Quantitative measurement, estimation, and prediction of precipitation remains one of the grand challenges in the hydrological and atmospheric sciences with far-reaching implications across the natural sciences. Although the roots of current research activity in this topic go back to the beginning of the twentieth century, advances in radar technology and in numerical modeling have provided the impetus for prolific research in the area of cloud and precipitation physics over the last 50 years. As radar rainfall measurements progressively became the staple of hydrometeorological observing systems, cloud and precipitation microphysics emerged as increasingly preeminent areas of research. Here we present a synthesis of the state of the science with respect to the physical dynamics of hydrometeors and, specifically, the transient processes that affect the temporal evolution of rainfall microstructure and that are directly relevant to the quantitative interpretation of radar rainfall measurements and explicit numerical simulations. The focus of our survey is on raindrop morphodynamics (equilibrium raindrop shape and raindrop oscillations), drop-drop interactions (bounce, coalescence, and breakup), and the dynamical evolution of raindrop size distributions in precipitating clouds.
 Precipitation is a fundamental process of the global water cycle and along with solar radiation one of the most important agents of variability in terrestrial hydrology. From natural hazards to ecosystems services and agriculture from water resources management to lifeline infrastructure, applications involving water depend ultimately on accurate measurement, estimation, and/or prediction of precipitation and precipitation rates at the spatial and temporal scales relevant for each end use. Yet, from the first rain gauge records in India to the network of rain gauges installed in Korea by King Sejong in the mid fifteenth century [Dooge, 1984] to the modern weather radars, measuring precipitation accurately remains the foremost challenge in hydrometeorology.
 Single-polarization or dual-polarization (henceforth polarimetric) weather radars have many advantages over rain gauge networks because of extended areal coverage, rapid access for real-time data, and relatively high spatial resolution [Atlas et al., 1984; Doviak and Zrnic, 1993; Uijlenhoet, 1999; Bringi and Chandrasekar, 2001]. The basis for radar measurements of rainfall is the backscatter behavior of hydrometeors integrated over a known cross section as described by the radar equation that expresses the theoretical representation of the relationship between radar characteristics and the characteristic size of the scatterers [Probert-Jones, 1962]. In the case of single-polarization radars for rainfall measurement the radar equation was sidelined early on in favor of the power law relationship between the rainfall rate, R, and the radar reflectivity, Z, that is the power scattered back to a radar antenna [Marshall et al., 1947; Battan, 1959]:
Here a and b are empirical constants, which should depend on the physics of precipitation at the local places of measurement, and may be expressed more generally in terms of the rainfall climatology as a function of season and location or, more specifically, the rainfall regime (e.g., stratiform, convective, and orographic) (see detailed discussions by Atlas et al.  and Rosenfeld and Ulbrich ). The ultimate goal of this survey is to produce a synthesis of our understanding of raindrop dynamics and rainfall microstructure that is directly relevant to microwave sensing of rainfall and for incorporation of a particle-based statistical representation of physical processes in clouds and precipitation systems in numerical models.
 Despite the wide appeal of equation (1) in that it directly relates radar observations to rainfall rate and despite its continued use, it is well established that the large spatial and temporal variability in rainfall rate and hydrometeor distributions among geographic locations and hydrometeorological regimes cannot be described in a generalized manner by such a simple relationship, especially in the case of single-polarization radar [Atlas et al., 1997]. In the case of polarimetric radars that transmit and receive horizontally and vertically polarized signals, the rainfall rate can be approximately estimated as the fourth moment of the retrieved probability distribution of different raindrop sizes (drop size distribution (DSD), see section 4 for details). Furthermore, the shape of raindrops is a critical factor in estimating the DSD and hence rainfall rate. Because raindrops are not exactly spherical, the radar reflectivities from horizontally (ZH) and vertically (ZV) polarized waves differ depending on the ratios of vertical (v) and horizontal (h) chords of raindrops, α = v/h. In fact, it is this dependency between the radar measurables and raindrop chord ratios that makes the DSD retrieval possible via polarimetric radars. Considerable research on shapes of raindrops at terminal velocities has been conducted since the beginning of the twentieth century [Lenard, 1904], and highlights of this research are addressed in section 2.
 Besides information on rainfall rate, various precipitation-related applications may require reliable quantitative DSD information at various heights to infer the three-dimensional structure of precipitating systems. However, the DSDs of natural rain evolve through height by multiple interactions among raindrops and with the surrounding environment, and therefore DSDs inferred (retrieved) from polarimetric radar measurements at a certain height are not representative of the DSDs at different heights. Collision-induced breakup and coalescence are commonly accepted as the main physical mechanisms of DSD evolution for moderate to heavy rainfall rates. Laboratory experiments to characterize these processes have provided the basis for stochastic numerical simulations of DSD evolution with height (sections 3 and 4).
2. RAINDROP MORPHODYNAMICS
 An isolated, relatively large raindrop falling at terminal velocity exhibits a marked flattening on the lower surface and smoothly rounded curvature on the upper surface (see Figure 1) rather than the conical taper it would be for the streamlined form popularly described as “teardrop” shape [McDonald, 1954]. (The terminal velocity of a raindrop falling through the atmosphere is the velocity at which the gravitational force pulling the drop downward is equal and opposite to the drag force acting on the drop. Owing to this balance of external forces, at terminal velocity the raindrop ceases to accelerate downward and falls at constant speed. A list of proposed formulations for raindrop terminal velocity is given in Appendix A.)
 Drop deformation increases with drop size, an effect that can be described in terms of the chord ratio, α [Magono, 1954]. Indeed, the chord ratio is the key measurable in remote sensing of rainfall by polarimetric radars based on the assumption that the chord ratios of raindrops increases predictably with drop size [see Bringi and Chandrasekar, 2001]. Therefore accurate knowledge of raindrop shape, in fact chord ratio, and elucidating its dependence on drop size has been one of the major goals in precipitation measurement research.
 A raindrop falling at terminal velocity is not always at steady state but may oscillate around its quiescent shape (henceforth referred to as equilibrium shape) because of the occurrence of interfacial instabilities. Thus the instantaneous chord ratios observed by polarimetric radars are determined by the combination of two components: (1) the equilibrium drop shape and (2) the oscillations around the equilibrium shape. In this section we present a brief overview of the research conducted on both components of raindrop morphodynamics beginning with the equilibrium drop shape and following with drop oscillations.
