Turbulence kinetic energy dissipation rates estimated from concurrent UAV and MU radar measurements

MU radar

CU DataHawk UAV

The ShUREX campaign and the characteristics of the CU DataHawk UAVs and onboard sensors are described by Kantha et al. (2017). The UAVs were equipped with a custom autopilot programmed to execute a preplanned trajectory near the MU radar. The UAVs could also be commanded to sample interesting atmospheric features revealed by the MU radar in near real time. For the present purpose, we only consider measurements performed during “vertical” ascents and descents. When moving up or down, the UAVs were flying along helical trajectories ~ 100–150 m in diameter at a typical vertical velocity rate of ~ 2 m s⁻¹. The maximum flight altitude was limited to ~ 4.0 km ASL by both battery capabilities and air traffic regulations. Among 41 flights performed during the 2016 campaign, 16 science flights provided (totally or partially) valuable data for comparisons with MU radar data. Hereafter, they will be denoted ‘FLT16-xx’, where ‘16’ refers to the year and ‘xx’ is the flight number. In 2017, 23 science flights were available for analysis.

The UAVs were equipped with a variety of sensors for atmospheric measurements (Kantha et al. 2017). Among these sensors, a commercial IMET sonde provided measurements of pressure, temperature and relative humidity (PTU) at 1 Hz. Velocity of the air flow relative to the UAV was measured by a fast-response Pitot-static tube and a differential pressure sensor, with the Pitot tube mounted at a height of 3 cm above the vehicle so as to project into the free stream above the aerodynamic boundary layer. This sensor was sampled at an effective rate of 400 Hz. At the nominal airspeed of 14 m s⁻¹, the digital resolution was 0.042 m s⁻¹ (see Kantha et al. 2017). In addition, a fast-response (< 1 ms) cold wire sensor was also available for temperature measurements sampled at 800 Hz.

There were too many flights to describe their characteristics in detail. Most of the flights had ascents and descents (denoted by ‘A’ and ‘D’ when necessary) sometimes separated by horizontal legs of various durations (e.g., FLT16-22 and FLT16-38). Unanticipated blocking of the Pitot tube (used also by the autopilot for flight control) by precipitation sometimes produced short time span (~ a few tens of seconds) downward motions during ascents (e.g., FLT16-05, FLT16-15). These sources of aberrant data points were manually removed.

The meteorological conditions were checked every day before deciding to launch UAVs or not since they cannot fly during rainy conditions and strong winds (> 10–15 m s⁻¹). The state of the lower atmosphere during the flights could be known from the available data without additional meteorological information. Indeed, among other things, the relative humidity measurements made by humidity sensors onboard UAV indicated flight in clouds and the radar images provided precise information on the vertical extent and evolution of the convective boundary layer when it exceeded the altitude of the first radar sampling gate. Therefore, turbulence associated with dry or saturated convections could be easily identified from the datasets and could be removed from statistics when focusing on stratified and clear air turbulence only. Actually, the weather was almost clear through the observations above the convective boundary layer.

Theoretical expressions of \(\varepsilon\) from radar data

Despite its apparent complexity, Eq. (3) has advantages with respect to Eq. (2). \(\varepsilon_{W}\) can be estimated solely from the radar data, while \(\varepsilon_{N}\) requires estimates of N (usually from balloon measurements) or standard climatological values as default values (e.g., Weinstock 1981; Deghan et al. 2014). In addition, \(\varepsilon_{W}\) can be used whatever the turbulence source may be (convective or shear flow instabilities), assuming that inertial turbulence is observed and \(L_{\text{out}} \gg\) 2a, 2b. Finally, \(\varepsilon_{N}\) requires, in principle, the estimation of moist \(N^{2}\) when air is saturated, because saturation modifies the background stability due to latent heat release. This additional difficulty does not seem to have been considered in the studies related to TKE dissipation rate estimates from ST radar data. However, we shall see that the accuracy of \(N^{2}\) is not an important issue because our analyses reveal a fundamental inadequacy of \(\varepsilon_{N}\). This conclusion goes beyond the problem of estimating \(N^{2}\) properly.

