Model formulation of drop breakup requires a set of analytic functions to describe the size distribution of water fragments that result from the collision of two raindrops of arbitrary diameter. The set of fragment distribution functions derived by Low and List in 1982 has provided the foundation for most of the recent modeling studies of raindrop collision. The formulas provide reasonably accurate approximations to histogram representations of laboratory data but produce distributions of drop fragments whose masses do not sum to the masses of the colliding drops. To correct the problem, new analytic expressions are derived using least squares fits with constraints on the total water mass content of the fragments and on the number of fragments produced by collision. Introduction of the mass conservation constraint reveals that, for drop collisions in which the mass of the larger colliding drop greatly exceeds that of the smaller drop, a certain feature is missing from both the histograms and the fits of Low and List. For such large-drop–small-drop collisions, fragmentation occurs only as filament-type breakup that results in very small satellites and leaves the large-drop mass nearly intact. From a theoretical point of view, then, the distribution of large-drop remnants must resemble a delta function rather than the broader distribution indicated by the Low and List histogram representation of the data. Mass-conserving fragment distributions derived for sheet- and disk-type breakup also differ from the distributions of Low and List but to a lesser degree. New fragment distribution functions for breakup have been computed, but only for collisions involving drops of widely disparate size. Preliminary tests show that when the new formulation is incorporated in the coalescence/breakup equation, numerical solutions produce equilibrium raindrop-size distributions that have a greater number of large drops but a 6% lower rainfall rate than have equilibrium distributions calculated in past studies.
Prediction of the raindrop size distribution is a primary objective in modeling the rainfall process. Accurate prediction rests on proper formulation of the equation that describes the rate of change in the drop number density due to the effects of collision-induced breakup and coalescence. While the general form of that equation is well established, constituent functions representing the coalescence efficiency and the distribution of collision-produced drop fragments now can only be approximated by empirical formulas that are subject to measurement and interpolation error. The most extensive laboratory measurements of raindrop fragmentation are those of Low and List (1982a,b, henceforth LLa and LLb) who photographed and analyzed the results of numerous collisions between pairs of drops. In the work of LLb, analytical formulas are developed to represent the average fragment number distributions for each of the 10 drop-size pairs used in the experiments. The distribution functions, which are formulated in terms of the diameters of the larger and smaller colliding drops, also serve as interpolating functions from which the fragment distribution can be obtained for all possible sizes of colliding drops.
When the formulas of LLa and LLb are incorporated in the coalescence/breakup equation, solutions yield raindrop-size distributions that approach a trimodal equilibrium form, with the most pronounced peaks occurring in the small-drop end of the spectrum. Some observed drop-size distributions have been found to exhibit multiple peaks not unlike those appearing in the model solutions (e.g., Asselin de Beauville et al. 1988; Zawadzki and de Agostinho Antonio 1988), but inaccuracies associated with the Joss–Waldvogel disdrometer used in those field studies were discovered by Sheppard (1990), who found that artificial peaks were produced through the instrument calibration. McFarquhar and List (1992) extended the work of Sheppard to show that an alternate calibration of the disdrometer tended to eliminate peaks found in the drop spectra obtained without the recalibration. While the existence of peaks in some observed drop spectra has been confirmed through the use of optical measuring devices (e.g., Willis 1984), verification of the peaks that appear in the model solutions has not been achieved.
In more recent work, Hu and Srivastava (1995) compared observed and model-generated drop-size distributions. By calculation of the slopes of the drop spectra in the large-drop range, it was determined that many examples of observed equilibrium drop-size distributions contain more large drops than are indicated by model solutions obtained using the LLa and LLb formulation. Other observations, for example, those of Sauvageot and Lacaux (1995), similarly show drop spectra with gradual slopes that seem to contradict the more rapid drop off in the large-drop concentrations implied by the model solutions. The findings of Hu and Srivastava prompted a structural stability analysis of the coalescence/breakup equation (Brown 1995) to see if small changes in the LLa and LLb interpolation formulas that describe coalescence and breakup might lead to substantial changes in the computed equilibrium drop-size distribution. The analysis indicated that small changes in the formulas of LLa and LLb would produce only small changes in solutions to the equation and that significant alteration of the coalescence/breakup formulation would be required in order for the model solutions to contain the numbers of large-sized drops that seem to occur in nature.
