Analysis of the nonlinear Schrodinger vortex reconnection is given in terms of coordinate-time power series. The lowest order terms in these series correspond to a solution of the linear Schrodinger equation and provide several interesting properties of the reconnection process, in particular the non-singular character of reconnections, the anti-parallel configuration of vortex filaments and a square-root law of approach just before/after reconnections. The complete infinite power series represents a fully nonlinear analytical solution in a finite volume which includes the reconnection point, and is valid for finite time provided the initial condition is an analytic function. These series solutions are free from the periodicity artifacts and discretization error of the direct computational approaches and they are easy to analyze using a computer algebra program.
This paper addresses reconnection of vortex tubes, with particular focus on the topology of the vortex lines (field lines of the vorticity). This analysis of vortex line topology reveals previously undiscovered features of the reconnection process, such as the generation of many small flux rings, formed when reconnection occurs in multiple locations in the vortex sheet between the tubes. Consideration of three-dimensional reconnection principles leads to a robust measurement of the reconnection rate, even once instabilities break the symmetry. It also allows us to identify internal reconnectio…
Reconnection plays a significant role in the dynamics of plasmas, polymers and macromolecules, as well as in numerous laminar and turbulent flow phenomena in both classical and quantum fluids. Extensive studies in quantum vortex reconnection show that the minimum separation distance δ between interacting vortices follows a 1/2 scaling. Due to the complex nature of the dynamics (e.g., the formation of bridges and threads as well as successive reconnections and avalanche), such scaling has never been reported for (classical) viscous vortex reconnection. Using the direct numerical simulation of t…
An insight into vortex reconnections in superfluids is presented making use of analytical results and numerical simulations of the Gross--Pitaevskii model. Universal aspects of the reconnection process are investigated by considering different initial vortex configurations and making use of a recently developed tracking algorithm to reconstruct the vortex filaments. We show that during a reconnection event the vortex lines approach and separate always accordingly to the time scaling $ δ\sim t^{1/2} $ with pre-factors that depend on the vortex configuration. We also investigate the behavior of …
This paper addresses the interaction of initially anti-parallel vortex tubes containing an axial flow that induces a twisting of the vortex lines around the tube axes, using numerical simulations. Vortex tube configurations with both the same and opposite senses of twist -- corresponding to same and opposite signs of kinetic helicity density -- are considered. It is found that the topology of the reconnection process is very different between the two cases. For tubes with the same sense of twist, the reconnection is fully three-dimensional (3D): vortex lines reconnect at a finite angle, and 3D…
Here we show that under quantum reconnection, simulated by using the three-dimensional Gross- Pitaevskii equation, self-helicity of a system of two interacting vortex rings remains conserved. By resolving the fine structure of the vortex cores, we demonstrate that total length of the vortex system reaches a maximum at the reconnection time, while both writhe helicity and twist helicity remain separately unchanged throughout the process. Self-helicity is computed by two independent methods, and topological information is based on the extraction and analysis of geometric quantities such as writh…
We investigate numerically the Navier-Stokes dynamics of reconnecting vortex rings at small $Re$ number. We find that reconnections are dissipative due to the smoothing of vorticity gradients at reconnection kinks and to the formation of secondary structures of stretched anti-parallel vorticity which transfer kinetic energy to small scales where it is subsequently dissipated efficiently. In addition, the relaxation of the reconnection kinks excites Kelvin waves which due to strong damping are of low wavenumber and affect directly only large scale properties of the flow.
We study reconnections of quantum vortices by numerically solving the governing Gross-Pitaevskii equation. We find that the minimum distance between vortices scales differently with time before and after the vortex reconnection. We also compute vortex reconnections using the Biot-Savart law for vortex filaments of infinitesimal thickness, and find that, in this model, reconnection are time-symmetric. We argue that the likely cause of the difference between the Gross-Pitaevskii model and the Biot-Savart model is the intense rarefaction wave which is radiated away from a Gross-Pitaeveskii reconn…
The study of vortex reconnections is an essential ingredient of understanding superfluid turbulence, a phenomenon recently also reported in trapped atomic Bose-Einstein condensates. In this work we show that, despite the established dependence of vortex motion on temperature in such systems, vortex reconnections are actually temperature independent on the typical length/time scales of atomic condensates. Our work is based on a dissipative Gross-Pitaevskii equation for the condensate, coupled to a semiclassical Boltzmann equation for the thermal cloud (the Zaremba-Nikuni-Griffin formalism). Com…
We consider the reconnection of two untwisted, linked quantised vortex rings in a Bose-Einstein condensate. We show that the reconnection is capable of transferring helicity, from the links present in the initial configuration, to a twisting of the resulting vortex ring, and hence a rotation of the phase plane along the vortex ring. As velocities in a quantum fluid are the gradient of the phase, a twisting of the phase along the vortices leads to an axial flow along the vortices. Hence one would expect that the dynamics of quantum turbulence are strongly influence by the helicity of the initia…
Vortex stretching in a compressible fluid is considered. Two-dimensional and axisymmetric cases are considered separately. The flows associated with the vortices are perpendicular to the plane of the uniform straining flows. Externally-imposed density build-up near the axis leads to enhanced compactness of the vortices - "dressed" vortices (in analogy to "dressed" charged particles in a dielectric system). The compressible vortex flow solutions in the 2D as well as axisymmetric cases identify a length scale relevant for the compressible case which leads to the Kadomtsev-Petviashvili spectrum f…
Reconnections between quantum vortex filaments in presence of trapped particles are investigated using numerical simulations of the Gross--Pitaevskii equation. Particles are described with classical degrees of freedom and modeled as highly repulsive potentials which deplete the superfluid. First, the case of a vortex dipole with a single particle trapped inside one of the vortices is studied. It is shown that the reconnection takes place at the position of the particle as a consequence of the symmetry breaking induced by it. The separation rate between the reconnecting points is compatible wit…
We employ detailed numerical simulations to probe the mechanism of flow reversals in two-dimensional turbulent convection. We show that the reversals occur via vortex reconnection of two attracting corner rolls having same sign of vorticity, thus leading to major restructuring of the flow. Large fluctuations in heat transport are observed during the reversal due to this flow reconfiguration. The flow configurations during the reversals have been analyzed quantitatively using large-scale modes. Using these tools, we also show why flow reversals occur for a restricted range of Rayleigh and Prand…
In low-temperature superfluid helium, viscosity is zero, and vorticity takes the form of discrete, vortex filaments of fixed circulation and atomic thickness. We present numerical evidence of three-dimensional inverse energy transfer from small length scales to large length scales in superfluid turbulence generated by a flow of vortex rings. We argue that the effect arises from the anisotropy of the flow, which favours vortex reconnections of vortex loops of the same polarity, and that it has been indirectly observed in the laboratory. The effect open questions about analogies with related pro…
In Part 1 of this work, we have derived a dynamical system describing the approach to a finite-time singularity of the Navier-Stokes equations. We now supplement this system with an equation describing the process of vortex reconnection at the apex of a pyramid, neglecting core deformation during the reconnection process. On this basis, we compute the maximum vorticity $ω_{max}$ as a function of vortex Reynolds number $R_Γ$ in the range $2000\le R_Γ\le 3400$, and deduce a compatible behaviour $ω_{max}\sim ω_{0}\exp{\left[1 + 220 \left(\log\left[R_Γ/2000\right]\right)^{2}\right]}$ as $R_Γ\right…
We demonstrate that a single reconnection of two quantum vortices can lead to creation of a cascade of vortex rings. Our analysis, motivated by the analytical solution in LIA, involves high-resolution Biot-Savart and Gross-Pitaevskii simulations. The latter showed that the rings cascade starts on the atomic scale, with rings diameters orders of magnitude smaller than the characteristic line spacing in the tangle. So created vortex rings may penetrate the tangle and annihilate on the boundaries. This provides an efficient mechanism of the vortex tangle decay in very low temperatures.
