^{1}and Lincoln D. Carr

^{1,2}

### Abstract

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 superfluidturbulence. 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 . 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.

The authors thank Paul Martin for useful discussions. This material is based in part upon work supported by the National Science Foundation (NSF) under Grant Nos. PHY-0547845 and PHY-1067973. L.D.C. acknowledges support from the Alexander von Humboldt foundation and the Heidelberg Center for Quantum Dynamics.

I. INTRODUCTION

II. BIOT-SAVART AND QUANTIZED VORTEX RINGS

III. CONVERSION TO ELLIPTIC FORM

IV. REDUCTION OF ELLIPTIC FORM TO CANONICAL ELLIPTIC INTEGRALS

V. ASYMPTOTICS FOR THE INCOMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND

VI. DISCUSSION AND CONCLUSIONS

### Key Topics

- Rotating flows
- 60.0
- Superfluids
- 19.0
- Vortex dynamics
- 15.0
- Mean field theory
- 12.0
- Rheology and fluid dynamics
- 6.0

## Figures

Global and local coordinate geometry. In subfigure (a) the vortex arc is depicted in where the circle parameterization, *C*, is composed of the solid line representing the vortex filament and a dashed line representing a continuation of the parameterization. These two regions are separated by the cutoff parameter *L*. This subfigure also shows the spherical decomposition of the field point **x** where γ_{1} is the azimuthal angle and γ_{2} is the polar angle associated with the spherical decomposition of **x**. Lastly, this subfigure shows the configuration of the Serret-Frenet local basis vectors , which, for ease of use, are oriented to correspond to the standard global basis vectors for . In subfigure (b) the projection of subfigure (a) onto the *x*–*y* plane is given and shows the polar decomposition of the filament point .

Global and local coordinate geometry. In subfigure (a) the vortex arc is depicted in where the circle parameterization, *C*, is composed of the solid line representing the vortex filament and a dashed line representing a continuation of the parameterization. These two regions are separated by the cutoff parameter *L*. This subfigure also shows the spherical decomposition of the field point **x** where γ_{1} is the azimuthal angle and γ_{2} is the polar angle associated with the spherical decomposition of **x**. Lastly, this subfigure shows the configuration of the Serret-Frenet local basis vectors , which, for ease of use, are oriented to correspond to the standard global basis vectors for . In subfigure (b) the projection of subfigure (a) onto the *x*–*y* plane is given and shows the polar decomposition of the filament point .

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