Convective and Turbulent Motions in Nonprecipitating Cu. Part I: Method of Separation of Convective and Turbulent Motions

1. Introduction

Shallow warm cumulus clouds (Cu) play an important role in the atmospheric radiation and moisture budgets (Norris 1998; Trenberth 2011; Stephens 2005; Zhu and Bretherton 2004; Von Salzen et al. 2005). These clouds are frequent over both oceans and continents and are responsible for the largest cloud feedback uncertainties in climate models (Neggers et al. 2007; Bony and Dufresne 2005; Vial et al. 2017). Small clouds change temperature gradients and humidity in the lower atmosphere, affecting, therefore, CAPE and intensity of deep convection and precipitation. Hence, an accurate calculation of radiative and microphysical properties of such clouds is of crucial importance for weather prediction and simulation of global circulation and climate, including climate changes.

The impacts of convective and turbulent motions on cloud microphysics and dynamics are different. Convective motions transport air mass: for instance, cloud updrafts, subsiding shells, compensating air settling can be attributed to convective motions transporting air and microphysical variables over distances comparable with the cloud size (Heus et al. 2009). Convective transport is described by “advective” terms in motion equations. Convective motions form the coherent structures, which sizes exceed the size of parent clouds.

In contrast, turbulent motions do not transport air mass to large distance, and their integral effect can be often described by the equation of turbulent diffusion, in which the laminar diffusion coefficients are replaced by their turbulent analogs. The main role of turbulence is to perform mixing inside clouds, as well as mixing between clouds and the surrounding air. We can crudely describe the physical role of convective motions as creating the adiabatic “skeleton” of cloud, while the role of turbulence is to smooth the cloud gradients and mix the cloud with the environment air, which causes the cloud processes to deviate from adiabaticity. These deviations naturally affect the microphysical cloud structure, droplet size distributions, etc. Turbulence-induced cloud dilution and droplet evaporation affect cloud geometry, cloud size, cloud life time, and other parameters.

In multiple convective parameterizations based on the flux method, the cloud is represented as an ascending bubble or plume, in which mass flux increases with height due to lateral entrainment related to the changes of the vertical velocity flux through the cross section of the cloud (Arakawa and Schubert 1974; Frank 1983; Tiedtke et al. 1988; Tiedtke 1989; Kain and Fritsch1990, 1993; Kain 2004; Arakawa 2004; Plant and Yano 2015). The same entrainment rate is used when effects of entrainment on thermodynamical and microphysical quantities are considered. According to the definition presented above, such entrainment should be of convective nature.

At the same time according to de Rooy et al. (2013), entrainment and mixing at cloud edges are of turbulent nature, i.e., turbulent entrainment. If there exists a directed (convective) entrainment at the cloud edges, both the convective and turbulent motions determine the internal cloud structure and other cloud properties. In particular, the diluted zones near cloud edges or slightly diluted cloud cores (Katzwinkel et al. 2014; Schmeissner et al. 2015) may form either due to the effect of turbulent diffusion [as it was assumed by Pinsky and Khain (2018, 2019, 2020a,b)], or/and due to directed mass entrainment of environment air into clouds.

It is clear that microphysical and dynamical cloud structures are closely related. We are going to investigate convective and turbulent motions and their role in formation of thermodynamic and microphysical structures of nonprecipitating Cu. We will use LES results of Cu simulation by a 10-m resolution 3D System for Atmospheric Modeling (SAM) with bin microphysics. We believe that the 10-m resolution of a model that produces realistic cloud structure allows investigating the mechanisms of cloud microphysics formation under the effects of both convective and turbulent motions. The 10-m grid spacing is much smaller than the external turbulent scale which in turn by order of magnitude smaller than the characteristic cloud size (Grabowski and Clark 1993).

Specifically, the general questions addressed in the study (consisting of several parts) are as follows:

• What are the properties of convective motions?

• To what extent the parcel models reproduce the structure of cloud “skeleton”?

• What are the properties of cloud turbulence? In which parts of cloud the turbulence can be considered locally homogeneous and isotropic?

• What are comparative contributions of convective and turbulent motions to mixing/entrainment process?

Also, some questions related to turbulent influence on microphysical processes will be addressed:

• What are the roles of convective and turbulent motions in formation of droplet size distributions (DSDs) and their properties?

• What is the role of small-scale turbulence on enhanced inertial coalescence and rain initialization in the convective clouds of different strengths?

To answer these questions, a physically grounded methodology should be developed to separate between convective and turbulent motions in clouds and their surroundings. Although convective and turbulent motions basically manifest themselves at different spatial scales, separation of motions into convective and turbulent ones is a complicated problem. First, convective motions can create strong gradients, i.e., can also contribute substantially to motions of small scales. Second, turbulent-like vortices with spatial scales of 100–200 m, which are comparable with cloud core width, may well exist, although some authors suggest the existence of natural separation between convective and turbulent motions due to self-organization process (Elperin et al. 2002; Boi et al. 2016). In Part I of the study we describe the method for separation between convection and turbulence motions, based on the wavelet decomposition technique. In the following parts this technique will be applied to answer the questions listed above.

