We follow the notation used in . They derived the following equation (their Eq. 7.31), which governs the evolution of the droplet radius distribution, v(r,t), subject to condensation: 1∂v(r)∂t=-∂∂rvdrdt. Here v(r)dr is the number of cloud droplets per unit mass of air with radii in the interval (r,r+dr). The condensational growth rate is dr/dt=ξ/r, where

ξ=S-1Fk+Fd, S=e/es(T) is the saturation ratio, e is the vapor pressure, es(T) is the equilibrium vapor pressure over a plane water surface at temperature T, Fk represents the thermodynamic term in the denominator that is associated with heat conduction, and Fd is the term associated with vapor diffusion . The effects of droplet curvature and solute on droplet condensational growth are usually considered to be negligible for activated droplets . However, in cloud chambers with low mean supersaturation, and therefore large droplet residence times, curvature and solute effects might become significant . Nevertheless, we neglect both effects in the governing equations but briefly address the consequences of doing so in Sect. .

To generalize this to the cloud chamber in the presence of aerosol injection (which produces new droplets at a steady rate) and sedimentation (which removes droplets that fall to the bottom of the chamber), we add two terms to Eq. () so that it becomes 2∂v(r)∂t=-∂∂rξvr-vuh+A(r), where u=k1r2 is Stokes' law droplet terminal velocity, h is the height of the chamber, and A(r) is the rate of production of (activated) droplets from the injected aerosol.

Analogous to Eq. (), the following equation governs the evolution of the squared radius distribution, w(s,t), subject to condensation: 3∂w(s)∂t=-∂∂swdsdt. Here w(s)ds is the number of cloud droplets per unit mass of air with s≡r2 in the interval (s,s+ds). The condensational growth rate is ds/dt=dr2/dt=2ξ. When this is substituted into Eq. (), the result is 4∂w(s)∂t=-2ξ∂w∂s, which has the form of the 1-D advection equation, with solution 5w(s,t)=w0(s-2ξt), where the initial condition w0(s) is an arbitrary function. The solution (Eq. ) states that the initial distribution of s=r2 simply translates at a rate 2ξ towards larger values of r2 without any change of shape.

To generalize Eq. () to the cloud chamber in the presence of aerosol injection and sedimentation, we add two terms to Eq. () so that it becomes 6∂w(s)∂t=-2ξ∂w∂s-wk1hs+B(s), where u=k1s is Stokes' law droplet terminal velocity and B(s) is the rate of production of (activated) droplets from the injected aerosol.

The probability that a droplet of radius r will fall out due to sedimentation in a small time interval Δt is (u/h)Δt=(k1r2/h)Δt. This can be derived as follows. We assume that the droplets are well-mixed, in which case the z coordinate of each droplet is a random variable. Droplets are well-mixed if the turbulent flow velocities are predominantly larger than the terminal velocities of the droplets, in which case the droplets generally move with the flow. As a fluid element approaches the bottom wall, its vertical velocity approaches zero. However, a droplet in this fluid element will continue to fall at its terminal velocity. In a small time interval Δt, the droplet will fall a distance Δz(r)=uΔt=k1r2Δt. Therefore, all droplets with z<Δz(r) will reach the bottom (“fall out”) during Δt. Because the droplets are well-mixed, a droplet's vertical coordinate z may have any value between 0 and h. Therefore, a droplet's probability of falling out during Δt is Δz(r)/h=(k1r2/h)Δt, as stated above.

derived governing equations for the distribution of r2 in the presence of supersaturation fluctuations, both with and without mean supersaturation, and in which the droplet residence time is a specified constant for all droplets, rather than depending on r2 as in Eq. (). also obtained analytical steady-state PDFs of r2 for these two governing equations.

derived analytical steady-state size distributions of rain and snow particles from a governing equation similar to Eq. () in which the rain and snow particles grow from cloud droplets by collection and are lost by precipitation. However, collection differs from growth by condensation in that collection reduces the number of particles as the particles grow. To represent both collection and precipitation realistically, included the dependence of fall speed on particle size.

In a steady state, Eq. () becomes 80=-ddrξvr-vk1hr2+A.

