Although the stratosphere is mostly stably stratified, breaking of gravity waves and instabilities cause turbulence and energy dissipation. This modifies the energy transport from the troposphere to the mesosphere. The amount of energy converted into heat is described by the turbulent energy dissipation rate ε. Moreover, turbulence is an important parameter for the vertical mixing of trace species. As in the stratosphere turbulent dissipation occurs on small scales of centimetres and below, measurements are technically challenging and therefore sparse e.g..

In order to enlarge the knowledge of turbulence by exploiting existing measurements available for large geographical areas and several years, the extraction of turbulence parameters such as ε from standard radiosonde data (vertical resolution 5 m) has been proposed . The evaluation uses the method developed by to detect static instabilities as a proxy for turbulence. Note that such a measurement is somewhat different from measuring the turbulent motions directly, as done by LITOS. For example, within an instability turbulence may have not yet been developed or, on the other hand, turbulence might be still active while the instability has already deceased. Additionally, turbulence may not be related to static instabilities at all.

The Thorpe analysis of unstable layers is done by comparing a measured potential temperature profile to an equivalent (statically) stable one obtained by sorting. This means that the order of the data points is changed upwards and downwards to yield a statically stable profile with monotonously increasing potential temperature. Precisely, the Thorpe displacement DT is defined by the vertical displacements needed for the sorting; i.e. if an air parcel at altitude zj is sorted to zk, then the Thorpe displacement at zj is DT(zj)=zj-zk. The Thorpe length is the root mean square of the Thorpe displacements taken over an unstable layer: LT=rms(DT). It describes the distance over which heavier air parcels are carried above lighter ones. use the Thorpe method for statistical analysis without computing dissipation rates.

The Ozmidov length scale LO=αεN3, where ε is the (kinetic) energy dissipation rate, N the Brunt–Väisälä frequency and α a numerical constant near unity, represents the vertical scales of the largest turbulent eddies . For the determination of the dissipation rate from a Thorpe analysis, the key assumption is a proportionality between Thorpe and Ozmidov lengths, LO∝LT. This relation has been extensively studied in the ocean, and the assumption is fulfilled to a good extent . But for the atmosphere there are only few examinations of the proportionality e. g.. With our new high-resolved instrument LITOS (Leibniz Institute Turbulence Observations in the Stratosphere) , the energy dissipation rate ε is obtained independent of LT by means of spectral analysis of wind fluctuations. Thus it is possible to check the relation LO∝LT.

Please note that our comparison involves two parameters: (a) evaluation method (Thorpe or spectral analysis) and (b) vertical resolution (low or high). We concentrate on results from high-resolved spectral analysis (as a very precise method of ε determination) and low-resolved Thorpe analysis. Such a Thorpe evaluation of radiosonde data has been proposed for extensive use . Note that call 1 Hz (5 m) high resolution, while we call it low resolution (compared to LITOS with 8 kHz). In principle, the Thorpe analysis can also be performed on data with higher resolution, as done, e.g., by for temperature data with a 50 Hz sampling rate; however, these data are rarely available compared to those of standard radiosonde. Furthermore, a kind of spectral analysis can be used to determine dissipation rates from low-resolution wind data , but this method depends on the absolute value of the wind velocity, which is not available for our measurements (see next section).

In Sect. , the measurement principle of LITOS and the determination of the energy dissipation rates with both methods are shortly reviewed. The independent measurements of LO and LT are compared in Sect. . Section  shows results for the energy dissipation rate ε from both a Thorpe analysis and our high-resolved spectral analysis. Conclusions are drawn in Sect. .

