1. Introduction

Cloud cover is one of the main parameters governing the radiative properties of the atmosphere and affecting global climate. Shallow cumulus clouds in the trade wind regions are at the heart of the long-standing uncertainty in climate sensitivity estimates. Therefore, understanding the key factors determining cloudiness in shallow convective regimes has become an important topic of investigation (Vial et al. 2017).

The prediction of cloud cover and cloud lifetime using numerical models represents significant difficulties because of the large diversity of model responses related to fundamental differences in description of the processes controlling fields of cumulus clouds (Cu) and their operation in models. In coarse-resolution models, small nonprecipitating Cu are irresolvable and values of liquid water content (LWC) and cloud cover are calculated using simple parameterizations or are prescribed a priori (Doms and Schattler 2002; Plant and Yano 2015). Cloud cover is often parameterized as a function of mean relative humidity of the atmosphere. Dynamical and thermodynamical effects of shallow Cu in such models are evaluated from cloud mass flux, which is calculated based on very simple cloud representation as a jet or plume and some closure hypotheses.

An efficient way to improve cloud parameterization in the coarse-resolution models is a statistical analysis of results of large-eddy simulations (LES), using a grid spacing of a several tens to several hundreds of meters (Siebesma et al. 2003; Kogan and Mechem 2014; Dagan et al. 2017; Khain P. et al. 2019). In LES, the evolution of Cu fields is described by a complicated system of dynamical and microphysical equations, containing multiple parameters whose particular effects on cloud evolution is not simple to evaluate. In such situation, dynamically simple models that allow analytic or semianalytic analysis of the factors determining the cloud lifetime may be useful.

Since effect of environment humidity is manifested through the mechanisms of entrainment and mixing, the observable high sensitivity of cloud parameters to this humidity indicates the important role of entrainment–mixing influencing cloud parameters and cloud lifetime. Analysis of experimental data shows that entrainment in warm Cu takes place at the lateral cloud boundaries (de Rooy et al. 2013).

Beginning with study by Baker et al. (1980) a great number of studies were dedicated to investigating process of mixing of cloudy and droplet-free volumes located within an isolated volume (e.g., Baker and Latham 1982; Burnet and Brenguier 2007; Lehmann et al. 2009; Korolev et al. 2016; Pinsky et al. 2016a,b). Specific features of such approach is the tendency of the solution to the final equilibrium state, in which drop size distributions (DSDs) depend on the mixing type (Khain and Pinsky 2018). Note that there is a principal difference between the problem of mixing within the closed volume and the mixing at the cloud edges. It is because the process of the mixing at cloud edges has no equilibrium stationary solution.

Effects of entrainment and mixing between clouds and its surrounding on DSDs were investigated also in many in situ measurements (Lu et al. 2014; Bera et al. 2016a; Kumar et al. 2017). In several studies detailed structure of the interface zone was analyzed using DNS. Abma et al. (2013) investigated the role of buoyancy forces on the formation of the subsiding shell around shallow cumulus clouds. Sherwood et al. (2013) analyzed differences in the entrainment of different quantities.

In more detail the evolution of the interface zone between cloud air and dry environmental air was theoretically investigated by Pinsky and Khain (2018, 2019a), who found that entrainment–mixing process near lateral cloud boundaries leads to formation of a cloud–environment interface zone consisting of the cloud dilution zone (where LWC decreases with increasing distance from cloud interior), and of the humid shell whose humidity exceeds that of the dry environment. The time evolution of the position and width of these zones is analyzed by solving a diffusion–evaporation equation for an open region in the vicinity of the cloud–dry air interface. Upon normalization, the problem is reduced to a one-parametric one, the governing parameter being the potential evaporation parameter $R=\left(S_2/A_2{q}_1\right)<0$, where S₂ < 0 is supersaturation in the cloud environment, q₁ is the liquid water mixing ratio of the cloud, and A₂ is a parameter slightly depending on temperature (see appendix B). It was shown that widths of the dilution zone and of the humid shell increase with time. At R < −1_, the interface between the dilution zone and the humid shell (i.e., the cloud edge) moves toward the cloud interior (the cloud dissolves). Pinsky and Khain (2018, 2019a) evaluated the speed of the interface zone propagation. The calculations were performed under the assumption that the vertical velocity is negligibly small, so condensation (evaporation) caused by updrafts (downdrafts) was not taken into account. Another simplification used in these calculations is the assumption that the cloud volume is infinite.

