22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Effects of electron-electron collisions on streamer initiation in air
E. Tam, J.J. Lowke and A.B. Murphy
CSIRO Manufacturing Flagship, Sydney, NSW, Australia
Abstract: In this paper, we present calculations of the formation of streamers including the
effects of electron–electron collisions. A two-dimensional model has been used to
determine electric field strengths and charge densities around a capped cylinder.
A comparison is made of the development of the streamers including and excluding the
dependence of the ionization coefficients on the electron density.
Keywords: lightning, streamer, modelling
1. Introduction
In recent years, aircraft manufacturers have been
motivated to produce aeroplanes largely made out of
carbon composite materials rather than aluminium, due to
the significant reduction in weight for comparable or
superior mechanical properties. These materials however,
have significantly reduced electrical conductivities, which
makes lightning protection more difficult. Aeroplanes are
struck by lightning about once every year on average, so a
thorough understanding of the initiation of lightning by
aircraft is important.
A long-standing question about lightning is the electric
field required for its initiation. The breakdown voltage of
air is frequently quoted as somewhere between 2.5 and
3.0 MV/m; however, no experimental measurements of
field strengths in natural lightning has found values even
approaching this range. In addition, the field required to
sustain a streamer has been found to be approximately
500 kV/m, or around 5 to 6 times less than predicted with
standard approaches.
A two-dimensional axisymmetric model has been
developed to predict discharge initiation. In this paper,
we calculate the effect of increased ionization coefficients
with increased electron densities from electron–electron
collisions. The calculations include the distortion of
background electric fields by the metal and dielectric
materials in the aircraft, and the initiation of corona
discharges on the aircraft. A finite-difference approach
incorporating a direct method is used to solve the system
of equations.
rate of this attachment is represented by 𝜂/𝑁, where 𝑁 is
the gas number density and the attachment coefficient 𝜂 is
the fraction of electrons attached in moving a unit
distance forward in the electric field. In regions of high
electric field, electrons are of sufficient energy to ionize
air molecules to form a new electron and a positive ion.
This rate is represented by 𝛼/𝑁, where the ionization
coefficient 𝛼 is the fraction of the electrons that produce a
new electron by ionization in moving a unit distance
forward in the electric field. Thus the net increase in
electrons in unit time at a given position is 𝜕𝑛𝑒 /𝜕𝜕 =
𝑛𝑒 (𝛼 − 𝜂)𝑊𝑒 , where 𝑊𝑒 is the electron drift velocity.
Additional terms are required in the electron continuity
equation to account for the recombination of electrons and
positive ions, and to represent the effects of convective
flow of the charged particles. Lastly, a diffusion term is
included where diffusion coefficient, 𝐷𝑒 , is estimated
using the Einstein–Smoluchowski relation, i.e., 𝐷𝑒 /𝜇 =
𝑘𝐵 𝑇/𝑒, where 𝜇 = 𝑊𝑒 /𝐸 is the electron mobility, 𝑘𝐵 is
the Boltzmann constant, 𝑒 is the charge of an electron and
𝑇 is the electron temperature.
Predictions of the onset of corona (streamer) discharges
at typical electric potentials are obtained by using a
control volume approach to solve the equations:
2. Model
In the atmosphere, there is a fairly continuous
production of free electrons, produced for example by
cosmic rays. We represent this effect by including a
constant production term 𝑆𝑒 . Assuming air has an
extremely low conductivity in the absence of an electric
field (e.g., 10−17 S/m) and knowing the mobilities of the
charged species, one can estimate the equilibrium charge
densities without the presence of an electric field; this is
our initial condition. Electrons in the atmosphere slowly
attach to form negative ions, in particular O−
2 ions. The
(3)
O-11-4
∇2 𝜙 =
𝑒
(𝑛 − 𝑛− − 𝑛𝑒 )
𝜖0 +
𝜕𝑛𝑒
= 𝛼𝑊𝑒 𝑛𝑒 − 𝜂𝑊𝑒 𝑛𝑒 − 𝛾𝑛𝑒 𝑛+
𝜕𝜕
− ∇ ⋅ (𝑾𝒆 𝑛𝑒 ) + ∇ ⋅ (𝐷𝑒 ∇𝑛𝑒 ) + 𝑆𝑒
𝜕𝑛+
= 𝛼𝑊𝑒 𝑛𝑒 − 𝛾𝑛+ (𝑛𝑒 + 𝑛− )
𝜕𝜕
− ∇ ⋅ (𝑾+ 𝑛+ ) + ∇ ⋅ (𝐷+ ∇𝑛𝑒 ) + 𝑆𝑒
𝜕𝑛−
= 𝜂𝑊𝑒 𝑛𝑒 − 𝛾𝑛− 𝑛+
𝜕𝜕
− ∇ ⋅ (𝑾− 𝑛− ) + ∇ ⋅ (𝐷− ∇𝑛− )
(1)
(2)
(4)
where 𝑡 is the time, 𝑛𝑥 is the number density of electrons,
positive and negative ions for 𝑥=𝑒, + and − respectively
and 𝛾 is the recombination coefficient.
