Publications

We consider flux equilibrium in dissipative nonlinear wave systems subject to external energy pumping. In such systems, the elementary excitations, or quasiparticles, can create a Bose-Einstein condensate. We develop a theory on the Bose-Einstein condensation of quasiparticles for various regimes of external excitation, ranging from weak and stationary to ultrastrong pumping, enabling us to determine the number of quasiparticles near the bottom of the energy spectrum and their distribution along wave vectors. We identify physical phenomena leading to condensation in each of the regimes. For weak stationary pumping, where the distribution of quasiparticles deviates only slightly from thermodynamic equilibrium, we define a range of pumping parameters where the condensation occurs and estimate the density of the condensate and the fraction of the condensed quasiparticles. As the pumping amplitude increases, a powerful influx of injected quasiparticles is created by the Kolmogorov-Zakharov scattering cascade, leading to their Bose-Einstein condensation. With even stronger pumping, kinetic instability may occur, resulting in a direct transfer of injected quasiparticles to the bottom of the spectrum. For the case of ultrastrong parametric pumping, we have developed a stationary nonlinear theory of kinetic instability. The theory agrees qualitatively with experimental data obtained using Brillouin light scattering spectroscopy during parametric pumping of magnons in room-temperature films of yttrium-iron garnet.

Nonlinear interactions are crucial in science and engineering. Here, we investigate wave interactions in a highly nonlinear magnetic system driven by parametric pumping leading to Bose-Einstein condensation of spin-wave quantamagnons. Using Brillouin light scattering spectroscopy in yttrium-iron garnet films, we found and identified a set of nonlinear processes resulting in off-resonant spin-wave excitationsvirtual magnons. In particular, we discovered a dynamically strong, correlation-enhanced four-wave interaction process of the magnon condensate with pairs of parametric magnons having opposite wave vectors and fully correlated phases.

We report an exact unique constant-flux power-law analytical solution of the wave kinetic equation for the turbulent energy spectrum, E(k)=C1εacs−−−−√/k, of acoustic waves in 2D with almost linear dispersion law, ωk=csk[1+(ak)2], ak≪1. Here ε is the energy flux over scales, and C1 is the universal constant which was found analytically. Our theory describes, for example, acoustic turbulence in 2D Bose-Einstein condensates (BECs). 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.

The appearance of spontaneous coherence is a fundamental feature of a Bose-Einstein condensate and an essential requirement for possible applications of the condensates for data processing and quantum computing. In the case of a magnon condensate in a magnetic crystal, such computing can be performed even at room temperature. So far, the process of coherence formation in a magnon condensate was inaccessible. We study the evolution of magnon radiation spectra by direct detection of microwave radiation emitted by magnons in a parametrically driven yttrium iron garnet crystal. By using specially shaped bulk samples, we show that the parametrically overpopulated magnon gas evolves to a state, whose coherence is only limited by the natural magnon relaxation into the crystal lattice.

Interaction between quasiparticles of a different nature, such as magnons and phonons in a magnetic medium, leads to the mixing of their properties and the formation of hybrid states in the areas of intersection of individual spectral branches. We recently reported the discovery of a new phenomenon mediated by the magnon-phonon interaction: the spontaneous bottleneck accumulation of magnetoelastic bosons under electromagnetic pumping of pure magnons into a ferrimagnetic yttrium iron garnet film. Here, by studying the transport properties of the accumulated magnetoelastic bosons, we reveal that such accumulation occurs in two frequency-distant groups of quasiparticles: quasiphonons and quasimagnons. They propagate with different velocities in different directions relative to the magnetization field. The theoretical model we propose qualitatively describes the double accumulation effect, and the analysis of the two-dimensional spectrum of quasiparticles in the hybridization region allows us to determine the wave vectors and frequencies of each of the groups.

This is a Reply to Sonins Comment [arXiv preprint arXiv:2003.09912, 2020] on Eltsov and Lvov [Pisma v ZhETF 111, 462 (2020)] in which we provide relation of the energy flux carried by the cascade to the amplitude of the excited Kelvin waves, important for analysis of future experiments.

