a. General solutions

Toward obtaining the desired form of the analytic solution, we seek a solution of the form f(r) ∼ r^p for small values of r and a solution of the form f(r) ∼ exp(−βr) for large values of r. For small values of r, we retain in (2.3) only the terms that are most singular in r:

View Expanded

Substituting f(r) = r^p into (2.4) yields

View Expanded

Note that since p does not depend on α, the vertical gradients df/dz may influence the tail of the size spectra but not the index p. For large values of r, we retain in (2.3) only the terms that decrease slowly with increasing r:

View Expanded

Substituting f(r) = exp(−βr) into (2.6), we obtain a quadratic equation for β with the following two solutions:

View Expanded

A physical solution to (2.6) requires that β > 0. Following Landau and Lifshitz (1958), the general solution to (2.3) has the form

f_1,2rc_1,2r^pβ_1,2rψ_1,2r(2.8)

Substituting (2.8) into (2.3), applying (2.2) and (2.7), and performing simple but tiresome manipulations, we obtain an equation for ψ, which for β = β₁ is

View Expanded

Converting to the dimensionless variable x = (2αμ/G)r and dx = (2αμ/G)dr, we obtain

View Expanded

This is a confluent hypergeometric equation (or Kummer equation), the solution to which, satisfying the conditions of finiteness, is the confluent hypergeometric function, F(a, b; c) (Landau and Lifshitz 1958):

View Expanded

The full solution for the droplet spectrum is then obtained from (2.8) and (2.11) to be

View Expanded

where c₁ is a normalization factor. For β = β₂, the equation for ψ becomes

View Expanded

This is also a hypergeometric equation, with the solution

View Expanded

and the complete droplet size distribution function is obtained from (2.8) and (2.14) to be

View Expanded

The constants c₁ and c₂ for the distribution functions (2.12) and (2.15) can be determined using the integrals with Kummer functions described in Landau and Lifshitz (1958) (see appendix A). For solution (2.12), we obtain

View Expanded

where Γ(x) is the Euler gamma function, andF(α, β, γ, x) is the Gaussian hypergeometric function. For (2.15) we obtain

View Expanded

Equations (2.16) and (2.17) define c₁ and c₂ for the distribution functions, normalized to the concentration N. From appendix A, we can also determine the mean radius and water content from (A.2) and (A.3) with n = 1, 3. Alternatively, if the liquid water content is specified (say, from a model prognostic equation), then the liquid water content can be used to determine the constants c₁ and c₂.

Note that the existence of the two solutions can describe bimodal droplet spectra that are sometimes observed in clouds (e.g., Warner 1969a) and in chamber mixing experiments (e.g., Baker et al. 1984). This feature of these solutions resembles Manton’s (1979) theory. The existence, positions, and relative strengths of the two modes are determined by the values of α, μ, G. The solution imposes some limitations on the relation among these parameters. In particular, the Kummer function can become negative or grow exponentially at large values of r for some combinations of α, μ, G.

b. Asymptotic solutions

The above expressions can be used for numerical calculations of the size spectra and their moments as described in this section. However, at least two asymptotic regimes exist where these solutions can be simplified. These asymptotic formulas can be obtained using the properties of the confluent hypergeometric function described in appendix A. Using these expressions we can consider two asymptotic regimes that simplify the solutions to (2.12) and (2.15). These regimes are determined by the term x = 2αμr/G, which is the third argument of the confluent hypergeometric functions in (2.12) and (2.15).

First asymptotic regime, x = 2αμr/G ≪ 1. Using (A.6), we obtain from (2.12) and (2.15) the following two solutions:

View Expanded

These expressions are the usual gamma distributions with index p. The condition x = 2αμr/G ≪ 1 implies that α should not exceed some value, α ≪ G/2μr_max, where r_max is the maximum radius for which the condition holds. To estimate this value of α, we assume r_max = 10 μm, N ∼ 200 cm⁻³, μ ∼ 1, and G ∼ 6.4 10⁻⁹. These values give an estimation of α < 0.3 km⁻¹, or for q_L ∼ 0.8 g m⁻³, a vertical gradient dq_L/dz < 0.25 g m⁻³ km⁻¹ (from 2.2).

Thus the asymptotic formulas (2.18), (2.19) can be valid in the center of the cloud layer near the maximum value of liquid water content or in any other region where the gradient in q_L is not very large. Note, however, that when α is very small, then μ ≫ 1, and from the condition of finiteness, only the second solution (2.19) is valid. We call this regime the “internal asymptotic regime,” as it is valid in the cloud interior far from the boundaries.

