An inverse turbulent cascade in a restricted two-dimensional periodic domain creates a condensate-a pair of coherent system-size vortices. We perform extensive numerical simulations of this system and carry out theoretical analysis based on momentum and energy exchanges between the turbulence and the vortices. We show that the vortices have a universal internal structure independent of the type of small-scale dissipation, small-scale forcing, and boundary conditions. The theory predicts not only the vortex inner region profile, but also the amplitude, which both perfectly agree with the numerical data.

Measurements of the energy spectrum and of the vortex-density fluctuation spectrum in superfluid turbulence seem to contradict each other. Using a numerical model, we show that at each instance of time the total vortex line density can be decomposed into two parts: one formed by metastable bundles of coherent vortices, and one in which the vortices are randomly oriented. We show that the former is responsible for the observed Kolmogorov energy spectrum, and the latter for the spectrum of the vortex line density fluctuations. PACS numbers: 67.25.dk, 47.32.C, 47.27.Gs Below a critical temperature, liquid helium becomes a two-fluid system in which an inviscid superfluid component coexists with a viscous normal fluid component. The flow of the superfluid is irrotational: superfluid vorticity is confined to vortex lines of atomic thickness around which the circulation takes a fixed value κ (the quantum of circulation). Superfluid turbulence [1, 2] is easily created by stirring either helium isotope ( 4 He or 3 He-B), and consists of a tangle of reconnecting vortex filaments which interact with each other and with the viscous normal fluid (which may be laminar or turbulent). The most important observable quantity is the vortex line density L (vortex length per unit volume), from which one infers the average distance between vortex lines, ≈ L −1/2 . Our interest is in the properties of superfluid turbulence and their similarities with ordinary turbulence.Experiments [3,4] have revealed that, if the superfluid turbulence is driven by grids or propellers, the distribution of the turbulent kinetic energy over length scales larger than obeys the celebrated k −5/3 Kolmogorov scaling observed in ordinary (classical) turbulence. Here k is the magnitude of the three-dimensional wavenumber (wavenumber and frequency are related by k = f /v, wherev is the mean flow). Numerical calculations performed using either the vortex filament model [5,6] or the Gross-Pitaevskii equation [7,8] confirm the Kolmogorov scaling. It is thought that the effect arises from the partial polarization of the vortex lines [1, 2, 9], but such effect has never been clearly identified. Another important experimental observation is that in both 4 He [10] and 3 He-B [11], the frequency spectrum of the fluctuations of L has a decreasing f −5/3 scaling typical of passive objects [6,12] advected by a turbulent flow. This latter result seems to contradict the interpretation of L as a measure of superfluid vorticity, ω = κL which is usually made in the literature [1,2,11,[13][14][15].In fact, from dimensional analysis, the vorticity spec- * andrew.baggaley@gla.ac.uk trum corresponding to the Kolmogorov law should increase with f (as f 1/3 ), not decrease. Since the vortex line density is a positive quantity, a better analogy is to the enstrophy spectrum: however in classical turbulence this spectrum is essentially flat [16,17], in disagreement with the helium experiments [10,11].The aim of this letter is to reconcile these two sets of observations (each separa...

We argue that the physics of interacting Kelvin Waves ͑KWs͒ is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear ͑partial differential͒ equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation ͑in which the energy exchange between KWs is absent͒, describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum. DOI: 10.1103/PhysRevB.81.104526 PACS number͑s͒: 67.25.dk, 47.37.ϩq, 45.10.Hj, 47.10.Df I. PHYSICAL BACKGROUND, METHODOLOGY, AND OVERVIEW OF RESULTS A. Kelvin waves (KWs) in superfluid turbulenceThe role of KWs in the dissipation of energy in zerotemperature quantum turbulence has long been discussed within the quantum-turbulence community. It is widely believed that KWs extend the transfer of a constant-energy flux from the fully three-dimensional Kolmogorov-type turbulence at large scales, through a crossover mechanism at scales comparable to the intervortex distance, to smaller scales via a local KW cascade on quantized vortices. Much theoretical work has been done recently, including the conjecture of a power-law scaling for the KW energy spectrum made by Kozik and Svistunov 1 in 2004 ͑hereafter the KS spectrum͒.Nevertheless there remain important unanswered questions in quantum turbulence: ͑a͒ what are the relative roles of KWs and the other processes, e.g., vortex reconnections, in the transfer of energy to small scales? ͑b͒ What are the dominant physical mechanisms in the classical-quantum crossover range? Two alternative scenarios were put forward for this range: first, one relying on the idea that the polarization of vortex tangles suppress vortex reconnections, which lead to a bottleneck hump, 2,3 and the second, implies that reconnections play an active role in removing the bottleneck. 4 ͑c͒ If the KWs do play a key role at small scales, what kind of interaction processes are important for the transfer of energy toward smaller scales? Is it the resonant wave-wave interactions or a linear process of wave number evolution due to a large-scale curvature and/or slow time dependence of the under...

We present a review of the latest developments in one-dimensional (1D) optical wave turbulence (OWT). Based on an original experimental setup that allows for the implementation of 1D OWT, we are able to show that an inverse cascade occurs through the spontaneous evolution of the nonlinear field up to the point when modulational instability leads to soliton formation. After solitons are formed, further interaction of the solitons among themselves and with incoherent waves leads to a final condensate state dominated by a single strong soliton. Motivated by the observations, we develop a theoretical description, showing that the inverse cascade develops through six-wave interaction, and that this is the basic mechanism of nonlinear wave coupling for 1D OWT. We describe theory, numerics and experimental observations while trying to incorporate all the different aspects into a consistent context. The experimental system is described by two coupled nonlinear equations, which we explore within two wave limits allowing for the expression of the evolution of the complex amplitude in a single dynamical equation. The long-wave limit corresponds to waves with wave numbers smaller than the electrical coherence length of the liquid crystal, and the opposite limit, when wave numbers are larger. We show that both of these systems are of a dual cascade type, analogous to two-dimensional (2D) turbulence, which can be described by wave turbulence (WT) theory, and conclude that the cascades are induced by a six-wave resonant interaction process. WT predicts several stationary solutions (non-equilibrium and thermodynamic) to both the long-and short-wave systems, and we investigate the necessary conditions required for their realization. Interestingly, the long-wave system is close to the integrable 1D nonlinear Schrödinger equation (NLSE) (which contains exact nonlinear soliton solutions), and as a result during the inverse cascade, nonlinearity of the system at low wave numbers becomes strong. Subsequently, due to the focusing nature of the nonlinearity, this leads to modulational instability (MI) of the condensate and the formation of solitons. Finally, with the aid of the the probability density function (PDF) description of WT theory, we explain the coexistence and mutual interactions between solitons and the weakly nonlinear random wave background in the form of a wave turbulence life cycle (WTLC).