Helical waves on quantized vortices

Due to quantum mechanical constraints, the vortices in superfluid helium-4 are line-like objects. This makes the study of hydrodynamics simpler. Rotational flow is possible only in the presence of these quantized vortices. In the zero temperature limit, superfluids resemble inviscid ideal fluids better than perhaps any other system. This makes them a convenient model system to study. Kelvin waves are helical perturbations of a vortex. Since any small perturbation of a straight vortex can be expressed as a sum of helical modes, they are the most basic excitations of a vortex. In superfluid turbulence Kelvin waves play a crucial role in the energy dissipation in the lowest temperatures. In classical fluids, helical vortices appear in wakes behind turbines and propellers. We study how Kelvine waves are generated due to an axial flow of the normal component. We show that the critical flow velocity for the amplification of a Kelvin wave depends both on the wavelength and the amplitude of the helix. We also study the interactions of nearby helical vortices in the absence of mutual friction. This work is also relevant to thin-cored helical vortices in classical fluids. We also consider the possible methods of identification of Kelvin waves for complicated vortex configurations. For classical fluids, helicity has proven to be a useful quantity. If the vorticity is restricted to vortex tubes, it is tied to the knottedness and twisting of the vortex tubes. One could think that for superfluids, where the vorticity is concentrated on line-like objects, helicity would also be a useful quantity. However, since a line cannot be twisted, it is not as straightforward to give a similar interpretation to the helicity. Helicity can be defined to the superfluids, but it turns out to be always zero. The main method used in this work is the vortex filament model. This model has turned out to be a valuable tool, since experiments provide only a limited amount of information about the vortex configurations. Besides the full model based on Biot-Savart law, we use the well-known local induction approximation. It is a useful method, but it has its limitations. The local induction approximation may be extended to include approximative non-local interactions. This model we used in our study on a recurrence phenomenon involving Kelvin waves. We found that the approximative model gives a good match with full Biot-Savart simulations for small amplitude Kelvin waves.