Reduced Order Modelling of Streamers and Their Characterization by Macroscopic Parameters by Colin A

Home , Macroscopic scale

Reduced order modelling of streamers and their characterization by macroscopic parameters by Colin A. Pavan BASc, University of Waterloo (2017) Submitted to the Department of Aeronautical and Astronautical Engineering in partial fulfillment of the requirements for the degree of Master of Science in Aeronautical and Astronautical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2019

Author...... Department of Aeronautical and Astronautical Engineering May 21, 2019

Certified by...... Carmen Guerra-Garcia Assistant Professor of Aeronautics and Astronautics Thesis Supervisor

Accepted by...... Sertac Karaman Associate Professor of Aeronautics and Astronautics Chair, Graduate Program Committee 2 Reduced order modelling of streamers and their characterization by macroscopic parameters by Colin A. Pavan

Submitted to the Department of Aeronautical and Astronautical Engineering on May 21, 2019, in partial fulfillment of the requirements for the degree of Master of Science in Aeronautical and Astronautical Engineering

Abstract Electric discharges in gases occur at various scales, and are of both academic and practical interest for several reasons including understanding natural phenomena such as lightning, and for use in industrial applications. Streamers, self-propagating ionization fronts, are a particularly challenging regime to study. They are difficult to study computationally due to the necessity of resolving disparate length and time scales, and existing methods for understanding single streamers are impractical for scaling up to model the hundreds to thousands of streamers present in a streamer corona. Conversely, methods for simulating the full streamer corona rely on simplified models of single streamers which abstract away much of the relevant physics. This disconnect highlights the need for a simplified model of individual streamers which captures the core dynamics but is scalable to ensembles of many mutually interacting streamers. In this work, several such models are developed. First, a 1.5D model of a single streamer was created wherein particles are treated one dimensionally and electric fields two dimensionally (axisymmetric). This model incorporates developments in modelling streamer processes such as photoionization that were not available in the days when 1.5D models were first invesitgated. Next, a 1.5D model was created with the governing equations solved in the reference frame of the streamer. The existence of such a quasi-steady frame has previously been hypothesized; this work gives a thorough evaluation of the validity of a steady-state streamer model and finds it to be a reasonable approximation on the time scale of electron motion. Based on the success of the quasi-steady model, a further simplification is made wherein streamers are characterized by a small set of macroscopic parameters: tip electric field, velocity, radius and background electric field. A simple model is developed relating these various properties and an efficient graphical representation of their interdependencies is presented.

Thesis Supervisor: Carmen Guerra-Garcia Title: Assistant Professor of Aeronautics and Astronautics

3 4 Acknowledgments

The author wishes to acknowledge following groups which have financially supported his education and the research presented in this work: The Boeing Company, through the Strategic Universities for Boeing Research and Technology Program; the MIT- Spain La Caixa Foundation Seed Fund through the MISTI Global Seed Funds grant program; and the MIT AeroAstro Vos fellowship. He would also like to thank Pro- fessor Manuel Martinez-Sanchez for many helpful discussions related to this work.

5 6 Contents

1 Introduction 21 1.1 Overview of electric discharges in gases ...... 21 1.1.1 Background ...... 21 1.1.2 Different discharge regimes and their transitions . . . . . 22 1.2 Streamers ...... 24 1.2.1 Description ...... 24 1.2.2 Past modelling efforts ...... 25 1.3 This work ...... 31 1.3.1 Motivation ...... 31 1.3.2 Outline ...... 32

2 1.5 dimensional model monstruction 33 2.1 General fluid model ...... 33 2.1.1 Governing equations and non-dimensionalization ...... 33 2.1.2 Transport coefficients ...... 36 2.1.3 Sources ...... 37 2.2 Modifications for 1.5D model ...... 41 2.2.1 Particle equations ...... 41 2.2.2 Electric field ...... 42 2.2.3 Photoionization ...... 49 2.2.4 Radial flux ...... 51 2.3 Anode mounted model ...... 52 2.3.1 Modifications to base 1.5D model ...... 52

7 2.3.2 Numerical methods considerations ...... 54 2.4 Quasi-steady streamer ...... 56 2.4.1 Modifications to base 1.5D model ...... 56 2.4.2 Solution scheme ...... 58

3 Analysis of 1.5D Model 67 3.1 Detailed analysis of model outputs ...... 67 3.1.1 Transient model ...... 67 3.1.2 Steady state model ...... 73 3.2 Applicability of quasi-steady model ...... 75 3.2.1 Existence of a quasi-steady frame ...... 77 3.2.2 Comparison of quasi-steady and transient solutions ...... 80 3.3 Comparison to published data ...... 81 3.4 Discussion ...... 85 3.4.1 Relative importance of diffusion and photoionization . . . . 85 3.4.2 Current ...... 87

4 Macroscopic quasi-steady streamer modelling 91 4.1 Derivation of model ...... 91 4.1.1 Electric field and characteristic dimension ...... 91 4.1.2 Radius ...... 92 4.1.3 Background field ...... 96 4.1.4 Graphical representation ...... 100 4.2 Comparison with 1.5D model ...... 102 4.2.1 Steady state ...... 102 4.2.2 Transient ...... 104 4.2.3 Comparison of all 3 models ...... 106 4.3 Comparison with published data ...... 106 4.3.1 Data sources ...... 106 4.3.2 Comparison and discussion ...... 111

8 5 Model extensions 115 5.1 Macroscopic propagation ...... 115 5.2 Negative streamers ...... 118

6 Conclusions 123 6.1 Summary of contributions ...... 123 6.2 Recommendations for future work ...... 125

A Additional derivations 127 A.1 Integrated absorption function ...... 127 A.2 Mirror charge ...... 128

B Additional model results 135 B.1 Transient model parameter variation ...... 135 B.2 Numerical accuracy analysis ...... 137

9 10 List of Figures

2-1 Electron transport coefficients: (L) experimental data from [1] (R) data from BOLSIG+...... 37 2-2 Peclet number based on ionization length for electrons as a function of normalized electric field...... 38 2-3 Comparison of kernel functions for different calculation methods. . . 48 2-4 Comparison of modified uniform cross-section to shell of charge. .48 2-5 Electric field magnitude with flux streamlines overlayed; adapted based on data from [2]...... 51 2-6 Geometry for anode-mounted streamer. Subscript a is the anode, subscript s is the streamer. d is the diameter of streamer or anode. . . . 53 2-7 Region of grid refinement; electric field of streamer shown for reference 55 2-8 Effect of truncating electric field ...... 59 2-9 Sparsity pattern of matrix ...... 60

2-10 Velocity calculation; note and on the axis correspond to ′ 푆푇,푐푎푙푐 푆푇 푆푇 and 0 respectively in the text...... 62 푆푇 2-11 Steady-state model iteration scheme ...... 65

3-1 Example output for transient model. The left two figures have been plotted at t=1.7, 4.9, 8.3 and 11.8 (non-dimensional time). Input pa-

rameters are: 푎0 = 0.15푚푚, 푅푎 = 5푎0, 휑푎/푅푎 = 4 ...... 68

11 3-2 Transient model output varying initial conditions. 푎0=0.25mm, 푅푎=5푎0

Colours refer to 휑푎/푅푎: red=2.5, green=3, black=3.5, blue=4. Linestyle refers to maximum of Gaussian initial seed. solid line=2, dashed line

(- - -) = 20, broken dash (− · −·) = 50, dotted (···)=100 ...... 70 3-3 Transient model output varying streamer radius and anode size with

fixed initial seed 푛0=20 and surface field 휑푎/푅푎 = 3. Colours refer

to 푅푎, measured in streamer radii: green=10, black=7.5, blue=5, magenta=2.5. Linestyle refers to streamer radius: solid line=0.15mm,

dashed line (- - -) = 0.25mm, broken dash (− · −·) = 0.35mm . . . . 71

3-4 Transient model output varying background charge density. 푎0=0.25mm,

푅푎=10푎0. Colours refer to 휑푎/푅푎: red=4,green=3.5, black=3, blue=2.5. Linestyle refers to charge density magnitude (퐶 in 휌 = 퐶/푟). solid line=0, dashed line (- - -) = 1, broken dash (− · −·) = 2, dotted (···)=4. 74

3-5 Example Output for Quasi-Steady Model. Model inputs: 푎0=0.15mm,

퐸∞=0.14 ...... 75 3-6 Example output for quasi-steady model; same case as figure 3-5 with enlargement of head area and particle densities on linear scale . . . . 76 3-7 Comparison of quasi-steady velocity (calculated by equation 3.2) to transient velocity (measured based on tip movement between time steps) 78 3-8 Comparison of velocity calculated by equation 3.2 (dashed lines) to

true velocity (solid lines) for two cases. (A) 푎0 = 0.35푚푚, 푅푎 = 5푎0

(B)푎0 = 0.25푚푚, 푅푎 = 10푎0 ...... 79 3-9 Comparison of transient model (black) to steady state model at various locations (chosen based on matching tip field). For transient model,

푎0 = 0.15mm, 푅푎 = 5푎0, 휑푎/푅푎 = 4 ...... 82 3-10 Same as figure 3-7, with the effect of diffusion and photoionization selectively neglected ...... 86 3-11 Comparison of electron source term magnitudes ...... 86 3-12 Example current analysis applied to transient model for streamer with

푎0 =0.25mm and 푅푎 = 5푎0, streamer tip at 35 ...... 89

12 3-13 Current ratio in streamer channel (10 radii behind head) and acceler-

ation of streamer. 푅푎 = 5푎0 for all ...... 90

4-1 Analysis of equation 4.4. See text for description of plots ...... 95 4-2 Control volume for charge conservation calculation ...... 96 4-3 Comparison of ratio of LHS to RHS for equation 4.11 ("Full Equation") and equation 4.12 ("Simple Equation") ...... 99

4-4 Evaluation of the value of 휅 ...... 101 4-5 Graphical representation of the solution of equation 4.4 and 4.15. Solid lines are constant effective radius (mm), dashed lines are constant background field (normalized by breakdown field) ...... 103

4-6 Figure 4-5 with results of steady state model overlayed ×. Steady state model results are for effective radius 0.15mm to 0.5mm increasing from bottom right to top left in steps of 0.05mm and backgroud field

increasing from 0.1E0 to 0.2E0 in steps of 0.01E0 from bottom left to top right ...... 105 4-7 Figure 4-6 with results of transient model overlayed. Markers are placed every 0.1 non-dimensional time along the curve. Colours corre-

spond to different transient model runs as described in text. × markers are steady state model results ...... 107 4-8 Comparison of transient model (black line on all graphs) to steady

state model at various locations. For transient model, 푎0 = 0.15, 푅푎 =

5, 휑푎/푅푎 = 4. The coloured markers on the graph on the left correspond to the steady-state model results shown on the right. Compari- son is made at the point where the tip field of the transient and steady state models are equal for equal radius streamers...... 109

13 4-9 Macroscopic parameter solution with published data. Solid contours

are radiative radius in mm (푟푟푎푑 = 1.4푎0), dashed contours are background field in 푘푉/푐푚. Published data labels are radiative radius.

Sources: ∙ Pancheshnyi et. al. [3], Luque et. al. [2], Veldhuizen et. al. [4], N Bagheri et. al. CWI group [5] ...... 112

5-1 Comparison of macroscopic propagation to full transient model. Ar- rows point to the initial condition; solid shifted lines are result of full

transient model (≈100 time steps), dashed shifted lines are result of macroscopic propagation model (single time step) ...... 117 5-2 Example of graphical characterization of negative streamers. Contours are for constant (effective) radius with labels in mm ...... 122

B-1 Modification of initial Gaussian seed. Fixed 푎0 = 0.25푚푚, 푉푎 = 25,

푅푎 = 10푎0; run 1 푛0 = 10, run 2 푛0 = 50, run 3 푛0 = 80, run 4 푛0 = 2,

run 5 푛0 = 25 ...... 136

B-2 Modification of streamer radius for large anode. Fixed 푅푎 = 10푎0,

푉푎 = 25, 푛0 = 25 ...... 136

B-3 Modification of streamer radius for small anode. Fixed 푅푎 = 2.5푎0,

푉푎 = 10, 푛0 = 25 ...... 137 B-4 Grid size convergence study. Results listed as "variable" are for a non- uniform mesh and have have their endpoints at the coarse and fine sizes of the grid. All others results are for uniform mesh...... 138

14 List of Tables

2.1 Derived characteristic values for 푎0 = 0.25푚푚 ...... 35 2.2 Photoionization coefficients for 3 exponential fit [6] ...... 41

15 16 Nomenclature

Variables

푎0 Streamer characteristic dimension/effective radius (m) 2 −1 퐷푗 Diffusion coefficient of species j(m s )

푑푒푙푒푐 Electrodynamic diameter of streamer (m)

푑푟푎푑 Radiative diameter of streamer (mm) 푒 Elementary charge (퐶) E Electric field (V m−1)

−1 퐸∞ Uniform background electric field (V m ) j Current density (A m−2) 푘 Boltzmann constant (J K−1)

−3 푛푗 Number density of species j (m ) 푝 Pressure (Torr) 푃 푒 Peclet number (dimensionless) Partial pressure of oxygen (Torr) 푝푂2

푝푞 Quenching pressure (Torr)

푞푗 Charge on a particle of species j (C) 푄 Total charge in a volume (C) r Location of an evaluation point (m) 푅 Equal to |r − r′| r′ Location of a source point (m)

푟푠푝ℎ Radius of a spherical anode (m)

17 푅푎 Radius of spherical anode (m)

푟푠 Radius of streamer at evaluation point (m) ′ Radius of streamer at source point (m) 푟푠 ˙ −3 −1 푆푗 Particle sourece term of species j (m s ) −3 −1 푆푝ℎ Photoioinzation rate (m s ) −2 −1 푆푇 Net photon production per unit tip area (m s ) 푡 Time (s) 푇 Temperature (K)

−1 v푗 Bulk velocity of species j (m s ) 푉 Velocity of streamer reference frame (m s−1)

푋푒,+,− Property X, evalutated for electrons, positive ions or negative ions respectively

푋0 Reference value of property X 푋ˆ Explicitly non-dimensional form of property X 푋* Explicitly dimensional form of property X

푋푐ℎ Property X evaluated in streamer channel 푋푘 Property X, evaluated on iteration k of an iterative scheme

푋푇 Property X evaluated at streamer tip

푧푇 Position of streamer tip (m) 훼 First Townsend coefficient (ionization coefficient) (m−1) 훽 Second Townsend coefficient (attachment coefficient)− (m 1) ∆ Small distance on the order of 훼−1 (m)

−1 휖0 Vacuum permitivity (F m ) 휇 Mobility of species j (m2 V−1 s−1)

−1 휈푖 Ionization frequency (s ) 휉 Photoionization efficiency (dimensionless) 휌 Charge density (Cm−3)

−3 휌1퐷 Charge density calculated by a 1D model (Cm ) 휏 Non-dimensional time (dimensionless) 휑 Potential (V)

18 휒 Photon absorption cross section (m2) 휔 Photon production coefficient (m−1) Ω A volume of integration 휕Ω Surface containing Ω

Acronyms

SS Steady-State, (also used as shorthand for a situation that is only technically quasi-steady) Trans Transient Model AMR Adaptive Mesh Refinement XD X Dimensional Model FVM/FEM Finite Volume Method/Finite Element Method RHS/LHS Right/Left hand side of equation PDE/ODE Partial/Ordinary differential equation

E푇 -V space A graphical format for characterizing streamers

19 20 Chapter 1

Introduction

1.1 Overview of electric discharges in gases

1.1.1 Background

Electric discharges in gases are a common occurrence seen in many different situ- ations. They range in scale, from localized emissions on the order of micrometers to impressive transient luminous events (TLEs) in the upper atmosphere which can extend for tens of or even hundreds of kilometers [7]. Large variations in time scale are also seen; streamer phase discharges occur in a matter of nanoseconds while DC glow discharges can be held in steady state indefinitely. Electric discharges are most notably encountered by humans as lightning, a phenomenon that has inspired a great body of research in its own right(see [8, 9]). Naturally occurring storms also produce discharges at a much smaller scale; discharges in the streamer corona regime have been a common sight to sailors for centuries, who termed it St. Elmo’s Fire [8, 10]. Formal studies into the mechanics of gas discharges originated in the 19th and 20th centuries, with significant contributions to understanding the streamer corona regime, a focus of this work, being made in the 1930s and 40s by Raether [11] and Loeb and Meek [12]. In the decades since, understanding of gas discharge phenomena has significantly increased with a large body of work devoted to the subject. A good introduction to the fundamental theory can be found in [10, 13].

21 Aside from their natural occurrence, electric discharges in gases have been explored for numerous applications across many different fields. There is a particular interest in use of streamer and glow coronas in industry as a non-thermal plasma source. This is due to their production of active species, whilst not significantly raising temperatures. In this work, a non-thermal plasma will refer to an ionized gas where the ions remain approximately at the temperature of the neutral gas while the electrons are heated to a temperature on the order of a few electronvolts. Applications include electrostatic precipitators [14], ozone production and gas cleaning/pollution control [15, 16], and semiconductor fabrication and surface processing [4, 17, 6]. Recently there has also been considerable interest in using these plasmas to modifying combustion reactions [18, 19]. Other applications use gas discharges to produce ions that may then be manipulated using electric fields to create an ionic wind for plasma-based actuators [20, 21] or propulsion [22]. A new field that has recently been investigated for non-thermal plasmas is medical and sterilization applications [23, 24]. Beyond these applied uses, there is an interest in studying gas discharges to better understand naturally occurring atmospheric events such as lightning and TLEs [25, 26, 27]. This is driven both by a desire to understand the fundamental processes, and from an engineering design perspective in the field of lightning strike protection. Past work has looked at glow corona discharges as they apply to grounded structures in storms such as lightning rods [28, 29] as well as larger scale discharges and their interactions with structures such as wind turbines and airborne vehicles [30, 31, 9]. Clearly there is a significant interest in understanding gas discharges from both a fundamentals and applications perspective. There are many complex phenomena involved and ample areas in which further research is required.

1.1.2 Different discharge regimes and their transitions

In general, an electric discharge involves the ionization of some initially insulating gaseous medium due to an applied electric field. The ionization process is governed primarily by collisional ionization driven by fast moving electrons in high electric fields. The minimum field required for a cascade ionization reaction is termedthe

22 breakdown field, 퐸0. Qualitatively speaking, this is the field in which a free electron is more likely to create an additional electron in a collision process than it is to be captured [27]. Discharges can broadly be separated into two categories based on the direction of electron motion. The discharge in which electrons ahead of the ionization front move towards the high field region is termed the positive discharge, while the situation in which electrons are repelled from the high field region is the negative discharge [27]. Both positive and negative discharges are seen at all scales of electric discharges. Discharges can occur in various regimes. For example, in laboratory discharges using non-uniform field geometries (e.g. point-to-plate) and atmospheric pressure air, glow coronae, streamer coronae, leaders (for long gaps) or arcs can be encountered. The discharge regime that ultimately forms will depend on the applied voltage level and its rate of rise, as well as the geometry and properties of the gaseous medium. These regimes all have very different structures and the transition between one to another remains poorly understood. One of the simplest structures is the glow corona, which consists of a very thin ionization region localized to a high voltage electrode [32, 33]. In the positive polarity, electrons are immediately absorbed and the region around the electrode is filled with positive ions that slowly drift away under the influence of the electric field. This positive space charge reduces the electric field, preventing further ionization until the positive ions have dispersed sufficiently [28, 34]. This can result in pulsations being observed in the glow at frequencies on the order of megahertz [32, 35]. Under certain conditions, the ionization region will separate from the high voltage electrode and begin to propagate under the influence of space charge induced electric fields. These ionization fronts mark the transition tothe streamer regime, and a a detailed description of their dynamics is discussed further in section 1.2. Streamers are inherently unstable structures and will branch and decay seemingly at random. An accurate description of the mechanics of these interactions remains an open question. This situation of multiple streamers interacting with one another is termed the streamer corona. It can consist of hundreds to thousands of individual streamer channels originating from a common stem.

23 As the number of streamers in the corona grows, the current in the common stem will increase until a critical threshold is reached at which point significant ohmic heating occurs. The heating rapidly increases the temperature (and hence conductivity) of the stem, and the streamer corona is said to have transitioned to a leader. Like streamers, leaders produce high electric fields due to their space charge and are able to propagate into regions where the ambient electric field is not itself large enough to cause breakdown in the gas. The main differences between the streamer and leader is the length scale (leaders are much larger in the same medium), and the importance of thermal effects to leaders but not to streamers. If the leader is able to bridge the gap between two electrodes, a spark will occur, with the excess static charge in one electrode flowing to the other. In smaller gaps, in which there is not enoughspace for the streamer corona to transition into a leader, the streamer will directly trigger a spark when it bridges the gap [13].

1.2 Streamers

1.2.1 Description

Streamers are filamentary channels of non-thermal ionized gas, capped by a region of high space charge resulting in a self propagating ionization front. To understand this description, it is instructive to break a streamer down into its core elements. The self propagating front is created by a region of unbalanced space charge in the "head" of the streamer. This unbalanced charge creates a high electric field, which accelerates electrons. In the positive streamer, electrons ahead of ionization front are accelerated towards the positive streamer head, while in the negative mode the electrons are expelled by the negative head [27]. As the electrons drift in the high electric field, they collide with neutrals producing secondary electrons. These electrons join the growing cascade, referred to as an electron avalanche. As the electrons move, they leave behind positive ions. In the positive streamer, these ions make up the new streamer head and thus the streamer propagates forward [10]. When the electrons reach the

24 streamer head, they balance the space charge produced by the excess positive ions, resulting in a quasi-neutral gas existing just behind the head in the streamer channel. In this region, any excess charge is segregated in a shell at the edge of the ionized channel due to electrostatic repulsion. Electrons in the channel are conducted by an axial field that is below the breakdown field for the gas. Ions, being muchmore massive, are almost stationary in this region at the time scales of streamer motion. One peculiarity of positive streamers is the need to seed primary electrons in front of the streamer in order to initiate the electron avalanche. This is typically assumed to occur due to photoionization, with the photons produced in large numbers by excited particles very near the streamer tip. Streamers in atmospheric pressure air and ambient fields below the breakdown threshold propagate at a speed on the order of the electron drift velocity near the streamer head, which is typically in the range 105 −106m/s [16, 10, 3]. This value has been shown to be dependent on the diameter of the streamer channel, which has been measured experimentally to be of order 10−1 − 1mm [3, 36, 4]. In general, streamers will propagate along electric field lines, but they are sensitive to perturbations and are commonly seen to zigzag and branch [37, 36]. The cause of this behaviour remains an open question as it often appears stochastic. Some explanations that have been proposed are random photon generation at the head [10], electron density fluctuations [38] and a Laplacian instability of the streamer boundary [39]. The collection of many of these streamers, often originating from a common stem, comprise the streamer corona or streamer tree. The streamer tree may contain hundreds or even thousands of individual streamers which mutually interact via their electric fields and common current stem [40].

1.2.2 Past modelling efforts

Attempts have been made to model streamer dynamics for many years. In general, they have increased in complexity in step with computing power. There are a number of factors which make the problem difficult. First, there are significant issues with resolving length scales. Steep gradients in the ionization region (orders of magnitude

25 smaller than the streamer radius) require very fine spatial resolution, while at the same time very fast electron drift velocities demand fine temporal resolution [14]. Streamers are observed to propagate over distances many times their radius, and detailed numerical simulations of long streamers quickly become intractable [41]. Models of varying fidelity have been constructed in the past, and a brief overview oftheseis presented below.

