Effect of ice crystal size and graupel velocity on thundercloud electrification

(1)

IndianJournalofRadio&SpacePhysics Vol.22,October1993,pp.313-319

\

t\oU.~~'~

~.~ ....

~ffect of ~e crystal size and graupel ~~l~ci'!y on th:.:.;.:u.:.;;n.;.de;.;r..;.cl;.;o.••u ••d ...•e...,le_c••trID_o_c_a_tion_

~athpal

I

DepartmentofPhysics,SahuJain(PG)College,Najibabad246

7~Y

"-. -arm-

~

~dJ.'-d1'.. ~a'lDber,W'n~"fe\'f'ear~vediP')'wil111,~erSitrvf~ttee~ee~""·29Mardtt99~*Mayf996 ,

fi

one-dimensional time dependent model for cloud electrification, which includes collision be-

~e;n ice crystal and graupel in the presence of cloud liquid water, has been develcped in order to test the importance of various charge transfer parameters. The laboratory measurements .;howed that the charge transfer depends on ice crystal. size and impact velocity. The generated electric fields have been calculated for various values of size dependency (a), velocity dependency (b), precipitation intensity (Po),concentration of ice crystals(n) and event probability(P).lt is found that the estimated electric field ⁽⁴x 105 Vm-I), within a time interval of about 1500 s, can be achieved for

Po ~ 30 mmh-I. The maximum electric field(Emu) is not sensitive to the values of a and b, but its rate of growth is faster ifaand b are larger. The obtained results have been discussed in the light of

various experimenta1 observations and. it has been found that CharginJ rates may be affected by the specified size and velocity dependencj{/'

(2..2-- ~ e.., t)

1 Introduction

A plausible theory of thundercloud electrification must explain the generation of an electric field inside itl-3, of the order of 4 x 105 Vm-I within a time interval of less than 1500s. Christian et aU have reported lightning flashes in thunderc- louds normally at precipitation intensities of about

lOmmh-l.

Based on the experimental findings, some investigators expressed the view that thundercloud acquires electrical charge principally from the charge separated during graupel formation4-8. It has been estimated that when ice particles with different velocities collide and rebound in presence of supercooled water, a charge transfer takes place, leaving the particles oppositely charged.

The sign of charge transfer depends on the temperature and cloud water content (CWC)8. The negative charging"Of a graupel on collision with ice crystals in presence of typical CWC may occur at temperature colder than - 15°C isotherm6,7.

In a thundercloud, the negatively charged graupels fall against updraught to form a negative charge region at its bottom. Th~ positively charged ice c:rystals are carried in the updraught to the topof cloud to form the upper positive charge centre. It establishes a downward directed electric field within the thundercloud. Some the-

oretical calculations for e~t:ri& field growth rates within a thundercloud VIa this charging process have been made by some investigators6,9-13. It emerges from experiments that the amount of charge transfer depends on the ice crystal size and on the impact velocity5,7,8,14,15.Keith and Saunders 14 found that for ice crystals up to a di- ameter ^(d) of 125 ^Jlm, the charge'transfer is proportional to ^d3,4 with a velocity dependence of 0,8. According to Gaskell and TIlingworth7, the charge transfer to a sublimating hailstone on collision of it with an ice crystal is proportional to

dl,7 with a linear velocity dependence. Marshall et

at.15 found a ^d2 dependence over a wide range of sizes. Jayaratne et aL8 found a d4 dependence with an impact velocity of 3.0 ^{IDS -}

I

for small ice crystals (d

<

125 Jlm). In the present work, therefore, we have attempted a mathematical formula- tion to determine the· effect of crystal size and velocity dependencies of reasonable range on charging rate within a thundercloud. The graupel size spectra and corresponding fall velocities under developint:; electric field conditions have also been taken into account.

2 Theory

A one-dimensional parallel plate capacitor model has been adapted for the present work.

(2)

314 INDIAN J RADIO&SPACEPHYS, OcrOBER1993

... (8) The instantaneous charging current ^(J) due to gra-

vitational separation of positively charged ice crystals and negatively charged graupels, is given by

J=-JH(t)-i(t) ... (1)

where, JH(t) is the current density due to negatively charged graupels and i^(t) is the leakage current density at instant

t

due to conduction and point discharge, etc. The current density due to positively charged ice crystals is small as com- pared to negatively charged graupels9. The expression for i (t) is given by

i(t)= 10-8 [exp{6.7xI0-6 ^E(t)}-1]/3 ^... (2) where, E (t) is the downward directed vertical electric field at time twithin a .thundercloud.

