Turbulence kinetic energy dissipation rate: assessment of radar models from comparisons between 1.3 GHz wind proﬁler radar (WPR) and DataHawk UAV measurements

. The WPR-LQ-7 is a UHF (1.3575 GHz) wind pro-ﬁler radar used for routine measurements of the lower tropo-sphere at Shigaraki MU Observatory (34.85 ◦ N, 136.10 ◦ E; Japan) at a vertical resolution of 100 m and a time resolution of 10 min. Following studies carried out with the 46.5 MHz middle and upper atmosphere (MU) radar (Luce et al., 2018), we tested models used to estimate the rate of turbulence kinetic energy (TKE) dissipation ε from the Doppler spectral width in the altitude range ∼ 0.7 to 4.0 km above sea level (a.s.l.). For this purpose, we compared LQ-7-derived ε using processed data available online (http://www.rish.kyoto-u.ac. jp/radar-group/blr/shigaraki/data/, last access: 24 July 2023) with direct estimates of ε ( ε U ) from DataHawk UAVs. The statistical results reveal the same trends as reported by

Abstract. The WPR-LQ-7 is a UHF (1.3575 GHz) wind profiler radar used for routine measurements of the lower troposphere at Shigaraki MU Observatory (34.85 • N, 136.10 • E; Japan) at a vertical resolution of 100 m and a time resolution of 10 min. Following studies carried out with the 46.5 MHz middle and upper atmosphere (MU) radar (Luce et al., 2018), we tested models used to estimate the rate of turbulence kinetic energy (TKE) dissipation ε from the Doppler spectral width in the altitude range ∼ 0.7 to 4.0 km above sea level (a.s.l.). For this purpose, we compared LQ-7-derived ε using processed data available online (http://www.rish.kyoto-u.ac. jp/radar-group/blr/shigaraki/data/, last access: 24 July 2023) with direct estimates of ε (ε U ) from DataHawk UAVs. The statistical results reveal the same trends as reported by Luce et al. (2018) with the MU radar, namely (1) the simple formulation based on dimensional analysis ε L out = σ 3 /L out , with L out ∼ 70 m, provides the best statistical agreement with ε U .
(2) The model ε N predicting a σ 2 N law (N is Brunt-Vaïsälä frequency) for stably stratified conditions tends to overestimate for ε U 5 × 10 −4 m 2 s −3 and to underestimate for ε U 5×10 −4 m 2 s −3 . We also tested a model ε S predicting a σ 2 S law (S is the vertical shear of horizontal wind) supposed to be valid for low Richardson numbers (Ri = N 2 /S 2 ). From the case study of a turbulent layer produced by a Kelvin-Helmholtz (K-H) instability, we found that ε S and ε L out are both very consistent with ε U , while ε N underestimates ε U in the core of the turbulent layer where N is minimum. We also applied the Thorpe method from data collected from a nearly simultaneous radiosonde and tested an alternative interpreta-tion of the Thorpe length in terms of the Corrsin length scale defined for weakly stratified turbulence. A statistical analysis showed that ε S also provides better statistical agreement with ε U and is much less biased than ε N . Combining estimates of N and shear from DataHawk and radar data, respectively, a rough estimate of the Richardson number at a vertical resolution of 100 m (Ri 100 ) was obtained. We performed a statistical analysis on the Ri dependence of the models. The main outcome is that ε S compares well with ε U for low Ri 100 (Ri 100 1), while ε N fails. ε L out varies as ε S with Ri 100 , so that ε L out remains the best (and simplest) model in the absence of information on Ri. Also, σ appears to vary as Ri −1/2 100 when Ri 100 0.4 and shows a degree of dependence on S 100 (vertical shear at a vertical resolution of 100 m) otherwise.

Introduction
The dissipation rate ε (m 2 s −3 or W kg −1 ) of turbulence kinetic energy (TKE) is an important variable for assessing the rate of change of TKE with time. This variable appears in a simplified expression of the ensemble-mean TKE budget equation (see, e.g., Stull, 1988, for a complete derivation and for its conditions of validity): where P is the shear production term and B the buoyancy flux term. Note that advection terms on the left-hand side Published by Copernicus Publications on behalf of the European Geosciences Union.
In addition to being rare, the above-mentioned studies did not aim to test the same radar models. Luce et al. (2018), hereafter denoted L18, assessed different models from comparisons with direct estimates of ε from air speed fluctuation measurements made from highly sensitive Pitot sensors aboard DataHawk UAVs and the VHF 46.5 MHz middle and upper atmosphere (MU) radar observations in the lower troposphere. One of the objectives of the present work is to show that the conclusions obtained from comparisons with the MU radar are also quantitatively valid for the WPR-LQ-7 (Imai et al., 2007), a UHF wind profiler routinely used at the Shigaraki MU Observatory. We also introduce and test another model expected to be valid for weakly stratified or strongly sheared conditions, i.e., low Richardson (Ri) numbers (Hunt et al., 1988;Basu and Holtslag, 2021) for which the static stability effects can be ignored. Ri is defined as N 2 /S 2 , where N 2 = (g/θ ) dθ/dz is the squared Brunt-Väisälä frequency (s −2 ), g ≈ 9.81 m s −2 is the gravitational acceleration, θ (K) is the potential temperature and S (s −1 ) is the vertical shear of the horizontal wind vector.
Section 2 introduces the expressions for ε used in the present paper with a focus on the newly introduced model in radar studies. Section 3 briefly describes the WPR-LQ-7 and the methods used for the comparisons. Section 4 describes the results for two turbulent layers, one of which was clearly produced by a Kelvin-Helmholtz (K-H; shear flow) instability because of the observation of S-shaped structures specific to this instability in both time-height MU and WPR-LQ-7 echo power cross-sections. The results of comparisons of ε values obtained from the different models applied to the two radars, DataHawk measurements, and a simultaneous radiosonde using the Thorpe sorting method of vertical potential temperature profiles are described for the two layers. Section 5 shows statistics on the consistency between the estimates of ε from the different models and the DataHawks and describes the dependence of the models on Ri. Finally, conclusions are given in Sect. 6.
2 The radar models of ε

The models tested by L18
The different models and their conditions of application have already been described by L18. Here, the expressions are simply reintroduced. Assuming a vertically pointing radar beam, the first expression is where σ 2 is an estimate of the variance w 2 of the vertical wind fluctuations produced by turbulence. L out has the dimension of a length scale and represents the scale of energy containing turbulent eddies if σ 2 is an unbiased estimate of w 2 . In practice, σ 2 is obtained after removing the non-turbulent contributions from σ 2 obs (e.g., Hocking, 1983;Nastrom, 1997;Hocking et al., 2016, and references therein). The practical method used in the present work is described in the Appendix of L18. The dissipation rate is expected to vary as σ 3 if σ and L out are independent or when the typical scale of turbulent eddies exceeds the dimensions of the radar volume, so that L out would mainly be a function of these dimensions (e.g., Frisch and Clifford, 1974;Labitt, 1979;Doviak and Zrnić, 1984;White et al., 1999).
