plasma-expert/reference/sources/liu-discharge-transitions-thesis.txt

=== PAGE 1 ===
Physics of Electrical 
Discharge Transitions 
in Air 
LIPENG LIU
DOCTORAL THESIS IN ELECTRICAL ENGINEERING 
STOCKHOLM, SWEDEN 2017
KTH ROYAL INSTITUTE OF TECHNOLOGY
SCHOOL OF ELECTRICAL ENGINEERING

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KTH Electrical Engineering
Physics of Electrical Discharge Transitions in Air
LIPENG LIU
Doctoral Thesis
KTH Royal Institute of Technology
School of Electrical Engineering
Stockholm, Sweden 2017

=== PAGE 4 ===
TRITA-EE 2017: 028
ISSN 1653-5146
ISBN 978-91-7729-348-4
Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan framlägges till 
offentlig granskning för avläggande av teknologie doktorsexamen onsdagen den 24 maj 
2017 kl. 10:00 i Kollegiesalen, Brinellvägen 8, Kungliga Tekniska Högskolan, Stockholm.
© Lipeng Liu, May 2017
Tryck: Universitetsservice US AB

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To the Lord
&
To the uncertainty and imperfection of life

=== PAGE 7 ===
Abstract
Electrical discharges with a variety of different forms (streamers, glow corona, leaders,
etc.) broadly exist in nature and in industrial applications. Under certain conditions, one 
electrical discharge can be transformed into another form. This thesis is aimed to develop 
and use numerical simulation models in order to provide a better physical understanding of
two of such transitions, namely the glow-to-streamer and the streamer-to-leader transitions 
in air.
In the first part, the thesis includes the two-dimensional simulation of the glow-to-
streamer transition under a fast changing background electric field. The simulation is 
performed with a fluid model taking into account electrons. An efficient semi-Lagrangian 
algorithm is proposed to solve the convection-dominated continuity equations present in
the model. The condition required for the glow-to-streamer transition is evaluated and 
discussed. In order to enable such simulations for configurations with large interelectrode 
gaps and long simulation times, an efficient simplified model for glow corona discharges 
and their transition into streamers is also proposed.
The second part of the thesis is dedicated to investigate the dynamics of the streamer-
to-leader transition in long air gaps at atmospheric pressure. The transition is studied with a
one-dimensional
thermo-hydrodynamic model and a detailed kinetic scheme for 
N2/O2/H2O mixtures. In order to evaluate the effect of humidity, the kinetic scheme 
includes the most important reactions with the H2O molecule and its derivatives. The 
analysis includes the simulation of the corresponding streamer bursts, dark periods and 
aborted leaders that may occur prior to the inception of a stable leader. The comparison 
between the proposed model and the widely-used model of Gallimberti is also presented.
Keywords: electrical discharges, transition, streamers, glow corona, leader discharges.

=== PAGE 8 ===
Sammanfattning
Elektriska urladdningar av olika former (streamers (från engelska), glöd-korona, ledare, 
etc.) förekommer i stor utsträckning i naturen och i industriella applikationer. Under vissa 
förhållanden kan en elektrisk urladdning omvandlas till en annan form av elektrisk 
urladdning. Denna avhandling syftar till att utveckla och använda numeriska 
simuleringsmodeller för att ge en bättre fysikalisk förståelse av två sådana övergångar, 
nämligen glöd-till-streamer- och streamer-till-ledar-övergångar, i luft.
I den första delen, avhandlas en tvådimensionell simulering av glöd-till-streamer-
övergången med ett hastigt föränderligt elektriskt fält i bakgrunden. Simuleringen utförs 
med en flödesmodell som tar hänsyn till elektronerna. En effektiv semi-Lagrangesk 
algoritm föreslås för att lösa de konvektionsdominerade kontinuitetsekvationerna i
modellen. Vidare utvärderas och diskuteras förutsättningarna för glöd-till-streamer-
övergången. För att möjliggöra sådana simuleringar i konfigurationer med stora 
elektrodavstånd och långa simuleringstider, föreslås också en effektiv och förenklad 
modell för glöd-korona-urladdningar samt deras övergång till streamers.
Den andra delen av avhandlingen är tillägnad att undersöka dynamiken i streamer-till-
ledar-övergångar över långa avstånd i luft, under atmosfäriskt tryck. Övergången studeras 
med en endimensionell termohydrodynamisk modell och en detaljerad kinetisk modell för 
blandningar av N2/O2/H2O. För att utvärdera effekten av luftfuktighet, innefattar den 
kinetiska modellen de viktigaste reaktionerna med H2O-molekylen och dess derivat. 
Analysen innefattar simuleringen av motsvarande streamer-kedjor, mörka perioder och 
avbrutna ledare som kan förekomma före starten av en stabil ledare. En jämförelse mellan 
den föreslagna modellen och den allmänt använda modellen av Gallimberti presenteras 
också.
Nyckelord: elektriska urladdningar, övergång, streamers, glöd-korona, ledarurladdningar.

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Acknowledgements
First and foremost, my deepest acknowledgement goes to my supervisor and one of my 
best friends, Marley Becerra. I feel deeply honoured to have such a great scientist as my 
mentor, not only regarding academic matters but also in life.
Second but also very important, I am very grateful for the financial support of the 
China Scholarship Council. In addition, the scholarship from the Foundation Ericson E.C. 
Fund and the Foundation Petersohns Minne to cover my travelling expenses when 
attending to several international conferences is also appreciated.
I enjoyed a lot the life in Stockholm and at KTH Royal Institute of Technology. As a
prestigious technical university, KTH provided valuable insight and invaluable assistance 
from international, experienced and well-recognized scientists and engineers. The adequate 
academic resources, harmonious interpersonal relationships, flexible schedules and the 
beautiful environment…all these elements make me feel relaxed, confident and energetic 
when pursuing my Ph.D. studies.
There are a lot of people I would like to acknowledge, my colleagues at the department 
of electromagnetic engineering of KTH, my friends in Stockholm, my sisters and brothers 
in the Immanuel Church in Stockholm, and my family. Without their support, 
companionship and encouragement, this thesis would have been impossible to finish. I will 
not list their names here, but I will keep them in my mind.
The scenes and memories in the last four years are always vivid. I remember when I 
came to Sweden and posted some pictures on the Internet with text ‘Happy Ph.D. life 
begins’. The post caused a heated discussion and was forwarded by thousands of people 
since most of them think the word ‘happy’ contradicts the word ‘Ph.D. life’. Yes, I have to 
admit that pursuing Ph.D is not easy. However, I really enjoy it and I feel so lucky and 
honoured. For me, it is like a journey abroad, meeting different people, doing different 
things and seeing different sceneries.
Happy Ph.D. life ends here, but it lasts forever in my heart.
Lipeng Liu
Stockholm, May 2017

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List of Publications
This thesis is based on the following papers, which will be referred to in the text by their 
roman numerals:
I.
L. Liu, M. Becerra, "An efficient semi-Lagrangian algorithm for 
simulation of corona discharges: the position-state separation method," IEEE 
Transactions on Plasma Science, volume 44, issue 11 (10 pp), 2016.
(doi: 10.1109/TPS.2016.2609504)
II.
L. Liu, M. Becerra, "Application of the position-state separation method 
to simulate streamer discharges in arbitrary geometries," IEEE Transactions 
on 
Plasma 
Science, 
volume
45, 
issue 
4
(9 
pp), 
2017.
(doi: 10.1109/TPS.2017.2669330)
III.
L. Liu, M. Becerra, "On the transition from stable positive glow corona 
to streamers," Journal of Physics D: Applied Physics, volume 49, issue 22
(13pp), 2016. (doi: 10.1088/0022-3727/49/22/225202)
Conference paper version presented at the 20th International Conference on 
Gas Discharges and their Applications, Orleans, France, 2014.
IV.
L. Liu, M. Becerra, "An efficient model to simulate stable glow corona 
discharges and their transition into streamers," Journal of Physics D: Applied 
Physics, volume 50, issue 10 (12pp), 2017. (doi: 10.1088/1361-6463/aa5a34)
V.
L. Liu, M. Becerra, "Gas heating dynamics during leader inception in 
long air gaps at atmospheric pressure," (23pp), submitted to Journal of Physics 
D: Applied Physics, 2017.
Conference paper version presented at the 21st International Conference on 
Gas Discharges and their Applications, Nagoya, Japan, 2016.
VI.
L. Liu, M. Becerra, "Two-dimensional simulation on the glow to 
streamer transition from horizontal conductors," presented at the 32nd
International Conference on Lightning Protection, Shanghai, China, 2014.
(doi: 10.1109/ICLP.2014.6973247)
VII. L. Liu, M. Becerra, "Two-dimensional simulation on the glow to 
streamer transition from lightning rods," presented at XIII International 
Symposium on Lightning Protection, Balneário Camboriú, Brazil, 2015.
(doi: 10.1109/SIPDA.2015.7339289)

=== PAGE 12 ===
The above papers are reprinted in the appendix with kind permission from the 
publishers: © IEEE 2014, 2015, 2016, 2017 and © IOP Institute of Physics 2016, 2017.
Other contributions of the author, not included in the thesis
VIII. L. Liu, M. Becerra, "A parallel projection method for the solution of 
incompressible Navier-Stokes equations based on position-state separation 
method," presented at the
27th
International Conference on Parallel 
Computational Fluid Dynamics, Montreal, Canada, 2015.

