Numerical study for production of space charge within the stratiform cloud

Numerical study for production of space chargewithin the stratiform cloud

A K Srivastava1,∗ and S N Tripathi∗∗

Department of Civil Engineering, Indian Institute of Technology Kanpur, Kanpur 280 016, India.1Current address: Indian Institute of Tropical Meteorology (Branch), Prof Ramnath Vij Marg,

New Delhi, India.∗e-mail: [email protected]

∗∗e-mail: [email protected]

One dimensional numerical model has been developed to predict the production of space chargeand variations in other electrical parameters within the low level stratiform type of cloud havingvery weak vertical motion. Non-linear coupled differential equations which govern ion concentra-tions, charged and neutral droplet concentrations and electric field were used. Symmetry has beenobserved in all the electrical parameters within the cloud. The magnitude of average positive ionconcentrations was observed to be high as compared to the negative ion concentrations, which isdue to low scavenging rate of positive ions than the negative ions, highly attributed to their mobili-ties. The rate of scavenging of ions affects the concentration of charged droplets, which eventuallyinfluence the electric field and thus the space charge density within the cloud. Maximum electricfield (Emax) was observed at middle of the cloud whereas minimum was observed at both the edgesof the cloud. Minimum electric field (Emin) was found to be equal and constant (∼27Vm−1) for anydrop concentration. Net positive and negative space charges were observed at the top and bottomof the cloud, respectively. The simulated results show some discrepancies to the natural condition,which are due to simulations made under some basic assumptions and limitations and that will beincorporated in the future studies for natural cloud condition.

1. Introduction

The Earth’s climate could be affected by changesin cloudiness caused by variations in the intensityof galactic cosmic rays (GCRs) in the atmosphere(Svensmark and Friis-Christebsen 1997; Carslawet al 2002; Siingh and Singh 2009 with referencestherein). The effect of cosmic rays on climate couldbe in three ways:

• through changes in the concentration of cloudcondensation nuclei,

• thunderstorm electrification, and• ice formation in cyclones.

The concentration of cloud condensation nucleidepends on the light ions produced during cosmic

ray ionization (Farrar 2000; Carslaw et al 2002).Svensmark et al (2007) based on a laboratoryexperiment in which a gas mixture equivalentto chemical composition of the lower atmospherewas subjected to UV light and cosmic rays,reported that the released electrons promotedfast and efficient formation of ultra-fine aerosolparticles which may grow to become cloud con-densation nuclei. Aircraft-based ion mass spectro-meter measurements in the upper troposphere haveshown an evidence for cosmic ray induced aerosolformation (Solanki 2002; Mouel et al 2005). Obser-vational studies show that the cosmic ray andcloud effect seems to be present largely in thelow-altitude clouds, because low-altitude cloudsexert a large net cooling effect on the climate;

Keywords. Stratiform cloud; space charge; electric field; attachment coefficient.

J. Earth Syst. Sci. 119, No. 5, October 2010, pp. 627–638© Indian Academy of Sciences 627

628 A K Srivastava and S N Tripathi

this determines the sign of the possible cosmic rayand cloud effect (Svensmark and Christensen 1997;Marsh and Svensmark 2000). Although the low-altitude clouds of limited vertical extent may onlycontain one phase of liquid water, it is a key para-meter in the numerical climate models used forfuture climate change predictions. Kirkby (2007)had presented an association of high GCR flux withcooler climate, and low GCR flux with a warmerclimate.

Clouds having both convective and stratiformtypes have different electrical properties. To dateseveral numerical schemes have been constructedto understand the electrification and the produc-tion of space charges within the convective clouds,which have strong vertical motion and strong elec-tric field (Chiu and Klett 1976; Masuelli et al 1997;Helsdon et al 2001; Altaratz 2003; Saunders 2008).However, no numerical studies, to the best of ourknowledge, have been carried out to understandthe production of space charge within the low levelstratiform type of cloud having very weak verticalmotion and weak electric field inside it.

Electrical structures of weakly electrified cloudssuch as stratus are less well known, which hasbeen a subject of investigation (Tinsley et al 2000;Harrison and Carslaw 2003; Tinsley and Yu 2004).Stratus is a low level stratiform cloud mainly com-posed of water droplets with little vertical thick-ness but tend to be spread out into a thin sheet,having maximum thickness of about 1 km fromtop to bottom and large aerial coverage. It hasbeen conjectured that inside the clouds, whereconductivity reduces by about an order of magni-tude with respect to clear air (Rust and Moore1974), large space charge (up to few tens tothousands elementary charges per cm−3) can accu-mulate (Tinsley 2000). The space charge, mostlythe difference between uni-polar charges, is impli-cated for droplet charging by various processesnotably amongst them is preferential scavengingof aerosol particles (electro-scavenging) (Tinsley2000). The space charge increases with the electricfield in the low-conductivity region within thecloud, thereby restoring the equilibrium verti-cal conduction current. Thus, although electricaleffects are much weaker in stratiform clouds thanin convective, they are certainly present and theelectric fields and charge densities are modulatedby cosmic rays (Carslaw et al 2002).

