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

We study the reconnection of vortices in a quantum fluid with a roton minimum by numerically solving the Gross-Pitaevskii (GP) equations. A nonlocal interaction potential is introduced to mimic the experimental dispersion relation of superfluid ${}^4\mathrm{He}$. We begin by choosing a functional shape of the interaction potential that allows us 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.

Figure 1

(a) Dispersion relations of ${}^4\mathrm{He}$ and its model as given by the nonlocal GP equation [Eq. (1)]: experimental results from neutron scattering experiments (black line, see Ref. [30]), its fit as proposed in Ref. [31] (red dash-dotted line), and our current model [Eq. (3)] with a raised roton minimum so as to avoid crystallization as explained in the text (blue dashed line). (b) Corresponding interaction potentials $V\left(r\right)/V_0$ entering in Eq. (1), with $V_0=21.96\text{eV}$ for the model of Ref. [31] and $V_0=0.15\text{eV}$ for our present model [Eq. (3)], used to obtain the theoretical dispersion curves of panel (a). We display in the inset the radial density profile of the stationary solution corresponding to the local GP equation [Eq. (10)] with a black dotted line, and the corresponding profile in the nonlocal case [Eq. (1)] with a blue dashed line. [(c), (d)] Color plots of the distribution of density, i.e., ${\left|\psi \right|}^2$, of the stationary vortex solution obtained as the solution of the relaxation problem given in Eq. (4). (c) With our raised roton gap [Eq. (3)] so as to avoid crystallization. (d) With the potential proposed in Ref. [31], which leads to a crystallization event.

Figure 2

(a) Radial distributions of the velocity field $\mathit{v}$ and probability current $\mathit{j}$ of the axisymmetric stationary solutions of the local [Eq. (10)] and nonlocal [Eq. (1)] GP equations. (b) Radial distributions of the associated pseudovorticity fields [Eq. (8)] and their comparison with a schematic fit provided in Eq. (9).

Figure 3

3D visualization of the superfluid density during reconnection, in the local (top) and nonlocal (bottom) models. A density threshold is applied for clarity, so that the bulk fluid density $\rho \sim 1$ is transparent. Four snapshots are represented along time, from left to right respectively: the initial condition $t=0$, the reconnection time $t=t_{\text{rec}}$, twice the reconnection time $t=2t_{\mathrm{rec}}$, and the end time of the simulation $t=t_{\text{end}}\approx 6t_0$.

Figure 4

Plots of the probability density functions (PDF) of various hydrodynamical quantities, in the local (left) and nonlocal (right) models. [(a), (b)] PDF of the superfluid densities ${\left|\psi \right|}^2$. High-density events $\rho >1$ are well represented in the nonlocal case (b), whereas they are absent in the local model (a). [(c), (d)] PDF of the probability current norm $j=\left|\mathit{j}\right|$ (see text for a precise definition). A $j^{-3}$ behavior is observed on the right tails of both models and comes from the midrange decay of $j$. A $j^{1.76}$ scaling is observable on both left tails. [(e), (f)] PDF of the pseudovorticity norm $w=\left|\mathit{w}\right|$ [Eq. (8)]. The observed power-law behaviors $P_w\sim w^{-1.4}$ are superimposed in both the local and nonlocal cases.

Figure 5

(a) Total length of the set of two vortices (see text), in the local (red) and nonlocal (blue) models. (b) Distance $\delta$ between the two vortices, in the local (red) and nonlocal (blue). Inset: squared distance ${\delta }^2$ between the two vortices for both models and their respective linear fit (dashed and dot-dashed lines). The slopes of the linear fits before and after reconnection are given in the text.

Figure 6

[(a), (e), (i)] Space-time maps of the local curvature of a vortex line $\kappa (s,t)$, in the local case for $g=128$ (a))and $g=21.345$ (e) and for the nonlocal model (i). The position of the principal curvature maximum is tracked along time. [(b), (f), (j)] Corresponding curvature profiles of the tracked maximum at five regularly spaced times during its propagation. Each profile is scaled back onto the initial condition at reconnection. Snapshot times are $t_1=t_{\text{rec}},t_2=t_{\text{rec}}+0.09,t_3=t_{\text{rec}}+0.17,t_4=t_{\text{rec}}+0.26$, and $t_5=t_{\text{rec}}+0.34$. We observe the growth of secondary curvature maxima in the nonlocal case (j). [(c), (g), (k)] Time evolution of the $\sigma ,\gamma ,c$ scaling parameters, which represent the width, height, and velocity of the curvature wave packet respectively [see Eq. (12)]. [(d), (h), (l)] Projections of the velocity vector of the local maximum of curvature onto the Frenet-Serret orthonormal basis $\mathit{T},\mathit{N},\mathit{B}$.