## 1. Introduction

For locally homogeneous, stationary and isotropic turbulence produced by shear flow instabilities in a stably stratified atmosphere, turbulence kinetic energy (TKE) dissipation rate *ε* and the structure function parameter for temperature

where *g* is the gravitational acceleration, *T* is the temperature, *N*^{2} = (*g*/*θ*)(*dθ*/*dz*) is the squared Brunt–Vaïsälä (buoyancy) frequency, *θ* is the potential temperature and the parameter *α* = (1/*β*_{θ})[(1 − *R*_{f})/*R*_{f}],^{1} where *β*_{θ} ≈ 3.0 is a universal constant and *R*_{f} is the flux Richardson number given by the ratio between buoyancy flux and turbulent energy production.

Kantha and Luce (2018) and Luce et al. (2019) tested this theoretical relationship in the free troposphere and in clear air conditions, from direct estimates of *ε* and *α* ≈ 0.17 (corresponding to *R*_{f} ~ 0.14) consistent with similar observations in the ocean (Kantha and Luce 2018).

Because the energy and flux budgets are different in the CBL due to different sources of instabilities and turbulence (mainly buoyancy-driven from the bottom due to solar heating), we expect Eq. (1) does not apply to CBL, at least for the well-mixed portion of the CBL, where *N*^{2} is expected to be nearly zero and in the lower part of the CBL, where it is negative. Luce et al. (2019) reported that the largest statistical disagreement between the direct and indirect estimates of *ε* using Eq. (1) were indeed observed inside the CBL. In addition to theoretical problems, the evaluation of Eq. (1) on individual profiles is made difficult by the fact that measuring local values of *N*^{2} is not easy in presence of thermal updrafts and downdrafts (see appendix of Luce et al. 2019).

The behavior of *ε* and *γ* to the sensible heat flux equation based on *K* theory. We tentatively express *γ* in terms of *ε* and

The derivations of the desired expression, based on the heat flux budget equation and on the parameterizations made by Deardorff (1972) for the well-mixed part of the CBL, are described in section 2. Section 3 describes briefly the instrumentation used for collecting observational data and presents the methods and criteria used for selecting CBL cases used for the present study. Section 4 describes the results of experimental evaluation of the expression proposed in section 2, from comparisons between UAV data and data disseminated in the literature, mainly airplane observations and large-eddy simulations (LES). These comparisons highlighted significant differences between temperature fluctuation intensity parameters, *θ*^{2}⟩, estimated from in situ measurements and those predicted by LES in the upper part of the well-mixed region and in the entrainment zone of the CBL. These differences suggest that the effects of entrainment on *θ*^{2}⟩ are more pronounced in the atmosphere than as depicted in LES results. Although inconsistencies between in situ observations and LES made comparison and interpretation difficult, we make the assumption that the in situ measurements are more representative of the CBL dynamics and structure than the LES model results. If this is true, then the Deardorff model of countergradient term would be quantitatively relevant, in spite of the deficiencies in its formulation. These issues are discussed in section 5.

## 2. Theory

### a. Sensible heat flux equation and countergradient terms

From the sensible heat flux equation, assuming horizontally homogeneous conditions, no subsidence, and a steady state under Boussinesq approximation, we have (e.g., Stull 1988)

The angle brackets ⟨⋅⋅⋅⟩ indicate spatial averages. The four terms on the right-hand side are as follows: *M* is the so-called mean gradient production term, *T* is the vertical turbulent transport term, *P* is the pressure gradient–potential temperature covariance term, and *B* is the buoyancy production term. The variables are as follows: Θ(*z*) is the background (mean) potential temperature, *p* is pressure perturbation, *θ*_{0} is the mean potential temperature in the CBL, and ⟨*θ*^{2}⟩(*z*) and ⟨*w*^{2}⟩(*z*) are the variance of turbulent potential temperature and vertical wind fluctuations, respectively. Terms *T* and *P* are third moments and must be parameterized by suitable closure schemes.

