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International audience; Developing numerical methods to simulate efficiently nonlinear fluid dynamics on universal quantum computers is a challenging problem. In this paper, a generalization of the Madelung transform is defined to solve quantum relativistic charged fluid equations interacting with external electromagnetic forces via the Dirac equation. The Dirac equation is discretized into discrete-time quantum walks which can be efficiently implemented on universal quantum computers. A variant of this algorithm is proposed to implement simulations using current noisy intermediate scale quantum (NISQ) devices in the case of homogeneous external forces. High resolution (up to N=217 grid points) numerical simulations of relativistic and nonrelativistic hydrodynamical shocks on current IBM NISQs are performed with this algorithm. This paper demonstrates that fluid dynamics can be simulated on NISQs, and opens the door to simulating other fluids, including plasmas, with more general quantum walks and quantum automata.

International audience; We study self-gravitating bosonic systems, candidates for dark-matter halos, by carrying out a suite of direct numerical simulations designed to investigate the formation of finite-temperature, compact objects in the three-dimensional (3D) Fourier-truncated Gross-Pitaevskii-Poisson equation (GPPE). This truncation allows us to explore the collapse and fluctuations of compact objects, which form at both zero temperature and finite temperature. We show that the statistically steady state of the GPPE, in the large-time limit and for the system sizes we study, can also be obtained efficiently by tuning the temperature in an auxiliary stochastic Ginzburg-Landau-Poisson equation. We show that, over a wide range of model parameters, this system undergoes a thermally driven first-order transition from a collapsed, compact, Bose-Einstein condensate to a tenuous Bose gas (that is not gravitationally condensed). By a suitable choice of initial conditions in the GPPE, we also obtain a binary condensate that comprises a pair of collapsed objects rotating around their center of mass.

In many simulations of turbulent flows the viscous forces $\nu\nabla^2 {\bf u}$ are replaced by a hyper-viscous term $-\nu_p(-\nabla^2)^{p}{\bf u}$ in order to suppress the effect of viscosity at the large scales. In this work we examine the effect of hyper-viscosity on decaying turbulence for values of $p$ ranging from $p=1$ (regular viscosity) up to $p=100$. Our study is based on direct numerical simulations of the Taylor-Green vortex for resolutions from $512^3$ to $2048^3$. Our results demonstrate that the evolution of the total energy $E$ and the energy dissipation $\epsilon$ remain almost unaffected by the order of the hyper-viscosity used. However, as the order of the hyper-viscosity is increased ,the energy spectrum develops a more pronounced bottleneck that contaminates the inertial range. At the largest values of $p$ examined, the spectrum at the bottleneck range has a positive power-law behavior $E(k)\propto k^\alpha$ with the power-law exponent $\alpha$ approaching the value obtained in flows at thermal equilibrium $\alpha=2$. This agrees with the prediction of Frisch et al. [Phys. Rev. Lett. 101, 144501 (2008)] who suggested that at high values of $p$, the flow should behave like the truncated Euler equations (TEE). Nonetheless, despite the thermalization of the spectrum, the flow retains a finite dissipation rate up to the examined order, which disagrees with the predictions of the TEE system implying suppression of energy dissipation. We reconcile the two apparently contradictory results, predicting the value of $p$ for which the hyper-viscous Navier-Stokes goes over to the TEE system and we discuss why thermalization appears at smaller values of $p$.

We study freely decaying quantum turbulence by performing high resolution numerical simulations of the Gross-Pitaevskii equation (GPE) in the Taylor-Green geometry. We use resolutions ranging from $1024^3$ to $4096^3$ grid points. The energy spectrum confirms the presence of both a Kolmogorov scaling range for scales larger than the intervortex scale $\ell$, and a second inertial range for scales smaller than $\ell$. Vortex line visualizations show the existence of substructures formed by a myriad of small-scale knotted vortices. Next, we study finite temperature effects in the decay of quantum turbulence by using the stochastic Ginzburg-Landau equation to generate thermal states, and then by evolving a combination of these thermal states with the Taylor-Green initial conditions using the GPE. We extract the mean free path out of these simulations by measuring the spectral broadening in the Bogoliubov dispersion relation obtained from spatio-temporal spectra, and use it to quantify the effective viscosity as a function of the temperature. Finally, in order to compare the decay of high temperature quantum and that of classical flows, and to further calibrate the estimations of viscosity from the mean free path in the GPE simulations, we perform low Reynolds number simulations of the Navier-Stokes equations., Comment: 15 figures, Supplemental material: Video M1 and M2

