Structure, dynamics, and reconnection of vortices in a nonlocal model of superfluids

Abstract

We study the reconnection of vortices in a quantum fluid with a roton minimum, by numerically solving the Gross-Pitaevskii (GP) equations. A non-local interaction potential is introduced to mimic the experimental dispersion relation of superfluid 4 He. We begin by choosing a functional shape of the interaction potential that allows to reproduce in an approximative way the so-called roton minimum observed in experiments, without leading to spurious local crystallization events. We then follow and track the phenomenon of reconnection starting from a set of two perpendicular vortices. A precise and quantitative study of various quantities characterizing the evolution of this phenomenon is proposed: this includes the evolution of statistics of several hydrodynamical quantities of interest, and the geometrical description of a observed helical wave packet that propagates along the vortex cores. Those geometrical properties are systematically compared to the predictions of the Local Induction Approximation (LIA), showing similarities and differences. The introduction of the roton minimum in the model does not change the macroscopic properties of the reconnection event but the microscopic structure of the vortices differs. Structures are generated at the roton scale and helical waves are evidenced along the vortices. However, contrary to what is expected in classical viscous or inviscid incompressible flows, the numerical simulations do not evidence the generation of structures at smaller or larger scales than the typical atomic size.

Interaction between active particles and quantum vortices leading to Kelvin wave generation

“…Similar theoretical computations can be straightforwardly performed in the case of non-local models of superfluids, that are more adequate for describing superfluid helium. In such models, the vortex density profile shows oscillations as a function of the distance to the core 41 , 42 , which could have some impact on the dynamics of small and light particles. In our derivation, we have neglected acoustic radiation and interaction of vortices with with sound waves.…”

Kolmogorov and Kelvin wave cascades in a generalized model for quantum turbulence

“…A generalized model that is able to describe more complex systems can be obtained by considering a nonlocal interaction between bosons. With proper modeling [24][25][26], density excitations exhibit a roton minimum in their spectrum as the one observed in He II [23]. It also describes well the behavior of dipolar condensates [32,33].…”

“…However, there have been some successful attempts to include such effects in the GP model. For instance, a roton minimum can be easily introduced in GP by using a nonlocal potential that models a long-range interaction between bosons [24][25][26]. The stronger interaction of helium can also be included phenomenologically by introducing high-order terms in the GP Hamiltonian.…”

“…Note that these terms can be derived as beyond mean field corrections [27]. Some generalized version of the GP model has been used to study the vortex solutions [28,29] and some dynamical aspects such as vortex reconnections [26]. Intuitively, for a turbulent superfluid, we can expect that such generalization of the GP model might be important at scales smaller than the intervortex distance and with less influence at scales at which Kolmogorov turbulence is observed.…”

Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity

“…Nevertheless, the results of the simulation depicted in Figure 4 show that E min is concave, and in that case Theorem 4 gives the orbital stability of the solitons illustrated in Figure 4. Finally, we consider the potential W a,b,c (ξ) = (1 + aξ 2 + bξ 4 )e −cξ 2 , (7.4) that it has been proposed in [9,57] to describe a quantum fluid exhibiting a roton-maxon spectrum such as Helium 4. Indeed, as predicted by the Landau theory, in such a fluid, the dispersion curve (3) cannot be monotone and it should have a local maximum and a local minimum, that are the so-called maxon and roton, respectively.…”