By neglecting *T* from arguments based on observations and by using the Rotta “return to isotropy” closure scheme for *P*,

where *τ* is a pressure relaxation time scale, Deardorff (1966, 1972) showed that Eq. (2) can be written as

where *K*_{H} = ⟨*w*^{2}⟩*τ* is the turbulent diffusivity and *γ*_{D} is a positive term called “countergradient term” given by

Equation (4) indicates that the sensible heat flux (based on *θ*) in the well-mixed part of the CBL is not only due to the usual local downgradient transport but also to a nonlocal convective transport *γ*_{D}. The nonlocal term *γ*_{D} expresses the vertical redistribution of the surface flux of *θ* upward by convective eddies, independently of the local gradient of *θ*. The term ∂Θ/∂*z* − *γ*_{D} can be seen as an apparent (negative) potential temperature gradient, which justifies the countergradient heat transport in the upper half of the CBL, where the temperature gradient is expected to be stable (e.g., Deardorff 1966; Stull 1988). However, it is worth noting that the Deardorff formulation ignores the turbulent transport term *T* in the vertical heat flux budget, even as it is trying to account for heat transport by large eddies in the CBL.

Deardorff’s approach was widely accepted, until numerical simulations of CBL dynamics from LES made evaluations of each term of Eq. (2) possible. Figure 1 shows vertical profiles of *M*, *T*, *P*, and *B* according to LES performed by Holtslag and Moeng (1991, hereafter referred as HM91) and Ghannam et al. (2017, hereafter referred as GH17) for strongly convective cases, in which buoyancy production of TKE dominates shear production.

The profiles shown in Fig. 1 are normalized by scaling variables used by similarity theory for strong convection. First, *z*_{i} is the CBL depth defined as the altitude of the minimum of (negative) heat flux, or, in practice as the height of the bottom of the capping temperature inversion. The depth of the well-mixed potion of the CBL is generally defined by 0.2 < *z*/*z*_{i} < 0.8. Also, *w*_{*} the Deardorff convective velocity scale and *θ*_{*} is the convective temperature scale defined as *w*_{*}*θ*_{*}=*wθ*_{s}, the surface sensible heat flux, that is, the main forcing for pure convection.

LES profiles from HM91 and GH17 are very consistent with each other (Fig. 1) and indicate that the turbulent transport term *T* cannot be neglected. HM91 proposed *z*/*z*_{i} < 0.8 where *b* ≈ 2 is an empirical value found by HM91 and a modified expression of Eq. (2): *P* = −⟨*wθ*⟩/*τ* − *C*_{2}*B* (e.g., Stull 1988, p. 222). Using Moeng and Wyngaard’s (1986) results, HM91 used *C*_{2} = 0.5, and got an alternative expression for Eq. (4):

with *K*_{H′} = ⟨*w*^{2}⟩*τ*/2 and

Equations (5) and (7) are similar in appearance but are based on different physical processes. Basically, Eq. (5) arises from the buoyancy production term *B* after neglecting the turbulent transport term *T*. Equation (7) arises from not neglecting *T* (see HM91, 1691–1692, for more details).

By applying *C*_{2} = 1/3, expected to be valid for isotropic turbulence (e.g., Stull 1988), a generalized expression of the countergradient term can be written:

The literature provides many other expressions for the generalized countergradient term, based on various closure schemes and approximations from LES (e.g., Abdella and McFairlane 1997; Tomas 2007; GH17).

Figure 2 shows vertical profiles of *γ*_{D} and *γ*_{HM} estimated from LES according to HM91 and calculated from profiles of variances shown by GH17 for the strongly convective case (called S1 in their paper). As noted by HM91, the two expressions of the countergradient term provide a similar behavior but there are substantial differences (e.g., a factor of about 2 around *z*/*z*_{i} ~ 0.7), which is clearly evident in Fig. 2.

### b. Estimation of γ_{D} from ${\mathit{C}}_{\mathit{T}}^{2}$ and ε

It is difficult to extract the wind velocity, especially its vertical component, accurately from UAV measurements and therefore the variance ⟨*w*^{2}⟩ cannot be obtained from UAV data. Therefore, Eqs. (5) and (7) for the countergradient term are of little use in our context. However, if we ignore the contribution of anisotropic eddies in the CBL then the ratio of the variances can be rewritten as

where

Equation (9) was applied by Gossard et al. (1998) for shear generated turbulence in a stably stratified background. However, it might not be indisputably valid in the present case. The main reason is that even if *w*^{2}⟩ slightly differ from ⟨*θ*^{2}⟩ or ⟨*υ*^{2}⟩ (with ⟨*w*^{2}⟩ ~ 2⟨*u*^{2}⟩ around the center of the well-mixed portion of the CBL and ⟨*u*^{2}⟩ > ⟨*w*^{2}⟩ at the edges) (e.g., Fig. 9 of Moeng and Sullivan 1994) indicating an anisotropic contribution to the variance likely due to the largest convective eddies in the well-mixed portion.

