### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Methodology
- 3. Results
- 4. Conclusions
- Acknowledgments
- References
- Supporting Information

[1] The probability density function of the drop diameter at the ground is investigated during stratiform and convective precipitation intervals at Darwin, Australia. We show how, after a renormalization procedure of the drop diameter, the empirical probability density functions of both types of precipitation collapse in a single curve, indicating the possible existence of an invariant distribution of the drop diameter at the ground.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Methodology
- 3. Results
- 4. Conclusions
- Acknowledgments
- References
- Supporting Information

[2] Precipitation is commonly classified in two main categories, *stratiform* and *convective*, according to the strength (convective indicating the stronger case) of the updraft motion generating atmospheric vapor condensation and eventually rainfall drops. Updraft velocities ∼2 m/s are commonly considered as the threshold separating the two cases [*Houze*, 1997]. In terms of the microphysical dynamics of drop formation, stratiform rain is characterized by aggregation and a small degree of riming, while convective rain by accretion and high degree of riming [*Steiner and Smith*, 1998]. It is worth noting that updraft velocities in dying convective cells can be of the order or smaller than 2 m/s. Thus stratiform rain can occur also in clouds of convective origin [*Houze*, 1997]. Moreover, a synoptic formation may present a mix of both types of precipitation [e.g., *Williams*, 2009].

[3] The separation between stratiform and convective precipitation is relevant for global circulation, climatological, hydrological models, and for a proper retrieval of rainfall rates from remote sensing measurements [*Steiner and Smith*, 1998]. Different criteria have been proposed for separating these two types of precipitation. Large (small) values of rainfall rate, *R*, are typically generated by convective (stratiform) rain. However there is a considerable overlap in the range of rainfall rates which may be generated by the two mechanisms, e.g., *Tokay and Short* [1996] report a range of 1–10 mm/h for 1 minute-resolution data of tropical rain.

[4] The process of aggregation, which is peculiar of stratiform precipitation, produces a thin horizontal (fixed altitude) layer of enhanced reflectivity in radar retrievals, the “bright band” [*Steiner and Smith*, 1998]. Reflectivity maps from vertically pointed radar can resolve the presence of the bright band, and help differentiating stratiform from convective precipitation [e.g., *Houze*, 1997; *Williams*, 2009]). Several criteria have indeed been developed during the years based on direct observations or statistical properties, yet the discrimination between the two types of precipitation remains somewhat “arbitrary” (see, e.g., *Lang et al.* [2003] for a review).

[5] If a large body of literature is dedicated to the difference between stratiform and convective precipitation, and in general to the time variability of the rainfall phenomenon, very few works explore the possibility of describing/capturing this variability in a parsimonious mathematical model. Efforts in this direction use 1) the “instantaneous” drop size distribution, _{I}(*D*), namely the number of drops per unit volume and diameter in an interval of time *I* which is usually the instrument time resolution [*Joss and Gori*, 1978]; 2) a rescaling by different factors of both the abscissa and the ordinate of _{I}(*D*) versus *D* to obtain the appearance of an approximately invariant shape [*Sekhon and Srivastava*, 1971; *Sempere Torres et al.*, 1994].

### 2. Methodology

- Top of page
- Abstract
- 1. Introduction
- 2. Methodology
- 3. Results
- 4. Conclusions
- Acknowledgments
- References
- Supporting Information

[8] As part of the Tropical Warm Pool International Cloud Experiment (TWPICE) campaign, RD69 JW disdrometer data at 1 minute time resolution were acquired for 97 consecutive days (from Nov. 4 2005 to Feb. 10 2006) in Darwin, Australia (12.45°S, 130.83°E). Drop diameters are separated in 20 different classes covering the range 0.3–5.6 mm and 1-min counts are corrected against the instrument dead time [*Sauvageot and Lacaux*, 1995]. Moreover, radar reflectivity maps are available for the time intervals 9 Nov. to 6 Dec. 2005, and 6 Jan. to 10 Feb. 2006, allowing for stratiform versus convective classification through the identification of the bright band. Time intervals with non zero precipitation, where a bright band is present, are classified as stratiform and those were the bright band is absent and a value of the reflectivity ≥40 DBz is observed as convective [e.g., *Houze*, 1997; *Uijlenhoet et al.*, 2003a]. A total of 19 stratiform and 33 convective time intervals were identified with this method (see auxiliary material for details). We complement this classification with the division of the entire data set in 6 categories of rainfall rate *R* as proposed by *Tokay and Short* [1996]. A unit time interval of observation, 1 minute in our case, belongs to the *very light* category if its rainfall rate *R* is ≤1 mm/h, *light* if 1 ≤ *R* ≤ 2 mm/h, *moderate* if 2 < *R* ≤ 5 mm/h, *heavy* if 5 < *R* ≤ 10 mm/h, *very heavy* if 10 < *R* ≤ 20 mm/h, and *extreme* if *R* > 20 mm/h. The *very light* category is almost exclusively occupied by stratiform precipitation, while the *extreme* category is almost exclusively occupied by convective precipitation. All the other classes are a mix of the two types, with convective precipitation dominating in the *heavy* and *very heavy* classes.

