Wilson, Alasdair David (2016) Finite difference simulations of neutral gas-MHD interactions in ...

227 Pages·2016·5.5 MB·English

Wilson, Alasdair David (2016) Finite difference simulations of neutral
gas-MHD interactions in partially ionized plasmas. PhD thesis.

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Finite Difference Simulations of
Neutral Gas-MHD Interactions in
Partially Ionized Plasmas
Alasdair David Wilson, M.Sci.
Astronomy and Astrophysics Group
SUPA School of Physics and Astronomy
Kelvin Building
University of Glasgow
Glasgow, G12 8QQ
Scotland, U.K.
Presented for the degree of
Doctor of Philosophy
The University of Glasgow
January 2016
Abstract
This thesis deals with the theoretical and numerical modelling of partially ionized
plasmas. The study of partially ionized plasmas is important in both astrophysical and
laboratory contexts and we present a novel finite difference approach to modelling a
magnetohydrodynamicplasmaandhydrodynamicgasaswellassomeinteractionterms
between them. In particular we model fluid limits of a collisional drag, momentum
coupling term and a critical velocity (Alfv´en ) ionization term.
Chapter 1 reviews the necessary background material relevant to this thesis. We
introduce relevant plasma parameters that are useful to help understand the regimes at
which MHD operates and which are later used when placing criteria on the conditions
required for, and the rate equations of, ionization. We introduce the fluid limit model
of non-resistive plasmas (ideal MHD), as well as theoretical models for gas-plasma
collisions and Alfv´en ionization.
Chapter 2 lays out the model of a linear finite difference gas-MHD momentum
coupling code (GMMC) that we modify by the addition of a fluid Alfv´en ionization
term. We explore the codes stability and fidelity and we explore Fresnel interference
patterns as a test scenario.
Chapter 3 lays out the model equations and derivation of a non-linear finite differ-
ence gas-MHD interactions code (GMIC).
Chapter 4 uses the code GMIC to explore the momentum coupling between the gas
and plasma fluids in a non-linear regime. We show that the presence of a frictional
drag term effects both fluids in a variety of simulated scenarios. Propagation of waves
and diffusion are effected significantly across a range of parameter space. We show the
formation of ‘plasmoids’ by interacting, momentum coupled waves. We see that the
momentum coupling has a somewhat similar effect on plasmas as a resistive term does.
Chapter 5 uses both the linear and non-linear codes to simulate Alfv´en ionization
in a variety of scenarios. With both codes we see that waves and flows can be sources
of ionization of the relative velocity between the two simulated fluids exceeds a pre-set
threshold. This ionization effects the dynamics of the system significantly, firstly the
extraction of kinetic energy in order to ionize introduces a directional and amplitude
dependant damping of waves; secondly by providing a source of new plasma that the
fluid is forced to react to. Also in this chapter we discuss how Alfv´en ionization might
play a role in astrophysical contexts, in particular, the solar photosphere and brown
dwarf atmospheres.
Chapter 6 deviates from the previous work to explore the possibility that dielec-
trophoretic forces may be able to stratify the dynamic atmosphere of the Sun. We
derive a simple expression for the DEP-force due to a neutral particle moving through
a magnetic field and becoming polarized and we examine the displacement caused
by this field and the ambipolar field. We also simulate the ability of a hypothetical
DEP-force to separate elements based on their polarizability to mass ratio.
Chapter 7 summarises the conclusion of the previous chapters and includes a brief
discussion about possible extensions to this work.
Contents
Preface ii
List of Tables viii
List of Figures ix
Acknowledgements xiii
1 Introduction 2
1.1 Characteristics of a Plasma . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Debye length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Plasma Parameter . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.3 Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 Cyclotron frequency . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2 Collisions in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1 Charged-neutral collisions . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Electron-ion interactions . . . . . . . . . . . . . . . . . . . . . . 15
1.2.3 Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3 Ionization Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3.1 Electron Impact Ionization . . . . . . . . . . . . . . . . . . . . . 19
1.3.2 Electrical Breakdown . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 Bulk Plasma Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.1 Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.4.2 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . 30
CONTENTS v
1.5 Modifications to MHD for a Partially Ionized Plasma . . . . . . . . . . 36
1.5.1 Momentum Coupling . . . . . . . . . . . . . . . . . . . . . . . . 37
1.5.2 Critical Velocity/Alfv´en Ionization . . . . . . . . . . . . . . . . 39
2 Linear Gas/MHD Interactions Code 49