2. The main principles and elements of the separation technique

The technics designed in this study is based on high-resolution simulation by SAM and wavelet decomposition idealized model of the vertical velocity field. SAM is a 3D cloud-resolving model that evolved from an LES model first developed by Khairoutdinov and Kogan (1999, 2000) and Khairoutdinov and Randall (2003). It is designed on the original Cartesian grid with a constant horizontal spacing and a variable vertical spacing. SAM is specially designed to simulate clouds at a very high resolution, which allows reproducing nearly the entire range of turbulent motions. The specific feature of the particular SAM version is utilization of spectral bin microphysics described in detail by Khain et al. (2004) and later modified by Khain et al. (2008). A nonprecipitating Cu under the thermodynamical conditions of BOMEX field experiment (Heus and Jonker 2008) was simulated. The cloud updraft width at the mature stage is around 400 m. The zone of downdrafts (subsiding shell) is of ~150-m width in agreement with the observations. Cu with similar geometrical parameters were also observed during Rain in Cumulus over the Ocean (RICO) (Gerber et al. 2008). All the figures related to the cloud are based on the data taken from the same simulation with SAM and belong to the mature stage of cloud evolution (t = 33 min) when cloud reaches its maximum height and intensity. For analysis we use 3D fields of air velocity, as well as thermodynamic and microphysical quantities outputting with increment 0.5 min. The fields were calculated within the volume of 512 × 512 × 320 grid points with 10-m grid spacing. Below we discuss separation of the vertical velocity component into convective and turbulent components. The two horizontal velocity components can be treated analogously. We represent the modeled field of the vertical velocity component as a set of 2D horizontal 512 × 512 slices w_i(x, y) with height increment of 10 m. An example of such a slice is shown in Fig. 1a. The 1D horizontal profile of vertical velocity along the dashed line passing through a velocity maximum parallel to the x axis is shown in Fig. 1b. One can see the region of air updraft and the subsiding shell surrounding it, embedded into the turbulent velocity field. The characteristic cloud diameter (determined by threshold LWC = 0.01 g kg⁻¹) is of about 800 m. This size is somewhat larger than the typical sizes of trade wind Cu. For instance, Heus and Jonker (2008) and Heus et al. (2009) reported the mean size of Cu in RICO of 500 m. At the same time, the size of simulated cloud as well as the maximum vertical velocity of ~6 m s⁻¹ are within the range of these parameters in small Cu (Gerber 2000; Gerber et al. 2001; Katzwinkel et al. 2014; Schmeissner et al. 2015). It is of interest that downdrafts exist inside the cloud, where downdraft speed reaches 2 m s⁻¹. All these values are typical of small nonprecipitating Cu (Heus and Jonker 2008; Gerber et al. 2008; Katzwinkel et al. 2014; Schmeissner et al. 2015). The widths of the updraft region and the subsiding shell are of ~350 m and 100–200 m, respectively. The subsiding shell has width of about half of the width of the updraft zone, in agreement with measurements made by Rodts et al. (2003).

(top) The SAM horizontal field of the vertical velocity and (bottom) its 1D cross section along the dashed line in Cu at t = 33 min and z = 1500 m. The field refers to the modeling of warm Cu in the Barbados Oceanographic and Meteorological Experiment (BOMEX; 1974), simulated by SAM with bin microphysics and 10-m spatial resolution. The magenta line in the top panel marks cloud boundary at LWC = 0.01 g kg⁻¹. Color bar indicates velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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(top) The SAM horizontal field of the vertical velocity and (bottom) its 1D cross section along the dashed line in Cu at t = 33 min and z = 1500 m. The field refers to the modeling of warm Cu in the Barbados Oceanographic and Meteorological Experiment (BOMEX; 1974), simulated by SAM with bin microphysics and 10-m spatial resolution. The magenta line in the top panel marks cloud boundary at LWC = 0.01 g kg⁻¹. Color bar indicates velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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(top) The SAM horizontal field of the vertical velocity and (bottom) its 1D cross section along the dashed line in Cu at t = 33 min and z = 1500 m. The field refers to the modeling of warm Cu in the Barbados Oceanographic and Meteorological Experiment (BOMEX; 1974), simulated by SAM with bin microphysics and 10-m spatial resolution. The magenta line in the top panel marks cloud boundary at LWC = 0.01 g kg⁻¹. Color bar indicates velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Note that Cu clouds in BOMEX have been simulated using SAM with the bin microphysics in several studies (e.g., Dagan et al. 2016, 2017; Heiblum et al. 2016; Khain et al. 2019). The specific feature of the present study is the utilization of 10-m resolution. The main effect of the higher resolution is detailed description of sharp gradients of microphysical values, especially at cloud edges allowing one to study variability of droplet size distributions. However, the main microphysical properties that could be compared with observations remained similar. For instance, vertical profile of effective radius is in a good agreement with that measured in RICO (Arabas et al. 2009; flight RF07 during which droplet concentration was the same as in the simulation), the ratio of effective radius and mean volume in simulations was ~1.1 and effective radius showed high horizontal uniformity in agreement with observations (see Khain et al. 2019; Eytan et al. 2019). Properties of droplet size distributions (modal radius, width, skewness, etc.) and their dependencies on the adiabatic fraction agree well with high-frequency measurements by Konwar et al. (2020, manuscript submitted to J. Atmos. Sci.) in a growing nonprecipitating Cu clouds. In this study we the focus on the development of the algorithm of separation between convective and turbulent motions.