2.1. Equilibrium Raindrop Shape
 On the basis of laboratory observations, Pruppacher and Beard  reported that the shape of drops falling at terminal velocities varied with the equivalent diameter, d (the diameter of a sphere having the same volume as the drop), as follows: (1) d < 0.25 mm, no detectable deformation from spherical shape; (2) 0.25 ≤ d < 1 mm, slight deformation into an oblate spheroid; and (3) d ≥ 1 mm, large deformation into the flattened raindrop shape described above. (Approximating a raindrop as an oblate spheroid, raindrop chord ratio is defined as the ratio of the drop's semiminor chord along the drop's fall axis (v) to the drop's semimajor chord perpendicular to the drop's fall axis (h), α = v/h.) Henceforth, drops will be categorized in three different classes: class I, cloud drops (d < 0.25 mm); class II, small raindrops (0.25 ≤ d < 1 mm); and class III, large raindrops (d ≥ 1 mm). Note that raindrops larger than a critical equivalent diameter, about 6–8 mm (depending on the turbulence level of the airstream), are unstable and are generally believed to be absent in nature [Blanchard, 1950; Komabayasi et al., 1964; Pruppacher and Beard, 1970; List and Hand, 1971; Pruppacher and Pitter, 1971].
 The physical-mathematical basis for the shape of a falling drop and its terminal velocity is given by the Navier-Stokes equations of motion for the air flowing past the drop as well as the motion of the water inside the drop subject to the appropriate dynamic and kinematic boundary conditions [Beard, 1976]. The dimensionless parameters that describe the underlying dynamics are the Reynolds number (Re), the Weber number (We), and the Strouhal number (St), defined by
respectively, where Ut is the velocity of the air flow (same as the terminal velocity of the drop), υ and ρa are the kinematic viscosity and density of the air, respectively, σ is the surface tension of water, and f is the frequency of vortex shedding in the drop wake (unsteady flow of alternating vortices downstream of the drop). Theoretical treatment of the Navier-Stokes equations for this flow problem may only be possible for very small drops when the terminal velocity Ut is very small and thus Re is very small. Consequently, the Weber number is very small (restoring force, surface tension dominates to keep the sphericity of the drops), and St = 0 (flow is not energetic enough to induce vortex shedding). For detailed explanations, please see Batchelor  and Landau and Lifshitz . Most raindrops observed in nature have relatively large sizes, and since the analytical study of Navier-Stokes equations meets with great difficulties, existing knowledge of the shape of the drop and the flow around it have been obtained mostly by observations and numerical calculations based on the balance of the dominant forces acting on an individual drop.
 For relatively small drops (classes I and II) the difference in hydrostatic pressure between the top and bottom of a falling drop is small. Similarly, the aerodynamic pressure difference between the upper and lower surfaces of the drop is small because of the absence or late separation of flow around the drop at small Reynolds numbers. Hence surface tension is the dominant physical control for smaller raindrops and tries to minimize the interfacial (air-liquid) area by shaping the drop surface into a sphere (a minimum surface-to-volume ratio) or an oblate spheroid. However, for larger drops (class III) the distances between the top and bottom of the drops are larger, and the effects of the difference in hydrostatic pressure become comparable to the surface tension effects. For such drops with large terminal velocities (Ut ≈ 4–10 m s−1) and large Weber numbers (We ≈ 0.25–10) the order of magnitude of destabilizing aerodynamic forces is comparable to the magnitude of the stabilizing surface tension forces. Furthermore, for flows around larger drops the Reynolds numbers are also large (Re ≈ 250–4000), and therefore the stabilizing viscous forces are overcome by inertia forces leading to flow separation around the drop and the establishment of a wake region behind it [see List and Hand, 1971]. This flow pattern leads to an aerodynamic pressure differential along the drop surface, which must be balanced by hydrostatic pressure and surface tension for a stable drop. Under these conditions the drop shape is given by Laplace's pressure balance equation relating the curvature at each point on the drop surface to the pressure difference across the interface, Δp:
where R1 and R2 are the principal radii of curvature.
 Many physical models for predicting equilibrium drop shapes based on the pressure balance equation given in (3) were proposed with different simplifying assumptions by various researchers over the years [e.g., Spilhaus, 1948; Imai, 1950; Savic, 1953; McDonald, 1954; Taylor and Acrivos, 1964; Pruppacher and Pitter, 1971; Green, 1975; Beard and Chuang, 1987]. In calculating the equilibrium drop shapes using equation (3), the main difficulty is the lack of information on the aerodynamic pressure distribution around an equilibrium-shaped raindrop. Following Savic , researchers try to overcome this deficiency partly by implementing the results of pressure measurements around a rigid sphere. However, this approach introduces errors that become more serious as the drop size (hence drop deformation) increases. To take into account the changes in the pressure distribution due to drop deformations away from sphericity, Beard and Chuang  introduced corrections to the pressure measurements taken around a rigid sphere, and they were successful in reproducing numerically the equilibrium shapes observed in laboratory experiments [Magono, 1954; Pruppacher and Beard, 1970]. Indeed, their model estimations of equilibrium raindrop chord ratios, α, are also in good agreement with the aircraft-based measurements of Chandrasekar et al.  and polarimetric radar estimates of Gorgucci et al. . Numerical studies with an extension to the aforementioned model including the effects of the surface electric field showed that drop shape has a highly nonlinear dependence on the electric field via the coupling between the surface electrostatic stress and the aerodynamic distortion [Chuang and Beard, 1990]. Further research is needed to describe aerodynamic pressure distribution and internal circulation effects using both experimental and numerical models.
 For inviscid fluids, small-amplitude raindrop oscillations can be described by the mathematical model of Rayleigh  for linear axisymmetric drop oscillations in vacuum, which was later extended to include the influence of a surrounding medium by Lamb . For the case of large-amplitude oscillations, A > 0.1(d/2), where A is the amplitude of the oscillations, nonlinear effects should be considered [see Lundgren and Mansour, 1988; Becker et al., 1991]. The drop oscillation frequency is given by
where ρ is the density of the drop and n is the order of spherical harmonic perturbations. For each harmonic n, there are m = n + 1 “degenerate” modes having unique spatial orientations. Following Beard and Kubesh , the (n, m) notation is used in this paper to designate oscillation modes.
