广义局部感应方程、椭圆渐近和超流Turbulence的模拟
Generalized Local Induction Equation, Elliptic Asymptotics, and Simulating Superfluid Turbulence
摘要：
我们证明了广义感应方程和广义局部感应方程（GLIE），它取代了常用的局部感应近似（LIA）来模拟涡线的动力学，从而超流湍流。我们表明，LIA是，事实上没有任何近似，在所有的弯曲涡丝的任何长度引起的速度场的一般特征。具体来说，LIA指出，弯曲涡丝诱导的速度场在副法向方向上是不对称的。直到一个潜在的长期，诱导的不可压缩的领域是由毕奥-萨伐尔积分，其中我们记得，有一个直接的类比流体力学和静磁学。系列近似的Biot-Savart被积函数表示对数发散的局部领域的副法线方向。虽然这在定性上是正确的，但LIA缺乏量化其小参数的指标。无论如何，LIA用于涡丝方法模拟量子化涡旋的自诱导运动。考虑到数值，我们表示的副正规领域的不完全椭圆积分，这是有效的$\mathbb{R}^{3}$。从这个和已知的扩展，我们得到GLIE，渐近的局部场点。像LIA一样，广义归纳法在局部场中显示出持续的副法线偏差，但与LIA不同的是，GLIE提供了截断余项的界限。作为一个应用程序，我们适应公式从涡丝方法的GLIE为将来使用这些方法。我们考虑的其他例子包括涡环，与超流$^4$ He和玻色-爱因斯坦凝聚体有关。
英文摘要：
We prove the generalized induction equation and the generalized local induction equation (GLIE), which replaces the commonly used local induction approximation (LIA) to simulate the dynamics of vortex lines and thus superfluid turbulence. We show that the LIA is, without in fact any approximation at all, a general feature of the velocity field induced by any length of a curved vortex filament. Specifically, the LIA states that the velocity field induced by a curved vortex filament is asymmetric in the binormal direction. Up to a potential term, the induced incompressible field is given by the Biot-Savart integral, where we recall that there is a direct analogy between hydrodynamics and magnetostatics. Series approximations to the Biot-Savart integrand indicate a logarithmic divergence of the local field in the binormal direction. While this is qualitatively correct, LIA lacks metrics quantifying its small parameters. Regardless, LIA is used in vortex filament methods simulating the self-induced motion of quantized vortices. With numerics in mind, we represent the binormal field in terms of incomplete elliptic integrals, which is valid for $\mathbb{R}^{3}$. From this and known expansions we derive the GLIE, asymptotic for local field points. Like the LIA, generalized induction shows a persistent binormal deviation in the local-field but unlike the LIA, the GLIE provides bounds on the truncated remainder. As an application, we adapt formulae from vortex filament methods to the GLIE for future use in these methods. Other examples we consider include vortex rings, relevant for both superfluid $^4$He and Bose-Einstein condensates.
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