The current density due to negatively charged graupels at instant

t

is given by

.. , (3)

where, the maximum radius ^Rmax for graupel is 3 X 10 -³ m, and the minimum size Rc is 5^X 10-5 m (Ref. 16).

The expression for the concentration of graupels per unit volume in

a

radius band of width

~ R centred around radius R, follows from that given by Marshall and Palmer17 for raindropslO as

for charge transfer to a graupel when an ice crystal rebounds from it

... (6) where,

A

is a constant which depends on particu- lar values of a and b used for size and velocity dependencies, respectively. The coefficients a, b and

A

are given in Table 1.

To solve Eq. (5), we make an approximation of replacing ^V(R,t) by ^V(R) which is the terminal fall velocity of a graupel of size R without consid- ering the effect of electrical forces. Following Rogers18 we have expressed analytically the aver- aged values of V (R) for different sizes of graupels in the ranges as given below

V(R)

=

k1R, for 5 x 10-5

<

^R~6 X 10-4 m V(R)=k2Rl/2, for6XlO-4 <R~2xlO-3m V(R)=k3Rl/2, for2X10-3 <R~3xlO-3m

... (7) where, ^k1=6400 S-1, k2= 160 ml12 s-1 and

k3

=

150 ml12 S-I.

The time dependent fall velocity V(R,t) at an instant t of a groupel of radius R in a downward directed vertical electric field ^(E) can be calculated from its equation of motion as

V(R, t)

=

[43fR3 pg 1.3

+

Q^{(R,t) E]/}

[6.7r,u R (CoRe124)R]

where, No

=

0.08 X108 m-4, X= 8200

PO-O.21 m-1, ^Po (mmh-1) is the precipitation in-

tensity, p =0.7 X 103"kgm-3 is the density of ice and Pw is the density of water.

The parameter Q^(R,t) in Eq. ⁽³⁾ is the charge at an instant t on a graupel of radius R. The ice crystals collide with graupels and then rebound, resulting in a charge transfer with an event probability ^(P). The rate of graupel charging, ^dQ(R,t)/

dt, is given by

dQ(R,t)/dt= 3fR2n[V(r,t) - v (r,t)J q(P) ... (5) where, n is the ice crystal concentration, V (R,t) the fall velocity at an instant t of a graupel of radius R, v(r,t) the ice crystal velocity considered.

to be negligible in comparison .with graupel veloc- itylO and

q

the electric charge acquired by a graupel per interaction with an ice crystal in presence of supercooled water droplets. The charge transfer depends on ice crystal size and impact velocity. Based on laboratory experiments, Mitzeva and Saunders5 have suggested the following equation

N(R) ~R= No (Pwlp)exp( - XR) ~R ... (4) where, ,u is the coefficient of viscosity of air in thundercloud and ^g is the acceleration due to gravity. The term ^(CDRe!24)R is the ratio of drag force to the viscous force acting on a graupel of radius R; CD is the drag' coefficient and Re is the Reynolds number. Substituting Eqs (4), (5) and (7) in Eq. (3), we get

JH(t)=yClt+~C2Et2 ... (9)

where,

~=

3fAn(p) p y= ^3fAn(p)

Table 1- Coefficient a,band A [Mitzeva and Saunders5]

a bA

2.0

2.8 8.9X 10-9 3.0

2.8 7.1 x10-5 4.0

2.80.57 2.0

1.0 6.4xlO-g 2.0

2.1 x2.0 lO-g 2.0

3.0 7.1X 10-9

j 1;1111 'II I

(3)

MATHPAL &AGARWAL: EFFECf OF ICE-CRYSTAL SIZE&GRAUPEL VELOCITY ON CLOUD ELECfRIFICATION 315

2000 t(s) 3303 3423 3583 4044

1SDO

n .10sni]

d •• '.00AIm

a-

^3.0

b •• 2.8

=O.3

-1 SOmm h 40mm hI 30mm hI 20mm h-1

500 1000

TIME,s

2 10

o s IE

10

>

0

^...J^W_ii:u1cct-UW...JW ₀₄3 10

Fig. I-Rates of growth of electric field for various values of precipitation intensity (Po)

Table 2- Values of 'maximum electric field Em •• and time tat Em•• for various values of Po (a= 3, b= 2.8, (p)= 0.3, n= 105

m-3 and d= 200 ,urn) Po (mmh-I) Em•• (XlO-5Vm-l)

50 4.45

40 4.23

30 3.95

20 3.57

maximum values of electric field ^(Emax) and the corresponding time to attain it for various values of Po. Figure 1 shows the rates of growth of electric field for various values of precipitation intensities.

