On the motion of slender vortex filaments

Abstract

Several approaches for slender vortex motion ͑the local induction equation, the Klein-Majda equation, and the Klein-Knio equation͒ are compared on a specific example of sideband instability of Kelvin waves on a vortex. Numerical experiments on this model problem indicate that all these equations yield qualitatively similar behavior, and this behavior is different from the behavior of a nonslender vortex with variable cross-section. It is found that the boundaries between stable, recurrent, and chaotic regimes in the parameter space of the model problem depend on the equation used. The boundaries of these domains in the parameter space for the Klein-Majda equation and for the Klein-Knio equation are closely related to the core size. When the core size is large enough, the Klein-Majda equation always exhibits stable solutions for our model problem. Various conclusions are drawn; in particular, the behavior of turbulent vortices cannot be captured by these approximations, and probably cannot be captured by any slender vortex model with constant vortex cross-section. Speculations about the differences between classical and superfluid hydrodynamics are also offered.

The Biot–Savart description of Kelvin waves on a quantum vortex filament in the presence of mutual friction and a driving fluid

“…Additional studies on the 'cut-off' method [12] have improved on the aforementioned analytical results. In addition to results for the Biot-Savart law, note that analytical results are common under the LIA [13], which is much simpler to solve in the helical case [14,15]). …”

Improved Thin-Tube Models for Slender Vortex Simulations

“…This approach has been first implemented in Zhou's study of Kelvin waves on a slender vortex [9]. Here, we implement a variant of the velocity correction approach which is summarized by…”

“…(8) remain unchanged, and the optimization operates on their output only. The attractive feature of method 2 (M2) is that it isolates the singular part of the element evolution equations and, as outlined in [9], enables the optimization of the numerical integration using operatorsplitting approaches. However, a disadvantage of M2 is that it requires explicit evaluation of higher order derivatives of the filament centerline.…”

Geometric, Stochastic and Algebraic Vortices

“…This means they cannot treat vortex reconnection or topology change. In addition, Zhou [44] has shown that these models can produce unphysical results because the core width is uniform along the vortex and does not change in time. Despite these shortcomings, the simplicity of slender vortices make them attractive mathematically.…”