Helicity, topology, and Kelvin waves in reconnecting quantum knots

Helicity is a topological invariant that measures the linkage and knottedness of lines, tubes, and ribbons. As such, it has found myriads of applications in astrophysics, fluid dynamics, atmospheric sciences, and biology. In quantum flows, where topology-changing reconnection events are a staple, helicity appears as a key quantity to study. However, the usual definition of helicity is not well posed in quantum vortices, and its computation based on counting links and crossings of centerline vorticity can be downright impossible to apply in complex and turbulent scenarios. We present a definition of helicity which overcomes these problems and which gives the expected result in the large-scale limit. With it, we show that certain reconnection events can excite Kelvin waves and other complex motions of the centerline vorticity, which slowly deplete helicity as they interact nonlinearly, thus linking the theory of vortex knots with observations of quantum fluids. This process also results in the depletion of helicity in a fully turbulent quantum flow, in a way reminiscent of the decay of helicity in classical fluids.

Figure 1

Renderings of the surface of zero phase for two knots in a quantum fluid. Top: Two linked rings, note the surface has one hole. Bottom: Trefoil knot, with three holes. The number of holes is associated with the number of turns the vector that lies on the surface perpendicular to the vortex does as it moves along the curve (a few of these vectors are shown).

Figure 2

Time evolution of the helicity for four quantum vortex configurations. At the top, snapshots of the configurations at different times are shown. The single ring only moves at constant speed. The two rings and the trefoil reconnect at times marked by the vertical arrows. When reconnection takes place between two perfectly antiparallel vortices (as in the two rings), helicity does not change. In the trefoil reconnection takes place simultaneously at three points and helicity changes abruptly at the time indicated by the red arrow; later it decays slowly to its final value. The (1,6)-torus knot deforms without reconnecting, and its helicity does not change.

Figure 3

Spatiotemporal spectrum for the two rings before (left) and after reconnection (right). The dashed (blue) line corresponds to the dispersion relation of sound waves, the solid (green) line to Kelvin waves, the dash-dotted line to sweeping with velocity $U_1$ (i.e., $\omega =U_{1}k$), and the dash-triple dotted line to sweeping with $\omega =U_{2}k$. Sweeping concentrates most of the power, and only one energetic mode with $k\approx 11$ may be compatible with the dispersion relation of Kelvin waves.

Figure 4

Spatiotemporal spectrum for the trefoil before (left) and after the reconnection (right). The dashed (blue) line corresponds to sound waves, the solid (green) line to Kelvin waves, and the dash-dotted line to sweeping with $\omega =Uk$. A broad range of modes compatible with the dispersion relation of Kelvin waves is excited after reconnection, and sound waves are visible at high frequencies.

Figure 5

Rendering of vortices at early times in a quantum ABC flow with helicity (spatial resolution of ${2048}^3$ grid points). The regularized helicity is equal to 3, matching the value expected for the classical flow at large scales. Normalizing by the quanta, the helicity of this flow is $\approx 480000{\mathrm{\Gamma }}^2$.

Figure 6

Evolution of helicity and all energy components during the decay of a quantum turbulent flow with ABC initial conditions using ${1024}^3$ grid points. Both helicity and the incompressible kinetic energy start to decay at around $t\approx 4$, in a correlated manner. The incompressible energy is redistributed into the other components, but the helicity is, in principle, dissipated.