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VISUALIZING VORTEX FILAMENTS THROUGH INTEGRABLE
PARTIAL DIFFERENTIAL EQUATIONS
SCOTT STRONG
Abstract. It has been said that vorticity acts as an internal ‘skeleton’ that
determines the structure of a fluid flow, which is important in understanding
turbulence.[1] A simple skeleton is one where vorticity has been ideally localized to curves in R3 , which are known as vortex filaments. Due to the complex
skeleton of classically turbulent flows, interest in vortex filaments have waned.
However, in the case of quantum turbulence, the vortical structure is confined
to what is essentially a tangle of filaments and it is in this regime where the
vortex filament model is accurate and realistic.[2]
By expressing the vorticity in the curvilinear Serret-Frenet frame, one can
conduct an asymptotic analysis of the local field and prove that the curved filament flows in its binormal direction.[3] Interestingly, in 1972 Hasimoto showed
that such filament flows can be mapped onto the focusing cubic nonlinear
Schrödinger equation (NLS).[4] Thus, solutions to NLS can, in theory, be used
to prescribe admissible geometries of vortex filaments. However, the problem
of recovering the filament geometry from the Serret-Frenet equations (SF),
which is a non-autonomous system of linear ODEs, is nontrivial.
While it is possible to numerically approximate solutions to SF to recover
the filament geometry, this tells us nothing about the properties of the inversion
of Hasimoto’s mapping. However, it is possible to reduce the SF problem
to a simpler 2 × 2 problem through the use of the famous SU (2) to SO(3)
map. From this one can study the NLS to vortex filament mapping and easily
visualize possible geometries.[5] We finish with a discussion of the asymptotic
integrability of vortex tubes, which we hope can be used for more appropriate
models of quantized vortices in superfluid flows.
References
[1] S. Childress. An introduction to theoretical fluid mechanics. Courant Lecture Notes. Courant
Institute of Mathematical Sciences, 2009.
[2] Makoto Tsubota and Michikazu Kobayashi. Chapter 1 energy spectra of quantum turbulence.
In M. Tsubota and W.P. Halperin, editors, Progress in Low TEMPERATURE PHYSICS:
QUANTUM TURBULENCE, volume 16 of Progress in Low Temperature Physics, pages 1 –
43. Elsevier, 2009.
[3] S. A. Strong and L. D. Carr. Generalized Local Induction Equation, Elliptic Asymptotics, and
Simulating Superfluid Turbulence. ArXiv e-prints, February 2011.
[4] H. Hasimoto. A soliton on a vortex filament. Journal of Fluid Mechanics, 51:477–485, 1972.
[5] P. G. Grinevich and M. U. Schmidt. Closed curves in Rˆ3: a characterization in terms of
curvature and torsion, the Hasimoto map and periodic solutions of the Filament Equation. In
eprint arXiv:dg-ga/9703020, page 3020, March 1997.
Department of Physics, Colorado School of Mines, Golden, CO 80401
E-mail address: [email protected]
Date: November 9, 2011.
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