Equation (3) or similar expressions were used by Gossard et al. (1982) and Chapman and Browning (2001), for example, using UHF radars at similar spatial resolutions as the MU radar and by McCaffrey et al. (2017) at vertical resolution of ~ 25 m.

Practical methods from radar data

The beam-broadening correction \(\sigma_{b}^{2}\) requires the knowledge of horizontal winds estimated from off-vertical beam data, and these winds may not be exactly those at the altitudes sampled by the vertical beam. It is another source of bias (Deghan and Hocking 2011), but difficult to correct in general. However, since the measurements were taken for low altitudes (< 4.5 km), this problem should be minimized here because the sampled altitude differences between the vertical and oblique directions do not exceed a few tens of meters.

Finally, Eq. (4) does not include correction due to gravity wave contributions (e.g., Naström 1997). Here, it is expected to be negligible: the dwell time (~ 25 s) should be sufficient for minimizing their contribution because it is a small fraction of internal gravity wave periods. The details of the practical procedure for estimating \(\sigma^{2}\) from Eq. (4) are given in “Appendix”.

A complete vertical profile of \(\sigma^{2}\) is calculated from time series of ~ 25 s in length every 6.144 s (overlapping of a factor 4) at a vertical resolution of 150 m (see Table 1). For comparison with UAV measurements, it must be realized that the UAV provides data only along a specific altitude versus time trajectory. Figure 1 shows the strategy used for reconstructing pseudo-profiles of \(\sigma^{2}\) along the UAV paths. Since the UAVs were flying in the vicinity of the MU radar, we calculated temporal averages of \(\sigma^{2}\) (\(\left\langle {\sigma^{2} } \right\rangle\)) over a few minutes only (4 min was arbitrarily selected) about the time-height location of the UAV (see Fig. 1), initially assuming that the UAV was flying directly over the radar, so it detected the same atmospheric structures at the same time as the radar. \(\left\langle {\sigma^{2} } \right\rangle\) was estimated for all altitudes sampled by the UAV, using a linear interpolation of the \(\sigma^{2}\) profiles (at 150-m resolution) at these altitudes. The same procedure was used for all other radar parameters (e.g., echo power, Luce et al. 2017). The height variations of the pseudo-profiles of \(\left\langle {\sigma^{2} } \right\rangle\) are thus due to a combination of the height and time variations of \(\sigma^{2}\). Figure 2 shows an example of pseudo-profiles of \(\left\langle {\sigma^{2} } \right\rangle\) during the ascent A1 and descent D1 of FLT16-15. The gray areas show the rms value of \(\sigma^{2}\) during the time averaging for A1 and D1, respectively.

The processing was then refined to account for the actual horizontal offset between UAV and radar by taking time lags due to wind advection into account, assuming frozen advection of the turbulent irregularities by the wind along the wind direction. This often provided higher correlation coefficients between \(\varepsilon_{U}\) and the radar-derived \(\varepsilon\) profiles, especially when the UAV was flying directly upstream of the radar. Yet, because the improvements were quite marginal, the procedure is not described in detail here. Note that time offsets could be avoided by flying in the beam of the radar, but the vehicle produces strong echoes that obliterate the turbulence measurements in the volume of interest, requiring a more complex analysis that considers neighboring times or altitudes (e.g., Scipión et al. 2016).

Figure 3a shows the histogram of the Doppler width \(2\sigma_{m}\) for all the available radar data surrounding the 16 UAV flights of ShUREX2016 in the height range 1.345–7.195 km ASL (corresponding to the first 40 radar gates). Similar statistics were obtained for 2017 data (not shown). It also shows the detection threshold (approximately ~ 0.2 m s⁻¹) for the radar configuration and processing method used. Figure 3b shows the corresponding histogram of \(2\sigma\) (i.e., the Doppler width after beam-broadening corrections). Due to estimation errors (especially when SNR is low), some \(\sigma\) values can be negative. They are not shown in Fig. 3b. Figure 3c shows the histogram corresponding to the values of \(2\sigma\) estimated along the UAV flight track (as shown in Fig. 2). The peaks around 0.2 m s⁻¹ are of course artificial and result from the minimum detection threshold of the radar. A bias is thus expected when comparing the lowest levels of radar-derived \(\varepsilon\) with \(\varepsilon_{U} .\) In addition, remaining small contaminations by various artifacts may still be present despite careful examination of the spectra (see “Appendix”). They can be a source of important biases for the lowest levels.