Second asymptotic regime, x = 2αμr/G ≫ 1. Using the limit (A.5), we obtain from (2.12) and (2.15) the following two solutions:

View Expanded

This asymptotic condition should be valid for all radii, and we can rewrite it as 2αμr_min/G ≫ 1. This criterion is satisfied when α is large enough, that is, when q_L is small and its vertical gradient is large. These conditions are typical near cloud boundaries. We can estimate the region of validity of this regime by approximating the q_L profile as a linear function of height: q_L = q_L0z, where z is height above cloud base and q_L0, is a reference liquid water content. Then α can be simplified as α = (1/q_L)(dq_L/dz) = 1/z. That is, α decreases rapidly with height above cloud base, and we can determine the following condition for z:

View Expanded

Using the values r_min = 1–1.5 μm, μ = 0.5–1, G = 1.3 × 10⁻⁸, we obtain z < 200–300 m, while for G = 2.6 × 10⁻⁸ we obtain z < 100–150 m. So this asymptotic regime can be valid within several hundred meters of the upper or lower boundary of the cloud (as long as these levels do not include a maxima in the liquid water content). We refer to this regime as the “boundary asymptotic regime” as it is valid near cloud boundaries.

The extent of applicability of the asymptotic solution in (2.20), (2.21) is further illustrated by the following example. As the value of μ in (2.7) should be real, the following condition should be satisfied by α:

View Expanded

This condition allows α to be either positive or negative, depending on the vertical velocity profile in the cloud. A positive value of α indicates that q_L increases with height in the cloud, while a negative value of α indicates that q_L decreases with height in the cloud. The value of α would be zero at the maxima of q_L. The existence of vertical profiles of q_L that increase with height to a maximum value, then decrease with height above this maximum is commonly observed (see section 3) and may be caused by 1) the corresponding vertical profiles of the vertical velocity in frontal stratiform or convective clouds, 2) the entrainment of dry air into the cloud through the cloud top or turbulent mixing with the dry air near low and upper boundaries with subsequent mixing in the cloud, and 3) the vertical profiles of radiative cooling near the cloud top in stratiform clouds.

The solution (2.20) is the usual gamma distribution, where p̃₁ = (p/2μ)(μ + 1) is the effective index of the gamma distribution. The second solution (2.21) satisfies the condition of finiteness at large r only if μ < 1 for α > 0. For the solution (2.21), the effective index p̃₂ = (p/2μ)(μ − 1) of the gamma distribution is negative. Size spectra of the form of the inverse power law have been observed for small ice crystals (e.g., Heymsfield and Platt 1984; Ryan 1996), although this form of the size distribution is not typically observed in water clouds. Calculations such as those presented in section 2c show that the second solution (2.21) may exist in the lower cloud layer but when added to (2.20), it influences the “tail” of the spectrum and does not produce the secondary mode. Thus we examine primarily the first solution.

To calculate the dropsize spectra from (2.18)–(2.21), we need to define the normalization constants c₂ and c₁. Using (A.7) from appendix A, we can evaluate c₂ from the normalization to the concentration N. For (2.19) we have

View Expanded

For the first solution f₁ of the boundary asymptotic solution (2.20), the normalization constant is

View Expanded

Using (2.19) and (2.24) for the interior asymptotic solutions and (2.20) and (2.25) for the boundary asymptotic solutions, we can easily calculate the dropsize spectra.

Using the expressions for the moments from appendix A, we can write the following expression for the liquid water content q_L and the relative dispersion of the spectrum σ_r:

View Expanded

c. Solutions with account for sedimentation and generalized gamma distributions

It can be shown that sedimentation influences the tail of the size spectrum. Accounting for sedimentation under the same assumptions as used in (2.3), the kinetic equation can written in the form

View Expanded

where α_l is the vertical gradient of the larger (sedimenting) droplets. We assume as usual that terminal velocity can be parameterized in the form

υrbr^ν(2.28)

Note that the solution for the small radii, ∼r^p, is the same as it was for (2.3) since the most singular terms by r do not depend on υ(r), thus the index p is determined by (2.5). Terminal velocity for droplets of r ∼ 10 μm is about 1 cm s⁻¹, and υ(r) ∼ 5 cm s⁻¹ for the droplets with r = 20 μm. Thus, sedimentation becomes essential for the droplets which are far enough to the right from the modal radius and have large radii. Hence, we seek an asymptotic solution to (2.28) and retain only the main terms with the slowest decrease with r. Then we can omit the terms with inverse powers of r and to rewrite (2.27) as

