1. Introduction

The investigations of turbulence in the atmosphere largely concern its effects on the structure and growth of the atmospheric boundary layer (e.g., Stull 1988; Garratt 1994; Nieuwstadt and Duynkerke 1996). Recent high-resolution large-eddy simulation (LES) models (Sass 2007; Heinze et al. 2015) provide important information on the second-moment balances in cloud-topped boundary layers under conditions of Barbados Oceanographic and Meteorological Experiment (BOMEX) and Dynamics and Chemistry of the Marine Stratocumulus field study (DYCOMS). High-resolution radar measurements were used by Pinsky et al. (2010) to study turbulent parameters of the cloud-top boundary layer.

Turbulence in individual clouds and its role in cloud dynamics and microphysics is understood to much less extent and remains a hot topic of cloud physics. Turbulence fluctuations are of random nature and require large statistic for analysis. To our knowledge, turbulent properties of individual clouds were not analyzed. This is especially true for convective clouds which are known as zones of enhanced turbulence, although turbulence intensity and its other parameters depend on cloud type and change substantially within clouds.

The role of turbulence in clouds is usually related to two processes: 1) turbulent mixing of clouds with dry surrounding air, as well as mixing inside clouds (e.g., Baker and Latham 1982; Lehmann et al. 2009; De Rooy et al. 2013; Pinsky and Khain 2018, 2019, 2020a,b; Desai et al. 2021) and 2) turbulence effect on microphysical processes such as condensation/evaporation (Sardina et al. 2018; Grabowski and Thomas 2021) and collisions between hydrometeors (Khain et al. 2007; Pinsky et al. 2008; Wang and Grabowski 2009; Saito and Gotoh 2018; Khain and Pinsky 2018). The turbulence also creates supersaturation fluctuations in clouds, which, according to some studies, may influence the formation of droplet size distribution (DSD) (Khain and Pinsky 2018). Despite a common opinion about the important role of turbulence in cloud thermodynamics and microphysics, its role remains to be the subject of intense study and debates (De Rooy et al. 2013; Khain and Pinsky 2018).

There are several problems impeding investigation of turbulence. First, there is a lack of accurate in situ measurements. While some aircraft measurements of turbulent parameters in the boundary layer were performed several tens of years ago (e.g., Grossman 1982), most measurements of cloud air velocity were performed with 1-Hz frequency which corresponds to the spatial resolution of 100 m (e.g., Burnet, and Brenguier 2007; Konwar et al. 2021). Second, a typical limitation of in situ measurements is the lack of necessary data (lack of statistics) concerning spatial variations of turbulent parameters. The reason is that airplane traverses are one-dimensional and do not cover an entire cloud. Besides, different points along the aircraft track correspond to different time instances. To obtain reasonable statistics, the data taken in multiple clouds along the aircraft travers were used. As a result, the turbulent parameters obtained (e.g., the turbulence kinetic energy dissipation rate ε) characterize some mean values in clouds obtained by averaging over different stages of cloud evolution (both development and decay) (Gerber et al. 2008; Pinsky et al. 2010; Katzwinkel et al. 2014; Siebert et al. 2015; Xue et al. 2016; Strunin and Strunin 2018; Feist et al. 2019). The common opinion is that ε in stratiform and small cumulus clouds (Cu) ranges from 10 to several tens of cm² s⁻³. For instance, according to Gerber et al. (2008) the mean turbulent kinetic energy dissipation rate averaged over seven Cu varied from ∼14 to ∼70 cm² s⁻³. In high-frequency measurements, Katzwinkel et al. (2014) and Schmeissner et al. (2015) evaluated the dissipation rate ε in small Cu about 100 m below cloud base using the estimated second-order structure function. The averaged dissipation rate was about 10 cm² s⁻³ as compared with 1.0 cm² s⁻³ outside of clouds. In some experimental studies the dissipation rate was evaluated using simplified expressions containing a priori given parameters.

Another issue is that no information on the vertical structure of turbulence in Cu is available. The values of ε in cumulonimbus clouds (Cb) are assumed to be several orders higher; however, no reliable measurements of turbulence were performed in such clouds (Khain and Pinsky 2018).

Cloud turbulence parameters were also calculated using high-resolution LES models and even direct numerical simulation (DNS) models (Benmoshe et al. 2012; Abma et al. 2013; Mellado et al. 2018). Often turbulent parameters are calculated using the equation for turbulence kinetic energy (TKE) with 1.5-order closure. In such studies, the turbulent parameters such as TKE, ε, and the turbulent diffusion coefficient are calculated using the concept of mixing length, which makes all turbulent parameters dependent on the model grid spacing. In cloud-resolving and large-scale models, the values of mixing length are on the order of one to several kilometers. In these cases, the turbulent diffusion terms do not characterize real turbulence, but play the role of the factor eliminating undesirable fluctuations (i.e., perform a smoothing). Besides, Reynolds/Rayleigh numbers in DNS do not match the corresponding numbers in the real atmosphere, which are several orders larger.

The uncertainties in the determination of turbulent parameters hinder understanding the role of turbulence in cloud microphysics. For instance, the well-known Damköhler parameter determining the mixing type (homogeneous versus inhomogeneous) depends on the characteristic mixing time which in turn depends on turbulent intensity (Lehmann et al. 2009; Korolev et al. 2016; Pinsky et al. 2016).

