扭曲磁通量管中的测地发电机混沌流和非Anosov映射
Geodesic dynamo chaotic flows and non-Anosov maps in twisted magnetic flux tubes
摘要：
最近Tang和Boozer [{ \textbf{Phys. Plasmas (2000)} }]研究了磁通量管（MFTs）中Alfven扭转中磁场发电机演化的各向异性，从局部李雅普诺夫指数出发，产生了度规张量。Thiffeault和Boozer [ \textbf{Chaos}（2001）]研究了Riemann曲率的消失如何约束混沌流的Lyapunov指数拉伸。采用Tang-Boozer-Thiffeault微分几何框架研究了填充螺旋混沌流的扭曲磁通量管对黎曼曲率张量的影响。当Frenet挠率为正时，黎曼曲率不稳定，而当时间为$t\to{\infty}$时，负挠率导致黎曼曲率稳定.这增强了MFT内部的发电机作用。黎曼度规依赖于沿极向和环向的径向随机流。Arnold、Zeldovich、Ruzmaikin和Sokoloff [ \textbf{JETP (1982)} ]应用Anosov流建立了基于Arnold猫图的均匀拉伸发电机流解。很容易表明，当随机径向流消失时，磁场也消失了，因为指数李雅普诺夫拉伸消失了。这是维希克反快发电机定理在磁通量管中应用的一个例子。研究了Arnold和扭曲MFT发电机的测地线流动。结果表明，由测地线方程可以得到一个恒定的随机径向流。在整个论文中，我们假设，合理的等离子体天体物理学假设是弱挠率。Gilbert [ \textbf{Proc Roy Soc A London (1993)}]也讨论了伪Anosov发电机流和映射
英文摘要：
Recently Tang and Boozer [{\textbf{Phys. Plasmas (2000)}}], have investigated the anisotropies in magnetic field dynamo evolution, from local Lyapunov exponents, giving rise to a metric tensor, in the Alfven twist in magnetic flux tubes (MFTs). Thiffeault and Boozer [\textbf{Chaos}(2001)] have investigated the how the vanishing of Riemann curvature constrained the Lyapunov exponential stretching of chaotic flows. In this paper, Tang-Boozer-Thiffeault differential geometric framework is used to investigate effects of twisted magnetic flux tube filled with helical chaotic flows on the Riemann curvature tensor. When Frenet torsion is positive, the Riemann curvature is unstable, while the negative torsion induces an stability when time $t\to{\infty}$. This enhances the dynamo action inside the MFTs. The Riemann metric, depends on the radial random flows along the poloidal and toroidal directions. The Anosov flows has been applied by Arnold, Zeldovich, Ruzmaikin and Sokoloff [\textbf{JETP (1982)}] to build a uniformly stretched dynamo flow solution, based on Arnold's Cat Map. It is easy to show that when the random radial flow vanishes, the magnetic field vanishes, since the exponential Lyapunov stretches vanishes. This is an example of the application of the Vishik's anti-fast dynamo theorem in the magnetic flux tubes. Geodesic flows of both Arnold and twisted MFT dynamos are investigated. It is shown that a constant random radial flow can be obtained from the geodesic equation. Throughout the paper one assumes, the reasonable plasma astrophysical hypothesis of the weak torsion. Pseudo-Anosov dynamo flows and maps have also been addressed by Gilbert [\textbf{Proc Roy Soc A London (1993)}
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