Figure 1 shows the existence of significant velocity fluctuations at different scales. We consider the pair of updraft–downdraft flow as the branches of “convective cell” coexisting with significantly fluctuating vertical velocity. These turbulent fluctuations make the structure of the clouds very complex, and the boundaries of the clouds very rugged.

The retrieval of convective (systematic) field from the total motion filed cannot be performed by averaging the results of simulations using slightly perturbed initial conditions. The matter is that perturbations of the initial conditions lead to different convective fields, i.e., to different clouds at different stages of their evolution, due to nonlinearity of cloud processes.

3. An idealized model of the vertical velocity field

In the model, the horizontal slices of vertical velocity component w(x, y), given at 128 × 128 grid points with 10-m grid spacing, is represented as a superimposition of the convective field W(x, y) (the updraft zone in the central area of the idealized “cloud” and air subsidence around it) and a random turbulent field w′(x, y). The values characterizing the velocity fields are chosen to be typical of Cu clouds. We construct the model velocity field according to Eq. (2). The goal is to separate the sum w(x, y) into convective and turbulent components which will be as close as possible to W(x, y) and w′(x, y), respectively.

a. Convective velocity field

The convective velocity is represented in the test by a field axisymmetric with respect to the center point [x₀, y₀], so W(x, y) = W(r), where $r=\sqrt{{\left(x-x_0\right)}^2+{\left(y-y_0\right)}^2}$. The positive velocity corresponds to the cloud core, while the negative velocity corresponds to the subsiding shell. We chose two types of convective velocity radial profiles to be tested in our analysis. The first profile has a very sharp jump (resembling a step) at the cloud edge. This profile is described by formula

$W\left(r\right)=\{\begin{array}{ccc}{w}_1& \text{if}& r\le L_1\\ w_2& \text{if}& L_1<r\le L_1+L_2\\ 0& \text{if}& r>L_1+L_2\end{array},$(4)

where w₁ > 0 and w₂ < 0. 2L₁ and L₂ are the widths of the cloud core and the subsiding shell, respectively. Heus and Jonker (2008) used a similar model in analysis of a subsiding shell surrounding Cu.

The second profile is known as “the Mexican hat” and is described by equation

$W\left(r\right)=w_1\left[1-{\left(r/L_1\right)}^2\right]\hspace{0.17em}\exp \left[-{\left(r/L_1\right)}^2/2\right],$(5)

where w₁ > 0, and 2L₁ is the width of the cloud core. This profile does not have such a sharp boundary between the cloud core and the subsiding shell as the jump profile given by Eq. (4). Examples of 2D images of the convective velocity fields and the corresponding profiles are shown in Fig. 2.

Examples of 2D images (128 × 128 grid points) of the convective velocity synthetic fields, and 1D cross sections through the centers of the areas. (a) The jump-like profile equation, Eq. (4), with parameter values w₁ = 5 m s⁻¹, w₂ = −0.5 m s⁻¹, L₁ = 125 m, and L₂ = 1250 m. (b) The “Mexican hat” profile, Eq. (5), with parameter values w₁ = 5 m s⁻¹ and L₁ = 125 m. The color bars indicate velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Examples of 2D images (128 × 128 grid points) of the convective velocity synthetic fields, and 1D cross sections through the centers of the areas. (a) The jump-like profile equation, Eq. (4), with parameter values w₁ = 5 m s⁻¹, w₂ = −0.5 m s⁻¹, L₁ = 125 m, and L₂ = 1250 m. (b) The “Mexican hat” profile, Eq. (5), with parameter values w₁ = 5 m s⁻¹ and L₁ = 125 m. The color bars indicate velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Examples of 2D images (128 × 128 grid points) of the convective velocity synthetic fields, and 1D cross sections through the centers of the areas. (a) The jump-like profile equation, Eq. (4), with parameter values w₁ = 5 m s⁻¹, w₂ = −0.5 m s⁻¹, L₁ = 125 m, and L₂ = 1250 m. (b) The “Mexican hat” profile, Eq. (5), with parameter values w₁ = 5 m s⁻¹ and L₁ = 125 m. The color bars indicate velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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b. Turbulent velocity field

A homogeneous and isotropic random turbulent velocity field is generated using code created by Constantine (2020). This code generates different realizations of 2D normally distributed random field with a given spatial correlation matrix. To design this matrix, we used Kolmogorov’s transversal structure function defined as $D_{w^{\prime }}\left(r\right)=D_{w^{\prime }}\left(\left|\overline{\mathbf{r}}\right|\right)=\langle {\left[w^{\prime }\left(\overline{\mathbf{r}}+{\overline{\mathbf{r}}}^{\prime }\right)-w^{\prime }\left({\overline{\mathbf{r}}}^{\prime }\right)\right]}^2\rangle$, where the angle brackets mean averaging, and $\overline{\mathbf{r}}$ is the vector on the x–y plane. For homogeneous isotropic turbulence, this function within the inertial turbulent subrange is presented as (Monin and Yaglom 1975)

