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Sergey Nazarenko

Publications and source records attributed to Sergey Nazarenko.

At least 73 records · Page 4Linked to original sources
arXiv2009

Energy cascades and spectra in turbulent Bose-Einstein condensates

We present a numerical study of turbulence in Bose-Einstein condensates within the 3D Gross-Pitaevskii equation. We concentrate on the direct energy cascade in forced-dissipated systems. We show that behavior of the system is very sensitive to the properties of the model at the scales greater than the forcing scale, and we identify three different universal regimes: (1) a non-stationary regime with condensation and transition from a four-wave to a three-wave interaction process when the largest scales are not dissipated, (2) a steady weak wave turbulence regime when largest scales are dissipated with a friction-type dissipation, (3) a state with a scale-by-scale balance of the linear and the nonlinear timescales when the large-scale dissipation is a hypo-viscosity.

nlin.CD↗
arXiv2009

Modulational instability of Rossby and drift waves and generation of zonal jets

We study the modulational instability of geophysical Rossby and plasma drift waves within the Charney-Hasegawa-Mima (CHM) model both theoretically, using truncated (four-mode and three-mode) models, and numerically, using direct simulations of CHM equation in the Fourier space. The linear theory predicts instability for any amplitude of the primary wave. For strong primary waves the most unstable modes are perpendicular to the primary wave, which correspond to generation of a zonal flow if the primary wave is purely meridional. For weaker waves, the maximum growth occurs for off-zonal inclined modulations. For very weak primary waves the unstable waves are close to being in three-wave resonance with the primary wave. The nonlinear theory predicts that the zonal flows generated by the linear instability experience pinching into narrow zonal jets. Our numerical simulations confirm the theoretical predictions of the linear theory as well as of the nonlinear pinching. We find that, for strong primary waves, these narrow zonal jets further roll up into Karman-like vortex streets. On the other hand, for weak primary waves, the growth of the unstable mode reverses and the system oscillates between a dominant jet and a dominate primary wave. The 2D vortex streets appear to be more stable than purely 1D zonal jets, and their zonal-averaged speed can reach amplitudes much stronger than is allowed by the Rayleigh-Kuo instability criterion for the 1D case. We find that the truncation models work well for both the linear stage and and often even for the medium-term nonlinear behavior. In the long term, the system transitions to turbulence helped by the vortex-pairing instability (for strong waves) and by the resonant wave-wave interactions (for weak waves).

nlin.CD↗
arXiv2009

Triple cascade behaviour in QG and drift turbulence and generation of zonal jets

We study quasi-geostrophic turbulence and plasma drift turbulence within the Charney-Hasegawa-Mima (CHM) model. We focus, theoretically and using numerical simulations, on conservation of {\em zonostrophy} and on its role in the formation of the zonal jets. The zonostrophy invariant was first predicted in \cite{perm,BNZ_invariant} in two special cases -- large-scale turbulence and anisotropic turbulence. Papers \cite{perm,BNZ_invariant} also predicted that the three invariants, energy, enstrophy and zonostrophy, will cascade anisotropically into non-intersecting sectors in the $k$-space, so that the energy cascade is "pushed" into the large-scale zonal scales. In the present paper, we consider the scales much less than the Rossby deformation radius and generalise the Fjørtoft argument of \cite{perm,BNZ_invariant} to find the directions of the three cascades in this case. For the first time, we demonstrate numerically that zonostrophy is well conserved by the CHM model, and that the energy, enstrophy and zonostrophy cascade as prescribed by the Fjørtoft argument if the nonlinearity is sufficiently weak. Moreover, numerically we observe that zonostrophy is conserved surprisingly well at late times and the triple-cascade picture is rather accurate even if the initial nonlinearity is strong.

nlin.CD↗
arXiv2008

Canonical Hamiltonians for waves in inhomogeneous media

We obtain a canonical form of a quadratic Hamiltonian for linear waves in a weakly inhomogeneous medium. This is achieved by using the WKB representation of wave packets. The canonical form of the Hamiltonian is obtained via the series of canonical Bogolyubov-type and near-identical transformations. Various examples of the application illustrating the main features of our approach are presented. The knowledge of the Hamiltonian structure for linear wave systems provides a basis for developing a theory of weakly nonlinear random waves in inhomogeneous media generalizing the theory of homogeneous wave turbulence.

