【作者】 代玉杰；

【导师】 刘金远；

【作者基本信息】 大连理工大学 ， 等离子体物理， 2009， 博士

【摘要】 空间和实验室等离子体中都存在宏观不稳定性,这些不稳定性有的在很短的时间内释放出巨大的能量,从而引起等离子体状态的急剧变化。扭曲不稳定性是最常见的由电流驱动的一种宏观不稳定性,其性质可用磁流体模型进行数值研究。在空间等离子体中,扭曲不稳定性可以用来解释太阳耀斑的能量释放。在实验室等离子体中,扭曲不稳定性被认为是导致托卡马克中锯齿振荡的重要原因之一。磁流体系统的边界条件对扭曲不稳定性具有重要的影响,对于环形系统下的扭曲不稳定性,目前已经研究的比较多了。但对于line-tied直圆柱位形,具有理想导体壁的扭曲不稳定性的某些性质,目前还不是十分清楚。为了研究这种不稳定性,最近人们做了两个实验:其中一个是在威斯康星大学做的研究两端封闭的直圆柱位形下,扭曲不稳定性的演化过程;另一个是在阿拉莫斯国家实验室做的研究磁重联的实验。在第二个实验中,直圆柱只有一端封闭,另一端是开放的。在理论方面,Evstatiev等提出了一种求解line-tied扭曲不稳定性的新方法,并对上述实验进行了数值模拟。但是,在Evstatiev等的文章中,没有考虑等离子体压强对扭曲不稳定性的影响。事实上,等离子体压强不能忽略,因为很多宏观不稳定性,例如气球模和交换模等都是由等离子体压强梯度驱动的。本文应用Evstatiev等提出的方法,数值模拟了等离子体压强对直圆柱位形下line-tied扭曲不稳定性的影响。并分别对等离子体压强为零、均匀等离子体压强和非均匀等离子体压强三种情形下的增长率和复本征谱进行了分析。本文还讨论了非理想磁流体中粘滞项和电阻项对line-tied扭曲不稳定性的影响,并从撕裂模等方面进行了分析,得到如下一些结论:1.等离子体压强为零时的增长率曲线是一组嵌套的曲线族,而且所有的曲线都交于点k=-1,γ=0,这是由本文给定的磁场位形的特殊性决定的。也就是说,点k=-1,γ=0是奇点,为了保证方程仍然成立,本征函数必然为零。2.对等离子体压强P₀=0.001和P_n=0.01模拟的结果表明,均匀等离子体压强使不稳定性的增长率减小了。另外,轴向波数k的范围并不随着均匀等离子体压强的变化而变化。不同压强下相对应的增长率曲线仍然交于同一点,这说明均匀等离子体压强没有使不稳定性的范围扩大。3.与等离子体压强为零时的情形比较,当等离子体压强非均匀时,所有的增长率曲线没有共同交点。轴向波数k的范围和增长率的最大值都增大了,即使当等离子体的β值仅为2%时,这种差别仍然很明显。这说明非均匀等离子体压强使不稳定性的范围增大了,很明显,这种变化是由等离子体的压强梯度引起的。4.粘滞作用会使增长率和二维本征函数的最大值都减小。这是因为粘性会使流体层之间的速度剪切减小,从而对磁流体系统有稳定作用,而且这种稳定作用对短波模式影响较大。粘性对不稳定性的稳定作用随着粘滞系数的减小而减小,当粘滞系数小于10^-5时,其对增长率的影响基本可以忽略。电阻率对磁流体的影响是导致撕裂模等不稳定性的发生,当电阻率小于10^-6时,可以不予考虑。5.本文的最后一章,应用能量原理分析了磁流体不稳定性。将最速下降法用在等离子体区域和真空区域对位能E的变分原理上,分析了D型托卡马克磁流体的不稳定性。更多还原

【Abstract】 There are many macro instabilities in astrophysical and laboratory plasmas, some of . them can release enormous energy within very short time and change the plasma state. Kink instability is a familiar instability which driven by current in all macro instabilities and can be studied with MHD model.Kink instability is used to explain the energy release in the solar flare. In laboratory plasmas, kink instability is a crucial component for sawtooth oscillations in tokamak.Kink instability is strongly depends on the system geometry and boundary conditions, and corresponding theory has been quite successful in predicting the behavior of toroidal plasma. It has not been thoroughly tested in cylindrical geometries for which the boundary conditions are line-tied at end plates.For further study of line-tied kink instability, two experiments have been made on cylindrical devices. The first investigates the evolution of line-tied kink modes at both ends of the machine at the University of Wisconsin, and the second is on the reconnection scaling experiment device line-tied only at one end plate at the Los Alamos National Laboratory.Evstatiev et al introduced a new method for analyzing line-tied kink modes in cylindrical geometry and simulated the experiments mentioned above. However, the effect of plasma pressure for kink instability is omitted in the paper of Evstatiev. In fact, Plasma pressure cannot be neglected any more if it is large enough, because many kinds of instabilities, such as ballooning mode and interchange mode are driven by its gradient.In this paper, the effect of plasma pressure for line-tied kink instability in cylindrical geometry is studied with the method introduced by evstatiev. Growth rate and complex spectra are analyzed for zero plasma pressure, uniform plasma pressure and heterogeneous plasma pressure. Effects of viscosity and resistivity in unideal MHD are analyzed for kink instability, tearing mode is studied too. The results are as follows:1. Curves of growth rate are nested and intersect the common point of k = -1,γ= 0 when plasma pressure is zero, which is a specific case for the magnetic components in this paper. That is to say, the point of k = -1,γ= 0 is a singularity, so eigenfunction must be zero for the balance of equation. 2. Growth rate decreases for the uniform plasma pressure P₀ = 0.001 and P₀ = 0.01. Scale of axial wave number k is not change for different uniform plasma pressure. Corresponding growth rate curves are intersecte the common point, which means the scale of instability is not extend when plasma pressure is uniform.3. Compared to zero plasma pressure, growth rate curves for heterogeneous plasma pressure are not intersect common point. The scale of axial wave number k and maximum value of growth rate are extend, the difference is distinct even plasma beta is 2% , which is caused by the gradient of heterogeneous plasma pressure.4. Maximum value of growth rate and 2-D eigenfunction decrease for the viscosity. It is because the viscosity can reduce velocity shear so MHD system is more stable, the effect is distinct for short wave mode. The stable effect for growth rate decreases with the decreases of viscosity and can be neglected when viscosity less than 10"5. Effect of resistivity for MHD system is that it will cause tearing mode, resistivity can be neglected if it is less than 10^-6.5. Enrgy principle is used to analyze MHD instability in the last chapter. Paths of steepest descent are used for the variational principle of potential energy in plasma and vacuum region, instability in D-shaped Tokamak is analyzed.更多还原