Active and finite-size particles in decaying quantum turbulence at low temperature
AActive and finite-size particles in decaying quantum turbulence at low temperature
Umberto Giuriato and Giorgio Krstulovic
Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire Lagrange, Nice, France
The evolution of a turbulent tangle of quantum vortices in presence of finite-size active particles isstudied by means of numerical simulations of the Gross-Pitaevskii equation. Particles are modeledas potentials depleting the superfluid and described with classical degrees of freedom following aNewtonian dynamics. It is shown that particles do not modify the building-up and the decay ofthe superfluid Kolmogorov turbulent regime. It is observed that almost the totality of particlesremains trapped inside quantum vortices, although they are occasionally detached and recaptured.The statistics of this process are presented and discussed. The particle Lagrangian dynamics isalso studied. At large time scales, the velocity spectrum of particles is reminiscent of a classicalLagrangian turbulent behavior. At time-scales faster than the turnover time associated to the meaninter-vortex distance, the particle motion is dominated by oscillations due to Magnus effect. Forlight particles a non-classical scaling of the spectrum arises. The particle velocity and accelerationprobability distribution functions are then studied. The decorrelation time of the particle accelera-tion is found to be shorter than in classical fluids, and related to the Magnus force experienced bythe trapped particles. a r X i v : . [ phy s i c s . f l u - dyn ] A p r I. INTRODUCTION
When a fluid is stirred, energy is injected into the system exciting structures at different scales. In particular, inthree dimensional classical flows, the energy supplied at large scales is transferred towards small scales in a cascadeprocess. Eventually, it reaches the smallest scales of the system, where dissipation acts efficiently. In presence ofa very large separation between the injection and dissipation scale, this cascade scenario proposed by Richardson,leads to a fully developed turbulent state that can be described by the Kolmogorov phenomenology [1]. Kolmogorovturbulence is expected to be universal, and it is in fact commonly observed in nature, industrial applications and inmore exotics flows such as superfluids.A superfluid is a peculiar flow, whose origin is a consequence of quantum mechanics. At finite temperature, asuperfluid is considered to be a mixture of two components: the normal fluid, that can be described by the Navier-Stokes equations, and the superfluid component with zero viscosity [2]. At very low temperatures, the normalcomponent can be neglected and the fluid becomes completely inviscid. As a consequence, an object moving at lowvelocities does not experience any drag from the fluid. However, when the object exceeds a critical velocity, quantumvortices are nucleated [3, 4]. Quantum vortices (or superfluid vortices) are the most fundamental hydrodynamicalexcitations of a superfluid. They are topological defects (and nodal lines) of the macroscopic wave function describingthe system, and as a consequence their circulation is quantized. In superfluid helium, the core size of quantum vorticesis of the order of 1˚A. Despite the lack of viscosity, quantum vortices can reconnect and change their topology (see forinstance [5–8]), unlike classical (prefect) fluids.When energy is injected in a low-temperature superfluid at scales much larger than the mean inter-vortex distance (cid:96) ,a classical Kolmogorov regime is expected. Such a behavior has been observed numerically [9–11] and experimentally[12, 13]. Indeed, at such scales the quantum nature of vortices is not important and the superfluid behaves like aclassical fluid. At the scales of the order of (cid:96) and smaller, the isolated nature of quantized vortices become relevant.The system keeps transferring energy towards small scales but through different non-classical mechanisms [14]. Anexample of such mechanisms is the turbulent Kelvin wave cascade. Kelvin waves are helical oscillations propagatingalong quantum vortices and the energy can be carried toward small scales thanks to non-linear wave interactions.This energy cascade has been successfully described in the framework of weak-wave turbulence theory [15, 16]. Theresulting theoretical prediction have been observed numerically in vortex-filament and Gross-Pitaevskii numericalsimulations [17–19].Flow visualization is certainly a fundamental issue in every fluid dynamics experiment. Among the techniqueswhich have been developed to sample a fluid, particle image velocimetry (PIV) and particle tracking velocimetry(PTV) are two of the most common methods [20]. The use of particles as probes has been also adapted to thestudy of cryogenic flows, in particular in superfluid helium He experiments [21], where micrometer sized hydrogenand deuterium particles have been used. For instance, hydrogen ice particles have been successfully employed tovisualize isolated or reconnecting vortex lines [22], as well as the propagation of Kelvin waves [23]. Moreover, theobservation of power-law tails in the probability density of the particle velocity is an important difference with respectto classical turbulent states [24–26]. Similar deviations from classical behaviors have been recently reported also forthe acceleration statistics [26, 27]. Particles in such experiments typically have a size that can rise up to severalmicrons, which is many orders of magnitude larger than the size of the vortex core in superfluid helium. For instance,the solidified hydrogen particles produced in the experiments [22, 23] are slightly smaller than 2 . µm , while in [25, 26]their size is between 5 µm and 10 µm . Although it has been seen that particles unveils the dynamics of quantumvortices, it is yet not clear how much they affect the dynamics of quantum turbulent flows.Several theoretical efforts have been made in the last decade in order to clarify what is the dynamics of particles ina superfluid and how particles interact with quantum vortices. For example, the vortex-filament (VF) method can becoupled with the classical hydrodynamical equations of a sphere, allowing to study different specific problems. Theinteraction between one particle and one vortex has been addressed [28, 29], as well the back-reaction of tracers in athermal counterflow [30, 31]. In the context of finite temperature superfluids, the spatial statistics of particles havebeen recently addressed in simulations of the Hall-Vinen-Bekarevich-Khalatnikov model [32].Finally, since the work of Winiecki and Adams [4], particles described by classical degrees of freedom have beenimplemented self-consistently in the framework of the Gross-Pitaevskii (GP) equation [33–37]. Although the GPmodel is formally derived for dilute Bose-Einstein condensates, it is considered as a general tool for the study ofsuperfluid dynamics at very low temperature. Indeed, unlike the VF method or the HVBK model, it naturallycontains quantum vortices as topological defects of the order parameter. It was found analytically and confirmednumerically that the GP model can reproduce the process of trapping of large active inertial particles by straightvortex lines [34], in accordance with hydrodynamical calculations [28, 29]. In this framework, the interplay betweenmany trapped particles and Kelvin waves has been also investigated [36].In the present work, we study the influence of particles on quantum turbulent flows at very low temperature byusing the GP model coupled with classical particles. In particular, we study the evolution of a free decaying superfluidturbulent vortex tangle loaded with finite-size active particles. We consider spherical particles of different masses andhaving a diameter up to 20 core sizes. Such size is about 1000 times smaller than the one of solidified particles usedin superfluid helium experiments. Nevertheless, it is slightly smaller than or comparable to the mean inter-vortexdistance in our simulations, similar to current experiments. We also study the different regimes of the turbulentevolution from the Lagrangian point of view. The paper is organized as follows. In Section II we describe the Gross-Pitaevskii model coupled with classical particles. We also review the standard properties of the model and give thebasic definitions used later to analyze the flow. We also describe the numerical method used in this work. Then, inSection III, we present our main results. In particular, in Section III A we address whether or not the presence ofparticles affects the scales of the flow at which Kolmogorov turbulence takes place. Section III B is devoted to thestudy of the particle dynamics inside the vortex tangle, to their trapping by vortices and to their dynamics at scaleslarger and smaller than the inter-vortex distance. Particle velocity and acceleration statistics are then presented inSection III C. Finally, Section IV contains our conclusions. II. MODEL FOR PARTICLES IN A LOW TEMPERATURE SUPERFLUIDA. Gross-Pitaevskii equation coupled with particles
We describe a superfluid of volume V at low-temperature by using the complex field ψ , which obeys the GPdynamics. We consider N p particles in the system. Each particle is characterized by the position of its center of mass q i and its classical momentum p i . The presence of a particle of size a p generates a superfluid depletion in a sphericalregion of radius a p . This effect is reproduced by coupling the superfluid field with a strong localized potential V p ,which has a fixed shape and is centered at the position q j ( t ).All the particles considered have the same size, as well as the same mass M p . The Hamiltonian of the system isgiven by H = (cid:90) (cid:126) m |∇ ψ | + g (cid:18) | ψ | − µg (cid:19) + N p (cid:88) i =1 V p ( | x − q i | ) | ψ | d x + N p (cid:88) i =1 p i M p , + N p (cid:88) i B. Numerical methods and parameters In the simulations presented in this work we solve the system (2-3) in a cubic periodic box of side L = 341 ξ with N c = 512 collocation points by using a standard pseudo-spectral method. We use a 4 th order Runge-Kutta schemefor the time-stepping and the standard 2 / c = 1 and ψ ∞ = 1.In order to produce a homogeneous and isotropic tangle of quantized vortex lines, we impose an initial Arnold-Beltrami-Childress (ABC) flow, following the procedure described in [38]. In particular, we use a superposition of k = 1 · π/L and k = 2 · π/L basic ABC flows: v ABC = v (1)ABC + v (2)ABC , with v ( k )ABC = [ B cos( ky ) + C sin( kz )]ˆ x + [ C cos( kz ) + A sin( kx )]ˆ y + [ A cos( kx ) + B sin( ky )]ˆ z, (5)and the parameters A = 0 . B = 0 . C = 0 . (cid:96) ( t = 0) ∼ ξ . As the flow is prepared by minimizingthe energy, most of the energy of the system is in the incompressible part of the energy and resulting form the vortexconfiguration.The ground state for the particles consists in a number of particles (we use N p = 200 and N p = 80) of the samesize and mass, randomly distributed in the computational box. Particles are initially at rest. This state is preparedusing the imaginary-time evolution of the equation (2). Then, the initial condition for the simulations is obtained bymultiplying the wave function associated with the ABC flow and the wave function associated to the particles groundstate. An example of an initial field containing particles is displayed in Fig.1.d.Because of the presence of a healing layer, the particle boundary is never sharp, independently of the functionalform of the potential V p . The superfluid field vanishes in the region where V p > µ and at the particle boundary thefluid density passes from zero to the bulk value ρ ∞ in approximately one healing length. The potential used to modeleach particle is a smoothed hat-function V p ( r ) = V (1 − tanh (cid:104) r − ζ a (cid:105) ) where the parameters ζ and ∆ a are set tomodel the particle. Their values are listed in Table I. In particular, ζ fixes the width of the potential and it is relatedto the particle size, while ∆ a controls the steepness of the smoothed hat-function. The latter needs to be adjusted inorder to avoid Gibbs effect in the Fourier transform of V p . Since the particle boundaries are not sharp, the effectiveparticle radius is defined as a p = (3 M / πρ ∞ ) , where M = ρ ∞ L (1 − (cid:82) | ψ p | d x / (cid:82) | ψ ∞ | d x ) is the fluid massdisplaced by the particle and ψ p is the steady state with just one particle. Practically, given the set of numericalparameters ζ and ∆ a , the state ψ p is obtained numerically with imaginary time evolution and the excluded mass M is measured directly. Particles attract each other by a short range fluid mediated interaction [33, 35], thus we use therepulsive potential V ij rep = γ (2 a p / | q i − q j | ) in order to avoid an overlap between them. The functional form of V ij rep is inspired by the repulsive term of the Lennard-Jones potential and the pre-factor γ is adjusted numerically so thatthe inter-particle distance 2 a p minimizes the sum of V ij rep with the fluid mediated attractive potential [33, 35]. Weexpress the particle mass as M p = M M , where M is the mass of the superfluid displaced by the particle. Namely,heavy particles have M > M < 1. In Table I all the parameters for the particles used inthe simulations presented in this work are reported. In the following, we will refer to each simulation specifying thesize and the mass of the particles used.Note that although the model (1) is a minimal model for implementing particles in the GP framework, we can notadd to the system an arbitrary number of particles. Indeed, since particles have a finite size, they occupy a volumeat the expense of the superfluid field and packing effects could become important if the filling fraction is too high.Moreover, the potential V p must be updated at each time step, which is numerically costly. Finally, note that the theevaluation of the force term (3) acting on particles requires to know the value of the fields at inter-mesh points. When TABLE I. Simulation parameters.Run N p a p M ζ ∆ a V /µ γ/µ I 0 – – – – – –II 200 4 . ξ . ξ . ξ . · − III 200 4 . ξ . ξ . ξ . · − IV 200 4 . ξ . ξ . ξ . · − V 200 4 . ξ . ξ . ξ . · − VI 80 10 . ξ . ξ . ξ . · − VII 200 10 . ξ . ξ . ξ . · − VIII 200 10 . ξ . ξ . ξ . · − IX 200 10 . ξ . ξ . ξ . · − the number of particles in the simulation is not large, the force f GP i ( q i ) = − ( V p ∗ ∇ ρ ) [ q i ] (3) can be computed withspectral accuracy using a Fourier interpolation. Such method has been used in [34–36], where the particle dynamics isextremely sensitive. In this work, the use of a Fourier interpolation for each particle is numerically unaffordable, due tothe large number of particles involved and the resolutions used. Instead, we use a fourth–order B–spline interpolationmethod, which has been shown to be highly accurate with a reduced computational cost [39] and particularly welladapted for pseudo–spectral codes. Indeed, the use of a Fourier interpolation to evaluate the three dimensionalforce for N p particles requires ∼ N p N c operations and evaluations of complex exponentials ( N c = 512 in the presentwork). Such cost quickly becomes too expensive at high resolutions and/or large number of particles. On the contrary,B–Spline interpolation requires just one Fast Fourier Transform of a field per component, and an interpolation usingonly four neighboring grid points per dimension [39]. Such scheme saves a factor ∼ N p of computational cost comparedto Fourier interpolation. Note that in the previous discussion we have not taken into consideration parallelizationissues, where local schemes (B–Splines) are much advantageous than global ones (Fourier transforms). Nevertheless,some issues with physical quantities at small scales arising from the B–spline interpolation are discussed in Appendix. III. PARTICLES IMMERSED IN A TANGLE OF SUPERFLUID VORTICES Superfluid turbulence in the context of the GP model has been largely studied [9, 11, 38, 40, 41]. In general, quantumturbulence develops from an initial state with a vortex configuration where the incompressible kinetic energy is mainlycontained at large scale. During the evolution, vortex lines move, interact among themselves and reconnect creatingcomplex vortex tangles. Through this process, sound is produced and incompressible kinetic energy is irreversiblyconverted into quantum, internal and compressible kinetic energy. Eventually, the compressible energy produced inthe form of acoustic fluctuations starts to dominate, thermalizes and acts as thermal bath providing an effectivedissipation acting vortices. As a consequence, vortices shrink and eventually disappear through mutual friction effectsfollowing Vinen’s decay law [19, 42]. In particular, it has been shown that the decrease of the incompressible kineticenergy behaves in a similar manner to decaying classical turbulence [9]. In order to make connection with decayingclassical Kolmogorov turbulence, the incompressible energy dissipation or dissipation rate is usually defined in thecontext of GP turbulence as (cid:15) = − d E Ikin d t . (6)As in decaying Navier-Stokes turbulence, in GP the most turbulent stage is achieved around the time when thisquantity is maximal. About this time, the classical picture holds and the incompressible energy spectrum satisfiesthe Kolmogorov prediction E Ikin = C(cid:15) / k − / , where C is the Kolmogorov constant which value has been found to be close to 1 in GP turbulence [11, 38, 41].The first purpose of this work is to check whether and to what extent the presence of particles in the system modifiesKolmogorov turbulence. We add to the ABC initial condition a number of randomly distributed particles and letthe system evolve under the dynamics (2-3). In Fig.1.a, b and c the three stages of the evolution (respectively initialcondition, turbulent vortex tangle and residual filaments in a bath of sound) are visualized in the case of 200 neutrallybuoyant particles of radius 4 ξ . Movies of this simulation and other with particles of a different size can be found in ( a ) We shall start our analysis by comparing the temporal evolution of global quantities. In Fig.2.a the time evolutionof the different components of the energy is displayed. Times are expressed in units of the large-eddy-turnover timedefined as T L = L/ v rms , where v rms = (cid:113) E Ikin ( t = 0) / t/T L . . . . . . E n e r g y / E t o t ( a ) E Ikin E Ckin E Q E int No particles N p = 200 , a p = 4 ξN p = 200 , a p = 10 ξN p = 80 , a p = 10 ξ t/T L . . . . . . (cid:15) / (cid:15) m a x ( b ) No Particles a p = 4 ξ, M = 1 , N p = 200 a p = 4 ξ, M = 0 . , N p = 200 a p = 4 ξ, M = 2 , N p = 200 a p = 10 ξ, M = 1 , N p = 200 a p = 10 ξ, M = 1 , N p = 80 FIG. 2. (color online) (a) Time evolution of the superfluid energy components in the cases with no particles (dashed line), 200small particles (dotted line), 200 large particles (solid line) and 80 large particles (dash-dotted line). (b) Incompressible energydissipation rate for different number of particles with different sizes and different masses (solid lines). Dash-dotted horizontallines of the corresponding colors indicate the value of the maximum of dissipation, obtained averaging over the shaded region.The dissipation is expressed in units of its maximum (cid:15) max in the case without particles. vortex tangle and L/ M = 1. The net transfer of incompressibleenergy towards compressible, quantum and internal energy is qualitatively unchanged in the various cases. The onlydifference is a slightly lower value of the incompressible energy in the case of large particles, in favor of the internalenergy of the superfluid. Such effect is more evident if the number of large particles is increased, and could be relatedto an increment of the filling fraction Φ, namely the fraction of the total volume occupied by the particles. In fact,for N p = 200 particles of radius a p = 4 ξ the filling fraction is Φ = 0 . N p = 80 particles of radius a p = 10 ξ it isΦ = 0 . 8% and for N p = 200 particles of radius a p = 10 ξ we have Φ = 2 . E GPP are negligible compared to the other energies throughout theduration of the simulations (data not shown).The dissipation rate of the incompressible kinetic energy is reported in Fig.2.b for particles of different masses anddifferent sizes. The dissipation increases in the early stages, when the energy begins to be transferred to the smallerscales, it reaches a maximum when all the scales are excited, and then starts to decay since no forcing is sustainingthe turbulence. We observe that the evolution of the dissipation is clearly not significantly modified by the presenceof particles. In particular, the value of the maximum of dissipation, which is the signature of the most turbulentstate reached by the tangle, is slightly lower only in the case where many large particles are moving in the system. Inparticular for this case, it is about 90% of (cid:15) max , the value measured in the case with no particles. The shaded regionin Fig.2.b represents the most turbulent time of the simulations. We consider that in this short stage, the system is ina quasi-steady state and we perform the temporal average of certain physical quantities in order to improve statisticalconvergence.Two other important quantity that is not affected much by the interplay between tangle and particles is the meaninter–vortex distance (cid:96) , whose time evolution is reported in Fig.3.a. The mean inter–vortex distance is then estimatedas (cid:96) = (cid:112) V /L v , where L v is the total vortex length in the system. This latter is estimated using the method introducedin [9], where L v is shown to be related to the proportionality constant between the incompressible momentum density J I ( k ) of the flow and the spectrum of a two-dimensional point-vortex J ( k ): L v π = (cid:80) k J I ( k ) (cid:82) J ( k ) d k . (7)The spectra of momentum densities are the angle average of the norm in Fourier space of the momentum density J = ρ v s , and the incompressible part is obtained projecting onto the space of divergence-free fields. We have checkedthe validity of this formula by using the vortex filament tracking method described in [44] at some checkpoints.In the turbulent regime, where the dissipation gets its maximum, the total length of the entangled vortices is alsolarger by a factor 4 compared to the initial condition, and the distance between the filaments is minimum. Thevalue (cid:96) min ∼ ξ of the inter–vortex distance in this regime will be used as a characteristic small length-scale of theKolmogorov turbulent regime. Such length is smaller than the diameter of the largest particles considered (2 a p = 20 ξ ),but nevertheless this has no appreciable repercussions on the behavior of the observables studied. Furthermore, as t/T L ‘ / ξ ( a ) No Particles a p = 4 ξ, M = 1 , N p = 200 a p = 4 ξ, M = 2 , N p = 200 a p = 10 ξ, M = 1 , N p = 200 a p = 10 ξ, M = 1 , N p = 80 k‘ min − − − − − E I ( k ) ( b ) 10 k‘ min E I ( k ) / ( (cid:15) / m a x k − / ) FIG. 3. (color online) (a) Time evolution of the mean inter–vortex distance for different numbers of particles of different sizesand different masses. (b) Incompressible energy spectrum for different numbers of particles of different sizes and differentmasses. (inset) Compensated incompressible energy spectrum. Solid lines refer to particles of size a p = 4 ξ , dashed lines referto particles of size a p = 10 ξ . Dotted line is the classical scaling (cid:15) max k − / . The spectrum is computed averaging over times inthe shaded