Essential Precipitation Physics for Dual Polarization Radar PDF

3 Essential Precipitation Physics for Dual Polarization Radar After observing a radar display for a sufficient amount of time, it becomes obvious to identify specific patterns and featu...

3 Essential Precipitation Physics for Dual Polarization Radar After observing a radar display for a sufficient amount of time, it becomes obvious to identify specific patterns and features in the dual polarization parameter space that are recurring among different precipitation events. The most notable pattern is likely the so-called bright band, a circular band of enhanced reflectivity often appearing in high-elevation plan position indicators (PPIs) during stratiform precipitation. The bright band is the result of the layered structure of the atmosphere during widespread precipitation, in combination with the change of the dielectric factor arising from the solid-to-liquid phase transition of precipitation particles and the geometry of the radar scan, with observations being collected on a conical surface intersecting the melting layer. This particular example illustrates the need for a comprehensive understanding that includes geometric, electromagnetic, and physical factors. To aid in this endeavor, this chapter is devoted to the introduction of fundamental concepts of cloud physics relevant to the interpretation of radar observations. 3.1 The Microscale Structure of Precipitation Clouds are a collection of particles suspended in air. The collection of particles (water droplets or ice crystals) remains stable as long as the particles are suspended. When precipitation particles form, they tend to fall outside of the cloud medium, and the system becomes unstable. In order to describe the time evolution of cloud and precipitation systems, their microscale structure can be characterized by specifying the phase, shape, size, and number concentration of the constituent particles. A convenient way to synthesize this information is by means of the particle size distribution (PSD). Given the well- established size–shape relation of liquid drops (Section 3.3), the spectra of clouds and raindrops can be easily represented by mathematical expressions considering only the size of the drops. In the case of ice particles, additional information regarding the shape and density of the particles is generally required to provide an appropriate description. 3.1.1 The Particle Size Distribution The PSD is the probability density function of particle sizes. For spherical particles, the size is represented by the diameter (D), although for more complex particles, https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.1 The Microscale Structure of Precipitation 63 different definitions of size may be adopted. For ice particles, the maximum dimension is used in some instances, or the equivalent melted-drop diameter (the equivalent diameter of the water drop to which the ice particle melts). The use of the equivalent melted diameter requires assumptions about the density–size relationship. The PSD is conventionally indicated by N (D), the distribution of particle sizes per unit volume per unit size interval. The number concentration may be expressed in different units, which usually vary depending on the context. Widely used units are cm−3 μm−1 (cloud droplets), m−3 mm−1 (raindrops), and cm−4 (snow particles). The PSD represents the physical link between the cloud and precipitation param- eters and the remote sensing observations within the sampling resolution volume. In addition to remote sensing, knowledge of the PSD is also important for applications in cloud physics, agriculture, numerical models, and hydrology. Figure 3.1 shows some characteristic size distributions of liquid and frozen precip- itation particles, spanning different size and concentration ranges. The PSD of rain (panel a) is conventionally referred to as the drop size distribution (DSD). The two DSDs representative of continental and maritime rain in panel (a) of Figure 3.1 are characterized by similar water content but different shapes. In general, the particular shape and time evolution of the size distribution can help elucidate important traits of the underlying precipitation mechanisms. For example, the longer right tail of the continental DSD reveals the dominance of the ice-phase processes in precipita- tion formation. In continental stratiform precipitation, the depositional growth of ice crystals (Bergeron process) followed by aggregation and subsequent melting below the freezing level may lead to larger raindrops, in comparison with the typical sizes attainable through warm rain processes in tropical regions (Section 3.5). In continental convective precipitation, most raindrops also originate from the melting of frozen particles, which in this case mainly consist of graupel and hail. One of the most popular analytical expressions for the PSD in remote-sensing applications is the two-parameter exponential distribution: N (D) = N0 e−ΛD , (3.1) where N0 is the concentration (intercept) parameter, and Λ is the size (slope) param- eter. For rain, Marshall and Palmer fitted an exponential model to raindrop size distributions observed in Canada. The model uses a fixed intercept parameter (N0 = 8 × 103 mm−1 m−3 ) and a slope parameter expressed as a function of the rain rate R (Λ = 4.1 R −0.21 mm−1 ), with D in mm and R in mm h−1. One of the limitations of the exponential form is the poor representation of the smallest drop sizes. However, given the high moments of the DSD involved in radar applications, this can be generally regarded as a minor inconvenience. As compared with raindrops, ice particles span a wide range of sizes, from tens of microns to a few centimeters. For ice aggregates larger than a few tenths of a millimeter, it has been shown that the PSD can also be well represented by the exponential function [24, 25], as depicted in panel (b) of Figure 3.1. Field and Heymsfield have shown how the growth of ice particles through aggregation translates into a typical scaling of the N0 − Λ relation. https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  64 Essential Precipitation Physics for Dual Polarization Radar Figure 3.1 Characteristic PSDs for rain (a), ice aggregates (b), and hail (c). The two distributions in panel (a) have the same liquid water content of ∼1.5 g m−3 but different shapes as a result of distinct precipitation-formation mechanisms. The light- and dark-gray ice particle distributions in panel (b) represent two successive stages of evolution through aggregation, showing the strong coupling of N0 − Λ (both parameters of the exponential PSD decrease with increasing aggregation). A notable advantage of the exponential form of the PSD is that it allows for simple calculation of any moment of the distribution:  ∞ Γ (n + 1) D n N(D)dD = N0 , (3.2) 0 Λn+1 where Γ is the gamma function (Γ (n + 1) = n! for n, an integer). This analytical formulation for the nth moment makes the exponential form especially convenient for use in theoretical work in general. Although eq. (3.2) requires an infinite upper limit of https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.1 The Microscale Structure of Precipitation 65 Table 3.1 Typical ranges of size and terminal fall velocity of precipitation particles. Precipitation Particle Size (mm) Velocity (m s−1 ) Small raindrops 0.5–2 2–6 Large raindrops 2–8 6–9 Pristine ice crystals 0.1–5 0.1–1 Snow 1–30 0.5–2 Graupel 1–20 2–20 Hail 10–100 10–50 integration, it is generally a good approximation for real cases with a finite maximum particle size because of the fast-decreasing tail of the distribution. A