Magnon BoseEinstein Condensates (BECs) and supercurrents are coherent quantum phenomena, which appear on a macroscopic scale in parametrically populated solid state spin systems. One of the most fascinating and attractive features of these processes is the possibility of magnon condensation and supercurrent excitation even at room temperature. At the same time, valuable information about a magnon BEC state, such as its lifetime, its formation threshold, and coherence, is provided by experiments at various temperatures. Here, we use Brillouin Light Scattering (BLS) spectroscopy for the investigation of the magnon BEC dynamics in a single-crystal film of yttrium iron garnet in a wide temperature range from 30 K to 380K. By comparing the BLS results with previous microwave measurements, we revealed the direct relation between the damping of the condensed and the parametrically injected magnons. The enhanced supercurrent dynamics was detected at 180 K near the minimum of BEC damping.

We report the experimental realization of a space-time crystal with tunable periodicity in time and space in a magnon Bose-Einstein condensate (BEC), formed in a room-temperature yttrium iron garnet (YIG) film by a microwave space-homogeneous magnetic field. The magnon BEC is prepared to have a well-defined frequency and nonzero wave vector. We demonstrate how the crystalline "density" as well as the time and space textures of the resulting crystal may be tuned by varying the experimental parameters: External static magnetic field, temperature, thickness of the YIG film, and power of the microwave field. The proposed space-time crystals provide an additional dimension for exploring dynamical phases of matter and can serve as a model nonlinear Floquet system, that brings in touch the rich fields of classical nonlinear waves, magnonics, and periodically driven systems.

Evolution of an overpopulated gas of magnons to a Bose-Einstein condensate and excitation of a magnon supercurrent, propelled by a phase gradient in the condensate wave function, can be observed at room temperature by means of the Brillouin light scattering spectroscopy in an yttrium iron garnet material. We study these phenomena in a wide range of external magnetic fields in order to understand their properties when externally pumped magnons are transferred towards the condensed state via two distinct channels: a multistage Kolmogorov-Zakharov cascade of the weak-wave turbulence or a one-step kinetic instability process. Our main result is that opening the kinetic instability channel leads to the formation of a much denser magnon condensate and to a stronger magnon supercurrent compared to the cascade mechanism alone.

We studied the transient behavior of the spin current generated by the longitudinal spin Seebeck effect (LSSE) in a set of platinum-coated yttrium iron garnet (YIG) films of different thicknesses. The LSSE was induced by means of pulsed microwave heating of the Pt layer and the spin currents were measured electrically using the inverse spin Hall effect in the same layer. We demonstrate that the time evolution of the LSSE is determined by the evolution of the thermal gradient triggering the flux of thermal magnons in the vicinity of the YIG/Pt interface. These magnons move ballistically within the YIG film with a constant group velocity, while their number decays exponentially within an effective propagation length. The ballistic flight of the magnons with energies above 20K is a result of their almost linear dispersion law, similar to that of acoustic phonons. By fitting the time-dependent LSSE signal for different film thicknesses varying by almost an order of magnitude, we found that the effective propagation length is practically independent of the YIG film thickness. We consider this fact as strong support of a ballistic transport scenario - the ballistic propagation of quasi-acoustic magnons in room temperature YIG.

We report preliminary results of the complementary experimental and numerical studies on spatiotemporal tangle development and streamwise vortex line density (VLD) distribution in counterflowing ⁴He. The experiment is set up in a long square channel with VLD and local temperature measured in three streamwise locations. In the steady state, we observe nearly streamwise-homogeneous VLD. Experimental second-sound data as well as numerical data (vortex filament method in a long planar channel starting with seeding vortices localized in multiple locations) show that the initial build-up pattern of VLD displays complex features depending on the position in the channel, but some tangle properties appear uniform along its length.