The event probability (P) is also an important parameter. The event probability is the fraction of all encOlmters the graupel possibility makes with crystals and then rebound resulting in a charge transfer8• The laboratory investigations by some workers show that the event probability varies with crystal and graupel size and also with impact velocity19,20.We suppose (P)=0.3 for our calculations, which is an average taken over a range of graupel sizes5,zo. However, in order to see how the charging process will be affected, we have made our calculations for other values of (P) also.

Figure 2 shows the rates of growth of electric field for various values of (P).We obtained almost the same values of maximum electric field for different values of (P), but the time required to attain them increases with reduction in (P). For ex-

f2X10-'

+e;b R(6+bln. exp (- XR) dR

6x10-4

f3XIO-' ]

+e~2b 2x 10-' R(S+2bln.exp (- XR) dR

3 Results and discussion

Calculations are made for rates of growth of electric field for various values of precipitation intensity Po for a

=

3, b

=

2.8 and ^(P)

=

0.3. If we assume an initial electric field E= 100 Vm -1, a precipitation intensity of 20 mmh - 1, the considered mechanism will be able to build up a maximum electric field of about 3.5 x 105 Vm -1 ir 4044s. However,. an increase in precipitation intensity increases both the rates of growth of. electric field build-up and the maximum field. For example, a precipitation intensity of 50 mmh -1 can generate a maximum electric field of about 4.5 x 105 Vm-1 in 3303 s. Table 2 shows the The rate of growth of the vertical electric field (E) within a thundercloud can be given by

dE/dt= JlEo ... (10)

where, ^Eo is the permittivity of free space.

A computer program was run to solve Eq. (10) for computing electric field E within a thundercloud for various values of parameters, i.e., a, b, (p), d, n and Po by imposing the initial condition that at t=0,E::= 100 Vm-1 and V=3 ms-1.

{3= 3n(p)/4pg

(4)

316 INDIAN J RADIO &SPACE PHYS, OCTOBER 1993

ample, we obtain a maximum electric field of about 4.5 x 105 Vm-I within 2642 sand 2000 s for (P)=0.6 and 0.8, respectively, at Po=50 mmh - ],a=2 and b= 2.8.

In order to see how the charging process will be affected by size dependence, we have made our calculations for different values of a. Figure 3 shows the rates of growth of electric field for various values of a. Surprisingly, for different values of a, although practically almost the same maxi-

n=101';3

d".200.urn

a'.

^2.0

b =2.8

Po-SOmmh-1

mum field is obtained, the time required to attain them increases with reduction in a. For example, we obtain a maximum field of about 3.95 x 105 Vm-I within 2882s and 1141s for

a= 2- and 4, respectively at Po =30 mmh-I

(b

=

2.8 and (P)

=

0.6). Table 3 shows the maximum values of electric field and the corresponding time to attain it for various values of Po and a

(taking b= 2.8).

Further, our calculations show that a reduction in velocity dependency ^(b) results in a slower de- velopment of electric field, though ultimately the same maximum electric field is obtained. Figure 4 shows the rates of growth of electric field for various values of b. For example, we obtain a maximum field of about 4.2 x 105 Vm -¹ within 3623s and 2682s for b = 1 and 3, respectively at Po =40 mmh -¹ (taking a=2). Table 4 shows the maximum values of electric field (Emu) and corre- sponding time to attain it for various values of Po

and b (taking a=2 and (P)=0.6). The calculations show that for b ~2, the field continues to grow beyond t=3000s for all precipitation intensities. Therefore, for such cases the value of the field reported in Table 4 is that obtained at t

=

30005, which is, of course, not the maximum value of the electric field. It is believed that any effective thundercloud charging process must be

lr'····

l

J

Fig. 2~Rates of growth of electric field for various values of rebound probability(P»)

Table 3-Values of maximum electric fieldE ••••and time tat Emu for various values ofPoand a (b= 2.8,(P)= 0.6, n= 105

m-3 and d= 200,urn)

a Emu(x 105Vm-1) t (s)

Po=50mmh-1

<p;>=0.3

2 4.45Z642

3

4.451661 4

4.451041 Po=40mmh-12

4.232742 3

4.231721 4

4.231081

Po =30mmh-12

3.952882 3

3.951821 4

3.951141 Po=20mmh-12

3.573243 3

3.572042 4

3.5712.81

2000 1500

500

2

10 _ o

Ie ¹⁰⁵

> ^ci^...J^W

~ 10"

n = 10 m 5 -3

d_p.== 200..umSOmm ,,1

W ]