The drop fragment measurements of LLa are presented in the form of histograms from which the analytic representations of the fragment number distributions of LLb are derived. The analytic expressions provide reasonable approximations to the drop fragment numbers produced by the experimental data. However, the fact that the water mass content of the derived fragment distribution is not in precise agreement with the combined mass of the colliding drops has been a matter of concern in application of the LLb formulas for prediction of the raindrop size distribution. In a study of the equilibrium drop size distribution, Valdez and Young (1985) used several techniques to add and subtract mass from the various parts of the LLb fragment distribution in order to achieve mass conservation. The difficulties led the authors to suggest that the usefulness of the LLb parameterization might be limited. Nevertheless, the coalescence/breakup model based on the formulas of LLa and LLb has been used in a variety of studies with mass conservation achieved through other artificial means. Brown (1986) and List et al. (1987) multiplied the fragment distribution function by an appropriate scaling factor to conserve the mass of each colliding-drop pair. Feingold et al. (1988) used the formulas of LLa and LLb without scaling but used a form of the coalescence/breakup equation that conserves the mass of the entire drop population in spite of mass gains and losses for individual colliding-drop pairs. It is found that the different procedures produce roughly the same equilibrium drop-size distribution for a given drop-size resolution. The similarity of the solutions further demonstrates the structural stability of the model equation, as minor differences in formulation produce only minor differences in the solution.
In this work it is shown that straightforward scaling of the fragment distribution function conceals certain inconsistencies in the formulas of LLb. Those inconsistencies are brought to light as incorporation of mass conservation in the basic data-fitting procedure reveals a physical requirement that involves significant modification of the fragment distribution formulas of LLb for certain types of drop collisions. The modified breakup formulation is shown to result in computed raindrop size distributions that are, in some respects, more consistent with observed drop spectra.
2. The fragment distribution function
The focus of this work is on the fragment distribution function and its role in determining the raindrop size distribution that results when collisions between drop pairs of all possible sizes are taken into account. The raindrop size distribution is obtained by solution of the coalescence/breakup equation in which the fragment distribution function is a key constituent. If the raindrop size distribution is represented by the number density n(m, t) for drops of mass m at time t, the coalescence/breakup equation has the form
where B and C represent the rates of change in n due to breakup and coalescence, respectively. In this model, the drop-size distribution is assumed to be spatially invariant. Integral forms for B and C are given by Gillespie and List (1978) and have been restated by others (e.g., Brown 1995). In the referenced papers, the fragment distribution function, which controls the breakup process, appears in the integrand of B as a function of the masses of the colliding drops. In this work, as in LLb, the fragment distribution function P is defined as a function of the colliding drop diameters in such a way that P(D; DL, DS)ΔD represents the average number of fragments of diameter D to D + ΔD produced by collision and subsequent breakup of two drops, the larger of diameter DL and the smaller of diameter DS. In accordance with LLb, P(D; DL, DS) is formulated as a weighted sum of three separate distribution functions, each corresponding to one of the three distinct types of breakup (filament, sheet, and disk) that were observed in their experiments. The function P then is written as
where the subscripts f, s, and d refer to filament, sheet, and disk types of breakup, respectively; R with subscript represents the proportion of fragmentation found to occur; and P with subscript the distribution function for the corresponding type of breakup. (For brevity, the dependence of the distribution functions on the colliding drop diameters DL and DS has been suppressed.) The fragment distribution function for each type of breakup is, in turn, broken down into a sum of normal- or lognormal-type distributions. If subscript b is used in place of f, s, or d to represent any of the three types of breakup, Pb can be written as
The distribution Pb,2 is nonzero only in the case of filament breakup. The domain of each Pb,i is restricted to the interval, D0 ≤ D ≤ DC, where D0 is the smallest drop diameter resolved in the experiments and DC is the diameter of the largest drop that can be created; that is,
so that the mass of a resulting fragment cannot exceed the sum of the masses of the colliding drops. For each distribution in (4), Hb,i is the height of the peak, μb,i is the drop diameter at which the peak occurs, and σb,i is approximately equal to the standard deviation of the truncated normal distribution that is cut off at D = DC. The peak of the skewed distribution in (5) occurs when ln(D) ≈ μb,3 (the median of Pb,3) so that the height of the peak is approximately equal to Hb,3eμb,3.
Following a collision, a portion of one or both of the colliding drops often can be identified. In (3), the normal distributions Pb,1 and Pb,2 describe the distributions of the large- and small-drop remnants, respectively; the lognormal function Pb,3 represents the distribution of satellite drops that are produced. In filament breakup, there always can be found a single remnant of each colliding drop, so that the fragment distribution has the full trimodal form given by (3). Due to the higher energy of the impact associated with sheet and disk breakup, however, the remnant of the small colliding drop is made unrecognizable, and all of its mass is included in the satellite distribution. Thus, Hs,2 and Hd,2 are taken to be zero, and the fragment distribution reduces to bimodal form.
