Editorial Type:
Article
Article Type:
Research Article

Vortex Structure of Head Bubble in Convective Cloud Starting Plume

Mark Pinsky aDepartment of Atmospheric Sciences, Hebrew University of Jerusalem, Jerusalem, Israel

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Eshkol Eytan bDepartment of Earth and Planetary Science, Weizmann Institute of Science, Rehovot, Israel

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Ehud Gavze aDepartment of Atmospheric Sciences, Hebrew University of Jerusalem, Jerusalem, Israel

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Alexander Khain aDepartment of Atmospheric Sciences, Hebrew University of Jerusalem, Jerusalem, Israel

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Online Publication:
17 Aug 2023
Print Publication:
01 Aug 2023
DOI:
https://doi.org/10.1175/JAS-D-22-0122.1
Page(s):
2091–2113
Free access

Abstract

A developing cumulus cloud (Cu) was modeled, and dynamic, thermodynamic, and microphysical properties of an ascending head bubble reproducing the upper part of a developing Cu were investigated. The data for analysis are taken from 10-m-resolution LES of trade wind Cu under BOMEX conditions. The detection of a rising bubble is carried out using wavelet filtering of the velocity fields and microphysical fields, while a low-frequency signal of the filtering is associated with the convective-scale structure of cloud. We substantiate and discuss the representation of the bubble as a vortex ring, and estimate the parameters of this vortex ring. The simplest Hill’s vortex was chosen as a model of a vortex ring inside cloud. Analytical approximations of the radial profiles of the vertical velocity and of conservative quantities (such as total water mixing ratio and liquid water potential temperature inside and outside the bubble) are obtained. The spatial structure of these quantities is investigated using analytical expressions. Analytical models for spatial distributions of liquid water content (LWC) and adiabatic fraction (AF) are also designed and analyzed. The results demonstrate the existence of a cloud core with high values of LWC and AF up to the height of 1800 m. The horizontally averaged value of the adiabatic fraction, calculated analytically using the Hill’s vortex concept, is evaluated as 0.39, which is the typical AF value in the upper parts of such Cu. The vertical profiles of different important quantities characterizing cloud structure are presented. The analysis performed in this study allows us to conclude that a rising vortex ring plays the dominating role in formation of the thermodynamic and microphysical structure of developing Cu.

Significance Statement

1) Dynamic and thermodynamic fields of a developing cumulus cloud simulated by high-resolution LES with spectral bin microphysics are separated into convective and turbulent components by means of the wavelet technique. 2) The analysis of convective component of the cloud revealed the existence of a vortex ring at the developing stage of the cloud and evaluate its parameters. 3) The analysis performed in this study allows us to conclude that a rising vortex ring plays the important role in formation of the thermodynamic and microphysical structure of developing Cu. 4) The study provides a novel insight into the cloud–environment interaction. 5) The approximating equations describing the vortex ring can be usefully applied for developing new schemes of convective parameterization.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Alexander Khain, alexander.khain@mail.huji.ac.il

Abstract

A developing cumulus cloud (Cu) was modeled, and dynamic, thermodynamic, and microphysical properties of an ascending head bubble reproducing the upper part of a developing Cu were investigated. The data for analysis are taken from 10-m-resolution LES of trade wind Cu under BOMEX conditions. The detection of a rising bubble is carried out using wavelet filtering of the velocity fields and microphysical fields, while a low-frequency signal of the filtering is associated with the convective-scale structure of cloud. We substantiate and discuss the representation of the bubble as a vortex ring, and estimate the parameters of this vortex ring. The simplest Hill’s vortex was chosen as a model of a vortex ring inside cloud. Analytical approximations of the radial profiles of the vertical velocity and of conservative quantities (such as total water mixing ratio and liquid water potential temperature inside and outside the bubble) are obtained. The spatial structure of these quantities is investigated using analytical expressions. Analytical models for spatial distributions of liquid water content (LWC) and adiabatic fraction (AF) are also designed and analyzed. The results demonstrate the existence of a cloud core with high values of LWC and AF up to the height of 1800 m. The horizontally averaged value of the adiabatic fraction, calculated analytically using the Hill’s vortex concept, is evaluated as 0.39, which is the typical AF value in the upper parts of such Cu. The vertical profiles of different important quantities characterizing cloud structure are presented. The analysis performed in this study allows us to conclude that a rising vortex ring plays the dominating role in formation of the thermodynamic and microphysical structure of developing Cu.

Significance Statement

1) Dynamic and thermodynamic fields of a developing cumulus cloud simulated by high-resolution LES with spectral bin microphysics are separated into convective and turbulent components by means of the wavelet technique. 2) The analysis of convective component of the cloud revealed the existence of a vortex ring at the developing stage of the cloud and evaluate its parameters. 3) The analysis performed in this study allows us to conclude that a rising vortex ring plays the important role in formation of the thermodynamic and microphysical structure of developing Cu. 4) The study provides a novel insight into the cloud–environment interaction. 5) The approximating equations describing the vortex ring can be usefully applied for developing new schemes of convective parameterization.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Alexander Khain, alexander.khain@mail.huji.ac.il
Keywords: Convective clouds; Turbulence; Wavelets

1. Introduction

In convective parameterization schemes used in large-scale and mesoscale numerical atmospheric models, simplified cloud representations are usually used. Convective clouds are represented typically as stationary plumes (jets) (Yano 2014a) that transport mass, moisture, and energy from the atmospheric boundary layer upward. To a lesser extent, the clouds were represented in large-scale models as rising bubbles (Ooyama 1971; Rosenthal 1973; Anthes 1977). The representation of a convective cloud as a steady-state convective plume was first introduced supposedly by Stommel (1947), while Scorer and Ludlam (1953) proposed representing convective clouds as a succession of buoyant bubbles. The representation of clouds as rising bubbles is widely used to investigate effects of entrainment, microphysical processes such as nucleation, drop size distribution, and rain formation (e.g., Warner 1969, 1970a; Pinsky and Khain 2002; Khvorostyanov and Curry 2005). A dynamically more advanced idea of developing clouds is representation the clouds as starting plumes consisting of a nonstationary jet capped by an ascending head bubble (Turner 1962; McCarthy 1974; Houze 2014; Pinsky et al. 2022).

Simplified representations of clouds assume the internal structure of clouds to be quite primitive, so that all cloud quantities assumed to be horizontally homogeneous. This assumption substantially simplifies the treatment of convective clouds in large-scale models, representing all the vertical velocities and other quantities as a weighted averaged of the values in environment and in clouds (Arakawa and Schubert 1974). However, the assumption of horizontal homogeneity crucially contradicts to observations which show high changes in vertical velocity, droplet size distributions, liquid water content (LWC), and adiabatic fractions (AF) across the clouds (e.g., Gerber 2000; Gerber et al. 2008; Katzwinkel et al. 2014; Konwar et al. 2021; Eytan et al. 2021). High vertical velocities in cloud cores decrease toward cloud edges, and are replaced by a subsiding shell at the periphery of clouds (Heus and Jonker 2008; Abma et al. 2013; Katzwinkel et al. 2014; Norgren et al. 2016; Nair et al. 2020). We realize that any parameterization is a “caricature” of the real processes (Yano 2014b). At the same time, such a “caricature” should reproduce major features of the real processes. The assumption about the horizontal homogeneity of clouds leads to the situation when the corresponding parameterizations cannot simultaneously predict LWC and cloud-top height (Warner 1970b; Zhao and Austin 2005). To renounce the assumption of horizontal cloud homogeneity, one requires a better understanding of the complicated processes of formation of cloud dynamic and microphysical structure.

The structure of a cloud is highly complicated and determined by motions of different spatial and time scales. To simplify the analysis of cloud structure, Pinsky et al. (2021) considered cloud motions as a sum of convective and turbulent motions. The impacts of convective and turbulent motions on cloud microphysics and thermodynamics are different. Convective motions transport air masses; for instance, cloud updrafts and subsiding shells attributed to convective motions transport air and microphysical quantities over distances comparable with the cloud size. Convective transport is described by the “advective” terms in motion equations. Convective motions can form coherent structures, whose sizes can be either smaller or larger than the size of the parent clouds. In contrast, turbulent motions do not transport air mass over large distances. Their integral effect is often described by the equation of turbulent diffusion. The main role of turbulence is to perform mixing inside clouds, as well as mixing between clouds and the surrounding air. We can crudely characterize the physical role of convective motions as creating the cloud “skeleton,” while the role of turbulence is to smooth the gradients of cloud variables and mix the cloud with the environment air, which causes the cloud processes to deviate from adiabatic ones.

Pinsky et al. (2021) developed a novel approach based on wavelet filtering, which allows us to separate convective and turbulent motions in isolated convective clouds. In particular, this separation was carried out for a Cu simulated using 10-m-resolution System Atmospheric Modeling (SAM) model with spectral bin microphysics (Khain et al. 2019; Khain et al. 2004; Khain and Pinsky 2018; Eytan et al. 2021). Pinsky et al. (2022) and Pinsky and Khain (2023) used this method to evaluate the properties of convective and turbulent motions. It was shown that turbulent motions cannot be responsible for formation of observed cloud inhomogeneity and, in particular, cannot form the observed inhomogeneity of vertical velocity, LWC, and AF. Pinsky et al. (2022) analyzed dynamic and thermodynamic structures of the cloud updraft zone (CUZ). The analysis showed that the influence of convective updraft on dynamic, thermodynamic, and microphysical properties of CUZ at the developing stage of trade wind Cu is much higher than that the lateral dynamic entrainment. Accordingly, the CUZ of a developing Cu resembles the starting plume consisting of an ascending head bubble with a tail behind it. CUZ is surrounded by a subsiding shell with a downdraft inside it. Pinsky et al. (2022) suggested that the head bubble could be represented as an ascending vortex ring, i.e., a cloud-starting plume is a rising vortex ring followed by the wake. The vortex creates an updraft airflow in the cloud center and a downdraft flow at cloud periphery and outside the vortex.