math-ph↗
arXiv2008

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) {\it angle-resonances} which cannot directly cascade energy but which can redistribute it among the initially excited modes and (2) {\it 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 1000 x 1000 modes, classify and quantify them and discuss consequences of the obtained cluster structure for the wavefield 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.

nlin.SI↗
arXiv2006

Gravity surface wave turbulence in a laboratory flume

We present experimental results for water wave turbulence excited by piston-like programmed wavemakers in a water flume with horisontal dimensions 6x12x1.5 meters. Our main finding is that for a wide range of excitation amplitudes the energy spectrum has a power-law scaling, $E_ω\sim ω^{-ν}$. These scalings were achieved in up to one-decade wide frequency range, which is significantly wider than the range available in field observations and in numerical simulations. However, exponent $ν$ appears to be non-universal. It depends on the wavefield intensity and ranges from about 6.5 for weak forcing to about 3.5 for large levels of wave excitations. We discuss our results in the context of the key theoretical predictions, such as Zakharov-Filonenko spectrum $ν=-4$, Phillips spectrum $ν= -5$, Kuznetsov's revision of Phillips spectrum (leading to $ν=-4$) and Nazarenko's prediction $ν=-6$ for weak turbulence in finite basins. We measured Probability Density Function of the surface elevation and good correspondence with the Tayfun shapeexcept values near the maximum which we attribute to an anisotropy and inhomogeneity caused by the finite flume size. We argue that the wavenumber discreteness, due to the finite-size of the flume, prevents four-wave resonant interactions. Therefore, statistical evolution of the water surface in the laboratory is significantly different than in the open ocean conditions.

nlin.CD↗
arXiv2006

Kelvin-wave turbulence generated by vortex reconnections

Reconnections of quantum vortex filaments create sharp bends which degenerate into propagating Kelvin waves. These waves cascade their energy down-scale and their waveaction up-scale via weakly nonlinear interactions, and this is the main mechanism of turbulence at the scales less than the inter-vortex distance. In case of an idealised forcing concentrated around a single scale k0, the turbulence spectrum exponent has a pure direct cascade form -17/5 at scales k>k0 and a pure inverse cascade form -3 at k<k0. However, forcing produced by the reconnections contains a broad range of Fourier modes. What scaling should one expect in this case? In this Letter I obtain an answer to this question using the differential model for the Kelvin wave turbulence. The main result is that the direct cascade scaling dominates, i.e. the reconnection forcing is more or less equivalent to a low-frequency forcing.

cond-mat.other↗
arXiv2006

Freely decaying turbulence and Bose-Einstein condensation in Gross-Pitaevski model

We study turbulence and Bose-Einstein condensation (BEC) within the two-dimensional Gross-Pitaevski (GP) model. In the present work, we compute decaying GP turbulence in order to establish whether BEC can occur without forcing and if there is an intensity threshold for this process. We use the wavenumber-frequency plots which allow us to clearly separate the condensate and the wave components and,therefore, to conclude if BEC is present. We observe that BEC in such a system happens even for very weakly nonlinear initial conditions without any visible threshold. BEC arises via a growing phase coherence due to anihilation of phase defects/vortices. We study this process by tracking of propagating vortex pairs. The pairs loose momentum by scattering the background sound, which results in gradual decrease of the distance between the vortices. Occasionally, vortex pairs collide with a third vortex thereby emitting sound, which can lead to more sudden shrinking of the pairs. After the vortex anihilation the pulse propagates further as a dark soliton, and it eventually bursts creating a shock.

nlin.CD↗
arXiv2006

Differential models for 2D turbulence

We present two phenomenological models for 2D turbulence in which the energy spectrum obeys a nonlinear fourth-order and a second-order differential equations respectively. Both equations respect the scaling properties of the original Navier-Stokes equations and it has both the -5/3 inverse-cascade and t -3 direct-cascade spectra. In addition, the fourth order equation has Raleigh-Jeans thermodynamic distributions, as exact steady state solutions. We use the fourth-order model to derive a relation between the direct-cascade and the inverse-cascade Kolmogorov constants which is in a good qualitative agreement with the laboratory and numerical experiments. We obtain a steady state solution where both the enstrophy and the energy cascades are present simultaneously and we discuss it in context of the Nastrom-Gage spectrum observed in atmospheric turbulence. We also consider the effect of the bottom friction onto the cascade solutions, and show that it leads to an additional decrease and finite-wavenumber cutoffs of the respective cascade spectra.

