Macroscopical model of streamer coronas around a spherical electrode

Abstract

We present a model for streamer coronas emerging from a spherical electrode at high electrostatic potential. By means of a macroscopic streamer model and approximating the corona as a set of identical streamers with a prescribed spatial distribution around the electrode, we establish that coronas more densely packed with streamers are slower and more efficient at screening the electric field inside the streamers. We also apply our model to investigate the electrostatic potential at the boundary of the corona sheath that surrounds a leader and we underline the relevance of the rise-time of the leader potential during a leader step.

Full text

Macroscopical model of streamer coronas
around a spherical electrode
M González
1,2
, F J Gordillo-Vázquez
1
and A Luque
1,3
1
Instituto de Astrofísica de Andalucía, IAA-CSIC Glorieta de la astronomía s/n, E-18008, Granada, Spain
2
Institut de Planétologie et d’Astrophysique de Grenoble (IPAG), Université Grenoble Alpes, F-38058,
Grenoble Cédex 9, France
E-mail: marta.gonzalez@univ-grenoble-alpes.fr and aluque@iaa.es
Received 21 May 2019, revised 11 September 2019
Accepted for publication 16 October 2019
Published 12 November 2019
Abstract
We present a model for streamer coronas emerging from a spherical electrode at high
electrostatic potential. By means of a macroscopic streamer model and approximating the corona
as a set of identical streamers with a prescribed spatial distribution around the electrode, we
establish that coronas more densely packed with streamers are slower and more efficient at
screening the electric field inside the streamers. We also apply our model to investigate the
electrostatic potential at the boundary of the corona sheath that surrounds a leader and we
underline the relevance of the rise-time of the leader potential during a leader step.
Keywords: streamers, corona, electric discharge
1. Introduction
When a sharp electrode such as a needle or a wire is subjected
to a high electric potential it ignites a type of electrical dis-
charge called corona. Depending on conditions such as
electrode geometry and rise time of the potential, the dis-
charge may exhibit a variety of forms [1,2]. A sufficiently
sharp electrode on which a sufficiently impulsive potential is
applied launches a corona composed of many thin filaments
called streamers that propagate due to electron impact ioniz-
ation in a high-field volume around their tips [3–5].
Streamer coronas exist on Earth in a variety of natural
and artificial conditions. They appear as sprites [6,7]or blue
jets [8–11]on the upper atmosphere and they form a sheath
around the hot, highly conducting leader core of a long spark
[12]or an advancing lightning channel [13]. In artificial
electrical systems, the undesired emergence of coronas can
cause severe negative effects like power loss or high-fre-
quency electromagnetic interferences. On the other hand,
streamer coronas have a variety of beneficial applications
such as water and gas cleaning, electrostatic precipitation,
ozone creation, or even inactivation of bacteria (see
e.g. [14,15]).
Single streamers have been widely studied from a
microscopical standpoint with Monte Carlo [16–18],fluid
[19–25], and hybrid [26]models. Other studies on specific
phenomena like branching [27,28], head-on collisions
between streamers [18,23,29,30], the effect of inhomo-
geneities of the medium [31,32]or the formation of luminous
structures inside the channels [33,34]are also available. The
interaction between streamers, however, is a very complex
matter and most of the works about it involve only two
streamers [35–39]. Up to our knowledge, only the works by
Naidis [35], Akyuz et al [40]and Luque and Ebert [41]have
simulated a system with more than two streamers, and in all
cases the underlying system was heavily simplified.
Recently we presented a macroscopical model for strea-
mers treating them as advancing imperfect conductors [42].
This model allowed us to explore long term channel proper-
ties like channel inhomogeneities and their role in the for-
mation of streamer glows. Here we apply our model to
investigate the properties of streamer coronas composed of
many interacting streamers emerging from a spherical elec-
trode. Our aim is to understand how properties of the corona
depend on the density of streamers that compose it. As we see
below, the corona model presented here is considerably
Plasma Sources Science and Technology
Plasma Sources Sci. Technol. 28 (2019)115007 (13pp)https://doi.org/10.1088/1361-6595/ab4e7a
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Original content from this work may be used under the terms
of the Creative Commons Attribution 3.0 licence. Any
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simplified but precisely for that reason it serves to build an
intuition about the corona’s complex dynamics.
This paper is structured in six sections, the first being this
introduction. In section 2we describe both the single streamer
and the streamer interaction model, explaining the underlying
assumptions and approaches. In section 3we describe the
numerical approximations and the implementation of the model.
In section 4we describe the results obtained in our simulations
for one configuration of experimental interest whereas in
section 5we investigate the potential drop within a corona.
Finally, in section 6we summarize the conclusions of our work.
2. Model description
In this section we start by briefly summarizing the streamer
model used as base for the streamer corona. Then we explain
how to add the interaction between several streamers and the
numerical implementation used in this work.
2.1. General description of the single streamer model
We start with a brief description of the main equations and
assumptions of the single streamer model, as thoroughly
described in [42]. The streamer is modeled as a symmetric
one-dimensional imperfectly conducting charged channel
attached to an spherical electrode of radius aand propagating
in the zdirection (see figure 1(a)). This system evolves
according to electrodynamic and chemical models that are
coupled through the electric field inside the streamer channel.
The transport of charge within the channel is determined
by charge conservation ∂
t
q=−∇ ·j, where qis the charge
density and jthe current density. Integrating this equation
across a plane Γperpendicular to the streamer channel we
obtain
()
l¶
¶=-
¶
¶t
I
z,1
where λis the linear charge density, and Iis the current
intensity:
()
òò
l==
GG
qSI j Sd, d. 2
z
Under the assumptions of negligible electron diffusion,
independence of mobilities from the electric field, and uni-
form charge-carrier densities across the channel, the current
intensity is proportional to the integral of the electric field on
the section of the streamer and (1)becomes a linear differ-
ential equation. The intensity Iis composed of three terms,
I=I
self
+I
bg
+I
surf
, that model the electric self-interaction
of the streamer, the influence of an electrode and background
field, and the effect of the surface current around the streamer
head as it propagates. This expression assumes that the
streamer radius is sufficiently small that charge transport in
the transversal direction is instantaneous and we can describe
it using a linear charge density λ.
The terms I
α
, where αä{self, bg, surf}, can be
expressed in an integral form for each zin the streamer:
() () () ( ) ()
ò
s
pl=¢¢¢
aa