The idea that the dynamics of the head bubble resembles a vortex ring is not new. Levine (1959) and Turner (1962) suggested that a rising buoyant bubble can be represented as a vortex ring (more precisely, as the Hill’s vortex, which belongs to the toroidal vortices family) (Alekseenko et al. 2007). Currently, the existence of vortex rings in Cu has been confirmed in both experimental studies (Damiani and Vali 2007; Wang et al. 2009; Wang and Geerts 2015) and large-eddy simulations (LES) of clouds (Zhao and Austin 2005; Sherwood et al. 2013; Romps et al. 2021; Eytan et al. 2022). Morrison et al. (2020) demonstrated a large convective system which served as a root of several bubbles arising at different altitudes, each of them having spherical vortex-like toroidal circulation. Morrison and Peters (2018) analyzed the ratio between the vertical velocity of the moist deep convective thermals and the maximum vertical velocity within them and compared this ratio with that predicted for Hill’s vortex. The existence of ring vortices seems typical for the developing stage of cloud evolution. Typically, rain forms at the mature or decaying stages, when the cloud structure changes, and cloud growth stops or is replaced by dissipation. The vortex ring structure changes dramatically or, more likely, the vortex dissipates altogether at dissipation stage.

Although the existence of vortex rings in cumulus clouds have been reported in many studies, the effect of the processes related to vortex rings on cloud microphysics remains uninvestigated. The vortex ring in a cloud is a convective-scale phenomenon observed against the background of turbulent-scale motions and fluctuations of different quantities. In the present study, we use the results of simulation of an isolated nonprecipitating Cu to detect such a vortex ring within the developing cloud and its nearest environment. We approximate it by a Hill’s vortex and investigate whether such approximation may provide reasonable dynamic, thermodynamic, and microphysical properties of the cloud.

2. Vortex rings in rising bubbles

a. SAM model and simulated case

In this study we use the same simulation of trade wind small Cu as that used in the study by Eytan et al. (2021) and Pinsky et al. (2021, 2022) where the simulation design is described in detail. Therefore, only a brief description is given here. We use SAM, a 3D LES model specially designed to simulate fine thermodynamic and microphysical cloud structures at high resolution, which allows direct reproducing of both convective and turbulent motions. The specific feature of the applied SAM version is using spectral bin microphysics (Khain et al. 2004; Khain and Pinsky 2018). The model describes mixed-phase microphysics and calculates size distribution functions of different types of hydrometeors including water drops. Drop size distributions (DSDs) are defined on a logarithmically equidistant (mass doubling) mass grid containing 33 bins, covering drops with radii from 2 μm to 4 mm. To describe cloud–aerosol interaction, a separate SD for aerosols is calculated using a grid that also contains 33 bins, the largest being of 2 μm radius.

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 (Khain and Sednev 1996). The changes in DSD resulting from collisions between drops are calculated by solving the stochastic collision equations using the method proposed by Bott (1998). Details of the method are presented in Khain and Sednev (1995). SAM with spectral bin microphysics has been successfully used for simulation of different types of clouds (Fan et al. 2011; Heiblum et al. 2016; Khain et al. 2019). The main parameters of the model design used in the present simulation are presented in Table 1.

Table 1.

The main parameters of the SAM model and simulations.
Table 1.

We simulated an isolated trade wind warm nonprecipitating Cu with a resolution of 10 m. Aerosols playing the role of cloud condensational nuclei are maritime. Their size distribution is represented by a sum of three lognormal distributions representing three different modes of aerosol size (Ghan et al. 2011). To study the vortex that appears inside Cu, we chose the simplest dynamic structure of the simulated Cu, which develops in the absence of a background horizontal wind, so the single source of cloud convection is the vertical temperature gradient. The cloud is triggered by a warm spherical bubble with superimposed random fluctuations. This kind of triggering clouds was proposed by Ovtchinnikov and Kogan (2000) and leads to formation of clouds with a wide cloud base.

The simulation uses the BOMEX-1974 sounding profiles (Fig. 1). One can see the presence of the temperature inversion layer at the height of 1400–2000 m over the surface, where the temperature increases by 0.25 K, while the humidity and relative humidity rapidly decrease. In this way, we simulate the Cu with the simplest convection structure.

Fig. 1.
Fig. 1.

Vertical profiles of (a) the temperature, (b) the mixing ratio of water vapor, and (c) the relative humidity of the air measured far from the cloud (from Pinsky et al. 2022).

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

b. Wavelet processing

A convective cloud is a spatially inhomogeneous and nonstationary object where zones of regular convective updraft and downdraft are superimposed by random turbulent motions. To retrieve the convective components of the complicated motions, we applied a 2D spatial wavelet filtering to separate convective motions from the turbulent ones, as described in detail by Pinsky et al. (2021, 2022). For this purpose, we represent the SAM-modeled fields of the velocity components and fields of different thermodynamic and microphysical quantities (at each output time with an increment of 0.5 min) as a set of horizontal slices with a height increment of 10 m. Every slice is filtered using both low-frequency and high-frequency wavelet filters. 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 in Cu can be very sharp, especially near cloud boundaries, the locality of filtration (i.e., ability of filter automatically changes its characteristics depending on the average gradient of the filtered quantity in every local area) is of major importance for keeping these gradients unchanged. Another approach to averaging (e.g., techniques base on FFT transforms) would lead to strong smoothing of the gradient zone, and thus to larger artificial broadening of the cloud interior region and to larger residual fluctuations in comparison with the wavelet filters.

Wavelets [detailed theoretical description can be found in Daubechies (1992)] are used in various fields of knowledge and applications. In particular, wavelets were applied in several studies to process atmospheric signals. Yano et al. (2001a,b) used wavelet processing for decomposition of convective-scale and mesoscale structures. Yano et al. (2004a) detected pulse-like events in the tropical atmosphere. Wavelet compression of deep moist convection was used in study by Yano et al. (2004b). Also, wavelet-based verification of precipitation forecast was applied in study by Yano and Jakubiak (2016). However, we do not know the studies in which wavelet decomposition is used in order to separate convective and turbulent motions in a single cloud except for our previous studies (Pinsky et al. 2021, 2022; Pinsky and Khain 2023). These studies are closely related to the presented study.

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 (Pinsky et al. 2021). Numerous simulations and statistical estimations allowed us to choose the wavelet type and wavelet parameters minimizing component distortion and the errors in component separation. The desirable properties of wavelet functions are orthogonality, possibility of fast wavelet transform, minimum possible asymmetry of wavelet functions, and maximum number of vanishing moments. As a result of justification procedure, we chose the wavelet “sym5” (having 10 coefficients) that demonstrates the best results.

The high efficiency of the wavelet method was demonstrated for both synthetic and DNS fields in this study. The natural meanings of such fine parameters as slopes of turbulent spectra, obtained in the study Pinsky and Khain (2023), also indicate the correct choice of filter parameters. The same set of parameters was used in our research of convective motions (Pinsky et al. 2022) and in this study of vortex structure in a single Cu cloud.

In the present study, low-frequency signals at the outlet of the wavelet filter correspond to the air vertical velocity W(x, y), as well as to the horizontal velocity components U(x, y) and V(x, y) of convective and subconvective scales which can form a vortex ring inside the head bubble. Figure 2 presents an example of unfiltered and filtered slices of velocity.

Fig. 2.
Fig. 2.

Examples of (a),(c) unfiltered and (b),(d) filtered slices of (a),(b) vertical velocity and (c),(d) horizontal velocity components. The red arrows in (c) and (d) mark 5 m s−1 velocity scale.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

We see a large difference between these two types of slices. The filtered vertical velocity field (Fig. 2b) clearly shows the central updraft zone surrounded by the downdraft zone forming a subsiding shell. The horizontal velocity field (Fig. 2d) demonstrates some regular nonaxisymmetric structure which is in contrast with the nonregular structure of the nonfiltered field (Fig. 2c). Comparison of Figs. 2c and 2d shows that the magnitudes of convective horizontal velocity components are of the same order and can be even a little smaller than the corresponding magnitudes of the turbulent components. However, it does not mean that turbulent velocity is larger than the convective one, since the vertical convective velocity component dominates. It exceeds fluctuations of turbulent velocity by factor of 3 (Fig. 2b). We believe that variations of the velocities shown on filtered slices are determined by the dominating convective processes, and thus can be used to analyze the main factors of convective cloud formation.

The similarly filtered fields of liquid water mixing ratio Ql(x, y), total water content Qt(x, y) and liquid water potential temperature Θl(x, y) are also used in our analysis. The latter are defined as
Ql(x,y)=⟨ql⟩,
Qt(x,y)=⟨qυ+ql⟩,
Θl(x,y)=⟨(T+γaz)(1−LwqlcpT)⟩,
where qυ, ql, and T are the water vapor mixing ratio, the liquid water mixing ratios (LWC), and the absolute temperature, respectively, simulated by SAM, and γa = 9.8 × 10−3 K m−1 is the dry adiabatic gradient (Pruppacher and Klett 1997; Khain and Pinsky 2018). Angle brackets in Eqs. (1)–(3) mean low-pass filtering by 2D wavelet filter. In more detail the filtering procedure and the choice of optimal filter parameters are described by Pinsky et al. (2021). The list of variables is presented in Table 2. The same optimal parameters that are used in filtering of the vertical velocity are applied to filtering of the horizontal velocity components to keep the continuity of the filtered velocity field. Also, the same parameters are applied in order to filter horizontal fields of thermodynamic and microphysical quantities because these fields have shapes similar to vertical velocity field.
Table 2.

List of symbols and units, where “nd” denotes nondimensional values.
Table 2.

