臺灣博碩士論文加值系統

摘要

　　電流片的兩側具有一對反平行磁場，此區域會發生磁場重聯的現象。磁場重聯會導致重聯層的形成，在重聯層中磁能會被轉換成電漿的動力能或熱能，並形成磁流體力學中的震波和不連續結構。在炙熱的電漿中，例如：日冕或是太陽風，由於熱傳導的時間尺度和Alfvén波模的時間尺度是相近或較小的，因此熱傳導效應必須被考慮。在本論文中，藉由一維的Riemann problem初值問題，假設在時間開始時，電流片的兩端具有反平行的磁場，來研究在磁場重聯層的熱傳導效應和磁場By分量的影響。在模擬中我們使用具熱傳導效應和磁阻抗的耗散性的磁流體力學程式(dissipative MHD)。

　　我們首先考慮在磁阻抗磁流體力學(resistive MHD)中磁場By分量對電流片發展過程的影響。電流片兩端的磁場初值設定如下，B(z) = −Bxotanh(z/delta) +By +Bz ，此處Bx0是反平行磁場的分量。在一個By=0的情況下，形成一對慢震波。而在By≠0時 (或是一個非常小的磁場By分量)，則形成一對慢震波和一對隨時變的中速震波(TDISs)。我們需注意的是TDIS並不存在By = 0的例子中，也就是說By = 0是一個特例。越過TDIS結構，電漿的密度和壓力增加而磁場減少。我們進一步發現越過TDIS的切向磁場旋轉角，Δphi，隨時間發展並逐漸達到最後值Δphi(final) = 90°–phi∞, 此處phi∞ = tan-1(By/Bx) 是越過電流片切向磁場旋轉角度的一半。慢震波下游的溫度及壓力隨著phi∞增加而漸減時而隨著電漿beta∞增加而漸增。

　　其次我們研究平行磁場的熱傳導分量對慢震波結構的影響。我們發現慢震波分為兩個部分：一個是等溫的主震波結構，一個是波前震波結構。越過主震波結構，電漿的密度，速度和磁場有明顯的躍遷，但溫度卻是連續的。由於熱流從下游流至上游，因此形成一個溫度平滑變化的波前震波結構。主震波下游的電漿密度隨時間增加而減少，溫度增加，而壓力是保持定值。我們可由修正的等溫Rankine-Hugoint條件去計算主震波下游的電漿密度、壓力、速度和磁場。同時也發現經過主震波結構後，存在一個溫度梯度的躍遷，這是為了滿足能量守恆。

　　最後我們研究同時具熱傳導條件和磁場By分量的條件下，慢震波結構和中速震波的演變。如同之前所討論，模擬結果形成一對慢震波和一對TDISs，而每一個慢震波包含兩個部分：一個等溫的主震波和一個波前震波結構。波前震波將會到達平衡態，並在主震波的前方維持一個固定寬度。一開始時，TDIS被侷限在波前震波結構中，而後會跑出波前震波的結構。隨磁場By分量的增加，波前震波前端的傳播速度減少，並較早達到一個穩定態，且慢震波下游的壓力和溫度均是減少。這些研究結果將可運用於在日冕區和太陽風的震波加熱。

Abstract

　Magnetic reconnection usually takes place in a current sheet that separates two plasma regions having an antiparallel magnetic field component. Magnetic reconnection can lead to the formation of reconnection layers, where magnetic energy is conversed to plasma kinetic/thermal energy and magnetohydrodynamic (MHD) shocks and discontinuities are formed. In a hot plasma, such as in the solar corona or solar wind, the heat conduction effects should be considered because the conduction timescale is comparable to or shorter than the Alfvén timescale. In this thesis, the effects of heat conduction and magnetic guide field By in the magnetic reconnection layer is studied by solving one-dimensional Riemann problem for the evolution of an initial current sheet with an antiparallel component of magnetic fields on the two sides. In the numerical simulations, a dissipative MHD code with heat conduction and magnetic resistivity is used.

　First, the By effects on the current sheet evolution is considered in the resistive MHD. The initial magnetic fields across the initial current sheet is set to be B(z) = −Bxotanh(z/delta) +By +Bz , where Bx0 is the antiparallel component. In the symmetric case with By = 0, a pair of slow shocks are formed. For By ≠ 0 cases (even for a very small By), it is found that a pair of slow shocks and a pair of time-dependent intermediate shocks (TDISs) are formed. Notice that TDIS is not present in the By = 0 case. It is apparent that the case with By = 0 is a singular case. The plasma density and pressure increase and the magnetic field decreases across TDIS. It is further found that the rotation angle of tangential magnetic field across TDIS, Δphi, develops with time and gradually reaches its final value. It obeys Δphi(final) = 90°–phi∞, where phi∞ = tan-1(By/Bx) is half of the rotation angle of tangential magnetic field across the initial current sheet. Both pressure and temperature downstream of the slow shock decrease with phi∞, and increase with plasma beta∞.

