电子磁流体中的磁场重联不稳定性的理论研究
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摘要
磁场重联是等离子体中的一种基本的输运机制，它是磁能转化成等离子体能量的一种最主要的机制，对等离子体加热和粒子的加速等都起到很大的作用。几乎在所有的快速，宏观的等离子体物理过程中，磁场重联都扮演着关键性的角色。撕裂模不稳定性是非稳念的磁场重联，它是托卡马克装置中最危险的不稳定性之一，会引起等离子体约束的大破裂。因此，研究磁场重联和撕裂模不稳定性是非常必要的。
磁场重联的产生机制一直是大家关注的热点问题，目前认为磁场重联是发生在电子惯性尺度范围内的，且电子惯性有可能起到很大的作用。为了更好的研究磁场重联的产生机制，我们采用了电子磁流体模型对磁场重联不稳定性的各种效应进行了研究。电子磁流体模型是描述发生在哨声波波频范围，电子惯性尺度范围的等离子体中的快速的小尺度现象，正适合于来更好的研究磁场重联的产生机制，另一方面，磁场重联不稳定性在这个尺度范围内的性质和在磁流体中的性质是有很大的不同的，且电子磁流体模型是比较简单的流体模型。因此，我们有必要在这个模型里对磁场重联不稳定性的各种效应进行重新的分析。本文，正是采用这个模型，对磁场重联不稳定性，特别是撕裂模不稳定性进行了研究。主要有以下三方面的内容：
1，对电子粘滞(μ_e)效应对电子磁流体中的磁场重联不稳定性进行的研究中，我们得到了描述在哨声波波频区域内，电子粘滞对磁场重联不稳定性影响的普遍的色散关系，并对磁场重联不稳定性的增长率在“常-ψ”近似和低波数区域分别进行了解析分析。我们发现在“常-ψ”近似下，增长率与电子粘滞的关系为γ∝μ_e~(1/4)(电子磁流体中的撕裂模不稳定性)；在低波数区域有γ∝μ_e~(1/8)。这都与电子粘滞对磁流体中的撕裂模的影响的关系不同。我们得到，当电子粘滞系数很小时，电子磁流体中磁场重联的增长率减小得比它在磁流体中慢得多，这就意味着电子磁流体中的磁场重联的速率更快。
2，我们采用修正的包含电子压强梯度效应的可压缩电子磁流体方程对撕裂模不稳定性进行分析，得到了电子压强梯度对撕裂模不稳定性的影响的色散关系。我们发现电子压强梯度对电子磁流体中的撕裂模不稳定性起到失稳的作用，会提高其增长率，这与其在磁流体中的撕裂模不稳定性的影响是完全相反的。在磁流体中，电子压强梯度对撕裂模不稳定性起到抑制作用。在分析中，我们发现，由于电子压强梯度的存在，电子磁流体中的撕裂模不稳定性发生了漂移，存在一支不稳定性模，一支纯振荡模，和其它的稳定模。对于压强梯度比较大的情况，在可压缩电子磁流体中，我们得到增长率与压强梯度(χ_(p0))的关系为γ∝d_e~2λ~(-2/3)χ_(p0)~(1/3)(λ表示可压缩效应)；而在不可压缩电子磁流体中，有γ∝d_e~(4/5)χ_(p0)~(3/5)(d_e电子惯性趋肤长度)。可以看到，可压缩和不可压缩流体中，增长率是有很大不同的。
3，我们利用准线性方法对非线性撕裂模不稳定进行研究，得到了电子粘滞对电子磁流体中的撕裂模不稳定性影响的统一时间演化表达式，包含了线性阶段，过渡阶段和非线性阶段。在推导中，我们避免使用磁面平均的方法，这是由于这种方法会把磁通演化方程和运动方程中的对流项平均掉。我们发现在指数增长的线性阶段和代数增长的非线性阶段之中，存在着一个短暂的过渡阶段。在这阶段里，撕裂模不稳定性的发展与其它两个阶段的发展是有很大的不同的。
Magnetic reconnection is a fundamental transport mechanism in plasma. During the process, magnetic energy is converted to hear, kinetic energy, and fast particle energy. It is probably the most important one for explaining releases of magnetic energy. Magnetic reconnection plays a crucial role in almost all rapid macroscale plasma precess. Tearing mode instability is the unsteady magnetic reconnection, which is one of the most dangerous instabilities in tokamak discharge and can cause disruption. Therefore, it is necessary to investigate magnetic reconnection and tearing mode instability.
