Abstract

Results of numerically investigated binary collisions of 32 drop pairs presented in Part I of this study are used to parameterize coalescence efficiencies and size distributions of breakup fragments of large raindrops.

In contrast to the well-known results of Low and List, it is shown that coalescence efficiencies Ec can be described best by means of the Weber number We yielding Ec = exp(−1.15We). The fragment size distributions gained from our numerical investigations were parameterized by fitting normal, lognormal, and delta distributions and relating the parameters of the distribution functions to physical quantities relevant for the breakup event. Thus, this parameterization has formally a substantial similarity to the one of Low and List, although no reference is made to breakup modes such as filament, disk, and sheet. Additionally, mass conservation is guaranteed in the present approach. The parameterizations from Low and List, as well as the new parameterizations, are applied to compute a stationary size distribution (SSD) from solving the kinetic coagulation–breakup equation until a time-independent state is reached. Although with the parameterizations of Low and List, the SSD shows an often-reported three-peak structure, with the new parameterizations the second peak vanishes completely.

1. Introduction

In meteorology the knowledge of raindrop size distributions is an important prerequisite in calculating related effects such as rainfall amount at ground or in evaluating remote sensing data as measured by meteorological radars, to mention only two applications. The mathematical formulation of a raindrop size spectrum as it is mostly used in cloud models is the exponential function of Marshall and Palmer (1948). However, this formulation is unable to reflect the decrease in raindrop size concentration for small raindrops as measured, for example, by Zawadski and de Agostinho Antonio (1988) or to capture some secondary maxima detected by Steiner and Waldvogel (1987). From the fact that the concentration of large raindrops continuously decreases with size, it is inferred that a process exists limiting the size of large raindrops. This implies that by some mechanism the drops break up whereby (small) fragment drops are released, which are then redistributed among a pre-existing drop spectrum. Consequently it can be expected that the modification of the drop spectrum in the manner described may have a strong impact on those cloud microphysical processes in which drops of any size are involved. Clearly this affects all types of clouds since very large drops, which are candidates for possible breakup, may be formed by pure coagulation (favorably in warm tropical clouds) or by melting of large ice particles occurring in mixed-phase midlatitude clouds.

Two disintegration mechanisms are commonly assumed to be responsible for breakup: hydrodynamic breakup and collision-induced breakup. With regard to single large drops, hydrodynamic breakup can take place (Pruppacher and Klett 1997), a parameterization of which has been given by Srivastava (1971). However, the most often supposed mechanism to effectively limit the sizes of large drops is collision-induced breakup (Hu and Srivastava 1995). This breakup mechanism has been thoroughly investigated by means of laboratory experiments decades ago. The most comprehensive analysis was performed by Low and List (1982a, hereafter LL82a). They provided parameterizations of coalescence efficiencies of large colliding drops as well as parameterizations of fragment size distributions (Low and List 1982b, hereafter LL82b). As pointed out in Schlottke et al. (2010, hereafter Part I), LL82a investigated only a very limited number of drop pairs and thus their results rely on a very weak basis.

In questioning whether the application of parameters derived from the experiments mentioned is able to account for the size distribution of large raindrops observed at ground level, on can refer to many numerical studies (e.g., Brown 1986; Tzivion et al. 1989; List and McFarquhar 1990), nearly all of which consider the parameters presented by LL82b. All the studies can explain the exponential shape of raindrop spectra but do so with differing degrees of reliability depending on the assumptions made with regard to the breakup parameters, such as extrapolating or interpolating them in regions not covered by the scant experimental data of LL82a. A feature common to all the simulations is the appearance of a type of undulation (superimposed to the exponential shape) that is presumed to reflect the predominance of a certain breakup mode (e.g., disk, sheet, or filament).

To elucidate the action of different breakup parameterizations, mostly stationary spectra are computed, taking into account the two opposing processes: collision with coalescence and collision with breakup. A reevaluation of the few data on fragment size distributions of LL82b was done by McFarquhar (2004), showing some differences in the shape of the stationary drop size distributions compared to computed older ones. This feature will be discussed at the end of this study.

