Helicity, topology, and Kelvin waves in reconnecting quantum knots

Abstract

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 and solar physics, in fluid dynamics, in atmospheric sciences, and in 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 vortex lines can be downright impossible to apply in complex and turbulent scenarios. We present a new definition of helicity which overcomes these problems. With it, we show that only certain reconnection events conserve helicity. In other cases helicity can change abruptly during reconnection. Furthermore, we show that these events can also excite Kelvin waves, which slowly deplete helicity as they interact nonlinearly, thus linking the theory of vortex knots with observations of quantum turbulence. Although helicity is perfectly conserved in barotropic ideal fluids, in real fluids [13,14] and in superfluids [15,16] vortex reconnection events, which alter the topology of the flow, can take place. It is unclear how well helicity is preserved under reconnection. As a few examples, experiments of vortex knots in water have shown that center line helicity remains constant throughout reconnection events [17], while theoretical arguments indicate that writhe (one component of the helicity) should be conserved in anti-parallel reconnection events [18], a fact later confirmed in numerical simulations of a few specific quantum vortex knots [19]. However, numerical studies of Burgers-type vortices indicate that helicity is not conserved [20]. While experiments studying helicity in quantum flows have not been done yet, the recent experimental creation of quantum knots in a Bose-Einstein condensate in the laboratory [21] is a significant step in that direction.Recently, quantum flows have been used as a testbed for many of these ideas [17,19], as vorticity in a quantum flow is concentrated along vortex lines with quantized circulation, and as these vortex lines can reconnect without dissipation. However, the lack of a fluid-like definition of helicity for a quantum flow requires complex topological measurements of the linking and knottedness of vortex lines [17], or artificial filtering of the fields [19] to prevent spurious values of helicity resulting from the singularity near quantum vortices. Moreover, helicity in quantum flows has an interest per se, as reconnection events in superfluid turbulence can excite Kelvin waves [22]. These are helical perturbations that travel along the vortex lines first predicted for classical vortices by Lord Kelvin (see [23]). Kelvin waves are believed to be responsible for the generation of an energy cascade [24,25] leading to Kelvin wave turbulence [26]. Possible links between helicity and the development of Kelvin wave turbulence have remain obscure as a result of the difficulties i...

Helicity within the vortex filament model

“…A method that allows helicity to be evaluated directly from the wavefunction of the Gross-Pitaevskii model of a superfluid has recently been presented in ref. 19. We note that the expression presented there reduces to the linking and writhe contributions only and, therefore, the observations made above would also be expected to apply to the definition of the helicity presented in that work.…”

“…In superfluids the quantum mechanical phase of the order parameter plays the role of the velocity potential. We note that similar choices have been suggested also by other authors16171819. We will illustrate that this choice of the spanwise vector coincides with working in the so-called Seifert frame in which case helicity becomes trivially zero2021.…”

“…In a microscopic model, such as that given by the Gross-Pitaevskii equation, they are made possible by the “quantum-pressure” term which acts to couple the incompressible and compressible modes of the system. Using the Gross-Pitaevskii model, the effect of reconnections on helicity has recently been analyzed in several papers1617181925.…”

Evolution of a superfluid vortex filament tangle driven by the Gross-Pitaevskii equation

“…Although unlikely, we show that rings can link creating a local (in time and space) fluctuation of helicity. It will be of great interest to repeat a similar analysis in a GP setting where the mean helicity of the flow is not zero, like the ABC flow introduced in [20] where linking and selflinking processes could be substantially enhanced. Overall, the results presented in this Letter confirm that some predictions traditionally associated to superfluid liquid Helium become important in weakly-interacting BECs at low temperature described by the GP model.…”

“…This quantity is the analogous to helicity in classical fluid dynamics, an important inviscid invariant. The linking number Lk takes integer values and gives information about the number of linked rings present in the system, whereas the writhe takes real values and its contribution comes from knots and KWs [20]. Figure 2b shows the temporal evolution of these three quantities.…”