=== PAGE 13 ===
Contents
1 
Introduction ........................................................................13 
1.1 
Examples of electrical discharge phenomena in air .......................... 14 
1.2 
Typical forms of electrical discharges in air ..................................... 15 
1.2.1 Fundamental processes in electrical discharges ........................................... 15 
1.2.2 Development of typical electrical discharges............................................... 16 
1.3 
Typical electrical discharge transitions in air.................................... 20 
1.4 
The motivation and context of this thesis.......................................... 21 
1.4.1 The motivation and aim................................................................................ 21 
1.4.2 The method and structure............................................................................. 23 
1.4.3 Author’s contribution................................................................................... 24 
2 
Towards an efficient numerical algorithm for corona 
discharge simulations .........................................................25 
2.1 
The simplest model for corona discharges in air............................... 26 
2.2 
Numerical challenges in solving the fluid model .............................. 27 
2.3 
An efficient numerical algorithm for corona discharges................... 29 
2.3.1 The position-state separation method........................................................... 29 
2.3.2 Applications to simulate glow and streamer discharges............................... 30 
3 
Physics of the glow-to-streamer transition in air ...............31 
3.1 
2D simulations of glow-to-streamer transition.................................. 32 
3.1.1 The formation of positive glow corona ........................................................ 32 
3.1.2 The mechanism of the glow-to-streamer transition...................................... 33 
3.2 
Efficient model for glow discharges considering the ionization layer
........................................................................................................... 34 
4 
Physics of the streamer-to-leader transition in air .............35 
4.1 
Dynamics of streamer-to-leader transition ........................................ 36 
4.2 
The effect of humidity on the streamer-to-leader transition.............. 37 
5 
Application case study: analysing unusual lightning strikes
............................................................................................39 

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5.1 
Observations of unusual lightning strikes ......................................... 40 
5.1.1 Lightning shielding failure in tall structures ................................................ 40 
5.1.2 Competition study of lightning receptors..................................................... 42 
5.2 
Effect of the glow-to-streamer transition in lightning strikes............ 43 
6 
Conclusions ........................................................................44 
7 
Future work ........................................................................46 
References ..................................................................................47 

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13
1
Introduction
"The eternal mystery of the world is its comprehensibility."
"If you can't explain it simply, you don't understand it well enough."
"Imagination is more important than knowledge."
Albert Einstein
Electrical discharges are usually produced under strong electrical fields where electron 
multiplication occurs. They can be localized in high electric field regions such as in the 
case of corona discharges, or can propagate in the medium as in lightning discharges. This 
thesis will not deal with electrical discharges in solids or liquids, but will focus on the most 
common gaseous medium: air. Generally, an electrical discharge in air can be viewed as 
plasma, however, not vise versa (e.g. a flame is plasma but not an electrical discharge).
The scientific research on electrical discharges in air started several hundreds of years 
ago. For example, the research of lightning is considered to have started with the American 
scientist Benjamin Franklin in 1746 when he conducted experiments on electricity [1]. His 
famous kite experiment in 1752 led him to define the sign of electrical charge and he 
concluded that the lower part of a thundercloud is usually negatively-charged [1]. The 
industrial applications of electrical discharges also date back to the 18th century. In 1770, 
English physicist Joseph Priestley discovered the erosive effect of electrical discharges, 
which led to the invention of electrical discharge machining technology [2]. Later in 1785, 
the Dutch chemist Martinus van Marum noticed that ozone can be produced by electrical 
sparking in oxygen [3]. Research on electrical discharges is not only an old, but also a 
prosperous subject with some discharge phenomena such as transient luminous events 
discovered only a few decades ago [4-6] and with emerging applications in industry [7-9].
One type of electrical discharge can be transformed into another form under certain 
conditions. The condition required for such a transition to take place is thus of great 
interest to investigate, not only from the theoretical point of view, but also from the 
perspective of engineering applications.
In the first part of this chapter (section 1.1-1.3), background regarding different forms 
of electrical discharges in air and their transitions are introduced. The second part (section 
1.4) is devoted to briefly describe the motivation and the structure of this thesis.

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14
1.1
Examples of electrical discharge phenomena in air
Electrical discharges in air widely exist in nature and industry. The most famous and 
common discharge phenomenon in nature is lightning, which is a rapid electrostatic 
discharge that usually happens during thunderstorms. Due to the electrification of 
thunderclouds, electrical discharges in nature occurs in several different ways, for example
in cloud-to-cloud and cloud-to-ground flashes and in upper-atmospheric lightning such as 
blue jets, gigantic jets and sprites [6]. Figure 1.1 illustrates the different phenomena 
associated to lightning at different altitude in the atmosphere.
On earth, lightning frequently strikes 40-50 times every second, of which about 25% 
correspond to cloud-to-ground lightning flashes [10]. Due to the flow of very large currents
(several tens of kA) within a short time, lightning can injure people and damage or disturb
directly or indirectly structures and their internal equipment [11]. Lightning is the second 
leading cause of weather-related death in the world [12]. In particular, lightning is a threat
to tall grounded structures such as buildings, ultra-high voltage (UHV) power transmission 
lines and wind turbines.
Figure 1.1 Conceptual sketch of different kinds of discharge phenomena in the atmosphere.
Part of the illustration of electrical discharges in the upper atmosphere is adapted from [6]. An
example of UHV power grid systems is shown to illustrate discharges commonly present in 
industrial applications.
Elve
Sprite
Gigantic jet
Blue jet
Cloud to ground 
lightning
Corona
Glow
െ
െ
െ
െ
+ +
+
+
+ +
െ
െെ
െ
െ
++
+

=== PAGE 17 ===
15
During thunderstorms, glow corona can be produced by towers (as illustrated in figure 
1.1), lightning rods, masts, chimneys and wind turbines due to the high electric field
induced at the tip of these structures. The corona discharge usually emits a faint glow of 
light with blue or violet colour that can only be seen in the dark. The glow generated from 
masts was noticed by sailors several hundreds of years before Benjamin Franklin’s 
electrical experiments. The sailors viewed glow corona as a sign from the patron saint of 
the sailors, St. Elmo and thus named it as St. Elmo's fire [11].
The most common electrical discharges in industry are corona discharges. For example, 
corona widely exists in high-voltage power transmission lines, as illustrated in figure 1.1.
These corona discharges are undesirable since they can cause power energy loss, audible 
noise and insulation damage [13]. On the contrary, corona discharges are also very useful 
in technological areas such as the ozone production, surface treatment, and pollution 
control [7]. In industry, electric arcs are another type of electrical discharges that play an 
important role. Although electric arcs are undesirable in electrical devices such as switches 
and circuit breakers, they are widely used in welding, lighting, electrical discharge 
machining [9], etc.
1.2
Typical forms of electrical discharges in air
1.2.1 Fundamental processes in electrical discharges
As illustrated in figure 1.1, there are different kinds of electrical discharges in the upper 
atmosphere (developing under low air density) such as sprites, blue jets, gigantic jets, and 
elves. These phenomena have different properties compared to electrical discharges under
atmospheric conditions. For example, sprites consist of thousands of growing channels 
with diameters of the order of tens to hundreds of meters [14]. On the other hand, they also 
share close similarities to discharges produced in the laboratory [15, 16]. For instance,
large-scale sprites are physically similar to small-scale streamer dischargers in air at 
atmospheric pressure [17, 18], blue jets emit a fan of streamers similar to the streamer 
corona zone in front of laboratory leaders [19], and gigantic jets have similar 
characteristics as leader discharges in laboratory [20]. This thesis will not discuss the 
upper-atmosphere discharges and their transitions in details, but will focus on the 
traditional discharges at atmospheric pressure.
Electrical discharges are plasmas consisting of six types of species: free electrons, 
atoms and molecules, excited atoms and molecules, positive ions, negative ions, and 
photons [21]. Among these species, free electrons are the most important specie which
dominates the discharge process due to their special features. For example, electrons drift 
with velocities two orders of magnitude faster than ions under the same electric field.
Figure 1.2 illustrates some typical fundamental processes where electrons are involved. 

