Publications

We suggest a new computer-assisted approach to the development of turbulence theory. It allows one to impose lower and upper bounds on correlation functions using sum-of-squares polynomials. We demonstrate it on the minimal cascade model of two resonantly interacting modes when one is pumped and the other dissipates. We show how to present correlation functions of interest as part of a sum-of-squares polynomial using the stationarity of the statistics. That allows us to find how the moments of the mode amplitudes depend on the degree of nonequilibrium (analog of the Reynolds number), which reveals some properties of marginal statistical distributions. By combining scaling dependence with the results of direct numerical simulations, we obtain the probability densities of both modes in a highly intermittent inverse cascade. As the Reynolds number tends to infinity, we show that the relative phase between modes tends to pi /2 and -pi/2 in the direct and inverse cascades, respectively, and derive bounds on the phase variance. Our approach combines computer-aided analytical proofs with a numerical algorithm applied to high-degree polynomials.

Never is the difference between thermal equilibrium and turbulence so dramatic, as when a quadratic invariant makes the equilibrium statistics exactly Gaussian with independently fluctuating modes. That happens in two very different yet deeply connected classes of systems: incompressible hydrodynamics and resonantly interacting waves. This work presents the first detailed information-theoretic analysis of turbulence in such strongly interacting systems. The analysis involves both energy and entropy and elucidates the fundamental roles of space and time in setting the cascade direction and the changes of the statistics along it. We introduce a beautifully simple yet rich family of discrete models with triplet interactions of neighboring modes and show that it has quadratic conservation laws defined by the Fibonacci numbers. Depending on how the interaction time changes with the mode number, three types of turbulence were found: single direct cascade, double cascade, and the first-ever case of a single inverse cascade. We describe quantitatively how deviation from thermal equilibrium all the way to turbulent cascades makes statistics increasingly non-Gaussian and find the self-similar form of the one-mode probability distribution. We reveal where the information (entropy deficit) is encoded and disentangle the communication channels between modes, as quantified by the mutual information in pairs and the interaction information inside triplets.

Polymer molecules in a flow undergo a coil-stretch phase transition on an increase of the velocity gradients. Model-independent identification and characterization of the transition in a random flow has been lacking so far. Here we suggest to use the entropy of the extension statistics as a proper measure due to strong fluctuations around the transition. We measure experimentally the entropy as a function of the local Weisenberg number and show that it has a maximum, which identifies and quantifies the transition. We compare the new approach with the traditional one based on the theory using either linear Oldroyd-B or nonlinear finite extensible nonlinear elastic polymer models.

We discover dramatic impact of vortex formation in the transverse component of the Poynting vector of the fundamental core mode in solid core micro - structured optical fibers on the energy dissipation. The vortices can reduce losses of the mode by several orders of magnitude under proper selection of the fiber parameters at a given wavelength.

Flows in fluid layers are ubiquitous in industry, geophysics, and astrophysics. Large-scale flows in thin layers can he considered two dimensional with bottom friction added. Here we find that the properties of such flows depend dramatically on the way they arc driven. We argue that a wall-driven (Couctte) flow cannot sustain turbulence, no matter how small the viscosity and friction. Direct numerical simulations (DNSs) up to the Reynolds number Re = 10(6) confirm that all perturbations die in a plane Couette flow. On the contrary, for sufficiently small viscosity and friction, perturbations destroy the pressure-driven laminar (Poiseuille) flow. What appears instead is a traveling wave in the form of a jet slithering between wall vortices. For 5 x 10(3)

Scrupulous measurements and detailed data analysis of the torque in a swirling turbulent flow driven by counter-rotating bladed disks reveal an apparent breaking of the law of similarity. Potentially, such breakdown could arise from several possible factors, including dependence on dimensionless numbers other than Re or velocity coupling to other fields such as temperature. However, careful redesign and calibration of the experiment showed that this unexpected result was due to background errors caused by minute misalignments which lead to a noisy and irreproducible torque signal at low rotation speeds and prevented correct background subtraction normally ascribed to frictional losses. An important lesson to be learnt is that multiple minute misalignments can nonlinearly couple to the torque signal and provide a dc offset that cannot be removed by averaging. That offset can cause the observed divergence of the friction coefficient C-f from its constant value observed in the turbulent regime. By significant modifications of the setup and conducting the experiment with one bladed disk and precisely aligned the disk, torque meter and motor shaft, we are able to conduct precise measurements close to the expected resolution at small torques at low rotation speeds and to confirm the similarity law in a wide range of Re, in particular, in low viscosity fluids.

