### Abstract

A new asymptotic equation is derived for the motion of thin vortex filaments in an incompressible fluid at high Reynolds numbers. This equation differs significantly from the familiar local self-induction equation in that it includes self-stretching of the filament in a nontrivial, but to some extent analytically tractable, fashion. Under the same change of variables as employed by Hasimoto (1972) to convert the local self-induction equation to the cubic nonlinear Schrödinger equation, the new asymptotic propagation law becomes a cubic nonlinear Schrödinger equation perturbed by an explicit nonlocal, linear operator. Explicit formulae are developed which relate the rate of local self-stretch along the vortex filament to a particular quadratic functional of the solution of the perturbed Schrödinger equation. The asymptotic equation is derived systematically from suitable solutions of the Navier-Stokes equations by the method of matched asymptotic expansions based on the limit of high Reynolds numbers. The key idea in the derivation is to consider a filament whose core deviates initially from a given smooth curve only by small-amplitude but short-wavelength displacements balanced so that the axial length scale of these perturbations is small compared to an integral length of the background curve but much larger than a typical core size δ=O(Re^{ -1 2}) of the filament. In a particular distinguished limit of wavelength, preturbation amplitude and filament core size the nonlocal induction integral has a simplified asymptotic representation and yields a contribution in the Schrödinger equation that directly competes with the cubic nonlinearity.

Original language | English (US) |
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Pages (from-to) | 323-352 |

Number of pages | 30 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 49 |

Issue number | 3 |

DOIs | |

State | Published - Apr 2 1991 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics