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Response to “Comment on ‘Motion of a helical vortex filament in superfluid 4
He under the extrinsic form of the local induction approximation”’ [Phys. Fluids26, 019101 (2014)]
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).
2. N. Hietala and R. Hänninen, “Comment on ‘Motion of a helical vortex filament in superfluid 4He under the extrinsic form of the local induction approximation’ [Phys. Fluids 25, 085101 (2013)],” Phys. Fluids 26, 019101 (2014).
5. J. Laurie, V. S. L’vov, S. Nazarenko, and O. Rudenko, “Interaction of Kelvin waves and non-locality of the energy transfer in superfluids,” Phys. Rev. B 81, 104526 (2010).
7. 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).
9. 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).
11. H. Adachi, S. Fujiyama, and M. Tsubota, “Steady state of counterflow quantum turbulence: Vortex filament simulation with the full Biot-Savart law,” Phys. Rev. B 81, 104511 (2010).
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I agree with the authors regarding their comments on the Donnelly-Glaberson instability for such helical filaments as those obtained in my paper. I also find merit in their derivation of the quantum LIA (local induction approximation) in the manner of the LIA of Boffetta et al. However, I disagree with the primary criticisms of Hietala and Hänninen. In particular, though they suggest LIA and local nonlinear equation modes are not comparable since the former class of models contains superfluid friction parameters, note that since these parameters are small one may take them to zero and consider a qualitative comparison of the models (which is what was done in my paper). Second, while Hietala and Hänninen criticize certain assumptions made in my paper (and the paper of Shivamoggi where the model comes from) since the results break-down when Ak → ∞, note that in my paper I state that any deviations from the central axis along which the filament is aligned must be sufficiently bounded in variation. Therefore, it was already acknowledged that Ak(=|Φ x |) should be sufficiently bounded, precluding the Ak → ∞ case. I also show that, despite what Hietala and Hänninen claim, the dispersion relation obtained in my paper is consistent with LIA, where applicable. Finally, while Hietala and Hänninen claim that the dispersion parameter should be complex valued, I show that their dispersion relation is wrong, since it was derived incorrectly (they assume the complex modulus of the potential function is constant, yet then use this to obtain a potential function with non-constant modulus).
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