The mechanism of onset of magnetic reconnection is one of the hot problems about magnetic reconnection. Now, it is believed that magnetic reconnection takes place at the scale of electron inertia skin depth, and the electron inertial probably plays an important role in magnetic reconnection. To investigate the mechanism of onset of magnetic reconnection more better, electron magnetohydrodynamics (EMHD) is used to study reconnection instability. The electron magnetohydrodynamic theory (EMHD) describes plasma phenomena in the whistler frequency regime and below the ion skin depth scale which is proper to investigate the onset of magnetic instability. The characters of magnetic reconnection in the whistler frequency regime and electron skin depth scale are different from that in MHD model, and EMHD theory provides a simple description of reconnection, it is needed to reexamine various aspects of magnetic reconnection which arc well-known in MHD model. In this article, we employ EMHD model to investigate reconnection instability, especially tearing mode instability. There are three aspects we investigated as follows:
1, The general dispersion relation of collisionless reconnection instability due to electron viscosityμ_e in the whistler frequency is derived. In the framework of electron magnetohydrodynamics (EMHD), the evolution of magnetic reconnection instability is studied, and the linear growth rates are obtained. The scaling laws of growth rate of reconnection instability with respect to electron viscosity in constant-ψ(used in the tearing mode) and low-k regimes are obtained respectively, and compare with those obtained in standard magnetohydrodynamic theory. In the constant-ψregime for 'tearing mode like' instability, the growth rate is proportional toμ_e~(1/4) ; while in low-k regime,it is proportional toμ_e~(1/8). It can be seen that the growth rates due to the electron inertial skin depth d_e are different in EMHD and MHD regimes. The EMHD growth rate decreases less rapidly for R→0, and the reconnection rate is more fast.
2, The general dispersion relation of tearing mode with electron pressure gradient effect in the whistler frequency is analytically derived in the framework of compressible electron magnetohydrodynamics (EMHD). It is shown that electron pressure gradient effect enhances the growth rate, which is contrary to it in MHD. In MHD, tearing mode instability is suppressed by pressure gradient effect. Due to the existence of electron pressure gradient, EMHD tearing mode becomes drift, and there are three modes: unstable mode, pure oscillating mode and stable modes. For the large electron pressure gradient (χ_(p0)). the growth rate of tearing mode in compressible EMHD fluid isγ∝d_e~2λ~(-2/3)χ_(p0)~(1/3) (λdenotes compressible effects); while in incompressible EMHD fluid,γ∝d_e~(4/5)χ_(p0)~(3/5)(d_e is electron inertial skin depth). The growth rate in the compressible EMHD fluid is much different from that in the incompressible EMHD fluid.
3, The general time evolution of tearing mode due to electron viscosityμ_e from linear to nonlinear phase in the framework of electron magnetohydrodynamics (EMHD) is derived by quasilinear theory. We did not used the flux-average method, since it would cancel the convective terms in the flux and motion equation. It is found that the linear phase and nonlinear phase are separated by a transition, during which the growth behavior is much different from that in the linear and nonlinear phases.