Figure 3 compares the unfiltered and filtered slices of thermodynamic and microphysical quantities. We see a large difference between these two types of slices. The filtered fields (Figs. 3b,d,f) clearly reveal spatial structures correlated (or anticorrelated) with the structure of the filtered vertical velocity (Fig. 2b). Therefore, we believe that the major horizontal variations of the thermodynamic and microphysical quantities shown on filtered slices are determined by the dominating convective motion. Some drawbacks of LWC fields filtering should be noted. In case a cloud adiabatic core exists, unfiltered fields of qt = qυ + ql should be close to the conservative fields in the vicinity of the core. This determines rather rigidly the vertical profile of LWC. The filtering of LWC fields can lead to an artificial deviation of this quantity from the adiabatic law and, therefore, to an underestimation of AF in the cloud core. For this reason, we use both unfiltered and filtered fields of LWC in the figures illustrating our analysis.

Fig. 3.
Fig. 3.

An example of (a),(c),(e) unfiltered and (b),(d),(f) filtered horizontal slices of the fields of thermodynamic and microphysical quantities: (a),(b) LWC, (c),(d) total water content, and (e),(f) liquid water potential temperature.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

c. Detection of the vortex ring: The head bubble center and velocities related to the bubble

Height–time dependence of the maximum convective vertical velocity Wmax(z, t) was investigated by Pinsky et al. (2022). This dependence is determined by the W maxima calculated using filtered horizontal slices at each height and time instances. The spatial coordinates of the points corresponding to Wmax are located inside the CUZ. The Wmax(z, t) field taken from Pinsky et al. (2022) is shown in Fig. 4a. The magenta curve (which is the convergence line of isocontours) shows the location of the maximum velocity points, and can be interpreted as a trajectory of some ascending object in the z–t plane. This object is nonsymmetric with respect to its trajectory in the vertical direction, is close to the cloud top and lengthened toward the cloud bottom. Figure 4a is consistent, therefore, with the concept of a starting plume represented by an ascending head bubble whose trajectory is shown by the magenta line and the wake below. The z–t trajectory is formed by the points of maximum vertical velocity generated by internal circulation inside the bubble. The lower velocities below the trajectory represent the wake. Analysis of the fields of different quantities taken in the area surrounding the magenta line allows us to investigate the dynamic, thermodynamic, and microphysical structure of the ascending head bubble and its vortex-like nature.

Fig. 4.
Fig. 4.

(a) Height–time field of Wmax(z, t). The magenta line shows the z–t trajectory of the rising bubble (from Pinsky et al. 2022). (b) Vertical profiles of the velocities related to the head bubble: the maximum vertical air velocity (black solid line and red circles) and the velocity of the cloud-top ascent (blue dotted line with circles). The cloud-top height is defined as the maximum height where LWC exceeds 0.01 g kg−1. The bubble updraft velocity, calculated by differentiating of z–t trajectory, is shown by dashed black line. The “equivalent” Hill’s vortex translation velocity equal to (2/5)Wmax⁡(section 3c) is shown by blue stars.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

The vertical profiles of the velocities related to the head bubble are shown in Fig. 4b. First of all, one sees a very good coincidence between the velocity Wmax(z) calculated as a maximum in the vertical columns (red circles), and the velocity Wmax(r) calculated as a maximum in the horizontal slices (solid black line) (dependence on r is shown in order to emphasize that it is taken in the horizontal plane). This coincidence of the vertical velocity profiles allows us to conclude that there is only one rising bubble, and not a bubble series as is sometimes assumed at the developing stage of cloud evolution (Houze 2014), at least in our simulations. The second comment is related to the profile of the bubble updraft velocity Wbubble calculated by the slope of the z–t trajectory (dashed black line in Fig. 4b). The line demonstrates an increase of the updraft velocity with height, which can be explained by an increase of the buoyancy force affecting the bubble. At the same time, the bubble upward velocity is close to the velocity of the cloud-top ascent (the cloud-top height is defined as a maximal height with LWC ∼ 0.01 g kg−1) up to the height of 1500 m, i.e., up to the lower part of the inversion layer seen in Fig. 1. Thus, the bubble dynamics determines the processes accompanying cloud-top ascent.

To better illustrate the inner dynamic structure of the bubble and the area surrounding it, the convective velocity fields in the x–z and y–z planes are shown at different time instances in Fig. 5. The centers of the panels are located on the z–t trajectory where the vertical velocity is maximum. An ascending stream is seen near the center. A pair of vortices rotating in mutually opposite directions are clearly seen. These two vortices are the projections of a vortex ring on the x–z and y–z planes.

Fig. 5.
Fig. 5.

Examples of the velocity field around the bubble in the (left) x–z and (right) y–z planes at times of (top) 29 and (bottom) 30.5 min. The red point in the center corresponds to the maximum vertical velocity Wmax(z, t) located on the z–t trajectory shown in Fig. 4a.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

The vortex ring represents the dynamics of the head bubble and creates flows’ convergence in the lower part of the bubble and their divergence in its upper part of the bubble. It actually determines the convective (coherent) structure of the cloud consisting of CUZ at the center and the subsiding shell at the periphery of developing cloud. As was discussed by Eytan et al. (2021) and Pinsky et al. (2022), the vortex ring affects the thermodynamic and microphysical structures of a developing cloud. Thus, using the wavelet techniques we detected the rising head bubble, picked up its trajectory, and qualitatively described its possible structure, assuming that it can be modeled by the simplest vortex ring.

3. Representation of the bubble dynamic structure by Hill’s vortex

The simplest vortex from the family of vortex rings is a spherical Hill’s vortex (Alekseenko et al. 2007). A possible application of this type of a vortex for description of the bubble rising in a convective cloud was discussed in several studies (Turner 1962; Sherwood et al. 2013; Pinsky et al. 2022). In this study, we also consider an ascending Hill’s vortex as a model of a developing cumulus cloud (at least of its upper part). To better understand the extent to which a Hill’s vortex can describe the dynamic structure of a rising bubble, we analyze the horizontal slices of the vertical velocity field, which were obtained in SAM simulations and underwent the wavelet filtering procedure as discussed in section 2.

a. Hill’s vortices describing rising bubble

Let us consider a spherical bubble of radius a rising at translation vertical velocity W0(z). This velocity can be defined as the velocity of the bubble surface and characterizes the moving of the bubble as a whole. The bubble center is located at the point (r = 0, z = zc), where r and z are the radial and vertical coordinates, respectively. Air circulation inside and outside the bubble is modeled by a Hill’s vortex centered at the point (r = 0, z = zc). The stationary solution for the vertical component W(r, z) and the radial components Ur(r, z) of the air velocity field induced inside and outside this Hill’s vortex in a fixed cylindrical coordinate system is described by the following equations (Alekseenko et al. 2007):

  • Inside the vortex, where R2 = r2 + (z − zc)2 < a2:
    W(r,z)=−3W04[4(ra)2+2(z−zca)2−103],
    Ur(r,z)=3W02a2r(z−zc).
  • Outside the vortex:
    W(r,z)=W0a322(z−zc)2−r2[(z−zc)2+r2]5/2,
    Ur(r,z)=3W0a32r(z−zc)[(z−zc)2+r2]5/2.

One can easily see that the vertical velocity of the upper and lower points of the bubble W(0, zc + a) and W(0, zc − a), respectively, are equal to W0.

For the sake of convenience, we also use normalized coordinates, r˜=r/a and z˜=(z−zc)/a, allowing simpler representation of the dependencies:
W(r˜,z˜)={3W02[53−(2r˜2+z˜2)],R˜2=r˜2+z˜2<1,W022z˜2−r˜2(z˜2+r˜2)5/2,R˜2=r˜2+z˜2>1,
Ur(r˜,z˜)={3W02r˜z˜,R˜2=r˜2+z˜2<1,3W02r˜z˜(z˜2+r˜2)5/2,R˜2=r˜2+z˜2>1.
The circulation described by Eqs. (5)–(9) is axisymmetric with respect to the vertical axis passing through the point (r = 0, z = zc). The two equations in Eq. (8), as well as the corresponding ones in Eq. (9), are matched at points on the bubble boundary which is a half circle.

The velocity field calculated using Eqs. (8) and (9) is illustrated in Fig. 6 using both a fixed coordinate system (Figs. 6a–c) and a coordinate system moving vertically upward with the Hill’s vortex (Fig. 6d).

Fig. 6.
Fig. 6.

Velocity fields in the Hill’s vortex: (a) vertical velocity, (b) radial velocity, (c) velocity vectors, and (d) velocity vectors depicted in the coordinate system moving with translation velocity. The normalized radius of the Hill’s vortex, equal to one, is shown by black semicircle.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

In the moving system, the vertical velocity is calculated as W(r˜,z˜)−W0. One can see an updraft on the left parts of the panels, and a downdraft on the right parts of the panels. The convergence zone on the lower parts of the panels, and divergence zone on the upper part of the panels are also seen. As Fig. 6d shows, in a moving coordinate system the velocity component normally directed at the vortex boundary is equal to zero at R˜2=r˜2+z˜2=1. It means that there are no fluxes directed inward the vortex or outward the vortex that cross its boundary, which is a characteristic feature of Hill’s vortices. However, the velocity field obeying Eqs. (8) and (9) has some specific features which are obviously in conflict with the real cloud properties. We discuss these inconsistencies in detail in section 3c.

b. Fitting the vertical velocity field in the horizontal plane

Now we consider the procedure of fitting the vertical velocity field and estimation of Hill’s vortex parameters using the LES data. For each point on the bubble z–t trajectory (shown in Fig. 4a by the magenta line) we designed a convective vertical velocity field in the horizontal plane. Then, assuming that this field can be described by the equations for a Hill’s vortex with the center at the maximum velocity point, we approximate the field using Eqs. (4) and (6) at z = zc, using the least mean square method. The approximating equations obtained this way are
W(r,zc)={W02[5−6(ra)2] ,r<a,−W02a3r3,r≥a.
In Eq. (10) r is the distance from model grid points to the bubble center. As a result of the approximation, we obtain estimations of the bubble radius a and the translation velocity W0. According to Eqs. (5) and (7), the radial component of the air velocity Ur(r, zc) = 0.