region. shown in Fig.3.c, the scaling of the incompressible energy spectrum E I ( k ) averaged around the maximum of dissipationis unaltered by particles in the system. The figure Fig.3.b displays the incompressible energy spectrum. It is apparentthat the scaling of the spectrum is always compatible with classical turbulence at scales larger than the inter–vortexdistance, and the way in which the energy is accumulated at smaller scales is not modified by the particles. In theinset of Fig.3.b the spectrum is compensated by with the Kolmogorov prediction E I ( k ) = C(cid:15) / k − / for classicalhydrodynamic turbulence. The dotted horizontal black line shows that the value of the constant C in the Kolmogorovlaw is a number of order 1 for superfluid turbulence.The only appreciable difference observed between the case with and without particles is that in the early stages ofthe evolution, the trapping of particle perturbs the vortex filaments and excite Kelvin waves. A comparison betweenthe volume renderings can be seen in the upper row of Fig.4. Such perturbations propagate during the evolution ofthe tangle. At the times when turbulence is developed, the details of the vortex configurations are completely different(see lower row of Fig.4). Nevertheless, the statistical properties of the system in this regime remain unchanged. Westress that the inter-vortex distance in quantum turbulence experiments lies typically in the range 10 − µm , whichis equal or slightly larger than the particle size [24, 25, 27]. In this sense, the simulations presented here are compatiblewith the experimental parameters. They thus support the believe that active particles have effectively no influenceon the typical development and decay of quantum turbulence. This numerical fact helps to validate past and futureexperiments that use particles as probes of superfluids.On the other hand, because of the lack of a Stokes drag in the system, particles cannot be treated as simple tracersof the superfluid velocity v s . Nevertheless, if they remain trapped inside the vortices they can track the evolution ofthe vortex filaments, which are the structures that effectively become turbulent. With the purpose of characterizingthis scenario, in the next section we investigate the motion of particles once they are immersed in a tangle of quantumvortices. B. Motion of particles in the superfluid vortex tangle Looking at the time evolution of the vortex tangle (see Fig.1 and movies in the Supplemental Material), the firstthing that is apparent is how particles quickly get trapped into vortex filaments. This dynamics is expected and ithas been studied in the case where vortices move slowly [34]. It is a consequence of the pressure gradients. However,it is less obvious if such behavior remains dominant when turbulence take place and reconnections become frequent.We study the evolution of particles and compute wether they are free or trapped by vortices. The temporal evolutionof the fraction of trapped particles is displayed in Fig.5.a for all runs. This measurement is made by computing thecirculation Γ = (cid:72) C v s · d x of the superfluid velocity v s along contours C encircling each particle, and counting for whichparticles it is different from zero. Specifically, we compute the circulation along many parallel square contours of side2( a p + ∆ x ) around each particle, where ∆ x is the grid spacing. If the circulation around at least one of these contoursis different from zero, the particle is considered as trapped [45]. For practical reasons, due to the parallelization of thenumerical code, we consider only contours perpendicular to the z axis of the computational box. As a consequence, ( a ) 10% of the particles with size 2 a p ∼ (cid:96) are always attached to at least two different filaments.Sometimes even more vortices pass simultaneously through the same particle, as it can be visualized in the volumeplot of Fig.5.c.Once a particle is trapped by a vortex, it can experience violent events, for instance during vortex reconnections. Insuch circumstances, such a particle could be detached and expelled from the vortex until it will eventually get trappedby another vortex of the tangle. We compute the probability density function (PDF) of the continuous time intervals0 . . . . . t/T L . . . . . . N tr a pp / N p ( a ) a p = 4 ξ, M = 1 , N p = 200 a p = 4 ξ, M = 0 . , N p = 200 a p = 4 ξ, M = 2 , N p = 200 a p = 10 ξ, M = 1 , N p = 200 a p = 10 ξ, M = 1 , N p = 80 . . . . 501 0 . . . . . t/T L . . . . . . N tr a pp / N p ( b ) a p = 4 ξ, M = 1 , Single trapped a p = 4 ξ, M = 1 , Double trapped a p = 10 ξ, M = 1 , Single trapped a p = 10 ξ, M = 1 , Double trapped a p = 10 ξ, M = 1 , Triple trapped − − − ∆ t trap /T L − − − − P D F ( d ) a p = 4 ξ, M = 0 . a p = 4 ξ, M = 1 a p = 4 ξ, M = 2 a p = 10 ξ, M = 1 ∝ t − . . . t/T L | Γ | / κ ( c ) FIG. 5. (color online) (a) Fraction of trapped particles as a function of time for different numbers of particles of differentsizes and different masses. (inset) The same for longer time in the case of 200 neutrally buoyant particles of size a p = 4 ξ . (b) Comparison between the fraction of multiply trapped particles as a function of time for neutrally buoyant particles. (c) Volume rendering of large particles ( a p = 10 ξ ) multiply trapped by quantum vortices. Vortices are rendered in red, sound inblue, particles in green. (d) Probability density function of the continuous time spent by particles inside vortices for differentspecies of particles. Dotted blue line corresponds to the same simulation of blue circles (particles with size a p and mass M = 1)but averaged over the full simulation times.) (inset) Absolute value of the circulation around a single particle of size a p = 4 ξ and mass M = 1 as function of time. The PDF is computed averaging over times in the shaded region. ∆ t trap spent by the particles inside vortices regime. The PDFs for particles of different sizes and masses are displayedin Fig.5.d. For all the species of particles examined, the probability distribution seems to follow roughly a power-lawscaling in time ∼ (∆ t trap ) − α , with α ∼ . 67. The PDF certainly vanishes much slower than an exponential decayat large ∆ t trap , which would typically result form a standard escape problem over energy barriers. We checked thatthe intermittency of the circulation and the shape of the trapping time PDF are not peculiar of the most turbulentregime, since they persist also at the late times of the simulations (see dotted blue line in Fig.5.d). Therefore, manyparticles spend a time at least of the order of the simulation time ( ∼ T L ) inside a vortex filament, i.e. the typicalescape time from the vortices is virtually infinite. This observation is exemplified in the inset of Fig.5.d, where theevolution of the circulation around a single small neutral particle is reported (the qualitative behavior is the same forthe other particles). It is also clear that the time spent by the particles with zero circulation around them (namelyfree from vortices) are short. Since we established that particles immersed in a tangle spend most of the time insidevortex filaments, in the following we study their motion once they get trapped.At large scales, the vortex tangle seems to behave as a classical hydrodynamic turbulent system. Therefore thefirst natural question is whether the particles can trace such large-scale fluctuations. In classical turbulence, it is wellknown that the Lagrangian velocity spectrum scales as (cid:10) | ˆ v p ( ω ) | (cid:11) = B(cid:15)ω − , (8)where B is a constant of order unity and ˆ v p ( ω ) is the Fourier transform of the Lagrangian particle velocity v p ( t )[46, 47]. Such scaling is valid in the inertial range 2 π/T L (cid:28) ω (cid:28) π/τ η , where τ η is the Kolmogorov time scale.In our case, we build an analogous of the Kolmogorov time scale under the