As described in , LITOS is a balloon-borne instrument which measures winds with high vertical resolution of millimetres. The wind sensor is a constant temperature anemometer (CTA), which facilitates the cooling effect on a heated wire of 5 µm diameter. To infer wind velocities from the anemometer voltage, a calibration in the same ambient conditions (pressure, temperature) is required. This is not possible for a balloon flight, as the pressure varies within several orders of magnitude during the flight. Nevertheless, we are only interested in the spectral form, and the absolute values are not important (see below); therefore we use the anemometer voltage for the analysis. The vertical resolution is obtained by applying a sample rate of 8 kHz with a balloon ascent rate of 5 ms-1. To date, three flights on large (∼10 000 m3) balloons have been performed, namely Balloon Experiments for University Students (BEXUS) 6, 8 and 12 in 2008, 2009 and 2011, respectively. They were launched at Kiruna (68∘ N, 21∘ E) in autumn. For BEXUS 6, the radiosonde data are partly disturbed so that it is not considered in this article.

The left panel in Fig.  shows an example of a time series of the anemometer voltage of the BEXUS 12 flight. Large-scale motions have already been removed by subtracting a spline. At altitudes with small variations (≲ 1 mV, e.g. from 10.28 to 10.3 km), the signal mainly shows instrumental noise; this corresponds to a calm region. Large fluctuations, as in the height range of 10.18 to 10.28 km, correspond to turbulence. Note that there is a substructure which divides the turbulent region into different patches. For the patch from 10.27 to 10.28 km (shaded in the graph), the power spectral density (PSD) is plotted in the right panel of Fig.  (blue curve). An inertial regime with a -5/3 slope and the transition to the viscous subrange with a -7 slope is identified. The part below ∼10-2 m spatial scale with approximately constant PSD corresponds to the instrumental noise level. As the transition to the inertial range is resolved, a fit of the model in the form given by is applied to the experimental data (red curve). This gives the inner scale l0, i.e. the transition from the inertial to the viscous subrange. In the example, l0=1.9×10-2m±4.5×10-3m (fit error). Note that l0 does not depend on the absolute value of the PSD, only on identifying the bend in the spectrum. From the inner scale, the energy dissipation rate is obtained by ε=c4ν3l04, where ν is the kinematic viscosity (derived from the radiosonde measurement of temperature and pressure on the same gondola), and c=5.7 . For the example in Fig. , ε=3.2×10-4Wkg-1 ± 3.0×10-4Wkg-1.

In order to obtain a vertical profile of energy dissipation rate, a sliding window of 5 s (roughly 25 m altitude) is used. For each window, ε is computed according to the procedure described above. For non-turbulent spectra, ε is set to 0. A spectrum is regarded as non-turbulent if the noise-level detection fails, if the inner scale l0 is not within the fit range, if ε has implausible values (less than 0 or greater than 100 W kg-1) or if the mean distance between the fit and the data is larger than a fixed threshold. That means the decision is made automatically based on a set of objective criteria. The resulting ε profile has a vertical resolution of ∼10 m due to the selected overlap.

The Thorpe analysis is performed similar to the procedure described in on data from a Vaisala RS92 radiosonde, which was on the same gondola as the CTA sensors. Moisture is handled using the routine given by . To this end, saturated regions are detected, and a composite potential temperature profile Θ* is computed by integration of ∂Θ/∂z using the moist buoyancy frequency within those saturated regions and the dry buoyancy frequency otherwise. The left panel of Fig.  shows the potential temperature profile for the BEXUS 12 flight. In the inset, the part from 15.48 to 15.80 km is magnified for better visibility of instabilities, which manifest as negative gradients of potential temperature. The Thorpe displacement is shown in the right panel of Fig. . Large displacements correspond to large vertical extents of unstable layers. To identify unstable layers and their vertical extension, the cumulative sum of the Thorpe displacement (which is negative within an unstable region and 0 within a stable one) is used. To select real overturns and discard negative potential temperature gradients originating from measurement noise, a statistical test is applied. To this end, the range of the potential temperatures within an inversion is compared with the range of a pure noise sample of the same length . We assume here that the standard deviation of instrumental noise in segments of 200 m is described by half the square of the standard deviation of first differences of the potential temperature after trend removal. For each detected unstable layer, the Thorpe length is computed according to Eq. ().

Only significant overturns with a 99 % percentile are used, discarding ∼45 % (BEXUS 8) and ∼30 % (BEXUS 12) of the inversions as noise-induced. The mean trend-to-noise ratio (TNR) is ξ¯=1.7 for the BEXUS 8 flight and ξ¯=4.1 for the BEXUS 12 flight.