Examples of the least square approximation at two different heights located below and inside the inversion layer are shown in Figs. 7a and 7b. The figure demonstrates a reasonably good quality of the vertical velocity field fitting performed using the least square approximating method. Some underestimation of maximum velocity (and hence of the translation velocities) is seen. The jump of the derivative sign ∂W(r, zc)/∂r, marked in Fig. 7a on the fitting line, represents the Hill’s vortex (and the bubble) boundary. This point corresponds to the minimum W(r, zc) in the agreement with the expression (10). One can see that the Hill’s vortex includes both the updraft zone and partly the downdraft zone.

Fig. 7.
Fig. 7.

Examples of W(r, zc) approximation at two different heights located below and inside the inversion layer. Only points located on x–y plane at the distance of r < 1000 m from the bubble center are used in the approximation. The jump of derivative ∂W(r, zc)/∂r seen on the fitting line and marked in Fig. 7a, corresponds to the Hill’s vortex boundary.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

The least squares approximation was used to design the vertical profiles of an “equivalent” Hill’s vortex (and the bubble) parameters. The vertical profiles of the bubble radius and vertical translation velocity are shown in Fig. 8 (solid black lines). The CUZ radii obtained from the fitting data are calculated by multiplication of the bubble radius by the coefficient equal to 5/6, in accordance with Eq. (10). The profiles of CUZ radius obtained by Pinsky et al. (2022) are also shown in Fig. 8a by blue line. A very good coincidence between these two profiles is clearly seen. The profile of the bubble translation velocity is compared to the profile calculated by dividing the maximum air velocity by coefficient 5/2, as shown in Fig. 8b [the theoretical coefficient 5/2 follows from Eq. (10) for the Hill’s vortex model]. The second profile is also shown in Fig. 4b. Maximum velocity in the vortex center turns out to be higher than it follows from approximation by Hill’s vortex. This effect can be attributed to the contribution of buoyancy in a real cloud as compared to nonbuoyant Hill’s vortex (Morrison and Peters 2018). Nevertheless, the approximation remains reasonable.

Fig. 8.
Fig. 8.

(a) Vertical profiles of the bubble radius and the CUZ radii. (b) Vertical profile of the translation velocity. Another profile in (b) is that of the maximum velocity calculated by dividing it by coefficient 5/2 is shown by dotted line.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

c. Which cloud features are reproduced by Hill’s vortices and which are not?

We realize that the stationary model of a Hill’s vortex is unable to quantitatively reproduce the results of SAM, even in the simplest case when wind shear and cloud rotation are not taken into account. However, the following properties of the dynamic structure related to the cloud and the rising bubble is satisfactorily reproduced.

  • The presence of the updraft flow at the center of a cloud and of a vortex ring, forming downdrafts at cloud periphery (Fig. 5). This structure rises together with the head bubble along the z–t trajectory (Fig. 4a).

  • If a bubble is represented by a round Hill’s vortex rising along the z–t trajectory at translation velocity W0, and its upper boundary corresponds to the cloud top, the relationship between the air velocities at its center Wmax and the velocity of the cloud top should obey the following relationships: Wbubble=Wtop=(2/5)Wmax⁡(r)=(2/5)Wmax⁡(z)=W0 [Eq. (8)]. Figure 4b demonstrates that most of these relationships are obeyed up to heights of 1600–1700 m well enough. Despite the fact that the relationships underestimate Wmax by ∼25%, these relationships can be used for different estimations.

  • The bubble radius a estimated by approximated Eq. (10), and the CUZ radius calculated from the fitting data as a×5/6 in accordance with Eq. (10), are in very good agreement with the results obtained by Pinsky et al. (2022) (Fig. 8a).

  • Very precise reproduction of Wmax(r) ∼ r−3 asymptotic dependence at the outer periphery of the vortices, following from Eq. (10) (Fig. 7).

  • The observed shapes of the radial profiles of the velocity field allows split it into two parts: the inner solenoidal flow (inside the bubble), and the outer flow (outside the bubble), in agreement with Hill’s vortex theory [Eqs. (8) and (9) and Fig. 6).

The following properties of a rising bubble are not reproduced satisfactorily by the model of round Hill’s vortex causing certain inconsistencies.

  • The asymmetry of the vertical velocity field with respect to the height level z = zc. This asymmetry is related to the existence of a wake below the rising bubble (Fig. 4a). Hill’s vortices do not contain such wakes. The vertical gradients of motion-related parameters such as buoyancy force, bubble mass, as well as bubble deformation can also cause vertical asymmetry.

  • Increased pressure zone just over the bubble and decreased pressure zone just below the bubble, related to the above-mentioned asymmetry. This asymmetry is illustrated by Fig. 9, obtained in SAM simulations (see also Morrison and Peters 2018). In contrast Hill’s vortex theory assumes pressure perturbances to be symmetrical with respect to the height level of z = zc.

  • The existence of normal velocities at the vortex boundaries, and in particular, of a nonzero radial velocity at z = zc. A nonzero radial velocity creates a weak dynamic entrainment of the air inwards the bubble (Pinsky et al. 2022). In contrast, according to Hill’s vortex theory, the normal component of the relative velocity at Hill’s vortex boundaries is equal to zero, i.e., Ur(r, zc) = 0 [Fig. 6d and Eqs. (5) and (7)].

  • A systematic excess of the maximum vertical air velocity over the values expected from Hill’s vortex theory (Fig. 8b). It is impossible to obtain simultaneously by way of the approximation both the size of the vortex and the maximum speed.

There are two main types of these inconsistencies: 1) a Hill’s vortex is spherical and is calculated for the potential external flow. A cloud flow is not potential. Also, buoyancy in real clouds plays a substantial role in formation of the maximum velocity, which can lead to the deviation of the vortex ring structure from that of Hill’s vortex (Morrison and Peters 2018). Hence, the deviations of the modeled structure from Hill’s vortex theoretical structure are natural. 2) The application of the filtering in the model for comparing it with the theoretical structure makes the results dependent on the extent of the filtration.

Fig. 9.
Fig. 9.

X–Z cross section in pressure perturbation field at 29 min. Pressure perturbation is positive above the cloud top and negative in the lower part of the toroidal vortex. The pink curve shows the cloud core with an adiabatic fraction exceeding 0.9.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

From the consideration outlined in section 3 it follows that the model of an ascending Hill’s vortex is the simplest model for the purpose of description of cloud dynamics at developing stage. One can expect that the vortex dynamical structure represents the major thermodynamic and microphysical cloud structures realistically, at least on the qualitative level. However, accurate description of the starting plume properties that would take into account the existence of a wake below the head bubble, requires more complicated models of rising vortices.

4. Conservative quantities inside and outside the bubble

It is customary to characterize liquid clouds by variables that are conservative with respect to the condensation/evaporation processes (Pruppacher and Klett 1997; Khain and Pinsky 2018). Among such variables there is the total water mixing ratio and the liquid water potential temperature described and defined in section 2a, widely used when analyzing clouds. In this section we analyze fields Qt(x, z) and Θl(x, z) of these quantities, defined by Eqs. (2) and (3), respectively. The fields were filtered by wavelet. As seen in Figs. 3d and 3f, these quantities have a pronounced horizontal structure correlated with the vertical velocity structure (Fig. 3b).

Suppose that C(r, z, t) is a conservative quantity obeying the budget equation for an incompressible flow:
∂C( r,z,t)∂t+div( uC )= ∂C∂t+u⋅grad( C )=0.
In Eq. (11), u is the air velocity vector. If the flow is steady, ∂C(r,z,t)/∂t=0 and Eq. (11) can be rewritten in the form
u⋅grad[ C( r,z ) ]=0.
Equation (12) shows that C(r, z) is constant along streamlines, and means that in a stationary case the conservative quantity gradient is directed perpendicularly to the velocity vector. Stationarity approximation (12) allows us to obtain some analytical results concerning the spatial structure of conservative quantities. The velocity Eqs. (8) and (9) for a rising bubble represented by a round Hill’s vortex also correspond to a steady flow regime. Therefore, for a round Hill’s vortex one can use Eqs. (8) and (9) written in normalized coordinates (r˜,z˜), and rewrite Eq. (12) in cylindrical coordinates as
{3W02r˜z˜∂C(r˜,z˜)∂r˜+3W02[53−(2r˜2+z˜2)]∂C(r˜,z˜)∂z˜=0,r˜2+z˜2<1,3W02r˜z˜(z˜2+r˜2)5/2∂C(r˜,z˜)∂r˜+W022z˜2−r˜2(z˜2+r˜2)5/2∂C(r˜,z˜)∂z˜=0,r˜2+z˜2>1.
The general solution of Eq. (13) is
C(r˜,z˜)={Φ1[r˜2(r˜2−53+z˜2)],r˜2+z˜2<1,Φ2[r˜−4/3(r˜2+z˜2)],r˜2+z˜2>1,
where function Φ1 is related to the interior of the bubble, whereas function Φ2 is related to the external flow. Both Φ1 and Φ2 are arbitrary differentiable functions. The solution can be verified by direct substitution of Eq. (14) into Eq. (13).
Equation (14) can be used to design an approximating radial profile of conservative quantity C(r˜,0) in the center horizontal plane of the bubble (z˜=0) at every point of the bubble trajectory zc (z = zc corresponding to z˜=0 in normalized coordinates):
C(r˜,0)=Φ1[r˜2(r˜2−53)], r˜2<1,
C(r˜,0)=Φ2{r˜2/3}, r˜2>1.
At the bubble boundary (r˜=1), the two solutions (15a) and (15b) should be matched as Φ1(−2/3,0)=Φ2(1,0).
The simplest linear approximation of Eq. (15a) and the hyperbolic approximation of Eq. (15b), obeying the matching conditions, are written as
C(r˜,0)={C0+C1r˜2(r˜2−53),r˜2<1,(C0−23C1)r˜−2/3,r˜2>1,
where C0 and C1 are constants calculated as a result of the least squares approximating procedure. Thus, function Φ1(x) is linear, and function Φ2(x) is hyperbolic tending to zero when x tends to infinity. In this case, the C(r˜,z˜) field inside and around the bubble is represented by the equations
C(r˜,z˜)={C0+C1[r˜2(r˜2−53+z˜2)],r˜2+z˜2<1,(C0−23C1)[r˜−4/3(r˜2+z˜2)]−1,r˜2+z˜2>1.
Equation (17) do not provide matching of solution all over the circumference r˜2+z˜2=1, but only at the point (r˜=1,    z˜=0). This drawback is the price to pay for the simplicity of the mathematical analysis. Besides, one can assume the boundary conditions at r˜→∞ only at the altitude of the vortex center. In principle, the wake behind the bubble should exist below it, so the conditions should reflect this fact. However, it is impossible in case the symmetric Hill’s vortex model is used. Equation (17) contains two parameters, C0 and C1, that were calculated by fitting the simulated profiles at each height level.

