# On a numerical model for extracting TKE dissipation rate from very high frequency (VHF) radar spectral width

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## Abstract

*N*is the buoyancy frequency, and

*c*

_{0}is a constant), derived from Weinstock (J Atmos Sci 35:1022–1027, 1978; J Atmos Sci 38:880–883, 1981) formulation, has been used extensively for estimating the turbulence kinetic energy (TKE) dissipation rate \(\varepsilon\) under stable stratification from VHF radar Doppler spectral width \(\sigma\). The Weinstock model can be derived by simply integrating the TKE spectrum in the wavenumber space from the buoyancy wavenumber \(k_{\text{B}} = \frac{N}{\sigma }\) to \(\infty\). However, it ignores the radar volume dimensions and hence its spatial weighting characteristics. Labitt (Some basic relations concerning the radar measurements of air turbulence, MIT Lincoln Laboratory, ATC Working Paper NO 46WP-5001, 1979) and White et al. (J Atmos Ocean Technol 16:1967–1972, 1999) formulations do take into account the radar spatial weighting characteristics, but assume that the wavenumber range in the integration of TKE spectrum extends from 0 to \(\infty\). The White et al. model accounts for wind speed effects, whereas the other two do not. More importantly, all three formulations make the assumption that

*k*

^{−5/3}spectral shape of TKE spectrum extends across the entire wavenumber range of integration. It is traditional to use Weinstock formulation for \(k_{\text{B}}^{ - 1} < 2a,2b\) (where

*a*and

*b*are radar volume dimensions in the horizontal and vertical directions) and White et al. formulation (without wind advection) for \(k_{\text{B}}^{ - 1} > 2a,2b\). However, there is no need to invoke these asymptotic limits. We present here a numerical model, which is valid for all values of buoyancy wavenumber \(k_{\text{B}}\) and transitions from \(\varepsilon \sim\sigma^{2}\) behavior at lower values of \(\sigma\) in accordance with Weinstock’s model, to \(\varepsilon \sim\sigma^{3}\) at higher values of \(\sigma\), in agreement with Chen (J Atmos Sci 31:2222–2225, 1974) and Bertin et al. (Radio Sci 32:791–804, 1997). It can also account for the effects of wind speed, as well the beam width and altitude. Following Hocking (J Atmos Terr Phys 48:655–670, 1986, Earth Planets Space 51:525–541, 1999), the model also takes into account contributions of velocity fluctuations beyond the inertial subrange. The model has universal applicability and can also be applied to convective turbulence in the atmospheric column. It can also be used to explore the parameter space and hence the influence of various parameters and assumptions on the extracted \(\varepsilon\) values. In this note, we demonstrate the utility of the numerical model and make available a MATLAB code of the model for potential use by the radar community. The model results are also compared against in situ turbulence measurements using an unmanned aerial vehicle (UAV) flown in the vicinity of the MU radar in Shigaraki, Japan, during the ShUREX 2016 campaign.

## Keywords

VHF radar MU radar Unmanned aerial vehicles (UAV) Turbulence kinetic energy (TKE) TKE dissipation rate Radar model Stably stratified flows Inertial subrange Buoyancy subrange Viscous subrange Troposphere Buoyancy scale Ozmidov scale Viscous scale Convective turbulence Convection Numerical model## Abbreviations

- ASL
above sea level

- CBL
convective boundary layer

- CU
Colorado University

- MCT
mid-level cloud-base turbulence

- MST
mesosphere stratosphere troposphere

- MU
middle and upper atmosphere

- ShUREX
Shigaraki UAV Radar Experiment

- TKE
turbulence kinetic energy

- VHF
very high frequency

- UAV
unmanned aerial vehicle

## Introduction

The dissipation rate \(\varepsilon\) of turbulence kinetic energy (TKE) is a fundamental parameter indicative of the strength of turbulence. With suitable assumptions, knowledge of \(\varepsilon\) allows the turbulent diffusion coefficient *K*, of great importance to mixing in the fluid column, to be determined. As such there is considerable interest in determining \(\varepsilon\) in both the atmosphere and the oceans. While the principal application of VHF wind profiling Doppler radars is for measuring the three components of wind velocity in the atmospheric column, they also have the capability to measure velocity fluctuations in the beam direction. The intensity of velocity fluctuations can be determined from the spectral width \(\sigma\) of the backscattered signal, after suitable corrections are applied to effects such as beam broadening and wind shear. The task then is to infer the dissipation rate from the spectral width. But first, the dissipation rate has to be related to turbulent velocity fluctuations. Considerable effort has been expended over the past four decades on the problem of extracting \(\varepsilon\) from \(\sigma\). A very useful summary of these efforts and the issues involved can be found in an excellent review by Hocking (1999).

