宇宙射線和磁流動力系統之不穩定性

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羅英奕(Ying-Yi Lo) 查詢紙本館藏

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論文名稱

宇宙射線和磁流動力系統之不穩定性
(Instability of the Cosmic Rays and MHD Waves System)

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摘要(中)

從歷史上，我們學到如三國時關公入千軍萬馬取上將首級如「探囊取物」、唐朝郭子儀「單騎退敵」，等等。這些人雖然為數不多，但卻扮演著關鍵性的角色。就如同自然界中，宇宙射線雖然在數量上很少，但卻有相當高的能量，而是否對宇宙有一定的影響？本論文以解析和數值模擬的方法，研究宇宙射線和電漿耦合的系統。
第一章提出一自洽的方程組，作為我們的模型。它們相互影響，彼此襯托，以流體的方式描繪出宇宙射線，電漿，磁場和波的關係。於第二章利用解析的方法，討論此模型的穩定性。除了對參數空間在一般及極限情況下外，也對第一章模型的穩態解，廣泛而深入的探究模型在線性微擾方面的種種。之前的研究已知由熱電漿、宇宙射線及反向艾爾文波（Alfvén waves）構成的宇宙射線-電漿系統，顯示系統對由宇宙射線驅動的磁聲（magnetoacoustic）不穩定性非常敏感。加入正、反向的艾爾文波能量方程後，形成一個四流體-即熱電漿和宇宙射線外，再加入正、反向的艾爾文波。此時，二階費米（Fermi）效應自然就會出現在這系統中。這系統能量交換有三種：（一）、波的能量由宇宙射線透過自我激發的效應產生，（二）、波則藉由二階費米加速將能量傳送給宇宙射線，（三）、宇宙射線壓力梯度與波的壓力梯度則傳遞能量給電漿；反之亦然。第三章則省略模型中波的效應，利用 MOCCT 數值的方法（對法拉第方程，是先求出電場），模擬在星系盤面上三維的 Parker 不穩定性。對宇宙射線擴散項先用 Biconjugate gradient stabilized （BICGStab）方法，再以修正的Lax-Wendroff 方法求得宇宙射線能量密度。我們在解 MHD 方程上用修正的 Lax-Wendroff 方法，而在針對解宇宙射線擴散方程上選擇了隱性（implicit）的 BICGStab 方法。所以整體而言，我們是利用混合（hybrid）數值方法。在非線性演化期間，我們發現了一些特性。當擴散係數遞減時，宇宙射線的壓力分佈從原來的均勻分佈變成集中在磁泡的足點附近；同時，宇宙射線的壓力梯度迫使磁泡的頂端變得比較大。於是，向下掉落的物質被壓力梯度所阻，因此減緩了不穩定的成長。此外，在演化末期，三維模擬由於交換不穩定（interchange instability）的參與，其結果與二維非常不同。

摘要(英)

In histroy, there are only few people who have very high positive energy, however, they trun the table in critical time. Analoginally, the cosmic rays with the energy density is larger than plasma and magnetic field, it should participate in and influence the evolutin of astronomical enviroments, despite they are rare. This thesis investigates a self-consistent hydrodynamical model, which comprises magnetized thermal plasma, cosmic rays, forward and backward propagating Alfvén waves.
Chapter 1 introduces our four-fluids model, begin from the cosmic ray transport equation, after frames trasformation and includes the relation between scattering frequencies and gwowth rate of Alfvén waves, intergal the momentum, then toward the hydrodynamics model.
In chpater 2, we study the stability of our model and discuss basic linearly and analytically. Prior rsearch indicated there is magneto-acoustic instability driven by cosmic ray and backward Alfvén wave excited by streaming instability. As the result by adding the forward Alfvén wave, the second order Fermi acceleration effect arises in our four-fluid model (i.e. cosmic ray, plasma, forward and backward Alfvén waves), spontaneousness. This cosmic-ray plasma and waves system exchange energy among cosmic ray, plasma and waves via: (1) waves gain energy from self-excite effect by cosmic ray; (2) the second order Fermi effect transfer energy from waves to cosmic ray, and (3) The work done by pressure gradient of cosmic ray and waves lead the plasma gain energy from cosmic ray and waves,vice versa.
In chapter 3, by using the MOCCT (Method of Characteristics/Constrained Transport) MHD code, we exploits a 3D numerical simulation, points on the Parker instability, but ingores the effects of self-gravity and waves. After sloved the diffusion term of cosmic ray energy equation via BICGStab (Biconjugate gradient stabilized) method, then obtained the convection trem and other MHD equations by modified Lax-Wendroff method. In general, we stduy 3D Parker instability including the cosmic ray effect with a hybrid numerical method.
During the epoach of non-linear stage, we found some characteristics: the cosmic ray pressure distribution is rather nonuniform. Cosmic rays tend to accumulate near the footpoint of the magnetic loop, and the cosmic ray pressure gradient force toward the top of the loop becomes larger. The falling motion of matter is then impeded by the cosmic ray pressure gradient force, and the growth rate of the Parker instability decreases. For 3D case, at near the end of evolution, due to the interchange mode participate in the system, the results are very different between 3D and 2D.