We perform a study of the evolution of helical quantum turbulence at different temperatures by solving numerically the Gross-Pitaevskii and the stochastic Ginzburg-Landau equations, using up to 40963 grid points with a pseudospectral method. We show that for temperatures close to the critical one, the fluid described by these equations can act as a classical viscous flow, with the decay of the incompressible kinetic energy and the helicity becoming exponential. The transition from this behavior to the one observed at zero temperature is smooth as a function of temperature. Moreover, the presence of strong thermal effects can inhibit the development of a proper turbulent cascade. We provide Ansätze for the effective viscosity and friction as a function of the temperature. Fil: Clark Di Leoni, Patricio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina. University of Rome Tor Vergata; Italia Fil: Mininni, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina Fil: Brachet, Marc E.. Universite Pierre et Marie Curie; Francia

We carry out extensive direct numerical simulations (DNSs) to investigate the interaction of active particles and fields in the two-dimensional (2D) Gross-Pitaevskii (GP) superfluid, in both simple and turbulent flows. The particles are active in the sense that they affect the superfluid even as they are affected by it. We tune the mass of the particles, which is an important control parameter. At the one-particle level, we show how light, neutral, and heavy particles move in the superfluid, when a constant external force acts on them; in particular, beyond a critical velocity, at which a vortex-antivortex pair is emitted, particle motion can be periodic or chaotic. We demonstrate that the interaction of a particle with vortices leads to dynamics that depends sensitively on the particle characteristics. We also demonstrate that assemblies of particles and vortices can have rich, and often turbulent spatiotemporal evolution. In particular, we consider the dynamics of the following illustrative initial configurations: (a) one particle placed in front of a translating vortex-antivortex pair; (b) two particles placed in front of a translating vortex-antivortex pair; (c) a single particle moving in the presence of counter-rotating vortex clusters; and (d) four particles in the presence of counter-rotating vortex clusters. We compare our work with earlier studies and examine its implications for recent experimental studies in superfluid Helium and Bose-Einstein condensates., Comment: 24 figures

The spectrally truncated, or finite dimensional, versions of several equations of inviscid flows display transient solutions which match their viscous counterparts, but which eventually lead to thermalized states in which energy is in equipartition between all modes. Recent advances in the study of the Burgers equation show that the thermalization process is triggered after the formation of sharp localized structures within the flow called "tygers." We show that the process of thermalization first takes place in well defined subdomains, before engulfing the whole space. Using spatio-temporal analysis on data from numerical simulations, we study propagation of tygers and find that they move at a well defined mean speed that can be obtained from energy conservation arguments. Fil: Clark Di Leoni, Patricio. University of Rome “Tor Vergata”; Italia. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina Fil: Mininni, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina Fil: Brachet, Marc E.. Université Paris Diderot - Paris 7; Francia

While in classical turbulence helicity depletes nonlinearity and can alter the evolution of turbulent flows, in quantum turbulence its role is not fully understood. We present numerical simulations of the free decay of a helical quantum turbulent flow using the Gross-Pitaevskii equation at high spatial resolution. The evolution has remarkable similarities with classical flows, which go as far as displaying a dual transfer of incompressible kinetic energy and helicity to small scales. Spatio-temporal analysis indicates that both quantities are dissipated at small scales through non-linear excitation of Kelvin waves and the subsequent emission of phonons. At the onset of the decay, the resulting turbulent flow displays polarized large scale structures and unpolarized patches of quiescense reminiscent of those observed in simulations of classical turbulence at very large Reynolds numbers., Comment: Fixed typos

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, fluid dynamics, atmospheric sciences, and 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 centerline vorticity can be downright impossible to apply in complex and turbulent scenarios. We present a definition of helicity which overcomes these problems and which gives the expected result in the large-scale limit. With it, we show that certain reconnection events can excite Kelvin waves and other complex motions of the centerline vorticity, which slowly deplete helicity as they interact nonlinearly, thus linking the theory of vortex knots with observations of quantum fluids. This process also results in the depletion of helicity in a fully turbulent quantum flow, in a way reminiscent of the decay of helicity in classical fluids. Fil: Clark Di Leoni, Patricio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina Fil: Mininni, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina Fil: Brachet, M. E.. Laboratoire de Physique Statistique ; Francia