=== PAGE 18 ===
16
For example, electrons can be produced by impact-ionization, photo-ionization, and 
detachment processes while they are lost through recombination with positive ions and
attachment with neutral molecules to form negative ions. As shown in figure 1.2, electrons 
are usually bound in different energy levels. Electrons in low energy levels can ‘jump’ into 
high energy levels (through excitation), or even become free electrons by collisions or by 
electromagnetic radiation. Electrons in high energy levels can also ‘return’ to low energy 
levels and emit photons (through quenching). For a detailed introduction of these processes
as well as the general kinetic theory of electrical discharges, the reader is referred to classic
books in the subject such as [22-25].
Figure 1.2 Sketch of typical fundamental processes of electrons in air. 
1.2.2 Development of typical electrical discharges
The research of electrical discharges has a long history. After Benjamin Franklin, an 
important step in the research of gaseous discharges was taken by English chemist and 
physicist Sir William Crookes, who invented vacuum tubes in 1875 [26]. Shortly after, 
British physicist John Sealy Townsend proposed the famous theory of Townsend 
discharges around 1900 [27] to explain the breakdown characteristics in short gaps at low 
pressures. However, there are several experimental observations in longer gaps at high 
pressure which cannot be explained by Townsend theory. For instance, experimental 
measurements in cloud chambers show that electron avalanches propagate with a velocity 
much larger than the electron velocity under the applied electric field. In addition, it was 
Photons
Photo-ionization
Free electrons
Detachment
Attachment
Recombination
Photons
Bound electrons
Energy

=== PAGE 19 ===
17
observed that the discharge can propagate not only towards the anode but also towards the 
cathode. These observations led scientists to define a different type of process: the streamer 
discharge. The theory describing streamers was proposed around 1940s, independently by 
L. B. Loeb and J. M. Meek [28, 29] and H. Raether [30].
Including streamers, all electrical discharges require free electrons to get started. In air, 
few free electrons ݊଴are produced by background radiation such as terrestrial radiation 
and cosmic rays [15, 31], through the reaction
M
୰ୟୢ୧ୟ୲୧୭୬
ሱۛۛۛۛۛሮMା+ ݁
(1.1)
where M denotes neutral molecules such as Nଶand Oଶ. The maximum electron density 
produced by background radiation at ground can be up to 10ସ cmିଷ[31].
Under high electric fields, the number of free electrons ݊௘can increase exponentially
with time ݐonce ionization frequency ݒ௜exceeds attachment frequency ݒ௔expressed as
݊௘(ݐ) = ݊଴exp[(ݒ௜െݒ௔)ݐ]
(1.2)
In dry air, ݒ௜> ݒ௔occurs when the reduced electric field ܧ/ܰis larger than 120 Td (1 Td 
= 10ଵ଻ V cmଶ), where ܧis the electric field and ܰthe number density of air [32]. Once the 
electric field is above the threshold when ݒ௜= ݒ௔, electron avalanches are produced. At 
atmospheric pressure (with ܰ= 2.5 × 10ଵଽ cmିଷ), the electric field threshold ܧis around
30 kV cmିଵ.
Figure 1.3 (a) shows a cloud chamber photograph of a single electron avalanche. If the 
net charge in the head of the avalanche is not sufficient to distort the electric field, the 
avalanche moves with the electron drift velocity [7]. If secondary electrons are produced 
during the lifetime of the avalanche, for example by photoionization as sketched in figure 
1.3 (b), the avalanche can grow quickly into a streamer. There is a minimum radius of the 
avalanche ݎ௦which is required for the streamer transition [33, 34]. At atmospheric 
pressure, ݎ௦ൎ0.2 mm [33]. Figure 1.3 (c) is a cloud chamber picture showing the 
transition from avalanches into a streamer.

=== PAGE 20 ===
18
Figure 1.3 (a) Typical cloud chamber photograph of a single electron avalanche, adapted from 
[30]; (b) Conceptual sketch of the electron avalanche development under a uniform electric 
field; and (c) Cloud chamber photograph showing the transition from avalanches into streamers
where the initial radius of the streamer ݎ௦is marked, adapted from [35].
The most common setup in the laboratory to produce electrical discharges is the point-
plate configuration, as illustrated in figure 1.4. This figure also illustrates different basic 
forms of electrical discharges at atmospheric pressure. As the voltage applied to the 
electrode increases, streamers can be firstly produced from electron avalanches as 
described in the previous subsection. These streamers are generally known as pre-onset 
streamers [26]. Depending on the applied voltage and gap distance, different forms of 
electrical discharges can be produced. In short gaps under high voltage, streamers can
reach the opposite electrode leading to streamer breakdown, which usually develops into
an electric arc. Electric arcs can be sustained if the applied voltage is maintained. If the 
produced streamers cannot bridge the gap, corona discharges will be formed. Under 
electric fields slowly changing in time (e.g. under DC voltages), the discharge is self-
sustained in a limited region around the electrode. Depending on the electrode and applied 
voltage, corona discharges usually have two typical modes, namely streamers with 
filamentary structures and homogeneous glow [36, 37]. In long air gaps (> 1 m), the 
current of a large number of branching filaments in a streamer can contract into distinct 
stems. Leader discharges can be incepted if the stem of a streamer reaches a temperature of
about 2000 K [23, 38, 39]. If the electrostatic conditions are sufficient, the leader channel 
acts as an elongation of the electrode since the electric field along the channel is rather 
low. Then, the channel can propagate into the gap by thermalizing air through the current 
collected from the streamer corona produced at its tip. The corona region ahead of the 
leader tip is also known as the streamer zone. Figure 1.5 shows a typical streak image of 
+
+
Anode
Cathode
+
+
+
+
+
+
+
+
+
+
+ +
+
- - - - -
- - -
- -
--
--
+
+
+
+
- --
- - -
- -
+ + +
E
݄ݒ
+
+  +
+
(a)      
(b)             
(c)
ݎ௦

=== PAGE 21 ===
19
positive leader propagation in a rod-plane gap in air, where the leader tip and the streamer 
zone can be clearly seen. Leader discharges are the most important breakdown mechanism 
in long air gaps (> 1m) [24]. Once the streamer at the tip of a leader channel reaches the 
opposite electrode, leader breakdown occurs. In this case, an electric arc can also be 
formed if the applied voltage is maintained.
Streamers and glow corona are non-thermal plasmas where the gas temperature is 
usually low [7]. For this reason, streamers and glows are also classified as cold plasmas.
However, the electronic temperature of cold plasmas is much larger than the gas 
temperature (translational temperature). Leader discharges and electric arcs are instead 
thermal plasmas which much higher gas temperature (> 2000 K).
Figure 1.4 Sketch of typical electrical discharges in non-uniform fields at atmospheric pressure.
Electric arc
Streamer
breakdown
Pre-onset
streamers
Streamer
waves
Glow
corona
Voltage high
Increase the
voltage
Point-plate configuration
applied by a voltage
ܷ
ݐ
ܷ
ݐ
Leader
breakdown
Streamer 
corona
Leader
Electric arc

=== PAGE 22 ===
20
Figure 1.5 Streak photograph of the propagation of a positive leader discharge in a rod-plane 
gap in air. Photograph reprinted from [40] with permission.
1.3
Typical electrical discharge transitions in air
Figure 1.6 illustrates typical electrical discharge transitions at atmospheric pressure marked 
as T1-T5. The short description of these transitions is given as
x
Avalanche-to-streamer transition T1: defines the formation of streamers as 
introduced in section 1.2.2. The electron density of the avalanche has to reach
about 10ଵସcmିଷat atmospheric pressure [33] for this transition to occur.
x
Streamer-to-glow transition T2: describes the formation of glow corona from 
pre-onset streamers, for example under a DC voltages as in [41]. Since the 
applied voltage is not sufficient to cause a streamer breakdown or a leader 
breakdown in the gap, glow corona discharges are restricted around the 
surface of the electrodes and are uniformly distributed, as shown in figure 1.4.
x
Glow-to-streamer transition T3: glow corona discharges can be transformed to 
other modes such as streamer bursts and breakdown streamers depending on 
the voltage amplitude and the geometry [36]. In this thesis, however, the glow-
to-streamer transition refers to the transition from glow corona to streamers 
under fast rising applied voltages, as shown in figure 1.4.
x
Streamer-to-leader transition T4: is defined by the formation of leader 
discharges from streamer corona. The streamer-to-leader transition takes place 
not only before the inception of a stable leader, but also during the leader 

=== PAGE 23 ===
21
propagation. In this thesis, more attention is focused on the first stage, i.e., the 
leader inception.
x
Breakdown-to-arc transition T5: describes the transition of any breakdown 
process into an arc discharge. Electric arcs in air at atmospheric pressure 
usually have much higher current and temperature (> 5000 K) than for leader 
channels.
Figure 1.6 Conceptual sketch of different forms of electrical discharges and their transitions 
under atmospheric pressure.
1.4
The motivation and context of this thesis
1.4.1 The motivation and aim
As mentioned before, electrical discharges have a long research history. However, our 
knowledge of electrical discharges is still limited and many questions are still unsolved. 
Let us take the most common and famous phenomenon of lightning as an example. Several 
hundreds of years have passed after Benjamin Franklin conducted his experiments on 
lightning. Nevertheless, one of the most basic and important questions of ‘how lightning is 
initiated’ is still unsolved [1]. Another basic process poorly understood is the attachment of 
Electron 
avalanche
Corona
Leader
Electric arc
Streamer
T1
T4
T5
Glow
T2
T3
Streamer 
zone
Leader 
channel
Anode
Cathode
Current
Current
Electric
field
Breakdown
Short
gap

