Analytic Solutions for Evolving Size Distributions of Spherical Crystals or Droplets Undergoing Diffusional Growth in Different Regimes

Motivated by simulations of slow-growing contrail cirrus, the solution of the diffusional growth equations for a population of spherical ice crystals or water droplets is reexamined. For forcing specified by the evolution of the total water content above saturation within a parcel (whether driven by vertical motions, radiative heating, turbulent mixing, etc.) three behavior regimes are identified: ‘‘very fast growth’’ that cannot equilibrate, ‘‘fast growth’’ with a narrowing size spectrum, and ‘‘slow growth’’ with a broadening spectrum. The boundaries between regimes, time scales involved, and evolution of the condensate mass, number, and supersaturation are determined. The slow-growth regime represents an example of ‘‘spectral ripening,’’ with crystal or droplet numbers falling in time because of surface tension effects. Surprisingly the diffusional growth equations for the size spectrum evolution can be solved exactly in this case: in appropriate coordinates the spectral shape becomes steady, crystal or droplet numbers fall as a forcing-dependent power law, and the mean particle mass grows linearly with time. Dependence on different physical variables, fluctuating forcing, and modifications due to kinetic theory corrections are all considered. In the limit of zero external forcing on the parcel the size-spectrum solution is mathematically equivalent to a classic result in the theory of Ostwald ripening of solid solutions. It is argued that the slow-growth regime may be important in the evolution of contrail cirrus and perhaps in setting upper limits on droplet number densities in stratiform boundary layer clouds. The theoretical results are compared with parcel model simulations for illustration and to study numerical issues in binned microphysics models.