Simultaneous measurements of \(\varepsilon\) by the radar and in situ turbulence sensors, both measuring the same volume, have been costly and hence infrequent (but see Bertin et al. 1997; Dehghan and Hocking 2011; Dehghan et al. 2014 and the references cited therein). As such, there are still some unresolved issues. However, quite recently, routine and inexpensive in situ measurements have been made possible through turbulence sensors deployed on small, unmanned aerial vehicles (UAVs, e.g., Scipion et al. 2016) flown near and above the MU radar in Shigaraki, Japan (Kantha et al. 2017; Luce et al. 2018). This has enabled us to revisit the problem of extracting \(\varepsilon\) from \(\sigma\). A brief summary of past work is also provided for completeness and context.

*N*:

*c*

_{0}is an empirical constant and

*N*, the buoyancy frequency indicative of the degree of stability of the fluid column, is given by:

*g*is the gravitational acceleration,

*z*is the vertical coordinate, and \(\varTheta\) is the potential temperature (assuming dry air, although water vapor effects can be easily incorporated through virtual potential temperature).

The extraction of the Doppler variance due to turbulence after correcting for beam-broadening and shear-broadening effects is outside the scope of the present work. Here, we assume that \(\sigma\) is correctly evaluated but can be affected by wave contributions. Equation (4) has been used extensively since, for estimating the turbulence kinetic energy (TKE) dissipation rates under stable stratification from VHF radar Doppler spectra, although there is still considerable debate as to the exact value of *c*_{0}. In a nice summary, Hocking et al. (2016) show that its empirical value appears to vary between 0.27 and 0.6, but recommend that \(0.5 \pm 0.25\) be used (their Eq. 7.55). However, as shown below, *c*_{0} depends on the Kolmogorov universal constant \(\alpha\) as well as the lower limit of integration of the turbulence wavenumber spectrum.

The primary dependence of \(\varepsilon\) is on \(\sigma_{w}\) (or equivalently \(\sigma\) when applied to radar data), since the dependence on *N* is much weaker. As such, Weinstock model yields \(\varepsilon \sim \sigma_{w}^{2}\) (or equivalently \(\sigma^{2}\)) behavior, whereas \(\varepsilon \sim\sigma_{w}^{3}\) (or equivalently \(\sigma^{3}\)) dependence based on some stratospheric measurements of turbulence is also evident in some cases (Chen 1974; Bertin et al. 1997). The latter behavior suggests \(\varepsilon \sim \frac{{\sigma^{3} }}{L}\), where *L* is a turbulence length scale (independent of *N*). It is possible that under certain conditions (Hocking 1999), *L* could remain constant giving rise to \(\varepsilon \sim\sigma^{3}\) behavior. Labitt (1979) proposed a model that yields \(\varepsilon \sim\sigma^{3}\) behavior.

*k*, the TKE dissipation rate \(\varepsilon\) and the Kolmogorov viscous scale \(\eta\):

*f*stands for function. One frequently used form for the spectrum in the universal range (e.g., Ogura 1958) is

*k*and the dissipation rate \(\varepsilon\), and not on the Kolmogorov viscous scale \(\eta\):

There has been an enormous amount of work done over the years to determine the precise value of \(\alpha\) and as a result, it is known to range between 1.53 and 1.65 (e.g., Gossard et al. 1984). The best-known empirical value at present is 1.65 (although 1.53 has also been used, e.g., Weinstock 1981; Hocking 1999). This value is very close to the often-cited (e.g., Ogura 1958) value of \(\left( {\frac{8}{9\kappa }} \right)^{2/3} \sim1.675\). Note that the Kolmogorov universal range consists of the inertial and viscous subranges and the transition between the two.

It is also important to keep in mind the fact that when the upper limit on the wavenumber in the integration of energy spectrum is taken as infinity, strictly speaking, Eq. (8) should be used instead of Eq. (10).

*k*

_{K}is the Kolmogorov wavenumber (Eq. 5). Note that the proportionality constant in Eq. (11) is not far from the value \(4\pi\). The inner scale is therefore

*k*

_{O}, thus suggesting the value of this ratio is close to 1.0, its precise value remains uncertain at present and should perhaps be determined by dedicated experiments in the future (personal communication by W. Hocking). The code allows for the user to prescribe the value of the proportionality constant

*f*

_{K}.

*R*itself is indicative of the extent of the ISR in the turbulence energy spectrum. Assuming

*f*

_{k}= 1, for

*Re*

_{b}~ 645, the ISR extends over a decade of wavenumbers (

*R*= 10) and for

*Re*

_{b}~ 13,900, over 2 decades (

*R*= 100). In the stably stratified regions of both the oceans and the atmosphere, the value of

*R*lies mostly but not entirely between 10 and 100.

*k*

_{O}(or equivalently scales larger than the Ozmidov length scale

*L*

_{O}), buoyancy forces begin to affect turbulence by tending to inhibit vertical motions. The turbulence spectral shape may deviate from that in the ISR. This buoyancy subrange (BSR) is thought to extend from the outer wavenumber

*k*

_{OUT}to the buoyancy wavenumber

*k*

_{B}defined as

*can*be defined as the reciprocal of

*k*

_{O}

*and k*

_{B}, respectively (without the factor \(2\pi\))

This is the definition that atmospheric scientists use (e.g., American Meteorological Society), although some studies (e.g., Weinstock 1981; Hocking 1999) define the buoyancy scale as \(2\pi /k_{\text{B}}\). The Ozmidov length scale is also often defined as \(2\pi /k_{\text{O}}\). There has been considerable and needless confusion regarding length scales we use in turbulence studies (see also comments in Section 3 of the review of the topic by Hocking 1999) because of the factor \(2\pi\), which can lead to serious discrepancies in the magnitudes of \(\varepsilon\) extracted from \(\sigma\).

However, there is a simple solution. The wavenumber spectrum is more fundamental to turbulence studies and as long as we deal with just wavenumbers, which involve no ambiguity whatsoever, and not arbitrarily defined length scales, there is no inconsistency in the derivations of the expressions for the dissipation rate (in terms of wavenumbers). Once that is done, wavenumbers can be converted to corresponding scales, whichever way they are defined. We do so and suggest that the radar community do the same, since there is usually some ambiguity as to how to define the corresponding length scales, i.e., with or without the factor \(2\pi .\) This has to do with the confusion between length scales and wavelengths in turbulence studies as we transition from the wavenumber space (see the excellent discussion in Hocking 1999, also personal communication by Hocking). Our own preference is to define turbulence length scales as inverse of wavenumbers.

*k*

_{OUT}<

*k*<

*k*

_{B}), where buoyancy forces affect turbulence, is thought to follow the law

*n*

_{b}is uncertain. Weinstock (1978) suggests that the value of

*n*

_{b}depends on the flux Richardson number, but is close to the ISR value of 5/3, while Lumley (1964; see also Sukoriansky and Galperin 2017) suggests

*n*

_{b}= 3. In any case, the Ozmidov wavenumber

*k*

_{O}(often called the buoyancy wavenumber in meteorology leading to needless confusion once again) is indicative of the transition from ISR to BSR in the spectrum (while buoyancy wavenumber

*k*

_{B}is indicative of the transition from turbulent motions to wave motions), and the spectrum in the wavenumber range comprising of both ISR and BSR can be modeled as:

*k*≫

*k*

_{OUT}, the ISR results: \(E \to \alpha \varepsilon^{2/3} k^{ - 5/3}\) and for

*k*≪

*k*

_{OUT}, we get the BSR

Note that the computer code has been written so that the user can input any values for *f*_{K} and *n*_{b}, although in the illustrative plots, we have used *f*_{K} = 1 and *n*_{b} = − 5/3.

*k*

_{B}(equivalently above the buoyancy length scale

*L*

_{B}) are due to wave motions and not turbulence. This must be taken into account in the derivations of \(\varepsilon\), since the radar does not discriminate between velocity fluctuations due to turbulence and wave motions. Keeping in mind the fact that wave motions can exist below the buoyancy wavenumber

*k*

_{B}, the integration of the turbulence energy spectrum

*E*(

*k*) over the wavenumber space

*k*

_{B}to \(\infty\) yields the energy resident in turbulent fluctuations and only turbulent fluctuations, i.e., the TKE:

*k*

_{B}to

*k*

_{Bragg}so that using

If non-isotropy of the components of TKE is acknowledged, there would be an appropriate factor multiplying \(\alpha\) on the right hand side of Eq. (29) (see Appendix B).