An ensemble of magnons, quanta of spin waves, can be prepared as a Bose gas of weakly interacting quasiparticles. Furthermore, the thermalization of the overpopulated magnon gas through magnon-magnon scattering processes, which conserve the number of particles, can lead to the formation of a Bose-Einstein condensate at the bottom of a spin-wave spectrum. However, magnon-phonon scattering can significantly modify this scenario and new quasiparticles are formed - magnetoelastic bosons. Our observations of a parametrically populated magnon gas in a single-crystal film of yttrium iron garnet by means of wave-vector-resolved Brillouin light scattering spectroscopy evidence a novel condensation phenomenon: A spontaneous accumulation of hybrid magnetoelastic bosonic quasiparticles at the intersection of the lowest magnon mode and a transversal acoustic wave.

A supercurrent is a macroscopic effect of a phase-induced collective motion of a quantum condensate. So far, experimentally observed supercurrent phenomena such as superconductivity and superfluidity have been restricted to cryogenic temperatures. Here, we report on the discovery of a supercurrent in a Bose-Einstein magnon condensate prepared in a room-temperature ferrimagnetic film. The magnon condensate is formed in a parametrically pumped magnon gas and is subject to a thermal gradient created by local laser heating of the film. The appearance of the supercurrent, which is driven by a thermally induced phase shift in the condensate wavefunction, is evidenced by analysis of the temporal evolution of the magnon density measured by means of Brillouin light scattering spectroscopy. Our findings offer opportunities for the investigation of room-temperature macroscopic quantum phenomena and their potential applications at ambient conditions.

We summarize recent experiments on thermal counterflow turbulence in superfluid He-4, emphasizing the observation of turbulence in the normal fluid and its effect on the decay process when the heat flux is turned off. We argue that what is observed as turbulence in the normal fluid is a novel form of coupled turbulence in the superfluid and normal components, with injection of energy on the scales of both the (large) channel width and the (small) spacing between quantized vortices. Although an understanding of this coupled turbulence remains challenging, a theory of its decay is developed which accounts for the experimental observations.

A nuclear capture reaction of a single neutron by ultracold superfluid He3 results in a rapid overheating followed by the expansion and subsequent cooling of the hot subregion, in a certain analogy with the big bang of the early universe. It was shown in a Grenoble experiment that a significant part of the energy released during the nuclear reaction was not converted into heat even after several seconds. It was thought that the missing energy was stored in a tangle of quantized vortex lines. This explanation, however, contradicts the expected lifetime of a bulk vortex tangle, 10-5-10-4 s, which is much shorter than the observed time delay of seconds. In this paper we propose a scenario that resolves the contradiction: the vortex tangle, created by the hot spot, emits isolated vortex loops that take with them a significant part of the tangle's energy. These loops quickly reach the container walls. The dilute ensemble of vortex loops attached to the walls can survive for a long time, while the remaining bulk vortex tangle decays quickly.

We study, numerically and analytically, the relationship between the Eulerian spectrum of kinetic energy, E _E(k, t), in isotropic turbulence and the corresponding Lagrangian frequency energy spectrum, E _L(ω, t), for which we derive an evolution equation. Our DNS results show that not only E _L(ω, t) but also the Lagrangian frequency spectrum of the dissipation rate ε L (ω, t) has its maximum at low frequencies (about the turnover frequency of energy-containing eddies) and decays exponentially at large frequencies ω (about a half of the Kolmogorov microscale frequency) for both stationary and decaying isotropic turbulence. Our main analytical result is the derivation of equations that bridge the Eulerian and Lagrangian spectra and allow the determination of the Lagrangian spectrum, E _L (ω) for a given Eulerian spectrum, E _E (k), as well as the Lagrangian dissipation, ε L (ω), for a given Eulerian counterpart, ε E (k) = 2 ν k 2 E E (k). These equations were derived from the Navier-Stokes equations in the sweeping-free coordinate system (intermediate between the Eulerian and Lagrangian frameworks) which eliminates the effect of the kinematic sweeping of the small eddies by the larger eddies. We show that both analytical relationships between E _L (ω) and E _E (k) and between ε L (ω) and ε E (k) are in very good quantitative agreement with our DNS results and explain how ε L (ω, t) has its maximum at low frequencies and decays exponentially at large frequencies.