nlin.CD↗
arXiv2006

Discreteness and its effect on the water-wave turbulence

We perform numerical simulations of the dynamical equations for free water surface in finite basin in presence of gravity. Wave Turbulence (WT) is a theory derived for describing statistics of weakly nonlinear waves in the infinite basin limit. Its formal applicability condition on the minimal size of the computational basin is impossible to satisfy in present numerical simulations, and the number of wave resonances is significantly depleted due to the wavenumber discreteness. The goal of this paper will be to examine which WT predictions survive in such discrete systems with depleted resonances and which properties arise specifically due to the discreteness effects. As in \cite{DKZ,onorato,naoto}, our results for the wave spectrum agree with the Zakharov-Filonenko spectrum predicted within WT. We also go beyond finding the spectra and compute probability density function (PDF) of the wave amplitudes and observe an anomalously large, with respect to Gaussian, probability of strong waves which is consistent with recent theory \cite{clnp,cln}. Using a simple model for quasi-resonances we predict an effect arising purely due to discreteness: existence of a threshold wave intensity above which turbulent cascade develops and proceeds to arbitrarily small scales. Numerically, we observe that the energy cascade is very ``bursty'' in time and is somewhat similar to sporadic sandpile avalanches. We explain this as a cycle: a cascade arrest due to discreteness leads to accumulation of energy near the forcing scale which, in turn, leads to widening of the nonlinear resonance and, therefore, triggering of the cascade draining the turbulence levels and returning the system to the beginning of the cycle. ~

math-ph↗
arXiv2006

Differential approximation for Kelvin-wave turbulence

I present a nonlinear differential equation model (DAM) for the spectrum of Kelvin waves on a thin vortex filament. This model preserves the original scaling of the six-wave kinetic equation, its direct and inverse cascade solutions, as well as the thermodynamic equilibrium spectra. Further, I extend DAM to include the effect of sound radiation by Kelvin waves. I show that, because of the phonon radiation, the turbulence spectrum ends at a maximum frequency $ω^* \sim (ε^3 c_s^{20} / κ^{16})^{1/13}$ where $ε$ is the total energy injection rate, $c_s$ is the speed of sound and $κ$ is the quantum of circulation.

cond-mat.stat-mech↗
arXiv2006

Sandpile behaviour in discrete water-wave turbulence

We construct a sandpile model for evolution of the energy spectrum of the water surface waves in finite basins. This model take into account loss of resonant wave interactions in discrete Fourier space and restoration of these interactions at larger nonlinearity levels. For weak forcing, the waveaction spectrum takes a critical $ω^{-10}$ shape where the nonlinear resonance broadening is equal to the Fourier grid spacing. The energy cascade in this case takes form of rare weak avalanches on the critical slope background. For larger forcing, this regime is replaced by a continuous cascade and Zakharov-Filonenko $ω^{-8}$ waveaction spectrum. For intermediate forcing levels, both scalings will be seen: $ω^{-10}$ at small and $ω^{-8}$ at large frequencies. The interval between these two scaling regions is characterised by strong avalanches.

nlin.CD↗
arXiv2005

Wave turbulence and vortices in Bose-Einstein condensation

We report a numerical study of turbulence and Bose-Einstein condensation within the two-dimmensional Gross-Pitaevski model with repulsive interaction. In presence of weak forcing localized around some wave number in the Fourier space, we observe three qualitatively different evolution stages. At the initial stage a thermodynamic energy equipartition spectrum forms at both smaller and larger scales with respect to the forcing scale. This agrees with predictions of the the four-wave kinetic equation of the Wave Turbulence (WT) theory. At the second stage, WT breaks down at large scales and the interactions become strongly nonlinear. Here, we observe formation of a gas of quantum vortices whose number decreases due to an annihilation process helped by the acoustic component. This process leads to formation of a coherent-phase Bose-Einstein condensate. After such a coherent-phase condensate forms, evolution enters a third stage characterised by three-wave interactions of acoustic waves that can be described again using the WT theory.

