Theoretical analysis of mixing in liquid clouds - Part 3: Inhomogeneous mixing

An idealized model of time-dependent mixing between cloud and non-cloud volumes is analyzed. Initial droplet size distribution (DSD) in cloud volume is assumed to be monodisperse. Both analytical investigation and parcel model investigation are used to study mixing processes and solve diffusion-evaporation equations. It is shown that the evolution of microphysical variables and the final equilibrium stage are unambiguously determined by two non-dimensional parameters. The first parameter, R, which is proportional to the ratio of the saturation deficit to the liquid water content in a cloud volume, determines whether the equilibrium stage is reached at 100 % relative humidity, or, rather, leads to a full evaporation of cloud droplets. The second parameter, Da, is the Damkölher number, which is equal to the ratio of the characteristic mixing time and phase relaxation time. This parameter (together with parameter R) determines whether mixing takes place according to a homogeneous or an inhomogeneous scenario. An analysis of the results obtained within a wide range of parameters R and Da is presented. It is shown that there is no pure homogeneous mixing, since the first stage of mixing is always inhomogeneous. Turbulent mixing between different volumes always starts as inhomogeneous and the mixing type can change during the mixing process. At any values of governing parameters, mixing leads to the formation of a tail of small droplets in the DSD and therefore to DSD broadening. The broadening depends on Da and the final DSD dispersion can be as large as 0.2 at large Da. The total duration of the mixing process varies from several to one hundred phase relaxation times, depending on R and Da. Delimitation between the types of mixing on the Da-R plane is carried out. The definitions of homogeneous and inhomogeneous mixings are reconsidered and clarified. The paper also compares the results of the current study with those obtained with classical mixing concepts.