This article introduces the Focus Issue on Two-Dimensional Turbulence appearing in Physics of Fluids (Volume 29, Issue 11, November 2017). Published by AIP Publishing.

Viscous electronics is an emerging field dealing with systems in which strongly interacting electrons behave as a fluid. Electron viscous flows are governed by a nonlocal current-field relation which renders the spatial patterns of the current and electric field strikingly dissimilar. Notably, driven by the viscous friction force from adjacent layers, current can flow against the electric field, generating negative resistance, vorticity, and vortices. Moreover, different current flows can result in identical potential distributions. This sets a new situation where inferring the electron flow pattern from the measured potentials presents a nontrivial problem. Using the inherent relation between these patterns through complex analysis, here we propose a method for extracting the current flows from potential distributions measured in the presence of a magnetic field.

This short note is written to call attention to an analytic approach to the interaction of developed turbulence with mean flows of simple geometry (jets and vortices). It is instructive to compare cases in two and three dimensions and see why the former are solvable and the latter are not (yet). We present the analytical solutions for two-dimensional mean flows generated by an inverse turbulent cascade on a sphere and in planar domains of different aspect ratios. These solutions are obtained in the limit of small friction when the flow is strong while turbulence can be considered weak and treated perturbatively. I then discuss when these simple solutions can be realized and when more complicated flows may appear instead. The next step of describing turbulence statistics inside a flow and directions of possible future progress are briefly discussed at the end.

Inelastic collapse of stochastic trajectories of a randomly accelerated particle moving in half-space z>0 has been discovered by McKean [J. Math. Kyoto Univ.2, 227 (1963)] and then independently rediscovered by Cornellet al.[Phys. Rev. Lett.81, 1142 (1998)]. The essence of this phenomenon is that the particle arrives at the wall at z=0 with zero velocity after an infinite number of inelastic collisions if the restitution coefficient beta of particle velocity is smaller than the critical value betac=exp(-pi/radic3). We demonstrate that inelastic collapse takes place also in a wide class of models with spatially inhomogeneous random forcing and, what is more, that the critical value betacis universal. That class includes an important case of inertial particles in wall-bounded random flows. To establish how inelastic collapse influences the particle distribution, we derive the exact equilibrium probability density function rho(z,v) for the particle position and velocity. The equilibrium distribution exists only at β

We investigate time irreversibility from the point of view of a single particle in Burgers turbulence. Inspired by the recent work for incompressible flows [Xu, Proc. Natl. Acad. Sci. USA 111, 7558 (2014)PNASA60027-842410.1073/pnas.1321682111], we analyze the evolution of the kinetic energy for fluid markers and use the fluctuations of the instantaneous power as a measure of time irreversibility. For short times, starting from a uniform distribution of markers, we find the scaling ([E(t)-E(0)]n)?t and (pn)?Ren-1 for the power as a function of the Reynolds number. Both observations can be explained using the "flight-crash" model, suggested by Xu et al. Furthermore, we use a simple model for shocks that reproduces the moments of the energy difference, including the pre-factor for (E(t)-E(0)). To complete the single-particle picture for Burgers we compute the moments of the Lagrangian velocity difference and show that they are bifractal. This arises in a similar manner to the bifractality of Eulerian velocity differences. In the above setting, time irreversibility is directly manifest as particles eventually end up in shocks. We additionally investigate time irreversibility in the long-time limit when all particles are located inside shocks and the Lagrangian velocity statistics are stationary. We find the same scalings for the power and energy differences as at short times and argue that this is also a consequence of rare "flight-crash" events related to shock collisions.

