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

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## Abstract

*L*

_{C}≈ 50–70 m. This empirical expression constitutes a simple way to estimate dissipation rates in the lower troposphere from MU radar data whatever the sources of turbulence be, in clear air or cloudy conditions, consistent with UAV estimates.

## Keywords

VHF radar Unmanned aerial vehicle Atmospheric turbulence Energy dissipation rate Outer scales of turbulence Doppler variance## Abbreviations

- AR
aspect ratio

- ASL
above sea level

- CBL
convective boundary layer

- CU
Colorado University

- CWT
cold wire temperature

- IMET
international met (systems)

- KOC
Kolmogorov–Obukhov–Corrsin

- MCT
mid-level cloud base turbulence

- ST
stratosphere troposphere

- MU
middle and upper atmosphere

- PTU
pressure temperature humidity

- ShUREX
Shigaraki UAV radar experiment

- SNR
signal-to-noise ratio

- TKE
turbulence kinetic energy

- UHF
ultra-high frequency

- VHF
very high frequency

- UAV
unmanned aerial vehicle

## Introduction

Turbulence kinetic energy (TKE) dissipation rate \(\varepsilon\) is a fundamental parameter indicative of the strength of turbulence. Dissipation rates of atmospheric turbulence can be potentially retrieved from stratosphere–troposphere (ST) radars operating in VHF (e.g., Hocking 1983, 1985, 1986, 1999; Fukao et al. 1994; Hocking and Hamza 1997; Nastrom and Eaton 1997; Li et al. 2016) and UHF bands (e.g., Sato and Woodman 1982; Cohn 1995; Bertin et al. 1997; Wilson et al. 2005 and references therein). Because ST radars can be used for detecting turbulence in the free atmosphere (above the atmospheric boundary layer), standard models are based on the assumption that turbulence results from shear flow instabilities in a stably stratified background (e.g., Fukao et al. 1994; Kurosaki et al. 1996; Nastrom and Eaton 1997). For such turbulence, the stable stratification limits the size of the largest turbulent eddies and damps vertical motions, leading to the definition of various outer scales of stratified turbulence (e.g., Weinstock 1978a, b, 1981). Additional key assumptions used to retrieve turbulence parameters from radar data are that isotropic turbulence, following the Kolmogorov–Obukhov–Corrsin (KOC) model, exists at smaller scales in the inertial subrange and that the Bragg wavelength of radar backscatter lies within this inertial subrange. Estimation of \(\varepsilon\) is therefore based on the measurement of radar Doppler spectral width assuming that part of the spectral broadening results from isotropic turbulent motions in the radar measurement volume (the so-called Doppler method). Indirect estimates of \(\varepsilon\) can also be obtained from the estimates of refractive index structure constant \(C_{n}^{2}\) from radar echo power (e.g., Gage and Balsley 1978; Cohn 1995; Hocking and Mu 1997; Hocking 1999) but that is beyond the scope of the present work.

The purpose of the present work is to show the results of comparisons between \(\varepsilon\) estimates made from middle and upper atmosphere (MU) radar data in the lower troposphere using existing formulations and direct in situ estimates of \(\varepsilon\) obtained from small unmanned aerial vehicles (UAVs) equipped with high-frequency sampling and fast-response Pitot (airspeed) sensors (Kantha et al. 2017). The potential of UAVs for characterizing turbulence properties was thoroughly described by Lawrence and Balsley (2013). Contrary to the radar technique which samples the atmosphere inside a volume at a fixed location, the UAV has the potential to probe all the space occupied by a turbulent layer/patch and may be a better tool for identifying the dimensions of a turbulent volume and for estimating outer scales.