Iz zzG zz z
4,d, 3
a
z
0
tip
where σ(z)is the conductivity of the streamer, ò
0
the vacuum
permitivity, and ()¢
a
G
zz,the kernel describing the influence
on zof the charges in
¢z
.
The chemical model contains 17 species coupled by 78
reactions, as described in [42]. It boils down to a system of
ordinary differential equations describing the evolution of the
Figure 1. Model geometry of the simulations with (a)a single streamer and (b)two identical interacting streamers, S
0
and S
1
.
2
Plasma Sources Sci. Technol. 28 (2019)115007 M González et al

densities
n
siof each charge carrier s
i
,iä{1, K,n
spec
}
according to a set of reactions r
j
,jä{1, K,n
reac
}. The
equation governing the evolution of each species s
i
is given
by:
()
()
å
¶
¶=
==
n
tAk n ,4
s
j
n
ij j
l
n
Ijl
1
,
1
,
i
j
reac in,
where A
i,j
is the net number of molecules of species s
i
created
each time reaction r
j
occurs (given by a difference on the
stochiometric coefficients in r
j
),k
j
is the rate coefficient of
reaction r
j
, and I(j,l)is the index of the reactant lof reaction
r
j
. The rate coefficients k
j
,jä{1, K,n
reac
}are functions of
the local electric field, which in [42]is approximated by the
field at the axis for each z. As we explain in section 3, in this
work we approximate instead the local field at each zby the
average of the field on the disk perpendicular to the streamer.
The propagation velocity of the streamer depends on the
streamer radius R
0
and the field at the tip (estimated imme-
diately outside the streamer, since it is discontinuous at the
tip)following the expression derived by Naidis [43]. This
expression is a relationship between the streamer radius, its
velocity and the background electric field. A nonlinear root
finder can be used to solve for the velocity given the other two
quantities. The main assumption behind Naidis’formula is
that the electron density undergoes multiplication by a fixed
factor within the area around the streamer head where the
electric field is above breakdown. For positive streamers, the
generation of a sufficient quantity of electrons outside this
active area due to photo-ionization is implicitly assumed in
the derivation.
The model includes the effect of the field ahead of the
streamer tip by imposing densities
n
s
0
iat the streamer tip, as
explained in [42]. Since streamers are initiated with a finite
length, the species densities are given initial conditions inside
the streamer body, which we set also as
n
s
0
i.
2.2. Additional streamers
Let us now consider the effect of additional streamers on the
model. Neighboring streamers influence each other directly
through their electrostatic interaction; to incorporate this in
our model we introduce two assumptions, in addition to those
in [42]and described in section 2.1:
•There is a perfectly symmetrical configuration of
streamers around a spherical electrode of radius a,so
we can consider that all the streamers are equal, i.e. that
the length, density charge, and other characteristics of all
the streamers S
i
,i>0 are the same as those of S
0
. This
allows us to concentrate on only one streamer, S
0
.
•The streamer radius R
0
is small compared to the
minumum distance between streamers. This assumption
allows us to approximate S
i
,i>0asaninfinitely thin
line of charge when we calculate its effect on S
0
.
Furthermore, we also neglect the variation within the
cross-section of S
0
of the electric field created by S
i
.To
summarize: we consider the finite width of the streamers
only for the interactions between separate points of the
same streamer but not for the interaction between
different streamers.
The geometry of the effect exterted on S
0
by a neigh-
boring streamer S
1
is sketched in figure 1(b). We calculate the
electric field at a point p
0
on streamer S
0
which, without loss
of generality, we consider to be on the zaxis. Thus, p
0
=(x
0
,
y
0
,z
0
)=(0, 0, z
0
)äS
0
.Asin[42], we only need the z
component of the electric field induced in p
0
by a point charge
q
1
located at p
1
=(x
1
,y
1
,z
1
)äS
1
. This is:
() [()]
()
p
=-
++-

