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LIGHTNING
Jernej Slanovec
Mentor: dr. Gorazd Planinšič
Seminar2: 22.10.2003
Jernej Slanovec
Lightning
Abstract
Lighting is a phenomenon, observed during thunderstorms. During the development
of large cumulonimbus clouds, a separation of charge occurs, which means that part
of the cloud obtains an excess negative charge, whereas another part acquires an
excess positive charge. These electrical differences lead to lightning.
In today’s presentation we will mostly try to describe the mechanism, which
enables the charge to be transferred from the cloud to the earth – the formation of
plasma channel, called a leader. Most of the atentiontion, however, will be devoted to
streamer propagation.
A streamer is also a plasma channel, but it is not capable of bridging long gaps
of air, which is the case in lightning. Numerous streamers in front of the leader
channel, however play a very important role, as we will see, and are therefore of
great importance.
Lightning discharges can be classified into positive and negative discharges and
the two mechanisms differ from eachother. We will look only at the mechanisms of a
positive lightning spark formation.
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Contents
1
2
3
4
5
Charge separation .................................................................................................................. 4
Basic stages of a lightning spark ........................................................................................... 5
Continuous streak photography ............................................................................................. 6
The leader stage ..................................................................................................................... 6
A positive leader.................................................................................................................... 9
5.1 A long streamer ............................................................................................................... 9
5.1.1 Current and field in the streamer ............................................................................ 15
5.2 The necessity of a streamer accompaniment................................................................. 17
5.3 Current in the leader ......................................................................................................... 17
5.4 Field in the leader channel ............................................................................................... 19
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1 Charge separation
The origin of charge separation is not yet fully understood. What seems to be a fact is
that lightning occurs in the violent mature stage of the cumulonimbus cloud.
Therefore we assume, that charge separation must be related to a rapid vertical
movements within the cloud. These tall clouds mainly form in the summertime, which
explains the lack of lightning in the winter.
An important fact seems to be the observation, that lightning rarely occurs before
the growing cloud penetrates the 5-kilometer level. This suggests, that the formation
of ice crystals in the upper, cooler regions of the cloud are of great importance for the
separation of the charge. Some cloud physicists believe, that charge separation
occurs during the formation of ice pellets. Experiments show, that as droplets begin
to freeze, positively charged ions are concentrated in the colder regions of the
droplets, whereas negatively charged ions are concentrated in the warmer regions.
So as the droplet freezes from the outside in, it develops a positively charged shell
and a negatively charged interior. As the interior begins to freeze, it expands and
shatters the outside shell. The smaller positively charged ice fragments are carried
upwards by turbulence, and the heavier core eventually caries it’s negative charge
toward the cloud base. As a result of numerous such events, the upper part of the
cloud is left with a positive charge and the lower portion of the cloud maintains an
overall negative charge with small positively charged pockets (figure 1.1).
Figure 1.1: The dipole model of the charge distribution in a storm cloud[1].
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Under such conditions a lightning may strike. Lightning discharges can be classified
into two main groups – intercloud discharges and ground strikes. The frequency
of the former is two to three times higher than that of the latter. Ground strikes can
further be divided into descending and ascending sparks, the direction of growth
being indicated by branches diverging downwards or upwards (figure 1.2).
Figure 1.2. A photograph of an ascending (left) and of a descending (right) lightning[1]
Lightning spark transports charge to the ground thus characterising its polarity:
negative or positive. About 90% of descending sparks carry a negative charge (are
negative) and about as many ascending sparks are positive. One can notice with a
naked eye, that sometimes a lightning spark flickers. This are multicomponent sparks
and they are usually negative. Positive sparks normally contain only one component.
2 Basic stages of a lightning spark
Now if we want to transport the charge from the cloud to the earth, we need some
sort of a conductive entity acting like a wire between two electrodes.