We can estimate

where *p*_{0} = 1000 hPa. In the CBL, *z* = 1000 m, (*p*_{0}/*p*)^{4/7} ≈ 1.06.

On the other hand,

where *c* is a universal constant (~2.1). Therefore, a relationship between *ε* and

or equivalently,

where 0.36 = 3/(4*c*). Equation (14) is the counterpart of Eq. (1) relating *ε* and *N*^{2}, because the heat flux is dominated by nonlocal transport effects produced by convective eddies in the well-mixed portion of the CBL. Thus, any formulation based only on local gradients would produce incorrect results (see Kantha and Luce 2018).

Now, we are faced with two issues related to Eq. (14). First, this equation was derived from Deardorff’s (1966) parameterization of heat flux budget, found later to be incorrect by numerical LES studies (HM91). Second, Eq. (13) is conditioned by the validity of Eq. (9). Therefore, we first examined whether Eq. (13) is indeed equivalent to Eq. (5) by comparing with *γ*_{D} estimated from ⟨*w*^{2}⟩ and ⟨*θ*^{2}⟩. This comparison can be made from published results in the literature, as long as *ε*, *w*^{2}⟩, and ⟨*θ*^{2}⟩ are all estimated. This we do in section 4.

## 3. Selection of CBL cases

The UAV deployment is described by Kantha et al. (2017). The UAVs were usually preprogrammed to move up and down up to a few kilometers above the ground, along spiraling ascents and descents in the vicinity of the 46.5 MHz MU radar (Fukao et al. 1990) and 1.35 GHz LQ7 wind profiler (Imai et al. 2007). High-resolution and low noise wind and temperature data were collected from high-frequency response pitot and cold wire temperature (CWT) sensors, respectively. The data processing used for retrieving pseudovertical profiles of energy dissipation rates and *ε* and

The time and duration of each of the 11 selected UAV flights are shown in Fig. 3. Each flight provided one to six vertical profiles of atmospheric parameters collected during ascents and descents. The profiles (Fig. 4) associated with the selected flight DH64 (four profiles around 1200 LT) show the following properties (note that in Figs. 4 and 5, the altitude is MSL and the ground altitude was 348 m MSL):

These characteristics are at least qualitatively consistent with those reported in the literature for CBL. Section 4 will emphasize the similarities and differences.

Vertical profiles of *N*^{2}, *ε* for a rejected flight (DH52, ~1500 LT) are shown in Fig. 5. Contrary to the selected case shown in Fig. 4, enhanced values of *N*^{2} can be seen around 1000 and 1800 m MSL below the capping inversion at 2400 m MSL. The *ε* profiles during DH52 exhibit much weaker values than during DH64 (except around ~1000 m MSL). They also show large variations with height (up to three orders of magnitude for *ε*) with remarkably similar tendencies between the ascent and descent. These features suggest that a stratified residual layer has started to form. This hypothesis can be strengthened from information provided by the radars (Figs. 6, 7). The time–height cross section of LQ7 radar echo power on 23 June 2017 shows the time evolution of CBLs developing on 20 and 23 June 2017. The enhanced echoes related to CBLs are clearly distinguishable at the bottom of each plot. On 20 June 2017, the echo power was enhanced over the entire CBL until ~1500 LT, especially during the three selected UAV flights (DH62, DH63 and DH64). On 23 June 2017, echo power in the CBL started to decrease from ~1400 LT with a more stratified appearance and a deep minimum just below an elevated layer of enhanced echo power generally assumed to be the signature of the CBL top (e.g., Kumar and Jain 2006). The formation of a residual layer, well before the sunset, may be explained by the presence of high-level clouds (decreasing the solar heating and therefore the surface heat flux), followed by precipitation from ~2000 LT.