$D_{w^{\prime }}\left(r\right)={\displaystyle \frac{4}{3}}\hspace{0.17em}C{\left(\epsilon r\right)}^{2/3},$(6)

where ε is given turbulent dissipation rate and coefficient C = 2. The transversal correlation function $B_{w^{\prime }}\left(r\right)$ relates with structure function $D_{w^{\prime }}\left(r\right)$ by the following equation (Monin and Yaglom 1975):

$B_{w^{\prime }}\left(r\right)=\langle w{\text{'}}^2\rangle -{\displaystyle \frac{1}{2}}\hspace{0.17em}{D}_{w^{\prime }}\left(r\right),$(7)

where 1/2⟨w′²⟩ is the turbulent kinetic energy of unit mass (TKE) related to the vertical velocity fluctuations. We also supposed that the correlation between velocities for sufficiently distant points is equal to zero. Using this condition and Eqs. (6) and (7) one obtains

$B_{w^{\prime }}\left(r\right)=\{\begin{array}{lll}\langle w^{\prime 2}\rangle -{\displaystyle \frac{4}{3}}{\left(\epsilon r\right)}^{2/3}\hfill & \text{if}\hfill & r\le r_0\hfill \\ 0\hfill & \text{if}\hfill & r>r_0\hfill \end{array},$(8)

where r₀ is set to allow the correlation function to be nonnegative. In this case, the scale r₀ (having the meaning of the external turbulent scale) can be calculated as

$r_0={\epsilon }^{-1}{\left({\displaystyle \frac{3}{4}}\hspace{0.17em}\langle w^{\prime 2}\rangle \right)}^{3/2}.$(9)

This form of the correlation function corresponds to Kolmogorov’s −5/3 slope of energy spectrum.

Equation (8) is used to calculate the correlation matrix between each pair of vertical velocities on the 2D grid mesh. This matrix containing 128⁴ correlations is the input information for the code. Using this matrix and a random number generator, different random realizations of normally distributed turbulent velocity field are generated to get statistically representative results. An example of realization of the turbulent velocity field is shown in Fig. 3. The colors in the figure indicate the magnitude and the direction of the vertical velocity (in m s⁻¹). The field looks stochastic and isotropic, and its characteristic spatial scale (the radius of correlation) is about 65 m.

A realization of the turbulent velocity synthetic field on 128 × 128 grid point mesh. The colors indicate the magnitude and the direction of the vertical velocity (in m s⁻¹). ε = 0.01 m² s⁻³, ⟨w′²⟩ = 1 m² s⁻², and r₀ ≈ 65 m.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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A realization of the turbulent velocity synthetic field on 128 × 128 grid point mesh. The colors indicate the magnitude and the direction of the vertical velocity (in m s⁻¹). ε = 0.01 m² s⁻³, ⟨w′²⟩ = 1 m² s⁻², and r₀ ≈ 65 m.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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A realization of the turbulent velocity synthetic field on 128 × 128 grid point mesh. The colors indicate the magnitude and the direction of the vertical velocity (in m s⁻¹). ε = 0.01 m² s⁻³, ⟨w′²⟩ = 1 m² s⁻², and r₀ ≈ 65 m.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Figure 4 shows estimations of the structure functions (left) and the energy spectra (periodograms) (right) of the turbulent field that is shown in Fig. 3, calculated along two orthogonal axes. The figures demonstrate an excellent coincidence between the statistically estimated functions characterizing the generated fields, and the theoretical functions. The coincidence of the blue and red curves demonstrates the isotropy of the generated turbulent velocity fields. Existence of the plateaus in the figures means cessation of turbulent kinetic energy increase as the spatial scales increase (with a decrease of wavenumbers). To calculate the structure functions and the spectra, we averaged 41 samples taken along lines parallel to x and y axes.

(a) The structure function estimations and (b) the energy spectra estimations (periodograms) of the vertical velocity field that is shown in Fig. 3. The blue and red curves correspond to the calculations along x axis and y axis, respectively. The dashed lines denote theoretical dependences. The proximity of the blue and red curves demonstrates the isotropy of the generated synthetic fields.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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(a) The structure function estimations and (b) the energy spectra estimations (periodograms) of the vertical velocity field that is shown in Fig. 3. The blue and red curves correspond to the calculations along x axis and y axis, respectively. The dashed lines denote theoretical dependences. The proximity of the blue and red curves demonstrates the isotropy of the generated synthetic fields.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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(a) The structure function estimations and (b) the energy spectra estimations (periodograms) of the vertical velocity field that is shown in Fig. 3. The blue and red curves correspond to the calculations along x axis and y axis, respectively. The dashed lines denote theoretical dependences. The proximity of the blue and red curves demonstrates the isotropy of the generated synthetic fields.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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c. Total velocity fields

The realizations of the total velocity fields calculated using Eq. (2) are shown in Fig. 5 for the convection models, described by the jump profile (Fig. 5a) and for the convection model described by the “Mexican hat” profile (Fig. 5b). The corresponding estimated energy spectra (periodograms) and estimated structure functions are shown in Figs. 6 and 7, respectively. Surprisingly, the energy spectra in both cases follow the −5/3 Kolmogorov’s law within the entire range of wavenumbers from maximum wavenumbers down to minimum wavenumbers. It makes it impossible to separate the convective and turbulent parts of the velocity using energy spectrum as a criterion. At the same time, the absence of plateaus in the structure functions and comparison of these functions with those shown in Fig. 4a show the possibility of applying the functions as a criterion for separation between convective and turbulent velocity components.