The leader stage represents the initiation and growth of a conductive plasma
channel – a leader – between the cloud and the earth or between two clouds. At the
moment the leader touches the ground or a grounded object a return stroke is
produced. During the travel from the cloud to the ground, the lightning leader tip
carries a high potential comparable to that of the cloud at the spark start, the
potential difference being equal to the voltage drop in the leader channel. After the
contact, the tip receives the ground potential and its charge flows down to the earth.
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The same thing happens with the other parts of the channel, possessing a high
potential. This unloading process occurs via a charge neutralization wave
propagating from the earth up through the channel. The wave velocity is comparable
to the velocity of light and is about 108 m/s. A high current flows along the channel
from the wave front towards the earth, carrying away the charge of the unloading
channel sites. The current amplitude is, on average, 30kA, reaching 200 – 250kA for
powerful lightning sparks. The transport of such a high current is accompanied by an
intense energy release. Due to this, the channel gas is rapidly heated and begins to
expand, producing a shock wave. The current rise in the return stroke can exceed
1011 A/s, producing a powerful electromagnetic radiation affecting the performance of
radio and TV sets.
3 Continuous streak photography
Lightning development can be investigated by continuous streak photographs. This
are images recorded on a continuously moving film.
Suppose, that a bright spot is moving down towards the ground with constant
velocity. Then an image recorded on a horizontally moving film (moving towards left),
would represent a sloping line (fig. 3.1a). If however, a bright channel is elongating
towards ground, the image will look like fig. 3.1b. If the velocity of film is known ( for
example 1 cm/s ), then the scale on horizontal axes can be replaced by time scale,
enabling one to calculate the velocity of the light source.
Fig. 3.1: Image display in streak photography : (a) point source, (b) elongating
channel[1]
4 The leader stage
Each lightning strike transports some charge to the ground. Therefore there must be
some sort of conducting entity in the space between the cloud and the earth. This
conducting entity is a plasma channel called a leader.
A leader is formed by ionization of air molecules by electron impact. If we want
to ionize an air molecule, we need a certain amount of energy (example - ionization
energy of nitrogen molecule : W(N2) = 27,10 eV). Electrons gain this energy from the
electric field. In air, each electron has a mean free path, before it collides into the
next air molecule. During this time, the electric field must provide the energy Wi ~
30eV to the electron, in order for a collision to produce a positive ion and an electron.
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Under normal conditions, the value of electric field, that can sustain this ionozation
process, is Ei ~ 30kV/cm (from this we can estimate a mean free path of an electron:
30eV
F l E e0 l Wi l
103 cm ) .
30kV / cm e0
Measurements of the field, produced by the charge separation in the cloud,
give the values 1 – 8 kV/cm for the cloud region, and at the earth, the storm field was
found to be 10 – 200 V/cm. In spite of such a low field a leader does propagate,
which means, that there is an intensive ionization occurring in its tip region, changing
the neutral air to a highly conductive plasma. This is possible because the leader
carries its own electric field induced by the space charge concentrated at the leader
tip and transported together with it. If the leader tip radius Rm is small enough, the
electric field in the space near the tip is locally enhanced thus making ionization
possible ( Em ( Rm ) Q 4 0 Rm2 Ei ). A rough analogy to this process is a metallic
needle connected with a thin wire to a high voltage source. If the needle is sharp
enough, the electric field in the vicinity of its tip will be very strong. Imagine now, that
the needle is falling down to the earth, pulling the wire behind it. The strong field
region, in which the air molecules become ionized, will move down together with the
needle.