more accurate representation of the PSD, in particular for the left portion of the size range (smallest particles), is attained using the three-parameter gamma distribution: N (D) = N0 D μ e−ΛD. (3.3) With respect to the exponential form, the additional multiplicative term D μ provides better control of the distribution “shape,” with the parameter μ being called the shape parameter. The gamma form in eq. (3.3) is widely used to represent the size distribu- tion of both cloud droplets and raindrops (panel [a] of Fig. 3.1), providing a simple tool to represent a wide range of rainfall conditions. The exponential form in eq. (3.1) can be regarded as a special case of the more general three-parameter gamma distribution, obtained by setting μ = 0 in eq. (3.3). Another special case of the gamma distribution is the power-law distribution (Λ = 0 in eq. [3.3]). Power-law relations have been used, for example, to represent the hail-size distribution by Auer (panel [c] of fig. 3.1). Depending on the composition, size, shape, and orientation, the precipitation parti- cles may fall with a characteristic velocity. The terminal fall velocity is defined as the constant speed reached by precipitation particles when the gravity force (downward) is balanced by the aerodynamic drag force (upward) exerted by the surrounding air. Table 3.1 lists the values of terminal fall velocity for common precipitation particles, showing a wide range of values going from approximately 0.1 m s−1 for ice crystals to approximately 50 m s−1 for the largest hailstones. 3.1.2 Measuring the Size Distribution: The Disdrometer Disdrometers are measuring devices specifically meant for the direct observation of the PSD. Different measuring systems have been developed, resulting in a variety of disdrometer types, including acoustic, displacement (Joss–Waldvogel disdrometer), optical, and image (2DVD) disdrometers. Some disdrometers, in addition to the size spectrum, can also measure the velocity of the particles, and some can measure the shape too. As an example, Figure 3.2 illustrates the principle of an optical laser dis- drometer. https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  66 Essential Precipitation Physics for Dual Polarization Radar Figure 3.2 Measuring principle of a laser disdrometer. The PSD is defined as the number of particles per unit volume per unit size interval, N (D) = nc fD (D), where nc is the number concentration (m−3 ), and fD (D) is the probability density function (mm−1 ). Note that the computation of the PSD from the observed disdrometer counts involves knowledge of the particles’ fall velocity Vt (D). Although the PSD is defined as the number of particles per unit volume per unit size interval (i.e., the drops in the cubic volume in Fig. 3.2), disdrometers often mea- sure the flux of particles during a given time interval (e.g., the number of particles falling through the laser beam during time Δt in Fig. 3.2) and then convert to drops concentration using assumed or measured fall velocities. This suggests how the actual measuring principle should be carefully considered when interpreting measured data. For example, in windy conditions may lead to unreliable estimation of the real drop concentration. https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.2 The Size Distribution of Raindrops 67 3.2 The Size Distribution of Raindrops The DSD is of great relevance for radar rainfall applications, as illustrated in Figure 3.3, which shows four different DSDs with the same radar reflectivity of 37 dBZ but rainfall rates ranging between 1 and 36 mm h−1. In fact, depending on how the total mass in the volume is distributed over different drop sizes, the rainfall rate may greatly vary and attain the largest values for the distributions dominated by small drops. This is a direct consequence of Z and R representing different moments of the DSD, which will be thoroughly discussed in Chapter 9. For a generic raindrop size distribution N (D), it is useful to define some special diameters. The median volume diameter D0 is the diameter such that half the total liquid water content W is contributed by diameters smaller than D0 and the other half by diameters larger than D0. Given the total water content resulting from the third moment of the DSD,  π Dmax W = × 10−3 ρ w D 3 N (D)dD [g m−3 ], (3.4) 6 Dmin where ρ w ≈ 1 g cm−3 is the water density and the drop’s diameter, D, is in mm, the median volume diameter is such that  D0 π 1 × 10−3 ρ w D 3 N (D)dD = W. (3.5) 6 Dmin 2 In addition to the median volume diameter, various mean diameters can be calcu- lated from the expression:  Dmax D D p+1 N (D)dD Dp =  min Dmax p. (3.6) Dmin D N (D)dD In particular, the mass-weighted mean diameter, D m ≡ D3 , (3.7) and the reflectivity-weighted mean diameter, D z ≡ D6. (3.8) When integrating a moment of the DSD as in the previous equations, the inte- gral is evaluated between the lower and upper limits Dmin and Dmax. In general, there is uncertainty about these limits, which is related to the particular way the DSD is measured. Any measuring device can evaluate the drop diameter within a specific limited range. For most disdrometers, the typical useful measurement range is ∼0.2–10 mm. Droplets smaller than 0.2 mm are difficult to measure with disdrometers https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  68 Essential Precipitation Physics for Dual Polarization Radar Figure 3.3 The same radar reflectivity can be observed from a unit volume containing different DSDs. All four represented DSDs show the same reflectivity but very different rainfall-rate values. Panels (a) and (b) represent monodisperse DSDs with ∼5000 1-mm drops (a) and 74 2-mm drops (b). A single 4-mm drop (c) can also produce the same 37-dBZ reflectivity but a greatly reduced rainfall rate. A typical exponential DSD is also shown for comparison (d). All drops are scaled 20:1 with respect to the 1-m3 volume. [28, 29]. However, the choice of Dmin has a negligible impact on the evaluation of the typically high-order moments considered in radar applications, and in general, Dmin is simply set to 0 mm. A proper selection of Dmax is more relevant for the evaluation of precipitation-related moments like W (third-order moment), R (3.67th- order moment), and Z (sixth-order moment). In many applications, it is quite common https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.2 The Size Distribution of Raindrops 69 to use a fixed maximum diameter in the 6- to 10-mm range (e.g., Dmax = 8 mm). An accurate estimation of the maximum diameter is related to the sample size of the available measurements. In fact, the true distribution of the drop diameters cannot be known because of the limited time-space physical sampling of any measuring device. In general, the estimation of the radar and precipitation variables is a statistical problem of estimating the characteristics of a population from a sample drawn from that population. The larger the sample size, the more accurate the estimate of the population characteristics will be. When dealing with actual disdrometer mea- surements having a limited sample size, a generally good approximation is to relate the maximum diameter to the mass-weighted mean diameter Dm. A widely used approximation is Dmax = 2.5Dm because drops larger than 2.5Dm are rarely observed in nature. It is also worth noting that for a gamma distribution with positive shape parameter μ (Section 3.2.2), the impact of the uncertainty on Dmax is greatly reduced as a result of the faster-decreasing tail of the distribution, with respect to the exponential form (Fig. 3.5). For greater ease of reading, the integral limits will be dropped from the integral symbols from now on, except where explicitly noted. 