The Gross–Pitaevskii equation (GPE) describes the dynamics of superflows and Bose–Einstein Condensates (BEC) at very low temperature. When a truncation of Fourier modes is performed, the resulting truncated GPE (TGPE) can also describe the correct thermal behavior of a Bose gas, as long as all relevant modes are highly occupied [M.J. Davis, S.A. Morgan, K. Burnett, Simulations of Bose fields at finite temperature, Phys. Rev. Lett. 87 (16) (2001) 160402]. We review some of our groupʼs recent GPE- and TGPE-based numerical studies of superfluid dynamics and BEC stability. The relations with experiments are discussed.

A short review is given of recent papers on the relaxation to (incompressible) absolute equilibrium. A new algorithm to construct absolute equilibrium of spectrally truncated compressible flows is described. The algorithm uses stochastic processes based on the Clebsch representation of the velocity field to generate density and velocity fields that follow by construction the absolute equilibrium stationary probability. The new method is shown to reproduce the well-known Gaussian results in the incompressible limit. The irrotational compressible absolute equilibrium case is characterized and the distribution is shown to be non-Gaussian. The high-temperature compressible spectra are found not to obey k2 scaling. Finally, oscillating behavior in constant-pressure variable-temperature relaxation is obtained, suggesting the presence of second sound.

Particles of low velocity, travelling without dissipation in a superfluid, can interact and emit sound when they collide. We propose a minimal model in which the equations of motion of the particles, including a short-range repulsive force, are self-consistently coupled with the Gross-Pitaevskii equation. We use this model to demonstrate the existence of an effective superfluid-mediated attractive interaction between the particles; and we study numerically the collisional dynamics of particles as a function of their incident kinetic energy and the length-scale of the repulsive force. We find a transition from almost elastic to completely inelastic (sticking) collisions as the parameters are tuned. We find that aggregation and clustering result from this sticking transition in multi-particle systems., 3 captioned figures, 5 videos, Supplemental Material at the end of the file and contains links to the videos

It is shown that the Truncated Euler Equations, i.e. a finite set of ordinary differential equations for the amplitude of the large-scale modes, can correctly describe the complex transitional dynamics that occur within the turbulent regime of a confined 2D Navier-Stokes flow with bottom friction and a spatially periodic forcing. In particular, the random reversals of the large scale circulation on the turbulent background involve bifurcations of the probability distribution function of the large-scale circulation velocity that are described by the related microcanonical distribution which displays transitions from gaussian to bimodal and broken ergodicity. A minimal 13-mode model reproduces these results., Comment: 6 pages, 2 figures

We present evidence of Kelvin excitations in space-time resolved spectra of numerical simulations of quantum turbulence. Kelvin waves are transverse and circularly polarized waves that propagate along quantized vortices, for which the restitutive force is the tension of the vortex line, and which play an important role in theories of superfluid turbulence. We use the Gross-Pitaevskii equation to model quantum flows, letting an initial array of well-organized vortices develop into a turbulent bundle of intertwined vortex filaments. By achieving high spatial and temporal resolution we are able to calculate space-time resolved mass density and kinetic energy spectra. Evidence of Kelvin and sound waves is clear in both spectra. Identification of the waves allows us to extract the spatial spectrum of Kelvin waves, clarifying their role in the transfer of energy. Fil: Clark Di Leoni, Patricio. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina Fil: Mininni, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina Fil: Brachet, M. E.. Ecole Normale Supérieure; Francia. Centre National de la Recherche Scientifique; Francia. Universite de Paris VI; Francia. Université Paris Diderot - Paris 7; Francia

The analyticity strip method is used to trace complex singularities in direct numerical simulations ofthe Kida–Pelz and Taylor–Green flows, performed with up to 2048 3 collocation points. Oscillations found in the Kida–Pelz energy spectrum are attributed to interferences of complex singularities. A generalized least-square fit that separates out the oscillations from the measure ofthe width ofthe analyticity strip is introduced. Using the available resolution, is found to decay exponentially in time up to t = 1.25. It is argued that resolutions in the range 16384 3 –32768 3

