1. Introduction

Atmospheric turbulence is a complex phenomenon, characterized by the presence of a plethora of scales (eddies) ranging from small viscosity-dominated to large, energy-containing structures. Turbulence may undergo space and time variations due to rapidly changing external conditions, it may be locally suppressed or enhanced. To describe characteristic features of turbulence, statistical theories are sought for. In this context, a number of recent research works (Vassilicos 2015; Mazellier and Vassilicos 2008; Bos and Rubinstein 2017; Bos et al. 2007; McComb et al. 2010) address the problem of the equilibrium Taylor law (Taylor 1935) and its failure in the presence of rapid changes of the system. A new, nonclassical, although universal scaling is introduced to describe the latter.

The general validity of the “equilibrium” equations, Eqs. (1) and (2), was coming into question, as laboratory studies by Antonia and Pearson (2000) and Burattini et al. (2005) revealed that C_ϵ varies considerably. This behavior was first explained by the dependence on the inflow and boundary conditions. In another work by Bos et al. (2007) it was shown based on theoretical arguments that C_ϵ takes different values in forced and decaying turbulence, even though the classical, Kolmogorov −5/3 form of the spectrum was assumed.

Rubinstein and Clark (2017) argued that turbulence characterized by the classical laws (1) and (2) should be called “equilibrium” turbulence. In such type of flow the energy spectrum takes a self-similar form and the classical −5/3 scaling holds (Kolmogorov 1941). Nonequilibrium states of a flow field appear after a sudden change of external conditions (e.g., forcing) when the system evolves toward another equilibrium (Mahrt and Bou-Zeid 2020; Wacławczyk 2021, 2022). In the nonequilibrium turbulence, the turbulence dissipation coefficient C_ϵ and the integral-to-Taylor-scale ratios are described by relations (3) and (4) and a nonequilibrium correction should be added to the spectrum. Bos and Rubinstein (2017) derived the form of this correction, as well as the nonequilibrium scaling laws for C_ϵ and $\mathcal{L}/\lambda$ based on theoretical arguments.

The nonequilibrium relations (3) and (4) are substantially different from their equilibrium counterparts in (1) and (2). Hence, classification of turbulence in a given area can be made on this basis. Previous research works which focus on such analysis concern laboratory or numerical experiments (e.g., decaying grid turbulence, turbulent boundary layer) under controlled conditions (Valente and Vassilicos 2015; Obligado et al. 2016; Cafiero and Vassilicos 2019; Obligado et al. 2022). The main idea of this work is to use this procedure to analyze data from in situ airborne measurements of atmospheric turbulence, where the flow conditions in the investigated region are unknown a priori and only limited information, that is 1D intersections of the flow field along the flight track are available. In principle, such analysis could be performed on any high-resolution observations of the atmospheric boundary layer. In this study we emphasize theoretical aspects of the method to be used; hence, we decided to analyze data already known from previous campaigns, but in a way that adds one more important aspect of turbulence. Our choice is the stratocumulus-topped boundary layer (STBL). Several research campaigns and subsequent data analyses were devoted to the study of the turbulence in STBL (see, e.g., Stevens et al. 2003; Malinowski et al. 2013; Siebert et al. 2010; Jen-La Plante et al. 2016). We focus on the data from the Azores Stratocumulus Measurements of Radiation, Turbulence and Aerosols (ACORES) campaign collected in the eastern North Atlantic (Siebert et al. 2021). Nowak et al. (2021) compared the properties of turbulence between the cases of coupled and decoupled STBLs [see Wood et al. (2015) for the definition of decoupling]. They suggested that the observed differences in inertial range scaling might be related to different stages of turbulence lifetime (e.g., development and decay) but did not study these mechanisms in detail. In this work, from the same dataset, we derive all turbulence statistics necessary to calculate C_ϵ, $\mathcal{L}/\lambda$, and Re_λ. The aim is to divide the flow area into regions where turbulence decays, develops or is in a stationary state, by investigating values of C_ϵ. Moreover, by plotting C_ϵ and $\mathcal{L}/\lambda$ as a function of Re_λ we want to identify whether turbulence is in or out of equilibrium. Additionally, by the curve fitting of the data, estimation of the initial (or equilibrium) Reynolds number becomes possible. We also propose a procedure to roughly estimate the characteristic time scale of the turbulence decay/development. To the best of the authors’ knowledge, this is the first application of such analysis in the study of atmospheric turbulence.

We perform a sensitivity study to show that all statistics, including ϵ, can be estimated from the wind velocity records, even in the case of nonequilibrium, provided that the resolution of the data is good enough and the flight segments are sufficiently long. These conditions are fulfilled by the ACORES data. Moreover, we extend analysis of Bos (2020) and estimate the upper bound of C_ϵ for flows with nonzero buoyancy.

The present paper is structured as follows. In the following section 2 formal definitions of the “equilibrium” and “nonequilibrium” turbulence are presented and the theoretical derivations of Bos and Rubinstein (2017) are recalled. In section 3 the investigated data are briefly presented. This is followed by the detailed description of the methods applied in the data analysis. Results are presented in section 4, and finally, conclusions and perspectives are discussed in section 5.

2. Theory of nonequilibrium turbulence

a. Nonequilibrium scaling laws

This section is devoted to the theory presented previously in the works of Bos and Rubinstein (2017) and Rubinstein and Clark (2017). Their derivations are briefly recalled here for the sake of clarity.