The second expression (e.g., Hocking, 1983Hocking, , 1999Hocking et al., 2016) is where C N is a constant. This expression is expected to be valid for turbulence in a stable stratification (N 2 > 0) whose outer scale is defined by the buoyancy length scale expressed as L B = w 2 /N = σ/N . Equation (3) is thus equivalent to ε N = C N σ 3 /L B . In a pioneering contribution, Hocking (1983) first derived Eq. (3) from the integration of the transverse 1D spectrum of vertical velocities over the inertial and buoyancy subranges to relate ε to w 2 . In its original derivation, the author assumed roughly equal contributions to w 2 from the inertial and buoyancy subranges. More recently, Hocking et al. (2016) proposed a more general expression by introducing a variable factor F , where F is the ratio of the buoyancy contribution to the inertial subrange contribution. This factor can vary from 0.5 to 1. It affects the value of the constant C N , and Hocking et al. (2016) recommend that a value of (0.5 ± 0.25) be used, which takes into account the variability of F , which is difficult to determine in practice. The results of comparisons using Eqs. (2) and (3) reported by L18 with DataHawk-derived ε showed that Eq. (2) provides the best overall statistical comparisons for L out ∼ 50-70 m. On the other hand, the analysis of the comparison results with ε N suggested that N is not a key parameter since the quality of the comparisons appeared to be independent of N.

The model for strongly sheared or/and weakly stratified flows
While it seems that the conditions of strong shear and weak stratification have not received much attention in the radar community, several studies showed that ε can be written as (Hunt et al., 1988;Schumann and Gerz, 1995) ε S = C S σ 2 S.
Note that we do not make the distinction between σ 2 and w 2 for simplicity. Equation (4) is equivalent to ε S = C S σ 3 /L H , where L H = σ/S is the Hunt length scale. Equation (4) can be interpreted as the fact that turbulent eddies are first stretched by shear before being affected by stratification in strongly sheared or weakly stratified flows. This concept was discussed by Hocking and Hamza (1997), but they did not mention the Hunt length scale and did not go further into it. Hunt et al. (1988) suggested that Eq. (4) can be valid up to Ri ∼ 0.5. Schumann and Gerz (1995) even proposed up to Ri ∼ 1 from large eddy simulations. Hunt et al. (1988) proposed C S ≈ 0.45 for neutral stationary boundary layers. Kaltenbach et al. (1994) found 0.54 < C S < 0.62 from large eddy simulations. By using Eq. (1) for a homogeneous shear layer in steady state, i.e., ε = P − B, and similarity theory, Basu and Holtslag (2021) re-evaluated the constant C S and provided a generalization of Eq. (4): with C S = 0.63. R f is the flux Richardson number. It is related to the turbulent Prandtl number Pr by R f = Ri/Pr.  found from direct numerical simulation (DNS) that Eq. (4) with C S ∼ 0.60 is valid up to Ri ∼ 0.2 at least. For 0 < Ri 0.25, C S (R f ) decreases from 0.63 to 0.60, using the analytical expression (Eq. 22) of Basu and Holtslag (2021) for Therefore, Eq.
(3) would be quantitatively equivalent to Eq. (5a) for Ri of the order of 1 despite the fact that the two approaches cannot be reconciled because, in essence, there is no contribution from an anisotropic buoyancy subrange in Eq. (5a). Equation (4) removes an inconsistency in Eq.
(3), since it wrongly indicates that ε → 0 when N → 0 for a given σ 2 . If S = 0, i.e., if the source of the instability that generates turbulence is removed, then ε = 0, which makes more sense. As discussed by Basu and Holtslag (2021, Sect. 6.2) and Basu and Holtslag (2022, their Appendix 1), the derivation of Eq. (5) does not consider the fact that the steadystate condition (also called "full equilibrium" (FE), Baumert and Peters, 2000) can only be reached for a single value of Richardson number Ri s , at least for large Reynolds numbers and large shear parameters ST L , where T L is the inertial timescale defined as TKE/ε (see, e.g., Mater and Venayagamoorthy, 2014). For Ri < Ri s , TKE increases at subcritical Ri, and for Ri > Ri s , TKE decreases (turbulence decays) at supercritical Ri (e.g Baumert and Peters, 2000). From large eddy simulation (LES) and DNS data, Schumann (1994) and Gerz et al. (1989) reported Ri S ≈ 0.13 for air, consistent with the value that can be deduced from Fig. 1 of Mater and Venayagamoorthy (2014). Schumann (1994) rewrote the TKE budget Eq. (1) as ∂TKE/∂t = (G − 1)(ε + B) where G = P /(ε + B) is called the growth factor. G = 1 for FE conditions. By assuming, for simplicity, that G only depends on Ri, Schumann proposed the empirical expression G(Ri) = G (1−Ri/Ri s ) 0 with G 0 = 1.47 ± 0.13 based on windtunnel data analysis. By using the same procedure as Basu and Holtslag (2021) from their Eqs. (10) to (12) but starting with ε = P /G − B, we get Equation (5b) can also be directly obtained by using Eq. (46a) in Eq. (10b) of Basu and Holtslag (2021). For 0 < Ri 0.25, C S = C S /G 1/2 (1 − GR f ) 1/2 increases from 0.52 to 0.70, i.e., ∼ 0.60 on average, for Ri s ≈ 0.13 and G 0 = 1.47. The Ri dependence of C S is thus only a small source of dispersion for low Ri values when comparing with other estimates. From the results of Baumert and Peters (2000) using a "structural equilibrium" approach (i.e., stationarity of ratios of turbulence characteristics) and based on laboratory and LES data (their Fig. 4), we can establish ε S = 0.15σ 2 S valid for Ri 0.25. This expression is obtained by com-H. Luce et al.: TKE dissipation rate: assessment of radar models bining L H /L B = Ri 1/2 , L E /L O = 4.2Ri 3/4 and L E /L B = 1.61 Ri 1/2 , where L E = θ 2 /(dθ/dz) and L O = ε/N 3 are the Ellison and Ozmidov length scales, respectively. The constant differs very significantly (by a factor of 3 to 4 less) from the aforementioned estimates. If we use L E /L O = 2.4 Ri 3/4 as proposed by Schumann (1994) for Ri ≤ 0.25, we get ε S = 0.44σ 2 S with the same L E /L B ratio. These expressions are more subject to experimental uncertainties and are thus not considered in this paper. In Appendix A, we propose an alternative derivation of Eq. (4), suggesting 0.45 ≤ C S ≤ 0.82. We retain the value of C S = 0.63 for the comparisons between the models.