=== PAGE 24 ===
22
lightning flashes to grounded objects [42]. Although numerous models have been 
developed to describe this process, the accurate simulation of the interaction of lightning 
flashes with structures on the ground is still challenging [43]. Since lightning attachment is 
a complex physical process, the existing models use rather crude approximations of the 
different electrical discharges involved in order to reach a practical quantitative evaluation. 
However, the simplifications assumed by these models, particularly those used to evaluate
the transitions between the different discharges, are still controversial. Hence, the debate 
on the effect of glow corona on the lightning attachment has not been concluded yet,
mainly due to the lack of understanding of its transition into streamers [44]. Furthermore,
quantitative estimates of the condition necessary for streamers to transform into leaders are 
still doubtful, especially when evaluating the attachment process after a first lightning 
strike [45]. Thus, an opportune project to contribute to the research on lightning and other 
applications is to investigate the physics behind these transitions.
This thesis aims to develop and use numerical simulation models in order to improve 
the physical understanding of two electrical discharge transitions in air at atmospheric 
conditions, namely the glow-to-streamer and the streamer-to-leader transitions marked 
respectively as T3 and T4 in figure 1.6. Even though other electrical discharge transitions 
are mentioned briefly in the text, they are outside the scope of this thesis. This is because 
other transitions either are rather well understood or require too much work to go one step 
further. For example, The first two-dimensional simulation of the avalanche-to-streamer 
transition in a uniform field was performed by Dhali and Williams in 1987 [46], followed 
by numerous simulations reported in the literature such as [47-52]. The transition from 
avalanches to a single filamentary streamer is rather well understood. It is widely accepted 
that the most important mechanism to provide secondary electrons in the avalanche-to-
streamer transition is photoionization by ultraviolet photons [53], as illustrated in figure 1.3
(b). In air where oxygen concentration is high (~21%), photons emitted by excited 
nitrogen can ionize oxygen molecules [51]. While in pure nitrogen or in nitrogen with 
extremely low oxygen concentration, it has been suggested that the predominant 
mechanism to provide photons is the Bremsstrahlung (deceleration radiation) process 
instead [54]. The next major breakthrough on the research on the avalanche-to-streamer 
transition might be the understanding of the branching mechanism. However, a systematic 
explanation for this problem is very difficult and challenging to accomplish [18, 52].
The first one-dimensional (1D) simulation of the streamer-to-glow transition under a 
sudden applied DC voltage was conducted by Morrow in 1997 [41]. In the simulation, 
Morrow observed ‘streamer-like’ ionizing waves were produced from a stable glow corona 
if the applied voltage was raised rapidly [41]. However, streamers have filamentary 
structures that cannot be described with a 1D model. Since the simulation of the transition 
is a multiscale problem which is extremely time-consuming even in 1D, Paper I and 
Paper II in the thesis have been aimed to develop a numerical algorithm to efficiently 

=== PAGE 25 ===
23
solve corona discharge models. Based on this algorithm, a detailed two-dimensional (2D)
model is used in Paper III to describe the physics of the glow-to-streamer transition. Since 
the evaluation introduced in Paper III is still impractical for the analysis of real objects in 
lightning attachment studies, Paper IV is intended to develop a simplified model which 
can properly take into account the relevant physical processes within the transition.
On the other hand, the streamer-to-leader transition occurs not only in front of the 
electrode before leader inception, but also at the tip of a propagating leader. Although the 
transition during the leader propagation is well understood [20, 55-57], the transition 
dynamics before the inception of a stable leader has been less studied. This is the main 
motivation of Paper V.
Paper VI and Paper VII introduce the first implementation of the model presented in 
Paper III towards the analysis the transition of glow corona into streamers initiated from 
shielding wires and lightning rods. These papers are aimed as a first step to the physical 
analysis of the effect of glow corona on lightning attachment, especially for unusual 
lightning strikes observed in UHV transmission lines and the lightning rods [44].
1.4.2 The method and structure
The main method to perform the study of these electrical discharge transitions is through 
numerical modelling and simulation.
Depending on different scenarios, different 
simplification and assumptions are used. Compared to the research through laboratory
experiments, the most obvious advantage of numerical simulations is that it can provide 
detailed information on the microscopic and transient parameters, which are very difficult 
to measure. However, the proposed numerical models need to be first validated by 
comparison with the measured macroscopic parameters reported in the literature such as 
the current-voltage characteristics before they are used.
The main content of the thesis is divided into additional six chapters. The 2D 
simulation of the glow-to-streamer transition is a challenging problem from the perspective 
of numerical techniques. To do this, Paper I and Paper II proposed an efficient numerical 
algorithm for corona discharge simulation. Chapter 2 introduces the numerical challenges 
in the numerical modelling of corona discharges and summarizes Paper I and Paper II.
Chapter 3 describes Paper III which deals with the physics of the glow-to-streamer 
transition. In addition, an efficient and simplified physical model for glow corona 
discharges proposed in Paper IV is also introduced. The dynamics of the streamer-to-
leader transition during leader inception presented in Paper V is summarized in Chapter 4.
Chapter 5 introduces Paper VI and Paper VII where the glow-to-streamer transition in the 
lightning attachment process is analysed. Conclusions and future work are presented in 
Chapter 6 and 7, respectively.

=== PAGE 26 ===
24
1.4.3 Author’s contribution
The author of the thesis is the first and communication author of Papers I-VII. The idea
and the solution algorithm for Paper I and Paper II were proposed by the author. The 
research questions and scientific approach for the remaining papers were proposed by the 
author and the supervisor. The development of all the computer code and the writing of
most part of the papers were performed by the author.

=== PAGE 27 ===
25
2
Towards 
an 
efficient 
numerical 
algorithm for corona discharge simulations
"GǀQJ\VKjQTtVKuˈEu[LƗQOuTtTu."
from L~Q\· (expressed in Chinese Pinyin)
"A workman must sharpen his tools if he wants to do his work well."
from Analects of Confucius (translation in English)
All the different forms of electrical discharges are essentially initiated from electron 
avalanches. They have multiscale properties not only in space (from nm to km) but also in 
time (from ns to s) [17]. The 3D structures of most discharges, such as the branching of 
streamers, make their modelling challenging. Feasible modelling of electrical discharge has 
to meet at least two conditions. First, the related physics have to be included, either with a 
complicated or a properly simplified model. Second, the model can be numerically solved 
within an acceptable time.
Electrical discharges are usually modelled in two different ways. The first one follows 
a kinetic or particle description such as in Monte Carlo or Boltzmann transport simulation,
which has a resolution into the particle [14, 52] or superparticle-level [58]. Generally, the
kinetic models are highly time-consuming [59]. The other approach is the fluid model,
which is computationally more efficient and therefore is widely used in the literature [7,
60]. The fluid model of gas discharges is defined by several continuity equations (to 
account for the development of the relevant species) coupled with Poisson’s equation (to 
account for the distortion of the electric field by the generated space charge) [7]. The fluid 
model for corona discharges is based on several important assumptions [7], including the 
local field approximation which assumes that the electron energy distribution function is in 
local equilibrium with the background gas. Theoretical analysis [48] and numerical 
experiments comparing particle and fluid models [59, 60] have shown that the assumptions 
of the fluid model generally holds and therefore it can be an alternative to the particle 
model [7].
In this chapter, the numerical challenges of solving the fluid model are described and 
an efficient numerical algorithm for corona discharges proposed in Paper I and Paper II is 
introduced.

=== PAGE 28 ===
26
2.1
The simplest model for corona discharges in air
For cold plasmas like corona discharges, the effect of air heating is usually neglected such 
that constant air temperature and pressure are assumed. An exhaustive description of the 
kinetics of corona discharges in dry air is difficult [61] and even more complex in humid 
air. Since the numerical simulation with a detailed kinetic scheme is very time-consuming, 
simplified models are usually used. The simplest model usually assumes that corona 
discharges are composed only of electrons, positive and negative ions, and excited species
considering averaged reaction rates [62]. The set of continuity equations describing these 
species in air is reprinted from Paper III as
߲ܰ௘
߲ݐ= ܵ୮୦+ (ߙെߟ)ܰ௘|ࢃ௘| െߚܰ௘ܰ௣+ ݇ௗܱଶ
כܱଶ
ିെ׏ ή [ܰ௘(ࢃ௘+ ࢝)]
(2.1)
߲ܰ௣
߲ݐ= ܵ୮୦+ ߙܰ௘|ࢃ௘| െߚܰ௘ܰ௣െߚܰ௣ܰ௡െ׏ ή ൣܰ௣(ࢃ௣+ ࢝)൧
(2.2)
߲ܰ௡
߲ݐ= ߟܰ௘|ࢃ௘| െ݇ௗܱଶ
כܰ௡െߚܰ௣ܰ௡െ׏ ή [ܰ௡(ࢃ௡+ ࢝)]
(2.3)
߲ܱଶ
כ
߲ݐ= ߙ௠ܰ௘|ࢃ௘| െ݇ௗܱଶ
כܰ௡െ݇௤ܱଶ
כܱଶെ׏ ή (ܱଶ
כ࢝)
(2.4)
where ݐis the time, ܰ௘, ܰ௣, ܰ௡, ܱଶand ܱଶ
כ are the number densities of electrons, positive 
ions, negative ions, oxygen molecules and metastable oxygen molecules, respectively. 
ࢃ௘, ࢃ௣, ࢃ௡are the drift velocities for electrons, positive ions and negative ions taking the 
background air as a reference. ࢝is the bulk velocity of background gas accounting for air 
flow. Diffusion of all the particles is neglected since it plays a negligible role. The symbols 
ߙ, ߟ, ߚ, ߙ௠denote the ionization, attachment, recombination coefficients and the rate of 
creation of metastable molecules, respectively. ݇ௗ, ݇௤are the detachment rate coefficient 
and quenching rate constant, respectively. ܵ୮୦is the photo-ionization rate. The transport 
parameters and reaction rates in air are summarized in the appendix of Paper III.
The continuity equations are fully coupled with Poisson’s equation expressed as
׏ ή ܧ= ݁
ߝ൫ܰ௣െܰ௡െܰ௘൯
(2.5)
where H is the permittivity of air, e is the electron charge and ܧis the electric field. There 
are several challenges in solving the above fluid model. In the next subsection, these 
challenges are described briefly.