Ep qzz
xy zz4.5
pz,1
0
01
121201
23 2
1
Now, given that ()( )==pxyzluuu,, , ,
xyz
11111
111, where
a
u1is the αcomponent of the unit vector in the direction
generating S
1
, and l
1
is the distance from p
1
to the origin, (5),
can be rewritten as:
() []
()
p
=-
+-

Ep qzlu
lz zlu42
.6
pz
z
z
,1
0
01
1
120201 132
1
Then the contribution of the whole streamer S
1
is:
() ()( )
[]
()
ò
p
l
=-
+-

Ep lz lu
lz zlu l
1
42
d, 7
zS
z
z
1, 0
0
11 0 11
120201 132 1
1
where λ
1
is the linear charge density in S
1
.
Now, the electric current generated in the disk of S
0
at z
0
is:
() () () () ()sp=Iz z RzEz,8
zz1, 0 0 0 0 21, 0
where, as in [42],() ( (( ) ))=- -
R
zR zzR1exp
0 tip 0 12 is a
smooth function that models the variation of the radius along
the streamer channel.
Since all the streamers are identical we drop the sub-
scripts in l,λ, and σand, for each streamer S
i
,i>0, define a
kernel
() ()( )
[]
()
p
=-
+-
Kz l Rz z lu
lz zlu
,2.9
ii
z
i
z
0
020
202032
Following the method of images, we add a virtual charge
¢
q(called image or mirror charge)to satisfy the boundary
conditions in the spherical electrode boundary. We need,
then, to add the effect of
¢
qlocated at a distance ¢<la
. Since
k¢=q
q
2, and
k¢=ll
2
with κ=a/lbeing a squeeze factor,
we derive the mirror kernel for each streamer S
i
with i>0:
() ( ) ()kk¢=-Kzl Kz l,,.10
mi i,0 0 2
From K
i
and K
m,i
we can define a kernel G
i
so the
contribution of streamer S
i
to the current intensity in z
0
äS
0
has the form of (3). Adding up all kernels we obtain the
following expression for the total current intensity Iat each
point z
0
äS
0
:
() () () ( ) ( )
ò
s
pl=¢¢¢

Iz zzG z z z
4,d, 11
a
z
00
0
tot 0
tip
where G
tot
is a kernel that now includes the self interactions of
streamer S
0
, the background and surface current, and the
contributions of all the surrounding streamers S
i
,i=1... n,
that is, =+++
å=
G
GGG G
i
n
i
tot self bg surf 1.
3
Plasma Sources Sci. Technol. 28 (2019)115007 M González et al

3. Numerical implementation
We use finite differences to solve the partial differential
equations from section 2. We apply a leap frog scheme to
couple the electrodynamical and the chemical parts of the
model, and a Crank–Nicolson discretization for both
equations (1)and (4). Figure 2shows the flowchart used to
implement the calculations. At each time tthe streamer length
is discretized in a set of cells C
i
,i=1... Lwith boundaries
z
i±1/2
, where the right boundary of cell Lis z
tip
. In the elec-
trodynamical step z
tip
moves forward with a velocity obtained
from the expression derived by [43]and we solve the charge
transport system. Some variables describing the system, like
conductivity, densities and electric field are defined at cell
boundaries (e.g. σ
i±1/2
), while the charge density λ
i
is con-
sidered constant within each cell i:
()
¶
¶=-
-+
q
tII,12
iii12 12
where q
i
is the total charge in the cell and I
i±1/2
are the
current intensities at the boundaries. The current intensity at
the boundary of cell jis given by (11), which we integrate
numerically in each cell. This way, we can define an inter-
action matrix M={m
ij
}, and the background component
vector b={b
i
}so at each time tthe current intensity at the
rightmost boundary of cell iis
() () () ( )
å
=+
+
=
Imtqtbt.13
i
j
L
ij ji12
1
For the single streamer model, each element m
ij
from the
interaction matrix is the cross-sectional current induced at the
boundary of cell iby an unitary charge in cell j, and each
element b
i
is the current induced at the boundary of cell iby
the background and electrode field. In this corona model, the
interaction matrix includes also the effect of the jth cell from
all the other streamers. With this expression, we cast (12)as a
subtraction of interaction matrices and background vectors
forming a linear system.
The electrodynamical system is particularly apt for par-
allelization, since each element of the interaction matrix can
be calculated independently. We have, thus, accelerated its
performance using CUDA (integrated with python using
pyCUDA). As for the chemical processes, the discrete system
to solve is sparse and nonlinear, and we solve it iteratively,
using Newton–Raphson. The solution of this system is
accelerated using Fortran and OpenMP.
Our previous work [42]showed that, in the conditions of
the model, the electric field at each point zof the streamer,
used to couple the electrodynamic and chemical parts of the
model, could be approximated by the field at the axis. We
have found that the goodness of this approach decays when
we increase the number of streamers. For this reason in this
work we instead calculated an average electric field across the
channel.
One point where we still need to calculate the field at the
axis is at the tip. The field at the axis in the single streamer
case is calculated using an expression analogous to (3)with a
different kernel G
ax
:
() ( ) ( ) ( )
ò
pl=¢¢¢