3.2.1 The Exponential Model for Raindrops For the exponential size distribution (eq. [3.1]) in rain, the slope parameter Λ is related to the median volume diameter D0 and to the mass-weighted mean diameter Dm. For a large enough maximum diameter (Dmax /D0 ≥ 2.5), it can be shown that ΛD0 = 3.67 (3.9) ΛDm = 4, (3.10) allowing us to rewrite eq. (3.1) as −3.67 DD N(D) = N0 e 0 [m−3 mm−1 ] (3.11) or D N (D) = N0 e−4 Dm [m−3 mm−1 ]. (3.12) 3.2.2 The Gamma Model for Raindrops Although the exponential model represents a good approximation for space-time- averaged size distributions, “instantaneous” DSDs (i.e., DSDs measured over a small time interval) can be better approximated by a three-parameter gamma model. It is common and useful to represent the gamma DSD in different forms to take advantage of specific properties, depending on the application. The most frequently used expression directly follows as an extension of the exponential form in eq. (3.1) and was presented in eq. (3.3). https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  70 Essential Precipitation Physics for Dual Polarization Radar For the gamma distribution, the median volume diameter and the mass-weighted mean diameter are related to both Λ and μ: ΛD0 = 3.67 + μ (3.13) ΛDm = 4 + μ. (3.14) Another form of the DSD is the classic probability density function: N (D) = nc fD (D); [m−3 mm−1 ], (3.15) where nc is the number concentration (units of m−3 ), and fD (D) is a probability density function with unit integral. For the special case of a gamma distriibution, fD (D) is expressed as (Fig. 3.4): Λμ+1 −ΛD μ fD (D) = e D ; [mm−1 ]. (3.16) Γ (μ + 1) Using this form, any nth moment of the DSD can be calculated with the following formula:  ∞ Γ (μ + 1 + n) 1 D n fD (D)dD =. (3.17) 0 Γ (μ + 1) Λn Although eq. (3.17) requires the integral limits to be (0, ∞), it still provides an approx- imation for the integral parameter of the DSD with real limits (Dmin , Dmax ) whenever Dmin /D0 is small and Dmax /D0 is large. Figure 3.5 shows example empirical fits on real disdrometer observations using the gamma model (solid lines) and the exponential model (dashed lines). (a) (b) Figure 3.4 (a) gamma density function for different values of the slope parameter Λ and shape parameter μ. When μ = 0, the distribution reduces to the exponential distribution (thick solid black line). (b) Same as panel (a), with y-axis in logarithmic scale. https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.2 The Size Distribution of Raindrops 71 (a) (b) Figure 3.5 N(D) versus D for 1-minute DSD measured by a laser disdrometer. Panel (a) shows a gamma fit with a positive shape parameter μ (the most frequent occurrence in real rainfall), whereas panel (b) shows a distribution fitted by a gamma with a negative μ. Note the overestimation of the large-drop concentration with the use of the exponential approximation (dashed line) for the distribution in panel (a) (μ > 0). 3.2.3 The Normalized Drop Size Distribution When comparing different distributions with varying water content, it is useful to introduce the concept of the normalized distribution. Equations (3.11) and (3.12) provide expressions where the drop diameter is normalized by either the characteristic median drop diameter or the mass-weighted mean diameter. In order to complete the normalization, the particle concentration needs to be considered next. Sekhon and Srivastava and Willis introduced the concept of the normalized distribution relying on the total water content W (eq. [3.4]) and the median volume diameter D0. Using eq. (3.17) to calculate the third moment of the DSD results in the following: π Γ (μ + 4) 1 W = × 10−3 ρ w nc 6 Γ (μ + 1) Λ3 π N0 Γ (μ + 4) = × 10−3 ρ w (3.18) 6 Λμ+4 π N0 Γ (μ + 4) = × 10−3 ρ w D0 μ+4, 6 (3.67 + μ)μ+4 where eq. (3.13) has been used, in addition to the relation between N0 and nc (eqs. [3.3], [3.15], [3.16]): Λμ+1 N0 = nc. (3.19) Γ (μ + 1) https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  72 Essential Precipitation Physics for Dual Polarization Radar For the case of an exponential (μ = 0), from eq. (3.18), N0 can be expressed in terms of the water content W and median volume diameter D0 as   (3.67)4 W N0 = 103. (3.20) πρ w D0 4 Expressing the intercept parameter N0 in terms of W and D0 in the exponential distribution suggests a possible way to normalize the more general gamma DSD. In fact, by dividing the expression in eq. (3.3) by (103 W )/(ρw D0 4 ), we can define a normalized distribution:   ρ w D0 4 Nnorm (D) = N0 D μ e−ΛD. (3.21) 103 W Substituting the expression of W from eq. (3.18) to eliminate N0 and using eq. (3.13) to normalize the particle diameter gives  μ 6 (3.67 + μ)μ+4 D −(3.67+μ) DD Nnorm (D) = e 0. (3.22) π Γ (μ + 4) D0 Finally, again multiplying eq. (3.22) by (103 W )/(ρw D0 4 ) leads to a third expression of the gamma DSD, that is, the normalized DSD:  μ D −(3.67+μ) DD N(D) = Nw f (μ) e 0 , (3.23) D0 where: 4   3 (3.67) W Nw = 10 (3.24) πρ w D0 4 is the normalized intercept parameter, and 6 (3.67 + μ)μ+4 f (μ) =. (3.25) (3.67)4 Γ (μ + 4) Comparing eqs. (3.24) and (3.20), it is clear that Nw is equal to the intercept parameter N0 of an equivalent exponential DSD with the same water content and the same median volume diameter as the gamma DSD. Using eq. (3.14) in eq. (3.18) to express the water content in terms of the mass- weighted mean diameter Dm , instead of D0 , an alternative formulation of the normal- ized DSD is given by Testud as  μ D D N (D) = Nw f (μ) e−(4+μ) Dm , (3.26) Dm https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.2 The Size Distribution of Raindrops 73 (a) (b) Figure 3.6 (a) Fifty gamma distributions, randomly selected after fitting over 40,000 1-minute DSDs measured by 14 laser disdrometers in Iowa. The plotted fits are different shades of gray based on the value of the water content (the corresponding rain rate associated with the observed DSDs varies between 1 and 80 mm h−1 ). (b) Same DSD fits as in panel (a) but plotted over the normalized axes. Note the “compression” toward D = D0 and N = Nw e−3.67 , which corresponds to the exponential distribution (μ = 0 in eqs. [3.23] and [3.25]). Note also the typical dominance of fitted distributions with a positive shape parameter μ. https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  74 Essential Precipitation Physics for Dual Polarization Radar where   44 W Nw = 10 3 (3.27) πρ w Dm 4 6 (4 + μ)μ+4 f (μ) =. (3.28) 44 Γ (μ + 4) Using the illustrated normalization approach, the DSD is expressed as a function of three independent and physically meaningful parameters. The median volume diam- eter D0 (or the mass-weighted mean diameter Dm ) and Nw have, respectively, the physical units of length and concentration, whereas the unitless parameter μ describes the shape of the DSD. On the other hand, in the standard form of the gamma DSD (eq. [3.3]), there is an implicit relation between μ and N0 , which has no intuitive physical meaning (N0 being expressed in units of m−4−μ ). Figure 3.6 shows an exam- ple of raindrop size distributions obtained after fitting a gamma distribution to dis- drometer observations, with the corresponding normalized distributions. 