 In the absence of external forcing, oscillations cannot be sustained for a long time, and viscous damping takes place. The decay associated with viscous damping can be parameterized by an exponential function of the form A = A0e−t/τ, where A0 is the maximum value of A, t denotes time elapsed, and the time constant, τ, is given by Beard et al. :
where μd is the dynamic viscosity of the drop. From equation (5) it is clear that the time constant for the viscous damping of higher harmonics is smaller, indicating a faster decay of these harmonics. Therefore the time-varying raindrop shapes are considered to be mainly determined by the fundamental (n = 2) and the first (n = 3) harmonics with little contribution from higher harmonics. These two harmonics yield a total of seven possible shape modes, out of which only two modes (transverse modes, (2, 1) and (3, 1)) are, in general, consistent with the mean chord ratio values obtained experimentally, which are larger than the chord ratio values of an equilibrium-shaped drop [see Beard and Kubesh, 1991]. Beard et al. [1989a] attributed the occurrence of transverse mode oscillations to the vortices detaching from alternate sides of the upper pole of the raindrop (asymmetric vortex shedding). The first experimental verification of the existence of such vortices in the near wake was documented recently by Saylor and Jones . In that communication they presented the two-dimensional structure of the vortices visualized downstream of the levitating water drops in a vertical wind tunnel. Although this finding shows strong support for the postulated forcing mechanism for the transverse mode raindrop oscillations, Saylor and Jones  reported only slight canting of the drops and the absence of transverse oscillations. They proposed that such canting of drops can be related to the observed asymmetric vortex shedding. However, another possible explanation for the observed canting of drops could be the effect of shear at the small scales, which is likely to occur in vertical wind tunnel experiments [see Andsager et al., 1999; Tokay and Beard, 1996].
Kubesh and Beard  hypothesized that lock-in resonance, that is, the coupling of drop oscillations and vortex shedding where vortices are triggered at suitable oscillation frequencies and modes, may be possible. As mentioned by Andsager et al. , similar resonance phenomena have been observed for other systems such as elastically mounted cylinders in laminar cross flow [Mihailovic et al., 1997], tethered spheres in a free stream [Jauvtis et al., 2001], etc. This lock-in resonance hypothesis for raindrops was based on good agreement between the measured drop oscillation frequencies [Beard and Kubesh, 1991] and the vortex shedding frequencies that were reported for rigid spheres in steady unbounded flow [Achenbach, 1974] and for liquid drops falling in an immiscible liquid (liquid-liquid experiments by Magarvey and Bishop [1961a, 1961b]). However, the frequency of vortex shedding in the wake of a rigid sphere and a raindrop, which may be distorted because of pressure perturbations acting on it, are not necessarily equal. Also, the Weber number for a raindrop falling in air at terminal velocity is different from that in the liquid-liquid experiments. Therefore these two problems are not dynamically similar, and one would expect different drop oscillation behaviors [see Barenblatt, 2003]. Furthermore, if a lock-in resonance mechanism exists as proposed, then vortex shedding characteristics should also be different because of possible differences in drop oscillation characteristics. Further progress toward a quantitative understanding of raindrop oscillations requires elucidating the interaction of shedding vortices and the drop itself. Such information necessitates a comprehensive study of the vortex-shedding behavior as well as its three-dimensional structure in the wakes of raindrops falling at terminal velocities [see Achenbach, 1974; Testik et al., 2005].
In equation (6), d is in centimeters. Outside this diameter range (i.e., d < 1.0–1.1 mm or d > 4.4 mm) they suggest the use of the equilibrium chord ratio parameterization based on the numerical results of Beard and Chuang :
where d is also in centimeters (see also further list of chord ratio formulations given in Appendix B). The use of equation (7) for the drop diameter range, d < 1.0–1.1 mm, can be based on several experimental observations that support the absence of drop oscillations for this diameter range. On the other hand, the applicability of (7) for d > 4.4 mm is rather controversial. Aircraft measurements reported by Chandrasekar et al.  did not show any evidence of detectable drop oscillations, and they supported the use of equilibrium chord ratio parameterization equation (7) for such drop sizes. By contrast, aircraft measurements of raindrop chord ratios at warmer temperatures (∼15°C) reported by Bringi et al.  showed a considerable deviation from their equilibrium values. Bringi et al.  argued that the absence of drop oscillations in the observations of Chandrasekar et al.  was due to the ice cores in partially melting drops. At present the scarcity of studies investigating oscillations of drops for d > 4.4 mm does not allow the drawing of a definite conclusion. Therefore further research is needed to address questions such as the following: (1) Do the raindrops of these sizes oscillate? (2) If they do, what is their oscillation behavior? (3) Under which environmental conditions do they oscillate?
3. DROP-DROP INTERACTIONS
 In studying rainfall the motivation to understand raindrop dynamics goes beyond that of studying isolated hydrometeors. Raindrops are formed and grow within clouds at very large concentrations (on the order of 103 to 104 drops per cubic meter), and the drop size distribution evolves over time as drops interact with each other and their environment. As raindrops fall through clouds, they interact with each other via collisions. These interactions are considered by many researchers as the main driving mechanism for the evolution of drop size distribution of raindrops, which has long been a subject of interest in cloud and precipitation physics [e.g., Langmuir, 1948; List and Gillespie, 1976; Valdez and Young, 1985; Brown, 1987; Srivastava, 1988; Beard and Ochs, 1993; Khain et al., 2000]. Over the years, numerous studies have been conducted to characterize/quantify raindrop collisions and their impact on rainfall DSDs [Magarvey and Geldart, 1962; McTaggart-Cowan and List, 1975; Low and List, 1982a; Ochs et al., 1995a; Beard et al., 2001]. Here our objective is to synthesize previous research with a focus on establishing the foundations for future investigations.