Traditional wave kinetics describes the slow evolution of systems with many degrees of freedom to equilibrium via numerous weak non-linear interactions and fails for very important class of dissipative (active) optical systems with cyclic gain and losses, such as lasers with non-linear intracavity dynamics. Here we introduce a conceptually new class of cyclic wave systems, characterized by non-uniform double-scale dynamics with strong periodic changes of the energy spectrum and slow evolution from cycle to cycle to a statistically steady state. Taking a practically important example - random fibre laser - we show that a model describing such a system is close to integrable non-linear Schrödinger equation and needs a new formalism of wave kinetics, developed here. We derive a non-linear kinetic theory of the laser spectrum, generalizing the seminal linear model of Schawlow and Townes. Experimental results agree with our theory. The work has implications for describing kinetics of cyclical systems beyond photonics.

The ability to send a wave to fetch an object from a distance would find a broad range of applications. Quasi-standing Faraday waves on water create horizontal vortices(1,2), yet it is not known whether propagating waves can generate large-scale flows-small-amplitude irrotational waves only push particles in the direction of propagation(3-5). Here we show that when waves become three-dimensional as a result of the modulation instability, a floater can be forced to move towards the wave source. The mechanism for this is the generation of surface vortices by waves propagating away from vertically oscillating plungers. We introduce a new conceptual framework for understanding wave-driven flows, which enables us to engineer inward and outward surface jets, stationary vortices, and other complex flows. The results form a new basis for the remote manipulation of objects on fluid surfaces and for a better understanding of the motion of floaters in the ocean, the generation of wave-driven jets, and the formation of Lagrangian coherent structures.

We describe ideal incompressible hydrodynamics on the hyperbolic plane which is an infinite surface of constant negative curvature. We derive equations of motion, general symmetries and conservation laws, and then consider turbulence with the energy density linearly increasing with time due to action of small-scale forcing. In a flat space, such energy growth is due to an inverse cascade, which builds a constant part of the velocity autocorrelation function proportional to time and expanding in scales, while the moments of the velocity difference saturate during a time depending on the distance. For the curved space, we analyze the long-time long-distance scaling limit, that lives in a degenerate conical geometry, and find that the energy-containing mode linearly growing with time is not constant in space. The shape of the velocity correlation function indicates that the energy builds up in vortical rings of arbitrary diameter but of width comparable to the curvature radius of the hyperbolic plane. The energy current across scales does not increase linearly with the scale, as in a flat space, but reaches a maximum around the curvature radius. That means that the energy flux through scales decreases at larger scales so that the energy is transferred in a non-cascade way, that is the inverse cascade spills over to all larger scales where the energy pumped into the system is cumulated in the rings. The time-saturated part of the spectral density of velocity fluctuations contains a finite energy per unit area, unlike in the flat space where the time-saturated spectrum behaves as k^-5/3.

The statistical properties of turbulence differ in an essential way from those of systems in or near thermal equilibrium because of the flux of energy between vastly different scales at which energy is supplied and at which it is dissipated. We elucidate this difference by studying experimentally and numerically the fluctuations of the energy of a small fluid particle moving in a turbulent fluid. We demonstrate how the fundamental property of detailed balance is broken, so that the probabilities of forward and backward transitions are not equal for turbulence. In physical terms, we found that in a large set of flow configurations, fluid elements decelerate faster than accelerate, a feature known all too well from driving in dense traffic. The statistical signature of rare "flight-crash" events, associated with fast particle deceleration, provides a way to quantify irreversibility in a turbulent flow. Namely, we find that the third moment of the power fluctuations along a trajectory, nondimensionalized by the energy flux, displays a remarkable power law as a function of the Reynolds number, both in two and in three spatial dimensions. This establishes a relation between the irreversibility of the system and the range of active scales. We speculate that the breakdown of the detailed balance characterized here is a general feature of other systems very far from equilibrium, displaying a wide range of spatial scales.

We consider developed turbulence in the Gross-Pitaevsky model, where a condensate appears due to an inverse cascade. Despite being fully turbulent, the system demonstrates nondecaying periodic oscillations around a steady state, when turbulence and condensate periodically exchange a small fraction of waves. We show that these collective oscillations are not of a predator-prey type, as was suggested earlier; rather, they are due to phase coherence and anomalous correlations imposed by the condensate.

We develop an analytic formalism and derive new exact relations that express the short-time dispersion of fluid particles via the single-time velocity correlation functions in homogeneous isotropic and incompressible turbulence. The formalism establishes a bridge between single-time Eulerian and long-time Lagrangian pictures of turbulent flows. In particular, we derive an exact formula for a short-term counterpart of the long-time Richardson law, and we identify a conservation law of turbulent dispersion which is true even in nonstationary turbulence.