Some basic properties of the various types of breakup allow formulation of relations that can be used in determining the parameters that appear in the normal and lognormal forms. Since there always remains a single large-drop remnant regardless of breakup type,
while the existence of a single small-drop remnant in the case of filament breakup results in the relation
For filament and sheet breakup, the peak of the large-drop remnant distribution occurs at the diameter of the drop before fragmentation. Thus, when b represents f or s,
while for the small-drop remnants in filament breakup,
It should be noted here that breakup is defined to occur when a collision results in two or more fragments, regardless of the fragment sizes. Under the definition, breakup includes events in which the colliding drops remain intact and in which partial coalescence brings about an increase in the mass of one of the colliding drops. Since a colliding drop need not experience a loss of mass, it is not unreasonable for the remnant distributions associated with filament and sheet breakup to be centered about the diameters of the colliding drops rather than at smaller values.
For the case of disk breakup, a large-drop remnant is not always clearly recognizable. Nevertheless, in accordance with LLb, (7) is assumed to hold to indicate the existence of a single large-drop remnant. The implications of this assumption have yet to be investigated. As for filament and sheet breakup, a normal distribution is used to describe the larger drop fragments resulting from disk breakup, but now the distribution is centered below DL at a value that depends on the diameters of both colliding drops. The procedure used in this work to determine the mode μd,1 is described in section 3c.
A further property of the breakup process is based not on the observations but on the physical principle that the total mass of the drop fragments resulting from the collision must be equal to the sum of the masses of the two colliding drops. Since drop mass is proportional to drop volume, the mass-conservation requirement can be written as
Relations (7)–(11) are taken to be basic properties of filament breakup that hold regardless of the sizes of the colliding drops, even though the first four relations are based on a limited number of observations rather than on physical principles. For the case of sheet breakup in which no small-drop remnant is recognizable, (7), (9), and (11) are applicable. For the case of disk breakup, only (7) and (11) are appropriate.
3. The data fitting procedure
a. Filament breakup
For the case of filament breakup, the nine parameters Hf,i, μf,i, and σf,1, i = 1, 2, 3, are determined first for each of the 10 drop-size pairs (DL, DS) used in the experiments. Then by the construction of formulas to interpolate the parameters over the two-dimensional region,
Pf can be calculated for arbitrary values of DL and DS as required for numerical solution of the coalescence/breakup equation. Figure 1 shows the triangular region defined by (12) with DM = 0.5 cm, the locations of the drop-size pairs used in the experiments of LLa, and the types of breakup found to occur for those drop-size pairs. It is seen that filament breakup can result regardless of the colliding-drop sizes.
For each experimental drop-size pair (DL, DS), nine relations are required to determine the nine parameters that describe the normal and lognormal distributions that comprise Pf (D). Since the basic properties of filament breakup provide only five applicable relations [(7)–(11)], the remaining relations must be supplied by requirements that arise from procedures to fit the function Pf (D) to the data for the given drop-size pair. LLa and LLb present their data in histogram form but do not describe the fitting procedure used to arrive at the analytic representations of the data. Figure 2 contains a reconstruction of one of the histograms (LLa, Fig. 11i) for the case in which DL = 0.40 cm and DS = 0.0395 cm.
In this work, least squares fits are used with integral relations to fit data values read from the LLa histograms. An integral relation, in addition to those of (7), (8), and (11), can be obtained through use of the average total fragment number F̄ given by LLb for each type of collision. In the case of filament breakup, each colliding drop produces a single, identifiable remnant accounted for in Pf,1 or Pf,2, so that the distribution of satellite drops satisfies the equation
where r, the number of colliding drop remnants, is equal to 2. The constraint, which is based only on original drop counts without size classification, reduces by one the number of least squares relations that depend on categorized histogram data.
To perform the least squares fit, values for the fragment distribution have been extracted from the LLa diagrams for drop diameters Dj, j = 1, . . . , J (J being a value between 17 and 33, depending on the particular drop-size pair), where the Dj include both the boundary points and midpoints of the histogram bins. If Dj represents a midpoint, the corresponding function value yj is assigned the histogram value for the bin; if Dj represents a boundary point, yj is taken to be the average of the histogram values for the two bins adjacent to the boundary point. The effect of the averaging procedure is to replace step-function values with values representative of a continuous, piecewise-linear function that can be better approximated by the normal and lognormal distributions. Least squares relations then can be written to minimize the sum of the squared errors at the data points with respect to any of the nine parameters. For example, minimization with respect to Hf,1 yields the relation
Best results have been obtained by selecting least squares relations that minimize root-mean-square error with respect to the heights of the peaks and with respect to the median of the satellite distribution. In the discussion to follow, a least squares relation will be denoted by the above equation number followed by the parameter being varied to minimize the error; thus, the above equation will be referenced as (14 − Hf,1).