Additional information from MU radar observations during DH64 and DH52 consistent with different stages of CBL evolution are shown in Figs. 7a and 7b, respectively. The bumpy structures in the echo power image around the altitude of 2000 m MSL are the signature of enhanced refractive index turbulence at the edges of the convective cells during DH64 (Fig. 7a). These enhanced echoes clearly delimit the top of a region of enhanced Doppler variance *σ*^{2} due to dynamic turbulence. In addition, strong vertical velocity *W* fluctuations (exceeding ±2 m s^{−1}) can be seen, for example, around 1215 LT, consistent with strong updrafts and downdrafts in the entrainment zone of the CBL.

In contrast, the radar echo power image for DH52 (Fig. 7b) reveals a nearly flat CBL top at an altitude consistent with the altitude indicated by the LQ7 radar (i.e., ~2500 m MSL). The echo pattern is more typical of stratified conditions. In addition, *σ*^{2} is weakly enhanced within nearly horizontal bands and the *W* disturbances are weak. These features confirm the different stages of the above two CBL events.

## 4. Results of analyses

### a. Mean profiles of UAV-derived parameters (N^{2}, ${\mathit{C}}_{\mathit{T}}^{2}$ , and ε)

The 26 selected vertical profiles of *N*^{2}, *z*/*z*_{i} are shown in Fig. 8. Normalization was made with constant scaling variables *z*_{i} ≈ 1500 m making use of surface measurements of solar heating, and coarse approximations for estimating other terms of radiative budget (see appendix 1 of Troen and Mahrt 1986). These scaling values are subject to some uncertainties and produce CBL heights slightly larger than the observed ones, but they are relatively common according to the literature. Part of the scatter in Fig. 8 can be due to normalization with constant values for

The mean profile of *N*^{2} estimated from the (dry) potential temperature (red curve) is qualitatively consistent with that expected for CBL but shows slightly positive values (~3 × 10^{−5} rad^{2} s^{−2} on average) everywhere in the well-mixed portion of the CBL (0.2 < *z*/*z*_{i} < 0.8). In the standard scheme of a CBL, the mean value of *N*^{2} is expected to be negative below *z*/*z*_{i} ~ 0.4 to 0.5 and positive above (e.g., Stull 1988; GH17). Humidity is not the cause of this discrepancy because similar values of *N*^{2} are obtained from the virtual potential temperature (blue curve) in the well-mixed portion of the CBL. The stability of the capping inversion (above *z*/*z*_{i} > 1) is however affected by humidity due the strong decrease of the mixing ratio at the CBL top. At the present time, we do not know if the larger mean values of *N*^{2} in the well-mixed part of the CBL are due to instrumental or/and atmospheric effects. The *N*^{2} estimates from UAV measurements can be affected by various biases, especially when vertical air velocities are not negligible with respect to the UAV ascent rate (as may be the case in CBL; see appendix of Luce et al. 2019). However, concurrent observations from radiosondes showed a similar mean tendency in the CBL (not shown). Also, Bélair et al. (1999) reported potential temperature profiles measured from radiosondes slightly steeper than expected from their numerical simulations (their Fig. 10).

The mean value of *z*/*z*_{i} ~ 0.5, so that the curve is almost symmetric with respect to the center of the CBL. The mean value of *z*/*z*_{i} ~ 1. Incidentally, profiles of

### b. Comparisons of normalized ${\mathit{log}}_{10}\left({\mathit{C}}_{\mathit{T}}^{2}\right)$ and ${\mathit{log}}_{10}\left(\epsilon \right)$ profiles with literature

The references used for the comparisons are given in the legends of the figures cited below. Some references have already applied polynomial fittings for deriving empirical models. When this is not the case, for ease of legibility, some scatterplots have been replaced by a polynomial fit, representative of the tendency of the distributions. Horizontal bars, representative of the variability of these distributions, are not shown, for legibility and because this variability was found sufficiently weak for giving credence to the observed tendency. The reader can find the original plots, sometimes plotted in log scale for altitude, in the indicated references.

Vertical profiles of

Vertical profiles of *z*/*z*_{i} ~ 0.7. In contrast, the airplane profile from Druilhet et al. (1983) is very similar to the UAV profile and is larger than LES profile by about a factor of 2–3 around *z*/*z*_{i} ~ 0.7. The profile described by Caughey and Palmer (1979) has characteristics intermediate to the UAV and LES profiles.

