Journal of Nuclear and Radiochemical Sciences, Vol. 8, No.1, pp. N1-N3, 2007

Notes

A Simple Rainout Model for Radon Daughters
Susumu Minato*
Radiation Earth Science Laboratory, 9-6 Yamaguchi-cho, Higashi-ku, Nagoya, 461-0024 Japan
Received: March 13, 2007; In Final Form: June 16, 2007
This paper presents a simplified time-independent model which calculates radioactivities of short-lived radon
daughters in rainwater as a function of rainfall rate for unit concentration of radon in cloud. The model incorporates
processes of impact collection for cloud droplets by raindrops, removal of raindrops from cloud and transport timeof raindrops from the cloud base to the ground as forms of semi-empirical formulas. Radon concentrations in rain
clouds are estimated from comparisons between model calculations and ground-level observations made over a
few years.

1. Introduction
Rn-222 is of considerable interest in studies of global circulation and the long range transport of pollutants. However, themeasurements in higher altitudes take much expense. It would
be quite valuable if 222Rn concentrations in cloud air could be
estimated from radon daughters contained in rainwaterobserved at ground level.
Several papers have so far been published describing incloud scavenging of radon daughters, which were briefly
reviewed by Fujinami et al.1 and Horng and Jiang.2 Among
those, only the following two papers deal with 222 Rn and its
daughters in three phases, i.e. cloud air, cloud droplets andraindrops, and take into account microphysical processes
occurring in clouds. One is the work of Minato3 and the otheris that of Ikebe et al.4 However, the former model does notinclude the removal process of raindrops from cloud, and thelatter assumes a constant removal rate of cloud droplets for any
type of rainfall event. These treatments may be considered tobe somewhat far from reality. The purpose of this paper is tobuild a more reasonable model by improving the models
described in the above two papers.
2. The model
Let us consider here a rain cloud under a steady state condition, where Rn (222Rn) and its daughters are uniformly distributed.
2.1. Cloud droplets. Since the half-life of RaA (218Po) isquite short compared to Rn, we can assume radioactive equilibrium between them in cloud air. Since the produced RaA
atoms attach to the cloud droplets immediately,4 the radon
daughters in cloud droplets per unit volume can be expressed
as follows.
aRn = λA nA + ψcnA ,λA nA = λBnB + ψcnB ,λBnB = λCnC + ψcnC .

(1)
(2)
(3)

Here, aRn is the radioactivity of Rn, nA, nB, and nC are the number of cloud droplets with RaA, RaB (214Pb), and RaC (214Bi),
respectively, λA , λB, and λC are the decay constants of RaA,RaB, and RaC, and ψc is the removal rate of cloud droplets by*Corresponding author. E-mail: willow@sf.starcat.ne.jp

raindrops.
Then, the activities of RaA, RaB, and RaC on cloud droplets
per unit volume in cloud take the forms,
aA = λA nA =λAaRn ,λA + ψc(4)aB = λBnB =λBλA nAλB=aA ,λB + ψcλB + ψc(5)aC = λC nC =λCλBnB
λC=aB ,λC + ψc
λC + ψc(6)2.2. Raindrops. The radon daughters in raindrops per unit
volume in the cloud are represented byψcnA= λA NA + ψrNA ,ψcnB + λA NA = λBNB + ψrNB ,ψcnC + λBNB = λC NC + ψrNC ,(7)(8)(9)

where NA, NB, and NC are the number of raindrops with RaA,RaB, and RaC, respectively, and ψr is the removal rate of raindrops from the cloud. From these equations, the activities ofRaA, RaB, and RaC in raindrops per unit volume can berewritten as
AA = λA NA =ψcλA nAψc=aA ,λA + ψr λA + ψr

AB = λBNB =ψcλBnB +λBλA NAψcλB=aB +AA , (11)λB + ψr
λB + ψr
λB + ψr

AC = λCNC =ψcλCnC +λCλBNB
ψcλC=aC +AB . (12)λC + ψr
λC + ψr
λC + ψr(10)

2.3. Ground level. The radon daughters under the cloudbase are calculated from the following equations.
dAA= – λA AA ,dt

(13)

dAB
= λB(AA – AB) ,dt

(14)

dAC
= λC (AB – AC) .
dt

(15)

© 2007 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 06/30/2007

N2MinatoJ. Nucl. Radiochem. Sci., Vol. 8, No. 1, 2007

The activities at ground level can be obtained by substituting
the transport time from the cloud base to the ground into thesolutions of eqs 13–15.
2.4. Radioactivity of rainwater. Now, in order to compare
the model mentioned above with ground-level observations, wehave to transform the radioactivities of raindrops per unit volume in cloud air, AA, AB, and AC expressed in units of Bq/m3 tothe activities of rainwater, αA , αB and αC , in units of Bq/mL
using the following relation,

αA =

ρρρ
AA , αB =AB , αC =AC ,LrLrLr(17)The collision rate for a single cloud droplet of radius R byraindrops between R' and R'+dR' is defined as
H(R) = ∫R' πR' 2 E(R', R)[v(R' )–v(R)]s(R' )dR'. (18)Here, E is the collision efficiency, v the velocities of cloud
droplets and raindrops and s the concentration of raindrops.3
Then, the removal rate can be calculated from the following
formula.