Since μf,1 = DL and μf,2 = DS, by virtue of (9) and (10), there remain seven parameters to be determined. A nested iterative procedure has been devised to solve the seven equations needed to determine those parameters. Integrals in the constraint relations are replaced by finite sums so that all equations are in algebraic form. In the outer iteration, approximating equations are used to write σf,1 in terms of Hf,1 and σf,2 in terms of Hf,2 according to the following procedure. The integral constraints (7) and (8) can be written as
where [G(σf,i)]−1 is held fixed over one outer iteration. The remaining five parameters Hf,1, Hf,2, Hf,3, μf,3, and σf,3 must be determined through solution of five equations. If the total number of satellite drops is specified through (13), there remain four degrees of freedom with a corresponding number of least squares equations needed to determine the parameters. If both the satellite number specification and the mass-conservation constraint (11) are used, only three least squares equations of the type (14) are required. Since the number of possible least squares and integral relations exceeds the number of unknown parameters, there is some flexibility in the choice of fit. In this work, the five selected relations have been solved by an extension of Newton’s method for systems of equations. The solution values for Hf,1 and Hf,2 are substituted in (15) to continue the iteration, which proceeds until convergence is reached.
b. Sheet breakup
For the case of sheet breakup, there are only six parameters needed to determine the (single) normal and lognormal distributions. Relations (9) and (7) specify the mode and integral of the large-drop remnant distribution. The remaining equations needed to construct the fragment distribution function can be supplied by any four of the five relations that represent mass conservation (11), the specification of satellite number (13) (with r = 1), or least squares requirements (14 − Hs,1, Hs,3, μs,3). As for the case of filament breakup, the standard deviation of the normal distribution for the large-drop remnant is found by iterative solution of (15). In this work, the least squares fits have been chosen to minimize the root-mean-square error with respect to Hs3 and μs3 when the mass-conservation constraint is used and with respect to Hs3, μs3, and Hs1 when mass conservation is not enforced.
c. Disk breakup
Disk breakup similarly requires the determination of six parameters to describe the normal and lognormal distributions, and relations analogous to those used for sheet breakup can be applied with one exception. For disk breakup, relation (9) no longer applies since the mode of the normal distribution describing the large-drop remnant occurs not at DL but at a drop diameter that depends on the sizes of both colliding drops. To deal with this complication of the least squares problem, the mean μd,1 was found by using the secant method (a variation of Newton’s method) in an outer iteration to minimize the root-mean-square error of the fit over the large-drop remnant portion of the fragment distribution. Equations directly analogous to those for sheet breakup then could be used to form the inner iterations for determination of the remaining disk-breakup parameters. The five equations included in the inner iterations were selected from the six that represent the fragment number constraints (7) and (13) (with r = 1), the mass-conservation constraint, and the least squares relations that minimize the root-mean-square error with respect to Hd,1, Hd,3, and μd,3. All three least squares relations then could be applied when mass conservation was not required. For the cases in which mass conservation was enforced, convergence was facilitated by fixing μd,1 at the value obtained from the fit obtained without mass conservation.
For disk breakup, the mass conservation constraint again was used in place of the least squares relation that minimizes error with respect to the parameter representing the height of the distribution of large-drop remnants. However, fits were obtainable for only two of the five experimental drop-size pairs that give rise to disk breakup. For the three remaining drop-size pairs, convergence failure of the solution procedure was found to occur as mass conservation could not be achieved for the specified total fragment number F̄ determined from the experimental data. In those cases, either the normal or lognormal distribution, each containing a limited number of fragments (1 and F̄ − 1, respectively), is centered about a drop size that is too small to accommodate the water mass necessary to achieve mass conservation. (The histogram data allow values for μd1 and μd3 to be determined only within narrow ranges.) The problem was overcome by abandoning the specification of F̄ and allowing the least squares procedure with mass conservation to converge to a solution containing a distribution of satellites whose numbers are able to account for the requisite water mass.