If the production of (activated) droplets from the injected aerosol occurs only for 0<r0<r<ra, and the loss due to sedimentation for r<ra is negligible, then we can integrate Eq. () from r=r0 to r=ra to obtain 0=-∫r0raddrξvrdr+∫r0raAdr, which becomes 0=-ξvr|r0ra+∫r0raAdr, then 0=-ξv(ra)ra-v(r0)r0+∫r0raAdr, and finally, using v(r0)=0, 9v(ra)ra=1ξ∫r0raAdr. Equation () allows us to consider the following ordinary differential equation (ODE) instead of Eq. () for ra<r<∞: 100=-ddrξvr-vk1hr2, with the boundary condition at r=ra given by Eq. (). When the supersaturation is steady and uniform, ξ is a constant, so we can write Eq. () as 110=-ddrvr-Cvr2, where C≡k1/(ξh) is a constant with units of (length)-4. The general solution to Eq. () is 12v(r)=Drexp⁡(-Cr4/4), where D is an integration constant with units of (mass)-1 (length)-2 which can be determined from v(ra)/ra, which in turn is given by Eq. (): 13D=v(ra)raexp⁡(Cra4/4).

Most of the solutions of the ordinary differential equations and integrals that appear in this study were obtained using Wolfram|Alpha .

One way to derive w(s)=w(r2) is analogous to that for v(r). Another way is to recognize that ρv=dN/dr and ρw=dN/ds=dN/dr2, where ρ is the air density and dN is the number of droplets per unit volume with r and r2 in the intervals [r,r+dr] and [r2,r2+dr2], respectively, from which we obtain dN/ρ=vdr=wdr2. Hence, 14w=vdr2/dr=v2r=D2exp⁡(-Cr4/4)=Gexp⁡(-Cs2/4), using Eq. (), s=r2, and G=D/2. Just as for the ODE (Eq. ), the corresponding solution (Eq. ) is valid only for r>ra>r0>0. Similarly, Eq. () is valid only for s>sa>s0>0.

As already noted, v(r)dr is the number of cloud droplets per unit mass of air with radii in the interval [r,r+dr]. Therefore, the number of cloud droplets per unit volume of air is 15N=ρ∫0∞v(r)dr=ρD∫0∞rexp⁡(-Cr4/4)dr=ρDπ2C, where v(r) is given by Eq. (). We see that N is related to both D and C. We can solve Eq. () for the integration constant D in Eq. (): 16D=N2Cρπ.

The number of cloud droplets per unit volume with radii larger than a is 17N(a)=ρ∫a∞v(r)dr=ρD∫a∞rexp⁡(-Cr4/4)dr=ρDπ2Cerfc(a2C/2)=Nerfc(a2C/2), where erfc(z)≡1-erf(z) is the complementary error function. From Eq. (), we obtain the fraction of the total number of droplets with radii larger than a, 18f(a)≡N(a)N=erfca2C2.

The PDF of the droplet radius distribution given by Eq. () is 19p(r)=ρv(r)N=2Cπrexp⁡(-Cr4/4). The PDF of the droplet squared radius distribution given by Eq. () is 20q(s)=ρw(s)N=Cπexp⁡(-Cs2/4). Both depend only on C. Figures  and display p(r) and q(s), respectively, for a supersaturation of 0.1 % and h=1 m.

By changing the independent variable from s≡r2 to the non-dimensional variable y≡sC/2, we obtain the non-dimensional PDF, 21Q(y)=2πexp⁡(-y2).

The median radius, r̃, is defined by ∫0r̃p(r)dr=0.5.

The cumulative density function (CDF) is the integral from 0 to R of p(r): 22I(R)≡∫0Rp(r)dr=1-f(R)=erfCR22, where f(R) is given by Eq. (). One can use Eq. () to determine C given R for any percentile I of the cumulative distribution function. In general, 23C=2R2erf-1(I). If given the median radius, r̃, then I=0.5 and C=2r̃2erf-1(0.5)≈0.953873r̃2, so that 24C≈0.909873r̃4.