The neglecting vertical velocity is a quite crude simplification, because it means neglecting adiabatic cooling/warming, which play the dominating role in cloud evolution and determine stages of cloud evolution. According to Katzwinkel et al. (2014), the evolution of Cu includes three stages: (i) actively growing clouds (w > 0_, positive buoyancy force), (ii) decelerated clouds (w > 0, negative buoyancy force), and (iii) dissolving clouds (w < 0, negative buoyancy force). We suppose that improvement of the parameterization of Cu in the coarse-resolution models should include description of all these stages. The active growth stage was analytically investigated by Pinsky and Khain (2019b, hereafter Part I) using a minimalistic cloud model taking into account droplet growth in updrafts and evaporation due to mixing with the dry environment. Analytical estimations of such important cloud characteristics as the adiabatic fraction, droplet concentration and the mean droplet volume radius were calculated as a function of spatial coordinates. The comparison with observations and LES model results showed that a minimalistic model can be useful to evaluate the effects of mixing on cloud structure. The duration of the first stage is determined largely by the atmospheric instability that, in turn, determines the updrafts and the maximum cloud depth.

The process of cloud dissipation is investigated to much less extent than cloud development. Cloud evolution at the dissolving stage was analyzed recently by Seeley et al. (2019), where the problem of long lifetime of tropical anvil clouds was investigated. The anvil clouds were described by a simple model of a cloud parcel, which was assumed uniform. Only integral effects of mixing and evaporation on cloud parcel was considered. The equation for total water mass fraction was derived using a “cloud growth” parameter, which has the meaning of the time needed to increase the cloud area by its initial value. This parameter was treated as a tuning parameter, which was chosen equal to 19 min. The analytical solution of the equation for total water mass fraction obtained in this study showed that “mixing only” cloud time is proportional to the parameter χ_c, which is inverse proportional to the potential evaporation parameter R.

Here we present Part II of the study, where we investigate the mechanisms determining dissolving of nonprecipitating warm Cu using a minimalistic model. At the cloud dissolving stage, buoyancy becomes negative, which leads to settling of the cloud body, accompanied by fast droplet evaporation. Subsaturation in a dissolving cloud is caused by the adiabatic temperature increase in downdrafts and turbulent mixing of clouds with the dry environment. Therefore, at the dissolving stage, two processes determine cloud dissipation: downdraft and mixing with the dry environment (scheme in Fig. 1). The comparative role of these processes is of high interest. Note that better understanding of factors affecting the cloud dissipation and estimation of the dissolving time is an important component of any Cu parameterization, in which cloud evolution and time dependence of cloud cover are predicted.

Schematic illustration of a dissolving Cu analyzed in this study.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Schematic illustration of a dissolving Cu analyzed in this study.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Schematic illustration of a dissolving Cu analyzed in this study.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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We analytically estimate the dissolving time of nonprecipitating Cu at the dissipation stage. This time is defined as the time of complete droplet evaporation. In contrasts to our previous studies (Pinsky and Khain 2018, 2019a), we assume here that cloud descends at velocity of w < 0. The evaluation is performed under the following simplification conditions:

• Only mechanisms influencing droplet evaporation (i.e., mixing with dry environment and the descent of the cloud volume leading to increasing in air temperature) are taken into account. Effects of droplet collisions and settling are neglected. The sedimentation is neglected because the radii of cloud droplets typically do not exceed 20 μm, having a terminal velocity on the order of 4 cm s⁻¹, that is, much smaller than downdraft velocity. The tail of the largest droplets in nonprecipitating clouds, which can appear as result of collisions, does not contain substantial LWC (otherwise the cloud would produce rain).

• Mixing caused by turbulence is carried out in the horizontal direction. The surrounding dry-air area is assumed to be infinitely large. The turbulent intensity is characterized by the turbulent coefficient, which is assumed uniform over space and time. Mixing changes microphysical properties in horizontal direction inside cloud in contrast with uniform cloud parcel model developed in other investigations (e.g., Seeley et al. 2019) Note that there are no reliable data about the changes of the turbulent coefficient in the horizontal direction. The assumption that turbulent coefficient is constant in the horizontal though the entire cloud is the natural simplification in the minimalistic model. Note that according to in situ observations, dissipation rate in small Cu is nearly constant (Katzwinkel et al. 2014). The vertical velocity of the air inside a cloud is uniform and does not change over time.