We study numerically optical turbulence using the particular example of a recently created, ultra-long fibre laser. For normal fibre dispersion, we observed an intermediate state with an extremely narrow spectrum (condensate), which experiences instability and a sharp transition to a fluctuating regime with a wider spectrum. We demonstrate that the number of modes has an impact on the condensate's lifetime. The smaller the number of modes, the more resistant is the condensate to perturbations. Experimental results show a good agreement with numerical simulations.

A passive scalar field was studied under the action of pumping, diffusion and advection by a 2D smooth flow with Lagrangian chaos. We present theoretical arguments showing that the scalar statistics are not conformally invariant and formulate a new effective semi-analytic algorithm to model scalar turbulence. We then carry out massive numerics of scalar turbulence, focusing on nodal lines. The distribution of contours over sizes and perimeters is shown to depend neither on the flow realization nor on the resolution (diffusion) scale r _d for scales exceeding r _d. The scalar isolines are found to be fractal/smooth at scales larger/smaller than the pumping scale. We characterize the statistics of isoline bending by the driving function of the Löwner map. That function is found to behave like diffusion with diffusivity independent of the resolution yet, most surprisingly, dependent on the velocity realization and time (beyond the time on which the statistics of the scalar is stabilized).

The multidisciplinary field of fluid mechanics is one of the most actively developing fieldsof physics, mathematics and engineering. In this book, the fundamental ideas of fluid mechanics are presented from a physics perspective. Using examples taken from everyday life, from hydraulic jumps ina kitchen sink to Kelvin–Helmholtz instabilities in clouds, the book provides readers with a better understanding of the world around them. It teaches the art of fluid-mechanical estimates and shows how the ideas and methods developed to study the mechanics of fluids are used to analyze other systems with many degrees of freedom in statistical physics and field theory. Aimed at undergraduate and graduate students, the book assumes no prior knowledge of the subject and only a basic understanding of vector calculus and analysis. It contains 32 exercises of varying difficulties, from simple estimates to elaborate calculations, with detailed solutions to help readers understand fluid mechanics.

A finite-amplitude propagating wave induces a drift in fluids. Understanding how drifts produced by many waves disperse pollutants has broad implications for geophysics and engineering. Previously, the effective diffusivity was calculated for a random set of small-amplitude surface and internal waves. Now, this is extended by Bhler & Holmes-Cerfon (J. Fluid Mech., 2009, this issue, vol. 638, pp. 526) to waves in a rotating shallow-water system in which the Coriolis force is accounted for, a necessary step towards oceanographic applications. It is shown that interactions of finite-amplitude waves affect particle velocity in subtle ways. An expression describing the particle diffusivity as a function of scale is derived, showing that the diffusivity can be substantially reduced by rotation.

A Reply to the Comment by Erik Lindborg.

We discuss fluctuation relations in simple cases of non-equilibrium Langevin dynamics. In particular, we show that, close to non-equilibrium steady states with non-vanishing probability currents, some of these relations reduce to a modified version of the fluctuation-dissipation theorem. The latter may be interpreted as the equilibrium-like relation in the reference frame moving with the mean local velocity determined by the probability current.

We present a mean-field model of cloud evolution that describes droplet growth due to condensation and collisions and droplet loss due to fallout. The model accounts for the effects of cloud turbulence both in a large-scale turbulent mixing and in a microphysical enhancement of condensation and collisions. The model allows for an effective numerical simulation by a scheme that is conservative in water mass and keeps accurate count of the number of droplets. We first study the homogeneous situation and determine how the rain-initiation time depends on the concentration of cloud condensation nuclei (CCN) and turbulence level. We then consider clouds with an inhomogeneous concentration of CCN and evaluate how the rain initiation time and the effective optical depth vary in space and time. We argue that over-seeding even a part of a cloud by small hygroscopic nuclei, one can substantially delay the onset and increase the amount of precipitation.