It has to be noted that the \(2\sigma\) values calculated along UAV flight tracks are not affected by estimation errors due to low SNR, because the UAVs did not exceed the altitude of 4.05 km ASL and SNR was always larger than 20 dB below this altitude. In addition, because UAVs flew during relatively weak winds (~ < 10–15 m s⁻¹), the beam-broadening effects were relatively weak. Consequently, the conditions were favorable to errors in \(2\sigma\) estimates being small and, in particular, very few negative values were obtained in the altitude range of the UAV measurements so that they should not affect the statistics.

Estimation of \(\varepsilon_{U}\) from Pitot sensor data

Experimental tests in outdoor flight showed that the flow acceleration over the UAV body did not damp the turbulent variations about the mean for the scales of interest, contrary to what it was expected from earlier tests in wind tunnel (which generated much smaller-scale turbulent fluctuations, not shown). Therefore, significant underestimations of energy dissipation rates from these effects are not expected.

Practical methods of estimations from Pitot sensor data

The problem of extracting the dissipation rate from UAV data is now reduced to that of identifying an inertial domain (when it exists) and estimating \(\beta\). Two different methods were applied with very similar results.

The first method consists in selecting an appropriate frequency band from spectra calculated from 5-s. time series chunks of Pitot data. A careful scrutiny of all the U frequency spectra shows that the highest probability to observe an inertial domain is found between 1 and 10 Hz. Two examples of typical spectra are shown in Fig. 4. At frequencies higher than 10 Hz, the spectra can be contaminated by noise when turbulence is weak (e.g., right panel of Fig. 4), and by artefacts (multiple peaks) mainly due to motor vibrations of the UAVs, especially during ascents (e.g., left panel of Fig. 4). The characteristics of these contaminations are specific to each UAV and flight, and they can also drift in time due to throttle variations. FLT16-15 was one of the most contaminated among the useful science flights. In practice, for the present purpose, we decided to estimate \(\beta\) from the spectral levels between 1.0 and 7.5 Hz. The spectral slopes between 1.0 and 7.5 Hz were estimated for all the time series of the 39 flights of ShUREX2016 and ShUREX2017. The corresponding histogram is shown in Fig. 5. The mean slope is − 1.64 (i.e., very close to the inertial slope − 5/3). The width of the distribution can be partly due to estimation errors when estimating slopes on individual spectra. Therefore, from a statistical point of view, the frequency band 1.0–7.5 Hz shows properties consistent with the existence of an inertial subrange.

The second method is based on the selection of spectral bands exhibiting a -5/3 slope in a frequency domain delimited by 0.1 and 40 Hz (arbitrarily) from spectra calculated from time series chunks 50 s in length. The width of the spectral bands is a constant 0.699 decade, e.g., log10(5 Hz)–log10(1 Hz), and 39 overlapping bands are used. For each of these bands, the spectral slope s is estimated from the calculation of the variances in two spectral “sub-bands” of identical relative logarithmic width. An inertial subrange is inferred when \(s = - \,5/3\, \pm \,0.25\) for at least 3 consecutive spectral bands. The numerical thresholds were chosen in order to fit, as far as possible, the results that would have been obtained from visual inspection of the spectra. In some cases, the criteria may appear too loose or too restrictive, but it appears to be efficient for rejecting most spectral bands affected by instrumental noise and contaminations. A more thorough description of the method and results is in preparation.

The above two methods were applied to ShUREX2016 and ShUREX2017 data and produced the same statistical results.