The datasets were collected during two field campaigns, called the Shigaraki UAV Radar Experiments (ShUREX), in May–June 2016 and June 2017 at the Shigaraki MU observatory in Japan. Kantha et al. (2017) described the instruments and configurations used during a previous ShUREX campaign in June 2015. The instrumental setup did not significantly change in 2016 and 2017, except for the use of higher-frequency sampling and lower-noise turbulence sensors. The CU DataHawk UAVs flew in the immediate vicinity of the radar (within a horizontal distance of ~ 1.0 km) and up to altitudes of ~ 4.0 km above the sea level (ASL). Selected data from 39 science flights (16 in 2016 and 23 in 2017) were used for the present study. “Instruments and data” section describes briefly the MU radar and DataHawk UAV and the observational configurations used during the campaigns. “Theoretical bases and practical methods of \(\varepsilon\) estimation” section presents the theoretical expressions used for retrieving TKE dissipation rates from radar and UAV data and describes in detail the practical methods. As summarized by Hocking (1999), the Doppler method leads to different analytical expressions according to hypotheses made on the properties of turbulence and according to the radar specifications. In the present work, we will focus on the most commonly used expressions without describing their derivations. The underlying hypotheses will be shortly recalled. More details can be found in Kantha et al. (2018) “Comparisons between \(\varepsilon_{U}\) and \(\varepsilon\) from the radar models” section presents the results of comparisons between the various dissipation rate estimates. These results are discussed in “Discussion” section, and conclusions are presented in “Conclusions” section.

## Instruments and data

### MU radar

MU radar parameters used during ShUREX 2016 and 2017 campaigns

Parameter | |
---|---|

Beam directions | (0°, 0°),(0°, 10°), (90°, 10°) |

Radar frequencies (MHz) | 46.00, 46.25, 46.50, 46.75, 47.00 |

Interpulse period (μs) | 400 |

Subpulse duration (μs) | 1 |

Pulse coding | 16-bit optimal complementary code |

Range resolution (m) | 150 |

Height sampling (FII) (m) | 5 |

Number of gates | 128 |

Coherent integration number | 32 |

Incoherent integration number | – (time series) |

Number of FFT points | 128 |

Acquisition time for one profile (s) | 24.57 |

Profile acquired every: (s) | 6.144 |

Nyquist frequency (Hz) | 2.60 |

Velocity aliasing (m s | 8.4 |

### 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 bases and practical methods of \(\varepsilon\) estimation

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

*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:

*a*, 2

*b*

^{1}we have (e.g., Hocking 1983, 1999, 2016):

*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.

*a*, 2

*b*. The White et al. (1999) formulation also considered the effects of the wind advection:

*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

*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”.

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).

^{−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.

*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.

## Comparisons between \(\varepsilon_{U}\) and \(\varepsilon\) from the radar models

### Estimation of \(L_{C}\)

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

- (1)
where clouds were observed,

- (2)
associated with cloudy or clear air convective boundary layers (CBL),

- (3)
associated with convective layers underneath clouds [mid-level cloud base turbulence (MCT) layer (e.g., Kudo et al. 2015)].

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

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

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

## Discussions

Our statistical results suggest that the TKE dissipation rates estimated from the MU radar operated at a range resolution of 150 m and a beam aperture of 1.32° are proportional to \(\sigma^{3}\), for all atmospheric conditions in the lower atmosphere (up to ~ 4.0 km). The \(\sigma^{3}\) dependence is consistent with the results reported by Chen (1974) who showed from data collected by airplanes that TKE dissipation rates are proportional to \(\left\langle {w^{\prime 2} } \right\rangle^{3/2}\) at stratospheric heights. To some extent, it is also consistent with results reported by Bertin et al. (1997), who performed similar studies for stratospheric heights with the UHF Proust radar and high-resolution balloon data. However, partly due to the small amount of data, they suggest both \(\sigma^{2}\) and \(\sigma^{3}\) dependences for the same dataset (their Figs. 7b, 8). Jacoby-Koaly et al. (2002) compared TKE dissipation rates estimated from UHF (1238 MHz) radar and airplane data in CBL at a vertical resolution of 150 m by using the White et al. formulation. They found good statistical agreements (at least when using data from oblique beams, because data collected from the vertical beam were contaminated by ground clutter). McCaffrey et al. (2017) also compared dissipation rates estimated in the planetary boundary layer from two UHF (449 MHz and 915 MHz) radars using the White et al. (1999) model with values estimated from sonic anemometers mounted on a 300 m tower. Their results also tend to confirm a \(\sigma^{3}\) dependence for turbulence in the CBL.