Measurements of atmospheric winds in the mesoscale range (10-500 km) reveal remarkably universal spectra with the $k^{-5/3}$ power law. Despite initial expectations of the inverse energy cascade, as in two-dimensional (2D) turbulence, measurements of the third velocity moment in atmosphere, suggested a direct energy cascade. Here we propose a possible solution to this controversy by accounting for the presence of a large-scale coherent flow, or a spectral condensate. We present new experimental laboratory data and show that the presence of a large-scale shear flow modifies the third-order velocity moment in spectrally condensed 2D turbulence, making it, in some conditions, similar to that observed in the atmosphere.

Cloud physics has for a long time been an important segment of atmospheric science. It is common knowledge that clouds are crucial for our understanding of weather and climate. Clouds are also interesting by themselves (not to mention that they are beautiful). Complexity is hidden behind the common picture of these beautiful and interesting objects. The typical school textbook definition that a cloud is 'a set of droplets or particles suspended in the atmosphere' is not adequate. Clouds are complicated phenomena in which dynamics, turbulence, microphysics, thermodynamics and radiative transfer interact on a wide range of scales, from sub-micron to kilometres. Some of these interactions are subtle and others are more straightforward. Large and small-scale motions lead to activation of cloud condensation nuclei, condensational growth and collisions; small changes in composition and concentration of atmospheric aerosol lead to significant differences in radiative properties of the clouds and influence rainfall formation. It is justified to look at a cloud as a composite, nonlinear system which involves many interactions and feedback. This system is actively linked into a web of atmospheric, oceanic and even cosmic interactions.

A review of recent experimental studies of turbulence suppression by mean flows in quasi-two-dimensional fluids is presented. Large-scale mean flows develop during spectral condensation of 2D turbulence as a result of the inverse energy cascade in spatially bounded flow. The spectral energy which is accumulated at the largest scale supports the mean flow which in turn affects turbulence. We show that such a flow can reduce the energy flux in the inverse energy cascade range via shearing and sweeping of the turbulent eddies. The former mechanism is more efficient at larger scales, while the latter acts on the smaller scales. Similar suppression of turbulence has been found in the presence of externally imposed flows. Turbulent (inverse energy) cascade is reduced in the presence of imposed flow, but still supports Kolmogorov-Kraichnan k −5/3 power law spectrum in the energy range.

The statistics of the nodal lines of scalar fields intwo-dimensional (2d) turbulence is found to be conformal invariantand equivalent to that of cluster boundaries in criticalphenomena. That allows for a rich variety of exact analyticresults, first time in turbulence studies. In particular, thestatistics of zero-vorticity lines in Navier-Stokes turbulence isfound to be equivalent to that of critical percolation. Thestatistics of the zero-temperature lines in surfacequasi-geostrophic (SQG) turbulence is found to be equivalent tothat of the isolines of a Gaussian (free) field.

We study the statistics of the relative separation between two fluid particles in a spatially smooth and temporally random flow. The Lagrangian strain is modeled by a telegraph noise, which is a stationary random Markov process that can only take two values with known transition probabilities. The simplicity of the model enables us to write closed equations for the interparticle distance in the presence of a finite-correlated noise. In one dimension, we are able to find analytically the long-time growth rates of the distance moments and the senior Lyapunov exponent, which consistently turns out to be negative. We also find the exact expression for the Cramér function and show that it satisfies the fluctuation relation (for the probability of positive and negative entropy production) despite the time irreversibility of the strain statistics. For the two-dimensional incompressible isotropic case, we obtain the Lyapunov exponent (positive) and the asymptotic growth rates of the moments in two opposite limits of fast and slow strain. The quasideterministic limit (of slow strain) turns out to be singular, while a perfect agreement is found with the already-known δ -correlated case.

The growth rate of small-scale density inhomogeneities (the entropy production rate) is given by the sum of the Lyapunov exponents in a random flow. We derive an analytic formula for the rate in a flow of weakly interacting waves and show that in most cases it is zero up to the fourth order in the wave amplitude. We then derive an analytic formula for the rate in a flow of waves and currents. Estimates of the rate and the fractal dimension of the density distribution show that the interplay between waves and currents is a realistic mechanism for providing patchiness of the pollutant distribution on the ocean surface.