Several thin layers of only 10 m or 20 m passed the significance test. We are aware that this is on the edge of radiosonde capability. These thin layers would to a large extent be suppressed if the instrument noise were set to the standard deviation of first differences divided by 2 as done by . However, LITOS also shows many thin layers. Hence, this procedure would result in many fewer coincident layers especially in the stratosphere and, by this, bias the comparison. Nevertheless, as we will describe below, our main results are independent from the procedure of noise calculation.

A plot of the Thorpe length for the BEXUS 12 flight is shown in the left panel of Fig. . Unstable layers take up 50 % of the altitude and can be found in the whole range. Large Thorpe lengths stand out, e.g., at 5 km, between 5 and 10 km and near 25 km altitude, corresponding to the large values of DT in the right panel of Fig. . Mean values of LT are 29 m in the troposphere and 22 m in the stratosphere; i.e. the Thorpe length is slightly larger in the less stable troposphere.

The Ozmidov scale is computed from the energy dissipation rate obtained by LITOS, using Eq. () and α=1. The Brunt–Väisälä frequency N is calculated from the radiosonde data as it only slowly varies with altitude. In that computation, the sorted potential temperature profile is used instead of the original data, because a background stratification is needed and an imaginary N should be avoided Sect. 3. The result for the BEXUS 12 flight is plotted in the right panel of Fig.  (cyan curve). According to LITOS, 53 % of the atmosphere is turbulent, i.e. ε>0 and hence LO>0.

Figure  shows the Thorpe length and the Ozmidov scale for the altitude range of 15.48 to 15.80 km. As LITOS computes ε on a constant grid independent of the layers, the substructure of larger turbulent layers can be seen (e.g. from ∼15.5 to 15.62 km), while the Thorpe length is a per-layer value by construction. From 15.706 to 15.789 km, only LITOS observes turbulence while the Thorpe method does not. The decrease of potential temperature is not significant, so the Thorpe method is blind for the turbulent motions.

In order to do a comparison between both length scales, the layers where both methods detect turbulence are selected. For BEXUS 8 (BEXUS 12), 86 % (69 %) of the significant, unstable layers are also detected by LITOS, and 90 % (88 %) of the layers detected by LITOS intersect with a significant, unstable layer. The energy dissipation rate (obtained from LITOS) is averaged over the layer (as detected by the Thorpe analysis of radiosonde data). Such mean values over a Thorpe layer will be denoted by averaging brackets 〈⋅〉. For each unstable layer, the resulting 〈ε〉 is plugged into Eq. () to infer an Ozmidov scale LO‾=〈ε〉/〈N〉3 for the layer. The blue bar plot in the right panel of Fig.  shows a graph for the BEXUS 12 flight. The layer near 5 km with large LT, e.g., is also seen in LO‾, albeit less pronounced. In contrast, at ∼10 km altitude the Ozmidov scale is larger than the Thorpe scale. Mean values of LO‾ are 15 m in the troposphere and 6 m (i.e. only half the value) in the stratosphere. Thus, in qualitative agreement with Eq. (), the less stable troposphere shows on average a larger Ozmidov scale, i.e. larger eddies. As the Thorpe length shows similar behaviour (see above), this generally supports the assumption of a relation between LT and LO.

In Fig.  Ozmidov scale LO‾ and Thorpe length LT are plotted against each other for those 136 (175) significant, unstable layers of the BEXUS 8 (BEXUS 12) flight where turbulence has been detected by LITOS. Both length scales are of the same order of magnitude, but no direct relation between them can be seen in either flight. The correlation coefficient between both is 0.32 for BEXUS 8 and 0.33 for BEXUS 12. Note that LT is limited by the resolution of the radiosonde (∼10 m). An analogous plot for the simultaneous layers obtained using the more restrictive noise estimation of contains similar scatter with no apparent correlation.