0 10

b = 2.8

1000 TIME,s

Fig. 3-Rates of growth of electric field for various values of a

1'1111111111 1

II 'I' ..~f II; IIlIllHHI: IHill.l1

(5)

MATIIPAL &AGARWAL: EFFECf OF ICE-CRYSTAL SIZE &GRAUPEL VELOCfIY ON CLOUD ElECfRIFICATlON 317

u

a:

•...

u

UJ J

-' 10

UJ

b= 3 b=1

n = ¹⁰⁵m3 d = 200Alm

-,

Po=^SOmm h

Q = ²

=(1.3

able to produce the observed field within a time nut exceeding 1500 s or so.

We assume the average concentration n

=

105 m -³ as an indicative value for our calculations. However, in order to see how the charging will be affected, we have made our calculations for other values of n also. Figure 5 repreSents the variation of electric field with time as a function of concentrations of ice crystals. Also, Fig. 6 shows the rates of growth of electric field for different crystal sizes (the corresponding concentrations of ice crystals are given inbracket18).

106

Fig. 4-Rates of growth of electric field for various values of b

Table 4- Values of maximum electric field Emax and time t at

Emux for various values of Po and b (a= 2.0. (P)=0.6, d=200.um and n=lO-' m-3. For the cases where the electric field growth continues even after 3000s, the values of field

are given at t = 3000s) bEma, (x lO' Vm") t (s)

Po= 50 mmh-I 1.0

4.403000 2.0

4.443000 2.8

4.452642 3.0

4.452562

Po=40 mmh-I1.0

4.183000 2.0

4.223000 2.8

4.232742 3.0

4.232682 Po= 30 mmh-I1.0

3.863000 2.0

~.923000 2.8

3.952882 3.0

3.952802 Po= 20 mmh-'1.0

3.433000

Z.O

3.503000 2.8

3.553000 3.0

3.563000

1200 1500

d=1OO~m (n=1.67lC104m3) d =40Alm (n=5 lC104 m3)

. S -3

d=10J.lm (n=4.17lC10 m )

600 900

TIME.s

300

I

E > ¹⁰5

g

^~^.' ¹⁰⁴

~r ^/\:n=l.'ml

^~ ^n-SX10~^ncm4m3 ^m3

u

^0:^•...

103

WI

d = ^2OOA.Jm

u -'

^UJUJ_b ₌ _2.8 Po^Q =^- SOmm²

n'

102r

=0.3

I

III

0

500150010002000 TIME,S

"j

; 104

0 -'

UJG:

u ii

•...

u

UJ...) UJ

Po= ¹⁰ mmh"' Q = 2

b = ^2.8

=0.3

Fig. 5 - Rates of growth of electric field for various values of ice crystal concentration ⁽ⁿ⁾

Fig. 6-Rates of growth of electric field for various values of ice crystal. size(d)and concentration (n)

2000 500 1000 1500

TlME.s

1021

o

(6)

318 _INDIAN _J_RADIO _&SPACE PHYS, OCTOBER 1993

Table 5 - Field measurements of thundercloud properties (Dye etal.22)

Temperature LWC V n

·C ( x 10 -4kg m -3) ms-I ( X104m -3)

Time hrsLT 1620 1632-1633 1635-1636 1640-1641 1647-1648

-5 -6 -6 -15

3.1 3.6

0.4

0.6

9 3 2 1

0.03 2.04 2.17 6.38

1.0X 102 1.0^X 103 1.5X 104 4.0X 103

Four instrumented aircrafts were used to inves- tigate the microphysical and electrical properties of a small thundercloud occurred in Montana21,22.

Some observations are summarized in Table 5. It was found that the graupels co-existed with 200 I.lm ice particles and little supercooled water.

The maximum ice particle concentrations in- creased to values of 6.4 x104 m-3. The observations show a very small electric field of about 100 Vm -1 as the cloud was just beginning to build up at 1620 hrs LT. The electric field increases to a value 8-15 kVm-1 at 1537 hrs LT.

The model calculations are shown in Fig. 7. We obtain an electric field of about 10.5 kVm-1 within 1500s at Po =10 mmh

-I.

It appears that the ice crystal-graupel interaction in presence of supercooled droplets may play an effective role for thundercloud electrification.