We derive the mode radius, r^, by expanding the derivative in Eq. () to obtain 0=dvdr-vr+Cvr3, then applying (dv/dr)r=r^=0 and solving for r^4: 25r^4=1C=ξhk1. The relationship between the supersaturation and the mode radius for h=1 m is shown in Fig. . This plot indicates that as the supersaturation increases by 4 orders of magnitude, from 0.001 % to 10 %, the mode radius increases from about 2 to 17 µm.

By writing Eq. () in the form r^2ξ=hk1r^2, we see from Eq. () that r^ is the droplet radius for which the timescale for droplet number growth due to condensation, r2/ξ, equals the timescale for droplet number depletion due to sedimentation, h/u=h/(k1r2).

The mean radius is 26r‾=∫0∞rp(r)dr=2Cπ∫0∞r2exp⁡(-Cr4/4)dr=2πΓ(34)C1/4, which depends only on C. Solve this for C1/4 to obtain C1/4=2πΓ(34)r‾, so 27C≈0.913893r‾4. Equations () and () imply that r̃r‾≈0.998898.

The mean radius of droplets with radii larger than a is 28r‾(a)=∫a∞rp(r)dr∫a∞p(r)dr=2Cπ∫a∞r2exp⁡(-Cr4/4)drf(a)=2πΓ(34,a4C4)C1/4erfca2C2, where f(a) is the fraction of the total number of droplets with radii larger than a and Γ(b,x) is the upper incomplete gamma function. Because ∫a∞p(r)dr=N(a)N, we can use N(a)N≡f(a)=erfca2C2, from Eq. (). The upper incomplete gamma function is defined here as Γ(b,x)≡∫x∞tb-1e-tdt. Note that the MATLAB^® upper incomplete gamma function is defined differently, as 1Γ(b)∫x∞tb-1e-tdt, and is called using gammainc(x,b,'upper'); note the reversed argument order.

The mean of the squared radius is 29r2‾=∫0∞sq(s)ds=Cπ∫0∞sexp⁡(-Cs2/4)ds=2πC, which depends only on C. Solve for C: 30C=4π(r2‾)2≈1.273240(r2‾)2. Equations () and () imply that 31r2‾r‾2≈1.180341. The mean of the squared radius of droplets with radii larger than a is 32r2‾(a)=∫a2∞sq(s)ds∫a2∞q(s)ds=Cπ∫a2∞sexp⁡(-Cs2/4)dsf(a)=2πCexp⁡(-a4C/4)erfca2C2.

The mean cubed radius is 33r3‾=∫0∞r3p(r)dr=2Cπ∫0∞r4exp⁡(-Cr4/4)dr=22πΓ54C3/4, which depends only on C. Solve Eq. () for C: 34C=4π2/3Γ54r3‾4/3≈1.635767(r3‾)4/3. Equations () and () imply that 35r3‾r‾3=πΓ54Γ343≈1.547460. The mean cubed radius of droplets with radii larger than a is 36r3‾(a)=∫a∞r3p(r)dr∫a∞p(r)dr=2Cπ∫a∞r4exp⁡(-Cr4/4)drf(a)=22πΓ54,a4C4C3/4erfca2C2.

The mean r4 is 37r4‾=∫0∞r4p(r)dr=2Cπ∫0∞r5exp⁡(-Cr4/4)dr=2C, which depends only on C. Solve Eq. () for C: 38C=2r4‾.

Equations () and () imply that 39r4‾r‾4=π22Γ(34)4≈2.188440.

The mean r4 of droplets with radii larger than a is 40r4‾(a)=∫a∞r4p(r)dr∫a∞p(r)dr=2Cπ∫a∞r5exp⁡(-Cr4/4)drf(a)=2C+2a2πCexp⁡(-a4C/4)erfca2C2.

The mean r5 is 41r5‾=∫0∞r5p(r)dr=2Cπ∫0∞r6exp⁡(-Cr4/4)dr=42πΓ(7/4)C5/4, which depends only on C. Solve Eq. () for C: 42C=4π2/5Γ74r5‾4/5≈2.365245(r5‾)4/5. The mean r5 of droplets with radii larger than a is 43r5‾(a)=∫a∞r5p(r)dr∫a∞p(r)dr=2Cπ∫a∞r6exp⁡(-Cr4/4)drf(a)=42πΓ(7/4,a4C4)C5/4erfca2C2.

