Dynamical constraints from field line topology in magnetic flux tubes

A topological constraint on the dynamics of a magnetic field in a flux tube arises from the fixed point indices of its field line mapping. This can explain unexpected behaviour in recent resistive-magnetohydrodynamic simulations of magnetic relaxation. Here, we present the theory for a general periodic flux tube, representing, for example, a toroidal confinement device or a solar coronal loop. We show how an ideal dynamics on the side boundary of the tube implies that the sum of indices over all interior fixed points is invariant. This constraint applies to any continuous evolution inside the tube, which may be turbulent and/or dissipative. We also consider the analogous invariants obtained from periodic points (fixed points of the iterated mapping). Although there is a countably infinite family of invariants, we show that they lead to at most two independent dynamical constraints. The second constraint applies only in certain magnetic configurations. Several examples illustrate the theory.

通量管中磁场动力学的一种拓扑约束源于其磁力线映射的不动点指数。这可以解释近期磁弛豫的电阻性磁流体动力学模拟中出现的意外行为。在此，我们提出了一般周期性通量管的理论，例如，它可代表一种环形约束装置或一个日冕环。我们展示了管的侧边界上的理想动力学如何意味着所有内部不动点的指数之和是不变的。这种约束适用于管内的任何连续演化，这些演化可能是湍流的和/或耗散的。我们还考虑了从周期点（迭代映射的不动点）得到的类似不变量。尽管存在可数无穷多个不变量族，但我们表明它们至多导致两个独立的动力学约束。第二个约束仅适用于某些磁位形。几个例子说明了该理论。