Growth Rate of Individual Cloud Droplets

Let us assume that the environmental profiles of temperature and water vapor are constant with time at T (z) and w, ( ) and are known. Then because the air parcel is moving with speed W, it will appear to an observer moving with the air parcel that the surrounding conditions are changing according to 

For a given environment and given entrainment rate e, we need one more equation to close the system, namely, an equation describing the liquid water mixing ratio wL of the drop population. This liquid water content can be calculated if the droplet size distribution is known, such as a simple integral over the distribution. We thus need to derive differential equations for the droplet diameter rate of change dDp/dt. These equations will link the cloud dynamics discussed here with the cloud microphysics discussed in the following sections. 

When cloud and fog droplets have diameters significantly larger than 1 pm, mass transfer of water to a droplet can be expressed by the mass transfer equation for the continuum regime (see Chapter 12) 

The diffusivity of water vapor in air is given as a function of temperature and pressure by 

Equation (17.60) neglects noncontinuum effects that may influence very small cloud droplets. These effects can be included in this equation by introducing a modified diffusivity D v, where (Fukuta and Walter 1970) 

FIGURE 15.13 Water vapor diffusivity corrected for noncontinuum effects and imperfect accommodation as a function of the droplet diameter at F = 283 K and p = 1 atm. 

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