　Second, the structure of slow shocks in the presence of a heat conduction parallel to the local magnetic field is studied. It is found that the slow shock consists of two parts: the isothermal main shock and foreshock. Significant jumps in plasma density, velocity and magnetic field occur across the main shock, but the temperature is found to be continuous across the main shock. The foreshock is featured by a smooth temperature variation and is formed due to the heat flow from downstream to upstream region. The plasma density downstream of the main shock decreases with time, while the downstream temperature increases with time, keeping the downstream pressure constant. It is shown that the jumps in plasma density, pressure, velocity, and magnetic field across the main shock are determined by the set of modified isothermal Rankine-Hugoniot conditions. It is also found that a jump in the temperature gradient is present across the main shock in order to satisfy the energy conservation.

　Third, the structure of slow shocks and intermediate shocks in the presence of a parallel heat conduction for an initial current sheet with By≠0 is studied. As before, a pair of slow shocks and a pair of TDISs are formed, and each slow shock consists of two parts: the isothermal main shock and the foreshock. The foreshock is found to reach a steady state with a constant width in the slow shock frame. The TDIS initially can be embedded in the slow shock’s foreshock structure, and then moves out of the foreshock region. With an increasing By, the propagation speed of foreshock leading edge tends to decrease and the foreshock reaches its steady state at an earlier time. Both the pressure and temperature downstream of the main shock decrease with increasing By. The present results can be applied to the shock heating in the solar corona and solar wind.

Contents

……………………………………………… Page

Abstract…………………………………………ii
Acknowledgments………………………………vi
Contents…………………………………………viii
List of Figures……………………………………xi
List of Tables……………………………………xii

Chapter 1 Introduction 1

1.1 Magnetic Reconnection 1
1.1.1 Basic Concept of Magnetic Reconnection 3
1.1.2 Petschek’s Model 5
1.2 Ideal MHD Equations 7
1.2.1 Linear MHD waves 8
1.2.2 Formation Process of Shock From Steepening Wave 12
1.3 MHD Shocks and Discontinuities 13
1.3.1 Rankine-Hugoniot Relations 13
1.3.2 Co-planarity Condition 19
1.3.3 Jump Solutions Across the Shock 20
1.4 Riemann Problem 24
1.5 Observations of Slow Shocks and Intermediate Shocks 27
1.6 Effects of Heat Conduction 31
1.7 Outline of the Thesis 32

Chapter 2 Effects of By Component on the Evolution of a Current Sheet 35

2.1 Introduction 35
2.2 Intermediate Shocks and Slow Shocks in the Resistive MHD Model 38
2.2.1 Resistive MHD Equations 40
2.2.2 Normalization 42
2.2.3 Numerical Scheme 42
2.2.4 Initial Conditions 45
2.3 Simulation Results 46
2.3.1 Case 1: without By Component (By ＝ 0) 47
2.3.2 Case 2: with By ≠ 0 (phi∞≠ 0) 48
2.3.3 Dependence on the Initial Rotation Angle (phi∞) 55
2.3.4 Plasma-Beta (β∞) Dependence 63
2.4 Summary 68

Chapter 3 Structure of Slow Shocks
with Heat Conduction 70

3.1 Introduction 70
3.2 Heat-Conduction MHD Simulation Model 71
3.2.1 Heat-Conduction MHD Equations 74
3.2.2 Normalization 75
3.2.3 Numerical Scheme 76
3.2.4 Initial Condition 83
3.3 Simulation Results 84
3.3.1 Case 1: Heat Conduction Effect
(K0 ≠ 0) 84
3.3.2 K0 Dependence 87
3.3.3 Plasma-Beta (β∞) Dependence 88
3.4 Modified Isothermal Rankine-Hugoniot Jump Conditions 92
3.5 Summary 98

Chapter 4 Structure of Intermediate Shocks and Slow Shocks with Heat Conduction 100

4.1 Introduction 100
4.2 Intermediate Shocks and Slow shocks in the Heat-Conduction MHD Model 101
4.2.1 Equations, Normalization, Numerical Scheme and Initial Conduction 101
4.3 Simulation Results 102
4.3.1 Case 1: K0 ≠ 0 without By Component
(By ＝ 0) 102
4.3.2 Case 2: K0 ≠ 0 with By Component
(By ≠ 0) 108
4.3.3 K0 Dependence 113
4.3.4 Dependence on the Initial Rotation
Angle (phi∞) 113
4.4 Summary 118

Chapter 5 Summary 120

5.1 Effect of By Component 121
5.2 Effects of Heat Conduction 122
5.3 Effects of Both By Component and Heat Conduction 124

Appendix A 126

References 129

Vita 138

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