tin space charge

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Production of space charge at the boundaries of layer clouds

Limin Zhou1,2,3,4 and Brian A. Tinsley2

Received 5 September 2006; revised 6 February 2007; accepted 8 February 2007; published 6 June 2007.

[1] The ionosphere-Earth current density Jz creates space charge at the upper andlower boundaries of layer clouds. This occurs because clouds have an order of magnitudelower conductivity than the clear air at the same altitude, and as the current densityflows through the boundaries, it creates a gradient of electric field that must be satisfied bythe accumulation of space charge, according to Gausss law. We have modeled theproduction of space charge and its partition between charges on droplets, aerosol particles,and ions and performed sensitivity tests for the variation of a number of relevantatmospheric parameters. We find typical droplet charges of 50100 elementary charges,positive at cloud top and negative at cloud base, consistent with recent observations.The charges are of sufficient magnitude to suggest measurable electrical effects onscavenging of ice-forming nuclei and cloud condensation nuclei. The results are relevantto the modeling of solar or internally forced changes of Jz and space charge on cloud

microphysics as a possible cause of small effects on weather and climate.Citation: Zhou, L., and B. A. Tinsley (2007), Production of space charge at the boundaries of layer clouds, J. Geophys. Res., 112,

D11203, doi:10.1029/2006JD007998.

1. Introduction

[2] The ionosphere is charged to a potential of about250 kV by the upward flow of about 1000 A of current fromthe total of highly electrified convective clouds around theglobe [Williams, 2005]. The return current is in the form of acurrent density, Jz, downward through the thickness of theglobal atmosphere to the land and ocean surface. Jz is in therange 16 pA m2, depending on the elevation of that

surface, the output of the tropical thunderstorm generators,the cloud cover and aerosol content in the atmosphericcolumn, the radioactive emanations from the land, thecosmic ray flux, and other inputs modulated by spaceweather such as energetic protons and electrons, and auroralcurrent systems that affect the ionospheric potential in the

polar caps [Tinsley and Zhou, 2006].[3] The conductivity within clouds is reduced compared

to that of clear air at the same altitude because of theattachment of atmospheric ions to cloud droplets. Thedecrease in s within clouds is by a factor of between 3and 30 compared to the clear air, according to the simu-lations of Griffiths et al. [1974]. Thus we have

Jz sE 1

where E is ambient electric field, which we take here to bepositive downward, and s is the conductivity. With constantcurrent density, as s is reduced, E must increase.

[4] As Jz passes through the upper and lower boundariesof clouds, it creates a gradient of E because of the gradientin sbetween cloudy and clear air. In turn, the gradient in Eentails the accumulation of space charge r (difference

between the total positive and total negative charge per unitvolume) according to Gausss law:

r E r=e0 2

Thus, assuming horizontal stratification,

r e0Jzd

dz

1

s

3

where z is the distance coordinate, measured verticallydownward.

[5] A layer of positive space charge is produced in thegradient of conductivity at the tops of clouds, (resistivity r=1/s is increasing in the direction of positive z and Jz),accompanied by a layer of negative space charge in thedecreasing resistivity at cloud base. Most of the spacecharge accumulates on droplets, because for the mixtureof air ions, aerosol particles and droplets in typical clouds,the droplets have the greatest radii of curvature and surfacearea. The remaining space charge is distributed between theaerosol particles and ions, depending on their relativeconcentrations. The total space charge is

r e n1 n2 g1SS1 g2SS2 pSNA 4

where e is the elementary charge (1.6 1019 C); n1 and n2are the concentrations of positive and negative ions; SS1

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, D11203, doi:10.1029/2006JD007998, 2007ClickHere

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1State Key Laboratory of Environmental Geochemistry, Institute ofGeochemistry, Chinese Academy of Sciences, Guiyang, China.

2W. B. Hanson Center for Space Sciences, University of Texas atDallas, Richardson, Texas, USA.

3Graduate School of the Chinese Academy of Science, Beijing, China.4 Now at Department of Geography, East China Normal University,

Shanghai, China.

Copyright 2007 by the American Geophysical Union.0148-0227/07/2006JD007998$09.00

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and SS2 are the concentrations of positively and negativelycharged aerosol particles with mean positive and negativecharges of g1 and g2 in units of e; SNA is the total dropletconcentration, summed over all values of the droplet radius

A; and p is the (algebraic) mean of the number ofelementary charges on both positively and negativelycharged droplets.

[6] Under steady state conditions the total downward

currentJz is constant, and is the sum of a downward currentdensity due to downward moving positive ions and adownward current density due to the upward movingnegative ions. The charged aerosol particles are taken asstationary (negligible mobility). Thus

Jz J1 J2 5

where

J1 s1E m1n1eE 6

and

J2 s2E m2n2eE 7

and where m1 and m2 are the mobilities of the positive andnegative ions.

[7] The relative amounts ofJ1 and J2 vary with distancebecause the relative amounts of n1 and n2 vary on accountof the so-called electrode effect, which creates the spacecharge. Just below the top of the cloud more positive ionscan flow downward than negative ions can flow upward,

because of the decreased ion concentration in the cloud.Similarly, more negative ions can flow upward just abovethe bottom of the cloud than positive ions can flow down.

[8] We have modeled the flow of current density through

a set of representative layer clouds, with representativeaerosol concentrations, in order to evaluate the amounts ofcharges on droplets and aerosol particles for equilibriumelectrostatic conditions. This is necessary in order to modelthe electrical effects on the scavenging by droplets of ice-forming nuclei (IFN) and cloud condensation nuclei (CCN).This scavenging appears to have significant effects on cloudcover and precipitation [Tinsley, 2000; Burns et al., 2007].

2. Construction of Model

[9] The continuity equation for the positive ions is

dn1

dt

q an1n2 b1Sn1 g1S2n1 1

e

r J1

4pD1n1X1A0

ANA

!pc

epc 18

and for negative ions is

dn2

dt q an1n2 b2Sn2 g2S1n2

1

er J2

4pD2n2X1A0

ANA

!pc

epc 19

where q is the ion pair production rate per unit volume, a isthe ion-ion recombination rate coefficient; S the concentra-tion of neutral aerosol particles; b1 and b2 are theattachment rate coefficients for the positive and negativeions, respectively, to the neutral particles; g1 and g2 are theattachment rate coefficients for positive ions to negativelycharged (concentration S2) particles and negative ions to and

positively charged (concentration S1) aerosol particles,

respectively; D1 and D2 are the diffusion coefficients forpositive and negative ions at the altitude of the cloud; andfrom work by Pruppacher and Klett[1997, section 18.3] thevalue ofp for the equilibrium condition of droplet chargingis given by

pc lnD1n1

D2n2

10

where c depends inversely on aerosol radius and in SI unitsis

c e2

4pe0AkT

11

where A is the mean droplet radius; k is Boltzmannsconstant, and T is the absolute temperature. The oppositesign for the divergence terms in (9) as compared to (8)arises because the negative charges move in the oppositedirection to Jz. The last terms in (8) and (9) are for thediffusion charging process. The drift charging processconsidered by Griffiths et al. [1974] can be shown to besmall compared to diffusion charging for the droplet sizesand electric fields that we have modeled.

[10] Equations (8) and (9) state that the rate of change ofion concentration (dn/dt) is the production rate, minus theloss rates due to ion-ion recombination and attachment,

minus the net flow out of the volume, minus the attachmentrates to droplets. The latter loss rates are for droplets withtheir equilibrium average number of charges p.