=== PAGE 29 ===
27
2.2
Numerical challenges in solving the fluid model
The numerical modelling of electrical discharges is an interdisciplinary task, which 
requires knowledge of plasma physics, computational fluid dynamics (CFD) and 
computational electromagnetics. For electrical discharge simulations, a suitable numerical 
method has to consider several aspects such as the accuracy and efficiency in solving the 
continuity and Poisson equations, the flexibility in handing irregular geometries, and the 
extensibility to high dimensions. In other words, a suitable numerical method should
x
be able to provide accurate and positivity-preserving solutions for the density 
profile of the modelled species when solving continuity equations since 
negative density solutions do not have any physical meaning;
x
be able to efficiently solve Poisson’s equation since it is highly coupled with 
the continuity equations and it is calculated at each time step within a
simulation;
x
be able to handle unstructured meshes since the geometries where electrical 
discharges present are often irregular such as point-to-plate configuration; and
x
be easily extended to high dimensions and other coordinate systems for 
example cylindrical coordinate system which is frequently used due to axis 
symmetry since 3D modelling is much more time-consuming.
There are several challenges when developing such a method. The first challenge
comes from the solution of continuity equations, which evaluate the variation of the 
density of the modelled species. In general form, it is expressed as:
߲ߩ
߲ݐ+ ׏ ή (࢛ߩ) = ܵ
(2.6)
where ߩand ࢛are the number density and velocity of the modelled specie. ܵaccounts for
the sources and sinks due to reactions with other species. In electrical discharges, charged 
species drift very fast under the electric field while the diffusion is much weaker. For such 
kind of convection-dominated problems, the diffusion is usually neglected.
It is a challenge to solve continuity equations accurately since very sharp gradients in 
density and velocity can also appear for example in the front of streamers (see Paper II).
Under these conditions, conventional numerical methods may encounter artificial 
numerical oscillations or excessive numerical diffusion. Figure 2.1 shows an example of 
the simulation results for a square test with the finite difference method (FDM) of first and 
second order. The square test in the field of CFD simply means simulating the drift of a 
square profile of density under a constant velocity field without any loss. It has been 

=== PAGE 30 ===
28
widely used in the CFD area since the analytic solution is straightforward while the 
accurate numerical solution is difficult due to the very sharp gradient at the edge of the 
square profile. As shown in figure 2.1, the first order upwind method has serious numerical 
diffusion compared to the analytic solution, while the second order scheme is less diffusive 
but has numerical oscillations.
Figure 2.1 Comparison between the simulated results and the analytic solution in a square test.
One idea to improve the numerical method is to combine the advantages of both low 
order and high order schemes, which was first introduced by Boris and Book in 1970s.
They developed the flux-corrected transport method (FCT) [63-65], which later was
successfully applied to the 2D simulation of the streamer propagation [46]. For a brief 
review of other numerical methods used in the literature for corona discharge simulation,
the reader is referred to the introduction in Paper I.
The second challenge comes from the efficient solution of Poisson’s equations. 
Different methods have been used in the literature, for example, the fast Fourier transform
algorithm [46], the symmetrical successive over-relaxation method [66], and the direct 
SuperLU solver [67]. However, it is challenging to use FDM or the finite volume method 
(FVM) to solve Poisson’s equations on unstructured meshes since the discretization is 
more complicated than for structured meshes [68]. The most suitable method to handle 
irregular geometries using unstructured mesh is the finite element method (FEM). FEM 
combined with FCT [69] and diffusive stabilization techniques [70, 71] have also been 
successfully employed in corona discharge simulations, such as in [72] and [73]. However, 
these methods are not inherently positivity-preserving and numerical oscillations can take 
place without extra imposed conditions.

=== PAGE 31 ===
29
2.3
An efficient numerical algorithm for corona
discharges
2.3.1 The position-state separation method
The aim of electrical discharge modelling is to calculate accurately the density of any 
specie in space at any time. In other words, the simulation task is finished once the species 
are solved as a function of time, position and state. When solving the continuity equation, 
challenges arise since both the time and space discretization of the density are mixed in 
one equation. To circumvent this, the transient solution of the position and the state of the 
density can be split into two subproblems: the state equation which describes the state 
change and the position equation which deals with the drift effect only. For example,
equation (2.6) can be divided into two different equations: the state equation
߲ߩ
߲ݐ= െߩ׏ ή ࢛+ ܵ
(2.7)
which deals with the variation of the variable ߩdue to convective acceleration and the 
reaction terms; and the position equation
߲ߩ
߲ݐ+ ࢛ή ׏ߩ= 0
(2.8)
which determines the transport of the variable ߩby considering the linear convection only.
The state equation (2.7) can be solved on an appropriate mesh (named as the reference 
mesh) to obtain the new density profile ߩכ at the next time step. This can be done with any 
conventional numerical method because the discretization of space and time is performed 
for different variables, i.e., ߩand ݑin equation (2.7) respectively. The position equation
(2.8) can be easily solved by integrating the ordinary differential equation
߲࢞
߲ݐ= ࢛
(2.9)
along the characteristic lines of the drift, resulting in a new mesh (named as the auxiliary 
mesh). The state on the reference mesh can be obtained by interpolation from the auxiliary 
mesh with the updated density ߩכ. For a detailed description of POSS, the reader is referred 
to Paper I.

=== PAGE 32 ===
30
2.3.2 Applications to simulate glow and streamer discharges
One of the challenges for POSS is that the used linear interpolation is not mass-conserving,
which means the solution has serious numerical diffusion if very small time step is used, as 
shown in Paper I. There are mass-conserving or shape-conserving algorithms available in 
the literature such as [74]. However, mass-conserving interpolation on unstructured meshes
is complicated and time-consuming. The efficiency of POSS will be significantly reduced 
if mass-conserving interpolation is used.
For glow corona discharges where the electric field changes slowly with time, large 
time step can be used for POSS and thus the numerical diffusion caused by the 
interpolation can be neglected. In Paper I, the POSS method has been successfully applied 
to simulate the formation of positive glow discharges in a 1D co-axial spherical 
configuration under a DC voltage. POSS is very efficient when simulating glow corona 
discharges since the required time step can be much larger than that restricted by the 
Courant–Friedrichs–Lewy (CFL) condition. Furthermore, POSS does not require the ‘flux 
correction’ procedure which is usually very time-consuming on unstructured meshes.
However, very small time steps have to be used for streamer simulation where electric 
fields change dramatically. In such a case, the chosen time step is determined by physical 
characteristic times instead of being limited by the stability of the numerical algorithm.
Under such conditions, POSS will encounter excessive numerical diffusion caused by the 
interpolation step, as shown in Paper I. In order to solve this problem, a multi-step 
interpolation strategy is introduced in Paper II. The idea is to use a small time step to 
capture the physical changes and use a larger time step for interpolation to avoid serious
numerical diffusion. Several reproducible streamer simulations in the literature are selected 
as benchmark tests to show that POSS combined with FEM is a competitive alternative 
method to simulate streamer discharges, especially in complex geometries. Although it is 
difficult to compare different methods used to simulate streamers in the literature, a general 
evaluation of different methods is possible. Paper II compares the total computation time 
used in different methods. It is shown that the computation time with POSS is significantly 
less than other approaches such as FEM-FCT and FVM-MUSCL (monotone upstream-
centered schemes for conservation law) [75]. Furthermore, POSS is more robust than other 
FEM method such as FEM-FCT since it is inherently positivity-preserving as shown in 
Paper I.

=== PAGE 33 ===
31
3
Physics 
of 
the 
glow-to-streamer 
transition in air
"Enter through the narrow gate; for the gate is wide and the road is easy that 
leads to destruction, and there are many who take it. For the gate is narrow 
and the road is hard that leads to life, and there are few who find it."
from Matthew 7-13,14
In industrial applications involving glow corona such as in ozone production, the generated 
discharge should be as homogeneous as possible to obtain a high collision rate between 
electrons and the background gas molecules [7]. In this way, the products yield can be 
increased and the power consumption reduced. [76]. For this reason, the glow-to-streamer 
transition has to be avoided. Glow discharges also occur in nature during thunderstorms as 
mentioned in section 1.1. The space charge generated by glow corona can significantly 
change the electric field distribution around grounded objects. As thunderstorms further 
develop, upward streamers and leaders can be subsequently initiated such that the shielding 
of the pre-existing glow space charge can play an important role. Thus, it is interesting to 
investigate the conditions required for the glow-to-streamer transition to occur.
The layer where intensive ionization occurs in front of the anode during corona 
discharges is usually difficult to simulate for long simulation times. One strategy to avoid 
the complexity of resolving the ionization layer is to use Kaptzov’s approximation [77],
which neglects the electron dynamics in the discharge and assumes a boundary condition 
to define the injection of unipolar ionic charges instead. The boundary condition forces the 
surface electric field to stay at the onset field once corona is initiated. Kaptzov’s 
approximation has been widely used to evaluate the effect of corona space charge on the 
initiation of streamers under fast changing background electric fields [44, 78-81].
In the first part of this chapter, the investigation of the glow-to-streamer transition 
without using Kaptzov’s approximation presented in Paper III is summarized. The second 
part is dedicated to introduce an efficient physical model for evaluating glow corona and 
the transition into streamers as proposed in Paper IV.