A second aspect of breakup—the production of relatively small fragment drops—may alter precipitation evolution markedly since these small fragment drops are fed back into a pre-existing cloud droplet size distribution. Depending on the sizes of the fragments, two effects may appear. If the fragment drops exceed a certain size (e.g., small raindrops—the remnants of the two colliding drops), they again participate in the coagulation process. If they are too small to support coagulation (i.e., they are cloud droplets), a dynamical aspect comes into play, namely that smaller drops are transported more effectively by updrafts to higher levels where these drops may commence and promote icing conditions; that is, there is increased growth of ice particles by riming. Hints that this idea is reliable can be extracted from the few simulations with (mostly two-dimensional) cloud models operating with spectral (bin) microphysics (Feingold et al. 1991; Reisin et al. 1998; Costa et al. 2000; Seifert et al. 2005). Note that these simulations use the balance equation for a drop size distribution, which contains in its complete form not only the collection integrals but also the breakup integrals wherein the breakup functions have to be specified. In almost all related investigations, the breakup functions are expressed by the parameterizations of LL82b in the case of collision-induced breakup and by that of Srivastava (1971) in the case of hydrodynamic breakup.

Recently Beheng et al. (2006), employing numerical calculations of collision-induced breakup (as described in Part I) and extending the number of drop pairs, have shown that distinct differences exist between LL82a and their results. This has been underpinned by the results presented in Part I of this investigation, which we briefly review here.

In Part I we presented the physical equations applied as well as specifics of their numerical solution, including boundary and initial conditions for simulating collision-induced breakup of 32 raindrop pairs. Eccentricity, ε has been considered as a relevant parameter controlling the collision outcomes such that for ε ≈ 0 head-on collisions and for ε ≈ 1 grazing ones are taken into account. Some details on budgets of kinetic energies and on the evolution of the surface area during the binary drop collision are discussed. Our main interest was in the collision outcomes, which were permanent coalescence and fragmentation (i.e., breakup of a temporarily coalesced system into several fragment drops). The appearances of different breakup modes (disk, sheet, and filament) were analyzed and related to eccentricity and Weber number. Further characteristics described were the total number of outcomes and examples of a few size distributions of the outcomes.

In this study we present parameterizations of the coalescence efficiency of large drops as well as of the size distributions of fragment drops, relying on a larger number of drop pairs than LL82a, whose diameters cover a larger range. Finally, computations of stationary drop spectra resulting from the application of the new parameterizations round out this study.

2. Some technical details

The diameters of the small and large drops (dS and dL, respectively) comprising the 32 drop pairs range between 0.035 cm ≤ dS ≤ 0.18 cm and 0.06 cm ≤ dL ≤ 0.46 cm. The drop pairs, diameters, and their diameter ratios are listed in Table 1. The first 10 drop pairs are identical to those investigated by LL82a and LL82b and the first 18 drop pairs are identical to those of Beheng et al. (2006). For each drop pair, six eccentricities are taken into account with discrete values of εi = 0.05, 0.2, 0.4, 0.6, 0.8, and 0.95 and eccentricity intervals Δεi = 0.1, 0.2, 0.2, 0.2, 0.2, and 0.1, respectively. Each case represents the collision outcome occurring for an individual annulus whose cross section is εiΔεi. For details, see Part I of this study.

Table 1.

Investigated drop pairs k with diameters dL and dS (cm) as well as dL/dS with the coalescence efficiencies Ec, the number of fragments due to breakup fk,b, and the total number of collision outcomes fk as derived from simulations. The last three columns contain the values derived from the new parameterizations.

Investigated drop pairs k with diameters dL and dS (cm) as well as dL/dS with the coalescence efficiencies Ec, the number of fragments due to breakup fk,b, and the total number of collision outcomes fk as derived from simulations. The last three columns contain the values derived from the new parameterizations.
Investigated drop pairs k with diameters dL and dS (cm) as well as dL/dS with the coalescence efficiencies Ec, the number of fragments due to breakup fk,b, and the total number of collision outcomes fk as derived from simulations. The last three columns contain the values derived from the new parameterizations.