As was mentioned above, utilization of 1.5 closure in LES models makes turbulent parameters dependent on model resolution and, therefore, this parameterization is not suitable for analysis of properties of cloud turbulence. An accurate approach to derive the turbulent properties is to analyze the velocity fields containing the major part of the turbulent range, simulated by LES of very high resolution. Cloud motions cover a wide range of scales. Crudely, these motions can be separated into convective and turbulent. The roles of these two types of motion are quite different. Convective motions transport air mass and determine the mass fluxes in clouds. In analytical and model studies, the transport by convective motions is described by equations of advection. Convective motions can be referred to as “coherent” with scales on the order of cloud size. Turbulent motions are stochastic, and the turbulent velocities being averaged over the temporal or spatial scale of convective motions should be equal or close to zero, i.e., turbulent motions do not transport mass. For instance, vertical mass flux in clouds (which is the major quantity in convective parameterizations) is of convective nature. In cloud studies, turbulent effects are typically described by the equation of turbulent diffusion. Turbulent motions are typically weaker than the convective ones.

In Pinsky et al. (2021, hereafter, Part I) the method of separation of motions in clouds into convective motions and turbulent motions by means of a wavelet technique was developed. The second part of the study (Pinsky et al. 2022, hereafter, Part II) was dedicated to the investigation of some properties of convective motions of Cu. It was shown that at the developing stage Cu resembles a so-called starting plume, consisting of a rapidly ascending bubble with a nonstationary jetlike tail. Entrainment fluxes of convective nature were also evaluated.

Here in Part III of the study we present and analyze turbulence characteristics of a warm nonprecipitating Cu at different stages of its evolution. For this purpose, a nonprecipitating trade wind Cu with 2400-m cloud top was simulated using 10-m-grid-spacing LES System Atmospheric Modeling (SAM; with spectral bin microphysics) under thermodynamic conditions observed in BOMEX. The turbulent velocity field was obtained using high-frequency wavelet filter of the cloud velocity field. Estimation and study of vertical dependencies and spatial–temporal variations of the main turbulent characteristics of center part of such Cu is the main object of the study.

2. Method of analysis

To better understand the analysis performed in this study, we briefly outline the method used and the results obtained in Part I and Part II of the paper. The analysis is carried out using LES—simulated data. A single trade wind warm nonprecipitating Cu was simulated using SAM, a 3D LES model first formulated by Khairoutdinov and Kogan (2000) and Khairoutdinov and Randall (2003). The SAM version used in this study was specially designed to simulate fine thermodynamic and microphysical cloud structures at high resolution (10 m), which allows direct reproducing of both convective and turbulent motions. The specific feature of the SAM version is the use of spectral bin microphysics (Khain et al. 2004; Khain and Pinsky 2018; Khain et al. 2019). The scheme directly calculates DSD functions at each grid point and each time step. DSDs are defined on logarithmically equidistant (mass doubling) mass grid containing 33 bins and completely covering all possible drop sizes including raindrops up to 8 mm in diameter. To describe cloud droplet formation and cloud–aerosol interaction as a whole, a separate SD for aerosols is calculated. The rates of diffusion growth/evaporation are calculated by solving a system of equations for supersaturation, together with the equations for diffusion growth/evaporation of droplets. The changes in DSDs resulting from collisions between drops are calculated by solving the stochastic collision equations (Khain and Pinsky 2018). The height (density) dependencies of collision efficiencies and collision kernels for drops were calculated offline (Pinsky et al. 2001).

The BOMEX-1974 sounding profiles (Stevens et al. 2001; Siebesma et al. 2003) are used in the simulations (Fig. 1). The temperature profile is typical of the trade wind zone with the inversion layer within the altitude range of 1400–2000 m, where the temperature decreases by 0.25°C and humidity and relative humidity rapidly decrease. For simplicity, no mean wind shear in the cloud environment is assumed.

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Vertical profiles of (a) temperature, (b) the mixing ratio of water vapor, and (c) the relative humidity of the air, measured far from the cloud (from Part II).

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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All stages of cloud evolution have been simulated (see Fig. 2). At the developing stage, the maximum liquid water content (LWC) increases until t_m = 32.5 min and reaches ∼3.3 g m⁻³. The cloud-top height first increases reaching 2400 m at t_m = 33.5 min. Later, after a short mature stage, all the parameters decrease indicating the decaying cloud stage. After 40 min, the cloud crumbled into several parts, including small volumes having, however, a significant LWC on the order of 1 g m⁻³. The total life time of the modeled cloud is equal to ∼43 min. Such cloud sizes and lifetimes are typical for comparatively large trade wind Cu (e.g., Gerber et al. 2008; Heus et al. 2009; Katzwinkel et al. 2014).

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Time dependence of maximum LWC (g m⁻³) (from Part II).

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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A cloud is a spatially nonuniform object where zones of regular convective updraft and downdraft are superimposed by random turbulent motions. To separate convective and turbulent motions, we applied a 2D spatial wavelet filtering, as described in Part I of the study. The specific feature of the wavelet analysis is the ability to perform filtration in localized areas of a long-lasting nonstationary signal. Since the gradients of convective velocity and other quantities can be very sharp, especially near cloud boundaries, the locality of filtration is of enormous importance to keep these gradients unchanged. An alternative application of averaging window (e.g., the Gaussian one) would lead to strong smoothing of the gradient zone, to larger artificial broadening of the cloud interior region and to larger residual fluctuations in comparison with the wavelet filter. Besides, the optimal width and of the Gaussian window strongly depend on the changes in the useful signal. Actually, the application of an averaging window would not allow us to separate convective and turbulent motions.

To justify the applicability of the wavelet method to cloud data processing and to optimize wavelet filtration, a synthetic vertical velocity field, obtained by summing up the deterministic convective and stochastic turbulent components, was modeled (see Part I). Numerous simulations and statistical estimations allowed us to choose the wavelet parameters minimizing component distortion and the errors in component separation (see Part I). The application of the optimum parameters allowed us to restore the initial deterministic convective and stochastic turbulent components. The high efficiency of the wavelet method in this study was demonstrated for both synthetic and DNS fields. The same set of parameters was used in our research of convective motions (Part II) and turbulent motions (this study) in a single Cu cloud. It should be stressed that the same set of parameters must be used for all the components of the velocity field in order to avoid strong disturbances of the continuity equation in analysis of LES-simulated clouds. Otherwise, one can expect some bias in statistical estimations of turbulent parameters.

