1. Introduction
Earlier theoretical studies of the size spectra of precipitating particles were directed toward explaining the exponential shape of the MP spectra and the evolution of its parameters. Golovin (1963), Scott (1968), Srivastava and Passarelli (1980), and Voloshchuk (1984) determined analytical solutions for the kinetic equation of condensation and coagulation with some idealized assumptions—homogeneous kernels of the coagulation integral and some non-Maxwellian models for the condensation growth rate; some of these solutions were in the form of exponential functions (1.1). Srivastava (1971) hypothesized that a balance exists between the collision–coalescence and spontaneous breakup of raindrops, which leads to the exponential MP spectra, but the derived slopes were distinctly steeper than the observed spectra. Passarelli (1978a, b) assumed that the snow spectra are described by the MP spectrum and found an analytical expression for the slopes via integral moments by solving the stochastic collection equation without taking into account the breakup process. Passarelli’s model with exponential spectra was further developed and generalized by a number of authors (e.g., Mitchell 1994; Mitchell et al. 1996). Verlinde et al. (1990) obtained a closed form for the analytical solution to the collection growth equation for the original size spectra described by the gamma distributions in (1.2).
Subsequently, it became clear that collisional rather than spontaneous breakup may be more important in restricting drop growth and the formation of the observed raindrop exponential spectra (e.g., see the review in Pruppacher and Klett 1997, hereafter PK97). Srivastava (1978) formulated a simplified model of collisional breakup with a fixed constant number of fragments as a variable parameter and developed a parameterization for raindrop spectra in the form of a general exponential but with time-varying Λ and N0. Low and List (1982a, b, hereafter LL82) developed a complex empirical parameterization of the fragment distribution function for collisional drop breakup. The parameterization of LL82 has been used in many numerical solutions of the stochastic coalescence–breakup equation to explain the mechanism of formation of the MP spectra and their slopes (e.g., Feingold et al. 1988; Hu and Srivastava 1995; Brown 1991, 1997; McFarquhar 2004; Seifert 2005).
These numerical solutions produced somewhat different equilibrium size spectra but with common features. The spectra were characterized by the following: a small-size region from ∼200 μm to ∼2 mm consisting of several peaks with shallow troughs between them; the first peak occurring near a diameter of ∼200–300 μm; an abrupt decrease at sizes smaller than 200 μm, determining the lower limit r0 of the large-size fraction; and a region beginning at ∼2–2.5 mm that comprises the MP exponential tail. McFarquhar (2004) refined the LL82 equations and emphasized that measurement and sampling problems impose uncertainties on the solutions, motivating more detailed laboratory studies and improved parameterizations.
Previous numerical studies focused on analyzing the positions of the peaks and values of the slopes but did not attempt to approximate the entire rain spectrum by gamma distributions or to determine the index pl, which is widely used in cloud models and remote sensing of rain and snow and typically is rather arbitrarily prescribed. Parameterization of the large particle size spectrum in the form of the gamma distribution used in (1.2) has been undertaken by many empirical studies that were directed toward the determination of the three parameters of the spectra, and in particular of the index pl. Ulbrich (1983) found a correlation between the type of rain and the index pl; pl < 0 was for orographic rain, indicating broad spectra, and 0 < pl < 2 was for thunderstorm rain, indicating narrower spectra. For widespread and stratiform rain, pl was more variable but mostly positive. Willis (1984) found the best value of pl ≈ 2.5 for raindrops from two hurricanes.
More recently, another type of pl dependence was suggested, that is, a Λ–pl relation, whereby Λ was expressed as a quadratic polynomial of pl or vice versa (e.g., Zhang et al. 2001, 2003a, b; Brandes et al. 2003). The validity of this parameterization was tested in direct simulations of convective rains with the cloud models using the LL82 kernel (e.g., Seifert 2005). The Λ–pl relation allows for reduction of the number of independent parameters in (1.1) to two but the general dependence of the index pl on the rain type described by Ulbrich (1983) is still unclear. A similar relation was suggested by Heymsfield (2003) for crystalline clouds.
