On the use of nonlinear boundary-value problems to estimate the cloud-formation potential of aerosol particles

This paper investigates the transient growth of aerosol particles in a humid environment. It seeks to explore the dependence of the fraction of droplet-forming particles on statistical properties of the distribution of dry particle diameters, as well as on the rate of temperature decay associated with vertical motion through the atmosphere. Low-dimensional, autonomous models are investigated using basic tools of dynamical systems analysis that establish the parameter-dependent existence and stability of families of equilibrium distributions of wet particle diameters. In the fully nonautonomous case, an original heuristic parameterization of the fraction of droplet-forming particles is derived in terms of a scalar, nonlinear boundary-value problem. To address the failure of the heuristic parameterization to account for the potential of a dynamic reversal of growth following an initial increase in particle diameter, a finely resolved, high-dimensional boundary-value formulation for the aerosol dynamics is investigated using methods of numerical continuation. In order to reduce the computational complexity of the numerical scheme, an adaptive asynchronous discretization algorithm is developed in which the state variables are partitioned across distinct temporal meshes. Results obtained from the proposed numerical scheme are compared with the estimated fractions of droplet-forming particles based on available approximate criteria in the literature and demonstrate relative errors as large as 20%. The analysis further uncovers a hitherto-unknown linear relationship between the arithmetic expectation and variance of a log-normal size distribution of dry diameters for fixed fractions of droplet-forming particles.