 The differential response of drops to the gravitational force is generally considered to be the main cause of raindrop collisions [Rogers and Yau, 1989] along with other factors such as electrical forces [Dayan and Gallily, 1975; Beard et al., 2004] and air turbulence [Pinsky et al., 1999; Falkovich et al., 2002] that are thought to play an important role primarily in collisions of cloud drops. Because larger raindrops have larger terminal velocities, as they fall they will catch up and interact with smaller raindrops in their paths. In some of these interactions, collisions will occur (see drop “b” in Figure 4a), whereas in others, smaller raindrops will be swept aside in the airstream around the larger drop (see drop “c” in Figure 4a). The success rate of these interactions is quantified by the collision efficiency (E1), defined as the ratio of the number of collisions to the number of small drops in the volume swept by the larger drop per unit time. Assuming that the flow field around the larger drop is symmetrical with respect to its vertical axis, the larger drop will collide with the smaller drops that lie in a circular area in its fall path defined by a critical diameter, say dcr, as shown in the schematic Figure 4a, which may be referred as the effective collision cross section [see also Wallace and Hobbs, 1977]. Considering this geometric setup, one can mathematically represent the collision efficiency in terms of dcr, d1, equivalent diameter of the larger drop, and d2, equivalent diameter of the smaller drop as follows:
The critical diameter dcr (and thus E1) is governed by the competing inertial, gravitational, and aerodynamic forces. Considering that external dimensional parameters governing collision of water drops at free fall are d1, d2, g, υ, ρ, and ρa, from dimensional analysis one can arrive at the following set of governing dimensionless parameters:
Of these parameters, the first one p = d2/d1 represents the diameter ratios of colliding drops, and noting that the term (gd1)1/2 represents the velocity scale for gravitational collision, the second dimensionless parameter can be interpreted as the Reynolds number for collision, Rec. Assuming constant water and air densities, only the first two dimensionless parameters are significant for analysis, that is, E1 = f(p, Rec) (see also similar dimensional arguments given by Manton  and Pruppacher and Klett ).
 Two different approaches have been used in the determination of collision efficiency values: (1) evaluation of forces between two drops [e.g., Hocking, 1959; Davis and Sartor, 1967; Klett and Davis, 1973] and (2) superposition of flow fields, assuming that each drop moves in a flow field generated by the other drop moving in isolation [e.g., Shafrir and Gal-Chen, 1971; Lin and Lee, 1975; Schlamp et al., 1976; Pinsky et al., 2001]. Of these two approaches the former relies on Stoke's and Oseen's equations and thus is valid only for small Re values (Re < 1, cloud drops), whereas the latter can be applied for a wider range of Re values (Re < 100 [see Pinsky et al., 2001]) including both cloud and small raindrop sizes. Using the collision efficiency values tabulated in Table 8.2 of Rogers and Yau , we constructed a graph to illustrate the dependency of the collision efficiency (E1) on diameter ratios of colliding drops (p) for different collision Reynolds numbers (Rec) (Figure 5). Figure 5 shows that for a fixed value of p, E1 values approach unity with increasing Rec (increasing d1 and d2 values). For larger Rec, E1 values are very sensitive to the changes in p and approach unity rather steeply. Thus E1 values for collisions between drop pairs with sizes d1 and d2 greater than, say, 0.1–0.2 mm (classes II and III) can be approximated by unity [see Gillespie and List, 1978; Rogers, 1989; Czys and Tang, 1995].
 Once the collision efficiency is known, the collision kernel, C(d1, d2), for a raindrop pair can be estimated and consequently the rate of collisions for a distribution of raindrops, CR (see section 4 on drop size distribution, N(d)), by using the following relationships:
Here Ut1 and Ut2 are the terminal velocities for drops with diameters d1 and d2, respectively. For an exponential drop size distribution (see section 4) with an infinite range of drop sizes (dmin = 0 and dmin = ∞) and assuming E1 ≈ 1 for precipitation-sized drops, Rogers  was able to solve equation (10) analytically using the two different parameterizations for terminal velocity (equations (A1) and (A2) given in Appendix A). Later, Czys and Tang  extended this solution by taking into account the finite size of the drop size spectrum.
 Raindrop collisions may result in several possible scenarios: (1) Colliding drops may bounce apart (bounce); (2) they may coalesce forming a single larger drop (coalescence); or (3) they may break into a number of smaller drops (breakup). Reliable estimates of the probability of occurrence of each scenario after a collision and their outcome are essential for predicting the evolution of raindrop size spectra induced by collision. Various studies were conducted in the past to identify the key governing mechanisms and their relative importance to the occurrence of each scenario and their outcome [e.g., List and Whelpdale, 1969; Montgomery, 1971; Ochs and Czys, 1987; Ochs et al., 1995b; Beard et al., 2001]. On the basis of these studies it was possible to identify key controlling factors of collision outcome: the sizes of colliding drops (d1, d2), the angle of impact (θ) (see the schematic illustrating the impact geometry given in Figure 4b), surface tension (σ), electric charges, relative humidity, ambient pressure, and temperature.
 When two drops approach each other very closely, air may be trapped between them. If the pressure in the gas film separating the drops deforms the drop surfaces sufficiently to transform the drops' kinetic energy into deformation energy before the gas film becomes thin enough for lubrication breakdown to occur [Gopinath and Koch, 2002], drops may bounce apart. Beard and Ochs  and Ochs and Beard  identified two distinct mechanisms that yield bouncing of interacting drops: (1) rebound mechanism, the restoring force of surface tension causing the drops to spring apart before the air film can drain, and (2) grazing bounce mechanism, the tangential velocity of the small drop carrying it past the large drop before the air film drains. Therefore the drop bouncing problem is primarily the problem of air film drainage between two deforming drops. Experiments showed that this drainage is hindered particularly when the two approaching drops are large enough to deform easily and local flattening of surfaces can strongly impede the expulsion of the intervening air [Gopinath and Koch, 2002]. For negligible electric charge and fixed physical properties the physical controls on the rate of air film drainage (and bounce efficiency that is the ratio of the number of bounces to the number of collisions) can be expressed by the collision Weber number (sometimes referred to as the impact energy),
where ΔUt = Ut1 − Ut2, the diameter ratio, p, and the impact angle, θ.
 Using experimental bounce efficiency values obtained by Ochs et al. [1995a], Beard and Ochs  provided a linear regression formula (see equation (C1) in Appendix C) in terms of Wec and p. Later, a similar formulation (equation (C2) in Appendix C) including the effects of reduced pressure on bounce efficiency was given by Beard et al. . Ochs et al. [1995a] have found out that low-to-intermediate relative electrical charge values of drops practically do not have an influence on the value of critical impact angle above which drop bounce can be expected. However, at higher charge values, increasing the relative charge eliminates the probability of bounce occurrence. In the same study it was also reported that bounce efficiency increases for the low relative humidity condition. This effect was attributed to a lower drainage of the air film between the drops at the reduced film temperature because of evaporative cooling.