The realizations of the total velocity synthetic fields calculated using Eq. (2) on a 128 × 128 grid point mesh for (a) the jump-like profile convection model, (4), and (b) the “Mexican hat” convection model, (5). The fields are obtained by summing the fields shown in Fig. 2 with the field shown in Fig. 3. The color bars indicate velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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The realizations of the total velocity synthetic fields calculated using Eq. (2) on a 128 × 128 grid point mesh for (a) the jump-like profile convection model, (4), and (b) the “Mexican hat” convection model, (5). The fields are obtained by summing the fields shown in Fig. 2 with the field shown in Fig. 3. The color bars indicate velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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The realizations of the total velocity synthetic fields calculated using Eq. (2) on a 128 × 128 grid point mesh for (a) the jump-like profile convection model, (4), and (b) the “Mexican hat” convection model, (5). The fields are obtained by summing the fields shown in Fig. 2 with the field shown in Fig. 3. The color bars indicate velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Estimated energy spectra (periodograms) for synthetic velocity field realizations shown in Fig. 5 for (a) the jump-like convection model, (4), and (b) for the “Mexican hat” convection model, (5). To calculate each spectrum, 41 samples taken along the lines parallel to x and y axes were averaged.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Estimated energy spectra (periodograms) for synthetic velocity field realizations shown in Fig. 5 for (a) the jump-like convection model, (4), and (b) for the “Mexican hat” convection model, (5). To calculate each spectrum, 41 samples taken along the lines parallel to x and y axes were averaged.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Estimated energy spectra (periodograms) for synthetic velocity field realizations shown in Fig. 5 for (a) the jump-like convection model, (4), and (b) for the “Mexican hat” convection model, (5). To calculate each spectrum, 41 samples taken along the lines parallel to x and y axes were averaged.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Estimated structure functions for velocity field realizations shown in Fig. 5 for (a) the jump-like convection model, (4), and (b) the “Mexican hat” convection model, (5). To calculate each structure function, 41 samples taken along the lines parallel to x and y axes were averaged.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Estimated structure functions for velocity field realizations shown in Fig. 5 for (a) the jump-like convection model, (4), and (b) the “Mexican hat” convection model, (5). To calculate each structure function, 41 samples taken along the lines parallel to x and y axes were averaged.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Estimated structure functions for velocity field realizations shown in Fig. 5 for (a) the jump-like convection model, (4), and (b) the “Mexican hat” convection model, (5). To calculate each structure function, 41 samples taken along the lines parallel to x and y axes were averaged.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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4. Choosing the optimal wavelet and tuning its parameters

For tuning we use 2D vertical velocity field realizations described in section 3 (see Fig. 5). Since we believe that the convective velocity model “Mexican hat” (5) is more realistic than that described by Eq. (4), we mainly used “Mexican hat” model (5), while (4) was used as an additional test. Totally we generated 100 random realizations of the vertical velocity.

First, we chose the type of wavelet and its order. Figures 8a and 8b show the dependences of the mean square difference between the convective velocities and their estimations δ and the correlation coefficient between turbulent velocities and their estimations R on the wavelet order for three wavelet types (called “symN,” “dbN” and “coif N”; see Misiti et al. 1997). One can see that according to both criteria (10) and (11), the wavelet “sym5” demonstrates the best results. In addition, this wavelet possesses several desirable properties (orthogonality, fast wavelet transforms, least asymmetry, four vanishing moments). However, in our case the difference between the optimal and not-optimal wavelets is not so large.

Dependences of (a) the mean square difference between the convective velocities and their estimations δ and (b) the correlation coefficient between turbulent velocities and their estimations R on the wavelet order for different wavelet types; dependences of (c) δ and (d) R on the number of steps used in the filtering procedure for wavelet “sym5”; dependences of (e) δ and (f) R on parameter α that controls thresholds of wavelet coefficients for wavelet “sym5.”

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Dependences of (a) the mean square difference between the convective velocities and their estimations δ and (b) the correlation coefficient between turbulent velocities and their estimations R on the wavelet order for different wavelet types; dependences of (c) δ and (d) R on the number of steps used in the filtering procedure for wavelet “sym5”; dependences of (e) δ and (f) R on parameter α that controls thresholds of wavelet coefficients for wavelet “sym5.”

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Dependences of (a) the mean square difference between the convective velocities and their estimations δ and (b) the correlation coefficient between turbulent velocities and their estimations R on the wavelet order for different wavelet types; dependences of (c) δ and (d) R on the number of steps used in the filtering procedure for wavelet “sym5”; dependences of (e) δ and (f) R on parameter α that controls thresholds of wavelet coefficients for wavelet “sym5.”

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Second, we chose the number of steps used in the filtering procedure for wavelet “sym5.” Figures 8c and 8d show the dependences of δ and R on the number of steps used in the filtering procedure. One can see that according to both criteria (10) and (11), the five-step procedure demonstrates the best results.

