A single filament with a variable core size parameter is used to model how a vortex tube breaks down in the Euler equations. The first singularity is a self‐similar collapse which brings two antiparallel pieces of filament together at a point. This pairing then quickly encompasses a finite fraction of the initial data and the arclength begins to grow faster than exponential. A local model should exist which would allow one to understand the stage of rapid stretching in terms of simpler processes.
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Turbulent Shear Flows I, edited by F. Durst (Springer, New York, 1977);
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We have not found an explicit proof that the Biot‐Savart law with cutoff is stable at short wavelengths. An earlier derivation of an instability for finite core vortices using just Biot‐Savart by
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The latter prescription is derived in Ref. 6 but we doubt its applicability to a fractal filament in which the radius of curvature is very different from the total length.
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W. T. Ashurst (private communication) and citations in Ref. 5.
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We mention in this context a curious run of ours with fixed and There was pairing but no fractal structure and L grew linearly in time. Our results for fixed are of course qualitatively different. We do not understand this very pronounced change in terms of known equilibrium properties of (2) but conjecture that one is seeing a jump in the entropy as a function of energy.
33.
U. Frisch (private communication) brought to our attention a simple counter example for Burger’s equation. See also the 1981 Les Houches Summer School Proceedings on Turbulence and Singularities (North‐Holland, Amsterdam, to be published).
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