The transition to dissipation in one-dimensional extended Hamiltonian systems with saddle-node bifurcations of stationary solutions is characterized. Three different systems are studied: (i) nonlinear Schrodinger flow past a localized obstacle; (ii) sine-Gordon pendulum chains forced by a local torque; (iii) electrically charged nonlinear Schrodinger flows. In case (i), no frequency gap is present in the dispersion relation. In contrast, in cases (ii) and (iii) a minimum frequency for propagating waves exists. In the gapless case, the growth rates of the unstable modes and the frequency of supercritical soliton emission are found to scale as the square root of the bifurcation parameter. No subcriticality is observed. In contrast, when a frequency gap is present, subcritical soliton emission takes place. Logarithmic and one-fourth power scaling laws are found, respectively, at the bottom and top of the subcriticality window.

We present an overview of the statistical properties of turbulence in two-dimensional (2D) fluids. After a brief recapitulation of well-known results for statistically homogeneous and isotropic 2D fluid turbulence, we give an overview of recent progress in this field for such 2D turbulence in conducting fluids, fluids with polymer additives, binary-fluid mixtures, and superfluids; we also discuss the statistical properties of particles advected by 2D turbulent fluids.

The detailed understanding of the intricate dynamics of quantum fluids, in particular in the rapidly growing subfield of quantum turbulence which elucidates the evolution of a vortex tangle in a superfluid, requires an in-depth understanding of the role of finite temperature in such systems. The Landau two-fluid model is the most successful hydrodynamical theory of superfluid helium, but by the nature of the scale separations it cannot give an adequate description of the processes involving vortex dynamics and interactions. In our contribution we introduce a framework based on a nonlinear classical-field equation that is mathematically identical to the Landau model and provides a mechanism for severing and coalescence of vortex lines, so that the questions related to the behavior of quantized vortices can be addressed self-consistently. The correct equation of state as well as nonlocality of interactions that leads to the existence of the roton minimum can also be introduced in such description. We review and apply the ideas developed for finite-temperature description of weakly interacting Bose gases as possible extensions and numerical refinements of the proposed method. We apply this method to elucidate the behavior of the vortices during expansion and contraction following the change in applied pressure. We show that at low temperatures, during the contraction of the vortex core as the negative pressure grows back to positive values, the vortex line density grows through a mechanism of vortex multiplication. This mechanism is suppressed at high temperatures.

Moments and probability density functions (PDF) of (absolute value) velocity increments jv.xC‘/ v.x/j in turbulence are linked by simple integral relations. It is shown that the steepest descent method can be applied to evaluate the integrals if the moments (the absolute value structure functions) obey multifractal scaling laws of the type hjv.x C ‘/ v.x/j n iD An‘ n .A double asymptotic relation then relates the moments to the PDF. The dominant (exponential) terms of the asymptotic relation naturally yield the Legendre transform that is at the core of the Parisi‐Frisch model of inertial-range intermittency. Using the asymptotic relation, the PDF can be reconstructed from the multifractal exponent spectrum n and the statistics of large scale moments. On the basis of experimental results, it is shown that moments are quantitatively represented by multifractal scaling laws and large scale Gaussian (or quasi-Gaussian) statistics. The large scale at which the statistics are Gaussian (or quasi-Gaussian) is determined from inertial-range data alone and is of the order of the integral scale for Taylor-scale Reynolds numbers R in the range (300‐2200). This representation of moments together with the double asymptotic relations is able to reconstruct quantitatively the experimental inertial-range PDF. Analytic expressions (She-Leveque and Log-normal) of scaling exponents are both shown to lead to reconstructed PDF with systematic deviations from experiment. ©1999 Published by Elsevier Science B.V. All rights reserved.