Part II of the study is dedicated to the analysis of convective motions and their influence on thermodynamic and microphysical characteristics in the cloud updraft zone (CUZ). Fields of U(x, y) at every time step with the 10 m altitude increment were used for this goal. Special conditions were imposed on analyzing W(x, y) 2D fields to confidently isolate CUZ and to eliminate small isolated volumes arising outside of CUZ during cloud development. It was shown that wavelet filtering allows us to distinctly highlight the large-scale convective velocity structure forming clouds skeleton. The cloud area occupied by CUZ contains the points of maximum convective updraft velocity. These points taken at each height level form the cloud core. It was found in Part II that these points correspond also to the maximum LWC, which is close to the adiabatic one up to the height of 1800 m.

To illustrate another main finding of Part II, we present Fig. 3 showing the height–time field of the maximum vertical convective velocity W_max(z, t). One can see that the cloud core characterized by W_max(z, t) does not behave like a stationary vertical jet [in this case, the contours of W_max(z, t) would form long horizontal bands]. At the same time, Fig. 3 allows us to detect some kind of a rising object leaving a trail at the developing stage of cloud. Therefore, Fig. 3 is consistent with the concept of a starting plume i.e., nonstationary jet capped by a rising head bubble whose z–t trajectory is shown by the magenta line in Fig. 3. This z–t trajectory is determined by W_max(z, t) at each height and each time instances. This z–t trajectory is presented in several figures of this paper.

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Height–time field of W_max(z, t). The magenta line shows z–t trajectory of the rising point parcel (from Part II).

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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In the present study we investigate the properties of fields of high-frequency wavelet signals corresponding to turbulent velocity u′(x, y), in order to analyze turbulent motions in the cloud area close to CUZ. Examples of two slices of vertical velocity W(x, y) and w′(x, y) in cloud at two different heights are shown in Fig. 4. The slices correspond to the time of maximally developed cloud (t = 33 min), and obtained both at the height region below the inversion zone (z = 1000 m; Figs. 4a,b) and in the middle of the inversion layer (z = 1700 m; Figs. 4c,d). The convective air motion (Figs. 4a,c) represents CUZ surrounded by a subsiding shell. In contrast, turbulent motion represented in Figs. 4b and 4d, consists of stochastic-like filaments of different sizes stretching out along the horizontal plane in different directions. The magnitudes of convective and turbulent velocities are comparable at both height regions. Inspection of Figs. 4b and 4d shows that sizes of filaments reach 200–250 m, and the velocity in these filaments can be as large as 2.5 m s⁻¹. Usually velocity fluctuations with sizes less than 100 m and magnitudes less than 1 m s⁻¹ are attributed to turbulence. Wavelet filters were configured and optimized for such a situation (see Part I). In this study, we will attribute all high-frequency velocity fluctuations exiting wavelet filter to turbulent ones, at the same time keeping in mind that the lowest-frequency filaments can have another nature, so their scales are intermediate between turbulence and convection. For example, such filaments can be interpreted as coherent structures formed by inverse energy cascades (Elperin et al. 2002). However, analysis described in section 5 does not reveal a statistically significant border between turbulence inertial interval and coherent structures interval on the energetic spectra.

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The horizontal fields of (a),(c) vertical convective velocity and (b),(d) vertical turbulent velocity in cloud at two different heights. The sample slices correspond to the time of maximally developed cloud (t = 33 min). The magenta point on the panels indicates the maximum of convective velocity.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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Design of the areas for analysis

As was discussed above, we determined the point corresponding to the maximum convective updraft velocity W_max(z, t) at each height and at each time instance. This point was identified with the cloud center (cloud core). Such maximums are shown in Figs. 4a and 4c by magenta points. Using this point we designed two mutually perpendicular bars of 600 m long and 400 m wide, as shown in Fig. 5. A rectangle of such size lies inside the CUZ, which makes analyzed data maximally uniform. We choose zones of maximum possible size within the CUZ to provide the maximum of estimated spectrum range and maximum statistical volume. The calculations of different turbulent moment functions are carried out separately along the x bar shown by dashed blue line (denoted by x index), and along the y bar shown by dashed red line (denoted by y index), while statistical averaging is carried out across both bars. Application of two perpendicular bars allows one to draw conclusions about horizontal isotropy/anisotropy of turbulence. Also, comparison of estimations, obtained using different bars increases the reliability of the results.

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Two perpendicular bars 600 m × 400 m centered at the point of the maximum convective updraft velocity, designed for estimation of turbulent characteristics. The two rectangles are moved in the space according to the center of the convection during cloud development.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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3. Sources of turbulence kinetic energy in cloud updraft zone

The buoyancy production term (BPT) of the TKE is written as BPT = 〈b′w′〉, where b′ = B − 〈B〉 is deviation of buoyancy acceleration from the mean buoyancy acceleration, which is calculated using the same wavelet filters as in the case of velocities. Total buoyancy acceleration B depends on thermodynamic and microphysical parameters of a cloud:

$B\approx g\left\{\frac{T-T_0}{T_0}+0.61(q_{\upsilon }-q_{\mathrm{\upsilon \hspace{0.17em}}0})-q_l\right\},$(3)

where g is gravity acceleration, T is absolute temperature in cloud, q_υ is water vapor mixing ratio, q_l is liquid water mixing ratio, and T₀ and q_υ0 are reference values measured far from the cloud. The SPT of the TKE is determined by turbulent velocity fluctuations and gradients of the mean (convective) velocity. In the tensor form it is written as $\text{SPT}=-\langle (\partial U_i/\partial x_j)u_i^{\mathbf{\prime }}{u}_j^{\mathbf{\prime }}\rangle$, where i, j = 1, 2, 3, x₁ = x; x₂ = y; and x₃ = z. Summation over repeated indices is implied. Wavelet techniques allow us to evaluate all nine components of SPT. To our knowledge, the relation between these sources in cumulus clouds is not known. Figure 6 demonstrates vertical profiles of BPT and SPT values, calculated along x-direction and y-direction bars. The comparison of profiles shows that BPT, usually increasing on average with the height, increase up to values of 0.013–0.014 m² s⁻³ and plays the main role in formation of turbulence at the developing and mature stage of cloud evolution. At the mature stage buoyancy production is minimum in the inversion layer at the height of 1500 m. Closeness of the BPT profiles along the two mutually perpendicular bars means isotropy of turbulence production in the horizontal plane. From the statistical point of view, normalized BPT is the correlation coefficient between fluctuations of the buoyancy force and fluctuations of the vertical velocity, and is a good measure of buoyancy–vertical velocity relationship. Calculations show that there is a large approximately constant correlation coefficient, equal to 0.5–0.6 up to the height of 1400 m, which characterizes strong influence of buoyancy on turbulence in the bottom and middle parts of the cloud. Higher, above the inversion layer, the correlation coefficient decreases down to 0.35, meaning a decrease of turbulence production by buoyancy.

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Vertical profiles of buoyancy production term (BPT) and shear production term (SPT), calculated along the x-direction and the y-direction bars in the cloud center zone at t = 33 min.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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Profiles of SPT demonstrate interesting behaviors. In the lower part of a cloud (lower than 1100 m) the SPT is negative. Negativity of SPT means transfer of kinetic energy from turbulent-scale motions to convective-scale motions. Although the negative shear production in atmospheric clouds has not been mentioned previously in academic literature, this phenomenon is known and justified by both measurements and DNS simulations for turbulent flows of different types (Gayen and Sarkara 2011; Ricardo et al. 2014; Cimarelli et al. 2019). We also see some similarity of our results with those reported by Smedman et al. (1997, 2004), who investigated the dynamics of the stable boundary layer near the surface. The observations analyzed in these studies show the presence of a strong narrow jet near the surface, leading to a suppression of turbulence (and surface fluxes), as well as preventing large eddies from penetrating downward. These results indicate a negative correlation between the wind shear in the jet and turbulent fluctuations, i.e., a negative SPT. In our case, we observe pronounced and concentrated vertical jet below the inversion layer. Assuming an analogy with the horizontal jet in the boundary layer, we can expect a decrease in the SPT due to convective-scale motions in this part of a developing cloud. Indeed, this result was reported in Part II of the study, dedicated to analysis of convective-scale motions in Cu. Above the inversion level, the jet begins to be smoother and SPT becomes positive, reflecting a typical situation when wind shears generate turbulence. Certainly, a more detailed analysis is required to clarify the reason of formation of the negative SPT zone. For instance, it is also possible to assume that the negative values of SPT are the result of spatial averaging, because the scales of turbulent and convective motions are not well separated and can even overlap. As a result, the averaging over a distance larger than the characteristic turbulent scale can lead to unexpected effects. In our opinion, it is less probable for this phenomenon to be related with process of self-organization in clouds.

At altitudes exceeding 1100-m SPT becomes positive and comparable in absolute value with BPT in the inversion zone at z = 1500 m. In general, Fig. 6 shows intensification of turbulence with height in Cu with the maximum within a few hundred meters below cloud top. It is possible that this increase in the SPT is related to the gradients related to the toroidal vortex at the top of the starting plume (Houze 2014) (the upper part of the cloud). This vortex creates significant velocity shears. Anyway, our results show that turbulence in Cu is caused largely by BPT.

Figure 7 presents more detailed information about the height–temporal changes of the BPT. The figure shows both the developing stage (20–33 min) and the dissipation stage (33–45 min) of the cloud. The figure shows that the buoyancy production term reaches its maximum near cloud top above the level of the maximum velocity. The buoyancy production (up to value of 0.016 m² s⁻³) is observed in both developing and the beginning of the dissipation stages in the zone of inversion layer. The maximum of the buoyancy production term at t ∼ 33 min at z ∼ 1400 m is apparently related to the local maximum of the mean buoyancy force in CUZ (Fig. 11 in Part II). At the time instance of the maximum cloud development, the BPT is negative only near cloud top at z ∼ 2200 m because ascending air volumes near cloud top become colder than surrounding at this altitude. The BPT remains positive until ∼35 min, then it rapidly decreases and becomes negative at the later stage of cloud evolution (t > 36 min). At this time, the correlation between buoyancy and fluctuations of the vertical velocity becomes negative tending to suppress turbulence. It means that mixing and entrainment in the decaying cloud make stratification within the cloud volume as stable as in the surrounding inversion zone. As a result, ascending volumes become colder, while descending volumes become warmer than their surroundings.

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Height–time field of the buoyancy production term. The magenta line shows z–t trajectory of cloud volume ascending at maximum convective velocity.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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4. Properties of turbulent motions in cloud updraft zone

a. Turbulence kinetic energy and anisotropy of turbulence

Along with the total TKE defined in Eq. (2), TKE related to individual components of turbulent velocity can be defined:

$E_u=\frac{1}{2}\langle {u^{\prime }}^2\rangle ;\hspace{1em}{E}_{\upsilon }=\frac{1}{2}\langle {{\upsilon }^{\prime }}^2\rangle ;\hspace{1em}{E}_w=\frac{1}{2}\langle {w^{\prime }}^2\rangle .$(4)

Besides energy assessments, these tree quantities also allow us to decide on how close the turbulence is to an isotropic one, and to estimate an anisotropy coefficient defined as (Verma et al. 2017; Verma 2018)

$A=\frac{\langle {u^{\prime }}^2\rangle +\langle {{\upsilon }^{\prime }}^2\rangle }{2\langle {w^{\prime }}^2\rangle }.$(5)

In case of purely isotropic turbulence this coefficient is equal to one.