3.3 Drop-Shape Models It has long been recognized on the basis of theoretical and experimental studies that raindrops have shapes resembling an oblate spheroid, with their symmetry axis aligned close to the vertical. The shape of a raindrop is primarily the result of the balance between surface tension and gravitational forces. Green proposed a simple model for approximating the shape of raindrops in terminal fall equilibrium, based on the balance between surface tension and hydrostatic forces. According to this model, the shape of a raindrop can be represented by an oblate spheroid; that is, it can be completely specified by the value of its axis ratio b/a, where b is the minor axis radius, and a is the major axis radius (Fig. 3.7). The volume-equivalent spherical diameter De = 2 (b)1/3 (a)2/3 is given in millimeters as follows :   1/2 σw (b/a)−2 − 2(b/a)−1/3 + 1 De = 103 × 2 , (3.29) gρ w (b/a)1/6 where σw = 0.07275 J m−2 is the surface tension of water, g = 9.81 m s−2 is the gravitational acceleration, and ρw = 997 kg m−3 is the water density. The axis ratio relation of the Green model is shown in Figure 3.7 as a thin solid black line: For ease of notation, the e subscript is dropped from De and D is used to represent the volume-equivalent diameter of raindrops. For applications not requiring very high accuracy, the axis ratio relation can be approximated by a linear relation. The commonly used linear formulation is the one given by Pruppacher and Beard , which is valid for diameter sizes ranging between 1 and 9 mm: b = 1.03 − 0.062D; 1 ≤ D ≤ 9 mm. (3.30) a https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.3 Drop-Shape Models 75 Figure 3.7 Axis ratio of raindrops as a function of the diameter. Some of the most relevant relations are illustrated, in addition to the aircraft observations of Chandrasekar et al.. Beard and Chuang elaborated an accurate numerical model of the raindrop shape that considers the effects of the aerodynamic pressure arising from the flow around the drop, in addition to the surface tension and the hydrostatic pressure as in the Green model. Figure 3.8 shows a representation of the Beard and Chuang model , evidencing the flattened base of the larger raindrops with respect to the simpler oblate spheroidal models. For most radar applications, it is generally sufficient to use a simplified spheroidal model with an axis ratio relation given by a polynomial fit to the Beard and Chuang model: b = 1.0048 + 0.00057D − 0.02628D 2 + 0.003682D 3 − 0.0001677D 4, (3.31) a which is valid for D between 0.5 and 7 mm. Many studies have shown that mean raindrop shapes are more spherical than the equilibrium values given by either the Green model or the Beard and Chuang model. The reason is related to the fact that raindrops falling in the free atmosphere are subject to both axisymmetric (oblate-prolate mode) and transverse-mode oscillations [36, 39, 40]. These oscillations have frequencies ranging from a few hundred hertz for the smaller raindrops (D ≈ 1 mm) to a few tens of hertz for the larger drops (D > 5 mm). https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  76 Essential Precipitation Physics for Dual Polarization Radar Figure 3.8 Representation of the Beard–Chuang drop-shape model (solid lines). The dashed contours represent the simplified spheroidal approximation given by eq. (3.31). Although the axisymmetric mode produces a symmetric scatter around the equi- librium axis ratio (two-sided variation about unity), with negligible effects on the resulting mean shape, the transverse-mode oscillations result in a one-sided scatter. This leads to an upward shift in the mean axis ratio relative to the equilibrium shape. In particular, for drops smaller than 4 mm, Andsager et al. provided an experimental fit to laboratory measurements in the form of a second-order polynomial (gray thin solid line in Fig. 3.7): b = 1.012 − 0.0144D − 0.0103D 2. (3.32) a According to this relation, drops with diameters smaller than 4 mm exhibit a mean axis ratio larger than that obtained at equilibrium. The laboratory results of Andsager et al. are in agreement with the aircraft observations previously studied by Chandrasekar et al. , who reported mean axis ratios higher than the equilibrium value for drops with diameters between 2 and 3.6 mm (+ symbol in Fig. 3.7). A frequent approach adopted in radar studies is the combination of eqs. (3.31) and (3.32), applying the Andsager et al. fit (eq. [3.32]) for D < 4 mm and the Beard–Chuang relation (eq. [3.31]) for D > 4 mm. More recently, Brandes et al. introduced a polynomial approximation that combines the observations of Pruppacher and Pitter , Chandrasekar et al. , Beard and Kubesh , and Andsager et al. : b = 0.9951 + 0.0251D − 0.03644D 2 + 0.005030D 3 − 0.0002492D 4. (3.33) a This relation agrees well with the fit provided by Andsager et al. for D < 4 mm, and it provides a good match between the observed and predicted radar variables obtained from scattering simulations on real DSDs. https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.3 Drop-Shape Models 77 3.3.1 Terminal Velocity of Raindrops Similar to the shape of the drops, the fall speed of raindrops is another quantity related to the drop size. Gunn and Kinzer performed measurements of the terminal fall velocity of drops in stagnant air, which have been the basis for several empirical formulations. Figure 3.9 shows the original measurements with two widely used fit- ting relations. Atlas and Ulbrich derived a power-law expression based on the measurement of Gunn and Kinzer : Vt (D) = 3.78D 0.67, (3.34) where the diameter D is expressed in millimeters and the fall velocity in meters per second. This widely used formulation provides a good approximation for drops up to ∼4 mm. The power-law relation (eq. [3.34]) has the notable advantage of facilitating the incorporation of the drops’ fall velocity into the expression for the rainfall rate (see Section 9.1 for more details):  R = 7.12 × 10−3 D 3.67 N (D)dD, (3.35) where R is expressed in millimeters per hour. Figure 3.9 Terminal fall velocity of raindrops at sea-level. Measurements by Gunn and Kinzer (circles) and fitting models by Atlas et al. , Atlas and Ulbrich , and Brandes et al.. https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  78 Essential Precipitation Physics for Dual Polarization Radar A more accurate fit to the measurements of Gunn and Kinzer is another widely used expression given by Atlas et al. : Vt (D) = 9.65 − 10.3 e−0.6D , (3.36) which is valid for D > 0.1 mm. Even closer fits to the Gunn and Kinzer data can be obtained using polynomial approximations, like the one given by Brandes et al. (dashed line in Fig. 3.9): Vt (D) = −0.1021 + 4.932D − 0.9551D 2 + 0.07934D 3 − 0.002362D 4. (3.37) The power-law approximation (eq. [3.34]) is a convenient formulation for develop- ing analytical relations involving the drops’ fall velocity. However, for real DSDs that include large drops (i.e., Dmax > 4 mm), eq. (3.36) or (3.37) are a more appropriate representation of the raindrops’ terminal velocities. Note that all above formulations for the drop’s fall velocity are valid in the standard atmosphere at the sea level. To get the drop’s fall velocity at higher altitudes, the right-hand side of the equations should be scaled by the correction factor (ρ0 /ρ)m , where ρ0 is the air density at sea level, ρ is the air density aloft and m ≈ 0.4. 