"E 4 ::: 10

o

...J

W iL

u

0:

t-

U 3

~ 10

w

Po= 10 mm h-1 b=2.8

=0·3

n = 2.04xl0 4 -3m d = ¹⁰⁰ jJm

300 600

TlME.s

900 1200 1500

4 Conclusion

In the present work, our emphasis is laid on a size distribution for grau'pels, their corresponding fall velocities under developing electric field and the charge on a graupel as a function of their size and velocity. Calculations have been made for a wide range of values of parameters Po, a, b, d, (P) and n.

It appears that the rates of growth of electric field are dependent on the values of the parame- ters a and ^b. It is seen from Tables 3 and 4 that the average rates of growth of electric field vary by about two and half orders of magnitude bet- ween a=2 and a=4 and about 1.25 orders of magnitude between b= 1 and 3. Calculations also show a strong effect of the size parameter on the charging rates. However, the maximum electric field is not very sensitive to the values of parame- ters a and b.

The adequate electric field may develop within reasonable time of 1500 s through this charging process in those thunderclouds for which Po~30mmh-1, and (P)=0.6, a=3 and b=2.8.

Fig. 7 - Rates of growth of electric field for various values of a

I! is found that higher values of precipitation in- tensities are required for lower values of b. If we take b = 1, then for ^a= 3 and (P)=0.6, the electric field acquires a value of about 4 x 105 Vm - ] within 1500 s for Po> 40 mmh-I. Further, higher values of precipitation intensities are required for lower values of ^(P). The estimated electric field 4 x 105 Vm -] is obtained within 1600s for Po =50 mmh-I (taking (P)=0.5, a=2 and b=2.8). It is, therefore, safe to conclude that this charging process may play an effective role for thundercloud electrification in a range of circumstances. How- ever, more complete field measurements of ice particle size, temperature, liquid water content, ice particle concentration, air motion and electrical properties are needed for better understand- ing onhundercloud electricity.

Acknowledgement

The financial assistance by CSIR, New Delhi, India, during the work is thankfully acknowl-

Ij 1111,II !

(7)

edged. The authors thank Dr N C Varshneya for his helpful suggestions.

MATHPAL & AGARWAL: EFFECf OF ICE-CRYSTAL SIZE & GRAUPEL VELoCITY ON CLOUD ELECfRIFICATION 319

11 Helsdon J H & Farley R D, J Geophys Res (USA), 92 (1987)5661.

12 Pradeep Kumar P & Mukku V N R, Indian J Radio &

< Space Phys,21 (1992) 229.

13Latham J & Dye J E, J Geophys Res (USA), 94 (1989)

References 13141.

I Winn W P, Schwede G W & Moore C B, J Geophys Res 14 Keith W D & Saunders CPR, J Geophys Res (USA), 94

(USA), 79 (1974) 1761. (1988) 13103.

2 Christian H, Holmes C R, Bollock J W, Gaskell W, IlIing-15 Marshall B J P, LathlilR J & Saunders CPR, Q J R worth A J & Latham J, J R Meteorol Soc (GB), 106~' MeteoroISoc(GB), 104(1978) 163.

(1980) 159. 16Griffiths R F& Latham J, J Atmos Sci (USA), 32 (1975)

3 Mason B J, Proc R Soc London Ser A (GB), 327 (1972) 958.

433. 17Marshall J S & Palmer W M, J Atmos Sci (USA), 5

4 SaundersCPR, Weather(GB),43 (1988)318. (1948)165.

5 Mitzeva R &Saunders C;P R,J Atmos &Terr Phys (GB), 18 Rogers R R, A short course in cloud physics (Pergamon

52 (1990) 241. Press, New York, USA), 1976,p.81.

6 Takahashi T,J Atmos Sci( USA),35 (1978) 1536. 19Keith W D, Ph D Thesis, University of Manchester, GB, 7 Gaskell W & Illingworth A J, Q J R Meteorol Soc (GB), 1987.

106 (1980) 841. 20 Kieth W D & Saunders CPR, Atmos Res (USA), 23

8 Jayaratne E R, Saunders CPR & Hallett J, QJ R Mete- (1989) 83.

orolsoc(GB),109(1983)609. 21 Gardiner B, Lamb D, Pitter R L, Hallet J&Saunder C P 9 Mathpal K C, Varshneya N C & Dass Narsingh, Rev R,J Geophys Res(USA), 90(1985)6079 .

•., Geophys Space Phys(USA), 18(1980)361. 22 Dye J E, Jones J J, Winn W P, Cerni T A, Gardiner B, 10 Mathpal K C & VarslIDeya N C, J Atmos & Terr Phys Lamb D, Pitter R L, Hallet J& Saunders CPR, J Geo-

(GB),44 (1982) 81. phys Res ( USA),91 (1986) 1231.