• The microphysical processes, such as droplet nucleation, droplet collisions and droplet settling, are not considered.

2. Brief description of the minimalistic model

The minimalistic model is described in detail in Part I of the study (see also appendix A). Here we present a brief description. A simplified geometry of a cloud is presented in Fig. 1. The cloud of the horizontal size of 2L at the beginning of the dissipation stage is characterized by initial temperature T₁, liquid water mixing ratio q₁, relative humidity of 100% (supersaturation over water is equal to zero S₁ = 0), and vertical velocity w directed downward. The cloud is surrounded by dry air with negative supersaturation (S₂ < 0). The cloud descends at velocity w₁ and simultaneously mixes with the surrounding air through turbulent diffusion characterized by turbulent coefficient K.

The analysis is performed for quantity Γ(x, t) defined as

$\mathrm{\Gamma }\left(x,t\right)=S\left(x,t\right)+A_{2}q\left(x,t\right),$(1)

where q(x, t) is the mixing liquid water ratio, A₂ is a coefficient depending on temperature (see Table B1 for definition). Taking into account a comparatively small vertical shift of clouds during the dissipation stage, A₂ is assumed constant. Quantity Γ(x, t) is conservative with respect to phase transition when motions in the horizontal direction are considered (see Part I). It is a sum of two terms, the second one shows what liquid mass can be potentially evaporated by mixing with surrounding, and the first term shows the ability of the environmental air to evaporate liquid detrained from the cloud. Parameter Γ(x, t) can be called available liquid content (normalized). The initial state of function Γ(x, 0) at t = 0 is given by a rectangle function (Fig. 2). Parameter Γ(x, 0) is positive inside the cloud and negative outside it:

$\mathrm{\Gamma }\left(x,0\right)=\{\begin{array}{c}{\mathrm{\Gamma }}_1=A_2{q}_1\text{if}\left|x\right|\le L\\ {\mathrm{\Gamma }}_2=S_2\text{if}\left|x\right|>L\end{array}.$(2)

The initial state of function Γ(x, 0) at t = 0.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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The initial state of function Γ(x, 0) at t = 0.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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The initial state of function Γ(x, 0) at t = 0.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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As seen from Eq. (4) there is no feedback between vertical velocity and Γ in the minimalistic model. It makes the model simple and allows to obtain analytical solution. Strictly speaking, vertical velocity is determined by combined effects of buoyancy and pressure forces (Khain and Pinsky 2018). In the present study these effects are taken into account implicitly without solving the equation for vertical velocity. Note that the prescription of a constant vertical velocity is typical for simple parcel models (Khain and Pinsky 2018).

3. Evolution of the liquid water mixing ratio

Although the accuracy of expression (6) decreases toward cloud edge, we apply Eq. (6) within the whole interval |x| ≤ L. Since q(x, t) ≥ 0, one can assume that the cloud boundaries correspond to Γ(x, t) = 0. The zone where Γ₂ < Γ(x, t) < 0 is a humid shell, that is, the zone of higher humidity (Part I).

Results of the minimalistic model are compared to the data reported by Katzwinkel et al. (2014) and Schmeissner et al. (2015). In these studies helicopter-borne observations of the cloud microphysical properties in shallow trade wind cumuli have been performed. The data was collected during the Cloud, Aerosol, Radiation and Turbulence in the Trade Wind Regime over Barbados (CARRIBA) project, in November 2010. Basic meteorological parameters (3D wind velocity vector, air temperature, and relative humidity), cloud condensation nuclei concentrations, and cloud microphysical parameters (droplet number concentration, size distribution, and liquid water content) were measured by the Airborne Cloud Turbulence Observation System (ACTOS), which is fixed by a 160-m-long rope underneath a helicopter flying with a true airspeed of approximately 20 m s⁻¹. More information about general conditions during CARRIBA can be found in the study by Siebert et al. (2013).