Attractive Bose-Einstein condensates are investigated with numerical continuation methods capturing stationary solutions of the Gross-Pitaevskii equation. The branches of stable (elliptic) and unstable (hyperbolic) solutions are found to meet at a critical particle number through a generic Hamiltonian saddle node bifurcation. The condensate decay rates corresponding to macroscopic quantum tunneling, two and three body inelastic collisions, and thermally induced collapse are computed from the exact numerical solutions. These rates show experimentally significant differences with previously published rates. Universal scaling laws stemming from the bifurcation are derived. [S0031-9007(99)08550-6]

We examine the scaling laws of magnetohydrodynamic (MHD) turbulence for three different types of forcing functions and imposing at all times the fourfold symmetries of the Taylor-Green (TG) vortex generalized to MHD; no uniform magnetic field is present and the magnetic Prandtl number is equal to unity. We also include pumping in the induction equation, and we take the three configurations studied in the decaying case in Lee et al. [Phys. Rev. E 81, 016318 (2010)]. To that effect, we employ direct numerical simulations up to an equivalent resolution of ${2048}^{3}$ grid points. We find that, similarly to the case when the forcing is absent, different spectral indices for the total energy spectrum emerge, corresponding to either a Kolmogorov law, an Iroshnikov-Kraichnan law that arises from the interactions of turbulent eddies and Alfv\'en waves, or to weak turbulence when the large-scale magnetic field is strong. We also examine the inertial range dynamics in terms of the ratios of kinetic to magnetic energy, and of the turnover time to the Alfv\'en time, and analyze the temporal variations of these quasiequilibria.

Superfluid turbulence is studied using numerical simulations of the nonlinear Schrodinger equation (NLSE), which is the correct equation of motion for superflows at low temperatures. This equation depends on two parameters: the sound velocity and the coherence length. It naturally contains nonsingular quantized vortex lines. The NLSE mass, momentum, and energy conservation relations are derived in hydrodynamic form. The total energy is decomposed into an incompressible kinetic part, and other parts that correspond to acoustic excitations. The corresponding energy spectra are defined and computed numerically in the case of the two-dimensional vortex solution. A preparation method, generating initial data reproducing the vorticity dynamics of any three-dimensional flow with Clebsch representation is given and is applied to the Taylor–Green (TG) vortex. The NLSE TG vortex is studied with resolutions up to 5123. The energetics of the flow is found to be remarkably similar to that of the viscous TG vortex. The...

The probabilistic reformulation of the multifractal model iii is obtained directly from the structure functions written as integrals over cumulative distribution functions (c.d.f.) by the steepest descent method. The saddle point being a function of scale, we perform a change of variable to obtain expressions that are asymptotically valid in the inertial range. Starting directly from the inertial range behavior of the c-d-f-, our algorithm yields values for the scaling exponents and codimension that are identical to those obtained from structure functions. Furthermore, a simple interpretation of multifractality in terms of global c-d-f- scaling is shown to collapse the inertial range c-d-f- into a single curve, directly related to the codimension. Our method determines a new length scale, larger than the integral scale, that gives a quantitative measure of the degree of multifractality of the data. Finally, some possible future applications are mentioned.

We undertake a systematic, direct numerical simulation (DNS) of the two-dimensional, Fourier-truncated, Gross-Pitaevskii equation to study the turbulent evolutions of its solutions for a variety of initial conditions and a wide range of parameters. We find that the time evolution of this system can be classified into four regimes with qualitatively different statistical properties. First, there are transients that depend on the initial conditions. In the second regime, power-law scaling regions, in the energy and the occupation-number spectra, appear and start to develop; the exponents of these power-laws and the extents of the scaling regions change with time and depended on the initial condition. In the third regime, the spectra drop rapidly for modes with wave numbers $k > k_c$ and partial thermalization takes place for modes with $k < k_c$; the self-truncation wave number $k_c(t)$ depends on the initial conditions and it grows either as a power of $t$ or as $\log t$. Finally, in the fourth regime, complete-thermalization is achieved and, if we account for finite-size effects carefully, correlation functions and spectra are consistent with their nontrivial Berezinskii-Kosterlitz-Thouless forms., 30 pages, 12 figures