In this paper, we have made an attempt todevelop a model for the production of space chargeand to understand the electrical parameters, suchas electric field within the low level stratiformcloud using non-linear coupled differential equa-tions, which govern the ion concentration, chargedand neutral droplet concentration and electric fieldvariation inside the cloud. The detailed model

Figure 1. Schematic diagram of production of uni-polarspace charge due to processes occurring in the presence ofcosmic rays along with the variation in conductivity withinthe stratiform cloud. Total air-Earth’s current density (Jz),penetrating from top of the cloud, is considered constantwith height.

description along with assumptions and initialboundary conditions are given in the next section.

2. Model descriptions

2.1 Physical understanding

Stratiform clouds could be electrified because theirpresence in the free atmosphere reduces conducti-vity (Rust and Moore 1974; Pruppacher and Klett1997) and the influence of space charge accumu-lation on their boundaries due to variations inair-Earth’s current density (Tinsley 2000), causingchanges in cloud microphysics. Figure 1 showsa schematic diagram based on the hypothesisproposed by Tinsley (2000) for the production ofuni-polar space charge within weakly electrifiedstratiform type of cloud. It shows total air-Earth’scurrent density (Jz = Jz1

+ Jz2), a product of

conductivity and electric field, penetrating fromtop of the cloud, is constant with height rangingfrom ∼1–4 pA m−2 into and out of each volume.However, Jz1

(positive current density) and Jz2

(negative current density) vary with height due torecombination and attachment of ions that occurin the same volume. Thus, the regions of net uni-polar space charge often exist on the upper andlower boundaries of the cloud. Schematic diagramalso shows variation in conductivity that reduceswithin the cloud (σc) as compared to the clear air(σa) due to scavenging of ions by their attachmentwith cloud droplets.

2.2 Theoretical formulation

A 1-D numerical model, to study the production ofspace charges and other electrical properties such

Production of space charge within the stratiform cloud 629

as electric field, has been developed for low levelstratiform clouds having weak electric field andweak vertical motion. The non-linear system hasbeen modeled using coupled differential equationsfor the rate of change of ion concentration, chargedand neutral droplet concentration and electric fieldvariation with height.

The time dependent conservation equation forpositive ions represented by 1 and negative ionsrepresented by 2 is given as:

dn1

dt= q − αn1n2 − β1(Sn1 + S2n1) −

1

e∇Jz1

, (1)

dn2

dt= q − αn1n2 − β2(Sn2 + S1n2) +

1

e∇Jz2

. (2)

Here n1,2 are positive (negative) ion concen-tration, q is the ion production rate, α is theion–ion recombination coefficient, β1,2 are posi-tive (negative) ion attachment coefficients withdroplets, S1,2 are charged droplet concentrationshaving unit positive (negative) charge, S is theneutral droplet concentrations, e is the electroniccharge and Jz1,2

is vertical current density.The above two equations are balanced by the

production of ions due to ionization from cosmicrays, i.e., q and loss of ions by various mecha-nism such as ion–ion recombination, which playa crucial role in ultra-fine particle formation inthe lower atmosphere (Mohnen 1977; Yu andTurco 2001), and ion-droplet attachment withinthe cloud (Griffiths et al 1974). The last termof equations (1) and (2) indicates flux of posi-tive (∇Jz1

) and negative (∇Jz2) ions, respectively

which produces an imbalance between both theions in a region. Direct measurement of total cur-rent density, Jz is difficult, therefore it is usu-ally derived from the simultaneous measurementsof conductivity and electric field and can beexpressed as:

Jz = Jz1+ Jz2

= σz1Ez + σz2

Ez. (3)

By such an approach, Jz is found to be constantvertically in the stable and fair weather conditions(Gringel et al 1986; Siingh et al 2007 and the refe-rences therein). In equation (3), σz1

and σz2are the

positive and negative ion conductivity, respectivelywhich can be expressed in terms of ionic numberdensity (n1, n2) and mobility of ions (µz1, µz2). Thetotal current density from equation (3) becomes:

Jz = (µz1n1e + µz2

n2e)Ez;

Ez =Jz

e

(

1

µz1n1 + µz2

n2

)

. (4)

Equation (4) can be used as an initial value inthe model for the electric field at cloud boundariesusing total current density (Jz) penetrating fromtop of the cloud.