Ez zGzzz
1
4,d, 14
a
z
ax
0
ax
tip
where
() ()( )
[() ( )] ()
p
¢= -¢
¢+ -¢
Gzz Rz z z
Rz z z
,.15
ax
2
2232
To calculate the influence of a streamer S
i
on the field at
the axis in z
0
äS
0
, we derive a kernel G
ax,i
from (7),
() ( )() ( )
ò
pl=

Ez Gzlll
1
4,d. 16
iSiax, 0
0
ax, 0
1
This expression can be combined with (14)to obtain the final
expression for the field at the axis for z
0
äS
0
:
()
() () ( )
()() ()
⎡
⎣
⎢⎤
⎦
⎥
ò
òå
pl
pl
=¢¢¢
=¢¢+¢¢
=


17
Ez zG zzz
zGzz G zz z
1
4,d
1
4,,d.
a
z
a
z
i
n
i
ax 0
0
ax,tot 0
0
ax 0
1
ax, 0
tip
tip
When z
0
=z
tip
, the integral of the field at the axis (17)is
bounded and, thus, convergent, but for z→z
0
the conv-
ergence order is increasingly large as z
0
approaches z
tip
.We
evaluate the field at a point slightly beyond the actual tip
z
0
=z
tip
+ε
tip
, where
e
=´-
R10 n
tip tip with n
tip
an order
factor (typically 3). This ensures that we use an appropriate
ε
tip
, close enough to represent the field at the tip, but far
enough that the resolution needed for the field estimation is
reduced. Even after this, we had to use resolutions of the
order of 10
5
to obtain results with enough precision, so
additional optimization was needed. We added a variable
change to expand the integration domain in the tip cell, so
with our new variable ξ, the field slope is not too steep. The
usual ξ=z
−n
variable change helped but not enough so we
opted for a variable change capturing the decay of the radius
close to the tip. We used the following expression:
() ( ) ( )
/
x=-
--
-
ze1, 18
12
zz
R
tip
0
which is the inverse of the smoothing decay factor
applied to the radius near the streamer tip,
=
fR
(
(( ) ))---zz R1exp tip 0 12
. The variable change in (18)
allowed us to use the same resolution as in the rest of the
streamer, heavily accelerating the code while ensuring that a
good and robust approximation of the field at the tip is
being used.
One final element in our model is the distribution of n
streamers in the spherical surface of the electrode. One pos-
sibility is to distribute them randomly with a constant density
per unit surface. However this often leads to pairs of strea-
mers that are too close and that in reality would merge into a
single one. Therefore we look for a distribution of inception
points that in some sense maximizes the minimum distance
between them, accounting for the repulsion of streamers as
well as for the possibility of merging.
4
Plasma Sources Sci. Technol. 28 (2019)115007 M González et al

The distribution of points on the surface of a sphere that
maximizes the minimum distance between points is a classical
open problem (see e.g. [44]), described as one of the problems of
the century by Smale [45].Theconfigurations of points solving
this problem are called spherical codes and have different
applications depending on the metric considered. Given the
electrostatic nature of our problem we opted for the so called
Thomson problem (as described in e.g. [46]), which aims to
determine the equilibrium positions of classical electrons on the
surface of a sphere, subject to their electrical repulsion. This
problem has been solved exactly for some numbers of particles,
where the equilibrium positions are the vertices of classic pla-
tonic solids and the distance between particles is constant.
However, these perfectly symmetrical configurations limit sig-
nificantly the number of streamers we can use, so we use quasi
symmetrical approximate solutions found numerically. In this
work, we use the database of solutions computed by [47],which
we apply to the distribution of streamers on the electrode sur-
face. We will use configurations with n=50, 100, 200, 400
streamer inception points, shown in figure 3.
Figure 2. Flowchart of the corona model. Calculations in the electrodynamic step accelerated with CUDA are marked in green, while Fortran
accelerated calculations in the chemical system are marked in pink.
5
Plasma Sources Sci. Technol. 28 (2019)115007 M González et al