Figure 6 shows examples of pseudo-vertical \(\varepsilon_{U}\) profiles in linear scales during the ascent (A1) and descent (D1) of FLT16-05 and FLT16-08 in altitude ranges covered by MU radar (i.e., above 1.345 km). The profiles are rather distinct during A1 and D1 of FLT16-05, but quite similar during A1 and D1 of FLT16-08. They clearly reveal altitude ranges with multiple peaks of enhanced TKE dissipation rates. These ranges are emphasized by the smoothed profiles shown by the solid and dashed black lines. The former was obtained by using a 30-point rectangular window applied to the time series sampled at 2.5 s (corresponding to 75 s averaging), and the latter by using a Gaussian averaging window. The (non-normalized) Gaussian function was taken as equal to \({ \exp }\left( { - \,z^{2} /2\alpha^{2} } \right)\), where \(\alpha = a/\sqrt 2 = 75/\sqrt 2 \,{\text{m}}\) in order to fit the characteristics of the expected range weighting function of the MU radar. The two methods provide very similar smoothed profiles. Therefore, the statistics of the comparison results should not depend on the method used for smoothing the \(\varepsilon_{U}\) profiles.

Estimation of \(L_{C}\)

The left panel of Fig. 7a shows \(\left\langle {\sigma^{2} } \right\rangle^{3/2}\) versus \(\varepsilon_{U}\) in logarithmic scale for 38 UAV flights (among 39) and all atmospheric conditions. One flight (FLT16-21) was not included due to oversight, and we used it afterwards as a test flight for confirming the statistical results obtained from the 38 flights. The horizontal dashed line in Fig. 7a shows the radar detection threshold (obtained from Fig. 3) and is approximately \(\left\langle {\sigma^{2} } \right\rangle^{3/2} = 0.001\). The thick solid line is a straight line of slope 1 and the dashed lines on both sides represent levels 3 times lower or higher. The scatter plot clearly indicates that \(\varepsilon_{U}\) appears to be proportional to \(\left\langle {\sigma^{2} } \right\rangle^{3/2}\) (especially for the largest values of \(\varepsilon_{U}\)), with relatively little scatter along the diagonal. For weak \(\varepsilon_{U}\) values (\(\varepsilon_{U} < \sim10^{ - 5} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\)), there is an important bias because the radar estimates are close to the minimum detectable levels and because residual contaminations might be present in the Doppler spectra despite careful data cleaning.

The histogram of \(\log_{10} \left( {L_{C} } \right)\) shown in the right panel of Fig. 7a displays a narrow peak for \(\left\langle {\sigma^{2} } \right\rangle^{3/2} > 0.01\) (or for \(\varepsilon_{U} > \sim10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) = \(0.1\,{\text{mW}}\,{\text{kg}}^{ - 1}\)) with a maximum near ~ 60 m and a mean (median) value of 75 m (61 m). The numerical values depend on the threshold on \(\left\langle {\sigma^{2} } \right\rangle^{3/2}\): the mean and median values of \(L_{C}\) increase if the threshold on \(\left\langle {\sigma^{2} } \right\rangle^{3/2}\) decreases. However, if the bias observed for \(\left\langle {\sigma^{2} } \right\rangle^{3/2} < 0.01\) is only due to instrumental effects, then \(L_{C}\) ~ 60–70 m should be representative of all \(\varepsilon_{U}\) levels.

Comparisons between \(\varepsilon_{U} \;{\text{and}}\;\varepsilon_{R}\)

Figure 8a shows examples of comparison results between \(\varepsilon_{U} {\text{ and }} \varepsilon_{R}\) profiles for FL16-T05 (A1) and FLT16-08 (A1). The agreements are remarkable. Note that there is about one order of magnitude of difference between the maximum values of TKE dissipation rates observed during FLT16-05 and FL16-T08. Figure 8b shows the result for FLT16-21 (A1). Since this flight was not included in the statistics leading to Eq. (9), these are absolute comparisons and it is the very first attempt at confirmation of Eq. (9). The agreement is again good, both in shape and levels, with a ratio of the two being much less than a factor 2 where \(\varepsilon_{U} \;{\text{and}}\;\varepsilon_{R}\) are maximum (above 2.5 km). The \(\varepsilon_{U}\) profiles fall within the range of variability of \(\varepsilon_{R}\) at almost all altitudes.