We investigate the ideal and incompressible magnetohydrodynamic (MHD) equations in three space dimensions for the development of potentially singular structures. The methodology consists in implementing the fourfold symmetries of the Taylor-Green vortex generalized to MHD, leading to substantial computer time and memory savings at a given resolution; we also use a regridding method that allows for lower-resolution runs at early times, with no loss of spectral accuracy. One magnetic configuration is examined at an equivalent resolution of 6144 3 points and three different configurations on grids of 4096 3 points. At the highest resolution, two different current and vorticity sheet systems are found to collide, producing two successive accelerations in the development of small scales. At the latest time, a convergence of magnetic field lines to the location of maximum current is probably leading locally to a strong bending and directional variability of such lines. A novel analytical method, based on sharp analysis inequalities, is used to assess the validity of the finite-time singularity scenario. This method allows one to rule out spurious singularities by evaluating the rate at which the logarithmic decrement of the analyticity-strip method goes to zero. The result is that the finite-time singularity scenario cannot be ruled out, and the singularity time could be somewhere between t = 2.33 and t = 2.70 . More robust conclusions will require higher resolution runs and grid-point interpolation measurements of maximum current and vorticity. Fil: Brachet, M. E.. Centre National de la Recherche Scientifique. Ecole Normale Superieur; Francia Fil: Bustamante, M. D.. University College Dublin; Irlanda Fil: Krstulovic, G.. Observatoire de la Cote D; Francia Fil: Mininni, Pablo Daniel. National Center for Atmospheric Research; Estados Unidos. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina Fil: Pouquet, A.. National Center for Atmospheric Research; Estados Unidos Fil: Rosenberg, D.. National Center for Atmospheric Research; Estados Unidos

The description in terms of hydrodynamical variables of a relativistic superfluid, modeled by a semiclassical wave equation, is given using a generalized Madelung transformation. The Galilean limit is shown (for both wavefunction and fluid variables) to be the well known Landau-Pitaevski model of superflows at T = 0 K. The special relativistic elementary classical acoustic and vortex excitations are explicited. A model for a relativistic self-gravitating superfluid is obtained by minimally coupling the wave equation to Einstein's gravity. The equations corresponding to a static star (using fluid variables) and an isotropic cosmology are derived.

We present a numerical study of the magnetic field generated by the Taylor-Green vortex. We show that periodic boundary conditions can be used to mimic realistic boundary conditions by prescribing the symmetries of the velocity and magnetic fields. This gives insight into some problems of central interest for dynamos: the possible effect of velocity fluctuations on the dynamo threshold, and the role of boundary conditions on the threshold and on the geometry of the magnetic field generated by dynamo action. In particular, we show that an axial dipolar dynamo similar to the one observed in a recent experiment can be obtained with an appropriate choice of the symmetries of the magnetic field. The nonlinear saturation is studied and a simple model explaining the magnetic Prandtl number dependence of the super- and subcritical nature of the dynamo transition is given.

A new mechanism of thermalization involving a direct energy cascade is obtained in the truncated Gross-Pitaevskii dynamics. A long transient with partial thermalization at small scales is observed before the system reaches equilibrium. Vortices are found to disappear as a prelude to final thermalization. A bottleneck that produces spontaneous effective self-truncation and delays thermalization is characterized when large dispersive effects are present at the truncation wave number. Order of magnitude estimates indicate that self-truncation takes place in turbulent Bose-Einstein condensates. This effect should also be present in classical hydrodynamics and models of turbulence.

Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to 40963. The results are analyzed in terms of the classical analyticity-strip method and Beale, Kato, and Majda (BKM) theorem. A well-resolved acceleration of the time decay of the width of the analyticity strip δ(t) is observed at the highest resolution for 3.7

A Comment on the Letter by Jeffrey Yepez et al., Phys. Rev. Lett. 103, 084501 (2009). The authors of the Letter offer a Reply.

