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首页> 外文期刊>Zeitschrift fur Naturforschung. A, A journal of physical sciences >Pattern Formation in Force-free Magnetic Fields and Beltrami Flows
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Pattern Formation in Force-free Magnetic Fields and Beltrami Flows

机译：无力磁场和Beltrami流中的模式形成

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摘要

Force-free magnetic fields and Beltrami flows, which are selenoidal vector fields and satisfy the condition that the field vector is everywhere parallel to its curl, have complex topological structures and usually show chaotic behavior. The lines of force are determined by the equations of a 3-D (dimensional) dynamical system. In the integrable case, all lines of force lie on some families of tori. If the integrable solution undergoes a small perturbation, most of the original tori still exist but undergo a slight distortion (KAM tori). Near the original heteroclinic cycles emerges a chaotic layer. By superposition of the basic solutions of force-free magnetic fields one can get very complicated pictures: a single line of force could be space filling within some subspace of a 3-D region, which has a fractional dimension and a positive Lyapunov exponent, i.e. one gets a chaotic line of force or a fractal. At the same time there are still ordered regions in the chaotic surroundings. Tubes of force which are tangled or self-knotted embed in the chaotic sea. The KAM tori can also be disrupted through resonances, leading to increased chaotic regions. Thus, the effect of nonlinear dynamics plays an important role in the pattern formation of force-free magnetic fields and Beltrami flows.
机译：无力磁场和Beltrami流是呈半球形的矢量场，并满足以下条件：场矢量在任何地方都与其卷曲平行，具有复杂的拓扑结构，通常显示出混沌行为。力线由3-D（三维）动力学系统的方程式确定。在可整合的情况下，所有力量都在一些花托家族上。如果可积解受到很小的扰动，则大多数原始花托仍然存在，但会发生轻微失真（KAM花托）。在原始的异质循环附近出现了一个混沌层。通过叠加无力磁场的基本解，可以得到非常复杂的图像：单个力线可以在空间中填充3-D区域的某些子空间，该空间具有分数维和正Lyapunov指数，即人们会得到混乱的力线或分形。同时，在混乱的环境中仍存在有序区域。纠结或自打结的力管嵌入混乱的海中。 KAM花托也可能通过共振而被破坏，从而导致混乱区域增加。因此，非线性动力学的影响在无力磁场和Beltrami流的图形形成中起着重要作用。

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