Direct and Inverse Cascades in Turbulent Bose-Einstein Condensates.

When a Bose-Einstein condensate (BEC) is driven out of equilibrium, density waves interact nonlinearly and trigger turbulent cascades. In a turbulent BEC, energy is transferred toward small scales by a direct cascade, whereas the number of particles displays an inverse cascade toward large scales. In this work, we study analytically and numerically the direct and inverse cascades in wave-turbulent BECs. We analytically derive the Kolmogorov-Zakharov spectra, including the log correction to the direct cascade scaling and the universal prefactor constants for both cascades. We test and corroborate our predictions using high-resolution numerical simulations of the forced-dissipated Gross-Pitaevskii model in a periodic box and the corresponding wave-kinetic equation. Theoretical predictions and data are in excellent agreement, without adjustable parameters. Moreover, in order to connect with experiments, we test and validate our theoretical predictions using the Gross-Pitaevskii model with a confining cubic trap. Our results explain previous experimental observations and suggest new settings for future studies.

Hamiltonian derivation of the point vortex model from the two-dimensional nonlinear Schrödinger equation.

We present a rigorous derivation of the point vortex model starting from the two-dimensional nonlinear Schrödinger equation, from the Hamiltonian perspective, in the limit of well-separated, subsonic vortices on the background of a spatially infinite strong condensate. As a corollary, we calculate to high accuracy the self-energy of an isolated elementary Pitaevskii vortex.

Self-similar evolution of wave turbulence in Gross-Pitaevskii system.

We study the universal nonstationary evolution of wave turbulence (WT) in Bose-Einstein condensates (BECs). Their temporal evolution can exhibit different kinds of self-similar behavior corresponding to a large-time asymptotic of the system or to a finite-time blowup. We identify self-similar regimes in BECs by numerically simulating the forced and unforced Gross-Pitaevskii equation (GPE) and the associated wave kinetic equation (WKE) for the direct and inverse cascades, respectively. In both the GPE and the WKE simulations for the direct cascade, we observe the first-kind self-similarity that is fully determined by energy conservation. For the inverse cascade evolution, we verify the existence of a self-similar evolution of the second kind describing self-accelerating dynamics of the spectrum leading to blowup at the zero mode (condensate) at a finite time. We believe that the universal self-similar spectra found in the present paper are as important and relevant for understanding the BEC turbulence in past and future experiments as the commonly studied stationary Kolmogorov-Zakharov (KZ) spectra.

Testing wave turbulence theory for the Gross-Pitaevskii system.

We test the predictions of the theory of weak wave turbulence by performing numerical simulations of the Gross-Pitaevskii equation (GPE) and the associated wave-kinetic equation (WKE). We consider an initial state localized in Fourier space, and we confront the solutions of the WKE obtained numerically with GPE data for both the wave-action spectrum and the probability density functions (PDFs) of the Fourier mode intensities. We find that the temporal evolution of the GPE data is accurately predicted by the WKE, with no adjustable parameters, for about two nonlinear kinetic times. Qualitative agreement between the GPE and the WKE persists also for longer times with some quantitative deviations that may be attributed to the combination of a breakdown of the theoretical assumptions underlying the WKE as well as numerical issues. Furthermore, we study how the wave statistics evolves toward Gaussianity in a timescale of the order of the kinetic time. The excellent agreement between direct numerical simulations of the GPE and the WKE provides a solid foundation to the theory of weak wave turbulence.

Energy Spectrum of Two-Dimensional Acoustic Turbulence.

We report an exact unique constant-flux power-law analytical solution of the wave kinetic equation for the turbulent energy spectrum, E(k)=C_{1}sqrt[ϵac_{s}]/k, of acoustic waves in 2D with almost linear dispersion law, ω_{k}=c_{s}k[1+(ak)^{2}], ak≪1. Here, ϵ is the energy flux over scales, and C_{1} is the universal constant which was found analytically. Our theory describes, for example, acoustic turbulence in 2D Bose-Einstein condensates. The corresponding 3D counterpart of turbulent acoustic spectrum was found over half a century ago, however, due to the singularity in 2D, no solution has been obtained until now. We show the spectrum E(k) is realizable in direct numerical simulations of forced-dissipated Gross-Pitaevskii equation in the presence of strong condensate.

Comment on "Theoretical analysis of quantum turbulence using the Onsager ideal turbulence theory".

