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

Abstract

Very recently, Shivamoggi [“Vortex motion in superfluid 4He: Reformulation in the extrinsic vortex-filament coordinate space,” Phys. Rev. B 84, 012506 (2011)]10.1103/PhysRevB.84.012506 studied the extrinsic form of the local induction approximation (LIA) for the motion of a Kelvin wave on a vortex filament in superfluid 4He, and obtained some results in a cubic approximation. Presently, we study the motion of helical vortex filaments in superfluid 4He under the exact fully nonlinear LIA considered in potential form by Van Gorder [“Fully nonlinear local induction equation describing the motion of a vortex filament in superfluid 4He,” J. Fluid Mech. 707, 585 (2012)]10.1017/jfm.2012.308 and obtained from the Biot-Savart law through the equations of Hall and Vinen [“The rotation of liquid helium II. I. Experiments on the propagation of second sound in uniformly rotating helium II,” Proc. R. Soc. London, Ser. A 238, 204 (1956)]10.1098/rspa.1956.0214 including superfluid friction terms. Nonlinear dispersion relations governing the helical Kelvin wave on such a vortex filament are derived in exact form, from which we may exactly calculate the phase and group velocity of the Kelvin wave. With this, we classify the motion of a helical Kelvin wave on a vortex filament under the LIA. The dispersion relations and results, which follow are exact in nature, in contrast to most results in the literature, which are usually numerical approximations. As such, our results accurately capture the qualitative behavior of the Kelvin waves under the LIA. Extensions to other frameworks are discussed.