3.3.2 Canting of Raindrops So far, we have assumed raindrops falling in still air with their symmetry axis along the vertical. Radio-wave-propagation studies revealed that raindrops, in addition to forming an anisotropic medium composed of highly oriented particles with their axis of symmetry close to the vertical , also produce cross-polarization between the orthogonal horizontal and vertical linear polarizations. This phenomenon led to the inference that the main symmetry axis of the raindrop may not be perfectly aligned with the vertical. The canting angle is defined as the angle between the projection of the drop’s symmetry axis on the polarization plane and the projection of the local vertical direction on the same plane. A uniform horizontal wind without a vertical gradient would not affect the orien- tation (canting) of the drop, which could still be regarded as falling within an inertial frame of reference; that is, the drop is moving at a constant speed (the terminal fall velocity) with zero net force acting upon it. Brussaard speculated that the observed canting angle of raindrops could be explained as the result of wind shear in the lower portion of the troposphere, where windspeed decreases towards the surface due to friction. In fact, approximating the horizontal wind U by a laminar flow varying with height and with fixed direction, he expressed the canting angle β as follows :   −1 sVt β = tan , (3.38) g where s = dU/dh is the vertical wind shear, and g is the gravitational acceleration. According to this simple model, the combination of two different forces acting on a raindrop of mass m (Fh = Vt sm and Fv = gm) leads the drop to assume a slanted ori- entation, with the rotational symmetry axis of the drop parallel to the relative airflow. The canting angle is also independent of the drop size because the mass m cancels https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.4 Precipitation Phases 79 out when taking the ratio Fh /Fv (eq. [3.38]). This implies that raindrops would take a preferential orientation with respect to the vertical within a given atmospheric volume which is subject to wind shear. However, Brussaard noted that the wind gradient is generally small at altitudes above 100 m, implying that the average canting angle should also be small (β < 3◦ ) at the heights typically sampled by a weather radar. A preferential average orientation in a raindrop population is actually rarely observed. Most measurements in rain can be explained by a symmetric distribution of the canting angles centered around zero. In general, regardless of the specific physical forcing, the observed canting-angle distribution of raindrops can be interpreted as a manifestation of the oscillatory behav- ior of the drops. As previously mentioned, drops tend to oscillate in two preferential modes : the axisymmetric mode oscillations between oblate and prolate shapes, and the asymmetric transverse mode (see, e.g., Fig. 3 in ). Although it has been shown that a strong coupling may exist between oscillation modes and eddy shed- ding in the drop’s wake , to this date, the physical mechanisms responsible for raindrop oscillations are still not completely understood. Whether these are related to extrinsic forcing, such as drops collisions, wind shear, and turbulence, or pro- moted by intrinsic aerodynamic forces is still to be clarified. In particular, further progress on the understanding of the nature of drop oscillation requires a clarification of the interactions between eddy shedding in the wake of the drop and the drop itself [41, 55]. More importantly for radar applications, it has been widely shown that regard- less of the underlying physics of oscillations or canting, the use of canting-angle distributions allows modeling the dual polarization observations. Numerous studies have been carried out assuming a narrow Gaussian distribution with zero mean and standard deviation in the range of 5–10◦. For example, Huang et al. analyzed the data set of artificially generated drops falling 80 m from a bridge during low-wind conditions described by Thurai and Bringi. They found that the distributions of the canting angle from the two disdrometer cameras were nearly symmetric about a mean of 0◦ , with a standard deviation of 7–8◦. A substantial agreement with these estimates is provided by the radar observations discussed by Bringi et al.. For applications such as electromagnetic scattering simulations of radar moments, it then appears reasonable to represent the canting-angle distribution assuming a Gaussian model with 0◦ mean and a standard deviation between 5◦ and 10◦. 3.4 Precipitation Phases Weather radars operating at 3–10 GHz (S, C, and X band) are generally referred to as precipitation radars, as opposed to the cloud radars operating at higher frequencies. The name originates from the sensitivity of the specific frequency range (correspond- ing to the 3- to 10-cm wavelength) to precipitation particles. Although the primary scope of weather radar is largely associated with the observation and quantification of rainfall, other nonliquid particles are routinely observed, such as ice crystals, dry and melting aggregates (snow), graupel, and hail. https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  80 Essential Precipitation Physics for Dual Polarization Radar Figure 3.10 Schematic illustration of possible paths to rain formation. The major physical interactions between cloud particles are represented with the corresponding typical trend of radar variables Zh and Zdr. The Wegener-Bergeron-Findeisen (WBF) process is a primary mechanism for ice growth in mixed-phase clouds, whereas the “both-grow” process indicates a coexistence regime (see Section 3.5.6). Clouds can be classified as liquid, glaciated (exclusively composed of ice particles), or mixed phase (containing both ice particles and supercooled water). Figure 3.10 shows a schematic representation of the main precipitation processes within a cumulus cloud. For each of the main processes, the typical trend of the polarimetric radar variables is indicated. A review of the main processes of precipitation formation and growth is provided in Section 3.5, with special emphasis on the impact on dual polar- ization radar moments. In addition to the fundamental role of temperature, the water-vapor pressure is the other important factor behind the processes of precipitation formation and growth. The water-vapor pressure (e) is the partial pressure exerted by the water vapor in thermodynamic equilibrium with its liquid (or solid) condensed phase. The amount of https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.4 Precipitation Phases 81 water vapor that can exist in a cloud decreases with temperature. A relative humidity of 100 percent is reached when the amount of water vapor is in equilibrium above a flat water surface. This is called the saturation vapor pressure with respect to water (es ). The water vapor saturation ratio is defined as e RH Sw = = , (3.39) es 100 where RH is the relative humidity. Analogously, the water-vapor saturation ratio with respect to ice is defined as e es Si = = Sw , (3.40) esi esi where esi is the saturation vapor pressure with respect to ice. The saturation vapor pressure with respect to ice is always lower than the saturation vapor pressure with respect to water (esi < es ; see Fig. 3.11) because of the stronger bonds between the adjacent molecules of the ice particles. As a consequence, the water vapor in a cloud may be saturated, at 100 percent relative humidity (Sw = 1), with respect to a water droplet, but at the same time, it would be supersaturated with respect to an ice particle (Si > 1). Figure 3.11 Phase diagram for water in the atmosphere. The solid (dashed) line indicates the saturation vapor pressure with respect to water (ice). https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  82 Essential Precipitation Physics for Dual Polarization Radar Both the saturation vapor pressure with respect to water (es ) and the saturation vapor pressure with respect to ice (esi ) increase with temperature as a result of the release of water vapor through evaporation and sublimation (Fig. 3.11). At subfreezing temperatures, for a given water vapor pressure e, when esi < e < es , the water vapor will attempt