紧黎曼流形中的伸缩环面发电机
Stretch-Twist torus dynamo in compact Riemannian manifolds
摘要：
早先Arnold，Zeldovich，Ruzmaikin和Sokoloff [ \textbf{JETP (1982)} ]计算了均匀拉伸环面变换的特征值，它导致发电机作用问题的第一个黎曼度量解。最近其他一些尝试获得黎曼度量表示发电机行动通过共形映射已进行[{ \textbf{Phys. Plasmas 14 (2007)} ]。早些时候，Gilbert [ \textbf{Proc. Roy. Soc. London A(2003)} ]通过发电机算子的本征值问题，研究了比Arnold等人通过产生Arnold猫映射的剪切而提出的发电机映射更现实的解决方案。本文给出了黎曼扭曲环面发电机流度规的本征值，它表示为流的极向分量与环向分量之比。这个结果是从欧拉方程得到的。在扭曲环面中，黎曼度量的本征值是${m}_{\pm}=\frac{1\pm{\sqrt{5}}}{2}$，这与Arnold得到的值非常接近。在粘性雷诺数为$Re\to{\infty}$的情况下，当接近环轴时，环流是不稳定的。在阿诺德发电机度量的特征值是$χ_{\pm}=\frac{3\pm{\sqrt{5}}}{2}$这是非常接近上述值。特征值决定了流动的极向和环形分量之间的速度比的增长率。环面中的弯曲流动遵循Chen等人的先前工作[ \textbf{Phys Fluids (2006)} ]。的$α$效果发电机被证明是一个二阶效应的环面曲率和速度流。循环发电机流和地图进行了讨论。
英文摘要：
Earlier Arnold, Zeldovich, Ruzmaikin and Sokoloff [\textbf{JETP (1982)}] have computed the eigenvalue of a uniform stretching torus transformation which result on the first Riemann metric solution of the dynamo action problem. Recently some other attempts to obtain Riemann metrics representing dynamo action through conformal maps have been undertaken [{\textbf{Phys. Plasmas 14 (2007)}]. Earlier, Gilbert [\textbf{Proc. Roy. Soc. London A(2003)}] has investigated a more realistic dynamo map solution than the one presented by Arnold et al by producing a shearing of the Arnold's cat map, by eigenvalue problem of a dynamo operator. In this paper, the eigenvalue of the Riemann twisted torus dynamo flow metric is obtained as the ratio between the poloidal and toroidal components of the flow. This result is obtained from the Euler equation. In the twisted torus, the eigenvalue of the Riemann metric is ${m}_{\pm}=\frac{1\pm{\sqrt{5}}}{2}$, which is quite close to the value obtained by Arnold. In the case the viscosity Reynolds number $Re\to{\infty}$, the torus flow is unstable as one approaches the torus axis. In Arnold's dynamo metric the eigenvalues are $χ_{\pm}=\frac{3\pm{\sqrt{5}}}{2}$ which are very close to the above value. Eigenvalues determine the growth rates of the velocity ratio between poloidal and toroidal components of the flow. The curved flow in torus follow previous work by Chen et al [\textbf{Phys Fluids (2006)}]. The $α$-effect dynamo is shown to be a second-order effect in the torus curvature and velocity flow. Loop dynamo flows and maps are also discussed.
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