The results to be presented refer to the coalescence efficiency Ec and the spectral number of outcomes k(D) for each drop pair k considered as mean values over every six simulations weighted by the appropriate eccentricity area. The diameter of fragment drops is D. It yields k(D) = k,b(D)(1 − Ec) + k,c(D)Ec, with the spectral number of breakup fragments being k,b(D) and that of coalesced drops k,c(D) = δ(DDc), where δ(DDc) denotes the Dirac delta function and Dc = (dL3 + dS3)1/3. All these variables are defined in detail in the appendix of Part I in a continuous form. The numerical evaluation of these variables has been performed using discrete values. For example, the coalescence efficiency Ec defined by

 
formula

[cf. Eq. (A4) of Part I] is evaluated by

 
formula

where in case of a pure coalescence α(εi) = 1 and α(εi) = 0 otherwise. Note that ∑ΔεiεiΔεi yields ½, which shows up in the appendix of Part I sometimes as its inverse (i.e., as a factor of 2).

Similarly, the eccentricity area-weighted spectral number of fragments k,b(D) defined as

 
formula

(cf. the appendix of Part I) is in discrete form resolved as

 
formula

with Dj and ΔDj derived from the results of the numerical simulations.

In case of the discrete formulation of k,c(D), the Dirac delta function is evaluated as 1 if the diameter of the coalesced drop Dc lies within the jth diameter interval |DjDc| ≤ ΔDj/2 and is set to zero otherwise.

3. A new parameterization of coalescence efficiency

In this section a parameterization is initially presented that relies on the numerical results described in Part I of this study. Thereafter the parameterization is compared to other ones available in literature. Note that most of the variables occurring in the text have been defined in Part I and will not be repeated here for brevity.

The considerations for deriving an appropriate parameterization for Ec follow the argumentation of LL82a stating that the coalescence efficiency is determined by the ability of the coalesced drop to dissipate the excess energy ΔET. After several attempts to combine ΔET and its included terms, namely collision kinetic energy (CKE) and ΔS, with various related breakup parameters, the following equation gives the best results:

 
formula

The correlation coefficient and root mean square error between the parameterization and the numerically obtained data are 0.92 and 0.11, respectively. Simulated and parameterized values for Ec are displayed in Table 1 for the 32 drop pairs considered here. It should be emphasized that the equation Ec = exp(−1.15ΔET/Sc) and the relation proposed by Bradley and Stow (1984) also fit the simulation results very well. A discussion of the appropriate variables needed to parameterize coalescence efficiency is given in the appendix.

Equation (5) is displayed in Fig. 1 together with the experimental data of LL82a and the simulation results of the present study. Note that the numerical data show a line like appearance that originates from using only some discrete eccentricity values as described above. Another presentation of Eq. (5) is given in Fig. 2 as interpolated isolines in a dLdS frame. Note that reasons for Ec approaching 0 and 1, respectively, are outlined in Part I in detail and will not be presented here again. It is emphasized that Eq. (5) attains a value of 1 for We approaching 0. This extreme case would occur, on the one hand, if the relative velocity between the colliding drops is nearly zero. But then both colliding drops would be of nearly the same size and this situation is presumably very rare. On the other hand, the highest Ec values appear also for collisions between drops with very small dS and with dL, covering the entire range investigated. Generally it can be expected that bouncing comes into play, which is hard to capture with the employed numerical method. This effect may lead to Ec values smaller than those from the simulations and from Eq. (5).

Fig. 1.

Coalescence efficiency Ec as a function of the Weber number We (cf. Fig. 9 of Part I).

Fig. 1.

Coalescence efficiency Ec as a function of the Weber number We (cf. Fig. 9 of Part I).

Fig. 2.

Isolines of Ec as a function of colliding drop pairs with diameters dL and dS with numerical values indicated by the color bar.

Fig. 2.

Isolines of Ec as a function of colliding drop pairs with diameters dL and dS with numerical values indicated by the color bar.

Next we compare this parameterization to the one given by LL82a. They derived

 
formula

for ΔET < 5.0 μJ and Ec = 0 otherwise, and they obtained the constants a = 0.778 and b = 2.61 × 106 J−2 m2 from fitting the equation to their experimental results. Equation (6) and the experimental results from LL82a and from our numerical simulations are shown in Fig. 3. As can be seen in the figure, the numerical data deviate from the LL82a relation considerably. This is valid for some small σΔET2/Sc values and also for large ones for which the LL82a formulation yields zero. The latter demonstrates that (as mentioned in Part I) the limit value of LL82a could not be confirmed by the present investigation. Furthermore, our numerical data show a large scatter in the displayed framework, suggesting that Eq. (6) is not appropriate to reflect these data.