Following the spectral approach proposed by Weinstock (1981), Eq. (4) with C S equal to C N ≈ 0.5 can also be obtained from the integration of the 1D Kolmogorov (−5/3 slope) scalar kinetic energy spectrum over a spherical shell of radius k H instead of k B , where k H (k B ) is the wavenumber corresponding to L H (L B ). For the context of radar measurements (e.g., Hocking et al., 2016), Eq. (4) can also be obtained from the integration of the 1D transverse vertical velocity spectrum with a −5/3 slope for large (horizontal) wavenumbers (k > k H ) and a zero slope for (k < k H ), both mathematical developments being equivalent.
Finally, ε S has the advantage that it can be evaluated entirely from the radar data, since the wind shear S can be estimated at the range and time resolutions of the radar, unlike ε N which requires N 2 to be obtained from in situ or radioacoustic sounding system measurements.
3 The WPR-LQ-7 and methods of comparisons with UAV data

The WPR-LQ-7
The WPR-LQ-7 is a 1.3575 GHz Doppler radar. It has a phased array antenna composed of seven Luneberg lenses of 800 mm diameter. Its peak output power is 2.8 kW. It can be steered into five directions sequentially (i.e., after fast Fourier transform (FFT) operations), vertical and 14.2 • off zenith toward north, east, south and west. The main radar parameters of the WPR-LQ-7 installed at Shigaraki MU Observatory since 2006 are given in Table 1.
The acquisition time for one profile composed of 80 altitudes from 300 m a.g.l. (∼ 684 m a.s.l.) every 100 m in each direction is 59 s after 18 incoherent integrations but for a total of 11.8 s of observations for each direction (due to the intertwining between the directions). The time series are processed by automatic algorithms to remove outliers (e.g., bats, birds, airplanes) and ground clutter as far as possible. Low signals near and below the detection thresholds are removed, and profiles of atmospheric parameters (echo power, radial winds, half-power spectral width, horizontal and vertical winds) averaged over 10 min are made available for routine monitoring (http://www.rish.kyoto-u.ac.jp/radar-group/ blr/shigaraki/data/, last access: 24 July 2023). Because of the high data quality control, the 10 min averaged data are used to retrieve ε with the objective to assess the routine data for further analysis. The 59 s resolution data and those collected by the MU radar at a time resolution (sampling) of 24.57 s (∼ 12.3 s) were used to help identify atmospheric structures from height-time signal-to-noise ratio (SNR) or echo power cross-sections, such as convective cells or Kelvin-Helmholtz (K-H) billows. Table 2 shows the acquisition time, the range and transverse resolutions of the WPR-LQ-7 for the altitude range of comparisons, and the range and resolutions of the MU radar for the data used in the present work.

The methods of comparison with
DataHawk-derived ε The DataHawk datasets were collected during two field campaigns, called the Shigaraki UAV Radar Experiment (ShUREX), in May-June 2016 and June 2017 at the Shigaraki MU observatory. The DataHawks were flying about 1 km away from the MU radar and the WPR-LQ-7.  described the instruments and configurations used during a previous ShUREX campaign in June 2015. The processing method used to retrieve ε from Pitot sensor data is not recalled here as it is described in detail by L18 for comparisons with MU radar data. The trajectories of the DataHawks being helicoidal upwards or downwards, pseudo-vertical profiles of ε at a vertical sampling of ∼ 5 m typically were obtained during the ascents and descents of the aircraft, from the ground up to a maximum altitude of ∼ 4.5 km. A total of 36 DataHawk datasets collected during the two campaigns provided 90 full or partial profiles used for the comparisons. Section 4 describes one of these flights with one full ascent (A1) and descent (D2) and one partial ascent (A2) and descent (D1). Three DataHawk flights collected during periods of precipitation contaminating the WPR-LQ-7 returns were rejected. The DataHawk-derived ε profiles were smoothed with a Gaussian window and resampled at the altitude of the radar gates to simulate the radar range resolution. The degraded DataHawk 100 m resolution profiles are hereafter denoted ε U . The WPR-LQ-7-derived ε L out , ε N and ε S profiles were computed at a time resolution of 10 to 30 min, i.e., by averaging up to three consecutive profiles that best correspond to each period of DataHawk ascent or descent. ε L out was calculated with L out = 70 m in accordance with the best agreement from comparisons with the MU radar (L18) and the statistics of L out shown in Sect. 5. The profiles of N 2 at a vertical resolution of 100 m were estimated from pressure and temperature profiles collected by the DataHawks. These profiles were used to obtain ε N (Eq. 3). ε S profiles were computed from Eq. (4) every 10 min from wind shear estimated from radar data and then averaged up to 30 min. 3.3 Estimation of ε from the Thorpe method applied to radiosonde data The Ozmidov length scale L O = ε/N 3 is commonly assumed to be proportional to the Thorpe length L T d 2 1/2 , where d is the Thorpe displacement in the so-called Thorpe layer. Then, we have L O = cL T and The literature is very divided on the value of c to use (0.25 < c < 4) (see Kohma et al., 2019, for a review). Wijesekera and Dillon (1997) showed large temporal variations of L T /L O from observations in the ocean. Large temporal variations were also reported from DNS depending on the stage and source of turbulence (e.g., Fritts et al., 2016). An intermediate value of c = 1 is sometimes used by default (e.g., Kantha and Hocking, 2011). However, Mater et al. (2013) showed that c ∼ 1 when the turbulent Froude number Fr = ε/(N TKE) is near unity (at the transition between shear-and buoyancy-dominated regimes). The basic N 2 for the Thorpe layers is generally estimated from the sorted potential temperature profile (N 2 s ) or from the rms value of the fluctuations defined as the difference between the measured and sorted profiles (N 2 rms ) (e.g., Smyth et al., 2001;Wilson et al., 2014).
Another scale, called the Corrsin length scale, is defined as L c = ε/S 3 . It is the counterpart of the Ozmidov length scale for shear flows under neutral stratification conditions. Similarly, assuming L c = c L T , we can write Equation (7) is thus a possible alternative to Eq. (6), when the Corrsin length scale is smaller than the Ozmidov length scale. These equations are coherent with the results of Mater et al. (2013), who showed that L T scales with (TKE) 1/2 /S in the shear-dominated regime and with (TKE) 1/2 /N in the buoyancy-dominated regime. Equation (7) can also be justified, and the parameter c can be estimated as follows.