Semi-analytical models

The simplest models of streamers try to avoid solving the coupled drift diffusion and electric field equations numerically and instead introduce simplified formulas based on scale length and order of magnitude arguments. Some of these were developed prior to the development of computers capable of adequately solving the equations, while others were motivated by a desire to characterize streamers in a simpler way. One of the earliest viable models was created by Gallimberti [37], improving on work done by Dawson and Winn [42]. This model groups all electron avalanches into a single equivalent avalanche and imposes boundary conditions motivated by particle number density considerations. This model was shown to give good agreement with experimental results, particularly in terms of streamer velocity, and indicated the existence of a minimum field in which a streamer can propagate consistent with the so called "stability field" quoted by many authors [4, 10]. Other contemporary authors of Gallimberti proposed a model based on perfectly conducting channels that showed some success; an overview of this model is given in [10, 13]. More complex models based on both these philosophies (pre-head avalanche dominated versus perfect conductor channel) have persisted as computational resources have increased. An interesting implementation of an avalanche based model in a more modern context is given by Naidis [43]. In this model, the equations governing electron dynamics are integrated in the region in front of the streamer in a frame moving at the instantaneous streamer velocity. With some reasonable assumptions on electric field shape and number densities informed by more complex models, Naidis’ modelis

26 able to give useful results at a fraction of the computation cost. This model will be discussed in more detail later in this work. Samusenko and Stishkov [44] also used a reference frame moving at the streamer velocity to reduce calculation complexity and develop an interesting model for streamer radius evolution.

1.5D Models

As computers became more powerful and widely available, it became viable to numerically solve the equations of particle motion, at least in one dimension. The electric field is inherently three dimensional due to the finite streamer radius, and therefore requires different treatment. These considerations give rise to the so-called "1.5di- mension" (1.5D) model, where the particle equations are solved by assuming densities are averaged across the streamer cross-section and the electric field is calculated based on axial symmetry. This approach was used throughout the 1980s and 90s [45, 41, 46]. One of the difficulties arising in these models is the need to specify some radial distribution of charge in the streamer channel. This is how the connection between 1D and 2D-axisymmetric is made. One method commonly employed is the method of discs, where the charge is distributed uniformly across the finite streamer channel radius [45]. Other authors have employed a ring method, wherein the excess space charge is segregated to a shell at the edge of the streamer [41]. A well known shortcoming of 1.5D models is the requirement to specify the streamer radius a priori, rather than solving it self consistently within the model [47, 48]. Some efforts have been made to address this issue (see [41, 46]), butan adequate 1.5D model of radial expansion has not yet been realized. Because of these issues, and the advent of better computational resources, recent decades have seen the 1.5D model neglected in favour of full 2D axisymmetric and 3D simulations. Never- theless, the 1.5D model is still able to provide valuable information. Aleksandrov and Bazelyan [41, 47] were motivated by the 1.5D model’s ability to simulate much longer streamers than higher dimensional simulations. Recent work by Luque et. al. [49] has revisited the concept of a 1.5D model. They were inspired by the observation that from a macroscopic perspective streamers appear to be relatively one dimensional

27 and the hope that a 1.5D model could be used as a simpler alternative to complex 3D simulations that have become the standard.

Axisymmetric and 3D single streamer models

The last decade or so has seen a proliferation of high fidelity single streamer models. Most attempt to solve the particle transport equations and coupled Poisson equation in 2D axisymmetric configurations [2, 15]. Others, in part motivated by adesire to understand branching employ a fully 3D domain [50, 51]. Non-fluid descriptions of plasma have also been used to model streamer dynamics; for example Li et. al. [52, 53] studied particle in cell (PIC) and hybrid models for negative streamers. Com- mercially available codes such as COMSOL Multiphysics is another tool that has been employed, showing good correspondence with dedicated solvers [5]. A review of several different fluid models was recently performed by Bagheri et. al. [5]. Readerswho are interested in learning more about the state of the art in detailed single streamer modelling are directed to that paper and the references therein. A discussion of detailed single streamer simulations must highlight the computational difficulties arising from the disparate time and length scales involved. Agood overview is given by Ebert et. al. [14]. In essence, the very thin ionization layer requires fine spatial resolution and the fast moving electrons force short time steps when using an explicit time marching scheme. This can quickly make the problem computationally intractable, especially in 3D. Many modern codes utilize some form of adaptive mesh refinement (AMR) to try and speed up computations. Teunissen et. al. [51] recently developed a model exploiting advances in both AMR and parallel computing technologies that was demonstrated to achieve orders of magnitude improvement in computation speeds over models of comparable accuracy [5].

Regime transition and branching

Many of the models discussed so far focus on single streamer propagation. They typically rely on some form of seed density and apply a sufficiently strong electric field to trigger the streamer to begin. One of the most popular geometries is the double

28 headed streamer [2, 54, 5], which creates a positive/negative streamer pair propagating in opposite directions. Other authors have investigated streamers mounted on an anode and usually impose some initial particle seed to trigger the streamer [55, 56]. In all these cases, the streamer formation is forced by the parameters of the problem. In many cases of interest, however, a streamer or glow regime are both possible and the transition between the two remains an open research question. One group that has investigated this transition is Liu and Becerra [34], who started with a model of a stable glow corona and introduced an instability to trigger a streamer. While they were able to generate a self-ionizing structure that detached from the anode, it cannot be considered a true streamer due to the 2D planar nature of the simulation. The glow to streamer transition requires capturing both ion and electron time scales, which are roughly two orders of magnitude different. To handle this, the same group developed some interesting methods worth mentioning here: position-state separation, based on the concept of operator splitting [57], and a hybrid steady/transient model which solves transients on the ion time scale and assumes electrons are instantaneously in steady state [58]. Another approach to streamer initiation can be seen in the work of Arrayas et. al. [59], who studied instability in planar ionization fronts. In terms of streamer initiation by branching, this is a problem that typically requires fully 3D simulations. There have been some attempts at studying it in a purely axisymmetric situation. For example, Luque and Ebert [38] were able to show the onset of an instability by introducing density perturbations into an axisymmetric model. Likewise, Liu and Pasko [54] showed the onset of a Laplcian instability in an axisymmetric streamer. Neither of these can be considered a true streamer branching event due to the 2D nature of the domains, but do offer some insight into the mecha- nisms causing it. Adequate models or streamer branching by necessity must be fully 3D. Papageorgiou et. al. [50] had some success with triggering a 3D instability and Li et al. [52, 53] were able to show the beginnings of branching in negative streamers using fully 3D simulations. Some of the best examples of simulating branching and multi-streamer interaction are contained in the recent work by Teunissen et. al. [51]. Note this is the same model discussed previously which used AMR and parallel pro-

29 cessing to drastically speed up computations, and this is part of the reason why it was able to capture the length and time scales required to see branching. Even in that work, however, the authors were limited to a small number of streamers and branching events, and significant work remains to be done to adequately describe the processes involved.

Streamer corona models

While substantial effort has been devoted to studying single streamer dynamics and even single streamer branching events, modelling the full streamer corona is a much more difficult task. The streamer corona is composed of hundreds to thousands of individuals streamers which mutually interact through their self-induced electric fields and common stem. This exacerbates the length scale issues seen in single streamer dynamics since now there are many more ionization regions to be resolved and large separations between them. Past works attempting to model these structures have been forced to make some substantial simplifications. One of the first studies of the streamer corona using modern computers was done by Akyuz et. al [60]. This model used a version of the Meek breakdown condition to describe the avalanches and perfectly conducting streamer channels rather than a full solution of the particle equations, with a finite element method (FEM) solver to handle the electric fields. More recently, Luque and Ebert [40] developed a streamer tree model that was able to support dozens of streamers and produced results resembling streamer coronas seen in laboratory experiments. This model too was forced to neglect solutions of the full particle fluid equations, but did attempt to incorporate a finite streamer conductivity to create more realistic electric fields. These types of models typically revert to heuristics for streamer branching and initiation, which motivates developing a more physics based understanding of these processes as discussed above.

30 1.3 This work

1.3.1 Motivation

It has been shown that a substantial body of work has been devoted to understanding streamer dynamics from theoretical point of view. The fundamental theory was constructed in the early 1940s, with numerical models starting to be constructed in the 1960s and 70s. The state of the art has really progressed in the last two decades due to widespread access to more powerful computers. The field has advanced to a stage where single streamers can be modelled with a relatively high degree of fidelity and consistency between various groups (see [5]). It is reaching the point where fully 3D simulations are becoming possible, allowing for investigation of streamer branching phenomena from a purely numerical perspective. As successful as these efforts have been, however, it would seem that the full streamer corona is still out of reach for direct numerical simulation. Approaching the problem from the other end are groups attempting to model the full corona, but who are forced to neglect important physics in order to make the problem tractable. The objective of this work is to approach the problem of streamer corona modelling from both perspectives. An accurate model capturing the core physics at the single streamer level is needed to be used as a building block for tree models such as [40]. Such a model should be as simple as possible, to allow extension to many streamer ensembles, whilst also being accurate enough to give practically useful results. Indeed, the authors of that paper have pursued the same goal in recent work [49], but the problem is far from solved. A fully 3D fluid model will be immediately ruled out since, as noted above, computers are far from being capable of extending such a model to a full corona. Likewise, an overly simplified model such as those proposed in [37, 60] should also be regarded sceptically due to them neglecting important dynamics at the individual streamer level. In this work, a middle ground is investigated, pulling inspiration from the 1.5D models of the 90s, which are relatively simple to solve with modern computers, as well as the quasi-steady frames considered by [43, 44]. The construction of simplified streamer corona building blocks will be

31 discussed, their validity and applicability analysed, and ultimately the path forward to a many-streamer model will be laid.

1.3.2 Outline

In chapter 2, this work begins with a discussion of the plasma fluid model typically used to describe streamer behaviour. This model is then reformulated for use in a 1.5D description of positive streamers. Two 1.5D positive streamer codes are developed; one which is the more conventional case of a streamer attached to an anode, and one which is developed in the reference frame of the streamer (a quasi-steady model). In chapter 3, these models are compared based on a number of metrics, including: qualitative behaviour of the streamer, comparison to previous models in the literature and correspondence between the transient and streamer reference frame models. This is followed in chapter 4 by the development of an even simpler model based on characterizing streamers by their macroscopic parameters. The three models are extensively compared both amongst themselves and against published data to verify their validity. Finally, in chapter 5, extensions on the models developed in the preceding chapters are discussed. These include looking at how the models developed herein might be used in a streamer corona model, as well as extending the streamer models to the negative polarity.

32 Chapter 2

1.5 dimensional model monstruction

2.1 General fluid model

2.1.1 Governing equations and non-dimensionalization

The governing equations for streamer propagation are the drift-diffusion equations for electrons and ions (equation 2.1) coupled with Gauss’ law for electrostatics (equation 2.2). For the model constructed in this work, only two species of ions will be considered: positive (+) and negative (-), both of which are singly charged. All reaction and transport coefficients are written as an average for all species present

−3 (see section 2.1.2). In equations 2.1, 푛푗, vj, 퐷푗 are the number density (푚 ), bulk 2 ˙ velocity (푚/푠) and diffusivity (푚 /푠) of species 푗 respectively. 푆푗 is the source term for species 푗 (푚−3/푠) and includes the effect of collisions and photoionization (see section 2.1.3). In equation 2.2, 휑 is the electric potential where −∇휑 = E and E is the electric field. 휌 is the excess charge density; for electrons and singly charged ions it is defined by 휌 = 푒(푛+ − 푛푒 − 푛−).

휕푛 푗 + ∇ · (푛 v − 퐷 ∇푛 ) = 푆˙ (2.1) 휕푡 푗 j 푗 푗 푗 휌 ∇ · (∇휑) = − (2.2) 휖0 For this work, the equations will be treated in a non-dimensional form. The

33 system of equations has three dimensions (length, time and electric charge) so it is possible to pick three independent parameters for non-dimensionalizing the equations. Note that mass is not considered a dimension in these equations since the drift- diffusion form of the particle equations neglects the inertia of the particles. The

chosen independent parameters are a characteristic length scale 푎0, velocity scale 푣0

and electric field 퐸0. The characteristic length will be treated later in this work as an effective radius of the streamer. The connection of this effective radius to aphysical radius will also be discussed in section 4.1.3. It should be noted that many other authors choose a length scale on the order of the inverse of the Townsend ionization coefficient, which roughly corresponds to the electron mean free path for ionization [14, 25, 61]. This scaling is more reasonable in full 3D simulations, but the imposed radius of the 1.5D model inherently makes the larger length scale a more obvious choice since it simplifies some other aspects of the model. The characteristic electric field is chosen to be the breakdown field, which is defined as the field inwhichthe rate of collisional ionization equals the rate of collisional attachment. In the drift- diffusion approximation (equation 2.1), the bulk velocity arises from acceleration in the electric field being damped by collisions. It is traditionally written asbeing

directly proportional to the electric field, with proportionality 휇, called the mobility,

푞푗 which itself varies with energy: vj = 휇푗Ej where 푞푗 is the charge on species j. Since |푞푗 | streamers propagate on the timescale of electron motion, the natural velocity scale is

the magnitude of the electron drift velocity in the breakdown field, 푣0 = 휇푒퐸0. It should also be noted that streamers exist in an ionized gas regime dominated by electron-neutral collisions. As such, the mechanics are dependent on the species present in, and density of, the ambient gas. The choice of this gas will affect the value

of all characteristic parameters. For this work, atmospheric pressure dry air (79% N2,

21% O2) at 300K will be assumed in all cases. Many of the properties of the gas are determined using BOLSIG+ [62] with cross section data for nitrogen and oxygen from [63, 64]. From this data, and using the definition that the breakdown field is the electric field at which the rate of ionization equals the rate of attachment, the electric breakdown field in air is taken to be about 118Td,or 29푘푉/푐푚 for the given

34 Table 2.1: Derived characteristic values for 푎0 = 0.25푚푚

Parameter Definition Approximate Value Units

휖0퐸0 17 −3 number density 푛0 = 6.4 × 10 푚 푒푎0 −3 charge density 휌0 = 푒푛0 0.10 퐶푚 time 휏 = 푎0 1.8 × 10−9 푠 푣0 2 diffusivity 퐷0 = 푎0푣0 34 푚 /푠 particle production 푛0 26 −3 −1 푆0 = 휏 3.5 × 10 푚 푠 Voltage 휑 = 푎0퐸0 730 푉

temperature and pressure. The characteristic velocity is taken to be 135km/s. The remaining dimensional parameters are given in table 2.1. Note that all parameters in table 2.1 have dependence on the characteristic length scale. This highlights one of the reasons why a length scale based on the mean free path is preferred in 3D simulations since it will not depend on the streamer geometry. For the 1.5D model though there are still advantages to using the geometric length; the typical values

listed in 2.1 are for 푎0 = 0.25mm. In non-dimensional form, the equations can be written in the form of equations 2.3

and 2.4 with the electric field defined by equation 2.5. In this instance, the notation 푥ˆ is used to emphasize that the variable 푥 has been divided by the characteristic value. The operator ∇ˆ refers to the gradient/divergence with respect to the non-dimensional length coordinate. For the remainder of this work, this will be omitted for simplicity and all variables should be assumed to be non-dimensional unless stated otherwise.

휕푛ˆ푗 ˆ (︁ ˆ ˆ )︁ ˆ˙ + ∇ · 푛ˆ푗 ^vj − 퐷푗∇푛ˆ푗 = 푆푗 (2.3) 휕푡ˆ

(︁ )︁ ∇ˆ · ∇ˆ 휑ˆ = −휌ˆ (2.4)

Eˆ = −∇ˆ 휑ˆ (2.5)

35 2.1.2 Transport coefficients

There are two important transport coefficients in the model: mobility, which determines the particle flux due to the electric field, and diffusivity, which determines the flux due to number density gradients. In the corona regime, ions can be considered cold and at the same temperature as the neutral gas, so it is generally acceptable to take their transport coefficients to be constant. Their large mass relative tothe electrons means that at streamer time scales they are almost stationary, so an approximate value for their transport parameters is sufficient. The ion mobility used in

this work is 2푐푚2/푉/푠 [49] with the diffusivity calculated using the Einstein relation (equation 2.6) at 300K.

퐷 푘푇 = (2.6) 휇 푒 The electrons on the other hand are not cold, and typically are accelerated to a few electronvolts of temperature. The transport coefficients are dependent on collision cross sections which are inherently a function of the electron energy. Because of the order of magnitude variation in electric field between the streamer tip and streamer channel, it is important that this dependence is accounted for. The BOLSIG+ solver [62] is used with cross sections taken from the LXCAT website. The SIGLO and Phelps databases [63, 64] are used to compute the electron transport coefficients in air at 300K for reduced electric fields in the range of 1-1000Td. A plot of the ratioof mobility to diffusion for air is shown in figure 2-1 comparing experimental dataand the data generated by BOLSIG+ used in this work. When running BOLSIG+, only electron-neutral collisions are considered. Typical electron number densities in streamers are on the order of 1020 m−3, which is approximately five orders of magnitude less than the number density of the background gas [14], so this is a reasonable assumption. It is useful to compute the relative magnitude of the diffusion to advective flux. This is analogous to the Peclet number encountered in traditional fluid dynamics and mass transfer problems [65]. The characteristic length of interest here is the mean free

36 Figure 2-1: Electron transport coefficients: (L) experimental data from [1] (R) data from BOLSIG+.

path of ionization, equal to the inverse of the Townsend ionization coefficient, since that is the governing parameter for the tip thickness (recall the discussion of possible characteristic length scales in the preceding section). The Peclet number based on

ionization length, 푃 푒훼 as a function of electric field is plotted in figure 2-2, indicating that the problem is advection (drift) dominated except at very high fields. Typical streamer tip fields are on the order of 5-6 times the breakdown field, so diffusionis expected to be non-negligible only very near the tip, although even there it is not dominant. For this reason, many authors choose to entirely neglect diffusion when solving the particle continuity equations [49, 43].

휇푒퐸 푃 푒푒,훼 = (2.7) 퐷푒훼

2.1.3 Sources

Collisional ionization and attachment

The primary source of charged particles in a streamer discharge is collisions between electrons and neutrals. This is the mechanism governing the electron avalanche formation in the streamer head. In streamer discharges, the ionizing collision rate is usually expressed in terms of the Townsend ionization coefficient, which is defined as the ratio of the collision frequency to the drift velocity, equation 2.8 [13], and has units of inverse length. As noted in section 2.1.1, this roughly corresponds to the inverse of the mean free path of electrons between ionization events. Air is an electronegative

37 1012

1010

108

106 Breakdown Field Breakdown

104 Convection Dominated

102

100

Diffusion Dominated 10-2 10-1 100 101

Figure 2-2: Peclet number based on ionization length for electrons as a function of normalized electric field.

gas and thus attachment is also significant. The Townsend attachment coefficient, 휂 is defined similarly, except with the collision frequency that for attachment. These were calculated using BOLSIG+ for the same conditions as the transport coefficients.

휈푖표푛 휈푎푡푡푎푐ℎ 훼 = 푒,푛 휂 = 푒,푛 (2.8) 휇푒|퐸| 휇푒|퐸| One item to note is the reactions that this work does not explicitly consider. These include non-ionizing inelastic collisions, different species of positive and negative ions, and charge exchange reactions amongst heavy species. The last item is unimportant because a common mobility is assumed for all ionized species and ions are relatively slow. As for the first two items, the BOLSIG+ solver has implicitly accounted for these effects in computing the transport and ionization coefficients. In the next section, photoionization will be discussed and photoionization is affected by both non-ionizing inelastic collisions and different ionized species. However, from the perspective of the electron and ion fluid equations it is relatively unimportant.

38 The common mobility for all ion species and lack of inertia in the governing equations means that implicitly accounting for the different species using BOLSIG+ is sufficient. The source terms due to collisions, as they appear in the continuity equations are shown by equation 2.9

˙ 푐표푙푙 ˙ 푐표푙푙 ˙ 푐표푙푙 (2.9) 푆푒 = 푛푒|푣푒|(훼 − 휂) 푆+ = 푛푒|푣푒|훼 푆− = 푛푒|푣푒|휂

Photoionization

For positive streamers in air, the electron avalanches fall towards the streamer head. This necessarily requires some method of seeding electrons far ahead of the streamer. While this is often done by applying a uniform background density, the generally accepted physical mechanism for it is photoionization. Almost all modern work on streamer modelling in air uses a photoionization model originally constructed by Zheleznyak et. al. [66] in 1982. This work has been summarized in numerous places (see [54, 67, 6]) and the reader is referred to one of those works for the details of the formulation. Since many different nomenclatures exist in the literature, it is usefulto present the equations here in the form they will be used throughout this work. The

photoionization at a point r is given by equation 2.10.

푝 휔 푔(푅) 푞 ′ ′ (2.10) 푆푝ℎ(r) = 휉 푆푖표푛(r ) 2 푑Ω ˆΩ′ 푝 + 푝푞 훼 4휋푅 A qualitative description is as follows. The pressure ratio describes the effect of excited molecules being quenched by collisions rather than emitting a photon (푝 is

the total gas pressure and 푝푞 is the quenching pressure). The term 휉 is the prob- ability of photon absorption leading to ionization and the ratio 휔/훼 is the ratio of photons produced per ionizing collision [68]. Recall that excitation was ignored in the section on collisional ionization, but it must be addressed explicitly when dis- cussing photoionization. The final term in the integral is the absorption function,

푔(푅), divided by the spherical area. This term accounts for the decrease in radiation

39 intensity between the emitting and absorbing volumes due to both wave divergence

and absorption. The integral over all space (Ω′) accounts for the non-local effect of photoionization; a photon produced anywhere in the domain at location r′ is capable of causing ionization at point r. The variable 푅 represents the linear distance between 푟 and 푟′. Note that in air, the nitrogen and oxygen both play important but different roles in photoionization. This was known to Zheleznyak et. al. [66] and is illustrated well by the work of Nijdam et. al. [36]. It has been proposed that nitrogen is the predominant photon emitting species in air, and that most photoelectrons are produced by oxygen absorbing photons of wavelength 98-102.5nm, which falls in the ultraviolet portion of the spectrum [66]. The absorption function has a semi-analytical form given by

equation 2.11, where 휒푚푖푛 and 휒푚푎푥 are absorption cross sections for oxygen as given in [66]. This formula was derived by averaging over the portion of the electromagnetic spectrum relevant to photoionization of air as identified above.

exp(−휒 푝 푅) − exp(−휒 푝 푅) 푔(푅) = 푚푖푛 푂2 푚푎푥 푂2 (2.11) 푅 ln(휒푚푎푥/휒푚푖푛) In the original work, this equation was shown to be in good agreement with low- pressure experimental results available at the time [69] and its applicability has since been verified up to atmospheric pressure dry air and, with appropriate modifications, humid air [68]. In the form of equation 2.10 with the absorption function given by equation 2.11, a volumetric integral must be performed to calulate the photoelectron production rate at every location. This quickly becomes computationally intractable. To reduce the numerical burden, methods were developed to convert the integral equation to a differential equation [67, 70, 6]. The work of Bourdon et. al. [6] givesa number of different approximations; in this work the three term exponential willbe used as given by equation 2.12 with coefficients listed in table 2.2.