Fig. 3.

Relation between coalescence efficiency and physical quantities according to the parameterization of LL82a (solid line), from measurements of LL82a (stars), and the present simulation (circles).

Fig. 3.

Relation between coalescence efficiency and physical quantities according to the parameterization of LL82a (solid line), from measurements of LL82a (stars), and the present simulation (circles).

Brazier-Smith et al. (1972) developed a parameterization equation for the coalescence efficiency and evaluated it successfully with experimental results for relatively small raindrops with diameters ranging from 150 to 750 μm. Such small raindrops have not been investigated in the present study. In their formulation, Brazier-Smith et al. (1972) defined coalescence efficiency as the ratio of a central region of coalescence to the overall region of impact, that is,

 
formula

with the critical impact (=eccentricity) parameter εcrit above which separation occurs. In general, the assumption of a central region of coalescence (εεcrit) and an outer region of separation (ε > εcrit) is confirmed by our simulations (cf. Fig. 12 of Part I). As explained in Part I, only in cases of very large values of CKE does the central region of coalescence disappear, leading to values of Ec = 0 (cf. drop pairs 9, 10, and 31). In contrast to the considerations on energies involved in the collision process as in Part I and here, Brazier-Smith et al. (1972) considered an additional energy contribution, namely the rotational kinetic energy (RKE). In a center-of-mass frame, RKE is given as

 
formula

with υr being the relative velocity of the approaching drops. Note that RKE depends explicitly on eccentricity ε. Brazier-Smith et al. (1972) assumed that separation will occur only if the rotational kinetic energy exceeds the surface free energy, that is, if the boundary between coalescence and separation is defined by the condition RKE = ΔS. This leads with Eq. (7) to Eq. (17) of Part I, which then reads

 
formula

with γ = dL/dS and We* = ρdsυr2/σ [cf. Eq. (12) of Part I]. Values of this expression, as well as the experimental results from LL82a and from our numerical simulations, are presented in Fig. 4 as a function of f (γ). As shown in this figure, the above relation overestimates the LL82a data as well as our simulations to a great extent. Thus, it is obvious that some important effects needed to describe coalescence efficiencies of large raindrops are not included in the parameterization equation of Brazier-Smith et al. (1972).

Fig. 4.

Relation between EcWe*, and f (γ) [cf. Eq. (9)] according to the parameterization of Brazier-Smith et al. (1972) (solid line), from measurements of LL82a (stars), and the present simulation (circles).

Fig. 4.

Relation between EcWe*, and f (γ) [cf. Eq. (9)] according to the parameterization of Brazier-Smith et al. (1972) (solid line), from measurements of LL82a (stars), and the present simulation (circles).

In concluding this section we state that so far no thorough physical reasoning exists to describe and explain the coalescence process of large drops in detail, especially considering eccentricity effects. So we are left with using empirical relations.

4. Parameterization of size distributions

In this section, a parameterization is presented for the size distributions of breakup fragments as a function of fragment diameter D. It is based on the results of the numerical simulations and has substantial similarity to the parameterization of LL82b.

As explained in section 2, the average number of simulated breakup fragments in the jth diameter interval is given as k,b(DjDj with the average spectral number k,b(Dj) of fragments with mean diameter Dj in the diameter interval ΔDj. For each case of an initial pair of drops of sizes dL and dS, we partitioned the collision outcome at most into four different diameter ranges of breakup fragments:

  • the first range (r = 1), containing drops with smallest diameters,

  • the second range (r = 2), which is representative of slightly larger drops,

  • the third range (r = 3), containing drops with diameters near dS, and

  • the fourth range (r = 4), characterized by drops with diameters near dL.

Note that here no discrimination with regard to breakup modes has been made, as was done by LL82b. The third and fourth ranges are always occupied. The first range shows up when very small satellite drops are calculated and the second range appears only in case of high collision kinetic energy, as shown in the following. Histograms for each collision of drop pairs k with diameters dL and dS are depicted in Figs. 5 and 6 as rectangles, except for drop pairs 11 and 30, which showed pure coalescence for every eccentricity (cf. Part I).

Fig. 5.