The statistical equilibria of the (conservative) dynamics of the Gross-Pitaevskii equation (GPE) with a finite range of spatial Fourier modes are characterized using a new algorithm, based on a stochastically forced Ginzburg-Landau equation (SGLE), that directly generates grand-canonical distributions. The SGLE-generated distributions are validated against finite-temperature GPE-thermalized states and exact (low-temperature) results obtained by steepest descent on the (grand-canonical) partition function. A standard finite-temperature second-order λ transition is exhibited. A mechanism of GPE thermalization through a direct cascade of energy is found using initial conditions with mass and energy distributed at large scales. A long transient with partial thermalization at small scales is observed before the system reaches equilibrium. Vortices are shown to disappear as a prelude to final thermalization and their annihilation is related to the contraction of vortex rings due to mutual friction. Increasing the amount of dispersion at the truncation wave number is shown to slow thermalization and vortex annihilation. A bottleneck that produces spontaneous effective self-truncation with partial thermalization is characterized in the limit of large dispersive effects. Metastable counterflow states, with nonzero values of momentum, are generated using the SGLE algorithm. Spontaneous nucleation of the vortex ring is observed and the corresponding Arrhenius law is characterized. Dynamical counterflow effects on vortex evolution are investigated using two exact solutions of the GPE: traveling vortex rings and a motionless crystal-like lattice of vortex lines. Longitudinal effects are produced and measured on the crystal lattice. A dilatation of vortex rings is obtained for counterflows larger than their translational velocity. The vortex ring translational velocity has a dependence on temperature that is an order of magnitude above that of the crystal lattice, an effect that is related to the presence of finite-amplitude Kelvin waves. This anomalous vortex ring velocity is quantitatively reproduced by assuming equipartition of energy of the Kelvin waves. Orders of magnitude are given for the predicted effects in weakly interacting Bose-Einstein condensates and superfluid ^{4}He.

Using computations of three-dimensional magnetohydrodynamic (MHD) turbulence with a Taylor-Green flow, whose inherent time-independent symmetries are implemented numerically, and in the absence of either a forcing function or an imposed uniform magnetic field, we show that three different inertial ranges for the energy spectrum may emerge for three different initial magnetic fields, the selecting parameter being the ratio of the Alfven to the eddy turnover time. Equivalent computational grids range from 128^3 to 2048^3 points with a unit magnetic Prandtl number and a Taylor Reynolds number of up to 1500 at the peak of dissipation. We also show convergence of our results with Reynolds number. Our study is consistent with previous findings of a variety of energy spectra in MHD turbulence by studies performed in the presence of both a forcing term with a given correlation time and a strong, uniform magnetic field. In contrast to the previous studies, however, the ratio of characteristic time scales here can only be ascribed to the intrinsic nonlinear dynamics of the flows under study., 11 pages, 8 figs, 2 tables

Direct numerical simulations of the three‐dimensional Euler equations at resolutions up to 2563 for general periodic flows and 8643 for the symmetric Taylor–Green vortex are presented. The spontaneous emergence of flat pancakelike structures that shrink exponentially in time is observed. A simple self‐similar model that fits these observations is discussed. Focusing instabilities similar to those leading to streamwise vortices in the context of free shear layers [J. Fluid Mech. 143, 253 (1984)], are expected to subsequently concentrate the vorticity and produce isolated vortex filaments. A finite time singularity for the Euler equation is not excluded as the result of interactions among these filaments.

The dynamics of the truncated Euler equations with helical initial conditions are studied. Transient energy and helicity cascades leading to Kraichnan helical absolute equilibrium at small scales, including a linear scaling of the relative helicity spectrum are obtained. Strong helicity effects are found using initial data concentrated at high wave numbers. Using low-wave-number initial conditions, the results of Cichowlas [Phys. Rev. Lett. 95, 264502 (2005)] are extended to helical flows. Similarities between the turbulent transient evolution of the ideal (time-reversible) system and viscous helical flows are found. Using an argument in the manner of Frisch [Phys. Rev. Lett. 101, 144501 (2008)], the excess of relative helicity found at small scales in the viscous run is related to the thermalization of the ideal flow. The observed differences in the behavior of truncated Euler and (constant viscosity) Navier-Stokes are qualitatively understood using the concept of eddy viscosity. The large scales of truncated Euler equations are then shown to follow quantitatively an effective Navier-Stokes dynamics based on a variable (scale dependent) eddy viscosity. © 2009 The American Physical Society. Fil: Krstulovic, G.. Laboratoire de Physique Statistique; Francia Fil: Mininni, Pablo Daniel. Universidad de Buenos Aires; Argentina. National Center for Atmospheric Research; Estados Unidos. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentina Fil: Brachet, M. E.. National Center for Atmospheric Research; Estados Unidos Fil: Pouquet, A.. National Center for Atmospheric Research; Estados Unidos