Figure 8a shows that TKE in the cloud center zone linearly increases with the height up to the level of 1600 m (up to the inversion zone) and then changes insignificantly. TKEs related to the horizontal components of the turbulent velocity are significantly lower than the TKE related to the vertical component. This result is expected since, as was mentioned above, we found that buoyancy fluctuations being the main source of turbulence act in the vertical direction. Wherein the values of E_u are very close to values of E_υ, which means the isotropy of turbulence in the horizontal plane. This result is also expected since initially the model formed a simple cloud structure with no horizontal shears of the basic motion field. Contours in Fig. 8b illustrate height–temporal changes of the TKE in the processes of development and decay of the cloud. One can see that at the developing stage turbulence is strongest near cloud top, above or at the level of the bubble having the maximum updraft velocity. After the bubble passes, the turbulence remains low below 1500 m until the cloud decay. This result shows that turbulence hardly plays any important role in the process of entrainment and mixing with environment within the lower (below inversion) part of cloud.

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(a) TKE profiles in the cloud center zone at t = 33 min. (b) Height–time field of TKE. The magenta line shows z–t trajectory of the cloud volume ascending at maximum convective velocity.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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There is a strong increase of TKE at the upper part of the cloud at t < 38 min at the cloud dissolving stage. This increase is supposedly related to the suppression of convective-scale motions in a stable layer, and transition of their energy to turbulent scales which is expressed by the increase of both BPT and SPT in a dissipating Cu. Another possible mechanism of turbulence generation is formation of high-frequency gravitational waves, caused by multiple fluctuations. At the dissolving stage BPT decreases and then becomes negative. It means that the main contribution to TKE should come from the SPT. We assume that this process can be described by implementation of the concept of turbulent potential energy that can convert to TKE (e.g., Zilitinkevich et al. 2013). This concept was introduced for the explanation of turbulence generation in the stable boundary layer.

The turbulence in solitary convective cloud should be anisotropic due to the presence of a preferable vertical direction. Turbulence fluctuations elongate along z axis, so the vertical profile of the anisotropy coefficient and its height–time field shown in Fig. 9 is less than one and increases from 0.35 at cloud base up to 0.5–0.7 at the inversion zone where turbulence becomes closer to isotropic one. These values agree well with the value of A = 0.73 obtained in DNS simulations of turbulent Rayleigh–Bénard convection (Verma et al. 2017; Verma 2018). Figure 6b shows that the turbulence is the most isotropy with A ∼ 0.8 at the dissolving stage of cloud evolution.

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(a) Profile of anisotropy coefficient in the cloud center zone. In purely isotropy 3D case this coefficient is equal to one. (b) Height–time field of the anisotropy coefficient. The magenta line shows z–t trajectory of the cloud volume ascending at maximum convective velocity.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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b. Spectrum of turbulent kinetic energy

A very important characteristic of turbulence is the distribution of kinetic energy of turbulent fluctuations over wavenumbers, i.e., energy spectrum (Monin and Yaglom 1975). We estimate individual spectrum components F_u(k), F_υ(k), and F_w(k) and then summarize them. For estimation of the spectra we used periodograms calculated along both x and y bars and then averaged across the corresponding bar. The amount of the spectra to be averaged across the bars determine the volume of statistics. The angle brackets in Eqs. (6a)–(6c) indicate statistical averaging. For example, components of spectrum along x bar are calculated as

$F_u(k)=\frac{2}{\mathrm{\Delta }xN}\langle {\left|{\displaystyle \sum _{n=0}^{N-1}{u}_n^{\mathbf{\prime }}\exp (-j2\pi nk\mathrm{\Delta }x)}\right|}^2\rangle ,$(6a)

$F_{\upsilon }(k)=\frac{2}{\mathrm{\Delta }xN}\langle {\left|{\displaystyle \sum _{n=0}^{N-1}{\upsilon }_n^{\mathbf{\prime }}\exp (-j2\pi nk\mathrm{\Delta }x)}\right|}^2\rangle ,$(6b)

$F_w(k)=\frac{2}{\mathrm{\Delta }xN}\langle {\left|{\displaystyle \sum _{n=0}^{N-1}{w}_n^{\mathbf{\prime }}\exp (-j2\pi nk\mathrm{\Delta }x)}\right|}^2\rangle ,$(6c)

$F_x(k)=F_u(k)+F_{\upsilon }(k)+F_w(k),$(6d)

where k = 1/L, 2/L, …, 1/2Δx is a set of wavenumbers, L = 600 m is the length of the bar, N = 61 is the number of samples along the bar, and Δx = 10 m is model resolution along the x axis. The angle brackets in Eqs. (6a)–(6c) indicate statistical averaging.