Kelvin waves propagating on quantum vortices play a crucial role in the phenomenology of energy dissipation of superfluid turbulence. Previous theoretical studies have consistently focused on the zero-temperature limit of the statistical physics of Kelvin-wave turbulence. In this letter, we go beyond this athermal limit by introducing a small but finite temperature in the form of non-zero mutual friction dissipative force; A situation regularly encountered in actual experiments of superfluid turbulence. In this case we show that there exists a new typical length scale separating a quasi-inertial range of Kelvin-wave turbulence from a far-dissipation range. The letter culminates with analytical predictions for the energy spectrum of the Kelvin-wave turbulence in both of these regimes.

We report the observation of the unusual behavior of induction decay signals in antiferromagnetic monocrystals with Suhl-Nakamura interactions. The signals show the formation of the Bose-Einstein condensation (BEC) of magnons and the existence of spin supercurrent, in complete analogy with the spin superfluidity in the superfluid He3 and the atomic BEC of quantum gases. In the experiments described here, the temperature of the magnon BEC is a thousand times larger than in the superfluid He3. It opens a possibility to apply the spin supercurrent for various magnetic spintronics applications.

We study the statistical and dynamical behavior of turbulent Kelvin waves propagating on quantized vortices in superfluids and address the controversy concerning the energy spectrum that is associated with these excitations. Finding the correct energy spectrum is important because Kelvin waves play a major role in the dissipation of energy in superfluid turbulence at near-zero temperatures. In this paper, we show analytically that the solution proposed by enjoys existence, uniqueness, and regularity of the prefactor. Furthermore, we present numerical results of the dynamical equation that describes to leading order the nonlocal regime of the Kelvin-wave dynamics. We compare our findings with the analytical results from the proposed local and nonlocal theories for Kelvin-wave dynamics and show an agreement with the nonlocal predictions. Accordingly, the spectrum proposed by L'vov and Nazarenko should be used in future theories of quantum turbulence. Finally, for weaker wave forcing we observe an intermittent behavior of the wave spectrum with a fluctuating dissipative scale, which we interpreted as a finite-size effect characteristic of mesoscopic wave turbulence.

In the Ref. (Lebedev and L'vov in J. Low Temp. Phys. 161, 2010, doi:10.1007/s10909-010-0215-2), this issue, two of us (VVL and VSL) considered symmetry restriction on the interaction coefficients of Kelvin waves and demonstrated that linear in small wave vector asymptotic, obtained analytically, is not forbidden, as Kosik and Svistunov (KS) expect by naive reasoning. Here we discuss this problem in additional details and show that theoretical objections by KS, presented in Ref. (Kozik and Svistunov in J. Low Temp. Phys. 161, 2010, doi:10.1007/s10909-010-0242-z), this issue, are irrelevant and their recent numerical simulation, presented in Ref. (Kozik and Svistunov in arXiv:1007.4927v1, 2010)is hardly convincing. There is neither proof of locality nor any refutation of the possibility of linear asymptotic of interaction vertices in the KS texts, Refs. (Kozik and Svistunov in J. Low Temp. Phys. 161, 2010, doi:10.1007/s10909-010-0242-z; arXiv:1006.0506v1, 2010). Therefore we can state again that we have no reason to doubt in this asymptote, that results in the L'vov-Nazarenko energy spectrum of Kelvin waves.