Spectral number of fragments (cm−1) as a function of fragment diameter (cm) for dL and dS as indicated. Black rectangles refer to the simulation results, blue lines to the parameterizations for 30 different drop pairs, and red lines to the parameterization of LL82b. The first 10 pairs are the same as in LL82b. In the cases of pair 11 and pair 30, the coalescence efficiency is 1.0; that is, in these cases no breakup occurs, so these data are omitted.

Fig. 5.

Spectral number of fragments (cm−1) as a function of fragment diameter (cm) for dL and dS as indicated. Black rectangles refer to the simulation results, blue lines to the parameterizations for 30 different drop pairs, and red lines to the parameterization of LL82b. The first 10 pairs are the same as in LL82b. In the cases of pair 11 and pair 30, the coalescence efficiency is 1.0; that is, in these cases no breakup occurs, so these data are omitted.

Fig. 6.

Continuation of Fig. 5.

Fig. 6.

Continuation of Fig. 5.

The mathematical formulation of the parameterization regarding the fragment size distributions follows to a large extent that of LL82b. Thus, for each range r, k,b(Dj) is approximated by Pr(D) = Nrpr(D), with Nr being the number of fragments belonging to range r and pr(D) a weighting function yielding ∫pr(D) dD = 1. Note that for convenience, the subscripts describing drop pair k and fragments only, index b, as in k,b, are omitted in the following development. Moreover, Pr(D) is used as abbreviation for Pr(D; dS, dL) with the pair dS, dL indicating the breakup event for a drop pair having diameters dS and dL.

a. First range (r = 1)

With regard to the first range r = 1 spectral fragment numbers are parameterized by a lognormal distribution reading

 
formula

where

 
formula

Now, the average spectral number P1(D) of fragments in the first range is given as

 
formula

It turns out that the parameters appropriate for parameterizing the numerical data are the mean, E, and the variance Var of the lognormal distribution. From the numerical results we derived E = D1 = 0.04 cm (i.e., a constant value) and Var = ΔD12/12, with ΔD1 parameterized by Eq. (16) below. Note that E and Var are linked to σ1 and μ1 by

 
formula
 
formula

In expressing N1 and ΔD1 by certain parameters relevant to breakup, it is found that

 
formula
 
formula

with the abbreviation CW = CKE × We and γ = dL/dS. As in the following, CKE is given in μJ and ΔD in cm. Simulated and calculated N1 and ΔD1 are correlated with correlation coefficients of 0.92 in both cases. The dependencies of simulated and calculated N1 and ΔD1 from γCW and CW, respectively, are shown in Fig. 7.

Fig. 7.

Parameters (left) N1 as a function of γCW and (right) ΔD1 as a function of CW from simulations as well as from the parameterization.

Fig. 7.

Parameters (left) N1 as a function of γCW and (right) ΔD1 as a function of CW from simulations as well as from the parameterization.

It should be emphasized here that drops with diameter ≈D1 are clearly raindrops. However, because of the distribution function [Eq. (10)] much smaller drops are also covered by this relation. Thus, a definite discrimination of drops in this range as cloud drops or (small) raindrops is not possible.

b. Second range (r = 2)

As in LL82b, in the second range r = 2 a normal distribution is used given by

 
formula

where

 
formula

Now, the average spectral number P2(D) of fragments in the second range is given as

 
formula

Here, μ2 = D2 and σ22 = ΔD22/12. Again, from the numerical results D2 is approximated as a constant fragment diameter D2 = 0.095 cm and N2 and ΔD2 as

 
formula
 
formula

In these cases, simulated and calculated N2 and ΔD2 are correlated with correlation coefficients of 0.99 in both cases. The dependencies of simulated and calculated N2 and ΔD2 from CW are shown in Fig. 8.

Fig. 8.

As in Fig. 7, but for parameters N2 and ΔD2.

Fig. 8.

As in Fig. 7, but for parameters N2 and ΔD2.

c. Third range (r = 3)

As in the second range, in the third range r = 3 the spectral fragment number is also characterized by a normal distribution:

 
formula

with μ3 = D3 and σ32 = ΔD32/12. In this case, the average spectral number P3(D) of fragments in the third range is

 
formula

From the numerical simulations it is found that the mean fragment diameter is approximately D3 = 0.9dS. The quantities N3 and ΔD3 are approximated as

 
formula

and

 
formula

Simulated and calculated N3 and ΔD3 are correlated with correlation coefficients 0.58 and 0.80, respectively. The dependencies of simulated and calculated N3 and ΔD3 from CW are shown in Fig. 9.