The third optimization of wavelet use is choosing of optimal thresholds for discarding the terms containing the noise component. Although mostly high-frequency terms are discarded, some low-frequency insignificant terms are also discarded (see the appendix). A parameter denoted as “α” controls thresholds which are the same for all steps of wavelet filtering procedure (Misiti et al. 1997). The dependences of δ and R on parameter α for five-step “sym5”-wavelet procedure are shown in Figs. 8e and 8f. The figure demonstrates that the value of α = 30 (corresponding to threshold equal to 6.1) is the best in our case.

Realizations of separated fields of convective and turbulent vertical velocities are shown in Fig. 9. In major details, Fig. 9a is similar to Fig. 2b showing the initial convective velocity field. The axisymmetric cloud core surrounded by a subsiding shell is clearly seen. The statistical structure of the turbulent velocity field, showing in Fig. 9b (isotropy and the characteristic spatial scales of inhomogeneities) corresponds to the initial one.

Realizations of the (a) separated synthetic convective velocity field and (b) turbulent synthetic velocity field on a 128 × 128 grid point mesh. Color bars indicate velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Realizations of the (a) separated synthetic convective velocity field and (b) turbulent synthetic velocity field on a 128 × 128 grid point mesh. Color bars indicate velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Realizations of the (a) separated synthetic convective velocity field and (b) turbulent synthetic velocity field on a 128 × 128 grid point mesh. Color bars indicate velocity values in m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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In simulations using the convective velocity field described by the jump-like profile Eq. (4) (Fig. 2a) we were unable to find suitable wavelets and parameters that would provide high-quality separation of convective and turbulent velocity fields. Due to very sharp gradients (jumps) between the convective zone of the updraft and the subsiding shell in these simulations, a residual trace of convective velocity is always visible in the turbulent velocity field after separation, as shown in Fig. 10. We believe, however, that such strong jumps of convective velocity are quite rare phenomena in real clouds.

Example of the turbulent velocity synthetic field presented on the 128 × 128 grid mesh after separation from the convective velocity field described by Eq. (4) and shown in Fig. 2a. The color bar indicates velocity values in m s⁻¹. The residual trace of the convective velocity field is seen. In this case, very sharp gradients (jumps) cannot be eliminated.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Example of the turbulent velocity synthetic field presented on the 128 × 128 grid mesh after separation from the convective velocity field described by Eq. (4) and shown in Fig. 2a. The color bar indicates velocity values in m s⁻¹. The residual trace of the convective velocity field is seen. In this case, very sharp gradients (jumps) cannot be eliminated.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Example of the turbulent velocity synthetic field presented on the 128 × 128 grid mesh after separation from the convective velocity field described by Eq. (4) and shown in Fig. 2a. The color bar indicates velocity values in m s⁻¹. The residual trace of the convective velocity field is seen. In this case, very sharp gradients (jumps) cannot be eliminated.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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5. Estimation of turbulent parameters

A separated field of turbulent velocity allows estimation of functions characterizing turbulence (structure functions, energy spectra), as well as the main turbulent parameters such as the fraction of TKE related to vertical velocity, ${\widehat{T}}_z$, turbulent dissipation rate $\widehat{\epsilon }$, external turbulent scale ${\widehat{r}}_0$, and turbulent diffusion coefficient $\widehat{K}$. As the input data for parameter estimations we use the mean square of turbulent velocity $\langle {\widehat{w}}^{\prime 2}\rangle$ and the estimated transversal structure function ${\widehat{D}}_{w^{\prime }}\left(r\right)$ (analog of function shown in Fig. 4a). An example of estimated transversal structure functions and a design of turbulent parameter estimations using these functions are illustrated in Fig. 11.

An example of estimated structure functions. The blue and red curves correspond to calculations along x and y axes, respectively. To calculate each structure function, 41 samples taken on parallel straight lines were averaged. The dashed line denotes the theoretical structure function. The dotted lines show the relationship between the structure function and turbulent parameters. The large point in the left bottom corner indicates the values used for estimation of turbulent dissipation rate. The actual coincidence of the blue and red curves demonstrates the isotropy of the separated field.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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An example of estimated structure functions. The blue and red curves correspond to calculations along x and y axes, respectively. To calculate each structure function, 41 samples taken on parallel straight lines were averaged. The dashed line denotes the theoretical structure function. The dotted lines show the relationship between the structure function and turbulent parameters. The large point in the left bottom corner indicates the values used for estimation of turbulent dissipation rate. The actual coincidence of the blue and red curves demonstrates the isotropy of the separated field.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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An example of estimated structure functions. The blue and red curves correspond to calculations along x and y axes, respectively. To calculate each structure function, 41 samples taken on parallel straight lines were averaged. The dashed line denotes the theoretical structure function. The dotted lines show the relationship between the structure function and turbulent parameters. The large point in the left bottom corner indicates the values used for estimation of turbulent dissipation rate. The actual coincidence of the blue and red curves demonstrates the isotropy of the separated field.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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To obtain statistically relevant results, we produced 100 realizations of the turbulent velocity field. The separation was carried out using the optimum wavelet “sym5” with the optimum parameters (see section 4). The mean values and the STD of the estimations are presented in Table 1. Since we used two estimations of the structure functions in two perpendicular directions (Fig. 11), some parameters were estimated twice (estimations 1 and 2). In our case, the ten percent underestimation of TKE entailed an underestimation of the external turbulent scale and the turbulent diffusion coefficient. Generally, the estimations can be considered as quite satisfactory.