3 푔(푅) ∑︁ = (푝 푅) 퐴 exp(−휆 푝 푅) (2.12) 푝 푂2 푗 푗 푂2 푂2 푗=1 The photoionization is incorporated into the particle continuity equations by di-

40 Table 2.2: Photoionization coefficients for 3 exponential fit[6]

−2 −2 −1 −1 j 퐴푗 (푐푚 푇 표푟푟 ) 휆푗 (푐푚 푇 표푟푟 ) 1 1.986 × 10−4 0.0553 2 0.0051 0.1460 3 0.4886 0.89

rectly including equation 2.10 as a source term for the electrons and positive ions. For this work, atmospheric pressure dry air is assumed in all cases (p=760Torr,

composition 21% O2, 79% N2). The quenching pressure is taken as pq=30Torr [66, 68], although it is noted that 60Torr is also a common choice [48, 70]. The term 휔 is 휉 훼 usually measured as a lumped parameter and assumed constant. It is a function of the reduced electric field, and in this work will be assumed to have a value of0.07 based on [66, 5].

2.2 Modifications for 1.5D model

2.2.1 Particle equations

Streamers can generally be treated as an axisymmetric phenomenon. This means all gradients in the azimuthal direction can be neglected. Expanding the differential operators in equation 2.3 in axisymmetric coordinates gives equation 2.13.

휕푛 휕 (︂ 휕푛 )︂ 1 휕 [︂ (︂ 휕푛 )︂]︂ 푗 + 푛 푣 − 퐷 푗 − 푆˙ = − 푟 푛 푣 − 퐷 푗 (2.13) 휕푡 휕푧 푗 푗,푧 푗 휕푧 푗 푟 휕푟 푗 푗,푟 푗 휕푟

For the 1.5D model, it is assumed that the particle densities vary in the axial (z) direction only. In the radial direction, the densities are assumed to be constant until there is a discontinuity at the streamer radius where the density drops to zero. Therefore, all parameters on the left hand side of equation 2.13 should be considered averages across the streamer cross section, where the averaging operation for a parameter 푋 is defined by equation 2.14 where 푟푠 is the radius of the streamer.

41 푟푠 2 (2.14) < 푋 >= 2 푋푟푑푟 푟푠 ˆ0 The term on the right hand side of equation 2.13 can be considered an additional source term caused by particles entering the streamer channel in the radial direction. In this work, this term will be considered negligible but may have implications for streamer radial expansion. The general 1D equation for particle conservation is then equation 2.15.

휕푛 휕 (︂ 휕푛 )︂ 푗 + 푛 푣 − 퐷 푗 = 푆˙ (2.15) 휕푡 휕푧 푗 푗,푧 푗 휕푧 푗 Details of the precise solution methods are discussed later in this work. In general, the spatial derivatives are handled using a second order finite volume scheme because of its conservativeness.

2.2.2 Electric field

Derivation of integral model

The electric field must be treated three dimensionally due to the finite radius ofthe streamer. A pure one dimensional model of the electric field would over-estimate the value since it neglects all gradients in the radial direction. In this case, the only component of the field that is relevant is the one along the axial direction. Instead of solving the Poisson equation in differential form, it is more efficient to solve in integral form using Green’s functions. For the 3D Laplacian operator, the Green’s function is −1 . Therefore, a quantity of charge ′ located at a position ′ creates a 4휋|r−r′| 휌푑Ω r potential field defined by equation 2.16.

휌푑Ω′ 푑휑 = − (2.16) 4휋|r − r′|

42 The total potential at a point r is the contribution from all charges in the domain, so an integral is taken over all source locations r′ to yield equation 2.17.

1 휌(r′) 휑(r) = − 푑Ω′ (2.17) 4휋 ˆ |r − r′|

The electric field in the axial direction is the negative gradient of the potential inthe axial direction. Since the integral is over the volume 푑Ω′ = 푟′푑푟′푑휃′푑푧′ (in cylindrical coordinates), which is independent of the coordinate z, the derivative can be brought inside the integral. The charge density is a function of r′, so is also independent of the coordinate z. For simplicity, define 푅 = |r − r′|. Performing these simplifications yields equation 2.20.

휕휑 1 휕 휌(r′) 퐸 (r) = − = 푑Ω′ (2.18) 푧 휕푧 4휋 휕푧 ˆ 푅 1 휕 (︂ 1 )︂ = 휌(r′) 푑Ω′ (2.19) 4휋 ˆ 휕푧 푅 1 휌(r′) 휕푅 = − 푑Ω′ (2.20) 4휋 ˆ 푅2 휕푧

The term 휕푅 can be interpreted as the cosine of the angle between the vector ′ and 휕푧 r−r the axial distance between those two points, 푧 − 푧′. This allows the partial derivative to be calculated, resulting in an electric field given by equation 2.21.

1 휌(r′)(푧 − 푧′) 퐸 (r) = − 푟′푑푟′푑휃′푑푧′ (2.21) 푧 4휋 ˆ 푅3 The charge density calculated from the solution to the particle continuity equations varies only with 푧′. By axial symmetry, it cannot have any 휃′ dependence. The choice of the radial distribution for 휌 is the connection between the 1D continuity and 2D axisymmetric electric field calculation. In general, the charge density canbe

′ ′ ′ ′ ′ ′ written as 휌(푟 , 푧 ) = 휌1퐷(푧 )푓(푟 , 푧 ) where 휌1퐷(푧 ) is the result of solving the particle continuity equations in 1D and 푓(푟′, 푧′) is some distribution function in the radial direction which might vary with axial position. The finite radius of the streamer means that beyond some distance ′ the charge density is zero. ′ is the non-dimensional 푟푠 푟푠

43 ′ radius of the streamer at location 푧 . Recall that 푎0 has previously been identified as an "effective radius", so presumably ′ will be equal to 1 for a constant radius 푟푠 streamer. However, the derivations that follow are also valid for a streamer with a variable radius so to keep the formulas general the arbitrary non-dimensional radius

′ will be used throughout this section. Also note that the averaging operation per- 푟푠 formed on the particle continuity equations (equation 2.14) imposes a condition on the normalization of the distribution function given by equation 2.22.

′ 2 푟푠 ′ ′ ′ ′ (2.22) 1 = ′2 푓(푟 , 푧 )푟 푑푟 푟푠 ˆ0 Using this form for the charge density, the electric field equation can be rewritten as equation 2.23 where the Kernel function 퐾 is defined by 2.24.

∞ ′ ′ ′ 퐸푧(r) = 휌1퐷(푧 )퐾(푧, 푧 )푑푧 (2.23) ˆ−∞

′ 2휋 푟푠 푓(푟′, 푧′) (푧 − 푧′) ′ ′ ′ ′ (2.24) 퐾(푧, 푧 ) = − 3 푟 푑푟 푑휃 ˆ0 ˆ0 4휋 푅 In this form, the terms contained in the 퐾 function are dependent solely on the geometry of the problem and chosen radial distribution, and are independent of the charge density. For computational efficiency, it is advantageous to precompute the 퐾 function meaning that during the numerical solution only a single integral must be repeatedly calculated.

Solution for particular particle distributions

Uniform charge density acting on axis Suppose that the charge density in the streamer is uniform up to a radius ′ and 0 beyond. This corresponds to a distribution 푟푠 function 푓(푟′, 푧′) = 1 which is clearly normalized correctly by equation 2.22. In this case, 푅 = √︀푟′2 + (푧 − 푧′)2 and the integral in 휃 can be performed immediately. The result is equation 2.25 which has the analytical solution equation 2.26. This is the so-called "disc method" [45, 71].

44 1 푟푠 (푧 − 푧′)푟′ ′ (2.25) 퐾푑푖푠푐 = − 3/2 푑푟 2 ˆ0 (푟′2 + (푧 − 푧′)2) [︃ ]︃ 1 푧 − 푧′ = − 푠푖푔푛(푧 − 푧′) (2.26) √︀ 2 ′ 2 2 푟푠 + (푧 − 푧 )

Shell of charge In reality, electrostatic repulsion is going to result in all excess charge accumulating in a very thin shell at the edge of the streamer channel. This

′ can be approximated by taking a radial distribution function ′ ′ 푟푠 ′ ′ 푓(푟 , 푧 ) = 2 훿(푟푠 − 푟 ) where is the Dirac delta function and the factor ′ comes from normalization 훿 푟푠/2 according to equation 2.22. In this case, there is again an analytical solution for the

퐾 function. This is referred to as the "ring method" [71, 41].

′2 [︃ ′ ]︃ 푟푠 (푧 − 푧 ) 퐾푠ℎ푒푙푙 = − (2.27) 4 ′2 ′ 2 3/2 (푟푠 + (푧 − 푧 ) )

Shell of charge, averaged over cross-section This technique was proposed by Luque et. al. [49]. It seeks to address the issue of a shell of charge underestimating short range interactions in the vicinity of the streamer head. The electric field in the axial direction is calculated as the average across the streamer cross section at the point of evaluation, rather than the value directly on the axis. This is written

mathematically as equation 2.28, where 퐾 is described by equation 2.29 and the radial distribution function 푓(푟′, 푧′) is the same as for the shell of charge. Note that the 푟, 휃 dependencies are contained inside 푅 (see below).

1 2휋 푟푠 ∞ ′ ′ ′ (2.28) ⟨퐸(r)⟩ = 2 휌1퐷(푧 )퐾(푧, 푧 , 푟, 휃)푑푧 푟푑푟푑휃 휋푟푠 ˆ0 ˆ0 ˆ−∞

′ 2휋 푟푠 푓(푟′, 푧′) (푧 − 푧′) ′ ′ ′ ′ (2.29) 퐾(푧, 푧 , 푟, 휃) = − 3 푟 푑푟 푑휃 ˆ0 ˆ0 4휋 푅 In this case, the integral in the angular direction 휃′ is not so simple. Even though the problem is axisymmetric, evaluating the field off the central axis means that the angular coordinates must be explicitly considered during the integration. Both the

45 source point and observation point must be written in their coordinate representation:

r′ = (푟′, 휃′, 푧′), r = (푟, 휃, 푧). This allows the separation distance to be written as equation 2.30 where in the last form of the equation the variable substitution 훾 = 휃−휃′ has been made.

푅 = √︀(푧 − 푧′)2 + (푟 cos 휃 − 푟′ cos 휃′)2 + (푟 sin 휃 − 푟′ sin 휃′)2

= √︀(푧 − 푧′)2 + 푟2 + 푟′2 − 2푟푟′(cos 휃 cos 휃′ + sin 휃 sin 휃′)

= √︀(푧 − 푧′)2 + 푟2 + 푟′2 − 2푟푟′ cos 훾 (2.30)

Once again, all the terms that are independent of particle density are grouped

into a Kernel function, 퐾푠,푎푣푔, defined by equation 2.31. Substituting in the shell of charge radial distribution function, the integral in the 푟′ direction is immediate. The change in variable to 훾 is used to replace 휃, but this results in 휃′ appearing in the bounds of the 훾 integration.

′ ′ ′2 푟푠 2휋−휃 2휋 (푧 − 푧 )푟푠 1 ′ 퐾푠,푎푣푔 = − 2 2 3/2 푑휃 푑훾푟푑푟 8휋 푟 ˆ ˆ ′ ˆ ′ 2 2 ′2 ′ 푠 0 −휃 0 ((푧 − 푧 ) + 푟 + 푟푠 − 2푟푟푠 cos 훾) (2.31)

The periodicity of the function with respect to 훾 on [0, 2휋] means that the value of 휃′ in the integral with respect to 훾 is irrelevant so it can be set to 0. This allows the 휃′ integral to be performed immediately, resulting in equation 2.32.

′ ′2 푟푠 2휋 (푧 − 푧 )푟푠 푟 퐾푠,푎푣푔 = − 푑훾푑푟 (2.32) 4휋푟2 ˆ ˆ ′ 2 2 ′2 ′ 3/2 푠 0 0 ((푧 − 푧 ) + 푟 + 푟푠 − 2푟푟푠 cos 훾)

Of the two remaining integrals, the one in r has an analytic solution. First let

′ and 2 ′2 ′ 2, both of which are constants with respect to . Then 푏 = 푟푠 cos 훾 푐 = 푟푠 +(푧−푧 ) 푟

46 add and subtract b to the numerator and complete the square in the denominator

′ ′2 2휋 푟푠 (푧 − 푧 )푟푠 푟 + 푏 − 푏 퐾푠,푎푣푔 = − 푑푟푑훾 (2.33) 2 2 2 2 3/2 4휋푟푠 ˆ0 ˆ0 ((푟 + 푏) + (푐 − 푏 ))

Making the substitution 푢 = 푟 + 푏 and separating the integrand into two terms lets the equation be written in a form with an easy to solve anti-derivative. The result is

a kernel function that still contains a single integral in 훾. Since 훾 is buried inside 푏, this equation is very difficult to solve analytically and it is most convenient toleave it in the form given by equation 2.34.

′ ′2 2휋 [︃ 2 ]︃ (푧 − 푧 )푟푠 1 푐 + 푏푟푠 퐾푠,푎푣푔 = − 푐 − 푑훾 (2.34) 4휋푟2 ˆ 푐2 − 푏2 2 2 1/2 푠 0 (푟푠 + 2푏푟푠 + 푐 ) Since this function is to be precomputed, it is relatively simple to solve the integral numerically.

Comparison of kernel functions

Equations 2.27 and 2.34 are equivalent to the interaction kernel functions 퐺푅 and given by Luque et. al. [49] (with an extra factor ′2 2 accounting for slightly 퐺퐶 −푟푠 /4휋푟푠 different ways in which the functions are used). A comparison of what these functions look like for a uniform streamer radius is shown in figure 2-3. The advantage of using equation 2.34 is that the short range interactions are captured, while still maintaining the physically correct positioning of the charge in the shell of the streamer. It is interesting to note that the computationally expensive equation for a shell of charge gives a very similar result to the simple uniform shell of charge divided by 2 as shown in figure 2-4. This fact is exploited since inmany cases it is much easier to work with with the uniform cross-section kernel than with the shell-averaged one. Figure 2-4 shows that this gives a reasonable approximation for streamers with a constant radius.

47 0.5 Uniform X-Section Shell of Charge 0.4 Averaged Shell of Charge

0.3

0.2

0.1

-0.1

-0.2

-0.3

-0.4

-0.5 -10 -8 -6 -4 -2 0 2 4 6 8 10

Figure 2-3: Comparison of kernel functions for different calculation methods.

0.25 Uniform X-Section / 2 Averaged Shell of Charge 0.2

0.15

0.1

0.05

-0.05

-0.1

-0.15

-0.2

-0.25 -10 -8 -6 -4 -2 0 2 4 6 8 10

Figure 2-4: Comparison of modified uniform cross-section to shell of charge.

48 2.2.3 Photoionization

Photoionization in 1.5D requires some special attention. The developments in simplifying the photoionization model of [66] discussed in section 2.1.3 were made by researchers looking to build higher dimensional models. Most existing 1.5D models simply impose a background density [41, 49]. As far as the author of this work knows, the simplifications to the model introduced in [67, 70, 6] have not been appliedto 1.5D models. One disadvantage of using the differential equation model is the need to impose boundary conditions which are not immediately clear (particularly in reduced dimensions). This is explored in depth in [6]. Therefore, it is desirable to adapt a simplified model of the absorption function for use with the integral formulation, which avoids the issues with boundary conditions and is relatively efficient in a reduced dimension model. The first simplification comes from observing that the integral in equation 2.10 in cylindrical coordinates has an analytical solution in the radial and polar directions when calculated along the streamer axis if the fit given by equation 2.12 is used. An implicit assumption here is that the quenching pressure ratio is a constant. This lets the photoionization equation be rewritten as equation 2.35. After grouping all terms that are (approximately) independent of the charged particle densities and electric field, the equation has the form of equation 2.36 where the

function 퐹 (푧 − 푧′) is defined by equation 2.37.

1 푝 휔 푔(푅) 푞 ′ ′ (2.35) 푆푝ℎ(r) = 휉 푆푖표푛(r ) 2 푑Ω 2 푝 + 푝푞 ˆΩ′ 훼 2휋푅 ∞ 1 푝푞 ′ ′ ′ 푆푝ℎ(푧) = 푆푖표푛(푧 )퐹 (푧 − 푧 )푑푧 (2.36) 2 푝 + 푝푞 ˆ−∞ ′ 2휋 푟푠 휔 푔(푅) ′ ′ ′ ′ ′ ′ (2.37) 퐹 (푧 − 푧 ) = 푓(푟 , 푧 )휉 2 푟 푑푟 푑휃 ˆ0 ˆ0 훼 2휋푅

The radial distribution function, 푓(푟′, 푧′) is defined in the same way as for the electric field. For the remainder of the derivation, it is assumed 푓(푟′, 푧′) = 1 which is consistent with the 1.5D disc method assumption since it implies a uniform ionization rate across the effective radius of the streamer. The result of the integration isgivenby

49 equation 2.38, where the characteristic length dimension has been explicitly written

out so that the constants 퐴푗 and 휆푗 maintain their dimensional definitions from table 2.2 and the partial pressure of oxygen is written in Torr1. The coordinate 푧 − 푧′ is still dimensionless, as is the function 퐹 itself. The steps to go from equation 2.37 to equation 2.38 are given in appendix A.1.

3 [︂ √ ]︂ ′ 휔 ∑︁ 퐴푗푎0 (︁ −휆 푎 푝 푟′ +(푧−푧′)2 −휆 푎 푝 |푧−푧′|)︁ 퐹 (푧 − 푧 ) = −휉 푝 푒 푗 0 푂2 푠 − 푒 푗 0 푂2 (2.38) 훼 푂2 휆 푗=1 푗

At this point, the photoionization is expressed identically to the electric field,

with the function 퐹 (푧 − 푧′) acting as the kernel function. This is already a significant simplification versus the full volume integral. Because of the very thin ionization region and exponential decay of the integrated absorption function, it can be simplified even further. The majority of ionization occurs in the tip of the streamer, in a region

of width 2∆, where ∆ is a few times the mean free path of electrons. At the same time, the photoionization source term is only significant relative to collisional ionization at

distances much greater that ∆ ahead of the streamer tip. Therefore, it can be assumed

′ that all photons are created exactly at the streamer tip, 푆푖표푛(푧 ) = 푆푇 훿(푧 − 푧푇 ),

where 훿 is the Dirac delta function. The total tip ionization, 푆푇 , is calculated by equation 2.39.

푧푇 +Δ ′ 푆푇 = 푛푒푣푒훼푑푧 (2.39) ˆ푧푇 −Δ Using this equation, the resulting photoionization model for 1.5D is equation 2.36. This equation does not have any differential or integral operators and can be calculated efficiently for any point along the streamer axis. The integral involvedin

calculating 푆푇 for this equation is evaluated once only, as opposed to the form of 1The photoionization length scale is governed by the photon penetration of the ambient gas. This is a different scaling than the electron mean free path (the true streamer characteristic length) and the streamer effective radius (the characteristic length scale in 1.5D). Equation 2.12 wasalso developed as a dimensional fit to experimental data. For these reasons, equation 2.38 hasthe characteristic length embedded in it in so many places and does not scale nicely with it.

50 Figure 2-5: Electric field magnitude with flux streamlines overlayed; adapted based on data from [2]. equation 2.36 where it must be calculated for every observation point.

1 푝푞 푆푝ℎ(푧) = 푆푇 퐹 (푧 − 푧푇 ) (2.40) 2 푝 + 푝푞

2.2.4 Radial flux

From higher dimensional streamer models, it is known that there can be large electric fields in the radial direction. This results in radial ionization waves and canresultin particles entering the streamer channel from the radial direction. In figure 2-52, the flux streamlines are shown in red on top of the electric field magnitude contours. It is seen that some of these streamlines enter the streamer in the radial direction, thus modifying the excess charge density. The effect of this radial flux is expected to be minor, but could contribute tothe radial growth of the streamer and modify the charge density in the streamer channel as well as the streamer current. It is possible to account for some of this effect using

2The author would like to acknowledge and thank A. Luque for providing the raw data used to construct this figure. The data was created using the model discussed in[2]

51 the radial component of the divergence operator in the particle continuity equations (the RHS of equation 2.13). Applying the averaging operation (equation 2.14) to the radial flux, neglecting diffusion and assuming no bulk radial velocity at r=0 givesa radial source term of equation 2.41.

2 푛˙ 푗,푟 = − 푣푟푛푗 (2.41) 푟푠 There are some practical problems with implementing this source term. The radial velocity can be estimated reasonably well by solving the electric field integral for a point on the streamer radius, but the number density at the interface is not so easy to determine and the model is very sensitive to it. In this work, the radial particle source term will be neglected, but the author has investigated it separately.

2.3 Anode mounted model

This section describes the construction of a 1.5D model originating from a spherical anode. The spherical anode was chosen as representative of a sharp tip since it concentrates the field, but still admits analytical solutions for the electric field using the mirror charge method. The model constructed here has similar geometry to those constructed by Babaeva and Naidis [56] and Luque et. al. [49]. Both of these works look at 1.5D models and how they compare to 2D models in similar geometry. The geometry is shown in figure 2-6.

2.3.1 Modifications to base 1.5D model

Boundary and initial conditions

The second order particle continuity equation for each species requires two boundary conditions to solve. The location of these depends on the situation. For negative ions and electrons, the upwind direction is ahead of the streamer so that is the natural location for boundary conditions to be applied. The conditions are no convective flux and no diffusive flux (equivalent to no density gradient) across this boundaryfar

52 Figure 2-6: Geometry for anode-mounted streamer. Subscript a is the anode, subscript s is the streamer. d is the diameter of streamer or anode. ahead of the streamer. This reflects the self-propagating nature of streamers; the seed electrons required to initiate the electron avalanche and created by the streamer itself. The positive ions move in the opposite direction, so the natural boundary condition to apply is at the anode surface. Assuming no positive ion emission, the boundary condition applied is no flux across the anode boundary. For a transient model, an initial condition is also required. A half-Gaussian seed of positive ions and electrons is placed with the maximum at the anode surface. The number density of seed ions and electrons was made equal to ensure no space charge at the beginning of the simulation but provide some electrons to start the ionization and photon production reactions. The anode was assumed to be instantly raised to high voltage and anode voltage rise-rates were not considered in this work (although they are known to affect streamer formation [57]). At one point, an additional positive space charge distribution was applied to the domain to resemble the leftover positive ions from a positive glow corona preceding the discharge. This space charge was assumed stationary throughout the discharge because of the low ion mobility, and therefore contributed only to the electric field calculation and not the particle continuity equations.

