Twisted Vortex State

We study a twisted vortex bundle where quantized vortices form helices circling around the axis of the bundle in a “force-free” configuration. Such a state is created by injecting vortices into a rotating vortex-free superfluid. Using continuum theory we determine the structure and the relaxation of the twisted state. This is confirmed by numerical calculations. We also present experimental evidence of the twisted vortex state in superfluid ${}^3\mathrm{He}\mathrm{\text{−}}\mathrm{B}$.

Figure 1

The formation of twisted vortex state. The vortices have their propagating ends bent to the sidewall of the rotating cylinder. As they expand upwards into the vortex-free state, the ends of the vortex lines rotate around the cylinder axis. The twist is nonuniform because boundary conditions allow it to unwind at the bottom solid wall. The figure gives a snapshot (at time $t=25{\Omega }^{-1}$) of a numerical simulation of 23 vortices initially generated near the bottom end ($t=0$). The parameters are $2\pi \Omega R^2/\kappa =86$, $\alpha =0.18$, and ${\alpha }^{\prime }=0.16$ [15], and $R/a=3\times {10}^5$, which corresponds to $T=0.4T_{\mathrm{c}}$ in ${}^3\mathrm{He}\mathrm{\text{−}}\mathrm{B}$ at 29 bar pressure. For time evolution see Ref. 16.

Figure 2

A snapshot of the numerically calculated vortex state expanding along $z$: (a) vortex configuration, (b) axial $v_z$, and (c) azimuthal $v_{\varphi }$ components of the superflow velocity. The velocities, plotted as functions of $z$, are averaged over the azimuthal angle and are shown at radii $r=nR/6$ with integer $n$. Note that $v_{\varphi }-\Omega r$ changes sign in (c) close to the center of the bundle, as predicted by Eq. (5). The simulation was started with 203 vortices and the picture was taken after a time interval of $60{\Omega }^{-1}$. The parameters are the same as in Fig. 1 except $2\pi \Omega R^2/\kappa =214$, $R/a=1.5\times {10}^5$. For clarity, $r$ and $z$ coordinates have different scales in (a).

Figure 3

NMR absorption signals as a function of time after injection of vortices at $t=0$. The two NMR absorption records in the main panel show the reduction in the counterflow peak and the overshoot in the Larmor region. The former is interpreted as the arrival of the vortex front and the rapid increase in $v_{\varphi }$. The latter arises from the axial flow velocity $v_z$, caused by twisted vortices, and the subsequent slow relaxation towards the equilibrium state. The inset shows the NMR line shapes in the initial vortex-free state $N=0$ and the final equilibrium vortex state $N=N_{\mathrm{eq}}$. The spectra are measured at constant temperature and have the same integrated absorption.

Figure 4

The time constant for the decay of the Larmor absorption (data points) compared to the decay of the twisted state according to the slowest mode of Eqs. (6, 7) (line). The line has $k=\pi /\ell$ as fitting parameter giving $\ell =18\mathrm{mm}$. For comparison, the half length of the cylinder is 55 mm and the detector coil is placed between 5 and 15 mm distance from the end plate [14].