Subharmonic Dynamo Action in the Roberts Flow
Subharmonic Dynamo Action in the Roberts Flow
复制标题
罗伯茨流中的分谐波发电机作用
DOI：
10.1080/03091920290004506
复制
发表时间：
2002
影响因子：
1.3
通讯作者：
K. Rädler
中科院分区：
文献类型：
--
作者：
F. Plunian;K. Rädler
本文讨论了Roberts流的发电机作用，即周期性地依赖于两个笛卡尔坐标X和Y，但与第三个笛卡尔坐标Z无关的流。具体地，考虑其中在X、Y和Z上是周期的磁场在XY平面中的周期长度是流的周期长度的整数倍的情况。使用了两种方法。首先，将控制磁场的方程归结为矩阵特征值问题，并对其进行数值求解。其次，通过在XY平面的适当区域上求平均，定义了平均磁场，导出了相应的方程，其中流动的感应效应表现为各向异性的f效应，并给出了解析解。这些结果对卡尔斯鲁厄发电机实验特别感兴趣，该实验使用的是罗伯茨型流动，该流动由圆柱形体积内的52个单元组成。为了检验基于平均场方法的自激预测的可靠性，对含有50个单元的矩形盒进行了类似的预测，并与利用上述特征值问题的直接解所得到的结果进行了比较。结果表明，简单的平均场方法通常低估了自激的要求。只有当盒的边长L、盒的高度H和自旋产生器的边长L满足$L、H、L时，相应的结果才与亚谐方法的结果一致。在附录B中，根据文中所用的次谐形式，对以前关于发电机的结果提出了一些评论。
The paper deals with the dynamo action of the Roberts flow, that is, a flow depending periodically on two cartesian coordinates, X and Y , but being independent of the third one, Z . In particular the case is considered in which the magnetic fields, which are periodic in X, Y and Z , have period lengths in the XY -plane being integer multiples of that of the flow. Two approaches are used. Firstly, the equations governing the magnetic field are reduced to a matrix eigenvalue problem, which is solved numerically. Secondly, a mean magnetic field is defined by averaging over proper areas in the XY -plane, corresponding equations are derived, in which the induction effect of the flow occurs as an anisotropic f -effect, and analytic solutions are given. The results are of particular interest for the Karlsruhe dynamo experiment, which works with a Roberts type flow consisting of 52 cells inside a cylindrical volume. In order to check the reliability of predictions concerning self-excitation based on the mean-field approach, analogous predictions are derived for a rectangular box containing 50 cells, and are compared with results obtained with the help of direct solutions of the eigenvalue problem mentioned. It turns out that the simple mean-field approach in general underestimates the requirements for self-excitation. The corresponding results agree with those obtained in the subharmonic approach only if the side length L of the box, its height H and the edge length l of a spin generator satisfy $ L \gg H \gg l $ . In Appendix B, some comments on previous results concerning $\cal {ABC}$ dynamos are made in the light of the subharmonic formalism used in the paper.