53 Electric field modifications

The anode is held at a constant voltage. Since the electric field is being solved using an integral formulation, the voltage on the anode must be imposed using a mirror charge. For simplicity, the model discussed herein will be restricted to spherical anodes since this is a situation which can physically create streamers, and admits a relatively simple solution for the mirror charge calculation. In spherical coordinates, a

spherical conducting surface of radius 푟푠푝ℎ can be kept at zero potential by cancelling a real charge located at with an imaginary "image charge" of 푟푠푝ℎ 푞 r : |r| > 푟푠푝ℎ 푞푖 = −푞 푟 푟2 located inside the sphere at position 푠푝ℎ . ri = 푟2 r The derivation of the Kernel function for the image charge in the presence of an anode is not so straightforward due to complications arising from the integrations. It is preferable to integrate over the same z domain as the real charge, since trying to integrate over a domain inside the spherical anode (the actual location of the image charge) introduces additional complexity. The mirror charge Kernel can be written in a form that is integrated over the same domain as the real charge as shown in appendix A.2. The result (for a uniform charge distribution across the channel) is 푟2 equation 2.42 where 푠푝ℎ ′. 푢 = 푧 − 푧

[︃ (︃ )︃]︃ (︁ )︁ (︁ )︁3 1 푟푠푝ℎ √︀ 2 2 푟푠푝ℎ 푢 퐾푖(푧) = − 푢 + 푟 − |푢| + − 푠푖푔푛(푢) (2.42) 2 푠 √︀ 2 2 2 푧 푧 푟푠 + 푢

2.3.2 Numerical methods considerations

Spatial derivatives and adaptive mesh refinement

The spatial derivatives in equation 2.15 are solved using a second order finite volume method. In both cases, the convective flux on the cell boundaries is solved usinga linear upwind difference scheme and the diffusive flux by a central difference scheme, both of which are second order accurate. The spatial grid is non-uniform, with smaller volumes placed in the vicinity of the streamer head. This required an adaptive mesh refinement scheme. In one dimension, this is relatively simple to implement. The

54 Figure 2-7: Region of grid refinement; electric field of streamer shown for reference

streamer tip position is determined by the location of maximum electric field and is a parameter already calculated at each step. Because of the known streamer structure, the region in which the mesh should be refined could be determined in reference to this tip location. The domain was refined for a distance of 5 radii ahead of the streamer tip and 3 radii behind it. Additional refinement was performed near the anode surface where there were also steep gradients. The region of refinement is shown in figure 2-7.

Temporal derivatives and stability

In the transient streamer, the temporal derivative is solved using a fourth order explicit Runge-Kutta scheme. The time scale for streamers is defined by the electron velocity. Since the flux term in the electron continuity equation is convection- dominated, the maximum time step can be estimated by using the Courant-Freidrichs- Lewy (CFL) condition for a pure convective flow. The CFL stability condition is given

by equation 2.43 where 퐶 is called the Courant number.

푢∆푡 퐶 = < 퐶 (2.43) ∆푧 푚푎푥

For an explicit 4th order Runge-Kutta scheme, 퐶푚푎푥 ≈ 2.78 for a stable solution [72]. However, the equations of this work have a large source term and non-constant velocity; this means that the simple CFL condition for a pure linear convective scheme

55 is inadequate. It has been found that to maintain stability after including sources, the time step must be such to keep a Courant number of about 0.25 at the tip of the streamer based on the convective electron flux.

2.4 Quasi-steady streamer

2.4.1 Modifications to base 1.5D model

Particle conservation equations

Suppose there exists a reference frame in which the streamer can be considered at

rest. Define this frame to be moving at a speed of 푉 relative to a stationary (lab) reference frame. The continuity equation written in this frame is then equation 2.44. In this frame, the streamer is quasi-steady, meaning that the structure is not changing on the time scales of relevance to the particle conservation equations. This allows for the time-derivative term to be dropped, recasting the PDE as an ODE with the axial coordinate the only dimension.

푑 (︂ 푑푛 )︂ 푛 (푣 − 푉 ) − 퐷 푗 = 푆˙ (2.44) 푑푧 푗 푗,푧 푗 푑푧 푗 A slightly different way of looking at this formulation that will be relevant later is found by rewriting the transient term by equation 2.45.

휕푛 휕푛 휕푧 푒 = 푒 (2.45) 휕푡 휕푧 휕푡 Substituting equation 2.45 into equation 2.15 and comparing with equation 2.44, the following equality can be identified: 휕푧 . Writing the equation 푉 = − 휕푡 = 푐표푛푠푡푎푛푡 this way highlights that the quasi-steady velocity is the velocity the reference frame is moving at. It also highlights that this velocity must be the same everywhere in the domain. This condition of the velocity being the same everywhere in domain is used in section 3.2 when investigating the validity of the steady state assumption.

56 Boundary and initial conditions

The quasi-steady streamer introduces an additional velocity to the convective flux term of the particle continuity equations (equation 2.44). This term is negative for all three particle species. For the electrons and negative ions, the upstream direction is unchanged since the dominant flux direction still dictates they be upwinded from ahead of the streamer. For the positive ions, the flux due to the moving reference frame is significantly larger than the convective flux due to the electric fieldand ends up reversing the upwind direction of the positive ions. As a result, all three particle species require boundary conditions to be imposed on the edge of the domain ahead of the streamer. The seed particles in the streamer are assumed to be created entirely by the streamer itself via photoionization, so far ahead of the streamer a no-flux boundary condition should be imposed. This boundary needs to beplaced far enough ahead that there is sufficient space for photoionization to produce an appropriate background electron density prior to the significant electric field increase (see section 3.1.2).

Electric field modifications

For the quasi-steady streamer, a semi-infinte domain is required. The streamer cannot be mounted to an anode as in the transient case as this would imply the anode moves with the streamer. Instead, the integration in 푧′ must be truncated at some point. The truncation ahead of the streamer must be made far enough ahead that the charge density is effectively zero. This corresponds to point 푧2 in equation 2.47. It has been found that approximately 15 streamer radii beyond the head is sufficient for this. Behind the streamer head, the charge density must asymptote to some constant value. Assume that this has occurred at some point 푧1. This allows the charge ′ density term to be removed from the integral for 푧 < 푧1. The resulting equation is 2.48. The remaining integral with infinite bounds has an analytic solution for kernel functions 2.26 and 2.27. For the case of the kernel function given by equation 2.34, at point 푧1 the gradients in charge density should be negligible which means short

57 range interactions are also negligible and the pure shell solution (without averaging)

′ is valid. The contribution from the main numerical integration on 푧 ∈ [푧1, 푧2] and the truncated part for a kernel with the form of equation 2.26 and a typical charge density profile for a streamer is shown in figure 2-8. From this figure, anestimate of the effective length scales of interaction can be made. The truncated partof the equation has a non-negligible contribution for about 20 streamer radii from the truncation point. The domain should therefore be large enough that the streamer head is at least this far from the truncation point.

∞ ′ ′ 퐸 = 휌1퐷퐾(푧 − 푧 )푑푧 (2.46) ˆ−∞ 푧1 푧2 ∞ ′ ′ ′ ′ ′ ′ = 휌1퐷퐾(푧 − 푧 )푑푧 + 휌1퐷퐾(푧 − 푧 )푑푧 + 휌1퐷퐾(푧 − 푧 )푑푧 (2.47) ˆ−∞ ˆ푧1 ˆ푧2 푧1 푧2 ′ ′ ′ ′ = 휌1퐷(푧1) 퐾(푧 − 푧 )푑푧 + 휌1퐷퐾(푧 − 푧 )푑푧 (2.48) ˆ−∞ ˆ푧1

2.4.2 Solution scheme

Numerical derivatives

In the steady state the particle densities can be solved simultaneously for a given electric field. Since the flux direction is known, a second order linear systemcan be constructed. A linear upwind difference scheme is used for the spatial differences with the upwind direction ahead of the streamer. A non-uniform mesh with grid refinement near the head of the streamer is used. Since the head is (approximately) stationary, no adaptive refinement is necessary. While is is possible to solve all three particle equations simultaneously via equation 2.49, the sparsity pattern of the matrix (see figure 2-9), which is easily converted to lower triangular, makes it moreefficient to solve sequentially by first calculating the electron densities and then the ions.An analysis of the grid size required for accurate solutions is given in appendix B.2.

58 Figure 2-8: Effect of truncating electric field

푀n = S (2.49) ⎡ ⎤ ne ⎢ ⎥ ⎢ ⎥ (2.50) n = ⎢n+⎥ ⎣ ⎦ n−

Velocity calculation

Looking at equation 2.44, for a given electric field it is possible to solve the particle densities for any value of the velocity. For the model to be useful, it should be able to calculate the velocity self consistently. Consider equation 2.44 written for electrons in terms of linear operator 휕푧 representing the differential operator, equation 2.51.

59 Figure 2-9: Sparsity pattern of matrix

1 푝푞 ′ 휕푧(푛푒(푣푒,푧 − 푉 ) − 퐷푒휕푧푛푒) = |푣푒,푧|(훼 − 휂)푛푒 + 퐹 (푧 − 푧푇 ) 푛푒|푣푒|훼푑푧 (2.51) 2 푝 + 푝푞 ˆ

To simply the following algebra, the coefficient on the photoionization term will

be lumped into a single parameter 퐵 = 1 푝푞 퐹 (푧 − 푧 ) which varies with z. The 2 푝+푝푞 푇 ionization and attachment terms, |푣푒,푧|훼 and |푣푒,푧|휂, will be written as ionization and attachment frequencies, 휈푖 and 휈푎 respectively. With these substitutions, the result is equation 2.52

−푉 휕 푛 = (휈 − 휈 )푛 + 퐵 휈 푛 푑푧′ − 휕 (푣 푛 ) + 휕 (퐷 휕 푛 ) (2.52) 푧 푒 푖 푎 푒 ˆ 푖 푒 푧 푒 푒 푧 푒 푧 푒 Applying the product rule for the differential operator gives equation 2.53

−푉 휕 푛 = 퐵 휈 푛 푑푧′ + [︀(휈 − 휈 ) − (휕 푣 ) − 푣 휕 + (휕 퐷 )휕 + 퐷 휕2]︀ 푛 (2.53) 푧 푒 ˆ 푖 푒 푖 푎 푧 푒 푒 푧 푧 푒 푧 푒 푧 푒

It is not immediately clear how to deal with the integral, but it turns out that it can be written as a linear operator using an integral transform. A more intuitive way to see the linearity is to look at the discrete case. First notice that the integral

60 is over the entire domain so it can be re-written as an inner product. For a discrete approximation, this is the same as a dot product as shown in equation 2.54.

퐵(푧) 휈 푛 푑푧′ = 퐵(푧) ⟨휈 |푛 ⟩ = B(휈 · n ) (2.54) ˆ 푖 푒 푖 푒 푖 e

Now the dot product can be broken up by instead calculating the dyadic product of B휈푖 first, yielding equation 2.55 which is linear in 푛푒.

B(휈푖 · ne) = (B휈푖) · ne (2.55)

By a slight abuse of bra-ket notation, define the operator |퐵⟩⟨휈푖| to be a linear operator that has the property:

(|퐵⟩⟨휈 |) |휓⟩ = 퐵(푧) 휈 휓푑푧′ 푖 ˆ 푖

In the discrete case, the operator is just an outer (dyadic) product as shown in equation 2.55 which is the motivation for using the bra-ket notation. In the continuous case it is an integral transform. Now, using this notation and taking the inverse of the differential operator, equation 2.53 can be written as equation 2.56.

−1 [︀ 2]︀ (2.56) 푉 |푛푒⟩ = −휕푧 (휈푖 − 휈푎) + |퐵⟩⟨휈푖| − (휕푧푣푒) − 푣푒휕푧 + (휕푧퐷푒)휕푧 + 퐷푒휕푧 |푛푒⟩

Observe that equation 2.56 is an eigenvalue equation where V is the eigenvalue of the operator defined by all the terms on the RHS and to the leftof |푛푒⟩. This means it should be possible to calculate the velocity self-consistently within the solution to the electron continuity equation. However, equation 2.56 is very cumbersome to deal with since, when the problem is discretized, the operator becomes a large dense matrix. Knowing that such a solution should exist is still very useful since it provides a mathematical basis for the actual solution method used. It (1) confirms that a self-consistent velocity solution is contained within the steady state equation and

61 Figure 2-10: Velocity calculation; note and on the axis correspond to ′ 푆푇,푐푎푙푐 푆푇 푆푇 and 0 respectively in the text. 푆푇

(2) identifies that only the particle density solution shape (but not magnitude) can be determined from this equation. Item (2) is because |푛푒⟩ is an eigenstate of the operator, which requires some coefficient to determine its magnitude. This is dealt with later; first, a more efficient solution procedure to find the velocity is described.

Assume the electric field is known. Pick an arbitrary value for 푆푇 = ⟨휈푖|푛푒⟩ and call this 0 . Next, choose a test value for the velocity, and solve equation 2.44. 푆푇 푉푡푒푠푡 From the result, it is possible to calculate ′ . This is then compared to 푆푇 = ⟨휈푖|푛푒⟩ 푆′ 0 . The ratio 푇 is a function of velocity, and the correct velocity for the assumed 푆푇 0 푆푇 electric field is the value for which the ratio is 1. This scheme is shown in figure 2-10.

Electric field calculation

It has been shown that it is possible to self-consistently determine the speed of the reference frame for a given electric field. Now the equation for electric field mustbe coupled with the particle continuity equations to find the true steady state solution. Recall that a particle density solution and unique velocity exist for any value of the input electric field according to equation 2.56. However, the particle density solution

62 can be multiplied by an arbitrary constant since an eigenfunction gives shape only. With the scheme discussed in the preceding paragraph, the arbitrary constant is the

value of 0 . The self-consistent solution of particle densities and electric field requires 푆푇 iteratively solving both of these in sequence until convergence is reached.

Define 퐸푘 to be the electric field used to solve the particle and charge densities ( 푘 and 푘) on iteration . Now let 푘 be the charge density calculated for 0 . 푛푗 휌 푘 휌0 푆푇 = 1 Then the true charge density is 푘 푘 푘. Now look at the electric field calculation. 휌 = 푆푇 휌0 Equation 2.57 shows that the electric field calculated on iteration 푘 + 1 is linearly dependent on the value of 푘 , the total tip ionization of the previous iteration. The 푆푇 criteria for choosing 푆푇 has now been converted to a condition on the electric field. If the electric field at a specific location (the tip) is specified for iteration 푘 + 1, then the value of 푘 is chosen to ensure this is realized 푆푇

퐸푘+1(푧) = 퐾(푧 − 푧′)휌푘(푧′)푑푧′ + 퐸 (푧) (2.57) ˆ ∞ = 푆푘 퐾(푧 − 푧′)휌푘푑푧′ + 퐸 (푧) (2.58) 푇 ˆ 0 ∞

At this point, the problem of finding a self-consistent solution has been converted to one of finding the correct tip electric field for the given quasi-steady streamer.

To do this, an observation from the model is invoked. If 푘+1 ∞, where ∞ is 퐸푇 ̸= 퐸푇 퐸푇 the final correct solution, then the streamer tip location will shift location between iteration and . Thus, ∞ is defined as the tip electric field for which the 푘 푘 + 1 퐸푇 location of the tip does not drift between iterations. Qualitatively, this makes sense. If the tip field is too low, then the rate of electron production is too slowanda larger ionization region is required. Since the far ahead boundary condition is fixed, this results in the tip location moving backwards. And vice versa. If the tip field is too high, then the rate of electron production is too high and the tip moves forward to shrink the ionization region. Quantitatively, it has been observed that only the correct value of the tip electric field will allow adequate convergence of the electric field shape as measured by the relative error metric, equation 2.59.

63 ⟨퐸푘|퐸푘+1⟩ 푅푒푙. 퐸푟푟 = (2.59) |퐸푘||퐸푘+1|

Iteration scheme and convergence

One other item to note is the stiffness of the problem. This comes about primarily because of the ionization and attachment coefficients’ strong non-linear dependence on the electric field, and requires very significant under-relaxation to be applied for stability. This is particularly problematic just behind the streamer tip where there is a sharp, almost discontinuous jump in the electric field across the tip. The code was written to adaptively adjust the numerical relaxation depending on the convergence rate, but would often require as much as 99% of the solution at the previous iteration to remain stable. The overall iteration scheme is illustrated in figure 2-11. The solution is considered converged when 3 simultaneous conditions are met:

1. The relative error in the electric field shape (defined by equation 2.59) is below

10−6

2. The relative error in the velocity has decreased to below 10−3

3. The absolute error in the tip field is below 10−3

Typical solutions would require >1000 iterations to adequately converge, but due to the 1.5D nature of the model this could be achieved quite quickly, taking as little as 5min running on a single computer core. Even the more difficult cases, requiring

> 104 rarely took more than an hour on the same system.

64 Figure 2-11: Steady-state model iteration scheme

65 66 Chapter 3

Analysis of 1.5D Model

3.1 Detailed analysis of model outputs

3.1.1 Transient model

Base model

The inputs to the transient model are a streamer effective radius (푎0), an anode radius

(푅푎) and a surface electric field, which is determined by the ratio of the anode voltage to the anode radius (휑푎/푅푎). An example output for the model is shown in figure 3-1. The left two subplots show the instantaneous particle density and electric field profiles for four instances in time. The top right plot shows the evolution of the macroscopic parameters (tip field and velocity) as a function of time. The bottom right plotshows the evolution of the streamer in E푇 -V space, which is a useful representation that is discussed further in chapter 4. Some observations about the streamer devlopment

1. The discharge undergoes an initial transient period before the initial conditions give way to a typical streamer structure. This is seen in the near anode region of the top left figure, and the steep gradients of the top right figure.

2. After the initial transient, the rate of change in macroscopic parameters (tip field and velocity) becomes much more gradual. The shape of the electric field

67 150 7 20

6 15 100 5 10 4 50 5 3

0 2 0 0 10 20 30 40 50 60 0 5 10 15

8 20

6 15

4 10

2 5

0 0 0 10 20 30 40 50 60 2 3 4 5 6 7

Figure 3-1: Example output for transient model. The left two figures have been plotted at t=1.7, 4.9, 8.3 and 11.8 (non-dimensional time). Input parameters are:

푎0 = 0.15푚푚, 푅푎 = 5푎0, 휑푎/푅푎 = 4

and density profiles (bottom left figure) also adopt a self-similar shape. This suggests the quasi-steady streamer is a good approximation.

3. The electric field near the anode maintains a value of approximately 1(equal to the breakdown). This is a shielding effect of the anode similar to what is seen in a glow corona. In high fields, extra electrons are produced and quickly absorbed by the anode leaving behind excess positive charge. This charge shields the anode, reducing the field. An equilibrium is reached only when the field on the surface equals the breakdown field.

68 Parameter sweep

The model was run for a series of different initial conditions (seed density), boundary conditions (anode voltage) and geometry (anode and streamer radii). Recall that the streamer has several characteristic lengths. The effective radius is essentially a characteristic length on the electric field produced by the streamer, but ionization and photoionization both have their own characteristic length scales. Furthermore, the background electric field has a characteristic length determined by the anode size. Therefore, varying the geometry will produce different results even in non- dimensional units. The effect of varying the initial and boundary conditions while holding geometry constant is shown in figure 3-2. Note that the the electric field from the anode is described by equation 3.1. This affects the scaling of the anode voltage;

for comparison of different sized anodes it is the value 휑푎/푅푎 that is significant since this determines the electric field at the surface of the anode. The anode radius then determines the rate at which the electric field decreases. The effect of varying the anode size and boundary condition are shown in figure 3-3. When looking at the figures, the top plot shows the streamer푇 inE -V space (same as bottom right graph of figure 3-1) and the bottom two graphs show the temporal evolution ofthetip field and velocity (same as top right graph of figure 3-1). Additional comparisons are shown in appendix B.1.

(︂ )︂2 휑푎푅푎 휑푎 푅푎 (3.1) 퐸푎푛표푑푒 = 2 = 푧 푅푎 푧 A few general trends to observe. First, varying the initial conditions does have some effect, but this becomes less significant as the streamer moves away fromthe anode and is not nearly as significant as modifying the voltage (see figure 3-2). This is as expected; initial conditions should be damped out, they are only there to initiate the streamer. As seen in figure 3-2, increasing the electric field at the surface (the

ratio 휑푎/푅푎) causes the streamer to move faster and have a higher tip field. From figure 3-3, increasing the size of the anode, which essentially controls the decay rate of the electric field as seen by the streamer, causes streamers to move much faster

69 Figure 3-2: Transient model output varying initial conditions. 푎0=0.25mm, 푅푎=5푎0 Colours refer to 휑푎/푅푎: red=2.5, green=3, black=3.5, blue=4. Linestyle refers to maximum of Gaussian initial seed. solid line=2, dashed line (- - -) = 20, broken dash (− · −·) = 50, dotted (···)=100

70 Figure 3-3: Transient model output varying streamer radius and anode size with

fixed initial seed 푛0=20 and surface field 휑푎/푅푎 = 3. Colours refer to 푅푎, measured in streamer radii: green=10, black=7.5, blue=5, magenta=2.5. Linestyle refers to streamer radius: solid line=0.15mm, dashed line (- - -) = 0.25mm, broken dash (− · −·) = 0.35mm

71 and have higher tip fields. This is because the streamer is in a higher field region for longer and gets accelerated more. Finally, observe that all the streamers of the same effective radius, regardless of the background field, seem to follow the same curvein

E푇 -V space after their initial transient. This implies that the function relating tip field and velocity is valid for all streamers of a fixed radius. This is discussedmore in depth in chapter 4.

Including space charge

During fast voltage ramps, the ionization on an electrode surface may immediately transition into a streamer. This is approximately the situation investigated in the preceding section. Under slower voltage ramps, or if the voltage creating a steady glow is suddenly increased, then a streamer will evolve in a region containing space charge. This is similar to the situation investigated in [57]. This space charge may also have been created by a previous steamer bursts since streamer discharges have been observed to pulsate on kilohertz frequencies [73]. In either case, the majority of the leftover space charge is positive ions which can be assumed stationary on the streamer propagation time scale. For this section, a spherical distribution of positive ions is assumed. The charge density decays at a 1/r rate up to a distance of 60 streamer radii (the edge of the domain). In reality the decay rate may be more like 1/r2; but for the purposes of the qualitative observations that will be made here the 1/r is sufficient. An example of the effects is shown in figure 3-4. Looking at figure 3-4, particularly the bottom two plots, a general trend canbe identified. As the background charge density increases, the streamer slows downand its tip field increases. Interestingly, the time it takes for the streamer to get pastthe initial transients (distance along the time axis to the peak in the bottom two plots) does not change significantly although the faster moving streamers will have moved much further in space in this amount of time. Notice also that the same common path in E푇 -V space is observed as in the case with no background density. In the bottom two plots, it would seem like all the streamers begin to converge to a common

72 value of tip field and velocity, but the domain is too short to conclude this definitively because the fast moving streamers leave the domain too soon.

3.1.2 Steady state model

The quasi-steady model takes as input an effective radius (푎0) and uniform background field퐸 ( ∞). It produces output as shown in figures 3-5 and 3-6. The figures show the same case, with the latter showing an enlargement of the streamer head region and a linear scale for the particle densities. Note that the streamer radius is the same as in figure 3-1. Based on the tip field and velocity, the streamer should be most similar to the transient streamer with its head at 푧/푎0 = 35 in the left two subplots of figure 3-1. Some observations about the steady state model:

1. The particle densities in the head region show a similar trend to the transient results, with the positive ions density increase leading the electron density increase. The excess charge density is maximized just behind these steep gradients.

2. The magnitude of the particle densities in the head region are of the same order of magnitude as in the transient model (more on this in section 3.3).

3. In figure 3-5, the exponential decay of the particles in front of the streamer can

be seen. At the very edge of the domain (푧/푎0 = 15), there is a noticeable change in slope. This slope is due to the no-flux boundary condition. However,

by 푧/푎0 = 10 the effect of this boundary condition is no longer present, meaning that the dynamics are controlled by the particle seeding from photoionization as intended.