Now, the last term of equations (1) and (2) canbe written as:

1

e∇Jz1

=1

e∇(µz1

n1eEz);

1

e∇Jz2

=1

e∇(µz2

n2eE2). (5)

For horizontal stratification and at steady state,the above equations can be written as:

1

e∇Jz1

=1

e

∂

∂z(Jz1

)=µz1n1

∂

∂z(Ez)+Ez

∂

∂z(µz1

n1);

1

e∇Jz2

=1

e

∂

∂z(Jz2

)=µz2n2

∂

∂z(Ez)+Ez

∂

∂z(µz2

n2).

(6)

Using the values from equation (6), equations (1)and (2) becomes:

dn1

dt= q − αn1n2 − β1(Sn1 + S2n1)

− µz1n1

∂

∂z(Ez) − Ez

∂

∂z(µz1

n1); (7a)

dn2

dt= q − αn1n2 − β2(Sn1 + S2n1)

+ µz2n2

∂

∂z(Ez) + Ez

∂

∂z(µz2

n2). (7b)

S1 and S2 are the number concentration of singlypositive and negative charge droplets, respectively.The time dependent balance equation for chargedparticles is depicted by Harrison and Carslaw(2003). However, by ignoring positive ions attach-ment to the positively charged droplets and nega-tive ions attachment to the negatively chargeddroplets, the balance equations for single positiveand single negative charge drops can be written inequations (8) and (9), respectively as:

dS1

dt= β1Sn1 − β2S1n2; (8)


dS2

dt= β2Sn2 − β1S2n1. (9)

The first term of both equations represent theproduction of positively and negatively chargeddroplets due to the attachment of positive andnegative ions with neutral droplets, respectivelywhereas the last term represents loss of posi-tively and negatively charged droplets due to theirattachment with negative and positive ions, respec-tively. S is the neutral droplet concentration withinthe cloud, which can also be written by usingabove two equations (8) and (9) as:

dS

dt= (β2S1n2 +β1S2n1)− (β1Sn1 +β2Sn2). (10)

The first bracket indicates the production ofneutral droplets by neutralizing the positively andnegatively charged droplets due to their attach-ment with negative and positive ions, respectively.However, the second bracket represents loss of neu-tral droplets due to their attachment with positiveand negative ions, respectively.

The characteristic of ions that relates directlyto their ability to take part in the flow of electriccurrent, is their mobility (µ), which is the aver-age velocity acquired by small ions as it movesunder the force exerted on it by the electric field.Ion mobility controls the rate of uptake of ions bythe aerosols and droplets. Mobility varies inverselywith the mass of cluster of ions (Harrison 1992);

µ =2 × 10−4

√

(M + 29/M), (11)

where M is the mass of ions.In the present cloud model, H+(H2O)m is used

as positive ion (µ1) and O−

2 (H2O)n as negative ion(µ2) where ‘m’ and ‘n’ are the hydrated watermolecules, vary in such a manner that both the ionmobilities are maximum at the edges of the cloudand decreasing inside with minimum at the middleof the cloud, as shown in figure 2.

The chemical differences between the species ofpositive and negative ions lead to some physi-cal asymmetries in the ion properties and causenegative ions to have greater mobility than posi-tive ions, i.e., µ2 > µ1 (Mohnen 1977). The magni-tude of the difference is typically ∼20% but varieswith humidity in response to changes in the ionhydration as well as ion composition (Harrison andCarslaw 2003). This asymmetry is central to thecloud electrification. In the present study, averagepositive ion mobility was observed to be ∼7% less

Figure 2. Variation in positive (µ1) and negative (µ2) ionmobilities inside the cloud.

as compared to the average negative ion mobilitywithin the cloud. The positive and negative ionmobility at the cloud boundaries were calculated tobe maximum ∼1.3 × 10−4 and 1.5×10−4 m2V−1s−1

using low hydrated water molecules, i.e., m = 4 andn = 1, respectively which approaches to minimum∼1.0 × 10−4 and 9.75 × 10−5 m2V−1s−1 using largehydrated water molecules, i.e., m = 6 and n = 3,respectively at middle of the cloud. The differencebetween both the ionic mobilities are high at theboundaries where negative ion mobility was ∼15%more than the positive ion mobility and low atthe middle of the cloud where negative ion mobi-lity was ∼4% more than the positive ion mobility.To get the mobility values at each point withinthe cloud, a second order polynomial was fitted forboth the ionic mobilities with step size (X) and thepolynomial equations are:

Y = 1.20648 × 10−4− 9.30144 × 10−8

× X + 9.30144 × 10−11× X2, (12a)

Y = 1.34547 × 10−4− 1.34673 × 10−7

× X + 1.34673 × 10−10× X2. (12b)

Equations (12a) and (12b) predict a decrease inboth positive and negative ion mobilities by ∼25%and 33%, respectively from cloud boundaries to themiddle of the cloud.