Figure 3 illustrates the evolution of q(x, t) in the course of cloud dissolution at different intensities of turbulent mixing, characterized by the turbulent coefficient, and at different downdraft velocities. As discussed in Pinsky and Khain (2019a), the value of turbulent coefficient K = 10 m² s⁻¹ produces the best agreement with observations in a developing Cu. In a dissolving Cu, the value of K is supposedly lower, because Cu often dissipate within the inversion zone and because downdrafts in dissolving clouds are typically weaker than updrafts in developing Cu (Katzwinkel et al. 2014; Schmeissner et al. 2015). According to in situ measurements analyzed in these studies, the subsidence velocities in dissolving Cu vary within the range of −2 ≤ w ≤ 0 m s⁻¹. The illustrations in Fig. 3 correspond to selected velocities of 0 m s⁻¹ (Figs. 3a–c), −0.5 m s⁻¹ (Figs. 3d–f), and −1 m s⁻¹ (Figs. 3g–i). Cloud width chosen is typical of small Cu (2L = 400 m).

Evolution of the cloud liquid water mixing ratio in the course of cloud dissolving at different values of the turbulent coefficients and different downdraft velocities. Other parameters of the simulations are: 2L = 400 m, q₁ = 1.25 g kg⁻¹, T = 20°C, and S₂ = −40%. The red contour corresponding to q(x, t) = 0 marks the cloud boundary.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Evolution of the cloud liquid water mixing ratio in the course of cloud dissolving at different values of the turbulent coefficients and different downdraft velocities. Other parameters of the simulations are: 2L = 400 m, q₁ = 1.25 g kg⁻¹, T = 20°C, and S₂ = −40%. The red contour corresponding to q(x, t) = 0 marks the cloud boundary.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Evolution of the cloud liquid water mixing ratio in the course of cloud dissolving at different values of the turbulent coefficients and different downdraft velocities. Other parameters of the simulations are: 2L = 400 m, q₁ = 1.25 g kg⁻¹, T = 20°C, and S₂ = −40%. The red contour corresponding to q(x, t) = 0 marks the cloud boundary.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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The red contours on the panels corresponding to q(x, t) = 0 can be regarded as the boundary between the cloud and its environment. The values of q(x, t) are of about 0.2–0.8 g kg⁻¹, which agree with the observations. The figure allows to compare the role of the two main factors determining cloud dissipation: the cloud-air subsidence and mixing with the dry surrounding air. One can conclude that both factors are important at the dissipation stage. The cloud subsidence is a more meaningful factor determining the decrease in LWC maximum. Indeed, at w₁ = 0 the cloud does not completely dissolve during 1200 s at any K ≤ 20 m² s⁻¹, while at w₁ = −1 m s⁻¹ the cloud completely dissolves at t = 600 s even at K = 1 m² s⁻¹. At the same time, mixing has a stronger impact at cloud shape [i.e., the dependence of q(x)] since it occurs unevenly inside the cloud stratum, being more intensive at cloud edge. The comparative effects of subsidence and mixing depend also on cloud width (as at the active growth stage as shown in Part I). In wider clouds, the effect of mixing on the cloud interior weakens, so the effect of subsidence becomes dominating. In contrast, in very small Cu (with 2L < 200 m) one can expect a stronger effect of mixing. This statement is illustrated in Fig. 4 where the evolution of the liquid water mixing ratio in the course of cloud dissolution is shown for different cloud widths ranging from 200 to 1000 m. The intensity of turbulence (K = 10 m² s⁻¹) and the subsidence velocity (w₁ = −0.5 m s⁻¹) are assumed similar in all cases. One can see that when cloud width is equal to 1000 m, the dissolution process requires more than 1100 s. The dissolving time decreases to 400 s (i.e., ~3 times) when the cloud width is reduced to 200 m. So, the impact of mixing strongly increases as the cloud width decreases.