In the present work, the \(\varepsilon_{N}\) model was found to be inadequate, although it has also been widely used with VHF radar observations (e.g., Hocking 1983, 1985, 1986, 1999, Fukao et al. 1994; Delage et al. 1997; Nastrom and Eaton 1997; Fukao et al. 2011, among many others). However, *on average*, and for stratified conditions, the model producing \(\varepsilon_{N}\) provides reasonable agreements with \(\varepsilon_{U}\), but it tends to overestimate (underestimate) TKE dissipation rates when turbulence is weak (large). Li et al. (2016) compared \(\varepsilon_{N}\) values obtained with the 53.5 MHz MAARSY radar at tropo-stratospheric heights with indirect estimates of dissipation rates from balloon data using a Thorpe analysis. Considering all the possible sources of discrepancies, the agreement was satisfying but the approach was likely not adapted for validating \(\varepsilon_{N}\) since the estimates from balloon data were themselves based on a model. Deghan et al. (2014) made studies similar to those presented here with a 40.68 MHz VHF radar and airplane observations mainly in the boundary layer. They also found reasonable agreements between \(\varepsilon_{N}\) and dissipation rates estimated from aircraft measurements (their Figs. 8, 12), sometimes with some noticeable difference in levels (a factor 5 in Fig. 12) and at coarser range resolution (500 m). In addition, they used standard values of \(N^{2}\) for some comparisons. This dataset may not have been sufficiently large for highlighting the biases produced by the model producing \(\varepsilon_{N}\) so that our respective results are not necessarily incompatible.

## Conclusions

- (a)
Maxima of \(\varepsilon\) (\(\sim10^{ - 5} - 10^{ - 2} \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) typically, or \(\sim 0.01 - 10 {\text{ mW}}\,{\text{kg}}^{ - 1}\)) were observed at the same altitudes and times by the UAV and the radar indicating that the same turbulence events were generally sampled by both instruments at a horizontal distance of 1.0 km or less. Conditions were thus favorable to test the standard models used for estimating \(\varepsilon\) from radar data.

- (b)
\(\varepsilon_{U}\) was found to be proportional to \(\sigma^{3}\), not \(\sigma^{2}\) as expected for stably stratified turbulence. The best agreement in turbulence levels was found by assuming that \(\varepsilon_{U} = K\sigma^{3}\) with \(K\sim0.016\). This surprisingly elementary model is equivalent to assuming a characteristic scale \(L_{C}\) of the order of 50–70 m. This scale is not necessarily related to an effective outer scale of turbulence because it seems to be appropriate for all cases, convectively or shear-generated turbulence, in deep convective layers or in stratified conditions. Therefore, it is likely not relevant to compare this scale (defined from dimensional analysis) with the dimensions 2

*a*, 2*b*of the radar sampling volume (2*b*= 150 m, 2*a*= [53 m–156 m] in the height range 1.3–4.0 km) for selecting the right model. More refined analyses are necessary in order to know if there are really consistency problems or not. From a pragmatic point of view, this simple model can be used for estimating \(\varepsilon\) solely from measurements of VHF radar Doppler spectral width. It is likely accurate enough for climatological studies without the need for additional measurements of \(N^{2}\), at least for low tropospheric altitudes. All \(\varepsilon_{U}\) values larger than \(\sim1.6\, \times \,10^{ - 4 } \,{\text{m}}^{2} \,{\text{s}}^{ - 3}\) ( \(\sim0.16{\text{ mW}}\,{\text{kg}}^{ - 1}\)) were found to be associated with weak radar aspect ratios (\(\left| {AR} \right| < 3\,{\text{dB}}\)), which can reasonably be interpreted as backscatter from isotropic turbulence. It is thus consistent and justifies the use of Doppler spectra from the vertical beam when not altered by ground clutter. It means that data from oblique beams do not need to be used for estimating \(\varepsilon\) from radar data, even as weak as those reported in the present study and even at a vertical resolution of 150 m. The use of Doppler spectra measured from oblique directions requires many more corrections and accurate knowledge of horizontal wind shear, which can be the cause of additional uncertainties when estimating the Doppler variance produced by turbulence. - (c)
The key finding described in (b) does not mean that the Weinstock (1981) model adapted by Hocking (1983) is always irrelevant for stratified conditions. But it is likely not suitable for lower troposphere observations at a range resolution of 150 m with the MU radar and for the observed range of \(\varepsilon\) values.