[11] Although a detailed treatment of ion attachment tothe aerosol would consider the polydispersive aerosol par-ticle size distribution, with multiple charges on the larger

particles [Hoppel, 1985; Hoppel and Frick, 1986; Yair andLevin, 1989], we represent the particles by a monodisper-sive size distribution, representing the dominant size cate-gory, with at most a single positive or negative charge. Thusb and g in each case apply to the radius of the dominant

particles. We will consider consequences of these approx-imations in the Discussion section.

[12] The continuity equations for the positively and

negatively charged aerosol particles are

dS1

dt b1Sn1 g2S1n2 12

and

dS2

dt b2Sn2 g1S2n1 13

D11203 ZHOU AND TINSLEY: SPACE CHARGE IN CLOUDS

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and at equilibrium when dS1/dt = dS2/dt = 0, we have

S1 Sb1n1

g2n2

14

S2 Sb2n2

g1n1

15

and with the total aerosol concentration ST = S1 + S2 + Swehave

S ST

1 b1n1

g2n2b2n2

g1n1

16

[13] The terms for divergence of J, with horizontalstratification, using (6) and (7) can be expressed as

1

er J1

1

e

@

@zJ1 m1n1

@

@zEE

@

@zm1n1 17

and

1

er J2

1

e

@

@zJ2 m2n2

@

@zEE

@

@zm2n2 18

with

E Jz

e

1

m1n1 m2n2

19

from (5), (6), and (7).[14] On substituting (14) and (15) in (8) and (9) we see

that on the right side of these equations, for equilibriumconditions, the sums of the third and fourth terms, (U), inboth cases are given by

U S b1n1 b2n2 20

The fifth terms are also equal, since at equilibrium

r J r J1 r J2 0 21

[15] If (10) is substituted into (8) and (9), we see that thesixth terms, (W), are also equal and are given by

W 4pD1D2n1n2X1A0

ANA

!ln D1n1=D2n2

D1n1 D2n222

[16] With the first terms already being equal, and also thesecond terms, in (8) and (9), we see that these two equationsare effectively the same, and only one of them can be usedfor evaluating n1 and n2, and subsequently evaluating S1, S2,S, and p. Either of the equations (8) or (9) can be trans-formed using (17) or (18), and (19) and (20), with S given

by (16), so that the only unknowns on the right side are n1

and n2. Thus (8) becomes

dn1

dt q an1n2 S b1n1 b2n2 m1n1

dE

dzE

d

dzm1n1

4pD1D2n1n2 ln D1n1=D2n2

D1n1 D2n2

X1A0

ANA

!23

[17] The other equation that determines the unique valuesofn1 and n2 (for specified values ofJand ST, and SNAA as afunction of height z through the cloud) is Gausss law(equation (2)). Making the assumption that only singlecharges can exist on the monodispersive aerosol particles,then g1 = g2 = 1, and SS1 = S1, and SS2 = S2. Then, using(4) and (10) and (11), equation (2) becomes

@E

@z

e

e0

"n1 n2 S1 S2

4pkTe0

e2

X1A0

ANA

!

ln D1n1=D2n2

#24

Substituting for S1 and S2 with (14) and (15), (24) becomes

@E

@z

e

e0n1 n2 S

b1n1

g2n2b2n2

g1n1

4pkT

e

X1A0

ANA

!ln D1n1=D2n2 25

[18] The electric field E and its derivative with distanceare determined in terms ofn1 and n2 by (19), and S is given

by (16), thus leaving n1 and n2 the only unknowns inequation (25), so that equations (23) and (25) can togetherdetermine a unique solution. The approach that was used in

our modeling was to assume approximate equilibrium con-ditions and make a first estimate of ion concentration as afunction of distance z by adding together (8) and (9) andtaking n1 = n2 = n0 and dn0/dt = 0 and using (20) and (16).The solutions to the quadratic equations yielded a value ofn0 for each value of z in the modeled cloud region.

[19] The electric field and its derivative with distance inthis first estimate can be calculated using n1 = n2 = n0 in(19) with a specified current density J.

[20] Next, make estimates of n1 and n2 by putting

n1 1 x n0 26

n2 1 x n0 27

where 1 < x < 1.[21] Then (26) and (27) are put into (25) so that it

becomes

@E

@z

e

e0

"2n0x S

b1g2

1 x

1 xb2g1

1 x

1 x

4pkTe0

e2

X1A0

ANA

!ln

D1

D2

1 x

1 x

#28


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In this equation x is the only unknown variable and theequation can be solved using an iterative numerical method.Thus we get the first estimate of n1 and n2, which representtheir initial values at the time of the assumed approximateequilibrium, i.e., as a function of altitude at time t = 0.

Starting with these initial n1 and n2 values, the evolution ofn1 and n2 with time can be calculated step by step byiterating (23) and (25) until convergence on a more preciseequilibrium is attained:

n1 t n1 t 1 dn1 t 1

dtDt 29

n2 t n2 t 1 dn2 t 1

dtDt 30

where n1(t 1) and n2(t 1) are obtained from the earlierestimate. In evaluating the derivatives with respect to zsmoothing over adjacent values was used to ensure stabilityin the iterations. It should be noted that this continuediteration in time does not simulate a change in n1 and n2from their values in cloud-free air. Rather, after the firstestimate of n1 and n2 we are already very close toequilibrium, and further iterations are to ensure an evenmore precise value.

[22] The values ofm1, m2, D1, and D2 depend on altitude,i.e., on the pressure and temperature of the atmosphere, andb1, b2, g1 and g2 depend on particle radius, (a), as well ason pressure and temperature. We used a model atmospherederived from the US Standard Atmosphere (U.S. StandardAtmosphere, 1976) as in work by Tinsley and Zhou [2006].We use values of m1 and m2 at standard sea level atmo-

spheric pressure and temperature (1013.25 hPa and288.15 K) of 1.4 104 and 1.9 104 m2 V1 s1.These values originated with Bricard [1965] and were saidto be for normal pressure and temperature, but aredescribed as being for STP by Shreve [1970] and

Pruppacher and Klett [1997, section 18.1]. However, thevariation of mobility with altitude given by these authorsimplies that the STP temperature is the standard sea leveltemperature of 288.15 K. We regard Bricards experimentalvalues, which depend on the differing molecular weights ofthe molecular clusters forming the positive and negative airions, as representative. They are dependent on atmosphericchemical processes; however, for cloud environments in the

troposphere we follow Pruppacher and Klett [1997] andconsider them representative. The relationship betweenmobility of ions and their diffusion coefficient D in air is[e.g., McDaniel and Mason, 1973]

D mkT=e 31

D varies with temperature, in part because the cross sectionfor collision of the ions with air molecules varies withcollision energy. To include this effect and we took thevariation of mobility with altitude, i.e., of mz with pressure

pz and temperature Tz at altitude z, referred to mo at pressurepo and temperature To as

mzm0

p0

pz

Tz

T0

1:5T0 120

Tz 12032

where the expression is derived in Appendix A. UsingBricards values at z = 0 km, the values ofm1 and m2 for 2

km are then 1.63 104 and 2.21 104 m2 V1 s1,respectively, and at 4 km 1.90 104 and 2.58 104 m2

V1 s1, respectively, and at 10 km 3.43 104 and 4.64 104 m2 V1 s1, respectively. The formula (32) differsfrom an alternative formula for variation of mobility withaltitude: mz/m0 = (p0/pz)(Tz/T0) that has often been used inatmospheric electricity literature. It evidently originatedwith Bricard [1965], and does not take into account thevariation ofD and ofm with temperature at constant density.This variation is implicit in the treatment by McDaniel and

Mason [1973] and in the present treatment. For furtherdiscussion of this point see Appendix A.