According to Rubinstein and Clark (2017) the term “equilibrium” turbulence, with C_ϵ = const refers to the case where turbulence energy spectrum takes a (possibly time-dependent) self-similar form:

$E(\kappa ,t)=k(t)L(t)\mathrm{\Phi }[\kappa L(t)],$(5)

where κ is the wavenumber, L(t) is a characteristic length scale (e.g., integral), and Φ is a function of a “fixed” form. All considerations are performed under the assumption of isotropy of the flow field. With the use of a spectral model those authors conclude the Kolmogorov K41 −5/3 scaling is the solution for a stationary flow. In the case of nonstationarity, a small, subdominant correction appears; however, the K41 inertial range can still be treated as an approximate solution. Equation (5) hence describes turbulence in a state of a possibly unsteady equilibrium. As shown by Bos et al. (2007) in such state C_ϵ does not depend on Re_λ; however, it can still take different values depending on whether turbulence is stationary or decaying. In fact, in the latter case the observed C_ϵ was ca. twice higher than in the forced case. This observation concerned final stages of decay in the grid turbulence, where the self-similar state was reached. Another conclusion follows for the region closer to the grid, where C_ϵ ≠ const and turbulence is out of equilibrium. According to Rubinstein and Clark (2017) such a state occurs when turbulence evolves rapidly from one self-similar state to another. Those authors propose to express the energy spectrum as a sum of equilibrium and nonequilibrium parts

$E(\kappa ,t)=E_0(\kappa ,t)+\tilde{E}(\kappa ,t)+\cdots ,$(6)

with $\tilde{E}\ll E_0$ and where the (possibly nonstationary) E₀(κ, t) satisfies the equilibrium relation Eq. (5). Bos and Rubinstein (2017) derived the Kolmogorov’s relation for E₀ in the inertial range

$E_0(\kappa ,t)=C_Kϵ{(t)}^{2/3}{\kappa }^{-5/3},$(7)

where C_K ≈ 1.5 is the Kolmogorov’s constant, and the following form of the nonequilibrium part

$\tilde{E}(\kappa ,t)=C_K^2\frac{2}{3}\hspace{0.17em}\frac{\dot{ϵ}(t)}{ϵ{(t)}^{2/3}}\hspace{0.17em}{\kappa }^{-7/3},$(8)

where $\dot{ϵ}$ is the time derivative of the dissipation rate. Equations (7) and (8) are assumed to be valid for a wavenumber range κ_L < κ < κ_η, see Fig. 1. Outside this range E(κ, t) was assumed to be zero. Explicit form of E(κ, t) was also derived in Yang et al. (2018), using somewhat different approach, which includes dissipative part of the spectrum.

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Model for spectral energy density in the equilibrium (black dashed line) and in nonequilibrium (solid red line) considered in Bos and Rubinstein (2017).

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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Derivation of the nonequilibrium dissipation scaling laws was further performed in Bos and Rubinstein (2017) by assuming that all the considered turbulence properties can be, analogously, decomposed into the equilibrium and nonequilibrium parts, that is, $ϵ={ϵ}_0+\widetilde{ϵ},$ $k=k_0+\tilde{k}$, $\mathcal{L}={\mathcal{L}}_0+\tilde{\mathcal{L}}$, and $C_{ϵ}=C_{ϵ0}+{\tilde{C}}_{ϵ}$ and calculated based on the equilibrium and nonequilibrium spectra, Eqs. (7) and (8), respectively. This, among other results, leads to the expressions (see appendix A for details)

$\frac{\tilde{k}}{k_0}\approx \frac{2}{9}\hspace{0.17em}\frac{\dot{ϵ}}{ϵ}\hspace{0.17em}\frac{k_0}{ϵ},$(9)

$\frac{\widetilde{ϵ}}{{ϵ}_0}\ll \frac{\tilde{k}}{k_0},$(10)

where the last inequality implies that $ϵ={ϵ}_0+\widetilde{ϵ}\approx {ϵ}_0$.

Using the Taylor series expansion, with $\tilde{k}/k_0$ being a “small” argument, those authors arrived at the following, approximate result for C_ϵ:

$\frac{C_{ϵ}}{C_{ϵ0}}\approx {\left(1+\frac{\tilde{k}}{k_0}\right)}^{-15/14}\hspace{0.17em}=\hspace{0.17em}{\left(\frac{{\text{Re}}_{\lambda 0}}{{\text{Re}}_{\lambda }}\right)}^{15/14},$(11)

where ${\text{Re}}_{\lambda 0}={\lambda }_0{\mathcal{U}}_0/\nu$ is the “equilibrium” Taylor-scale-based Reynolds number. Equation (11) is very close to the experimental result in (3), as the power coefficients n and m in Eq. (3) are close to 1. Similarly, the ratio $\mathcal{L}/\lambda$ derived by Bos and Rubinstein (2017) reads

$\frac{\mathcal{L}}{\lambda }\approx \frac{{\mathcal{L}}_0}{{\lambda }_0}{\left(1+\frac{\tilde{k}}{k_0}\right)}^{-1/14}\hspace{0.17em}=\hspace{0.17em}\frac{C_{ϵ0}}{15}\hspace{0.17em}{\text{Re}}_{\lambda 0}^{15/14}{\left(\frac{1}{{\text{Re}}_{\lambda }}\right)}^{1/14}.$(12)

The power coefficient 1/14 is small and within the accuracy of experimental results, Eq. (4) is valid.

We note here that in Bos and Rubinstein (2017) the case of decaying turbulence is studied, as such setting is usually investigated in laboratory experiments. However, those considerations remain valid for another form of nonequilibrium, where the production is locally large and turbulence develops. In such case $\dot{ϵ}/ϵ$ in the Eqs. (8) and (9) is positive and so is the nonequilibrium part of the spectrum and $\tilde{k}$. In atmospheric turbulence both $\dot{ϵ}/ϵ<0$ and $\dot{ϵ}/ϵ>0$ can be observed. In the former case, Eq. (9) predicts ($1+\tilde{k}/k_0$) < 1 and it follows from Eq. (11) that

$C_{ϵ}=C_{ϵ0}{\left(1+\frac{\tilde{k}}{k_0}\right)}^{-15/14}>C_{ϵ0}.$

When turbulence develops, we have ($1+\tilde{k}/k_0$) > 1 and obtain

$C_{ϵ}<C_{ϵ0}.$

All considerations of Bos and Rubinstein (2017) were performed under the assumptions of homogeneity and local isotropy. These assumptions are usually employed to calculate turbulence dissipation rate from in situ measurements performed by aircrafts, where it is also assumed that the energy spectrum follows the −5/3 law. We also use the assumption of local isotropy in our analyses; however, following Bos and Rubinstein (2017) we account for possible deviations from the Kolmogorov’s scaling due to nonstationarity. The next step, left for further work, would be to account for inhomogeneity and anisotropy. This could be done along the line of the studies of Chen and Vassilicos (2022), where transport equations for the second-order structure functions are considered. We note that the nonequilibrium scaling relations (11) and (12) are also present in the nonhomogeneous flows, like, e.g., turbulent boundary layers or wakes, cf. Obligado et al. (2016, 2022), Nedić et al. (2017); hence, we could assume that the dependence of C_ϵ and $\mathcal{L}/\lambda$ on Re_λ will not be much affected by anisotropy in the atmospheric boundary layer flows, although the proportionality constants may be somewhat different.

b. Estimation of C_ϵ from turbulence models

Bos (2020) proposed a rough estimate of the upper bound of C_ϵ, based on common turbulence models. For this, Eq. (9) was used:

$\frac{\tilde{k}}{k_0}\approx \frac{2}{9}\hspace{0.17em}\frac{\dot{ϵ}}{{ϵ}_0}\hspace{0.17em}\frac{k_0}{{ϵ}_0},$(13)

where ϵ was replaced by ϵ₀, as the nonequilibrium correction $\widetilde{ϵ}/ϵ$ is small. The term $\dot{ϵ}$ can be estimated from the k–ϵ closure equations [cf. Pope (2000) and appendix B], which model the time evolution of the kinetic energy and ϵ. It is to note, however, that the transport equation for ϵ is purely empirical and its form cannot be derived from first principles; therefore, it should be used only due to a lack of better options.

We extend here the analysis by Bos (2020) by including the buoyancy into the considerations. We assume the possibly simplest form of k–ϵ model for buoyant flows, as suggested by Rodi (1987) and assume that transport terms of ϵ are negligible in comparison to other components in the ϵ equation. With this we have

$\frac{\dot{ϵ}}{{ϵ}_0}=\frac{{ϵ}_0}{k_0}\left(A_1\hspace{0.17em}\frac{P}{{ϵ}_0}-A_2+A_3\hspace{0.17em}\frac{G}{{ϵ}_0}\right),$(14)

where P is the shear production of the turbulence kinetic energy, G is the buoyancy production (see appendix B), and the model constants take the following values: A₁ = 1.44, A₂ = 1.92, and A₃ = 1.14. It should be noted that the latter appears to be highly case dependent and different values and modifications of the model were reported in the literature. The one used here was suggested by Baum and Caponi (1992).

Substituting (14) into (13) and (11) we obtain

$\begin{array}{c}\frac{C_{ϵ}}{C_{ϵ0}}\approx {\left(1+\frac{\tilde{k}}{k_0}\right)}^{-15/14}\\ ={\left[1+\frac{2}{9}\left(A_1\hspace{0.17em}\frac{P}{{ϵ}_0}-A_2+A_3\hspace{0.17em}\frac{G}{{ϵ}_0}\right)\right]}^{-15/14}.\end{array}$(15)

We will now introduce the flux Richardson number

${\text{Ri}}_f\hspace{0.17em}=\hspace{0.17em}\frac{-G}{P},$

which is negative for the unstable and positive for the stable stratification, such that Eq. (15) now reads

$\frac{C_{ϵ}}{C_{ϵ0}}\approx {\left[1+\frac{2}{9}(A_1-A_3{\text{Ri}}_f)\frac{P}{{ϵ}_0}-\frac{2}{9}\hspace{0.17em}{A}_2\right]}^{-15/14}.$(16)

There has been a lot of discussions in the literature about the existence of a certain critical Richardson number above which turbulence is suppressed (Galperin et al. 2007; Zilitinkevich and Baklanov 2002). We refer here to the work of Grachev et al. (2013), who report that the Kolmogorov’s −5/3 law fails when Ri_f exceeds a critical value 0.2–0.25. However, some small-scale “supercritical” turbulent motions can still survive up to Ri_f = 0.5. We can now consider different values of P/ϵ₀ and Ri_f and estimate corresponding C_ϵ/C_ϵ0 from Eq. (16). Ri_f = 0.5 will be considered as the upper limit of the flux Richardson number. As reported by Bos (2020) for the case G = 0, when the production is equal to the dissipation P = ϵ₀

$C_{ϵ}\approx 1.1C_{ϵ0},$

which seems to be a good estimate, as the k–ϵ model considered predicts stationary state for production slightly exceeding dissipation. For the case without buoyancy, C_ϵ is bounded from above by the value (Bos 2020)

$C_{ϵ}\approx 1.8C_{ϵ0},$

which correspond to the case with no shear production P/ϵ = 0. This follows exactly from Eq. (15), with G/ϵ = 0. We estimated the ratio C_ϵ/C_ϵ0 for nonzero buoyancy from Eq. (16). Results are presented in Fig. 2.

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Values of C_ϵ/C_ϵ0 estimated from Eq. (16).

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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Generally when turbulence becomes locally stronger, C_ϵ is smaller than its equilibrium counterpart. The opposite is true when turbulence becomes locally weaker. It is to note that the value C_ϵ = 1.8C_ϵ0 remains an upper bound also in the presence of negative buoyancy flux (positive Ri_f), which follows from the fact that (A₁ − A₃Ri_f) in Eq. (16) remains positive for the maximal value of Ri_f = 0.5. It could, however, be possible that some limited regions of zero turbulence production P and negative buoyancy G < 0 would be present. We can expect turbulence would be in a strong nonequilibrium, decaying fast in such zones.

Larger values of C_ϵ were obtained, e.g., in the study of Nedić et al. (2017) from direct numerical simulations data of boundary layer flows at relatively low Reynolds number. However, the derivations of Bos and Rubinstein (2017) were performed for simplified high-Re form of the spectrum, cf. Fig. 1. Hence, the results may not be valid at low Re, where the dissipative part of the spectrum is nonnegligible. Low values of Re_λ are unlikely to be found in the atmospheric boundary layer turbulence; hence, we would rather expect C_ϵ to remain bounded from above by 1.8C_ϵ0.