- (d)
\(\varepsilon_{W}\), based on the formulation proposed by Frisch and Clifford (1974) and subsequent authors, is also suitable since it predicts a \(\sigma^{3}\) dependence. However, it is a slight underestimate with respect to \(\varepsilon_{U}\) (\(\varepsilon_{U} \sim2.2 \varepsilon_{W}\) in average) but is not biased.

- (e)
Finally, \(\varepsilon_{R}\) and \(\varepsilon_{W}\) were found consistent with \(\varepsilon_{U}\) in convective regions (such as MCT and CBL) while \(\varepsilon_{N}\) failed to reproduce the correct levels in the core and at the edges of the convectively generated turbulent layers. This is an additional argument for avoiding a systematic use of \(\varepsilon_{N}\) in the free atmosphere. Nevertheless, if used, climatological values of \(N^{2}\) seem to be preferable to measured values.

## Footnotes

## Notes

### Authors’ contributions

HL performed all the radar and UAV data processing with assistance from HH and DL. LK led the ShUREX campaign and participated in the analysis and synthesis of the study results. DL was responsible for collection of the UAV data and AD provided the useable UAV data of ShUREX2017. All authors read and approved the final manuscript.

### Acknowledgements

This study was 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. The MU radar belongs to and is operated by RISH, Kyoto University. The authors thank T. Mixa, M. Yabuki, R. Wilson and T. Tsuda for their cooperation during the campaigns and thank N. Nishi (Fukuoka University) for his kind assistance during the revision of the manuscript and his supply of meteorological data used for reviewers’ reply.

### Competing interests

The authors declare that they have no competing interests.

### Funding

This study was 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. It was also partly supported by the US National Science Foundation (Grant No. AGS 1632829). The MU radar belongs to and is operated by RISH, Kyoto University.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## References