We offer a new example of conformal invariance (local scale invariance) far from equilibrium-the inverse cascade of surface quasigeostrophic (SQG) turbulence. We show that temperature isolines are statistically equivalent to curves that can be mapped into a one-dimensional Brownian walk (called Schramm-Loewner evolution or SLEκ). The diffusivity is close to κ=4, that is, isotemperature curves belong to the same universality class as domain walls in the O(2) spin model. Several statistics of temperature clusters and isolines are shown to agree with the theoretical expectations for such a spin system at criticality. We also show that the direct cascade in two-dimensional Navier-Stokes turbulence is not conformal invariant. The emerging picture is that conformal invariance may be expected for inverse turbulent cascades of strongly interacting systems.

We study experimentally and theoretically how waves affect the distribution of particles floating on a liquid surface. According to the Archimedes' law the weight of floating particle is equal to the weight of displaced liquid. This law is not quite precise for small floating objects. An additional force generating by surface tension pulls a hydrophilic particle deeper into the water so that the mass of the displaced liquid exceeds the particle mass. This mass mismatch makes the floating particle effectively inertial that may lead to the clustering of particles when they move by the surface waves. Here we report the results of our recent work showing that particles gather in the nodes or antinodes of a standing surface wave depending on the sign of capillarity effect. Experimentally measured rate of particles gathering is shown to be proportional to the square of the wave amplitude which agrees with the theoretical model.

Turbulent flows, with their irregular behavior, confound any simple attempts to understand them. But physicists have succeeded in identifying some universal properties of turbulence and relating them to broken symmetries.

The simplicity of fundamental physical laws manifests itself in fundamental symmetries. Although systems with an infinite number of strongly interacting degrees of freedom (in particle physics and critical phenomena) are hard to describe, they often demonstrate symmetries, in particular scale invariance. In two dimensions (2D) locality often extends scale invariance to a wider class of conformal transformations that allow non-uniform rescaling. Conformal invariance enables a thorough classification of universality classes of critical phenomena in 2D. Is there conformal invariance in 2D turbulence, a paradigmatic example of a strongly interacting non-equilibrium system? Here, we show numerically that some features of a 2D inverse turbulent cascade show conformal invariance. We observe that the statistics of vorticity clusters are remarkably close to that of critical percolation, one of the simplest universality classes of critical phenomena. These results represent a key step in the unification of 2D physics within the framework of conformal symmetry.

The problem of droplet growth by condensation in a turbulent flow of nearly saturated vapour is addressed theoretically and numerically. We show how the presence of an underlying turbulent velocity field induces a correlation between droplet trajectories and supersaturation. This leads both to the enhancement of the droplet growth rate and to a fast spreading of the droplet size distribution.

We find the probability distribution of the fluctuating parameters of a soliton propagating through a medium with additive noise. Our method is a modification of the instanton formalism (method of optimal fluctuation) based on a saddle-point approximation in the path integral. We first solve consistently a fundamental problem of soliton propagation within the framework of noisy nonlinear Schrödinger equation. We then consider model modifications due to in-line (filtering, amplitude and phase modulation) control. It is examined how control elements change the error probability in optical soliton transmission. Even though a weak noise is considered, we are interested here in probabilities of error-causing large fluctuations which are beyond perturbation theory. We describe in detail a new phenomenon of soliton collapse that occurs under the combined action of noise, filtering and amplitude modulation.

The growth of density inhomogeneities was investigated in a flow of wave turbulence. The conditions for a nonzero sum of the Lyapunov exponents was found which provides for an exponential growth of density inhomogeneities. The leading order contribution came from the pair of waves having coinciding frequencies but producing different stokes drifts. It was observed that nonzero Lyapunov exponents appear in two and three dimensions both for potential and solenoidal waves.

The logarithmic renormalization predicted by Kraichnan (1971) for the direct cascade of enstrophy in the inertial range of two-dimensional turbulence has been observed in a numerical simulation. A moderate resolution allows for a very long time integration that provides very good statistics. Deviations from Gaussianity in the vorticity probability distribution are observed.

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e., to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in nonequilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.

We present a consistent method to calculate the probability distribution of soliton parameters in systems with additive noise. Even though the noise is weak, we are interested in probabilities of large fluctuations (generally non-Gaussian) which are beyond perturbation theory. Our method is a development of the instanton formalism (method of optimal fluctuation) based on a saddle-point approximation in the path integral. We first solve a fundamental problem of soliton statistics governed by a noisy nonlinear Schrödinger equation. We then apply our method to optical soliton transmission systems using signal control elements (filters and amplitude and phase modulators).

