LITOS (Leibniz-Institute Turbulence Observations in the Stratosphere) is a balloon-borne instrument used to observe small-scale fluctuations in the stratospheric wind field . The wind measurements are taken with a constant temperature anemometer (CTA) which has a precision of a few cms-1. It is sampled with 8 kHz yielding a sub-millimetre vertical resolution at 5 ms-1 ascent rate. Thus the inner scale of turbulence is typically covered. A standard meteorological radiosonde (Vaisala RS92 or RS41) is used to record atmospheric background parameters. LITOS was launched three times as part of a ∼ 120 kg payload from Kiruna (67.9∘ N, 21.1∘ E) within Balloon Experiments for University Students (BEXUS) 6, 8, and 12 in 2008, 2009, and 2011, respectively . The second generation of the small version of the instrument is an improvement on the one described by and consists of a spherical payload of ∼ 3 kg weight. It is suspended ∼ 180 m below a meteorological rubber balloon. Two CTA sensors are mounted on booms protruding at the top of the gondola. The instrument was launched several times from the IAP's site at Kühlungsborn (54.1∘ N, 11.8∘ E), e.g. on 27 March 2014, 6 June 2014, and 12 July 2015.

In this paper, flights are only taken into account when data from more than one CTA sensor on the same gondola are available. Summarised, the data analysis is performed in three steps. First, the dissipation rate is retrieved similarly to the procedure described by . Then the ε values from both sensors are compared to detect sections where one sensor is possibly affected by the wake of ropes. Finally, the remaining spectra are manually inspected to sort out cases for which both sensors potentially have been affected. Another source of artificial turbulence is the wake of the balloon . Typically, the wake influences both sensors similarly and cannot be detected by the above methods. Therefore, we limit our analysis to flights and altitude regions, where wake effects do not play a role due to sufficient wind shear that brings the payload out of the balloon's wake.

The details of the retrieval are as follows: the data of the ascent are split into windows with depths of 5 m altitude with 50 % overlap. In each window, the mean value is subtracted, and the periodogram is computed, which is an estimation of the power spectral density (PSD). The periodogram is smoothed with a Gaussian-weighted running average. The instrumental noise level is detected and subtracted. Initially, turbulence is assumed to occur in each window and thus the algorithm attempts to fit the model for fully developed turbulence in the form given by and to the observed spectrum (see Eq.  in Appendix ). If the fit succeeds, the inner scale l0 is obtained. This leads to the energy dissipation rate ε given by ε=cl04ν3l04, where ν is the kinematic viscosity (known from the radiosonde measurement) and cl0 is a constant depending on the type of sensor. The determination of cl0 for our sensor configurations is described in Appendix . Non-turbulent (or disturbed) spectra manifest in bad fits which are sorted out with the following set of criteria:

The noise level detection fails, which usually means that the noise is not white; i.e. the periodogram is disturbed at small scales.

Sometimes a sensor has been located in the wake of a rope supporting the gondola and the other sensor has not, causing the ε values of both sensors to differ by up to 5 orders of magnitude. To sort out such sections, altitude bins for which the dissipation rate from both sensors deviates by more than a factor of 15 are discarded. For the flights with a small payload, the remaining spectra have been inspected manually for sections where both sensors have been affected by the rope wake, and those that look suspicious have been taken out. A spectrum is regarded as wake-affected if it has a plateau in PSD near 10 cm spatial scale, which is estimated to be the extent of a Kármán vortex street originating from the lines supporting the gondola. This problem of wake effects from the ropes does not occur for the BEXUS flights, where the sensors were placed further away from the supporting lines. For all other altitude bins the average of both sensors is taken.

On the other hand, for the BEXUS flight the distance between the balloon and the payload was only 50 m, i.e. comparatively small. Thus, the payload flew through the wake of the balloon for a considerable duration of the flight. Therefore, only limited altitude sections with large wind shears are considered for this flight.