4. Results
In this section we describe the results obtained in a config-
uration which has similarities to laboratory conditions,
although the need of symmetry of our model makes it
impossible to recreate exact experimental configurations. All
the initial conditions and simulation parameters are shown in
table 1: we consider a positive streamer corona emerging
from a spherical electrode that is live with a potential of
50 kV. The electrode radius is 2 cm, resulting in mean
streamer separations at the electrode’s surface varying from
about 3.5 mm to about one centimeter, as reflected in table 2.
Since our model does not account for streamer inception, we
have to initiate the streamers already with a finite length,
which we chose as 1 cm.
4.1. Propagation of a single streamer compared to a streamer
inside a corona
Before describing the outcome from the many-streamers
model let us briefly summarize the results obtained with our
model for single streamers [42].Infigure 4we plot the
electric field created by a single streamer advancing from an
electrode with the initial conditions from table 1. The plot
shows the electric field against the streamer length for all
recorded times, where time is indicated by a color scale. The
initial field of the streamer is that of the electrode, and it is
screened in the interior of the streamer channel. The field at
the tip increases with time, and when it is sufficiently high,
the tip of the streamer starts advancing. The propagation,
in turn, softens both the screening in the streamer channel and
the increase of the field at the tip. Increasing the potential of
the electrode, the leading ionization, or the radius of the
streamer channel will lead to faster propagation.
The right panel of figure 4shows the evolution of a
symmetrical corona with 200 streamers with the same initial
conditions (see table 1)as the single streamer model. Quali-
tatively, the general evolution of the streamer is similar
regardless of the number of streamers: screening of the field
in the streamer channel and increase of the field at the tip,
which triggers propagation. Quantitatively, however, the
effect of the additional streamers in the corona stands out
when we compare both panels from figure 4: the additional
streamers contribute to a better screening of the electric field
and lead to a lower electric field at the tip. The corona with
200 streamers, in figure 4, shows a field inside the channel
approximately an order of magnitude lower than in the single
streamer case. The field at the tip is also lower for 200
streamers although the differences are less than a factor 2.
These differences are enough, however, for the total propa-
gation distance to be substantially smaller, around a factor 6.
4.2. Dependence on the streamer density
Now that the general evolution of the streamer corona has
been described let us generalize our results to a varying
density of streamers. Figure 5shows the status of simulations
with different amounts of streamers at final time, =t10 ns.
The top left panel of figure 5shows the electric field in
the streamer channel for coronas with 1, 50, 100, 200, and
400 streamers, each shown in a different color. In all the
simulations, the field shows a similar profile to that of a single
streamer, which is to be expected given that due to the
symmetry in our corona, all the streamers in the simulation
Figure 3. Configurations of n=50, 100, 200 and 400 streamers in a
spherical electrode following a Thompson distribution.
Table 1. Simulation parameters for the positive corona emerging
from a spherical electrode.
Variable Value Units
Electrode radius, a2cm
Electrode potential, V50 kV
Electron density at the tip, ne
010
20
electrons m
−3
Streamer radius, R
0
1mm
Streamer initial length 1 cm
Background chemistry humid air —
Pressure atmospheric —
Final time 10 ns
Table 2. Density of streamers at the electrode and approximate mean
distance between them when considering a spherical electrode of
radius =a2c
m
. The density is d=n/4πa
2
and the mean distance
is d
−1/2
.
Number of streamers, nSurface density Average separation
50 -
10 m
4
2
10 mm
100 ´-
2
10 m
42 7mm
200 ´-
4
10 m
42 5mm
400 ´-
810m
42 3.5 mm
6
Plasma Sources Sci. Technol. 28 (2019)115007 M González et al

