A General Theory for Estimating Ice Crystal Projections from Gamma Distributed Particles: Implications for In Situ Observations, Model Corroboration, and Model Parameterization Development

Ice microphysical processes and the particle observations that result from these processes depend upon 2-D projected ice particle properties. Microphysics schemes use particle projected areas to estimate particle fallspeeds and collection kernels during riming and aggregation whereas in situ cloud probe image projections are often used to derive power-law relations that dictate how particle properties vary across the size spectrum. Newer particle property-based microphysics schemes (Hashino and Tripoli, 2007, J. Atmos. Sci., Jensen et al., 2017, J. Atmos. Sci., Chen and Tsai, 2016, J. Atmos. Sci.) predict multiple size distribution moments such that particles are represented as spheroids. This first order approximation of particle shape allows for planar crystals to be represented as oblate spheroids whereas columnar crystals can be represented as prolate spheroids. However, both the microphysical models and the parameterizations derived from in situ observations do not use information regarding how various particle orientations change projected quantities. Because of this deficiency, we have developed a theoretical method for including particle orientations when projecting modeled ice spheroidal shape. This helps to inform model parameterization development and provides a consistent basis for comparing model output with in situ derived ellipse fit projection data. Model particle shape distributions can either be predicted using particle property approaches or diagnosed using traditional bulk approaches. We derive analytic equations from 3-D spheroids to their 2-D projections using eccentricity quantities, and we validate our theoretical approach through comparison with a Monte Carlo approach. In general, the inclusion of particle orientations act to make observed projections appear more circular as others have shown before. However, our new bulk projection tests show that orientation distributions that include sines or cosines lead to simple linear features in the projected eccentricity distributions. Furthermore, bulk distributions of projected aspect ratio frequently have singularities at unity whereas the distributions of eccentricity or second eccentricity always converge. This theoretical approach demonstrates that associated uncertainties with fallspeed orientation distributions can create bimodalities in the observed projection distributions for certain Gaussian-like orientations. This suggests that observed in situ aspect ratio bimodalities could be driven by orientation alone. Moreover, the approach can be used to provide a basis for parameterizing the growth of aggregates in particle property models.