View Expanded

We seek a solution of the form

frβr^λ(2.30)

which is typical of generalized gamma distributions. Substituting the relations

View Expanded

into (2.29) after simple manipulations we obtain the following relation:

View Expanded

If λ > 1, then 2(λ − 1) = 2λ − 2 > λ − 2, and 2(λ − 1) > λ − 1. Hence we retain in (2.32) at large r only the main powers of r, the largest terms are the first and fifth and we have

G²kβλ²r^2(λ−1)α_lbr^υ(2.33)

This equation can be satisfied only if the power law indices and coefficients are equal, thus we obtain two conditions:

View Expanded

The first condition relates the power law of the generalized gamma distribution with the power law of the terminal velocity. In particular, for the Stokes particles when υ(r) ∼ r², it follows from (2.34) that υ = 2, λ = 2, and f(r) ∼ exp(−βr²). For the larger droplets with r ∼ 100–500 μm, when υ(r) ∼ r (e.g., Pruppacher and Klett 1997) and υ = 1, we obtain λ = 3/2 and f(r) ∼ exp(−βr^3/2). For the next region of the larger droplets of precipitation when terminal velocity υ(r) ∼ r^1/2 (500 μm < r < 2.5 mm), we have λ = 5/4 and f(r) ∼ exp(−βr^5/4). So the relation (2.34) for λ has clear physical meaning: the slower increase in terminal velocity with radius (the smaller power law) is associated with slower decrease of size spectrum with radius and the slower process of the droplet washout through precipitation.

a. Interpretation of the solution parameters

As shown in the previous section, the index of the gamma distribution, p, is determined from (2.5) for the first solutions (2.18) and (2.19) in the cloud interior (near the maximum liquid water content) or for the case when μ ∼ 1. The index of the second solution in the asymptotic regime (2.20) near the cloud boundaries is determined by the equation

View Expanded

with p defined by (2.5) and μ defined by (2.7). Note that p̃₁ = p for μ = 1.

To interpret the gamma-distribution solutions, we first analyze the index p and the parameter G, and then conduct a more complete analysis of the indices p̃₁ and p̃₁. The parameters G in (2.1a) and p in (2.5) can be written in the following form:

View Expanded

where k is the eddy diffusion coefficient and we have introduced the vertical gradient of the adiabatic liquid water content

View Expanded

As was discussed in section 4 of Part I, the effective diffusion coefficient in the space of radii is k_r = Gk. Equation (3.2) for G shows that 1) the diffusion coefficient k_r is proportional to the mean radius r, which thereby is a “mixing length in the space of radii” (similar to the turbulent coefficient k proportional to the mixing length); and 2) at any given height z, k_r is inversely proportional to the liquid water ratio χ (the ratio of q_L to the adiabatic value of q_L).

Equation (3.3) for the index p also has a clear physical meaning. The indices p (and p̃₁) are expressed in (3.3) as the product of the three factors: 1) dynamical, w_ef/k;2) adiabatic, χ = q_L/q_L,ad; and 3) height, z. The first factor shows that the index p increases as the vertical velocity increases, narrowing the size spectrum, and p decreases as the turbulence coefficient increases, broadening the spectra. Smaller values of the adiabatic liquid water ratio χ are associated with smaller values of p and wider spectra. This is important for the interpretation of the vertical dependence of the drop spectral broadening, since in most cases χ decreases with height above cloud base but increases with cloud width (Pruppacher and Klett 1997; see Fig. 2-22). The adiabatic factor causes decrease of p and broadening of size spectra with increasing height; in particular, it explains the spectral broadening observed by Warner (1969a) and many others. The third factor in (3.3), z, causes an increase in p with height and narrowing of the spectra. Equation (3.3) provides a very simple tool for estimation of the vertical dependence of the spectral breadth from measured or calculated profiles of q_L, r, w_ef, k, and thermodynamic parameters in a cloud.

The relationships described above are consistent with observations in clouds (e.g., Mason 1971; Pruppacher and Klett 1997; Cotton and Anthes 1989). Thus, (3.3) and (3.1) establish an analytical relationship between the index p, and hence drop spectra dispersion, on the meteorological characteristics of the atmosphere.