Previous research has revealed some fundamental properties of the size spectra of precipitating particles and has shed some light on the mechanisms of their formation. However, direct application of these findings in cloud models and remote sensing retrievals meets the following problem: owing to the complexity of collision–breakup kernels, to our knowledge, only numerical solutions of the stochastic coalescence–breakup kinetic equation have been obtained for realistic representations of the gravitational kernel. The numerical solutions require small time steps of 0.1–1 s, are rather time consuming, and do not provide simple analytical parameterizations for the indices and slopes of the exponential and gamma distributions that are needed in cloud and climate models and remote sensing retrievals.
In Khvorostyanov and Curry (1999a, b, hereafter KC99a and KC99b, respectively) and Khvorostyanov and Curry (2008, hereafter Part I), gamma distributions were derived for the small-size fraction as the solutions of the kinetic equation of stochastic condensation. The goal of this paper (Part II) is to obtain analytical solutions of the stochastic kinetic equation for precipitating cloud particles and to explain observed variations in the size spectra that can be used to parameterize the size spectra for modeling and remote sensing applications. The paper is organized as follows: In section 2, the basic stochastic kinetic equation is given, and assumptions and simplifications are described. Section 3 presents the general solutions and four particular cases are considered: the size range where fall speed is a linear function of particle size, the size range where fall speed is proportional to the square root of particle size, conditions where coagulation growth is dominant, and the subcloud layer with no small-size fraction. In section 4, a physical interpretation is given for these solutions and they are illustrated for a crystalline cloud. Conclusions are formulated in section 5.
2. Basic equation and assumptions
As in Part I, we consider either pure liquid or crystalline clouds; the mixed phase is not considered here. The entire size spectrum of the drops or crystals consists of two size fractions: small, fs(r), and large, fl(r), with the boundary radius at r = r0 between the two size fractions, where a minimum is usually observed in the size spectra composed of both fractions. Based on this minimum, the value r0 ∼ 30–50 μm can be assumed for the liquid phase (see, e.g., Fig. 4.3 in Cotton and Anthes 1989), and ∼30–80 μm (or a maximum dimension of D0 ∼ 60–160 μm) for the crystals, as is illustrated in section 4. The functions fs(r) and fl(r) correspond to the bulk categorization of the condensed phase into cloud water and rain for the liquid phase and cloud ice and snow for the crystalline phase, respectively.
A continuous collection approximation is assumed for (∂fl/∂t)col,ls. In this approximation, only the collision–coalescence between the particles of the different fractions of the spectrum, fs(r) and fl(r), is considered; that is, small particles are collected by large particles. The continuous collection approximation is usually used for evaluation of the accretion rate of the large-size fraction as in Kessler (1969) and subsequent works. If it appears in the formulation of the kinetic equations, the corresponding term (∂fl/∂t)col,ls is usually written without derivation by analogy with the Maxwellian growth, with the growth rate of individual particles (dr/dt)coag or (dm/dt)coag defined in the continuous collection approximation (e.g., Cotton and Anthes 1989; PK97). This approach is intuitively clear, but its accuracy and relation to the full Smoluchowski stochastic collection equation is not clear.
If the cloud water content of the large-size fraction is small enough, the last four terms in (2.1)—(∂fl/∂t)col,g, (∂fl/∂t)col,l, (∂fl/∂t)br,g, and (∂fl/∂t)br,l for collection and breakup—are also small and can be neglected, which is usually done in most bulk cloud models. This situation corresponds to sufficiently small concentrations Nl of the large drops, so that collisions among large particles are not frequent. With increasing Nl, water content, and rain or snow intensity, the error of this approximation may increase and these terms should be accounted for; this is especially important for convective clouds. Srivastava (1982) and Feingold et al. (1988) evaluated the coalescence and breakup terms analytically, but the collision and breakup kernels were specified to be constant (independent of radii), and the solution was expressed via Bessel functions that are difficult to analyze analytically.
