| 摘要: | 摘要 撕裂模不穩定性可發生於核融合電漿、太陽表面、地球磁層頂及磁尾等及天文系統中。撕裂模不穩定性透過磁力線重連的過程將不同方向之磁力線結合形成磁島，並將電流片所儲存之磁能轉換為磁流體動能與熱能。本研究之目的即是以壓力均向與非均向磁流體的觀點對撕裂模之不穩定性做一有系統的分析與研究。藉由線性及非線性的二維磁流體模式之數值計算，分別探討不同熱力條件下，磁雷諾數、擾動波長、電漿 beta、沿電流片方向之磁場分量 及壓力非均向性等參數對撕裂模不穩定性之影響。由線性計算，我們可得出撕裂模不穩定性之特徵模解，而非線性計算部分，為了避免人為的任意初始擾動干擾非線性不穩定之發展，我們先經由線性計算篩選出合適的特徵模解做為非線性計算之初始擾動，並於非線性的模式中由線性擾動自然發展至非線性階段。 壓力均向電漿中之撕裂模不穩定性的線性研究結果顯示，在固定其他參數的情況下，最快的線性成長率會隨著磁雷諾數的值增加而減小，但電漿 beta及磁場分量 則對撕裂模不穩定性之成長率沒有明顯的影響。在非線性的研究中則發現，較小的磁雷諾數雖然對應較大的線性成長率，但在非線性系統中則無法造成撕裂模不穩定性發展；只有適當的磁雷諾數範圍下，撕裂模不穩定性才能夠順利成長並達到非線性飽和。我們將比較非線性撕裂模不穩定性之結果與Hau及Sonnerup[1999]由人造衛星觀測與理論模式得出之地球磁層頂磁島結構。 由於壓力非均向性為太空環境中無碰撞電漿之基本特性，我們因此對壓力非均向電漿中之撕裂模不穩定性做深入的探討，所使用之能量方程為較雙絕熱定律廣義之Double-Polytropic Laws。除了研究擾動波長、磁雷諾數、電漿 beta及磁場分量 對撕裂模不穩定性之效應外，不同的壓力非均向性 以及熱力條件 ， 的影響更是此部分研究的重點。線性的研究結果顯示，在某些參數條件下，最快的線性成長率會隨著磁雷諾數的值增加而增加，而電漿 beta及磁場分量 對撕裂模不穩定性成長率則有相當大之影響，這與壓力均向撕裂模不穩性之特性不同。我們發現，線性成長率會隨著 ,電漿 beta之增加與 ， 值之減小而增加，此結果與磁鏡不穩定性之發生條件吻合。在某些參數情況下，撕裂模不穩定性的成長率可達Alfvén時間尺度，且除了純粹成長的情形外，更有伴隨著週期性震盪的成長形式。此種震盪式成長的撕裂模不穩定性，其磁力線結構會隨著震盪的週期時而發展為O型的封閉磁力線（即磁島），時而壓縮成X形磁力線，兩種型態的磁力線結構隨著震盪而交互呈現，其磁場擾動與電漿密度擾動會呈現反相關（或異常的正相關）之慢速或磁鏡波模結構。在非線性的研究中，我們探討的重點為震盪型的撕裂模不穩定性能否能存在於實際的非線性系統，此問題在現有的壓力非均向磁流體文獻中並未探討過。模擬結果顯示，在適當的線性成長率及震盪週期情況下，擁有震盪現象的撕裂模不穩定性能在非線性演化中震盪成長並迅速達到飽和，且在發展過程中能使電漿加速至Alfvén速率。分析顯示，磁鏡波於X型與O型磁場結構上所形成的磁瓶，為造成撕裂模不穩定性震盪與成長率顯著增加之主要原因。 Abstract The resistive tearing-mode instabilities of Harris sheet magnetic field configuration are studied based on two-dimensional, resistive, compressible MHD models with isotropic and anisotropic pressures, respectively. Different energy closures along with various parameter regimes of magnetic Reynolds number Rm, plasma beta, pressure anisotropy , magnetic By and the ratio of wavelength to the layer thickness are explored to give a systematic analysis. The linear growth rate and the eigenmode structures are calculated from the linear numerical models. The linear solutions are then used as initial perturbations of the nonlinear numerical models to allow the full evolution of resistive tearing-mode instability. The linear calculations of isotropic resistive tearing-mode instability show that for the range of Rm=10-105 the fastest growth rate increases with decreasing Rm and only slightly increases with increasing plasma beta and By but is not sensitive to the equation of state. While the nonlinear calculations of isotropic tearing-mode instability show that only for large Rm, where the diffusion time is much larger than the linear growth time, the magnetic island may possibly grow substantially and become saturated. For small Rm the plasmoids either diminish in the late stage or do not have apparent growing, that is, the linear analysis is not meaningful for large resistivity cases. The calculations are compared to the magnetic island structure at earth’s magnetopause reconstructed from the single-spacecraft data by Hau and Sonnerup [1999]. The anisotropic resistive tearing-mode instability is studied within the framework of gyrotropic MHD theory for which the standard CGL or double adiabatic laws are replaced with the more generalized double-polytropic equations to incorporate various thermodynamic states of collisionless plasmas. The linear calculations show that the dependence of linear growth rate on the energy equations, plasmas beta as well as the magnetic By component is much more pronounced than that in isotropic plasmas; in particular, the growth rate is larger for smaller and as well as for larger and , a tendency in accordance with the mirror instability criterion based on the double-polytropic MHD model. For certain parameter regime, the growth rate may even increase with increasing magnetic Reynolds number, a result in contrast to the isotropic tearing instability. For sufficiently large pressure anisotropy of , the eigenmode is not purely exponential but contains oscillation with the overall growth rate much larger than the isotropic case. Cases of large growth rate on the order of Alfvén time scale are associated with oscillatory slow-mode structures that may exhibit negative as well as positive density-magnetic field correlation as predicted by Hau and Sonnerup [1993] being one of the anomalous behaviors associated with slow-mode waves in anisotropic plasmas. The nonlinear evolution shows for the first time that the oscillatory solutions may develop in the nonlinear anisotropic MHD model; in particular, the X-line and O-line take place alternatively with magnetic islands growing and saturated with oscillation period of about ten Alfvén transit time. The growth rate is on the order of Alfvén time scale and the plasma flow velocity may reach the Alfvén speed. The presence of mirror-mode wave may result in magnetic bottle structure on top of the X and O lines that is responsible for the enhanced growth rate and oscillation of magnetic reconnection. |
