Vortex Stretching as a Mechanism for Quantum Kinetic Energy Decay

A pair of perturbed antiparallel quantum vortices, simulated using the three-dimensional Gross-Pitaevskii equations, is shown to be unstable to vortex stretching. This results in kinetic energy $K_{\nabla \psi }$ being converted into interaction energy $E_I$ and eventually local kinetic energy depletion that is similar to energy decay in a classical fluid, even though the governing equations are Hamiltonian and energy conserving. The intermediate stages include the generation of vortex waves, their deepening, multiple reconnections, the emission of vortex rings and phonons, and the creation of an approximately $-5/3$ kinetic energy spectrum at high wave numbers. All of the wave generation and reconnection steps follow from interactions between the two original vortices. A four vortex example is given to demonstrate that some of these steps might be general.

Figure 1

Three-dimension isosurfaces with $\rho =0.05$. Nearly the entire vortices, which extend from $y=-16\pi$ to $y=16\pi$, are shown in (a). The pair is propagating towards smaller $x$. In (b), only $y>0$ is shown to allow us to see into the vortices through the $x\mathrm{\text{−}}z$ symmetry plane as they are starting to reconnect across the $x\mathrm{\text{−}}y$ dividing plane. Simultaneously, kinks form on each vortex in $y$. These kinks are the source of waves that propagate out along the vortices after reconnection.

Figure 2

(a) $t=4.5$. After reconnection, the waves deepen until second reconnections occur, allowing vortex rings to separate from the origin vortices. Further incipient reconnections, which lead to the formation of additional vortex rings, can be seen along the vortices. (b) $t=45$. View showing all the vortex rings for $y>0$. Multiple vortex rings start separating from the original antiparallel pair starting at $t=12$ and propagate to larger $y$. By $t=45$ some have left the system.

Figure 3

Estimates of the line length compared to changes in the interaction and kinetic energies. (a) Analysis over the full domain. There is strong $E_I$ and vortex line growth ($L$) for $0.5<t<6$. For $6<t<25$ both $K_{\nabla \psi }$ and $L$ decrease. For $T>30$ the global kinetic energy $K_{\nabla \psi }$ grows again. This is associated with the accumulation of energy for $y>4\pi$. (b) Only the first $y$ quadrant, the original interaction region. Here too, the initial length and $E_I$ grow at the expense of $K_{\nabla \psi }$. Later $K_{\nabla \psi }$, half the total Hamiltonian $0.5H$ and $L$ decrease. (c)–(e) Distributions of the $E_I$ with respect to density at $t=0.5$, 6, 30 to show how energy appears to flow from $K_{\nabla \psi }$ to $E_I$ to waves. Spectra: (f) by $t=48$, $K_{\nabla \psi }(k_y)$ has an enhancement high wave number regime compared to $E_I$. Spectra in the other directions have similar trends but are less distinct. For comparison $k^{-3}$ and $k^{-5/3}$ lines are drawn to demonstrate this effect, which could be indicating some type of downscale energy cascade in $K_{\nabla \psi }$.

Figure 4

Four colliding rings, the entangled state generated, the final relaxed state, and the time dependence of the kinetic and interaction energies plus a measure of line length in the inner region that contained the original vortices. In this case the measure of line length, the volume where $\rho <0.1$, tracks the interaction energy $E_I$ more closely than the kinetic energy.