[23] The values ofb1, b2, g1 and g2 were obtained fromthe theory of Keefe et al. [1968] and Hoppel and Frick[1986]. These formalisms include the effects of imagecharge as well as three-body trapping of the ions, withdiffusion bringing the ions close enough to the neutral orcharged aerosol particle to be captured. Table 1 gives thevalues ofb1, b2, g1 and g2 for 4 km and 10 km for particleswith a = 2 108 m; 4 108 m; and 10 108 m.

3. Results

3.1. General Features

[24] Figure 1 shows a set of input and output parametersfrom one run of the model. The model consists of five

Table 1. Values ofb1, b2, g1, and g2 Used in the Simulations at 2 km, 4 km, and 10 km Altitude for the Aerosol Particle

Radii 0.02 mm, 0.04 mm, and 0.10 mm

a, mm b1, 1012 m3 s1 b2, 10

12 m3 s1 g1, 1012 m3 s1 g2, 10

12 m3 s1

zc = 2 km0.02 0.61 0.86 2.36 3.130.04 1.53 2.16 3.32 4.550.10 4.55 6.38 6.23 8.65

zc = 4 km0.02 0.63 0.88 2.56 3.370.04 1.61 2.27 3.64 4.980.10 4.90 6.88 6.83 9.49

zc = 10 km0.02 0.70 0.96 3.06 3.910.04 1.93 2.70 4.84 6.490.10 6.62 9.34 9.72 13.47


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regions, divided according to the droplet concentration. Asin Figure 1a, the first 2 m, with z increasing downward,contain aerosol and ionization, but no cloud droplets. Thenext region is the cloud upper boundary layer, of variablethickness, where the droplet concentration increases to its

maximum value, after which there is a layer of arbitrarythickness (for convenience set at 11 m) to represent auniform cloud interior, with constant droplet concentrationequal to the maximum. Then the lower boundary layer is ofvariable thickness, with droplet concentration decreasing tozero, and a lower droplet free layer that is set at 2 mthickness. Since the interior is uniform, the results for the

boundary layers are applicable to clouds with interiors thatare uniform and of any thickness, not just the 11 mthickness used here. To avoid discontinuities in the numer-ical integration, the droplet concentration in the boundarylayers follows a sin2 (p(z z1)/(2zb)) variation, where zb is

the thickness of the boundary layer, and z1 is either edge ofthe cloud. There is little difference in the results using othersmooth monotonic variations. The total aerosol concentra-tion ST is kept constant throughout the cloud, and the meandroplet radius A = 10 mm in all runs.

[25] In Figure 1 the cloud is at the altitude of 4 km; withion production q = 8.0 106 m3 s1. The current density

Jz = 3.0 1012 A m2; the maximum droplet concentra-

tion (SNA)max = 2.0 108 m3 (and thus SANA = A SNA);

the aerosol concentration ST = 5.0 109 m3; the (mono-

dispersive) aerosol radius 4 108 m (0.04 mm), and eachboundary layer is 10 m thick. In subsequent runs sensitivitytests are made by varying these parameters.

[26] We will examine the mean ion concentration (n0), thestrength of electric field (E), the concentrations of positiveion and negative ions (n1, n2) and their difference (n1 n2),the concentration of positively charged and negatively

Figure 1. Profiles of parameters of a simulation of space charge production in a cloud, as a function ofdistance z measured downward through it: (a) droplet concentration, (b) mean ion concentration,(c) electric field, (d) divergence of field, (e) ion concentration difference (n1 n2), (f) charged aerosol

particle concentration difference (S1 S2), (g) mean droplet charge, (h) charged aerosol concentrations(S1 and S2), and (i) ion concentrations (n1 and n2). For other parameters for this simulation, see text. InFigures 1b1g, the solid line is for the initial estimate, and the dotted line is for 600 s simulation. InFigures 1h and 1i, the solid and dotted lines are for the initial estimates, and the dashed and dash-dottedlines are for 600 s. In Figure 1h, the solid and dashed lines are for positively charged aerosol particles,and the dotted and dash-dotted lines are for the negatively charged particles. In Figure 1i, the solid anddashed lines are for the positive ions, and the dotted and dash-dotted lines are for the negative ions. Thethin solid lines are the zero references.


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charged aerosol particles (S1, S2) and their difference (S1 S2), the mean number of elementary charges on the droplets(p) in the cloud layer as a function of layer thickness for upto 600 s iteration time. We will also examine the effect of Jz;(SNA)max; ST; a; the thickness of the cloud boundary layers

zb; and height of the cloud zc on the various elements ofspace charge that are generated by the flow of Jz into thecloud.

3.2. Charge Distribution in the Cloud Layer

[27] Figures 1b 1g shows relevant parameters as a func-tion ofz; with initial values (t = 0) after the first estimate ofthe parameters shown as solid lines, and values after 600 sshown as dotted lines. Figure 1b gives no; Figure 1c gives

E; Figure 1d gives the derivative of the electric field@E

@z;

Figure 1e gives (n1 n2); Figure 1f gives (S1 S2); andFigure 1g gives p (as a net mean charge p can be positiveor negative). Figure 1h shows S1 as solid and dashed linesfort= 0 and t= 600 s and S2 as dotted and dash-dotted linesfor the same times. Figure 1i shows n1 and n2 with the sameline designations.

[28] From Figures 1a and 1b it can be seen that with theincrease of droplet concentration with depth into the cloudin the upper boundary layer the ion concentration n0decreases because of ions attaching onto the droplets. Inthe middle part of the cloud the constant droplet concen-tration ensures thatn0 remains constant until the decrease ofdroplet concentration in the lower boundary layer causes n0to increase back to its original value. In Figure 1c the electricfield E rises in the upper boundary layer, is constant in the

uniform part of the cloud, and decreases in the lower boundarylayer, which causes the positive peak of

@E

@zin the upper

boundary layer and negative peak of@E

@zin the lower boundary

layer (see Figure 1d). In accordance with equation (28),@E

@zdetermines the space charge distribution in the cloud.

Although the electric field derivatives reach essentiallyequal and opposite amplitudes in the upper and lower

boundary layers, this is not true for all the components ofthe space charge, because the mobility of the negative ionsis greater than that of the positive ones, and more negative

Figure 2. Profiles for variation of vertical current density (Jz) for 600 s simulation. Dashed lines are forJz = 4 10

12 A m2; solid lines for 2 1012 A m2; and dotted lines are for 1 1012 A m2. Thin

solid lines are the zero references. The parameters plotted are (a) mean ion concentration, (b) electricfield, (c) divergence of field, (d) ion concentration difference, (e) charged aerosol concentrationdifference, and (f) mean droplet charge.


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aerosol particles are formed, leaving more positive ionsthan negative. Thus in Figures 1i and 1e it can be seen thatn1 > n2 in the cloud-free areas and in the uniform centralsection, with the differences being about 1.8 108 m3

and 1 107 m3, respectively. (Note the logarithmic scale inFigure 1i and the linear scale in Figure 1e). In the upper

positive space charge regions the magnitude of the (n1 n2)difference reaches 4 108 m3, but in the lower negative spacecharge region the difference only reaches 1.1 108 m3.

[29] The related asymmetry in S1 and S2 can be seenin Figures 1h and 1f. In the cloud-free and central regionsS1 < S2, with the differences being about 1.8 10

8 m3

and 5 10

7

m

3

, respectively. In the upper space chargeregion the difference (S1 S2) is positive and reaches amaximum of 1.1 109 m3, then gradually reduces to theconstant small negative value in the central region, while inlower space charge region it gradually approaches its great-est negative value of 1.25 109 m3, and then rapidlyincreases up to the cloud-free value.

