Dynamics of quantized vortex filaments under a local induction approximation with second-order correction

Abstract

We study a second-order local induction approximation (LIA) for the dynamics of a single open quantized vortex filament (such as those filaments arising in superfluid helium). While for a classical vortex filament, this second-order correction can be interpreted as a correction due to the inclusion of axial flow within a filament core, in the quantized filament case, this second order correction can be viewed as a correction due to variable condensate healing length. We compare the evolution of the decay rate, transverse velocity, and rotational velocity of Kelvin waves along vortex filaments under this model to that of the first order LIA of Schwarz for quantized vortex filaments, as well as to a corresponding nonlocal model involving Biot-Savart integrals for the self-induced motion of the vortex filament. For intermediate wavenumbers, the second-order model solutions show improved agreement with the nonlocal Biot-Savart model, due to an additional control parameter. We also consider the stability of Kelvin waves under the second-order corrections; these results allow us to understand the Donnelly-Glaberson instability in the context of the second-order model. The second-order corrections tend to stabilize the resulting solutions, in agreement with what was previously found from the nonlocal Biot-Savart formulation, yet still permit a local description of the vortex filament in terms of a partial differential equation (akin to the first-order LIA) rather than an integro-differential equation.