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Philips S. Brown Jr.

Abstract

The coalescence/breakup formulas introduced by Low and List included a new formulation of the coalescence efficiency and a new formulation of the fragment distribution function P, which is written as a weighted sum of three distribution functions, each corresponding to one of three types of fragmentation (filament, sheet and disk). The purpose of this work is to examine in detail the effects of each main component of the Low and List formulas upon the evolution of the drop spectrum. Compared to an earlier formulation, the Low and List coalescence efficiency tends to distribute more water mass from the small-drop range to the 1argee-drop range. The individual processes of filament and sheet breakup tend to produce single peaks in the small-drop end of the spectrum, while disk breakup tends to produce a bimodal distribution with peaks near drop diameters 1 and 2.3 mm. Disk breakup is the dominant type of fragmentation in determining the response time of the system and in causing the destruction of very large-size drops. The combined breakup processes produce a trimodal equilibrium drop distribution. Sheet breakup reinforces the small-drop peak at D = 0.23 mm established by filament breakup to form the principal mode. Sheer and disk breakup act in combination to form the less-prominent secondary and tertiary modes.

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Philip S. Brown Jr.

Abstract

Past work has provided thorough analysis of the coalescence/breakup process in a “box model” setting in which the drop size distribution is assumed invariant with height. In this work, the analysis is extended to examine the coalescence/breakup process in a one-dimensional shaft model setting that allows vertical variation of the drop size distribution due to sedimentation. The objectives are to gain a better understanding of the rain process and to acquire the knowledge necessary to parameterize the shaft model solutions. When vertical variation is taken into account, the steady-state form of the model equation describes the rate of change in the drop size distribution with fall distance for a fixed input condition at the shaft top. This equation is formally quite similar to the box model equation describing temporal evolution of the drop spectrum, and many characteristics of the box model solutions carry over to the steady-state, shaft model solutions, but with fall distance replacing time as the independent variable. Both solutions, for example, approach the same trimodal equilibrium form. Some important differences do exist, however. Analysis of the box model and shaft model equations reveals that the roles of coalescence and breakup are reversed in determining the rate at which the solutions approach equilibrium, and that the final adjustment to equilibrium is slightly different in the two cases. Further comparison of the box and shaft models shows that the water mass and water mass flux reverse roles as conserved quantities. In spite of these differences, the strong similarities in the equations allow direct adaptation of a box model parameterization to describe the steady-state, shaft model solutions.

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Philip S. Brown Jr.

Abstract

Analytic solution of the linearized coalescence-breakup equation is used as a basis for parameterizing the evolving drop-size distribution. The linearized coalescence-breakup equation is formulated using only a small number of drop-size bins for the sake of computational efficiency but at the sacrifice of considerable detail. The low-resolution analytic solutions are then enhanced with detail provided by a high-resolution representation of the equilibrium distribution. In this step, the assumption is made that the drop distribution approaches high-resolution equilibrium form in a manner consistent with the temporal behavior of the low-resolution analytic solution. The low-resolution analytic solution and the high-resolution enhancement comprise the parameterization. Two specific parameterizations, based on two-bin and four-bin model solutions, are presented. The two-bin parametric solution is easy to compute but approaches equilibrium monotonically and thereby fails to undergo the pronounced fluctuation that characterizes high-resolution model results. The four-bin parametric solution involves more computation but produces an evolving drop distribution that closely resembles the fluctuating distributions obtained by detailed numerical solution of the coalescence-breakup equation.

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Philip S. Brown Jr.

Abstract

An analysis of the structural stability of the coalescence/breakup equation is performed to determine the degree to which changes in the equation's formulation can affect the solution. The work was motivated by speculation in various quarters that the currently used coalescence/breakup formulation should be adjusted to achieve better agreement between solutions and field observations. Both analytical procedures and numerical experiments, in which hypothetical changes in the rate coefficients are assumed, show the coalescence/breakup equation in its current formulation to be structurally stable. Not only do small changes in the rate coefficients produce negligible change in the solutions, but even large changes in the rate coefficients fail to destroy the fundamental behavior of the system in that all solutions continue to approach a unique equilibrium. Moderate-sized perturbations of the coefficients are found to have only minor influence on the solutions unless the coalescence and breakup efficiencies, constituents of the rate coefficients, are perturbed in an opposite sense to reinforce the individual effects. Only with such changes in the formulation is the equilibrium solution found to be altered to a significant degree.