to return to equilibrium, through deposition onto the ice surface of the crystals. Deposition occurs whenever ice crystals exist in a cloud supersaturated with respect to ice, by means of water vapor diffusion. If a mixed-phase cloud is supersatu- rated also with respect to water (e > es ), then both ice crystals and supercooled drops may grow at the same time (Section 3.5.6). The phase diagram of water in Figure 3.11 provides an indication of the funda- mental phase transitions. When cloud droplets rise above the freezing level, they do not readily freeze, as confirmed by the common observation of liquid water droplets in the clouds well above the freezing level (supercooled liquid drops). In fact, the common knowledge of water freezing below 0◦ C cannot be readily applied to water in the atmosphere. The experience of water freezing below 0◦ C is valid for bulk water, whereas water in the clouds is distributed over many small droplets. This particular fragmentation of the available water in tiny spherical particles increases the resistance to freezing. Small droplets of pure water freeze spontaneously at temperatures of around −40◦ C (homogeneous freezing). However, the presence in the atmosphere of particles acting as ice nuclei provides a substrate on which water molecules can stick, thus favoring freezing at much warmer temperatures (heterogeneous freezing). Heterogeneous freezing occurs when a solid aerosol particle acts as an ice nucleus, favoring the freezing of the liquid drops at temperatures as warm as −10◦ C. Below −15◦ C , a large number of ice crystals will typically be observed. In general, the water vapor pressure in mixed-phase clouds remains close to the saturation over water. In fact, the excess water vapor would lead to the formation of new precipitation particles, with a corresponding decrease in the water-vapor pressure. The phase of the precipitation particles is a fundamental factor influencing the radar backscattered power. Figure 3.12 shows the normalized radar cross-section for water and ice at two widely used operating radar frequencies (S and X bands). Radar echoes from the upper portion of ice clouds are generally weak as a result of the combined effect of the lower water-vapor content at high altitudes and the lower dielectric factor of ice (|Ki |2 = 0.19) compared with that of water (|Kw |2 = 0.93). Typical values of reflectivity from pristine ice crystal distributions do not exceed 15 dBZ, whereas crystal aggregates may reach ∼30–35 dBZ. Higher reflectivities at subfreezing tem- perature normally require the presence of graupel or hail particles (Fig. 3.13). 3.5 Precipitation Processes Precipitation-formation mechanisms are traditionally classified as either warm-rain or cold-rain (snow) processes. Warm rain refers to clouds where all constituent particles are liquid drops. Cold-rain processes take place below 0◦ C and may involve both glaciated and liquid (supercooled) particles. The region of the cloud where liquid and https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.5 Precipitation Processes 83 Figure 3.12 Normalized radar cross-section of spherical water (solid line) and ice particles (dashed line) for X-band (10-GHz, left y-axis) and S-band (3-GHz, right y-axis) radar frequencies. In the Rayleigh region (small diameters), the backscattered power from ice particles is approximately 7 dB lower than the power scattered from water drops with the same dimension. This may affect the detectability of cloud regions dominated by small pristine ice crystals. Note the different behavior between the X band and S band for large diameters. At the X band, ice particles with a diameter above ∼2 cm (e.g., large hail) may produce a stronger return with respect to an equivalent water particle. ice particles are present simultaneously is called the mixed-phase region and may extend between the freezing level (0◦ C ) and −40◦ C , which represents the lower limit where water can exist in a supercooled state. Warm rain is prevalent in the tropical region, mainly over the oceans, whereas cold rain is most frequent at middle and high latitudes. Condensation of water vapor in the atmosphere generally occurs in a rising air mass as a consequence of adiabatic cooling. Water drops are characterized by a relatively high surface tension, which is reflected in the characteristic spherical shape of cloud droplets and small raindrops. For a droplet to form from condensation of the water https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  84 Essential Precipitation Physics for Dual Polarization Radar Figure 3.13 Illustration of the dominant ice particle types for varying ranges of reflectivity (Zh ) and ice water content (IWC). (The estimation of the IWC and snowfall rate is discussed later in Section 8.5.2.). vapor, a rather high vapor pressure is necessary to prevail over the surface-tension force. Smaller drops have greater curvature and require larger vapor pressure to keep water molecules from evaporating (curvature effect). In a pure water-vapor atmosphere, a relative humidity of several hundred percent would be needed for the water vapor to condense into a cloud drop (homogeneous nucleation). Such large values of relative humidity never occur in the atmosphere. In fact, the abundant presence of cloud condensation nuclei (CCN) makes it much easier for cloud droplets to form (heterogeneous nucleation). The CCN are suspended wettable aerosol particles, with a typical size of ∼1 μm, thus lowering the equilibrium vapor pressure of the liquid drops by decreasing the curvature effect (the particles are larger compared with droplet embryos) and through molecular-scale effects (in the case of water-soluble particles). The presence of CCN allows the formation of cloud droplets when the relative humidity barely exceeds 100 percent. Given the role of the aerosol particles in the equilibrium supersaturation ratio, it is clear how the size and composition of the CCN have a crucial role in the composition of the cloud DSD and the subsequent development of rainfall. Significant differences exist between maritime and continental clouds as a con- sequence of the different air masses where they develop. Continental air masses are generally richer in CCN with respect to marine air masses. Marine aerosols have a number concentration of the order of 103 cm−3 , whereas the concentration of continental aerosols can reach ∼105 cm−3. However, a higher percentage of maritime https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.5 Precipitation Processes 85 aerosols (10–20 percent) can serve as CCN because of the large presence of water- soluble particles like sea salts, in contrast with only ∼1 percent for continental aerosols. This peculiar difference, considering the observed similar amount of liquid water content in maritime and continental clouds, implies that maritime clouds are generally composed of fewer but larger cloud droplets in comparison with continental clouds (see panel a in fig. 3.1). Once formed, the cloud droplets can grow by diffusion of the surrounding water vapor (growth by condensation). The rate of growth is linearly dependent on the ambient supersaturation and is inversely proportional to the drops’ radius (see later eq. 3.53). The slower growth of larger droplets leads to a narrowing of the DSD, hindering the development of precipitation (fewer collisions and therefore fewer chances of coalescence). Analogously to the role of the CCN in the condensation of water droplets, ice-crystal formation in cold clouds is favored by the presence of foreign particles acting as ice nuclei, as mentioned in the Section 3.4. For ice