[30] Figure 1g shows the mean number of charges on thedroplets in the cloud. In the upper space charge region themean charge is positive and its amount sharply rises tomaximum value of about 90e and then gradually decreasesto 0.17e in the central region (i.e., only slightly more

positive than negative charged droplets). In the lower space

charge region the mean droplet charge is negative, graduallyapproaching the peak value of about 90e and then rapidlydecreasing in (absolute) value to near zero again.

[31] After 600 s iteration time only small differences inthe parameters are found as compared to the initial estimateof equilibrium, as is apparent in all the panels of Figure 1.These differences do not increase significantly after morethan 600 s of iteration. The differences are partly due to thesmoothing, and partly (as in the uniform middle section)from the approximation m1 = m2 and n1 = n2 in the initialestimate forn0. Simulations for t = 2 s are closer to that oft = 600 s. We consider that compared to other uncertainties

in the model the t = 0, t = 2 s, and t = 600 s results areequivalent.

3.3. Effect of the Current Density Jz

[32] From equation (19) reduction ofJz will reduce Eand@E

@zand affect the space charge distribution. Figures 2a2f

show the results for 600 s iteration, with three differentcurrent density values. The dashed lines are for Jz =4 1012 A m2; the solid lines for 3 1012 A m2; andthe dotted lines for 2 1012 A m2. The zero level inFigures 2c2f is shown as a thin solid line.

Figure 3. Profiles for variation of droplet concentration. Dashed lines are for the maximum dropletconcentration of 4 108 m3; solid lines are for 2 108 m3; and dotted lines are for 1 108 m3. The

parameters that are plotted are the same as in Figure 2.


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[33] The other parameters are: zc = 4 km; q = 8 106

m3 s1; (SNA)max = 2 108 m3; ST = 5 10

9 m3; andzb = 10 m. In Figure 2a it can be seen that changing Jz didnot significantly change n0 in the cloud layer. In Figure 2ban increase in Jz causes a proportionate increase of electricfield (E) throughout the cloud. In Figures 2b and 2c the

electric field gradient@E

@zscales as E and Jz, and causes the

space charge to scale proportionately. So in Figures 2d2fthere are increases in (n1 n2), (S1 S2), and p in the

boundary layer due to increases of@E

@z, especially in p and

(S1 S2) that carry most of the space charge.3.4. Effect of Cloud Droplet Concentration

[34] Enhancement of the concentration of cloud dropletsincreases the attachment surface for ions, reduces the ionconcentration and conductivity, and increases the electricfield and its gradients and the space charge. Figures 3a3fshows the results after 600 s iteration with three values ofdroplet concentration; the dotted lines are for (SNA)max1 108 m3; solid lines for 2 108 m3; and dashedlines for 4 108 m3. The other parameters are: zc = 4 km;

Jz = 3 1012 A m2; q = 8 106 m3; ST = 5 10

9 m3:a = 4 108 m; zb = 10 m.

[35] In Figure 3a the increases in droplet concentrationcauses decreases of n0 in the cloud layer. In Figures 3b and3c the decreased conductivity causes higher electric field E

in the cloud layer, and corresponding increases in@E

@z. In

Figure 3d with higher droplet concentration the curve for(n1 n2) peaks at higher values, and closer to the edge ofthe cloud. This applies also to (S1 S2) as shown inFigure 3e and to p as shown in Figure 3f. For the uniformcentral region of the cloud (n1 n2) decreases because ofthe increasing droplet concentration and decreasing n0.

3.5. Effect of Thickness of Boundary Layer

[36] The boundary layer thickness zb determines @E@z

,

which affects all the components of the space charge.Figures 4a 4f show the results for 600 s iteration; thedashed lines are for zb = 5 m; dotted lines for 10 m; andsolid lines for 50 m. The other parameters are: zc = 4 km;q = 8 106 m3; Jz = 3 10

12 A m2; (SNA)max = 2 108 m3; ST = 5 10

9 m3; a = 4 108 m. In Figure 4a,the increase of boundary layer thickness causes a less rapiddecrease of n0 with distance into the upper boundary layer,and a less rapid increase coming out of the lower one. So in

Figures 4b and 4c there are lower values of@E

@zfor the

Figure 4. Profiles for variation of boundary layer thicknesses. Dashed lines are for thicknesses of 5 m;dotted lines are for 10 m; and solid lines are for 50 m. The parameters that are plotted are the same as inFigure 2.


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greater boundary thicknesses. In Figure 4d in the upperboundary layer the value of (n1 n2) decreases as the layerwidens, while in the lower boundary layer (n1 n2) shows anegative peak only for the 5 m and 10 m layers. For the 50 mthick boundary layer the lower negative peak disappears.This is because the weaker space charge is overcome by theeffect of the greater negative ion mobility. In Figure 4e thewider boundary layer causes lower (S1 S2) within it.Similarly in Figure 4f the wider boundary layer results in areduction in p within it. However, theory requires, and thefigures indicate that the integrated space charge with depththrough either of the boundary regions remains constant.

The theoretical reason is that integrated space charge iseffectively a surface charge ss = eo(Ei Eo) to accommo-date the change in electric field from Eo outside the cloud to

Ei in the uniform interior of the cloud.[37] For the 50 m case in Figure 4f the small negative

excursion at the very top of the upper space charge layerappears to be an anomalous feature. However, it is also dueto the preponderance of the effects of negative ion mobilityexceeding positive mobility. The relatively slow increase ofspace charge and thus of the ratio n1/n2 with distance intothe cloud (see (n1 n2) for the 50 m case in Figure 4d)creates a small region where the droplet concentration is

nonzero and n1/n2 is less than the inverse of the ratio D1/D2 =m1/m2 = 1.4/1.9. Under these circumstances, (D1n1)/(D2n2) < 1 ,and from equation (10) p is negative.

3.6. Effect of Altitude of Cloud

[38] There are several parameters which increase with thealtitude of the cloud, zc, and affect the space chargecomponents: the ion mobility and the ion-aerosol attach-ment rate coefficients as noted earlier, and the ion produc-tion rate, q, which increases with altitude up to about 15 km[Tinsley and Zhou, 2006]. For 10 km q is taken as 3.9 107

m3; for 4 km 8.0 106 m3 and for 2 km 4.5 106 m3.

The other parameters are Jz = 3 10

12 A m

2; (SNA)max =2 108 m3; ST= 5 10

9 m3; a = 4 108 m; zb = 10 m.Figures 5a5f shows the results; the dashed lines are for10 km; dotted lines for 4 km; and solid lines for 2 km.

[39] In Figure 5a the mean ion concentration at 10 km ishigher than that at 4 km which is higher than 2 km, duemainly to the higher ion production rates at higher altitudes.In Figure 5b the enhancement of the ion concentration andmobility at the higher altitudes causes the decrease of theelectric field E. Then in Figure 5c in the space charge

region,@E

@zshows a marked decrease with the increase in

Figure 5. Profiles for variation of cloud altitude. Dashed lines are for 10 km altitude; dotted lines are for4 km; and solid lines are for 2 km. The parameters that are plotted are the same as in Figure 2. The

waviness near 13 m and 23 m in Figure 5c for the 2 km case is due to an instability in the computing.


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cloud height. So in Figures 5e and 5d for the 10 kmsimulation the tendency for the predominantly negativeion mobility to produce negative (S1 S2) and positive(n1 n2) has comparable effects to that of the space charge,and so in Figure 5f p is negative, or only slightly positive.