The rate of uptake of ions by cloud droplets isknown as collision rate, which leads to charging ofthese droplets. It is normally expressed in terms ofattachment coefficient, which varies with height asa function of ionic mobility. The attachment coeffi-cient, primarily, depends on droplet size and thenumber concentration of ions of both polarities.


It is also said to depend on the magnitude ofcharges carried by the droplets. However, whencloud droplets act as collectors of ions, the collec-tion current is independent of the droplet chargefor typical conditions (Pruppacher and Klett 1997,pp 798). So, the charged droplets of either signand the neutral droplets should all collect ions atthe same rate that only depends on the propertiesof the ions. Attachment coefficients, β1,2 for posi-tive (negative) ions with cloud droplet of radii ‘ac’were given by Griffiths et al (1974); Chiu (1978);Pruppacher and Klett (1997):

β1,2(ac) = 4πacD1,2, (13)

where D1,2 = (µ1,2KT/e) is the diffusion coefficientof positive (negative) ions.

Thus, equation (14) becomes:

β1,2(ac) =4πacKTµ1,2

e, (14)

where K is the Boltzmann constant and T is thetemperature.

The radius of cloud droplets of average size ‘ac’can be computed from a relationship given by Chiu(1978):

ac =

(

3

4

ρlcπρwS

)1/3

, (15)

where ρ is the density of air (∼1.239 kg m−3), lcis the liquid water content within the cloud, ρw isthe density of water (∼1000 kg m−3) and S is thedrop concentration within the cloud. Equation (16)reveals that the average drop radii will decreasewith increase in drop number concentration withinthe cloud.

The majority of free charges in the fair weatherpart of the atmosphere arise from natural ioniza-tion, e.g., cosmic rays (Chalmers 1967; Siingh andSingh 2009), which is more dominant activity awayfrom the surface. The space charge, in the fairweather, is the difference between the concentra-tion of positive and negative ions. However, withinthe clouds most of the positive and negative ionswill be in the form of charged droplets (Tinsley2000). Therefore, space charge, in the above case,will be the sum of differences between the con-centration of positive and negative ions and theconcentration of positively and negatively chargeddroplets, which is given as:

ρ = e[(n1 − n2) + (S1 − S2)]. (16)

Variations in electric field (E) and spacecharge (ρ) are coupled by Poisson’s equation(∇ · E = (ρ/ε)); where ε is the permittivity ofvacume (∼ 8.85419e − 12F m−1). Thus, assuminghorizontal stratification, the Poisson’s equation canbe written as:

∂Ez

∂z=

ρ

ε=

e

ε[(n1 − n2) + (S1 − S2)]. (17)

To develop a 1-D numerical model to under-stand the electrical properties and the productionof space charge inside the stratiform cloud, wehave solved equations (7a), (7b), (8), (9), (10) and(17) simultaneously, using Gauss–Seidel iterativemethod (Jain 1984) to get solution for enough timeto come to a steady state. The methodology ofsolving these equations will be discussed in the nextsection, which also includes basic assumptions andthe boundary conditions of the model.

2.3 Methodology

The above equations were discretized usingpurely implicit finite difference scheme, whichenables the integration of a differential equationnumerically by evaluating the values of the func-tion at finite number of points. All the equationswere integrated for sufficient time until steadystate condition is reached. Finite difference scheme,applied here, has been solved using false tran-sient method in time step, i.e., the steady statetime computed by the program is independent ofactual time. The origin of this method is Taylorseries expansion, which assumes that the func-tion is smooth, i.e., continuous and differentiable.Time derivative is discretized using forward differ-ence and space derivatives with backward/forwarddifference usually called forward time backwardspace (FTBS) and/or forward time forward space(FTFS) method of discretization (Ghoshdastidar1998).

The model presented here is for stable cloudlayer, i.e., no vertical or horizontal motions existwithin the cloud. The basic assumption made whilesolving these equations is: cloud contains mono-dispersed droplets (i.e., all with the same radius),representing the dominant size category, with atmost a single positive and/or negative charge.Thus, β (ion attachment coefficient with droplet)in each case apply to the radius of the dominantdroplets. The boundary conditions of the modelincludes:

• equal and maximum positive and negative ionconcentration,


Table 1. Typical values of the atmospheric parameters usedto calculate the initial boundary conditions of model.

Atmosphericcomponents Typical values References

q ∼1 × 107 ion Schonland (1953)pair m−3 s−1

α ∼1.6 × 10−12 m3 s−1 Schonland (1953)

Jz ∼3 pA m−2 Harrison (2005)

Table 2. Average drop concentrationsand the corresponding radii within thecloud for a constant liquid water con-tent ∼0.3 g kg−1 within stratus cloud(Chiu 1978).