*k*

_{B}≪

*k*

_{Bragg}so that

*k*

_{B}from Eq. (17) yields

*c*

_{0}~ 0.47. As such, there is no ambiguity in the value of

*c*

_{0}, since it is tied to the Kolmogorov universal constant, if and only if the lower limit on integration is strictly enforced as equal to the buoyancy wavenumber, which delineates wave motions from turbulent motions. Application to radar spectral width data gives Eq. (4), the widely used form of Weinstock (1981) model.

*L*

_{B}= 1/

*k*

_{B}):

*c*

_{0}~ 0.44 if \(\alpha = 1.65\), but 0.47 if \(\alpha = 1.53\) as in Hocking (1999). Invoking Eq. (3), we get the Weinstock model (Eq. 4). If we had used

*k*

_{O}as the lower limit of integration,

*c*

_{0}would have been 0.61.

The closeness of Eqs. (39) and (33) is merely a happenstance.

*c*

_{0}= 0.47. The relative closeness of the two wavenumbers tends to downplay the influence of the exact spectral shape in the BSR portion of the spectrum.

However, the radar measurement volume is finite, and the Weinstock formulation does not take into account the resulting radar spatial weighting characteristics. On the other hand, Labitt (1979) and White et al. (1999) formulations do (see “Radar epsilon model” section), but provide numerical and not analytical solutions. However, both integrate the wavenumber spectrum of the backscattered radar signal from 0 to \(\infty\), whereas the Weinstock model integrates the turbulence spectrum from the buoyancy wavenumber *k*_{B} to \(\infty\). As summarized by Hocking (1999), these three Doppler methods lead to different formulas and results, which has been a source of some confusion. Also, all three formulations make the assumption that the *k*^{−5/3} spectral shape representative of the inertial subrange (ISR) of the turbulence kinetic energy spectrum extends across the entire wavenumber integration range. This ignores the potential presence of the buoyancy subrange (BSR) within the integration range. Also, strictly speaking, the upper limit on wavenumber should be *k*_{Bragg} and not \(\infty\), although the difference is quite small. Finally, the observed spectral width due to velocity fluctuations in the beam direction may have contributions from wave motions and not just turbulence. We address these issues in the next section.

## Radar epsilon model

*a*and

*b*are radar volume dimensions in the horizontal and vertical. Because the radar beam measures velocity fluctuations transverse to the horizontal wind advecting turbulence past it, the value of

*C*

_{K}can be less than 1 (see Appendix A of Hocking 1999), the exact value of depending on the type of turbulence (see Appendix B). However, this issue has been ignored thus far and all previous derivations in radar literature, including Labitt (1979), have assumed

*C*

_{K}= 1. We have used

*C*

_{K}= 0.873, appropriate to shear-generated turbulence, in this paper, keeping the transverse nature of radar measurements (for more details see Appendix B). The precise value of

*C*

_{K}is still uncertain and requires further studies.

*E*(

*k*) appropriate to the ISR, which yields

This is what we will call the original Labitt method, which assumed *k*^{−5/3} spectral shape to exist over the entire wavenumber range of interest, from *k* = 0 to \(\infty\), and ignored the effect of wind advection (see Hocking 1999).

*x*-direction. Note that \(L = V_{\text{H}} \Delta t\), where

*V*

_{H}is the wind speed in the horizontal direction and \(\Delta t\) is the dwell duration (duration for collecting the time series), which is 24.57 s for MU radar during the ShUREX campaigns. Using Eqs. (42)–(43) in Eq. (47), we get Eq. (2.14) of White et al. (1999):

*x*=

*kb,*\(\theta\) and \(\varphi\), we get

Henceforth, we will not present results from White et al. formulation (Eq. 51), since they are very close to the generalized Labitt formulation (Eq. 50) and it is hard to discern any difference between the two in the plots. Note that *C*_{K} = 1 in original Labitt and White et al. derivations.

Note that both White et al. (1999) and Labitt (1979) formulations integrate right through the wavenumber *k*_{B} and therefore are accounting for velocity fluctuations due to wave motions below *k*_{B} (albeit with spectral shape corresponding to ISR). A related minor issue with both formulations is that the upper limit should be *k*_{Bragg}, but this is not important since it makes little difference. On the other hand, since the lower limit is not *k*_{B}, the wavenumber delineating turbulent and wave motions, integration is carried out right through *k*_{B} and so the results, not surprisingly do not involve the buoyancy scale 1/*k*_{B} (or equivalently the buoyancy frequency *N*), whereas Weinstock formulation integrates down to *k*_{B} only and so its results involve *N* explicitly.