Because of an instability in computing@E

@zfor the 2 km case,

the iteration was stopped just beyond the first estimate ofn1and n2, i.e., for just 2 s; however, for the 4 km and 10 kmcases, the results are indistinguishable from iterations for

600 s.

3.7. Effect of Aerosol Concentration

[40] The rate of attachment of ions to aerosols is directlyproportional to their concentration, affecting the distributionof charges on aerosols, ions, and droplets. Figures 6a6fshow the results; the dotted lines are for ST = 5 10

7 m3;the dash-dotted lines for 5 108 m3; the dashed lines for5 109 m3; and the solid lines for 6 1010 m3, fromsimulations for 600 s. The other parameters are zc = 4 km;

Jz = 3 1012 A m2; (SNA)max = 1 10

8 m3 (this issmaller than in previous cases); a = 4 108 m, zb = 10 m.

[41] In Figure 6a, there exits a larger difference in n0between the curves forST= 6 10

10 m3 and 5 109 m3,where attachment to aerosol particles dominates theion recombination, than between those of 5 109 m3

and 5 108 m3, and between those of 5 108 m3 and5 107 m3,where attachment is becoming less importantcompared to ion-ion recombination outside the cloudyareas, and less important compared to attachment to dropletsinside them. The higher electric field (E) curve for thehigher aerosol concentration cases reflects the lower ion

concentrations, but there is little difference in @E@z

for the

four aerosol conditions, as illustrated in Figures 6b and 6c.In Figure 6d the lower aerosol concentration can be seen toallow a greater net positive charge do develop in the upper

boundary layer, and a greater net negative charge in thelower boundary layer, while for the highest aerosol concen-tration of 6 1010 m3, the effects of space charge are verysmall, and there is only a small net positive ion excessthroughout the cloud layer. In Figure 6e the differences inthe amounts of positively and negatively charged aerosolvaries approximately as the total aerosol concentration.

Figure 6. Profiles for variation of aerosol concentration. Dotted lines are for total concentration ofaerosol particles 5 107 m3; dash-dotted lines are for 5 108 m3; dashed lines are for 5 109 m3;

and solid lines are for 6 1010

m3

. The parameters that are plotted are the same as in Figure 2.


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Figure 6f shows the mean charges on the droplet for the fourvalues of aerosol concentration, and it can be seen that formost of the boundary layer, the effect of aerosol concentra-tion on the droplet charge is small except for very highaerosol concentrations.

[42] The same small negative excursion in p at cloud topis present, as in Figure 4f, for the lowest aerosol concen-tration. In this case the effect of greater negative mobilityappears, not because of a wider boundary layer, but becausen1/n2 is closer to unity for the lowest aerosol concentrations,and the lower peak droplet concentration extends the regionof initial very low droplet concentration.

3.8. Effect of (Monodispersive) Aerosol Radius

[43] As noted in section 2, the aerosol size affects the ion-aerosol attachment coefficients (b, g) according to thecomplex formalism of Keefe et al. [1968], Hoppel [1985],and Hoppel and Frick [1986], and these in turn affect the

partition of the space charge between ions, charged aerosol particles and droplets. Figures 7a7f show model results;the dashed lines are for a = 10 108 m; the solid line isfor 4 108 m; and the dotted line for 2 108 m.The other parameters are: are zc = 4 km; q = 8 10

6 m3;Jz = 3 10

12 A m2; (SNA)max = 2 108 m3; ST = 5

109 m3; zb = 10 m.

[44] Generally, smaller particles have smaller attachmentcoefficients, and in Figure 7a, the reduction of aerosolradius causes the reduction of attachment coefficient andincrease in concentration of n0. As a result, in Figures 7band 7c, the reduction of aerosol radius causes the decreaseof the strength of the electric field, E. However, as in earlier

cases,@E

@zis essentially unchanged in the three cases. In

Figures 7d, 7e, and 7f the reduction of aerosol size andattachment diverts the space charge into the ions as itdecreases it in the charged aerosol particles, and increasesit slightly in the charged droplets.

3.9. Effect of the Difference in Ion Mobilities

[45] As has been mentioned earlier, the effect of thenegative ion mobility being greater than the positive ionmobility has caused the magnitude of the maximum positivevalue of p in the positive space charge region to exceed themagnitude of the maximum negative value of p in thenegative space charge region. Also, in the absence of spacecharge, the value of (S1 S2) is negative, while the value of(n1 n2) is positive. This can be seen for the droplet-freeregions above and below the cloud in Figures 1e, 1f, 1h, and1i and in Figures 3d, 3e, 4d, 4e, 5d, 5e, 6d, 6e, 7d, and 7e.In the uniform central region of the cloud the effects are not

Figure 7. Profiles for variation of (monodispersive) aerosol radius. Dashed lines are for radius 10 108 m;solid lines are for 4 108 m; and dotted lines are for 2 108 m. The parameters that are plotted are

the same as in Figure 2.


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as readily apparent, on account of the greatly decreased ionconcentration, but in Figure 6f, with low droplet concen-tration and a wide range of aerosol concentrations, p showsa small positive value (2.3e) for the largest ST of 6 10

10

m3, and a barely detectable negative value (0.5e) for thesmallest ST of 5 10

7 m3.[46] To illustrate these effects better, and minimize the

space charge, the model results in Figure 8 are for (SNA)max =1 108 m3; q = 1 107 m3; Jz = 1 10

12 A m2; withthe dotted lines for ST = 5 10

9 m3; the dashed line for1 1010 m3; and the solid line for 3 1010 m3. The other

parameters were: zc = 4 km; a = 4 108 m, zb = 10 m. Withthe expanded scale for p in Figure 8f it can be seen that forthe highest aerosol concentration (solid line) the meandroplet charge in the central region rises to about 1.6e, andat the extreme lower edge of the cloud, even in the presenceof some residual space charge, the average droplet chargerises to about 2.4 elementary charges.

[47] The parameters that determine the droplet charge aregiven in equations (10) and (11), so that with D1/D2 = 1.4/1.9, a positive value ofp will be present when n1D1 > n2D2,i.e., when n1/n2 > 1.357. This is the case in the centralregion for all of the aerosol concentrations in Figure 8.

While (n1 n2) is greater for the lower aerosol concen-trations, the ratio n1/n2 is greater for the highest, on accountof the lower values ofn1 and n2 (thus ofn0 in Figure 8a). InFigure 6f, n1D1 > n2D2 in the space charge free regions forthe largest aerosol concentration (6 1010 m3) but not forthe smallest (5 107 m3).

[48] Also, there are small regions of near-zero dropletconcentration and values of n1/n2 < 1.357, which givenegative p at the upper edges of the cloud. This occurs forthe ST= 5 10

9 m3 case in Figure 8f and also for the 10 kmcase in Figure 5f and the 5 107 m3 and 5 108 m3 casesin Figure 6f. This requires low aerosol concentrations or highion production rates, so that n1/n2 ! 1, enhanced by theabsence of large amounts of positive space charge, whichotherwise bring n1/n2 > 1.357 near the upper cloud edges.

4. Discussion

4.1. Dependence of Space Charge on Jz and on DropletConcentration

[49] Jz is controlled by the ionosphere potential Vi and thelocal atmosphere columnar resistance R, so that Jz is given

by Vi/R. There are variations in Vi on many timescales, from

Figure 8. Profiles to illustrate the consequences of the difference between positive and negative ionmobilities, with reduced ion production rate, reduced droplet concentration, and reduced current density,as described in the text. The results are for three values of total aerosol particle concentration. Dottedlines are forST = 5 109 m3; dashed lines are for 1 1010 m3; and solid lines are for 3 1010 m3.The parameters that are plotted are the same as in Figure 2.