4. In figure 3-6 it is possible to see that the gradients in number density and electric field have become very small a short distance behind the streamer head.This is a notably different behaviour than the transient case which has the anode influencing things far downstream.

73 Figure 3-4: Transient model output varying background charge density. 푎0=0.25mm, 푅푎=10푎0. Colours refer to 휑푎/푅푎: red=4,green=3.5, black=3, blue=2.5. Linestyle refers to charge density magnitude (퐶 in 휌 = 퐶/푟). solid line=0, dashed line (- - -) = 1, broken dash (− · −·) = 2, dotted (···)=4.

74 Electrons Positive Ions Negative Ions Net Charge 100

10-5 -30 -25 -20 -15 -10 -5 0 5 10 15

0 -30 -25 -20 -15 -10 -5 0 5 10 15

Figure 3-5: Example Output for Quasi-Steady Model. Model inputs: 푎0=0.15mm, 퐸∞=0.14

5. In figure 3-6 it is apparent that the steep gradient behind the streamer tipis well resolved; recall from figure 2-7 that there are 400 grid points evenly spaced between -1 and 1.

3.2 Applicability of quasi-steady model

At this point, a quasi-steady model of a streamer has been proposed and constructed, but it is unclear whether such a model is physically representative of actual streamer dynamics. Naidis [43] and Samusenko et. al. [44] assumed the existence of such a quasi-steady frame without investigating the validity of this assumption. Many experimental works record streamer velocity based on photographic evidence and simply report a mean velocity [16, 4] but experimental investigations of how streamer proper-

75 120 Electrons 100 Positive Ions Negative Ions 80 Net Charge

0 -5 -4 -3 -2 -1 0 1 2 3 4 5

Figure 3-6: Example output for quasi-steady model; same case as figure 3-5 with enlargement of head area and particle densities on linear scale

76 ties vary are scarce. Pancheshnyi et. al. [3] do provide some experimental evidence of an approximately steady streamer, although their data showed that this assumption breaks down at lower pressures. Detailed streamer simulations, including all six of those presented in [5], indicate streamer properties and velocity vary with time. The key question to be answered as to the applicability of a steady state model is: What is the rate of macroscopic property variation relative to the characteristic ionization time scale?. If the macroscopic properties are changing slowly relative to this time scale, than an instantaneous quasi-steady state is a valid assumption. However, if the macroscopic properties change on a time scale comparable to the particle dynamics time scale, it is not a valid assumption. In this section, the applicability of the steady state model will be investigated by comparing the 1.5D transient and 1.5D models.

3.2.1 Existence of a quasi-steady frame

As noted in section 2.4.1 a steady state frame exists if and only if every point in the domain is moving at the same velocity. This condition is clearly violated near the anode, which is why the model is only quasi-steady. However, if it is assumed that the streamer behaviour is dominated by the region just in front of and including the streamer tip (consistent with avalanche models such as [37, 43]), then it is sufficient to only look at this region. Start with equation 2.44 and rearrange in a way that isolates velocity to get equation 3.2

(︂푑푛 )︂−1 [︂ 푑 (︂ 푑푛 )︂ ]︂ 푉 = 푒 푛 푣 − 퐷 푒 − 푛 |푣 |(훼 − 휂) − 푆˙ (3.2) 푑푧 푑푧 푒 푒 푒 푑푧 푒 푒 푝ℎ The transient model results allow equation 3.2 to be evaluated at every point in the domain at an instant in time. This calculation was performed at a moment in time for the transient model as shown in figure 3-7. There are two key observations from this figure: (1) the result of equation 3.2 is approximately the same everywhere in the high-field region and (2) the result of this calculation is in excellent agreement with the true velocity of the streamer model. Note that the oscillations in the quasi- steady velocity are in part due to calculating the derivatives numerically and are most

77 Figure 3-7: Comparison of quasi-steady velocity (calculated by equation 3.2) to transient velocity (measured based on tip movement between time steps) . severe in regions where the denominator in equation 3.2 is small. (1) and (2) imply that a quasi-steady assumption is valid at this moment in time. Another question that must be asked is when does the quasi-steady assumption apply. It was noted previously that the transients in the anode-mounted streamer model are most significant when the streamer is near the anode and evolving from the initial conditions. Only after some amount of time, when the initial conditions have ceased to matter, is the quasi-steady assumption valid. This analysis is shown in figure 3-8. Equation 3.2 was solved for the high field region ahead of thestreamer and the average value recorded for each time step. Figure 3-8 indicates that the velocity from equation 3.2 matches the velocity calculated based on tip movement quite well after the initial transient, indicating that the quasi-steady assumption is a good one, and that the initial transient is significant only for about 10 streamer radii of propagation (although reasonable agreement can be found after as few as 5 radii).

78 Figure 3-8: Comparison of velocity calculated by equation 3.2 (dashed lines) to true velocity (solid lines) for two cases. (A) 푎0 = 0.35푚푚, 푅푎 = 5푎0 (B)푎0 = 0.25푚푚, 푅푎 = 10푎0

79 3.2.2 Comparison of quasi-steady and transient solutions

Having validated the existence of a quasi-steady reference frame, it is necessary to check that the steady state solution resembles the instantaneous transient solution. To do this, the transient model on an anode was run over a domain extending 60 streamer radii from the surface of the anode. The steady state model was then run for the same radius streamer and a variety of background fields between 0.1 and 0.2 in non-dimensional units (which is a typical range for transient model channel fields). The transient model and steady state model were compared at the point where the tip field of the transient streamer matched that of the steady state model. Theresults are shown in figure 3-9. In each of the four cases shown, the top graph is theelectric field profile and the bottom graph is the charge density profile. Observations from this figure are:

1. The shape of the electric field and density near the head are nearly identical in both the transient and steady state streamer models; the two profiles overlap so well that it is difficult to distinguish the two.

2. Far ahead of the streamer tip, after the electric field has dropped below the breakdown field, the electric field of the transient and steady state modelsdi- verge. This is because the steady state model has a constant background field imposed while the transient streamer is in a Laplacian field from the anode. This is not a concern because the deviation occurs outside of the high-field region, and therefore is not going to significantly effect the results.

3. Behind the streamer head, the electric field shows good agreement between the transient and steady state models until the anode is approached. This indicates that the anode effects are well shielded, and the electric field profileis controlled primarily by the space charge in the streamer itself. It also supports the assertion that the uniform background field of the quasi-steady model can be associated with the channel field of the transient model.

4. The charge density between the steady state and transient models begins to

80 deviate in the streamer channel. This is thought to be caused in part by the anode effects and in part by immobile ions, which cannot relax as theydo on the infinite time scale of the quasi-steady model. These effects resultina different charge density profiles, and are partly responsible for the longtime- scale evolution of the streamer macroscopic parameters (discussed further in section 3.4.2).

Overall the agreement in electric field and charge density profiles is excellent in the region near the streamer head. Since this is the region that governs most streamer dynamics, it can be concluded that the quasi-steady streamer is a valid representation of the instantaneous state of the streamer when the two are compared based on matching tip electric field and effective radius.

3.3 Comparison to published data

There is a large body of literature available for comparing streamer models. For this section, the following references will be used for comparison:

(A) Babaeva and Naidis [71]. This reference is chosen because it applies a similar geometry (spherical anode) to both a 2D and 1.5D model. It also includes photoionization and calculates electric fields using a mirror charge approach, and overall is probably the most similar model to the one constructed in this work.

(B) Pancheshnyi et. al. [3]. This reference is chosen because it contains both experimental data and simulations that agree well with it. Such comparisons are rare, so this is a useful source to include. The geometry is a needle to plane gap with an approximately uniform field of 7kV/cm far from the needle tip.It looks at multiple pressures, but only the near-atmospheric cases (680 Torr and above) will be considered for comparison.

(C) Bagheri et. al. [5]. This reference summarizes the results of six modern 3D/2D axisymmetric streamer codes and can be considered an overview of the current

81 (a) SS model with 퐸∞ = 0.19 (b) SS model with 퐸∞ = 0.18

(c) SS model with 퐸∞ = 0.16 (d) SS model with 퐸∞ = 0.14 Figure 3-9: Comparison of transient model (black) to steady state model at various locations (chosen based on matching tip field). For transient model, 푎0 = 0.15mm, 푅푎 = 5푎0, 휑푎/푅푎 = 4

82 state of the art. Case 2 and 3 in that work will be considered; the former imposes a uniform background density and the latter includes photoionization. The domain of the model is parallel plates with a uniform electric field of magnitude 15kV/cm applied and a Gaussian charge seed. A double headed streamer develops, although the seed is placed closer to the anode.

The model constructed in this work emphasizes streamer macroscopic parameters more so than precise details of the model, so these will be the basis of comparison. The parameters to be investigated are: electric field shape and magnitude, electron number density, velocity and current.

Electric field All three reference models and the 1.5D model of this work (both transient and steady state) show the characteristic peak at the streamer head, with a slow decay in front of the streamer and an almost discrete jump behind the tip. One of the interesting comparison comes from looking at the two reference models attached to anodes (particularly A and also B) and comparing them to the 1.5D model of this work. All show the same increase of the electric field to approximately the breakdown field very close to the anode before it decreases significantly in the channel. Inthe channel, model A and B show an electric field on the order of the so-called stability field, ≈5kV/cm. In this works’ 1.5D transient model, a similar value is observed at the beginning, although as the streamer progresses this begins to decrease. This is somewhat seen in (A), but the domain is too short to make any conclusion. From figure 12 in (C), a similar internal value is seen and it appears to be quite constant. As for tip field magnitude, all reference models except case 3 in (C) give avalueon the order of 5-6 times the breakdown field which is consistent with this works’ 1.5D transient model.

Electron number density Typical number densities for the streamer head in at-

mospheric pressure air are expected to be on the order of 1020푚−3 [14]. A number density of this order of magnitude is seen for all reference models. Taking into ac-

count the non-dimensionalization for a radius of 푎0 = 0.15푚푚, 푛0 for figure 3-1 and

83 3-5 is 1.06 × 1018푚−3 which, when multiplied by the non-dimensional number densities observed in those figures (of order 100), gives a number density of the same order as the reference models. Reference model (A) also gives information on the individual species and total charge density. By comparison with figure 3-5 (after re-dimensionalizing) these are also in good agreement.

Velocity Velocity is known to be very dependent on streamer radius [16, 43]. It is therefore not appropriate to compare velocities without also carefully accounting for streamer size. This is easier to do using a a method of comparison developed later in this work. The comparison of published velocities to the 1.5D model is done in section 4.3. To advance the result from that section, the velocity-radius relationship is in quite good agreement.

Current Both reference (A) and (B) give information on current. Current density of the transient model is shown later in this work for streamers of radius 0.25mm (see section 3.4.2). From that section, the current density is on the order of 15 for a streamer moving at 3, and on the order of 150 for a streamer moving at 15. In dimensional units and using an effective area based on the effective radius, this works out to a current of 40mA for a streamer moving at 0.4mm/ns and 400mA for a streamer moving at 2mm/ns. The latter is quite similar to the simulation results given in (A) for a streamer of radius 0.2mm (which is moving ≈50% faster and has a current ≈50% greater). In (B), experimental observations for a streamer similar to the slower case were made. That streamer had a peak current on the order of 75mA and a velocity of 0.4-0.6mm/ns. Considering the differences in the models, the difficulty in matching exact cases and the wide spread between the slow andfast cases, the current calculated by the 1.5D model in this work is in very good agreement with the reference sources. Based on the above comparisons, it appears that the 1.5D models, both transient and quasi-steady, that were developed in this work are in good agreement with published data. The intention of the 1.5D model is to be a simplified representation of a

84 streamer that captures the macroscopic parameters to some first order approximation while still being very computationally efficient. The above analysis suggests that this goal is accomplished.

3.4 Discussion

3.4.1 Relative importance of diffusion and photoionization

In section 2.1.2, it was argued based on the non-dimensional Peclet number that diffusion is negligible. In chapter 4, diffusion is completely neglected in all the formulas so it is interesting to look at the the results of the 1.5D model and see what the effect of diffusion actually is. Likewise in chapter 4 it is argued that including photoioinization is important so the effects on neglecting it should also be considered. Todo this, the analysis done to construct figure 3-7 is repeated, except this time neglecting either diffusion or photoioinization in equation 3.2. The results are shown infigure3- 10. Note that when diffusion is neglected, the only change is very near the tipof the streamer and even then it is relatively minor. This is consistent with figure 2-2 and the Peclet number analysis. When photoionization is neglected, there is minimal changes near the tip but ahead of the streamer there are significant changes because the contribution of ionizing collisions has become very small (due to decreasing 푛푒) and photoionization becomes the dominant mechanism. The effect of photoionization can also be studied by directly looking attheion- ization rate in figure 3-11. As seen in this figure, the collisional ionization rateis dominant by several orders of magnitude at the streamer tip. In the streamer channel, the attachment of electrons is the most dominant source term, and far ahead of the streamer it is the photoionization rate that is dominant. The photoionization becomes dominant approximately 1 streamer radius ahead of the streamer, although this is because the collisional ionization rate has decayed very rapidly due to decreasing electron density. In terms of total number of electrons produced in the domain, it is the collisional ionization that dominates by a lot (this is shown in section 4.1.3).

85 Figure 3-10: Same as figure 3-7, with the effect of diffusion and photoionization selectively neglected

Figure 3-11: Comparison of electron source term magnitudes

86 3.4.2 Current

By summing the continuity equations for the electrons, positive ions and negative ions multiplied by their respective charge, one obtains the current conservation equation. In the quasi-steady reference frame, this is written as equation 3.3.

휕휌 = −∇ · (j − 휌V) = 0 (3.3) 휕푡 From this equation, two terms can be identified. The first is the particle current, j. This is due to the motion of the charged particles under the influence of the electric field. It is dominated by the contribution from the electron flux. The otherterm, 휌V will be referred to as the steady-state current. This is an apparent current caused by the motion of the reference frame. If the streamer were truly quasi-steady, then the two would be equal. If they are not, that implies some net charge accumulation or depletion in the streamer. When the net charge in the streamer increases, the tip field increases and the streamer will speed up, and vice versa in the case of netcharge decreasing. Equation 3.3 provides an interesting way of analyzing the results of the transient streamer model. First off, it can be used as a check on the validity of the steady state assumption similar to equation 3.2. The steady state assumption is valid anywhere in the domain where the particle current and steady state current are equal (the velocity is directly measured from the transient model). Secondly, it can be used to determine how fast the streamer macroscopic parameters are changing. If the currents are significantly different in the streamer channel, then the charge storedin the streamer is changing quickly and the macroscopic parameters should be changing quickly. On the other hand, if the currents are similar then the stored charge is roughly constant and the macroscopic parameters should be relatively constant. An example of the current analysis is shown in figure 3-12. In figure 3-12a, the current ratio is approximately 1 everywhere except near the anode while in figure 3-12b it begins to deviate a bit a few radii behind the tip. The macroscopic parameters corresponding to the first case change slower than they do in the second case.

87 This comparison of the channel current ratio to the rate of macroscopic parameter change is highlighted better in figure 3-13. The current ratio was measured 10radii behind the streamer head. The acceleration is calculated based on the rate of change of velocity of the transient model. As seen in the figure, when the current ratio is further from unity, the magnitude of the acceleration is greater. Also note that the current ratio is less than one for a streamer with negative acceleration. This means that when the steady-state current is greater than the particle current the streamer slows down. The qualitative explanation for this is that the streamer is moving too fast to replenish its charge; therefore the net charge decrease, causing the tip field to decrease and the streamer to slow down.

88 100 Particle Current 80 Steady-State Current

0 0 10 20 30 40 50 60 70

1.5

0.5

0 0 10 20 30 40 50 60 70

(a) Slow changing properties, 휑푎/푅푎 = 3

600 Particle Current Steady-State Current 400

200

0 0 10 20 30 40 50 60 70

1.5

0.5

0 0 10 20 30 40 50 60 70

(b) fast changing properties, 휑푎/푅푎 = 5 Figure 3-12: Example current analysis applied to transient model for streamer with

푎0 =0.25mm and 푅푎 = 5푎0, streamer tip at 35

89 1.2 4 1.2 4

1.15 3 1.15 3

1.1 2 1.1 2

1.05 1 1.05 1

1 0 1 0

0.95 -1 0.95 -1

0.9 -2 0.9 -2

0.85 -3 0.85 -3

0.8 -4 0.8 -4 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20

(a) 푎0 = 0.15푚푚, 휑푎/푅푎 = 3 (b) 푎0 = 0.25푚푚, 휑푎/푅푎 = 3

1.2 4 1.2 4

1.15 3 1.15 3

1.1 2 1.1 2

1.05 1 1.05 1

1 0 1 0

0.95 -1 0.95 -1

0.9 -2 0.9 -2

0.85 -3 0.85 -3

0.8 -4 0.8 -4 2 4 6 8 10 12 14 2 3 4 5 6 7 8

1.2 4 1.2 4

1.15 3 1.15 3

1.1 2 1.1 2

1.05 1 1.05 1

1 0 1 0

0.95 -1 0.95 -1

0.9 -2 0.9 -2

0.85 -3 0.85 -3

0.8 -4 0.8 -4 2 3 4 5 6 7 2 2.5 3 3.5 4 4.5

(e) 푎0 = 0.15푚푚, 휑푎/푅푎 = 5 (f) 푎0 = 0.25푚푚, 휑푎/푅푎 = 5 Figure 3-13: Current ratio in streamer channel (10 radii behind head) and acceleration of streamer. 푅푎 = 5푎0 for all

90 Chapter 4

Macroscopic quasi-steady streamer modelling

4.1 Derivation of model

4.1.1 Electric field and characteristic dimension

The electric field emerging from the head of a streamer is assumed to haveashape given by equation 4.1. This shape was motivated by the Laplacian solution for a

constant potential sphere. 푧 and 푧푇 are non-dimensionalized in the standard way, and 푏 is a parameter describing the decay rate of the electric field with units of length. By fitting this equation to the electric field calculated in the 1.5D quasi-steady streamer

model, the value of 푎0/푏 is found to be 0.6 and very consistent (see figure 4-1 in the next section).

퐸 − 퐸 퐸 = 푇 ∞ + 퐸 (4.1) 푎0 2 ∞ ( 푏 (푧 − 푧푇 ) + 1) In [71, 43], it is identified that the electric field decays to half its maximum value over a distance of 1/4 the radiative diameter of the streamer. Using this observation, it is possible to relate the effective radius used for modelling in this work, 푎0, to a physically meaningful radius1. First observe that equation 4.1 decays to half its

91 √ 2−1 maximum value at a distance 푧 − 푧푇 = . For 푎0/푏 = 0.6, this gives a width of 푎0/푏 about 0.7 in non-dimensional length units. Comparing this to what the width of the high-field region should be, the streamer characteristic dimension can be related to the radiative radius by equation 4.2

푑 푟 푟 0.7푎 = 푟푎푑 = 푟푎푑 푎 = 푟푎푑 (4.2) 0 4 2 0 1.4

4.1.2 Radius

The dependency of the streamer radius, tip field and velocity in quasi-steady state was investigated by Naidis [43]. The derivation presented here builds off that work and incorporates additional terms that have been found to be of importance. The model starts with the continuity equation for electrons in one dimension ahead

of the streamer, equation 2.44 on 푧 ∈ [푧푇 , 푧0] where 푧0 is defined as the location where 퐸 = 1, the edge of the so-called active region. In the form below, the equation is written non-dimensionally except for the Townsend ionization and attachment coefficients, which are written in dimensional form (identified by a superscript 훼*, 휂*). In equation 4.3, diffusion is neglected based on the analysis in section 3.4.1. The equation is then integrated to given an explicit equation for the characteristic length, equation 4.4.

푑 (푛 푣 − 푛 푉 ) = 푎 |푣 |푛 (훼* − 휂*) + 푆˙ (4.3) 푑푧 푒 푒 푒 0 푒 푒 푝ℎ [︃ ]︃ [︂ 푧푇 * * ]︂−1 (︂ )︂ (︂ )︂ 푧푇 ˙ (훼 − 휂 )|푣푒| 푛푒,푇 푉 − 푣푒,푇 푆푝ℎ/푛푒 푎0 = 푑푧 ln + ln − 푑푧 ˆ푧0 푣푒 − 푉 푛푒,0 푉 + 1 ˆ푧0 푣푒 − 푉 (4.4) This equation is similar to that originally derived by Naidis, although includes an extra term for photoionization; the significance of this term is discussed below. The denominator and second term in the numerator can be solved directly using

1In the 1.5D model, the effective radius is used to describe the extent of the channel of ionized gas. Because of the form chosen for the radial distribution function in the electric field calculation (section 2.2.2), the effective radius of this work will not necessarily be the same as the electrodynamic radius of a physical streamer, despite similar definitions.

92 equation 4.1 and given a tip field and velocity. The first term in the numerator is estimated in [43] to be approximately 8. Since it is available, it is useful to use the 1.5D model to investigate the different terms in the equation 4.4. The results ofthis analysis are shown in figure 4-1 and described in detail below. Note that the steady state model runs used to generate figure 4-1 span a wide range of effective radiiand background fields. Radii are varied from 0.15mm to 0.5mm in steps of 0.05mm and the background field is varied from 0.1 to 0.2 times the breakdown field insteps of 0.01. Note that the value along the x-axis is simply the run number; the lines connecting the different points should not be interpreted as any trend and areonly included to avoid cluttering the graph with hundreds of data markers.

Top Left: This figure shows the value of the denominator of equation 4.4.The line labelled "1.5D SS Model" performs the integration using the output of the 1.5D quasi-steady model. The line labelled "Simple Model" uses the electric field given by equation 4.1 and the tip field and velocity from the steady state model to determine the integrand. Overall, the curves are pretty similar. The discrepancy is due to the electric field not being perfectly described by equation 4.1. In reality, the maximum of the electric field is a turning point, whereas equation 4.1 has some non-zero derivative

at 푧 = 푧푇 . Because of the high sensitivity of the ionization coefficient to the electric field, particularly near the tip, the simple model ends up underestimating thevalue of the integral

Top Right: This figure shows the value of the three terms in the numerator of equation 4.4 calculated based on the 1.5D quasi-steady model. Term 1 is the logarithm of the number density ratio. Naidis identifies the ratio to be on the order 103 −104 and takes a value of 8 for the logarithm [43]; the model developed in this work agrees with these numbers although falls on the higher end of the range. Term 2 is the logarithm of the the ratio of the difference of the electron drift velocity to the steady state velocity at the tip versus at the end of the active region. This term can be directly calculated given a tip field and velocity, but it is relatively unimportant compared

93 to the others. Finally, term 3 is the (magnitude) of the term containing the integral of the photoionization rate. This term is typically in the range 2 to 3 and has a negative sign in front of it, so is subtracted from term 1 and 2. In the derivation of Naidis, this term is ignored but in this work it is found to be non-negligible due to the very small number density near the end of the active region. The difference of term 1 and 3 (the terms that cannot be calculated given a tip field and velocity) has an average value of 6.9. Interestingly, this corresponds to the logarithm of the lower limit of Naidis’ estimate of the number density ratio. For the remainder of this work (excluding the rest of figure 4-1), this value will be rounded down to 6. The reason for rounding down is to compensate for the simplified model under-estimating the value of the denominator (see description of top left graph)

Bottom Left: This figure shows the result of the radius calculation. The "actual" radius is the input given to the 1.5D SS model. The line labelled "1.5D SS Model" is the radius calculated using equation 4.4 and the results of the 1.5D quasi-steady model. It is unsurprising that this line falls almost exactly on the line describing the true radius, since the steady state model is solving the same governing equation in 1D. The overlapping lines does provide a good validation that (1) equation 4.4 is formulated correctly, (2) diffusion is indeed negligible and (3) the steady state modelis converging to the correct tip field/velocity solution. The line labelled "Simple Model" solves equation 4.4 using equation 4.1 and a fixed value of 6.9 for the difference of term 1 and 3 per above. As seen from this figure, the simple model over-estimates the radius slightly but in general provides a very good estimate considering its simplicity. The overestimation is primarily due to the denominator (see top left figure).