[9] Rainfall is a stochastic phenomenon: its variability (at the drop level) is manifested in the different drop counts, and different distributions of drop diameters observed for each measurement time interval. This variability is not just the trivial one associated with the fluctuations of a stationary process, e.g., *Konstinski and Jameson* [1997] show that the count process can be described by a Poisson mixture. If in addition to this, one considers the probability density of drop diameter, at the ground, to be a fixed functional form with parameters, θ(*t*), variable in time *t*, i.e., *p*_{G}(*D*) = *p*_{G}(*D*, θ(*t*)), then it is possible to realistically mimic rainfall rate time series [*Smith*, 1993]. With this same “philosophy”, we consider a renormalization procedure [*Ignaccolo et al.*, 2009] which operates as follows. A time series of disdrometer counts covering an interval of time *T* is divided in renormalization time intervals of length *I* < *T* (e.g., *I* is the time resolution of the instrument). For each time interval we calculate the mean *μ*_{I} and the standard deviation *σ*_{I} of the drop diameters. We then apply the transformation

We thus assume that the time series {*D*_{k}} of observed drop diameters derives from a stochastic process which would be stationary if it were not for a variable mean and a variable standard deviation. Equation (1) has the aim of removing this non-stationarity. To do so the length of the renormalization time interval *I* must be smaller than the characteristic time (“stability” time) *T*_{S} during which the average and standard deviation of the drop diameters can be considered constant. Once the renormalization procedure is applied we plot the probability density at the ground, *p*_{G}(*D*_{R}), of the renormalized diameter. If equation (1) captures the actual dynamical properties of rainfall phenomena, we expect the probability density *p*_{G}(*D*_{R}) obtained from different rainfall time series to be somewhat similar. For the details and issues related to the practical implementation of this renormalization procedure we refer the reader to the auxiliary material.

### 3. Results

- Top of page
- Abstract
- 1. Introduction
- 2. Methodology
- 3. Results
- 4. Conclusions
- Acknowledgments
- References
- Supporting Information

[10] For each stratiform and convective precipitation time interval we calculate the empirical probability density function, *p*_{G}(*D*), for the mid points *D*_{j} of the disdrometer diameter classes as the ratio *n*_{j}/(Δ_{j}N): Δ_{j} and *n*_{j} are respectively the size and the count of the *j*–th diameter class, and *N* is the total number of drops in the time interval. Then we calculate the sample average and sample standard deviation of *p*_{G}(*D*) for both the stratiform and convective populations. The results are reported in Figure 1a. The two averaged probability densities are statistically different as the convective one lies outside the *σ*–bounds (average ± standard deviation) of the stratiform average for diameters larger than 1.5 mm. In the same range the average stratiform probability density is almost superimposed to the lower *σ*–bound of the convective one. We now apply the renormalization procedure described by equation (1) to both the stratiform and convective populations, choosing as length of renormalization time interval *I* the time resolution: 1 min. The results are shown in Figure 1b. We see how the stratiform and the convective probability density functions *p*_{G}(*D*_{R}) for the renormalized drop diameter virtually coincide.

[11] To further validate this result, we repeat the same analysis for the 6 categories of rainfall rate proposed by *Tokay and Short* [1996]: *very light*, *light*, *moderate*, *heavy*, *very heavy*, and *extreme*. The results are depicted in Figures 1c and 1d. The probability densities *p*_{G}(*D*) of observing a diameter *D* are different for all rate categories, while the probability densities *p*_{G}(*D*_{R}) for the renormalized diameter *D*_{R} “collapse” in a single curve, with the exception of the *extreme* category curve which is markedly different in the left tail: *D*_{R} < −1. This is due to the inability of the disdrometer to measure drops of small diameter for drop counts in the range 400–600 and up to a value of 1200–1300 which are typical for this category. In fact the dead time correction has no effects on null raw counts. The percentage of 1–minute time intervals in the *extreme* category which have a null count in the 1st class is ≃69%, while the percentage for a null count in both the 1st and the 2nd class is ≃26%.