The aforementioned ratios L E /L O = 4.2 Ri 3/4 and L E /L O = 2.4 Ri 3/4 , found for weakly stratified flows (Ri ≤ 0.25) by Baumert and Peters (2000) and Schumann (1994), respectively, may be representative of L T /L O = 1/c(Ri) for flows free of gravity wave motions. Indeed, L E = L T is obtained if dθ/dz is a constant, which implies the absence of gravity waves. By introducing the expressions of L T /L O = 1/c(Ri) into Eq. (6), we obtain Eq. mental data, which is very consistent with the other values.
In Appendix B, we show that c = 0.28 can be found from an alternative approach based on the inference of the turbulent Froude number from L E /L O for weakly stratified flow condition (Garanaik and Venayagamoorthy, 2019). Equations (6) and (7) are equivalent if Eq. (5) is written as Equations (4) and (7) have in common that they are formally independent of N 2 when Ri 0.25. If they were both to be confirmed by experimental analysis, they would constitute a coherent whole.
4 Two case studies  Figure 1b shows the corresponding cross-section of MU radar echo power (dB) at vertical incidence and a time resolution of ∼ 12 s after doing range imaging with Capon processing (e.g., Luce et al., 2017) in the altitude range [1.275-7.0 km]. Radar echoes from a DataHawk, called "DH35" in reference to the flight numbering, are visible after ∼ 14:30 and before ∼ 15:40 LT on both images. They are the signatures of two ascents ("A1", "A2") and two descents ("D1", "D2") of a DataHawk. Four red segments emphasize them in Fig. 1b. Incidentally, radar echoes from another DataHawk (DH36) can be noted after 16:30 LT. A Vaisala RS92-SGP radiosonde, called "V6", was launched at 14:51 LT from the observatory. Its time-height position is indicated by the blue line in Fig. 1b. It roughly coincides with A1. An approximately 800 m deep enhanced echo power layer with S-shaped structures, signature of Kelvin-Helmholtz billows, is clearly visible in both images in the altitude range [3.0-4.0 km] until ∼ 16:00 LT at least. The layer is denoted by "KHI" on the images. DH35 crossed the layer four times during A1, D1, A2 and D2 between 15:00 and 15:30 LT. DH35 sampled the most obvious case of K-H instability during the entire two campaigns. Since a necessary condition for the development of K-H billows is Ri < Ri c = 0.25 at the critical level, their observation suggests that it was fulfilled when sampled by the instruments. Another focus will be given to a turbulent layer between 2100 and 2500 m, sampled twice by DH35 during A1 and D2, even though it is not clearly visible in the radar echo power images (Fig. 1). For this layer, Ri is expected to be 1 according to various estimates and layer properties described in Sect. 4.2.
The analysis of these two cases is done to illustrate the differences in the ε behavior of the three different radar mod-els applied to two radars, possibly at different Richardson numbers (Ri 0.25 and 1), compared to the DataHawkderived ε.

The K-H layer
4.1.1 Comparisons between DataHawk-derived ε, ε L out , ε N and ε S Figure 2a shows the four DataHawk-derived ε profiles during A1, D1, A2 and D2 (dotted black lines) and the profiles of ε L out , ε N and ε S in the height range  m obtained from the WPR-LQ-7 data (solid red, blue and green lines, respectively) and MU radar data (dashed red, blue and green lines, respectively). Figure 3a, b and c show the same information for the three models but separately. For clarity, because they are virtually identical during D1, A2 and D2, the radar-derived ε profiles are shown for A1 (15:00-15:20 LT) only. The sources of errors and uncertainties on radar estimates are multiple (e.g., Dehghan and Hocking, 2011), and confidence intervals for each individual estimate are difficult or even impossible to establish. However, the consistency between the estimates from successive independent segments of data and concordant temporal evolution as shown in Fig. 4 and Table 3 gives some credence to the significance of the comparisons. Table 3a and the corresponding figure, Fig. 4, show ε, ε L out , ε N and ε S values averaged over the depth of the K-H layer for A1, D1, A2 and D2. The DataHawk-derived ε values peak in the range of the K-H layer and vary little during the ascents and descents over ∼ 30 min: typically ∼ 2 mW kg −1 . During A1, the DataHawk-derived ε profile shows a narrower peak between 3200 and 3600 m. The Data-Hawk may have sampled a thinner region of the K-H layer (∼ 400 m), perhaps associated with the edge of a K-H billow. This could also be the case for V6 as the Thorpe analysis suggests a ∼ 300 m deep layer at the altitude of ∼ 3.3 km (and an additional thinner layer around the altitude of ∼ 3.6 km). If we exclude the difference in layer depth during A1, ε L out and ε S estimated from both radars coincide very well with DataHawk-derived ε both in shape and levels during A1, D1, A2 and D2, with very similar variations in time (Table 3a and Fig. 4), indicating that the two radar models are satisfactory and are equivalent in these circumstances. In contrast, the ε N profiles exhibit the worst agreement with DataHawk-derived ε near the center of the K-H layer where they show a minimum (Figs. 2a and 3b, solid and dashed blue lines). This feature is similar to the one reported by L18 (their Fig. 12) for a turbulent layer generated by a convective instability at a mid-level cloud base. Table 3a and Fig. 4 confirm that ε N values are lower than the other estimates by a factor of 2 to 3 approximately during A1, D1, A2 and D2 for both radars. This disagreement, occurring repeatedly on the two radars, confirms the inadequacy of the ε N model for this layer.

Comparison between DataHawk-derived ε and ε T
The altitude and depth of the turbulent layers identified by the Thorpe method from V6 and ε T (Eq. 6) with c = 1 are shown by the dots and the solid vertical segments, respectively, in grey in Fig. 2a and in magenta in Fig. 3d. In Fig. 3d, ε T values (Eq. 7) with c = 1, c BP = 0.24 and c Sc = 0.41 are also shown for the K-H layer at 3.33 km and the turbulent layer at 2.37 km discussed in Sect. 4.2. N 2 s and N 2 rms for the K-H layer are 7.7 × 10 −6 and 7.1 × 10 −6 s −2 , respectively, i.e., 7.4 × 10 −6 s −2 on average. Because L T = 130 m in the K-H layer, we obtain ε T ≈ 0.35 mW kg −1 , which is about 7 times lower than DataHawk-derived ε (2.4 mW kg −1 ) (Table 3a). The hypothesis that V6 passed through the K-H layer in a region where ε was much lower is not consistent with the low variability (stationarity) of the dissipation rates estimated from DataHawk and radar data for more than 40 min (see Table 3a and Fig. 4). We therefore assume that ε T must be ≈ 2.4 mW kg −1 . To achieve this condition with Eq. (6), we must have c = 2.6.