 For two drops approaching each other, once the air film has thinned locally to a thickness smaller than 0.0001 mm, the magnitude of attractive van der Waal's forces will likely be of sufficient strength to induce coalescence, especially with the help of small, random surface perturbations [Pruppacher and Klett, 1997; Pigeonneau and Feuillebois, 2002]. The process of coalescence is responsible for the growth of cloud drops to form precipitation-sized drops as well as the spreading of the raindrop size distribution to larger sizes. Quantitatively, the probability of occurrence of coalescence is generally represented by the coalescence efficiency, E2, which is the ratio of the number of coalescences to the number of collisions. The stochastic growth of drops is modeled by the coalescence kernel, K = C(d1, d2) E2, and the coalescence rate can be obtained by replacing C by K in equation (10). For drops with diameters smaller than about 0.1 mm, it is usually assumed that coalescence efficiency is unity [Rogers and Yau, 1989]. This assumption is supported by the recent experimental studies of Beard et al. , which showed that coalescence efficiency for minimally charged drops of class I (with d1 = 0.1–0.2 mm) and p = 0.5–1.0 is larger than 0.91–0.95. Therefore the coalescence rate of such small drops can be taken as identical to the collision rate. On the other hand, for larger drop pairs (precipitation-sized drops) the collision efficiency can be assumed to be unity as discussed previously, whereas the coalescence efficiency is always less than one. In this context the far-reaching impact is that understanding of raindrop size evolution and growth is critically dependent on the accurate estimation of coalescence efficiency.
 For negligible electric charge and fixed physical properties, coalescence efficiency can also be studied in terms of Wec, p, and θ. Information on coalescence efficiency values mainly comes from experimental studies. These studies are usually conducted using two different experimental techniques: (1) A large drop is held in a fixed position, and small drops are directed at it [e.g., Whelpdale and List, 1971; Levin et al., 1973; Levin and Machnes, 1977]. (2) A large drop falls through a cloud of drops, or both drops are at free fall [e.g., Beard and Ochs, 1983; Ochs et al., 1995a]. Beard and Ochs  point out the widespread disagreement between the collision efficiency values obtained by these two different techniques and recommend the use of results from experiments utilizing the second technique, because it replicates the natural conditions for raindrop collision and coalescence (see also arguments given by Pruppacher and Klett [1997, p. 595]).
 The coalescence efficiency values for colliding precipitation drops that are at free fall were studied experimentally for a wide range of parameters. Ochs and Beard  studied a low-impact energy range (0.1 < Wec < 1.6) for accretion of cloud drops (class I) by small precipitation drops (class II drops with d1 = 0.2–0.8 mm and p < 0.2). By using calculated collision efficiency values they inferred the coalescence efficiency values between 0.54 and 0.82. Note that experimentally measured efficiencies, usually referred to as collection efficiencies, are, in fact, the product of collision and coalescence efficiencies (E1E2) since the two effects cannot be strictly separated. Collisions in the high-impact energy range (Wec > 10) (d1 = 1.8–4.4 mm and p < 0.56) were investigated by Low and List [1982a]. They reported coalescence efficiency values between 0.21 and 0.65. Recently, Ochs et al. [1995a] and Beard and Ochs  studied collisions in the moderate-impact energy range (1.14 < Wec < 9.60) for class II drops (d1 = 0.55–0.95 mm and 0.47 < p < 0.73) and inferred coalescence efficiency values between 0.15 and 0.55 from their observations. Using such laboratory-based measurements conducted by utilizing either of the experimental techniques described above, various semiempirical parameterizations for coalescence efficiency as a function of dimensionless parameters Wec and p have been proposed. A comprehensive list of these parameterizations is given in Appendix C.
 Other factors affecting the coalescence efficiency were also investigated to some extent. For example, Ochs and Czys  and Czys and Ochs  showed that collisions of class II drops charged with same polarity (d1 = 0.68 mm and d2 = 0.38 mm), typical in nature, always coalesced for angles of impact smaller than a critical value of 43° for their range of experimental parameters. Assuming that raindrops can be represented by the conducting spheres, Beard and Ochs  scaled charge-induced coalescence by the electric field between the drops and proposed a semiempirical parameterization of coalescence efficiency as a function of the drop charge based on their experimental results. Czys  investigated temperature effects on coalescence efficiency for collisions of class II drops (d1 = 0.7 mm and d2 = 0.6 mm) falling freely at terminal velocity. These experiments revealed that the coalescence efficiency increased from approximately 0.42 for mean drop temperatures of 10°–20°C to about 0.81 for mean drop temperatures of 2°–10°C; that is, coalescence efficiency increased as drop temperature decreased. Ochs et al. [1995b] showed experimentally that relative humidity does not affect the coalescence efficiency, although it does alter the other types of collision outcome (i.e., bounce and breakup). Beard et al.  studied the effect of reduced pressure (545 and 745 hPa) on coalescence of negligibly charged class II drops (0.4 ≤ d1 and d2 ≤ 0.85 mm) and reported that reduction in pressure promoted contact of drops, thereby reducing bounce and increasing coalescence and/or breakup. On the other hand, List and Fung  reported no effect of reduced pressure (500 hPa) on coalescence or breakup of colliding drop pairs of class III drops with diameters of 2.61 and 1.17 mm.
 Experimental evidence shows that during the early stages of coalescence the coalescing drop-drop system is often highly unstable and may break up if sufficient energy is supplied to overcome the surface tension forces [Pruppacher and Klett, 1997]. Therefore the ratio of the collision kinetic energy (CKE) to the surface energy (SE) becomes an important criterion in determining whether colliding drops coalesce or break up. On the basis of physical arguments one may expect that breakup will occur if CKE dominates over the SE for the large drop (SE1). The underlying premise is that deterioration of the small drop may lead to coalescence as well as breakup, whereas the deterioration of the large drop will lead to breakup only. The CKE relative to the mass center of the drop pair (see the arguments given by Low and List [1982a]), SE1, and their ratio (dimensionless energy (DE)) can be expressed as follows:
Therefore occurrence of breakup is promoted for increasing values of DE. Using the experimental results presented in Table 3 of Low and List [1982a] and if the effects of impact angle are neglected, we estimate the critical value of DE above, where only breakup (absence of coalescence) takes place as DEcr ≈ 1.5.