In a recent paper, Tanogami [Phys. Rev. E 103, 023106 (2021)2470-004510.1103/PhysRevE.103.023106] proposes a scenario for quantum turbulence where the energy spectrum at scales smaller than the intervortex distance is dominated by a quantum stress cascade, in opposition to Kelvin-wave cascade predictions. The purpose of the present Comment is to highlight some physical issues in the derivation of the quantum stress cascade, in particular to stress that quantization of circulation has been ignored.

Theory of anisotropic superfluid 4He counterflow turbulence.

We develop a theory of strong anisotropy of the energy spectra in the thermally driven turbulent counterflow of superfluid 4He. The key ingredients of the theory are the three-dimensional differential closure for the vector of the energy flux and the anisotropy of the mutual friction force. We suggest an approximate analytic solution of the resulting energy-rate equation, which is fully supported by our numerical solution. The two-dimensional energy spectrum is strongly confined in the direction of the counterflow velocity. In agreement with the experiments, the energy spectra in the direction orthogonal to the counterflow exhibit two scaling ranges: a near-classical non-universal cascade dominated range and a universal critical regime at large wavenumbers. The theory predicts the dependence of various details of the spectra and the transition to the universal critical regime on the flow parameters. This article is part of the theme issue 'Scaling the turbulence edifice (part 2)'.

Direct Evidence of a Dual Cascade in Gravitational Wave Turbulence.

We present the first direct numerical simulation of gravitational wave turbulence. General relativity equations are solved numerically in a periodic box with a diagonal metric tensor depending on two space coordinates only, g_{ij}≡g_{ii}(x,y,t)δ_{ij}, and with an additional small-scale dissipative term. We limit ourselves to weak gravitational waves and to a freely decaying turbulence. We find that an initial metric excitation at intermediate wave number leads to a dual cascade of energy and wave action. When the direct energy cascade reaches the dissipative scales, a transition is observed in the temporal evolution of energy from a plateau to a power-law decay, while the inverse cascade front continues to propagate toward low wave numbers. The wave number and frequency-wave-number spectra are found to be compatible with the theory of weak wave turbulence and the characteristic timescale of the dual cascade is that expected for four-wave resonant interactions. The simulation reveals that an initially weak gravitational wave turbulence tends to become strong as the inverse cascade of wave action progresses with a selective amplification of the fluctuations g_{11} and g_{22}.

Phase transition in time-reversible Navier-Stokes equations.

We present a comprehensive study of the statistical features of a three-dimensional (3D) time-reversible truncated Navier-Stokes (RNS) system, wherein the standard viscosity ν is replaced by a fluctuating thermostat that dynamically compensates for fluctuations in the total energy. We analyze the statistical features of the RNS steady states in terms of a non-negative dimensionless control parameter R_{r}, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale ℓ_{f} and the total energy E_{0}. For small R_{r}, the RNS equations are found to produce "warm" stationary statistics, e.g., characterized by the partial thermalization of the small scales. For large R_{r}, the stationary solutions have features akin to standard hydrodynamic ones: they have compact energy support in k space and are essentially insensitive to the truncation scale k_{max}. The transition between the two statistical regimes is observed to be smooth but rather sharp. Using insights from a diffusion model of turbulence (Leith model), we argue that the transition is in fact akin to a continuous second-order phase transition, where R_{r} indeed behaves as a thermodynamic control parameter, e.g., a temperature. A relevant order parameter can be suitably defined in terms of a (normalized) enstrophy, while the symmetry-breaking parameter h is identified as (one over) the truncation scale k_{max}. We find that the signatures of the phase transition close to the critical point R_{r}^{★} can essentially be deduced from a heuristic mean-field Landau free energy. This point of view allows us to reinterpret the relevant asymptotics in which the dynamical ensemble equivalence conjectured by Gallavotti [Phys. Lett. A 223, 91 (1996)PYLAAG0375-960110.1016/S0375-9601(96)00729-3] could hold true. We argue that Gallavotti's limit is precisely the joint limit R_{r}→[over >]R_{r}^{★} and h→[over >]0, with the overset symbol ">" indicating that those limits are approached from above. The limit therefore relates to the statistical features at the critical point. In this regime, our numerics indicate that the low-order statistics of the 3D RNS are indeed qualitatively similar to those observed in direct numerical simulations of the standard Navier-Stokes equations with viscosity chosen so as to match the average value of the reversible thermostat. This result suggests that Gallavotti's equivalence conjecture could indeed be of relevance to model 3D turbulent statistics, and provides a clear guideline for further numerical investigations involving higher resolutions.

Turbulence of Weak Gravitational Waves in the Early Universe.