It is traditional to use Weinstock formulation for \(k_{\text{B}}^{ - 1} < 2a,2b\) and White et al. formulation (without wind advection and therefore *L* = 0) for \(k_{\text{B}}^{ - 1} > 2a,2b\), where the buoyancy length scale given by \(k_{\text{B}}^{ - 1}\) is taken to be the size of the largest eddies permitted under stable stratification (e.g., Fukao et al. 1994; Kantha and Hocking 2011; Luce et al. 2018). However, these are the two asymptotic limits. Since White et al. formulation requires numerical integration anyway, it is not any more difficult to use numerical integration of the generalized Labitt formulation, but with proper integration limits. A major advantage is that the numerical model allows for inclusion of wind advection and radar weighting characteristics, with no need for approximations whatsoever.

*n*

_{b}= 4/3 is the spectral slope in the wave region (

*f*

_{K}= 0.6–1, \(\gamma_{K} = 1\)). This form is used for completeness, although there is rarely any need to invoke VSR and so for all practical purposes

This form has general applicability. If we put terms in the first curly bracket in Eq. (54) equal to 1 and take limits *x*_{L} = 0 and \(x_{U} = \infty\), we get the generalized Labitt formulation (Eq. 50). In addition, if we put terms in the third square bracket equal to 1 (equivalently *L* = 0), the influence of wind advection is ignored. On the other hand, putting terms in the second curly bracket equal to 1 (equivalent to putting *a,* \(b = \infty\)) yields the Weinstock model, for which the lower limit must be *x*_{L} = *k*_{B}*b*. (Using *x*_{L} = 0 as the lower limit makes the integral go to \(\infty\) thus giving \(\varepsilon = 0\).)

The numerical model presented in this paper integrates Eq. (54) in 3 segments: 1) universal range (VSR and ISR) (*k* = *k*_{O} to *k* = *k*_{Bragg}), 2) BSR (*k* = *k*_{B} *to k* = *k*_{O}) and 3) Beyond BSR when wave contributions are included (*k* = 0 *to k* = *k*_{B}), so that *x*_{L} = *0* and *x*_{U} = *k*_{Bragg}*b*. However, the principal problem is that the Ozmidov wavenumber *k*_{O} is not known *a priori*, since it depends on the yet to be determined \(\varepsilon\). So an iterative procedure is necessary, using an initial guess value for \(\varepsilon\) from one of the standard formulations (e.g., White et al. 1999).

*b*= 75 m, and \(\lambda = 6.4516\) m (but can be done for any radar). The two-way half-power beam half-width is 1.32°, which determines the parameter

*a*as a function of altitude above ground level (AGL). For the results in Fig. 1, the altitude has been set to 2 km, the average altitude of UAV measurements. The buoyancy frequency

*N*is kept fixed at 0.0121 s

^{−1}, a value appropriate to the troposphere. The wind speed

*V*

_{H}is put to zero, for simplicity.

The left panel shows \(\varepsilon\) plotted against \(\sigma\), while the right panel shows \(\varepsilon\) plotted against the buoyancy scale 1/*k*_{B} (The two plots have equivalent information.) The red line shows the Weinstock formulation (Eq. 1 with *N* = 0.0121 s^{−1}), and the black line shows the general Labitt formulation denoted as G Labitt. Note that because wind = 0, the general Labitt formulation reduces to the original Labitt formulation. These two form the two asymptotic limits discussed earlier and the numerical model (blue line) transitions from one to the other quite nicely. The blue line parallels the red line for low values of \(\sigma\) and therefore 1/*k*_{B}, but transitions toward the red line at high values of \(\sigma\) and 1/*k*_{B}. This is simply because when 1/*k*_{B} < 2*a*, 2*b*, the dissipation rate \(\varepsilon \sim\sigma^{2}\) (Weinstock 1981, but with *N* ~ constant), but transitions to \(\varepsilon \sim\sigma^{3}\) behavior when 1/*k*_{B} > 2*a*, 2*b* (Chen 1974). Therefore, when the radar volume spatial characteristics are taken into account, both behaviors become feasible. Note that the value of *C*_{K} has been put equal to 0.873. This is the reason the numerical model values do not asymptote to Weinstock and Labitt values. They would have if *C*_{K} were to have been chosen equal to 1.0 (see Appendix B for details).

