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## Abstract

Using results of laboratory experiments on collision-induced raindrop breakup, Low and List (LL) developed a parameterization describing the fragment size distribution (FSD) produced by collisions of raindrops. An equilibrium raindrop size distribution (ED) is approached when this parameterization is used in numerical models of steady rain. Since scant observational evidence of such EDs exists, the need for a careful examination of the parameterization's foundation is evident. Using LL's experimental observations, an alternate parameterization is developed that alleviates three shortcomings of the original scheme, namely, ensuring mass conservation, the use of adequate uncertainty analysis, and the use of a more physical basis for deriving parameterized relationships. FSDs generated by raindrop collisions are represented by combinations of lognormal, Gaussian, and modified delta distributions for each of the three breakup types (filament, sheet, and disk) observed. The mode, width, and height of these distributions are calculated for the 10 colliding drop size combinations used in the LL experiments; uncertainty estimates for these parameters are determined using a bootstrap method, a technique that randomly chooses results of individual collisions. Relations giving the mode, width, and height in terms of the diameters of arbitrary large and small colliding raindrops are then determined so that the FSD from any raindrop collision can be predicted.

Simulations with a time-dependent box model are conducted using an exponential Marshall–Palmer distribution, with a rain rate *R* of 54 mm h^{−1} as input. A bimodal ED, with peaks at 0.26 and 2.3 mm, is approached numerically from the opposing forces of coalescence and breakup. The nature of the ED differs from that found in previous studies using the LL parameterization, which had three peaks at diameters of 0.26, 0.91, and 1.8 mm. Simulations that produced EDs from consideration of only specific breakup types showed that filament breakups were mainly responsible for the production of the peak at 0.26 mm, and that the small drop peaks associated with sheet and disk breakup had too small of an amplitude compared to the small drop peak produced by filament breakup and occurred at diameters separated too far from each other to produce a third peak in the ED generated from all breakup types acting simultaneously.

There are substantial differences in total number of raindrops, *N,* for the ED associated with this parameterization (8.6 × 10^{3} m^{−3}) to the *N* associated with that of LL (4.1 × 10^{3} m^{−3}); however, differences in *R* (51.1– 50.5 mm h^{−1}) and radar reflectivity factor, *Z* (4.7 × 10^{4} cf. 3.5 × 10^{4} mm^{6} m^{−3}), are less significant. Large differences between *Z* and *N* values associated with the Marshall–Palmer distribution (8.0 × 10^{4} mm^{6} m^{−3} and 4.0 × 10^{3} m^{−3}) also exist. Bimodal distributions with peaks at 0.26 and 2.3 mm are consistently realized in a series of Monte Carlo simulations, where fit coefficients are randomly chosen from the surface of equally realizable solutions, but the height of the small drop peak can vary from 5.1 × 10^{3} to 1.2 × 10^{4} m^{−3} per logarithmic coordinate, and that of the large drop peak from 4.2 × 10^{2} to 6.7 × 10^{2} m^{−3} per logarithmic coordinate. Although *N* can vary from 6.2 × 10^{3} to 1.1 × 10^{4} m^{−3} between simulations, variations in *R* (48.4 to 53.1 mm h^{−1}) and *Z* (4.1 × 10^{4} to 5.5 × 10^{4} mm^{6} m^{−3}) are less significant. These results hence place a limit on the certainty with which the collision-induced breakup of raindrops can be predicted.

## Abstract

Using results of laboratory experiments on collision-induced raindrop breakup, Low and List (LL) developed a parameterization describing the fragment size distribution (FSD) produced by collisions of raindrops. An equilibrium raindrop size distribution (ED) is approached when this parameterization is used in numerical models of steady rain. Since scant observational evidence of such EDs exists, the need for a careful examination of the parameterization's foundation is evident. Using LL's experimental observations, an alternate parameterization is developed that alleviates three shortcomings of the original scheme, namely, ensuring mass conservation, the use of adequate uncertainty analysis, and the use of a more physical basis for deriving parameterized relationships. FSDs generated by raindrop collisions are represented by combinations of lognormal, Gaussian, and modified delta distributions for each of the three breakup types (filament, sheet, and disk) observed. The mode, width, and height of these distributions are calculated for the 10 colliding drop size combinations used in the LL experiments; uncertainty estimates for these parameters are determined using a bootstrap method, a technique that randomly chooses results of individual collisions. Relations giving the mode, width, and height in terms of the diameters of arbitrary large and small colliding raindrops are then determined so that the FSD from any raindrop collision can be predicted.

