Theoretical analysis of mixing in liquid clouds – Part IV: DSD evolution and mixing diagrams

Evolution of droplet size distribution (DSD) due to mixing between cloudy and dry volumes is investigated for different values of the cloud fraction and for different initial DSD shapes. The analysis is performed using a diffusion– evaporation model which describes time-dependent processes of turbulent diffusion and droplet evaporation within a mixing volume. Time evolution of the DSD characteristics such as droplet concentration, LWC and mean volume radii is analyzed. The mixing diagrams are plotted for the final mixing stages. It is shown that the difference between the mixing diagrams for homogeneous and inhomogeneous mixing is insignificant and decreases with an increase in the DSD width. The dependencies of the normalized cube of the mean volume radius on the cloud fraction were compared with those on normalized droplet concentration and found to be quite different. If the normalized droplet concentration is used, mixing diagrams do not show any significant dependence on relative humidity in the dry volume. The main conclusion of the study is that traditional mixing diagrams cannot serve as a reliable tool for analysis of mixing type.


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This study is Part 4 of series of papers dedicated to investigation of turbulent mixing 19 between cloud and environmental volumes. Korolev  small droplets, i.e. to DSD broadening. It was also shown that the relative humidity in the 57 initially dry volume rapidly increases due to both water vapor diffusion and evaporation of 58 penetrating droplets. As a result, the effective radii in the initially dry volume rapidly 59 approach the values typical of cloudy volume. At the same time, the liquid water content 60 remains significantly lower than that in the cloudy volume during much longer time than 61 required for the effective droplet radius to grow.

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In the present study (Pt4) we continue investigating the turbulent mixing between an 63 initially dry volume and a cloudy volume. The focus of the study is investigation of DSD  turbulent coefficient is evaluated as in Monin and Yaglom (1975).
In Eq. (1)  is the turbulent kinetic energy dissipation rate and 0.2 C  is a constant (Monin 96 and Yaglom, 1975), Boffetta and Sokolov (2002 Geometry of mixing and the initial conditions 101 The conceptual scheme presenting mixing geometry and the initial conditions used in the 102 following analysis are shown in Figure 1.  The initial liquid water mixing ratio in the cloudy volume is equal to  initial liquid water mixing ratio 2 0 w q  . Therefore, the initial profiles of these quantities 116 along the x -axis are step functions: The initial profile of droplet concentration is shown in Fig. 1b. This is the simplest 123 inhomogeneous mixing scheme, wherein mixing takes place only in the x -direction, and the 124 vertical velocity is neglected.

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Since the total volume is adiabatic, the fluxes of different quantities through the left and 126 right boundaries at any time instance are equal to zero, i.e.
where v q is the water vapor mixing ratio.

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To investigate of mixing process for different initial DSD, we assume that DSD in the cloud 131 volume can be represented by a Gamma distribution: where 0 N is an intercept parameter,  is a shape parameter and  is a slope parameter of   can be written as follows: One can see that function ( , ) xt  depends on three independent parameters 2 1 w Aq , 2 S and  .

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This function does not depend on the shape of the initial DSD in the cloud volume. In the final 175 state when  t , ( , ) xt  is : 177 Therefore, ( ) t    depends on the cloud fraction and the initial values of liquid water 178 mixing ratio in the cloud volume and the relative humidity in initially dry volume. (1 ) The solution of Eq. (12) is Eq. (13) means that in the course of evaporation, distribution () Combining Eqs. (12) and (14) yields The equation system (15-17) for distribution ( , , )  g x t should be solved under the following 233 initial condition      At the initially wide DSD (Figure 4), the radii of the DSD maximum do not change. It 274 means that at the initial RH= 80%, mixing and evaporation leads to a fast saturation of the 275 initially dry volume, after which the peak radius remains unchanged.       this, we used the parcel model proposed by Korolev (1995) that describes evaporation by 395 means of equations with temperature-dependent parameters. Figure 9 shows the mixing 396 diagrams plotted for initial narrow and wide DSD cases.  The comparison of the left and the right panels in Fig. 9 shows that the differences

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As regards the wide DSD case, the difference between the curves corresponding to 424 different mixing type is negligible (Fig. 9, right)   difference becomes negligibly small.

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The cloud fraction  is a predefined parameter and is not determined from observations.