Note that the assumption (2.9) is not mandatory for the analytical solutions obtained below; we simply use the fact, based on HS95, that the sum of these four terms caused by interactions only within the large fraction can be much smaller (because of mutual compensation) than the term (∂fl/∂t)col,ls caused by interactions between the small and large fractions. When the numerical models provide new information about the relative values of the terms in the coagulation–breakup equation, this assumption and its analytical solutions can be modified accordingly.
3. Solutions for the large-size fraction taking into account diffusion growth and coagulation
We seek analytical solutions for the general case and then for particular cases that include the size range where fall speed is a linear function of particle size, where fall speed is proportional to the square root of particle size, under conditions where coagulation growth is dominant, and in the subcloud layer with no small-size fraction.
a. General solution
This is the general solution to the kinetic Eq. (3.1) for the large-size fraction at r > r0, taking into account the condensation and continuous collection. For application in bulk microphysical models, the integral Jl1 can be evaluated numerically at any value of w, k, S, αl, or qls; the fall speeds υ(r) for the drops accounting for nonsphericity and for various crystal habits can be evaluated as continuous functions of r following Böhm (1989), Mitchell (1996), and Khvorostyanov and Curry (2002, 2005, hereafter KC02 and KC05, respectively).
Tabulated analytical expressions for these integrals exist only for very limited values of Bυ (Gradshteyn and Ryzhik 1994). Therefore, we will illustrate the general solution for three particular cases: 1) when υ(r) is a linear function of r, 2) when the condensation rate is much smaller than the collection rate and can be neglected, and 3) when υ(r) is proportional to r1/2.
b. Fall speed as a linear function of particle size
The equations for S < 0 are not applicable at r = R, since then
The physical meaning of the parameter R and of the solutions (3.14)–(3.16) is clear from (3.13a): (r/R)2 =
If in (3.17b) the parameter (2U + 1) < 0, then the index pl is positive and the spectra represent the typical gamma distributions. If (2U + 1) > 0, then the index pl is negative; that is, these generalized gamma spectra represent a product of the inverse power law of the Heymsfield and Platt (1984) type (hereafter, the HP spectrum) and the exponential Marshall–Palmer spectrum. Which functional type dominates depends on the combination of parameters. The condition w > αlk usually takes place in (3.17b), and the term with w mostly determines pl. As discussed in Part I, the effective w decreases with increasing scales of spatiotemporal averaging. At sufficiently large scales, w becomes small, pl tends to zero, and the size spectrum (3.17a) tends to an MP distribution. This prediction of our model coincides with the observations (e.g., Joss and Gori 1978) and statistical theories of the MP spectra (e.g., Liu 1993). However, the local spectra can be narrower than those of MP and may be better described by gamma distributions like (3.17a), as discussed in the introduction. Which type of the spectrum is preferable and what might the relationship be between βl and pl is still a subject of discussion in the literature; see section 4.
c. Quasi–power law for the terminal velocity and
d. Fall speed as υ(r) = Aυr1/2
e. Solution for subcloud layer
4. Interpretation of the solutions
In this section, a physical interpretation is given for the solutions in section 3, and the solutions are illustrated with calculations for a crystalline cloud.
a. General analysis
As an example, we can estimate from (4.1) the slope βl0 for the snow size spectra using data from Passarelli (1978b). Taking Ec ∼ 0.5–1, the snowflake bulk density ρw ∼ 0.1–0.2 g cm−3, the water content of the small-size fraction as qls ∼ 0.1 g m−3, and the thickness of the layer ∼0.5–2 km, we obtain βl0 ≈ 40–160 cm−1, or Λ = βl0/2 ≈ 20–80 cm−1. This is within the range of values of Λ = 10–100 cm−1 given by Passarelli (1978b), Houze et al. (1979), Platt (1997), and Ryan (2000).