The histograms in the top and in the right axes in Fig.  show the distributions for LO‾ and LT, respectively, for the composite dataset of BEXUS 8 and BEXUS 12. The maximum for the Thorpe length is slightly larger than for the Ozmidov scale. The decrease towards large scales is similar for both lengths. At small scales LT is limited by the resolution of the radiosonde which produces the cut-off at 10 m, while the histogram for LO‾ shows a continuous decrease.

In contrast to our measurements, an approximate proportionality between Thorpe and Ozmidov lengths is observed in the ocean e.g.. For example, find that most of their data fall between LO=4LT and LO=(1/4)LT, with a range from 10-2 m to 102 m. This is the basis for applying the Thorpe analysis on atmospheric data. Several authors e.g. have inferred energy dissipation rates from the Thorpe analysis by plugging LO=cLT into Eq. () and solving for ε, thus getting ε=c2LT2N3. Knowledge about the constant c2 is very limited (see discussion below), and only provide some information based on stratospheric data. With our high-resolved wind data we can determine this constant independently. found the distribution of the ratio LT/LO to be lognormal, which implies (LO/LT)2=c2 to be lognormal as well. Logarithmic histograms of (LO‾/LT)2 for the BEXUS 8 and BEXUS 12 flights are presented in Fig. . The red curves display the most likely normal distributions for the logarithmic data. They show a sufficient agreement to the histograms and are both centred around ∼0.1. The distribution of values is fairly broad: the full width at half maximum (FWHM) spans ∼1.9 orders of magnitude.

We have calculated the same histograms also for the layers that are extracted using the more restrictive noise estimation of . Even if here only 36 (15) coincident layers are detected for BEXUS 8 (BEXUS 12), the distribution of (LO/LT)2 is of similar width. The most probable values are slightly lower but agree within the uncertainty given by the distribution of the data.

The relation between the energy dissipation rate ε and the length scales discussed above involves the Brunt–Väisälä frequency; see Eqs. () and (). Thus the ε values computed from LITOS via spectral analysis and the ones from the radiosonde via Thorpe analysis have to be compared separately.

The left panel of Fig.  shows an altitude profile of energy dissipation rates obtained from the Thorpe analysis of the radiosonde on BEXUS 12, assuming LO=cLT with c2=0.3 as in . In the right panel, the altitude profile of energy dissipation obtained from LITOS is plotted in cyan, while the blue bars depict the mean values 〈ε〉 over the unstable layers detected by the Thorpe analysis, for comparability. On average, the values are of the same order of magnitude, and the profiles have a similar structure. The large dissipation near ∼10 km in the CTA data does not stand out in the Thorpe analysis. The mean value over all significant, unstable layers from Thorpe, 1 mWkg-1, is larger than the one from LITOS, 0.3 mWkg-1. For BEXUS 8 the averages are 3 mWkg-1 from Thorpe and 2 mWkg-1 from LITOS. That fits the fact that the used value for c2, 0.3, is larger than the one obtained from our own data, c2=0.1 (cf. Fig. ). If the whole ε profile (not only the unstable layers detected by Thorpe) is taken into account, the average dissipation rate obtained by LITOS is 0.4 mWkg-1 for BEXUS 12 and 2 mWkg-1 for BEXUS 8. To get a closer look, the deviation of 〈ε〉 from LITOS to ε inferred from the Thorpe analysis indicated by the ratio 〈εLITOS〉/εThorpe of the blue and green curves in Fig.  is plotted in Fig. . It reveals a large range of 5 orders of magnitude. Overall, for 71 % (BEXUS 8: 64 %) of the layers, ε inferred from the Thorpe analysis is larger than the value from the spectral analysis. That the ratio is sometimes larger and sometimes smaller than unity illustrates that the most likely value of c2 does not contain the whole information, but the width of the distribution is important. The correlation coefficient between 〈εLITOS〉 and εThorpe is 0.06 (BEXUS 8: 0.39). Due to the influence of the Brunt–Väisälä frequency, pronounced peaks in LO or LT (Fig. ) do not necessarily correspond to large ε (Fig. ), e.g. at ∼5 km.

