黎曼空间中作为薄磁通量绳的发电机和反发电机
Dynamos and anti-dynamos as thin magnetic flux ropes in Riemannian spaces
摘要：
给出了磁流体力学中的两个磁反发电机的例子。首先是一个3D度量共形相关的阿诺德猫快速发电机度量：${ds_{A}}^{2}=e^{-λz}dp^{2}+e^{λz}dq^{2}+dz^{2}$示出呈现非发电机的行为，其中的磁场随时间呈指数衰减。曲率随着z坐标无限制地增加而衰减。一些黎曼曲率分量，如$R_{pzpz}$，也经历耗散，而分量$R_{qzqz}$无限制地增加。剩余的曲率分量$R_{pqpq}$在圆环曲面上是恒定的。在等离子体天体物理学中可能有用的另一个反发电机是细磁通量绳或扭曲的磁性细通量管，它们也表现为反发电机，因为它也随时间衰减。该模型基于磁扭转磁通管的黎曼度规，其中轴具有Frenet曲率和挠率。由于在最后一个例子中，绳子轴的Frenet扭转几乎为零，或者可能的发电机几乎是平面的，它满足Zeldovich定理，该定理指出平面发电机不存在。这个结果的拓扑结构的改变可能会导致一个真正的发电机讨论。
英文摘要：
Two examples of magnetic anti-dynamos in magnetohydrodynamics (MHD) are given. The first is a 3D metric conformally related to Arnold cat fast dynamo metric: ${ds_{A}}^{2}=e^{-λz}dp^{2}+e^{λz}dq^{2}+dz^{2}$ is shown to present a behaviour of non-dynamos where the magnetic field exponentially decay in time. The curvature decay as z-coordinates increases without bounds. Some of the Riemann curvature components such as $R_{pzpz}$ also undergoes dissipation while component $R_{qzqz}$ increases without bounds. The remaining curvature component $R_{pqpq}$ is constant on the torus surface. The other anti-dynamo which may be useful in plasma astrophysics is the thin magnetic flux rope or twisted magnetic thin flux tube which also behaves as anti-dynamo since it also decays with time. This model is based on the Riemannian metric of the magnetic twisted flux tube where the axis possesses Frenet curvature and torsion. Since in this last example the Frenet torsion of the axis of the rope is almost zero, or the possible dynamo is almost planar it satisfies Zeldovich theorem which states that planar dynamos do not exist. Changing in topology of this result may result on a real dynamo as discussed.
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