Consider the following atmospheric conditions typical of stratus clouds: γ_s = 6 K km⁻¹, k = 5 m² s⁻¹, and w_ef = 1–10 cm s⁻¹. As discussed in Part I, the value of w_ef can be estimated from the relation w_ef ∼ w′q^′_L/q_L. The values of the flux w′q^′_L measured in Arctic stratus presented in Curry (1986) are always positive (except near cloud base). For the upper deck observed on 28 June 1980, the value of w′q^′_L was determined to be 25–55 × 10⁻⁴ g m⁻³ m s⁻¹ at z = 818 m with corresponding q_L ∼ 0.12–0.2 g m⁻³. This gives an estimate w_ef ∼ 12–20 cm s⁻¹. Note that for the same deck, the covariances w′w′ ∼ 0.1–0.2 m² s⁻¹ (Curry et al. 1988), that is, w′ ∼ (w′w′)^1/2 ∼ 33–45 cm s⁻¹, which is larger than the previous estimate for w_ef. This is in agreement with the discussion of the covariances in Part I, which showed that the correlation function w′f′ or w′S′ due to the phase shift between them by τ_f should be smaller than the autocovariance w′w′; here w_ef should be smaller than w′. However, w_ef should be still much greater than the synoptic-scale average w_syn ∼ 0.5–1 cm s⁻¹.

The values of parameters p and G calculated from (3.2) and (3.3) are presented in Tables 1 and 2. Table 2 shows that the values of p for the “interior cloud” solution (2.18) and (2.19) range from 2 to 8 for w_ef = 10 cm s⁻¹ in a cloud with mean radius r = 5 μm and droplet concentration N = 1–4 × 10² cm⁻³, typical for continental, summertime Arctic stratus clouds (Curry 1986) or some types of marine clouds near the coast (Noonkester 1984). This value of p corresponds to dispersions of the spectra σ_r ∼ 0.33–0.58 (Table 3), which is close to values observed in the central layers of such clouds. For values of r = 10 μm, typical of marine stratus clouds, the values of p increase; however, since typical drop concentrations are usually N = 1–2 10² cm⁻³ for these clouds, values of p would not normally exceed p = 8–16 for this cloud type and would be less for smaller values of w_ef. Thus, from Table 2 we see that the spectra narrows with increasing droplet concentration and mean radius. If the value of the turbulence coefficient is k = 10 m² s⁻¹ or the maximum effective vertical velocity is 5 cm s⁻¹, the values of p in Table 2 would be twice as small.

These tables show that for w_ef = 1–2 cm s⁻¹, as could be evaluated from the large-scale divergence (i.e., with the averaging over the large areas), the values of p are too small (p < 1 for r = 5 μm), and the dispersions are too large compared to the local observations. The problem of correspondence of the local size spectra and those averaged over the large cloud areas has been widely discussed in the literature, and it was found that the scale of averaging influences the index p: the local (measured over the horizontal scales L_x ∼ 30–100 m) spectra are usually narrow, with p = 6, σ_r = 0.38 to p = 10, σ_r = 0.30 (e.g., Aleksandrov and Yudin 1979; Curry 1986). However, the dispersions increase and the indices p decrease to p ∼ 1–2 (σ_r = 0.58–0.7) with increasing scale of averaging, (L_x ∼ a few kilometers) or with averaging over many clouds (e.g., Mason 1971; Sedunov 1974). This effect of averaging is clearly seen from the data by Noonkester (1984) for marine stratocumulus shown in Table 4, which illustrates the increase in σ_r (artificial “broadening” of the spectra) with averaging over the scale 6.44 km (σ_r,all) as compared to the local spectra averaged over 0.529 km (σ_r,loc). This impact of averaging on the width of the size spectra is described by (3.3) since p ∼ w_ef, which decreases with increasing scale of averaging. Note, however, that the cloud optical and radiative properties and the rate of precipitation formation are determined by the local (L_x ∼ 50–100 m) size spectra, hence the local values p = 6–10 with σ_r = 0.3–0.4 are representative and should be used for the parameterizations.

Since the large-scale models with the horizontal grid spacing ∼100–400 km produce w_ef ⩽ 1–2 cm s⁻¹ we address the problem of the subgrid parameterization of w_ef. One possible solution to this problem was suggested by Dmitrieva-Arrago and Akimov (1998) in a general circulation model with prognostic equations for q_L and dewpoint deficit, and a prescribed value of p. The subgrid value of w_ef for stratiform clouds is defined in this GCM as the minimum velocity required to support the falling droplets with radii smaller than some threshold value, say, 40 μm. Assuming Stokes’s velocities gives w_ef ≈ 20 cm s⁻¹, which is comparable to the estimation above from the covariances. With k ∼ 10 m² s⁻¹, values in Table 2 would give reasonable values of p ∼ 5–10, so the method developed here can be used in such GCMs with parameterizations of subgrid w_ef similar to Dmitrieva-Arrago and Akimov (1998).