Fairall (1987) has already reported that LES underpredicts *R* defined as the ratio of entrainment (top) heat flux to the surface heat flux, for modeling entrainment effects. The value of *R* is ~−0.2 for conventional LES of CBL (strong convection). For this value, the LES profile is consistent with the Fairall model (dashed red and pink curves of Fig. 10). For smaller (more negative) values of *R* (−0.5, −0.7), *R* = −0.5). Observations from Wyngaard and Lemone (1980) show a large dispersion and do not seem to be consistent with any of the other profiles. Therefore, they are not considered for the rest of this work. The domain of validity of the Wyngaard and Lemone model (red dotted line) does not extend above the well-mixed part of the CBL (i.e., *z/z*_{i} < 0.8).

### c. Comparisons between profiles of $\langle {w}^{2}\rangle $ and $\langle {\theta}^{2}\rangle $

As for *ε*, the vertical profiles of normalized vertical velocity variance *θ*^{2}⟩_{n} (Caughey and Palmer 1979; Lenschow et al. 1980; Druilhet et al. 1983; Therry and Lacarrère 1983) (Fig. 12). All the LES ⟨*θ*^{2}⟩_{n} profiles for strong convection show a trend similar to normalized *z*/*z*_{i} ~ 0.7. All the observed ⟨*θ*^{2}⟩_{n} profiles show larger values, especially near the entrainment zone, but observed ⟨*θ*^{2}⟩_{n} values converge toward LES ⟨*θ*^{2}⟩_{n} values near the top and bottom of the CBL. As a result, the observed ⟨*θ*^{2}⟩_{n} profiles show a lower variability with height. The mean LES ⟨*θ*^{2}⟩_{n} (blue curve) and observed ⟨*θ*^{2}⟩_{n} (pink curve) profiles vary by over a factor of 5 and 2, respectively, in the well-mixed part of the CBL (0.2 < *z*/*z*_{i} < 0.8). However, LES results do seem to provide better agreement with laboratory experiments reported by Willis and Deardorff (1974).

Comparisons between the mean observed ⟨*θ*^{2}⟩_{n}, LES ⟨*θ*^{2}⟩_{n}, and model ⟨*θ*^{2}⟩_{n} profiles obtained for strong, moderate, and weak convection are shown in Fig. 13. The various LES and models agree very well with one another for each of the convection intensity, but none of the LES and model profiles fit the mean observed profile. This indicates that the discrepancies between LES/models and observations cannot simply be explained away by improper scaling, and that some other issue exists. The validation of Eq. (14) is thus made difficult by these substantial differences between observations and numerical simulations.

### d. Attempt at validation of γ_{D} from Eq. (13)

To the authors’ knowledge, Caughey and Palmer (1979) (Figs. 4, 5, 9, 10) and Druilhet et al. (1983) (Figs. 3, 4) are the sole studies providing profiles of (normalized) *ε*, *w*^{2}⟩, and ⟨*θ*^{2}⟩ from the same datasets. Profiles of *ε*, *z*/*z*_{i} < 0.3 but the two profiles diverge at higher altitudes: *γ*_{Dn} values from Eq. (13) exceed the normalized values from Eq. (5) by a factor of 3 in the entrainment zone (*z*/*z*_{i} > 0.8). However, the two profiles obtained from Druilhet et al.’s data are very similar, giving some credence to the theoretical derivations proposed in section 2b. In addition, they coincide very well with the UAV profile indicating that the CBL cases studied by Druilhet et al. (1983) were likely comparable with those we observed, and not Caughey and Palmer (1979). The reason for this discrepancy is not known.