Simulations with a time-dependent box model are conducted using an exponential Marshall–Palmer distribution, with a rain rate *R* of 54 mm h^{−1} as input. A bimodal ED, with peaks at 0.26 and 2.3 mm, is approached numerically from the opposing forces of coalescence and breakup. The nature of the ED differs from that found in previous studies using the LL parameterization, which had three peaks at diameters of 0.26, 0.91, and 1.8 mm. Simulations that produced EDs from consideration of only specific breakup types showed that filament breakups were mainly responsible for the production of the peak at 0.26 mm, and that the small drop peaks associated with sheet and disk breakup had too small of an amplitude compared to the small drop peak produced by filament breakup and occurred at diameters separated too far from each other to produce a third peak in the ED generated from all breakup types acting simultaneously.

There are substantial differences in total number of raindrops, *N,* for the ED associated with this parameterization (8.6 × 10^{3} m^{−3}) to the *N* associated with that of LL (4.1 × 10^{3} m^{−3}); however, differences in *R* (51.1– 50.5 mm h^{−1}) and radar reflectivity factor, *Z* (4.7 × 10^{4} cf. 3.5 × 10^{4} mm^{6} m^{−3}), are less significant. Large differences between *Z* and *N* values associated with the Marshall–Palmer distribution (8.0 × 10^{4} mm^{6} m^{−3} and 4.0 × 10^{3} m^{−3}) also exist. Bimodal distributions with peaks at 0.26 and 2.3 mm are consistently realized in a series of Monte Carlo simulations, where fit coefficients are randomly chosen from the surface of equally realizable solutions, but the height of the small drop peak can vary from 5.1 × 10^{3} to 1.2 × 10^{4} m^{−3} per logarithmic coordinate, and that of the large drop peak from 4.2 × 10^{2} to 6.7 × 10^{2} m^{−3} per logarithmic coordinate. Although *N* can vary from 6.2 × 10^{3} to 1.1 × 10^{4} m^{−3} between simulations, variations in *R* (48.4 to 53.1 mm h^{−1}) and *Z* (4.1 × 10^{4} to 5.5 × 10^{4} mm^{6} m^{−3}) are less significant. These results hence place a limit on the certainty with which the collision-induced breakup of raindrops can be predicted.

## Abstract

Pulsed input of raindrop packages at the top of one-dimensional shafts, followed by coalescence and breakup during the fall produces, when integrated over time of the whole rainfall, three-peak drop-size distributions (3PDs) at all levels throughout the shaft. The 3PDs are different from the three-peak equilibrium distributions (3PEDs) that develop with steady sources of rain. However, for long pulse lengths, the 3PD becomes closer to the 3PED; for short pulse lengths, the large-drop peak is not very prominent for the input of Marshall-Palmer distributions. The diameters corresponding to the maximum concentrations are approximately the same as those of the 3PED. For 3PED input, the biggest differences between the number concentrations for the 3PED and 3PDs occur for pulse lengths between 240 and 600 s while the radar reflectivity concentrations of 3PDs steadily approach that of 3PEDs with increasing pulse length. All trends are the same whether a 3PED or a Marshall-Palmer distribution is used as input.

In nature, the raindrops often arrive at the ground in packets, with the largest drops followed by progressively smaller ones. This, and the presence of 3PDs at the ground, when the drop-size distributions were integrated over time, were observed in Malaysia by List et al.

## Abstract

Pulsed input of raindrop packages at the top of one-dimensional shafts, followed by coalescence and breakup during the fall produces, when integrated over time of the whole rainfall, three-peak drop-size distributions (3PDs) at all levels throughout the shaft. The 3PDs are different from the three-peak equilibrium distributions (3PEDs) that develop with steady sources of rain. However, for long pulse lengths, the 3PD becomes closer to the 3PED; for short pulse lengths, the large-drop peak is not very prominent for the input of Marshall-Palmer distributions. The diameters corresponding to the maximum concentrations are approximately the same as those of the 3PED. For 3PED input, the biggest differences between the number concentrations for the 3PED and 3PDs occur for pulse lengths between 240 and 600 s while the radar reflectivity concentrations of 3PDs steadily approach that of 3PEDs with increasing pulse length. All trends are the same whether a 3PED or a Marshall-Palmer distribution is used as input.

In nature, the raindrops often arrive at the ground in packets, with the largest drops followed by progressively smaller ones. This, and the presence of 3PDs at the ground, when the drop-size distributions were integrated over time, were observed in Malaysia by List et al.

