December 1967 George 8. Freier a43
THE RELAXATION TIME OF AIR IN THUNDERSTORMS
GEORGE D. FREIER
School of Physics, University of Minnesota, Minneapolis, Minn.
ABSTRACT
The concept of how the conducting atmosphere in various states of ionization can lead t o various relaxation times
for changes in the observed electric field is investigated. It is concluded that the observed relaxation times can bc
short, and that the short relaxation times will exist even though Ohms law is not strictly obeyed in relations between
the current density and the electric field.
1. INTRODUCTION
If the charge carriers in a conduction current are pro-
duced by strong electric fields, some observers argue that
the concept of conductivity is no longer strictly operative
and that short relaxation times do not necessarily imply a
large conductivity. Many of the concepts of electrical
conduction, however, are still operative, and they can
lead to a relaxation time for the observed changes in the
electric field as described by Gunn [6]. Conductivity is
defined through Ohms law as j=XE where j is the current
density and E is the electric field. If ,is in any way a
function of E, there will no longer be a linear relation
between j and E, and it is the purpose of this paper to
investigate how the non-linear relation may be reflected
in the observed relaxation time.
2. CONDUCTIVITY VARYING WITH TIME
From the microscopic point of view the current density,
j, is written as
j=n+e+v++n-e-v-,
where n, is the number of ions per unit volume, e, is
their charge, and vf is the velocity. Usually v is propor-
tional to the electric field E, and the expression leads to tt
well-defined microscopic concept of conductivity ; namely,
A=( n+e+w++ n - e -w- )
where wi is the dynamic mobility of the ion due to the
force field produced by E.
The values of n* are determined by kinetic equilibrium
between production rates per unit volume, p, the recom-
bination of the ions, and the combination of the ions with
other immobile particles, which may change the dynamic
mobility of the ions by many orders of magnitude. If any
of these quantities change, the value of n, will change,
but it will take some time for this to be realized. As long
as there are ions present in the field, the ions will ne\\-
tralize the charges which produce the electric fields.
Rather than consider how the electric conductivity
may vary with the electric field, it will be better to con-
sider how the conductivity varies with time even though
the time variation may be caused by field changes.
If there is a charge Q which produces an electric field
E in a conducting medium, then
where A is the area surrounding the charge. Since the
present discussion involves a study of time variation,
it will be assumed that there is spatial symmetry, in which
case the equation reduces t o
d&- --~=AQ. dt -
The solution then is
Q=Qo exp ( 4 d d t )
where Q0 is the value of the charge at zero time. If X
varies with time, it will be possible to investigate the
deviation from simple exponential decay.
For the purposes of this analysis it will suffice to consider
the ion production and recombination equation in the
form
dn clt =p--cyn2-qiVn
where it is assumed that the number of positive and
negative ions are equal, and they recombine with the
recombination coefficient -cy while all attachment to more
immobile particles of volume density, W, takes place
with the attachment coeEcient 7. The accepted value
for 01 is 1.6X10-6 ~111.~ set.-', and Chalmers [2] has an
844 MONTHLY WE
average value for q of 2X1W6 ~m .~ set.-' Some of the
cases for the solutions of this equation in a thunderstorm
have been discussed by Freier [5].
I n a region of space where N is much less than n, the
equilibrium value for n is d&. At the 500-mb. level of
pressure in the earths atmosphere, cosmic rays produce
ions a t the rate of q=15.4 ion pairs ~m .-~ set.-' The
corresponding equilibrium value of n, is 3100 ~m .-~, and
if the singly charged ions had a mobility of w =l cm.*
set.-' v.-, the conductivity would be X=%new=8.9X
set.-' A charge placed in this medium would decay with a
relaxation time of ~=l /h X =9 0 sec. If the source of
cosmic radiation were suddenly removed, the number of
ions would decrease from the equilibrium value as
I n a time t=l/dG, n would decrease to half the original
equilibrium value, and for the above values of p and a
this characteristic time would be 200 sec. For times much
less than 200 sec., ._
and a charge placed in the medium would decay with an
observed relaxation time of 90 sec.