[12] If the renormalization procedure of equation (1) identifies an invariant property, *p*_{G}(*D*_{R}), of rainfall dynamics, the average drop diameter *μ*_{I}, standard deviation *σ*_{I}, and drop count *N*_{I} can be used to describe the variability of the phenomenon. The relationships between these 3 “variability” parameters for each rainfall rate category are investigated using the scatter plots of the 3 couples (*μ*_{I}, *σ*_{I}), (*μ*_{I}, *N*_{I}), and (*σ*_{I}, *N*_{I}). The results are shown in Figure 2. For each couple and rain category the center of mass is calculated (points in Figure 2). The spreading of points around the center of mass is quantified by the upper *σ*–perimeter (lines in Figure 2). This quantity is calculated spanning the plane around the center of mass with a 5° step. For each point inside the cone, the distance from the center of mass is calculated to obtain the corresponding cone average distance *μ*_{°} and the cone standard deviation distance *σ*_{°}. Finally, we define the upper *σ*–bound inside the cone as the point laying on the bisector of the cone whose distance from the center of mass is equal to *μ*_{°} + *σ*_{°}. The line connecting all the upper *σ*–bounds is the upper *σ*–perimeter. Figure 2 shows that the relationship between the center of mass for different rainfall rate categories of the couple (*μ*_{I}, *σ*_{I}) is approximately linear, while it is a more complicated function for the couples (*μ*_{I}, *N*_{I}), and (*σ*_{I}, *N*_{I}). However the *σ*–perimeters exhibit a considerable overlap between adjacent rainfall rate categories in the case of the (*μ*_{I}, *σ*_{I}) plot, and a distinct separation in the case of the (*μ*_{I}, *N*_{I}), and (*σ*_{I}, *N*_{I}) plots. The dependence among the parameters *N*_{I}, *μ*_{I}, and *σ*_{I} is expected as a dependence among the three parameters (intercept, shape and scale) of the gamma distribution, commonly used to fit instantaneous drop diameters concentration, has been reported several times in literature [e.g., *Uijlenhoet et al.*, 2003a].

[13] As final consistency check, we address the question of the stability time *T*_{S} of the moving average and standard deviation and compare it to the length of the renormalization interval *I* used: 1 min. Figure 3a shows three extracts of the sequence of average drop diameter *μ*_{I}. These extracts are representative of the three types of average diameter variability observed in the entire Darwin data set. Type I) smooth and intense variability (extract I). The average diameter sequence resembles a “smooth” function with variations of ∼0.4 mm occurring in a time range of ∼10 minutes. Note that 0.4 mm is a “large” variation corresponding to ∼1/3 of the observed range of average drop diameters since the range [0.4, 1.2] mm contains ∼90% of all observed 1–min average diameters, Figure 3b. Type II) moderately noisy and moderate variability (extract II). In this case the difference between two consecutive average diameters oscillates in a random fashion in the range [−0.1, 0.1], and large variations (∼0.4 mm) of the average diameter occur in a time range of ∼30 min. Type III) noisy/smooth and intense variability (extract III). This type is like type I except for the presence of patches of large variations (0.4 mm or more) in a very brief time interval (∼2 min). Similar curves are obtained for the standard deviation sequence as there is an almost linear relationship between average and standard deviation (Figure 2a). Therefore, the results depicted in Figure 3a indicates that the stability time *T*_{S}, the time for which both average and standard deviation of drop diameters can be considered constant, changes along the record with a minimum around ∼2 minutes, which is double the length of the renormalization interval used. Figure 3b also supports this conclusion as we compare, using the entire Darwin data set, the observed frequency *P*(*μ*_{I}) and *P*(*σ*_{I}) of the average *μ*_{I} and the standard deviation *σ*_{I} for lengths of the renormalization interval ranging from 1 to 5 minutes. We see how both probabilities are similar for a length of 1 and 2 minutes while their agreement get progressively worse if lengths of 3, 4, and 5 minutes are used.

### 4. Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Methodology
- 3. Results
- 4. Conclusions
- Acknowledgments
- References
- Supporting Information

[14] The results presented indicate that the renormalization procedure, equation (1), eliminates the difference in drop diameter distributions between convective and stratiform precipitation. In particular, the collapse observed using either the stratiform versus convective classification or the rainfall rate classification indicates that the drop diameters can be considered to be stemming from probability distributions which differ only in their means and standard deviations. Thus

where *p*_{G}^{I}(*D*) and *p*_{G}(*D*_{R}) are the probability density, at the ground, of the drop diameter *D* and the renormalized drop diameter *D*_{R}, for each renormalization time interval *I*. The function in equation (2) is the “invariant” probability density at the ground of the renormalized drop diameter. More observations at different locations on the Earth's surface are needed to prove the general validity of this renormalization procedure and the invariance of the function . Nevertheless, in Darwin (AU), rainfall dynamics can be split into two contributions: an invariant one described by the density (*D*_{R}), and one describing the variability of the phenomenon via the parameters (*N*_{I}, *μ*_{I}, *σ*_{I}): the drop count, the average diameter, the standard deviation of the drop diameter inside the renormalization interval.

### Supporting Information

- Top of page
- Abstract
- 1. Introduction
- 2. Methodology
- 3. Results
- 4. Conclusions
- Acknowledgments
- References
- Supporting Information

Auxiliary material for this article contains all the details about the processing of the data set used in the manuscript as well details on particular subsets of the data set.

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