Equation (4.1) provides an explanation for some observed peculiarities of βl0 in crystalline clouds. The analyses performed by Houze et al. (1979), Platt (1997), and Ryan (2000) show that βl0 increases by about an order of magnitude when temperature T decreases from 0° to −50°C (Fig. 3 in Platt 1997). The same analysis shows that the ice water content decreases in this temperature range, although somewhat faster (Fig. 4 in Platt 1997). According to (4.1), βl0 ∼ q−1ls, and this may explain the observed increase in βl0 with decreasing T and qls. The slower increase with T by βl0 relative to the decrease in qls may be caused by decreasing cloud thickness at lower T, that is, vertical gradients of IWC and αl in the numerator of (4.1). This temperature dependence of βl0 or Λ may cause the height dependence observed by Passarelli (1978b), who measured Λ ≈ 65 cm−1 at z = 3.35 km (T = −20°C) and Λ ≈ 24 cm−1 at z = 2.55 km (T = −12°C). This increase in Λ (or βl0) with decreasing height can also be a consequence of the qls(T) dependence.
An interesting feature of the exponential MP spectra is that the range of slopes is similar for both liquid and crystalline particles. Marshall and Palmer (1948) give an equation Λ = 41R−0.210 (cm−1), with R0 being the rainfall rate (mm h−1), yielding βl0 = 2Λ = 130–50 cm−1 for R0 = 0.1–10 mm h−1. An estimate from (4.1) with Ec ∼ 0.5, ρw ∼ 1 g cm−3, qls ∼ 1 g m−3, and αl ∼ 1 km−1 yields βl0 = 80 cm−1, which is in the middle of the range of βl0 values determined for the MP spectra, and hence (4.1) can be applicable for MP spectra in liquid clouds.
The estimates from (4.2a) and (4.2b) with typical cloud parameters show that the value of cβ1 has a smaller magnitude and is opposite in sign to cΛ1Z; hence, this relation is determined mostly by the coefficient cβ2. This is in agreement with (4.3), which predicts a nearly linear relation except for very high values of Λ, and with (4.4), where the power of Λ is 0.8 and the relation is close to linear. Thus, (4.2a) and (4.2b) predict positive correlations between pl, Λ, and the vertical velocities, and are in agreement with experimental data and parameterizations. The increases in pl and Λ with increasing w predict narrower spectra in stronger updrafts and broader spectra in downdrafts, which is similar to the effects of stochastic condensation described in KC99a,b and Part I. However, it should be emphasized that this analysis is just an illustration of possible applications of these analytical solutions and should be used with caution. The slope αl in (4.1) and the pl–Λ relation in (4.2a) and (4.2b) are based on solutions with the presence of qls. In vigorous downdrafts, in the subcloud layer, or near the surface where qls ∼ 0, the solutions from section 3d can be used. Then (3.29)–(3.32) show that the slopes βl0(r) or Λ(r) can be expressed as polynomials of pl (equal to 1 in this case). These equations and the asymptotic analysis show that Λ(r) is a 2nd-order polynomial of pl if υ(r) ≪ |w| and the slope is (3.32); such parameterization was suggested in Zhang et al. (2003a, b), and in Brandes et al. (2003), and a 3- or 2.5-order polynomial if υ(r) ≫ |w| and the slope is (3.31).
Unfortunately, data on vertical velocities, turbulence coefficient, and presence of the small-size fraction are absent in the cited papers, which precludes a more detailed comparison. A verification of the relations (4.2a) and (4.2b) would require simultaneous measurements of w, qls, qll, turbulence coefficient, and the size spectra. However, these analytical solutions are consistent with the general findings from the experimental observations: since the slopes and indices are expressed through related quantities, this leads to the existence of the pl–Λ relations. At the same time, the solutions in section 3 for various particular cases show that these relations cannot be universal, but should depend on the altitude and position of the measured spectra in clouds or below the cloud base, and specifically on the sign of w, the values of k and αl, and the presence of qls.
b. Example calculations for a crystalline cloud
The properties of snow spectra in a crystalline cloud are illustrated here in more detail. We select a generic case, chosen for illustration to mimic the profiles in similar clouds simulated in Khvorostyanov, Curry et al. (2001) and in Khvorostyanov and Sassen (2002) using a spectral bin model. The profiles for this case of qll(z) and αl(z) are shown in Figs. 1a and 1b along with the IWC of the small-size fraction qls and the ice supersaturation, which are the same as described in Fig. 1 of Part I. The temperature decreases from about −5°C at the lower boundary to −60° at 12 km.
