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

From a dimensional analysis, the dissipation rate (assuming isotropy) can be inferred from:

$$\varepsilon \sim\left\langle {w^{\prime 2} } \right\rangle^{3/2} /L$$

where \(\left\langle {w^{\prime 2} } \right\rangle\) is the variance of vertical wind fluctuations and *L* is a typical scale of the turbulent eddies. A rms value \(\sigma\) of radial turbulent velocity fluctuations can be obtained from the measured Doppler spectral width after removing non-turbulent contributions to the spectral broadening (see “Appendix”) (e.g., Hocking 1986; Fukao et al. 1994; Naström 1997; Dehghan and Hocking 2011). Similarly to the above expression, we can write:

$$\varepsilon_{R} = \sigma^{3} /L_{c}$$

(1)

where \(L_{c}\) has the dimension of a turbulence scale. Expression (1) is only indicative, but it will be first used in order to see if a particular value of \(L_{c}\) emerges from our dataset.

In practice, two main expressions are used for estimating \(\varepsilon\) from the Doppler spectral width, based on more elaborated models. When the outer scale \(L_{\text{out }}\) of turbulence is small compared to the horizontal and transverse dimensions of the radar sampling volume, 2*a*, 2*b*^{Footnote 1} we have (e.g., Hocking 1983, 1999, 2016):

$$\varepsilon_{N} = C\sigma^{2} N$$

(2)

where *C* is a constant (= \(0.5 \pm 0.25)\) according to Hocking (2016). *C* = 0.47, sometimes used in the literature, was applied for producing the figures. The parameter \(N\) is the Brunt–Väisälä frequency. Expression (2) has been established for characterizing turbulence in stratified conditions only [whereas expression (1) is always valid]. Various definitions of outer scales of stably stratified turbulence have been proposed in order to obtain dissipation rate expressions in the form of (2) (e.g., Weinstock 1978a, b, 1981). Expression (2) is virtually identical to the theoretical expression given by Weinstock (1981) obtained by integrating the spectrum of inertial turbulence down to the buoyancy wavenumber \(k_{\text{B}} = N/\sqrt {\left\langle {w^{\prime 2} } \right\rangle }\) so that \(\varepsilon \approx 0.5\left\langle {w^{{{\prime }2}} } \right\rangle N\). Hocking (2016) makes use of the one-dimensional transverse spectrum [expression (7.42)] whose integration, by including additional contribution from the buoyancy subrange, leads to an estimate of vertical wind fluctuation variance \(\sigma^{2}\) supposed to be measured by the radar. By doing so, Expression (2) is obtained with various values of *C*, coincidently close to the coefficient 0.5 of the Weinstock model. Kantha et al. (2018, this issue) used this approach with different conceptual models of turbulence and different definitions of turbulence scales and even generalized it to expressions including the radar volume effects. However, it seems that a definitive modeling is still an open issue.

The alternative approach proposed by, e.g., Frisch and Clifford (1974) and Labitt (1979) considers the role of spatial low-pass band filter played by the radar volume, valid if \(L_{\text{out}} \gg\) 2*a*, 2*b*. The White et al. (1999) formulation also considered the effects of the wind advection:

$$\varepsilon_{w} = \left( {\frac{4\pi }{1.6}} \right)^{3/2} \frac{{\sigma^{3} }}{{I^{3/2} }}$$