On the other hand, estimating ε T (Eq. 7) requires the retrieval of S, but there is no prescribed method to compute the vertical shear of horizontal wind from balloon data in the Thorpe layers. Here, we estimated S from the difference of the wind vectors at the extremities of the Thorpe layer and from a linear interpolation of the zonal and meridional wind components in the Thorpe layer. We found S = 0.013 and 0.010 s −1 , respectively, i.e., 0.0115 s −1 on average, so that Ri ≈ 0.055. This value is close to the mean value ( Ri = 0.09) obtained at a vertical resolution of 20 m (Fig. 2b). The relevance of ε T ≈ 0.35 mW kg −1 obtained with c = 1 from Eq. (6) can be tested from Eq. (8) with c = c Ri −3/4 = 1 using c BP = 0.24 and c Sc = 0.41. We get Ri = 0.15 and Ri = 0.30, respectively. These values are significantly larger than 0.055. For S = 0.0115 s −1 and c = 1, we get ε T ≈ 25.7 mW kg −1 , i.e., about 11 times larger than DataHawk-derived ε. We must have c = 0.31 to be consistent with the DataHawk-derived ε value. c Sc = 0.41 and c BP = 0.24 (and c = 0.28 found in Appendix B) reasonably meet the necessary correction, giving credence to the validity of Eq. (7). As a corollary, for Ri = 0.055, we would get c = c Ri −3/4 = 2.7, i.e., the required value of c for Eq. (6) to be valid. The various estimates of ε T for the Thorpe layer are shown in Fig. 3d. Although based on a fragile hypothesis (ε from Thorpe analysis of the radiosonde data is equal to DataHawk-derived ε), Eq. (7) appears to be more adapted than the standard model (Eq. 6). It also has the major advantage over Eq. (6) that c is a true constant, at least when Ri < 0.25, although its value remains to be defined more precisely.   Table 4a shows the Hunt, Corrsin, buoyancy and Ozmidov length scales for the K-H layer calculated from WPR-LQ-7-derived and MU-radar-derived ε, σ and S during A1, D1, A2 and D2. N 2 is computed from balloon data at the radar resolutions (100 m for the WPR-LQ-7 and 150 m for the MU radar). Once the scales are calculated, they are averaged over the altitude range 3000-3600 m of the K-H layer to compare them with the Thorpe length. All the radar-derived scales reveal the same behaviors between the segments A1, D1, A2, and D2 and do not show substantial differences between the radars, reinforcing the reliabil-  Table 3. Mean values of TKE dissipation rates (mW kg −1 ) according to the different models and instruments for the K-H layer (a) and for the turbulent layer (TL) between 2100 and 2500 m (b).

Comparison of turbulence scales estimated from radar data
(a) K-H < ε U > < ε L out > < ε S > < ε N > < ε L out > < ε S > < ε N > ε T MU radar MU radar MU radar LQ-7 LQ-7 LQ-7  a This low value is due to a MU-radar-derived wind shear about 2 times smaller than LQ-7-and balloon-derived wind shear. It is doubtful and will affect L H and L C in Table 4. b (sorted/rms).  Table 3 for the K-H layer (3000-3600 m) sampled four times (top) and the layer between 2100 and 2500 m sampled twice during A1 (segment no. 1) and D2 (segment no. 4) (bottom). The horizontal black line at 0.5 mW kg −1 separates the two cases, and each radar-derived value has been successively shifted to the right by 0.025 with respect to segment numbers for clarity. Appendix B (c = 0.28) and the needed value of 0.31. We obtain ( L H / L B ) 2 = Ri = 0.19 and c /c 4/3 = Ri = 0.28. Both radar estimates are significantly larger than Ri estimated from balloon data with the Thorpe analysis but are close to Ri = 0.33 estimated from balloon data at the vertical resolution of 100 m (Fig. 2b). The quantitative disagreements result mainly from comparisons between estimates made with different techniques (radar and in situ) and resolutions. Nevertheless, it is interesting to note that these comparisons tend to corroborate the conclusions obtained from in situ measurements alone (Sect. 4.1.2).

The turbulent layer between 2100 and 2500 m
Using the same methods as for the K-H layer, we obtain Ri = 0.75 (14.5) at a vertical resolution of 20 (100) m from V6 data (Fig. 2b), Ri ≈ 2.0 from Thorpe analysis of V6 in the altitude range [2100-2500 m] (not shown) and Ri = 4.6 from N 2 calculated at a vertical resolution of 100 m from V6 data and S calculated from WPR-LQ-7 data during A1 and D2 (Table 4b). Therefore, the Richardson number strongly varies according to the method and data used, but all the estimates are consistent with a Ri value significantly larger than for the K-H layer and likely larger than 1 (Sect. 4.1). Therefore, the weakly stratified condition (Ri < 0.25) for which the alternative Eqs. (4) and (7) are valid is likely not verified for this layer. DataHawk-derived ε is about 1 order of magnitude lower than for the K-H layer: ∼ 0.2 mW kg −1 on average (Table 3b). The mean value is less reliable during D2. Many values of DataHawk-derived ε are missing because the algorithm did not detect a −5/3 subrange in the velocity spectra (see L18). Figure 2a, Table 3b and Fig. 4 show that ε L out and ε N and their mean values derived from both radars and ε T are close to each other (within a factor of less than ∼ 2) and are very consistent with DataHawk-derived ε. All the radar and DataHawk estimates together show a tem- Table 4. Mean values of Hunt, Corrsin, buoyancy and Ozmidov length scales for the K-H layer (a) and for the turbulent layer (TL) between 2100 and 2500 m (b).   Table 3).