關鍵字(中)

★ 不穩定性
★ 磁流動力
★ 宇宙射線

關鍵字(英)

★ instability
★ MHD
★ cosmic ray

論文目次

Chinese Abstract i
Abstract iii
Chinese Acknowledgments v
Acknowledgments vii
Contents ix
1 Model 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Scattering of Cosmic Rays . . . . . . . . . . . . . . . . 3
1.1.2 Streaming Instability . . . . . . . . . . . . . . . . . . . 5
1.2 The Cosmic Ray Transport Equation . . . . . . . . . . . . . . 5
1.2.1 Transform to The Original Frame of Reference . . . . . 7
1.2.2 Relation Between Scattering Frequencies and Growth
Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Toward The Hydrodynamics . . . . . . . . . . . . . . . . . . . 11
1.3.1 Complete Set of Equations . . . . . . . . . . . . . . . . 13
1.4 Nonlinear Test Particle Picture . . . . . . . . . . . . . . . . . 15
1.5 Steady State Solution . . . . . . . . . . . . . . . . . . . . . . . 18
2 Instability Analysis of Cosmic Rays And Waves System 1
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.1 The Concept of Stability . . . . . . . . . . . . . . . . . 23
2.2 Methods to Study Instability . . . . . . . . . . . . . . . . . . . 26
2.2.1 Energy Principle . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 Interchange Instability . . . . . . . . . . . . . . . . . . 29
2.2.3 Normal Mode Analysis . . . . . . . . . . . . . . . . . . 30
2.2.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.5 Generalized Hermite-Biehler theorem . . . . . . . . . . 32
2.2.6 Routh-Hurwitz Stability Criterion . . . . . . . . . . . . 33
2.2.7 Sylvester matrix . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Linear Stability Analysis on The Four-
uid Model . . . . . . . 12
2.4 Results I { Special Case . . . . . . . . . . . . . . . . . . . . . 39
2.4.1 Nonlinear Test Particle Picture Case . . . . . . . . . . 39
2.4.2 Perpendicular perturbations (mu = 0) Case . . . . . . . 48
2.4.3 Unidirectional wave system (P+w0 = 0 or P-w0 = 0) . . . 49
2.4.4 Large cosmic ray pressure (Pc0 >> (P+w0 + P-w0)) . . . . . 50
2.4.5 Vanishing cosmic ray pressure (Pc0 ~ 0) . . . . . . . . 52
2.4.6 Large wavenumber (k lD) . . . . . . . . . . . . . . . 52
2.4.7 Vanishing wavenumber (~k ~ 0) . . . . . . . . . . . . . 53
2.5 Results II General cases . . . . . . . . . . . . . . . . . . . . 54
2.5.1 (e-; ec)-plane . . . . . . . . . . . . . . . . . . . . . . . 54
2.5.2 (e-; ~k)-plane. . . . . . . . . . . . . . . . . . . . . . . . 54
2.5.3 (e-; mu)-plane . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5.4 (mu; ~k)-plane . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5.5 (mu; betaw)-plane . . . . . . . . . . . . . . . . . . . . . . . 55
2.5.6 ( betas; beta w)-plane . . . . . . . . . . . . . . . . . . . . . . 56
2.5.7 (~nug0; betaw)-plane . . . . . . . . . . . . . . . . . . . . . . 56
2.6 Results III { An Example . . . . . . . . . . . . . . . . . . . . 57
3 3D Simulation of Parker Instability with Cosmic Ray 65
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1.1 Lax-Wendroff Method . . . . . . . . . . . . . . . . . . 68
3.1.2 Time Splitting And BICGstab Method . . . . . . . . . 69
3.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . 73
3.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 76
3.3 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.4 Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.1 kappa = 200 . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.2 kappa = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4 Summary 97
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
A Linearized Equations for System 107
B Coefficient for Polynomial 111
C Criteria for real roots of a quartic equation 115
D Lax-wendroff Metods 117
E BICGstab Method 119

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指導教授

高仲明(Chung-Ming Ko)

審核日期

2009-7-15

推文