3. Data and methods

To verify, whether the presented above analysis can be applied to atmospheric turbulence, we used high-resolution data collected in July 2017 during the ACORES campaign in the eastern North Atlantic around the island of Graciosa (Siebert et al. 2021). Measurements were performed with the Airborne Cloud Turbulence Observation System (ACTOS) (Siebert et al. 2006) mounted 170 m below the BO-105 helicopter. We analyze the vertical wind velocity w corrected for platform motion and attitude (cf. Nowak et al. 2021) provided by the ultrasonic anemometer–thermometer (Siebert and Muschinski 2001). The sensor was mounted on ACTOS and has the sampling frequency of 100 Hz. The standard deviations due to uncorrelated noise for velocity measurements are 0.002 m s⁻¹. Relatively low velocity of the helicopter (∼20 m s⁻¹) and high frequency of the sensor allows us to reliably resolve small scales of turbulence down to ∼0.5 m. The further analysis is based on the vertical velocity component only, as the horizontal components still showed some influence of platform attitude and motion which could not be completely removed with standard procedures. These issues caused problems with Reynolds decomposition and subsequent analysis. We note that Obligado et al. (2022) compared estimates of C_ϵ with $\mathcal{U}$ based on turbulence kinetic energy with estimates where only one component of velocity was used to calculate $\mathcal{U}$. In spite of a slight shift of the data points due to anisotropy of the flow in the turbulent boundary layer, the scaling remained unaffected. We will use here the latter methodology and define $\mathcal{U}=\sqrt{⟨w^2⟩}.$

Helicopter flights during ACORES were performed over the ocean inside the 10 km × 10 km square adjacent to the Graciosa island. The typical flight duration was 2 h. The trajectory included vertical profiles up to 2000 m and horizontal legs of the length of several kilometers. In our analysis we use data from the horizontal segments for the same flights as selected in Nowak et al. (2021): flight 5 on 8 July 2017 and flight 14 on 18 July 2017, distinctive by the stratocumulus presence and STBL stratification (considerably well mixed in flight 5, considerably decoupled in flight 14). Detailed analysis of turbulence statistics based on these data was performed in Nowak et al. (2021). The study of nondimensional dissipation coefficient and $\mathcal{L}/\lambda$ ratio performed in this work provides, additionally, estimation of turbulence temporal tendencies in the two types of STBL.

a. Modified spectra

Due to finite frequency of ACTOS turbulence sensor, the wind velocity is measured with a spectral cutoff. As mentioned, ACORES data have the spatial resolution of ∼0.5 m, which is still larger than the typical size of the smallest Kolmogorov’s eddies (∼0.001 m to ∼0.01 m). With this, part of the energy spectrum remains unresolved and to recover value of ϵ indirect methods, which rely on the Kolmogorov’s hypotheses, should be used.

In this subsection we show that in spite of the presence of the nonequilibrium correction, the turbulence kinetic energy dissipation rate can still be calculated from the one-dimensional spectra by fitting to the −5/3 slope, provided that the fitting range is moved toward larger wavenumbers. This is possible, because the nonequilibrium correction affects the one-dimensional spectra only slightly and its influence is negligible for large wavenumbers. Hence, we can still estimate ϵ from the measured velocity signals even in the presence of the spectral cutoff at the Nyquist frequency f_s/2 = 50 Hz.

In the isotropic turbulence, relation between one-dimensional longitudinal spectra and the spectral energy density reads (Pope 2000)

$E_{11}({\kappa }_1,t)={\displaystyle {\int }_{{\kappa }_1}^{\infty }\frac{E(\kappa ,t)}{\kappa }\left(1-\frac{{\kappa }_1^2}{{\kappa }^2}\right)d\kappa };$(17)

E(κ, t) is described in Eq. (6) and its equilibrium and nonequilibrium parts are given in Eqs. (7) and (8):

$E(\kappa ,t)=C_Kϵ{(t)}^{2/3}{\kappa }^{-5/3}+C_K^2\frac{2}{3}\frac{\dot{ϵ}}{{ϵ}^{{}^{2/3}}}{\kappa }^{-7/3}.$(18)

The ratio $\dot{ϵ}/ϵ\approx \dot{ϵ}/{ϵ}_0$ can be estimated from Eq. (14) for selected values of P/ϵ and G/ϵ and the energy spectrum can be determined if k₀ and ϵ₀ are known. Plots of the one-dimensional spectra E₁₁, calculated numerically from Eq. (17) after substituting Eq. (18), corresponding to the case where P/ϵ = 2, G/ϵ = 0 (developing turbulence with the production twice larger than the dissipation) and P/ϵ = 0, G/ϵ = 0 (decaying turbulence, no turbulence production) are presented in Fig. 3. We used k₀ = 0.06 m² s⁻² and ϵ₀ = 5 × 10⁻⁴ m² s⁻³, which are sample values (to the order of magnitude) measured in the STBL.

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One-dimensional spectra calculated for the case P/ϵ = 2, G/ϵ = 0 (developing turbulence with the production twice larger than the dissipation): solid red line; P/ϵ = 0, G/ϵ = 0 (decaying turbulence, no turbulence production): dot–dashed blue line; compared with K41 spectrum: dashed black line.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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To estimate TKE dissipation rate, we assume that within a certain range of wavenumbers the energy spectrum follows the −5/3 law

$E_{11}=C_{11}{ϵ}^{2/3}{\kappa }^{-5/3},$(19)