An important feature of the solutions for p obtained here is that although (3.3) p includes many parameters that vary by several orders of magnitude, the expression for p leads to a dimensionless quantity of order 1–10 for a wide variety of stratus cloud types, and the relative dispersion is related now directly to the meteorological factors (w, γ_d, k), the properties of the cloud (N, r), and fundamental atmospheric constants.

For convective clouds, typical values of vertical velocity are w ∼ 2.0–5.0 m s⁻¹; that is, 20–30 times larger than values used in Table 2 for stratiform clouds. Although the values of the turbulence coefficient for cumulus can be about 10–20 times larger than those for stratus clouds, this shows that the values of p should be larger and spectra narrower in cumulus, which is expected since the processes in the vigorous updrafts are closer to the regular condensation.

An example is given using data presented by Austin et al. (1985) of microphysical measurements in shallow convective clouds obtained simultaneously at two levels by two aircrafts separated by 800 m in height. The size spectra at lower level in the middle of the cloud have the r = 7–8 μm, σ_abs = 0.6–1.0 μm, σ_r = 0.2–0.3, and p = 13–25. At the higher level, near the top, r = 8–8.5 μm, σ_abs = 0.5–0.7 μm, σ_r = 0.17–0.21, and p = 22–35. So the size spectra narrows with height, in contrast to the convective cloud observed by Warner (1969a). A similar estimation can be obtained for convective clouds; assuming w_ef = 2 m s⁻¹, k = 30 m² s⁻¹, χ = 0.7, and z = 700 m, we obtain p =35, σ_r = 0.17. While this is a very rough estimation, it shows that this approach gives the correct order of magnitude for convective clouds.

Equation (3.3) shows that the size spectra may become broader or narrower with increasing height, depending on the behavior of ζ = (w_ef/k)χ. If ζ decreases with height faster than z, say as z^−3/2, we obtain p ∼ zζ ∼ z^−1/2, so p decreases and dispersion increases with height, in agreement with observations by Warner (1969a) near cloud bottom. For ζ decreasing with height slower than z, such as closer to cloud tops, the situation is reversed, that is, p increases and dispersions decrease with height.

Note that the entrainment processes in cumulus or near the tops of stratocumulus can be easily accounted for in this approach by addition into the right-hand side of the kinetic equations (2.1), (2.3) of the term −τ⁻¹_m × f, with τ_m being the characteristic mixing time. Then the two solutions (2.7) for β_1,2 are modified,

View Expanded

and can give the bimodal distributions. Equation (3.4a) shows that with decreasing mixing time, τ⁻¹_m increases and leads to the larger separation of the modes and to broader spectra. Recall that one of Manton’s (1979) results was that the size spectra of each single mode was too narrow and only their bimodal superposition could ensure the dispersion growing with height at a rate comparable to the measured values (Warner 1969a). A bimodal superposition of the gamma distributions considered here may also provide sufficient dispersion for the case of cumulus clouds.

Equation (3.3) for p shows also that there is a temperature dependence of the drop size spectra. Liquid water clouds can occur over a large range of temperatures, particularly since supercooled liquid clouds are observed at temperatures as low as −35° to −39°C (Pruppacher and Klett 1997). According to (3.3), drop size spectra narrow at colder temperatures. When the temperature decreases, the value of the moist adiabatic lapse rate γ_s approaches the dry adiabat γ_d; therefore the difference (γ_d − γ_s) in the denominator of (3.3) decreases and p increases. Hence, the size spectra narrows with decreasing temperature. For example, for a low-level cloud at T = −20°C, the difference (γ_d − γ_s) ≈ 1.5 K km⁻¹ instead (γ_d − γ_s) ≈ 4 K km⁻¹ at T = 0°C, so the values of p in Table 2 should be 2.7 times larger for T = −20°C. This increase may be partially compensated by a decrease in w_ef.

This temperature dependence explains narrowing of the spectra with decreasing temperature observed in Arctic summertime clouds (Herman and Curry 1984; Tsay and Jayaweera 1984). In the summertime Arctic altostratus (T = −12° to −15°C), r = 3.8 μm and r₉₅ = 9 μm, while in stratus (T = 1°–6°C), r = 4.5–6 μm, and r₉₅ = 12–18 μm, where r₉₅ is the right boundary of the interval containing 95% of the droplets. This narrowing of the spectra with decreasing temperature leads to both the relative colloidal stability of upper clouds (as narrowing of the spectra prevents coagulation) and the decreasing probability of droplet freezing (which is proportional to the drop volume).

