HMF has a peak at 18 km (-5 × 10-3 N m-2), 24.2 km (-2.5 × 10-3 N m-2) and 32 km (-2 × 10-3 N m-2), remaining negative (about -1 × 10-3 N m-2) up to 58 km. On the right one, HMF attains relative maxima at 17.7 km (3.1 × 10-3 N m-2), 23 km (3.1 × 10-3 N m-2) and 29.6 km (2.1 × 10-3 N m-2), becoming even negative above 38 km. The average HMF profile for the whole area is positive in the lower stratosphere, with peaks of 0.82 × 10-3 N m-2 (23 km) and 0.65 × 10-3 N m-2 (30 km) and negative above 31.3 km with minima of -0.92 × 10-3 N m-2 (32.6 km) and -0.79 × 10-3 N m-2 (44 km).

These values are more than 10 times smaller in magnitude than those observed during the idealized storm up to 17 × 10-3 N m-2 (23 km) and -8 × 10-3 N m-2 (43 km), while vertical distribution of positive and negative maxima is in very good agreement. On the other hand, they are consistent with previous model results presented in Sect. 1.2.

If we look at the instantaneous energy (Fig. 10, right image), we arrive to similar conclusion. The intensity of instantaneous real case wave energy is, however, less underestimated than HMF. With respect to the idealized case at t=70 min, real case energies (Ep and Ek) on 21 October at 23:00 UTC are between 3 and 4 times smaller. Total wave energy is increased by about 58 % with respect to no rain conditions (21 October at 08:00 UTC), from 6.67 (Ep= 3.55, Ek= 3.12) to 10.56 (Ep= 4.83, Ek= 5.73) J kg-1.

Here we compare the potential energy calculated by WRF for the two study cases of 21–25 August and 19–24 October 2012, with lidar measurements collected during the months of August and October 2012. For model results, we average over the whole time period the instantaneous potential energy relative to the single OHP grid box. Lidar data are collected only in case of clear sky in a narrow time window of about 3 h.

The variance method, computed for each night, provides one average profile per day from data collected approximatively in the 19:00–22:00 UTC time window. The variance is computed with Nz=20, between 30 and 50 km, and Nz=40 between 50 and 85 km. Nz defines the average wavelength selected by the band-pass vertical filter used. Nz=20 corresponds to a band-pass filter centred at about 3.6 km (with a spectral interval between 2.4 and 5.8 km at half maximum), while Nz=40 corresponds to a band-pass filter centred at about 7.1 km (with a spectral interval between 5.1 and 11.3 km at half maximum). Single lidar profiles (provided every 160 s) have been integrated over ∼ 26.7 min Nt=10 and then averaged over the 3 h time window. Stratospheric convective waves are deep (wavelengths of ∼ 10km). This corresponds to the part of the spectrum of gravity waves we intend to capture with lidar, using a spectral window centred at 7.1 km in the upper stratosphere, consistent with the order of magnitude of GW wavelength reproduced in our simulations. The integration time used limits the shortest period which can be measured (∼ 52 min).

To reduce statistical uncertainties due to the small number of available lidar data for the 21–25 August and 19–24 October time period (just 3 and 2 daily profiles, respectively), we consider all measurements collected during August and October for a total of 14 and 13 daily profiles, respectively. Lidar data are then supposed to describe, to a certain extent, the average effect of monthly atmospheric dynamics on GW energy. Error bars indicate the confidence level, with respect to the temporal variability, in both WRF and lidar data.

Figure 11 shows in red WRF mean Ep profiles together with lidar monthly averages (blue), for August (left) and October (right). The green lines represent the conservative growth rate, while the black horizontal line at 58 km of altitude indicates the beginning of the sponge layer. During August, lidar data show that Ep mostly follows a conservative increases from 34 to 62 km of altitude, with a punctual but sensible energy drop between 52 and 54 km. In October, lidar measurements show that vertical energy transport is very efficient and conservative only between 36 and 44 km. Above this altitude there is a strong departure of Ep from the conservative growth rate up to 52 km, where Ep starts again to increase with increasing altitude but much less than in August. In the last four altitude levels, above 57 km, the average Ep attains 15.4 J kg-1 in August and 9.7 J kg-1 in October.

WRF seems to capture the main Ep feature revealed by the lidar system. In August, model results show an exponential increase of energy with altitude above 30 km in August, while in October there is a clear energy loss between 44 and 50 km, somewhat coincident with that seen in the lidar profile.

In Fig. 12, we present the scatterplot of WRF and lidar data for each month. In August (left image), the reduced chi square (i.e. divided by the degrees of freedom, df) is very close to 1. This value indicates a good agreement between the two data sets, with respect to the error variance. In October, where the energy vertical profile is much more variable with altitude, the reduced chi square is slightly higher but still close to the unity and equal to 1.67. However, despite a relatively good linear correlation between the two data set, WRF seems to underestimate systematically lidar values by a factor of 3 (October) and 4 (August).

OHP is located very close to Alps and mountain waves can have a strong impact on the atmospheric energy budget measured by lidar, which works in clear-sky conditions. At the same time, WRF is capable of resolving mountain waves and their energy is supposed to be fully captured, at least in the inner domain. However, U and V wind profiles shown in Fig. 9 indicate that over the study region wind inversion at the tropopause prevents the largest part of mountain waves from propagating upward. Hence, orographic GWs are not supposed to contribute consistently to the stratospheric energy budget near OHP. Outer domains, however, have a coarser resolution. Mountain and convective GWs occurring outside the inner borders are only partially resolved and their energy can not totally propagate into the inner domain. During the study period, large weather perturbations occur all around Europe (Fig. 6). Over the OHP, the lack of energy contribution from convective (and eventually orographic) GWs generated outside inner boundaries can be a leading factor of WRF energy underestimation with respect to lidar data.