If by contrast, the original source strength were due
t o corona discharge in strong electric fields such that
p=104 ~m .-~ set.-', then the equilibrium values for n,
would be 7.9X lo4 ~m .-~, the corresponding conductivity
for small ions would be X=%.3X10-2 set.-', and the
relaxation time for a charge in the medium would be 3.5
sec. If the corona discharge suddenly stopped due t o an
electrical discharge which reduced the field causing the
corona, the number of ions would reduce to half the origi-
nal value in a time of 8.0 sec. An observer studying the
relaxation time of the electric field due t o a charge placed
in the medium would find a relaxation time of 3.5 sec.,
even though the production rate had gone to zero. Ohms
law in a most strict sense would not hold during the
entire process, yet the short relaxation time would be
indicative of a large conductivity.
In a Lhunderstorm the environment is altered by the
presence of N larger particles per unit volume which
can capture the small ions (Fletcher 141). The electric
field may or may not be involved in the capture of ions.
If the electric field is involved we probably have a case
of hyperelectrification described by Gum [7] and if the
field is not involved we can use the recombination co-
efficient, q, given above.
For the case for which the electric field plays no role
it may be assumed for simplicity that N>>n. The
equilibrium value for n is then nl=p/qN. If N=5X IO3
ATHER REVIEW Vol. 95, No. 12
CM.-~ and if q is determined by cosmic ray ionization,
then the equilibrium value is nl= 15.4/(2X (5X lo3)
= 1.5X IO3 ~m .-~ The corresponding electrical conductiv-
ity would be determined almost entirely by the small
ions which have the equilibrium value nl and thus give
a value for X=4.4X10-4 set.-' and a relaxation time of
180 sec. Wilson [12] and Gunn [6] believed that a situation
somewhat similar to this exists in the thunderstorm cloud,
If corona discharge could increase q to lo4 ~r n .-~ set.-'
and N were also increased t o lo5 ~m .-~, then the equilib-
rium value for, small ions would be nl=5X1O4 ~m .-~,
and the values for X and'^ would be respectively 1.4X10-
set.-' and 5.7 sec. These values are close to those given
by Latham and Mason [lo] for conductivity of the air
in a thunderstorm where there is corona from the drops.
If the corona discharge were suddenly stopped for the
case where N>>n, the value of the number of small
ions would decrease a t the rate
The small ion density, n, would decrease by a factor of
e in 1/qN=5 sec. This time is now less than the relaxa-
tion time calculated above, so that the obserOed relaxation
rate for a charge in the medium would deviate seriously
from a simple exponential form.
The other case to be considered for thunderstorms is
that where the hyperelectrification described by Gunn
[7] is used to capture ions. If the production of ions is by
corona discharge from the drops, the conductivity will
most likely be unipolar, and consequently the drop may
become charged by the corona produced ions. When H
spherical drop of radius a and charge Q is placed in an
electric field, it will have a current
flowing to it. Q, is the maximum charge (Qm=3a2En)
that the drop can have and still have field lines terminate
on it. If there are N drops per ~m .~ the number of ions
captured per unit time in a unit volume is 3E0a2mnw
(1 - &/&,I N.
The electric field at the opposite side of the drop where
field lines again originate is E= 3E0( 1 + &/Qm), and
it is here that corona would take place. N drops per ~r n .~
would give a source strength of ions, q, such that a t
equilibrium
q=3Eoa27rn~( 1 -Q/Qm)N.
The number of droplets, N , per unit volume is limited by
the water content, W , of the cloud. If p is the mass den-
sity of water, the equilibrium condition may then be
written as
_-- q-9 - Eo - (1-Q/Qm)2W. n 4 a p
December 1967 George D. Freier 845
For a field of 1 stat. v./cm., a water content of 5 gm. m.-3,
drops of 10-3-cm. radius, and an ion mobility of 300 cm.2
set.-' (stat. v.)- this becomes
a= 3.4( 1 - Q/Qm) sec. - n
The charge on the drops can change the relation between
the production rate and the equilibrium number of ions,
and for a given charge on the drops n must be large if p
is large.
In the case of hyperelectrifka tion the charged droplets
can also be moved by the electric field to give a conduction
current. The electric force on the droplet would be Ed&,
and this force can be in equilibrium with the viscous
force 6~aqv if drops are sufficiently small so that gravi-
tational forces may be neglected. The current density
due t o motion of the drops in the field is then
where v is the velocity of the drop and q is the viscosiLy
of the air. N is limited by the liquid water content and
& may be expressed as a fraction, f, of the maximum
charge. When these substitutions are made the conduction
current density due t o the motion of the drops becomes
The viscosity of air is 1.8x10-4 dyne sec. cm.-2so that with
a field of one stat. v.cm.- the current density is j=3f2
X 10-3e.s.u. cm.- set.-' which is quite large for f near unity.