4. Results of the fits
a. Filament breakup
Filament breakup is of special interest since incorporation of the mass conservation constraint can alter significantly the derived form of the fragment distribution function. The cases most affected are those in which the large-drop mass greatly exceeds the small-drop mass (in particular, the cases for which DL ≥ 0.18 cm and DS ≤ 0.0395 cm). Figure 1 shows that for such collisions, in which DL ≫ DS, filament breakup is the only type of fragmentation to occur. To illustrate the extent of the effect, the fragment distribution that results from filament breakup of drop pairs with DL = 0.40 cm and DS = 0.0395 cm is examined in detail. The histogram for this case (Fig. 2) given by LLa has drop-size categories, or bins, of about 0.005 cm in width in the small-drop range and 0.01 cm in the large-drop range. The distributions centered about drop diameters DL and DS represent the large- and small-drop remnants, while the distribution with peak near drop diameter 0.025 cm represents the satellite drops. One condition used by Low and List in choosing their bin sizes was to require that in the neighborhood of a peak each bin would contain a significant number of fragments. Such a criterion places a lower limit but not an upper limit on the width of the bins. Other considerations similarly affect the choice of bin width. For nominal colliding-drop diameters (DL, DS), variations in the actual sizes of the drops used in breakup experiments are unavoidable. Consideration of such variations along with uncertainties in measuring the fragment masses from drop-collision photographs encourage the use of bins wide enough to guarantee containment of the actual drop sizes. It will be shown, however, that a stringent upper limit placed on the bin width can be of considerable importance in determining an appropriate fragment distribution function.
Application of the data-fitting procedure described in the preceding section has been carried out using fragment number constraints (7), (8), and (13), mode designations (9) and (10), and the least squares relations (14 − Hf,1, Hf,2, Hf,3, μf,3). The resulting fit, shown by the solid curve in Fig. 2, is nearly identical to the fit of LLa (their Fig. 11i). While mass conservation was not imposed in the derivation of the fit shown in Fig. 2, the water mass content of the fragment distribution, which is computed to be 0.0321 g, differs only by 4.2% from the value 0.0335 g that represents the combined mass of the colliding drops. Modest scaling of the fitted distribution would then yield the correct water mass content at the sacrifice of causing error in the fragment number.
The failure to incorporate the mass conservation constraint in the fitting procedure, however, masks an inherent problem in the construction of the histogram that leads to a large error in the fragment distribution function. Figure 3 shows the fit obtained by replacing (14 − Hf,1) with the mass conservation requirement (11). This substitution causes the histogram values describing the large-drop remnant to have negligible influence on the overall fit. Now the three parameters describing the large-drop normal distribution are essentially determined by three conditions that are independent of the histogram shape: the location of the mode at DL; the number of large-drop remnants (=1); and the specification of total water mass, nearly all of which is contained in the large-drop remnant. The truncated normal distribution that describes the large-drop remnant is seen to have an extremely small standard deviation and commensurately high peak Hf,1 that is, in fact, cut off in the figure as the distribution Pf,1 extends far beyond the bounds of the graph. For such collisions in which DL ≫ DS and in which the satellite drops have diameters less than DS, the distribution representing the large-drop remnant is necessarily a narrow spike of the type shown in Fig. 3 and not of the broader type distribution shown in Fig. 2. When DL = 0.40 cm and DS = 0.0395 cm, the maximum size of a drop resulting from collision would be DC = 0.4001 cm. Furthermore, if all the satellites (whose combined mass is about 2 × 10−5 g) were created from mass supplied by the large drop and if the small-drop remnant were to have no more than double the original small-drop mass, the large-drop remnant would have diameter at least 0.3996 cm. Accordingly, the normal distribution representing the large-drop remnant should be expected to have a standard deviation of order 0.0001 cm. This estimate is consistent with the fit shown in Fig. 3, in which σf,1 is calculated to be 0.00018 cm.
The fragment distributions associated with filament breakup for all 10 drop-size pairs have been determined. Figure 4 shows the distributions obtained by the procedure used to construct the fit shown in Fig. 2. For comparison, Fig. 5 shows the corresponding fits obtained by the mass-conserving procedure just described. It is seen that for small-drop diameters greater than or equal to 0.0715 cm, the fits are roughly similar, but for DS = 0.0395 cm, the distribution of large-drop remnants is significantly different when mass conservation is taken into account. Just as in Fig. 3, the extremely high peaks of the mass-conserving distributions have been cut off at the upper limits of the vertical axes. Table 1 contains values of Hf,1 determined with and without the mass-conservation constraint for each of the 10 drop-size pairs used in the experiments of LLa. It is seen that the mass-conservation constraint brings about an order of magnitude increase in the height of the large-drop remnant distribution when DL ≫ DS. For the cases in which the extremely high peaks occur, convergence of the iterative solution described in section 3a was achieved only through use of many iterations, high-precision arithmetic, and initial guesses fairly close to the solution values.