Table 1.

Estimations of turbulent parameters.

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6. Examples of SAM-simulated velocity field separation

Figure 12 shows the separated field of convective velocity (Fig. 12a) and the field of turbulent velocity (Fig. 12b) in a cloud simulated using SAM at t = 33 min and z = 1500 m, about 1 km above the cloud base. The initial field of vertical velocity to be separated is shown in Fig. 1. The “sym5” wavelet and its optimal parameters obtained (section 4, Fig. 8) were used for the separation. Figure 12a shows a convective updraft with the center point with coordinates x₀ = 2500 m and y₀ = 2620 m. This updraft area with a characteristic size of 400 m, lying entirely inside the cloud is surrounded by a subsiding shell. The maximum updraft velocity is 5 m s⁻¹, while the downdraft velocity in the subsiding shell is about 1 m s⁻¹. These values of cloud updrafts and downdrafts in subsiding shells are typical for growing Cu (e.g., Katzwinkel et al. 2014). The turbulent velocity field is spotty with the maximum spot size of about 180 m long. The amplitude of turbulent velocity fluctuations is about 2 m s⁻¹ which is significant.

The separated SAM fields of (a) convective velocity and (b) turbulent velocity in a cloud at t = 33 min and z = 1500 m. The magenta lines mark the cloud boundary at LWC = 0.01 g kg⁻¹. Color bars indicate velocity values in m s⁻¹. The initial nonseparated field is shown in the Fig. 1.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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The separated SAM fields of (a) convective velocity and (b) turbulent velocity in a cloud at t = 33 min and z = 1500 m. The magenta lines mark the cloud boundary at LWC = 0.01 g kg⁻¹. Color bars indicate velocity values in m s⁻¹. The initial nonseparated field is shown in the Fig. 1.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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The separated SAM fields of (a) convective velocity and (b) turbulent velocity in a cloud at t = 33 min and z = 1500 m. The magenta lines mark the cloud boundary at LWC = 0.01 g kg⁻¹. Color bars indicate velocity values in m s⁻¹. The initial nonseparated field is shown in the Fig. 1.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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The estimation of turbulent parameters is in some respects different from Eqs. (12)–(15) used in section 5, for two reasons. First, the simulated cloud does not occupy the entire computational domain. The actual domain is about 5 × 5 km², which is even larger in the horizontal plane than that shown in Figs. 1 and 12. The air is motionless at points located far from the cloud. Indeed, Fig. 12b clearly shows that the turbulence is intense only within the cloud and the nearest cloud surrounding. In this particular case, the area of intense turbulence has several hundred meters in diameter. To calculate TKE one has to use the points inside or near the cloud, in contrast to the idealized model described in section 3, where the vertical velocity field occupies the entire domain as shown in Figs. 5 and 9. The boundaries of the cloud are shown by magenta lines in Fig. 12.

The second reason is that the assumption that structure function scaling is equal to 2/3 [Eq. (6)] in accordance with the theory of homogeneous and isotropic turbulence, may be erroneous and leads to wrong results. Therefore, instead of Eqs. (12)–(15) we used the method developed by Pinsky et al. (2010) based on the formula for structure function that is more general than Eq. (6): $D_{w^{\prime }}\left(r\right)=A{r}^{\beta }$. Coefficient A and slope β are calculated using the mean square approximation of the structure function, as shown in Fig. 13 (the details of approximation are given in the caption to Fig. 13). Since we estimate the structure function in two perpendicular directions, we provide two approximations. To exclude the influence of large scales, only several first points (several tens of meters) of the structure function are used in the approximations.

Structure functions estimations of the field shown in Fig. 12b and their approximations. Blue and red curves correspond to calculations along x and y axes, respectively. Seven lags (70-m interval) were used in order to approximate the structure functions. To calculate each structure function, 41 samples taken along lines parallel to x and y axes were averaged.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Structure functions estimations of the field shown in Fig. 12b and their approximations. Blue and red curves correspond to calculations along x and y axes, respectively. Seven lags (70-m interval) were used in order to approximate the structure functions. To calculate each structure function, 41 samples taken along lines parallel to x and y axes were averaged.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Structure functions estimations of the field shown in Fig. 12b and their approximations. Blue and red curves correspond to calculations along x and y axes, respectively. Seven lags (70-m interval) were used in order to approximate the structure functions. To calculate each structure function, 41 samples taken along lines parallel to x and y axes were averaged.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0127.1