where C₁₁ ≈ 0.49. Linear least squares fit procedure in the selected range of wavenumbers is performed to estimate the coefficient and calculate ϵ. The above formula is no longer true in the case of modified spectra; however, if the fitting range is moved toward larger wavenumbers, the influence of $\tilde{E}$ should be relatively small, because this term decreases faster with κ than E₀. We used here the range of wavenumbers [1 − 10] m⁻¹. Our reference case will be P = ϵ, G = 0, for which we assume that E₁₁ follows Eq. (19). For this case the linear least squares fit procedure provides ϵ_ref = 4.55 × 10⁻⁴ m² s⁻³, which is somewhat smaller than the value prescribed a priori (equal to ϵ₀ = 5 × 10⁻⁴ m² s⁻³). This follows from the fact that Eq. (14) predicts $\dot{ϵ}/ϵ$ somewhat smaller than zero for P = ϵ. We next modified P/ϵ and G/ϵ and calculated the corresponding ϵ in the same range. Results of ϵ/ϵ_ref and C_ϵ/C_ϵ0 determined from Eq. (15) are given in Table 1. As it is seen, even in the case of production being locally 3 times larger than the dissipation, the overprediction of ϵ is around 10%. Analogously, in case of no production ϵ is underpredicted by 7%. Most of the values of C_ϵ/C_ϵ0, calculated for ACORES data, are within the range 0.6C_ϵ0–1.8C_ϵ0 (see section 4). Hence, as it follows from the above estimation, we expect the over/underprediction of ϵ due to modification of the spectra should be less than 10% for the chosen fitting range. To minimize this error it is desirable to move the fitting range toward possibly largest resolved wavenumbers, but such that the effect of aliasing due to finite sampling frequency is still negligible. Wacławczyk et al. (2020) argued that ϵ estimates based on the structure function approach are better in such case. This is also confirmed in the experience of Siebert et al. (2006) and Nowak et al. (2021). For this reason, in the subsequent analysis we estimate ϵ using the transverse second-order structure function approach, by fitting the data to the formula D₂ ≈ C₂(ϵr)^2/3, where C₂ = 2.8. Figure 4 presents structure functions estimated from airborne measurements in the decoupled STBL. The wind velocity was recorded at height 230 m in a region of large turbulence production and at 990 m (in a region inside the cloud with locally weak turbulence production). As it is seen S₂ profiles clearly deviate from the Kolmogorov’s scaling at high r.

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One-dimensional transverse structure functions estimated from airborne measurements, from the vertical wind velocity component recorded at height 280 m (region of large turbulence production): solid red line; at height 990 m (region inside the cloud with locally weak turbulence production): dot–dashed blue line; compared with K41 spectrum: dashed black line. Vertical lines mark the fitting range.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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Table 1

Parameters estimate for different values of P/ϵ and G/ϵ.

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c. Effects of Reynolds decomposition

To calculate turbulence statistics the instantaneous velocity should first be decomposed into the mean and fluctuating parts. Here, the ensemble average is approximated by the space average along the flight track. The size of averaging window used for detrending, AW_D, should be much larger than the characteristic turbulence length scale but much smaller than the length associated with the mean changes. In practice, in the atmospheric turbulence, the presence of large-scale convective motions, internal waves, and changes of atmospheric conditions along the flight track makes the choice of the averaging window difficult. In this work we perform the analysis for vertical velocity components, for which the changes of the mean component were of lesser magnitude. In Nowak et al. (2021), AW_D = 50 s, which correspond to, approximately 1 km length, were used for the horizontal segments. Such window was approximately 10 times larger than the estimated integral turbulence length scales which were of order 100 m.

Size of the averaging window for detrending AW_D affects mostly the estimation of the integral length scale and the turbulence kinetic energy. We investigated how the values of dimensionless dissipation C_ϵ determined from the formula

$C_{ϵ}=\frac{ϵ\mathcal{L}}{{\mathcal{U}}^3}$(22)

are sensitive to the choice of AW_D. The integral length scale $\mathcal{L}=L_{11}$ was calculated from the transverse two-point correlation coefficient as described in appendix C, ϵ was estimated from the second-order structure function in the fitting range [0.6–6] m, which corresponds to the range of wavenumbers considered in section 3a. Finally $\mathcal{U}$ was calculated as the standard deviation of the vertical component of fluctuating velocity w.

The analysis is performed for the subcloud segment F05LEG307 from flight 5, characterized by low values of C_ϵ, which suggests nonequilibrium, developing turbulence, a part of subcloud segment F14LEG287 from flight 14, where turbulence is close to equilibrium and cloud segment F14LEG992 from flight 14 with C_ϵ > C_ϵ0. For the first two segments the conditions remain approximately constant along the flight track. In the third segment, larger variations of turbulence properties along the flight track are observed; however, the mean value of C_ϵ still exceeds the equilibrium value, see section 6. After detrending another averaging window AW_S should be chosen for the calculation of statistics. This window should be sufficiently long to reduce the bias and the random errors to acceptable levels (Lenschow et al. 1994). In the following subsection, we estimate the size of AW_S sufficient to estimate C_ϵ with a good accuracy. Here, we perform averaging over the whole considered parts of the segments, that is, AW_S = 270 s for flight 5 and AW_S = 300 s for both segments of flight 14. These lengths correspond to at least $\approx 30\mathcal{L}$ for the largest calculated $\mathcal{L}$.

Results are presented in Table 2. As can be seen, the larger AW_D is chosen, the larger resulting $\mathcal{L}$ and $\mathcal{U}$ are. By increasing AW_D even further we would get slower growth and eventually convergence of $\mathcal{L}$ and $\mathcal{U}$. In spite of the dependence of $\mathcal{L}$ and $\mathcal{U}$ on AW_D, changes of C_ϵ are much smaller. E.g., once AW_D is reduced from 75 to 50 s, C_ϵ changes by 0.01 or 0.02. If AW_D is further reduced, the value of C_ϵ becomes close to equilibrium. This would be expected, as both $\mathcal{L}$ and $\mathcal{U}$ are influenced by large scales, which are filtered out if AW_D is too much reduced. We decided to use AW_D = 50 s, the same as in Nowak et al. (2021), for further analysis. Such value still allows us to detect differences in C_ϵ levels, and, with moderate $\mathcal{L}$, the size of averaging windows for the calculation of statistics, AW_S, could be taken small enough to observe variations of turbulence properties along flight tracks.

Table 2

Estimates of turbulence statistics for different AW_D.

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d. Effect of averaging windows

Keeping AW_D = 50s constant we next investigated influence of the results to the choice of AW_S, that is, the window used to calculate statistics from the previously detrended signal. The analysis is performed for the first half of F14LEG287 from flight 14 in the decoupled STBL, where C_ϵ remains approximately constant. The largest AW_S = 200 s corresponds to, approximately $32\mathcal{L}$, or ≈4 km. We moved the window by 5 s (≈100 m) along the flight track and calculated turbulence statistics. Such procedure allows us to detect possible variations of atmospheric conditions along the flight track. For this, it would be optimal to have possibly small AW_S, as otherwise changes of C_ϵ are averaged out. On the other hand, AW_S should be large enough to calculate statistics with a good accuracy. We gradually decreased AW_S to 150, 100, and 50 s which corresponds to, approximately $25\mathcal{L}$, $16\mathcal{L}$, and $8\mathcal{L}$, respectively. Results are presented in Fig. 5. Standard deviations std(C_ϵ) for the investigated segment increase with the decreasing AW_S and equal, respectively, std(C_ϵ) = 0.036, 0.045, 0.064, and 0.21. The last number is clearly unacceptable (which is also visible in Fig. 5), as it should be possible to detect changes of C_ϵ on the order of 0.1 to draw conclusions about the state of turbulence in the investigated region. We decided to further use such averaging window AW_S which correspond to the length larger than $20\mathcal{L}$.