b. Sheet breakup
The experimental results of LLa and LLb indicate that sheet breakup does not occur for those collisions in which the diameter of the large colliding drop greatly exceeds that of the small drop. Accordingly, in contrast to the case of filament breakup, the derived fragment distribution functions for sheet breakup experience no major (order-of-magnitude) adjustments in the heights of the peaks through application of the mass-conservation constraint. Nevertheless, the effect on the form of the fragment distribution is notable for two of the seven experimental drop pairs for which sheet breakup was found to occur. For the colliding drop pair with (DL, DS) = (0.46 cm, 0.10 cm), inclusion of the mass-conservation constraint alters the large-drop remnant portion of the distribution in much the same way for the case of sheet breakup as for filament breakup. The standard deviation is narrowed and the height increased by about a factor of 3. For the colliding drop pair with (DL, DS) = (0.18 cm, 0.0715 cm), mass conservation has only a minor effect on the derived fragment distribution for filament breakup. For sheet breakup, however, the peak is increased by about a factor of 2.5 as shown in Figs. 6a,b. The values of Hs,1 computed with and without the mass-conservation constraint are shown in Table 2 for each of the seven drop-size pairs for which sheet breakup was found to occur in the experiments of LLa and LLb.
c. Disk breakup
According to LLa and LLb, disk breakup results only from collisions between pairs of large drops, as indicated in Fig. 1. As for sheet breakup, then, wide disparity in the masses of the colliding drops does not occur, and the imposition of mass conservation does not produce the severe narrowing of the large-drop remnant distribution found in the case of filament breakup. It was noted in section 3c that certain inconsistencies in the histograms prevented specification of the total fragment number in deriving a mass-conserving fit of the data of LLa and LLb. For those cases in which the total fragment number could be specified, however, substitution of the mass conservation constraint for one of the least squares relations was found to have little effect on the resulting fit. In those two cases, mass conservation was nearly achieved without benefit of the constraint. In the remaining three cases, least squares fits with fragment number specification were found to produce fragment distribution functions that violate the mass conservation condition by considerable amounts. Figure 7 shows the fragment distributions for the colliding drop size pair (DL, DS) = (0.46 cm, 0.10 cm) derived using the least squares error minimization relations detailed in section 3c with 1) the fragment number specification F̄ = 14.58 and 2) the mass conservation constraint. For this drop-size pair, the differences that result from applying the alternate conditions are most pronounced. The distribution shown in Fig. 7a has water mass content 0.041 g, which is about 20% less than the 0.051 g contained in the colliding drop pair. The distribution of Fig. 7b has the correct water mass content but contains 20.55 fragments, compared to the observed average of 14.58 reported by LLa. In spite of the disparity, LLa report a large variance in the fragment number for this case, with the result that the value of 20.55 produced by the fitting procedure lies well within one standard deviation of the observed average F̄. Table 3 contains the values of Hd,1 computed with and without the mass-conservation constraint for the five drop-size pairs for which disk breakup was found to occur.
5. Effects of the mass conserving fit on solution of the coalescence/breakup equation
The results of the structural stability analysis of Brown (1995) indicated that only large changes in the coalescence efficiency and/or the fragment distribution functions of LLa and LLb would be enough to bring about significant change in solutions of the coalescence/breakup equation. The results of this work show that very large changes in the fragment distribution function should, in fact, be made for a limited range of colliding drop sizes. To provide a rough test of the effect of the mass conserving fragment distributions on the solutions of (1), a procedure has been devised to allow calculation of the modified fragment distribution component Pf,1 for arbitrary drop-size pairs (DL, DS). Use of the modified form of Pf,1 in conjunction with the LLb formulas to describe all the remaining fragment distribution components allows calculation of P(D; DL, DS) for the thousands of colliding drop-size pairs required for high-resolution numerical integration of the breakup integral in (1). The procedure for determining Pf,1 makes use of a two-dimensional, quadratic expression in DL and DS to represent the peak value Hf,1; σf,1 is then found by iterative solution of (15). The formula for Hf,1, described in detail in the appendix, is constructed from the 10 values given in the last column of Table 1. The revised form for Pf,1 then is “patched” into the formulas of LLb for the region R defined by the inequalities 0.18 cm ≤ DL and D0 ≤ DS ≤ 0.0715 cm. Linear interpolation is used over a narrow band of drop sizes within this region to provide a continuous transition between the LLb values and the modified values of Hf,1. While quadratic interpolation is an appropriate means of estimating Hf,1 between data points, extrapolation outside the region spanned by those points can lead to unrealistic function values. The formulas given by LLb for the various normal distribution parameters have been used to prevent such unreliable extrapolations outside region R.