particles, however, the formation process is more efficient than the corresponding process of cloud drops as a result of the supersaturation of water vapor with respect to ice (Fig. 3.11). Heterogeneous freezing generally does not take place until the temperature drops below −10◦ C , so it is not uncommon for warm clouds to have tops extending to subfreezing temperatures as cold as −15◦ C. Upon the formation of water droplets and ice crystals in the cloud, other processes come into play for the subsequent development of precipitation. The main mechanism for the onset of precipitation is the aggregation of particles of different sizes and fall speeds. The aggregation of water drops is referred to as the collision-coalescence process (warm rain). At subfreezing temperatures, in addition to the aggregation of different ice particles, accretion plays a major role, especially in convective systems, and refers to the capture of supercooled drops by an ice-phase precipitation particle (cold-rain process). The aggregation processes do not affect the overall water content in the cloud (unlike condensation/evaporation and deposition/sublimation) but may lead to precipitation through a redistribution of the mass between the smaller and the larger particles. 3.5.1 Collision-Coalescence In the warm-rain process, the precipitation formation occurs through the collision- coalescence of liquid drops. As reported in Table 3.1, cloud drops have size diameters up to approximately 0.1 mm and a negligible fall speed as a result of the equilibrium between the gravitational force and the frictional resistance of the air. Larger drops are called drizzle (0.1–0.25 mm) and raindrops (D > 0.25 mm). These drops have a non- negligible terminal velocity that increases with size, thus contributing to precipitation. When drops with different fall speeds are present inside a cloud, collision between pairs of drops becomes possible. Upon collision, drops may bounce off each other (no change to the DSD), break up (Section 3.5.2), or merge, leading to the formation of a new, larger drop (coalescence). https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  86 Essential Precipitation Physics for Dual Polarization Radar Collision-coalescence is the rain-formation mechanism in warm rain, prevalent in the tropics, whereas it is generally of minor importance in middle and higher latitudes. However, even if the primary rain-formation processes at higher latitudes rely on the crucial role of ice crystals, collision-coalescence still takes place in the lower portion of the cloud and can significantly reshape the raindrop size distribution below the melting layer. This reflects in a modulation of the vertical profile of precipitation and dual polarization radar variables (see examples in Section 3.5.4). The simplest method to compute the growth rate of liquid drops by collision- coalescence is based on the continuous-growth model. Considering a drop with radius R falling within a population of smaller droplets with radius r (r < R), the collision volume dV is given by the collisional area of both the large and the smaller drop, multiplied by the difference in terminal fall speed: dV = π(R + r)2 [Vt (R) − Vt (r)] , (3.41) where Vt is the drop’s terminal fall speed. For a continuous population of drops with number concentration N(r), the volume growth rate is obtained by the integration with upper limit R:  R dV 4 = π (R + r)2 r 3 N (r)E(R,r) [Vt (R) − Vt (r)] dr, (3.42) dt 3 0 where E(R,r) is the collection efficiency, that is, the product of collision efficiency (the fraction of the water drops in the path of a falling larger drop that collide with the larger drop) and coalescence efficiency (the fraction of all collisions that results in the actual merging into a single larger drop). In terms of the drop’s radius, eq. (3.42) can be rewritten as    dR π R R+r 2 3 = r N (r)E(R,r) [Vt (R) − Vt (r)] dr. (3.43) dt 3 0 R If R r and Vt (R) Vt (r), eq. (3.43) can be approximated by dR EW = Vt (R), (3.44) dt 4ρ w where E is the average collection efficiency, ρ w is the water density, and W is the liquid water content (mass per unit volume) of all the droplets with radius r < R. The variation of the drop’s radius with height is given by dR dR dt dR 1 = = , (3.45) dz dt dz dt w − Vt (R) where w is the updraft velocity. If w is negligible, then dR EW =− , (3.46) dz 4ρ w which suggests that the drop’s radius should decrease with height (i.e., increase approaching the surface). https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.5 Precipitation Processes 87 Although the continuous-growth model can reasonably reproduce the average raindrop-spectrum broadening, it is not able to explain the rapid development of precipitation from the cloud stage. Using eq. (3.45), Bowen showed the impact of the collection efficiency, cloud water content, and updraft intensity on the growth of cloud droplets by coalescence. From Bowen’s calculations, the time required for the cloud droplets to reach a millimeter-size raindrop is approximately 1 hour. This time frame is at least a factor of 2 longer than common observations of rain developing from the cumulus clouds, which takes about 15–20 minutes. This discrepancy led researchers to address the growth by coalescence as a statistical problem involving a large ensemble of discrete particles subject to random fluctuations and local variations in droplet concentration. Stochastic collision-coalescence models provide a better understanding of the process by which warm cloud precipitation develops in relatively short times. In the stochastic models (e.g., that by Telford ) the onset of precipitation is due to the discrete collisions between drops, as opposed to the continuous-growth models, and in particular to the extraordinary growth of few large “fortunate” drops that fall in a region with a high concentration of smaller droplets. Turbulent fluctuations may also increase the effective fall path of some large drops, contributing to a further broadening of the cloud DSD. The collision-coalescence process is particularly effective for raindrop size distri- butions with a large liquid water content mainly contributed by small drops. In this case, the large drops are able to grow very rapidly while falling through the population of smaller drops. 3.5.2 Drop Breakup Breakup and coalescence are the two possible outcomes of a collision between drops (drops may also bounce off upon collision, although this has no effect on the DSD). For warm rain, the combination of breakup and coalescence is the main physical mechanism driving the evolution of the DSD. Being tightly related, the two processes are in general considered together in numerical modeling, as part of the stochastic collection-breakup equation (see, e.g., Prat and Barros ). Just like collision-coalescence, breakup is a mass-redistribution process that affects the drop-size spectrum and determines the maximum size of raindrops. There are two main mechanisms for drop breakup: (1) aerodynamic breakup arising from hydrody- namic instability due to the flow around large drops and (2) collisional breakup. Col- lisional breakup has been recognized to be the most important mechanism, whereas aerodynamic breakup becomes relevant only for drops with a diameter larger than ∼3 mm. When D > 6 mm, raindrops are unstable and tend to have a short life, although undisturbed water drops can survive falling in air and reach a maximum size of ∼8 mm before breakup due to aerodynamical deformation. Collisional breakup between one large drop and one smaller drop occurs every time the collisional kinetic energy is able to overcome the surface tension, as opposed to being dissipated by the viscous motion of water molecules inside the coalesced drop. From laboratory studies, Low and List identified three main geometric https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  88 Essential Precipitation Physics for Dual Polarization Radar shapes assumed by the drops after initial contact (filaments, sheets, and disks), which are associated with specific statistical distributions of the fragment sizes. These results led to a widely used parameterization of the collision, coalescence, and breakup pro- cess in numerical cloud modeling [66, 67]. 