A theoretical framework is developed to estimate the supersaturation in liquid, ice, and mixed-phase clouds. An equation describing supersaturation in mixed-phase clouds in general form is considered here. The solution for this equation is obtained for the case of quasi-steady approximation, that is, when particle sizes stay constant. It is shown that the supersaturation asymptotically approaches the quasi-steady supersaturation over time. This creates a basis for the estimation of the supersaturation in clouds from the quasi-steady supersaturation calculations. The quasi-steady supersaturation is a function of the vertical velocity and size distributions of liquid and ice particles, which can be obtained from in situ measurements. It is shown that, in mixed-phase clouds, the evaporating droplets maintain the water vapor pressure close to saturation over water, which enables the analytical estimation of the time of glaciation of mixed-phase clouds. The limitations of the quasi-steady approximation in clouds with different phase composition are considered here. The role of phase relaxation time, as well as the effect of the characteristic time and spatial scales of turbulent fluctuations, are also discussed.

The process of collective diffusional growth of droplets in an adiabatic parcel ascending or descending with the constant vertical velocity is analyzed in the frame of the regular condensation approach. Closed equations for the evolution of liquid water content, droplet radius, and supersaturation are derived from the mass balance equation centered with respect to the adiabatic water content. The analytical expression for the maximum supersaturation S max formed near the cloud base is obtained here. Similar analytical expressions for the height z max and liquid water mixing ratio q max corresponding to the level where S max occurs have also been obtained. It is shown that all three variables S max , q max , and z max are linearly related to each other and all are proportional to w 3/4 N 21/2 , where w is the vertical velocity and N is the droplet number concentration. Universal solutions for supersaturation and liquid water mixing ratio are found here, which incorporates the dependence on vertical velocity, droplet concentration, temperature, and pressure into one dimensionless parameter. The actual solutions for S and q can be obtained from the universal solutions with the help of appropriate scaling factors described in this study. The results obtained in the frame of this study provide a new look at the nature of supersaturation formation in liquid clouds. Despite the fact that the study does not include a detailed treatment of the activation process, it is shown that this work can be useful for the parameterization of cloud microphysical processes in cloud models, especially for the parameterization of cloud condensation nuclei (CCN) activation.

[1] In the present study a new method of calculating droplet concentration near cloud base is proposed. The ratio of maximum supersaturation S max to the liquid water mixing ratio when S max is reached near cloud base is found to be universal, and it does not depend on the vertical velocity w and droplet number concentration N. It is found that S max depends on vertical velocity as S max ∝ w 3/4 and on droplet concentration as S max ∝ N À1/2 . The droplet concentration calculated using the simple approach agrees well with exact solutions obtained numerically using high precision parcel models.Comparison with the results of other parameterizations is presented. It is demonstrated that the approach proposed in the study can be applied to an arbitrary form of activation spectra or any CCN size distribution given either analytically or by tables. Moreover, it can be applied for the cases when the CCN size spectrum changes with time. Temperature dependencies of S max and related quantities are analyzed.