=== PAGE 34 ===
32
3.1
2D simulations of glow-to-streamer transition
3.1.1 The formation of positive glow corona
In order to assess the mechanism of the glow-to-streamer transition, it is worth to first 
understand the dynamics of glow corona discharges. The theory of positive glow corona 
was not well understood until the end of last century when Australian scientist Richard 
Morrow performed a pioneering 1D simulation [41]. The general dynamics of such a
transition under a sudden positive DC voltage is summarized as follows:
x
As the applied voltage to the inner conductor (anode) exceeds the onset 
voltage, the air close to the anode is ionized and pre-onset streamers are 
produced.
x
Electrons and negative ions are absorbed by the anode while positive ions 
move to the outer conductor (cathode).
x
As positive ions drift away from the anode, the electric field around the anode 
increases sufficiently to ionize again the nearby air, forming a new space 
charge layer.
x
The above-described process is repeated until a stable glow corona discharge
is produced, as shown in figure 3.1.
Figure 3.1 Sketch of the cross section view of a positive glow corona discharge under DC applied 
voltage in a coaxial cylindrical configuration.
Inner conductor
Ionization layer
Space charge layer
Outer conductor

=== PAGE 35 ===
33
Morrow extended the FD-FCT to a non-uniform mesh in order to simulate a stable 
glow corona in a spherical coaxial configuration with a 2 cm long air gap [82].
Nevertheless, such a 1D simulation took several days to finish for the several microseconds 
required to reach a stable glow [50]. The speed up the simulation of glow corona discharge 
including the ionization layer has been the main motivation to develop the POSS method 
earlier introduced in Chapter 2.
3.1.2 The mechanism of the glow-to-streamer transition
Morrow observed in his numerical experiments that (1D) streamer-like ionizing waves
were produced from a stable glow corona if the applied voltage was raised rapidly [41].
However, streamers have filamentary structures that cannot be described with such a 1D
model. As a first approach, Paper III performs a 2D simulation of the glow-to-streamer 
transition without Kaptzov’s approximation. The POSS method proposed in Paper I is 
used to handle the difficulties associated to the convection-dominated continuity equations
in the simulation.
In Paper III, the generation of glow corona under DC voltage is first simulated. Once 
the glow corona under DC voltage is formed, the applied voltage is raised with a constant 
dV/dt rate. Since the space charge generated by glow corona in a coaxial cylindrical 
configuration is uniformly distributed, the transition to filamentary streamers cannot be 
produced unless either physical or numerical instabilities are included in the model. In 
Paper III, three different types of instabilities are taken into account. It is shown that these 
instabilities do not change the critical dV/dt required for the transition when filamentary 
streamers are observed. The basic mechanism of the glow-to-streamer transition is 
described as follows:
x
As the applied voltage is increased, the time for new produced positive ions to 
drift away is reduced.
x
These ions accumulate around the surface of inner conductors, intensifying the 
local distortion of the space charge and the electric field caused by the 
introduced instability.
x
The inhomogeneity of the electric field in turn further increases the distortion 
of the space charge due to increased ionization.
x
The homogeneity of the layered structure of glow corona is destroyed by the 
formation of streamers.
One of the most interesting conclusions of Paper III is that streamers are easier
incepted from blunt corona generating electrodes than from sharp ones. This is because the 
space charge drifts faster for sharper electrodes and thus the applied voltage has to be 
increased at a faster rate for the space charge to start accumulating.

=== PAGE 36 ===
34
3.2
Efficient model for glow discharges considering 
the ionization layer
The simulation of positive glow corona discharges with the fully-coupled physical model 
(FPM) introduced in section 2.1 is extremely time-consuming, even in 1D. First, a very 
small time step is required by the FPM to resolve electrons in the ionization layer since the 
electrons drift two orders of magnitude faster than ions. Second, a finer mesh is also
required to discretize the ionization layer, further increasing the computational cost.
One strategy to simplify the simulation of corona discharges is to neglect the ionization 
layer and to use Kaptzov’s approximation instead. Due to its simplicity, Kaptzov’s 
approximation has been frequently in the literature [44, 83-85]. However, Paper III shows 
that Kaptzov’s approximation does not hold under fast changing background electric fields.
Based on the detailed simulation of corona discharges with the FPM as presented in 
Paper III, it was found that a simplified physical model (SPM) for glow corona discharges 
can be formulated due to the following facts:
x
Electron avalanches only take place in a well-defined layer where ionization 
exceeds attachment;
x
The electrostatic conditions in the computation region are mainly defined by 
the ionic space charge since the density of electrons is more than two orders of 
magnitude smaller than for ions;
x
Electrons are more than two orders of magnitude faster than ions; and
x
The source terms of photo-ionization ܵ୮୦, electron-ion recombination ߚܰ௘ܰ௣
and negative ions detachment ݇ௗܱଶ
כܱଶ
ିin the continuity equation for electrons 
(equation (2.1)) are several orders of magnitude smaller than the effective 
ionization (ߙെߟ)ܰ௘|ࢃ௘| in the ionization layer.
These facts allow us to assume that electrons reach quasi-steady state, i.e. 
డே೐
డ௧= 0
within the characteristic time of ion drift. It has to be emphasized that the quasi-steady 
state approximation for electrons here used is only valid for stable glow corona discharges.
Paper IV proposed the SPM to simulate glow corona discharges and their transition 
into streamers. The model is validated by performing comparisons with the FPM and with 
experimental data available in the literature for air under atmospheric conditions. It is 
shown that the SPM can obtain estimates similar to those calculated with the FPM and 
those measured in experiments but using significantly less computation time.

=== PAGE 37 ===
35
4
Physics 
of 
the 
streamer-to-leader 
transition in air
"A theory is a supposition which we hope to be true, a hypothesis is a 
supposition which we expect to be useful; fictions belong to the realm of art; if 
made to intrude elsewhere, they become either make-believes or mistakes."
George Johnstone Stoney
Leader discharges exist in long air gap laboratory discharges [24, 39], troposphere
lightning [1, 40] and upper atmosphere lightning such as blue jets and gigantic jets [16,
55]. A leader is a highly ionized, conductive and thermal channel with a temperature 
ranging between 2000 and 6000 K [23]. Leader discharges in the length of 1~15 m can be 
produced in laboratory with high impulse voltages where detailed observations and 
measurements can be obtained. Laboratory experiments are important since a specific 
measurement with sufficient space and time resolution of a natural lightning event or a ‘jet’ 
is very difficult. The leaders in much larger scales (1-100 km) are believed to have similar 
characteristics as the leaders produced in the laboratory [16, 20, 40, 55]. In long air gap 
discharges in laboratory, extensively studied by the Les Renardières group in 1970s [86-
89], the development of a positive leader discharge can be described as follows. First, the
first streamer corona is incepted as the applied voltage increases. The electric field 
produced by the corona space charge counteracts the Laplacian field around the electrode, 
resulting in a dark period. Then, several secondary streamer discharges (streamer bursts)
with dark periods in between may occur depending on the recovery of the electric field as 
space charge drifts into the space and the applied voltage increases. Second, a leader 
channel segment can be initiated if the gas temperature of any streamer stem reaches the 
critical value of about 2000 K. Third, the leader may continue propagating into the gap if 
the electrostatic conditions in front of the newly formed leader are sufficient. Otherwise, it
will be aborted. Finally, leader breakdown takes place once the streamer corona at the 
leader tip reaches the opposite electrode, which is usually known as the ‘final jump’ [86].
The streamer-to-leader transition occurs in front of the electrode before the inception of 
leaders as well as at the head of a propagating leader. Although the transition during the 
leader propagation is well understood [20, 55-57], the transition dynamics before the
inception of a stable leader has been less studied. This is the most important motivation to 
conduct the research presented in Paper V.

=== PAGE 38 ===
36
4.1
Dynamics of streamer-to-leader transition
In the previous studies of the streamer-to-leader transition during leader propagation [20,
55-57], a 1D thermo-hydrodynamic model was used. The model describes the cross section 
of the streamer stem with a 1D radial coordinate system by neglecting axial variations and 
assuming a constant current flowing in the axial direction. The radial electric field is 
neglected while the axial field is computed from the current and conductivity of the cross 
section using Ohm’s law. Several features of the streamer-to-leader transition during leader 
propagation from these studies can be summarized as: (1) the stem of the streamer corona 
has to reach temperatures larger than about 1 500~2 000 K in order to initiate a leader 
discharge. (2) The stem is heated by the sum of the current produced by the streamers 
within the streamer zone through Joule heating. (3) The heating process is governed by the 
contraction of thermal channel which is triggered by a thermal-ionizational instability.
Several additional modifications to the thermo-hydrodynamic models available in the 
literature are made in Paper V according to the facts described as follows. First, the 1D 
model has limitations to estimate the density of charged species during the dark periods 
due to the axial variation of the electric field in front of the electrode. These variations 
cause changes of density for electrons and ions along the axial direction which cannot be 
calculated. Second, experiments with Schlieren photography [90] have recently shown that 
a single solitary stem is not necessarily formed before a leader is incepted under switching 
voltage waveforms. Instead, several stems connected to the electrode can be produced by a 
streamer, through which the streamer current is shared. Third, the initial condition of 
previous studies [20, 55-57] usually assumes a fixed electron peak density ( 2 ×
10ଵସ cmିଷ) and the simulation results are extremely sensitive to the initial radius since the 
current density of the stem changes significantly with the initial radius.
In Paper V, the analysis of the streamer-to-leader transition includes the simulation of 
the corresponding streamer bursts, dark periods and aborted leaders that may occur. The 
simulations are performed using as input the time-varying discharge current in two 
laboratory discharge events reported in the literature [90], which are used as case studies. 
The initial condition is defined according to the inception electric field instead of using a 
fixed electron peak density. During the dark period after the streamer stops propagating,
the density of all the charged species are set to low background levels such that no joule 
heating occurs during the dark period. Since the electric field does not affect the energy 
relaxation by neutral species in the gas, their chemistry dynamics can be simulated during 
the dark period. Moreover, the corona current in this simulation is simply divided by the
number of stems (assumed to be electrically similar) according to Schlieren photography 
[90]. In Paper V, excellent agreement between the estimated and experimental thermal 
radius for a 1m rod-plate air gap discharge has been found.
Another interesting conclusion found in Paper V is that the gas at the axis has to reach 

