In a laboratory cloud chamber that is undergoing Rayleigh–Bénard convection, supersaturation is produced by isobaric mixing. When aerosols (cloud condensation nuclei) are injected into the chamber at a constant rate, and the rate of droplet activation is balanced by the rate of droplet loss, an equilibrium droplet size distribution (DSD) can be achieved. We derived analytic equilibrium DSDs and probability density functions (PDFs) of droplet radius and squared radius for conditions that could occur in such a turbulent cloud chamber when there is uniform supersaturation. We neglected the effects of droplet curvature and solute on the droplet growth rate. The loss rate due to fallout that we used assumes that (1) the droplets are well-mixed by turbulence, (2) when a droplet becomes sufficiently close to the lower boundary, the droplet's terminal velocity determines its probability of fallout per unit time, and (3) a droplet's terminal velocity follows Stokes' law (so it is proportional to its radius squared). Given the chamber height, the analytic PDF is determined by the mean supersaturation alone. From the expression for the PDF of the radius, we obtained analytic expressions for the first five moments of the radius, including moments for truncated DSDs. We used statistics from a set of measured DSDs to check for consistency with the analytic PDF. We found consistency between the theoretical and measured moments, but only when the truncation radius of the measured DSDs was taken into account. This consistency allows us to infer the mean supersaturations that would produce the measured PDFs in the absence of supersaturation fluctuations. We found that accounting for the truncation radius of the measured DSDs is particularly important when comparing the theoretical and measured relative dispersions of the droplet radius. We also included some additional quantities derived from the analytic DSD: droplet sedimentation flux, precipitation flux, and condensation rate.

In a laboratory cloud chamber, such as the

Although the resulting equilibrium droplet size distributions (DSDs) have been extensively measured in the

In this study, we assume that (1) droplets grow subject to a uniform mean supersaturation, (2) the effects of droplet curvature and solute on the droplet growth rate can be neglected, and (3) droplets fall relative to the turbulent flow at their Stokes' fall speed (for example, they are not affected by turbophoresis or thermophoresis). In Sect. 1, we derive the equations which govern the evolution of the droplet radius and squared radius distributions, including the loss rate due to sedimentation. In Sect. 2, we show how the equilibrium radius distribution is realized by using a Monte Carlo method and compare the results to those that are obtained analytically in later sections. In Sect. 3, we derive the analytic equilibrium solutions for the distributions and probability density functions (PDFs) of radius and of squared radius and from these obtain expressions for the median and mode radii. In Sect. 4, we derive the first five moments of the radius from the analytic equilibrium PDFs, including moments for truncated DSDs (those with positive lower limits). In Sect. 6, we use statistics from a set of measured DSDs to check for consistency with the analytic DSD. We also demonstrate the importance of taking into account a non-zero truncation radius when comparing theoretical moments to moments from a measured but truncated DSD. In Sect. 7, we present some additional quantities derived from the analytic DSD: droplet sedimentation flux, mean and PDF of the droplet residence time, precipitation flux, and condensation rate. Finally, Sect. 8 contains the conclusions.

Our initial goal is to develop and solve the equations that govern the equilibrium droplet radius distribution under conditions that might be found in the

We follow the notation used in

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. (

Analogous to Eq. (

To generalize Eq. (

The probability that a droplet of radius

The steady-state (equilibrium)
radius distribution,

A Monte Carlo method for solving Eq. (

Figure

Figure

Figures

We now derive the analytic equilibrium solutions for the distributions of

In a steady state, Eq. (

If the production of (activated) droplets from the injected aerosol occurs only for

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

One way to derive

As already noted,

The number of cloud droplets per unit volume with radii larger than

The PDF of the droplet radius distribution given by Eq. (

PDF of the droplet radius distribution given by Eq. (

PDF of the droplet squared radius distribution given by Eq. (

By changing the independent variable from

The median radius,

The cumulative density function (CDF) is the integral from 0 to

We derive the mode radius,

The mode radius versus the supersaturation for

By writing Eq. (

The mean radius is

The mean radius of droplets with radii larger than ^{®} upper incomplete gamma function is defined differently, as

The mean of the squared radius is

The mean cubed radius is

The mean

Equations (

The mean

The mean

Mean radius,

This study was motivated by the question of whether fluctuations in supersaturation are needed to explain the steady-state DSDs measured in the Michigan Tech turbulent cloud chamber (

We use statistics from a set of 11 DSDs with a wide range of droplet number concentrations

Do we expect the neglect of droplet curvature and solute effects in the analytic DSDs
to significantly affect the comparison of analytic and measured DSDs?
The distributions of the dry diameters of the injected NaCl aerosol particles for the measured DSDs are approximately lognormal, with a mode diameter of 40 to 60 nm and a standard deviation of about 30 nm

Because the PDF of the equilibrium droplet radius distribution, Eq. (

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

Knowing

In Fig.

