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Comment on “Motion of a helical vortex filament in superfluid 4
He under the extrinsic form of the local induction approximation” [Phys. Fluids25, 085101 (2013)]
1. R. A. Van Gorder, “Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation,” Phys. Fluids 25, 085101 (2013).
4. R. M. Ostermeier and W. I. Glaberson, “Instability of vortex lines in the presence of axial normal fluid flow,” J. Low Temp. Phys. 21, 191 (1975).
6.In our notation we have included the factor 1/4π in γ and kept the units, since Van Gorder's notation is only partly dimensionless due to the units in a0. This has no effect on any of the equations below.
10. C. F. Barenghi, R. J. Donnelly, and W. F. Vinen, “Thermal excitation of waves on quantized vortices,” Phys. Fluids 28, 498 (1985).
11. R. J. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, 1991), p. 214.
14. R. A. Van Gorder, “Fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He,” J. Fluid Mech. 707, 585 (2012).
15. R. A. Van Gorder, “Self-similar vortex dynamics in superfluid 4He under the Cartesian representation of the Hall-Vinen model including superfluid friction,” Phys. Fluids 25, 095105 (2013).
17. B. K. Shivamoggi and G. J. F. van Heijst, “Motion of a vortex filament in the local induction approximation: Reformulation of the Da Rios–Betchov equations in the extrinsic filament coordinate space,” Phys. Lett. A 374, 1742 (2010).
19. R. A. Van Gorder, “Integrable stationary solution for the fully nonlinear local induction equation describing the motion of a vortex filament,” Theor. Comput. Fluid Dyn. 26, 591 (2012).
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We comment on the paper by Van Gorder [“Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation,” Phys. Fluids25, 085101 (2013)]. We point out that the flow of the normal fluid component parallel to the vortex will often lead into the Donnelly–Glaberson instability, which will cause the amplification of the Kelvin wave. We explain why the comparison to local nonlinear equation is unreasonable, and remark that neglecting the motion in the x-direction is not reasonable for a Kelvin wave with an arbitrary wavelength and amplitude. The correct equations in the general case are also derived.
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