Shown in Fig. 2 are the vertical profiles of the slopes βl0 and Λ calculated for this case from (4.1). The generalized empirically derived slope Λ for crystalline clouds from Platt (1997) is shown in Fig. 2b for comparison. The calculated slopes increase with decreasing temperature, although not linearly as predicted by the generalized experimental Λ but somewhat faster, especially above 7 km, since αl and Λ are inversely proportional to qls, which decreases upward nonlinearly at these heights. However, the general agreement of the calculated and experimental curves is fairly good, both in magnitude and vertical gradients. This indicates that if a large ensemble of values of qls and qll, measured at various temperatures, are used to calculate αl and Λ, then the results would converge to the experimental curve shown in Platt (1997).
An example of size spectra at ice subsaturation at heights of 4.8–6 km is shown in Fig. 3. The small-size fraction (Fig. 3a) was calculated with the generalized gamma distributions from section 4 of Part I. In the spectral region from 6 to about 40–50 μm, the spectra are almost linear in log–log coordinates, close to the Heymsfield–Platt inverse power laws with the indices increasing toward cloud top (to colder temperatures). At 50–130 μm, the effect of the exponential tail dominates and the spectra have a maximum at ∼100–130 μm that decreases with height.
The spectra of the large-size fraction (Fig. 3b) are calculated by using (3.18)–(3.20). The spectra plotted in log-linear coordinates are nearly linear, that is, close to the Marshall–Palmer exponents. The size-dependent slope βl(r) slightly decreases with r as predicted by the second term in (3.19), but the departure from linearity is small, and the slope is determined mostly by βl0. The composite spectra obtained by matching the small- and large-size fractions at r0 = 72 μm (Fig. 3c) are seen to be bimodal. One can see that the calculated composite spectra and the experimental spectrum from Platt (1997) shown in Fig. 3d are in good agreement, having minima and maxima at similar positions (note the difference in radii and diameter scales in horizontal axes). The experimental and calculated values of fl can be compared using the relation 1 m−4 = 10−9 L−1 μm−1; the maximum ∼1011 m−4 in Fig. 3d corresponds to ∼102 L−1 μm−1, which is comparable to the maximum in Fig. 3c. In calculations here, the first bimodality still occurs within the small-size fraction. If the matching point was located at greater r0 ∼ 120–150 μm, there would be the second region of bimodality at r0 due to the different slopes of the small- and large-size fractions; the bimodality is often observed in this region (e.g., Mitchell 1994; Mitchell et al. 1996), and polymodal spectra are also often observed (Sassen et al. 1989; Poellot et al. 1999).
The spectra in the 7.5–8.7-km layer with positive supersaturation are depicted in Fig. 4. The indices of the small-size fraction (Fig. 4a) are positive (see Part I), and the spectra are monomodal gamma distributions with maxima at r ∼ 30–50 μm. The portion of the spectra from ∼10 to 50–60 μm in log–log coordinates is almost linear; that is, it obeys the power law with indices slightly increasing with height. The large-size fraction (Fig. 4b), as seen in log-linear coordinates, represents the Marshall–Palmer distributions. The slopes are much steeper than in the lower layer and increase upward; that is, the large-size spectra also become narrower at colder temperatures; these features are mostly due to the smaller qls and the dependence βl0 ∼ q−1ls. The composite spectra matched at 90 μm (Fig. 4c) exhibit features of bimodality, but weaker than in the lower layer. Now the bimodality occurs between the large- and small-size fractions rather than within the small-size fraction as was the case in Fig. 3 at S < 0. For comparison, given in Fig. 4d are the average size spectra from Lawson et al. (2006) measured in cirrus clouds at three temperatures. One can see that the calculated spectra (Fig. 4c) are similar to the observed spectra; in particular, they exhibit similar bimodality, become narrower at lower temperatures, and the bimodality decreases and vanishes with increasing height. The reason for this is the decrease with the height of IWC of the small fraction, the slowing down of the accretion, and diminution of the large fraction. This analysis is consistent with the observations (Sassen et al. 1989; Mitchell 1994; Platt 1997; Ryan 2000; Poellot et al. 1999) that bimodality is more pronounced in the lower layers.