behave as streamer S
0
(as explained in section 2). Even with
these assumptions, the effect of additional streamers in the
corona is clear: the electric field decreases as the number of
streamer increases, both the peak field and the field along the
streamer channel. Since the streamer velocity increases with a
higher field at the tip [43], as the number of streamer
increases, the propagation velocity decreases. This is sum-
marized in figure 6, where we plot the mean velocity of
coronas with varying number of streamers.
A special case is the simulation with n=400 streamers.
In this case the streamers barely propagate at all within 10 ns
because the electric field at the tip is too low for sustained
streamer advance. What this case tells us is that there is limit
for the density of streamers in a corona imposed by electro-
static considerations. In our case the maximum number of
streamers was about 400 but this quantity depends on the
electrode potential and radius. Likely, the state in which a
perfectly symmetrical distribution of many streamers stops is
physically unstable and, due to small differences in the length
of the streamers, an actual corona would continue its propa-
gation with a lower density as a result of leaving behind the
shortest streamers.
Turning back to figure 5, its upper right panel shows the
charge density in the streamer channel against streamer length
at final time for the same simulations with the same code
color as the upper left panel. For simulations with n<400,
the charge density increases with z, but the peak is reached
within the streamer body, slighly before the streamer tip,
where it decreases as it is invested in streamer propagation.
The charge density in the simulation with 400 streamers
decreases near the electrode and then steadily increases with
z, reaching the peak at z
tip
. In general, the charge density is
lower as we increase the number of streamers, both the peak
and near the electrode. However, the positive slope of the
charge density increases with n.
The lower left panel from figure 5shows the electron
density in the streamer channel against z. For the simulations
with significant propagation, n<400, the electron density
increases with z, and the peak is reached at the tip, where the
leading electron density is =-
n
10 m
020 3 as per our bound-
ary condition. Both the electron density near the electrode and
its slope decrease with increasing n.
The observations detailed in this section lead us to the
first and main conclusion of this work: the higher the density
of streamers in a corona, the slower is their propagation and
the lower is their internal field. To understand this principle
let us consider two configurations with different streamer
densities d
1
and d
2
with d
1
<d
2
. Suppose that at a given time
the single-streamer charge density is the same in both con-
figurations. In the configuration with d
2
the electric field is
more screened due to the neighboring streamers. Since this
implies a lower current, after some propagation it leads to a
lower charge density. Therefore in general the charge density
per streamer is lower in dense coronas. Since the electric field
at the tip of each streamer results mostly from this charge
Figure 4. Time evolution of the electric field in the streamer channel against z. The left panel shows the results for the simulation with a single
streamer and the right panel shows the results for a simulation of a corona with 200 streamers distributed as explained in the text. Different
colors show the electric field at different times, according to the colorbar.
7
Plasma Sources Sci. Technol. 28 (2019)115007 M González et al

density, the propagation of dense coronas is also slower. This
process is combined with another principle described in [42]:
slower streamers have lower internal electric fields. These
lower electric fields in our case explain the depletion of
electrons observed in the lower panel of figure 5: this is due to
three-body attachment, which is more efficient for lower
fields.
4.3. Electric field screening
Let us now focus on the evolution at a fixed point in the
streamer channel. Figure 7shows the electric field (top
panels), and the electron density (bottom panels)for two fixed
points in the streamer. We chose =
z
2c
m
(left panels)and
=
z
3c
m
(right panels), which are the two extremes of the
streamer at initial time. From a qualitative standpoint, all the
points in the streamer channel have an evolution similar to
that of the electrode boundary, and all the points beyond the
initial length of the streamer that are reached by it will behave
in a similar fashion to the initial tip.
The top panels of figure 7show the electric field at the
electrode boundary =
z
2c
m
(top left panel), and at the initial
tip =
z
3c
m
(top right panel). At the electrode boundary, in
coronas with any number of streamers, the electric field
initially decays at a rate that is faster for higher streamer
densities. For n<100 the field remains quasi stationary at the
minimum value, but for denser coronas the field slightly
increases with time, with a steeper slope for larger amounts of
streamers. At =
z
3c
m
, which initially is the tip of the
streamer, the field increases until a value close to -
10 V m
71
is
reached, and then the streamer expansion starts and the field
decreases. The evolution of the channel field, then, becomes
similar to that of an inner point. The curves for the corona
with 400 streamers are jagged due to its marginal
Figure 5. Electric field (top left panel), charge density (top right panel), and electron density (bottom left panel)against length at final time
(=
t
10 ns). Each color in each plot represents the simulation of a corona with a different number of streamers, as shown in the legend in the
bottom right panel. Note that the maximum electron density (indicated here by a dashed line)is an input parameter of our model.
Figure 6. Streamer mean propagation velocity against number of
streamers.
8
Plasma Sources Sci. Technol. 28 (2019)115007 M González et al