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C_ϵ/C_ϵ0 calculated for different AW_S from part of signal from flight 14, F14LEG287. Horizontal coordinate x denotes the position of the center of the averaging window.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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4. Results: Analysis of turbulence states in STBL

We first address and compare two horizontal flight segments of comparable, low altitudes of 287 m (decoupled STBL, F14LEG287) and 307 m (coupled STBL, F05LEG307). In the former case the averaging window AW_S = 150 s was used, in the latter, the window was increased to AW_S = 170 s due to larger length scales. The window was moved every 5 s (≈100 m) along the flight track and each time turbulence statistics were calculated. Figure 6a presents values of C_ϵ as a function of the position of the center of the averaging window. In the figure two parallel lines of a constant C_ϵ0 = 0.45 and C_ϵ = 0.8 are plotted. The first one would correspond to the equilibrium, stationary case. We note that this equilibrium value may in fact differ slightly for different flows, as it is weakly influenced by the large turbulence structures and their scaling (Valente and Vassilicos 2012; Bos et al. 2007). This influence was not taken into account in the theoretical considerations in section 2, as it was assumed that the energy density has sharp spectral cutoffs at κ_L and κ_η. Moreover, possible flow anisotropy may also play a role and modify the proportionality constant. We choose C_ϵ0 = 0.45 as it is the value obtained for F14LEG287 at small AW_D = 15 s, see Table 2, for which nonequilibrium effects were mostly filtered out, as it was discussed in section 5d. The second value, C_ϵ = 0.8 corresponds to a freely decaying turbulence with P = 0, where the equilibrium, self-similar state was reached (see Table 1).

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C_ϵ as a function of the position of the center of the averaging window. Red rectangles mark nonequilibrium region. Data for decoupled STBL: (a) F14LEG287, (b) F14LEG448, (c) F14LEG992.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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As it is seen in Fig. 6a, initially, C_ϵ ≈ C_ϵ0. As the averaging window is moved further along the flight track, a moderate increase of C_ϵ is first observed, followed by a sharp decrease toward smaller values. It would suggest that a weak turbulence decay, followed by a region of strong turbulence production was detected. In the analogous figure for the coupled STBL, Fig. 7a, only C_ϵ < C_ϵ0 are detected, which can be again interpreted as a region of strong turbulence production.

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As in Fig. 6, but for coupled STBL: (a) F05LEG307, (b) F05LEG553, (c) F05LEG819.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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We next compared two subcloud flight segments: F14LEG448 in the decoupled STBL of height 448 m and F05LEG553 in the coupled STBL of height 553 m. In Nowak et al. (2021) nearly zero buoyancy production at those levels was reported. The weak turbulence production is manifested through larger average values of C_ϵ, both in decoupled and coupled case, cf. Figs. 6b and 7b, respectively. Insufficient turbulence production at this altitude may lead to decoupling, when the eddies fail to mix air over the entire depth of the STBL, and consequently STBL separates into two parts: cloud driven at the top and surface driven at the bottom, cf. Nowak et al. (2021).

Finally, we consider in-cloud segments F14LEG992 in the decoupled STBL and F05LEG819 in the coupled STBL. The signals were recorded at heights 992 and 819 m. In both cases values of C_ϵ vary considerably, cf. Figs. 6c and 7c; however, they are somewhat larger in the decoupled case which indicates weaker turbulence production and is again in line with the observations of Nowak et al. (2021). The results of C_ϵ can be compared with experimental estimations of Zheng et al. (2021) in high-Re turbulence behind a grid, where C_ϵ first decreased in the production region and next increased in the nonequilibrium decay region toward value C_ϵ ≈ 0.8. We also mention here the direct numerical simulation study of Gallana et al. (2022), who calculated C_ϵ in stably and unstably stratified flows. In the former case C_ϵ increased toward C_ϵ ≈ 0.8, indicating turbulence suppression and in the unstable conditions decreased toward C_ϵ ≈ 0.4, below the equilibrium value. In STBL turbulence is manly driven by radiative and evaporative cooling at the cloud top, leading to instability. However, boundary layer is also capped from above by a stably stratified layer. This may explain large variations of C_ϵ along the in-cloud flight tracks.

To classify whether turbulence is in its equilibrium or nonequilibrium state, we calculated the length scales and the local Reynolds numbers and plotted $\mathcal{L}/\lambda$ and C_ϵ as a function of Re_λ. The data points were compared with the theoretical predictions (lines). For C_ϵ, in the equilibrium case in high-Re atmospheric flows, we should obtain C_ϵ = const. In the nonequilibrium, dependence of C_ϵ on Re_λ should be described by Eq. (11). As we do not have the knowledge about the equilibrium value Re_λ0, the coefficient in Eq. (11) was calculated from the linear least squares algorithm. In the lowest flight segment F14LEG287 in decoupled STBL statistics calculated at x < 4500 m follow, approximately the equilibrium line C_ϵ = const (black dots in Fig. 8a). However, when the position of the center of the averaging window exceeds 4500 m the nonequilibrium states can be detected (red stars in Fig. 8a). The values of C_ϵ calculated for coupled STBL for F05LEG307 decrease slightly with Re_λ, which suggests the presence of nonequilibrium scaling (cf. Fig. 9a).