poral decrease by a factor of 3 to 4 in the 40 min between A1 and D2 (Fig. 4), giving credence that the agreements between the various estimates during A1 and D2 are not fortuitous. The temporal decrease of ε is consistent with a decaying turbulence when Ri > 1. ε S shows the largest discrepancies with DataHawk-derived ε, perhaps because the model is not valid for large Ri values. L H and L C (136 m and 202 m, respectively) exceed L B = 60 m and L O = 56 m, which are close to L T (64 m) (Table 4b). Therefore, L H and L C should not be the turbulence scales to consider. From the Thorpe analysis, N 2 ≈ 1.4 × 10 −5 s −2 and S ≈ 0.0026 s −1 (Ri ≈ 2). From Eq. (6) with c = 1, ε T ≈ 0.2 mW kg −1 (Table 3b), i.e., very close to the mean value of DataHawkderived ε or only 2 times lower than the value during A1. It is consistent with the fact that L T can be assimilated to L O . From Eq. (7) with Ri = 2, c Sc = 0.41 and c BP = 0.24, we obtain ε T ≈ 0.012 and 0.004 mW kg −1 , respectively, which is much less than 0.2 mW kg −1 . The various estimates of ε T are shown in Fig. 3d. As expected ε T fails because it is expected to be valid for Ri < 0.25 only. (2) Figure 5 shows the histogram of log 10 (L), where L = σ 2 3/2 /ε U for σ 2 3/2 > 0.01 as in L18 obtained from the WPR-LQ-7 from data collected during 36 flights (corresponding to 90 profiles). The peak of the distribution has a mean (median) value of 67 m (71 m). These values are almost identical to those obtained from comparisons with MU radar, i.e., 75 m (61 m) (Fig. 7a of L18). This result seems to indicate that the empirical expression ε L out with L out = 70 m is not specific to the MU radar but at least to any radar with similar resolution volume (Table 2). However, the acquisition time of the MU radar and WPR-LQ-7 data differs by a factor of ∼ 2.5 (Table 2). It is likely fortunate that the statistical values of ε L out (and thus σ ) coincide so well.
5.2 Statistical evaluation of the models from comparisons with ε U Figure 6a, b and c show the scatter plots of log 10 (ε U ) vs log 10 (ε L out ), log 10 (ε S ) and log 10 (ε N ) with L out = 70 m, the shear estimated from WPR-LQ-7 data and N from DataHawk data at a height resolution of 100 m. Figure 6d and e show the results assuming a constant shear ( S = 7.7 m s −1 km −1 ) and a constant N ( N 2 = 6.7 × 10 −5 s −2 ). Of course, Fig. 6d and e differ only in the constant 0.64/0.5.
The correlation coefficients are fortuitously ∼ 0.66 for all the cases except for ε N for which the correlation is ∼ 0.60 only. This is an additional clue of the inadequacy of ε N . The red lines show the results of linear regressions after rejecting dissipation rate values smaller than 1.6 × 10 −5 m 2 s −3 as in L18, even though the quantitative threshold has no reason to be the same since the comparison methods differ. The slope of the regression line between log 10 (ε L out ) and log 10 (ε U ) is ∼ 1.0 (Fig. 6a), confirming the statistical σ 3 dependence of ε when no discrimination is made on the con- Figure 6. Scatter plots of log 10 (ε U ) vs (a) log 10 ε L out , (b) log 10 (ε S ), (c) log 10 (ε N ), (d) log 10 (ε S ) with constant S and (e) log 10 (ε N ) with constant N. The red lines are the result of a line regression (whose slope value is indicated in the panel) for ε U > 1.6 × 10 −5 m 2 s −3 , and R is the correlation coefficient. ditions under which turbulence occurs. The regression slope obtained with log 10 (ε N ) or log 10 (ε S ) for S or N equal to a constant (Fig. 6d, e) is 0.60, i.e., close to 2/3, as expected because the two models vary as σ 2 . However, the regression slope between log 10 (ε U ) and log 10 (ε N ) with measured N (Fig. 6c) is significantly lower than 0.66 (0.50), and the regression slope between log 10 (ε U ) and log 10 (ε S ) with measured S (Fig. 6b) is significantly larger than 0.66 (0.73). The regression slope between log 10 (ε U ) and log 10 (ε N ) using MU radar data was 0.55 (L18), i.e., virtually identical to the present case (Fig. 6c). All the regression slopes depend on the quantitative threshold on ε, and, in Fig. 6a, it varies from ∼ 0.9 to ∼ 1.1 for different thresholds, excluding small values. However, all other slope estimates vary in concert, so that the observed trends remain valid for a different threshold.
1. A regression slope between log 10 (ε U ) and log 10 (ε S ) that is closer to 1 than the slope between log 10 (ε U ) and log 10 (ε N ) indicates that ε S provides estimates more consistent with ε L out than ε N . Figure 7a and b show a comparison between the radar models, i.e., log 10 (ε L out ) vs log 10 (ε N ) and log 10 (ε L out ) vs log 10 (ε S ), respectively, for ε U > 1.6×10 −5 m 2 s −3 . The red lines are the results for constant N and constant S. The blue lines show the results of a linear regression. The blue and red lines obviously coincide (slope = 0.60) for ε N . A slope of 0.77 is obtained with ε S , indicating a greater equivalence between ε L out and ε S , as expected from Fig. 6b. Consequently, our results suggest that ε S is more relevant than ε N and should be used instead of ε N for operational use if the empirical model ε L out is not chosen.
