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David L. Mitchell

Abstract

Using a form of the stochastic collection equation, conservation equations for the first and second moments of the mass were parameterized to yield a height dependent one-dimensional snow growth model for unrimed stratiform snowfall. Snow-size distributions were represented by the form N(D) = N 0 D ν exp(−λD), and solutions for λ and N 0 were obtained. The spectral parameter ν allows the concentration of the smaller ice particles to deviate from the exponential form and controls the degree of subexponential or superexponential behavior. The sub- and superexponential spectra analyzed in this study had ν values of 1 and −1, respectively.

A number of simple analytical relationships was developed that describes various properties of size distributions, regardless of the particle type involved. A method was developed for obtaining the three parameters of the size distribution used in the model from measured size distributions. In addition, an expression was derived to relate the two λ of an exponentially parameterized and a nonexponentially parameterized size distribution.

The effect of sub- and superexponential spectra on the evolution of snow-size spectra by vapor diffusion and aggregation was examined using a steady state, fixed snowfall rate profile. Diffusional growth rates of individual ice crystals (no aggregates) were relatively low when the size distribution was constrained to be superexponential in form. This resulted in steeper spectra (smaller crystal sizes) and higher ice-crystal number concentrations. The diffusional growth rate of individual ice crystals for subexponential spectra was relatively high. Subexponential spectra were characterized by broader distributions and lower ice crystal number concentrations. Aggregation was the only growth process that substantially increased ice particle sizes for superexponential spectra, while both vapor diffusion (in the upper cloud) and aggregation (in the mid-to-lower cloud) contributed substantially to size increases for subexponential spectra.

An expression for the aggregation efficiency was formulated. The primary factors governing aggregation appear to be the aggregation efficiency, the ice particle number concentration and the mean diameter. The expression may be useful in larger numerical cloud models. Mean aggregation rate constants were determined for sub- and superexponential spectra, and for exponential spectra. The mean aggregation rate constant for superexponential spectra was approximately 50% greater than for subexponential spectra.

Finally, it was found that the degree of subexponential behavior predicted when ν = 1 was consistent with that observed at various levels in stratiform clouds. However, better measurements are needed to substantiate this finding.

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David L. Mitchell

Abstract

A Lagrangian time-dependent three-dimensional model was developed that predicts the evolution of ice particle size spectra in cirrus clouds in terms of the growth processes of vapor diffusion and aggregation, as well as the cloud updraft profile. This was done by deriving moment conservation equations from a form of the ice particle number density equation, and parameterizing the moment conservation equations. Size distributions were parameterized by the form N(D) = N 0 D v exp(−λ D), where λ and N 0 are functions of the growth processes. Growth by diffusion and aggregation were formulated to depend on the percentages of spatial and columnar crystal habits at a given level in the cloud.

Two cirrus cloud field studies were simulated by the model, where the model was run in the one-dimensional, height-dependent mode that assumes steady-state, horizontally homogeneous ice water contents. Predicted and observed ice particle size spectra compared favorably. This implied that ice particles at lower levels evolved from higher levels. Although cirrus often exhibit banded layers, as found in both case studies, it appears that evaporating ice falling from a higher layer will initiate ice evolution in moist, lower layers.

Model simulations indicated cloud updrafts have the general effect of increasing ice particle concentrations by increasing their residence time in the cloud. Size sorting occurs, by which the smaller ice particles are preferentially retained, and mean ice particle sizes decrease.

The model indicates that transitions in ice crystal habit may strongly influence the evolution of ice particle size spectra. A transition from columnar to spatial habits produces a shift toward smaller but more numerous ice crystals. This has also been observed in field studies.

It could not be determined whether aggregation was an important ice particle growth process, since model simulations using both high and low aggregation efficiencies yielded similar agreement with field data.

The model described is fundamentally analytical and is computationally efficient. It may be used in large-scale models and may be useful in describing cloud-radiative interactions.