The ionization and attachment coefficients are each a
function of the electric field; so 𝛼/𝑁, 𝜂/𝑁 and 𝑾 all
1
depend on 𝐸/𝑁. The first two of these variables are
shown in Fig. 1. The values are taken from the theoretical
predictions of [1], and are consistent with other
theoretical investigations and also with experimental
values of both ionization and attachment coefficients [2].
Recombination coefficients and ion mobilities were taken
from [1].
Fig. 1. Ionization and attachment coefficients, normalized
to the number density, as a function of the ratio of electric
field strength and number density in dry air. Top and
right-hand scales indicate values at 1 atmosphere.
With the ionization and attachment coefficients shown in
Fig. 1, an electric field strength of ~2.5 MV/m is required
for the formation of lightning discharge. Ionization
coefficients in air at low electric fields, e.g., 500kV/m at
1 bar, are effectively zero, because electrons of energies
sufficient to cause ionization are removed, for example,
by electron energy losses producing vibrational excitation
of nitrogen. However, for high electron densities, the
number of electrons of sufficient energy for ionization is
increased due to the electron energy distribution function
becoming more Maxwellian due to electron–electron
collisions. The modified ionization coefficients from [3]
are of particular interest as discharges are possible for
much lower electric fields. Thus we compare the results
using the ionization coefficients from [1] with using
ionization coefficients based on the ones calculated by
Petrova et al. in [3], which are presented in Fig. 2.
3. Results
The thundercloud is represented by imposing a potential
of –100 MV at the upper plane of the computational
domain. Frequently, thunderclouds have been calculated
to be at higher potentials, for example at –500 MV. The
height of this potential surface is taken to be at 2 km.
Aircraft usually fly at higher altitudes, but most lightning
strikes to aircraft occur close to take-off or landing.
We choose the model aircraft to be a vertical cylinder
of radius 25 cm and length 50 m. We choose the cylinder
to be vertical as this will be a position for which there is
maximum enhancement of the background electric field at
the tips of the cylinder. The length of 50 m is
representative of large commercial aircraft. The radius of
25 cm is chosen to approximate the radii of curvature
occurring at wing tips and the tail of aircraft. The ratio of
the cylinder's length (measured from its centre to its tip)
of 25 m, divided by its radius of 25 cm, gives an aspect
ratio of 100.
There are three stages in the development of the electric
field. First, there is the background field from the
thundercloud of 50 kV/m produced from the 100 MV
postulated for the cloud potential at a height of 2 km.
Second, there is distortion by the cylinder, representing
the aircraft fuselage, as illustrated in Fig. 3, whereas yet
there are no electrical charges in the air at the high field
tips of the cylinder. Third, there are distortions due to the
space charge produced by the discharge.
Fig. 3. Electric field at an early stage (t ~1 ns), showing
distortion of the background field by the cylinder. Note
the scale is logarithmic in this figure and the background
field is 50 kV/m.
Fig. 2. Ionization coefficients normalised to the number
density as a function of 𝑬/𝑵 for electron densities
ranging from 4×10-10 to 4×10-4 m-3.
2
Fig. 3 shows the electric field strength in the immediate
surroundings of the conductive cylinder. The electric
field strength inside the conductor is calculated to be to be
less than 10 V/m, significantly lower than the average
field strength of 50 kV/m. The electric fields in the gas in
the immediate surroundings of the cylinder tips can be
seen to be significantly enhanced, with field strengths of
O-11-4
the order of 5 MV/m, in agreement with an approximate
analytic prediction.
Fig. 4a shows, in more detail, the electric field in the
vicinity of the top tip of the cylinder shown in Fig. 3
(again after ~1 ns). The peak electric field in this figure
exceeds 3 MV/m, so streamer formation is expected as the
electric field exceeds the breakdown field of air. Fig. 4b
shows typical calculated field distributions after the space
charge effects have become important (at ~19 ns in the
figure) using the ionization coefficients presented in
Fig. 1. This field distribution changes with time as the
discharge develops and becomes larger, there being a
significant maximum of the electric field at the leading
edge of the discharge.
