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- Author or Editor: Philip S. Brown Jr. x
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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.
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.
Abstract
A parameterization of raindrop coalescence and breakup has been extended to include evaporation. The parameterization is developed through analysis of accurate numerical solutions of the coalescence/breakup/evaporation equation. Modeled drop size distributions are found to evolve first toward a trimodal form characteristic of the equilibrium distribution that occurs when only collisional processes are at work. With sustained evaporation, the trimodality disappears and a unimodal-type drop size distribution emerges. The results imply that the trimodal form occurs when collisional processes are dominant but that a unimodal distribution prevails as the water mass is reduced. The mass reduction causes collisions to become infrequent and allows evaporation to deplete the small-sized raindrop population. When subjected to continued evaporation, the coalescence/breakup equilibrium itself undergoes a transition from trimodal to unimodal form, and it is this evolving form toward which all other drop size distributions converge. In the transition, the liquid water content decreases exponentially with a time constant of 300 S −1 s, where S is the saturation deficit; furthermore, the shape of the evaporating distribution is determined by the ratio of the liquid water content to the saturation deficit. The parameterization procedure makes use of the analysis results in order to describe system behavior.
Abstract
A parameterization of raindrop coalescence and breakup has been extended to include evaporation. The parameterization is developed through analysis of accurate numerical solutions of the coalescence/breakup/evaporation equation. Modeled drop size distributions are found to evolve first toward a trimodal form characteristic of the equilibrium distribution that occurs when only collisional processes are at work. With sustained evaporation, the trimodality disappears and a unimodal-type drop size distribution emerges. The results imply that the trimodal form occurs when collisional processes are dominant but that a unimodal distribution prevails as the water mass is reduced. The mass reduction causes collisions to become infrequent and allows evaporation to deplete the small-sized raindrop population. When subjected to continued evaporation, the coalescence/breakup equilibrium itself undergoes a transition from trimodal to unimodal form, and it is this evolving form toward which all other drop size distributions converge. In the transition, the liquid water content decreases exponentially with a time constant of 300 S −1 s, where S is the saturation deficit; furthermore, the shape of the evaporating distribution is determined by the ratio of the liquid water content to the saturation deficit. The parameterization procedure makes use of the analysis results in order to describe system behavior.
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(λ j ) < 0 so that the equilibrium drop distribution is asymptotically stable. The exponential basis functions feature a wide range of decay rates, and since Im (λ j ) ≠ 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.
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(λ j ) < 0 so that the equilibrium drop distribution is asymptotically stable. The exponential basis functions feature a wide range of decay rates, and since Im (λ j ) ≠ 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.
Abstract
Numerical solutions of the coalescence/breakup equation often produce drop size distributions that move away from equilibrium before turning back. In particular, distributions evolving from Marshall–Palmer form rapidly overshoot their equilibrium position, reverse direction, and then settle slowly toward equilibrium. To explain such reversals in the changing drop distribution, an analysis has been performed using a simple model that contains only three drop-size categories. From several basic properties of the coalescence/breakup process, it is shown that the distribution is forced to develop an excess (deficit) of both large and small drops balanced by a deficit (excess) of medium-sized drops. The ratio of excess water mass in the small-drop category to that in the large-drop category approaches a constant value as the distribution approaches equilibrium. In an effort to achieve this proportion, a reversal occurs in some part of the drop distribution.
Abstract
Numerical solutions of the coalescence/breakup equation often produce drop size distributions that move away from equilibrium before turning back. In particular, distributions evolving from Marshall–Palmer form rapidly overshoot their equilibrium position, reverse direction, and then settle slowly toward equilibrium. To explain such reversals in the changing drop distribution, an analysis has been performed using a simple model that contains only three drop-size categories. From several basic properties of the coalescence/breakup process, it is shown that the distribution is forced to develop an excess (deficit) of both large and small drops balanced by a deficit (excess) of medium-sized drops. The ratio of excess water mass in the small-drop category to that in the large-drop category approaches a constant value as the distribution approaches equilibrium. In an effort to achieve this proportion, a reversal occurs in some part of the drop distribution.
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.
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.
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.
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.
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.
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.
Abstract
A finite-difference representation of the stochastic collection equation similar to the one proposed by Berry and Reinhardt (1974) has been analyzed for stability. Two sources of instability have been identified: the use of low precision in calculating the gain and loss integrals, and the use of droplet spectra having high liquid water content and small variance. Means of dealing with both sources of instability are presented. Use of mass density per unit log radius is found to offer no advantage with regard to stability over the use of number density as the dependent variable.
Abstract
A finite-difference representation of the stochastic collection equation similar to the one proposed by Berry and Reinhardt (1974) has been analyzed for stability. Two sources of instability have been identified: the use of low precision in calculating the gain and loss integrals, and the use of droplet spectra having high liquid water content and small variance. Means of dealing with both sources of instability are presented. Use of mass density per unit log radius is found to offer no advantage with regard to stability over the use of number density as the dependent variable.
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.
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.
Abstract
For some droplet distributions the equations governing the process of droplet growth by condensation constitute a stiff system, i.e., a system containing small as well as large time constants. Solution of such systems by standard numerical techniques often requires the use of extremely small time steps in order to maintain stability or convergence. This paper contains a description and analysis of an efficient, easy-to-apply solution procedure that provides stability without practical restriction on the size of the time step.
Abstract
For some droplet distributions the equations governing the process of droplet growth by condensation constitute a stiff system, i.e., a system containing small as well as large time constants. Solution of such systems by standard numerical techniques often requires the use of extremely small time steps in order to maintain stability or convergence. This paper contains a description and analysis of an efficient, easy-to-apply solution procedure that provides stability without practical restriction on the size of the time step.