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David L. Mitchell

Abstract

Based on the stochastic collection equation, height- and time-dependent snow growth models were developed for unrimed stratiform snowfall. Moment conservation equations were parameterized and solved by constraining the size distribution to be of the form N(D)dD = N 0 exp(−&lambdaD)dD, yielding expressions for the slope parameter, λ, and the y-intercept parameters, N O, as functions of height or time. The processes of vapor deposition and aggregation were treated analytically without neglecting changes in ice crystal habits, while the ice particle breakup process was dealt with empirically.

The models were compared against vertical profiles of snow-size spectra, obtained from aircraft measurements, for three case studies. The predicted spectra are in good agreement with the observed evolution of snow-size spectra in all three cases, indicating the proposed scheme for ice particle aggregation was successful. The temperature dependence of aggregation was assumed to result from differences in ice crystal habit. Using data from an earlier study, the aggregation efficiency between two levels in a cloud was calculated. Finally, other height-dependent, steady-state snowfall models in the literature were compared against spectra from one of the above case studies. The agreement between the predicted and observed spectra regarding these models was less favorable than was obtained from the models presented here.

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David L. Mitchell

Abstract

It was found that the anomalous diffraction approximation (ADA) could be made to approximate Mie theory for absorption and extinction in water clouds by parameterizing the missing physics: 1) internal reflection/refraction, 2) photon tunneling, and 3) edge diffraction. Tunneling here refers to processes by which tangential or grazing photons beyond the physical cross section of a spherical particle may be absorbed. Contributions of the above processes to extinction and/or absorption were approximated in terms of particle size, index of refraction, and wavelength. It was found that tunneling can explain most of the difference between ADA and Mie theory for water clouds in the thermal IR.

The modified ADA yielded analytical expressions for the absorption and extinction efficiencies, Q abs and Q ext, which were integrated over a gamma size distribution to yield expressions for the absorption and extinction coefficients, β abs and β ext. These coefficients were expressed in terms of the three gamma distribution parameters, which were related to measured properties of the size distribution: liquid water content, mean, and mass-median diameter. Errors relative to Mie theory for β abs and β ext were generally ⩽10% for the effective radius range in water clouds of 5–30 μm, for any wavelength in the solar or terrestrial spectrum. For broadband emissivities and absorptivities regarding terrestrial and solar radiation, the errors were less than 1.2% and 4%, respectively. The modified ADA dramatically reduces computation times relative to Mie theory while yielding reasonably accurate results.

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David L. Mitchell

Abstract

Although the use of an effective radius for radiation transfer calculations in water clouds has been common for many years, the export of this concept to ice clouds has been fraught with uncertainty, due to the nonspherical shapes of ice particles. More recently, a consensus appears to be building that a general definition of effective diameter D eff should involve the ratio of the size distribution volume (at bulk density) to projected area. This work further endorses this concept, describes its physical basis in terms of an effective photon path, and demonstrates the equivalency of a derived D eff definition for both water and ice clouds. Effective photon path is the unifying underlying principle behind this universal definition of D eff.

Simple equations are formulated in terms of D eff, wavelength, and refractive index, giving monochromatic coefficients for absorption and extinction, β abs and β ext, throughout the geometric optics, Mie, and Rayleigh regimes. These expressions are tested against Mie theory, showing the limitations of the use of D eff as well as its usefulness.

For water clouds, the size distribution N(D) exhibits relatively little dispersion around the mean diameter in comparison with ice clouds. For this reason, a single particle approximation for β abs based on D eff compares well with β abs predicted from Mie theory, providing a new and efficient means of treating radiation transfer at terrestrial wavelengths. The D eff expression for β ext agrees well with Mie theory only under specific conditions: 1) absorption is substantial or 2) absorption occurs in the Rayleigh regime, or 3) size parameter x e ≳ 50, where x e = πD eff/λ. Since the D eff expressions for β abs and β ext are single particle solutions, it is not surprising that agreement with Mie theory is best when the size distribution dispersion is reduced, approaching the single particle limit.