Figure 9a, b shows the results of comparisons in linear and logarithmic scales for all the 16 ShUREX 2016 and 23 ShUREX 2017 science flights, respectively, concatenated in chronological order. As expected, the largest discrepancies appear in logarithmic scale for small values (due to the minimum detectable value of \(\varepsilon_{R} \sim0.016\, \times \,0.001\, = \,1.6 \, \times \,10^{ - 5} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\)). Elsewhere, there is almost a one-to-one correspondence between all the peaks. Therefore, both the UAV and the radar detected the same turbulence events with similar \(\varepsilon\) intensities (assuming the validity of Eq. 9).

These events (and clouds) were often associated with the largest values of TKE dissipation rates, but they constituted only 23% of the overall dataset. Thus, the datasets used for the analysis (Fig. 7a) contained mainly turbulence in clear air conditions outside regions potentially affected by cloud dynamics and CBL. Figure 7b shows the results after excluding the convective turbulence events. They do not strongly differ from those shown in Fig. 7a, indicating that the observations made from the overall datasets are also representative of the free atmosphere, in the absence of convection. In particular, the \(\sigma^{3}\) dependence is still observed, but with slightly smaller values of \(L_{C}\). The histogram of \(L_{C}\) (right panel of Fig. 7b) seems to have a double-peak distribution. The smaller one may not be representative (because likely due to residual contaminations, see left panel of Fig. 7b). The maximum of the larger distribution is around \(L_{C} \sim50\,{\text{m}}\). Finding a smaller value for stratified conditions only is not surprising since the convective layers are much deeper and should be associated with larger characteristic scales. The difference between the two estimates (60 and 50 m) is not very large, however. We will keep 60 m for the subsequent comparisons.

TKE dissipation rates and isotropy of the radar echoes

Figure 10 shows the scatter plot of \(\varepsilon_{R}\) and \(\varepsilon_{U}\) versus the radar echo power aspect ratio AR (dB) defined as \(\log 10\left( {P_{v} } \right) - \left( {\log 10\left( {P_{N} } \right) + \log 10\left( {P_{E} } \right)} \right)/2,\) where \(P_{v}\), \(P_{N}\) and \(P_{E}\) are echo power measured in the three radar beam directions (vertical, North and East, 10° off zenith). Echo powers were estimated and averaged over 4 min along the UAV flight tracks in the same way as \(\left\langle {\sigma^{2} } \right\rangle\) (see Fig. 1). A small aspect ratio (say, < 3 dB in absolute value) is generally considered as a signature of scattering from turbulence, which is isotropic or nearly so. This is especially true when the averaging time is short [aspect ratio is a statistical parameter that requires time averaging and horizontal homogeneity hypothesis]. It is striking to note that almost all values of \(\varepsilon_{U}\) > \(1.6\, \times \,10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) are associated with \(\left| {AR} \right|\, < \,3\,{\text{dB}}\), i.e., with isotropic echoes. This result is extremely consistent with the fact that the layers of enhanced TKE dissipation rates are associated with relatively deep layers of turbulence, isotropic at the Bragg scale. In such layers, the radar echoes are weakly (or even not) affected by specular reflectors. This result justifies a posteriori the use of data from the vertical beam for estimating \(\left\langle {\sigma^{2} } \right\rangle\). Incidentally, it is also an additional clue suggesting that the radar and the UAVs indeed detected the same, most prominent, turbulence events. For \(\varepsilon_{U}\) < \(1.6 \, \times \,10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) (i.e., for the weakest values), AR can be significantly larger than 3 dB. It is difficult to know if this property is due to anisotropic turbulence (consistent with weak turbulence) or due to the coarse resolution of the radar (thin layers of isotropic turbulence surrounded by stable layers within the radar volume).