We study the statistics of relative distances R(t) between fluid particles in a spatially smooth random flow with arbitrary temporal correlations. Using the space dimensionality d as a large parameter we develop an effective description of Lagrangian dispersion. We describe the exponential growth of relative distances 〈R²(t)〉 ∝ exp2λ̄t at different values of the ratio between the correlation and turnover times. We find the stretching correlation time which determines the dependence of 〈R₁R₂〉 on the difference t₁ - t₂. The calculation of the next cumulant of R² shows that statistics of R² is nearly Gaussian at small times (as long as d ≫ 1) and becomes log-normal at large times when large-d approach fails for high-order moments. The crossover time between the regimes is the stretching correlation time which surprisingly appears to depend on the details of the velocity statistics at t≪τ. We establish the dispersion of the ln(R²) in the log-normal statistics.

We consider the tails of probability density functions (PDF) for different characteristics of velocity that satisfies Burgers equation driven by a large-scale force. The saddle-point approximation is employed in the path integral so that the calculation of the PDF tails boils down to finding the special field-force configuration (instanton) that realizes the extremum of probability. We calculate high moments of the velocity gradient ∂_xu and find out that they correspond to the PDF with In[P(∂_xu)] ∝ - (-∂_xu/ℜe)^3/2 where ℜe is the Reynolds number. That stretched exponential form is valid for negative ∂_xu with the modulus much larger than its root-mean-square (rms) value. The respective tail of PDF for negative velocity differences w is steeper than Gaussian, In P (w) ∼ - (w/u_rms)³, as well as single-point velocity PDF In P (u) ∼ - (|u|/u_rms)³. For high velocity derivatives u (k) = ∂^k_xu, the general formula is found: In P (|u^(k)|) ∝ -(|u^(k)|/ℜe^k)^3/(k+1).

The steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described at scales less than the correlation length of the flow and larger than the diffusion scale. The probability distribution of the scalar is expressed via the probability distribution of the line stretching rate. The description of the line stretching can be reduced to the classical problem of studying the product of many matrices with a unit determinant. We found a change of variables which allows one to map the matrix problem into a scalar one and to prove thus a central limit theorem for the statistics of the stretching rate. The proof is valid for any finite correlation time of the velocity field. Whatever be the statistics of the velocity field, the statistics of the passive scalar in the inertial interval of scales is shown to approach Gaussianity as one increases the Peclet number Pe (the ratio of the pumping scale to the diffusion one). The first n

For Kraichnan’s problem of passive scalar advection by a velocity field delta correlated in time, the limit of large space dimensionality d ≫ 1 is considered.Scaling exponents of the scalar field are analytically found to be ζ 2n = n ζ 2-2(2-ζ 2)n (n -1)/d, while those of the dissipation field are μ n = -2(2-ζ 2)n (n -1)/d for orders n ≪d.The refined similarity hypothesis ζ 2n = n ζ 2 + μ n is thus established by a straightforward calculation for the case considered.

Three-dimensional turbulence of incompressible fluid is described by using Clebsch canonical variables. This reveals the families of new local integrals of motion so that there are additional cascade spectra besides the energy cascade. A weakly anisotropic spectrum of developed turbulence is shown to be as universal as isotropic Kolmogorov spectrum. The correlation functions of three-dimensional incompressible turbulence approach their isotropic values in the inertial interval so that the share taken by the anisotropic parts of velocity correlators decrease with the wavenumber as k^{- 2 3}, which satisfactorily fits the experimental data. The complementarity of the turbulence description in Clebsch and velocity variables is demonstrated.

Evolution of decay turbulence of capillary waves on a deep water is considered in the framework of the isotropic kinetic equation. It is shown that the evolution comprises two stages. During the first stage an arbitrary localized large-scale wave distribution explosively evolves into a small-scale Kolmogorov spectrum. The second stage starts at the moment the Kolmogorov spectrum reaches dissipative scales. The characteristic time of this stage is much longer (up to thousand times) than that of the first one. The energy distribution is close to the Kolmogorov spectrum and decays by a self-similar law.