 Neck breakup results from a glancing contact during which a water neck/bridge joining the two drops forms. The small drop does not appear to affect the large drop except in the immediate vicinity of the point of contact. After separation the large and small drops are still substantially intact, but additional smaller drops (fragments, sometimes referred as satellites) form as a result of the breakup of the neck. In Figure 6, pictures showing the dynamical evolution of neck breakup taken by a high-speed camera with 1000 frames per second are shown. These experiments (later, see also Figures 7 and 8) were conducted in a 5 m tall rain tower. For high-energy collisions, Low and List [1982a] documented the occurrence of necks usually as the small drop hits the large one near the outer edge. However, they also noted that regardless of the point of initial contact, as the CKE decreased and the size ratio became larger (hence smaller DE), the neck mode of separation into two or more fragments remained the only way of breakup.
 Sheet breakup occurs when the small drop hits the larger one in such a position that it tears off one side of the large drop. Subsequently, the bulk of the large drop starts rotating about the point of impact, while an extending film or sheet of water forms from the impact area. The small drop often disappears in this sheet, and the large drop becomes strongly distorted. By disintegration of this sheet a number of fragments form (see Figure 7). For sheet breakup to occur, higher CKE values than for neck breakup are required.
 Disk breakup occurs when the small drop hits the large drop near the center. Following the collision, the two drops temporarily coalesce. and a disk of water begins to spread out from the point of impact where the small drop becomes incorporated. During this process, increased drag force acting on the disk-shaped water body causes a rapid deceleration. Once the disk reaches its maximum extent, the outer fringe sheds drops, and then the whole disk gradually disintegrates into a relatively large number of fragments (see Figure 8). Disk breakup is limited to collisions involving higher values of CKE. Dissipation of this high CKE is attempted through conversion into surface energy by selecting a disk shape, which also yields energy dissipation through increased air drag, and by internal viscosity through oscillations.
 Bag breakup occurs in a similar fashion as disc breakup except that now a toroid forms with a thin film of water at its center. Subsequently, the film blows up into a bag because of the increased pressure differential between the stagnation points. Eventually, the bag shatters into a large number of drops. To our knowledge the only study that documented the occurrence of this type of breakup is by McTaggart-Cowan and List . In that study it was concluded that a dead center hit is required to produce this very rare type of breakup (<0.5% of the collisions observed in their experiments).
 The foremost objectives of drop breakup studies can be expressed in terms of two questions: (1) What is the conditional probability of occurrence (or physical conditions required) for each breakup type? (2) What is the outcome of each breakup type in terms of fragment size distribution?
 In this context, List et al.  studied neck breakup for the collision of class III drops with diameters ranging from 2.0 to 4.5 mm with velocity differences of the drop pairs equal to those observed in nature. They reported that 3 to 12 fragments (on average, 4.2) resulted from these interactions, the smaller number occurring when interacting drops maintain their original identity but loose mass to form a small satellite drop. Later, from a study of drops colliding at an angle (d1 and d2 = 0.3–1.5 mm), Brazier-Smith et al.  also reported similar observations. However, McTaggart-Cowan and List  and Low and List [1982b] pointed out later the inadequacy of these previous experimental setups. They argued that in order to reproduce the physics of collision process in a laboratory environment correctly, the colliding drops must be at terminal velocities in the same direction as the gravity vector. Taking this consideration into account in their experiments, McTaggart-Cowan and List  formed the first data bank for modeling the collision-induced breakup in numerical studies on the evolution of raindrop size spectra. On the basis of the breakups studied for collisions of six different raindrop pairs of class III (with d1 = 3.0, 3.6 and 3.8 mm and d2 = 1.0 and 1.8 mm), occurrence probability of each breakup type was reported as 27%, 55%, and 18% for neck, sheet, and disk, respectively. The range of drop sizes studied was extended by Low and List [1982a] to form a single data bank spanning a wide range of raindrop sizes including both class II and class III drops (d1 and d2 = 0.395–4.6 mm). Subsequently, Low and List [1982b] proposed breakup parameterizations based on these experimental data. The overall fragment number distribution, Q(d; d1, d2), for a colliding drop pair of (d1, d2) was parameterized by weighing the results of each breakup type according to the individual contribution and the probability of occurrence as
where Qn,s,d(d; d1, d2) represents the fragment number distributions for each breakup type (subscripts n, s, and d denote neck, sheet, and disk, respectively) fitted as sums of normal and lognormal distributions and Rn,s,d is the ratio of the number of breakups of type n, s, and d to the total number of colliding drops that break up. Later, corrections and improvements to the proposed parameterizations by Low and List [1982b] were introduced by List et al. , Brown [1997, 1999], and McFarquhar [2004a] (see explanations given in section 4).
where d (mm) is the drop diameter, N0 (m−3 mm−(1+μ)) is the concentration scaling parameter, Λ (mm−1) is the slope coefficient, and η is the distribution shape factor [Steiner et al., 2004]. A special case of this parameterization for η = 0 leads to the exponential DSD, where N0 becomes the intercept parameter. A well-known special form of the exponential DSD is the Marshall-Palmer distribution [Marshall and Palmer, 1948] that considers a fixed N0 value; see Figure 9. Smith  argues that when moment methods are used to determine parameters for the fitted DSDs, the experimental uncertainties tend to be greater than the differences in important bulk characteristics of rainfall, such as intensity or radar reflectivity factor, between the resulting gamma DSDs and corresponding simpler exponential DSDs. Consequently, Smith concluded that it makes little practical difference whether exponential or gamma DSD functions are employed. Nevertheless, Haddad et al.  argued that the three parameters (N0, η, and Λ) of the gamma distribution cannot be estimated independently from radar observations and proposed a new parameterization based on the mean and relative deviation of mass-weighted drop diameters along with rain rate.