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Parameters estimated using Eqs. (16)–(19) are presented in Table 2 and are typical of small Cu. For instance, Gerber et al. (2008) evaluate the dissipation rate in clouds with radius of about 400 m at z = 1700 m as 70 cm² s⁻³; Katzwinkel et al. (2014) evaluated the maximum values of the dissipation rates along the traverses as 100 cm² s⁻³. Dissipation rate changes with height and reaches the maximum values around the levels of the maximum vertical velocity which in such Cu is of 5–6 m s⁻¹ (as was mentioned above). The maximum vertical velocity obtained in the simulation agrees with this value. It is noteworthy that there is an excellent coincidence of the two structure functions (Fig. 13) and the estimations of turbulent parameters, obtained based on these functions. Coincidence of the structure functions means local isotropy in the horizontal plane. The slopes of the structure functions are close to 2/3, which is the theoretical slope for homogeneous and locally isotropic turbulence (Monin and Yaglom 1975). However, we are not sure that turbulence characteristics correspond to homogeneous and isotropic turbulence in the entire cloud. It is possible and expected that near the surface the slope of the turbulent spectrum deviates from the −5/3 law (which corresponds to the slope of the structure function deviation from 2/3). However, according to observations in clouds, the −5/3 law seems to be typically obeyed (MacPherson and Isaac 1977; Mazin et al. 1989; Pinsky and Khain 2003). The detailed analysis of turbulent properties in Cu will be performed in the future study.

Table 2.

Estimations of turbulent parameters in the modeled cloud at t = 33 min and z = 1500 m.

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7. Discussion and conclusions

The goal of the present study is to separate between convective and turbulent motions in clouds using the wavelet decomposition technique. We expect that these motions have different impacts on cloud dynamics and microphysics. To develop the method and to prove its efficiency, an idealized field of vertical velocity was generated that is a sum of the convective velocity field and turbulence velocity field. The first component is the “classical” velocity of a convective updraft surrounded by a subsiding shell. The second component is the homogeneous and isotropic turbulence velocity field formed by the generator of random normal fluctuations with a given structure function obeying turbulent laws. A wavelet technique with tuned parameters was developed that allowed to separate the complex convective-turbulent velocity field into convective and turbulent components which are close to convective and turbulent motions in real clouds. It was shown, however, that an accurate separation is impossible in case the convective motions are characterized by jump-like boundaries with very sharp gradients. Analysis of such zones requires ultrahigh (a few meters) resolution simulations and special consideration.

The method of separation was applied to the field of vertical velocity in a developing Cu simulated using SAM with bin microphysics. The method proved efficient even in the case when a cloud (together with its turbulent fluctuations) covers only a certain part of the computational area; i.e., the turbulence cannot be considered homogeneous within the entire computational area. High efficiency and benefits of the separation method have been demonstrated. It was shown (at z = 1500 m in our case) that large-scale convective motions look like “Mexican hat” [Eq. (5)] with the maximum of 6 m s⁻¹ in the cloud center and a subsiding shell with downdrafts up to 2 m s⁻¹. This structure of the obtained large-scale convective motions means that our choice of the idealistic model of convective speed used for optimization was correct. It also means that the proposed method allows to obtain proper results concerning the dynamical structure of Cu, relevant to modern concepts in cloud physics.

The separation allowed to obtain “pure” turbulent components of velocity and to evaluate the basic parameters of turbulence within clouds, such as turbulent kinetic energy, the dissipation rate of kinetic energy, and external turbulence scale. The turbulence velocity fluctuations and dissipation rates were found to be quite significant, so the maximum fluctuations reached a few meters per second. Such values correspond to those observed in Cu in zones of strong updrafts. It is remarkable that the retrieved turbulent velocity field has many features of homogeneous and isotropic turbulence obeying −5/3 law, and the structure function is close to the 2/3 law. The values of dissipation rate were evaluated as about 125 cm² s⁻³. These values are close to evaluations in Cu at the mature stage of their evolution at heights where updrafts and velocity gradients reach their maximum. The values of the effective turbulent diffusion coefficient were evaluated as several meters squared per second. Previously, we applied such values, alongside with others, in the low-parametric cloud models (Pinsky and Khain 2018, 2019, 2020a,b) in investigations of the impact of mixing on cloud microphysical structure in the vicinity of its lateral boundaries.

In this study, we focused on a method of velocity fields separation without analyzing their relation to the cloud structure. In reality, parameters of both the convective velocity field and the turbulent velocity field change with height and time. Therefore, detailed investigations of both fields, obtained under different thermodynamic and aerosol conditions, are required for better understanding and description of thermodynamic and microphysical processes in clouds.

The 10-m resolution may be not fine enough to describe the areas of strong gradients of different quantities (boundary of cloud core, cloud–clear air interface, etc.). But we do not expect a significant qualitative improvement in case higher resolution is used. It is because the contribution of small low energetic vortices with scales below 10 m should be little. At the same time, the computational cost increases dramatically.

We suppose that separation of turbulent and convective motions in clouds might not be achieved in some cases due to overlapping convective and turbulence scales. It is also possible that there exist motions representing “convective type” turbulence, or motions that are intermediate between convection and turbulence. A motion of an intermediate scale can arise as a result of instability of convective motions (Grabowski and Clark 1993), or just due to formation of comparatively small warm or cold plumes.

The next parts of the study will be dedicated to detailed analysis of convective and turbulent structures of Cu at different stages of cloud evolution. In our further study we intend to apply the procedure developed in the present study in order to separate all velocity components (u, υ, and w) into convective and turbulent parts. Separate analysis of the convective and turbulent fields will allow to evaluate the rate of convective entrainment and to compare it with turbulent entrainment and with measuring data.