2. We checked that generating a normal random distribution of N or S with a mean and standard deviation similar to the observed distributions produced a regression slope close to 2/3, similar to Fig. 6d and e. Therefore, the observed slopes with measured S and N (Fig. 6b, c) must reveal a statistical dependence of σ with 1/N and S, respectively. The equivalence between ε L out and ε S described in Sect. 4 for the K-H layer implies that σ is simply proportional to S (σ = 0.64 L out S) if L out is constant. There is a canonical value of L out (∼ 70 m), but since L out is not constant and is unknown (and can vary by 2 orders of magnitude at least; Fig. 5), the correlation between σ and S can only be established for a fixed value of L out (and for any other variable on which σ depends). Figure 8 shows the same information as Fig. 7 but after dividing by σ 2 to remove the self-correlation between the variables and to show the relationship between log 10 (σ/L out ) and log 10 (0.5N ) (Fig. 8a) and between log 10 (σ/L out ) and log 10 (0.64S) (Fig. 8b). The two scatter plots show negative and positive correlation coefficients (−0.36 and 0.12, respectively). The correlations are weak but significant according to the P value. If no threshold on ε U is applied, the correlation coefficients are −0.26 and +0.22. This suggests that σ increases to some extent as S increases and N decreases. It is quite intuitive, but, to our knowledge, this is the first study to suggest and highlight this. The results may reveal a Richardson number dependence. Figure 9a shows scatter plots of σ vs Ri 1/2 100 , where Ri 100 (S 100 ) now explicitly refers to the Richardson number (shear) calculated at the vertical resolution of 100 m. The red (black) dots show the results without and with a threshold on S 100 (S 100 > 5 m s −1 km −1 ), respectively. The (negative) correlation coefficient is slightly stronger with the threshold (−0.34 instead of −0.21). The high values of Ri 100 are mainly associated with a weak shear (S 100 < 5 m s −1 km −1 ) and with the largest variability in σ (Fig. 9a). However, this property may not be significant because the uncertainty on Ri increases as the shear tends to zero. Therefore, we focus on the scatter plot obtained with the threshold on the shear (black dots). It seems to show a linear dependence between log 10 (σ ) and log 10 (Ri −1/2 100 ), at least down to log 10 (Ri 1/2 100 ) ≈ −0.2, i.e., for Ri 100 < 0.4. Attempts of linear regression analysis do not confirm the linear trend, likely due to the strong dispersion and weak correlation. However, the time series obtained from the concatenation of all the profiles of log 10 (Ri −1/2 100 ) and log 10 (σ ) after removing their mean values reveal a more obvious dependence between the two variables (Fig. 9b). The curves reveal similar variations and dynamics, especially for records [0 − 200], compatible with σ 2 inversely proportional to Ri 100 , at least to a first approximation. For log 10 (Ri 1/2 100 ) −0.2, i.e., for low values of Ri 100 (< 0.4), log 10 (σ ) appears to have very little dependence on log 10 (Ri 1/2 100 ). If meaningful, it would be consistent with the fact that N does not play a significant role for low Ri values, as suggested by the ε S model. Figure 10 shows the scatter plots of log 10 (ε L out /ε U ), log 10 (ε N /ε U ) and log 10 (ε S /ε U ) vs log 10 (Ri 100 ) applying two thresholds on ε U : 1.6 × 10 −5 m 2 s −3 in Fig. 10a, b and c and 5.0 × 10 −4 m 2 s −3 in Fig. 10d, e and f. The latter is introduced to analyze the dependence of the results on the levels of considered dissipation rates. The red and blue curves show the values averaged in bandwidths of 0.3 from log 10 (Ri 100 ) = −1.7. For ε U > 1.6 × 10 −5 m 2 s −3 , the mean curves of log 10 (ε L out /ε U ) and log 10 (ε S /ε U ) do not reveal a significant dependence on log 10 (Ri 100 ), at least up to log 10 (Ri 100 ) ∼ 1, and are almost identical and close to 0 (Fig. 10a, c). Therefore, the applicability of the two models does not seem to depend significantly on the Richardson number on average. For ε U > 5.0 × 10 −4 m 2 s −3 , the curves produced by two models remain close and almost unchanged for log 10 (Ri 100 ) < 0 (Fig. 10d, f). However, the mean values of log 10 (ε S /ε U ) now tend to decreases as log 10 (Ri 100 ) increases. Therefore, when log 10 (Ri 100 ) > 0, ε S tends to underestimate ε U when ε U exceeds ∼ 5.0 × 10 −4 m 2 s −3 and inversely when ε U < 5.0 × 10 −4 m 2 s −3 . The fact that we experimentally observe that ε S is not appropriate for large values of Ri 100 is consistent with the expected domain of applicability of the model, even if it is not clear why it is in this way. For log 10 (Ri 100 ) < 0, log 10 (ε N /ε U ) is less than 0 and decreases as log 10 (Ri 100 ) decreases for both thresholds (Fig. 10b, e). This experimental observation is a confirmation of the inadequacy of ε N when the Richardson number is low. The results with ε L out are difficult to interpret when log 10 (Ri 100 ) > 0. The model is consistent with ε S when ε U > 1.6×10 −5 m 2 s −3 (Fig. 10c) and seems to be more consistent with ε N than with ε S when ε U > 5.0 × 10 −4 m 2 s −3 (Fig. 10e, f). It may be vain to interpret the properties of this model, since it is only an empirical model for which L out = 70 m only represents a canonical value of a function of multiple variables, including the shear and N.

Conclusions
The objective of this work was to test the suitability of TKE dissipation rate models based on Doppler radar spectral width measurements from comparisons with in situ estimates (ε U ) derived from high-resolution Pitot tube measurements aboard DataHawk UAVs. We showed the following: 1. The models applied to the 46.5 MHz MU VHF radar by L18 produce statistically identical results on the 1.357 GHz WPR-LQ-7.
a. The empirical model ε L out = σ 3 /L out with L out ∼ 70 m (as for the MU radar) provides the best statistical agreement with ε U , at least for ε 2 × 10 −5 m 2 s −3 (Table 2). If L out really depends on the size of energy containing eddies, it is then independent of radar parameters (assuming σ 2 is a true indication of w 2 in both radars). b. The model ε N predicting a σ 2 N law for stably stratified conditions fails to reproduce ε U . The biases Figure 7. Scatter plots of (a) log 10 ε L out vs log 10 (ε N ) and (b) log 10 ε L out vs log 10 (ε S ) for ε U > 1.6 × 10 −5 m 2 s −3 . The red lines show the results for constant N (a) and constant S (b). The blue lines show the results of a linear regression (whose slope value is indicated in the panel).
are nearly quantitatively identical to those obtained with the MU radar: ε N tends to overestimate when ε U 5 × 10 −4 m 2 s −3 and to underestimate when ε U 5 × 10 −4 m 2 s −3 .
2. Applying ε N to both radars to a turbulent layer attributed to a K-H instability with Ri < 0.25 strongly underestimates ε U in the core of the layer when N 2 is minimum. On the other hand, in agreement with the statistical results, ε N provided values consistent with the other estimates in a turbulent layer, likely associated with larger Ri ( 1). These two observations are rather consistent with the domain of validity of ε N according to the theoretical derivations (Eq. 5), leading to the newly introduced expression of ε S expected to be valid for weakly stratified or strongly sheared conditions (e.g., . 3. The application of ε S to the K-H layer (Ri < 0.25) using both radars leads to a good agreement with ε U . Its application to the turbulent layer associated with larger Ri slightly underestimates ε U , again in accordance with Eq. (5).
4. The statistical comparisons between ε S and ε U using all data show much better agreement than between ε N and ε U , although a bias of the same nature as that observed with ε N is also noted but to a lesser degree. Empirical ε L out remains the most consistent model compared with ε U . L out ∼ 70 m is likely a canonical value that results from all the hidden contributions of the various param- Figure 9. (a) Scatter plots of log 10 Ri 1/2 100 vs log 10 (σ ) for ε U > 1.6 × 10 −5 m 2 s −3 without the threshold on the shear (grey) and for S 100 > 5 m s −1 km −1 . (b) The corresponding time series of log 10 Ri −1/2 100 (grey) and log 10 (σ ) (black) after subtracting their mean. eters that a most general (and unknown) model should include.