assumptions that the dissipation rate1 (cid:15) max is the only important physical parameter in the classical turbulence regime and that the Kolmogorov turbulentcascade ends at the inter–vortex distance (cid:96) min . Therefore, we define the smallest time scale of the classical turbulenceregime as τ (cid:96) = (cid:0) (cid:96) /(cid:15) max (cid:1) / , and we expect classical turbulent phenomenology to hold for times τ (cid:96) (cid:28) t (cid:28) T L .In Fig.6 the measurement of the frequency spectrum of the particle velocity (cid:10) | ˆ v p ( ω ) | (cid:11) = (cid:10) | (cid:82) ˙ q ( t ) e − iωt d t | (cid:11) duringthe turbulent regime is shown for different species of particles, compensated with the classical scaling (cid:15) max ω − . Note | ω | τ ‘ / π − − − ω (cid:10) | ˆ v p ( ω ) | (cid:11) / (cid:15) m a x ( b ) ∝ | ω | − M = 0 . M = 0 . M = 1 | ω | τ ‘ / π − − − ω (cid:10) | ˆ v p ( ω ) | (cid:11) / (cid:15) m a x a p = 4 ξ a p = 10 ξ ( a ) ∝ | ω | − M = 1 , late M = 1 , straight M = 1 M = 2 M = 0 . M = 0 . FIG. 6. (color online) Frequency spectrum of the particle velocity for particles of different masses and different sizes, compen-sated with the prediction for the Lagrangian spectrum in classical turbulence ∝ (cid:15)/ω : (a) small particles with a p = 4 ξ ; (b) large particles with a p = 10 ξ . Dash-dotted gray line is the frequency spectrum of a single small particle trapped in a straightvortex slightly perturbed. Dotted lines of corresponding colors are the prediction for the particle natural frequency Ω p . Dashedred line is the scaling due to vortex reconnection or Kelvin waves ∝ | ω | − . Dashed golden line is the spectrum evaluated atlate times in the simulation (6 T L < τ < T L ). that the average which defines the spectrum is meant over different realizations. In numerics we average over all theparticles trajectories during the turbulent regime. At frequencies ω < τ (cid:96) / π the spectra approach a plateau of valueone, confirming that particles sample well the flow and their behavior is described by the standard classical turbulencepicture at large scales. Note that the classical temporal inertial range of our simulations is pretty small, since T L ∼ τ (cid:96) .For comparison, we also present the velocity spectrum of a particle of size a p = 4 ξ and mass M = 1, computed in atemporal window at much later times, when Kolmogorov turbulence has decayed and only few vortices are left. Notethat a ω − scaling of the Lagrangian velocity spectrum has been also observed in numerical simulations of the vortexfilament model [48], although not in the Kolmogorov inertial range and not related to the energy dissipation rate norto Kolmogorov turbulence.As expected, in our simulations no Kolmogorov scaling is observed at small time scales. Indeed, one of the moststriking features of quantum turbulence is the crossover between the classical Kolmogorov regime and the physicstaking place at scales smaller than the mean inter–vortex distance. Unlike classical turbulence (see for instance [46]),there is still a non trivial scaling at time scales shorter than τ (cid:96) . Such difference is a consequence of the quantumnature of the system, here manifested by the presence of quantized vortices.When a particle is trapped by a vortex, the superfluid flow turns around it. As a consequence, while the particlemoves, it experience a Magnus force. This lift force is simply expressed as F Magnus = ρ ∞ a p Γ × ( ˙ q − v s ), where thecirculation vector Γ is oriented along the vortex filament and the superfluid velocity v s contains the contributions ofthe mean flow and the vortex motion [36, 49]. Magnus effect induces a precession of the particle about the filamentwith the characteristic angular velocity Ω p = 32 ρ ∞ a p M effp Γ , (9)where the particle effective mass M effp = M p + M = ( M + ) M takes into account the added mass effect due tothe mass of the superfluid displaced by the particle M . As mentioned in [36], for current experiments with hydrogenparticles in superfluid helium, this frequency is of order 10 − M effp ¨ q = F Magnus implies the following expression for the frequency spectrumof the particle velocity: (cid:10) | ˆ v p ( ω ) | (cid:11) = Ω Γ ( ω − Ω p ) (cid:10) | Γ × ˆ v s ( ω ) | (cid:11) . (10)2Independently of the external superfluid velocity, the expression (10) predicts that the spectrum (cid:10) | ˆ v p ( ω ) | (cid:11) must bepeaked around the natural frequency of trapped particles ω = Ω p . Such behavior has been studied in detail in the caseof particles trapped inside slightly perturbed straight vortex filaments [36]. The spectrum of this simple configurationis also reported for comparison in Fig.6.a for a small particle of relative unit mass. A clear bump in the frequencyspectrum, corresponding to Ω p , is still visible when particles are immersed in a complex quantum vortex tangle. Forthe large particles, the presence of a peak is less evident because the natural frequency is lower, and therefore a longersampling (in time) would be necessary to resolve it properly (2 π/ Ω p = 0 . T L for the particles of size a p = 10 ξ andmass M = 1). Moreover, as large particles are multiply trapped by many vortices, the resulting motion is certainlymore complex than a precession with a single characteristic angular frequency of one single vortex. The broadness ofthe peak around the Magnus frequency for the small particles in Fig.9.a, could be also related to this fact.At small time scales, a different scaling of the velocity spectrum is observed for the light particles, now in agreementwith (cid:10) | ˆ v p ( ω ) | (cid:11) ∝ | ω | − . This behaviour is consistent with the fact that at scales smaller than the inter–vortexdistance, the typical velocities of a superfluid turbulent tangle are supposed to scale as v fast ( t ) ∝ (cid:112) κ/ | t − t | , becausethe circulation becomes the only relevant physical parameter and the motion of vortices is dominated by their mutualadvection and reconnections. In this scenario, if particles are sufficiently light to be able to follow the fast vortexdynamics, we can substitute (cid:10) | ˆ v p ( ω ) | (cid:11) ∼ ˆ v ( ω ) ∝ κ | ω | − . Another effect that could contributes to the same resultis the attraction of particles by the vortices, since the scaling in time of the particle-vortex distance is the same ofvortex reconnection [34]. Note that for the heaviest particles such fast scaling is absent, since their reaction is probablytoo slow to be sensible to the fast fluctuations of the tangle. C. Particle velocity and acceleration statistics Unlike classical turbulence, where the statistics of the one-point particle velocity v is known to be Gaussian [1],experiments in superfluid helium using hydrogen and deuterium particles as tracers have reported long tails, with a v − power–law scaling in their velocity distribution [24–26]. Such scaling has been related to the singular velocityfield of quantized vortices [50, 51]. At low temperatures, as Stokes drag is negligible, particles should not move withthe superfluid flow and such scaling can be understood as a consequence of quantum vortex reconnections sampledby trapped particles [7, 24]. Furthermore, in reference [25], by using particle tracking velocimetry in counterflowturbulence, it was shown that while varying the sampling scale, the velocity PDFs continuously change from Gaussianstatistics to power-law tails, the crossover taking place at scales of the order of the inter–vortex distance. In this lastsection we present measurements of particle velocity and acceleration statistics within the GP-P model.We start the discussion by presenting the Eulerian velocity field. Formally, the velocity of the superfluid is simplygiven by ∇ φ . This field contains the density fluctuations, as well as the divergence of the vortex velocity flow closeto its core. This divergence leads to the well