The spectra calculated using Eqs. (6) are normalized to the sum of spectral values, so TKE is calculated as $E_x=\sum F_x(k)$. The spectra along the y bar are calculated and normalized analogously. The averaged periodogram forms a very good estimate of the spectru since it does not distort the spectrum shape, i.e., does not introduce a bias, on the one hand, and generates only small random errors, on the other hand. Examples of estimated spectra are presented in Fig. 10. Of course, the estimations of spectra correspond to the inertial turbulence range and possibly to the scales larger than the external turbulent scale. Visually many spectra obtained at different time instances at different height look quite acceptable and similar, so the presented spectrum can be considered as typical. One can see that both F_x(k) and F_y(k) TKE spectra are very close to one another. It means the presence of turbulence isotropy in the horizontal plane at all wavenumbers. The slopes of estimated spectra in log–log coordinates are close to −11/5 at large wavenumbers and decrease at smaller ones. Separating wavenumber is equal to ∼1.15 × 10⁻² m⁻¹ that corresponds to the spatial scale of about 87 m.

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Examples of estimated spectra along the x-bar F_x(k) and along the y-bar F_y(k) at t = 33 min and z = 1200 m (solid lines). Dotted lines show the corresponding spectra F_u(k) estimated along the x bar (blue curve) and the y bar (red curve). Slopes −5/3 and −11/5 are shown by black dashed line and black dotted line, respectively.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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The slopes of the spectra are related to types of turbulence. Two types of turbulence phenomenology were discussed in previous studies (Verma 2018). In Kolmogorov’s homogeneous and isotropic turbulence (Monin and Yaglom 1975) where turbulence is forced at scales larger than the external scale, the slope of spectra is equal to −5/3. In this case, the kinetic energy flux from large scales to smaller ones is a constant and equal to turbulent dissipation rate ε. In buoyancy-driven turbulence (Verma et al. 2017; Verma 2018) the buoyancy affords forcing at all scales, and the kinetic energy flux decreases with the increase of wavenumbers as k^−4/5, so the slope of the spectra is equal to −11/5. This scaling known as the Bolgiano–Obukhov scaling (Obukhov 1959; Procaccia and Zeitak 1989) can be also expected in cloud turbulence where buoyancy plays a dominating role.

Figure 10 also shows F_u(k) component of TKE spectrum, related to fluctuations of u′ velocity component. As could be expected from Fig. 8a, illustrating turbulence anisotropy, the values of total TKE spectrum exceed the values of F_u(k) by a factor of 3.

In Fig. 11 we demonstrate a height dependence and height–time contours of spectra slopes, calculated using approximation of averaged periodogram (Fig. 10) within the wavenumber range of [0.01–0.05] m⁻¹ (the range of spatial scales of [20–100] m corresponding to the inertial turbulent range). The slope profiles, calculated along x bar and y bar and shown in Fig. 11a are in a good agreement with each other, reflecting the horizontal isotropy of turbulence. The slopes slightly decrease with height from −1.7 down to −2.2, remaining basically between the values of −11/5 and −5/3. In the lower part of cloud, the slopes are closer to the −5/3 Kolmogorov scaling than to the −11/5 Bolgiano–Obukhov scaling, while above ∼800 m the −11/5 Bolgiano–Obukhov scaling is estimated. This result is in accordance with the weak buoyancy at the lower part of cloud showing in Figs. 6 and 7.

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(a) Height dependence of spectra slopes, calculated using approximation of averaged periodogram within the wavenumber range of [0.01–0.05] m⁻¹. (b) Height–time field of spectrum slope; light blue zone corresponds to the Bolgiano–Obukhov scaling. Red zones correspond to the −5/3 Kolmogorov scaling. The magenta line shows z–t trajectory of cloud volume ascending at maximum convective velocity.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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Figure 11b shows that in the process of lifting of cloud volume at maximum convective velocity (the lifting of the head bubble in the starting plume that manifests itself in the vicinity of the magenta line in Fig. 11b) the spectrum slope demonstrates mainly the −11/5 Bolgiano–Obukhov scaling for the heights larger than 800 m. This reflects decisive impact of buoyancy on turbulence formation (see also Fig. 6). One can see that at the dissolving stage of cloud development the spectrum slope becomes closer to the Kolmogorov’s −5/3 with the decrease in the BPT. The mechanisms of turbulence generation at the dissolving stage are not so clear and require further investigation.

c. Turbulence kinetic energy flux (turbulent dissipation rate)

Calculated slopes of TKE spectra allow one to evaluate TKE flux in the wavenumber space. The flux can either vary over wavenumbers or not. In the latter case which corresponds to Kolmogorov’s homogeneous and isotropic turbulence, the TKE spectrum in the inertial subrange is described by the formula

$F(k)=C{\epsilon }^{2/3}{k}^{-5/3},$(7)

where C ≈ 1.5 is Kolmogorov’s constant, and ε is turbulent dissipation rate equal to the constant kinetic energy flux. Using Eq. (7) the quantity

$\langle \epsilon \rangle =\frac{1}{k_2-k_1}{\displaystyle {\int }_{k_1}^{k_2}{C}^{-3/2}{k}^{5/2}{F}^{3/2}(k)dk}$(8)

can be interpreted as the mean value of the kinetic energy flux within the considered wavenumber subrange, [k₁ ÷ k₂] or as the mean value of turbulent dissipation rate for an arbitrary shape of spectrum F(k) if the subrange falls inside turbulent inertial range. Formula (8) and its interpretation is used in our analysis. The height profiles of mean turbulent dissipation rates calculated along the x bar 〈ε〉_x and along the y bar 〈ε〉_y using mean periodograms in [0.01–0.05] m⁻¹ (wavenumber subrange), as well as the mean values of kinetic energy fluxes 〈ε₂〉_x and 〈ε₂〉_y calculated using total wavenumber range, are shown in Fig. 12a.

