Solutions to the population balance equation for cloud hydrometeors

Clouds constitute an ensemble of a huge number of particles. A general approach for representing the system is the use of size or mass distributions, leading to a population balancing equation (PBE). Since solving this equation is quite challenging and computationally expensive, numerical weather prediction models often use so-called bulk schemes, based on general moments of the underlying distribution, and assuming a fixed functional form for the particle size distribution, from which evolution equations for the moments are derived. These schemes are much simpler and computationally cheaper, though they overly constrain the evolution of the size distribution.

In this work, we directly address the problem of solving the PBE for the mass distribution of atmospheric hydrometeors. We present analytical solutions for idealized cases and reduce more complex scenarios to systems of ordinary differential equations, which can be numerically integrated. These analytical solutions can be compared to the standard bulk schemes, testing the accuracy of the latter. Future work will involve extending these solutions to more complicated regimes and validating the results against empirical field measurements of the hydrometeor size distributions.