## Abstract

The release of multiple pulses of rain with durations of 120 s < *T* < 600 s and pulse repetition periods of 240 s < *&tau* < 1200 s at the top of one-dimensional shafts, followed by drop coalescence and breakup during the fall, leads to the arrival of drops in overlapping packages at the ground, with the largest drops first in the individual packages. When the drop spectra are averaged over time, three-peak distributions (3PDs) are found. More frequent and shorter pulses produce 3PDs that resemble three-peak equilibrium distributions (3PED) for steady warm rain closer than less frequent longer pulses because overlapping pulses lead to more interactions between large and small drops. When a 3PED is released at shaft top, it disappears because different size drops fall at different speeds, and then must be recreated by drop interactions.

These multiple pulses replicate trends measured during nonsteady tropical rain better than single pulses. A “pulse interference diameter,” based on the parameterization of single pulse rain, describes how much interaction occurs between pulses.

## Abstract

The release of multiple pulses of rain with durations of 120 s < *T* < 600 s and pulse repetition periods of 240 s < *&tau* < 1200 s at the top of one-dimensional shafts, followed by drop coalescence and breakup during the fall, leads to the arrival of drops in overlapping packages at the ground, with the largest drops first in the individual packages. When the drop spectra are averaged over time, three-peak distributions (3PDs) are found. More frequent and shorter pulses produce 3PDs that resemble three-peak equilibrium distributions (3PED) for steady warm rain closer than less frequent longer pulses because overlapping pulses lead to more interactions between large and small drops. When a 3PED is released at shaft top, it disappears because different size drops fall at different speeds, and then must be recreated by drop interactions.

These multiple pulses replicate trends measured during nonsteady tropical rain better than single pulses. A “pulse interference diameter,” based on the parameterization of single pulse rain, describes how much interaction occurs between pulses.

## Abstract

The shapes and single-scattering properties of small, irregular, quasi-spherical ice crystals, with equivalent radii between approximately 8 and 90 *μ*m and size parameters from about 90 to 1000, are studied using two-dimensional images measured by a cloud particle imager in midlatitude cirrus during the 2000 Cloud Intensive Operation Period conducted over the Atmospheric Radiation Measurement program's Southern Great Plains site. A statistical shape analysis of the ice crystal images is carried out to obtain size-dependent relative standard deviations of radius and correlation functions of logradius, which together define the shape statistics of the sample ice crystals. The former describes the overall variation in the lengths of radius vectors defining the particle surface from a given origin, whereas the latter describes correlations of lengths of radius vectors as functions of angular distance between them. The logradius is essentially the natural logarithm of radius. There is no strong dependence of the shapes of these particles on size. The retrieved correlation functions resemble a power-law correlation function closely, suggesting that the shape statistics based on the power-law correlation function can be used as a first approximation for the shapes of small, quasi-spherical ice crystals in cirrus. By using the retrieved shape statistics as input, a Gaussian random sphere geometry previously used to describe the shapes of desert dust particles is used to generate model particles that obey the retrieved shape statistics. The single-scattering properties of these particles are then computed at a wavelength of 550 nm using ray optics. The scattering simulations give asymmetry parameters around 0.76, slightly smaller than values suggested from previous studies using Chebyshev polynomials to describe the shapes of quasi-spherical ice crystals. Assuming a flux accuracy criterion at the top of the atmosphere of ±5% is needed, this difference in asymmetry parameter is significant.

## Abstract

The shapes and single-scattering properties of small, irregular, quasi-spherical ice crystals, with equivalent radii between approximately 8 and 90 *μ*m and size parameters from about 90 to 1000, are studied using two-dimensional images measured by a cloud particle imager in midlatitude cirrus during the 2000 Cloud Intensive Operation Period conducted over the Atmospheric Radiation Measurement program's Southern Great Plains site. A statistical shape analysis of the ice crystal images is carried out to obtain size-dependent relative standard deviations of radius and correlation functions of logradius, which together define the shape statistics of the sample ice crystals. The former describes the overall variation in the lengths of radius vectors defining the particle surface from a given origin, whereas the latter describes correlations of lengths of radius vectors as functions of angular distance between them. The logradius is essentially the natural logarithm of radius. There is no strong dependence of the shapes of these particles on size. The retrieved correlation functions resemble a power-law correlation function closely, suggesting that the shape statistics based on the power-law correlation function can be used as a first approximation for the shapes of small, quasi-spherical ice crystals in cirrus. By using the retrieved shape statistics as input, a Gaussian random sphere geometry previously used to describe the shapes of desert dust particles is used to generate model particles that obey the retrieved shape statistics. The single-scattering properties of these particles are then computed at a wavelength of 550 nm using ray optics. The scattering simulations give asymmetry parameters around 0.76, slightly smaller than values suggested from previous studies using Chebyshev polynomials to describe the shapes of quasi-spherical ice crystals. Assuming a flux accuracy criterion at the top of the atmosphere of ±5% is needed, this difference in asymmetry parameter is significant.