To quantify the stability of the atmosphere, the gradient Richardson number Ri=N2/S2 is used, which is the ratio of the squared Brunt–Väisälä frequency N2 and the square of the vertical shear of the horizontal wind S2. The Brunt–Väisälä frequency can be written as N2=gΘdΘdz, where Θ is the potential temperature and g is the acceleration due to gravity. The wind shear is defined as S2=dudz2+dvdz2, where u and v are the zonal and meridional wind components, respectively. The Richardson number represents the ratio of buoyancy forces (which suppress turbulence) to shear forces (which generate turbulence). According to a theory for plane-parallel flow established by and , turbulence occurs below a critical Richardson number of Ric=1/4. The general applicability of that criterion was recently questioned based on measurements e.g. and model simulations e.g.. Often the shear is not strictly horizontal so that the theory by and is not applicable, as pointed out by . To take into account slanted shear, proposed a concept of slantwise instability. However, the Richardson number is still useful as an estimation of stability. The Richardson number also depends on the scale on which it is computed . Usually, computing Ri on a smaller scale yields locally smaller numbers, since for a computation on larger scales an average over regions with small and large Ri is obtained. In this study Ri is retrieved from the radiosonde measurements. In order not to dominate the derivatives by instrumental noise, the potential temperatures and winds are smoothed with a Hann-weighted running average over 150 m prior to differentiation with central finite differences.

Mesoscale numerical simulations are performed with the Weather Research and Forecasting (WRF) model, version 3.7 . Two nested domains with horizontal resolutions of 6 and 2 km and time steps of 15 and 5 s, respectively, are applied. In the vertical direction 138 terrain following levels with stretched level distances of 80 m near the surface and 300 m in the stratosphere are used and the model top is set to 2 hPa (about 40 km altitude) for the BEXUS flights and 5 hPa (about 32 km altitude) for the flights from Kühlungsborn. At the model top a 7 km-thick Rayleigh damping layer is applied to prevent wave reflections ; i.e. the top of the damping layer is the model top. Physical parameterisations contain the rapid radiative transfer model longwave scheme , the Goddard shortwave scheme , the Mellor–Yamada–Nakanishi–Niino boundary layer scheme , the Noah land surface model , the WRF single-moment 6-class microphysics scheme WSM6; and the Kain–Fritsch cumulus parameterisation scheme . The initial and boundary conditions are supplied by ECMWF (European Centre for Medium-Range Weather Forecasts) operational analyses on 137 model levels with a temporal resolution of 6 h. In the WRF a temporal output interval of 1 h is used, data interpolated along the flight track are output with an interval of 5 min. Simulations are initialised 5 to 6 h before the launch time of the balloon. The computation of turbulent kinetic energy (TKE) is done by the boundary layer scheme and described in . It is based on a prognostic equation which is solved additionally to the equations of motion and which includes transport, shear production, buoyancy production and dissipation terms. Shear and buoyancy terms include deformation and stability effects of the resolved flow and are related to turbulent motions by the horizontal and vertical eddy viscosities. The equation operates on the scale of the grid size.

In this paper WRF simulations are used to get an overview of the meteorological situation. showed that regions of GW breaking can be simulated by WRF simulations with horizontal grid distances of 2 km and a similar model set-up by means of convective overturning and reduced Richardson numbers. Here, the TKE output from the model is also used to identify regions of intensified turbulent mixing in the atmosphere along the balloon flight tracks. This can be a hint that observed turbulence was caused by large-scale GW breaking. It is not intended to quantitatively compare observed dissipation rates with simulated regions of enhanced TKE values.

The BEXUS 12 flight was launched from Kiruna on 27 September 2011 at 17:36 UT.

Figure a and b show atmospheric conditions observed by the radiosonde on board the payload. Temperatures decreased up to the tropopause at 10.3 km, excepting some small inversion layers. Above, there was a sharp increase in temperature known as tropopause inversion layer (TIL) . Higher up, temperatures slightly decreased. Winds came from the north-west near the surface and reversed between ∼ 6 and 10 km. The reversal caused nearly the opposite wind direction at 9 km altitude compared to 5 km, and a change of sign in both wind components. It further entailed strong wind shear below the tropopause, causing low Richardson numbers (below the critical number of 1/4). Above the tropopause the wind field showed signatures of gravity wave activity with short wavelengths and no obvious altitude-dependent structure. In the stratosphere, Richardson numbers were generally larger than in the troposphere.

Figure d depicts observed dissipation rates. Each blue cross corresponds to an altitude bin classified as turbulent (as described in Sect. ). The orange curve depicts a Hann-weighted running average over 500 m. Please note that large sections in the troposphere and stratosphere are subject to wake influence (marked grey) due to the small distance of only 50 m between the payload and the balloon. These sections are generally not discussed here. Between 9 and 10 km there was a thick layer with high dissipation. As described above, this altitude region featured low Richardson numbers caused by high wind shears. Thus turbulence was presumably induced by dynamic instability. Additionally, at this altitude a wind reversal was observed which caused filtering of gravity waves with phase velocities equal to the background winds (if present). Most probably, these high dissipation rates are not caused by wake because calculations show that the gondola was outside the wake in this altitude section due to the large wind shear. Furthermore, the dissipation rates are even larger than typical wake turbulence.

