Decay of homogeneous two-dimensional quantum turbulence

We numerically simulate the free decay of two-dimensional quantum turbulence in a large, homogeneous Bose-Einstein condensate. The large number of vortices, the uniformity of the density profile, and the absence of boundaries (where vortices can drift out of the condensate) isolate the annihilation of vortex-antivortex pairs as the only mechanism which reduces the number of vortices, $N_{\mathrm{v}}$, during the turbulence decay. The results clearly reveal that vortex annihilation is a four-vortex process, confirming the decay law $N_{\mathrm{v}}\sim t^{-1/3}$ where $t$ is time, which was inferred from experiments with relatively few vortices in small harmonically trapped condensates.

Figure 1

The condensate's density ${\left|\psi \right|}^2$ vs $x,y$ during the decay of quantum turbulence without dissipation ($\gamma =0$, left) and with dissipation ($\gamma =0.01$, right). Vortex locations are inferred by an algorithm which identifies the $2\pi$ phase winding and the associated density depletion. We mark the location of vortices with a positive circulation with a red circle and those with a negative circulation using a blue square.

Figure 2

The condensate's density ${\left|\psi \right|}^2$ vs $x,y$ during the decay of quantum turbulence without dissipation ($\gamma =0$, left) and with dissipation ($\gamma =0.01$, right) at times (a) $t=7.5\times {10}^3$, (c) $t=2.5\times {10}^4$, (e) $t=1\times {10}^5$, (b) $t=1\times {10}^3$, (d) $t=1\times {10}^4$, and (f) $t=5\times {10}^4$. The small holes in these density plots are the vortices.

Figure 3

The total vortex number, $N_{\mathrm{v}}$, plotted vs time, $t$. The solid black curve, dashed blue curve, and dot-dashed red curve correspond to ensemble-averaging ten simulations without dissipation ($\gamma =0$) and with dissipation ($\gamma =0.0025$ and $\gamma =0.01$), respectively. Notice the more rapid decay induced by increasing dissipation.

Figure 4

Log-log plots of the data presented in Fig. 3, with corresponding best fits plotted as red dashed lines (position adjusted for clarity). The left panel corresponds to the case $\gamma =0$, the central panel corresponds to $\gamma =0.0025$, and the right panel corresponds to $\gamma =0.01$. Alternative theoretical fits (discussed in the text) are plotted as dot-dashed lines.