It is the leader, that determines the characteristics of a lightning spark. It can
start somewhere in the cloud and then propagate downwards toward the earth. This
is then called a descending leader and about 90% of descending leaders are
negative leaders. However, constructions over 200m high and those in mountainous
regions suffer mostly from ascending lightning. An ascending leader is initiated by a
charge induced by the electric field of a storm cloud in a conducting, vertically
extending grounded object. Contrary to descending, ascending leaders are mainly
positive – 90%. Another feature which distinguishes positive from negative leaders is
the way in which they grow. A positive leader develops in a continuous manner
whereas a negative leader grows in a stepwise manner (figure 4.1). Each step in a
negative leader is 10 – 200 m long with an average step of 30m, and the time interval
between the two steps is 30 – 90 s. The averige velocity of leader propagation, is
equal in both cases, and is about 3105 m/s.
FFFFFFFFFFFFFFFFF
Figure 4.1. A schematic streak picture of a positive ascending (a) and a negative descending (b) lightning leader ;[1]
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Figure 4.2. Streak photographs of a positive (left) and a negative (right) laboratory leader ; [1]
The mechanism of a positive leader differs from that of a negative leader. We will try
to describe the basics of positive leaders.
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5 A positive leader
The wave mechanism of spark formation was suggested in the 1930s. The channel
thus formed was called a streamer. We will show, however, that a streamer is not
capable of bridging long gaps between a cloud and the earth. It requires huge
voltages to grow several meters in length. Typically – for a positive streamer – the
relationship is U min Ecr l where Ecr is about 500kV/m. So for a distance of 3km the
required potential drop between the cloud and the earth would have to be at least
1.5GV, which is well outside the values typical even for a powerful lightning
(~100MV). Long gaps of air are broken down by a more complex structure – a leader.
However, a streamer is a crucial part in the structure of a leader.
During the leader process, a hot plasma channel (T= 5000 – 6000K; electron
temperature can be higher Te ~10000K ) is travelling through the gap. Numerous
streamers start at high frequency from the leader tip, as from a high voltage
electrode, and form a kind of fan. They fill up a volume of several cubic meters in
front of the tip (figure 5.1). This region is known as the streamer zone of a leader.
The current sum of all the streamers provides energy for heating the leader channel
thus maintaining its highly conductive plasma state.
The streamer zone is filled up with charges of streamers that are being formed
and those that have died. As the leader propagates, the zone travels together with its
tip, so that the leader channel enters a space filled with a space charge. A charged
leader cover is thus formed, holding most of the charge. It is this charge that changes
the electric field in the space around the developing spark and lightning. It is
neutralized on contact of the leader channel with the earth, creating a powerful
current impulse characteristic of the return stroke of a spark.
Figure 5.1. Two photographs and a scheme of a positive leader[1]
5.1 A long streamer
Let us consider a well developed streamer, which has started from a high voltage
anode and is travelling towards a grounded cathode (figure 5.2 ). The front portion of
the streamer is shown schematically in figure 5.3 together with axial distribution of the
longitudinal field E, electron density ne, maximum achievable density nc, and a
difference between the densities of positive ions and electrons.
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This situation usually arises in nature, when the cloud above the earth is
negatively charged. If then there is a high enough tower located in the vicinity, with a
metallic rod extending vertically, the storm field is high enough to induce a positive
charge in the rod (electrons run out of the rod into the ground). The rod acts like an
anode and numerous streamers start to propagate upwards from this rod. This is how
an ascending lightning begins.
streamer
Figure 5.2. A schematic cathode-directed
streamer of length l : U0(x) external field
potential; U(x) potential along the conductive
streamer axis;[1]
l
Figure 5.3. A schematic representation of the front
portion of a cathode directed streamer and
qualitative distributions of the electron density ne,
the density difference n+ - n- (space charge), and
longitudinal field E along the axis.[1]
The strong field near the tip is created by its own charge, and decreases
approximately as E = Em (rm/r)2 . The radius at which the field is maximum is termed
the tip radius rm and it approximately coincides with the initial radius of the cylindrical
channel extending behind the tip. The strong field region in front of the tip is the site
of ionization of air molecules by electron impact.