Bottom Right: This figure shows the value of 푎0/푏 calculated by fitting equation 4.1 to the electric field determined in the 1.5D quasi-steady model. The value is remarkably consistent at 0.6.

94 104 5 10 1.5D SS Model Simple Model 8 4

6 term 1 3 term 2 4 term 3

2 2

1 0 0 50 100 0 50 100

0.6 1

0.5 0.8

0.4 0.6

0.3 0.4 Actual 0.2 1.5D SS Model 0.2 Simple Model 0.1 0 0 50 100 0 50 100

Figure 4-1: Analysis of equation 4.4. See text for description of plots

95 Figure 4-2: Control volume for charge conservation calculation

4.1.3 Background field

The model described up to this point is quite similar to that proposed by [43]. One notable shortcoming of that model is that it yields a family of solutions for a given radius. To pin down a specific streamer requires choosing an additional independent parameter. Based on the quasi-steady model, the obvious choice for this is the channel field. In the quasi-steady streamer, the channel field is the same as theimposed background field, but it is observed in finite length streamers that the internal field of the streamer drops below the background field. The argument for the channel field being an important parameter for a quasi-steady streamer is obvious from a simple charge conservation consideration. The rate of charge produced in the streamer head must equal the rate of charge advected along the streamer channel for a steady-state solution to exist. This can be expressed using Reynold’s Transport Theorem, which in this case is equivalent to integrating the charge conservation equation. The situation is shown schematically in figure 4-2 and described mathematically by equation 4.5. Note that throughout this derivation, all values (including Townsend coefficients) are dimensionless unless explicitly identified otherwise with an asterix.

푑푄 휕휌 = 푑Ω + j · ^n푑퐴 = 0 (4.5) 푑푡 ˚Ω 휕푡 ‹훿Ω With immobile ions and comparable electron and ion number densities, the current

96 is dominated by the electron current. Likewise, the charge conservation equation is approximately governed by the divergence of the electron current. This is shown in equation 4.6 and 4.7.

∑︁ j = 푞푘푛푘vk ≈ −푒푛푒ve (4.6)

휕휌 = −∇ · j ≈ 푒∇ · (푛 v ) (4.7) 휕푡 푒 e Now assume particle motions are predominantly 1D, consistent with the other models in this work, and use equation 2.44 to rewrite the RHS of equation 4.7 using the electron continuity equation to get equation 4.8.

휕휌 [︂ 휕푛 ]︂ ≈ 푒 푛 |푣 |(훼 − 휂) + 푆 + 푉 푒 (4.8) 휕푡 푒 푒 푝ℎ 휕푧 Substituting the values for current and rate of change in charge density into equation 4.5 and solving the integrals in 1 dimension gives equation 4.9, where the subscript 푐ℎ refers to the value being evaluated in the streamer channel (at position 푧1).

An implicit assumption has been made here that the integral over [푧1, 푧2] is approx-

imately the same as an integral over [푧푇 − ∆, 푧푇 + ∆]; in general this is valid since very little ionization occurs outside of the tip region.

푧2 (4.9) (푛푒(푉 − 푣푒))푐ℎ = (푛푒|푣푒|(훼 − 휂) + 푆푝ℎ) 푑푧 ˆ푧1 Now observe that part of the integral on the RHS can be rewritten using the notation developed in equation 2.39. The photoionization term is also written in its simplified representation developed in equation 2.36. The result is equation 4.10. Observe that the last term on the RHS is just the negative ion flux on the boundaries, and recall that the negative ion flux is governed by the steady state velocity, with the electric field driven drift almost negligible. This term is then moved to theLHSto give equation 4.11.

97 (︂ 푧2 )︂ 푧2 1 푝푞 (4.10) (푛푒(푉 − 푣푒))푐ℎ = 푆푇 1 + 퐹 (푧 − 푧푇 )푑푧 − 푛푒푣푒휂푑푧 2 푝 + 푝푞 ˆ푧1 ˆ푧1

(︂ 푧2 )︂ 1 푝푞 (4.11) (푛푒(푉 − 푣푒) + 푛−푉 )푐ℎ = 푆푇 1 + 퐹 (푧 − 푧푇 )푑푧 2 푝 + 푝푞 ˆ푧1 This equation can be simplified further. From the results of the steady state model, it is known that the negative ion number density in the channel is roughly 2 orders of magnitude less than the electron density, so the term 푛−푉 can be neglected. The ratio of the number of ions produced by photoionization to those produced by collisions (the integral term on the RHS) is also much less than 1 so this term can also be safely neglected. The result is equation 4.12.

(4.12) (푛푒(푉 − 푣푒))푐ℎ = 푆푇

Both equation 4.11 and equation 4.12 were applied to the quasi-steady model to verify their applicability. This was done by looking at the ratio of the LHS to the RHS. A ratio of 1 implies exact equality, meaning that the equation is perfectly satisfied. The result is shown in figure 4-3 for a series of different steady statemodel runs with the same data set used in figure 4-1. From this figure, it is clear thatthe full equation is satisfied to within << 1%; this makes sense since the only physics neglected in the full equation present in the 1.5D model is ion advection by the electric field, diffusion and collisional ionization far from the tip, all of which are small.The simplified equation is shown to be satisfied to within about 1-2%, which is consistent with neglecting terms two orders of magnitude smaller than the dominant term. This verifies that equation 4.12 is a good approximation for current conservation inthe streamer channel. For equation 4.12 to be useful to characterize streamers by macroscopic parameters, it must be reduced to a form where the only independent variables are tip field, velocity and effective radius (the same inputs as equation 4.4 plus its output). Todo this, 푆푇 and the number densities are normalized using the value at the tip of the

98 1.05

0.95

Full Equation Simple Equation 0.9 0 10 20 30 40 50 60 70 80 90

Figure 4-3: Comparison of ratio of LHS to RHS for equation 4.11 ("Full Equation") and equation 4.12 ("Simple Equation")

99 streamer since the electric field at the tip is an allowable input. As with the derivation for the effective radius, the Townsend coefficient is again written in dimensional form to allow the effective radius to be used as an input. The subscript 푇 refers to the value being evaluated at the tip, and the subscript 푐ℎ refers to the channel.

푛 푧2 푛 |푣 |훼* * 푒,푇 푒 푒 (4.13) 푉 − 푣푒,푐ℎ = |푣푒,푇 |훼푇 푎0 * 푑푧 푛푒,푐ℎ ˆ푧1 (푛푒|푣푒|훼 )푇 Now, all the terms on the RHS that are not directly known from the inputs to this equation are grouped into a single parameter, 휅, defined by equation 4.14. This parameter is evaluated using the 1.5D quasi-steady model outputs. The data set used is the same as for figures 4-1 and 4-3, which spans a wide field of radii and background fields. As shown in figure 4-4, the value is fit very well by asecondorder polynomial constrained to pass through the origin where the independent variable is the drift velocity of electrons in the streamer channel.

푧2 * 푛푒,푇 푛푒|푣푒|훼 (4.14) 휅 = * 푑푧 푛푒,푐ℎ ˆ푧1 (푛푒|푣푒|훼 )푇 At this point, the current conservation equation has been converted into a form where the channel electric field can be solved implicitly given a streamer tip field, velocity and effective radius. The equation for this is equation 4.15. Note thatthis derivation is for the positive streamer, so all electric fields in equation 4.15 are positive and the direction of the electron motion has been taken into account when changing the sign on the second term of the LHS.

* 푉 + (휇푒퐸)푐ℎ = (휇푒퐸훼 )푇 푎0휅((휇푒퐸)푐ℎ) (4.15)

4.1.4 Graphical representation

Equation 4.4 and 4.15 provide an implicit way to characterize a streamer. In the 1.5D model that has been constructed, the independent variables are the radius and background field, with velocity and tip field being outputs. As written, the equations

100 0.05

Fitted Curve: =-0.18467v2 +0.22188v +0 0.048 ch ch R2=0.9969

0.046

0.044

0.042

0.04

0.038

0.036

0.034

0.032

0.03 0.15 0.2 0.25 0.3

Figure 4-4: Evaluation of the value of 휅

101 flip the inputs and outputs. A useful way to interpret these equations is byplotting their solution on a contour map. This is shown in figure 4-5. Using this figure, given any two macroscopic parameters (velocity, tip field, background field or radius), the other two can be determined.

4.2 Comparison with 1.5D model

4.2.1 Steady state

The macroscopic parameter model was constructed by a combination of directly solving the governing equations, and relying on trends discerned from the 1.5D steady state model. It is now important to verify that the simplified macroscopic parameter model gives good agreement with the steady state model. The steady state model was run for a matrix of input parameters. The effective radius was varied from 0.15mm to 0.5mm in steps of 0.05mm. The background field was varied from 0.1 to 0.2 in steps of 0.01 (non-dimensional field units). This data set is the same group of data usedto construct figures 4-1, 4-3 and 4-4. The range spans the domain in which streamers in atmospheric pressure air commonly occur. Note that the background field range also includes the so called "stability field" in air, which is generally taken to bearound 5kV/cm [10, 15, 40] or 0.17 in the non-dimensional units of this work. While there is some doubt as to the physical implications of this value, it is commonly quoted in the literature and many detailed streamer codes have internal fields close to this value. Note that a comparison to other published results is given later in this work in section 4.3. Figure 4-5 with the steady state model results overlayed is shown in figure 4-6. Overall, the trends between the 1.5D model and the simplified model are consistent. The radius and background field contours have a similar shape for both the 1.5D model and the simple model. Looking first at the radius, the shape of the curves very closely matches the 1.5D data and the magnitude is in very good agreement. As for the background field, the general trend of increasing from bottom left to topright

102 20

18 0.9 0.8 0.7 0.6 0.5 16 0.45 0.4 0.35 0.3 0.24

0.25 14

0.22 12

0.2 0.2 10

0.18 8

0.16 0.15 6 0.14

0.12 4 0.1 0.1 0.08 2 0.06

0.05 0 4 4.5 5 5.5 6 6.5 7

Figure 4-5: Graphical representation of the solution of equation 4.4 and 4.15. Solid lines are constant effective radius (mm), dashed lines are constant background field (normalized by breakdown field)

103 is consistent between the models. The slope of the contours in the simplified model does not quite match the slope of a curve through the 1.5D data, but on close inspection this is found to be mostly due to the sensitivity of equation 4.15 to the radius. To illustrate this, consider the points corresponding to a radius of 0.25mm and 0.3mm. In general, when these points lie directly on top of the corresponding simplified model radius contour, the background field is very well estimated. For specific examples see the points at 0.25mm radius, 0.18 background field and0.3mm radius, 0.14 background field. In light of this observation, the difference in slopes is not too concerning. In general the background field is estimated to an absolute accuracy of about 0.02 in non-dimensional units, or about 20% relative accuracy in the worst case, which is pretty reasonable considering the simplicity of the model. Furthermore, when solving the radius contours, a constant background field must be imposed when calculating the electric field. For the purposes of the simple graphical representation, it is not necessary to iterate this value to find a converged solution since changing the background field for the radius calculation has only a minor effect. However, a minor change in the radius contour does cause a noticeable change in the intersection with the background field.

4.2.2 Transient

In section 4.2.1, it was shown that the macroscopic parameter model is a relatively accurate approximation of the quasi-steady 1.5D simulation. Also, in section 3.2 it was shown that the quasi-steady model is valid when compared to the fully transient model. This leads one to expect that the macroscopic parameter map will also be applicable to understanding the transient model. It was also shown in section 3.1.1

that the E푇 -V relation follows contours that look very similar to those seen in figure 4-

5, so one might expect the plot of the transient model results in E푇 -V space to follow the contours of constant radius in figure 4-5, crossing the contours of varying channel field as the streamer moves away from the anode and the background field decreases. A plot of six transient model runs is overlayed on figure 4-5 in figure 4-7. These

runs are the same cases used in figure 3-13, specifically 푎0 ∈ {0.15푚푚, 0.25푚푚} and

104 20

18 0.9 0.8 0.7 0.6 0.5 16 0.45 0.4 0.35 0.3 0.24

0.25 14

0.22 12

0.2 0.2 10

0.18 8

0.16 0.15 6 0.14

0.12 4 0.1 0.1 0.08 2 0.06

0.05 0 4 4.5 5 5.5 6 6.5 7

Figure 4-6: Figure 4-5 with results of steady state model overlayed ×. Steady state model results are for effective radius 0.15mm to 0.5mm increasing from bottom right

to top left in steps of 0.05mm and backgroud field increasing from 0.1E0 to 0.2E0 in steps of 0.01E0 from bottom left to top right

105 휑푎/푅푎 ∈ {3, 4, 5} with a fixed anode radius of 푅푎 = 5푎0 and initial density of 20. From figure 4-7 it is clear that transient model is well captured by the macroscopic parameter formulation. The model starts with an initial transient, but rapidly reaches the E푇 -V relationship expected based on its radius. At this point, it starts to slowly move along the curve of constant radius but at a much slower rate than the initial transient. The rate of motion along the curve determines the validity of the quasi- steady assumption. Faster motion means the quasi-steady assumption is less valid. The rate of this motion along the curve is closely related to the analysis given in section 3.4.2.

4.2.3 Comparison of all 3 models

To highlight the convergence of all three models discussed in this work (1.5D transient, 1.5D quasi-steady and macroscopic parameter) a side-by-side comparison is useful. This is done by comparing figures 3-9 and 4-7 simultaneously as shown in figure 4- 8. The transient and steady state models of the same effective radius are compared at the instant where their tip-fields are equal. The four images in figure 4-8show that all three models are in very good agreement. This means that the macroscopic parameter model is a good representation of the 1.5D quasi-steady model and that the quasi-steady model is a good representation of the instantaneous state of the transient model. Hence, the macroscopic parameter model is a good representation of the instantaneous state of the streamer.

4.3 Comparison with published data

4.3.1 Data sources

The macroscopic parameter model is to be compared with other reference data to verify its applicability in describing not just the 1.5D model developed in this work, but a variety of experimental data and data from more detailed simulations. Note that in experiments, it is often possible to measure speed and radius through luminosity

106 20

18 0.9 0.8 0.7 0.6 0.5 16 0.45 0.4 0.35 0.3 0.25 0.2

12 0.24

0.15

0.22 10

0.2 8 0.18

0.1 6 0.16

0.14 4 0.12 0.1 0.08 2 0.06 0.05

0 4 4.5 5 5.5 6 6.5 7 7.5 8

Figure 4-7: Figure 4-6 with results of transient model overlayed. Markers are placed every 0.1 non-dimensional time along the curve. Colours correspond to different transient model runs as described in text. × markers are steady state model results

107 (a) Comparison with SS model at 퐸∞ = 0.19

(b) Comparison with SS model at 퐸∞ = 0.18

108 (c) Comparison with SS model at 퐸∞ = 0.16

(d) Comparison with SS model at 퐸∞ = 0.14 Figure 4-8: Comparison of transient model (black line on all graphs) to steady state model at various locations. For transient model, 푎0 = 0.15, 푅푎 = 5, 휑푎/푅푎 = 4. The coloured markers on the graph on the left correspond to the steady-state model results shown on the right. Comparison is made at the point where the tip field of the transient and steady state models are equal for equal radius streamers.

109 based techniques but not the electric fields of individual streamers. Recall that inthis work, an effective radius was used, which was estimated to be 1/1.4 times the radiative radius. The radius measured in the experimental work is close to the radiative radius by nature of the measurement technique and the one measured in the simulation work is typically the electrodynamic radius. The ratio of the electrodynamic radius to the radiative radius is generally taken to be 2 based on [3], although from figure 12 in that same work it can be seen that this rule may not apply for smaller streamers. The reference data and how they are interpreted is listed below.

1. Pancheshnyi et. al. (2005) [3]. This work contains experimental measurements of streamer radius, diameter and current in a point-to-plane discharge gap at various pressures. These experiments are compared with a simulation that showed good agreement with the velocity and current, as well as radius at higher pressures. Because of this agreement, it is assumed that the electric fields determined by the model are representative of the experimental results. The experimental data for P=680, 710 and 740 Torr are used for (radiative) radius and velocity, with the tip field taken from simulation. The tip field in all cases is about 600Td, which was converted to kV/cm using the pressure in each case. Note that this work gives data for many lower pressures, but since the model constructed in this work has been particularized for atmospheric pressure air it does not make sense to compare to the other cases without a rigorous consideration of pressure scaling.

2. Luque et. al. (2008) [2]. This work contains simulations of streamers in a needle plane geometry. The authors provide a matrix of results making it very easy to get the tip field-velocity-radius data. A variety of points were sampled from the data for a streamer propagating in a 7mm gap with a 10.5kV and a 14kV voltage applied.

3. Veldhuizen et. al. (2002) [4]. This work focused on experimental observations and also used a point-to-plane gap with separation 20mm and open to ambient air. The authors record a range of velocities between 0.14mm/ns and 0.3mm/ns

110 when 12.5kV was applied and up to 1.2mm/ns when 25kV was applied. Based

on images, they measure a (radiative) diameter of 0.16±0.03mm. For the tip field, the authors use an interesting estimate based on the inflection pointofthe curve of Townsend ionization coefficient versus reduced electric field and givea value of 520Td which they convert to 170kV/cm2.

4. Bagheri et. al. (2018) [5]. This work contained a comparison of several different streamer codes. Data from the CWI group produced for the case 3 (photoionization included) comparison was sampled at various points. This model was originally described in [51] and was discussed briefly in the introduction because of its use of AMR and parallel processing. The data for this model is for the positive half of a double-headed streamer in air. Two things to note here: (1) [5] compared multiple different codes with reasonably good agreement, sothe CWI data can be considered representative of the output of many contemporary codes and (2) the electric fields in case 3 (which included photoionization) were significantly lower than in cases with constant background particle densities and most other publications.

4.3.2 Comparison and discussion

The reference data discussed in the preceding section are plotted on top of the macroscopic parameter solution in figure 4-9. To aid in comparison, it was easiest towork in dimensional units. Note that the radius listed in this figure is the radiative radius which is equal to 1.4푎0, or half the electrodynamic radius as appropriate. Overall, the tip field-velocity-radius dependence predicted by the macroscopic parameter model is in very good agreement with a wide range of published data. This is significant considering that both experimental results and simulations were considered. It is particularly interesting to see that, with a couple exceptions, the radius is in very good agreement despite the different definitions used and the fact that the simplified model uses an effective radius. The exceptions are the CWIgroup

2There may be a error in the original work, as 520Td at typical ambient conditions is closer to 130kV/cm. The 170kV/cm value will be used in this work.

111 2.5

0.9 0.8 0.7 0.6 0.5 0.45 0.4 0.35 0.3 2 0.38 0.25 0.36

0.33 7 1.5 0.2 0.28 6.4 0.08 0.23 5.8

1 5.2 0.15 4.7 0.24 0.22 0.2 0.21 4.1 0.18 0.19 0.17 0.25 0.17 3.5 0.5 0.15 0.1 0.25 0.23 0.13 2.9 0.08 2.3 1.7 0.08

0 120 140 160 180 200 220

Figure 4-9: Macroscopic parameter solution with published data. Solid contours

are radiative radius in mm (푟푟푎푑 = 1.4푎0), dashed contours are background field in 푘푉/푐푚. Published data labels are radiative radius. Sources: ∙ Pancheshnyi et. al. [3], Luque et. al. [2], Veldhuizen et. al. [4], N Bagheri et. al. CWI group [5]

112 data from [5], which has already been identified to have unusually low tip fields, and the data from [4], which has an uncertain tip field and applies this field to awide range of streamers. It is also encouraging to see that this radius agreement spans a range of tip fields and velocity, which indicates the simplified model is appropriate throughout the range atmospheric pressure streamers exist in. Recall also that much of the published data was captured for needle-plane geometries while the macroscopic parameter model is based on a steady state assumption. It was shown in section 4.2.2 that the macroscopic parameter model (a quasi-steady formulation) was in pretty good agreement with the transient 1.5D model created in this work. Figure 4-9 shows that the macroscopic parameter model also agrees with more detailed transient simulations, which further supports the hypothesis that quasi-steady streamers are a good approximation. One aspect of the macroscopic parameter model that is difficult to verify is the streamer channel field. First off, this value is impossible to measure in the experimental results; it is typically assumed to simply be the stability field, ≈5kV/cm (this is the assumption made in [4]). Secondly, this value varies inside the streamer and the simulation data often does not report it. For the cases where it can be estimated (for example, from figure 13 in [3]), a value of 5kV/cm does seem reasonable. Theprob- lem with using this value for direct comparison with the quasi-steady model is the short domain all the reference data is simulated on. Nevertheless, taking as a typical value for the internal field of 5kV/cm, it would appear the macroscopic parameter underestimates this parameter in most cases. There are a couple possible explanations for this. The first is the short simulation domain of the reference model; from the 1.5D model constructed in this work, it seems that the internal field decreases as the streamer moves away from the anode (see figure 3-9). Once the streamer is further away from the anode the quasi-steady approximation becomes better. Sec- ond, as it was discussed in section 4.2.1 the background field calculation is extremely sensitive to the radius calculation. Small changes to the radius calculation can move the channel field contours significantly. Finally, and perhaps most likely, is thefact that the model is only quasi-steady. This is somewhat related to the first point. The

113 model essentially assumes an infinite uniform background field in which the streamer can propagate forever. In a decreasing background field, the streamer typically slows down because more charge is removed via the streamer channel than is created in the head. This requires a larger channel field than would be present in a a steady streamer with the same tip field and velocity (see discussion of current in section 3.4.2) The last thing that is important to emphasize is the model simplicity. The steady state model used to produce data points in figure 4-6 is 1.5D and is able to rapidly look at many different cases (to run all 88 cases took about 12 hours on a single computer core). Creation of the macroscopic parameter contour map itself takes a matter of seconds and spans an even larger parameter space. Contrast this with the reference data used. These used complicated experimental setups or detailed simulations in multiple dimensions. Despite this large difference in complexity, the simple models presented in this work actually show very good agreement with both experiments and detailed simulations.

114 Chapter 5

Model extensions

5.1 Macroscopic propagation

The motivation for constructing the simplified models in this work was the possibility of using them to model the entire streamer corona. Initial efforts were made in doing this but a working model has not been realized at the current time; this section will discuss the progress made, difficulties encountered, and potential paths forward.

