High mode number stability of an axisymmetric toroidal plasma
High mode number stability of an axisymmetric toroidal plasma
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DOI：
10.1098/rspa.1979.0001
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发表时间：
1979-02
期刊：
影响因子：
--
通讯作者：
J. Connor;R. Hastie;J. B. Taylor
中科院分区：
文献类型：
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作者：
J. Connor;R. Hastie;J. B. Taylor
在研究受磁场约束的等离子体的稳定性时，最重要的几种振荡模式是平行于磁场的长波和垂直于磁场的短波。然而，这些特性与具有剪切的环形磁场中周期性的要求相冲突。这一冲突可以通过将计算转化为无周期约束的无限域计算来解决。这个变换是全面研究轴对称等离子体在大环向波数n时磁流体动力学稳定性的起点。(小的n值可以用直接的数值计算来研究，但当n大时这是不可能的。)对于n>；1，在问题中有两个不同的长度尺度，并围绕一个象素表示建立了系统的近似，形式上是1/n的展开。在最低阶，每个磁面的振荡是解耦的，并得到一个局部本征值。然而，在这个最低的顺序中，模式结构并没有完全确定。在高阶得到第二本征值方程，它完成了对模式结构的确定，并将低阶理论的局部本征值与问题的真实本征值联系起来。这一高阶理论表明，不稳定模集中在具有最小局部本征值的表面附近，真实本征值接近最低局部本征值，最不稳定的高n模出现在n-gt；00处。因此，只需解一个常微分方程解的局域理论，通常足以确定任意轴对称等离子体对高模振荡的稳定性。
In the investigation of stability of a plasma confined by magnetic fields some of the most important modes of oscillation are those with long wavelength parallel to the magnetic field and short wavelength perpendicular to it. However, these characteristics conflict with the requirement of periodicity in a toroidal magnetic field with shear. This conflict can be resolved by transforming the calculation to one in an infinite domain without periodicity constraints. This transformation is the starting point for a full investigation of the magnetohydrodynamic stability of an axisymmetric plasma at large toroidal wave number n. (Small values of n can be studied by direct numerical computation but this fails when n is large.) For n > 1 there are two distinct length scales in the problem and a systematic approximation is developed around an eikonal representation, formally as an expansion in 1/n. In lowest order the oscillations of each magnetic surface are decoupled and a local eigenvalue is obtained. However, the mode structure is not fully determined in this lowest order. In higher orders a second eigenvalue equation is obtained which completes the determination of the structure of the mode and relates the local eigenvalue of the lower order theory to the true eigenvalue for the problem. This higher order theory shows that unstable modes are localized in the vicinity of the surface with the smallest local eigenvalue, that the true eigenvalue is close to the lowest local eigenvalue and that the most unstable high n modes occur for n-> oo. Hence the local theory, which involves no more than the solution of an ordinary differential equation, is normally adequate for the determination of stability of any axisymmetric plasma to high mode number oscillations.