Average drop Drop radiiconcentration (m−3) (µm)

1.0e8 9.5

5.0e8 5.6

1.0e9 4.5

• total drop concentration are zero at top andbottom of cloud, consequently,

• total charged and neutral droplet concentrationare also zero at both the cloud boundaries.

n1,2(0, t) = n1,2(N, t) =( q

α

)1/2

, (18)

E(0) = E(N) =Jz

e(µ1n1 + µ2n2), (19)

S1,2(0, t) = S1,2(N, t) = 0, (20)

S(0, t) = S(N, t) = 0, (21)

where ‘0’ and ‘N ’ represents first and last boundaryof the cloud. S is the total drop concentration whenthey are uncharged at time t = 0, which we haveassumed to be varied sinusoidally within the cloudas S = S0(sin(π∆z/Z))2 with maximum concen-tration at middle of the cloud. S0 is the maximumdrop concentration at middle of the cloud; Z is thetotal thickness of the cloud (1000 m in the presentsimulation) and ∆z is the small increment (1 m).Typical value of the atmospheric parameters, usedto calculate the initial boundary conditions for ionconcentration and electric field, is given in table 1.

3. Results

The variations in ionic concentration, within thecloud, are extensively attributed to the variations

Figure 3. Variation in total drop concentrations within thecloud at different drop sizes.

in drop number concentration and their sizesthrough scavenging due to attachment process thatinfluence the concentration of charged droplets andthus the production of space charge and electricfield (Harrison and Carslaw 2003). The experimen-tally determined cloud droplet number concentra-tions are to be below 1.0e8 m−3 and rarely above1.0e9 m−3 within the stratiform clouds (Andersonet al 1994; Gillani et al 1995). In the presentstudy, we have considered three different sets ofcalculation assuming three different average dropconcentrations 1.0e8, 5.0e8 and 1.0e9 m−3 havingmono-dispersed drops of average drop radii ∼9.5,5.6 and 4.5µm, respectively, which were calculatedfrom equation (16) keeping constant liquid watercontent ∼0.3 gm kg−1 within stratus cloud (Chiu1978) and shown in table 2.

3.1 Total drop concentration

The drop concentration within the cloud is oneof the important parameter, which influences thecloud electrification. The variation in total dropconcentrations within the cloud having three dif-ferent drop sizes (mono-dispersed) are shown infigure 3, which indicates maximum drop concen-tration at middle of the cloud. It also reveals thatthe large average drop concentrations are havingsmaller drop sizes and vice-versa.

3.2 Total ion concentration

The variation in ion concentrations within thecloud is shown in figure 4 for three differentaverage drop concentrations, as mentioned before.The magnitude of average positive and negativeion concentrations were observed to be ∼8.18e8and 8.16e8 m−3, respectively within the cloud fordrop concentration 1.0e8 m−3 having drop radii


Figure 4. Variation in ionic concentrations within the cloudhaving different drop concentrations at different sizes.

∼9.5µm. Significant decrease in bi-polar ionic con-centrations was observed by ∼42% and 54% as thedrop concentrations increased by 5 and 10 timeswith significant decrease in drop radii ∼41% and52%, respectively. In all the three cases, the aver-age positive ion concentrations were observed to behigher than the negative ion concentrations withinthe cloud, which is due to smaller attachment

coefficient of positive ions as compared to thenegative ions.

The attachment coefficient of ions to the clouddrops highly depend on the mobility of ions,drop sizes and also on the drop concentrations(equations 14 and 15). It generally decreases withdecreasing drop sizes (not shown here). The rate ofion removal or scavenging within the cloud highlydepends on their attachment coefficients and dropconcentrations apart from drop size. Thus, theresults obtained (figure 4) reveal a decrease in ionscavenging for larger size drops (∼9.5µm), whichis mainly attributed to the presence of significantlylower drop concentrations (1.0e8 m−3). Therefore,although the attachment coefficient is high, in theabove case, ion scavenging rate is low. Figure alsoshows a significant increase in ion scavenging ratedue to increased average drop concentrations by 5and 10 times with significant decrease in drop radii∼41% and 52%, respectively.

The concentration of both ions decreased up toabout middle of the cloud where it crosses eachother, above which the average positive ion con-centrations were observed to be ∼3%, 9% and12% more than that of the average negative ionconcentrations having average drop concentrations1.0e8, 5.0e8 and 1.0e9 m−3 with radii ∼9.5, 5.6 and4.5µm, respectively; however, opposite was truenear the bottom of the cloud. In this case, higherpositive ions near the top and higher negative ionsnear the bottom of the cloud were observed due tounequal loss rates of two ions. Production of bothions are same but loss of negative ions is more nearthe cloud top and vice-versa is true near the cloudbottom.

