Van Gorder Robert A
Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Apr;87(4):043203. doi: 10.1103/PhysRevE.87.043203. Epub 2013 Apr 23.
We provide a formulation of the local induction approximation (LIA) for the motion of a vortex filament in the Cartesian reference frame (the extrinsic coordinate system) which allows for scaling of the reference coordinate. For general monotone scalings of the reference coordinate, we derive an equation for the planar solution to the derivative nonlinear Schrödinger equation governing the LIA. We proceed to solve this equation perturbatively in small amplitude through an application of multiple-scales analysis, which allows for accurate computation of the period of the planar vortex filament. The perturbation result is shown to agree strongly with numerical simulations, and we also relate this solution back to the solution obtained in the arclength reference frame (the intrinsic coordinate system). Finally, we discuss nonmonotone coordinate scalings and their application for finding self-intersections of vortex filaments. These self-intersecting vortex filaments are likely unstable and collapse into other structures or dissipate completely.
我们给出了笛卡尔参考系（外在坐标系）中涡旋丝运动的局部感应近似（LIA）的一种表述形式，该表述形式允许对参考坐标进行缩放。对于参考坐标的一般单调缩放，我们推导了一个关于控制LIA的导数非线性薛定谔方程平面解的方程。我们通过应用多尺度分析，在小振幅情况下对该方程进行微扰求解，这使得我们能够精确计算平面涡旋丝的周期。微扰结果与数值模拟结果高度吻合，并且我们还将此解与在弧长参考系（内在坐标系）中得到的解联系起来。最后，我们讨论了非单调坐标缩放及其在寻找涡旋丝自交点方面的应用。这些自相交的涡旋丝可能是不稳定的，会坍缩成其他结构或完全消散。