Model philosophy

Before attempting to incorporate branching or multi-streamer interactions, it is first necessary to develop a model of a single streamer that can be moved forward in time based on its macroscopic parameters. This is done by first initiating a streamer using the anode-mounted streamer model and then moving this streamer forward based on the quasi-steady model. The quasi-steady model is known to be inadequate very near the anode, which is why the transient model must be used to initiate it. With the quasi-steady model results and the macroscopic parameter model, it is possible to specify a unique velocity for any combination of tip field and radius. This velocity can then be used to propagate the streamer forward for some finite time step on the order of the time scale of variation of macroscopic parameters. This time scale is significantly longer than the time scale of particle motion. The results of thequasi- steady model can then be used to determine particle densities and electric fields at

115 the new tip location, which would allow the streamer to be initialized for the next time step. Looking forward to ensembles of multiple streamers; their interaction will be governed by a few effects. Most significant is the electric fields; the strong fields created at the tip of one streamer will influence the behaviour of another streamer. These effects include attraction/repulsion (depending on the orientation) and influencing the ionization rate. For this reason, it is very important that a macroscopic model of many streamers accurately captures the electric field. Another mechanism of interaction occurs when streamers branch. This results in a common stem of ionized gas. This is a different situation than a streamer mounted on the anode and will require appropriate modifications to the boundary conditions of the models discussed inthis work. A final possible mechanism of interaction is photoionization, since that isa non-local effect. As noted previously, many authors hypothesize that stochastic photoelectron production may play a role in branching and this effect will be increased the more streamers are present. It will be noted that considering the effects of photoionization in a large 3D volume is a substantial undertaking, so it in general would be preferable to use stochastic methods rather than consider the photonic interactions of many streamers.

Preliminary implementation

Preliminary attempts have been made at realizing the first stage of the above model: a single streamer propagating purely based off its macroscopic parameters. This model has shown some successes and some areas for improvement. One of the successful areas was in propagating the streamer using the macroscopic model over short time steps. First, the streamer was initialized using the full transient model. Next, it was propagated forward for 0.05 in non-dimensional time which required approximately 100 time steps (based on the CFL stability condition). Then, the same initial conditions were propagated forward using a model based on the macroscopic properties and a quasi-steady assumption. This was done over the same time interval, but in a single step. The results were then compared as shown in figure 5-1. Three curves are

116 Figure 5-1: Comparison of macroscopic propagation to full transient model. Arrows point to the initial condition; solid shifted lines are result of full transient model (≈100 time steps), dashed shifted lines are result of macroscopic propagation model (single time step) shown on the figure: the initial condition (labelled lines), the result of propagating the streamer forward using the full transient model (solid lines shifted to right) and the macroscopic propagation model result (dashed lines). From this figure, it appears that the macroscopic propagation model is able to replicate the full transient solution quite well. The discrepancy between the solid and dashed lines is very minor. This suggests that over the time interval considered, the streamer really does move forward as a self-similar, quasi-steady structure. As the time interval was increased, the models began to diverge. This indicates that the characteristic time for macroscopic parameter variation in this case is about 0.05. As for the areas of the model requiring improvement, the first is the anode connection. The anode is inherently stationary thus the quasi-steady assumption leading to the macroscopic propagation model does not apply. This necessitates some transition region between the quasi-steady region and the stationary region. This transition region is very difficult to capture. In figure 5-1, it was placed at the pointwhere the electric field had zero gradient since this is the boundary condition imposedin the quasi-steady model. While this gave some promising results, the non-uniform background field created by the anode caused problems. The other major concern is the stiffness of the particle continuity equations arising from the highly non-linear source terms. Slight changes in the electric field cause huge differences in the solution

117 to the particle continuity equations. This is why the quasi-steady model had to be so heavily relaxed (see section 2.4.1). In figure 5-1, while the electric fields from the macroscopic model and full transient model may look indistinguishable at first glance, closer inspection reveals some minor differences which get amplified on the next time step, causing model instability.

Next steps

The model discussed in the preceding section attempted to match single streamer propagation using the quasi-steady model. It had some success, particularly in showing that the quasi-steady propagation is valid on short time scales, consistent with the analysis of section 3.2. The problems it encountered had to do with trying to match details of the electric field and particle densities too precisely. Going forward, itis proposed to stop trying to match these precisely and move directly on to a streamer tree model. Luque and Ebert [40] developed such a model which was largely based on macroscopic streamer parameters. It is proposed that, with the refined methods of characterizing streamers by macroscopic properties developed in this work, a model such as that could be improved. Furthermore, the results of the quasi-steady model in this work and how it relates to the macroscopic parameters would allow one to "peek into" the mechanics of the individual streamers in such a tree model in a way that otherwise would not be possible.

5.2 Negative streamers

The entirety of this work so far has been focused on positive streamers. This is in part due to the author’s interest in applying the work to certain projects pertaining to triggered lightning where the positive streamer corona typically precedes the negative one. Much of the existing literature also focuses on the positive polarity, but there is also considerable interest in understanding negative streamer mechanics. This is a natural next step for extending the models discussed in this work, but there are differences that need to be taken into account.

118 Main differences of positive and negative streamers

At first glance, negative streamers appear simpler than positive streamers. Themo- bile charge carriers (electrons) move in the same direction as the streamer and seed the electron avalanches originating at streamer head [13]. This means that an additional method of seeding electrons ahead of the streamer is not necessary. Photoionization will of course still be present, and many authors have looked at how positive and negative streamers differ when this effect is included [54, 2]. In the positive streamer, the electron avalanches originate primarily from photoelectrons, but in the negative case the existing electron accumulation in the streamer head provides the main source. Photoionization is still able to seed secondary electrons, and its importance increases at lower pressures due to excited state quenching scaling differently than other streamer properties [70, 15]. Another major difference is the electric field required for streamer propagation. This is estimated to be two to three times the field required for the positive streamer [56, 15]. Both sources cited attribute this difference in part to the the dependence of streamer propagation on its radius. Something similar can be seen in the present work. Consider equations 4.2 and 4.12; the positive and negative streamers will have a different sign for the velocity of the electrons which will have a considerable effect on the evaluation of these equations. A more in depth re-calculation is given below. Many of the works studying the different behaviour of positive and negative streamers under the same conditions employ the double-headed streamer model [2, 54, 52]. Typically this model is created with parallel plates and some charge seed placed in between. The charge is then allowed to evolve in the uniform background field; the electron front moving towards the anode is the negative streamer andthe positive ion front moving towards the cathode is the positive streamer.

Implications for modelling negative coronas

While not a primary consideration of this work, the transient model of negative streamers should be relatively simple to implement; it just requires flipping the upwind

119 direction of the particles and modifying boundary conditions. Special care needs to be taken at the cathode surface boundary because of the "cathode-fall" region where electrons are repelled [74]. The quasi-steady model requires more work to modify correctly. Recall that in section 2.4.1 the ionization rate at the streamer tip, 푆푇 , plays an important role in establishing the correct solution. This is because photoioniztion seeds electrons in the positive streamer, and the result is linear in that ionization seed. For the negative streamer in atmospheric pressure, the seeding of the avalanches comes from the electrons in the head of the streamer, with the photoelectrons acting as a secondary effect. The implication for the quasi-steady model is that equation 2.57 will likely need to be modified to be linear in a Dirichlet electron boundary condition. This implementation has not been completed at this time, but is an interesting avenue of further exploration. The next question is whether a negative streamer can be characterised graphically in the same manner as a positive one. In the original paper by Naidis [43], this was shown to be possible using a similar model as presented in chapter 4. A naive approach to adapting the model of chapter 4 for the negative streamer involves neglecting the photoionization term, keeping the same number density ratio and reversing the electron velocities as appropriate. The output of such a model is shown in figure 5- 2. This figure requires refinement, but even with the simple assumptions itbegins to show some interesting trends. For one, there is a minimum velocity at which the streamer can propagate (this is caused by the second term in the numerator of equation 4.2 becoming complex for 푉 ≤ 1). This has a very simple explanation; a velocity of 1 in non-dimensional units implies the streamer is moving at the electron velocity in the breakdown field. The streamer must have a tip field greater than one to cause ionization, so the electrons at the tip are moving at least at this speed. Since the electrons define the tip location in the negative streamer, the streamer cannot possibly be moving slower than them. Detailed models of positive and negative streamers such as [2, 56] indicate that negative streamers typically have lower tip fields than positive streamers, suggesting physically realizable streamers lie on the left side figure 5-2. These lower tip fields were attributed in [2] to positive streamers typically

120 having smaller radii and hence less curvature. In [56] a slightly different analytical formulation for the relationship between tip field, radius and velocity was used; the conclusion of which was that for the same radius and velocity a positive streamer should have a higher tip field. Comparing figure 5-2 to figure 4-5, particularly at smaller radii and velocities, this is found to be the case with the model of this work as well1. It was also shown by Naidis [43] that his model, similar to the one presented here, agreed well with experiment. While a rigorous analysis is not performed in this work, the above comparisons indicate that a macroscopic model of negative streamers should be possible, and that even a simple implementation shows correct trends.

1 As specific examples, for 푎0 = 0.25, 푉 = 8 a positive streamer has a tip field magnitude of 5.5퐸0 while the negative streamer is closer to 5.0퐸0. For 푎0 = 0.1, 푉 = 4 a positive streamer has tip field magnitude of 6.5퐸0 while a negative streamer is around 4퐸0

121 20

0.9 0.8 0.7 0.6 0.5 0.4 16 0.45 0.35 0.3 0.25 0.2

0.15

10 0.1

0.05 6

0 -3 -3.5 -4 -4.5 -5 -5.5 -6 -6.5 -7 -7.5 -8

Figure 5-2: Example of graphical characterization of negative streamers. Contours are for constant (effective) radius with labels in mm

122 Chapter 6

Conclusions

6.1 Summary of contributions

This work investigated methods of modelling positive streamers in atmospheric pressure air. As discussed in chapter 1, there has been a significant effort in the academic community to understand these structures; however, most of the existing literature focuses on intricate models of single streamers. These models do not easily extend to many-streamer corona models because of the many different length and time scales involved. The few works that have attempted to model the full streamer corona are forced to abstract away relevant physics which occurs at the level of individual streamers. In this work, three different models were developed aimed at striking a balance between simplicity and physical accuracy. In chapter 2, 1.5D models of positive streamers were developed and then in chapter 3 an in-depth analysis of their applicability was performed. First, a 1.5 dimensional model of a streamer emerging from a spherical anode was constructed. To build this model, a simplified expression for the photoionization was developed which allowed the process to be included with minimal computational burden. This model was shown to be in good agreement with other reference models existing in the literature. Next, the 1.5D model was modified so that the governing equations could be solved in the reference frame of the streamer to produce a quasi-steady model. The validity of the quasi-steady assumption was investigated in a number of different

123 ways. It was shown that the governing equation of the quasi-steady streamer yields the same velocity as measured in the transient model. It was also shown that the current conservation equation can be used to evaluate the time scales on which the steady state calculation is valid. Finally, the quasi-steady model was shown to yield similar profiles for the particle densities and electric field as the full transient model. Overall, it is concluded that a model constructed in the quasi-steady frame is able to accurately describe the instantaneous state of a streamer. The quasi-steady model yielded a unique solution given just two input parameters: an effective radius for the streamer and a uniform background electric field. This solution determined both macroscopic parameters (velocity and tip electric field) as well as microscopic details of the streamer (electric field and particle profiles). In light of this observation, the macroscopic parameter model was developed in chapter 4 to help characterize the instantaneous (quasi-steady) state of streamers. As evidenced by figure 4-5, within the realm of interest for physical streamers any combination oftwo macroscopic properties (effective radius, background field, velocity and tip field)is sufficient to uniquely determine the streamer instantaneous state. The macroscopic characterization, which was developed in part based on results of the quasi-steady model, was shown to do a good job of characterizing both the 1.5D transient model of this work, as well as simulation and experimental results found in the literature. In chapter 5 of this work, extensions of the models were discussed. These included applying the quasi-steady and macroscopic parameter models to propagate a streamer in a way that did not require solving the full governing equations, and an extension of the model to negative streamers. A preliminary implementation of both these extensions was made, with initial results showing that they are promising avenues for future study. In summary, it has been shown that models of streamers based on simplifying assumptions such as instantaneously steady propagation and macroscopic parameter characterization do a fair job of capturing the physics of much more detailed models. These simplified models are natural building blocks for more complex streamer corona models since they provide valuable information on the relationships between

124 the macroscopic streamer properties, without the computational burden typically associated with resolving the disparate length and time scales of single streamers. At the same time, the ability to relate the macroscopic properties to specific microscopic behaviour using the quasi-steady model gives a level of insight not typically available in other macroscopic streamer models. It is hoped that the models presented herein will provide a much needed bridge between understanding individual streamers on the microscopic scale and the behaviour of larger, macroscopic-scale structures such as streamer coronae.

6.2 Recommendations for future work

There are four main avenues recommended for further exploration. Two of these have already been discussed briefly in chapter 5, but they will be re-stated and summarized here for completeness. The areas of study can be broadly summarized as follows:

Multi-Streamer Interactions The most important area requiring further study is the extension of the model to study multiple streamers. It has been noted repeatedly throughout this work that the primary goal of developing the reduced order models is to create building blocks for use in full streamer corona models. Now that the building blocks have been constructed and verified, the next step is to utilize them to study the streamer corona in a way that is not feasible with previous models. A preliminary discussion of how one might do this is found in section 5.1, but a full implementation will be a significant undertaking and is outside the scope of this work.

Negative Streamers As discussed briefly in chapter 5, it should be possible to extend the models developed in this work to negative streamers. In that chapter, a preliminary implementation was given but it still requires much refinement and verification. Negative streamers are of in interest in many applications, soifthe simplified quasi-steady and macroscopic parameter models can be shown to extend to the opposite polarity it will greatly enhance the versatility of the modelling methods developed in this work.

125 Extra degrees of freedom Many detailed streamer simulations have shown that the streamer radius is not constant, but changes as the streamer propagates. Recalling the discussion in chapter 1, the inability to self-consistently evolve the radius is the most significant shortcoming of the 1.5D model. It is unclear just how important correctly evolving the radius will be to capturing the core properties of a streamer corona, but for consistency this is something that must be investigated. Just as it is proposed to evolve the streamer tip field and velocity in time, the radius isa macroscopic property that should be allowed to vary in the same way. This would be an important addition to the current macroscopic parameter model.

Reduced order models of Branching/Initiation Related to the pursuit of a full streamer corona model, it is important to develop an understanding of non- axisymmetric affects, such as those leading to branching, initiation or streamers not travelling in a straight line. In the model of a full streamer corona, these would likely be implemented at a level above individual streamers. Nevertheless, a reduced order model of these effects that incorporates the physics governing the processes will be necessary to realize an accurate model of multiple streamer interactions. 1.5D models are probably not the optimum method of understanding these, but a similar philosophy of striking a balance between accuracy and physical correctness should be used.

126 Appendix A

Additional derivations

A.1 Integrated absorption function

Note that this derivation is done with all dimensional variables since the coefficients of the absorption function fit, equation 2.12, are dimensional. This affects the lengths and the pressures.

′ 2휋 푟푠 ′ ′ ′ 휔 푔(푅) ′ ′ ′ 퐹 (푧 − 푧 ) = 푓(푟 , 푧 )휉 2 푟 푑푟 푑휃 ˆ0 ˆ0 훼 2휋푅 Assume 푓(푟′, 푧′) = 1 and use equation 2.12 for the absorption function. The photoionization is evaluated at the 푟 = 0 axis so there is no angular dependence and the 휃′ integral can be done immediately to cancel the 2휋 in the denominator.

′ 푟푠 ′ 휔 푔(푅) ′ ′ ′ 퐹 (푧 − 푧 ) = 휉 2 푟 푑푟 푑휃 ˆ0 훼 푅

3 2 ∑︁ 푔(푅) = (푝푂2 푅) 퐴푗 exp(−휆푗푝푂2 푅) 푗=1 Now write out 푅2 and find the Jacobian with respect to 푟′

푅2 = 푟′2 + (푧 − 푧′)2

푅푑푅 = 푟′푑푟′

127 Now substitute 푔(푅) into the integral and change the integration variable. One of the 푅 in the denominator is cancelled by 푅 in the absorption function and the other is cancelled by the change in variables.

√ ′2 ′ 2 3 푟푠 +(푧−푧 ) ′ 휔 2 ∑︁ 퐹 (푧 − 푧 ) = √ 휉 푝푂2 퐴푗 exp(−휆푗푝푂2 푅)푑푅 ˆ ′ 2 훼 (푧−푧 ) 푗=1 Move all the constants with respect to 푅 outside the integral: √ 3 [︃ ′2 ′ 2 ]︃ 푟푠 +(푧−푧 ) ′ 휔 2 ∑︁ 퐹 (푧 − 푧 ) = 휉 푝푂2 퐴푗 √ exp(−휆푗푝푂2 푅)푑푅 훼 ˆ ′ 2 푗=1 (푧−푧 )

The resulting integral is now trivial:

3 [︂ √ ]︂ ′ 휔 ∑︁ 퐴푗 (︁ −휆 푝 푟 +(푧−푧′)2 −휆 푝 |푧−푧′|)︁ 퐹 (푧 − 푧 ) = −휉 푝 푒 푗 푂2 푠 − 푒 푗 푂2 훼 푂2 휆 푗=1 푗 As written, this has units of inverse length. As a final step, use the characteristic length scale to non-dimensionalize all variables with length units. The non- dimensional parameters are now explicitly identified using a hat. See the footnote in the main text about the unusual scaling of this equation.

3 [︂ √ ]︂ ′ 휔 ∑︁ 퐴푗푎0 (︁ −휆 푎 푝 푟^ +(^푧−푧^′)2 −휆 푎 푝 |푧^−푧^′|)︁ 퐹ˆ(ˆ푧 − 푧ˆ ) = −휉 푝 푒 푗 0 푂2 푠 − 푒 푗 0 푂2 훼 푂2 휆 푗=1 푗

A.2 Mirror charge

This section gives the derivation for a mirror charge Kernel for the disc method in spherical geometry. First, it is necessary to derive the disc charge solution in spherical coordinates; in the body of this report it was solved in cylindrical coordinates. Assume that the discs of charge are centered about the polar axis of a sphere so that the problem retains azimuthal symmetry. First define the charge to be located at point

f ′ = (푟′, 휃′, 훾′) where 휃′ is the polar angle and 훾′ is the azimuthal angle (using 휑, the normal notation, is avoided to prevent confusion with electric potential). Define

128 r0 = (푟0, 휃0, 훾0) as a point on the surface of the cylinder of charge. For a cylinder of ′ charge of fixed radius 푟푐, the components of r0 and r can be related as follows:

′ 푟푐 = 푟0 cos 휃0 tan 휃 푟 cos 휃 푑푟 = 0 0 푑휃′ 푐 cos2 휃′ Now, a differential planar area can be defined:

(︂푟 cos 휃 )︂2 푑퐴′ = 푟 푑푟 푑훾′ = 0 0 tan 휃′푑휃′푑훾′ 푐 푐 cos 휃′ The potential field created by a point on the disc as seenat r is then (recall 푅 = |r′ − r|):

−휌(r′) 푑휑(r) = 푑퐴′ 4휋푅 Integrate this over all space, assume that the charge density only varies along the polar axis and group all known terms into a kernel function 푘. Note that 푟0 and 휃0 are functions of 푧′, the distance along the polar axis, only.

휑(r) = 휌(푧′)퐾푑푧′ ˆ −1 tan 휃′ 1 퐾 = (푟 cos 휃 )2 푑휃′푑훾′ 4휋 0 0 ¨ cos2 휃′ 푅 Now write out the equation for 푅 in full:

푅 = √︀푟2 + 푟′2 − 2푟푟′ cos 휃 cos 휃′ − 2푟푟′ sin 휃 sin 휃′ cos(훾 − 훾′)

First, the integral in 훾′ will be handled. The equation for this is:

2휋 푑훾′ 2휋 푑훾′ = √︀ 2 ′2 ′ ′ ′ ′ ′ ˆ0 푅 ˆ0 푟 + 푟 − 2푟푟 cos 휃 cos 휃 − 2푟푟 sin 휃 sin 휃 cos(훾 − 훾 )

This is simplified somewhat using trigonometric identities. By azimuthal symmetry,

129 any observation plane is equivalent so choose the plane where 훾 = 0.

2휋 푑훾′ 2휋 푑훾′ = √︀ 2 ′2 ′ ′ ′ ′ ′ ˆ0 푅 ˆ0 푟 + 푟 − 2푟푟 cos(휃 − 휃 ) − 2푟푟 sin 휃 sin 휃 (cos(훾 ) − 1)

Now group the terms that are constant in 훾′ as:

푏2 푎2 = 푟2 + 푟′2 − 2푟푟′ cos(휃 − 휃′)푏2 = 푟 = 2푟푟′ sin 휃 sin 휃′푐2 = 푎2 + 푏2

휃, 휃′ ∈ [0, 휋] so 푎, 푏, 푐 are all real. Substituting these into the equation gives a much simpler form of the equation:

2휋 푑훾′ 2휋 푑훾′ = √︀ 2 ′ ˆ0 푅 ˆ0 1 − 푐 cos 훾

Using the trigonometric identity cos 훾′ = 1 − 2 sin2(훾′/2), this can be converted into a complete elliptic integral of the first kind, which is defined as:

휋/2 푑훼 퐸푙푙푖푝푡푖푐퐾(푚) = 2 ˆ0 1 − 푚 cos 훼 The result also turns out to be the Kernel function for a ring of charge (which has a fixed 휃′),

2휋 푑훾′ 4 (︂−2푏2 )︂ 퐾푟푖푛푔(푟, 휃, 0) = = 퐸푙푙푖푝푡푖푐퐾 2 ˆ0 푅 푎 푎 Note that 푏 = 0 when 휃′ = 0. For the method being used here, all that matters is the potential along the axis so this simplification can be applied. The complete elliptic integral evaluated for an argument 0 is just 휋/2, meaning:

2휋 퐾푟푖푛푔(푟, 0, 0) = √ 푟2 + 푟′2 − 2푟푟′ cos 휃′ Now, the radial distance from the sphere’s center should be written in terms of

휃′. The equation for this is:

130 푟 cos 휃 푟′ = 0 0 cos 휃′ Note that this substitution prior to the next integration is part of what makes the calculation of the image charge for the disk non-trivial. Before proceeding to the the image charge kernel derivation, derivation for the real charge Kernel will be completed to show that it is identical to the result derived in the main body of the work. At this point the kernel function (for the real charge) is:

′ −1 2 tan 휃 1 ′ 푘 = (푟0 cos 휃0) 푑휃 2 ′ √︁ 2 2 ˆ cos 휃 2 (︀ 푟0 cos 휃0 )︀ (︀ 푟0 cos 휃0 )︀ ′ 푟 + cos 휃′ − 2푟 cos 휃′ cos 휃 The following substitutions are made:

2 ′ ′ ′ 푟 − 2푟푟0 cos 휃0 푥 = cos 휃 푑푥 = − sin 휃 푑휃 푛 = 2 (푟0 cos 휃0) Which results in

−1 1 푑푥 퐾 = 푟0 cos 휃0 √ 2 2 2 ˆcos 휃0 푥 1 + 푛푥 The integral is solved analytically, giving a final result:

−1 [︂√︁ ]︂ 퐾(푟, 0, 0) = 푟2 + 푟2 − 2푟푟 cos 휃 − √︀푟2 + (푟 cos 휃 )2 − 2푟푟 cos 휃 2 0 0 0 0 0 0 0

Now, write this in terms of the axial coordinate ′. Note that 2 ′2 2 where 푧 푟0 = 푧 + 푟푠 ′ 푟푠 is the radius of the streamer (or disc of charge) and 푧 = 푟0 cos 휃0.