The distributions of electron densities, for the
conditions of Fig. 4b are shown in Fig. 5a. In this figure,
the values of the densities in the metal are set to zero for
clarity, as the electron density is several orders of
magnitude larger in the metal than in the gas.
Fig. 5. Electron densities at t ~19 ns at the top (positive)
end of the cylinder using the ionization coefficients
presented in (a) Fig. 1, and (b) Fig. 2.
Fig. 4. (a) Electric field at an early stage (t ~1 ns) near
the top cap of a cylinder, showing distortion of the
background field by the cylinder. (b) As for (a), but at
t ~ 19 ns, using ionization coefficients presented in Fig. 1.
O-11-4
The maximum electron densities in Fig. 5a occur in the
regions of maximum electric field strength in Fig. 4b.
When the field drops below 2.5 MV/m, there is a rapid
transition from net electron production to net electron loss
due to attachment of electrons to the neutral gas
molecules. Electron attachment to neutrals will reduce
the brightness of a stepped leader.
Fig. 5b shows the electron density at the top of the
cylinder at the same time as Fig. 5a, but using the
ionization coefficients presented in Fig. 2. Higher
electron densities can be seen in Fig. 5b compared to
3
Fig. 5a, as well as a slight increase in the streamer length
from ~5 cm to ~7 cm. This is to be expected as the
ionization coefficients in Fig. 2 are always larger than
those presented in Fig. 1, provided the electron density is
greater than zero.
restricted to the speed of the electrons, the streamers
would have a maximum length of ~ 0.2 mm. However, as
shown in Figs. 5 and 6, this is clearly a massive
underestimate, since the streamer length reaches 10 and
30 cm in the respective figures. This indicates that the
growth of the streamers is predominately due to the
production of charges rather than the charged species
rearranging themselves at a rapid pace. In areas where
the electric field is large, the charged species production
is also large, which quickly leads to a large density of
charged species, so the charged species do not need to
travel as far to influence the electric field. Thus we see in
Fig. 5, where the electrons move away from the
production zone, a small increase in streamer size occurs
when the modified ionization coefficient is used. In
Fig. 6, however, the electrons move in the same direction
as the streamer, and thus the charge production region of
the streamers always has an elevated electron density,
strongly enhancing the ionization coefficient, leading to
much faster growth.
4. Discussion and conclusions
We have presented the results of simulations of the
initiation of streamer discharges from a capped cylinder,
of dimensions similar to that of an aeroplane fuselage, in
an electric field representing that between a thundercloud
and ground.
The calculations were made in two
dimensions because of the available computational
resources. This means that the streamers are effectively
conical in shape.
Nevertheless, they illustrate the
interaction between electric fields and charged species
that lead to corona initiation and streamer progress.
We have obtained significantly higher charge densities
than were possible in a previous study [4], in which
values of the ionization coefficient were capped to avoid
numerical instabilities. The higher charge densities allow
us to more accurately predict the rapid formation of
streamers that occurs in nature.
We have shown the importance of taking into account
the dependence of ionization coefficients on electron
density that arises when electron–electron collisions are
considered. This leads to higher electron densities and
more rapid streamer growth, particularly from the
negative end of the cylinder.
Fig. 6. Electron densities at t ~19 ns at the bottom
(negative) end of the cylinder using ionization
coefficients presented in (a) Fig. 1, and (b) Fig. 2.
In Fig. 6, we present the electron densities for the cases
corresponding to those in Fig. 5, but for the bottom
(negative) end of the cylinder. Here the difference
between the two different ionization coefficients is far
more obvious. The stark difference is a consequence of
the direction in which the electrons move in relation to the
streamer front.
The drift velocity of the electrons with the calculated
electric fields is of the order of 104 m/s. Within the time
frame of ~20 ns, assuming the streamer growth is
4
5. References
[1] J.J. Lowke. J. Phys. D: Appl. Phys., 25, 202 (1992)
[2] A.V. Phelps. Excitation and Ionization Coefficients.
Gaseous Electronics V. (New York: Pergamon
Press) (1987)
[3] T. Petrova, H. Ladouceur and A. Baronavski. Phys.
Rev. E, 76, 066405 (2007)
[4] J.J. Lowke. IEEE Trans. Plasma Sci., 32, 4 (2004)
O-11-4