The whole set of simultaneous correlation functions describing steady vorticity cascade is obtained from the Euler equation by a straightforward procedure. Nonlocality of the cascade provides for a large logarithmic parameter that enables one to obtain a universal set of the correlation functions of the vorticity omega in the inertial interval: [omega(n)(r1)omega(n)(r2)] is-proportional-to ln2n/3(L/\r1 - r2\), with L being the scale of the external pump.

It is shown how viscosity increases turbulence level in the inertial interval by suppressing turbulent transfer.

Обычно в утверждение об универсальности спектра развитой турбулентности вкладывают следующий смысл: в интервале масштабов, промежуточных между источником и стоком, турбулентность изотропна и распределение энергии по масштабам зависит от единственного внешнего параметра—потока энергии в к-пространстве [1—3]. В соответствии с этой гипотезой были получены изотропные однонараметрические спектры (называемые обычно колмогоровскими) как для вихревой гидродинамической [1—4], так и для волновой турбулентности в гидродинамике, физике плазмы, акустике [5]. Следует указать, однако, что во всех случаях взаимодействие и волн, и вихрей, помимо энергии, сохраняет суммарный импульс. Любой же реальный источник турбулентности является анизотропным и несимметричным, что приводит к появлению ненулевого импульса системы. Переносящие малый поток импульса К стационарные поправки 5пъ к несущим поток энергии Р слаботурбулентным колмогоровским решениям пк были построены в работе [6]. Для волн со степенным законом дисперсии сок эти так называемые дрейфовые поправки имеют простой вид Зик/ид^(Кк) сол/(Ле2)«" со $0*(шк/к)^ ка~ гсо $ 0к,(1) обусловленный тем, что у=(Кк) сок/(Рк2) является единственным безразмерным параметром, который можно составить из рассматриваемых величин.

A critical analysis of the conformal approach to the theory of 2D turbulence is delivered. It is shown, in particular, that conformal minimal models cannot give a general turbulent solution, which should provide for constant fluxes of all vorticity integrals of motion.

A steady Kolmogorov-like spectrum of turbulence is found as an exact solution of the kinetic equation for inertial-gravity waves. The spectrum obtained satisfactorily fits the results of atmospheric observations for mesoscale motions (from hundred to thousand kilometers).

The problem of interaction locality in k space is studied in a diagrammatic perturbation approach for the Navier-Stokes equation in quasi-Lagrangian variables. Analyzing the whole diagram series we have found an exact relation between the asymptotic behavior of the triple-correlation function of velocities that governs the energy transfer over scales and the double-correlation function giving the energy distribution. Namely, at rR, we obtain S3(r,R,R-r)S2(R)(r/R) [S3(r)/S2(r)]R2ζ-1r22-ζ, where ζ2 is the static exponent of double-velocity moment. This relation between two different physical quantities (in principle, measurable independently) is accessible to an experimental check. Also, this relation allows us to describe an energy exchange between distant scales in k space: For any steady spectrum carrying constant energy flux, the interactions of the given k-eddies with large (k1k) and small eddies (k2k) are shown to decrease by the same law with the distance in k space, such as (k1/k)22-ζ and (k/k2)22-ζ. It means a balance of interactions for such a spectrum. Considering, in particular, the multifractal picture of developed turbulence, we analyze the range of exponents h of the velocity field [δv(r)rh] which provides the locality of interaction in the k space. It is shown that the condition of infrared locality of interaction (with larger k1-eddies) could give only the upper restriction for the exponent. The upper limit thus found (hmax=1) coincides with the boundary exponent of singularity of energy dissipation. As far as an interaction locality in the ultraviolet limit (k2k) is concerned, we prove that any reasonable dimension function D(h) provides locality whatever small h is considered.

Weak turbulence of waves with a weakly damped dispersion law is discussed. In the stationary case, small anisotropic additions to an isotropic spectrum of the Kolmogorov type are found. It is shown that a small anisotropic source located in a region of small k leads to an essentially anisotropic spectrum in a region of large k. Strongly anisotropic stationary spectra are found. [Russian original - Zh. Eksp. Teor. Fiz. 80, 592-596 (February 1981)]