Figure 1

(Color online) Snapshots of the vortex ring of radius $R=0.1\mathrm{cm}$ perturbed by $N=10\text{Kelvin}$ waves of various amplitude $A$ taken during the motion of the vortex. In the left panel (a) the amplitude of the Kelvin waves is small, $A∕R=0.05$, but the perturbed vortex ring (red color) already moves slower than the unperturbed vortex (blue color). In the center panel (b) the Kelvin waves have large amplitude, $A∕R=0.35$, and the perturbed vortex ring moves backwards (negative $z$ direction) on average. The top right panel (c) shows the top $\left(xy\right)$ view of the large amplitude vortex at $t=0\mathrm{s}$ (blue) and $t=26\mathrm{s}$ (red, outermost). For comparison, a nondisturbed vortex is shown with dashed line (green). The lower right panel (d) gives the averaged location of the ring as a function of time. From top to bottom the curves correspond to $A∕R=0.0,0.05,0.10,\dots ,0.35$.Reuse & Permissions

Figure 2

(Color online) Average translational velocity of the vortex ring as a function of the initial oscillation amplitude $A∕R$. Velocity is scaled by the velocity of the unperturbed ring, $v_{\text{ring}}$. The dash-dotted line corresponds to $N=20$, solid line to $N=10$, and the dashed line to $N=6$ in Eq. (6). Critical amplitudes, above which the velocities become negative, are $A∕R=0.085$, 0.17, and 0.32, respectively.Reuse & Permissions

Figure 3

(Color online) Curve drawn by the vortex at $y=0$ plane. Here the $z$ coordinate is the coordinate relative to the average location of the vortex and $N=10$ in Eq. [6]. In the top left panel (a) the amplitude is $A∕R=0.05$ and in the top right panel (b) $A∕R=0.20$. In both panels only the first $30\sec$ are shown. The thickness of the plotted curve arises from the chaotic motion rather than initial transient. The bottom panel (c) corresponds to $A∕R=0.50$ and we have drawn the curve for the first $90\sec$. The time step between the markers is $2\mathrm{ms}$; it is apparent that at large amplitudes the vortex is far from a sinusoidal helix and that the rotational speed at $y=0$ plane varies significantly.Reuse & Permissions

Figure 4

(Color online) Main figure: Angular frequency of Kelvin waves $\omega$ relative to the value ${\omega }_0$ obtained in the small amplitude limit $A∕R=0.001$ and presented as a function of the wave amplitude $A∕R$. The dashed line is for $N=6$, the solid line for $N=10$, and the dash-dotted line for $N=20$. The inset shows the same when plotted as a function of $A∕\lambda$, where $\lambda$ is the wavelength of the Kelvin wave. The additional dotted line is the result obtained for straight vortex when using a wavelength of $0.1\mathrm{cm}$ together with periodic boundary conditions and using 25 periods above and below to numerically determine the vortex motion.Reuse & Permissions

Figure 5

(Color online) The observed vortex length compared with the initial length $L_0=2\pi \sqrt{R^2+N^2{A}^2}$ is illustrated by solid (blue) lines and plotted as function of time in case of $N=10$. For comparison, the dashed (red) lines show the fluctuations in energy that are due to numerical errors and which can be reduced by increasing the space resolution. With increasing amplitude of oscillations the parameters for $A∕R$ shown are 0.20, 0.30, 0.40, and 0.50.Reuse & Permissions