Comparisons between \(\varepsilon_{U}\) and \(\varepsilon_{N}\), \(\varepsilon_{W}\)

On the one hand, \(\varepsilon_{N}\) was estimated from Eq. (2) by calculating \(N^{2} = g/T\left( {{\text{d}}T/{\text{d}}z + \varGamma_{a} } \right)\) (\(\varGamma_{a} = 0.001 \text{ K m}^{-1}\)  ) from IMET and CWT data at the radar range resolution (150 m) by applying a low-pass filter with a cutoff of 300 m on the temperature profiles. Examples of \(N^{2}\) (150 m) profiles estimated from IMET and CWT data for FLT16-05 (A1) and FLT16-08 (A1) are shown in Fig. 11. The \(y\) reveal that \(N^{2}\) is positive everywhere during FLT16-05 (A1) and negative in the height range 2.6–3.0 km during FLT16-08 (A1).

On the other hand, \(\varepsilon_{W}\) can be estimated from Eq. (3) by performing a numerical integration of I for each altitude z (without \(L_{H}\)).

The top panel of Fig. 12 shows the superposition of \(\varepsilon_{U} , \;\varepsilon_{R} ,\; \varepsilon_{N} \;{\text{and}}\;\varepsilon_{W}\) profiles for FLT16-05 (A1) and FLT16-08 (A1). The \(\varepsilon_{W}\) profiles appear to be similar to the \(\varepsilon_{R}\) profiles but are underestimated, typically by a factor ~ 2 with respect to \(\varepsilon_{R}\) (and \(\varepsilon_{U}\)) at almost all altitudes and for both cases. The \(\varepsilon_{N}\) profiles exhibit more complex features. For FLT16-05 (A1), \(\varepsilon_{N}\) shows a good agreement in levels and shape with \(\varepsilon_{U}\). Therefore, from the analysis of a single profile, it can be concluded that \(\varepsilon_{N}\) provides reasonable agreement (at least in linear scale). However, for FLT16-08 (A1), \(\varepsilon_{N}\) is not defined in the altitude range (2.6-3.0 km) where \(\varepsilon_{U} ,\; \varepsilon_{R} \;{\text{and}}\;\varepsilon_{W}\) are found to be maximum because \(N^{2}\) is negative. \(\varepsilon_{N}\) is maximum on both sides due to \(N^{2}\) enhancements at the edges (Fig. 11). Here, \(\varepsilon_{N}\) provides radically different information on the turbulent state of the atmosphere between the altitudes of 2.6 km and 3.5 km. The corresponding time-height cross sections of radar echo power at high resolution and at vertical incidence, shown in the bottom panel of Fig. 12, permits us to better understand the differences between FLT16-05 and FLT16-08. During FLT16-05, stratified conditions were clearly observed but were quickly changing between ascent (A1) and descent (D1) (so that \(\varepsilon_{U}\) profiles were distinct during A1 and D1, see Fig. 6a). During FLT16-08, the UAV crossed a MCT layer in the altitude range 2.6–3.5 km. MCT layers are basically generated by a convective instability at the cloud base due to evaporative cooling from sublimating precipitation in the sub-cloud layer (e.g., Kudo et al. 2015). For such conditions, \(N^{2} < 0\) can be observed as is the case here, and \(N^{2}\) can be enhanced at the edges due to turbulent mixing. Therefore, \(\varepsilon_{N}\) fails to reproduce \(\varepsilon_{U}\) because it is not defined when \(N^{2} < 0\) in the core of the layer and because the model is inapplicable to convective turbulence.

Figure 13a, b shows information similar to Fig. 9a, b for all flights and all ascents and descents including \(\varepsilon_{N}\) (green line) and \(\varepsilon_{W}\) (black line), \(\varepsilon_{R}\) and \(\varepsilon_{U}\) being shown as red and blue lines, respectively. The figure confirms the tendencies shown in Fig. 12. The series of \(\varepsilon_{W}\) values reveals very similar features as the series of \(\varepsilon_{R}\) and \(\varepsilon_{U}\) with a slight but systematic underestimation. \(\varepsilon_{N}\) estimates are apparently consistent with the other estimates but discrepancies, such as those shown in Fig. 12, are difficult to see.