observed v − scaling of velocity PDF [50, 52, 53]. The PDF of ∇ φ isdisplayed in Fig.7. We turn now to analyze the particle velocity PDFs. We compute the velocity PDFs for all runs, − . − . − . . . . . v i − h v i i ) /c − − − − − P D F ( a ) 0 1 2 M . . σ v / c − v i − h v i i ) /σ v − − − − − P D F ( b ) a p = 4 ξ , M = 2 a p = 4 ξ , M = 1 a p = 4 ξ , M = 0 . a p = 4 ξ , M = 0 . a p = 10 ξ , M = 1 a p = 10 ξ , M = 0 . a p = 10 ξ , M = 0 . Gaussian ∝ | v i | − ∇ φ FIG. 7. (a) Probability density function of the single component particle velocity, for different species of particles. Dottedgolden line the Eulerian velocity field ∇ φ , corresponding to the simulation without particles at the time 1 . T L . The data forthe particles are averaged in time between t = 1 . T L and t = 1 . T L . (inset) Standard deviation of the particle velocity as afunction of the particle mass. (b) The same of (a ) but with the velocities normalized by the standard deviation σ v . Dottedlines are gaussian, dash-dotted line is a power-law scaling | v i | − . in the turbulent regime. Data is filtered with a Gaussian convolution in order to smooth out the noisy oscillations3at frequencies ω < ω noise = 50 (2 π/τ (cid:96) ) (see Appendix A). In Fig.7 the PDF of the single component velocity isplotted for all the species of analyzed particles. In Fig.7.a, velocities are expressed in terms of the speed of sound c , whereas in Fig. 7.b they are normalized by their root–mean–squared values. The root–mean–squared values aredisplayed in the inset of Fig.7.a as a function of the mass for the two particle sizes. It is apparent from Fig.7.b, thatthe particle statistics exhibits a Gaussian distribution. Note that Gaussian velocity statistics were also observed inthermal counterflow simulations of the vortex filament method with tracers particles [30]. The absence of power-lawtails could be a consequence of weak statistical sampling of large velocity fluctuations due to the low number ofparticles present in the system and/or by compressible effects of the GP model. We will comment more about this inthe Discussion section.We would like to remark here that high frequency fluctuations are strongly sensitive to numerical artifacts. In theAppendix, inspired by the experimental results of reference [25], we have computed the velocity PDFs of the velocityfluctuations filtered at a given frequency ω c . The frequency was varied from values lower to larger than 2 π/τ (cid:96) . Forone simulation we have compared two different interpolation methods to evaluate the force term in Eq.(3) needed todrive the particles. It turns out that for the fourth–order B–spline method, the velocity PDFs start to develop tailswhile the filtering scale is varied, that eventually lead to a v − scaling. However, when using Fourier interpolation,that is an exact evaluation (up to spectral convergence of the pseudo–spectral code) of the force term, the PDFs donot develop any tail and remain Gaussian. We have decided to keep this example with spurious numerical effectsin the Appendix, as it might be useful for future numerical studies and data analysis of similar problems. We havechecked that the results presented in the paper are independent of the interpolation scheme.We turn now to study the acceleration statistics. As displayed in Fig.8.a, the PDF of the acceleration presents somedeviations from a Gaussian distribution at large values. The norm of the acceleration has also an exponential tail for − a i − h a i i ) /σ a − − − − P D F ( a ) a p = 4 ξ , M = 2 a p = 4 ξ , M = 1 a p = 4 ξ , M = 0 . a p = 4 ξ , M = 0 . a p = 10 ξ , M = 0 . Gaussian | a | /σ | a | − − − − − P D F ( b ) χ e −| a | /σ | a | Log-Normal − | a | − h log | a |i ) /σ log | a | − FIG. 8. (color online) (a) Probability density functions of the single component particle acceleration. (b) Probability densityfunctions of the norm of the particle acceleration. Dotted line is a gaussian, dashed line is a χ distribution and dash-dotted lineis an exponential tail e −| a | /σ | a | . (inset) Probability density functions of the logarithm of the norm of the particle acceleration.Dashed golden line is a log–normal distribution. | a | > σ | a | , as displayed in Fig.8.b. The core of the PDF in this case is a χ distribution, which is expected for thenorm of a vector with Gaussian components. In classical Lagrangian turbulence, the norm of the particle accelerationis observed to obey a log–normal distribution [54]. In the inset of Fig.8.b, we compare our data with such distribution.For the lightest and smallest particle, the small accelerations appears to be more probable than in the classical case.Note that, as pointed out in [54], small values of the acceleration are very sensible to experimental (numerical) errors.By contrast, the large accelerations are less probable than a log-normal distribution. This observation is compatiblewith classical numerical calculations in the framework of the viscous vortex filament model, in which it has beenshown that, because of inertia, solid particles undergo less rapid changes of velocity than fluid particles [55].Finally, in Fig.9, we show the two–point correlator of the particle acceleration, defined as: ρ a ( t ) = (cid:104) a i ( t ) a i ( t + t ) (cid:105) − (cid:104) a i ( t ) (cid:105) (cid:104) a i ( t + t ) (cid:105) σ a ( t ) σ a ( t + t ) . (11)In classical Lagrangian turbulence, the decorrelation time t a (such that ρ a ( t a ) = 0) is related to the Kolmogorovtime scale t a = 2 τ η [56]. This is not the case in quantum turbulence. Figure 9.a displays the autocorrelation ρ a ( t )for all the simulations. It is apparent that the acceleration decorrelates much faster than τ (cid:96) , the equivalent of theKolmogorov time scale in our system. This fact is a consequence of the myriad of physical phenomena taking place4 . . . . t/τ ‘ − . . . . . . . ρ a ( t ) ( a ) a p = 4 ξ , M = 2 a p = 4 ξ , M = 1 a p = 4 ξ , M = 0 . a p = 4 ξ , M = 0 . a p = 10 ξ , M = 1 a p = 10 ξ , M = 0 . a p = 10 ξ , M = 0 . . 00 0 . 25 0 . 50 0 . 75 1 . 00 1 . 25 1 . t Ω p − . . . . . . . ρ a ( t ) ( b ) 0 1 2 M t a Ω p FIG. 9. (color online) Acceleration two–point correlator, plotted versus time normalized by the dissipation time scale τ (cid:96) (a) , andby the Magnus natural frequency. 1 / Ω p (b) Markers indicate the time of acceleration decorrelation t a . (inset) t a normalizedby 1 / Ω p as a function of the particle relative mass. at smaller scales. As most particles are trapped by vortices, they oscillate at the Magnus frequency Ω p in Eq.(9). Iftime is normalized by Ω p (9), then t a Ω p becomes of order 1, at least for the heaviest particles (see Fig.9.b and theinset therein). For the lightest particles the decorrelation time is even lower, meaning that they are sensible to othermechanisms, like reconnection events between vortex filaments and Kelvin waves excitations at even smaller scales. IV. DISCUSSION In this work we used the Gross-Pitaevskii model to study free decaying quantum turbulence at zero temperaturein presence of finite size active particles. We considered different families of spherical particles having sizes smallerthan and of the order of the mean inter-vortex distance. We first performed a standard analysis of the observablescommonly used for studying Kolmogorov turbulence, such as the energy decomposition, the temporal evolution ofmean energy, the rate of incompressible kinetic energy and the mean inter-vortex distance. Although particles areactive and get captured by vortices generating Kelvin waves, there is not a significant impact at scales larger thanthe inter-vortex distance, where Kolmogorov turbulence takes place. Monitoring the motion of the particles in thesystem, we confirmed their tendency to remain trapped into vortex filaments during the evolution of the tangle, withintermittent episodes of detachment and recapture. This behavior