(3)

where

$$I \propto \int\limits_{0}^{\pi /2} {{\text{d}}\phi \int\limits_{0}^{\pi /2} {\sin^{3} \theta \; \times \left( {b^{2} \cos^{2} \theta + a^{2} \sin^{2} \theta + \frac{{L_{H} }}{12}\sin^{2} \theta \cos^{2} \phi } \right)^{1/3} {\text{d}}\theta } }$$

and \(L_{H} = VT\), where \(V\) is the mean horizontal wind speed during the dwell time *T*. It is important to note that expression (3) is based on the hypothesis that the radar is sensitive to the three-dimensional longitudinal spectrum of turbulence (see Doviak and Zrnic’ 1993, p. 398). Therefore, Eqs. (2) and (3) are not the asymptotic forms (for \(L_{\text{out}} \ll\) 2*a*, 2*b* and \(L_{\text{out}} \gg\) 2*a*, 2*b,* respectively) of a more general expression. The \(\varepsilon\) estimates from Eqs. (2) and (3) (i.e., \(\varepsilon_{N}\) and \(\varepsilon_{W}\), respectively) will be compared with those derived from UAV data, hereafter noted \(\varepsilon_{U}\) in “Comparisons between \(\varepsilon_{U}\) and \(\varepsilon\) from the radar models” section.

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 \(L_{\text{out}} \gg\) 2*a*, 2*b*. 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 Doppler variance due to turbulent motions was estimated from the Doppler spectra by applying:

$$\sigma^{2} \approx \sigma_{m}^{2} - \sigma_{b}^{2}$$

(4)

where \(\sigma_{m}^{2}\) is the Doppler variance measured *at vertical incidence*, and \(\sigma_{b}^{2}\) is the variance due to beam-broadening effects.\(\sigma^{2}\) was used in order to obtain \(\varepsilon_{R}\), \(\varepsilon_{N}\) and \(\varepsilon_{W}\). Equation (4) is very simple compared to the expressions provided by Naström (1997) and Dehghan and Hocking (2011), because only data from the vertical beam are used. At VHF, data collected at vertical incidence are usually avoided because the radar echoes can be strongly affected by (non-turbulent) specular reflectors so that the spectral width is reduced and \(\sigma^{2}\) is biased (e.g., Tsuda et al. 1988). However, Eq. (4) has a great advantage, since shear-broadening effects are null or negligible when using a vertical beam. Even though the theoretical effects due to shear-broadening when using data collected at oblique incidences are well-established, the corrections remain challenging in practice, because they require accurate estimates of wind shears, and the wind shear profiles estimated at the radar range resolution may not be representative of shear profiles at higher resolutions (e.g., Figure 5 of Luce et al. 2018). The use of data at vertical incidence will be justified a posteriori in “Comparisons between \(\varepsilon_{U}\) and \(\varepsilon\) from the radar models” section.

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

The basics for retrieving \(\varepsilon_{U}\) were described by Kantha et al. (2017). Frequency spectra (Eq. 5 of Kantha et al. 2017) were estimated from variance-conserving, Hanning-weighted time intervals of 5 s duration (corresponding to 2000 points since the effective sampling rate was 400 Hz) every 2.5 s (corresponding to a successive time interval overlap of 50%). Assuming local isotropy and stationarity of turbulence and using the frozen-advection Taylor hypothesis, the theoretical Kolmogorov 1D power spectral density is of the form (Tatarski 1961; Hocking 1983):

$$S_{U} \left( f \right) = 0.55\varepsilon^{2/3} \left( {\frac{{\bar{U}}}{2\pi }} \right)^{2/3} f^{ - 5/3}$$

(5)

[the coefficient 0.55 holds for motions parallel to the mean relative wind]. \(\bar{U}\) is the mean relative wind (airspeed). Assuming that the calculated spectrum \(\hat{S}_{U} \left( f \right)\) shows an inertial domain (at least in a frequency band), the spectral data will have the frequency dependence:

$$\hat{S}_{U} \left( f \right) = \beta f^{ - 5/3}$$

(6)

An experimental value of \(\varepsilon_{U}\) can be obtained by estimating \(\beta\) by fitting spectral data, and equating Eqs. (5) and (6) (e.g., Frehlich et al. 2003; Siebert et al. 2006):

$$\varepsilon_{U} = \frac{2\pi }{{\bar{U}}}\left( {\frac{\beta }{0.55}} \right)^{3/2}$$

(7)

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.