3.3 Total charged and neutral dropconcentration

Because total drop concentrations within the cloudare considered to vary sinusoidally with altitude,it is maximum at the middle and minimum at thecloud boundaries (shown in figure 3 for differentsize of droplets). Consequently, the ion concentra-tion will significantly reduce up to middle of thecloud due to their attachment to the cloud droplets,as already discussed in figure 4. The ion removalprocess will thus increase the charged drop con-centrations, i.e., positive (S1) and negative chargedrops (S2) as shown in figure 5 with dash and dash-dot lines, respectively, and consequently decreasein total drop concentrations (S) within the cloud atsteady-state. The steady-state uncharged dropletsare termed as neutral droplet concentrations (So),which is shown in the same figure with solid line.Average positive, negative and neutral drop con-centrations were observed ∼32, 35 and 33% of thetotal drop concentrations.


Figure 5. Variation in positively charged, S1 (dash line),negatively charged, S2 (dash-dot line) and neutral, So (solidline) drop concentrations within the cloud having differentdrop concentrations of different sizes.

In contrast to the average positive and negativeion concentrations within the cloud, average posi-tive charge drops were observed to be less as com-pared to the average negative charge drops. This isdue to the low magnitude of attachment coefficientof positive ions as compared to that of negativeions, as discussed before. The magnitude of average

Figure 6. Variation in electric field within the cloud havingdifferent drop concentrations at different sizes.

positive and negative charged drops were observed∼3.2e7 and 3.5e7 m−3 within the cloud, contain-ing average drop concentrations 1.0e8 m−3 havingdrop radii ∼9.5µm, which increases by ∼5 and 10times as the average drop concentrations increasedby 5 and 10 times with significant decrease in dropradii ∼41% and 52%, respectively. The concentra-tion of both the charged droplets were observed tobe increased up to about middle of the cloud whereit crosses each other and showing similar beha-viour as obtained in case of ionic concentrationsnear top and bottom of the cloud. The remaininguncharged droplets, termed as neutral, were alsoobserved within the cloud, which compensate thetotal drop concentrations.

3.4 Electric field

An enhancement in the production of chargeddroplets associated with ion removal will conse-quently increase the electric field within the cloudto compensate vertical current density, as shownin figure 6. Minimum electric field (Emin) wasobserved ∼27V m−1 at both the edges of cloud,which is the constant initial value goes into themodel, calculated using various initial parametersalready discussed earlier. Maximum electric field(Emax) was observed at middle of the cloud whereit was ∼260V m−1 in the case of cloud havingaverage drop concentrations 1.0e8 m−3 with radii∼9.5µm. It increases significantly by ∼3 and 5times as the average drop concentrations increasedby 5 and 10 times with significant decrease in dropradii ∼41% and 52%, respectively. Variations inelectric field inside the cloud are in accordancewith the observed electric field variations in non-thunderstorm cloud as reported by MacGormanand Rust (1998, pp. 46–47).


Figure 7. Variation in uni-polar space charge within thecloud having different drop concentrations at different sizes.

3.5 Uni-polar space charge

Regions of dominance of ions and charged particlescreate a gradient in the air conductivity inpresence of non-zero current density, which leadto the presence of space charge. The atmosphericelectrical conductivity is generally increased expo-nentially with altitude in fair weather condition.However, it decreases inside the cloud (and aerosol)layers where ions are scavenged by cloud drops(and aerosol particles), which are highly associatedwith electric field as discussed earlier, so the regionsof net uni-polar space charge will often exist on theupper and lower surfaces of the layers (Harrisonand Carslaw 2003).

The present simulation results the variationsin uni-polar space charge within the cloud con-taining different drop concentrations at differentsizes, which are shown in figure 7. The aver-age net positive and net negative space charges(∼2.6 × 107 e m−3) were observed at the top andbottom of the cloud, which contains average dropconcentrations 1.0e8 m−3 of radii ∼9.5µm. Thespace charge was observed to be increased by∼3 and 5 times as the average drop concentra-tions increased by 5 and 10 times with significantdecrease in drop radii ∼41% and 52%, respectively.Both positive and negative space charges wereobserved to be maximum at ∼250 and 750 m alti-tude level, respectively inside the cloud (from itstop) containing different drop concentrations ofdifferent sizes.

4. Discussions and implications

to the natural cloud

An attempt has been made to develop a numericalmodel for the production of space charge and to

understand the behaviour of other electrical para-meters within the stratiform cloud. Though thepresent simulated results show some discrepanciesto the natural conditions, which could be due tosimulations made under some basic assumptionsand limitations to solve the complex non-linearcoupled differential equations, it is crucial to studythese electrical parameters to understand theirimpacts on non-thunderstorm clouds, and there-fore conceivably on climate, via cloud microphysi-cal processes (Harrison 2005).