The streamer moves forward as a wave. Let us try to describe the process of
elongation (once the streamer is already well developed ) in a discontinuous manner
– step by step :
In front of the tip there is a high field region. Electrons are accelerated toward the
tip, thereby ionizing air molecules. The number of ionized molecules depends, of
course, on the initial number of electrons which in turn trigger this avalanche
process. The initial electrons necessary for this are generated by the streamer in
advance. What happens is that not all the molecules, hit by an electron, are
ionized. In our case, N2 molecules get exited by an electron impact and as they
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emit photons, O2 molecules, whose ionization potential is lower than that of N 2,
get ionized, thus providing the initial electron density n0 of about 105 – 106 cm-3
at a distance of 0.1 – 0.2 cm from the tip (figure 5.4).
N 2* N 2 ; O2 O2 e
Fig. 5.4. Initaial conditions, which
lead to streamer elongation;
Each of these electrons now gains energy from the strong field, ionizing air
molecules thus triggering an electron avalanche. The electron density in front of
the tip increases by many orders in magnitude (n ~ 1014cm-3).So now we have a
number of free electrons (and positive ionized air molecules) in front of the tip
(figure 5.5). We also estimate, that so far they hadn’t moved too much towards
the tip. This free electrons are now attracted to the tip and move, in our case, to
the left.
Fig. 5.5. The initial electron have
triggered an »avalanche «
ionization process in front of the
tip and the number of electrons
there has greatly increased;
The old positive tip has been neutralized and a new tip has been formed (figure
5.6). The tip has actually moved by much more than any single electron has, as
is typical of any movement described as wave movement – individual particles in
the system move by a little whereas the effect is transferred over longer
distances.
Fig. 5.6. The old tip has been neutralized
and the new tip has been formed (exposed);
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Considering what we have just said about streamer propagation, one can naturally
raise a question, how does rain affect the formation of a lightning spark. A leader
namely propagates because of the fact, that each single streamer in front of the
leader tip propagates. So if there are water droplets in the air, this certainly affects
the whole process.
Even though the moist air is a much better conductor than dry air is, it would
seem, that droplets in the air hinder the charge flow; they have to evaporate. So if
we look at this from this point of view, clearly there has to be an excess of energy
available in moist and wet conditions as compared to dry conditions, since a certain
amount of energy is needed for evaporation of water. We can estimate this energy, if
we know the amount of water that evaporates during the leader development. The
region in the air, that is surely free of water, is the leader channel.
In the cloud, there is allways water present ( c~ 3g/m3) and then there is also
some amount of water due to rainfall, which varies along the height of the cloud and
is greatest at the base of the cloud. Since the intesity of rainfall varies between 1 –
30 kg/(m2 h) , we can take an approximate value of 10 kg/(m 2 h), that is 2.8 g/(m2 s).
The velocity of average droplets falling towards ground is 10 m/s, so that the density
of “falling” water is F ~0.3 g/m3 . The total density of water in the air is therefore
about 3.3 g/m3 and the energy, necessary for evaporation of this water is about
8103 J/m3 .
The leader (channel) reaching from the cloud to the earth, is about 3km long
and ~ 10cm thick; so the volume of air free of water is V~ 0.1 23103 m3 = 30 m3
(this is a large estimation of a leader channel volume; we can therefore say V~10m 3
). This means, that the required energy is about 105 J. A leader finds itself mostly
outside of the cloud (F/c ~0.1) and therefore the better estimate of the energy,
necessary for evaporation of water; is probably about WE ~ 104 J.
The value of WE is larger than the energy Wi, necessary for ionization. An
average leader transports about q ~ 10C of charge to the ground. If we take Wi0
~30eV as the ionization energy of a single air molecule, then Wi ~q/e0Wi0=300J.
The total energy, used for ionization of air molecules is, however probably higher.
The charge that actually flows to the ground doesn’t give us the total number of air
molecules, that had been ionized during the leader development, since some ions
recombine with electrons, and so these electrons can’t reach the earth. So it is
plausible to estimate the total ionization energy to Wi ~103 J, which is so same ten
times smaller than WE .