Note that the spectra calculated at ice sub- and supersaturation are somewhat different. The experimental spectra, however, are usually presented without information about supersaturation, and may have been obtained from mixtures sampled in both sub- and supersaturated layers. This precludes a more detailed comparison at the present time and indicates that simultaneous measurements of the size spectra and supersaturation are desirable.
Figure 5 shows the slopes βle(r) and size spectra calculated with (3.29) and (3.30) for the subsaturated subcloud layer where the small-size fraction has been evaporated. At large and small r, the behavior of βle(r) is determined by (3.31) and (3.32), respectively. The slopes rapidly increase with radius, but the rate of this growth βle(r)/dr decreases at large r, which is determined by the increasing contribution from the second term with υ(r) in (3.30). This results in a rapid decrease in fl(r) toward the larger values of r. This feature has been observed in liquid clouds (e.g., Willis 1984) and, as Fig. 5 shows, can also be pertinent for subcloud layers of crystalline clouds. Since we consider here an example with spherical particles and asymptotic υ(r) ∼ r1/2 (Bυ = ½), the asymptotic behavior of the spectra is fl ∼ exp(−cl1Aυr5/2) as described in section 3d. For some crystal habits like aggregates or plates, the power Bυ can be much smaller than ½ and closer to 0 (e.g., Mitchell 1996; KC02, KC05); then, the decrease in fl can be much slower and the tails in the subcloud layer much longer.
5. Conclusions
The stochastic kinetic equations for the size spectra of liquid and crystalline precipitating particles are solved analytically for various assumptions. These solutions and their functional dependencies are used to explain and interpret observations and empirically derived expressions for rain and snow size spectra such as the Marshall–Palmer distribution. The major results of this work are summarized below.
The general solution of the stochastic kinetic equations for the large-size fraction (precipitating particles) is characterized by the product of an exponential term and a term that is an algebraic function of radius. The argument of the exponent consists of a slope of the Marshall–Palmer type and an additional integral that depends on the condensation and accretion rates, vertical velocities, turbulence coefficient, terminal velocity, and vertical gradient of the liquid (ice) water content. The algebraic function is inversely proportional to the sum of the condensation and accretion rates and depends on the super- or subsaturation, terminal velocity, and collection efficiency.
Several practically important particular cases are considered: (a) terminal velocity as a linear function of radius, (b) terminal velocity as a square root function of radius, (c) an accretion growth rate much greater than the condensation growth rate, and (d) a subcloud evaporation layer with a very small or absent small-size fraction. The general solution is substantially simplified for these cases. The exponential part tends to the Marshall–Palmer exponent with the slope βl0, but contains additional terms that make the slope radius-dependent and nonlinear, causing the spectra to decrease with radius faster than the MP exponent as observed in many experiments. This may influence the spectral moments, for example, the radar reflectivity and the relations between reflectivity and precipitation rates. The radius dependence of the algebraic function is weaker than that of the exponent, converts for sufficiently large radii to the power law, and also allows the spectra to have the form of gamma distributions with the slope βl0 and index pl, which can be positive or negative. However, these gamma distributions are different from those obtained for the small-size fraction (cloud particles) described in Part I.
A simple expression is derived for the slope βl0 via four parameters: βl0 is proportional to the relative gradient of the liquid (ice) water content of the large-size fraction and to the water or ice density, and it is inversely proportional to the collection efficiency and liquid (ice) water content qls of the small-size fraction. All of these parameters are available in cloud-scale and large-scale models, and these dependencies provide reasonable explanations for the observed features of βl0 with variations of each parameter. In particular, the inverse dependence, βl0 ∼ q−1ls, provides an explanation of the observed strong inverse temperature dependence of βl0 since qls in general decreases with decreasing temperature.