Evolution of the cloud liquid water mixing ratio in the course of cloud dissolving at different cloud sizes: (a) 2L = 1000 m, (b) 2L = 400 m, and (c) 2L = 200 m. Other parameters of the simulations are: w₁ = −0.5 m s⁻¹, K = 10 m² s⁻¹, q₁ = 1.25 g kg⁻¹, T = 20°C, and S₂ = −5%. The red contour corresponding to q(x, t) = 0 marks the cloud boundary.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Evolution of the cloud liquid water mixing ratio in the course of cloud dissolving at different cloud sizes: (a) 2L = 1000 m, (b) 2L = 400 m, and (c) 2L = 200 m. Other parameters of the simulations are: w₁ = −0.5 m s⁻¹, K = 10 m² s⁻¹, q₁ = 1.25 g kg⁻¹, T = 20°C, and S₂ = −5%. The red contour corresponding to q(x, t) = 0 marks the cloud boundary.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Evolution of the cloud liquid water mixing ratio in the course of cloud dissolving at different cloud sizes: (a) 2L = 1000 m, (b) 2L = 400 m, and (c) 2L = 200 m. Other parameters of the simulations are: w₁ = −0.5 m s⁻¹, K = 10 m² s⁻¹, q₁ = 1.25 g kg⁻¹, T = 20°C, and S₂ = −5%. The red contour corresponding to q(x, t) = 0 marks the cloud boundary.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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In all the examples presented in the Figs. 3 and 4, the cloud width decreases with time. As was shown by Pinsky and Khain (2018, 2019a), narrowing/broadening of clouds due to lateral mixing (at w = 0) is determined by the potential evaporation parameter $R=\left(S_2/A_2{q}_1\right)<0$. In our case $R=\left[-0.4/\left(223\times 0.001\hspace{0.17em}25\right)\right]=-1.46<-1$, which corresponds to cloud narrowing. Figure 5 illustrates evolution of q(x, t) at R > −1. The relative humidity in the cloud environment was set equal to 95%, so potential evaporation parameter value is $R=\left[(0.95-1)/\left(223\times 0.001\hspace{0.17em}25\right)\right]=-0.18>-1$.

Evolution of the cloud liquid water mixing ratio in the course of cloud dissolving at potential evaporation parameter R = −0.18 > −1 and different turbulent coefficients. Other parameters of the simulations are: w₁ = −0.5 m s⁻¹, 2L = 400 m, q₁ = 1.25 g kg⁻¹, T = 20°C, and S₂ = −40%. The red contour corresponding to q(x, t) = 0 marks the cloud boundary.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Evolution of the cloud liquid water mixing ratio in the course of cloud dissolving at potential evaporation parameter R = −0.18 > −1 and different turbulent coefficients. Other parameters of the simulations are: w₁ = −0.5 m s⁻¹, 2L = 400 m, q₁ = 1.25 g kg⁻¹, T = 20°C, and S₂ = −40%. The red contour corresponding to q(x, t) = 0 marks the cloud boundary.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Evolution of the cloud liquid water mixing ratio in the course of cloud dissolving at potential evaporation parameter R = −0.18 > −1 and different turbulent coefficients. Other parameters of the simulations are: w₁ = −0.5 m s⁻¹, 2L = 400 m, q₁ = 1.25 g kg⁻¹, T = 20°C, and S₂ = −40%. The red contour corresponding to q(x, t) = 0 marks the cloud boundary.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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One can see that in this case the cloud width increases with time independently of the turbulent coefficient [in accordance with results obtained by Pinsky and Khain (2018, 2019a)]. Closer to the moment of complete evaporation of cloud droplets, the horizontal profile of the liquid water mixing ratio q(x) becomes nonmonotonic with an additional maximum located near cloud edges. Such nonmonotonic profiles are sometimes measured in real clouds (Gerber et al. 2008; Bera et al. 2016a,b). We attribute this effect to the following. Within the cloud core, subsaturation is formed largely by air subsidence. Near cloud edge, the impact of mixing becomes stronger and depends on the RH of the surrounding air. If RH of the surrounding air is lower than the RH formed by subsidence, the LWC further decreases near the cloud edges (as in Figs. 3 and 4). If the RH of the surrounding air exceeds the RH that formed by subsidence, mixing prevents fast droplet evaporation within the dilution zone of the subsiding cloud. Figure 5 shows, therefore, that the condition of cloud broadening/narrowing derived by Pinsky and Khain (2018) for a motionless cloud is also applicable to weak subsidence conditions. However, at R ~ −1 the cloud is monotonously narrowing like at R < −1_. Thus, the condition becomes inapplicable because downdraft strongly influences the movement of cloud boundaries, causing the cloud to narrow with time. For example, at w₁ = −0.5 m s⁻¹, R ≈ −0.7 is the boundary value separating the regimes of temporal broadening and narrowing of the cloud over time.

The cloud structure and its effect on the humidity of the environment air is further illustrated in Fig. 6 showing the horizontal profiles of quantity Γ(x)/A₂ at t = 600 s.