−1 [︁ √ ]︁ 퐾(푟, 0, 0) = √︀푧2 + 푧′2 − 2푧푧′ + 푟2 − 푧2 + 푧′2 − 2푧푧′ 2 푠 Re-write using difference of squares and go back to the full potential equation

−1 [︁ ]︁ 휑(푧) = 휌(푧′) √︀(푧 − 푧′)2 + 푟2 − |푧 − 푧′| 푑푧′ 2 ˆ 푠

131 Finally, take the negative gradient to get the electric field. By azimuthal symmetry, the gradient of 휑 along the polar axis is purely along the axis.

[︃ ′ ]︃ 1 ′ 푧 − 푧 ′ ′ E(푧) = 퐸푧(푧) = 휌(푧 ) − 푠푖푔푛(푧 − 푧 ) 푑푧 ˆ √︀ ′ 2 2 2 (푧 − 푧 ) + 푟푠 This is exactly the same result as was derived in cylindrical coordinates. Having verified that the derivation is consistent, it will now be shown how to modify itto account for image charges. The image charge is imposed by cancelling a real charge of q located at with a charge of 푟푠푝ℎ located at a radial distance r : 푟 > 푟푠푝ℎ 푞푖 = − 푟 푟2 푠푝ℎ along the same line. Start with general equation for the potential 푟푖 = 푟

−1 tan 휃′ 1 휑(r) = 휌(푧′) (푟 cos 휃 )2 푑휃′푑훾′푑푧′ ˚ 4휋 0 0 cos2 휃′ 푅

푟 Now substitute with 푠푝ℎ where ′ 푟0 cos 휃0 . 휌 휌푖 = −휌 푟′ 푟 = cos 휃′

1 tan 휃′ 1 휑 (r) = 휌(푧′)푟 푟 cos 휃 푑휃′푑훾′푑푧′ 푖 ˚ 푠푝ℎ 4휋 0 0 cos 휃′ 푅 The integration in ′ remains unchanged, except that ′ in the result becomes ′, 훾 푟 푟푖 the location of the image charge.

′ 푟푠푝ℎ ′ tan 휃 1 ′ ′ 휑푖(푟, 0, 0) = 푟0 cos 휃0 휌(푧 ) 푑휃 푑푧 2 ˆ ˆ cos 휃′ √︀ 2 ′2 ′ ′ 푟 + 푟푖 − 2푟푟푖 cos 휃

푟2 푟2 cos 휃′ Substitute ′ 푠푝ℎ 푠푝ℎ and with some re-arrangement find 푟 = ′ = 푖 푟 푟0 cos 휃0

휃0 ′ ′ 푟푠푝ℎ 푟0 cos 휃0 ′ tan 휃 푑휃 ′ 휑푖(푟, 0, 0) = 휌(푧 ) 푑푧 2 푟 ˆ ˆ cos 휃′ √︂ (︁ )︁2 0 푟푠푝ℎ 2 2 ′ 1 + (푟 − 2푟푟0 cos 휃0) cos 휃 푟푟0 cos 휃0 푠푝ℎ

Apply the substitution of:

(︂ )︂2 ′ ′ ′ 푟푠푝ℎ 2 푥 = cos 휃 푑푥 = − sin 휃 푑휃 푛 = (푟푠푝ℎ − 2푟푟0 cos 휃0) 푟푟0 cos 휃0

132 This puts the equation in exactly the same form for a Kernel function as was

derived for the real charge, simply with a different value for 푛 and different coefficient on the integral:

푟 푟 cos 휃 1 푑푥 푘(푟, 0, 0) = 푠푝ℎ 0 0 √ 2 2 2 푟 ˆcos 휃0 푥 1 + 푛푥 Follow through by solving the integral and using the value of n:

1 푟 [︁√︁ √︁ ]︁ 퐾(푟, 0, 0) = 푠푝ℎ 푟2 (푟2 − 2푟푟 cos 휃 ) + (푟푟 )2) − 푟2 (푟2 − 2푟푟 cos 휃 ) + (푟푟 cos 휃 )2) 2 푟2 푠푝ℎ 푠푝ℎ 0 0 0 푠푝ℎ 푠푝ℎ 0 0 0 0

Now change from r to z and replace 푟0 and cos 휃0 the same as for the real charge.

1 푟 [︁√︁ √︁ ]︁ 퐾(푧, 0, 0) = 푠푝ℎ 푟2 (푟2 − 2푧푧′) + 푧2(푧′2 + 푟2)) − 푟2 (푟2 − 2푧푧′) + (푧푧′)2) 2 푧2 푠푝ℎ 푠푝ℎ 푠 푠푝ℎ 푠푝ℎ

Identifying terms that can be simplified using difference of squares:

1 푟 [︁√︁ √︁ ]︁ 퐾(푧, 0, 0) = 푠푝ℎ (푟2 − 푧푧′)2 + 푧2푟2 − (푟2 − 푧푧′)2 2 푧2 푠푝ℎ 푠 푠푝ℎ And then factoring out 푧:

⎡√︃ 2 √︃ 2⎤ 1 푟 (︂푟2 )︂ (︂푟2 )︂ 퐾(푧, 0, 0) = 푠푝ℎ 푠푝ℎ − 푧′ + 푟2 − 푠푝ℎ − 푧′ 2 푧 ⎣ 푧 푠 푧 ⎦

푟2 Now group a common term 푠푝ℎ ′ 푢 = 푧 − 푧

1 푟 [︁ ]︁ 퐾(푧, 0, 0) = 푠푝ℎ √︀푢2 + 푟2 − |푢| 2 푧 푠 The potential from the image charge is then:

1 푟 [︁ ]︁ 휑 (푧, 0, 0) = 휌(푧′) 푠푝ℎ √︀푢2 + 푟2 − |푢| 푑푧′ 푖 2 ˆ 푧 푠 Now differentiate with respect to z to get the electric field:

133 [︃ (︃ )︃ ]︃ (︁ )︁ (︂ 푟2 )︂ 1 ′ 푟푠푝ℎ √︀ 2 2 푟푠푝ℎ 푢 푠푝ℎ ′ 퐸푖,푧(푧) = 휌(푧 ) − 푢 + 푟 − |푢| + − 푠푖푔푛(푢) − 푑푧 2 푠 2 √︀ 2 2 2 ˆ 푧 푧 푢 + 푟푠 푧

Which gives the image charge Kernel function which is integrated over the same domain are the real charge Kernel.

[︃ (︃ )︃]︃ (︁ )︁ (︁ )︁2 1 푟푠푝ℎ √︀ 2 2 푟푠푝ℎ 푢 퐾푖(푧) = − 푢 + 푟 − |푢| + − 푠푖푔푛(푢) 2 푠 √︀ 2 2 2 푧 푧 푢 + 푟푠

134 Appendix B

Additional model results

B.1 Transient model parameter variation

135 Figure B-1: Modification of initial Gaussian seed. Fixed 푎0 = 0.25푚푚, 푉푎 = 25, 푅푎 = 10푎0; run 1 푛0 = 10, run 2 푛0 = 50, run 3 푛0 = 80, run 4 푛0 = 2, run 5 푛0 = 25

Figure B-2: Modification of streamer radius for large anode. Fixed 푅푎 = 10푎0, 푉푎 = 25, 푛0 = 25

136 Figure B-3: Modification of streamer radius for small anode. Fixed 푅푎 = 2.5푎0, 푉푎 = 10, 푛0 = 25 B.2 Numerical accuracy analysis

With any numerical methods problem, it is important to perform a grid-independence study to ensure that the solution is adequately converged. This was done on the steady state model. The study was performed using a constant grid size and then once with the variable grid size used as the baseline case. The results are shown in figure B-4. As shown in the figure, the variable mesh gives results almost as good as a uniform mesh with the finer size; this confirms that the region of refinement is adequate. The right end of the curves gives the smallest grid size that could be practically used. The electric field calculation pre-computes a dense matrix with the electric field Kernel function, so this became a limiting factor. Despite this limit, the results appear to be pretty well converged. It is estimated by the shape of the curve that the smallest grid size is converged to within 0.05 of its final value for both tip field and velocity. Looking at the baseline case with variable grid size, it seems to give results reasonably close to the converged case. It is estimated that the error in the velocity and tip field is no more than 0.1 absolute error in their respective non-dimensional units.

137 Figure B-4: Grid size convergence study. Results listed as "variable" are for a non- uniform mesh and have have their endpoints at the coarse and fine sizes of the grid. All others results are for uniform mesh.

138 Bibliography

[1] W. Roznerski and J. Mechlińska-Drewko. The ratio of lateral diffusion coefficient to mobility for electrons in oxygen and dry air. Journal of Physics D: Applied Physics, 12(11), 1979.

[2] Alejandro Luque, Valeria Ratushnaya, and Ute Ebert. Positive and negative streamers in ambient air: Modelling evolution and velocities. Journal of Physics D: Applied Physics, 41(23):234005, 2008.

[3] S. Pancheshnyi, M. Nudnova, and A. Starikovskii. Development of a cathode- directed streamer discharge in air at different pressures: Experiment and comparison with direct numerical simulation. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 71(1):1–12, 2005.

[4] E M Van Veldhuizen and W R Rutgers. Pulsed positive corona streamer propagation and branching. Journal of Physics D: Applied Physics, 35(17):313, sep 2002.

[5] B. Bagheri, J. Teunissen, U. Ebert, M. M. Becker, S. Chen, O. Ducasse, O. Eich- wald, D. Loffhagen, A. Luque, D. Mihailova, J. M. Plewa, J van Dijk,and M. Yousfi. Comparison of six simulation codes for positive streamers inair. Plasma Sources Science and Technology, 27(9):095002, sep 2018.

[6] A. Bourdon, V. P. Pasko, N. Y. Liu, S. Célestin, P. Ségur, and E. Marode. Efficient models for photoionization produced by non-thermal gas discharges in air based on radiative transfer and the Helmholtz equations. Plasma Sources Science and Technology, 16(3):656–678, 2007.

[7] Gregory D. Moss, Victor P. Pasko, Ningyu Liu, and GeorgioÅ§ Veronis. Monte Carlo model for analysis of thermal runaway electrons in streamer tips in transient luminous events and streamer zones of lightning leaders. Journal of Geo- physical Research: Space Physics, 111(2):1–37, 2006.

[8] Vernon Cooray. An Introduction to Lightning. Springer Netherlands, Dordrecht, 2015.

[9] V A Rakov and M A Uman. Lightning: Physics and Effects. Cambridge Univer- sity Press, 2006.

139 [10] Y P Raizer. Gas Discharge Physics. Springer-Verlag Berlin Heidelberg, Berlin, 1991.

[11] H. Raether. Die Entwicklung der Elektronenlawine in den Funkenkanal. In Ergebnisse der Exakten Naturwissenschaften, pages 73–120. Springer Berlin Hei- delberg, Berlin, Heidelberg, 1937.

[12] Leonard B. Loeb and John M. Meek. The mechanism of spark discharge in air at atmospheric pressure. I. Journal of Applied Physics, 11(6):438–447, 1940.

[13] A A Fridman and L A Kennedy. Plasma physics and engineering. Taylor & Francis Group, Boca Raton, FL, 2nd edition, 2011.

[14] U Ebert, C Montijn, T M P Briels, W Hundsdorfer, B Meulenbroek, A Rocco, and E M van Veldhuizen. The multiscale nature of streamers. Plasma Sources Science and Technology, 15(2):S118–S129, 2006.

[15] Jianqi Qin and Victor P. Pasko. On the propagation of streamers in electrical discharges. Journal of Physics D: Applied Physics, 47(43), 2014.

[16] T. M.P. Briels, J. Kos, G. J.J. Winands, E. M. Van Veldhuizen, and U. Ebert. Positive and negative streamers in ambient air: Measuring diameter, velocity and dissipated energy. Journal of Physics D: Applied Physics, 41(23), 2008.

[17] Annemie Bogaerts, Zhaoyang Chen, and Renaat Gijbels. Glow discharge modelling: from basic understanding towards applications. Surface and Interface Analysis, 35(7):593–603, jul 2003.

[18] Andrey Starikovskiy. Physics and chemistry of plasma-assisted combustion. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 373(2048):20150074, aug 2015.

[19] Sergey V. Pancheshnyi, Deanna A. Lacoste, Anne Bourdon, and Christophe O. Laux. Ignition of propane-air mixtures by a repetitively pulsed nanosecond discharge. IEEE Transactions on Plasma Science, 34(6):2478–2487, 2006.

[20] Eric Moreau. Airflow control by non-thermal plasma actuators. Journal of Physics D: Applied Physics, 40(3):605–636, 2007.

[21] B. Jayaraman, Y.C. Cho, and W. Shyy. Modeling of Dielectric Barrier Discharge Plasma Actuator. In 38th AIAA Plasmadynamics and Lasers Conference, number June, Miami, FL, jun 2007. American Institute of Aeronautics and Astro- nautics.

[22] Haofeng Xu, Yiou He, Kieran L. Strobel, Christopher K. Gilmore, Sean P. Kelley, Cooper C. Hennick, Thomas Sebastian, Mark R. Woolston, David J. Perreault, and Steven R.H. Barrett. Flight of an aeroplane with solid-state propulsion. Nature, 563(7732):532–535, 2018.

140 [23] M. G. Kong, G. Kroesen, G. Morfill, T. Nosenko, T. Shimizu, J. Van Dijk, and J. L. Zimmermann. Plasma medicine: An introductory review. New Journal of Physics, 11, 2009.

[24] Gregory Fridman, Gary Friedman, Alexander Gutsol, Anatoly B. Shekhter, Vic- tor N. Vasilets, and Alexander Fridman. Applied plasma medicine. Plasma Processes and Polymers, 5(6):503–533, 2008.

[25] Ute Ebert, Sander Nijdam, Chao Li, Alejandro Luque, Tanja Briels, and Eddie van Veldhuizen. Review of recent results on streamer discharges and discussion of their relevance for sprites and lightning. Journal of Geophysical Research: Space Physics, 115(A7):1–13, 2010.

[26] T. M P Briels, E. M. van Veldhuizen, and U. Ebert. Positive streamers in ambient air and a N2:O2 mixture (99.8: 0.2). IEEE Transactions on Plasma Science, 36(4 PART 1):906–907, 2008.

[27] I. Gallimberti, G. Bacchiega, Anne Bondiou-Clergerie, and Philippe Lalande. Fundamental processes in long air gap discharges. Comptes Rendus Physique, 3(10):1335–1359, 2002.

[28] Marley Becerra. Glow corona discharges and their effect on lightning attachment: Revisited. 2012 31st International Conference on Lightning Protection, ICLP 2012, 2012.

[29] N. C. Nguyen, C. Guerra-Garcia, J. Peraire, and M. Martinez-Sanchez. Compu- tational study of glow corona discharge in wind: Biased conductor. Journal of Electrostatics, 89:1–12, 2017.

[30] J Anderson Plumer. Laboratory Test Results and Natural Lightning Strike Ef- fects: How Well do They Compare. In International Conference on Lightning Protection, Vienna, Austria, 2012.

[31] P. Lalande, A. Bondiou-Clergerie, and P. Laroche. Analysis of Available In-Flight Measurements of Lightning Strikes to Aircraft. In International Conference on Lightning and Static Electricity, Toulouse, France, 1999. SAE International.

[32] R. Morrow. The theory of the Positive Glow Corona. Journal of Physics D: Applied Physics, 30:3099–3114, 1997.

[33] R. T. Waters and W. B. Stark. Characteristics of the stabilized glow discharge in air. Journal of Physics D: Applied Physics, 8(4):416–426, 1975.

[34] Lipeng Liu and Marley Becerra. On the transition from stable positive glow corona to streamers. Journal of Physics D: Applied Physics, 49(22), 2016.

[35] Yu S. Akishev, M. E. Grushin, A. A. Deryugin, A. P. Napartovich, M. V. Pan’kin, and N. I. Trushkin. Self-oscillations of a positive corona in nitrogen. Journal of Physics D: Applied Physics, 32(18):2399–2409, 1999.

141 [36] S. Nijdam, F. M.J.H. Van De Wetering, R. Blanc, E. M. Van Veldhuizen, and U. Ebert. Probing photo-ionization: Experiments on positive streamers in pure gases and mixtures. Journal of Physics D: Applied Physics, 43(14), 2010.

[37] I. Gallimberti. A computer model for streamer propagation. Journal of Physics D: Applied Physics, 5(12):2179–2189, 1972.

[38] A. Luque and U. Ebert. Electron density fluctuations accelerate the branching of positive streamer discharges in air. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 84(4):1–7, 2011.

[39] Manuel Arrayás, Ute Ebert, and Willem Hundsdorfer. Spontaneous branching of anode-directed streamers between planar electrodes. Physical Review Letters, 88(17):1745021–1745024, 2002.

[40] Alejandro Luque and Ute Ebert. Growing discharge trees with self-consistent charge transport: The collective dynamics of streamers. New Journal of Physics, 16, 2014.

[41] N. L. Aleksandrov and E. M. Bazelyan. Simulation of long-streamer propagation in air at atmospheric pressure. Journal of Physics D: Applied Physics, 29(3):740– 752, 1996.

[42] G. A. Dawson and W. P. Winn. A model for streamer propagation. Zeitschrift für Physik, 183(2):159–171, 1965.

[43] G. V. Naidis. Positive and negative streamers in air: Velocity-diameter relation. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 79(5):3–6, 2009.

[44] A. V. Samusenko and Yu. K. Stishkov. Model of development of the predischarge streamer in air-based on the long-line equation. High Temperature, 49(6):803– 814, 2011.

[45] R. Morrow. Theory of negative corona in oxygen. Physical Review A, 32(3):1799– 1809, sep 1985.

[46] Jing Ming Guo and Chwan Hwa John Wu. Streamer radius model and its as- sessment using two-dimensional models. IEEE Transactions on Plasma Science, 24(6):1348–1358, 1996.

[47] N. L. Aleksandrov and E. M. Bazelyan. Step propagation of a streamer in an electronegative gas. Journal of Experimental and Theoretical Physics, 91(4):724– 735, 2002.

[48] S. V. Pancheshnyi, S. V. Sobakin, S. M. Starikovskaya, and A. Yu. Starikovskii. Discharge dynamics and the production of active particles in a cathode-directed streamer. Plasma Physics Reports, 26(12):1054–1065, dec 2000.

142 [49] A. Luque, M. González, and F. J. Gordillo-Vázquez. Streamer discharges as ad- vancing imperfect conductors: Inhomogeneities in long ionized channels. Plasma Sources Science and Technology, 26(12), 2017.

[50] L Papageorgiou, A C Metaxas, and G E Georghiou. Three-dimensional numerical modelling of gas discharges at atmospheric pressure incorporating photoionization phenomena. Journal of Physics D: Applied Physics, 44(4):045203, feb 2011.

[51] Jannis Teunissen and Ute Ebert. Simulating streamer discharges in 3D with the parallel adaptive Afivo framework. Journal of Physics D: Applied Physics, 50(47), 2017.

[52] Chao Li, Ute Ebert, and Willem Hundsdorfer. Spatially hybrid computations for streamer discharges : II. Fully 3D simulations. Journal of Computational Physics, 231(3):1020–1050, feb 2012.

[53] Chao Li, Jannis Teunissen, Margreet Nool, Willem Hundsdorfer, and Ute Ebert. A comparison of 3D particle, fluid and hybrid simulations for negative streamers. Plasma Sources Science and Technology, 21(5):055019, oct 2012.

[54] Ningyu Liu and Victor P. Pasko. Effects of photoionization on propagation and branching of positive and negative streamers in sprites. Journal of Geophysical Research: Space Physics, 109(A4):1–18, 2004.

[55] S. V. Pancheshnyi, S. M. Starikovskaia, and A. Yu Starikovskii. Role of photoionization processes in propagation of cathode-directed streamer. Journal of Physics D: Applied Physics, 34(1):105–115, 2001.

[56] Natalya Y. Babaeva and George V. Naidis. Dynamics of positive and negative streamers in air in weak uniform electric fields. IEEE Transactions on Plasma Science, 25(2):375–379, 1997.

[57] Lipeng Liu and Marley Becerra. An Efficient Semi-Lagrangian Algorithm for Simulation of Corona Discharges: The Position-State Separation Method. IEEE Transactions on Plasma Science, 44(11):2822–2831, 2016.

[58] Lipeng Liu and Marley Becerra. An efficient model to simulate stable glow corona discharges and their transition into streamers. Journal of Physics D: Applied Physics, 50(10), 2017.

[59] Manuel Arrayas, Santiago Betelu, Marco A. Fontelos, and Jose L. Trueba. Fin- gering from Ionization Fronts in Plasmas. Society for Industrial and Applied Mathematics, 68(4):1122–1145, 2008.

[60] Mose Akyuz, Anders Larsson, Vernon Cooray, and Gustav Strandberg. 3D simulations of streamer branching in air. Journal of Electrostatics, 59(2):115–141, 2003.

143 [61] Sebastien Celestin. Study of the dynamics of streamers in air at atmospheric pressure. PhD thesis, École Centrale Paris, 2008.

[62] G. J.M. Hagelaar and L. C. Pitchford. Solving the Boltzmann equation to obtain electron transport coefficients and rate coefficients for fluid models. Plasma Sources Science and Technology, 14(4):722–733, 2005.

[63] A. V. Phelps and L. C. Pitchford. SIGLO Database. http://www.lxcat. laplace.univ-tlse.fr, 2013.

[64] Lawton and Phelps. PHELPS database. http://www.lxcat.laplace. univ-tlse.fr, 2013.

[65] H. K. Versteeg and W. Malalasekera. An Introduction to Computational Fluid Dynamics. Pearson Education Ltd, Harlow, Essex, England, 2nd edition, 2007.

[66] M B Zheleznyak, Mnatsakanian A Kh, and S V Sizykh. Photoionization of nitrogen and oxygen mixtures by radiation from a gas discharge. High Temperature, 20(3):357–362, 1982.

[67] P. Ségur, A. Bourdon, E. Marode, D. Bessieres, and J. H. Paillol. The use of an improved Eddington approximation to facilitate the calculation of photoionization in streamer discharges. Plasma Sources Science and Technology, 15(4):648–660, 2006.

[68] G. V. Naidis. On photoionization produced by discharges in air. Plasma Sources Science and Technology, 15(2):253–255, 2006.

[69] G. W. Penney and G. T. Hummert. Photoionization Measurements in Air, Oxy- gen, and Nitrogen. Journal of Applied Physics, 41(2):572–577, feb 1970.

[70] Alejandro Luque, Ute Ebert, Carolynne Montijn, and Willem Hundsdorfer. Pho- toionization in negative streamers: Fast computations and two propagation modes. Applied Physics Letters, 90(8), 2007.

[71] N. Yu Babaeva and G. V. Naidis. Two-dimensional modelling of positive streamer dynamics in non-uniform electric fields in air. Journal of Physics D: Applied Physics, 29(9):2423–2431, sep 1996.

[72] Karen Willcox and Qiqi Wang. 16.90 Computational Methods in Aerospace Engineering. https://ocw.mit.edu, 2014.

[73] Eric Moreau, Pierre Audier, Thomas Orriere, and Nicolas Benard. Electrohy- drodynamic gas flow in a positive corona discharge. Journal of Applied Physics, 125(13), 2019.

[74] P. A. Vitello, B. M. Penetrante, and J. N. Bardsley. Simulation of negative- streamer dynamics in nitrogen. Physical Review E, 49(6), 1994.

144