For ice clouds, it is demonstrated that the D eff expressions for β abs and β ext are probably inadequate for most applications, at least at terrestrial wavelengths. This is due to the bimodal nature of ice particle size spectra N(D) with relatively high concentrations of small (D < 100 μm) ice crystals. These small crystals have relatively low absorption efficiencies, causing the N(D)-integrated β abs to be lower than β abs based on D eff. This difference in N(D) dispersion between water and ice clouds makes it desirable to use an explicit solution to the absorption and extinction coefficients when calculating the radiative properties of ice clouds. Analytical solutions to the integral definitions of β abs and β ext are provided in the , which may not be too computationally expensive for many applications.

Most schemes for predicting ice cloud radiative properties are founded on the assumption that the dependence of β abs and β ext on the size distribution can be described solely in terms of D eff and ice water content (IWC). This assumption was tested by comparing the N(D) area-weighted efficiencies for absorption and extinction, Q abs and Q ext, for three N(D) that have the same IWC and D eff, but for which N(D) shape differs. Analytical solutions for β abs and β ext were used, which explicitly treat N(D) shape, over a wavelength range of 1.0 to 1000 μm. For a chosen D eff value, uncertainties (percent differences) resulting only from N(D) shape differences reached 44% for Q abs, 100% for Q ext, and 48% for the single scattering albedo ω o for terrestrial radiation. This sensitivity to N(D) shape has implications relating to the formulation of schemes predicting ice cloud radiative properties, as well as satellite remote sensing of cloud properties.

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David L. Mitchell

Abstract

Based on boundary layer theory and a comparison of empirical power laws relating the Reynolds and Best numbers, it was apparent that the primary variables governing a hydrometeor's terminal velocity were its mass, its area projected to the flow, and its maximum dimension. The dependence of terminal velocities on surface roughness appeared secondary, with surface roughness apparently changing significantly only during phase changes (i.e., ice to liquid). In the theoretical analysis, a new, comprehensive expression for the drag force, which is valid for both inertial and viscous-dominated flow, was derived.

A hydrometeor's mass and projected area were simply and accurately represented in terms of its maximum dimension by using dimensional power laws. Hydrometeor terminal velocities were calculated by using mass- and area-dimensional power laws to parameterize the Best number, X. Using a theoretical relationship general for all particle types, the Reynolds number, Re, was then calculated from the Best number. Terminal velocities were calculated from Re.

Alternatively, four Re–X power-law expressions were extracted from the theoretical Re–X relationship. These expressions collectively describe the terminal velocities of all ice particle types. These were parameterized using mass- and area-dimensional power laws, yielding four theoretically based power-law expressions predicting fall speeds in terms of ice particle maximum dimension. When parameterized for a given ice particle type, the theoretical fall speed power law can be compared directly with empirical fall speed-dimensional power laws in the literature for the appropriate Re range. This provides a means of comparing theory with observations.

Terminal velocities predicted by this method were compared with fall speeds given by empirical fall speed expressions for the same ice particle type, which were curve fits to measured fall speeds. Such comparisons were done for nine types of ice particles. Fall speeds predicted by this method differed from those based on measurements by no more than 20%.

The features that distinguish this method of determining fall speeds from others are that it does not represent particles as spheroids, it is general for any ice particle shape and size, it is conceptually and mathematically simple, it appears accurate, and it provides for physical insight. This method also allows fall speeds to be determined from aircraft measurements of ice particle mass and projected area, rather than directly measuring fall speeds. This approach may be useful for ice crystals characterizing cirrus clouds, for which direct fall speed measurements are difficult.

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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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David L. Mitchell and W. Patrick Arnott

Abstract

This study builds upon the microphysical modeling described in Part I by deriving formulations for the extinction and absorption coefficients in terms of the size distribution parameters predicted from the microphysical model. The optical depth and single scatter albedo of a cirrus cloud can then be determined, which, along with the asymmetry parameter, are the input parameters needed by cloud radiation models.