propagation; as explained above this evolution is unstable and
not to be expected in actual coronas.
In the upper-left panel of figure 7we appreciate two
clearly differentiated regimes for the streamer base: (a)a fast,
roughly exponential, decay lasting for about 1 ns and (b)an
approximately constant field afterwards.
The fast exponential decay (a)can be characterized by
fitting the electric field decay to () ( ) ( )t=-
E
tE t0exp
zz ,
where the parameter τrepresents the characteristic screening
time. We applied this fit to times <<t
0
0.6 ns and the
resulting parameters τare plotted in figure 8. For comparison
that figure also shows the dielectric relaxation time (some-
times also called Maxwell relaxation time)computed as
τ
M
=ò
0
/σ, where ò
0
is the vacuum permitivity and σthe
electrical conductivity. Figure 8shows the Maxwell relaxa-
tion time using two approaches: with the inner channel con-
ductivity as a red line, and with the average corona
conductivity as a green line. For the average corona con-
ductivity we have weighted the channel conductivity with the
fraction of the surface covered by streamers,
n
Ra4
0
22, where
=
R
1m
m
0is maximum the radius of each streamer and
=
a
2c
m
is the radius of the electrode. Figure 8shows that
the dielectric relaxation time is a reasonably good approx-
imation for this transitory regime, although it shows sig-
nificant deviations for large streamer densities.
Regime (b), which is more representative for the typical
propagation of a streamer corona, sets in once the interior
field is mostly screened. In that case there is a strong anti-
correlation between high fields and high conductivities in the
corona volume and therefore the dielectric relaxation
Figure 7. Electric field (top panels), and electron density (bottom panels)against time at fixed z. The left plots show the variables at
=
z
2c
m
, the electrode boundary, while the right plots show the results for the initial tip of the streamer, =
z
3c
m
. Each color in each plot
represents a simulation with a different number of streamers, while the rest of the initial parameters are those in table 1.
Figure 8. Characteristic decay times at the electrode boundary of the
electric field (dots), and Maxwell relaxation time calculated from the
conductivity, for the channel value in each streamer (red), and for the
corona average (green)against number of streamers.
9
Plasma Sources Sci. Technol. 28 (2019)115007 M González et al

calculated with a mean conductivity is a poor estimate for the
evolution of the electric field. A careful investigation of this
regime is beyond the scope of this paper and we leave it for a
future work.
Turning back to figure 7, its lower two panels show the
temporal evolution of the electron density at the two points
that we considered. In both cases, the electron density
decreases with time, although the decrease is larger when we
approach the electrode. At any point in the streamer the
depletion of electrons is faster as we increase the number of
streamers in the corona. As explained above, this is due to the
faster rate of three-body electron attachment at lower electric
fields.
5. Potential drop within a streamer corona
As an impulsive corona propagates away from an electrode, it
carries away part of the electrode potential. How efficiently
this is performed depends on the strength of field-screening
inside each of the streamers and, as we discussed above, this
is heavily influenced by the streamer density in the corona. In
this section we investigate this process.
The main motivation for this part of our study is the
acceleration of electrons ahead of a lightning leader. This has
been proposed as a mechanism for the generation of
Terrestrial Gamma-ray Flashes (TGFs)[48,49], which are
intense bursts of energetic radiation connected to intra-cloud
lightning processes [50,51]. According to the thermal-run-
away model of TGFs [52,53], the high electric field at the tip
of streamers pushes electrons from the bulk of the energy
distribution into a runaway regime where they accelerate
further and create relativistic-runaway electron avalanches
(RREA)[54]in the electric field generated by a leader. A key
magnitude in this process is the total potential available for
the acceleration of electrons from the streamer tips to a
position far-away from the leader [55].
Let us first consider the value of this potential in the
simulations described above, where the electrode is at a
potential =
V
50 kV
0. The potential at the streamer tips is
V
1
=V
0
−ΔV, where the potential drop ΔVis
() ( )
ò
D=VEzzd, 19
a
ztip
with abeing the electrode radius, and E(z)the electric field at
a distance zfrom the center of the electrode.
The left panel from figure 9shows the evolution of the
potential drop at the corona boundary, calculated using (19).
The right panel shows the drop at z
tip
at =t10 ns against the
number of streamers. As in section 4.3, we appreciate two
regimes in the evolution of the electric screening: a fast decay
from the background field where the corona acts as an aver-
age air conductivity followed by a proper corona where the
Figure 9. Potential drop (V)between the electrode and the streamer tips. The left panel shows the evolution with time of the potential drop at
the corona boundary, where colors indicate simulations with different n. The right panel shows the potential drop at the corona boundary and
at final time =
t
10 ns against the number of streamers.
10
Plasma Sources Sci. Technol. 28 (2019)115007 M González et al

