Vortex Multiplication in Applied Flow: A Precursor to Superfluid Turbulence

A surface-mediated process is identified in ${}^3\mathrm{He}\mathrm{\text{−}}B$ which generates vortices at a roughly constant rate. It precedes a faster form of turbulence where intervortex interactions dominate. This precursor becomes observable when vortex loops are introduced in low-velocity rotating flow at sufficiently low mutual friction dissipation at temperatures below $0.5T_{\mathrm{c}}$. Our measurements indicate that the formation of new loops is associated with a single vortex interacting in the applied flow with the sample boundary. Numerical calculations show that the single-vortex instability arises when a helical Kelvin wave expands from a reconnection kink at the wall and then intersects again with the wall.

Figure 1

The probability that vortex multiplication will start from a remnant vortex, when rotation is increased from ${\Omega }_{\mathrm{i}}$ to ${\Omega }_{\mathrm{f}}$, plotted vs temperature (bottom axis) and mutual friction parameter $q=\alpha /(1-{\alpha }^{\prime })$ (top axis). No vortices are formed above $0.5T_{\mathrm{c}}$. Below $0.45T_{\mathrm{c}}$ vortex formation always leads to turbulence. At intermediate temperatures only part of the runs result in vortex formation.

Figure 2

Vortex formation with a slow precursor which suddenly develops to rapid turbulence. The number of vortices $N(t)$ is recorded at the top and bottom of the sample. The solid line is the average slow rate $\dot{N}$ while the dashed lines denote the rapid turbulence. Initially the sample is in the equilibrium vortex state at ${\Omega }_{\mathrm{i}}=0.05\mathrm{rad}/\mathrm{s}$ with $N\approx 37$ vortices and a few vortices connecting to the cylinder wall (left inset). Rotation is then increased to a new stable value ${\Omega }_{\mathrm{f}}$, which is reached at $t=0$. Three runs with different ${\Omega }_{\mathrm{f}}$ are shown. During the ramp to ${\Omega }_{\mathrm{f}}$, $N\approx \mathrm{const}$ while the flow builds up and compresses the vortices in a central cluster (right inset). The precursor is attributed to an instability of the single vortices, which extend across the counterflow region and end at the cylindrical boundary.

Figure 3

Simulation of the measurements in Fig. 2. The total number of separate vortices $N(t)$ vs time is followed in a cylinder of radius 3 mm and length 10 mm. The calculation is started from a vortex ring of 2 mm radius in the azimuthal plane. The ring is unstable in azimuthal flow and generates via the Kelvin wave instability [3, 4] tens of vortices in a rapid burst, which gives the configuration shown in the inset. After this initial burst slow vortex formation at constant average rate $\dot{N}$ starts. Here each new vortex is produced from Kelvin waves expanding on an isolated vortex (as shown in the inset) which is blown up to ringlike shape and then reconnects at the boundary. The later rapid growth in vortex number is dominated by intervortex interactions.

Figure 4

As a model of the precursor, the collision of a vortex ring with a plane wall (at $z=0$) is calculated. A ring of 0.5 mm radius is initially above the wall, with the plane of the ring tilted by $\pi /10$ from the $x=0$ plane. In the frame of reference of this figure, uniform flow of the normal component at $v_{\mathrm{n}}=3\mathrm{mm}/\mathrm{s}$ is applied in the $x$ direction. The different contours show the ring at 0.05 s intervals. The sharp kinks at the wall reconnections induce Kelvin waves on the loop. These are able to grow in the applied flow only on one of the two legs formed in the reconnection (here on the right). The largest wave reconnects with the wall and a new loop is then separated. The conditions for the growth of Kelvin waves are fulfilled in such collisions only in the underdamped temperature regime: The inset on the top shows the probability that a new vortex is created when the original ring is initially placed at $z=1\mathrm{mm}$ with random orientation. This probability depends on the ratio $q=\alpha /(1-{\alpha }^{\prime })$, shown on the top axis, rather than on $\alpha$ or ${\alpha }^{\prime }$ separately.