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a diurnal variation in universal time to the Milankovichtimescales of orbital variation [Tinsley and Zhou, 2006].These variations are in response to both external andinternal forcings (cosmic ray and other space weatherinputs, and tropical climate change). R can vary by tensof percent in response to cosmic ray flux changes over the11-year solar cycle in the high geomagnetic latitude regions.Associated with these must be comparable variations in Jz.

[50] In most cases considered here, most of the attach-ment of ions within the clouds occurs on droplets ratherthan on aerosol particles, and so the mean ion concentrationn0 is determined mainly by the droplet concentration SNA asin equations (8) and (9). So in equation (19) (m1n1 + m2n2)depends approximately inversely on the droplet concentra-tion, and either a higher droplet concentration or a larger Jz

produces a proportionately larger E and larger@E

@zin the

boundary layer, as is illustrated in Figures 2c and 3c. Then from

equation (25), high@E

@zwill cause proportionately larger space

charge to appear, mainly on the droplets but with smaller

amounts on the charged aerosol particles and ions, as inFigures 2e and 2f and 3e and 3f.[51] Thus the enhancement ofJz in clouds with moderate to

high droplet concentration can increase the mean charge on thedroplets and aerosol particles (positive in the upper boundarylayer and negative in the lower layer). A 50% variation of Jzcan cause up to 40% variation of charge on the droplets and

particles. These variations can affect the collision process between the droplet and the particles that constitute ice-forming nuclei and condensation nuclei [Tinsley et al., 2000,2001, 2006], and may affect cloud cover and climate.

4.2. Dependence of Space Charge on Thickness ofBoundary Region

[52] The models presented here have been for stablecloud layers. However, localized downward (upward) airmotion in the upper (lower) boundary layer causes anarrowing of the layer, as seen, for example, in the sharpupper surfaces of convective clouds. A downdraft or updraftalso inserts higher-conductivity air into the cloud layer, thathas lower conductivity, and thus acts to increase the electricfield, in the same way as a mountain peak concentrateselectric field around its summit [see Reiter, 1992, pp, 158162]. This would increase Jz and the droplet charges. Thenarrowing of the boundary layer increases the gradient of

conductivity and of@E

@zas can be seen in Figure 4c and also

produces larger droplet and aerosol charges, as in Figures 4f

and 4e. The measurements of droplet charges in thickaltostratus and stratocumulus clouds by Beard et al.[2004] showed largest positive average droplet charges(p $ 85e) in downdrafts at cloud top and largest negativeaverage values (p $ 65e) in updrafts at cloud base.

[53] Harrison and Carslaw [2003, Figure 5] discussedthe inferred space charge from electric field measurementsmade by Reiter [1992, p. 197] above and below a stablestratus cloud in a valley, with the measurements being madeon instruments suspended below a cable car, ascendingthrough the cloud. Reiter showed electric field gradientsconsistent with conductivity gradients above and below the

visible boundaries of the cloud, which he described asvery thin stratocumulus cloud that from his sketch hada total thickness of about 50 m. The electric field andconductivity gradients imply space charge up to 120 mabove the upper visible boundary of the cloud. There is littledifference between the average conductivity or electric fieldwithin the designated cloud and a point 50 m above theupper surface. Below the cloud the conductivity increased

and electric field decreased uniformly over a distance of$190 m to the fair weather values. It appears that the spacecharge near the boundaries of this very thin cloud was notdetermined by gradients in conductivity caused by attach-ment of ions to cloud water droplets, but caused bygradients in the concentration of aerosol or haze or pre-condensation particles below and above the cloud. Asummary of a large number of similar valley stratus cloudsfrom the instrumented cable car is given by Reiter [1992,Figure 4.38] where he shows that a thickness of 500 m ofthese clouds is needed to produce an increase in electricfield (or reduction in conductivity) by a factor of two, andthat a thickness of less than about 40 m has no effect.

[54] Thus this cloud is not representative of the clouds we

have modeled here, or those observed by Beard et al.[2004]; clouds with relatively high liquid water contentand droplet concentrations 100 150 cm3, where thecharge is present on droplets inside the cloud boundaries.In particular, the width of the space charge regions near the

boundaries in the valley stratus is evidently much greaterthan is the case in dense stratocumulus clouds, whereupdrafts and downdrafts appear to produce the hard edgescharacteristic of considerably narrower boundary regions.

4.3. Dependence of Space Charge on Altitude of Cloud

[55] In the troposphere (except for the lowest 1 or 2 kmover land) the ion production rate is determined by thecosmic ray flux, which increases rapidly with altitude

[Tinsley and Zhou, 2006]. In the present model the ion production rate at 10 km is 4.9 times larger than that at4 km. The mobility also increases with altitude. Then theconductivity in and around the cloud at high altitude isconsiderably larger, and the electric field and its gradientsconsiderably smaller, than that at low altitude, as in

Figures 5a 5c. In equation (25), smaller@E

@zin the boundary

layers for the high cloud causes smaller net charge on the

aerosol and droplets, compared to the low-altitude case, as inFigures 5e and 5f. Thus a given value of Jz causes muchgreater accumulation of space charge on droplets andaerosol particles for low clouds as compared to high clouds,

and greater amounts of electroscavenging, as suggested byTinsley et al. [2000, 2001, 2006]. This greater amount ofcharging of low clouds compared to high clouds may bea factor in explaining the observations of Marsh andSvensmark[2000] that the greatest response of clouds overthe solar cycle is found in low clouds.

4.4. Dependence of Space Charge on AerosolConcentration and (Monodispersive) Radius

[56] The effect of changing aerosol concentration ismainly on ion concentration outside the cloud and on itsfringes (where the droplet concentration tends to zero)


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except when the aerosol concentration is so high thatattachment to aerosol dominates over attachment to dropletsin the space charge region. Since concentrations of 108 to109 m3 are typical of the 410 km level of the atmosphere[Tinsley and Zhou, 2006], the result of Figure 6, show thatvariations of aerosol concentration, except in heavily pol-luted situations, have a relatively minor effect on dropletcharges in the space charge region.

[57] The effect of reducing the (monodispersive) aerosolradius is, on the whole, similar to that of a reduction inaerosol concentration, as can be seen by comparing Figures6 and 7. This is on account of the reduced attachment rateswhen either concentration or size is reduced. There aredepartures from this similarity in the areas outside and onthe fringes of the cloud, where differences in the ratio ofattachment coefficients of positive and negative ions, de-

pendent on aerosol particle radius [Hoppel and Frick, 1986]have significant effects.

4.5. Effect of a Polydispersive Aerosol Size Distribution

[58] To consider how the present results would be affectedby the presence of a polydispersive aerosol size distribution

instead of the monodispersive one used, we consider theresults of Hoppel and Frick [1986], whose formalism wasused in the present simulations, and the results of Yair and

Levin [1989]. Hoppel and Frick [1986] considered a two-component aerosol dominated by a monodispersivecomponent with a = 0.02 mm, and used values ofm1 andm2 of 1.20 10

4 and 1.35 104 m2 V1 s1,respectively.

[59] With the attachment to particles dominating over ion-ion recombination, they obtained a ratio n1/n2 = 1.310. Withthis ion concentration ratio considered as fixed and deter-mining the charging of a second monodispersive compo-nent, with radii up to 1 mm, they showed that for particleswith radii greater than 0.02 mm there would be more

positively than negatively charged particles; that is, the particles would have a mean charge p that was positive.Also, the larger particles had larger mean charges. Thus the

present simulations, where the large particles are 10 mmradius droplets, are consistent with their results.