## Abstract

Equilibrium raindrop size distributions with three peaks in number concentration (3PED) result from different breakup types and coalescences. Filament breakup produces drops in the small-drop peak of the 3PED and sheet breakup in the medium-drop peak, while disk breakup does not significantly affect the 3PED. Drops in the large-drop peak are produced by all types of interaction.

A forty-dimensional phase space for number concentration, each dimension representing a different drop size, shows how equilibrium is approached and maintained. When reduced to seven dimensions, the complex processes of moving drops in and out of the peaks at equilibrium are clearly identified. Small drops are lost by coalescence, and filament breakup is the major sink of medium drops.

The introduction of a measure for the difference between two spectra, the metric, shows that, starting from a Marshall–Palmer distribution with a rainrate of 54 mm h^{−1}, the distribution moved farther away from the 3PED between 120 s and 350 s before approaching the 3PED continuously. A three-dimensional model, the diameter ranges under the peaks being the dimensions, shows that this occurs because many small drops are produced by filament breakup replacing the initial deficit of small drops with a surplus, which subsequently is depleted.

There is no evidence for any oscillation of the raindrop spectra about its equilibrium value. It appears that a significant population of both 1 mm and 2 mm drops are needed before a 3PED can occur, but a 3PED can be safely assumed to develop once they are present.

## Abstract

Equilibrium raindrop size distributions with three peaks in number concentration (3PED) result from different breakup types and coalescences. Filament breakup produces drops in the small-drop peak of the 3PED and sheet breakup in the medium-drop peak, while disk breakup does not significantly affect the 3PED. Drops in the large-drop peak are produced by all types of interaction.

A forty-dimensional phase space for number concentration, each dimension representing a different drop size, shows how equilibrium is approached and maintained. When reduced to seven dimensions, the complex processes of moving drops in and out of the peaks at equilibrium are clearly identified. Small drops are lost by coalescence, and filament breakup is the major sink of medium drops.

The introduction of a measure for the difference between two spectra, the metric, shows that, starting from a Marshall–Palmer distribution with a rainrate of 54 mm h^{−1}, the distribution moved farther away from the 3PED between 120 s and 350 s before approaching the 3PED continuously. A three-dimensional model, the diameter ranges under the peaks being the dimensions, shows that this occurs because many small drops are produced by filament breakup replacing the initial deficit of small drops with a surplus, which subsequently is depleted.

There is no evidence for any oscillation of the raindrop spectra about its equilibrium value. It appears that a significant population of both 1 mm and 2 mm drops are needed before a 3PED can occur, but a 3PED can be safely assumed to develop once they are present.

## Abstract

Knowledge of ice crystal particle size distributions (PSDs) is critical for parameterization schemes for atmospheric models and remote sensing retrieval schemes. Two-dimensional in situ images captured by cloud imaging probes are widely used to derive PSDs in term of maximum particle dimension (*μ*m and *μ*m are caused by the strong dependence of sample volume on particle size, whereas differences for

## Abstract

Knowledge of ice crystal particle size distributions (PSDs) is critical for parameterization schemes for atmospheric models and remote sensing retrieval schemes. Two-dimensional in situ images captured by cloud imaging probes are widely used to derive PSDs in term of maximum particle dimension (*μ*m and *μ*m are caused by the strong dependence of sample volume on particle size, whereas differences for

## Abstract

We welcome the opportunity to correct the misunderstandings and misinterpretations contained in Yano’s comment that led him to incorrectly state that Wu and McFarquhar misunderstood the maximum entropy (MaxEnt) principle. As correctly stated by Yano, the principle itself does not suffer from the problem of a lack of invariance. But, as restated in this reply and in Wu and McFarquhar, the commonly used Shannon–Gibbs entropy does suffer from a lack of invariance for coordinate transform when applied in continuous cases, and this problem is resolved by the use of the relative entropy. Further, it is restated that the Wu and McFarquhar derivation of the PSD form using MaxEnt is more general than the formulation by Yano and allows more constraints with any functional relations to be applied. The derivation of Yano is nothing new but the representation of PSDs in other variables.