Figure 14 shows the histogram of differences (in logarithmic scales) between \(\varepsilon_{R}\) and \(\varepsilon_{U}\) (left panel), between \(\varepsilon_{W}\) and \(\varepsilon_{U}\) (middle panel), and between \(\varepsilon_{N}\) and \(\varepsilon_{U}\) (right panel), when \(\varepsilon_{N}\) is defined (i.e., \(N^{2} > 0)\) for \(\varepsilon \, > \,1.6 \times10^{ - 5} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\). The histogram for \(\log_{10} \left( {\varepsilon_{R} /\varepsilon_{U} } \right)\) shows a peak close to 0 (by definition). It shows the narrowest peak (minimum standard deviation) among the three estimates. From a statistical point of view, \(\varepsilon_{R}\) is thus the best estimate relative to \(\varepsilon_{U}\). The histogram for \(\log_{10} \left( {\varepsilon_{W} /\varepsilon_{U} } \right)\) shows a slightly wider distribution with a peak around -0.40 indicating an underestimation by a factor ~ 2.5 on average. The histogram for \(\log_{10} \left( {\varepsilon_{N} /\varepsilon_{U} } \right)\) shows a relatively wide distribution at ~ 0.16 indicating a slight overestimation of 1.44 on average. However, this result, favorable to \(\varepsilon_{N}\), hides an important bias. Figure 15 shows the results of regression and correlation analyses between \(\varepsilon_{U}\) and the three radar estimates assuming errors on both variables, and after rejecting dissipation rates smaller than \(1.6 \, \times \,10^{ - 5} \, {\text{m}}^{2} \,{\text{s}}^{ - 3}\) (because of the radar detection threshold and residual contaminations for weak values). The slopes of the regression line are close to 1, 0.91 and 0.92, between \(\log_{10} \left( {\varepsilon_{U} } \right)\) and \(\log_{10} \left( {\varepsilon_{R} } \right)\) and between \(\log_{10} \left( {\varepsilon_{U} } \right)\) and \(\log_{10} \left( {\varepsilon_{W} } \right)\), respectively, confirming reasonably, the \(\sigma^{3}\) dependence of TKE dissipation rates. The corresponding correlation coefficients are also nearly identical, at 0.8 and 0.78, respectively. As reported above, the main statistical difference between \(\varepsilon_{R}\) and \(\varepsilon_{W}\) is the slight bias in levels (factor ~ 2.5 with \(\varepsilon_{U}\)). In contrast, the regression analysis between \(\log_{10} \left( {\varepsilon_{U} } \right)\) and \(\log_{10} \left( {\varepsilon_{N} } \right)\) leads to a slope of 0.51 and the correlation coefficient is significantly smaller (0.69). Therefore, as expected, the model producing \(\varepsilon_{N}\) is not suitable, because it predicts a \(\sigma^{2}\) dependence of \(\varepsilon\). Although correct in average, the model tends to overestimate \(\varepsilon\) for \(\varepsilon_{U} < \sim4 \times10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) and to underestimate \(\varepsilon\) for \(\varepsilon_{U} > \sim4 \times10^{ - 4} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\). Note that detection threshold effects cannot explain the largest biases for small values, since the associated values were rejected by the regression analysis (but not in Fig. 13). The scatter plot obtained with a constant value of \(N^{2}\) equal to the mean value for the stratified regions sampled by the UAVs \( ( = 1.47 \, \times \,10^{ - 4} \text{ s}^{-2}), \) is superimposed to the bottom panel of Fig. 15 (red dots). The regression slope is very similar but the correlation coefficient is improved (0.8). Therefore, the tendency shown by the model producing \(\varepsilon_{N}\) does not depend on an accurate estimate of \(N^{2}\).