=== PAGE 39 ===
37
a temperature much larger than the critical value (of 2000 K) to initiate a stable leader that
can propagate into the gap. This is because the gas temperature can drop due to very strong 
convection losses taking place soon after the streamer-to-leader transition. If the 
temperature after the drop falls below the critical value, the leader is aborted since the 
thermalization cannot be sustained. On the contrary, the leader can propagate if the gas 
temperature after the transition is higher than 2000 K after the convection loss.
4.2
The effect of humidity on the streamer-to-leader 
transition
At standard temperature and pressure (STP) conditions, the concentration of water 
molecule (H2O) can reach up to 3% (~22 g mିଷ). It seems that such a low percentage of 
H2O can hardly affect the whole discharge processes. However, experiments indicate that 
humidity does play an important role [88, 91].
In order to investigate the effect of humidity, Paper V proposed a detailed kinetic 
scheme for N2/O2/H2O mixtures. The kinetic scheme includes the most important 
reactions with the H2O molecule and its derivatives, resulting in a scheme with 45 species 
and 192 chemical reactions. The effect of humidity on the electronic power partitioning 
and the vibrational energy relaxation are also discussed and included in the model.
It has been suggested in the literature that humidity plays a significant role on the 
thermalization of air through the V-T (vibrational-translational) relaxation [39]. However, 
the simulations in Paper V show that the V-T relaxation has a weak effect on the gas 
heating due to two main reasons. First, humidity weakly increases V-T relaxation and this 
effect becomes weaker in the following discharges. Second, the V-T relaxation power has 
a minor effect in the energy balance before a leader is formed since it is several orders of 
magnitude smaller than other energy sources during most of the streamer-to-leader 
transition. However, this conclusion is based on the assumption that humidity does not 
affect the current density of a stem. Even though it is known that humidity reduces the total 
charge injected by streamers [39, 88], there is unfortunately no experimental or theoretical 
knowledge about the effect of humidity on the current density of stems. Figure 4.1 shows 
an example of the photograph of streamer corona discharges in dry and humid air 
condition. As it can be seen, humidity plays an important role in streamer corona 
discharges [91]. The observations indicate that further studies on the formation of the 
streamer stem are required to fully assess the effect of water content on the streamer-to-
leader transition.

=== PAGE 40 ===
38
Figure 4.1 Influence of humidity on streamer corona discharges in a 0.9 m rod-plane gap. (a) 
5 g mିଷ(b) 32 g mିଷ. Images adapted from [91] with permission.
Laboratory experiments have shown that humidity can significantly reduce the duration 
of the dark period [39]. Paper V indicates that humidity weakly influences the dynamics 
of the stem as long as the same initial conditions and input discharge current are used in 
the simulation. Thus, the effect of humidity on the dark period appears to be mainly
explained by the reduction of the electrostatic shielding produced by the streamer space 
charge.
In Paper V, the developed model is also compared with the widely-used model of 
Gallimberti. The model proposed by Gallimberti was derived considering several 
simplifying assumptions, for example, the electric field of the stem was assumed constant,
the radial variations of the chemistry and the gas flow were neglected and the vibrational-
translational relaxation was simplified with an equivalent time constant as a function of 
temperature and humidity only. However, the simulation and analysis performed in 
Paper V show that the assumptions used by the model of Gallimberti do not hold when 
evaluating the streamer-to-leader transition.
(a)
(b)

=== PAGE 41 ===
39
5
Application 
case 
study: 
analysing 
unusual lightning strikes
"Physics is, hopefully, simple. Physicists are not."
"The science of today is the technology of tomorrow."
Edward Teller
As mentioned in section 1.1, lightning is a threat to tall grounded structures such as 
buildings, UHV power transmission lines and wind turbines. The belief that lightning was 
so powerful that only gods and goddesses could generate and control it dominated early 
civilizations [92]. Since the mid-eighteenth century, science has helped to explain the 
nature and formation of lightning [93]. From then on, different lightning protection 
methods are used to protect these structures against lightning strikes. For example, power 
transmission lines are protected by shielding lines (earth lines) and tall buildings are 
protected by lightning rods. However, it is found that these devices sometimes can fail to 
protect a structure. This is generally known as a lightning shielding failure.
Thunderclouds are usually negatively charged and produce a background electric field 
ܧ௕up to for example 20 kV mିଵnear the ground [94]. During thunderstorms, high
voltages can be induced at the tips of tall grounded objects. As a result, positive glow 
corona can be initiated as it has been mentioned in section 1. With the presence of a 
downward lightning leader approaching these glow generating objects, upward streamers 
can be initiated (glow-to-streamer transition), followed by the inception of upward 
lightning leaders (streamer-to-leader transition).
It is widely known that the space charge generated by the glow corona can weaken and 
smooth the electric field around corona-generating surfaces. It is of great interest to know 
the effect of space charge on the glow-to-streamer transition, which has been previously 
studied with 1D models [56, 78-80, 95-101] and 2D models [44, 83] using Kaptzov’s 
approximation. One of the motivations of Paper III is to investigate the effect of space 
charge without using Kaptzov’s approximation, i.e., with consideration of the ionization 
layer. The idea has been used to analyse the effect of space charge on the glow-to-streamer 
transition for horizontal conductors and lightning rods in Paper VI and Paper VII.
In this chapter, the work of Paper VI and Paper VII is summarized. Unusual lightning 
strikes to tall grounded structures are discussed.

=== PAGE 42 ===
40
5.1
Observations of unusual lightning strikes
5.1.1 Lightning shielding failure in tall structures
The electrogeometric method (EGM) [102] has been widely used to evaluate the lightning 
protection of transmission lines due to its simplicity and fair agreement with early field 
observations [103]. The EGM method calculates the geometric exposure zone of a 
conductor to downward lightning leaders according to the prospective return stroke peak
current (ܫ୮). The exposure zone is determined by an arc with radius ݎୱfrom the conductor 
surface, as shown in figure 5.1. The radius ݎୱis calculated with an empirical formula
expressed as ݎୱ= ܽܫ௣
௕, where ܽand ܾare coefficients tuned based on field observations.
Figure 5.1 Sketch of the cross section view of a typical UHV transmission lines used in Japan 
and the lightning exposure zone calculated from EGM.
As shown in figure 5.1, shielding wires are generally arranged on the outside of phase 
conductors forming a negative protection angle [102]. In this way, the shielding wires 
should be able to protect well the phase conductors according to the EGM, at least for the 
upper phase lines as shown in figure 5.1. However, shielding failures for ultra-high voltage 
power transmission lines have been observed as shown in figure 5.2. Similar shielding 
failures have also been reported for tall towers. For example, figure 5.3 shows that the 
lightning termination at the tip of a tower sometimes can fail to protect the tower itself 
[40].
108 m
92 m
71 m
50 m
19 m
16 m
Downward 
lightning 
leader
Ƚ
Upper phase
Shielding lines
Middle
phase
Lower
phase
ݎୱ
ݎୱ

=== PAGE 43 ===
41
Figure 5.2 Lightning stroke to the upper phase of a UHV transmission line (operated at 500 
kV). Photographs were taken on July 22, 2000 (top) and July 9, 1998 (bottom), respectively.
Photographs reprinted from [104] with permission © IEEE 2007. The cross section view of 
the tower shown in the top image is given in figure 5.1.
Figure 5.3 The lightning stuck the Ostankino Television Tower over 200 m below its top. 
Photograph reprinted from [40] with permission.