## Abstract

We welcome the opportunity to correct the misunderstandings and misinterpretations contained in Yano’s comment that led him to incorrectly state that Wu and McFarquhar misunderstood the maximum entropy (MaxEnt) principle. As correctly stated by Yano, the principle itself does not suffer from the problem of a lack of invariance. But, as restated in this reply and in Wu and McFarquhar, the commonly used Shannon–Gibbs entropy does suffer from a lack of invariance for coordinate transform when applied in continuous cases, and this problem is resolved by the use of the relative entropy. Further, it is restated that the Wu and McFarquhar derivation of the PSD form using MaxEnt is more general than the formulation by Yano and allows more constraints with any functional relations to be applied. The derivation of Yano is nothing new but the representation of PSDs in other variables.

## Abstract

Peaks in number density observed by many authors using Joss-Waldvogel disdrometers are almost certainly due to the manner in which a best-fit curve was chosen to represent the calibration data. The extreme sensitivity to small changes in the calibration curve was demonstrated with extensive observations of Malaysian tropical rain, which for best-fit calibrations showed multiple peaks; however, when a linear interpolation between calibration points was applied, only single peaks were seen in this first, recalibrated dataset. Hence, field evidence for multiple peak equilibrium distributions obtained in numerical models should be reconsidered. On the other hand, the rainfall and radar reflectivity, calculated from best fits, differ by less than 4% from that based on linear interpolation between calibration points.

## Abstract

Peaks in number density observed by many authors using Joss-Waldvogel disdrometers are almost certainly due to the manner in which a best-fit curve was chosen to represent the calibration data. The extreme sensitivity to small changes in the calibration curve was demonstrated with extensive observations of Malaysian tropical rain, which for best-fit calibrations showed multiple peaks; however, when a linear interpolation between calibration points was applied, only single peaks were seen in this first, recalibrated dataset. Hence, field evidence for multiple peak equilibrium distributions obtained in numerical models should be reconsidered. On the other hand, the rainfall and radar reflectivity, calculated from best fits, differ by less than 4% from that based on linear interpolation between calibration points.

## Abstract

During the March 2000 Cloud Intensive Operational Period, the University of North Dakota Citation executed spiral descents through midlatitude cirrus of a nonconvective origin over the Atmospheric Radiation Measurement Program’s Southern Great Plains site. Aggregates of bullet rosettes (ABRs) observed during the descents using a cloud particle imager are used to derive a relationship between the length *L* and width *W* of bullets that are the fundamental components of the ABRs, which is given by *W* = 7.14*L*
^{0.455}, where 100 *μ*m ≤ *L* ≤ 600 *μ*m. To derive a representation of an aggregate of bullet rosettes, six bullet rosettes, each of which is composed of six bullets of the same size but each of which has different-sized bullets from the other bullet rosettes, are attached together randomly without overlap. Using a geometric ray-tracing method, the phase function, asymmetry parameter *g*, and single-scattering albedo of the representation of ABR (rABR) and the component bullets and bullet rosettes are calculated at wavelengths *λ* of 0.55, 0.64, 1.38, 1.62, 2.11, and 3.78 *μ*m. As the aspect ratio of the component bullets increases, the forward scattering increases by up to 1.3% and the lateral and backward scattering decrease by up to 8.9% and 10.2%, respectively, for a bullet rosette at a nonabsorbing *λ* (0.55 *μ*m). For longer *λ*, light absorption decreases the rate at which these scatterings change with aspect ratio. The shape of the aggregates also affects the scattering properties. The rABR constructed here scatters up to 4.4% (7.0%; 20.4%) and 34.2% (11.1%; 32.7%) more light in the lateral and backward directions, respectively, and 1.2% (1.3%; 2.4%) less in the forward direction in comparison with the component bullets (component bullet rosettes; equivalent projected area bullet rosette), resulting in up to 2.5% (1.6%; 3.8%) decrease in *g* at 0.55 *μ*m. In addition, as the aspect ratio and number of attached bullets in ABRs increase, *g* increases by up to 1.8% and decreases by up to 2.0% at 0.55 *μ*m, increases by 2.0% and decreases by 0.3% at 2.11 *μ*m, and increases by 1.1% and decreases by 0.5% at 3.78 *μ*m, respectively. As an implication for remote sensing studies, the difference in the bidirectional reflectance distribution function calculated using the rABR and a bullet rosette is shown to vary by up to 107% at moderately absorbing (2.11 *μ*m) wavelengths and by up to 35% and 28% at nonabsorbing (0.55 *μ*m) and strongly absorbing (3.78 *μ*m) wavelengths.