Fig. 9.

As in Fig. 7, but for parameters N3 and ΔD3.

Fig. 9.

As in Fig. 7, but for parameters N3 and ΔD3.

d. Fourth range (r = 4)

The spectral fragment numbers in the fourth range r = 4 are expressed differently from those of the other ranges. Here we consider the Dirac delta function δ(DD4), the integral of which is

 
formula

and the average spectral number P4(D) of drops, P4(D) = N4δ(DD4). This follows from the numerical simulations showing that, in general, N4 = 1. To ensure mass conservation, the diameter D4 is derived from with M3,4 calculated as the residual of the masses [=third moments of the fragment size distributions functions (indicated by the first subscript)] of the two initial drops minus the masses of the drops from ranges 1, 2, and 3 (second subscript):

 
formula

where the moments M3,r on the rhs are given according to their range-specific distribution functions assumed by

 
formula
 
formula
 
formula

Note that in this way a single drop remains with a diameter only slightly smaller than dL.

Now, the overall spectral number of breakup fragments is given as

 
formula

and is displayed in Figs. 5 and 6 for all 30 cases of different drop pairs of sizes dL and dS (note that in two cases, drop pairs 11 and 30, the coalescence efficiency is one; that is, no breakup fragments appeared in the simulations as mentioned earlier). According to section 2, the average number of resulting drops is given by

 
formula

which is equivalent to Eq. (A9) of Part I.

To be consistent with the nomenclature of Part I, we now reintroduce the subscript k denoting drop pair k, such that the average number fk,b of all breakup fragments for each drop pair is then

 
formula

which is given in Table 1 together with the numbers of the breakup fragments gained from the numerical simulations. The overall number fk of all resulting drops is then fk = fk,b(1 − Ec) + Ec, which is equivalent to Eq. (A8) of Part I and is also given in Table 1 together with the overall number of drops from the numerical simulations. An approximate relation of fk as a function of CKE is given in section 4e of Part I.

As can be seen in Figs. 5 and 6, the parameterization very well fits the numerically obtained results for the position and amplitude of the maxima in ranges 1 to 3 of fragment diameter. However, differences compared to the parameterization of LL82b can also be easily seen. An exception is the current parameterization in case of pair 32. There the small diameter peak is not recovered. The opposite is true for pair 15 where a small-scale peak emerges from the current parameterization but is not found in the numerical results. These discrepancies, however, seem to be of minor importance.

It should be mentioned here that McFarquhar (2004) arrived at a parameterization that is an extension and improvement compared to that of LL82b. In McFarquhar (2004) uncertainties associated with the experimental data of LL82a and those related to this parameterization were investigated and quantified. Moreover, relations for fragment size distributions were derived that were valid for arbitrary small and large colliding drops. In the present study, uncertainties with respect to our numerical experiments would only involve eccentricity spacing and spatial resolution. But the immense computational costs prevented a finer spacing. With regard to spatial resolution, the reader is referred to results presented in Part I where an assessment of our parameterization can be inferred from the correlation coefficients given in the text.

5. Stationary drop size distributions

Having derived new parameterizations of the coalescence efficiency (section 3) and the fragment size distribution (section 4), the question arises of how the new breakup parameterizations influence the shape of a size distribution resulting from solving the kinetic coagulation–breakup equation (KCBE) until a stationary state is reached (i.e., the production of larger drops by collision/coalescence equals the loss of large drops by breakup). Note that evaporation effects are omitted here, as in Part I. The KCBE used and its numerical solution method are described in detail by Seifert et al. (2005) and Pruppacher and Klett (1997). The breakup kernel B(x, y), depending on drops with masses x and y, is defined by B(x, y) = K(x, y)(1 − Ec)/Ec, with the collection kernel for gravitational coagulation K(x, y) = π[r(x) + r(y)]2|υ(x) − υ(y)|Cc(x, y)Ec(x, y), where r and υ are the radii and the terminal fall velocities, Cc(x, y) is the collision efficiency and Ec(x, y) is the coalescence efficiency for the drops under consideration. For specification of the different variables the reader is referred to Seifert et al. (2005). The fragment size distribution function appearing in the breakup integrals of the KCBE, termed P(x; x′, x″) (with drop masses as independent variables), is equivalent to Eq. (32) where P(D) ≡ P(D; dS, dL).