=== PAGE 44 ===
42
5.1.2 Competition study of lightning receptors
In 1990s, Moore and his team conducted a series of field experiments aiming to investigate
the performance of Franklin rods [105, 106]. The lightning rods with different radii were 
installed on about 3300 m high mountains in New Mexico, US. The lightning rods are 
arranged with several meters distance between each other, as show in figure 5.4 (a). Their
results show that none of the sharp rods with diameter D < 1 cm or too blunt rods with 
diameter D > 5 cm was struck in seven summer thunderstorm seasons. On the contrary, all 
lightning strikes were received by moderate blunt rods, as shown in figure 5.4 (b). These
field observations are counter-intuitive because sharp-tipped rods are generally viewed as 
more efficient lightning receptors since the electric field around them is higher and 
strongly non-uniform. 
Figure 5.4 (a) Photograph of the experimental setup and (b) photograph of six blunt lightning 
rods used in the field tests conducted by Moore et al [105]. The images are reprinted from 
[105] with permission.
D = 1.27 cm
D = 1.90 cm
D = 2.54 cm

=== PAGE 45 ===
43
5.2
Effect of the glow-to-streamer transition in
lightning strikes
Section 5.1 presented several field observations showing that lightning can strike grounded 
structures in an unusual way, especially when they are very tall (ذ 100 m) or they are 
installed on high buildings or mountains. There are only a few explanations to unusual 
lightning shielding failures in the literature. For example, the shielding failure of the TV 
tower shown in Figure 5.3 has been attributed to the stochastic nature of lightning [40]. In 
order to qualitatively complement the existing analyses of such observations, a first 
evaluation of the glow-to-streamer transition in the attachment of lightning to grounded 
objects has been presented in Paper VI and Paper VII.
Thus, UHV transmission lines are modelled as perfectly cylindrical, coaxial and 
grounded conductors in Paper VI. Since bundle conductors are usually used in UHV 
transmission lines to reduce the energy loss due to corona discharge, the glow-to-streamer 
transition from a scaled bundle conductor is firstly studied. It is found that the bundle 
conductors could be viewed as a single conductor with the equivalent geometric mean 
radius of the wire configuration when evaluating the condition required for the glow-to-
streamer transition. As concluded in Paper III, it is easier for streamers to be incepted 
from blunt corona generating electrodes than for sharp ones. Thus, it becomes easier for 
the glow-to-streamer transition to take place from the phase conductors since the geometric 
equivalent radius is significantly larger than the physical conductor radius, as estimated in 
Paper VI. However, this conclusion is based on several simplifications as noticed in 
Paper VI. In reality, the asymmetry of the transmission lines, the protrusions on the 
conductors, the wind or rain, the operating voltage, and the 3D geometry of the 
transmission lines and the downward lightning leaders might also play an important effect 
on the conditions for the glow-to-streamer transition to take place.
Similar analysis applies to the case of the lightning shielding failure to the TV tower
shown in Figure 5.3. In Paper VII, a scaled lightning rod with cylindrical body and 
hemispherical tip (as used in Moore’s experiments [105]) is modelled. The numerical 
simulations show that streamers can be incepted from the body of a lightning rod under the 
influence of downward stepped leaders, even a distant one. Thus, it can be easier for 
streamers to be incepted from the body of a grounded tower rather than from its tip.
Similar to the above explanations, the observations taken by Moore et al can also be 
partially explained by the effect of glow corona. The photographs in figure 5.4 (b) show
that lightning can also strike to the body of the rod as implied and predicted by the 
simulations in Paper VII. However, further theoretical or experimental work is required to 
assess a full understanding of the observations.

=== PAGE 46 ===
44
6
Conclusions
In this thesis, the work of Papers I-VII is summarized aiming to provide a better physical 
understanding of electrical discharge transitions in air. The main work and conclusions are 
listed as below.
In Paper I and Paper II, an efficient semi-Lagrangian algorithm referred to as the 
position-state separation method (POSS) is proposed for the simulation of corona
discharges. Several benchmark tests are conducted to demonstrate the low computational 
cost, robustness, and high-resolution of POSS to solve convection-dominated continuity 
equations. For the simulation of corona discharges where the velocity field is weakly 
changing in time, the solution with POSS is not restricted by the CFL condition when 
solving the continuity equations. Therefore, a time step significantly larger than that for 
explicit Eulerian methods can be used. POSS can also be used to simulate filamentary 
streamer discharges where electrical field changes dramatically in space and time. Without 
flux correction and combined with a finite element method, POSS is easy to be 
implemented on arbitrary geometries. In summary, POSS is an accurate, efficient and 
stable alternative method to simulate electrical discharges.
In Paper III, the 2D numerical simulation of the glow-to-streamer transition under a
fast changing background electric field is presented. It is found that the surface electric 
field of a glow corona generating electrode deviates from the onset electric field. 
Therefore, Kaptzov’s approximation does not hold and the ionization layer should be 
considered. During the glow-to-streamer transition, the electronic current increases 
significantly by at least two orders of magnitude within several hundreds of nanoseconds. 
The more glow corona space charge is generated from the electrode, the higher critical rate 
of rise of the applied voltage is required for the glow-to-streamer transition. Thus, it is 
easier for streamers to be incepted from blunt corona generating rods than from sharp ones.
In Paper IV, a simplified physical model (SPM) for simulation of glow corona and its 
transition into streamers is proposed. The SPM is verified by comparisons with the fully 
coupled physical model (FPM) and validated with experimental results available in the 
literature for discharges in air under atmospheric conditions. The SPM is proposed as a 
computationally efficient alternative to calculations of glow corona discharges based on 
Kaptzov’s approximation. It is shown that the SPM can obtain similar results compared 
with the FPM for stable glow corona and its transition into streamers. With an efficient 
segregated numerical strategy to handle electrons, the SPM is three orders of magnitude 
faster than the FPM. This enables the efficient simulation of glow corona and the transition 
into streamers considering the ionization layer, even for configurations with large 
interelectrode gaps and for long simulation times.
In Paper V, the dynamics of streamer-to-leader transition prior to the leader initiation 

=== PAGE 47 ===
45
in long air gap discharges is investigated with a thermo-hydrodynamic model and a 
detailed kinetic scheme of N2/O2/H2O mixtures. It is found that although a small 
percentage of water molecules can accelerate the vibrational-translational relaxation to 
some extent, this effect leads to a negligible temperature increase during the streamer-to-
leader transition. It is also found that the gas temperature should significantly exceed 
2000 K for the transition to lead to the inception of a propagating leader. Otherwise, the 
strong convection loss produced by the gas expansion during the transition causes a drop in
the translational temperature below 2000 K, aborting the incepted leader. Furthermore, it is 
shown that the assumptions used by the widely-used model of Gallimberti do not hold 
when evaluating the streamer-to-leader transition.
In Paper VI and Paper VII, 2D simulations on the glow-to-streamer transition are 
performed for horizontal conductors and lightning rods, respectively. It is suggested that it 
is easier for streamers to be initiated from corona generating bundle conductors than from 
single conductors. It is shown that a bundle conductor could be viewed as a single 
conductor with the equivalent geometric mean radius of the wire configuration when 
evaluating the critical rate of rise of the background electric field during thunderstorms. It 
is also concluded that the glow space charge generated by lightning rods cannot hinder 
streamers to be incepted under the fast changing background electric field produced during 
thunderstorms. For example, even with the presence of a distant downward stepped leader, 
streamers can be incepted from the body of the lightning rod. Paper VI and Paper VII
indicate that glow corona generated from the tall grounded structures plays a significant 
role in the attachment of lightning to structures.

=== PAGE 48 ===
46
7
Future work
Paper I and Paper II proposed an efficient numerical algorithm to simulate electrical 
discharges. It has been combined with the finite element method, aiming to make it capable 
to handle arbitrary geometries. There are several possible further improvements for POSS
that can be done. First, second order shape functions can be used in finite element 
formulation of POSS to reduce the unknowns. Second, the problem of mass-conservation 
requires a rigorous discussion from the mathematic point of view. Third, the efficiency of 
interpolation can be further improved in later studies.
Paper IV proposed an efficient model for glow corona discharges which can predict 
the self-oscillations in current produced by positive glows. Several experiments in air and 
at atmospheric pressure are used as benchmarks to verify the model. Even though the 
assumptions used in the model are independent of the gas medium and gas pressure, the 
comparison between simulations with available experiments under other conditions
(different gas and pressure) in the literature is required. Moreover, the simplest fluid model 
is used in Paper IV, which means that all the positive ions have the same mobility and 
reaction rates with other species. The next step is to use a fluid model with a more 
elaborated kinetic scheme as in Paper V to further investigate the kinetic dynamics of 
glow corona discharges. In addition, the humidity effect on the glow corona discharges can 
be assessed in the future.
In Paper V, the dynamics of the streamer-to-leader transition is evaluated for two 
discharge events in a 1 m rod-plate gap. In the future, the model can be used to investigate
the charge dynamics of such a transition in a general case, for example, to calculate the 
injected charge required to initialize leaders in nature. In Paper V, it is suggested that
humidity plays an important role in the avalanche-to-streamer transition which was poorly 
understood. The next step is thus to perform the simulation of streamer filaments with a 
detailed kinetic scheme as proposed in Paper V.
Since electrons can be considered, the model proposed in Paper IV can be applied in
the future for other applications where the widely used Kaptzov’s approximation may not 
hold. For instance, in hybrid UHV AC/DC transmission lines, the space charge injected 
into the space is not monopolar and thus it is not straightforward to apply Kaptzov’s 
approximation.
In Paper VI and Paper VII, the simulations of glow-to-streamer transition were 
conducted for scaled configurations. The next step is to perform simulations for real cases 
with the efficient model proposed in Paper IV. For example, both shielding lines and 
phase lines can be simulated together in a 2D Cartesian coordinate system. In this way, the 
space charges injected from those conductors are coupled together and a more complete 
evaluation of the glow-to-streamer transition can be performed.

=== PAGE 49 ===
47
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