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C_ϵ as a function of Re_λ. Solid black lines: equilibrium scalings C_ϵ0 = 0.45 and C_ϵ1 = 1.8C_ϵ0; dashed red line: nonequilibrium scaling, Eq. (11); calculated statistics: symbols. Decoupled STBL: (a) F14LEG287, (b) F14LEG448, (c) F14LEG992.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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As in Fig. 8, but for coupled STBL: (a) F05LEG307, (b) F05LEG553, (c) F05LEG819. In (c) color coded are different local equilibrium states.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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Figures 8b and 9b refer to the flight segments F14LEG448 and F05LEG553 below the cloud. Figure 8b suggests that part of the signal in the decoupled STBL is a record of equilibrium turbulence, while statistics of another part fit the nonequilibrium formulas. For the coupled STBL the statistics follow rather the equilibrium predictions, cf. Fig. 9b. A large scatter of results is visible in Fig. 9b such that it is not easy to draw conclusions. Note that the flight segment F05LEG553 is relatively short and so is the range of the detected Re_λ values. Equilibrium scaling may suggest that in spite of the weak turbulence production, the changes of atmospheric conditions are relatively slow, such that the turbulence spectra already relaxed to the self-similar form (5). Similar, nearly constant levels of C_ϵ were reported, e.g., by Neunaber et al. (2022) in the study of turbulence in wakes behind turbines.

Particularly interesting is the study of the in-cloud segments F14LEG992 and F05LEG819. In the decoupled STBL most of the points follow the nonequilibrium scaling relations, cf. Fig. 8c. Exceptions are several points corresponding to large C_ϵ ≈ 1.8C_ϵ0, which would indicate regions of no turbulence production and/or stable stratification. In the coupled STBL on the other hand, in spite of large variations of C_ϵ, the points seem to follow three different lines of constant C_ϵ, namely, C_ϵ ≈ 0.8, C_ϵ ≈ 0.6, and C_ϵ ≈ 0.4, cf. Fig. 9c. We may call such state of the system as a quasi-equilibrium state, which means that the changes of external conditions are slow enough, such that the system undergoes through several local equilibria.

We note here that for certain points the assignment to equilibrium or nonequilibrium region may be somewhat arbitrary. This concerns especially those points which represent statistics calculated at the boundary between the equilibrium and nonequilibrium region. Sometimes this assignment becomes more definite when the dependence of the length scale ratio $\mathcal{L}/\lambda$ on Re_λ is studied, see Figs. 10 and 11. In the equilibrium case, the statistics should follow Eq. (2), while in the nonequilibrium $\mathcal{L}/\lambda$ and Re_λ are inversely proportional and follow Eq. (12). Both functional dependencies are plotted in Figs. 10 and 11 as the black and red lines, respectively. We note that we did not perform another curve fitting here, the constants are those estimated for data from Figs. 8 and 9 for respective flight segments and divided by 15, as follows from the Eqs. (11) and (12). The good agreement confirms validity of the theoretical derivations of Bos and Rubinstein (2017). After the classification of statistics in Figs. 8–11 the nonequilibrium parts were marked in Figs. 6 and 7 (as red rectangles).

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$\mathcal{L}/\lambda$ as a function of Re_λ. Solid black line: equilibrium scaling, Eq. (2); dashed red line: nonequilibrium scaling, Eq. (12); calculated statistics: symbols. Decoupled STBL: (a) F14LEG287, (b) F14LEG448, (c) F14LEG992.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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As in Fig. 10, but for coupled STBL: (a) F05LEG307, (b) F05LEG553, (c) F05LEG819. In (c) color codes are different local equilibrium states.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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Nonequilibrium scaling relations were found in many laboratory studies, as examples we mention here the works of Zheng et al. (2021), which concerns high-Re flows behind a grid, and of Nedić et al. (2017), which concerns boundary layer flows.

We next considered separately F14LEG143 of the decoupled STBL of altitude 143 m, as there are no measurement data of comparable height for the coupled STBL. As it is seen in Fig. 12, two region of C_ϵ ≠ C_ϵ0 can be distinguished. The part on the left, where C_ϵ > C_ϵ0 suggests turbulence decay. Statistics calculated at the final part of the signal, from F14LEG143, with C_ϵ < C_ϵ0, imply on the other hand the dominant role of the production. On the middle part of the curve in Fig. 12, C_ϵ oscillates around the equilibrium value of 0.45. These equilibrium values are marked as black dots in Fig. 13a. Apparently, they do not depend on Re_λ. Moreover, the ratio $\mathcal{L}/\lambda$ increases with Re_λ, as predicted by the equilibrium relation (2), see Fig. 13b. However, the remaining points, marked as red and blue stars in Fig. 13, do not follow one nonequilibrium line. We can argue that they correspond to two distinct regions, with different initial conditions and different Re_λ0. By using the least squares algorithm we calculated two coefficients of the nonequilibrium relation (11),

$A=C_{ϵ0}{\text{Re}}_{\lambda 0A}^{15/14},B=C_{ϵ0}{\text{Re}}_{\lambda 0B}^{15/14},$

where the subscript A refers to the points with C_ϵ > C_ϵ0 (red stars in Fig. 13) and B refers to the points with C_ϵ < C_ϵ0 (blue stars in Fig. 13). We found Re_λ0B ≈ 1.2Re_λ0A. Assuming C_ϵ0 ≈ 0.45 it is possible to recover the initial-state Re_λ0 from the calculated A and B coefficients. We found Re_λ0A ≈ 7500 and Re_λ0B ≈ 9000 as the equilibrium (initial) Reynolds numbers in the respective regions. We next plotted the red points for rescaled coordinate ${\text{Re}}_{\lambda }^{\prime }=1.2{\text{ Re}}_{\lambda }$ and found, as expected, that they follow the same nonequilibrium curve as the blue points, see Fig. 14a. The same can be done for the relation $\mathcal{L}/\lambda$; however, in this case $\mathcal{L}/\lambda \sim {\text{Re}}_{\lambda }^{-1/14}$ and to obtain the same coefficient we have to plot the data for differently rescaled ${\text{R}{\text{e}}^{\prime }}_{\lambda }={(1.2)}^{15}{\text{Re}}_{\lambda }$. Results are plotted in Fig. 14. They again confirm the validity of the analytically derived formulas in (11) and (12).

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As in Fig. 6, but for the decoupled STBL, F14LEG143.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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(a) C_ϵ as a function of Re_λ, (b) $\mathcal{L}/\lambda$ as a function of Re_λ. Decoupled STBL, F14LEG143. Solid black lines: equilibrium scaling; dashed red line and dot–dashed blue lines: nonequilibrium scaling, Eq. (11); calculated statistics: symbols.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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As in Fig. 13, but for rescaled Re_λ.