Solution of the coalescence/breakup equation (1) has been carried out using the method of Bleck (1970) as extended by List and Gillespie (1976). A total of 34 bins were used to resolve the drop spectrum as a discrete function of drop mass m; bin boundaries mi were defined by the recursion relation mi+1 = 21/2 mi, i = 0, . . . , 34, where m0 is the mass of a drop of diameter D0 (=0.1 mm), the minimum drop size detectable in the experiments of LLa. Since there exists no basis for including in the model any droplets of diameter less than D0 that might be created through breakup, such droplets are neglected in calculation of the drop-size distribution. The degree of resolution used here has been shown to provide numerical solutions in reasonable agreement with those obtained using a very high degree of refinement in representing the drop-size range (Hu and Srivastava 1995, their Fig. 2).
For an initial Marshall–Palmer distribution with rainfall rate of 100 mm h−1 (and corresponding liquid water content of 4.17 g m−3), the solution to (1) has been advanced in time until the drop-size distribution converges to an equilibrium. Convergence is considered to be achieved when the solution values for log10n(m, t) remain unchanged to four significant digits over a 60-s time interval. Figure 8 shows the equilibrium solution produced by the LLa and LLb formulation (solid curve) and by the modified formulation of breakup that takes account of mass conservation (dashed curve). For each case, the breakup coefficients have been scaled in the manner described by Brown (1986). The scaling is necessary with the LLa and LLb formulation to guarantee mass conservation for all drop-size pairs. Scaling remains necessary even for a modified formulation of breakup that guarantees mass conservation at the experimental data points in order to extend the conservative property to the region between data points as well. Since scaling remains necessary, it would seem that application of the mass-conservation constraint at only a small number of points would provide little improvement over results obtained through use of the formulas of LLb. However, significant adjustment of the solution occurs since imposition of the mass-conservation constraint in deriving the fragment distribution function at the experimental data points results in distributions of dramatically different form. Weighted averages of the new forms are determined by the interpolation formulas for the region between data points; scaling then adjusts only the size of the resulting fragment distribution whose shape can differ significantly from that prescribed by the formulas of LLb.
The modified formulation is seen to produce a flattened drop spectrum that contains a greater number of large-sized drops than is produced by the LLa and LLb model. The increased number of large drops can be attributed to the fact that the modified formulation inhibits the spread of the large-drop number concentration into the adjacent bin of smaller-sized drops when collisions take place between very large and very small drops. For such collisions the large-drop size is virtually unchanged, with the result that the breakup differs little from bounce in its effect on the large colliding drop.
According to Hu and Srivastava (1995), observations of Zawadzki and Antonio (1988), Blanchard and Spencer (1970), and others indicate that, for cases of high rainfall rate, there can occur near-equilibrium drop-size distributions of the approximate form
where N0 and Λ are parameters independent of drop diameter D. A plot of log10(N0e−ΛD) then is a straight line with slope −Λ log10e. The observational data indicate slopes corresponding to values of Λ ≈ 20–25 cm−1. Similar values of Λ were derived from disdrometer data by Sauvageot and Lacaux (1995) for heavy-rain events in which collisional processes shape the drop spectrum. Linear least squares fits of the model-generated drop size distributions of Fig. 8 yield approximating exponential distributions with Λ ≐ 35 cm−1 for the Low and List model and Λ ≐ 31 cm−1 for the modified version, so that the mass-conserving formulation brings the slope of the model solutions into closer agreement with those of the observed drop spectra referenced above. (Higher-resolution solutions would result in steeper slopes for both formulations.) Though the modified formulation results in an increased number of drops of diameter greater than 2.3 mm, the rainfall rate associated with the new spectrum is actually reduced by about 6% due to depletion of drops in the 1.5–2.3-mm diameter range. The depletion of those drops also limits the increase in the reflectivity factor (which is 47.6 dBZ for the Low and List equilibrium solution) to less than 0.2% in spite of the gain in population of drops greater than 2.3 mm in diameter.
The modified formulation of breakup causes the peaks in the equilibrium distribution to become less pronounced than those found in the Low and List solutions. The result is an equilibrium form that appears more nearly exponential. Calculation of the Pearson correlation coefficient that measures the strength of the linear relationship between D and log10n(D) yields the value −0.99 for the modified solution, compared to the value −0.97 for the Low and List solution. (Exact linearity would produce a coefficient of ±1.) Accordingly, both the slope and the overall shape of the solution produced by the mass-conserving formulas are in somewhat better agreement with some observations.