In an earlier study, Zhou and Tinsley (2007)also modeled the production of space charge andits partition between charges on droplets, aerosolparticles and ions and perform its sensitivity forthe variations of different electrical parameters.They have considered mono-dispersive aerosol sizedistribution, with at most a single positive andnegative charge on aerosol. They have also consi-dered the droplet concentrations of mono-dispersesize distribution and reported typically 50–100 ele-mentary charges on both positively and negativelycharged droplets. Though most of the attachmentof ions within the clouds occurs on droplets ratherthan on aerosol particles, we have considered onlydroplet concentrations of mono-disperse size distri-bution having single positive and negative chargesin the present study. A discussion on the impli-cations of assumptions considered in the presentstudy to the natural condition have been given inthe next section by comparing simulated resultswith those obtained by Zhou and Tinsley (2007)using a cloud model and also with few availableobservational results.

4.1 Effects of single and multiple dropletcharges

Enhancement in the concentration of clouddroplets within cloud increases the ions attach-ment, which reduces the ion concentrations andresults in an increase in the charged dropconcentrations within the cloud as clearly seenin figures 4 and 5, respectively. A decrease inthe ionic concentrations reduces ionic conducti-vity within the cloud which, in turn, results inan increase in electric field (figure 6). The aboveresults help in producing positive and negativespace charges at top and bottom boundaries ofthe cloud, respectively, which increases signifi-cantly with an increase in cloud drop concentra-tions (figure 7). The magnitude of the above resultsdiffers significantly from the results obtained byZhou and Tinsley (2007, figure 3), which is clearlya consequence of considering mono-disperse sizedistributions of single positive and single nega-tive charged aerosols along with multiple chargeddroplets in a cloud model having comparatively less


cloud layer thickness. Nevertheless, the nature ofvariations in the electrical parameters was found tobe nearly similar to the present simulated results.To understand the percentage difference in themagnitude of the electrical parameters obtainedfrom the present simulation and simulation doneby Zhou and Tinsley (2007), a comparison in themagnitude of electric field was done for an identi-cal case for the droplet concentrations of 1.0e8 m−3.The simulated electric field value in the presentstudy shows a decrease of ∼30% from the valueobtained by Zhou and Tinsley (2007).

In an earlier study, Griffiths et al (1974) haveshown that the effect of droplet charge (variedover a wide range) and its distribution over thedroplets on normalized conductivity (the ratioof the in-cloud value to the clear-air value atthe same altitude) is significantly very small inthe weakly-electrified clouds. However, the onlysignificant effect exercised by cloud charge is tointroduce an asymmetry into the process of ionicimmobilization. Zhou and Tinsley (2007) showedthat 50% variations in air-Earth’s current density(Jz) can cause up to about 40% variations in chargeon droplets and aerosols. These variations cancertainly influence the collision processes betweenthem and may affect cloud cover and climate.Earlier, Tinsley et al (2006) have numericallydemonstrated that charges on aerosol particlesand/or cloud droplets modify the droplet–particlecollision efficiencies involved in scavenging, andthe droplet–droplet and particle–particle collisionefficiencies involved in coalescence of dropletsand particles, even in weakly-electrified clouds andaerosol layers. Apart from the model estimatedresults, there are also a few observational evi-dences for the enhancement of space charge withinthe non-thunderstorm cloud. In the earlier studies,Clark (1958); Imyanitov and Chubarina (1967)have observed the space charge density typically∼0.5 − 1.0 × 108 e m−3, which are nearly compa-rable to the present model estimated values. On theother hand, Tinsley (2000) estimated the spacecharge density at the interface of top of the cloudand found very large magnitude of charge density(∼109

− 1010 e m−3), which are due to his assump-tion of a very thin region (∼1 − 10m) above thecloud where space charge accumulation occurs(Harrison and Carslaw 2003).

The discrepancy in the present study of spacecharge density and the other electrical para-meters, apart from the layer thickness, could alsodue to the electrical parameter’s dependence onseveral factors, including the stability of cloudlayer in terms of turbulent mixing (not includedin the present model simulation), ionization rate(q) and the total air-Earth current density (Jz)(Harrison and Carslaw 2003), which are considered

Table 3. Variation in space charge withinthe cloud at varying cloud thickness.

Cloud thickness ρ+(ρ+Max)

(m) (e m−3)

1000 8.07e7 (1.27e8)

500 1.61e8 (2.45e8)

100 5.85e8 (7.64e8)

50 7.77e8 (1.0e9)

as constant under steady condition and takenas the typical values (table 1) for calculatinginitial boundary conditions of the model. Apartfrom the above factors, cloud drop concentra-tions and their sizes are the other important para-meters that can also profoundly affect the variouselectrical parameters within the cloud, which isnot mentioned in the earlier studies (Clark 1958;Imyanitov and Chubarina 1967; Tinsley 2000) andconsidered slightly different from the present simu-lations by Zhou and Tinsley (2007) in the cloudmodel. It can change the total ionic and chargeddrop concentrations and thus modulate the totalspace charge density by changing electric fieldwithin the cloud.