磁场重联的产生机制一直是大家关注的热点问题，目前认为磁场重联是发生在电子惯性尺度范围内的，且电子惯性有可能起到很大的作用。为了更好的研究磁场重联的产生机制，我们采用了电子磁流体模型对磁场重联不稳定性的各种效应进行了研究。电子磁流体模型是描述发生在哨声波波频范围，电子惯性尺度范围的等离子体中的快速的小尺度现象，正适合于来更好的研究磁场重联的产生机制，另一方面，磁场重联不稳定性在这个尺度范围内的性质和在磁流体中的性质是有很大的不同的，且电子磁流体模型是比较简单的流体模型。因此，我们有必要在这个模型里对磁场重联不稳定性的各种效应进行重新的分析。本文，正是采用这个模型，对磁场重联不稳定性，特别是撕裂模不稳定性进行了研究。主要有以下三方面的内容：
1，对电子粘滞(μ_e)效应对电子磁流体中的磁场重联不稳定性进行的研究中，我们得到了描述在哨声波波频区域内，电子粘滞对磁场重联不稳定性影响的普遍的色散关系，并对磁场重联不稳定性的增长率在“常-ψ”近似和低波数区域分别进行了解析分析。我们发现在“常-ψ”近似下，增长率与电子粘滞的关系为γ∝μ_e~(1/4)(电子磁流体中的撕裂模不稳定性)；在低波数区域有γ∝μ_e~(1/8)。这都与电子粘滞对磁流体中的撕裂模的影响的关系不同。我们得到，当电子粘滞系数很小时，电子磁流体中磁场重联的增长率减小得比它在磁流体中慢得多，这就意味着电子磁流体中的磁场重联的速率更快。
2，我们采用修正的包含电子压强梯度效应的可压缩电子磁流体方程对撕裂模不稳定性进行分析，得到了电子压强梯度对撕裂模不稳定性的影响的色散关系。我们发现电子压强梯度对电子磁流体中的撕裂模不稳定性起到失稳的作用，会提高其增长率，这与其在磁流体中的撕裂模不稳定性的影响是完全相反的。在磁流体中，电子压强梯度对撕裂模不稳定性起到抑制作用。在分析中，我们发现，由于电子压强梯度的存在，电子磁流体中的撕裂模不稳定性发生了漂移，存在一支不稳定性模，一支纯振荡模，和其它的稳定模。对于压强梯度比较大的情况，在可压缩电子磁流体中，我们得到增长率与压强梯度(χ_(p0))的关系为γ∝d_e~2λ~(-2/3)χ_(p0)~(1/3)(λ表示可压缩效应)；而在不可压缩电子磁流体中，有γ∝d_e~(4/5)χ_(p0)~(3/5)(d_e电子惯性趋肤长度)。可以看到，可压缩和不可压缩流体中，增长率是有很大不同的。
3，我们利用准线性方法对非线性撕裂模不稳定进行研究，得到了电子粘滞对电子磁流体中的撕裂模不稳定性影响的统一时间演化表达式，包含了线性阶段，过渡阶段和非线性阶段。在推导中，我们避免使用磁面平均的方法，这是由于这种方法会把磁通演化方程和运动方程中的对流项平均掉。我们发现在指数增长的线性阶段和代数增长的非线性阶段之中，存在着一个短暂的过渡阶段。在这阶段里，撕裂模不稳定性的发展与其它两个阶段的发展是有很大的不同的。
Magnetic reconnection is a fundamental transport mechanism in plasma. During the process, magnetic energy is converted to hear, kinetic energy, and fast particle energy. It is probably the most important one for explaining releases of magnetic energy. Magnetic reconnection plays a crucial role in almost all rapid macroscale plasma precess. Tearing mode instability is the unsteady magnetic reconnection, which is one of the most dangerous instabilities in tokamak discharge and can cause disruption. Therefore, it is necessary to investigate magnetic reconnection and tearing mode instability.