Horizontal profiles of function Γ(x)/A₂ at t = 600 s calculated for different values of w₁ and K. (a)–(c) RH₂ = 60%. (d)–(f) RH₂ = 95%. Other parameters of the simulations are 2L = 400 m, q₁ = 1.25 g kg⁻¹, and T = 20°C. The dashed black line in (a), (b), (d), and (e) separates the region containing liquid water from the area of high humidity.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Horizontal profiles of function Γ(x)/A₂ at t = 600 s calculated for different values of w₁ and K. (a)–(c) RH₂ = 60%. (d)–(f) RH₂ = 95%. Other parameters of the simulations are 2L = 400 m, q₁ = 1.25 g kg⁻¹, and T = 20°C. The dashed black line in (a), (b), (d), and (e) separates the region containing liquid water from the area of high humidity.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Horizontal profiles of function Γ(x)/A₂ at t = 600 s calculated for different values of w₁ and K. (a)–(c) RH₂ = 60%. (d)–(f) RH₂ = 95%. Other parameters of the simulations are 2L = 400 m, q₁ = 1.25 g kg⁻¹, and T = 20°C. The dashed black line in (a), (b), (d), and (e) separates the region containing liquid water from the area of high humidity.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Positive values of quantity Γ(x, z)/A₂ most likely correspond to q(x, t) > 0, while the area of negative values is the humid shell around the cloud. The negative tails of the curves correspond to the values of subsaturation (S < 0) of the environment air, normalized on A₂. Panels in the figure differ by the relative humidity of the environment air (RH₂ = 60% for Figs. 6a–c and with RH₂ = 95% for Figs. 6d–f) and by the downdraft velocity (0 m s⁻¹ for Figs. 6a,d, −0.5 m s⁻¹ for Figs. 6b,e, and −1 m s⁻¹ for Figs. 6c,f). The dash black lines in Figs. 6a, 6b, 6d, and 6e separate the region containing liquid water from the area free of liquid water but having high humidity, while Figs. 6c and 6f correspond to the situation when liquid water has completely evaporated.

One can see that at RH₂ = 60% (Figs. 6a,b) clouds are narrower than they had been initially (2L = 400 m), whereas at RH₂ = 95% (Figs. 6d,e) clouds are wider than initially, although in both cases the interface zone broadens due to turbulent diffusion. This phenomenon was explained above. The increase in the turbulent coefficient and especially in the subsidence velocity accelerates cloud dissolution. The moment of total liquid water evaporation can be seen in Fig. 6b for K = 20 m² s⁻¹ (black curve). The curve shows that the last water evaporates at the center of the cloud. At a downdraft speed of −1 m s⁻¹, after 10 min, the profile Γ(x)/A₂ has maxima at cloud edges at |x| > L. This result shows that cloud dissolution at high environment humidity can lead to the appearance of several spots of higher humidity separated by distances exceeding the size of the initial cloud.

These simulations were performed under the condition of uniform downdraft velocity inside a dissolving cloud. When the subsidence is more intensive in the central part of the cloud (see, e.g., Katzwinkel et al. 2014), cloud dissolution can lead to formation of several small clouds along the periphery of the parent cloud. This effect interpreted as cloud breakup may have in impact on cloud cover.

4. Estimation of dissolving time

To analyze the issue of the dissolving time t_dis, we should establish the time that corresponds to complete droplets evaporation. Since the LWC maximum of is usually reached at the cloud center, at least at comparatively low RH of the environment air (at x = 0—see Figs. 3–6), one can use the condition Γ(0, t_dis) = 0 to estimate cloud dissolving time. It is noteworthy that dissolution of a cloud does not mean a complete disappearance of its remains. The area of increased humidity remains in place of the cloud and its surrounding as shown in Figs. 6c and 6f.

To illustrate the impact of different parameters on the dissolving time, Eq. (12) was solved for different values of the potential evaporation parameter R, downdraft velocity w₁ and turbulent coefficient K. The corresponding dependences are presented in Figs. 7 and 8 .