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Philip S. Brown Jr.

Abstract

Computational problems have been reported in the literature regarding application of Bleck's method to the collision-breakup equation. It is found that straightforward calculation of the Bleck integrals can involve an inordinate number of iterations to achieve convergence. The sources of the problem have been identified, and detailed modifications are given to provide rapid convergence.

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Philip S. Brown Jr.

Abstract

The computational stability of a numerical scheme used to solve the stochastic coalescence equation is investigated. A numerical analysis leads to the derivation of a simple stability criterion for determining an allowable time step for explicit numerical solution of the coalescence equation. The time step is found to depend upon the size and location of the spectral peak, the largest droplet category included, and the arrangement of the spectral grid points. The availability of the derived results can lead to more efficient application of present schemes used in large–scale moist convection models as well as in studies involving microphysics only. Similar stability criteria for more accurate numerical approximations can be obtained using the procedure described; such criteria are to be derived in forthcoming work.

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Philip S. Brown Jr.

Abstract

Models or the coalescence/breakup process yield drop number distributions that approach equilibrium but the number density often is not a monotonic function of time. In some cases, the small-raindrop portion of the distribution rapidly attains high concentration levels before settling back toward an equilibrium position. An eigenanalysis of the coalescence/breakup equation is performed to gain an understanding of the solution behavior near equilibrium. The analysis reveals that the departure of the solution from equilibrium can be expressed as a linear combination of basis functions of the form e λj where Re(λj) < 0 so that the equilibrium drop distribution is asymptotically stable. The exponential basis functions feature a wide range of decay rates, and since Im (λj) ≠ 0 in some cases, the functions provide evidence of oscillations in the drop spectrum. It is shown that one particular damped oscillation can combine with a rapidly decaying transient to describe very well the nonmonotonic behavior characteristic of model-generated drop spectra. While the physical mechanism behind the oscillation is not yet understood, the initial reversal in the small-drop peak may be explained as rapid response due to filament breakup followed by a slower response due to coalescence. A particular sequence of observed raindrop distributions is found to exhibit a reversal in the spectral peak similar to that produced by the model.

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Philip S. Brown Jr.

Abstract

An analysis of the Low and List drop-breakup formulation has uncovered several computational problems that arise in calculating both the fragment distribution function and the Bleck expansion coefficients that appear in the discrete coalcacence/breakup equation. Special procedures have been developed to deal effectively with these problems. The discrete coalescence/breakup equation has been solved using the Low and List breakup formulation and the earlier List and Gillespie formulation. Comparison shows that the Low and List model solutions approach equilibrium more slowly than do the earlier model solutions; moreover, the Low and List equilibrium drop spectra exhibit a bimodality in the small-drop end of the spectrum. The longer time constants associated with the Low and List equations ease somewhat the severe computational stability problem associated with List and Gillespie equations.

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Philip S. Brown Jr.

Abstract

Coalescence and collision-induced breakup of water drops are the two basic drop-interaction processes governing warm-rain development. In this work, recently derived model results are used to construct a parameterization of drop-spectrum evolution as an initial Marshall-Palmer-type drop distribution undergoes the effects of coalescence and breakup. The parameterization, as developed so far, is designed to represent with accuracy only the large-drop portion of the spectrum. The initial distributions are of the form of N = N 0e−AD where Λ and N 0 are constants, and D denotes raindrop diameter (mm). During its evolution, the large-drop portion of the spectrum maintains exponential form, approaching an equilibrium with Λ=3.6 mm−1. Analysis pertaining to the parameterization has led to the discovery that initial spectra with Λ>3.6 mm−1 rapidly just to a quasi-stationary state before slowly approaching the true equilibrium.

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Philip R. A. Brown

Abstract

A technique for the measurement of the ice water content (IWC) of cirrus clouds is described. The IWC is obtained by the measurement of the total water content (TWC) and the subtraction of the saturation specific humidity with respect to ice at the ambient pressure and temperature. The method is independent of the measurement of the crystal size spectrum and also of any assumptions about the bulk densities of various crystal habits. Examples of IWC measurements made during the International Cirrus Experiment are presented and compared with conventional measurements from a 2D optical array probe. The prime sources of error are the accuracy of the calibration of the TWC probe and the occurrence of subsaturated air, which invalidates one of the main principles of the technique.

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