b. Applications of the solution

To illustrate the analytical solutions and evolution of the droplet spectra below the maximum in q_L, we consider the following characteristics typical of low stratus clouds and close to the Arctic stratus observed on 28 June 1980, upper layer (Curry 1986). The vertical profile of q_L is approximated by a parabolic profile of the form q_L(z) ∼ q_L,max × (z/H)^a × (1 − z/H)^b, with H being the cloud depth; q_L and the parameter α are presented in Fig. 1. Cloud boundaries are at 700 and 1100 m and the maximum value q_L,max = 0.4 g m⁻³ is located at ∼1000 m. The vertical profile of the mean radius is chosen such that r = 2.5 μm at the cloud bottom and linearly increases to r = 6 μm at the cloud top. The vertical profile of the droplet concentration N calculated using these values of q_L and r is also presented in Fig. 1 and exhibits the fastest increase above the base with much slower increase in midcloud. Such profiles are also in agreement with the results of large eddy simulations of stratocumulus (Feingold et al. 1994), averaged over the domain.

To investigate the effect of vertical velocity, turbulence, and radiation on the drop size spectra, we examine four different cases, using the definition of effective vertical velocity w_ef = w_rad + w_dyn derived in Part I, section 2a. The four cases are the following: 1) w_rad is determined by accounting for the longwave cooling rate only, w_dyn = 1 cm s⁻¹, and k = 5 m² s⁻¹; 2) w_rad is determined by accounting for longwave cooling only, w_dyn = 10 cm s⁻¹, and k = 5 m² s⁻¹; 3) w_rad = 0, w_dyn = 15 cm s⁻¹, and k = 5 m² s⁻¹; and 4) w_rad is determined by accounting for longwave cooling only, w_dyn = 10 cm s⁻¹, and k = 10 m² s⁻¹. These variations allow examination of the individual effects of w, k, and radiation on the drop size spectra.

Radiative heating rates are calculated as described in Khvorostyanov (1995) using input variables consistent with the observed Arctic stratus clouds from Fig. 1. Calculated profiles of radiative heating rates presented in Fig. 2 show maximum longwave cooling of −120 K day⁻¹ and solar heating of ∼40 K day⁻¹, which is in a good agreement with those determined from measurements of this cloud deck (Herman and Curry 1984; Curry 1986). Corresponding radiative-effective vertical velocities (Fig. 3) exhibit strong maxima at heights 50–100 m below cloud top, equivalent to updrafts of 5–8 cm s⁻¹ for the longwave cooling and downdrafts of −3 cm s⁻¹ for solar heating, and almost vanish in the lower half of the cloud, where the dynamic velocity plays the major role. The value of parameter α (Fig. 1) shows a strong maximum near cloud base, sharply decreases upward in the lowest 50 m of the cloud, then continues to decrease much more slowly until it reaches zero at a height of 1000 m and becomes negative above this level. The values of parameter μ calculated from (2.7) and shown in Fig. 4 exhibit a monotonic increase upward from about 0.2–0.6 near cloud bottom to values μ = 4–5.5 within 200–300 m above cloud base, with values of μ significantly increasing above 900 m where radiative-effective velocities and their gradients increase. The indices p of the general solutions (Fig. 5) for the case with w_dyn = 1 cm s⁻¹ (synoptic scale) do not exceed 1–2 in the whole cloud but reach 20 in the thin layer of maximum radiative cooling near 1050 m. The values of p for w_dyn = 10–15 cm s⁻¹ are much greater and closer to the local observations. They also exhibit an increase from the cloud bottom with the values 5–15 in midcloud depth, reach maximum values of 20–35 near the maximum q_L, and then decrease upward toward the cloud top. For the cases with w_rad > 0, this increase is especially rapid near the maximum q_L where w_rad is maximum.

The indices p̃₁ of the gamma distribution for the first solution (2.20) calculated with (3.1) (Fig. 6) also increase upward from the cloud bottom but more slowly than the indices p. This is explained by the behavior of the parameter μ, which is greater for the larger vertical velocities and smaller turbulence coefficient; that is, μ shows the opposite behavior from parameter p in (3.3), thereby partially counteracting the effects of vertical velocities and turbulence on p̃₁ and size spectra as seen from (3.1). As discussed in section 2, the asymptotic solutions (2.20) with indices p̃₁ are valid only in the lower layer of ∼100–200 m, while the larger indices p (Fig. 5) are more representative of the spectral dispersions in the upper half of the cloud. It is seen from Figs. 5 and 6 that values of both p and p̃₁ increase with increasing vertical velocity and decrease with increasing turbulence.