ψc =∫C (R)H(R)dR ,∫C (R)dR(19)where c is the concentration of cloud droplets, which is given
in Reference 3 as a function of the mean radius of cloud droplets.
Numerical calculations for the above equations yield the following empirical formula.

ψc = (0.028–0.036e–0.19<R>)P 0.83 ,Table 1 gives the model calculated results. Figure 1 shows
the whole year data of short-time observations made during
rainfall events at Nagoya1 along with a few events data collected in Taiwan2 and at Kashiwazaki.6 The two curves in thefigure represent fits of the model calculation given in Table 1
to the almost maximum and minimum specific activities. In
this way, we can estimate Rn concentrations inside clouds fromground-level observations of rainwater radioactivities using(16)where ρ is the density of rainwater in g/mL, and L r is the watercontent of raindrops in the cloud in g/m3, which can be derived
from the Marshall-Palmer formula 5 in g/m 3 as a function of
rainfall rate P in mm/h,
L r = 0.089P 0.84 .

3. Calculated results and comparisons

TABLE 1: Model calculated rainfall rate dependence ofspecific activity of rainwater at ground level per 1 Bq/m3of 222Rn in cloud air
Rainfall rate

RaB+RaC/Bq mL-1/mm h-1WinterSummer

Whole

0.1
0.2
0.3
0.5
0.8
12
358
10
20
30
50
70

6.49
5.50
4.83
3.94
3.12
2.74
1.73
1.26
0.817
0.533
0.432
0.220
0.146
0.0871
0.0617

11.4
9.93
8.90
7.45
6.04
5.38
3.52
2.63
1.76
1.18
0.967
0.512
0.349
0.213
0.154

9.45
8.16
7.26
6.01
4.82
4.28
2.75
2.04
1.35
0.892
0.728
0.379
0.256
0.155
0.111

Conditions; average cloud droplet radius=11 µm, cloud baseheight=2 km, and cloud top height= 10, 5, and 7.5 km for winter, summer, and whole year, respectively.

(20)

where ψc is the removal rate of cloud droplets per minute and<R> the mean radius of cloud droplets in µm.
Next is to evaluate ψr , i.e. the removal rate of raindrops perminute. The rainfall rate is defined as follows.
P = ψr L r D.

(21)

Here, D is the cloud depth, i.e. the difference between cloud
top and cloud base. Then, we obtainψr = 0.187

P 0.16D(22)

where D is in units of km.
Finally, we need an average velocity of raindrops under thecloud < v > to calculate the transport time from the cloud baseto the ground, which can be derived from the following relation.
P = < v >L r .

(23)

Namely,< v > = 0.187P0.16 ,where <v> is in units of km/min.

(24)

Figure 1. Comparison of momentary specific activities during rainfall observed at ground level with model calculations for the corresponding 222Rn concentrations in cloud air using the parameter forwhole year given in Table 1

A Simple Rainout Model for Radon Daughters

J. Nucl. Radiochem. Sci., Vol. 8, No. 1, 2007

N3

Figure 2. Comparison of specific activities averaged over a single rainfall event observed at ground level3 with model calculations for the corresponding 222Rn concentrations in cloud air using the parameters in Table 1 for (a) winter and (b) summer. Vertical bars represent one standard
deviation.

activities observed at Nagoya for a few years.3 The two curvesin the figure correspond nearly to one standard deviation (1σ).
Thus, the model evaluated ±1σ ranges of Rn concentrations inclouds amount to 0.60–1.50 Bq/m3 for a period of December–
April and 0.13–0.50 Bq/m 3 for May–November. In order toconfirm the validity of the method presented in this paper, let
us compare these values with observed data.
Figure 3 shows the data observed over Eurasian and North
American continents and over Hawaii, which were summarized
by Piliposian and Appleby.7 Although Nagoya is located at aland-sea interface, it may be all right to use the data taken overthe continents for December–April, since continental air
masses are dominant in this season. On the other hand, thedata of Hawaii would be useful in interpreting the evaluatedresult for May–November when maritime air masses sometimes move towards the Japanese Islands.
References

Figure 3. Comparison of observed 222Rn concentrations over continents and a large ocean7 with model evaluated ranges (±1σ) for thosein clouds, based on the data shown in Figure 2.

Table 1.
Figure 2 shows the seasonal data of single rainfall averaged(1) N. Fujinami, S. Esaka, and S. Minato, Nucl. Geophys. 7,359 (1993).
(2) M. C. Horng and S. H. Jiang, Radiat. Meas. 37, 603 (2003).
(3) S. Minato, J. Radioanal. Chem. 78, 199 (1983).
(4) Y. Ikebe, K. Saijo, T. Iida, and H. Nagai, Res. Lett. Atmos.
Electr. 5, 17 (1985).
(5) J. S. Marshall and W. McK. Palmer, J. Meteor. 5, 165(1948).
(6) H. Sakaue, F. Maruta, H. Fujimaki, and S. Tonouchi,
Annual Report of Niigata Prefectural Radiation Monitoring
Center. 1, 50 (2003), (in Japanese).
(7) G. T. Piliposian and P. G. Appleby, Continuum Mech.
Thermodyn. 15, 503 (2003).