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J. Math. Phys. 53, 033102 (2012); http://dx.doi.org/10.1063/1.3696689 (12 pages)

# Generalized local induction equation, elliptic asymptotics, and simulating superfluid turbulence

Scott A. Strong1 and Lincoln D. Carr1,2

2Ruprecht-Karls-Universität Heidelberg, Physikalisches Institut, Philosophenweg 12, D-69120 Heidelberg, Germany

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(Received 8 March 2011; accepted 5 March 2012; published online 22 March 2012)

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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 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 4He and Bose-Einstein condensates.

© 2012 American Institute of Physics

### Article Outline

1. INTRODUCTION
2. BIOT-SAVART AND QUANTIZED VORTEX RINGS
3. CONVERSION TO ELLIPTIC FORM
4. REDUCTION OF ELLIPTIC FORM TO CANONICAL ELLIPTIC INTEGRALS
5. ASYMPTOTICS FOR THE INCOMPLETE ELLIPTIC INTEGRAL OF THE FIRST KIND
6. DISCUSSION AND CONCLUSIONS

## KEYWORDS and PACS

### PACS

• Vortices and turbulence

• Dynamic properties of condensates; excitations, and superfluid flow

• Dynamic properties of condensates; collective and hydrodynamic excitations, superfluid flow

## PUBLICATION DATA

### ISSN

0022-2488 (print)
1089-7658 (online)

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