We can also estimate the energy that is used for the heating of the leader
channel. The radius of the channel depends on the potential difference between the
cloud and the earth and it varies between r0 = 0.1 cm – 1 cm (this is the radius before
thermal expansion; after expansion r ~ 5 cm , as in the estimation above). The
volume of air to be heated is so about (2r0)2 l ~ 22 10-4 3 103 m3 ~ 1 m3 , which
corresponds to about 1kg of air. If T~5000K, then Q = mcpT ~1103
5000=5106J.
From this we can conclude, that rain probably doesn’t affect the leader
formation much, since a lot more energy is needed for heating as for ionization and
evaporation of rain droplets. However, it can be, that only a consideration of Wi and
WE is of should be of some importance, since ionization is essential for leader
propagation. And since the two energies are comparable, this could suggest, that
lightning strikes might “prefer” dry conditions.
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Now let us consider a fast streamer so that the calculation of electron production
can ignore the slight drift of electrons from a given site for the short tome the wave
passes by. In this case we can write:
ne
i ne
t
t
rm
nc
dx
exp i dt exp i
0
0
n0
Vs
(1)
where i is the frequency of electron ionization of molecules (figure 5.7); n0 the
density of free electrons at t = 0 at the site of interest ; nc the density of free
electrons after the wave passes by;
Its time integral has been transformed to the integral over coordinate x,
corresponding to the coordinate system moving together with the wave. Due to the
sharp increase of the ionization frequency with the field, the region where the field is
not much less than its maximum contributes the most to the electron production. This
region of the wave is of the same order as the tip radius rm. So we can write an
approximate expression for the streamer velocity Vs :
im rm
,
(2)
im i Em … ionization frequency at field Em
Vs
ln nc n0
Figure 5.7. Ionization frequency of air molecules by electron impact under normal conditions[1]
The quantities Em and rm which determine Vs are interrelated by the tip potential Ut.
For an isolated conductive sphere with uniformly distributed charge Q’ we have
U rm Em Q 4 0 rm . A streamer looks more like a cylinder with a hemispherical end.
It can be shown, that in a long perfect conductor of this shape, the tip charge is
Q 2 0 rmU t
(3)
and the field at the tip front point is related to the potential by
U t 2 Em rm
(4*)
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(* infact, Ut should be replaced by Ut = Ut – U0(l), figure 5.2)
An estimate of plasma density nc behind the tip can also be obtained. The electron
density in the strong field region increases as ne n0 exp( imt ) for the time t rm / Vs .
During this period of time, the electron density rises to its final value nc n0 exp( im t )
and the electron drift towards the channel with velocity Ve e Em exposes the positive
charge of the new tip. The electron charge that flows though a unit cross section over
time t is
t
0
jdt
t
0
t
t
e E n
dq
q
dt n e vdt ee Em n0 exp( imt ) e m c
D
0
0
dtS
im
S
(5)
It leaves behind a positive charge of the same surface density. The effective
thickness of a positively charged layer is x rm (without proof), so that its field is
approximately equal to Em D / 0 as is the case for an evenly charged plate. By
substituting D from (eq. 5), we get an estimate:
nc 0 im / ee
(6)
The least convincing part in the streamer theory seems to be the issue
concerning the streamer tip radius (or the maximum field Em, as they are interrelated
by (eq. 4) ). It is likely that their values are established under the action of selfregulation mechanism related to proportionality Vs i(Em) .
im U t / 2 Em
im rm
From eq.(2) and eq.(4): Vs
ln nc n0 ln nc n0
If, at constant tip potential Ut, the tip radius turns out to be too small, the channel
front end will not only move forward but it will also expand, since the strong lateral
field will trigger the process of ionization in radial direction. The value of rm will rise
while that of Em will fall (according to (eq. 4)).
Suppose, on the contrary, that the radius rm is too large and the field is too low.
Here we are comparing the current maximum field E to the value Em, which we
believe to be “the right one” for the
streamer propagation. The tip looks
like a semisphere and any slight
plasma protrusion out of this surface
will locally enhance the field just in
front of the protrusion, since this
protrusion will have a smaller radius
of its tip. The ionization rate will
greatly increase there (figure 5.7),
and the protrusion will run forward on
it’s own (eq. 2) as a channel of a
smaller radius. But now again, the
channel radius of this runaway
protrusion will expand if it is too
small. So this two mechanisms somehow regulate the tip radius.
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Numerical simulations show, that the streamer’s choice of maximum field seems to
be Em 150kV – 170kV/cm. The tip radius then varies with the tip potential,
approximately satisfying equation U t 2 Em rm .
Example:
Em =170kV/cm (in air),
im 1.1 1011 s-1 ,
e 270 cm2 / Vs ,
rm = 0.1cm (corresponding to Ut = 34kV),
n0 106 cm-3 nc 2 1014cm-3
Vs 5 106 m/s
This is in aggrement with experiments, where Vs = 106 –107 m/s.
5.1.1 Current and field in the streamer
The streamer starts to develop at the anode and then elongates. Its channel is under
high potential which changes from the anode potential Ua at the starting point to a
certain value Ul at the channel end, close to the tip potential Ut (the difference
between Ul and Ut is about Emx <<Ut , where x<<rm is the effective thickness of
a positively charged layer). The channel is electrically charged, since the potential at
any point x along it is higher than the unperturbed potential of the space U0(x)
created by electrode charges in the absence of a streamer.
Assume first, that the channel is a perfect conductor. The capacitance of a long
solitary conductor is
C 2 0l / ln(l / r )
(7)
and its charge is Q CU , because a perfect conductor is under only potential U . The
average capacitance per unit length is
Cl
2 0
C
l ln(l / r )
(8)
and it varies (slightly) with l and r.
As an approximation justifiable by calculations, we shall use the capacitance per
unit length (eq. (8)) and apply it to the real streamer channel. The charge per unit
length is
( x) Cl U ( x) U 0 ( x)
2 0 U ( x) U 0 ( x)
ln(l / r )
(9)
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When a channel elongates by dl, its new portion acquires charge ldl. Index l will
denote parameters of the front channel end , x = l. Therefore
2 0 Ul U 0 (l )Vs
q
(10)
lVs
,
Ut Ul
t
ln(l / r )
Example: at l = 1m, r = 0.1cm, Vs = 5 106m/s and Ul Ut 34 kV we get il =1.37A.
il
Here we must also mention that the current il near the channel end is lower than that
of the tip, because the charge per unit tip length t Q / rm 2 0 Ut U0 (l ) is larger
than in the channel ( remember that for a conductor of the shape in question we
have Q 2 0 rmU t … eq. 3). So the current it much exceeds il . This current
perturbation, however, has a local character, and is caused by the “displacement” of
the old tip from its previous position into its new position. It cannot be detected by
current registration from the anode side. If a current detector were placed at the site
of a newly born portion of the channel, it would register current i it for a very short
period of time t rm /Vs 10-9s ; then the current would decrease to il .
We can now estimate the longitudinal field Ec in the channel behind the
streamer tip. The current behind the tip is conduction current il Sj rm2enc e Ec .
im rm
By equating this expression to (eq.10) and using Vs
and
ln(nc / n0 )
Ut U 0 (l ) U 2Em rm with Ut = Ul , we get:
il
2 0 UVs
2 0 U im rm
rm 2enc e Ec
ln(l / rm )
ln(l / rm ) ln(nc / n0 )
Ec
{ 2U / rm 4 Em ,
2 0 U im
ln(nc / n0 ) ln(l / rm ) e nc e rm
4 Em
ln(nc / n0 ) ln(l / rm )
(11)
0 im /(ee ) nc }
For a 1m streamer, the product of logarithms in the denominator is close to 130.