3.5.3 Evaporation of Raindrops When the environmental air is below water-vapor saturation (RH < 100 percent), for example, when dry air is present below the cloud base, the raindrops start to evap- orate by the process of water-vapor diffusion. This is the opposite of condensation, which is water-vapor diffusion toward the raindrops. Recall, condensation allows for continuous growth in a saturated environment (RH ≥ 100 percent). From a radar point of view, evaporation is of great interest because of the well-recognizable dual polarization signatures that may show up in rain. Depending on the level of environ- mental subsaturation, very dry atmosphere below the cloud base may have RH as low as ∼30 percent. On the other hand, the relative humidity in rain may barely exceed 100 percent, and there are typically other growth processes going on, such as collision- coalescence, that have a larger impact on the raindrop size distribution. For this reason, condensation does not lead to distinct radar signatures in rain. Besides the direct impact on the DSD and precipitation evolution, it is worth noting that evaporation is also very important for its influence on the storm’s dynamics. In fact, evaporational cooling below the cloud base may enhance downdrafts, producing damaging winds, and develop a cold pool at the surface, which plays a fundamental role in the storm’s development and propagation. The rate of diffusion, that is, the rate of mass increase (condensation) or decrease (evaporation), can be approximated by the gradient of the water-vapor density around the drop of radius R : dm   = 4πRfv Dv ρv (∞) − ρv (R) , (3.47) dt where m is the mass of the drop, Dv is the diffusion coefficient for water vapor in air, fv is a ventilation factor, and ρ v is the water-vapor density (g m−3 ) in the environment (∞) and at the drop’s surface (R). In the condensation of water vapor on a drop, latent heat is released to the envir- onment, whereas during evaporation, the latent heat is absorbed by the drop. In both cases, it is assumed that the latent heat is transported by diffusion away from the droplet. This results in a balanced equation that is analogous to eq. (3.47): dm L = 4πRfv Ka [T (R) − T (∞)] , (3.48) dt where L is the latent heat of vaporization, Ka is the coefficient of thermal conductivity of air, T (∞) is the ambient temperature, and T (R) is the temperature at the drop’s surface. The ventilation factor fv in eqs. (3.47) and (3.48) is required to account for the altered diffusion of water vapor and heat for a drop falling through the surrounding air https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.5 Precipitation Processes 89. Combining eqs. (3.47) and (3.48) results in the expression for the two unknowns ρv (R) and T (R): ρ v (∞) − ρv (R) Ka = , (3.49) T (R) − T (∞) Dv L where the fraction on the right-hand side varies weakly with temperature and pressure. Considering the equation of state for an ideal gas (es = ρvs Rv T , with Rv being the gas constant for water vapor), and assuming that the vapor density at the drop’s surface can be approximated by the saturation vapor density (ρv (R) ≈ ρvs ), these combine to give a second expression relating ρ v (R) and T (R): es (T ) ρ v (R) ≈. (3.50) Rv T (R) This approximation neglects the effects of the drop’s curvature and the solution (the water drop is assumed to be sufficiently diluted), both of which are only relevant for small cloud droplets. A numerical solution for ρv (R) and T (R) from eqs. (3.49) and (3.50) eventually allows us to calculate the change in the drop’s mass by either condensation or evaporation. An alternative analytical approximation for the growth by condensation is as fol- lows : dR Sw − 1 R = fv , (3.51) dt ρ w (Fk + Fd ) where Sw = e/es is the ambient saturation ratio, and Fk and Fd are, respectively, the heat-conductivity term and the vapor-diffusion term, defined as   L L Rv T Fk = − 1 ; Fd = , (3.52) K a T Rv T Dv es (T ) where ρw is the liquid water density. Considering precipitation in calm air, the change in drop diameter for a given fall distance is dD/dH = (dD/dt)(1/Vt ), where H is the distance of fall below a given altitude. Equation (3.51) can then be written in terms of the vertical variation of the drop diameter: dD Sw − 1 Vt D = 4fv. (3.53) dH ρ w (Fk + Fd ) Equation (3.53) shows that the evaporation is particularly important for small drops, given the inverse relation between the derivative dD/dH and the drop size. For the same reason, the impact of evaporation increases as the process progresses, and the size of the drop is further reduced. The opposite happens with condensation, which is more relevant at the beginning of the process, for small cloud droplets. As the drops start to grow by condensation, the process progressively loses importance, in favor of the collision-coalescence mechanism, which is needed to produce precipitation- size drops. https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  90 Essential Precipitation Physics for Dual Polarization Radar D top 0.5 1 2 3 4 6 3.0 2.5 2.0 Height km 1.5 1.0 RH = 50% RH = 85% 0.5 0.0 0 1 2 3 4 5 6 Diameter mm Figure 3.14 Modification of the drop size with height by evaporation for different initial diameters Dtop. A uniform relative humidity of 50 percent (black solid line) and 85 percent (gray dashed line) is assumed through the 3-km layer. The temperature is set to 0◦ C at the top of the layer (3000 m) and increases downward, following a dry adiabatic lapse rate (9.8◦ C km−1 ). The impact of evaporation is illustrated in Figure 3.14, where the modification of the drop size with height is represented for several initial drop diameters and two saturation ratios (Sw = 0.5 and Sw = 0.85). The smaller drops evaporate much faster than the large drops. For the 3-km-depth atmospheric layer, drops with an initial diam- eter of 0.5 mm evaporate before reaching the surface, even with a relative humidity barely below 100 percent. On the other hand, 1-mm-sized drops may or may not reach the surface, depending on the environmental subsaturation conditions, whereas larger drops are, in general, marginally affected by evaporation. The application example in Section 3.5.4 examines in more detail how evaporation affects different initial DSDs and the associated impact on the radar variables. 3.5.4 Simplified Examples of Warm-Rain Processes Associated with Dual Polarization Radar Signatures The basic equations governing the warm-rain processes described in the previous sections can be used as a basis for simplistic models to help understand how individual rain processes may affect the radar variables. The following examples are intended to show the characteristic radar response to individual processes, rather than to simulate radar observations under realistic atmospheric conditions where multiple processes and interactions are taking place at the same time. https://doi.org/10.1017/9781108772266.004 Published online by Cambridge University Press  3.5 Precipitation Processes 91 The radar variables are evaluated according to the Rayleigh–Gans appro