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Comment on "Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation"

We comment on the paper by Van Gorder ["Motion of a helical vortex filament in superfluid ${}^4$He under the extrinsic form of the local induction approximation", Phys. Fluids 25, 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 wave length and amplitude. The correct equations in the general case are also derived.