The mass coordinate x is discretized using a logarithmic grid defined by xi+1 = xi21/β with β = 3 (i.e., the mass doubles every third bin). Initially an exponential distribution of Marshall and Palmer (1948) is assumed, reading f (d, t = 0) ≡ f0(d) = N0 exp(−λd), with N0 = 8 × 106 m−4 and λ = 4.1 × 103R−0.21 m−1, with diameter d in m and rain rate R in mm h−1, where R = 54 mm h−1 is prescribed according to McFarquhar (2004). The KCBE is integrated until stationarity is achieved, which needs about a 2-h simulation time.

In Fig. 10 stationary drop size distributions (SSDs) are shown. They were obtained by applying the parameterization originally developed by LL82b (LL curve) and also calculated by the new parameterization (NP curve). In addition, the SSD of McFarquhar (2004) is included. Clearly seen are the three peaks of the LL curve occurring at 0.26, 0.80, and 1.95 mm. It should be noted that the choice of the discretization parameter β has an influence on the numerical result, as was pointed out by Hu and Srivastava (1995), but the overall appearance of the curve (e.g., the three-peak structure of the LL curve) is conserved.

Fig. 10.

Stationary drop size distributions f (d) (m−3 mm−1) as a function of drop diameter d (mm) for the cases indicated. See text for details.

Fig. 10.

Stationary drop size distributions f (d) (m−3 mm−1) as a function of drop diameter d (mm) for the cases indicated. See text for details.

The NP curve, however, shows some remarkable differences compared to the LL one. In comparison to the LL curve, one recognizes for the small-diameter peak a slight shift of the NP curve toward about 0.35 mm but nearly the same amplitude. However, a striking difference concerns the intermediate peak of the LL curve, which has completely disappeared with the new parameterization such that a bimodal structure then shows up. The pronounced plateau of the LL curve at middle diameters has in case of the NP curve been moved to smaller diameters with a faint maximum at 1.6 mm. Furthermore, the slope of the NP curve at largest drop diameters is strongly diminished compared to the LL curve. In effect, this means that under stationary conditions more very large drops are existent. The slope of the NP curve for drop diameters larger than about 2 mm is estimated to 22 cm−1, which fits very well to that derived by Hu and Srivastava (1995) from surface measurements at high rainfall rates. Moreover, the slopes of nearly 500 raindrop spectra measured in Karlsruhe (Germany) during 8 years when rain rate was higher than about 10 mm h−1 (Seifert et al. 2005) vary between 15 and 18 cm−1, which is also in fair agreement with the calculated one. However, one must be careful with this interpretation as it is not clear that the measured spectra are stationary ones.

It is interesting to note that the SSD curve calculated by McFarquhar (2004) (MF curve) agrees with the NP curve in that the middle peak also vanishes. In comparison to the LL curve, one notices that the maximum at the small-diameter part of the SSD is approximately at the same position but has a greater amplitude and the slope at the large diameter end is comparable. Despite the gross agreement between the NP and the MF curves, differences are also visible. Between the smallest diameter and a diameter of about 1.6 mm, the MF curve shows considerably larger values, which indicates that in this range the number density is higher than that covered by the NP curve. For diameters larger than about 2.8 mm the NP representation yields more large raindrops than does the MF case. The qualitative agreement between the NP and MF curves, however, is astonishing since the procedure applied by McFarquhar (2004) has its roots in an extrapolation of the LL82b parameterization (with its scant database) to arbitrary drop pairs whereas the present study is based on numerical calculations considering a very wide range of drop diameters as presented and discussed in Part I.

Finally, it is stressed that based on the shape of the SSDs (NP and MF curves) shown in Fig. 10, these spectra may mathematically be better described by a combination of at least two gamma functions instead of an exponential function (cf. Ulbrich 1983).