We study decaying magnetohydrodynamics (MHD) turbulence stemming from the evolution of the Taylor-Green (TG) flow generalized recently to MHD, with equal viscosity and magnetic resistivity and up to equivalent grid resolutions of 2048^3 points. A pseudo-spectral code is used in which the symmetries of the velocity and magnetic fields have been implemented, allowing for sizable savings in both computer time and usage of memory at a given Reynolds number. The flow is non-helical, and at initial time the kinetic and magnetic energies are taken to be equal and concentrated in the large scales. After testing the validity of the method on grids of 512^3 points, we analyze the data on the large grids up to Taylor Reynolds numbers of ~2200. We find that the global temporal evolution is accelerated in MHD, compared to the corresponding neutral fluid case. We also observe an interval of time when such configurations have quasi-constant total dissipation, time during which statistical properties are determined after averaging over of the order of two turn-over times. A weak turbulence spectrum obtains which is also given in terms of its anisotropic components. Finally, we contrast the development of small-scale eddies with two other initial conditions for the magnetic field and briefly discuss the structures that develop, and which display a complex array of current and vorticity sheets with clear rolling-up and folding., Comment: 16 pages, 10 figures. Submitted to GAFD

The incompressible Navier-Stokes equations are numerically integrated on a Cray-2 machine with the periodic Taylor–Green initial data. Using a spectral method taking advantage of the symmetries of the flow, a resolution of 8643 and corresponding high Reynolds numbers (Rλ = 140) are obtained Visualisations of the resulting turbulent flow show that the turbulent activity is strongly correlated with low-pressure zones. We demonstrate that this behavior of the pressure field is linked to the fact that the square vorticity is spatially more concentrated than the energy dissipation.

Sound emission produced by the interaction of several vortices in a two-dimensional homogeneous system obeying the nonlinear Schrödinger (NLS) equation is considered. The radiation effect is explicitly computed in terms of assumed vortex motion. The results are applied to a simple test case of two corotating vortices. The prediction is compared to the result of numerical simulations of the NLS equation. The numerical data give support to the estimate of radiation.

Since the first calculations of Lilly I ,it has been recognized that high resolutions are required to properly simulate the small scale inertial range of incompressible two-dimensional Navier-Stokes turbulence 2 . Preliminary calculations at a 5122 resolution presented by Orszag 3 showed that when the large scale Reynolds numDer is increased from ll00 to 25000, a distinct change is observed from a k -4 energy spectrum predicted by Saffman 4 to a spectrum -3 roughly proportional to k which is expected if an enstrophy cascade develops 5"6'7. Spectral simulations whith 10242 collocation points were more recently reported 8'9. In these calculations, aliasing was removed byspectral truncation at a wavenumber k M = 341. To achieve this resolution on a i megaword Cray-i computer, the "sparse mode technique" was implemented, i.e. the stream function has a Fourier representation

We propose two sets of initial conditions for magnetohydrodynamics (MHD) in which both the velocity and the magnetic fields have spatial symmetries that are preserved by the dynamical equations as the system evolves. When implemented numerically they allow for substantial savings in CPU time and memory storage requirements for a given resolved scale separation. Basic properties of these Taylor-Green flows generalized to MHD are given, and the ideal non-dissipative case is studied up to the equivalent of 2048^3 grid points for one of these flows. The temporal evolution of the logarithmic decrements, delta, of the energy spectrum remains exponential at the highest spatial resolution considered, for which an acceleration is observed briefly before the grid resolution is reached. Up to the end of the exponential decay of delta, the behavior is consistent with a regular flow with no appearance of a singularity. The subsequent short acceleration in the formation of small magnetic scales can be associated with a near collision of two current sheets driven together by magnetic pressure. It leads to strong gradients with a fast rotation of the direction of the magnetic field, a feature also observed in the solar wind., 8 pages, 4 figures

The analyticity strip method is used to trace complex singularities in direct numer- ical simulations of the Taylor-Green flows, performed with up to 2048 3 colloca- tion points. No indication of finite-time real singularity is found. Simulations are also carried out beyond the time at which the truncated equations cease to approx- imate the original Euler equations. Kolmogorov-like turbulence is then obtained during an intermediate regime of the spontaneous relaxation of (time-reversible) spectrally-truncated Euler equations towards absolute equilibrium.