An example of the size spectra in the lower part of the cloud calculated with (2.20) is presented in Fig. 7. The shape and the general behavior of the spectra are in reasonable agreement with those typically observed in clouds. The calculated size spectra can be compared to those measured on 28 June 1980 in stratus (Herman and Curry 1984, their Table 2). There were five aircraft passes through this cloud deck at various altitudes. Although the measured spectra are slightly different at various times, they exhibit some common features. 1) In lower 100–150 m, the spectra are monomodal, with distinct maxima in the size bins at 1.6 and 3.1 μm, and rapidly decrease in the bins 4.7–6.3 μm. 2) At higher altitudes, the maxima in the first bins become smaller but still exist, and the new maxima occur in the bins 6.3–9.4 μm with the minimum at 4.7 μm, so the mean radius grows with height and the spectra exhibit features of bimodality. We see that calculated spectra (Fig. 7) generally reproduce the shape of the measured spectra:(i) the fast increase of f(r) in the region from 0 to 1–2 μm caused by the high value p ∼ 5–10, (ii) slower exponential tail, (iii) displacement of the maximum to the lager radii with height, and (iv) simultaneous decrease of the relative dispersion σ_r upward (see Table 4 for the measured σ_r and Figs. 8 and 9 for calculated σ_r). The feature of bimodality is not reproduced in Fig. 7, as accounted for by only one mode in the analytical solutions. This bimodality is probably caused by the mixing with the entrained dry air from above and can be reproduced in this model by using a superposition of the solutions with the two exponents β_1,2 in (3.4a) with appropriate mixing time τ_m.

c. Comparison with some previous theories and observations

The theory of stochastic condensation and the stochastic kinetic equations have been criticized by Warner (1969b), Bartlett and Jonas (1972), and others, who modeled the impact of turbulence on the size spectrum as random fluctuations of a cloud parcel using Lagrangian parcel models. They found that turbulence within a cloud is unable to account for the observed drop size distributions. It was noted by Stepanov (1975) and Voloschuk and Sedunov (1977) that the kinetic equation under some conditions does not cause further broadening of the spectra with time. We consider here in more detail the correspondence between the approach based on the kinetic equation and the Lagrangian parcel models. We will show that there is no contradiction between these two approaches and that Warner (1969b) and Bartlett and Jonas (1972) considered a particular case of stochastic condensation.

As we see from the low-frequency form of the kinetic equation (2.1), the effect of turbulence vanishes when the right-hand side of the equation is zero, and (2.1) converts into the equation of regular condensation (2.1). The right-hand side of (2.1) can be zero if either

View Expanded

Thus, condition (3.6) can be regarded as equivalent to the condition of zero turbulence (3.5). Equation (3.6) can be rewritten as

View Expanded

Multiplying (3.7) by (4π/3)ρ_wr³ and integrating, we obtain

View Expanded

The last approximate relation is based on the fact that r² is usually only slightly (by 10%–20%) larger than r².

To interpret (3.8), we express G in terms of the vertical gradient of the adiabatic liquid water content defined by (3.4). Substituting (3.4) into the first of the relations (3.2), we obtain

View Expanded

Thus, the parameter G relates the vertical gradient of the adiabatic liquid water content to the effective area of the droplets per unit volume; that is, G characterizes the rate of absorption by the droplets of vapor excess released during adiabatic lifting. Substituting (3.9) for G into (3.8) and incorporating (2.2), we obtain the following relation:

View Expanded

Equation (3.10) shows that the condition (3.6) corresponds to the adiabatic q_L or, as shown by (3.5) and (3.6), this is equivalent to k = 0. So we see that turbulence does not influence the drop size spectrum if the vertical profile of q_L corresponds to the adiabatic value.

In the simulations made by Warner (1969b) and Bartlett and Jonas (1972) using Lagrangian parcel models, moist adiabatic ascent was assumed and adiabatic liquid water contents were derived. This corresponds to the absence of turbulence, therefore small broadening of the spectra was obtained in these parcel models (probably due to the numerical dispersion). Hence, there is no contradiction between the treatment of stochastic condensation process as described here and the Lagrangian parcel models, which correspond to the condition of equilibrium (3.6). This condition along with the physical meaning of parameter G as described by (3.9) means that any vertical displacement in the cloud is accompanied by a change in the dropsize spectrum. Displacements of parcels in regions of the clouds that have adiabatic liquid water profiles will not lead to the broadening of the drop size spectra, and the droplet radius will be a function of the height and the cloud will be in some equilibrium state. However, liquid water content in a cloud is typically much less than its adiabatic value and condition (3.10) is not satisfied. This departure of the cloud liquid water content from the adiabatic profile increasing upward will result in the evolution of the size spectra according to the kinetic equation of stochastic condensation. Since the ratio χ = q_L/q_L,ad usually decreases upward, the applicability of the parcel models without entrainment and with adiabatic gradients becomes more limited with increasing height above cloud base.