Figure 10 illustrates approximation of Qt(r) at two height levels (Figs. 10a,c) and the corresponding Qt(r˜,z˜) fields inside the bubble and its nearest space (Figs. 10b,d). Equations (16) and (17) were used for the approximation and field design, respectively. The approximations made for the ΔQt = Qt(r) − Qt(∞) quantity, where Qt(∞) is the total water mixing ratio taken far from cloud at the height of the bubble center, appear quite satisfactory, which indicates the validity of the solution (17). The substantial scattering of calculated radial profiles is explained by asymmetry of the simulated cloud. One can see that in the center of the bubble the total water content is maximum, which can be explained by the presence of a water vapor flux from the cloud base. The total water content decreases monotonically from the center to the periphery down to the values characterizing cloudless air space. This behavior of the radial profile is directly related to the vortex air dynamics in the bubble. The approximated radial profiles in Figs. 10a and 10c correlate with the radial profiles of the vertical velocity shown in Fig. 7. The Qt(r˜,z˜) fields shown in Figs. 10b and 10d are symmetric with respect to the normalized height z˜=0, due to the symmetricity of the Hill’s vortex model. Inside the bubbles, the contour lines depicting these fields correspond to the velocity streamlines shown in Fig. 6c by arrows. At the same time, outside the bubbles the fields Qt(r˜,z˜) look unchanged as Qt(r˜,z˜) deviations from Qt(∞) are small. The difference in Qt(r˜,z˜) magnitudes shown in Figs. 10b and 10d is explained by the decrease of Qt(∞) with height (Fig. 10b corresponds to z = 970 m, while Fig. 10d corresponds to z = 1480 m).

Fig. 10.
Fig. 10.

Examples of (a),(c) approximation of the total water content difference Qt(r) − Qt(∞) at two height levels and (b),(d) the corresponding Qt(r˜,z˜) fields inside the bubble and in its nearest surrounding. The normalized radius of the Hill’s vortex, equal to one, is shown in (b) and (d) by black semicircle.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

Figure 11 illustrates the approximation of another conservative quantity, namely, the liquid water potential temperature difference ΔΘl = Θl(r) − Θl(∞) at the same height levels as shown in Fig. 10 (Figs. 11a,c), and the corresponding Θl(r˜,z˜) fields inside the bubble and its nearest surrounding (Figs. 11b,d). Θl(∞) is the temperature taken far from the cloud. Equations (16) and (17) were also used for the approximation and field design. In the center of the bubble, the liquid water potential temperature is at minimum value, which is explained by the highly active condensation process in the cloud center [Eq. (3)]. The liquid water potential temperature monotonically increases from the center to the periphery up to the values of cloudless air space. Such spatial structure is explained by the presence of a vortex ring in the bubble. Θl(r˜,z˜) field is also symmetric with respect to normalized height z˜=0, as Qt(r˜,z˜).

Fig. 11.
Fig. 11.

Examples of (a),(c) approximation of the liquid water potential temperature difference Θl(r) − Θl(∞) at two height levels and (b),(d) the corresponding Θl(r˜,z˜) fields inside the bubble and its nearest surrounding. The normalized radius of the Hill’s vortex, equal to one, is shown in (b) and (d) by black semicircle.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

The possible reason of the large dispersion of Θl(r) − Θl(∞) and Qt(r) − Qt(∞) is the cloud asymmetry. It means that air volumes located at the same distance from the cloud core at the same time are at different distances from the cloud edges, and have different Θl(r) and Qt(r). Since entrainment increases with height, the spreading also increases with height. The same increase in the spreading with height takes place in case of vertical velocity.

Figures 11a and 11c demonstrate a negative correlation between Θl(r˜,z˜=0) and W(r˜) (the latter seen in Fig. 7, illustrating the approximation of the vertical velocity). In addition, many features of Qt(r˜,z˜) fields, mentioned in the analysis above, such as correspondence to velocity streamlines and the lack of matching at the bubble boundaries, are valid for the Θl(r˜,z˜) fields. At the same time, the fields in Figs. 10b and 10d clearly show the cloud core is located in the central part of a cloud. The natural assumption about conservatism of the quantities along the streamlines suggests an idea of the kind of structures that should exist outside the vortex ring during its ascent. As was mentioned above, the approximation of conservative quantities ΔQt and ΔΘl using Eq. (16) is carried out for each height level, so the obtained constants C0(zc) and C1(zc) are functions of the bubble center height. The quality of the approximation can be evaluated by comparing the vertical profiles of maximum values of ΔQt_max (or minimum values of ΔΘl_min) with the corresponding profiles of the approximating coefficient C0(zc) from Eq. (16). The profiles are shown in Fig. 12.

Fig. 12.
Fig. 12.

Vertical profiles of (a) the total water mixing ratio difference ΔQt(z) and (b) the liquid water potential temperature difference ΔΘl(z). The profiles obtained by fitting using Eq. (16) are shown by dotted lines.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

One can see a good approximation by an equation, following from the Hill’s vortex concept, up to the height 1600 m in case of ΔQt(z) and up to 1800 m in case of ΔΘl(z). In this regard, we notice that the updraft velocity profiles of the approximating Hill’s vortex, shown in Fig. 4b, are also in good mutual agreement up to the height 1600–1800 m. Since altitudes of 1600–1800 m are located within the inversion layer (Fig. 1), the increasing differences between the profiles seen in Fig. 12 can be attributed, to a large extent, to the influence of inversion on the vortex ring structure.

The analysis performed in section 4 indicates that the Hill’s vortex model captures basic features of the thermodynamic cloud structure caused by convective-scale motions. The influence of turbulent mixing between a cloud and surrounding air was excluded as much as possible from our analysis because of the filtration, so the vortex ring is most probably the main factor responsible for the cloud structure.

5. LWC inside bubble and adiabatic fraction

The most important quantity related to condensation/evaporation in a cloud is LWC ql(r, z) inside the cloud, which is usually compared to the adiabatic profile of LWC, qad(z) (Gerber 2000; Gerber et al. 2008; Khain et al. 2019; Konwar et al. 2021; Eytan et al. 2021, 2022). The adiabatic LWC profile reflecting the balance of liquid water in a slowly rising saturated adiabatic volume is a function of temperature and humidity profiles taken far from cloud (Pinsky et al. 2022, appendix A). This profile can be considered the background profile characterizing the environment where a cloud develops. Both measurements and LES show that adiabatic LWC is larger or equal to LWC in clouds, i.e., qad(z) ≥ ql(r, z). Accordingly, the ratio of these two quantities, called the adiabatic fraction, lies within the range of 0≤AF(r,z)=ql(r,z)/qad(z)≤1. Typical averaged AF values in Cu vary within the range of AF ∼ 0.4–0.6 (Gerber et al. 2008; Khain and Pinsky 2018; Katzwinkel et al. 2014). The AF values in the cloud core are close to unity. It is generally accepted that the values of adiabatic LWC and of AF rapidly fall to zero within the interface zone near cloud edge due to turbulent mixing. As was stated in the introduction, the cause of the AF decrease within a cloud body is the subject of the analysis. Pinsky et al. (2022) suggested that formation of AF in Cu is the result of circulation related to a vortex ring. In this section we show that the vortex ring inside a cloud can lead to decrease of LWC in the rising bubble and can explain the results of the observations and LES.