5. The equivalence between ε S and ε U for the K-H layer associated with a low Ri necessarily implies that σ is proportional to S: σ ∼ 0.64 L out S. For all the layers with the same value of L out , then σ linearly depends on S. This is a necessary condition if agreement is observed with two models that predict a σ 3 and a σ 2 dependence. For a wide distribution of log 10 (L out ) as in Fig. 5, which includes values for all Ri, this linear dependence should be strongly "blurred" because L out is variable, and Ri is not necessarily low. Moreover, an additional source of dispersion is that the wind shear calculated at the radar resolution S 100 and at a time resolution of 10-30 min is not necessarily the most effective shear to be considered because S is a scale-dependent parameter (in the same way as Ri). As a result, a very weak, but yet significant, correlation between σ and S 100 was found (Fig. 8). This weak correlation is responsible for the better agreement obtained with ε S than with ε N . However, because the results are based on a limited amount of data and because some degree of coincidence cannot be ruled out, further studies are necessary to analyze the dependence between σ and S 100 , in particular under more suitable conditions (i.e., low Ri).
6. Reciprocally, σ has a statistical degree of dependence on 1/N , as revealed by the poorer statistical agreement between ε N and ε U , leading to a regression slope of less than 2/3 (0.50, almost identical to 0.55 obtained from MU radar data by L18) (Fig. 6).
with the fact that N should not affect turbulence when the Richardson number is low (Eq. 4).
8. The analysis of the three models, ε L out , ε N and ε S , with ε U vs Ri 100 (Fig. 10) confirms the good agreement between (ε L out , ε U ) and between (ε S , ε U ) and the inadequacy of ε N for Ri 100 1. The underestimation of ε N increases as Ri 100 decreases. The results for Ri 100 1 are more difficult to interpret and more puzzling, but ε S and ε L out lead to comparable results and do not show substantial bias as a function of Ri 100 . In any case, all results involving large Ri (> 1) must be taken with caution because the turbulence may be intermittent. In principle, the interpretation of the results should consider this.
9. The classical model ε T = c 2 L 2 T N 3 (Eq. 6) based on the equivalence between the Thorpe length L T and the Ozmidov length scale L O (c = 1) fails to reproduce DataHawk-derived ε in the K-H layer for which Ri is expected to be less than 0.25. Although the disagreement can be due to several factors (e.g., an inappropriate choice of c, horizontal inhomogeneity), it can also be due to the fact the model involves the Ozmidov length scale defined for a turbulence affected by the stable stratification. In essence, L T cannot be related to L O anymore by a constant if the stratification effects can really be neglected for low Ri. Therefore, an alternative approach using the Corrsin length scale L C instead of L O was introduced, leading to ε T = c 2 L 2 T S 3 (Eq. 7), compatible with studies showing a Ri 3/4 dependence of L T /L O for Ri < 0.25. Contrary to c, c is a true constant (with respect to Ri) for low Ri. It is worth noting that Eqs. (7) and (4) form a coherent pair of models independent of N for a weak stratification or strongly sheared flows. Using values of c deduced from the literature, ε T provides estimates consistent with DataHawkand radar-derived ε (expect ε N ) for the K-H layer. On the other hand, ε T fails for a decaying turbulent layer (Ri > 1) as the model is not expected to be valid for Ri > 0.25. ε T with c = 1 shows a better agreement with DataHawk-and radar-derived ε (including ε N ), coherent with the fact that the stable stratification should affect the turbulence for large Ri. These results need to be confirmed by statistical studies.

Appendix A
The turbulent Froude number Fr = ε/N k, where k refers to TKE for a simple and standard notation, is often used to characterize turbulent mixing (e.g., Ivey and Imberger, 1991). A strong and weak stratification is associated with Fr < 1 and Fr > 1, respectively. T L = k/ε is called the inertial timescale and is a characteristic time of TKE dissipation. The corresponding timescales associated with the turbulence production by the wind shear and with the conversion into potential energy are S −1 and N −1 , respectively. When the stratifica-tion is weak, i.e., when N T L = Fr −1 < 1, then T L (dissipation time) and S −1 (production time) should be of the same order, i.e., the shear parameter ST L = O(1), for stationary turbulence. Several studies (e.g., Mater and Venayagamoorthy, 2014, and references therein) reported a critical value for weakly stratified and stationary flows: Assuming isotropy, k = 3/2 w 2 , where w 2 is the vertical velocity variance (assumed to be σ 2 in the paper). We obtain ε = 0.45 w 2 S.
with α ≈ 0.7. Note that this coefficient is deduced from their Fig. (3) using the linear trend shown by the authors. They did not explicitly refer to this value. By using the definitions Fr = ε/N k (see Appendix A) and L O = ε/N 3 , Eq. (B1) can be re-written as ε = (Fr/α) 3 L 2 T N 3 .
Equation (B4) provides a way to relate the isotropic turbulent length scale L k defined as the scale of the largest eddies weakly affected by the buoyancy and the shear to the Thorpe length: L k ∼ 1.5 L T . It also provides an expression of the master length scale L M defined as (2k) 3/2 /B 1 ε (e.g., Mellor and Yamada, 1982) with 11.9 ≤ B 1 ≤ 27.4 (Table 2, Basu and Holtslag, 2021). We obtain L M = 4.24/B 1 L T . On the other hand, for strongly stratified conditions (Fr < 1), we can write, from Fig. (3) of Garanaik and Venayagamoorthy (2019), with α ≈ 0.6 according to the figure, or, purposely, 0.66 = 2/3. We obtain ε = 3/2 FrL 2 T N 3 , For k = 3/2 w 2 , then, w 2 = L 2 T N 2 or If k = β w 2 , then L B = 1/ (α β)L T . Equation (B5) demonstrates that the equivalence (at least the proportionality) between the buoyancy length scale and the Thorpe length is valid for strongly stratified conditions only (i.e., for conditions for which stability affects the vertical motions before being affected by the wind shear). For a weak stratification, the vertical TKE cannot be fully converted into potential energy because the parcels cannot move vertically over a length of L B (but L H only). Therefore, the basic stability does not intervene anymore in the variance of the vertical velocity fluctuations as in the case of a neutral stratification.
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Financial support. This research was partially supported by JSPS KAKENHI (grant no. JP15K13568) and the research grant for Mission Research on Sustainable Humanosphere from Research Institute for Sustainable Humanosphere (RISH), Kyoto University. Lakshmi Kantha acknowledges partial support from U.S. Office of Naval Research (ONR) MISO/BoB DRI (grant no. N00014-17-1-2716). Partial support to Dale Lawrence and Abhiram Doddi was provided by NSF Grant PDM 2128444 "New Pathways to Enhanced Turbulence and Mixing via Kelvin-Helmholtz Instability Tube and Knot Dynamics".
Review statement. This paper was edited by Wiebke Frey and reviewed by two anonymous referees.