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Yangang Liu and John Hallett

Abstract

Observed turbulence and fluctuating microphysical properties of clouds lead the authors to assume that a cloud droplet size distribution results from a large number of random events associated with turbulence and to consider a droplet system with fluctuating cloud droplet size distributions constrained by conservation laws. This assumption in turn suggests multiplicity rather than uniqueness of cloud droplet size distributions, that is, different cloud droplet size distributions occurring with different probability. The authors argue from a system point of view that two characteristic cloud droplet size distributions can be identified without knowing the specific probability of occurrence. The maximum likelihood cloud droplet size distribution is obtained by applying Shannon’s maximum entropy principle; the minimum likelihood cloud droplet size distribution is obtained by studying the functional relationship between a cloud droplet size distribution and the corresponding energy change to form such a droplet population. The maximum and minimum likelihood cloud droplet size distribution for an ideal droplet system with conserved mass are derived to be, respectively, a Weibull distribution and a delta distribution. The unique properties of the two characteristic cloud droplet size distributions are associated with observed cloud droplet size distributions and ones predicted by the uniform condensation model. These associations suggest that the lack of agreement between cloud droplet size distributions predicted by the condensation model and those observed in real clouds may be a result of trying to compare two totally different characteristic cloud droplet size distributions of the same droplet system. The present study discusses the maximum and minimum likelihood cloud droplet size distributions and their relationship to observed and model-predicted cloud droplet size distributions. The proposed theory sets in a new context the discrepancy between observed and model-predicted cloud droplet size distributions and also provides an explanation for the scale dependence of observed microphysical properties.

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Yangang Liu and Peter H. Daum
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Leon D. Rotstayn and Yangang Liu

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Observations show that an increase in anthropogenic aerosols leads to concurrent increases in the cloud droplet concentration and the relative dispersion of the cloud droplet spectrum, other factors being equal. It has been suggested that the increase in effective radius resulting from increased relative dispersion may substantially negate the indirect aerosol effect, but this is usually not parameterized in global climate models (GCMs). Empirical parameterizations, designed to represent the average of this effect, as well as its lower and upper bounds, are tested in the CSIRO GCM. Compared to a control simulation, in which the relative dispersion of the cloud droplet spectrum is prescribed separately over land and ocean, inclusion of this effect reduces the magnitude of the first indirect aerosol effect by between 12% and 35%.

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Yangang Liu and Peter H. Daum

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Various commonly used Kessler-type parameterizations of the autoconversion of cloud droplets to embryonic raindrops are theoretically derived from the same formalism by applying the generalized mean value theorem for integrals to the general collection equation. The new formalism clearly reveals the approximations and assumptions that are implicitly embedded in these different parameterizations. A new Kessler-type parameterization is further derived by eliminating the incorrect and/or unnecessary assumptions inherent in the existing Kessler-type parameterizations. The new parameterization exhibits a different dependence on liquid water content and droplet concentration, and provides theoretical explanations for the multitude of values assigned to the tunable coefficients associated with the commonly used parameterizations. Relative dispersion of the cloud droplet size distribution (defined as the ratio of the standard deviation to the mean radius of the cloud droplet size distribution) is explicitly included in the new parameterization, allowing for investigation of the influences of the relative dispersion on the autoconversion rate and, hence, on the second indirect aerosol effect. The new analytical parameterization compares favorably with those parameterizations empirically obtained by curve-fitting results from simulations of detailed microphysical models.

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Gang Liu, Yangang Liu, and Satoshi Endo
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Gang Liu, Yangang Liu, and Satoshi Endo

Abstract

Surface momentum, sensible heat, and latent heat fluxes are critical for atmospheric processes such as clouds and precipitation, and are parameterized in a variety of models ranging from cloud-resolving models to large-scale weather and climate models. However, direct evaluation of the parameterization schemes for these surface fluxes is rare due to limited observations. This study takes advantage of the long-term observations of surface fluxes collected at the Southern Great Plains site by the Department of Energy Atmospheric Radiation Measurement program to evaluate the six surface flux parameterization schemes commonly used in the Weather Research and Forecasting (WRF) model and three U.S. general circulation models (GCMs). The unprecedented 7-yr-long measurements by the eddy correlation (EC) and energy balance Bowen ratio (EBBR) methods permit statistical evaluation of all six parameterizations under a variety of stability conditions, diurnal cycles, and seasonal variations. The statistical analyses show that the momentum flux parameterization agrees best with the EC observations, followed by latent heat flux, sensible heat flux, and evaporation ratio/Bowen ratio. The overall performance of the parameterizations depends on atmospheric stability, being best under neutral stratification and deteriorating toward both more stable and more unstable conditions. Further diagnostic analysis reveals that in addition to the parameterization schemes themselves, the discrepancies between observed and parameterized sensible and latent heat fluxes may stem from inadequate use of input variables such as surface temperature, moisture availability, and roughness length. The results demonstrate the need for improving the land surface models and measurements of surface properties, which would permit the evaluation of full land surface models.