Because the PDF of the equilibrium droplet radius distribution, Eq. (), depends only on C≡k1/(ξh), the moments of the PDF also depend only on C. The dependence of the first five moments on C are given by Eqs. (), (), (), (), and (). Measurements of one or more moments would allow one to determine C.

However, measured DSDs are often truncated due to a lack of detectability of small cloud droplets or difficulty in differentiating unactivated aerosol particles from small cloud droplets. To deal with such DSDs, we derived the dependence of the first five moments of the droplet radius on C and the truncation radius, a. These are given by Eqs. (), (), (), (), and (). With these, one can determine C from a moment and the DSD's truncation radius.

Knowing C, one can solve for the supersaturation, S-1, given k1, h, and the thermodynamic parameter (Fk+Fd)-1. If the droplets fall at their Stokes' fall speeds, then k1 is Stokes' fall speed parameter, which is known. However, if the droplet fall speeds are affected by turbophoresis or thermophoresis, for example, then the fall speed parameter may not be known. Even if the actual fall speed parameter is unknown, it is still useful to calculate the supersaturation from C using k1 equal to Stokes' fall speed parameter. We call this the “nominal supersaturation.”

In Fig.  we plotted the mean radius, [r], mean squared radius , [r2], and mean cubed radius, [r3], versus the nominal mean supersaturation for h=1 m for DSDs with no truncation (blue) and for DSDs truncated at r=2.5 µm (red). The black dots indicate the nominal mean supersaturation values implied by [r], [r2], and [r3] obtained from 11 measured DSDs truncated at r=2.5 µm . The inferred nominal mean supersaturations range from 0.008 % to 0.6 %. The vertical green lines pass through the nominal mean supersaturation values implied by the measured [r2] values and allow a visual assessment of the consistency of the supersaturation values implied by the three measured moments for each of the 11 DSDs.

If each DSD measured in the Π chamber were determined by the mean supersaturation alone, we would expect all three of the moments from a DSD to imply the same nominal mean supersaturation. However, even if moments of the analytic PDF derived in this study are consistent with the corresponding measured moments, that would not prove that supersaturation fluctuations were absent. It could be that the effects of supersaturation fluctuations on the PDF are nearly the same as those of the mean supersaturation and are therefore difficult to discern. Or it could be that the effects are small despite the fluctuations being significant due to a low correlation between the fluctuations of supersaturation and droplet radius .

Figure  quantifies the degree of consistency of the three measured moments with the corresponding derived moments for truncation radii ranging from 0 to 3 µm. For each of the 11 DSDs, we used the supersaturation values implied by each of the three moments to calculate the mean and standard deviation of the implied supersaturation. We then calculated the average coefficient of variation of the implied supersaturation, which is plotted versus truncation radius in Fig. . The average coefficient of variation exhibits a pronounced minimum at r≈2.3 µm, which is nearly the same radius as the reported truncation radius (r=2.5 µm). Such agreement is expected if (1) the derived PDF is similar to the measured PDF and (2) the actual truncation radius is about 2.5 µm. The value of the average coefficient of variation at the truncation radius is a measure of the degree of consistency of the three measured moments with the corresponding derived moments. The value obtained (∼2.5%) could be compared to values obtained using other PDFs, such as ones that include the effects of supersaturation fluctuations.

In Fig. , the minimum value of the average coefficient of variation is less than 25 % of the no-truncation value, which demonstrates that it is essential to consider the truncation radius when comparing theoretical moments to moments from a measured but truncated DSD. Figure  in Sect.  adds further support to this conclusion.