*L*

_{B}< 8 m) and UAV data with \(\varepsilon < 1.1\) × 10

^{−5}m

^{3}s

^{−2}(UAV sensor noise threshold) are omitted. The models mentioned in Fig. 1 are also shown. The magenta line is given by

*L*= 25 m, and is in better agreement with UAV data than both the Weinstock and Labitt models. Luce et al. (2018) also suggest that Eq. (55) with

*L*= 25 m, best fits the ShuREX 2016 UAV data. The Labitt (1979) formulation (and White et al. 1999) yields \(\varepsilon \sim\sigma^{3}\) behavior for ALL values of \(\sigma\), simply because the lower limit on integration in these formulations is zero and not

*k*

_{B}. Note that, like in Fig. 1, because wind = 0, the general Labitt formulation reduces to the original Labitt formulation.

The upper wavenumber limit has some impact. If *k*_{Bragg} and not \(\infty\) is imposed as the upper limit, the blue line (model), instead of overlapping the red line (Weinstock) at low values of \(\sigma\), would deviate increasingly from it as \(\sigma\) decreases, with \(\varepsilon\) values somewhat higher (not shown) than those given by Weinstock formulation.

*B*

_{1}~ 16.6 (Kantha and Carniel 2009). For stably stratified conditions, \(\frac{{\overline{{w^{2} }} }}{{q^{2} }}\) becomes a function of the gradient Richardson number \(Ri = \frac{{N^{2} }}{{S^{2} }}\) (

*S*is the mean shear) (e.g., Kantha and Carniel 2009). Observational data suggest that \(\frac{{\overline{{w^{2} }} }}{{q^{2} }}\) ~ 0.13–0.15 in the

*Ri*range 0–0.25 of interest in stably stratified flows (Kantha and Carniel 2009, their Fig. 5). Taking an average value of 0.14, Eq. (56) becomes

Equating (55) and (57), it is seen that \(L = 0.4\ell\), and therefore, the turbulence length scale is around 60 m, on the average, in ShUREX 2016 measurements.

*k*

_{Bragg}. Figure 3 shows the effect of wind speed and altitude at the MU radar site. The acquisition time DT is ~ 30 s. Weinstock (red line) and original Labitt model (black line) for AGL (above ground level) altitude 2 km and zero wind speed are also shown for comparison. Results for 4 cases are shown. The blue line is the model result for AGL altitude of 2 km and zero wind speed, the same as the blue line in Fig. 1. The magenta line is for AGL altitude of 10 km and zero wind speed. The cyan line is for AGL altitude of 2 km and 10 m/s wind speed. The green line for AGL altitude of 10 km and wind speed of 10 m/s is nearly indistinguishable from and hence overwrites the cyan and magenta lines. It is interesting to note that all these values fall above the Weinstock values. These simulations therefore demonstrate the importance of not only imposing proper integration limits, but also accounting for the altitude AGL and wind speed for a particular radar, when deriving \(\varepsilon\) from \(\sigma\). The use of the two asymptotic limits (Weinstock and White et al.) is therefore not always justified.

We have provided model results only for the MU radar, since that was where in situ UAV measurements were made. It is also not possible to explore here the large parameter space that governs the \(\varepsilon\) estimates of the numerical model. Instead, the Appendix provides a MATLAB code suited to exploring the parameter space for any radar.

## Correcting radar spectral width for wave contributions

## Extension to convective mixing

While we focused on turbulence in stably stratified flows so far, the general Labitt formulation can also be extended to convective mixing of any type seen in the atmospheric column, by recognizing that in this case, the ISR extends over the entire wavenumber range of interest. Neither the Ozmidov wavenumber *k*_{O} nor the buoyancy wavenumber *k*_{B} is relevant. The stratification is unstable (*N*^{2} < 0), and these wavenumbers are undefined. Thus, the lower integration limit in Eq. (50) can be taken as *k*_{D}*b*, (where *k*_{D} = *1/D, D* being the depth of the convective layer), since no wave motions are feasible in unstably stratified fluid column. Equations (44) and (50) constitute the proposed spectral model for convective mixing. The resulting TKE dissipation rate \(\varepsilon\) exhibits \(\sigma^{3}\) behavior (Eq. 55), with the length scale *L* dependent on a variety of factors including the depth of the layer *D* and parameters 2*a* and 2*b*. More details can be found in the Appendix. The model is applicable to the convective boundary layer (CBL) and mid-level cloud-base turbulence (MCT, e.g., Kudo et al. 2015), as well as convective mixing produced by cloud-top radiative cooling.