We study the statistical properties of an ensemble of weak gravitational waves interacting nonlinearly in a flat space-time. We show that the resonant three-wave interactions are absent and develop a theory for four-wave interactions in the reduced case of a 2.5+1 diagonal metric tensor. In this limit, where only plus-polarized gravitational waves are present, we derive the interaction Hamiltonian and consider the asymptotic regime of weak gravitational wave turbulence. Both direct and inverse cascades are found for the energy and the wave action, respectively, and the corresponding wave spectra are derived. The inverse cascade is characterized by a finite-time propagation of the metric excitations-a process similar to an explosive nonequilibrium Bose-Einstein condensation, which provides an efficient mechanism to ironing out small-scale inhomogeneities. The direct cascade leads to an accumulation of the radiation energy in the system. These processes might be important for understanding the early Universe where a background of weak nonlinear gravitational waves is expected.

Derivation of the Biot-Savart equation from the nonlinear Schrödinger equation.

We present a systematic derivation of the Biot-Savart equation from the nonlinear Schrödinger equation, in the limit when the curvature radius of vortex lines and the intervortex distance are much greater than the vortex healing length, or core radius. We derive the Biot-Savart equations in Hamiltonian form with Hamiltonian expressed in terms of vortex lines,H=κ(2)/8π∫(|s-s'|>ξ(*))(ds·ds')/|s-s'|,with cutoff length ξ(*)≈0.3416293/√(ρ(0)), where ρ(0) is the background condensate density far from the vortex lines and κ is the quantum of circulation.

Wave-turbulence description of interacting particles: Klein-Gordon model with a Mexican-hat potential.

In field theory, particles are waves or excitations that propagate on the fundamental state. In experiments or cosmological models, one typically wants to compute the out-of-equilibrium evolution of a given initial distribution of such waves. Wave turbulence deals with out-of-equilibrium ensembles of weakly nonlinear waves, and is therefore well suited to address this problem. As an example, we consider the complex Klein-Gordon equation with a Mexican-hat potential. This simple equation displays two kinds of excitations around the fundamental state: massive particles and massless Goldstone bosons. The former are waves with a nonzero frequency for vanishing wave number, whereas the latter obey an acoustic dispersion relation. Using wave-turbulence theory, we derive wave kinetic equations that govern the coupled evolution of the spectra of massive and massless waves. We first consider the thermodynamic solutions to these equations and study the wave condensation transition, which is the classical equivalent of Bose-Einstein condensation. We then focus on nonlocal interactions in wave-number space: we study the decay of an ensemble of massive particles into massless ones. Under rather general conditions, these massless particles accumulate at low wave number. We study the dynamics of waves coexisting with such a strong condensate, and we compute rigorously a nonlocal Kolmogorov-Zakharov solution, where particles are transferred nonlocally to the condensate, while energy cascades towards large wave numbers through local interactions. This nonlocal cascading state constitutes the intermediate asymptotics between the initial distribution of waves and the thermodynamic state reached in the long-time limit.

Wave turbulence in quantum fluids.

Wave turbulence (WT) occurs in systems of strongly interacting nonlinear waves and can lead to energy flows across length and frequency scales much like those that are well known in vortex turbulence. Typically, the energy passes although a nondissipative inertial range until it reaches a small enough scale that viscosity becomes important and terminates the cascade by dissipating the energy as heat. Wave turbulence in quantum fluids is of particular interest, partly because revealing experiments can be performed on a laboratory scale, and partly because WT among the Kelvin waves on quantized vortices is believed to play a crucial role in the final stages of the decay of (vortex) quantum turbulence. In this short review, we provide a perspective on recent work on WT in quantum fluids, setting it in context and discussing the outlook for the next few years. We outline the theory, review briefly the experiments carried out to date using liquid H2 and liquid (4)He, and discuss some nonequilibrium excitonic superfluids in which WT has been predicted but not yet observed experimentally. By way of conclusion, we consider the medium- and longer-term outlook for the field.

Weak Alfvén-wave turbulence revisited.

Weak Alfvénic turbulence in a periodic domain is considered as a mixed state of Alfvén waves interacting with the two-dimensional (2D) condensate. Unlike in standard treatments, no spectral continuity between the two is assumed, and, indeed, none is found. If the 2D modes are not directly forced, k(-2) and k(-1) spectra are found for the Alfvén waves and the 2D modes, respectively, with the latter less energetic than the former. The wave number at which their energies become comparable marks the transition to strong turbulence. For imbalanced energy injection, the spectra are similar, and the Elsasser ratio scales as the ratio of the energy fluxes in the counterpropagating Alfvén waves. If the 2D modes are forced, a 2D inverse cascade dominates the dynamics at the largest scales, but at small enough scales, the same weak and then strong regimes as described above are achieved.

