Rayleigh-Taylor instability and vortex rings in coupled Gross-Pitaevskii equations

The Rayleigh-Taylor instability is a gravitational instability in two fluids where the heavier fluid is set over the lighter fluid. The instability occurs both in classical fluids and quantum fluids. We numerically study the Rayleigh-Taylor instability using coupled Gross-Pitaevskii equations for two-component Bose-Einstein condensates. We carry out numerical simulations that the heavier component is set in a torus initially which is surrounded by the lighter component. When the torus falls, the Rayleigh-Taylor instability develops and a sagging pattern appears. This instability is investigated for the two cases with and without a vortex ring inside the torus. We find that a vortex ring suppresses the instability when the radius of the torus is small.

Figure 1

Growth rate $\lambda \left(k\right)$ of Eq. (3) for $\mathrm{\Delta }\rho g=0.2$, $n=1$, $\gamma =2$, and ${\rho }_{+}+{\rho }_{-}=2$.

Figure 2

(a) Two snapshots of the patterns of $|{\varphi }_{-}|>0.8$ at $t=5$ and 25 for $k=8\pi /100$, $R=5$, and $\delta =0.5$. The initial condition is a cylinder without a vortex. (b) Two snapshots of the patterns of $|{\varphi }_{-}|>0.8$ at $t=5$ and 25 for $k=8\pi /100$, $R=5$, and $\delta =0.5$. The initial condition is a cylinder with a vortex. (c) Snapshots of the patterns of $|{\varphi }_{-}|>0.8$ at $t=5,10,\cdots ,25$ in the space of $(x,z)$ at a cross section at $y=25$ (left) and $y=37.5$ (right) corresponding to (a). (d) Snapshots of the patterns of $|{\varphi }_{-}|>0.8$ at $t=5,10,\cdots ,25$ in the space of $(x,z)$ at cross sections at (left) $y=25$ and (right) $y=37.5$ corresponding to (b).

Figure 3

Growth rates $\lambda \left(k\right)$ for cylinders with (a) $R=5$ without a vortex line, (b) $R=5$ with a vortex line, and (c) $R=3$ without a vortex line. $\mathrm{\Delta }\rho g=0.2$.

Figure 4

(a) Two snapshots of regions satisfying $|{\varphi }_{-}|>0.8$ at $t=5$ and 25 for $\mathrm{\Delta }\rho g=0.2$, $R=12$, $a=4$, $n=2$, and $\delta =0.5$. (b) Two snapshots of regions satisfying $|{\varphi }_{-}|>0.8$ at $t=5$ and 25 when the vortex ring is initially set inside the torus. (c) Two snapshots of the vortex ring at $t=5$ and 25.

Figure 5

(a) Two snapshots of regions satisfying $|{\varphi }_{-}|>0.8$ for $R=8$, $a=4$, and $\mathrm{\Delta }\rho g=0.2$. (b) Two snapshots of regions satisfying $|{\varphi }_{-}|>0.8$ for $R=8$ in the case with a vortex ring. (c) Two snapshots of the vortex ring in (b).

Figure 6

Projections on the $(x,y)$ plane of the vortex rings at $t=15$ for $R=12$ against the perturbations of (a) $n=3$ and (b) $n=4$.

Figure 7

$\lambda \left(n\right)$ of $n=2$, 3, 4, 5, and 6 at $\mathrm{\Delta }\rho g=0.2$ for (a) $R=12$, (b) $R=10$, and (c) $R=8$.

Figure 8

Time evolutions of the $z$ coordinate $Z$ for (a) $R=12$, (b) $R=8$, and (c) $R=6$. Time evolution of the width $W$ for (d) $R=12$, (e) $R=8$, and (f) $R=6$.

Figure 9

(a) Time evolutions of $\delta {\varphi }_{-2}$ at $\mathrm{\Delta }\rho g=0.2$ and 0.08 for $R=8$ in the case without a vortex ring. (b) Critical values of $\mathrm{\Delta }\rho g$ for several $R$ values. The upper line corresponds to the case with a vortex ring, and the lower line corresponds to the case without a vortex ring.