WRF model simulations were performed for the time and place of the flight. To show that these produced reasonable results, model winds and temperatures interpolated along the flight trajectory are plotted in Fig. a along with the radiosonde profiles. Observed and modelled results compare very well; the only difference is that the radiosonde data contain signatures from small-scale gravity waves which WRF cannot resolve. In Fig. , model snapshots at the middle of the ascent are shown. Panel (a) depicts horizontal winds at 850 hPa. Westerly winds flowed over the Scandinavian mountains, which are expected to have excited mountain waves. Another potential source of gravity waves is geostrophic adjustment. Bending stream lines are visible, e.g. over the Scandinavian mountains, west of the flight track. Panel (b) presents a vertical section of horizontal winds and potential temperatures. It demonstrates that the jet (∼ 7 to 10 km altitude) had a local structure and involved strong wind shears.

With a grid resolution of 2 km WRF can resolve waves with horizontal wavelengths larger than about 10 km. These waves can be seen, for example, in the vertical winds, which are used as a proxy. This quantity is plotted in Fig. c. Strong wave-like patterns are visible especially over the Scandinavian mountains, which correspond to the mountain wave excitation mentioned above. Weaker wave patterns are visible near the flight trajectory, downstream of the mountains. Between roughly x=400km and x=550km, the wave patterns change at tropopause height (approximately 10 km altitude): above, there is less amplitude than below. This is ascribed to the wave breaking and filtering mentioned before. Filtering means catastrophic breaking of waves; i.e. a wave that is filtered is annihilated. Further upwards the amplitude increases slowly.

Waves can propagate over considerable distances and times. Therefore it is not sufficient to look at potential sources in the vicinity of the flight track. Even if sources are found, the waves may have propagated to other places (away from the point of interest), while waves from sources outside the domain may have propagated to the location of observation. For resolved waves the model takes care of these issues. Waves seen in the WRF at the location of the flight may have travelled from remote places, yet the important information is not their origin, but that they were present during the measurement.

To trigger turbulence, wave breaking is necessary. Such events are triggered by dynamic or convective instabilities or by wave–wave interactions e.g.. In the WRF, the breakdown to turbulence is parameterised by solving a prognostic equation for TKE, which is based on production terms due to shear and buoyancy obtained from the resolved flow. TKE is plotted in Fig. d. It peaks near 10 km height at the location of the flight. This corresponds nicely to the intense turbulent layer observed by LITOS. It is reproduced in the WRF due to the shear instability on scales resolved by the model, highlighting the geophysical significance of the layer.

A small LITOS payload of second generation was launched from Kühlungsborn on 27 March 2014 at 10:10 UT. It was carried by a comparatively small (3000 g) balloon and a 60 m dereeler.

Figure a shows temperatures smoothed over 15 data points (∼ 150 m) as well as zonal and meridional winds. The smoothing is necessary because for this flight the temperature measurement is perturbed by radiation effects as the radiosonde was incorporated in the main payload; these effects get worse with increasing altitude. Temperatures decreased up to the tropopause at 9 km. Between 9 and ∼ 30 km altitude they stayed nearly constant and started to increase further upwards. Winds were easterly and turned northerly above ∼ 20 km altitude. A strong southeasterly jet was present between ∼ 6 and 10 km height. Superposed are signatures of small-scale gravity waves. Wind shears originating from the jet may have excited turbulence and/or waves. The effect of the shear is visible as a layer with enhanced dissipation at this altitude (see below). Richardson numbers are shown for altitudes below 9.4 km only because they involve derivatives of the temperature profile, which was disturbed by radiation effects as described above.

Dissipation rates are presented in Fig. d. The data below 650 m altitude are affected by the unwinding of the dereelers while the data above the tropopause are subject to wake influence. Therefore, these are discarded and not shown in the plot. Dissipation rates varied over several orders of magnitude within small altitude ranges (typically a few 10 m). The running average shows some structure in the troposphere, e.g. a few layers that are standing out with larger rates. Most prominently this can be seen near 8 km. That is in the same altitude as the wind shear due to the jet, which speaks for shear-induced turbulence. Precisely, there were two turbulent layers from 7.5 to 7.9 km and from 8.1 to 8.3 km height; within both, Richardson numbers were below 1 and partly below 1/4. Other sheets with large dissipation were detected, e.g. near 6.1 km and around 3.0 km altitude.

To validate the corresponding WRF simulations, winds and temperatures interpolated to the flight track are plotted in Fig. a. They agree very well with the radiosonde data.