In a concurrent work, Villois et al. 2020 reported the evidence that vortex reconnections in quantum fluids follow an irreversible dynamics, namely vortices separate faster than they approach; such time-asymmetry is explained by using simple conservation arguments. In this work we develop further these theoretical considerations and provide a detailed study of the vortex reconnection process for all the possible geometrical configurations of the order parameter (superfluid) wave function. By matching the theoretical description of incompressible vortex filaments and the linear theory describin…
We investigate a single vortex reconnection event in superfluid helium at finite temperatures using the vortex filament model. The reconnection induces Kelvin waves which strongly increase energy dissipation. We evaluate the mutual friction dissipation from the reconnection and show that the dissipation power has universal form, which is seen by scaling both time (measured from the reconnection event) and power by the mutual friction parameter alpha. This observation allows us to conclude that the Kelvin-wave cascade is not important in the energy dissipation process within the range alpha > 0…
We statistically study vortex reconnections in quantum fluids by evolving different realizations of vortex Hopf links using the Gross--Pitaevskii model. Despite the time-reversibility of the model, we report a clear evidence that the dynamics of the reconnection process is time-irreversible, as reconnecting vortices tend to separate faster than they approach. Thanks to a matching theory devised concurrently in Proment and Krstulovic (arXiv:2005.02047), we quantitatively relate the origin of this asymmetry to the generation of a sound pulse after the reconnection event. Our results have the pro…
Collisions in a beam of unidirectional quantized vortex rings of nearly identical radii $R$ in superfluid $^4$He in the limit of zero temperature (0.05 K) were studied using time-of-flight spectroscopy. Reconnections between two primary rings result in secondary vortex loops of both smaller and larger radii. Discrete steps in the distribution of flight times, due to the limits on the earliest possible arrival times of secondary loops created after either one or two consecutive reconnections, are observed. The density of primary rings was found to be capped at the value $500{\rm \,cm}^{-2} R^{-…
We study the relaxation of a topologically non-trivial vortex braid with zero net helicity in a barotropic fluid. The aim is to investigate the extent to which the topology of the vorticity field -- characterized by braided vorticity field lines -- determines the dynamics, particularly the asymptotic behaviour under vortex reconnection in an evolution at high Reynolds numbers 25,000. Analogous to the evolution of braided magnetic fields in plasma, we find that the relaxation of our vortex braid leads to a simplification of the topology into large-scale regions of opposite swirl, consistent wit…
CEA, CNRS, SPEC, Université Paris-Saclay（法国原子能与替代能源委员会，法国国家科学研究中心，物理、化学与材料科学实验室，巴黎萨克雷大学）；Department of Mathematics and Statistics, McMaster University（麦克马斯特大学数学与统计系）；Univ. Lille, CNRS, ONERA, Arts et Metiers ParisTech, Centrale Lille（里尔大学，法国国家科学研究中心，法国航空航天研究院，巴黎国立高等工艺学院，里尔中央理工学院）
From a new anti-parallel initial condition using long vortices, three-dimensional turbulence forms after two reconnection steps and the formation of at least one vortex ring. The long domain is needed in order to accommodate the multiple reconnections, which enhance vortex stretching rates and the generation of small-scale vortex structures within the vortex rings. In addition to making the initial vortices very long, new features introduced with this initial condition are an initial profile less likely to shed vortex sheets and an improved method for mapping the direction of the vorticity ont…
The energy spectrum of the 3D velocity field, induced by collapsing vortex filaments is studied. One of the aims of this work is to clarify the appearance of the Kolmogorov type energy spectrum $E(k)\varpropto k^{-5/3}$, observed in many numerical works on discrete vortex tubes (quantized vortex filaments in quantum fluids). Usually, explaining classical turbulent properties of quantum turbulence, the model of vortex bundles, is used. This model is necessary to mimic the vortex stretching, which is responsible for the energy transfer in classical turbulence. In our consideration we do not appe…
Experimental and simulational studies of the dynamics of vortex reconnections in quantum fluids showedthat the distance $d$ between the reconnecting vortices is close to a universal time dependence $d=D[κ|t_0-t|]^α$ with $α$ fluctuating around 1/2 and $κ=h/m$ is the quantum of circulation. Dimensional analysis, based on the assumption that the quantum of circulation $κ=h/m$ is the only relevant parameter in the problem, predicts $α=1/2$. The theoretical calculation of the dimensionless coefficient $D$ in this formula remained an open problem. In this Letter we present an analytic calculation o…
We report on the analysis of the root mean square curvature as a function of the numerical resolution for a single reconnection of two quantized vortex rings in superfluid helium. We find a similar scaling relation as reported in the case of decaying thermal counterflow simulations by L. Kondaurova et al. There the scaling was related to the existence of a Kelvin-wave cascade which was suggested to support the L'vov-Nazarenko spectrum. Here we provide an alternative explanation that does not involve the Kelvin-wave cascade but is due to the sharp cusp generated by a reconnection event in a sit…