The mechanism of onset of magnetic reconnection is one of the hot problems about magnetic reconnection. Now, it is believed that magnetic reconnection takes place at the scale of electron inertia skin depth, and the electron inertial probably plays an important role in magnetic reconnection. To investigate the mechanism of onset of magnetic reconnection more better, electron magnetohydrodynamics (EMHD) is used to study reconnection instability. The electron magnetohydrodynamic theory (EMHD) describes plasma phenomena in the whistler frequency regime and below the ion skin depth scale which is proper to investigate the onset of magnetic instability. The characters of magnetic reconnection in the whistler frequency regime and electron skin depth scale are different from that in MHD model, and EMHD theory provides a simple description of reconnection, it is needed to reexamine various aspects of magnetic reconnection which arc well-known in MHD model. In this article, we employ EMHD model to investigate reconnection instability, especially tearing mode instability. There are three aspects we investigated as follows:
1, The general dispersion relation of collisionless reconnection instability due to electron viscosityμ_e in the whistler frequency is derived. In the framework of electron magnetohydrodynamics (EMHD), the evolution of magnetic reconnection instability is studied, and the linear growth rates are obtained. The scaling laws of growth rate of reconnection instability with respect to electron viscosity in constant-ψ(used in the tearing mode) and low-k regimes are obtained respectively, and compare with those obtained in standard magnetohydrodynamic theory. In the constant-ψregime for 'tearing mode like' instability, the growth rate is proportional toμ_e~(1/4) ; while in low-k regime,it is proportional toμ_e~(1/8). It can be seen that the growth rates due to the electron inertial skin depth d_e are different in EMHD and MHD regimes. The EMHD growth rate decreases less rapidly for R→0, and the reconnection rate is more fast.
2, The general dispersion relation of tearing mode with electron pressure gradient effect in the whistler frequency is analytically derived in the framework of compressible electron magnetohydrodynamics (EMHD). It is shown that electron pressure gradient effect enhances the growth rate, which is contrary to it in MHD. In MHD, tearing mode instability is suppressed by pressure gradient effect. Due to the existence of electron pressure gradient, EMHD tearing mode becomes drift, and there are three modes: unstable mode, pure oscillating mode and stable modes. For the large electron pressure gradient (χ_(p0)). the growth rate of tearing mode in compressible EMHD fluid isγ∝d_e~2λ~(-2/3)χ_(p0)~(1/3) (λdenotes compressible effects); while in incompressible EMHD fluid,γ∝d_e~(4/5)χ_(p0)~(3/5)(d_e is electron inertial skin depth). The growth rate in the compressible EMHD fluid is much different from that in the incompressible EMHD fluid.
3, The general time evolution of tearing mode due to electron viscosityμ_e from linear to nonlinear phase in the framework of electron magnetohydrodynamics (EMHD) is derived by quasilinear theory. We did not used the flux-average method, since it would cancel the convective terms in the flux and motion equation. It is found that the linear phase and nonlinear phase are separated by a transition, during which the growth behavior is much different from that in the linear and nonlinear phases.
引文
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84 S. Dastgeer, A. Das, P. Kaw and P. H. Diamond, Phys. Plasma 7, 571(2000).
85 D. Biskamp, E. Schwartz, A. Zeiler, A. Celani and J. F. Drake, Phys. Plasma 6, 751(1999).
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125 Xiaogang Wang and A. Bhattacharjee, Phys. Plasmas 2, 171(1995).
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127 Bo Hu, R. Betti and J. Manickam, Phys. Plasmas 13, 112505(2006).
128 Glenn Bateman, Canh N. Nguyen, Arnold H. Kritz and Franco Porcelli, Phys. Plasmas 13, 072505(2006).
129 W. X. Ding, D. L. Brower, B. H. Deng, A. F. Almagri, D. Craig, G. Fiksel, V. Mirnov, S. C. Prager, J. S. Starff, and Svidzinski, Phys. Plasma 13, 112306(2006).
130 C. Angioni, T. P. Goodman, M. A. Henderson and O. Sauter, Nucl. Fusion 43, 455(2003).
131 G. Y. Fu, W. Park, H. R. Strauss, J. Breslau, J. Chen, S. Jardin and L. E. Sugiyama, Phys. Plasma 13, 052517(2006).