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 The Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2.1 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Finite Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.3.1 Lax-Wendroff Scheme . . . . . . . . . . . . . . . . . . . . . . . . 58
2.4 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4.1 Stability Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.5 Deviations from MHD . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.5.1 Momentum Coupling . . . . . . . . . . . . . . . . . . . . . . . . 71
2.5.2 Alfv´en Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.6 Example Result: Interference Fringes . . . . . . . . . . . . . . . . . . . 76
2.6.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 77
2.6.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.7 Limitations of GMMC . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3 Non-Linear Gas/MHD Interactions Code 90
3.1 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.1.1 Conservative Form MHD . . . . . . . . . . . . . . . . . . . . . . 92
3.1.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.1.3 Flux Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.2 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2.1 Lax-Friedrichs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3.2.2 Richtmyer Two-Step . . . . . . . . . . . . . . . . . . . . . . . . 104
3.2.3 MacCormack . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
CONTENTS vi
3.2.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.4 The Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.4.1 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.5 Gas Plasma Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4 Momentum Coupling in a Partially Ionized Plasma 114
4.1 Cylindrical bombs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.1.1 Positive Gaussian Cylindrical Bomb . . . . . . . . . . . . . . . . 115
4.1.2 Negative Gaussian Density Cylindrical Bomb . . . . . . . . . . . 121
4.2 Coupled Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5 Ionization in a Gas-MHD Plasma 136
5.1 Critical Velocity Ionization in a Linear Regime . . . . . . . . . . . . . . 136
5.1.1 Bulk Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.1.2 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.1.3 Cylindrically Symmetric Waves . . . . . . . . . . . . . . . . . . 148
5.2 Critical Velocity Ionization in a Non-Linear Regime . . . . . . . . . . . 154
5.2.1 Ionization from Gas Bomb . . . . . . . . . . . . . . . . . . . . . 154
5.2.2 Ionization from Double Bombs . . . . . . . . . . . . . . . . . . . 159
5.2.3 Interplay with Momentum Coupling . . . . . . . . . . . . . . . . 164
5.3 Astrophysical Contexts . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
5.3.1 Brown Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.3.2 Solar Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.3.3 Epoch of Recombination . . . . . . . . . . . . . . . . . . . . . . 174
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6 Dielectrophoresis in Astrophysical Plasmas 178
6.1 DEP-FFF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
CONTENTS vii
6.2.1 p/m Ratio of Select Elements . . . . . . . . . . . . . . . . . . . 183
6.2.2 Solar Lorentz Field . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.2.3 Ambi-polar Field . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
7 Conclusions 199
7.1 Further Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Bibliography 205
List of Tables
6.1 Selected atomic characteristics . . . . . . . . . . . . . . . . . . . . . . . 188
6.2 Selected molecular characteristics . . . . . . . . . . . . . . . . . . . . . 189
List of Figures
1.1 The equilibrium ionization fraction of hydrogen . . . . . . . . . . . . . 24
1.2 The Paschen curve for an arbitrary gas . . . . . . . . . . . . . . . . . . 26
1.3 Electrical schematic of a homopolar CIV device . . . . . . . . . . . . . 41
1.4 Burning voltage against discharge current for a homopolar device . . . 42
1.5 Schematic showing Alfv´en ionization mechanism . . . . . . . . . . . . . 43
2.1 Time evolution of the 1-dimensional wave equation solved using a FTCS
scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.2 Time evolution of the 1-dimensional wave equation solved with an LF
scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.3 The time evolution of the 1 dimensional wave equation solved with a
LW scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.4 Flowchart showing the process for incorporation of an Alfv´en ionization
subroutine into a finite difference code . . . . . . . . . . . . . . . . . . 77
2.5 The spatial envelope, A for a set of discrete wave sources that will form
m
Fresnel interference fringes. . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.6 Plasma density map showing interference patterns in an uncoupled gas-
plasma mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.7 Vector diagram of the magnetic field in the uncoupled case . . . . . . . 81
2.8 Plasma density map showing interference patterns in a coupled gas-
plasma mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.9 Vector diagram of the magnetic field in the coupled case . . . . . . . . 84