In this paper, the first extensive examination of the relation between the Thorpe length LT and Ozmidov scale LO for stratospheric conditions was performed, using the new high-resolution instrument LITOS and a radiosonde on the same gondola. Therewith, the assumption for computing energy dissipation rates ε from a Thorpe analysis of standard radiosondes, namely the proportionality LO∝LT, was checked. In our data no obvious relation between LO‾ and LT can be seen, particularly no proportionality. The proportionality “constant” used in radiosonde analyses, c2=(LO‾/LT)2, shows a very broad distribution with a width of ∼2 orders of magnitude. This is also reflected in the large deviation of ε values up to a factor of ∼3000 obtained with both methods. Nevertheless, although the values for individual layers are highly variable, the mean of c2 is 0.1 for both BEXUS 12 and BEXUS 8, which is close to 0.3 used by , who reviewed oceanic measurements to obtain that value. obtained c2=1.0 by a comparison of radiosonde data to radar measurements. used c2=1.32 (c=1.15) referring to a French thesis; this value was obtained from selected thick stratospheric layers (>200m) with statistically homogeneous turbulence. However, in those publications no data basis, distribution width or error is given. Recently, reported a few case studies of turbulent layers in the troposphere detected simultaneously by radar and balloon; using their reported estimates of LT and LO leads to values of c2 between 0.1 and 1.6.

One reason for discrepancies from the proportionality LO∝LT may be that large overturns might change significantly during the time the sensor needs to fly through the layer, such that the sorting procedure then no longer makes sense. Furthermore, direct numerical simulations by indicate that LO/LT is not constant but rather depends on the age of turbulence.

Some turbulent layers are not detected at all by the Thorpe analysis. Those are not associated with a (significant) negative gradient of potential temperature, which is necessary for detection by the Thorpe method. Not all turbulence is related to static instabilities. Even if initially a negative potential temperature gradient may have occurred, it is removed by the turbulent motions which outlive the instability; such fossil turbulence cannot be detected by the Thorpe method. Apart from that, turbulent layers may be too thin to be observed with the relatively coarse vertical resolution of the radiosonde. On the other hand, some unstable layers detected by the Thorpe analysis are not observed by LITOS. An explanation is that the static instability may not yet have led to turbulent motions. In these cases, a correspondence between both measurements is not expected.

Not all layers are detected by both systems. Of the significant layers detected by the Thorpe analysis, 86 % (BEXUS 8) and 69 % (BEXUS 12) are also detected by LITOS. For BEXUS 12, the mean thickness of significant, unstable layers as detected by the Thorpe analysis is 53 m. The mean thickness of those significant layers also detected by LITOS is 63 m; that of significant layers not detected by LITOS is only 31 m. That means that the simultaneous detection depends on the size of the layer; mainly thin layers are detected by only one method. But as this only applies to ≲30 % of the layers, and those layers were taken out of the comparison, the bias for our results should be small.

For LITOS, the detection limit for l0 on small scales (i.e. high frequencies or large ε) is given by the sampling rate. This limit has been encountered in a few cases for small regions where the inertial range extends further than the Nyquist limit of 4 kHz. For large scales (i.e. low frequencies or small ε), the detection limit is determined by the trend removal and the window length. As a reasonable part of the inertial range has to be resolved to enable a fit, the limit is estimated to ∼1 m. The maximal identified l0 values of 10 cm and 8.7 cm for BEXUS 8 and BEXUS 12, respectively, were far below this limit. That means that these limitations do not affect the results.

To date, we have only two flights with data usable for the analysis presented in this paper, namely BEXUS 8 and BEXUS 12, which both took place at polar latitudes near autumn equinox. Of course they cannot represent the whole variability of the stratosphere. Nevertheless, although there are differences between both flights, such as dissipation rates being on average 1 order of magnitude higher for BEXUS 8, these are not relevant for the results discussed above. More flights with our new high-resolution instrument are planned to broaden the data basis.

Our results question the applicability of the Thorpe analysis for the extraction of energy dissipation rates for individual turbulent layers. Nevertheless, statements in the statistical mean seem to be possible. Further research on the relation between Thorpe and Ozmidov lengths and the temporal evolution of turbulence is necessary.