a. Derivation of approximating equations for LWC

We derive the equation for LWC and the adiabatic fraction under an assumption the supersaturation inside a rising adiabatic bubble being equal to zero. This assumption is suitable for our goals since it simplifies the analysis, and because supersaturation is small, especially in small Cu, and decreases above the cloud base in case of comparatively low accelerations of the vertical velocity (Pinsky et al. 2013, 2014). The assumption is quite accurate for cases of high aerosol (and droplet) concentration, as in our LES simulation. The equations that are derived will be associated with the quantity Ql obtained after the wavelet filtering. The transition of water vapor to the liquid phase in these conditions is described by
dQldt=∂Ql(r,z,t)∂t+u⋅grad[Ql(r,z,t)]=A1A2dzdt=A1A2W,
where Ql(r, z, t) is the liquid water mixing ratio; u(t, r, z) is a nondivergent air velocity vector field [div(u) = 0], and A1=[g/(RaT)][LwRa/(cpRυT)−1] and A2=1/qυ+Lw2/(cpRυT2) are coefficients considered constant inside the bubble. The definitions of other variables can be found in Table 2. The term (A1/A2)W in the right-hand side of Eq. (18) describes conversion of water vapor into liquid water at W > 0, and evaporation at W < 0. The integration of this term over time results in a linear adiabatic LWC profile qad(z)∼(A1/A2)z, where z is the height above the cloud-base level (Khain and Pinsky 2018). It should be remembered that the Eq. (18) is written in a nonnormalized coordinate system. As earlier, we neglect turbulence effects in this case.
We derive an approximation for LWC using the stationary solution of Eq. (18) and the stationary solution for the Hill’s vortex equation [Eqs. (4) and (5)] applying the latter as a model of a vortex ring inside the bubble. The stationary approach leads to the equation that has the same sense as Eq. (12):
u⋅grad[Ql(r,z)]=A1A2W(r,z).
In contrast with Eq. (12), the right-hand side of Eq. (19) is not equal to zero, reflecting the presence of an LWC source. The components of the velocity field u in the fixed cylindrical coordinate frame are determined by Eqs. (4) and (5). Equation (19) is valid inside Hill’s vortex r2 + (z − zc)2 < a2. Since the component of velocity, normal to the vortex boundaries, is equal to zero at r2 + (z − zc)2 = a2 (section 3), we do not consider the external LWC field. However, this statement does not mean that the solution below disregards the effects of noncloudy air. As droplets tend to evaporate in downdrafts, it will be shown below that some fraction of the subsiding shell that coincides with a part of the Hill’s vortex is noncloud.
The general solution of Eq. (19), valid inside the bubble is
Ql(r,z)=A1A2(z−zc)+Φ3{r22[3r2−5a2+3(z−zc)2]},r2+(z−zc)2<a2,
where Φ3 is an arbitrary differentiable function. The right-hand side of the solution (20) is the sum of two terms. The first one is related to the adiabatic increase of LWC inside the bubble with height. The second relates to the influence of the Hill’s vortex on LWC distribution inside the bubble. This term is symmetric with respect to the central horizontal plane z = zc. In the central horizontal plane of the bubble, the solution takes the form
Ql(r,zc)=Φ3{r22(3r2−5a2)},r2<a2.
Equation (21) shows that at the center of the bubble Ql(0, zc) = Φ3(0). In contrast to Eqs. (14) and (15), Eqs. (20) and (21) are written in a regular nonnormalized coordinate system.
Now we suggest that there is no liquid water outside the bubble at z = zc, so at r = a the solution (21) obeys the condition Φ3(a, zc) = 0. The simplest linear approximation of solution (21) is Φ3(x)=E0+E1x=E0+(E1r2/2)(3r2−5a2), where E0 and E1 are certain coefficients. This solution obeys the zero boundary condition Φ3(a, zc) = 0 if E0 and E1 are related by the ratio of E0 = E1a4. Therefore, the simplest linear solution obeying this condition is written as
Ql(r,zc)={E0[1+r2(3r2−5a2)2a4],r2<a2,0,r2≥a2,.
The height profile of LWC maximum E0(z) was estimated using the least squares approximation of quantity Ql(x, y) defined by Eq. (1). The approximation is carried out at every height level in the same way it was done for the vertical velocity profiles in section 3b. Figure 13 compares the obtained approximated profile with the adiabatic one, and with the profile of LWC ql at the points of maximum convective vertical velocity.
Fig. 13.
Fig. 13.

Comparison of the approximated profile E0(z) with the adiabatic one and with the profile of LWC taken at the points of maximum convective vertical velocity.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

The profile of LWC ql was later obtained using the unfiltered LWC field, an example of which is demonstrated in Fig. 3a. One can see a good agreement between the profiles up to height of 1800 m. Despite the fact that some values of E0(z) exceed the adiabatic ones, which is physically hardly possible, the result allows us to conclude that there is a small undiluted cloud core in the vortex ring center.

b. Rough estimate of the adiabatic fraction

As it was done in section 4, we generalize the linear solution, (22), in order to describe LWC and AF spatial distributions inside the bubble. Since in the bubble center r = 0, z = zc, LWC is approximately equal to the adiabatic one E0≈(A1/A2)zc (Fig. 13), spatial distribution of LWC inside the bubble can be roughly represented as
Ql(r,z)=A1A2(z−zc)+A1A2zc{1+r2[3r2−5a2+3(z−zc)2]2a4},r2+(z−zc)2<a2.
This equation shows that on the vertical axis crossing the center of the bubble, LWC corresponds to the linear adiabatic profile Ql(0,z)=(A1/A2)z, where z is measured from the cloud base. At the lowest point of the bubble LWC is equal to Ql(0,zc−a)=(A1/A2)(zc−a), whereas at the highest point LWC is equal to Ql(0,zc+a)=(A1/A2)(z+ca). These estimations show that LWC is not symmetric with respect to z = zc, in contrast to the case of conservative values. The asymmetry is caused by the existence of sources of the LWC.
From Eq. (23), the adiabatic fraction inside the bubble AF(r, z) can be represented as
AF=Ql(r,z)A1z/A2=A1A2z+A1A2zcr2[3r2−5a2+3(z−zc)2]2a4A1A2z=1+zczr2[3r2−5a2+3(z−zc)2]2a4.
Then, using normalized variables r˜ and z˜ defined in section 3a, and taking into consideration that z=zc+az˜, we transform this equation into the form
AF(r˜,z˜)=1−zczr˜2[5−3(r˜2+z˜2)]2=1−12(1+azcz˜)−1r˜2[5−3(r˜2+z˜2)].
Multiplier [1+(a/zc)z˜]−1 determines a certain asymmetry of the adiabatic fraction field with respect to the horizontal plane near the cloud base. At heights of several hundred meters over the cloud base and higher where zc>3a, this multiplier become close to one. This leads to a symmetry of AF(r˜,z˜) field, as seen from Eq. (24b):
AF(r˜,z˜)=1−12r˜2[5−3(r˜2+z˜2)].
Equations (24) have two major drawbacks. The first one is that in some areas located close to the bubble boundaries, where the downdrafts inside the bubble take place, Eq. (24) yields a negative solution. The negative values of AF should be interpreted as zero LWC in the subsiding shell of a cloud. The other drawback related to this feature, reflecting a certain roughness of Eq. (24a), is the jump of AF at the bubble boundaries, seen in Fig. 14.
Fig. 14.
Fig. 14.

A spatial distribution of the adiabatic fraction inside the bubble. (a) The radial profile of the adiabatic fraction AF(r˜,0) in the horizontal plane crossing the bubble center (z˜=0). (b) AF(r˜,z˜) field inside the bubble located close to the cloud base. (c) AF(r˜,z˜) field inside the bubble located at a significant height from the cloud base. The normalized radius of the Hill’s vortex, equal to one, is shown in (b) and (c) by black semicircles.

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

The spatial distribution of the adiabatic fraction inside the bubble, AF(r˜,z˜), calculated using Eq. (24a) is illustrated in Fig. 14. Figure 14a presents the radial profile of the adiabatic fraction AF(r˜,0) in the horizontal plane crossing the bubble center (z˜=0). Near the bubble axis, the maximum values of AF are close to one and then monotonically decrease down to zero toward the bubble border. Figures 14b and 14c show spatial distributions of the adiabatic fraction AF(r˜,z˜) inside the bubble. The distribution is pronouncedly asymmetric with respect to the plane z˜=0 at low altitudes (Fig. 14b), and becomes more symmetric as the height increases above the cloud base, as in Fig. 14c. All the panels demonstrate the presence of a cloud core containing the highest LWC close to the adiabatic one. The core located in the bubble center has a width on the order of 20% of the bubble diameter, i.e., about 4% of the bubble cross-sectional area. This value is close to the estimations made by high-resolution LES in the zone of the vortex ring (Eytan et al. 2022).

The results shown in Fig. 14 have a direct relationship to the mechanisms of cloud–environmental interaction. Figure 13 shows that a part of the vortex ring located in the area of the subsiding shell is occupied with the cloud. The appearance of the noncloud air is a result of total evaporation of droplets in the air leaving the cloud in the upper part of the vortex. Following the streamlines, this noncloud air penetrates the cloud boundary in the lower part of the vortex. Humidity of the noncloud air in the subsiding shell should be high, which leads to high humidity around the cloud. This conclusion is supported by high-frequency measurements (Gerber 2018). The relative humidity of the air penetrating a cloud is an important parameter that determines the droplet sizes near cloud edges. The conclusion about high air humidity is irrelevant in case of low-frequency measurements, since the subsiding shell is too narrow.

As regards the variables which are conservative in the moist adiabatic processes (Figs. 10 and 11), their values at the top of the vortex ring are equal to their values at the lower part of the vortex. This fact is directly related to the problem of the origin of entrained air in cumulus clouds (Paluch 1979; Heus et al. 2009; de Rooy et al. 2013). Paluch (1979) suggested vertical mixing in clouds and entrainment of environmental air from cloud top is a major factor of cloud formation. Many other measurements and LES indicate that entrainment takes place through the lateral boundaries. Analysis of Figs. 11 and 14 suggests that the properties of the surrounding air remain unchanged while it descends in the subsiding shell from the cloud top, and this phenomenon accounts for the surrounding air entrainment into clouds. However, the entrained air may not belong to the vortex ring. This air experiences the impact of detrained air at the level of an ascending cloud top, and moves out of the cloud remaining nearly at the same altitude. Then this air can enter the cloud when it experiences the impact of the entrainment flux caused by the vortex ring, when the ring ascends to a distance equal to its vertical size.

c. Top-hat adiabatic fraction inside CUZ

To conclude this section, we compare the top-hat liquid water adiabatic fraction inside the cloud updraft zone, estimated in Pinsky et al. (2022) with that obtained using the concept of a Hill’s vortex. For this purpose, we first calculate the CUZ radius using Eq. (24b) at height level z˜=0. The following fourth-order equation is to be solved:
AF(r˜,0)=1−12r˜2(5−3r˜2)=0.
A suitable solution of this equation is r˜0=2/3.
Then, the top-hat adiabatic fraction AF¯ is calculated by integrating AF(r˜,0) inside a circle of radius r˜0, in the same way it was done by Pinsky et al. (2022). The integration is carried out in polar coordinates:
AF¯=1πr˜02∫0r˜0∫02πr˜[1−12r˜2(5−3r˜2)]dαdr˜=1−54r˜02+12r˜04.
Substituting r˜0=2/3 into Eq. (26) one obtains the top-hat liquid water adiabatic fraction estimation:
AF¯=718≈0.39.
The comparison of this result with previously reported ones is presented in Fig. 15 showing the profiles of the adiabatic fraction obtained in LES by Pinsky et al. (2022) (black dotted line), and the profile of the constant value equal to 0.39 [Eq. (27)]. One can see a good agreement between the results.
Fig. 15.
Fig. 15.