Triple cascade behavior in quasigeostrophic and drift turbulence and generation of zonal jets.

We study quasigeostrophic (QG) and plasma drift turbulence within the Charney-Hasegawa-Mima (CHM) model. We focus on the zonostrophy, an extra invariant in the CHM model, and on its role in the formation of zonal jets. We use a generalized Fjørtoft argument for the energy, enstrophy, and zonostrophy and show that they cascade anisotropically into nonintersecting sectors in k space with the energy cascading towards large zonal scales. Using direct numerical simulations of the CHM equation, we show that zonostrophy is well conserved, and the three invariants cascade as predicted by the Fjørtoft argument.

Gravity wave turbulence in wave tanks: space and time statistics.

We present the first simultaneous space-time measurements for gravity wave turbulence in a large laboratory flume. We found that the slopes of k and omega wave spectra depend on wave intensity. This cannot be explained by any existing theory considering wave turbulence as the result of either breaking events or weakly nonlinear wave interactions. Instead, we show that random waves and breaking or coherent structures appear to coexist: The former show themselves in a quasi-Gaussian core of the probability density function and in the low-order structure functions, and the latter in the probability density function tails and the high-order structure functions.

Resonant interactions of nonlinear water waves in a finite basin.

We study exact four-wave resonances among gravity water waves in a square box with periodic boundary conditions. We show that these resonant quartets are linked with each other by shared Fourier modes in such a way that they form independent clusters. These clusters can be formed by two types of quartets: (1) Angle resonances which cannot directly cascade energy but which can redistribute it among the initially excited modes and (2) scale resonances which are much more rare but which are the only ones that can transfer energy between different scales. We find such resonant quartets and their clusters numerically on the set of 1000x1000 modes, classify and quantify them and discuss consequences of the obtained cluster structure for the wave-field evolution. Finite box effects and associated resonant interaction among discrete wave modes appear to be important in most numerical and laboratory experiments on the deep water gravity waves, and our work is aimed at aiding the interpretation of the experimental and numerical data.

Gravity wave turbulence in a laboratory flume.

We present an experimental study of the statistics of surface gravity wave turbulence in a flume of a horizontal size 12 x 6 m. For a wide range of amplitudes the wave energy spectrum was found to scale as Eomega approximately omega(-nu) in a frequency range of up to one decade. However, nu appears to be nonuniversal: it depends on the wave intensity and ranges from about 6 to 4. We discuss our results in the context of existing theories and argue that at low wave amplitudes the wave statistics is affected by the flume finite size, and at high amplitudes the wave breaking effect dominates.

A mathematical model for evaluation of maternal cell contamination in cultured cells from spontaneous abortions: significance for cytogenetic analysis of prenatal selection factors.

OBJECTIVE: To develop a mathematical model for more precise estimation of the incidence of chromosomal abnormalities and the sex ratio among spontaneous abortions masked by maternal cell contamination. DESIGN: Retrospective analysis. SETTING: Academic medical center. PATIENT(S): One hundred twelve samples of spontaneous abortion with a "46,XX" karyotype and 97 parents with aborted embryos. INTERVENTION(S): The presence of Y chromosome DNA in native tissues of "46,XX" spontaneous abortions was detected by amelogenin locus analysis. Detection of aneuploidies in noncultured tissues of "46,XX" abortions was performed by microsatellite DNA analysis and confirmed by fluorescence in situ hybridization. MAIN OUTCOME MEASURE(S): Accuracy of cytogenetic evaluation of spontaneous abortions. RESULT(S): Y chromosome DNA was revealed in 16% of the embryos with a "46,XX" karyotype. According to the mathematical model proposed, the frequency of chromosomal abnormalities in a sample of 478 abortions increased from 54.6% to 60.3%, and the sex ratio in embryos with normal karyotype changed from 0.66 to 1.02. The experimental validation of the model has shown that the observed and expected incidences of chromosomal abnormalities in "46,XX" abortions were in good agreement. CONCLUSION(S): Maternal cell contamination clearly affects the incidence of registered chromosomal abnormalities and the sex ratio in spontaneous abortions. Correction for maternal cell contamination should be taken into account before invoking biological explanations of sex ratio bias and might be useful to include in diagnostic reporting.