For larger fields thir current would be considerably
greater. If this current dissipates other cloud charges
a relaxation time of T=E,/~TNQV would be observed
and for the above choice of values this would yield
7=(26.3/f2) sec.
There seems t o be evidence that hyperelectrification can
lead to corona and short relaxation times but nothing
very quantitative can come from the present knowledge
of the processes involved. This laboratory is making
studies of the processes described above, but we find the
measurements difficult to make. With corona discharge in
a field of condensed water droplets the space charge gives
such a large divergence to the electric field that it is
difficult to know just where in the region the above con-
ditions are satisfied.
This large divergence of the electric field would also be
present in a thunderstorm so that selected conditions
could prevail only in a relatively small region of space.
However, the process described is one in which the depo-
sition of charge could progress through the medium in the
direction of the field and thus deposit charge in large
regions of the thunderstorm volume.
If there were no corona in the thunderstorm cloud the
above equations would have Q=O and f =O so that the
droplets would be efficient sinks for all ions in the presence
of an electric field. I n this case the conductivity of the
air would be very small and relaxation times would be
very long. This author thinks that this possibility is
incompatible with observations of relaxation times made
in thunderstorm processes.
After a lightning discharge it seems that the electric.
sufficiently weak so that hyper-
be considered in the early stages
e early stages of field regeneration
elaxation times for field recovery should and observed
then reflect processes of ordinary ion attachment.
3. CONDUCTION BY EDDY DIFFUSION
Turbulence of the air may also play some role ii; the
decay of charge and the associated fields. In the cases of
conduction as discussed above, the detailed mechanism
is one of forced molecular diffusion which leads to a
relation between the molecular diffusion constant and the
ion mobility, Damely, w=eD/kT, where w is the ion
mobility, D is the molecular diffusion constant, k is
Boltzmanns constant, and T is the absolute temperature.
Kennard [9] shows how the forced diffusion can a1n.n~-h
lead to an equivalent density gradient. The settling of :L
fine precipitate such as found in Perrins experiment as
described by Block [l] is a case where the forced diffusion
due to gravity is in equilibrium with the diffusion due to
density gradients. The diffusion of oxygen and nitrogen
in the earths atmosphere destroys gravitational
stratification.
The statistical arguments for random walk in inolecrilai.
diffusion processes are given by Joos [8] and lead to the
result
.
bn X b2n b2n
bt-2T  a ~~-~ bs2 _-
where X z is the mean square displacement for a particle
in the time T when subject to a concentration grndient
bnpx. I t appears that similar statistical argiirnents rntiy
be made for eddy diffusion, in lvhich case the mean sqtliire
displacements are io5 times greater tllian for rno1eciil:ir
diffusion (Sutton [I]]).
If the electrical force on :L ptirticle is F =E e , \\liere E
is the electric field, the particle \rill be accelerated iiiitil
it encounters nn interaction \\-here it $1 eh t i p eueip-.
Repeated ucceleriitions and interactions lead to ii drif I
velocity V, which is proportional to F or V=KF=KeE,
where K is a constant. This drift velocity will prodiice ti
flux of particles equal to nV=nKF. If this electrical forced
diffusion mere in equilibrium with diffusion Iroduced by
concentration gradients the fluxes mould be equal and
opposite so that the first integral of equation (1) a t
equilibrium would be
(2 ) bn
bX -D -=nKF=nKeE.
279-151 0 - 67 - 3
846 MONTHLY WEATHER REVIEW Vol. 95, No. 12
The quantity Ke would be the mobility, w , of the ion,
and D is the coefficient of diffusion. In the case of molecular
diffusion the ions would have a Boltzman distribution in
the electrostatic potential, V , given by n-no exp( - Ve/kT),
and in this case
1 b n e bV eE
n dx kT dx kT
where the electric field is given bv E=-dV/dx. The
dynamic mobility of this ion in molecular diffusion is then
w = Ke = e D/ Tk.