A central question in the dynamics of vortex lines in superfluids is dissipation on approaching the zero temperature limit T→0. From both NMR measurements and vortex filament calculations, we find that vortex flow remains laminar up to large Reynolds numbers Reα∼103 in a cylinder filled with He3-B. This is different from viscous fluids and superfluid He4, where the corresponding responses are turbulent. In He3-B, laminar vortex flow is possible in the bulk volume even in the presence of sizable perturbations from axial symmetry to below 0.2Tc. The laminar flow displays no excess dissipation beyond mutual friction, which vanishes in the T→0 limit, in contrast with turbulent vortex motion where dissipation has been earlier measured to approach a large T-independent value at T0.2Tc.

We numerically study two-dimensional quantum turbulence with a Gross-Pitaevskii model. With the energy initially accumulated at large scale, quantum turbulence with many quantized vortex points is generated. Due to the lack of enstrophy conservation in this model, direct energy cascade with a Kolmogorov-Obukhov energy spectrum E(k)k^-5^/3 is observed, which is quite different from two-dimensional incompressible classical turbulence in the decaying case. A positive value for the energy flux guarantees a direct energy cascade in the inertial range (from large to small scales). After almost all the energy at the large scale cascades to the small scale, the compressible kinetic energy realizes the thermodynamic equilibrium state without quantized vortices.

We argue that the physics of interacting Kelvin Waves (KWs) 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 differential) 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 absent), 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.

We advance our prior energy- and flux-budget (EFB) turbulence closure model for stably stratified atmospheric flow and extend it to account for an additional vertical flux of momentum and additional productions of turbulent kinetic energy (TKE), turbulent potential energy (TPE) and turbulent flux of potential temperature due to large-scale internal gravity waves (IGW). For the stationary, homogeneous regime, the first version of the EFB model disregarding large-scale IGW yielded universal dependencies of the flux Richardson number, turbulent Prandtl number, energy ratios, and normalised vertical fluxes of momentum and heat on the gradient Richardson number, Ri. Due to the large-scale IGW, these dependencies lose their universality. The maximal value of the flux Richardson number (universal constant ≈0.2-0.25 in the no-IGW regime) becomes strongly variable. In the vertically homogeneous stratification, it increases with increasing wave energy and can even exceed 1. For heterogeneous stratification, when internal gravity waves propagate towards stronger stratification, the maximal flux Richardson number decreases with increasing wave energy, reaches zero and then becomes negative. In other words, the vertical flux of potential temperature becomes counter-gradient. Internal gravity waves also reduce the anisotropy of turbulence: in contrast to the mean wind shear, which generates only horizontal TKE, internal gravity waves generate both horizontal and vertical TKE. Internal gravity waves also increase the share of TPE in the turbulent total energy (TTE = TKE + TPE). A well-known effect of internal gravity waves is their direct contribution to the vertical transport of momentum. Depending on the direction (downward or upward), internal gravity waves either strengthen or weaken the total vertical flux of momentum. Predictions from the proposed model are consistent with available data from atmospheric and laboratory experiments, direct numerical simulations and large-eddy simulations.

New techniques, both for generating and detecting turbulence in the helium superfluids ³ He-B and ⁴ He, have recently given insight in how turbulence is started, what the dissipation mechanisms are, and how turbulence decays when it appears as a transient state or when externally applied turbulent pumping is switched off. Important simplifications are obtained by using ³ He-B as working fluid, where the highly viscous normal component is practically always in a state of laminar flow, or by cooling ⁴ He to low temperatures where the normal fraction becomes vanishingly small. We describe recent studies from the low temperature regime, where mutual friction becomes small or practically vanishes. This allows us to elucidate the mechanisms at work in quantum turbulence on approaching the zero temperature limit.

We ask what determines the (small) angle of turbulent jets. To answer this question we first construct a deterministic vortex-street model representing the large-scale structure in a self-similar plane turbulent jet. Without adjustable parameters the model reproduces the mean velocity profiles and the transverse positions of the large-scale structures, including their mean sweeping velocities, in a quantitative agreement with experiments. Nevertheless, the exact self-similar arrangement of the vortices (or any other deterministic model) necessarily leads to a collapse of the jet angle. The observed (small) angle results from a competition between vortex sweeping tending to strongly collapse the jet and randomness in the vortex structure, with the latter resulting in a weak spreading of the jet.