Considine and Curry (1996) used a simple statistical model to derive a drop size spectra that has the shape of a modified gamma distribution. As such, the p index is not directly comparable to the drop size spectra developed here. However, the parameters of Considine and Curry’s drop size distribution are vertical velocity variance, temperature, pressure, the lapse rate in the cloud, and the lapse rate of a rising parcel. These parameters include essentially the same meteorological variables used in the drop size spectra developed here. Considine and Curry’s results implied that if the lapse rate within the cloud is equal to the saturated adiabatic lapse rate, then the spectrum becomes infinitely narrow, approaching a monomodal distribution. This is consistent with our result that a cloud with adiabatic liquid water content will have an infinitely narrow spectrum and the spectrum will not be influenced by turbulent motions. Considine and Curry also found that the drop size spectra broadens as the vertical velocity variance increases, which is consistent with our results. This general agreement with Considine and Curry supports some of the assumptions used in their derivation. However, the present derivation provides a sounder theoretical justification for some of these assumptions. The drop size distribution developed here is easier to apply, since that derived by Considine and Curry requires knowledge of the parcel lapse rate that depends on the supersaturation ratio (which is not easily determined).

One of the most important quantities that characterizes the effect of turbulence on the size spectra is the relative dispersion σ_r = (1 + p)^−1/2. Its vertical profiles calculated with the values of p̃₁ from Fig. 6 are given in Fig. 8 for the four cases considered above. The dispersion for the cases with w_dyn = 10–15 cm s⁻¹ decreases from 0.45–0.6 above cloud base to 0.3–0.4; that is, the size spectra in low stratus are narrowing with height above cloud base. These values are in good agreement with many observations, but the dispersions with w_dyn = 1 cm s⁻¹ are 0.55–0.85, much larger than those observed locally. Note that the effect of decrease of σ_r upward is opposite to that observed above the base in convective clouds where dispersions increase upward (Warner 1969a); however, it is in a good agreement with the observation in stratus clouds (Noonkester 1984; Curry 1986) and in the upper halves of some convective clouds (e.g., Austin et al. 1985). The data from Table 4 show that the relative dispersions have maxima near cloud top and bottom and minimum in the middle cloud, near the maximum of q_L. The calculated values of σ_r are similar to the measured values shown in Table 4.

The indices p̃₁ corresponding to the asymptotic solution (2.20) characterize the breadth of the size spectra only in the lower layer, while the indices p can serve as a measure of the spectral breadth in the whole cloud. The dispersions calculated with the indices p from Fig. 5 are shown in Fig. 9. These dispersions exhibit minima of 0.2–0.3 near maximum values of q_L, and asymmetry of σ_r in vertical with larger dispersions near the base than in the upper layer, in agreement with observations shown in Table 4.

Hence, the analytical solutions derived in section 2 provide an explanation of the observed dispersions in clouds along with their possible different behavior with height. We have to emphasize that these are only approximate estimations of the dispersions based on several simplifications. The more accurate evaluation of σ_r should be based on the numerical solution of the complete kinetic equation with more realistic account for the profiles of k(z), gradients ∂f/∂z, etc.

The technique described above can be used to study covariances of moments of the distribution function with various quantities as was done based on observations by Curry (1986). The simplest is the covariance of liquid water content, q_L, with vertical velocity. Using (3.24) (see Part I) for u^′_if′ in the low-frequency approximation (kⁿ_ij = k_ij) and the assumption of horizontal homogeneity (only the term ∂/∂z is kept), multiplying it by (4π/3)ρ_wr³, and integrating over radii, we obtain

View Expanded

The last integral in (3.11) is equal to −3Nr². Including the expression (3.9) for G, the covariance (3.11) is written as

View Expanded

The expression (3.12) shows that covariance of q_L with vertical velocity should be positive for the cases when adiabatic gradient of q_L is greater than the vertical gradient of q_L and is negative for the reverse situation. The first situation is met almost always, which causes positive correlation. In particular, this positive correlation of q_L and vertical velocity was described by Curry (1986) based on observations in Arctic stratus, which means that vertical gradients of q_L were less than adiabatic, in accord with the present theory.