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(a) The height profiles of the mean dissipation rate, calculated along the x-bar 〈ε〉_x and along the y-bar 〈ε〉_y in the [0.01–0.05] m⁻¹ wavenumber subrange (solid lines) and of the mean kinetic energy flux calculated along the x-bar 〈ε₂〉_x and along the y-bar 〈ε₂〉_y calculated using the total wavenumber range (dotted lines). (b) Height–time field of the mean turbulent dissipation rate. The magenta line shows z–t trajectory of the cloud volume ascending at the maximum convective velocity.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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One can see that the profiles of turbulent dissipation rate 〈ε〉_x and 〈ε〉_y are very close to the corresponding profiles of mean kinetic energy flux 〈ε₂〉_x and 〈ε₂〉_y. The profiles calculated along x bar and y bar are also close to one another, which again, shows the horizontal isotropy of turbulence. The values of 〈ε(z)〉 monotonically increase with height up to the value of 17 cm² s⁻³ at z = 1800 m and then remain unchanged. These not large values agree well with the values evaluated from in situ measurements in small trade wind cumuli performed near cloud tops of these clouds (Gerber et al. 2008; Katzwinkel et al. 2014). However, none of the measurements provide a vertical dependence of 〈ε(z)〉. Contours in Fig. 12b illustrate a strong increase of the turbulent dissipation rate at the cloud dissolving stage at the upper part of the cloud. At the lower part of the cloud, the turbulent dissipation rate changes only slightly until 38 min. This behavior are similar to behavior of TKE shown in Fig. 8b.

d. Turbulent diffusion coefficient

The turbulent diffusion coefficient is an important characteristic of cloud turbulence. It is used to evaluate the effect of turbulent mixing between a cloud and its environment, as well as mixing inside a cloud. Turbulent mixing between convective cloud and the nearest environment in the horizontal direction is of the greatest interest because it is assumed to influence the cloud life cycle (Houze 2014). It may affect the broadening or narrowing of a cloud (Pinsky and Khain 2018, 2019). Turbulent mixing in vertical direction can be one of the main factors influencing the thermodynamic and microphysics near upper Cu boundary.

Following Benmoshe et al. (2012), we estimate the turbulence diffusion coefficient from two equations formulated by Zilitinkevich (1970):

$K=C_{k}l{E}_{\text{turb}}^{1/2};\hspace{1em}\langle \epsilon \rangle =\frac{C{E}_{\text{turb}}^{3/2}}{l},$(9)

where E_turb is a fraction of TKE related to horizontal and vertical turbulent velocity fluctuations within the inertial subrange (in our case, the minimal wavenumber in the inertial subrange is equal to ∼1.15 × 10⁻² m⁻¹), l is a mixing length, and 〈ε〉 is the mean value of turbulent dissipation rate estimated in the previous section. Coefficients C_k and C are equal to 0.2 and 0.93, respectively (Benmoshe et al. 2012). The equations for turbulence diffusion coefficient obtained from Eq. (9) is

$K=0.186\frac{E_{\text{turb}}^2}{\langle \epsilon \rangle }.$(10)

Figure 13a demonstrates height profiles of turbulent diffusion coefficients K_x(z) and K_y(z) calculated using Eq. (10) along x bar and y bar, respectively. The coefficient monotonically increases with height up to value of 24 m² s⁻¹ at the level of 1600 m (middle of the inversion layer) and then does not change significantly. Isotropy in horizontal plane clearly manifests itself in these profiles. The values of turbulent diffusion coefficient are close to one, measured in nonlarge clouds at the dissipation stage of their development (Strunin and Strunin 2018). Contours in Fig. 13b illustrate a strong increase of turbulent diffusion coefficient at the cloud dissolving stage at the upper part of the cloud. At the lower part of cloud, K(t) changes only slightly until 38 min. These behaviors are similar to behaviors of TKE and 〈ε〉 shown in Fig. 8b and 12b, respectively. The reasons for the enhanced turbulent coefficient within the inversion layer in the presence of buoyancy-caused fluctuations are similar to those of high TKE and were discussed above.

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(a) The height profiles of turbulent diffusion coefficient, calculated along the x-bar K_x(z) and along the y-bar K_y(z). (b) Height–time field of turbulent diffusion coefficient. The magenta line shows z–t trajectory of the cloud volume ascending at the maximum convective velocity.

Citation: Journal of the Atmospheric Sciences 80, 2; 10.1175/JAS-D-21-0223.1

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5. Conclusions

In Part I of the study we showed that it is possible to separate in-cloud air motions into convective motions responsible for directed mass transport, and turbulent motions whose effects are associated with diffusion processes. We supposed that convective motions with scales on the order of a cloud size form a cloud “skeleton,” while turbulent motions are responsible for processes of mixing. Accordingly, complicated cloud processes can be represented by a combined effect of motions of these two types. To explore this idea, a nonprecipitating trade wind Cu with 2400-m cloud top observed in BOMEX was simulated using 10-m-grid-spacing LES (SAM with spectral bin microphysics). The velocity fields and microphysical fields were separated into convective and turbulent components using a wavelet filtering. In Part II, the properties of convective motions of this Cu were investigated and the ability of low parametric parcel models to reproduce realistic cloud structure was evaluated.

It is widely believed that turbulence plays an important role in cloud–environment interaction and in formation of microphysical structure of clouds and precipitation formation. At the same time, only a few studies investigated turbulence in clouds due to reasons mentioned in the introduction. As a result, we have few sporadic observed data related to cloud turbulence. As regards to numerical simulations, the typically applied parameterization of cloud turbulence is based on the subgrid processes, and the resulting turbulent parameters such as turbulent coefficient and TKE turn out to be dependent on the finite grid resolution. This is Part III of the study in which we present, for the first time, a systematic investigation of cloud turbulence in the zone of cloud updrafts. Using results of 10 m resolution simulation of a cumulus clouds under BOMEX conditions, we calculated height and time dependences of the main turbulent parameters such as TKE, spectra of turbulence, dissipation rate, and the turbulent coefficients at different stages of cloud development.