- Bertin F, Barat J, Wilson R (1997) Energy dissipation rates, eddy diffusivity, and the Prandtl number: an in situ experimental approach and its consequences on radar estimate of turbulent parameters. Radio Sci 32:791–804CrossRefGoogle Scholar
- Chapman D, Browning KA (2001) Measurements of dissipation rate in frontal zones. Q J R Meteorol Soc 127:1939–1959CrossRefGoogle Scholar
- Chen WY (1974) Energy dissipation rates of free atmospheric turbulence. J Atmos Sci 31:2222–2225CrossRefGoogle Scholar
- Cohn SA (1995) Radar measurements of turbulent eddy dissipation rate in the troposphere: a comparison of techniques. J Atmos Ocean Technol 12:85–95CrossRefGoogle Scholar
- Dehghan A, Hocking WK (2011) Instrumental errors in spectral-width turbulence measurements by radars. J Atmos Sol Terr Phys 73:1052–1068CrossRefGoogle Scholar
- Dehghan A, Hocking WK, Srinivasan R (2014) Comparisons between multiple in situ aircraft turbulence measurements and radar in the troposphere. J Atmos Sol Terr Phys 118:64–77CrossRefGoogle Scholar
- Delage D, Roca R, Bertin F, Delcourt J, Crémieu A, Masseboeuf M, Ney R (1997) A consistency check of three radar methods for monitoring eddy diffusion and energy dissipation rates through the tropopause. Radio Sci 32:757–767CrossRefGoogle Scholar
- Doviak RJ, Zrnic' DS (1993) Doppler radar and weather observations. Academic, San Diego, CA, p 562Google Scholar
- Frehlich R, Meillier Y, Jensen MA, Balsley BB (2003) Turbulence measurements with the CIRES tethered lifting system during CASES-99: calibration and spectral analysis of temperature and velocity. J Atmos Sci 60:2487–2495CrossRefGoogle Scholar
- Frisch AS, Clifford SF (1974) A study of convection capped by a stable layer using Doppler radar and acoustic echo sounders. J Atmos Sci 31:1622–1628CrossRefGoogle Scholar
- Fukao S, Sato T, Tsuda T, Yamamoto M, Yamanaka MD (1990) MU radar—new capabilities and system calibrations. Radio Sci 25:477–485CrossRefGoogle Scholar
- Fukao S, Yamanaka MD, Ao N, Hocking WK, Sato T, Yamamoto M, Nakamura T, Tsuda T, Kato S (1994) Seasonal variability of vertical eddy diffusivity in the middle atmosphere. 1. Three-year observations by the middle and upper atmosphere radar. J Geophys Res Atmos 99:18973–18987CrossRefGoogle Scholar
- Fukao S, Luce H, Mega T, Yamamoto MK (2011) Extensive studies of large-amplitude Kelvin–Helmholtz billows in the lower atmosphere with the VHF middle and upper atmosphere radar (MUR
*)*. Q J R Meteorol Soc 137:1019–1041CrossRefGoogle Scholar - Gage KS, Balsley BB (1978) Doppler radar probing of the clear atmosphere. Bull Am Meteorol Soc 59:1074–1093CrossRefGoogle Scholar
- Gossard EE, Chadwick RB, Neff WD, Moran KP (1982) The use of ground-based Doppler radars to measure gradients, fluxes and structure parameters in elevated layers. J Appl Meteorol 21:211–226CrossRefGoogle Scholar
- Hocking WK (1983) On the extraction of atmospheric turbulence parameters from radar backscatter Doppler spectra. I. Theory. J Atmos Terr Phys 45:89–102CrossRefGoogle Scholar
- Hocking WK (1985) Measurement of turbulent energy dissipation rates in the middle atmosphere by radar techniques: a review. Radio Sci 20:1403–1422CrossRefGoogle Scholar
- Hocking WK (1986) Observations and measurements of turbulence in the middle atmosphere with a VHF radar. J Atmos Terr Phys 48:655–670CrossRefGoogle Scholar
- Hocking WK (1999) The dynamical parameters of turbulence theory as they apply to middle atmosphere studies. Earth Planets Space 51:525–541. https://doi.org/10.1186/BF03353213 CrossRefGoogle Scholar
- Hocking WK, Röttger J, Palmer RD, Sato T, Chilson PB (2016) Atmospheric radar. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Hocking WK, Hamza AM (1997) A quantitative measure of the degree of anisotropy of turbulence in terms of atmospheric parameters, with particular relevance to radar studies. J Atmos Sol Terr Phys 59:1011–1020CrossRefGoogle Scholar
- Hocking WK, Mu PKL (1997) Upper and middle tropospheric kinetic energy dissipation rates from measurements of
*C*_{n}^{2}—review of theories, in situ investigations, and experimental studies using the Buckland Park atmospheric radar in Australia. J Atmos Sol Terr Phys 59:1779–1803CrossRefGoogle Scholar - Jacoby-Koaly S, Campistron B, Bernard S, Bénech B, Ardhuin-Girard F, Dessens J, Dupont E, Carissimo B (2002) Turbulent dissipation rate in the boundary layer via UHF wind profiler Doppler spectral width measurements. Bound Layer Meterol 103:3061–3389CrossRefGoogle Scholar
- Kantha L, Lawrence D, Luce H, Hashiguchi H, Tsuda T, Wilson R, Mixa T, Yabuki M (2017) Shigaraki UAV-Radar Experiment (ShUREX 2015): an overview of the campaign with some preliminary results. Prog Earth Planet Sci 4:19. https://doi.org/10.1186/s40645-017-0133-x CrossRefGoogle Scholar