The profiles of LES *γ*_{Dn} and *γ*_{HMn} (Fig. 2) are superimposed on the profiles of observed *γ*_{Dn} (Fig. 14b) in Fig. 15. One apparent paradox stands out: LES *γ*_{HMn} (solid blue) fits observed *γ*_{Dn} (black, green, and red curves) much more accurately than LES *γ*_{Dn} (dotted blue). The poor agreement between observed *γ*_{Dn} and LES *γ*_{Dn} was expected due to the differences between observed ⟨*θ*^{2}⟩_{n} and LES ⟨*θ*^{2}⟩_{n} around *z*/*z*_{i} ~ 0.7 (see Fig. 12). The good agreement between LES *γ*_{HMn} and observed *γ*_{Dn} is more puzzling, but it may suggest that the two counter gradient terms *γ*_{HM} and *γ*_{D} are in practice quantitatively similar, despite different physical mechanisms. Since LES ⟨*w*^{2}⟩_{n} and observed ⟨*w*^{2}⟩_{n} do not differ substantially (see Fig. 11), LES *γ*_{HMn} also fit observed *γ*_{HMn}.

The paradox may be tentatively resolved from the following arguments:

*γ*_{HMn} can be rewritten as

Also,

Equations (15) and (16) are equivalent if the term in the square brackets in (16) can be approximated by a constant equal to 2. As mentioned earlier, observed ⟨*θ*^{2}⟩_{n} varies over a factor of 2 only in the well-mixed part of the CBL (0.2 < *z*/*z*_{i} < 0.8) (see Fig. 12). Its mean value is about 2.8 (0.45 in log_{10} scale). Thus,

for the mean values of *z*_{i} used in the present study. The generalized expression (8) would read observed *γ*_{Gn} ≈ 1.2*γ*_{Dn}. The numerical coefficients strongly depend on the values of the scaling variables, but observed *γ*_{Dn} and *γ*_{Gn} would nearly be proportional to LES *γ*_{HMn}, in any case.

These derivations are relevant only if the Eq. (15) for *γ*_{HMn}, based on closure schemes recalled in section 2a, is still valid despite changes of ⟨*θ*^{2}⟩_{n}, that is, changes of the buoyancy production term *B* of the heat flux budget. An increase of the term *B* must imply a decrease of (*M* + *P* + *T*) [see Eq. (1)] for equilibrium. The closure expression used by HM91 for the term *T* may not be valid anymore. Interestingly, a decrease of *T* above *z*/*z*_{i} > 0.3 tends to make it closer to 0, which would be in agreement with the hypothesis made by Deardorff (1966) for deriving Eq. (5). A decrease of *M* without changes of ⟨*w*^{2}⟩ implies an increase of *d*Θ/*dz* (or *N*^{2}) with respect to the level expected from LES. It is at least qualitatively consistent with what we observed in Fig. 8a. These assertions are, of course, rather speculative and require additional in situ and LES studies for validation.

### e. Application of models of countergradient terms

The mean profiles of normalized *ε* (in log scale) reconstructed from

are compared with measured profiles in Fig. 16. In Eq. (18c) *γ*_{D}(Dr83) is the Deardorff countergradient term estimated from variances provided by Druilhet et al. (1983). As expected, the reconstructed profiles from *γ*_{D}(LES) agree poorly with the measured profiles. In contrast, the agreement is satisfying when using *γ*_{HM}(LES) or *γ*_{D}(Dr83). The absolute level of the countergradient term would be about *γ*_{D} ~ 0.21 × 10^{−3} to 0.37 × 10^{−3} Km^{−1} in the well-mixed region where it is roughly constant. These values compare quite well with the values reported by Deardorff (1972) (*γ*_{D} ~ 0.31 × 10^{−3} to 0.52 × 10^{−3} Km^{−1}) from airplane observations.

## 5. Summary and conclusions

We have proposed a theoretical expression [Eq. (14)] relating the TKE dissipation rate *ε* to *γ*_{D} for sensible heat flux (section 2). This expression, expected to be valid in the well-mixed central region of the CBL within the framework of Deardorff’s hypotheses, is *γ*_{D} from observational and LES data available in the literature and by considering an alternative expression for the countergradient term *γ*_{HM} from HM91, which was based on a different physical argument. These investigations have led to a cascade of puzzling results, which we summarize below:

## Acknowledgments

The authors thank T. Mixa, R. Wilson, and T. Tsuda for their cooperation during the campaigns. This study was partially supported by JSPS KAKENHI Grant JP15K13568 and the research grant for Mission Research on Sustainable Humanosphere from Research Institute for Sustainable Humanosphere (RISH), Kyoto University. Partial support to DL was provided by NSF Grant Instabilities, Dynamics and Energetics accompanying Atmospheric Layering (IDEAL) AGS 104163

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