Statistical collapse of stratiform and convective drop diameter distributions at the ground



[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

[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, equation imageI(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 equation imageI(D) versus D to obtain the appearance of an approximately invariant shape [Sekhon and Srivastava, 1971; Sempere Torres et al., 1994].

[6] Following this idea some studies compare “renormalized” instantaneous drop size distributions for convective and stratiform precipitation, where the two types of precipitation are separated using rainfall rates [e.g., Maki et al., 2001; Testud et al., 2001; Willis, 1984] or radar reflectivity [e.g., Campos et al., 2006]. However, while Willis [1984] and Testud et al. [2001] conclude that the average curves for stratiform and convective precipitation are somewhat similar, Maki et al. [2001] and Campos et al. [2006] reach the opposite conclusion.

[7] The possibility of finding a renormalization procedure resulting in an “universal” drop size distribution is attractive for many reasons. For modeling purposes the existence of an universal distribution allows to split rainfall dynamics into two contributions: an invariant component and a contribution which can be modeled by few parameters describing the variability of the phenomenon. In this manuscript we adopt a renormalization procedure which is different in spirit and substance from that of Sekhon and Srivastava [1971]: 1) We consider the probability density pG(D) of observing, at the ground, a drop diameter in the range D, D + dD instead of the concentration equation image(D). 2) Instead of rescaling separately (by multiplication of two different factors) the diameter D and the concentration equation image(D), we make a change of variable, DDR, and plot the probability density pG(DR) versus DR: no rescaling of pG(DR) is adopted. We use data from a RD69 Joss–Waldvogel (JW) disdrometer to show that the probability density for the renormalized diameter is, within the statistical significance, invariant in stratiform and convective intervals, and also independent from a classification based on the rainfall rate range.

2. Methodology

[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., pG(D) = pG(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 μI and the standard deviation σI of the drop diameters. We then apply the transformation

equation image

We thus assume that the time series {Dk} 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) TS 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, pG(DR), of the renormalized diameter. If equation (1) captures the actual dynamical properties of rainfall phenomena, we expect the probability density pG(DR) 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

[10] For each stratiform and convective precipitation time interval we calculate the empirical probability density function, pG(D), for the mid points Dj of the disdrometer diameter classes as the ratio nj/(ΔjN): Δj and nj 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 pG(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 pG(DR) for the renormalized drop diameter virtually coincide.

Figure 1.

The probability densities at the ground pG(D) and pG(DR) for the drop diameter D and renormalized drop diameter DR. (a and b) The 19 stratiform and 33 convective precipitation time intervals considered in this study (see Table S1 of auxiliary material): squares and solid lines indicate respectively the mean and the σ–bounds of the stratiform population, while circles and dashed lines indicate those of the convective population. (c and d) The 6 different rain rate classes used for rainfall classification: squares indicate the very light class, upward triangles the light class, downward triangles the moderate, diamonds the heavy class, pentagons the very heavy class, and circles the extreme class.

[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 pG(D) of observing a diameter D are different for all rate categories, while the probability densities pG(DR) for the renormalized diameter DR “collapse” in a single curve, with the exception of the extreme category curve which is markedly different in the left tail: DR < −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, pG(DR), of rainfall dynamics, the average drop diameter μI, standard deviation σI, and drop count NI 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 (μI, σI), (μI, NI), and (σI, NI). 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 (μI, σI) is approximately linear, while it is a more complicated function for the couples (μI, NI), and (σI, NI). However the σ–perimeters exhibit a considerable overlap between adjacent rainfall rate categories in the case of the (μI, σI) plot, and a distinct separation in the case of the (μI, NI), and (σI, NI) plots. The dependence among the parameters NI, μI, and σI 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].

Figure 2.

Scatter plots of the couples (a) (μI, σI), (b) (μI, NI), and (c) (σI, NI), as a function of rainfall rate class. The center of mass and the upper σ–perimeter are indicated as follows: square and solid line for the very light class, upward triangle and long-dashed line for the light class, downward triangle and short-dashed line for the moderate class, diamond and dotted line for the heavy class, pentagon and dashed-dotted line for the very heavy class, and circle and wavy line for the extreme class.

[13] As final consistency check, we address the question of the stability time TS 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 μI. 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 TS, 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(μI) and P(σI) of the average μI and the standard deviation σI 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.

Figure 3.

(a) The variation of the average diameter μI for three different extracts from the Darwin data set. The numbers reported on y-axis indicate the value of the average diameter μI along the solid lines. Each subsequent dashed line indicates an increase of 0.4 mm. Finally, the length of the renormalization interval I is 1 minute. (b) The plot of the observed frequencies, for the entire Darwin data set, P(μI) and P(σI) of the average μI and standard deviation σI of drop diameters for different lengths of the renormalization interval I. The squares refer to a length of 1 minute. The color-filled curves indicate the difference from the 1 minute observed frequency for a length of 2 (darkest), 3, 4, and 5 (brightest) minutes. Note that in making Figure 3b, we consider only intervals for which all the single 1 minute subintervals are renormalization time intervals at the 1 minute resolution. This procedure is adopted to eliminate the influence of the on-off intermittency of the rainfall phenomenon.

4. Conclusions

[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

equation image

where pGI(D) and pG(DR) are the probability density, at the ground, of the drop diameter D and the renormalized drop diameter DR, for each renormalization time interval I. The function equation image 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 equation image. Nevertheless, in Darwin (AU), rainfall dynamics can be split into two contributions: an invariant one described by the density equation image(DR), and one describing the variability of the phenomenon via the parameters (NI, μI, σI): the drop count, the average diameter, the standard deviation of the drop diameter inside the renormalization interval.


[15] We are thankful to C. R. Williams and the National Oceanic and Atmospheric Administration for the public availability of the data set. M. I. thankfully acknowledges the support of the Army Research Office (USA) and of Accademia Nazionale dei Lincei (Italy) through “B. Segre” scholarship. C. D. M. thankfully acknowledges Comune di Milano (Italy) through BIODESCESA project.