Therefore, the field in the channel is Ec 5.2 kV/cm ( Em = 170kV/cm).
Within the theory accuracy, this value does not contradict the average measured
channel field of 5kV/cm necessary to support the streamer.
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5.2 The necessity of a streamer accompaniment
We have shown that the field in the channel behind the streamer tip is too low for
ionization of air molecules by an electron impact (~5kV/cm). The only thing that could
preserve the high conductive plasma state is high temperature of the channel.
However, it can be shown, that the current in the channel is incapable of doing that.
The volume (radius) of the channel is too great and can only be heated by a few
degrees ( T<10K). The Joule heat release into the channel could raise the
temperature sufficiently (T 5000 – 6000K) only for a smaller channel radius rm. We
already mentioned, however, that then the lateral field becomes far too strong and
ionization occurs in the radial direction, thus increasing the channel radius.
Without the ionization process in the channel, the electrons are lost due to
recombination and attachment (a 10-7s) to oxygen molecules. Plasma decays and
the streamer looses its connection with the anode and dies. Fast streamers,
supported by megavolt voltages, are capable of elongating to l 1m in cold air
without loosing much of their connection with the original electrode. Lightning sparks,
on the other hand, bridge gaps of d 3 km, and the formation of the plasma channel
– a leader, crossing the gap, takes up to 0.01s.
Thus, a key condition for a long-term spark development is the formation of a
thick space-charge cover around it, having the same sign as the channel potential.
The charge reduces the field on the channel surface, depriving the channel of its
ability to expand due to ionization. It is only a channel with a small cross section that
can preserve the ability to be heated.
According
to
the
Gaussian
theorem,
with
Er /(2 0 r )
2 0 U
, the field Er at the lateral surface of the channel with a
Cl U ( x) U 0 ( x)
ln(l / r )
small radius r reaches values U / r ln(l / r ) 1 10MV / cm only for such a
structureless channel as a streamer, and lateral ionization expansion immediately
follows.
The space charge of a streamer zone and leader cover, having the same sign as
that of the channel potential, greatly reduces the field at the channel surface.
Roughly, owing to the field redistribution by space charge, the huge potential (MV)
now drops across a much longer length R of the streamer zone and the charge cover
radius, rather than across a length nearly as short as the channel radius r. In this
case, the field scale is a moderate magnitude U/R but not U/r, because even a
laboratory spark has R 1m, and therefore ELateral ~ MV/m ~ 10kV/cm.
5.3 Current in the leader
The way in which a leader grows in length is in a way similar to the one of a
streamer. Currents of all streamers starting from the leader tip are summed up,
heating the spark channel. This total current charges the region in front of the tip
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(pulls out the electrons), neutralising the charge of the old tip, and when a new tip is
formed, the spark elongates by a length of about the tip length. Part of the streamer
zone appears to be behind the tip, transforming to a new cower for the newly born
leader portion.
Although a leader has a more complex structure than a streamer, the capacitance
per unit length of a leader system (the channel plus a cover) will be described by the
same formula (eq. 8) if l is substituted by leader length L and the conducting channel
radius r by cover radius R, the actual radius of a charged volume. Similarly, the
current iL at the leader channel front is related to the tip potential and leader velocity
VL by the same expression (eq.10)
2 0 U L U 0 ( L)VL
(12)
iL LVL
.
ln( L / R)
Like in a streamer, the linear capacitance of a semispherically shaped streamer zone
is Cl 2 0 . The tip current flowing into the streamer zone
(13)
it 2 0 Ut U0 ( L)VL
is by a factor of ln(L/R) higher than iL , again like in a streamer. But since the leader
logarithm is closer to unity (at least for a laboratory leader: L 10 m and R 1 m) the
currents iL and it do not differ that much. A typical laboratory leader has i iL it
1A , U 1MV, and from (eq. 12) VL 2 104m/s which is close to numerous
measurements, in which VL (1 – 2.2) 104 m/s. This is of course much less than the
streamer velocity, since a streamer must “propel” itself away from the leader tip.