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David L. Mitchell, Yangang Liu, and Andreas Macke

Abstract

A new radiation scheme, suitable for two-stream radiation transfer models, was developed for cirrus clouds. Analytical expressions were derived for the extinction and absorption coefficients and the asymmetry parameter. These are functions of the ice particle size distribution parameters, ice particle shapes, and wavelength. The ice particle shapes considered were hexagonal plates and columns, bullet rosettes, and planar polycrystals. These appear to be the principal crystal types found in cirrus clouds. The formulation of radiative properties accounts for the size distribution projected area and the distance radiation travels through ice particles. For absorption, refraction and internal reflection of radiation were parameterized.

By assuming an idealized cirrus cloud, the dependence of the single scatter albedo, reflectance, and emissivity on wavelength, ice particle shape, and size distribution was demonstrated. Reflectance and emissivity exhibited a strong dependence on ice particle shape, with planar polycrystals and bullet rosettes often being twice or more reflective than hexagonal columns and plates.

The radiation scheme was tested with microphysical and radiation measurements from two cirrus cloud field studies. It was shown for both case studies that, by matching observed and predicted albedo-emissivity curves, the radiation scheme could predict the observed mean ice particle size and ice water path (IWP), provided the dominant ice particle shape was known or inferred. Retrieved IWP values differed from measurement-derived values by ≤15% for the first case study and 18% on average for the second case study. Hence, it may be feasible to retrieve realistic IWP estimates from satellite data for a given ice particle shape.

Other radiation schemes have not been able to explain the second case study, which was characterized by relatively high albedos. These high albedos appeared to result from unusually small hexagonal plate crystals having asymmetry parameter values similar to those of cloud droplets.

An improved treatment of the asymmetry parameter was not the primary reason for the good agreement between theory and observations. Rather, key factors appeared to be improved treatments of ice particle photon path, projected area and mass, and the omission of certain physical processes included in Mie theory that may not be appropriate for ice particles.

The radiative properties were predicted from analytical expressions, making this scheme useful for predicting radiative properties in large-scale models without excessive increases in computation time.

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Yangang Liu, Peter H. Daum, and John Hallett

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A systems theory has previously been developed by Liu and Hallett to interpret droplet size distributions in turbulent clouds by utilizing ideas from statistical physics and information theory. The present paper generalizes that systems theory to allow for varying fluctuations. The generalized theory provides a self-consistent theoretical framework for a wide range of fluctuations. It reduces to that presented previously when liquid water content is conserved, and becomes consistent with the uniform growth models for nonturbulent, adiabatic clouds. The theory indicates that there exists an important characteristic scale, defined as the saturation scale, beyond which droplet size distributions do not change with further increases in averaging scale, but below which droplet size distributions strongly depend on the scale over which they are sampled and are therefore ill-defined without an adequate specification of scale. It is further demonstrated that the saturation scale and the details of scale dependence depend on the level of fluctuations; stronger fluctuations lead to larger saturation scales and stronger scale dependency of droplet size distributions. The potential scale mismatch leads to issues regarding the comparability between models and observations, and the direct coupling of numerical models of different scales, which in turn underscores the significance of understanding and quantifying the scale dependence of droplet size distributions. The importance of fluctuations suggests the need to measure and analyze turbulence simultaneously and at the same scales with measurements of droplet size distributions in order to provide a practical limit to the sample size required to reach the saturation scale, and to specify the effect of turbulence. The ideas presented in this paper have general applications to fields where fluctuations exist.

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Yangang Liu, Peter H. Daum, R. McGraw, and R. Wood

Abstract

Existing Sundqvist-type parameterizations, which only consider dependence of the autoconversion rate on cloud liquid water content, are generalized to explicitly account for the droplet concentration and relative dispersion of the cloud droplet size distribution as well. The generalized Sundqvist-type parameterization includes the more commonly used Kessler-type parameterization as a special case, unifying the two different types of parameterizations for the autoconversion rate. The generalized Sundqvist-type parameterization is identical with the Kessler-type parameterization presented in Part I beyond the autoconversion threshold, but exhibits a more realistic, smooth transition in the vicinity of the autoconversion threshold (threshold behavior) in contrast to the discontinuously abrupt transition embodied in the Kessler-type parameterization. A new Sundqvist-type parameterization is further derived by applying the expression for the critical radius derived from the kinetic potential theory to the generalized Sundqvist-type parameterization. The new parameterization eliminates the need for defining the driving radius and for prescribing the critical radius associated with Kessler-type parameterizations. The two-part structure of the autoconversion process raises questions regarding model-based empirical parameterizations obtained by fitting simulation results from detailed collection models with a single function.

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