is independent of the vortex line density. We alsofound that particle can be easily captured simultaneously by several quantum vortices.We also studied turbulence from the Lagrangian point of view. In particular, we computed the power spectra of theparticle velocities. At large scales the particle dynamics is compatible with the one of Lagrangian tracers in classicalturbulence, while at short time scales the Magnus precession around the filaments caused by the vortex circulationis dominating the motion. Such information can be extracted consistently both in the frequency spectrum of thevelocity and in the decay time of the correlation of the acceleration. Furthermore, if particles are light enough, fasterfrequencies are also excited. This suggests (as intuitively expected) that light particles can be more sensitive to thesmall scale fluctuations of the flow.Finally we investigated the particle velocity statistics. The distribution of the particles velocity is Gaussian, incontrast with the power law scaling | v i | − recently observed in superfluid helium experiments [24, 25]. There areseveral reasons explaining why power-law tails are absent in our simulations. Firstly, since the simulation of eachparticle has an important numerical cost, the number of particles is restricted only to a couple of hundreds. Due tothis issue, vortex reconnections might be unlikely sampled by the sparse distribution of particles. Note also that, asparticles have a finite size, increasing their number keeping the size of the system constant will increase substantiallythe filling fraction. In this case, turbulence could be even prevented by the presence of particles. Although interesting,this limit is out of the scope of this work. Secondly, the GP-model is compressible and particle moving at largevelocities are slowed down by vortex nucleations. This certainly reduces large velocity fluctuations, perhaps limitingthe development of power-law tails. It would be interesting to address such issues in generalized GP models, includinga roton minimum and high-order non-linearities. Moreover, our simulations are by definition at zero temperature andparticles do not follow the singular superfluid velocity field because of the lack of viscosity in the system. Indeed,in the GP model the pressure gradients that drive the particle dynamics are always regular because of the vanishingdensity at the vortex cores, unlike other models as the vortex filament method. As a consequence, the divergence of5the superfluid velocity along the vortex lines can not be experienced by the particles. Conversely, at finite temperaturethe superfluid and the normal component can be locked thanks to mutual friction. In this case, since particles wouldsample the normal fluid velocity because of Stokes drag, they might be able to sample the 1 /r flow around a quantumvortex. Finally, we observed that fast velocity fluctuations are highly sensitive to interpolation and filtering methodsthat could even lead to power-law tails. These tails are completely spurious, and special care is needed while analyzingnumerical or experimental data. Appendix A: Numerical artifacts on the particle velocity statistics. Comparison between B–spline andspectral interpolation methods As explained in the main text, we evaluate the force f GP i = − ( V p ∗ ∇ ρ ) [ q i ] (3) at the particle position q i usinga B–spline interpolation method [39] at each time step. Such method is precise and computationally cheap, butit turns out to present some issues that we have to take care of. In order to check the reliability of the method,we re-run a simulation using Fourier interpolation for one species of particles in the time window correspondingto the turbulent regime. Fourier interpolation is exact in the sense that uses the information of the full threedimensional field, that is resolved with spectral accuracy (i.e. discretization errors are at most exponentially smallwith the number of discretization points). The numerical cost of this method is the one of one Fourier transform (perparticle). In Fig.10 the velocity and acceleration spectra computed using B–spline and Fourier interpolation methodsare compared. Clearly, the B–spline interpolation introduces non-physical fast oscillations, but at the frequencies | ω | τ ‘ / π − − − − − − − (cid:10) | ˆ v p ( ω ) | (cid:11) ( a ) Ω p B-splineFourier | ω | τ ‘ / π − − − − (cid:10) | ˆ a p ( ω ) | (cid:11) ( b ) FIG. 10. (color online) Velocity spectra (a) and the acceleration spectra (b) for particles of size a p = 10 ξ and mass M = 0 . . T L < t < . T L . ω < ω noise = 50(2 π/τ (cid:96) ) the behavior of the spectra is unchanged. Nevertheless, some differences in the features ofparticle statistics are still visible at fast timescales once the noise is filtered out.We use a Gaussian convolution to perform a filtering of the velocity time series for each particle in the frequencywindow ω c < ω < ω noise , where ω c is a variable infrared cut-off frequency. Then we compute the PDF of the filteredvelocity for different values of ω c . Such PDFs are shown in Fig.11 comparing the simulations in which Fourier andB–spline interpolation are used for the same species of particle. Surprisingly, only in the latter case we observe power-law tails for the fast oscillations distributions. Such PDFs are similar to the ones observed experimentally [24, 25],but in the present case, they are just a consequence of numerical artifacts. ACKNOWLEDGMENTS U.G. acknowledges J.I. Polanco for fruitful discussions. The authors were supported by Agence Nationale de laRecherche through the project GIANTE ANR-18-CE30-0020-01. Computations were carried out on the M´esocentreSIGAMM hosted at the Observatoire de la Cˆote d’Azur and the French HPC Cluster OCCIGEN through the GENCI6 − . − . . . . v i − h v i i ) /σ v − − − − − − P D F Fourier ( b ) − − v i − h v i i ) /σ v − − − − − − P D F B-spline ( a ) ω c = 16 . τ ‘ / πω c = 13 . τ ‘ / πω c = 10 . τ ‘ / πω c = 7 . τ ‘ / πω c = 4 . τ ‘ / πω c = 1 . τ ‘ / πω c = 0 . τ ‘ / πω c = 0 . τ ‘ / πω c = 0 . τ ‘ / π Gaussian ∝ ( v i − h v i i )) − FIG. 11. (color online) Probability density function of the velocity filtered in the frequency window ω c < ω < ω noise fordifferent values of ω c . Data refer to particles of size a p = 10 ξ and mass M = 0 . 13. Dotted line is a gaussian distributionand dash-dotted line is a power-law scaling 0 . 002 ( v i − (cid:104) v i (cid:105) ) − . The data are averaged over particles and over the times1 . T L < t < . T L . Different PDFs are shifted for visualization. (a) Particle force interpolated with B–spline method. (b) Particle force interpolated with Fourier method. allocation A0042A10385. [1] U. Frisch and A. N. Kolmogorov, Turbulence: the legacy of AN Kolmogorov (Cambridge university press, 1995).[2] R. J. Donnelly, Quantized vortices in helium II , Vol. 2 (Cambridge University Press, 1991).[3] T. Frisch, Y. Pomeau, and S. Rica, Transition to dissipation in a model of superflow, Phys. Rev. Lett. , 1644 (1992).[4] T. Winiecki and C. S. Adams, Motion of an object through a quantum fluid, EPL (Europhysics Letters) , 257 (2000).[5] J. Koplik and H. Levine, Vortex reconnection in superfluid helium, Phys. Rev. Lett. , 1375 (1993).[6] G. P. Bewley, M. S. Paoletti, K. R. Sreenivasan, and D. P. Lathrop, Characterization of reconnecting vortices in superfluidhelium, Proceedings of the National Academy of Sciences , 13707 (2008).[7] A. Villois, D. Proment, and G. Krstulovic, Universal and nonuniversal aspects of vortex reconnections in superfluids, Phys.Rev. Fluids , 044701 (2017).[8] L. Galantucci, A. W. Baggaley, N. G. Parker, and C. F. Barenghi, Crossover from interaction to drivenregimes in quantum vortex reconnections, Proceedings of the National Academy of Sciences , 2644(1997).[10] A. W. Baggaley, J. Laurie, and C. F. Barenghi, Vortex-density fluctuations, energy spectra, and vortical regions in superfluidturbulence, Phys. Rev. Lett. , 205304 (2012).