[60] Yair and Levin [1989] took the simulations to agreater level of complexity, by calculating the distributionof charges as a function of particle size for several poly-dispersive aerosols, where the ion asymmetry ratio, n1/n2, isdetermined in a self consistent way by the varying attach-ment coefficients as a function of particle size and thevalues ofm1 and m2. As in the work of Hoppel and Frick[1986], for a situation where the dominant particles in thedistribution have a radius comparable to the ion mean free

path, and the temperature is about that of the loweratmosphere, then with D2 larger than D1 the dominant small

particles charge negatively on average. If the concentrationof aerosol particlesis relatively high so that (S1 + S2) ) (n1 + n2)with negligible ion-ion recombination, then the formalismleads to an ion asymmetry ratio n1/n2 that is greater than

D2/D1, the inverse of the diffusion coefficient ratio, so thatD1n1 > D2n2.

[61] For the larger particles in a polydispersive distribu-tion the mean number of charges on each tends (with

increasing radius compared to mean free path), to thelimiting value given by equations (10) and (11):

p 1:157 107 A ln D1n1= D2n2 33

for the atmospheric temperature at 4 km altitude.[62] Thus, with D1n1 > D2n2, the mean number of charges

on the larger particles will be positive, although the spacecharge remains zero. This also requires that the larger

particles have a small enough concentration so that evenwhen they take up multiple positive charges, their share ofthe positive charge does not reduce the ion asymmetry ratio

below the inverse of the diffusivity ratio.[63] The work of Hoppel and Frick [1986, Figure 9]

provides a guide as to the particle radius with which thislimit is approached. They show results for the equilibriumion ratio n2/n1, for a monodispersive aerosol for whichalmost all of the charge is on the particles, and a negligibleamount on the ions, and where there is no space charge (sothat p ! 0). In this case the aerosol radius for which thelimit of equation (33) is approached is when n2/n1approaches the ratio D

1

/D2

(0.89 in this case). This limit,for charge on the particle as a function of radius given by(33), is met within about 1% for particles of radius 1 mm orgreater.

[64] This means that for mixtures of droplets with a polydispersive aerosol, provided that the total charge onthe larger aerosol particles is small compared to that on thedroplets, then the charges on the droplets and larger aerosol

particles are all given, to a degree of approximation, whichimproves with aerosol size, by equation (33). The aboveresult, i.e., that the charges are approximately proportionalto the radii, applies whether or not the actual ion asymmetryratio is caused by space charge.

[65] For our simulations, when the 10 mm radius droplets

have charges of positive or negative magnitude of 50e, thento a good approximation the charges on aerosol particles of1, 2, and 5 mm would be 5e, 10e, and 25e, respectively. Forthese particles, and for somewhat smaller ones, the chargesare sufficient to significantly increase the rate of them beingelectroscavenged by droplets [Tinsley et al., 2001, 2006].For the subset of these particles with properties of contactice nuclei, the resulting primary ice nucleation would tendto increase rates of precipitation in cold clouds. Also, suchscavenging of the larger CCN would narrow the CCN sizedistribution, and this would subsequently narrow the dropletsize distribution in subsequent cycles of cloud formation,and tend to reduce rates of precipitation in both warm andcold clouds.

4.6. Other Effects

[66] The simulations presented in Figures 1 8 were fortotal aerosol concentration ST constant with height z, withonly SNA varying in the cloud boundary regions. Theseresults suggest that for isolated layers of aerosol, positiveand negative space charge, of magnitude proportional to Jz,would similarly develop at the upper and lower boundaries.At least for the upper boundary of the tropospheric regionwhere upward mixing of aerosols from the surface takes

place (the top of the mixing, or tropospheric boundary


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layer) observations of electric field variations show thepresence of positive space charge [Sagalyn and Faucher,1954; Reiter, 1992].

[67] Multiple layers of aerosol distributed in altitude aresometimes observed, especially after volcanic eruptions,and generally multiple layers of clouds are common. Jz,will flow through such layers, producing space charge ineach. It is not necessary that the layers be distinct; vertical

gradients in ion production or in the concentration ofaerosol particles, or of droplets within the clouds, will alsoresult in space charge.

[68] The aerosol in the atmosphere in general is poly-dispersive, and thus the effects described in section 4.5would apply, with the larger aerosol particles tending tocharge positive, and the smaller ones negative, with thedistribution of charge between them depending on theamount of space charge. Such differential charging, if largeenough, may affect the rates of coagulation of small aerosol

particles onto larger ones, with the effects depending on themagnitude of Jz.

[69] For ultrafine aerosol particles near the boundaries ofclouds, where there is a supply of volatiles from evaporating

droplets that can allow ultrafine particles to grow largeenough to act as cloud condensation nuclei, the presence ofspace charge, depositing like charges on the ultrafine

particles and droplets, may act to preserve these particlesagainst scavenging on droplets. This protection and en-hancement of the small CCN component, together with thenarrowing of the distribution due to the removal by electro-scavenging of large CCN, would tend increase dropletconcentrations and reduce droplet radii in subsequent cyclesof cloud formation, reduce rates of coagulation and precip-itation, and also increase cloud cover (the Twomey effect)[see Tinsley, 2004].

[70] Where evaporation of charged cloud droplets isoccurring at the boundaries of clouds, the presence of spacecharge not only generates the charge on the droplets beforeevaporation, but serves to slow the decay of the charge onthe residual aerosol particle, beyond the initial decay timeconstant of about 10 min. The final charge is determined

by the amount of space charge, and as noted earlier, couldbe $5e for 1 mm particles. Thus the evaporation residue hascharges of tens of elementary charges for periods of 10 min,and depending on radius, decaying down to several ele-mentary charges for as long as the space charge persists.These evaporation residues are considered to be efficientice-forming nuclei [Beard, 1992] and this characteristic,along with the electrically enhanced scavenging, may con-tribute to significant increases the rates of contact icenucleation [Tinsley et al., 2001].

5. Conclusions

[71] The variable flow of ionosphere-Earth current den-sity Jz generates variable amounts of space charge near theupper and lower boundaries of clouds. We have modeled the

production of space charge and its partitioning betweendroplets, aerosol particles, and atmospheric ions, for a rangeof relevant atmospheric variables. The results give averagecharges on droplets of 50 to 100 elementary charges;

positive at cloud top and negative at cloud base, whichsupports the recent measurements of Beard et al. [2004].

The effects of moderate downdrafts at cloud top andupdrafts at cloud base may be necessary to provide a narrowenough boundary layer together with a more concentratedcurrent density, to generate the observed charges.

[72] When taken with models of electrical effects oncloud microphysics, the results are consistent with obser-vations of meteorological responses to processes affecting

Jz, and suggest that the accumulation of space charge in

layer clouds may affect cloud microphysics sufficiently tocause measurable effects on weather and climate.