6. Summary and outlook

In this study new parameterizations of coalescence efficiency and fragment drop size distributions have been presented. For the coalescence efficiency a simple parameterization in terms of the Weber number is proposed. However, as discussed in the appendix, this relation is an empirical one that does not consider any eccentricity effect. The parameterization of the fragment drop spectra follows formally that of LL82b but does not refer to their breakup-mode specific reasoning. Furthermore, the effect of applying the new parameterizations on the temporal evolution of a drop size spectrum is investigated. The drop size spectrum is calculated until stationarity is accomplished (i.e., formation of large drops by coagulation is counteracted by the reduction of large drops due to breakup). In contrast to the LL82b parameterization, yielding an intermediate peak at a diameter of about 0.8 mm, the present results do not confirm this mode. Also, the slope of the drop size spectrum in the large drop diameter range is markedly reduced compared to that obtained by applying the parameterization of LL82b. This behavior is supported by surface measurements in strong rain as reported by Hu and Srivastava (1995) and Seifert et al. (2005), but this statement has to be taken with care.

The overall appearance of the stationary size spectrum is astonishingly in qualitatively good agreement with the results of McFarquhar (2004), who applied a parameterization based on an extrapolation of the experimental data of LL82a, whereas the present results rely on a completely different basis, namely numerical simulations of collision-induced breakup by means of an advanced CFD code (cf. Part I of this study).

Finally, it is stressed that the parameterizations are easy to implement in existing models that numerically solve the kinetic coagulation–breakup equation and may also be used to improve coarser models describing cloud microphysical processes in terms of moment equations as derived by Seifert and Beheng (2006).

A minor drawback of this investigation is that the presented parameterization of coalescence could not be based on a comprehensive physical model and parameters comprising all effects controlling this mechanism, especially the dependency on eccentricity. This is left to future work.

Acknowledgments

The authors gratefully acknowledge support by the German Science Foundation (DFG) under Grants BE 2081/7-1 and WE 2549/17-1 in the framework of the priority program Quantitative Precipitation Forecast. Greg McFarquhar kindly provided the data for presenting his curve in the last figure. We also express our gratitude to Dr. C. Ancun whose hospitality has contributed essentially to the preparation of this study.

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APPENDIX

Remarks on the Choice of the Appropriate Variables for Parameterizing Coalescence

Several variables have been defined in Part I and here with regard to the coalescence process; among them are ΔET, CKE, and We. In seeking for appropriate variables for parameterizing coalescence, using any one or a combination of such variables is possible. It turns out that ΔET and CKE alone are not adequate parameters for parameterization. But consideration of the Weber number We or the variable ΔET/Sc seems to be an appropriate measure [cf. Eq. (5)]. Both include a dependency on the surface energies ST or Sc. Assuming the surface energy of the coalesced system Sc as defined by Sc = πσ(dL3 + dS3)2/3 [cf. Eq. (9) of Part I], however, is problematic insofar as it is here tacitly assumed that the drop system during the collision event has a spherical shape with an effective diameter de = (dL3 + dS3)1/3. This is in contradiction to the numerical results shown in Figs. 4 and 5 of Part I. Evidently the coalesced system has a constant volume but mostly an irregular shape such that the surface energy developing during collision should better be formulated by Si = σA, with A = surface area and the subscript i indicating an intermediate surface state of the colliding system. Because the temporal development of A is complex, it seems that one cannot put that evolution in a simple formula. Also, taking into account RKE [Eq. (8)] that explicitly depends on eccentricity, the condition by Brazier-Smith et al. (1972) is not consistent with our numerical results (cf. Fig. 4). This may be due to the dependency on the Weber number We* instead of We. For a discussion of consideration of We or We*, the reader is referred to section 4d of Part I.

In conclusion it can be stated that no appropriate physical or geometrical parameter(s) or combinations are available from which a reliable parameterization of coalescence efficiency can be derived.

Since this problem does exist and no definite (and simple) description (especially of the evolving surface area during collision) is actually at hand, we have restricted the choice of suitable parameters to those used in this study.

Footnotes

* Current affiliation: North Rhine-Westphalia State Agency for Nature, Environment, and Consumer Protection, Recklinghausen, Germany

Corresponding author address: Dr. Klaus D. Beheng, Institut für Meteorologie und Klimaforschung, Karlsruher Institut für Technologie, 76131 Karlsruhe, Germany. Email: klaus.beheng@kit.edu