Since current models produce equilibrium solutions in relatively short times, while naturally occurring equilibrium drop spectra are difficult to find except in high rainfall-rate situations, it is suspected that the model solutions contain time constants that are too small. For the solutions shown in Fig. 8, the largest time constants are 301 s for the mass-conserving formulation and 272 s for the Low and List model. The rain rates are 79 and 84 mm h−1, respectively. (The time constants are inversely proportional to the water mass content, which is 4.17 g m−3 for the case under consideration.) Accordingly, the modified solution provides a slower approach to equilibrium and perhaps a slightly better representation of reality.
6. Summary and conclusions
In this work, new fitting procedures have been developed to arrive at fragment distribution functions that approximate data provided by LLa in histogram form. Two basic procedures are compared, one that employs a least squares fit with constraints on the fragment number and another that employs a least squares fit with constraints on both fragment number and water mass content of the fragments. Comparison of the fits shows that for the case of filament breakup, in which the mass of the larger colliding drop far exceeds that of the smaller colliding drop, extremely large differences occur in the parameter values characterizing the normal distribution of the large-drop remnant. Fitting the peak of the normal distribution by least squares gives a relatively low, broad distribution (in accordance with the histogram); fitting by the imposition of mass conservation results in a narrow distribution that resembles a delta function. The disparity of the fits brings to light the need for consideration of certain physical constraints in the choice of bin widths used in construction of the histograms. For collisions between very large and very small drops, an appropriate histogram bin size for the large-drop remnant can be found by estimating the maximum amount of water mass that may be transferred from the large drop to both the small drop and the satellites in the breakup event.
The constraint on water mass has been incorporated at the expense of dropping the least squares relation that fits the peak of the large-drop remnant distribution to the corresponding peak in the histogram representation of the data. This approach to introducing mass conservation is most appropriate when the combined mass of the small-drop remnant and of the satellite drops represents only a small fraction of the mass of the large-drop remnant (i.e., when DL ≫ DS). In such cases, the height and standard deviation of the normal distribution describing the large-drop remnant are largely determined by the total water mass and the fact that there always remains one large-drop remnant. For cases in which the combined mass of the small-drop remnant and of the satellite drops is a significant fraction of the mass of the large-drop remnant, other ways of adjusting the histogram data to achieve mass conservation could be justified.
Of the three types of breakup (filament, sheet, and disk) identified by LLa, only filament breakup is found to occur when DL ≫ DS. Accordingly, the influence of the mass-conservation constraint on the derived fragment distribution function is most significant for those cases of filament breakup and is less important for other filament-producing collisions and for collisions resulting in sheet or disk breakup. A rough test of the effect of the mass-conserving fragment distribution on the derived raindrop size distribution has been obtained by use of an interpolation formula to represent the extremely high peak of the normal distribution of the large-drop remnant that results from filament breakup for the limited range of drop-size pairs where DL ≫ DS. When the resulting fragment distribution function replaces the LLb formulation in the coalescence/breakup equation, the computed drop-size distribution is found to have less pronounced peaks, more large drops, and more nearly exponential form. In spite of the increased number of large-sized drops, a decrease in the population of drops in the 1.5–2.3-mm range results in a reduction in rainfall rate and negligible increase in the reflectivity factor.
Due to the difficulty of the laboratory work that would be required to supply substantially more raindrop collision data, it is likely that further analysis will be required to draw as much information as possible from existing data. Future work will be aimed at the derivation of mass-conserving fragment distribution functions that are suitable over the entire raindrop-size domain, are consistent with the data of LLa and LLb, and incorporate new laboratory data of Beard et al. (1995) for small-raindrop collisions. It is the implicit assumption that a revised breakup formulation will lead to model-generated raindrop size distributions that are in better agreement with field observations.
This work was supported by the Division of Atmospheric Science of the National Science Foundation under Grant ATM-9424423. Computer support was provided by the Trinity College Computing Center and by the National Center for Atmospheric Research, which is sponsored by the National Science Foundation. The author is grateful to Prof. Harvey S. Picker of the Trinity College Physics Department for helpful discussions and to Jason A. Walde for performing the computer graphics work for this project.
A Formula for the Large-Drop Remnant Peak Resulting from Filament Breakup
Several formulas for evaluating the peak of the large-drop remnant distribution were constructed from the 10 values of Hf,1 given in Table 1 for the case in which mass conservation has been applied in derivation of the fragment distribution function for filament breakup. The particular formula used in obtaining the drop spectrum (dashed curve) shown in Fig. 8 is quadratic in both DL and DS with coefficients generated by a least squares fit package. The formula can be written as
(Here, DL and DS are in cm and Hf,1 has units cm−1.) The low-degree basis functions contained in (A1) yield a nonoscillatory fit that closely approximates the data.
Corresponding author address: Philip S. Brown Jr., Mathematics Department, Trinity College, Hartford, CT 06106.