Mathematical modelling of oscillations and wake waves in plasma

310 Pages·2019·36.876 MB·English

by  Chizhonkov, Evgeniĭ Vladimirovich

Mathematical Aspects of
Modelling Oscillations and
Wake Waves in Plasma

Mathematical Aspects of
Modelling Oscillations and
Wake Waves in Plasma
E.V. Chizhonkov
Translated from Russian by V.E. Riecansky
CRC Press
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Contents
Preface ix
Part I: Free plasma oscillations
1. Introductory information 1
1.1. What is breaking? 1
1.2. Physical model and basic equations 5
1.3. About initial conditions 13
1.4. About boundary conditions 16
1.5. Bibliography and comments 18
2. Plane one-dimensional non-relativistic electron oscillations 23
2.1. Problem statement in Eulerian and Lagrangian variables 23
2.2. Axial solutions 25
2.3. ‘Triangular’ solutions 36
2.3.1. Simple solutions 37
2.3.2. Composite solutions 38
2.4. Numerical–analytical method 40
2.5. Bibliography and comments 44
3. Plane one-dimensional relativistic electron oscillations 48
3.1. Problem statement in the Eulerian and Lagrangian
variables 48
3.2. Theoretical background of breaking 50
3.2.1. Quadratic frequency shift 51
3.2.2. Violation of the property of invariance 54
3.3. Method in Lagrangian variables 55
3.4. Scenario of development and completion of oscillations 57
3.5. Method in the Eulerian variables 62
3.6. Artificial boundary conditions 65
3.6.1. Full damping of oscillations 66
3.6.2. Linearization of the original equations 67
3.6.3. Accounting for the weak nonlinearity of the original
equations 68
3.6.4. Deterioration of the approximation at the boundary 69
3.7. Bibliography and comments 71
vi Contents
4. Cylindrical one-dimensional relativistic and non-
relativistic electron oscillations 76
4.1. Problem statements in Eulerian and Lagrangian variables 76
4.2. Analytical studies 82
4.2.1. Axial solution 82
4.2.2. Perturbation method 87
4.3. Finite difference method 89
4.3.1. Auxiliary designs. Splitting into physical processes 89
4.3.2. Construction of difference schemes 91
4.3.3. Process scenario 94
4.5. Calculation of axial solutions 103
4.5.1. Free non-relativistic oscillations 103
4.5.2. Forced relativistic oscillations 110
4.6. About spherical oscillations 118
4.6.1. Problems formulation 118
4.6.2. Axial solution 122
4.6.3. Perturbation method 125
4.6.4. For numerical modelling 127
4.7. Bibliography and comments 129
5. Influence of ion dynamics on plane one-dimensional
oscillations 132
5.1. Formulation of the problem 132
5.2. Scaling equations and difference scheme 136
5.3. Axial solution 140
5.4. Calculation results 144
5.5. Bibliography and comments 148
6. Plane two-dimensional relativistic electron oscillations 151
6.1. Formulation of the problem 151
6.2. Asymptotic theory 153
6.3. Difference scheme 157
6.3.1. Difference equations in the internal nodes of the grid 159
6.3.2. Implementation of the artificial boundary conditions 161
6.4. Numerical experiments 163
6.4.1. General remarks 163
6.4.2. Calculations with circular symmetry 165
6.4.3. Quasi-one-dimensional model 168
6.4.4. Small deviation from circular symmetry 173
6.4.5. Significant difference from circular symmetry 176
6.5. Bibliography and comments 180
Contents vii
Part II: Plasma wake waves
7. Introductory information 182
7.1. Source equations 182
7.2. The case of an arbitrary pulse velocity 186
7.2.1. Equations in scalar form 186
7.2.2. New coordinates and quasistatics 187
7.2.3. Equations in dimensionless variables 188
7.2.4. Equations in convenient variables 189
7.3. The basic formulation of the problem 192
7.3.1. Nonlinear statement 192
7.3.2. Linearized formulation 194
7.4. ‘Slow’ pulse 196
7.4.1. Linearized equations 196
7.4.2. Auxiliary Cauchy problem 197
7.4.3. Numerical–asymptotic method 201
7.5. Bibliography and comments 203
8. Numerical algorithms for the basic problem 208
8.1. Difference method I 208
8.1.1. Construction of a difference scheme 208
8.1.2. Study of schemes in variations 211
8.1.3. The algorithm for implementing the difference scheme I 214
8.2. Difference method II 216
8.2.1. Construction of a difference scheme 216
8.2.2. Study of schemes in variations 218
8.2.3. Algorithm for the implementation of difference scheme II 219
8.3. Difference method III (Linearization method) 221
8.3.1. Setting the task in a convenient form 221
8.3.2. Preliminary transformations 223
8.3.3. Difference method III in the linear case 225
8.3.4. Difference method III in the nonlinear case 226
8.4. Projection method 228
8.4.1. Setting the problem in a convenient form 228
8.4.2. Description of the projection method 229
8.4.3. Numerical implementation of the projection method 232
8.5. Numerical experiments and comparison methods 233
8.6. Bibliography and comments 239
9. Additional research 246
9.1. Axial wake wave solution 246
9.1.1. Formulation of the ‘truncated’ problem 246
9.1.2. Numerical algorithm for solving the ‘truncated’ problem 250
9.1.3. Calculation results 251
9.2. Accounting for the dynamics of ions in the wake wave 257
viii Contents
9.2.1. Problem statement in physical variables 257
9.2.2. Statement of the problem in convenient variables 260
9.2.3. Solution method 262
9.2.4. Calculation results 266
9.3. Elliptical pulse 268
9.3.1. Formulation of the problem 268
9.3.2. Difference scheme and solution method 273
9.3.3. Calculation results 277
9.4. Bibliography and comments 279
Conclusion 281
References 284
Index 291
Preface
This book is fully devoted to one of the most popular areas in
modern mathematics – numerical modelling. However, its appearance
is due to a qualitative, one might say revolutionary, change in the
situation in another field of science – in physics. The fact is that at
present there is an urgent need to rethink ideas about the interaction
of electromagnetic fields with matter. Let us give a weighty argument
in favour of this statement from the popular science article ‘Why do
we need high-intensity laser pulses?’ [43] written by L.M. Gorbunov,
an outstanding specialist in the field of plasma physics (1934–2007).
The simplest atom is the hydrogen atom. In it, a single electron
moves around the nucleus (proton). The electric field, due to which
the two particles are held near each other, is about 5∙109 V/cm. This
is a very strong field compared to those found in everyday life that
surrounds us. For example, the breakdown of such a good insulator,
like mica, occurs at 2∙106 V/cm (the field here is one thousand times
weaker!). The classical theory, in essence, is a ‘theory of small
perturbations’, that is, it assumes a small external field compared
to fields that keep the atomic systems in equilibrium. But in recent
years lasers have been created that generate high-intensity ultra-short
light pulses. During experiments with them, new physical phenomena
were discovered, the possibilities of using pulses in various fields,
ranging from nuclear physics and astrophysics to medicine, are
analyzed. These pulses have a duration of less than 1 picosecond
(that is, less than 10−12 s). Their length in space is less than 300
microns, which is less than a third of a millimeter. The wavelength
of radiation is usually about 1 micron, and it belongs to the infrared
range. Tens or hundreds of wavelengths are stacked on the pulse
length. The energy that carries such an pulse can reach hundreds
of joules, and power – up to 1015 watts. This value is called the
petawatt. It far exceeds the total capacity of all power plants in the
world. If such a pulse is focused on a pad with a radius of 10 μm,