[11] V. Shukla, P. D. Mininni, G. Krstulovic, P. C. di Leoni, and M. E. Brachet, Quantitative estimation of effective viscosityin quantum turbulence, Phys. Rev. A , 043605 (2019).[12] J. Maurer and P. Tabeling, Local investigation of superfluid turbulence, Europhysics Letters (EPL) , 29 (1998).[13] J. Salort, C. Baudet, B. Castaing, B. Chabaud, F. Daviaud, T. Didelot, P. Diribarne, B. Dubrulle, Y. Gagne, F. Gauthier,A. Girard, B. H´ebral, B. Rousset, P. Thibault, and P.-E. Roche, Turbulent velocity spectra in superfluid flows, Physics ofFluids , 125102 (2010), https://doi.org/10.1063/1.3504375.[14] W. F. Vinen, Decay of superfluid turbulence at a very low temperature: The radiation of sound from a kelvin wave on aquantized vortex, Phys. Rev. B , 134520 (2001).[15] V. S. L’vov and S. Nazarenko, Weak turbulence of kelvin waves in superfluid he, Low Temperature Physics , 785 (2010),https://doi.org/10.1063/1.3499242.[16] J. Laurie, V. S. L’vov, S. Nazarenko, and O. Rudenko, Interaction of kelvin waves and nonlocality of energy transfer insuperfluids, Phys. Rev. B , 104526 (2010).[17] G. Krstulovic, Kelvin-wave cascade and dissipation in low-temperature superfluid vortices, Phys. Rev. E , 055301(R)(2012).[18] A. W. Baggaley and J. Laurie, Kelvin-wave cascade in the vortex filament model, Phys. Rev. B , 014504 (2014).[19] A. Villois, D. Proment, and G. Krstulovic, Evolution of a superfluid vortex filament tangle driven by the gross-pitaevskiiequation, Phys. Rev. E , 061103(R) (2016).[20] F. Toschi and E. Bodenschatz, Lagrangian properties of particles in turbulence, Annual Review of Fluid Mechanics ,375 (2009), https://doi.org/10.1146/annurev.fluid.010908.165210. [21] W. Guo, M. La Mantia, D. P. Lathrop, and S. W. Van Sciver, Visualization of two-fluid flows of superfluid helium-4,Proceedings of the National Academy of Sciences , 4653 (2014).[22] G. P. Bewley, D. P. Lathrop, and K. R. Sreenivasan, Superfluid helium: Visualization of quantized vortices, Nature ,588 (2006).[23] E. Fonda, D. P. Meichle, N. T. Ouellette, S. Hormoz, and D. P. Lathrop, Direct observation of kelvin waves excited byquantized vortex reconnection, Proceedings of the National Academy of Sciences , 4707 (2014).[24] M. S. Paoletti, M. E. Fisher, K. R. Sreenivasan, and D. P. Lathrop, Velocity statistics distinguish quantum turbulencefrom classical turbulence, Phys. Rev. Lett. , 154501 (2008).[25] M. L. Mantia and L. Skrbek, Quantum, or classical turbulence?, EPL (Europhysics Letters) , 46002 (2014).[26] M. La Mantia and L. Skrbek, Quantum turbulence visualized by particle dynamics, Phys. Rev. B , 014519 (2014).[27] M. La Mantia, D. Duda, M. Rotter, and L. Skrbek, Lagrangian accelerations of particles in superfluid turbulence, Journalof Fluid Mechanics , R9 (2013).[28] Y. A. Sergeev and C. F. Barenghi, Particles-vortex interactions and flow visualization in 4he, Journal of Low TemperaturePhysics , 429 (2009).[29] C. F. Barenghi, D. Kivotides, and Y. A. Sergeev, Close approach of a spherical particle and a quantised vortex in heliumii, Journal of Low Temperature Physics , 293 (2007).[30] Y. Mineda, M. Tsubota, Y. A. Sergeev, C. F. Barenghi, and W. F. Vinen, Velocity distributions of tracer particles inthermal counterflow in superfluid he, Phys. Rev. B , 174508 (2013).[31] E. Varga, C. F. Barenghi, Y. A. Sergeev, and L. Skrbek, Backreaction of tracer particles on vortex tangle in helium iicounterflow, Journal of Low Temperature Physics , 215 (2016).[32] J. I. Polanco and G. Krstulovic, Inhomogeneous distribution of particles in coflow and counterflow quantum turbulence,Phys. Rev. Fluids , 032601 (2020).[33] V. Shukla, M. Brachet, and R. Pandit, Sticking transition in a minimal model for the collisions of active particles inquantum fluids, Phys. Rev. A , 041602(R) (2016).[34] U. Giuriato and G. Krstulovic, Interaction between active particles and quantum vortices leading to kelvin wave generation,Scientific Reports , 4839 (2019).[35] U. Giuriato, G. Krstulovic, and D. Proment, Clustering and phase transitions in a 2d superfluid with immiscible activeimpurities, Journal of Physics A: Mathematical and Theoretical , 305501 (2019).[36] U. Giuriato, G. Krstulovic, and S. Nazarenko, How do trapped particles interact with and sample superfluid vortexexcitations?, arXiv e-prints , arXiv:1907.01111 (2019), arXiv:1907.01111 [cond-mat.other].[37] A. Griffin, S. Nazarenko, V. Shukla, and M.-E. Brachet, The vortex-particle magnus effect, arXiv preprint arXiv:1909.11010(2019).[38] P. Clark di Leoni, P. D. Mininni, and M. E. Brachet, Dual cascade and dissipation mechanisms in helical quantumturbulence, Phys. Rev. A , 053636 (2017).[39] M. A. T. van Hinsberg, J. H. M. Thije Boonkkamp, F. Toschi, and H. J. H. Clercx, On the efficiency andaccuracy of interpolation methods for spectral codes, SIAM Journal on Scientific Computing , B479 (2012),https://doi.org/10.1137/110849018.[40] N. Sasa, T. Kano, M. Machida, V. S. L’vov, O. Rudenko, and M. Tsubota, Energy spectra of quantum turbulence:Large-scale simulation and modeling, Phys. Rev. B , 054525 (2011).[41] G. Krstulovic, Grid superfluid turbulence and intermittency at very low temperature, Phys. Rev. E , 063104 (2016).[42] W. F. Vinen, Mutual friction in a heat current in liquid helium ii iii. theory of the mutual friction, Proceedings of theRoyal Society of London. Series A. Mathematical and Physical Sciences , 493 (1957).[43] N. G. Berloff and P. H. Roberts, Capture of an impurity by a vortex line in a bose condensate, Phys. Rev. B , 024510(2000).[44] A. Villois, G. Krstulovic, D. Proment, and H. Salman, A vortex filament tracking method for the gross–pitaevskii modelof a superfluid, Journal of Physics A: Mathematical and Theoretical , 415502 (2016).[45] The circulation measured with this method is subjected to a numerical error coming from the grid spacing. Such error isremoved in post-processing, knowing that Γ can only be an integer multiple of κ . Furthermore, extremely high values of Γhave been removed since they are related to the ill-defined situation in which a topological defect is placed at the boundary C .[46] P. K. Yeung, Lagrangian characteristics of turbulence and scalar transport in direct numerical simulations, Journal of FluidMechanics , 241?274 (2001).[47] H. Tennekes, J. L. Lumley, J. Lumley, et al. , A first course in turbulence (MIT press, 1972).[48] D. Kivotides, Y. A. Sergeev, and C. F. Barenghi, Dynamics of solid particles in a tangle of superfluid vortices at lowtemperatures, Physics of Fluids , 055105 (2008), https://doi.org/10.1063/1.2919805.[49] L. Kiknadze and Y. Mamaladze, The magnus (kutta-jukovskii) force acting on a sphere, arXiv preprint cond-mat/0604436(2006).[50] A. C. White, C. F. Barenghi, N. P. Proukakis, A. J. Youd, and D. H. Wacks, Nonclassical velocity statistics in a turbulentatomic bose-einstein condensate, Phys. Rev. Lett. , 075301 (2010).[51] M. S. Paoletti and D. P. Lathrop, Quantum turbulence, Annual Review of Condensed Matter Physics , 213 (2011),https://doi.org/10.1146/annurev-conmatphys-062910-140533.[52] A. W. Baggaley and C. F. Barenghi, Quantum turbulent velocity statistics and quasiclassical limit, Phys. Rev. E ,067301 (2011). [53] V. Shukla, M. Brachet, and R. Pandit, Turbulence in the two-dimensional fourier-truncated gross–pitaevskii equation, NewJournal of Physics , 113025 (2013).[54] N. Mordant, A. M. Crawford, and E. Bodenschatz, Three-dimensional structure of the lagrangian acceleration in turbulentflows, Phys. Rev. Lett. , 214501 (2004).[55] D. Kivotides, C. F. Barenghi, A. J. Mee, and Y. A. Sergeev, Interaction of solid particles with a tangle of vortex filamentsin a viscous fluid, Phys. Rev. Lett. , 074501 (2007).[56] P. K. Yeung and S. B. Pope, Lagrangian statistics from direct numerical simulations of isotropic turbulence, Journal ofFluid Mechanics207