Figure  shows that the inferred nominal mean supersaturations range from 0.008 % to 0.6 %. It is of interest to compare the range of the inferred mean supersaturations to the range of critical supersaturations for the measured injected aerosol size distributions. We noted above that for a mode diameter of 60 nm and a standard deviation of 30 nm, 99 % of the aerosol particles have a diameter less than about 170 nm. The critical supersaturation for a NaCl particle with a dry diameter of 170 nm is 0.052 %. In other words, about 1 % of the injected aerosols would be activated with a mean supersaturation of 0.052 %. What does this imply for the six DSDs in Fig.  with inferred mean supersaturations that are considerably less than 0.052 %? There are several possibilities, and they are not mutually exclusive.

Neglecting droplet curvature and solute effects in the analytic DSD governing equation produces significant underestimates of the inferred supersaturations. It could be that once curvature and solute effects are included in the droplet growth equation, the inferred mean supersaturations for all 11 measured DSDs will be large enough to activate at least the largest of the injected aerosols. To investigate this possibility, we used the droplet growth equation, both with and without the curvature and solute terms included, in the Monte Carlo model described in Sect.  to calculate mean droplet radius versus supersaturation for 100 supersaturation values (Fig. ). With the curvature and solute terms included, the equation for the droplet growth rate becomesrdrdt=(S-1)-ar+br3Fk+Fd,where -a/r is the curvature term, and b/r3 is the solute term . Figure  shows that the mean droplet radius is smaller when these terms are included, for the same fixed supersaturation. This is due to the slower initial growth of the droplets. The differences in mean radius are largest for supersaturations slightly larger than the critical supersaturation.

How do the curvature and solute terms affect the inferred supersaturation? For a given droplet radius, the inferred supersaturation is larger with solute and curvature terms included. In our specific case, Fig.  suggests that a measured DSD (r>2.5 µm only) with a mean radius of about 4.4 µm or larger could have been activated and grown with a fixed supersaturation of 0.055 %. Figure  shows that this requirement excludes the measured DSDs with the five smallest mean radii.

The standard deviation is the square root of the variance. The variance of the radius is 44σ2=∫0∞(r-r‾)2p(r)dr=2πC1-Γ342π≈0.1724016C, which is easily obtained from the identity 45σ2=r2‾-r‾2, using Eqs. () and (). The variance of the radius for droplets with radii larger than a is 46σ(a)2=r‾s(a)-r‾(a)2=2πCexp⁡(-a4C/4)f(a)-Γ34,a4C42f(a)2π.

The relative dispersion of droplet radius, σ/r‾, is obtained from Eqs. () and (): 47σr‾=2πC1-Γ342π1/22πΓ34C1/4=π1/41-Γ342π1/2Γ34≈0.4246653. The relative dispersion for a truncated DSD is obtained from Eqs. () and ().

Figure  displays the relative dispersion of the radius versus droplet number concentration, nd. The measured values of dispersion are from DSDs truncated at r=2.5 µm . The calculated values of dispersion used the average C implied by the three measured moments for each of the 11 DSDs. They were obtained by assuming either DSDs truncated at r=2.5 µm (red dots) or not truncated (black dots) and used Eqs. () and () or (), respectively, with h=1 m. The calculated relative dispersion is constant (≈0.425) for no truncation but is in good agreement with the measured values (which range from about 0.2 to about 0.4) when DSD truncation is accounted for. This is a dramatic example of the importance of considering the truncation radius when comparing theoretical moments to moments from a measured but truncated DSD. When confronted with these measurements of relative dispersion versus droplet number concentration, concluded that the results show that relative dispersion decreases monotonically with increasing droplet number density and attempted to explain the results theoretically.

The variance of the squared radius is 48σs2=∫0∞(s-r2‾)2q(s)ds=2C1-2π≈0.7267605C, which is obtained from the identity 49σs2=r4‾-(r2‾)2, using Eqs. () and (). The variance of the squared radius for droplets with radii larger than a, σs2(a), can be obtained from Eqs. () using Eqs. () and ().

The relative dispersion of the squared radius, σs/r2‾, is obtained from Eqs. () and (): 50σsr2‾=2C1-2π1/22πC=π21-2π1/2≈0.7555106. The relative dispersion of the squared radius for a truncated DSD can be obtained using Eqs. (), (), and ().