If each DSD measured in the

Average over the 11 DSDs of the coefficient of variation of the nominal mean supersaturation values implied by the three measured moments versus the truncation radius.

Figure

In Fig.

Mean droplet radius versus supersaturation from the Monte Carlo model: with solute and droplet curvature effects (dotted lines) and without (solid lines) for all droplets (blue) and excluding droplets with radii

Relative dispersion of the radius versus droplet number concentration.
The measured values of dispersion are from DSDs truncated at

Figure

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.

Large supersaturation fluctuations occur only near the bottom and top boundaries of the cloud chamber, as is typical of Rayleigh–Bénard convection

Droplet growth in the chamber for these DSDs is primarily or entirely due to supersaturation fluctuations throughout the cloud chamber.
In this case, the analytic DSD solution, which assumes that there are no supersaturation fluctuations, is not valid.

The standard deviation is the square root of the variance.
The variance of the radius is

The relative dispersion of droplet radius,

Figure

The variance of the squared radius is

The relative dispersion of the squared radius,

Liquid water content (g m

The droplet sedimentation flux,
the number of droplets that exit the chamber due to sedimentation per unit area and time, is

The precipitation flux,
the mass of liquid water that exits the chamber due to sedimentation per unit area and time, is

The mean droplet residence time,

We noted in Sect.

Mean droplet residence time versus supersaturation for

To derive the PDF of droplet residence times,

PDF of droplet residence times for 0.1 % supersaturation and

To derive the condensation rate of a population of droplets, d

One can easily verify that d

In a laboratory cloud chamber, such as the

Because supersaturation is difficult to measure when cloud droplets are present, it has not been generally possible to determine the magnitudes of the mean supersaturation and the supersaturation fluctuations in the Pi chamber under cloudy conditions. Therefore, it also has not been generally possible to directly determine the relative contributions of mean and fluctuating supersaturation to the measured droplet PDFs.

We derived analytic PDFs of droplet radius and squared radius for conditions that could occur in a turbulent cloud chamber in which there is uniform supersaturation,
droplet curvature and solute effects on droplet growth are negligible,
and a balance exists between droplet formation (activation) and loss (due to fallout).
The loss rate due to fallout is based on three assumptions.
(1) The droplets are well-mixed by turbulence, in which case the

It should be emphasized that it is only the supersaturation that directly determines the droplet radius PDF.
A cloud chamber undergoing Rayleigh–Bénard convection is analogous to an ascending parcel:
in both cases a forcing process continually increases the supersaturation, while droplet growth decreases it.
For an ascending parcel, the forcing process is adiabatic cooling, while for a cloud chamber, it is turbulent fluxes of sensible heat and water vapor from the walls.
In both cases, the timescale for condensation to decrease supersaturation is the phase relaxation timescale, which depends inversely on droplet number concentration and mean radius

We demonstrated how the equilibrium radius distribution is realized by using a Monte Carlo method and compared the results to some of those that were obtained analytically. A notable feature is the wide PDF of droplet residence times. This PDF determines the width of the DSD when there is uniform supersaturation: all droplets grow at the same rate, so the greater a droplet's residence time, the larger it gets and the more it contributes to the large-droplet tail of the PDF.

From the analytic equilibrium PDFs of radius and of squared radius, we obtained expressions for the median and mode radii.
We also derived the first five moments of the radius from the analytic equilibrium PDFs, including moments for truncated DSDs (those with positive lower limits).
We used statistics from a set of measured DSDs to check for consistency with the analytic PDF.
The droplet number concentrations of the measured DSDs ranged from 14 to 3000 cm

We found that accounting for the truncation radius of the measured DSDs
is particularly important when comparing the theoretical and measured relative dispersions of the droplet radius.
We showed that the monotonic decrease in the measured relative dispersion reported by

Finally, we presented some additional quantities derived from the analytic DSD: droplet sedimentation flux, precipitation flux, and condensation rate.

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

The author declares that there is no conflict of interest.

The impetus for this study arose during a sabbatical visit to the Cloud Physics Laboratory at Michigan Technological University. The author thanks Raymond Shaw in particular for facilitating the author's visit.

This paper was edited by Hang Su and reviewed by two anonymous referees.