Dependence of dissolving time on parameter R at different values of turbulent coefficient K and different downdraft velocities w₁: (a) 0, (b) −0.5, and (c) −1 m s⁻¹. Other parameters of the simulations are: 2L = 400 m, q₁ = 1.25 g kg⁻¹, and T = 20°C.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Dependence of dissolving time on parameter R at different values of turbulent coefficient K and different downdraft velocities w₁: (a) 0, (b) −0.5, and (c) −1 m s⁻¹. Other parameters of the simulations are: 2L = 400 m, q₁ = 1.25 g kg⁻¹, and T = 20°C.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Dependence of dissolving time on parameter R at different values of turbulent coefficient K and different downdraft velocities w₁: (a) 0, (b) −0.5, and (c) −1 m s⁻¹. Other parameters of the simulations are: 2L = 400 m, q₁ = 1.25 g kg⁻¹, and T = 20°C.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Dependence of dissolving time on downdraft velocity at different values of turbulent coefficient K. (a) RH₂ = 60%. (b) RH₂ = 95%. Other parameters of the simulations are 2L = 400 m, q₁ = 1.25 g kg⁻¹, and T = 20°C.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Dependence of dissolving time on downdraft velocity at different values of turbulent coefficient K. (a) RH₂ = 60%. (b) RH₂ = 95%. Other parameters of the simulations are 2L = 400 m, q₁ = 1.25 g kg⁻¹, and T = 20°C.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Dependence of dissolving time on downdraft velocity at different values of turbulent coefficient K. (a) RH₂ = 60%. (b) RH₂ = 95%. Other parameters of the simulations are 2L = 400 m, q₁ = 1.25 g kg⁻¹, and T = 20°C.

Citation: Journal of the Atmospheric Sciences 77, 3; 10.1175/JAS-D-19-0118.1

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Analysis of the figures shows the following:

• Reasonable times of cloud dissolving (e.g., t_dis < 15 min) can be obtained only if the downdraft velocity exceeds about 0.5 m s⁻¹. Such values of downdrafts are typical in dissolving clouds (Katzwinkel et al. 2014; Schmeissner et al. 2015). At lower downdrafts, the dissolving time becomes unrealistically long. Therefore, the downdraft velocity is the major parameter causing the cloud dissolving time.

• Turbulent intensity characterized by the turbulent coefficient affects the dissolving time to some extent. Changing the value of the turbulent coefficient from 1 m² s⁻¹ (not intensive mixing) to 20 m² s⁻¹ (very intense mixing) leads to a reduction the dissolving time twice, especially at small downdraft velocities.

• Cloud dissolving time increases rapidly with decreasing R, which is proportional to the ratio of the amount of water vapor that should be added in order to saturate the unit volume of the environment air to the amount of liquid water available for evaporation in a unit cloud volume. Increasing values of R hinder the evaporation of remaining liquid water in the cloud. Therefore, the dryer the environment air, the faster a cloud dissipates.

Thus, subsidence of droplets and their evaporation in downdraft is the main reason of cloud dissipation at least for wide clouds. At the same time mixing with dry environment can substantially influence dissolving stage of small Cu.

5. Discussion and conclusions

A minimalistic analytical model allowing analysis of the dissipation stage of nonprecipitating convective clouds is proposed. The model considers cloud dissolution as a result of two processes: turbulent mixing with dry environment and cloud volume settling. The temporal changes of the spatial structure of a cloud and its immediate environment in the course of cloud dissolving are analyzed. The analyzed analytical equation differs from the equation derived in Part I of the paper by the initial conditions and the direction of cloud volume motion. An equation for cloud dissolving time is obtained in frame of this study.

Despite the model simplicity, it exhibits significant advantage over a great number of cloud parcel models, because it describes horizontal inhomogeneity caused by entrainment–mixing process. The important evidence of model validity is the realistic evaluation of lifetime (~10–15 min) at observed velocities of subsidence. Parabolic-like profiles of LWC in small clouds and trapezoidal-like form in large clouds also seem quite reasonable.

The comparison of the role of two main factors determining cloud dissipation, that is, cloud-air subsidence leading to an increase in temperature, and mixing with dry surrounding air, shows that both factors are important at the dissipation stage. However, the subsidence is a more meaningful factor determining a decrease in LWC maximum in the course of the cloud volume descent, while mixing has a greater impact on the cloud shape since it acts stronger near cloud borders than inside a cloud.

Comparative effect of downdrafts depends on the subsiding velocity. The effect of downdrafts dominates, when velocity is strong enough (>1 m s⁻¹). In the case of small Cu with zero or even positive vertical velocities, cloud lifetime is determined by mixing with environmental air: the smaller cloud and the lower air humidity in cloud environment, the stronger effect of mixing is.