Liquid water content (g m-3), the mass of droplets per unit volume of air, is L=ρD∫0∞m(r)rexp⁡(-Cr4/4)dr. Use m(r)=ρL4/3πr3, the mass of a droplet of radius r, and Eq. () to obtain 51L=ρDρL43π∫0∞r4exp⁡(-Cr4/4)dr=ρLN82π3Γ(5/4)C3/4=ρLN43πr3‾. It is interesting that Eq. () is the same as for a monodisperse DSD with r3 replaced by r3‾.

The droplet sedimentation flux, the number of droplets that exit the chamber due to sedimentation per unit area and time, is 52Fsed=ρD∫0∞u(r)rexp⁡(-Cr4/4)dr=Nk1r2‾, where u(r)=k1r2 is the droplet terminal velocity. This result says that the droplet sedimentation flux is the same as if all droplets fell at the speed of one with the root-mean-square droplet radius.

The precipitation flux, the mass of liquid water that exits the chamber due to sedimentation per unit area and time, is P=ρ∫0∞u(r)m(r)v(r)dr=N∫0∞u(r)m(r)p(r)dr. Substitute for u(r) and m(r) to obtain 53P=Nk1ρL43π∫0∞r5p(r)dr=Nk1ρL43πr5‾.

The mean droplet residence time, τ‾, is given by Eq. (1.45) in : 54τ‾≡hNF, where F is the total droplet flux, including the fluxes due to turbophoresis and thermophoresis. We assume that F=Fsed so that 55τ‾=hNFsed=hk1r2‾=πh4k1ξ1/2. This follows from using Eq. () for Fsed and Eq. () for r2‾. The mean residence time in this case depends upon the chamber height, the Stokes fall speed coefficient, and the supersaturation. Figure  shows τ‾ versus the supersaturation for h=1 m. Figure  shows that the range of nominal mean supersaturations inferred from the measured moments is 0.008 % to 0.8 %. Figure  indicates that τ‾ decreases from about 900 to 90 s over this range of actual supersaturations.

We noted in Sect.  that the actual fall speed parameter, k′1, is unknown. However, it could be determined from measurements of τ‾, r2‾, and N by using Eqs. () and (): k1′=hNτ‾r2‾.

To derive the PDF of droplet residence times, R(τ), we start with the probability for a droplet of radius r to fall out in a small time interval, dt, which we derived in Sect. , then use r2=2ξt to obtain k1r2hdt=2k1ξhtdt≡btdt. This means that during a time interval dt, a fraction btdt of the droplets falls out. If n(t) is the number of droplets injected at t=0 that remain at time t, then dndt=-btn, which has the solution n(t)=n(0)exp⁡(-bt2/2). The distribution of droplet residence times is therefore dn(t)/dt, which we normalize to obtain the PDF of droplet residence times, 56R(τ)=bτexp⁡(-bτ2/2). We verified that the mean droplet residence time obtained from Eq. () agrees with Eq. (). Figure  displays the PDF of droplet residence times, R(τ), for 0.1 % supersaturation and h=1 m. In Fig.  (in Sect. ), we compared R(τ) from a Monte Carlo method and R(τ) from Eq. () for 0.1 % supersaturation.

To derive the condensation rate of a population of droplets, dq‾/dt (mass of water condensed per mass of dry air per unit time), start with the condensation rate for a single droplet, dmdt=ρL4πξr, where we used dr/dt=ξ/r. Then 57dq‾dt=∫0∞dmdtv(r)dr=D∫0∞ρL4πξr2exp⁡(-Cr4/4)dr=ρL4πξρN2Γ(3/4)πC1/4.

One can easily verify that dq‾/dt=P/(ρh) using P from Eq. () and r5‾ from Eq. (), along with C≡k1/(ξh) and Γ(7/4)/Γ(3/4)=3/4. An equivalent form that is not specific to a particular DSD was derived by : 58dq‾dt=Nρ∫0∞dmdtp(r)dr=NρL4πξρ∫0∞rp(r)dr=ρL4πξρNr‾. One can use Eq. () to show that Eq. () is equal to Eq. ().