In the case of eddy diffusion, the same arguments could
be used to derive equations (1) and (2) with the coefficient
of eddy diffusion, D, replacing the coefficient of molecular
diffusion, D.
This author has not been able to find a statistical
relation for (l/n)dn/bz in the case of eddies and turbulent
motion. If this value exists and if it is sufficiently large,
an ion could have a large mobility due to processes of
eddy diffusion.
The experiments of Colgate [3] imply a large electrical
conductivity in the cloudy region of a diffusion cloud
chamber where there could be turbulent motion due to
bubbling of carbon dioxide escaping from water. One
would conclude from the experiment that eddy diffusion
gave a large electrical conductivity.
The author has done experiments in which charges were
placed on well-insulated stator blades of an electric field
mill. The charge could be conducted from these plates
through the surrounding air. When the field mill is shielded
from the surroundings, it will then measure its own field,
and the decay rate can be measured.
The experiment was done by running the mill contin-
uously in some cases and by running it intermittently
in other cases. Very little difference was noted in the decay
rate or relaxation time even though in the case of contin-
uous running there would be considerable turbulence
around the blades throughout the decay period. In all
cases the relaxation time corresponded to that given by
molecular conductivity, and it was concluded that eddy
diffusion plays a very minimum role in charge transport.
--=---=-
4. CONCLUSIONS
Even though electric fields in thunderstorms may pro-
duce corona discharges intermittently to give a non-linear
relationship between the current density and the electric
field, the relaxation times for the electric field give a true
indication of whether or not there is a large value of the
electrical conductivity. The relaxation times for the
changes in the electric field may be altered by the charac-
teristic times for the ions to recombine or attach to more
immobile particles. As long as the electric field is suf-
ficiently large to maintain a corona discharge within the
cloud, the equilibrium value of small ions can be main-
tained at a sufficiently high level to give a high electrical
conductivity, and this high electrical conductivity can
persist for times which are as long or, in most cases, longer
than the relaxation time of the medium for field changes.
A short relaxation time for the electric field changes
necessarily implies a large value for the electrical con-
ductivity of the medium.
It is not clear whether eddy diffusion can contribute to
the electrical conductivity.
ACKNOWLEDGMENT
The author is grateful to the National Science Foundation for
grant GP-4996, under which this work was done.
REFERENCES
1. E. Block, The Kinetic Theory of Gases, Methuen and Co., Ltd.,
London, 1924, 178 pp. (see pp. 97-105).
2. J. A. Chalmers, Atmospheric Electricity, Pergamon Press,
London, 1957, 327 pp. (see p. 73).
3. S. A. Colgate, Enhanced Drop Coalescence by Electric Fields
in Equilibrium with Turbulence, Journal of Geophysical
Research, vol. 72, No. 2, Jan. 1967, pp. 479-487.
4. N. H. Fletcher, The Physics of Rainclouds, Cambridge Univer-
sity Press, Cambridge, 1962, 386 pp. (see p. 93).
5. G. D. Freier, Conductivity of Air in Thunderstorms, Journal
of Geophysical Research, vol. 67, No. 12, Nov. 1962, pp. 4683-
4691.
6. R. Gunn, Electric Field Generation in Thunderstorms, Journal
of Meteorology, vol. 11, No. 2, Apr. 1954, pp. 130-138.
7. R. Gunn, The Hyperelectrification of Raindrops by Atmos-
pheric Electric Fields, Journal of Meteorology, vol. 13, No. 3,
June 1956, pp. 283-288.
8. G. Joos, Theoretical Physics, G. E. Stechert and Co., New York,
1932, 748 pp. (see p. 563).
9. E. H. Kennard, Kinetic Theory of Gases, McGraw-Hill Book
Co., New York, 1938, 483 pp. (see pp. 201-204).
10. J. Latham and B. J. Mason, Electrical Charging of Hail
Pellets in a Polarizing Electric Field, Proceedings of the
Royal Society of London, vol. 266, Series A, 1962, pp. 387-401.
11. 0. G. Sutton, Atmospheric Turbulence, Methuen and Co., Ltd.,
London, 1949, 107 pp. (see p. 36).
12. C. T. R. Wilson, ((A Theory of Thundercloud Electricity,
Proceedings of the Royal Society of London, vol. 236, Series A,
1956, pp. 297-317.
[Received June 23, 1967; revised August i4) 19673