Lightning leaders exhibit higher values of current. We can only guess about the
values of descending leaders or make estimates. Ascending leader currents, on the
other hand, are not difficult to measure and there have been many measurements of
this kind. Normally, a current detector is mounted on top of a tower dominating the
locality. The current nearly always rises in time. At the moment an ascending leader
starts its travel, its current is lower than 10A, whereas at the end of the travel, it may
rise to 200 – 600 A, with an average value of about 100 A. The current rises because
the leader experiences grater and greater potential difference when approaching the
2 0 U L U 0 ( L)VL
cloud iL LVL
. It is the outer potential U0(L) that grows
ln( L / R)
(actually it falls, since the cloud is usually negatively charged) as the leader gets
nearer to the cloud. Besides, the leader also goes up with an increasing velocity VL .
A combination of these factors rises the current.
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Jernej Slanovec
Lightning
5.4 Field in the leader channel
There are no direct experimental data on the state of a lightning leader channel.
Therefore one has to rely on information derived from laboratory spark experiments.
Streak photographs were taken continuously of a leader propagating from a rod
anode to a grounded plane ( figure 5.8). Pulses of voltage U0 were applied to gaps of
various length d. By measuring the streamer zone length Ls in the photographs at the
moment the zone touched the grounded electrode and assuming the average zone
field to be Ecr = 4.65 kV/cm, one can find the leader tip potential Ut = EcrLs and
evaluate the average field in the leader channel as EL = (U0 - Ut)/L , where L = d – Ls
is the channel length (table 1).
Fig. 5.8: A streak photograph of a
positive leader; streamer zone has
just reached the ground cathode.[1]
d [m]
U0 [MV]
Ls [m]
L [m]
Ut [MV]
EL [V/cm]
5
10
15
1.3
1.9
2.2
2.3
3.2
3.6
2.7
6.8
11.4
1.1
1.5
1.7
750
590
440
Table 1. Leader paramater derived from experimental data.(after [1])
It is clear that the field in the leader channel drops with increasing length. In a leader
3 km long, the field in the channel is thought to be ~ 10 V/cm.
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Jernej Slanovec
Lightning
Conclusion
In the seminar I tried to present some basic features of lightning, because lightning is
a phenomenon observed many times by everyone. Nowadays it is somehow
“understood” by peple in a sence: “Lightning strikes because of the charged clouds!”.
This “explanation” is far from describing even the basic picture. For me the main
problem in trying to console my curriosity was, how a lightning can propagate through
air, and in general, how a spark can cross gaps of air. That is why the central part of
the seminar is about leaders and streamers.
Perhaps one question often raised is, wheather lightning strikes could serve as
a power supply. The answer is: “No!”. A very quick calculation can convice us of that.
The voltage between the cloud and the earth can hardly exceed 100MV, and the
transported charge is less than 100C. Maximum energy release is so 1010 J, which is
less than a family cottage consumes in a year for heating, illumination and other
needs. Still more, only a small portion of this energy can be utilized, since most of it is
dissipatated in the atmospfere. All storms send to the earth an average of 4 – 5
lightning sparks per square kilometer over a year, providing a power of less than
1kW/km2. In a country of 500km 400 km, this is about 200MV, which is a very small
value compared with the electrical power produced by an industrial country. If we
also imagine the complicated infrastructure (overground nets?) that would be needed
for utilization of such energy , we come to realize, that the idea is not a very good
one.
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Jernej Slanovec
Lightning
Reference
Bazelyan, E. M. and Raizer, Y. P. : Lightning Physics and Lightning Protection (Bristol
[England], Philadelphia: Institute of Physics Pub., 2000)
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