Superflow passing over a rough surface: Vortex nucleation

The transition from a free vortex superflow to a dissipative vortex flow in a superfluid remains an open problem, in particular because in real experiments, microscopic asperities on the walls play a major role since the relevant scales in a superfluid motion are basically of atomic scale. Here we model a superflow using the Gross-Pitaevskii mean-field equation in a domain with a rigid sinusoidal boundary. We demonstrate, analytically and numerically, the existence of a well-defined critical velocity for vortex nucleation. Additionally, we discuss the intrinsic mechanism of vortex nucleation by the appearance of a solitary wave characterized by a depletion of the wave function near the boundary.

Figure 1

Sketch of the domain of integration of the Gross-Pitaevskii Eq. (1). The bottom interface is taken as a sinusoidal profile (5). Periodic boundary conditions along the $x$-axis are imposed. The top boundary obeys a von Neumann boundary condition. Finally, a flow with velocity $v_0$ is imposed via a Galilean boost, which transforms Eqs. (1) to (6), as explained in the text.

Figure 2

Result of the numerical simulations. Snapshots for three different flow velocities. Row 1 shows the instantaneous value of ${\left|\psi (x,y)\right|}^2$, whereas the second row shows the phase $\varphi (x,y)=\arg \psi (x,y)$. The velocities correspond to (a) $v_0=0.8c$, (b) $v_0=0.85c$, and (c) $v_0=0.9c$, and the snapshots are taken at (a) $t=300$, (b) $t=300$, and (c) $t=360$. The numerical simulations are done for a system of ${256}^2$ points, with $dx=0.125$ and a time step $dt=0.001$. The undulation is characterized by $\epsilon =5dx$ and $\alpha =2\pi /\left(256dx\right)$, thus $\alpha \epsilon =0.1227$.

Figure 3

Plot of the quantity ${\mathrm{\Omega }}^{\left(2\right)}/\kappa$ vs time for three distinct flow velocities: (a) for $v_0=0.65$; (b) for $v_0=0.7$; and (c) for $v_0=0.8$. Each plot contains three snapshots of the superfluid density ${\left|\psi \right|}^2$ in the space near the undulated profile. The corresponding times of the snapshot are shown by a vertical segmented purple line. The numerical simulations are done for a system of ${128}^2$ points, with $dx=0.5$ and a time step $dt=0.01$. The undulation is characterized by $\epsilon =5dx$ and $\alpha =2\pi /\left(128dx\right)$, thus $\alpha \epsilon =0.245$. Additionally, we add for each case a movie displaying the vortex nucleation dynamics in the Supplemental Material [37].

Figure 4

Critical Mach number as a function of $\alpha \epsilon$. (a) Plots $M_c=v_0^{\left(c\right)}/c$ vs $\alpha \epsilon$ in linear scale, and (b) plots $1-M_c$ vs $\alpha \epsilon$ in log-log scale. The numerical simulations provide the lower bound $v_c^{(-)}/c$ critical Mach number (segmented line) and the upper bound $v_c^{(+)}/c$ critical Mach number (continuous line) for different numerical parameters accordingly to Table 1: (i) $dx=0.25$ and $N_x=1024,N_y=256$ (orange); (ii) $dx=0.125$ and $N_x=N_y=512$ (red); (iii) $dx=0.25$ and $N_x=512,N_y=256$ (purple); (iv) $dx=0.5$ and $N_x=N_y=256$ (magenta); (v) $dx=0.125$ and $N_x=N_y=256$ (green); and (vi) $dx=0.5$ and $N_x=N_y=128$ (blue). Additionally, the data are compared with three theoretical curves: the cyan curve corresponds to Falkovich–von Kármán theory [Eq. (35)], the black segmented curve corresponds to the first-order critical velocity (26), and the full black curve corresponds to the third-order Rayleigh-Janzen critical velocity (A3).

Figure 5

Results of numerical simulations. Snapshots of the embryo at $t=300$ time units. (a) ${\left|\psi (x,y)\right|}^2$ and (b) $\arg \psi (x,y)$. The modulus shows a slight depletion near the boundary, and the phase of the wave function (b) shows a step phase gradient emulating a shock wave. The simulations were done for $dx=0.125, N_x=N_y=256$, a flow velocity $v_0=0.8375, \epsilon =3dx=0.375, \alpha =\frac{2\pi }{N_{x}dx}=0.19635$, and $\alpha \epsilon =0.0736$.

Figure 6

Result of the numerical simulations for the anatomy of the vortex nucleation process. Snapshots for three different times for a flow velocity $v_0=0.85$. Row 1 shows the instantaneous value of ${\left|\psi (x,y)\right|}^2$, whereas the second row shows the phase $\varphi (x,y)=\arg \psi (x,y)$. The time instants correspond to (a) $t=232.5$, (b) $t=242.5$, and (c) $t=245$. In the snapshot (b1) we sketch the forefront in red and the backfront in orange. The numerical simulations are done for a system of ${256}^2$ points, with $dx=0.25$ and a time step $dt=0.0025$. The undulation is characterized by $\epsilon =10dx$ and $\alpha =2\pi /\left(256dx\right)$, thus $\alpha \epsilon =0.2455$.