Narrowing/broadening of clouds due to lateral mixing with dry air in the course of cloud dissolution is determined by the potential evaporation parameter R. We presented two examples, in the first example, R = −1.46 < −1, which corresponds to cloud narrowing. In the second example, R = −0.18 > −1 and the cloud first broadens and then narrows. Both examples are in accordance with the results obtained earlier by Pinsky and Khain (2018). We also defined dimensionless parameter β ≥ 0 [see Eq. (11)], which characterizes the comparative contribution of subsidence and mixing to cloud dissolution. The higher mixing intensity is, the smaller the value of β. In a case when R is close to −1, parameter β, which is proportional to the downdraft velocity, strongly impacts the movement of cloud boundaries, causing the cloud to narrow with time.

The dissolution of cloud does not mean complete destruction of its remnants. A region of higher humidity remains in place of the cloud. The area and location of the humid air remaining after a complete cloud dissolving depend on the environment humidity. At high humidity, cloud dissolving leads to formation of several regions of higher enhanced humidity. We suppose that inhomogeneous subsidence within a cloud can lead to formation of several clouds in the vicinity of the parent cloud, which can be interpreted as breakup of clouds. This breakup is not caused by a horizontal wind shear, but by a combined effect of subsidence and mixing.

The dissolving time is determined as the time of total droplets evaporation. The main parameter determining the dissolving time is the downdraft velocity that should exceed 0.5 m s⁻¹ in order to provide reasonable values of this time (t_dis < 15 min). Turbulent intensity also affects the dissolving time; for example, the change of the turbulent coefficient value from 1 m² s⁻¹ (not intensive mixing) to 20 m² s⁻¹ (very intensive mixing) reduces the dissolving time by half. Yet another parameter affecting the dissolving time is R. The larger the value of R, the more difficult it is for the remaining liquid water in the cloud to evaporate. Therefore, the lower the LWC and the humidity of the environment air, the faster a total cloud dissolves.

Part I and Part II of this study show that the following parameters affect evolution of nonprecipitating Cu: the humidity and temperature of the environment, LWC at the cloud top (in case of cloud dissolution), cloud width, the subsidence velocity, and the turbulent coefficient. We believe that these parameters will be useful for analysis and classification of observed data and results of LES. We also believe that the presented simple solution can be useful for analysis of the cloud dissipation stage and may be included into procedures of parameterization of cloud cover caused by nonprecipitating or slightly precipitating Cu.

We would like to stress substantial difference in the relative role of entrainment–mixing and evaporation/condensation at developing and decaying stages. The entrainment–mixing process affects cloud by dilution at the cloud edges. In developing clouds, cloud updrafts tend to support supersaturation, and in the case of wide clouds, the effect of mixing on the cloud core is small, keeping the cloud core undiluted. At the same time, descending of the entire cloud affects cloud microphysics not only at cloud edges, but inside the cloud core as well. As a result, when subsidence is fast enough, cloud settling plays dominating role in cloud decay determining the dissolving time.

Note that the finding about the important role of cloud subsidence does not contradict to the role of cloud dilution, especially for very thin clouds in dry environment.

At present we cannot propose detailed algorithm of Cu parameterization. Note that concepts of shallow Cu parameterization in many coarse-resolution models (Plant and Yano 2015) use the assumption of immediate formation and immediate dissipation of clouds. The minimalistic model allows one to consider time changes of structure of particular clouds that can be used to create more advanced parameterizations. We would like to stress that one of the main tasks of this study is to find governing parameters causing cloud evolution and determining cloud lifetime that can be used for such parameterization. Some of such parameters (like the environment humidity and temperature) are calculated in the coarse-resolution models. Other parameters and their distributions can be determined from LES of cloud ensembles forming at these environment humidity and temperatures (e.g., Khain et al. 2019). Note also that cloud ensembles are often in a quasi-stationary state. It means that number of dissipating clouds during a sufficiently large time is approximately equal to the number of newly born and growing clouds, so the dissolving time may be useful to evaluation of general properties of cloud ensembles.

In the present study we considered nonprecipitating clouds because most small Cu are nonprecipitating. Parameterizations of shallow Cu in coarse-resolution models also assume that these clouds are not precipitating. At the same time in some clean maritime regions small Cu produce raindrops. The question arises: how does raindrop formation affect the cloud lifetime? It is not simple question. On one hand, rain formation and rainfall decreases the time during which droplets are suspended in the cloud. On the other hand, droplets in precipitating clouds are larger than in nonprecipitating, so they require more time to evaporate. Nonprecipitating clouds penetrate higher into the inversion layer and experience stronger evaporation by mixing with dry air and a stronger effect of negative buoyancy. A special investigation using high-resolution bin microphysics cloud models is required to answer this question.