Simple analytical expressions are also derived for the power indices pl of the gamma distributions (shape parameters), which are expressed via the coefficients of the terminal velocity, the slopes βl0, the vertical velocity, and the turbulence coefficient.
Based on these expressions for βl0 and pl, a βl0–pl relation is derived as a second-order polynomial for the case with terminal velocity proportional to radius; this relation is similar to the empirical parameterizations based on radar and disdrometer data. The coefficients of this relation are expressed via the vertical velocity, the turbulent coefficient, and the cloud liquid or ice water content.
These analytical solutions for the spectra of the large-size fraction and its parameters provide explanations for the observed dependencies of the spectra on the temperature, turbulence, vertical velocities, liquid water or ice water content, and other cloud properties. The results are illustrated with calculations for a crystalline cloud. These analytical expressions can be used for parameterization of the size spectra and their related quantities (e.g., optical properties, radar reflectivities) in bulk cloud and climate models and in remote sensing techniques. The solutions have been presented for liquid-only and ice-only size spectra. The treatment of mixed-phase clouds would require simultaneous consideration of the small- and large-size fractions of the drop and crystal spectra by accounting for their interaction (e.g., using the Findeisen–Bergeron process and transitions among the fractions) and is planned for the future work. Further work is needed to test the assumptions made in section 2 and to evaluate these expressions using observations.
Acknowledgments
This research has been supported by the Department of Energy’s Atmospheric Radiation Measurement Program and NASA’s Modeling and Parameterization Program. Paul Lawson and Brad Baker are thanked for providing the data on the experimental size spectra and useful comments. The authors are grateful to three anonymous reviewers for their useful remarks, which helped to improve the text. Jody Norman is thanked for help in preparing the manuscript.
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APPENDIX A
Derivation of Kinetic Equation in Continuous Collection Approximation
APPENDIX B
Evaluation of the Integrals in Section 3b for υ(r) = Aυr
APPENDIX C
Evaluation of the Integrals in Section 3d for υ(r) = Aυr1/2
Profiles for calculations of the spectra: (a) the IWC of the large fraction (solid circles), of the small fraction in the baseline case (open circles), and of the small fraction with doubled IWC (open diamonds); (b) parameter αl; and (c) ice supersaturation.
Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2485.1
(a) Exponential slope βl0 and (b) Λl0 of the size spectra of the large-size fraction calculated with the IWC of the small-size fraction qls (circles) and 2qls (diamonds) as described in Part I and shown in Fig. 1. (b) The generalized experimental slopes for crystalline clouds from Platt (1997) and Ryan (2000) are recalculated from the temperature to the height with an average lapse rate of 6.5 °K km−1, which is plotted with triangles.
Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2485.1
Crystal size spectra vs radius r (L−1 μm−1) at ice subsaturation at the heights (km) indicated in the legend; temperature decreases from −14.9°C at 4.8 km to −22.15°C at 6.0 km. Spectra for the (a) small fraction, (b) large fraction, (c) composite, and (d) the generalization by Platt (1997) of the observed crystal size spectra by long dimension D (m−4) in the temperature interval −25° to −30°C. Note the different normalizations of the spectra and the axes scales: in (a)–(c) by radius, and in (d) by diameter.
Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2485.1
Crystal size spectra vs radius (L−1 μm−1) at ice supersaturation at the heights indicated in the legend (a) for the small fraction extended to 200 μm, (b) for the large fraction, (c) for the composite spectra, and (d) for the experimental spectra in cirrus clouds from Lawson et al. (2006) at three indicated temperatures.
Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2485.1
(a) Slopes and (b) size spectra in the subcloud layer with zero IWC of the small-size fraction qls at the heights (km) indicated in the legend.
Citation: Journal of the Atmospheric Sciences 65, 7; 10.1175/2007JAS2485.1