nlin.CD↗
arXiv2005

Wave Turbulence

In this paper we review recent developments in the statistical theory of weakly nonlinear dispersive waves, the subject known as Wave Turbulence (WT). We revise WT theory using a generalisation of the random phase approximation (RPA). This generalisation takes into account that not only the phases but also the amplitudes of the wave Fourier modes are random quantities and it is called the ``Random Phase and Amplitude'' approach. This approach allows to systematically derive the kinetic equation for the energy spectrum from the the Peierls-Brout-Prigogine (PBP) equation for the multi-mode probability density function (PDF). The PBP equation was originally derived for the three-wave systems and in the present paper we derive a similar equation for the four-wave case. Equation for the multi-mode PDF will be used to validate the statistical assumptions about the phase and the amplitude randomness used for WT closures. Further, the multi-mode PDF contains a detailed statistical information, beyond spectra, and it finally allows to study non-Gaussianity and intermittency in WT, as it will be described in the present paper. In particular, we will show that intermittency of stochastic nonlinear waves is related to a flux of probability in the space of wave amplitudes.

math-ph↗
arXiv2005

Joint statistics of amplitudes and phases in Wave Turbulence

Random Phase Approximation (RPA) provides a very convenient tool to study the ensembles of weakly interacting waves, commonly called Wave Turbulence. In its traditional formulation, RPA assumes that phases of interacting waves are random quantities but it usually ignores randomness of their amplitudes. Recently, RPA was generalised in a way that takes into account the amplitude randomness and it was applied to study of the higher momenta and probability densities of wave amplitudes. However, to have a meaningful description of wave turbulence the RPA properties assumed for the initial fields must be proven to survive over the nonlinear evolution time, and such a proof is the main goal of the present paper. We derive an evolution equation for the full probability density function which contains the complete information about the joint statistics of all wave amplitudes and phases. We show that, for any initial statistics of the amplitudes, the phase factors remain statistically independent uniformly distributed variables. If in addition the initial amplitudes are also independent variables (but with arbitrary distributions) they will remain independent when considered in small sets which are much less than the total number of modes. However, if the size of a set is of order of the total number of modes then the joint probability density for this set is not factorisable into the product of one-mode probabilities. In the other words, the modes in such a set are involved in a ``collective'' (correlated) motion. We also study new type of correlators describing the phase statistics.

math-ph↗
arXiv2004

Probability densities and preservation of randomness in wave turbulence

Turbulence closure for the weakly nonlinear stochastic waves requires, besides weak nonlinearity, randomness in both the phases and the amplitudes of the Fourier modes. This randomness, once present initially, must remain over the nonlinear evolution time. Finding out to what extent is this true is the main goal of the present Letter. For this analysis we derive an evolution equation for the full probability density function (PDF) of the wave field. We will show that, for any statistics of the amplitudes, phases tend to stay random if they were random initially. If in addition the initial amplitudes are independent variables they will remain independent in a coarse-grained sense, i.e. when considered in small subsets which are much less than the total set of modes.

math-ph↗
arXiv2003

"Noisy" spectra, long correlations and intermittency in wave turbulence

We study the k-space fluctuations of the waveaction about its mean spectrum in the turbulence of dispersive waves. We use a minimal model based on the Random Phase Approximation (RPA) and derive evolution equations for the arbitrary-order one-point moments of the wave intensity in the wavenumber space. The first equation in this series is the familiar Kinetic Equation for the mean waveaction spectrum, whereas the second and higher equations describe the fluctuations about this mean spectrum. The fluctuations exhibit a nontrivial dynamics if some long coordinate-space correlations are present in the system, as it is the case in typical numerical and laboratory experiments. Without such long-range correlations, the fluctuations are trivially fixed at their Gaussian values and cannot evolve even if the wavefield itself is non-Gaussian in the coordinate space. Unlike the previous approaches based on smooth initial k-space cumulants, the RPA model works even for extreme cases where the k-space fluctuations are absent or very large and intermittent. We show that any initial non-Gaussianity at small amplitudes propagates without change toward the high amplitudes at each fixed wavenumber. At each fixed amplitude, however, the PDF becomes Gaussian at large time.

math-ph↗