We investigate the evolution of vortex-surface fields (VSFs) in compressible Taylor--Green flows at Mach numbers ($Ma$) ranging from 0.5 to 2.0 using direct numerical simulation. The formulation of VSFs in incompressible flows is extended to compressible flows, and a mass-based renormalization of VSFs is used to facilitate characterizing the evolution of a particular vortex surface. The effects of the Mach number on the VSF evolution are different in three stages. In the early stage, the jumps of the compressive velocity component near shocklets generate sinks to contract surrounding vortex su…
In this Comment we would like to emphasize that in Phys. Rev. B 90, 104506 (2014) the calculated energy spectrum takes into account only the small interaction (cross) term and, additionally, this term is only calculated at the instant when the two vortices reconnect. The majority of the kinetic energy is contained in the self-energy term which has a characteristic spectrum of $1/k$. If this, and the additional average over time, is taken into account the suggested Kolmogorov type $k^{-5/3}$ spectrum is likely not visible in the kinetic energy spectrum which contains both terms. Therefore, we f…
At high Reynolds number, the interaction between two vortex tubes leads to intense velocity gradients, which are at the heart of fluid turbulence. This vorticity amplification comes about through two different instability mechanisms of the initial vortex tubes, assumed anti-parallel and with a mirror plane of symmetry. At moderate Reynolds number, the tubes destabilize via a Crow instability, with the nonlinear development leading to strong flattening of the cores into thin sheets. These sheets then break down into filaments which can repeat the process. At higher Reynolds number, the instabil…
Within Density Functional Theory, we have investigated the coalescence dynamics of two superfluid helium nanodroplets hosting vortex lines in different relative orientations, which are drawn towards each other by the Van der Waals mutual attraction. We have found a rich phenomenology depending on how the vortex lines are oriented. In particular, when a vortex and anti-vortex lines are present in the merging droplets, a dark soliton develops at the droplet contact region, which eventually decays into vortex rings. Reconnection events are observed between the vortex lines or rings, leading to th…
Kinetics of merging and breaking down vortex loops is the important part of the whole vortex tangle dynamics. Another part is the motion of individual lines, which obeys the Biot-Savart law in presence of friction force and of applied external velocity fields if any. In the present work we evaluate the coefficients of the reconnection rates $A(l_{1},l_{2},l)$ and $B(l,l_{1},l_{2})$. Quantity $A$ is a number (per unit of time and per unit of volume) of events, when two loops with lengths $l_{1}$and $l_{2}$ collide and form the single loop of length $ l=l_{1}+l_{2}$. Quantity $% B(l,l_{1},l_{2})…
The conjecture that helicity (or knottedness) is a fundamental conserved quantity has a rich history in fluid mechanics, but the nature of this conservation in the presence of dissipation has proven difficult to resolve. Making use of recent advances, we create vortex knots and links in viscous fluids and simulated superfluids and track their geometry through topology changing reconnections. We find that the reassociation of vortex lines through a reconnection enables the transfer of helicity from links and knots to helical coils. This process is remarkably efficient, owing to the anti-paralle…
Vortices and axisymmetric vortex rings are considered in the framework of the subcritical nonlinear Schrodinger equations. The higher order nonlinearity present in such systems models many-body interactions in superfluid systems and allows one to study the effects of negative pressure on vortex dynamics. We find the critical pressure for which the straight-line vortex becomes unstable to radial expansion of the core. The energy of the straight-line vortices and energy, impulse and velocity of vortex rings are calculated. The effect of a varying pressure on the vortex core is studied. It is sho…
The evolution and self-reconnection of a perturbed trefoil vortex knot is simulated, then compared to recent experimental measurements (Scheeler et al. 2014a). Qualitative comparisons using three-dimensional vorticity isosurfaces and lines, then quantitative comparisons using the helicity. To have a single initial reconnection, as in the experiments, the trefoil is perturbed by 4 weak vortex rings. Initially there is a long period with deformations similar to the experiment during which the energy, continuum helicity and topological self-linking number are all preserved. In the next period, on…
We present a systematic derivation of the Biot-Savart equation from the Nonlinear Schrödinger equation, in the limit when the curvature radius of vortex lines and the inter-vortex distance are much greater than the vortex healing length, or core radius. We derive the Biot-Savart equations in Hamiltonian form with Hamiltonian expressed in terms of vortex lines, $$ H= \frac{κ^2}{8 π}\int_{|{\bf s}-{\bf s}'|>ξ_*} \frac{d{\bf s} \cdot d {\bf s}'}{|{\bf s}-{\bf s}'|} \,,$$ with cut-off length $ξ_* \approx 0.3416293 /\sqrt{ρ_0},$ where $ρ_0$ is the background condensate density far from the vortex l…
We present a rigorous derivation of the point vortex model starting from the two-dimensional nonlinear Schr{ö}dinger equation, from the Hamiltonian perspective, in the limit of well-separated, subsonic vortices on the background of a spatially-infinite strong condensate. As a corollary, we calculate to high accuracy the self-energy of an isolated elementary Pitaevskii vortex, for the first time.
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 …
