黎曼流形上周期流中的共形映射及其对拉伸-扭曲和折叠的抑制
Conformal maps in periodic flows and in suppression of stretch-twist and fold on Riemannian manifolds
摘要：
保形发电机映射的例子已经在前面介绍过[物理等离子体\textbf{14}（2007）]，其中在黎曼流形中的扭曲磁通量管中获得了快速发电机。本文证明了在Floquet条件下，保形映射导致快速发电机的指数拉伸或李雅普诺夫指数、保形因子之间的一致性。在黎曼平坦流形中可以得到展开共形发电机映射，因为在这里，黎曼曲率起着折叠的作用。以前，Oseledts [地球物理学天体物理学流体动力学\textbf{73}（1993）]已经表明，在一个紧凑的黎曼流形上的两个环面的扭曲和untwisting轨道的数量引起快速发电机作用的增长。本文将共形薄磁通量管的拉伸约束为零，以求得无拉伸非发电机的共形因子。由于薄通量管可以被认为是一个扭曲或untwisting双环面映射，它表明，untwisting，弱扭转，和非拉伸共形环面映射不能支持一个快速发电机行动，边际发电机被获得。这是一个反快速发电机定理的例子，除了Vishik和Klapper和Young [Comm Math Phys \textbf{173}（1996）]在理想的高传导流中给出的定理之外。从黎曼曲率张量，它表明，新的共形非发电机，实际上是奇异的，因为一个接近磁通量管轴。因此，共形映射抑制拉伸方向和扭曲，导致快速发电机的作用，而黎曼平坦的展开流形有利于非快速发电机。
英文摘要：
Examples of conformal dynamo maps have been presented earlier [Phys Plasmas \textbf{14}(2007)] where fast dynamos in twisted magnetic flux tubes in Riemannian manifolds were obtained. This paper shows that conformal maps, under the Floquet condition, leads to coincidence between exponential stretching or Lyapunov exponent, conformal factor of fast dynamos. Unfolding conformal dynamo maps can be obtained in Riemann-flat manifolds since here, Riemann curvature plays the role of folding. Previously, Oseledts [Geophys Astrophys Fluid Dyn \textbf{73} (1993)] has shown that the number of twisted and untwisting orbits in a two torus on a compact Riemannian manifold induces a growth of fast dynamo action. In this paper, the stretching of conformal thin magnetic flux tubes is constrained to vanish, in order to obtain the conformal factor for non-stretching non-dynamos. Since thin flux tube can be considered as a twisted or untwisting two-torus map, it is shown that the untwisting, weakly torsion, and non-stretching conformal torus map cannot support a fast dynamo action, a marginal dynamo being obtained. This is an example of an anti-fast dynamo theorem besides the ones given by Vishik and Klapper and Young [Comm Math Phys \textbf{173}(1996)] in ideally high conductive flow. From the Riemann curvature tensor it is shown that new conformal non-dynamo, is actually singular as one approaches the magnetic flux tube axis. Thus conformal map suppresses the stretching directions and twist, leading to the absence of fast dynamo action while Riemann-flat unfolding manifolds favors non-fast dynamos.
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