- Kantha L, Luce H, Hashiguchi H (2018) A note on an improved model for extracting TKE dissipation rate from VHF radar spectral width. Earth Planets Space
**(in press)**Google Scholar - Kudo A, Luce H, Hashiguchi H, Wilson R (2015) Convective instability underneath midlevel clouds: comparisons between numerical simulations and VHF radar observations. J Appl Meteorol Clim 54:2217–2227CrossRefGoogle Scholar
- Kurosaki S, Yamanaka MD, Hashiguchi H, Sato T, Fukao S (1996) Vertical eddy diffusivity in the lower and middle atmosphere: a climatology based on the MU radar observations during 1986–1992. J Atmos Sol Terr Phys 58:727–734CrossRefGoogle Scholar
- Labitt M (1979) Some basic relations concerning the radar measurements of air turbulence. MIT Lincoln Laboratory, ATC working paper no. 46WP-5001Google Scholar
- Lawrence DA, Balsley BB (2013) High-resolution atmospheric sensing of multiple atmospheric variables using the DataHawk small airborne measurement system. J Atmos Ocean Technol 30:2352–2366CrossRefGoogle Scholar
- Li Q, Rapp M, Schrön A, Scneider A, Stober G (2016) Derivation of turbulent energy dissipation rate with the middle atmosphere alomar radar system (MAARSY) and radiosondes at Andoya, Norway. Ann Geophys 34:1029–1229Google Scholar
- Luce H, Hassenpflug G, Yamamoto M, Fukao S (2006) High-resolution vertical imaging of the troposphere and lower stratosphere using the new MU radar system. Ann Geophys 24:791–805CrossRefGoogle Scholar
- Luce H, Kantha L, Hashiguchi H, Lawrence D, Yabuki M, Tsuda T, Mixa T (2017) Comparisons between high-resolution profiles of squared refractive index gradient M
^{2}measured by the MU radar and UAVs during the ShUREX 2015 campaign. Ann Geophys 35:423–441CrossRefGoogle Scholar - Luce H, Kantha L, Hashiguchi H, Lawrence D, Mixa T, Yabuki M, Tsuda T (2018) Vertical structure of the lower troposphere derived from MU radar, unmanned aerial vehicle, and balloon measurements during ShUREX 2015. Prog Earth Plan Sci 5:29. https://doi.org/10.1186/s40645-018-0187-4 CrossRefGoogle Scholar
- McCaffrey K, Bianco L, Wilczak J (2017) Improved observations of turbulence dissipation rates from wind profiling radars. Atmos Meas Technol 10:2595–2611CrossRefGoogle Scholar
- Naström GD (1997) Dopplser radar spectral width broadening due to beamwidth and wind shear. Ann Geophys 15:786–796CrossRefGoogle Scholar
- Nastrom GD, Eaton FD (1997) Turbulence eddy dissipation rates from radar observations at 5–20 km at White Sands Missile Range, New Mexico. J Geophys Res Atmos 102:19495–19505CrossRefGoogle Scholar
- Sato T, Woodman RF (1982) Fine altitude resolution observations of stratospheric turbulent layers by the Arecibo 430 MHz radar. J Atmos Sci 39:2546–2552CrossRefGoogle Scholar
- Scipión DE, Lawrence DA, Milla MA, Woodman RF, Lume DA, Balsley BB (2016) Simultaneous observations of structure function parameter of refractive index using a high-resolution radar and the DataHawk small airborne measurement system. Ann Geophys 34:767–780. https://doi.org/10.5194/angeo-34-767-2016 CrossRefGoogle Scholar
- Siebert H, Lehmann K, Wendisch M (2006) Observations of small-scale turbulence and energy dissipation rates in the cloudy boundary layer. J Atmos Sci 63:1451–1466CrossRefGoogle Scholar
- Tatarski I (1961) Wave propagation in a turbulent medium. Translated by R. A. Silvermann. Graw-Hill, New YorkGoogle Scholar
- Tsuda T, May PT, Sato T, Kato S, Fukao S (1988) Simultaneous observations of reflection echoes and refractive index gradient in the troposphere and lower stratosphere. Radio Sci 23:655–665CrossRefGoogle Scholar
- Weinstock J (1978a) On the theory of turbulence in the buoyancy subrange of stably stratified flows. J Atmos Sci 35:634–649CrossRefGoogle Scholar
- Weinstock J (1978b) Vertical turbulence diffusion in a stably stratified fluid. J Atmos Sci 35:1022–1027CrossRefGoogle Scholar
- Weinstock J (1981) Energy dissipation rates of turbulence in the stable free atmosphere. J Atmos Sci 38:880–883CrossRefGoogle Scholar
- White AB, Lataitis RJ, Lawrence RS (1999) Space and time filtering of remotely sensed velocity turbulence. J Atmos Sci 16:1967–1972Google Scholar
- Wilson R, Dalaudier F, Bertin F (2005) Estimation of the turbulent fraction in the free atmosphere from MST radar measurements. J Atmos Ocean Technol 22:1326–1339CrossRefGoogle Scholar
- Yamamoto M, Sato T, May PT, Tsuda T, Fukao S, Kato S (1988) Estimation error of spectral parameters of mesosphere–stratosphere–troposphere radars obtained by least squares fitting method and its lower bound. Radio Sci 23:1013–1021CrossRefGoogle Scholar

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