Citation: Journal of the Atmospheric Sciences 79, 10; 10.1175/JAS-D-22-0028.1

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5. Conclusions

The main purpose of this work was to show that certain information about the temporal variations of turbulence and the budget of the turbulence kinetic energy in the STBL can be acquired by analyzing statistics of wind velocity based on in situ measurement data. The key indicators are the nondimensional dissipation coefficient C_ϵ and the integral-to-Taylor-scale ratio $\mathcal{L}/\lambda$. C_ϵ takes largest values, up to around C_ϵ ≈ 0.8 when turbulence decays, reaches its equilibrium C_ϵ ≈ C_ϵ0 ≈ 0.45 in the stationary state and becomes smaller C_ϵ < C_ϵ0 when turbulence becomes stronger. We showed here that the upper limit C_ϵ ≈ 1.8C_ϵ0 should not be exceeded even under a stable stratification, due to the presence of the critical Richardson number, above which turbulence is suppressed.

Moreover, as it was discussed in the previous works (Bos and Rubinstein 2017) two different states of turbulence system can be identified: equilibrium, where the turbulence kinetic energy spectrum has a self-similar shape, (5), and a nonequilibrium where a correction to the spectrum should be considered; see Eq. (6). The equilibrium state can possibly be nonstationary; however, the changes of external conditions (forcing) are slow enough such that the system has enough time to relax and the energy spectrum takes the self-similar form (5). To detect the nonequilibrium states we studied dependence of C_ϵ and $\mathcal{L}/\lambda$ on the Taylor-based Reynolds number Re_λ. In the equilibrium C_ϵ ≈ const and $\mathcal{L}/\lambda$ increases with increasing Re_λ. In the nonequilibrium turbulence a different, although universal scaling is present, namely, $C_{ϵ}\sim {\text{Re}}_{\lambda }^{-15/14}$ and $\mathcal{L}/\lambda \sim {\text{Re}}_{\lambda }^{-1/14}$. This method complements measures of nonstationarity and nonequilibrium reported by Mahrt and Bou-Zeid (2020, see also references therein), who mention the lack of an inertial subrange and results of direct production–dissipation balance estimates. The latter are not available from horizontal flight segments, as derivatives in the vertical direction cannot be calculated. However, as it was discussed in section 2b, the production–dissipation balance is related to the ratio C_ϵ/C_ϵ0.

In this work we focus on the analysis of airborne measurement data, which often suffer from various deficiencies, related to finite frequencies of the sensors or short available averaging windows (Wacławczyk et al. 2017; Akinlabi et al. 2019). We showed that despite these deficiencies both C_ϵ and $\mathcal{L}/\lambda$ can still be estimated from the ACORES data (Nowak et al. 2021). In particular, to estimate ϵ it is crucial to move the fitting range toward smaller scales (larger wavenumbers) to minimize the influence of nonequilibrium correction to the spectrum. This −7/3 correction is generally subdominant and influences larger scales. We also showed that the indicator functions can be estimated with a sufficient accuracy if the size of the averaging window used to calculate statistics is larger than $15\mathcal{L}$. This allows us to obtain C_ϵ and $\mathcal{L}/\lambda$ for horizontal flight segments. Results compare very well with the theoretical predictions of Bos and Rubinstein, which concerns both dependence on local Re_λ as well as on the initial conditions Re_λ0, which is shown by a proper rescaling of the data. Moreover, with a curve fitting of the data we were able to recover values of Re_λ0.

The analysis performed here allows for additional comparison of turbulence in the coupled and decoupled STBL’s, previously studied by Nowak et al. (2021). In both cases, C_ϵ is smaller at lower altitudes due to strong shear and/or buoyancy production. On the other hand, in the subcloud parts C_ϵ increases above C_ϵ0 which suggests that in these regions turbulence decays. Large variations of C_ϵ were detected inside the cloud in both cases. Apparently, along the flight track, regions where turbulence production surpass or is in balance with the dissipation alternate with areas of turbulence decay. Here, the main difference between the coupled and decoupled STBL was observed. Namely, in the former, turbulence seems to undergo a series of equilibrium states, where C_ϵ ≈ const and $\mathcal{L}/\lambda \sim {\text{Re}}_{\lambda }$, while in the latter case a clear nonequilibrium is observed. This may be a consequence of a different flow organization in the latter case, with smaller and faster large-scale circulations (Nowak et al. 2021). Moreover, at all the investigated altitudes, mean C_ϵ was higher in the decoupled than in the coupled case which indicates the weaker turbulence production in the decoupled STBL.

In this work only limited amount of data from two ACTOS flights were investigated. In future, more thorough analysis to better characterize turbulence in the coupled and decoupled STBL is foreseen. The same analysis can be repeated for various measurement data in the atmospheric boundary layers where the impact of transient states is nonnegligible (cf. Mahrt and Bou-Zeid 2020), e.g., to study the collapse of turbulent convective daytime boundary layer (El Guernaoui et al. 2019; Wang et al. 2020; Lothon et al. 2014), in and between various types of the clouds in order to better understand their dynamics and even in clear air turbulence to characterize its dynamic properties.

Another interesting aspect is the relation of the present study to the K62 theory (Kolmogorov 1962), which takes into account temporal and spatial variations of the instantaneous dissipation rate. This phenomenon is known as the “internal intermittency” and affects the scaling of higher-order structure functions. A careful analysis of the fine-scale structure of turbulence performed by Siebert et al. (2010) based on ACTOS data revealed the intermittency coefficient in cloud turbulence is consistent with values obtained in laboratory experiments. A question to be asked is how the nonequilibrium structure functions are affected by the small-scale intermittency and whether temporal variations of statistics considered here can be linked to the premises of the K62 theory.

The presence of nonequilibrium in the stratocumulus clouds indicate that the common turbulence closures like the Smagorinsky LES model may fail to predict the dynamics of such clouds correctly. In particular, this may influence the prediction of boundary layer decoupling or cloud dissipation. A nonequilibrium turbulence model may largely improve these predictions. This is another direction of future studies.

Data availability statement.

All data from the ACORES study are available from the authors on request.