Figure  depicts WRF results for the time of the flight. Panel (a) shows horizontal winds at 850 hPa, which were easterly or south-easterly. In panel (b) horizontal winds are depicted as altitude section, showing that the strong jet did not have much structure in a horizontal direction, while the sharp vertical structure is reproduced as observed by the radiosonde. Panel (c) shows a vertical profile of vertical winds. Wave patterns are visible, which stretch over the whole altitude range. Particularly, a superposition of a wave with long vertical wavelength (λz≈8km) and nearly horizontal phase fronts and waves with short horizontal wavelength (10 to 20 km) and phase fronts in the vertical can be seen. Figure d shows the TKE. Outside the boundary layer there is an enhancement near 7.5 km altitude. It corresponds nicely to a thick, strong turbulent layer in the measurement by LITOS between ∼ 7 and 8.5 km height. Within this observed turbulent layer, which in fact consists of several layers, Richardson numbers are smaller than 1 almost everywhere and at times smaller than 1/4.

A night-time flight with LITOS was launched from Kühlungsborn on 11/12 July 2015, at midnight local time (22:01 UT on 11 July). A dereeler of 180 m (with a 3000 g balloon) was used for payload suspension, making balloon wake effects negligible for this flight. The radiosonde was positioned 60 m below the main payload to avoid disturbances of the temperature sounding.

The observed background parameters are depicted in Fig. a and b. Westerly winds prevailed up to ∼ 19 km altitude, whereas above winds came from the east. This change in direction was not associated with a significant wind shear because velocities were small in that altitude region. A jet is visible at about 10 km height. Superposed on the winds are signatures of small-scale gravity waves. Above the tropopause at 11.3 km altitude there was a small tropopause inversion layer. Higher up temperatures remained rather constant up to ∼ 20 km, where they started to increase.

Richardson numbers were typically lower than for the other flights, indicating less stability. There are several layers where the Richardson number is below the critical limit of Ric (1/4). These layers are relatively thin.

Energy dissipation rates (data below 550 m are excluded due to disturbances from the launch procedure) showed a strong patchy structure, with enhanced dissipation at, for example, ∼ 2.0, 3.8, 7.2, 8.9, 11.0, 12.1, and 14.3 km. These layers of intense turbulence mostly corresponded to Richardson numbers smaller than Ric=1/4, or at least to Ri<1. But particularly in the lower stratosphere between 11 and 15 km, turbulence also occurred for high Richardson numbers. It should be kept in mind that the Richardson number depends on the scale on which it is computed e.g.. A higher resolution (i.e. computing Ri on smaller scales) may result in locally smaller Ri numbers, because the computation on large scales yields a kind of average. Similarly, in large eddy simulations found larger Richardson numbers for smaller model resolutions (i.e. larger scales). Here, due to measurement noise a smoothing over 150 m has been applied before computing Ri, determining the resolution. However, this issue cannot explain the whole discrepancy. In simulations of gravity waves, found instabilities and onset of turbulence for Richardson numbers both smaller and larger than 1/4. He noted that the theory by and is not applicable to his simulations because the gravity wave phase propagation and thus the wave-induced shear is slanted. In the real atmosphere, waves usually propagate at a tilt (i.e. the shear is not orthogonal to the altitude axis). has already discussed slantwise static instabilities created by gravity waves. He developed a wave period criterion for turbulence by comparing the e-folding time of the (slantwise) instability with the period of the wave. Turbulence is more likely to occur for slantwise static instability than for vertical static instability. In the light of these comments, the violation of the Richardson criterion for the LITOS measurements is comprehensible.

Above ∼ 15 km altitude, hardly any turbulence was detected; only a few thin turbulent layers were observed. Thus above 15 km the average dissipation rate (for which no turbulence is counted as zero) was only 0.01 mWkg-1, while below 15 km it was 0.64 mWkg-1.

Results from corresponding WRF simulations are depicted in Fig. . Horizontal winds at the 850 hPa level were mainly westerly. The altitude section shows that the strong jet did not have much variation in the horizontal direction. Vertical winds reveal wave patterns that are particularly intense around the tropopause and gradually become weaker near ∼ 15 km, with less amplitude above. This drop in wave amplitude is at the same altitude as the drop in observed dissipation. The TKE has enlarged values around 3 km altitude and near the tropopause; however the enhancement is small at the flight path. Correspondingly, the thickness of the strong turbulent layers detected by LITOS is relatively small, meaning that these dissipative layers are potentially not resolved in the model.