The specific features that distinguish this study are 1) direct estimation of turbulence parameters for single isolated convective cloud from the resolved velocity field using results of 10-m-resolution LES model; 2) separating the resolved flow into turbulence and nonturbulence flows by means of a novel method allowing to carry out estimation of different turbulent parameters with sufficient statistical accuracy. We would like to stress that the goal of this study is to get the first general insight into the turbulent structure of an isolated Cu. The turbulent characteristics such as spectra were obtained using points chosen in the horizontal cross sections of cloud updraft zone (CUZ). Therefore, the turbulent parameters calculated in the study can be considered as horizontally averaged parameters of CUZ.

The main results and conclusions obtained in the study can be summarized as the following.

• The main source of turbulence in clouds is the buoyancy production term (BPT). Like BPT the shear production term (SPT) increases with height and reaches its maximum values on the order of 10⁻² m² s⁻³ near cloud top. The possible reason of the SPT maximum near cloud top is the existence of the Hill’s vortex which likely leads to increase in the vertical shears of the horizontal velocity. In agreement with the behavior of BPT and SPT, turbulence in the lower cloud part (below the inversion level) is weak and hardly affects the process of mixing and entrainment.

• The fact that BPT is larger than SPT determines many properties of cloud turbulence. For instance, the turbulence is nonisotropic, so that vertical component of TKE is substantially larger than its horizontal components. The second consequence of BPT being larger than STP is the result that the turbulence spectrum obeys largely the −11/5 Bolgiano–Obukhov scaling. The classical Kolmogorov −5/3 scaling dominates in the low part of a cloud and largely at the dissolving stage of cloud evolution. The deviation of the scaling from the −5/3 law is related to the fact that buoyancy contributes to the generation of turbulence at all scales, including scales of the inertial subrange, where the Kolmogorov’s −5/3 law does not suggest any influx of the turbulent energy.

• The analysis of the TKE spectra F(k) clearly shows two wavenumber ranges with different slopes. The range of larger k (that can be associated with the turbulent inertial subrange) is docked to the range with much lower slope at smaller wavelengths. The transition is around the scale of 90 m, that conditionally can be associated to the external turbulent scale. Although we do not exclude the possibility of the existence of turbulent vortices larger than 87 m, we have not, however, found in Cu the sharp scale of separation between convective and turbulent motions.

• In case of the −11/5 Bolgiano–Obukhov scaling, it is difficult to directly apply the concept of the dissipation rate which, according to the Kolmogorov law, should be constant within the whole inertial range. However, using the spectra obtained we evaluated some “effective” turbulent dissipation rate which makes the same sense as the Kolmogorov’s rate and coincides with the latter in case the slope of the spectrum is equal to −5/3. The “effective” dissipation rate increases with height from nearly zero at cloud base up to 17 cm² s⁻³ near cloud top. These values are consistent with the dissipation rates evaluated in small Cu from the high-frequency measurements which vary from 5 to 50 cm² s⁻³ (Gerber et al. 2008; Katzwinkel et al. 2014; Schmeissner et al. 2015). The calculated values seem to be reasonable. We should remind that all turbulent characteristics, estimated in the study, are averaged over a significant area of zone of cloud updrafts, so we do not consider possible horizontal variability of these characteristics. The fact that the results are consistent with high-frequency measurements speaks in favor of the fact that the parameters of wavelet filtration are chosen correctly, i.e., that they are close to the optimum ones. The size of the external turbulent scale of about 87 m, obtained in the present study, agrees well with existing concepts that the size of the maximum turbulent eddies is an order of magnitude smaller than the size of the cloud (Benmoshe et al. 2012, and references in that study).

• Finally, the coefficient of turbulent diffusion was found to increase with height and ranged from 5 m² s⁻¹ near cloud base to 24 m² s⁻¹ near cloud top. These values are typically assumed for the nonprecipitating BL Cu (Pinsky and Khain 2018, 2019; Khain and Pinsky 2018). It is interesting that the values of TKE, dissipation rate and the coefficient of turbulent diffusion reach their maxima at the dissolving stage when cloud top descends down to the inversion layer. An increase in the turbulence intensity at the dissipation stage indicates that a significant part of convective kinetic energy converts to TKE at this stage. According to the results of the study, BPT is the dominating source of cloud turbulence at the developing stage. STP supposedly dominates at the decaying stage.

We are aware of the fact that this study presents only a first general picture of the turbulent structure of a single cloud at different stages of its evolution. We do not present spatial distribution of the turbulent parameters in the horizontal direction. To perform such analysis, larger statistics (e.g., higher-frequency output data) is required. An interesting question is why TKE turned out to be maximum within the inversion layer at the decaying stage, while BPT term becomes negative. We believe that this can be a result of the SPT effect which should be increased during the formation of gravity waves. We intend to analyze the issue in future investigations.

We conclude Part III with a comment concerning a possible role of turbulence in the process of entrainment and mixing of a cloud with its surrounding. According to survey of De Rooy et al. (2013), the lateral entrainment takes place due to turbulence. Due to turbulent diffusion, the dry air from surrounding can penetrate the cloud at a distance $L=\sqrt{Kt}$. Assuming K = 10 m² s⁻¹ and 200 s is the time of intense cloud growth (see, e.g., Figs. 8b, 11b, 12b), we obtain the estimation of L ∼ 40 ÷ 50 m. This is the characteristic width of the interface zone near cloud edge with high gradients of all microphysical values (Gerber et al. 2008). So turbulence can be responsible for the formation of the narrow interface zone as well as for certain smoothing of cloud variables inside clouds also at scales of a few tens of meters. These evaluations show that the microphysical structure inside a cloud is mainly determined by convective motions. This topic is partially discussed in Part II, but requires more detail analysis.