4.2 Effects of thickness of cloud layer

The model presented here is for the stable cloudlayer. However, localized downward (and upward)winds near the top (and bottom) of the cloudcan cause narrowing of the cloud layer and thusincreased gradient of all the electrical parameterswithin the cloud (Zhou and Tinsley 2007). Zhouand Tinsley (2007) have reported that variationsin vertical air-Earth’s current density (Jz1

and Jz2)

due to variations in relative amounts of positiveand negative ion concentrations (n1 and n2) cre-ate space charges at the cloud boundaries. Con-sequently, more positive ions can flow downwardthan the negative ions flow upward from the topof the cloud, because of the decreased ion concen-tration within the cloud. Similarly, more negativeions can flow upward from the bottom of the cloudthan positive ions flow down.

To understand the dependence of space chargeon thickness of cloud layer, simulations have beendone for the variations of space charge within thecloud by varying cloud thickness. Simulations weredone for a single case of cloud, containing dropletconcentrations of 5.0e8 m−3 having drop radii of∼5.6µm. Results are shown in table 3 where thevalues outside the brackets indicate net averagepositive space charge at top of the cloud; the valuesin the brackets indicate maximum positive spacecharge within the cloud. Similar magnitude withopposite sign, i.e., negative space charge, was


observed at bottom of the cloud. Table 3 clearlyreveals that the magnitude of space charge highlydepends on cloud layer thickness. A significantincrease in space charge with narrowing cloud layerthickness was observed in the present simulation.Nearly similar feature in the nature of variationof space charge within the cloud was reportedearlier by Tinsley (2000); Harrison and Carslaw(2003); Zhou and Tinsley (2007). Other electricalparameters within the cloud except electric fieldshow similar behaviour as discussed above for spacecharge. However, a significant decrease in electricfield was observed with decrease in cloud thicknessas can be understood from equation (18).

5. Conclusions

• Numerical simulations of the production of spacecharge and the variations in other electrical para-meters within the low level stratiform cloud havebeen done by assuming mono-disperse dropletdistribution, with at most a single positiveand/or negative charge.

• The magnitude of average positive ion concen-trations was observed to be more than the aver-age negative ion concentrations within the clouddue to low scavenging rate of positive ions thanthe negative ions, which is highly attributed totheir mobilities. Consequently, low concentrationof average positive charge drops were observed ascompared to the average negative charge drops.On the other hand, the concentration of aver-age net positive ions and net positively chargeddrops were observed to be higher near the topof cloud than the concentration of average netnegative ions and net negatively charged drops;however, reverse was observed near the cloudbottom.

• The magnitude of maximum electric field (Emax)was observed ∼260V m−1 at middle of the cloudhaving average drop concentrations 1.0e8 m−3

with radii ∼9.5µm, which increases significantlyby ∼3 and 5 times as the average drop concen-trations increased by 5 and 10 times with sub-sequent decrease in drop radii ∼41 and 52%,respectively. However, the magnitude of mini-mum electric field (Emin) was observed to be con-stant (∼27V m−1) at both the edges of cloud forany drop concentrations.

• Net positive space charge density was observedat the top and negative at the bottom ofthe cloud, which was attained ∼2.6 × 107 e m−3

when cloud containing the average drop concen-trations 1.0e8 m−3 having drop radii ∼9.5µm.The space charge density was observed to beincreased by ∼3 and 5 times as the average dropconcentrations increased by 5 and 10 times with

significant decrease in drop radii ∼41 and 52%,respectively.

• All the above electrical parameters are highlysensitive to the thickness of the cloud. Exceptelectric field within the cloud, other electri-cal parameters show significant increase withnarrowing the cloud thickness. However, a signi-ficant decrease in electric field was observed withdecrease in cloud depth.

Acknowledgements

This work was supported through a grant fromISRO RESPOND program. The authors thankProf. B A Tinsley from the University of Texasat Dallas, Richardson, Texas, USA for his valu-able discussions/suggestions in constructing theequations for this study. Authors are also thank-ful to Dr Devendraa Siingh from Indian Instituteof Tropical Meteorology, Pune for his valuable sug-gestions. Authors are grateful to the anonymousreviewer for the constructive and valuable com-ments/suggestions, which helped to improve thescientific content of the manuscript.

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MS received 15 December 2009; revised 8 April 2010; accepted 1 July 2010

Numerical study for production of space charge within the stratiform cloud

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