Knowledge of ice crystal particle size distributions (PSDs) is critical for parameterization schemes for atmospheric models and remote sensing retrieval schemes. In-situ observations are commonly used to obtain PSDs and other cloud microphysical properties. In particular, two-dimensional in situ images captured by cloud imaging probes are widely used to derive PSDs in term of their maximum particle dimension (Dmax). In the second chapter, different definitions of Dmax for non-spherical particles recorded by 2D probes are compared. It is shown that derived PSDs can differ by up to a factor of 6 for Dmax < 200 μm or Dmax > 2 mm. The large differences for Dmax < 200 μm are caused by the strong dependence of sample volume on particle size, whereas differences for Dmax > 2 mm are caused by the small number of particles detected. Derived bulk properties can also vary depending on the definition of Dmax because of discrepancies in the definition of Dmax used to characterize the PSDs and that used to describe the properties of individual ice crystals. For example, the mass-weighted mean diameter can vary by 2 times, the ice water content (IWC) by 3 times, and the mass-weighted terminal velocity by 6 times. Therefore, a consistent definition of Dmax should be used for all measurements and single particle properties. As an invariant measure with respect to the orientation of particles in the imaging plane for 2D probes, the diameter of the smallest circle enclosing the particle (DS) is recommended as the optimal definition of Dmax. If the 3D structure of a particle is observed, then the technique can be extended to determine the minimum enclosing sphere. The ice clouds in various weather systems from polar to equator have been sampled using aircraft equipped with in-situ probes in the past several decades and plenty of datasets are available, thus the comparison of observed and modeled PSDs using the parameters of gamma distribution function is investigated next to evaluate and potentially improve the numerical modeling of ice clouds. The Weather Research and Forecasting (WRF) model is used to represent cloud microphysical features observed in a mesoscale convective system (MCS) sampled on 20 May 2011 during the Mid-latitude Continental Convective Clouds Experiment. Inter-comparison studies are conducted with 3 different spectral bin microphysics schemes: the Caltech-NCAR-NOAA Bin scheme (CNNB), the Fast Spectral Bin Model (FSBM) and the University of Pecs and NCAR Bin scheme (UPNB). The simulated ice cloud PSDs and their variability are compared against those measured in-situ with a two-dimensional cloud probe and a high volume precipitation spectrometer installed on the University of North Dakota Citation aircraft in the trailing stratiform region behind the MCS. The observed and simulated PSDs are fit to gamma distribution functions using the incomplete gamma fit (IGF) routine to determine the intercept (N0), slope (μ) and shape (λ) parameters. The dependence on environmental conditions of the gamma distribution parameters as ellipsoids of equally realizable solutions in the parameter phase space (N0, μ, λ) is compared between the three bin schemes and the in-situ observations. Statistically significant differences in PSDs are found among the three bin schemes and between the simulations and observations, including in the median PSD form, the natural variability of PSDs under similar environmental conditions and the dependence of PSDs on temperature. Assumptions about the particle properties (such as mass/terminal velocity-dimensional relations, etc.) and the representations of microphysical processes, such as nucleation, diffusional growth and aggregation growth, in different bin schemes are investigated to explain the differences between models and in-situ observation.Based on modeling limitations in the above comparison, a final aspect of this work investiagtes the shape of observed PSDs that cannot be captured by state-of-the-art bin-resolving schemes. Several analytical forms of cloud PSDs have been used in numerical modeling and remote sensing retrieval studies of clouds and precipitation, including exponential, gamma, lognormal, and Weibull distributions. However, there is no satisfying physical explanation as to why certain distribution forms preferentially occur instead of others. Theoretically, the analytical form of a PSD can be derived by directly solving the general dynamic equation, but no analytical solutions have been found yet. Instead of using a process level approach, the use of the principle of maximum entropy (MaxEnt) for determining the analytical form of PSDs from the perspective of a system is examined. MaxEnt theory states that the probability density function with the largest information entropy among a group satisfying the given properties of the variable should be chosen. Here, the issue of variability under coordinate transformations that arises using the Gibbs/Shannon definition of entropy is identified, and the use of the concept of relative entropy to avoid these problems is discussed. Focusing on cloud physics, the four-parameter generalized gamma distribution is proposed as the analytical form of a PSD using the principle of maximum (relative) entropy with assumptions on power law relations between state variables, scale invariance and a further constraint on the expectation of one state variable. The four-parameter generalized gamma distribution is very flexible to accommodate various type of constraints that could be assumed for cloud PSDs. The exact constraints and distribution parameters need to be further determined using in-situ datasets and idealized numerical models for potential applications in numerical models and remote sensing retrievals.
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Observation and simulation of mid-latitude ice clouds
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