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Waves on vortex filaments

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  • Mathematics


NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Various problems concerning waves on vortex filaments are considered. The local force balance method introduced by Moore and Saffman for the calculation of the induced velocity at a point of a vortex filament with arbitrary structure and shape is used to examine the effect of axial flow on the stability of trailing vortices and vortex rings. It is found that the effect is small in both cases. The method is extended to study the stability of vortex rings carrying electric charges, which are possible models for vortices in liquid helium. Two cases are considered--the conducting ring and the uniformly charged ring. In each of these cases it is found that the velocity of a charged ring is smaller than an uncharged one, and if the charges are strong enough, the ring may reverse its direction of motion. Furthermore, the charged ring becomes unstable when the charge effect is comparable to the vorticity effect. The motion and stability of a buoyant vortex ring are also considered. It is shown that a heavy ring travelling in the direction of gravity decelerates, thins and expands, while a light ring accelerates, fattens and contracts. The heavy ring is stable to disturbances of the centerline, but the light ring is unstable with a growth rate independent of wave number. Intrinsic equations governing the curvature [...] and torsion [...] of a vortex filament are obtained. They form a set of coupled nonlinear integro-partial differential equations. By retaining only the leading order term in the singularity of the Biot-Savart integral, which corresponds to the localized induction hypothesis introduced by Arms and Hama, these can be reduced to a single nonlinear Schrodinger equation for the complex variable [...] where s is the arclength. A complete set of steady state solutions for this equation is obtained. This includes the straight vortex, the helical vortex, the vortex ring, and a solitary wave form, all being limiting cases of a general periodic wave structure. A modified scheme is introduced to resolve an apparent nonuniformity of the solitary wave solution in the limit [...]. Non-local effects (effects of the regular part of the Biot-Savart integral) are examined by means of an asymptotic expansion of the intrinsic equations in the small parameter [...] where a is the core radius of the filament. It is shown that even in the tail ends of the solitary wave where the local effect fails to dominate, the solitary wave solution exists to [...].

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