Profiles of the adiabatic fraction obtained by Pinsky et al. (2022), superimposed the value presented in Eq. (27).

Citation: Journal of the Atmospheric Sciences 80, 8; 10.1175/JAS-D-22-0122.1

The value of the adiabatic fraction presented in Eq. (27) is a typical value measured in clouds (Gerber et al. 2008; Schmeissner et al. 2015). It should be noted that extrapolation of AF¯ value to the heights near the top and the bottom of a cloud is doubtful because of incorrectness of Eq. (23) for these areas. Also, the lateral entrainment of dry air into a bubble is not taken into account when using the Hill’s vortex model. These drawbacks lead to an underestimation of AF¯. Moreover, evaluations of AF and AF¯ based on the vortex ring concept cannot be applied to zones near the cloud base of a developing cloud, since a developed vortex ring with flow divergence in the upper part and flow convergence in the lower part arises at least several hundred meters above the cloud base. Below this level, the air ascending from cloud base has high values of AF (Gerber 2000; Gerber et al. 2008; Eytan et al. 2021). We also suppose that the vertical asymmetry of AF is related to closeness of the vortex to the cloud base where application of Hill’s vortex concept is problematic.

6. Discussion and conclusions

The study is aimed at analysis of dynamic, thermodynamic, and microphysical structures of an ascending bubble detected in LES simulations of a developing Cu. Representation of a developing cloud as a starting plume consisting of a head bubble followed by a nonstationary jet was found useful (Pinsky et al. 2022). The data for analysis are taken from 10-m-resolution LES of a trade wind cumulus cloud developing in BOMEX conditions. The circulation inside and outside the rising bubble was detected using wavelet filtering, as the low-frequency signal of the filtering corresponds to the convection motion. Spatial changes of thermodynamic and microphysical parameters related to convection are also highlighted due to wavelet filtering.

The novelty of the present work is in attempt to interpret the trade-cumuli dynamics in terms of the vortex-ring dynamics. The present study is based on representation of the rising bubble as a vortex ring, specifically as a round Hill’s vortex which is the simplest one for analysis. It has been proven the vortex ring plays the dominating role in formation of the horizontal cloud structure. Toroidal motions could be considered to be coherent structures within clouds, and the turbulence is riding on those motions and minimizing the heterogeneity. The major results obtained can be summarized as follows:

  • Hill’s vortex model is a reasonable model for describing the cloud dynamics of the head bubble at cloud developing stage. The updraft velocity and the size of the vortex, estimated in this study, are in good agreement with similar values obtained in the previous CUZ study (Pinsky et al. 2022). However, the model cannot describe the properties of the bubble wake, which require a more complicated model of an asymmetric rising vortex.

  • The analytical approximations for the radial profiles of the total water mixing ratio and the liquid water potential temperature inside and outside bubble are obtained. The vortex model describes reasonably well the changes of these conservative quantities both in cloud interior and cloud surroundings. The difference in the values between the interior and the surroundings is maximum in the center of cloud, and becomes less pronounced toward cloud periphery.

  • The vortex ring includes both the ascending cloudy air and some fraction of the descending noncloudy air (the subsiding shell). In this way the toroidal circulation contributes substantially to the process of entrainment and detrainment, i.e., to the process of cloud-surrounding interaction. The vortex ring determines cloud top, while the velocity of the ascending vortex determines the velocity of the cloud-top growth. For these reasons, cloud top ascends like a linear front despite the substantial radial differences in the vertical velocities along the radial direction.

  • The spatial distributions of LWC and the adiabatic fraction inside the bubble were analyzed, allowing us to conclude on the decisive role of the vortex ring circulation in formation of cloud microphysical structure. The Hill’s vortex model is a reasonable model of vortex ring, explaining the radial decrease of LWC and AF from the maximum adiabatic values in cloud center down to zero value in the subsiding shell.

The results demonstrate the existence of adiabatic cloud core with high values of LWC and AF, located in the cloud center up to the height of 1800 m. The circulation inside the vortex ring tends to decrease the size of the core: we found that the relative area of the undiluted core is a few percent, which agrees with LES results. The core seems to play the dominating role in cloud formation, being the source of the updrafts, which determine cloud development. Such a relatively small undiluted core indicates that the air from the subsiding shell penetrates the cloud over distances of several hundred meters. Penetration of such depth cannot be explained by turbulent diffusion, but only convective-scale vortex rings. The presence of a thin core also indicates that undiluted air penetrates the cloud up to higher altitudes, and the maximum LWC is in the core. It should be noted that an undiluted core is typically observed in deep convective clouds (e.g., Heymsfield et al. 1978). In small Cu, the area of the core is small enough to remain unnoticed in in situ measurements.

  • The top-hat value of the adiabatic fraction, calculated analytically using the Hill’s vortex concept, is equal to 0.39, which is a typical value measured in trade wind clouds.

  • The vortex ring with an inherent subsiding shell of humid air accounts for the high relative humidity of the air entrained through the cloud edges. In in situ measurements with low frequency, the humidity of the air entrained into the cloud is highly underestimated. The long-term discussion about the type of turbulent mixing at cloud edges (homogeneous versus inhomogeneous) (Devenish et al. 2012; Jensen et al. 1985; Korolev et al. 2016; Pinsky et al. 2016) suggests that the cloud–environment mixing volumes have substantially different relative humidity (RH). In case the RH of the entrained air is high, the types of mixing become undistinguishable.

  • The contribution of latent heat can be evaluated by the difference in the vertical velocities in cloud core from those in the center of the retrieved Hill’s vortex. The nonbuoyant Hill’s vortex has ∼20% lower maximum velocity.

In general, the analysis performed in this study allows us to conclude that a rising vortex ring plays an important role in formation of radial Cu thermodynamic and microphysical structure at the developing stage of cloud evolution. Since any influence of turbulent mixing between cloud and surrounding air was excluded, it was found that a vortex ring is the main factor forming Cu cloud structure. The findings stress the importance of the adiabatic motions.

We believe that the obtained results concerning the distribution of conservative values within a vortex ring resolve the problem of the origin of entrained air in cumulus clouds (Paluch 1979; Heus et al. 2009; de Rooy et al. 2013). Paluch (1979) suggested that entrainment in clouds take place in the vertical direction from the cloud top, so the values of the conservative quantities within a cloud can be obtained by interpolation between the cloud top and the cloud base. However, multiple measurements and LES indicate that entrainment takes place through the lateral boundary. The existence of a vortex ring allows one to propose a more natural explanation of Paluch finding. Indeed, the value of conservative variables in the cloud core are equal to the values at cloud base. The air diverging above cloud top (due to the vortex ring circulation) remains at the same level or slightly descends outside of the cloud where the vertical velocity is very low. As the cloud continues growing this air penetrates the cloud at the level of enhanced entrainment (lower part of the vortex ring).

Thus, in the presence of a vortex ring, the interpolation of the conservative variables between cloud base and cloud top in the vertical direction can be approximately similar to the horizontal interpolation between the cloud core and the close environment. This process requires additional analysis.

We have to add a comment concerning the existence of the TV in a developing convective cloud. We believe that any convective cloud contains vortices of the scales smaller than the scale of the cloud itself. In the present study we carried out wavelet filtering in an attempt to choose only the most pronounced vortex that determines the lateral entrainment in the entire small Cu cloud. Application of such filtering allowed to separate convective-like and turbulent scales. Supposedly, it is possible to find one such large vortex in a deep convective cloud as well. However, such vortex will contain many subvortices, as shown, for instance, by Grabowski and Clark (1991, 1993). However, such subvortices will be larger than the external turbulent scale and will characterize actually small clouds arising at the top of the deep cumulus making the whole cloud look like a cauliflower.

We realize that representation of a vortex ring in clouds by a Hill’s vortex has serious drawbacks, since the real vortex ring is more complicated and has a close relationship to its wake. The present study can be considered as the first attempt to show the importance of vortex rings arising in the upper part of the developing cloud. At the same time, the approximating equations obtained in the study can be usefully applied in current schemes of convective parameterization. It is clear that the present study does not explain the process of entrainment and detrainment because of the condition of zero normal velocity at the vortex boundaries. At the same time, the approximating equations obtained in the study can be usefully applied in current schemes of convective parameterization. All the results obtained in this study are applicable in structure analysis of both simplest trade wind clouds and other Cu, developed in absence of horizontal wind shear conditions. We also believe that the techniques used for retrieval of vortex ring applied in the study are also suitable for analysis of clouds developing within wind shear conditions.

Acknowledgments.

This research was supported by the Israel Science Foundation (Grants 2635/20, 1449/22), the Office of Science (BER), and partially supported by Grants DE-SC008811, DE-SC0014295, and ASR DE-FOA-1638 from the U.S. Department of Energy Atmospheric System Research Program.

Data availability statement.

The dataset on which this paper is based is too large to be retained or publicly archived with available resources. Documentation and methods used to support this study are available from Pinsky et al. (2021).

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    • Export Citation

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    • Search Google Scholar
    • Export Citation

  • Yano, J.-I., 2014b: Formulation structure of mass-flux convection parameterization. Dyn. Atmos. Oceans, 67, 1–28, https://doi.org/10.1016/j.dynatmoce.2014.04.002.

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    • Export Citation

  • Yano, J.-I., and B. Jakubiak, 2016: Wavelet-based verification of the quantitative precipitation forecast. Dyn. Atmos. Oceans, 74, 14–29, https://doi.org/10.1016/j.dynatmoce.2016.02.001.

    • Search Google Scholar
    • Export Citation

  • Yano, J.-I., M. W. Moncrieff, X. Wu, and M. Yamada, 2001a: Wavelet analysis of simulated tropical convective cloud systems. Part I: Basic analysis. J. Atmos. Sci., 58, 850–867, https://doi.org/10.1175/1520-0469(2001)058<0850:WAOSTC>2.0.CO;2.

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    • Export Citation

  • Yano, J.-I., M. W. Moncrieff, and X. Wu, 2001b: Wavelet analysis of simulated tropical convective cloud systems. Part II: Decomposition of convective-scale and mesoscale structures. J. Atmos. Sci., 58, 868–876, https://doi.org/10.1175/1520-0469(2001)058<0868:WAOSTC>2.0.CO;2.

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  • Yano, J.-I., R. Blender, C. Zhang, and K. Fraedrich, 2004a: 1/f noise and pulse-like events in the tropical atmospheric surface variabilities. Quart. J. Roy. Meteor. Soc., 130, 1697–1721, https://doi.org/10.1256/qj.03.42.

    • Search Google Scholar
    • Export Citation

  • Yano, J.-I., P. Bechtold, J.-L. Redelsperger, and F. Guichard, 2004b: Wavelet-compressed representation of deep moist convection. Mon. Wea. Rev., 132, 1472–1486, https://doi.org/10.1175/1520-0493(2004)132<1472:WRODMC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Zhao, M., and P. H. Austin, 2005: Life cycle of numerically simulated shallow cumulus clouds. Part II: Mixing dynamics. J. Atmos. Sci., 62, 1291–1310, https://doi.org/10.1175/JAS3415.1.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Vertical profiles of (a) the temperature, (b) the mixing ratio of water vapor, and (c) the relative humidity of the air measured far from the cloud (from Pinsky et al. 2022).

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Examples of (a),(c) unfiltered and (b),(d) filtered slices of (a),(b) vertical velocity and (c),(d) horizontal velocity components. The red arrows in (c) and (d) mark 5 m s−1 velocity scale.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    An example of (a),(c),(e) unfiltered and (b),(d),(f) filtered horizontal slices of the fields of thermodynamic and microphysical quantities: (a),(b) LWC, (c),(d) total water content, and (e),(f) liquid water potential temperature.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    (a) Height–time field of Wmax(z, t). The magenta line shows the z–t trajectory of the rising bubble (from Pinsky et al. 2022). (b) Vertical profiles of the velocities related to the head bubble: the maximum vertical air velocity (black solid line and red circles) and the velocity of the cloud-top ascent (blue dotted line with circles). The cloud-top height is defined as the maximum height where LWC exceeds 0.01 g kg−1. The bubble updraft velocity, calculated by differentiating of z–t trajectory, is shown by dashed black line. The “equivalent” Hill’s vortex translation velocity equal to (2/5)Wmax⁡(section 3c) is shown by blue stars.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Examples of the velocity field around the bubble in the (left) x–z and (right) y–z planes at times of (top) 29 and (bottom) 30.5 min. The red point in the center corresponds to the maximum vertical velocity Wmax(z, t) located on the z–t trajectory shown in Fig. 4a.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Velocity fields in the Hill’s vortex: (a) vertical velocity, (b) radial velocity, (c) velocity vectors, and (d) velocity vectors depicted in the coordinate system moving with translation velocity. The normalized radius of the Hill’s vortex, equal to one, is shown by black semicircle.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Examples of W(r, zc) approximation at two different heights located below and inside the inversion layer. Only points located on x–y plane at the distance of r < 1000 m from the bubble center are used in the approximation. The jump of derivative ∂W(r, zc)/∂r seen on the fitting line and marked in Fig. 7a, corresponds to the Hill’s vortex boundary.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    (a) Vertical profiles of the bubble radius and the CUZ radii. (b) Vertical profile of the translation velocity. Another profile in (b) is that of the maximum velocity calculated by dividing it by coefficient 5/2 is shown by dotted line.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    X–Z cross section in pressure perturbation field at 29 min. Pressure perturbation is positive above the cloud top and negative in the lower part of the toroidal vortex. The pink curve shows the cloud core with an adiabatic fraction exceeding 0.9.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Examples of (a),(c) approximation of the total water content difference Qt(r) − Qt(∞) at two height levels and (b),(d) the corresponding Qt(r˜,z˜) fields inside the bubble and in its nearest surrounding. The normalized radius of the Hill’s vortex, equal to one, is shown in (b) and (d) by black semicircle.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Examples of (a),(c) approximation of the liquid water potential temperature difference Θl(r) − Θl(∞) at two height levels and (b),(d) the corresponding Θl(r˜,z˜) fields inside the bubble and its nearest surrounding. The normalized radius of the Hill’s vortex, equal to one, is shown in (b) and (d) by black semicircle.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Vertical profiles of (a) the total water mixing ratio difference ΔQt(z) and (b) the liquid water potential temperature difference ΔΘl(z). The profiles obtained by fitting using Eq. (16) are shown by dotted lines.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    Comparison of the approximated profile E0(z) with the adiabatic one and with the profile of LWC taken at the points of maximum convective vertical velocity.

  • <sc>Fig</sc>. 14.
    View in gallery
    Fig. 14.

    A spatial distribution of the adiabatic fraction inside the bubble. (a) The radial profile of the adiabatic fraction AF(r˜,0) in the horizontal plane crossing the bubble center (z˜=0). (b) AF(r˜,z˜) field inside the bubble located close to the cloud base. (c) AF(r˜,z˜) field inside the bubble located at a significant height from the cloud base. The normalized radius of the Hill’s vortex, equal to one, is shown in (b) and (c) by black semicircles.

  • <sc>Fig</sc>. 15.
    View in gallery
    Fig. 15.

    Profiles of the adiabatic fraction obtained by Pinsky et al. (2022), superimposed the value presented in Eq. (27).

Fig. 1.

Vertical profiles of (a) the temperature, (b) the mixing ratio of water vapor, and (c) the relative humidity of the air measured far from the cloud (from Pinsky et al. 2022).
Fig. 2.

Examples of (a),(c) unfiltered and (b),(d) filtered slices of (a),(b) vertical velocity and (c),(d) horizontal velocity components. The red arrows in (c) and (d) mark 5 m s−1 velocity scale.
Fig. 3.

An example of (a),(c),(e) unfiltered and (b),(d),(f) filtered horizontal slices of the fields of thermodynamic and microphysical quantities: (a),(b) LWC, (c),(d) total water content, and (e),(f) liquid water potential temperature.
Fig. 4.

(a) Height–time field of Wmax(z, t). The magenta line shows the z–t trajectory of the rising bubble (from Pinsky et al. 2022). (b) Vertical profiles of the velocities related to the head bubble: the maximum vertical air velocity (black solid line and red circles) and the velocity of the cloud-top ascent (blue dotted line with circles). The cloud-top height is defined as the maximum height where LWC exceeds 0.01 g kg−1. The bubble updraft velocity, calculated by differentiating of z–t trajectory, is shown by dashed black line. The “equivalent” Hill’s vortex translation velocity equal to (2/5)Wmax⁡(section 3c) is shown by blue stars.
Fig. 5.

Examples of the velocity field around the bubble in the (left) x–z and (right) y–z planes at times of (top) 29 and (bottom) 30.5 min. The red point in the center corresponds to the maximum vertical velocity Wmax(z, t) located on the z–t trajectory shown in Fig. 4a.
Fig. 6.

Velocity fields in the Hill’s vortex: (a) vertical velocity, (b) radial velocity, (c) velocity vectors, and (d) velocity vectors depicted in the coordinate system moving with translation velocity. The normalized radius of the Hill’s vortex, equal to one, is shown by black semicircle.
Fig. 7.

Examples of W(r, zc) approximation at two different heights located below and inside the inversion layer. Only points located on x–y plane at the distance of r < 1000 m from the bubble center are used in the approximation. The jump of derivative ∂W(r, zc)/∂r seen on the fitting line and marked in Fig. 7a, corresponds to the Hill’s vortex boundary.
Fig. 8.

(a) Vertical profiles of the bubble radius and the CUZ radii. (b) Vertical profile of the translation velocity. Another profile in (b) is that of the maximum velocity calculated by dividing it by coefficient 5/2 is shown by dotted line.
Fig. 9.

X–Z cross section in pressure perturbation field at 29 min. Pressure perturbation is positive above the cloud top and negative in the lower part of the toroidal vortex. The pink curve shows the cloud core with an adiabatic fraction exceeding 0.9.
Fig. 10.

Examples of (a),(c) approximation of the total water content difference Qt(r) − Qt(∞) at two height levels and (b),(d) the corresponding Qt(r˜,z˜) fields inside the bubble and in its nearest surrounding. The normalized radius of the Hill’s vortex, equal to one, is shown in (b) and (d) by black semicircle.
Fig. 11.

Examples of (a),(c) approximation of the liquid water potential temperature difference Θl(r) − Θl(∞) at two height levels and (b),(d) the corresponding Θl(r˜,z˜) fields inside the bubble and its nearest surrounding. The normalized radius of the Hill’s vortex, equal to one, is shown in (b) and (d) by black semicircle.
Fig. 12.

Vertical profiles of (a) the total water mixing ratio difference ΔQt(z) and (b) the liquid water potential temperature difference ΔΘl(z). The profiles obtained by fitting using Eq. (16) are shown by dotted lines.
Fig. 13.

Comparison of the approximated profile E0(z) with the adiabatic one and with the profile of LWC taken at the points of maximum convective vertical velocity.
Fig. 14.

A spatial distribution of the adiabatic fraction inside the bubble. (a) The radial profile of the adiabatic fraction AF(r˜,0) in the horizontal plane crossing the bubble center (z˜=0). (b) AF(r˜,z˜) field inside the bubble located close to the cloud base. (c) AF(r˜,z˜) field inside the bubble located at a significant height from the cloud base. The normalized radius of the Hill’s vortex, equal to one, is shown in (b) and (c) by black semicircles.
Fig. 15.

Profiles of the adiabatic fraction obtained by Pinsky et al. (2022), superimposed the value presented in Eq. (27).
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