## Abstract

During the March 2000 Cloud Intensive Operational Period, the University of North Dakota Citation executed spiral descents through midlatitude cirrus of a nonconvective origin over the Atmospheric Radiation Measurement Program’s Southern Great Plains site. Aggregates of bullet rosettes (ABRs) observed during the descents using a cloud particle imager are used to derive a relationship between the length *L* and width *W* of bullets that are the fundamental components of the ABRs, which is given by *W* = 7.14*L*
^{0.455}, where 100 *μ*m ≤ *L* ≤ 600 *μ*m. To derive a representation of an aggregate of bullet rosettes, six bullet rosettes, each of which is composed of six bullets of the same size but each of which has different-sized bullets from the other bullet rosettes, are attached together randomly without overlap. Using a geometric ray-tracing method, the phase function, asymmetry parameter *g*, and single-scattering albedo of the representation of ABR (rABR) and the component bullets and bullet rosettes are calculated at wavelengths *λ* of 0.55, 0.64, 1.38, 1.62, 2.11, and 3.78 *μ*m. As the aspect ratio of the component bullets increases, the forward scattering increases by up to 1.3% and the lateral and backward scattering decrease by up to 8.9% and 10.2%, respectively, for a bullet rosette at a nonabsorbing *λ* (0.55 *μ*m). For longer *λ*, light absorption decreases the rate at which these scatterings change with aspect ratio. The shape of the aggregates also affects the scattering properties. The rABR constructed here scatters up to 4.4% (7.0%; 20.4%) and 34.2% (11.1%; 32.7%) more light in the lateral and backward directions, respectively, and 1.2% (1.3%; 2.4%) less in the forward direction in comparison with the component bullets (component bullet rosettes; equivalent projected area bullet rosette), resulting in up to 2.5% (1.6%; 3.8%) decrease in *g* at 0.55 *μ*m. In addition, as the aspect ratio and number of attached bullets in ABRs increase, *g* increases by up to 1.8% and decreases by up to 2.0% at 0.55 *μ*m, increases by 2.0% and decreases by 0.3% at 2.11 *μ*m, and increases by 1.1% and decreases by 0.5% at 3.78 *μ*m, respectively. As an implication for remote sensing studies, the difference in the bidirectional reflectance distribution function calculated using the rABR and a bullet rosette is shown to vary by up to 107% at moderately absorbing (2.11 *μ*m) wavelengths and by up to 35% and 28% at nonabsorbing (0.55 *μ*m) and strongly absorbing (3.78 *μ*m) wavelengths.

## Abstract

Using the scaling of the coalescence and breakup equation, it is shown that for equilibrium distributions, and in particular for three-peak equilibrium distributions (3PEDs), the mean free time (*τ*) and mean free path (*λ*) between raindrop collisions are both inversely proportional to the rainrate and that the total number of raindrop collisions varies with the square of the rainrate. This quantifies previous results, which showed that equilibrium is more rapidly approached for more intense rainrates. For Marshall–Palmer (MP) distributions, more collisions occur for rainrates less than approximately 40 mm h^{−1} than for 3PEDs and fewer for greater rainrates because the relative number of large drops increases with rainrate. For rainrates around 50 mm h^{−1}, there are barely any differences in the behavior of *λ* and *τ* between MP distributions and 3PEDs.

## Abstract

Using the scaling of the coalescence and breakup equation, it is shown that for equilibrium distributions, and in particular for three-peak equilibrium distributions (3PEDs), the mean free time (*τ*) and mean free path (*λ*) between raindrop collisions are both inversely proportional to the rainrate and that the total number of raindrop collisions varies with the square of the rainrate. This quantifies previous results, which showed that equilibrium is more rapidly approached for more intense rainrates. For Marshall–Palmer (MP) distributions, more collisions occur for rainrates less than approximately 40 mm h^{−1} than for 3PEDs and fewer for greater rainrates because the relative number of large drops increases with rainrate. For rainrates around 50 mm h^{−1}, there are barely any differences in the behavior of *λ* and *τ* between MP distributions and 3PEDs.