 The assumed form of DSD is critical for remote sensing of rainfall using weather radars. Therefore an accurate representation of the form of the DSD is essential for rainfall estimation. In addition, a reliable capability of DSD retrieval is important for studying precipitation processes and to validate microphysical parameterizations in numerical models [Brandes et al., 2004]. The form of DSD has been mainly inferred using data from field experiments. Since the initial “retrieval” of DSD by Marshall and Palmer  using dyed filter paper, increasingly sophisticated measurement techniques and technologies have been utilized, each having their own pros and cons. Valuable continuous and automatic drop-sized measurements became available with the introduction of disdrometers [i.e., Joss and Waldvogel, 1967] for ground level measurements and particle measuring systems installed on an aircraft [Chandrasekar et al., 1988; Bringi et al., 1998] for measurements at different elevations. However, these measurement techniques have considerable limitations due to small sampling areas in the case of disdrometers and the fact that aircraft traversing the rain shaft disturb the cloud environment as well as the lack of temporal data at a particular spatial position from such airborne measurements. These limitations can be partly overcome via remote sensing albeit at the expense of increased measurement uncertainty. Fukao et al.  and Wakasugi et al.  demonstrated that VHF/UHF (very high frequency and ultrahigh frequency) radars pointing vertically upward (i.e., radar profilers) can simultaneously detect two distinct echoes, one from precipitation particles and the other from clear air turbulence, making the estimation of the DSD at different distances possible. Significant progress has been achieved in the estimation of kinematic and microphysical vertical structure of precipitating clouds with dual-frequency and Doppler radar profilers [Williams, 2002; Rajopadhyaya et al., 1999; Meneghini et al., 1992]. On the other hand, dual-Doppler polarimetric radars make DSD and vertical air motion (and mass flux) estimation possible at different horizontal distances from the radar, thus providing for three-dimensional views of storm structure [e.g., Cifelli et al., 2002]. For further details the reader is kindly referred to an extensive reference list on polarimetric radars given by Bringi and Chandrasekar . High spatial and temporal DSD estimation capabilities gained by dual-frequency dual-Doppler VHF/UHF and polarimetric radars make them promising tools for research toward universal DSD characterization for different hydrometeorological regimes. In the case of spaceborne observatories a milestone was achieved in 1997 when the precipitation radar, a single-frequency Ku band sensor in the TRMM satellite (Tropical Rainfall Measuring Mission) along with a radiometer (TRMM Microwave Imager), was launched [Kummerow et al., 1998; Iguchi et al., 2000; Meneghini et al., 1997]. The development of algorithms using data from multisensor, multifrequency, multipolarization Doppler radars (more degrees of freedom) to estimate multiple parameters of the DSD is currently a topic of active research in anticipation of the Global Precipitation Measurement Mission and related research activities [e.g., Kuo et al., 2004]. The inverse problem, which is converting radar measurables into DSD information, is rather challenging and is mentioned only briefly in this review; the focus is on ground-based radar. Extensive treatment of this topic can be found elsewhere in the literature [e.g., Bringi and Chandrasekar, 2001; Meneghini and Kozu, 1990].
 DSD retrievals with UHF/VHF radar data have been attempted with two methods: a method fitting model Doppler spectrum with the observed Doppler spectrum using convolution and a method deriving DSD directly from the observed Doppler spectrum using deconvolution [Kim et al., 2000]. On the other hand, DSD retrievals with polarimetric radar data have been attempted by estimating the parameters characterizing the assumed form of DSD. For example, for a gamma-type DSD (see equation (1)), which is characterized by three parameters (N0, η, and Λ), the retrieval method requires three different independent measurements/relations. For estimation of these parameters, Gorgucci et al.  and Bringi et al.  utilize a method that relies on measurements of radar reflectivity at horizontal polarization (Zh), differential reflectivity (Zdr), and specific differential phase (KDP), whereas Zhang et al.  and Brandes et al.  utilize a method that relies on measurements of Zh, Zdr, and an empirical constraining relationship between μ and Λ derived from ground-based disdrometer observations. Brandes et al.  points out that the (Zh, Zdr, and KDP) method is inadequate for lower rain rates, and Illingworth and Blackman  criticize the retrieval of three parameters in gamma DSD with the former method because of the possible redundancy among the Zh, Zdr, and KDP parameter set (see also the response of Bringi et al. ). On the other hand, the relationship between η and Λ is highly empirical and lacks a physical basis, which raises concerns with regard to its generalized (i.e., case independent) applicability.
 The uncertainties associated with DSD estimation from polarimetric radar data using current methods are considerable, and to improve DSD retrieval methods, further research through radar-disdrometer comparisons is needed. However, such comparisons are not trivial, since DSD estimations from polarimetric radar data at a certain height will not correspond to the measured DSD at the ground level by a disdrometer. Therefore these comparisons should also take into account DSD evolution with height, which is driven by various complex processes such as collision-induced drop breakup/coalescence, evaporation/condensation, etc.
 Understanding the DSD evolution with height is the key for developing the capability of extending/extrapolating polarimetric radar DSD estimations to different heights. Note that such a capability has important applications such as soil erosion studies [Fox, 2004; Fornis et al., 2005], air pollution studies [Mircea et al., 2000], telecommunications [Panagopoulos and Kanellopoulos, 2002], etc., besides improving DSD retrieval methods as discussed above. The evolution of DSD in clouds is shaped by various processes such as drop breakup/coalescence, evaporation/condensation, size sorting due to updraft/downdraft, etc. (for a qualitative description on the effects of individual processes on DSD evolution, see Rosenfeld and Ulbrich ). Because of the complexity a theoretical underpinning of the DSD evolution has not been well established and is not likely to be in the near future. Numerical simulations based on laboratory models of governing processes offer a path for virtual integration of such processes.