potential drop increases with time. The key result here is that
even with 50 streamers (an average separation between
streamers of 1 cm at the electrode)the potential drop is less
than 10% of the total potential. A higher streamer density
implies even smaller potential drops. This means that almost
the full potential of the electrode is transferred to the
boundary of the corona.
To extend to our results to a situation closer to a lightning
leader we run a simulation with the parameters listed in table 3.
As our purpose is only to obtain a qualitative understanding of
the corona dynamics around the leader, we mimicked the leader
tip as a spherical electrode with radius 2 cm on which we apply a
potential () ( ( ))=--
V
tV tt1exp
0rise
with =
V
1M
V
0and
=t100 ns
rise . This dependence is an approximation to the rise
of the potential at the leader tip after a leader step. Given the
higher potential, we consider that the radius of the streamer is in
this case 5 mm. We consider here n=45 streamers, resulting in
a density »´ -
d
910m
32
at the leader surface. These para-
meters approximate the characteristics of a streamer burst around
a laboratory leader (see e.g. [56]).
Figure 10 shows the results of the simulation. The left
panel shows the electric field against zwhereas the right panel
shows the potential drop against time. When we compare the
electric field with the configuration shown in figure 4, dif-
ferences are obvious due mainly to the higher potential and
the finite rise time. Initially, the channel field in the streamer
from figure 10 is low, the initial potential being too low. Soon
afterwards propagation starts and, due to the continuing leader
potential increase, despite the screening inside the channel,
the field at the leader boundary is significative. The field at the
streamer tip remains below ´-
2
10 V m
71
due to the velocity
of the streamer, which averages ~´ -
7.6 10 m s
61
.
The right panel of figure 10 shows the potential drop at
the corona boundary against time. Due to the increase in the
leader potential, the streamers do not screen the field effi-
ciently and at the end of the simulation, =t100 ns, when the
leader potential is »
V
0.6 M
V
, around half of that potential
is spent within the streamer corona. An electron starting at the
Table 3. Initial conditions for the leader configuration.
Variable Value Units
Number of streamers 45 —
Leader tip radius 2 cm
Potential rise time, t
rise
100 ns
Electrode (leader)potential, V
0
1mV
Electron density at the tip, ne
010
20
electrons m
−3
Streamer radius, R
0
5mm
Streamer initial length 1 cm
Background chemistry humid air —
Pressure Atmospheric —
Final time 50 ns
Figure 10. Left panel: Electric field against zcolor-coded for all times. Right panel: Potential drop (ΔV)at the corona boundary against time.
This simulation was done with the initial conditions listed in table 3, simulating a corona around a leader tip.
11
Plasma Sources Sci. Technol. 28 (2019)115007 M González et al

leader tip posseses around half of the leader potential avail-
able for acceleration.
With a finite potential rise time the electric field inside
the streamers is established by the competion between
screening and the increasing electrode potential. On the right
panel of figure 10 we also show the potential drop of
a simulation with a faster rise time =t50 ns
rise , which is
significantly lower than that for =t100 ns
rise . A proper
comparison, however, has to take into account the different
values of the streamer length and applied potential at a given
time. In our simulations we found that the average inner field
for =t100 ns
rise stabilizes around ´
4
.5 10
5
Vm
−1
whereas
for =t50 ns
rise it reaches ´
3
10
5
Vm
−1
. This shows the
relevance of the potential rise-time in the dynamics of strea-
mers emerging from an electrode or a leader tip.
6. Conclusions
In this work we developed a macroscopical model for strea-
mer coronas where, to allow for the simulation of many
streamers, we simplified the corona as a symmetric config-
uration of straight streamer channels. Nevertheless, the model
provides intuition about the effect of additional streamers on
the overall evolution of the corona. It also provides semi-
quantitative estimates of macroscopical corona properties. By
means of this model we reached the following conclusions:
•As we increase the density of streamers in the corona, the
screening of the field also increases. The electric field at
the tip of the streamers decreases, which leads to slower
streamer propagation for large numbers of streamers.
•Some macroscopical corona properties such as the
timescale of electric field screening exhibit a clear
collective behavior. These properties cannot be under-
stood from the dynamics of a single streamer but rather
derive from the interactions between many of them.
•The potential drop in a streamer corona is smaller for
streamer coronas with larger amounts of streamers. To
estimate the available potential for acceleration of
electrons ahead of a leader corona we need to consider
the rise time of the leader potential.
Further work on the modeling of streamer coronas should aim
at removing some of the strongest simplifications in the
present model. One particularly desirable objective would be
to remove the unrealistical symmetry between all streamers
and allowing different propagation speeds.
Acknowledgments
This work was supported by the European Research Council
(ERC)under the European Union H2020 program/ERC grant
agreement 681257 and by the Spanish Ministry of Science
and Innovation under projects FIS2014-61774-EXP and
ESP2017-86263-C4-4-R. This project has also received
funding from the European Unionʼs Horizon 2020 research
and innovation program under the Marie Skłodowska-Curie
grant agreement SAINT 722337. The authors acknowledge
financial support from the State Agency for Research of the
Spanish MCIU through the ‘Center of Excellence Severo
Ochoa’award for the Instituto de Astrofísica de Andalucía
(SEV-2017-0709).
ORCID iDs
A Luque https://orcid.org/0000-0002-7922-8627
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