## Concluding remarks

*B*

_{1}~ 16.6 and \(\ell\) is the turbulence macroscale indicative of the scale of the energy-containing eddies and corresponds roughly to the wavenumber of the peak of the spectrum (and NOT necessarily the largest eddy size). This is simply the Taylor–Prandtl hypothesis, where it is assumed that while TKE dissipation occurs at viscous scales, the dissipation rate itself is independent of viscosity and depends instead only on the energy in the energy-containing large scales (~

*q*

^{2}) and the energy-containing eddy turnover timescale (\(\sim\;\ell /q\)). Therefore, the scale

*L*is proportional to the turbulence macroscale \(\ell\), irrespective of the source of turbulence, but the proportionality constant depends on the ratio of \(\frac{{\overline{{w^{2} }} }}{{q^{2} }}\), which is 1/3 for isotropic turbulence, roughly 1/2 for convective turbulence and less than 1/4 for stable stratification (e.g., Kantha 2003).

Comparing Eq. (61) to Weinstock (1978, 1981) formulation \(\varepsilon = 0.47\frac{{\left( {\overline{{w^{2} }} } \right)^{3/2} }}{{L_{\text{B}} }}\) for stable stratification, it is clear that formulations of that type are essentially equivalent to assuming that the turbulence macroscale \(\ell\) is proportional to the buoyancy scale *L*_{B}. This of course is not always true.

In situ measurements of TKE dissipation rates during the ShUREX 2016 campaign by UAV-borne turbulence sensors deployed in the immediate vicinity of the MU radar (Kantha et al. 2017; Luce et al. 2018) suggest that Eq. (61), with *L* ~ 25 m and therefore \(\ell\) ~ 60 m, is in better agreement with observed data than the Weinstock model (Eq. 4, with *N* = 0.0121 s^{−1}) or the conventional Labitt (Eq. 50) and White et al. (Eq. 51) models. The reason and significance of this result is not clear at this point and requires further study (see the companion paper Luce et al. 2018, this special issue, for details and discussion of this issue).

Finally, we note that the numerical model can also be applied to any turbulence measurements in stably stratified flows, where \(\overline{{w^{2} }}\) is available (e.g., Weinstock 1981 or Chen 1974), along with *N*, not just radar data. Conversely, if the TKE dissipation rate is known, it is possible to infer TKE. This is of particular interest in the oceans, where it is relatively simple to measure \(\varepsilon\) using microstructure probes, but TKE measurements are much harder. To some extent, this applies to UAV measurements in the atmosphere also, since once again, \(\varepsilon\) measurements are straightforward (e.g., Kantha et al. 2017; Luce et al. 2018), but TKE measurements are more difficult, because of the problems in inferring ambient wind velocity accurately enough from measurements of UAV velocity relative to the wind and relative to the ground.

## Notes

### Authors’ contributions

LK was responsible for the formulation of the modified radar model. HL collected and analyzed the MU radar and UAV data and assisted LK in the numerical model formulation. HH oversaw the ShUREX campaign. All authors have read and approved the final manuscript.

### Acknowledgements

LK thanks the Japanese Society for Promotion of Science (JSPS) for providing partial funding for the ShUREX 2016 campaign. LK’s interest in the topic and motivation for this work was triggered by the excellent review by Hocking (1999), who is widely regarded as an authority on the topic. The authors thank Prof. Hocking for drawing our attention to the fact that the radar measures velocity fluctuations in the radial direction, which are measurements transverse to the wind advecting turbulence past the radar beam, and therefore, the variance of the vertical velocities measured by the radar is not the same as that obtained from integrating the conventional TKE spectrum. The hospitality of RISH personnel and director Prof. Tsuda to their visitors is exemplary.

### Competing interests

The authors declare that they have no competing interests.

### Availability of data and materials

Data are not yet available because they are still being analyzed for follow-on studies and papers.

### Funding

This study was supported by JSPS KAKENHI Grant Number 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.

### Publisher’s Note

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

## Supplementary material

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