 The stochastic coalescence/breakup equation constitutes the mathematical basis for these numerical models:
 Typically, coalescence and breakup are taken into account using laboratory-based coalescence efficiencies and fragment probability distribution functions [Brazier-Smith et al., 1972; McTaggart-Cowan and List, 1975; Low and List, 1982a, 1982b]. Various modeling studies [e.g., Valdez and Young, 1985; Hu and Srivastava, 1995; McFarquhar, 2004a, 2004b] have shown that an equilibrium DSD is approached after sufficient evolution time given steady state conditions and the absence of other forcing. Numerical simulations based on the results of Low and List [1982a, 1982b] exhibited equilibrium DSDs with three peaks located at specific diameters (see Figure 10) [Valdez and Young, 1985; List et al., 1987; List and McFarquhar, 1990a; Brown, 1994]. Multipeak DSD observations by different researchers [Steiner and Waldvogel, 1987; Asselin de Beauville et al., 1988; Zawadzki and Antonio, 1988; List, 1988] using Joss-Waldvogel-type disdrometers were taken as validation for the calculated three-peak equilibrium DSDs. Later, Sheppard  showed that instrument-related difficulties produced artificial peaks at locations where the observational peaks had been reported, indicating that much of the previous observational evidence for the calculated three-peak equilibrium DSDs might be merely coincidental [McFarquhar, 2004a]. However, the possibility of multipeak equilibrium DSDs has not been completely abandoned. Limited multipeak DSD observations with other instruments [Willis, 1984; Garcia-Garcia and Gonzalez, 2000; Atlas and Ulbrich, 2000] motivated some researchers to reexamine and improve the Low and List [1982b] parameterizations by ensuring mass conservation after drop collisions in the interpretation of experimental results [Brown, 1997] and by explicitly accounting for uncertainty associated with extrapolating data from the 10 original size combinations to arbitrary sizes of colliding raindrops [McFarquhar, 2004a]. Numerical studies with the modified parameterizations also concluded that given a sufficient amount of time under steady rain conditions, a multipeak equilibrium DSD will be reached. Yet, in contrast with the results of numerical simulations, the existence of an equilibrium DSD itself with or without peaks has not been substantiated by observations under natural conditions. Furthermore, analytical solutions of raindrop collision rates (see section 3) by Rogers  and Czys and Tang  have underlined the improbability of achieving collision-controlled equilibrium distributions in realistic times except perhaps in intense rain (tropical convection). In particular, Jameson and Kostinski  questioned the adequacy of numerical simulations of DSD evolution and argued that the discrepancies between results of simulations and observations may be due to the absence of idealized “control volumes” in the atmosphere that would allow drops to interact sufficiently to yield an equilibrium DSD. Furthermore, they pointed out that exclusion of three-dimensional spatial variability of natural rain, that is, the assumption that rainfall is spatially homogeneous, may also be an important reason for these discrepancies.
 The assumption of spatially homogeneous rain DSDs is also intrinsically embedded in quantitative rainfall estimation based on the relative probabilities of drops of different sizes alone. Lovejoy and Schertzer  discuss the presence of highly variable clusters of raindrops over a range of observed scales. They proposed that rainfall is statistically inhomogeneous and displays multifractal properties. Recent field observations [Jameson and Kostinski, 1998, 2000] support this “patchy” character of rainfall composed of clusters of elementary DSDs over a range of scales, whereas the traditional interpretation of measured DSDs is statistical mixtures of these patches [Jameson and Kostinski, 2001]. One of the key implications of recognizing spatial heterogeneity in rain structure is that rainfall estimates depend on the resolution of the measurement instruments used, and thus the nature of the statistical mixture varies with resolution. This result raises a question similar to that formulated by Jameson and Kostinski : What is the appropriate DSD for measuring rainfall as a function of sensor resolution?
 Despite great advances during the second half of the twentieth century, characterization of rainfall processes at multiple scales and rainfall measurement remains elusive. Knowledge building toward quantitative prediction requires sustained efforts on many fronts including, for example, coupled studies of VHF/UHF radars and multisensor remote sensing at multiple scales and cloud-resolving models with explicit particle-based formulations of microphysical processes. Powerful new technologies from chip-scale integrated sensing systems to high-speed imaging open new opportunities for hydrometeor-scale, process-oriented laboratory and field studies that will allow more accurate modeling of the dynamical evolution of rainfall microstructure. We suggest that the shift from an Eulerian parcel-based description of cloud microphysics to a Lagrangian particle-based framework is now inevitable and therein lies great opportunities for discovery and progress.
Appendix A:: TERMINAL VELOCITY FORMULATIONS
 Some of the proposed terminal velocity formulations for drops falling in air are summarized below:
where c1 = 4000 s−1 for the size range 0.08 ≤ d ≤ 1.5 mm [Rogers, 1989].
where d should be in centimeters and Ut should be in cm s−1 for the size range 0.6 ≤ d ≤ 5.8 mm [Atlas et al., 1973].
where μa is dynamic viscosity of air and ρa is density of air for drop sizes d ≤ 0.06 mm [Rogers and Yau, 1989].
where c2 = 1500 cm1/2 s−1 and ρa,0 is the reference density of 1.20 kg m−3 corresponding to dry air at 101.3 kPa and 20°C for the size range 1.2 ≤ d ≤ 4.0 mm [Rogers and Yau, 1989].
where Ut should be in cm s−1, z is altitude in kilometers, d should be in millimeters for the size range 0.3 ≤ d ≤ 6.0 mm for International Commission on Air Navigation standard atmosphere [Best, 1950].
where Ut should be in cm s−1, d should be in centimeters, and ρa,z is air density at the altitude z [Lhermitte, 1990].
Appendix B:: CHORD RATIO FORMULATIONS
 Some of the proposed chord ratio formulations are summarized below:
where x = (a1 − a2)1/3 − (a1 + a2)1/3, a1 = (a22 + 0.00441)1/2, a2 = 0.0946β − 0.319, β = ln(d2/0.002) + 0.44ln(d1/0.4), and d1 and d2 are given in millimeters for the size ranges 0.1 ≤ d1 ≤ 1 mm and 0.002 ≤ d2 ≤ 0.064 mm [Pruppacher and Klett, 1997].
 This research was supported in part by NSF grant ATM 97-530093 and NASA grant NNGO04GP02G to the second author and by the Pratt School of Engineering at Duke University. The first author was a postdoctoral associate at Duke University when this work was conducted. The pictures shown in Figures 6, 7, and 8 were taken during the summer internship of Nikhil Nadkharni in the Barros laboratory then at Harvard University. Inspiration and motivation for this work has been sustained over the years by the intellectual generosity and grace of many colleagues who took the time to listen and share their wisdom.
 The Editor responsible for this paper was Daniel Tartakovsky. He thanks three anonymous technical reviewers and one anonymous cross-disciplinary reviewer.