Through the use of anomalous diffraction theory, analytical expressions were developed describing the absorption and extinction coefficients and the single scatter albedo as functions of size distribution parameters, ice crystal shapes (or habits), wavelength, and refractive index. The extinction coefficient was formulated in terms of the projected area of the size distribution, while the absorption coefficient was formulated in terms of both the projected area and mass of the size distribution. These properties were formulated as explicit functions of ice crystal geometry and were not based on an “effective radius.”

Based on simulations of the second cirrus case study described in Part I, absorption coefficients predicted in the near infrared for hexagonal columns and rosettes were up to 47% and 71% lower, respectively, than absorption coefficients predicted by using equivalent area spheres. This resulted in single scatter albedos in the near infrared that were considerably greater than those predicted by the equivalent area sphere method. Reflectances in this region should therefore be underestimated using the equivalent area sphere approach.

Cloud optical depth was found to depend on ice crystal habit. When the simulated cirrus cloud contained only bullet rosettes, the optical depth was 142% greater than when the cloud contained only hexagonal columns. This increase produced a doubling in cloud albedo. In the near-IR, the single scatter albedo also exhibited a significant dependence on ice crystal habit. More research is needed on the geometrical properties of ice crystals before the influence of ice crystal shape on cirrus radiative properties can be adequately understood.

This study provides a way of coupling the radiative properties of absorption, extinction, and single scatter albedo to the microphysical properties of cirrus clouds. The dependence of extinction and absorption on ice crystal shape was not just due to geometrical differences between crystal types, but was also due to the effect these differences had on the evolution of ice particle size spectra.

The ice particle growth model in Part I and the radiative properties treated here are based on analytical formulations, and thus represent a computationally efficient means of modeling the microphysical and radiative properties of cirrus clouds. Although phase functions and asymmetry parameters for different ice particle shapes are not treated here and need to be better characterized, this work may advance our ability to simulate complex microphysical and radiative processes in large-scale models.

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David L. Mitchell and Andrew J. Heymsfield

Abstract

Recent work on the terminal velocity of ice crystal aggregates suggests that their “Re–X” relationship may not be well predicted by current theory. This study examines possible reasons for this departure from theory, and develops a new Re–X relationship appropriate for ice crystal aggregates. The methodology of Khvorostyanov and Curry was applied to this new relationship to formulate power-law expressions for all ice particle types.

Fall speed differences between the Khvorostyanov and Curry approach and the approach described here were as large as 50% for aggregates and 30% for single crystals. This was primarily due to the following: 1) surface roughness coefficients used in the former were appropriate for rigid spheres and liquid drops but not for ice crystals and 2) the relationship between Reynolds number Re and Best number X at high Re is better described for aggregates by adding a second term to the Re–X governing equation, as done in this work.

The corrections and improvements described here may be critical to the calculation of snowfall rates, to the modeling of the aggregation process, and for interpreting Doppler radar measurements during snowfall events. Since most of the size distribution mass is generally associated with aggregates below cloud base, an accurate treatment of aggregate fall speeds is needed for determining snowfall rates.

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David L. Mitchell, Renyi Zhang, and Richard L. Pitter

Abstract

The masses, dimensions, and habits of over 2800 natural ice particles precipitating from orographic winter storms in the central Sierra Nevada were obtained using photomicrographs. Ice particles that could be unambiguously classified were used to generate empirical expressions relating snow particle masses and dimensions. Many of the ice particle types had not been investigated previously. The influence of riming and aggregation on ice particle masses was examined. When possible, comparisons are made between these results and those of other experimental observations. By incorporating these mass-dimensional relationships into an expression for the ice mass content in a snowstorm, it was possible to estimate the mass fraction of the fresh snowpack resulting from accreted supercooled cloud water. The results from two storms analyzed suggest that about 30 to 40 percent of the deposited snow is composed of accreted cloud water during moderately rimed snowfall.

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