Appendix A: Variation in Ion Mobility WithAltitude

[73] The ions diffuse through the air and are accelerated by the ambient electric field between collisions with airmolecules. We consider the case appropriate to the motionof ions in the fair weather electric field, where the concen-tration of ions is small compared to that of the air mole-cules, and the energy gained from the electric field, betweencollisions, is small compared to the thermal energy. Theions gain an increment of energy between collisions, but

lose it as the collisions randomize and thermalize the ionmotion. The ion gains a drift velocity v small compared tothe mean thermal velocity. The mobility is m defined as

m vE A1

where E is the applied electric field. The increments ofvelocity gained between collisions, which average to v,depend both on the mean thermal velocity and on the meanfree path. In turn, the mean free path depends on thecollisional cross section for ion collisions with airmolecules, and thus on the effective size of ions andmolecules in the collisions, which vary with the relativevelocity between the ions and molecules, and thus on thetemperature. The mean free path depends inversely on theconcentration of air molecules. These factors are alsocharacteristic of diffusion of one species of gas throughanother, and it was first shown by Townsend [1900], andmore recently by Chapman and Cowling [1970, section19.12] and McDaniel and Mason [1973, section 5-1-A] that

m eD= kT A2

where D is the diffusion coefficient for ions through air, e isthe elementary charge, and k is Boltzmanns constant. It wasshown by Chapman and Cowling [1970, section 14.5] thatfor diffusion of two gases through each other, where the

molecular cross sections vary similarly with collisionenergy, the diffusion coefficient D and the coefficient ofviscosity h are related by

D const h=r A3

where r is the gas density, determined by pressure (P) andtemperature:

r P=T A4

[74] For molecules considered as rigid elastic spheres ofdiameter s and mass m, Chapman and Cowling [1970,


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section 12.1], using different symbols, give the first approximation to has

h5

16s2kmT

p

12

A5

which gives

h

ho

T

To

12

A6

so that using (A2), (A3), and (A4), and for rigid elasticspheres,

m

mo

Po

P

T

To

12

A7

where the subscripts (o) refer to reference conditions, whichcould be STP (T = 273.15 K, P = 1013.25 hPa) as used in

physics and chemistry, or T = 288.15 K, P = 1013.25 hPa,

as used in standard atmospheres. The diffusion coefficientsand viscosities are difficult to derive theoretically, as it isnecessary to allow for the variation of collision cross sectionwith collision energy. At a higher level of accuracy,empirical determinations of D orhwith temperature, fitted

by theoretical expressions, are used.[75] An improvement on the rigid elastic spheres model

is Sutherlands model [ Bircumshaw and Stott, 1929;Montgomery, 1947; Chapman and Cowling, 1970, section12.32], where

h

ho

T

To

32 To S

T SA8

where Sis Sutherlands constant; an adjustable parameter toensure agreement with experiment. S is determined byviscosity measurements, and depends on the species of gasinvolved. Assuming that the cross section for ions in airvaries in a similar way with temperature as that of othergases in air, we adopt the value for S for air, given by

Rogers and Yau [1989] of S = 120 K, consistent with thevalue from Bircumshaw and Stott[1929]. Then, using (A2),(A3), and (A4),

m

mo

Po

P

T

To

32 To S

T SA9

[76] At this level of approximation, an alternative ap-proach would be to use the empirically determined temper-ature variation of the diffusion coefficient [ Boynton and

Brattain, 1929]

D

Do

Po

P

T

To

nA10

with the value of n varying with the species of gas. Fordiffusion of water vapor in air the value ofn = 1.81 has been

obtained [Montgomery, 1947; List, 1966]. The use of thisexpression in (A2) gives

m

mo

Po

P

T

To

0:81A11

which agrees to better than 1% with (A9) over the range

200320 K.[77] Strictly speaking, at the highest level of accuracy amore complex fit to empirical data for D should be moreappropriate than a fit to data for h, especially for very lowand very high temperatures [McDaniel, 1989, section 1.6].However, for the temperature range considered here, ade-quate accuracy is obtained by using (A9).

[78] A completely different approach to specifying thevariation of ion mobility with height is based on the conceptof reduced mobility, where a value for m is determinedexperimentally, for specified conditions of pressure P andtemperature T, and reduced to STP, i.e., to a value of moat STP, by means of the formula

mo PPoToTm A12

[79] As discussed by McDaniel and Mason [1973, section1.3], this corrects only for the variation of mobility with gasconcentration (number density), and is applicable for meas-urements at a temperature close enough to STP that thevariation of mobility with temperature (at constant density)can be ignored. The reversal of the use of this equation, togive a value ofm in terms ofmo, P, and T; that is,

m Po

P

T

Tomo A13

would only be valid for a small range of temperature aroundTo, as in work by Bricard [1965]. To use it for a theoreticalmodel of ion mobility throughout the troposphere, strato-sphere, and lower mesosphere as was done by Shreve[1970], is not appropriate.

[80] One could argue that the assumption implicit inequation (A9), that the collision cross section of air ionswith air molecules behaves like that of other gases diffusingthrough air, leads to uncertainties in the temperature varia-tion such that there is little improvement in accuracy to begained by using (A9) rather than (A13). However, there areadditional problems with regard to the results of Shreve[1970]. He made an exact calculation of mobility withrespect to altitude using (A13), together with the temper-

atures and pressures vs. altitude of the U.S. StandardAtmosphere (1962), and values of positive and negativemobility m1 and m2 at STP. These STP values werefrom work by Bricard [1965] and were 1.4 104 m2 V1

s1 and 1.9 104 m2 V1 s1, respectively. Shreve thengenerated the fitted expressions, m1(z) = 1.4 10

4 exp(0.14z) m2 V1 s1 and m2(z) = 1.9 10

4 exp (0.14z) m2

V1 s1 where z is in km, and is the altitude above the zeroat sea level in the standard atmosphere. These expressionsshow that the values from Bricard were used by Shreve forthe temperature at zero altitude in the standard atmosphere


16 of 17

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17/17

(288.15 K) rather than the STP temperature of 273.15 K,and his designation as STP values is misleading.

[81] An additional concern is that Shreves expressionsfor the altitude variation of mobility were fitted to the wholealtitude range from 0 to 60 or 70 km, whereas the exactcalculation departs significantly from the log linear fittedcurve near the 10 km region and the 50 km region.

[82] We have recalculated the height variations of the

mobilities m1(z) and m2(z) for the pressures and temperaturesof the U.S. Standard Atmosphere (1976) using the Bricard[1965] values for m1 and m2 at zero altitude, and equation(A9) for the variation. The linear fits to the curves are givenfor three height ranges, to better fit the nonlinearity in thecurves of the calculated variation:

z< 10 km; m1 z 1:40 104 exp 0:111z m2 V1 s1

m2 z 1:90 104 exp 0:111z m2 V1 s1

10 < z < 50 km m1 z 4:25 104

exp 0:154 z 10 m2 V1 s1

m2 z 5:76 104

exp 0:154 z 10 m2 V1 s1

50 < z < 60 km m1 z 2:01 104

exp 0:099 z 50 m2 V1 s1

m2 z 2:72 104

exp 0:099 z 50 m2 V1 s1

[83] The above height variations are appropriate for usewith standard atmosphere models. They do not, however,consider the changing composition and structure of the ionsas a function of absolute humidity [Fujioka et al., 1983] or

changes in the amounts of trace constituents such as HNO3,NH3, and H2SO4, which can replace H2O molecules in thecluster surrounding the ion. Such processes are involved inthe process of ion mediated nucleation [Yu and Turco,2001], especially in the downward branch of the strato-spheric Brewer-Dobson circulation, where the air can be-come supersaturated in H2SO4 vapor [Tinsley and Zhou,2006].

[84] Acknowledgments. This work has been funded by grant ATM-0242827 by the U.S. National Science Foundation. Limin Zhou has beensupported in part by a fellowship from the Chinese Academy of Sciences.

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L. Zhou, Department of Geography, East China Normal University, 3663North Zhongshan R. D., Shanghai, China.

D11203 ZHOU AND TINSLEY: SPACE CHARGE IN CLOUDS D11203

tin space charge

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