In this Letter, we suggest a simple and physically transparent analytical model of pressure driven turbulent wall-bounded flows at high but finite Reynolds numbers Re. The model provides an accurate quantitative description of the profiles of the mean-velocity and Reynolds stresses (second order correlations of velocity fluctuations) throughout the entire channel or pipe, for a wide range of Re, using only three Re-independent parameters. The model sheds light on the long-standing controversy between supporters of the century-old log-law theory of von Kàrmàn and Prandtl and proposers of a newer theory promoting power laws to describe the intermediate region of the mean velocity profile.

We revise the theory of superfluid turbulence near the absolute zero of temperature and suggest a differential approximation model for the energy fluxes in the k-space, ε _HD(k) and ε _KW(k), carried, respectively, by the collective hydrodynamic (HD) motions of quantized vortex lines and by their individual uncorrelated motions known as Kelvin waves (KW). The model predicts energy spectra of the HD and the KW components of the system, Ε _HD(k) and Ε _KW(k), which experience a smooth crossover between different regimes of motion over a finite range of scales. For an experimentally relevant range of Δ ≡ ln (l/ a) (l is the mean intervortex separation and a is the vortex core radius) between 10 and 15 the total energy flux ε = ε _HD(k) + ε _KW(k) and the total energy spectrum Ε(k) = Ε _HD(k) + Ε _KW(k) are dominated by the HD motions for k HD = const.: Ε(k) ∝ k ^-5/3 for smaller k and tends to equipartition of the HD energy Ε(k) ∝ k ² for larger k. This bottleneck accumulation of the energy spectrum is milder than the one predicted before in (L'vov et al. in Phys. Rev. B 76:024520, 2007) based on a model with sharp HD-KW transition. For Δ = 15, it results in a prediction for the effective viscosity ν ≃ 0.004κ (κ is the circulation quantum) which is in a reasonable agreement with its experimental value in ⁴He low-temperature experiment ≈0.003κ (Walmsley et al. in Phys. Rev. Lett. 99:265302, 2007). For k>2/l, the energy spectrum is dominated by the KW component: almost flux-less KW component close to the thermodynamic equilibrium, Ε ≈ Ε _KW ≈ const at smaller k and the KW cascade spectrum Ε(k) → Ε _KW(k) ∝ k ^-7/5 at larger k.

An analytical model for the time-developing turbulent boundary layer (TD TBL) over a flat plate is presented. The model provides explicit formulae for the temporal behavior of the wall-shear stress and both the temporal and spatial distributions of the mean streamwise velocity, the turbulence kinetic energy and Reynolds shear stress. The resulting profiles are in good agreement with the DNS results of spatially-developing turbulent boundary layers at momentum thickness Reynolds numbers equal to 1430 and 2900 [5-7]. Our analytical model is, to the best of our knowledge, the first of its kind for TD TBL.

We suggest a way of rationalizing intraseasonal oscillations of Earth's atmospheric flow as four meteorologically relevant triads of interacting planetary waves, isolated from the system of all of the rest of the planetary waves. Our model is independent of the topography (mountains, etc.) and gives a natural explanation of intraseasonal oscillations in both the Northern and the Southern Hemispheres. Spherical planetary waves are an example of a wave mesoscopic system obeying discrete resonances that also appears in other areas of physics.

In this Letter, we discuss the parametric instability of the texture of homogeneous (in time) spin precession and explain how the spatial inhomogeneity of the texture may change the threshold of the instability in comparison with the idealized spatial homogeneous case considered in our JETP Letter 83, 530 (2006). This discussion is inspired by the critical comment of I.A. Fomin (JETP Lett., this issue) related to the above questions. In addition, we considered here the results of direct numerical solution of the full Leggett-Takagi equation of motion for magnetization in ³He-B and experimental data for the magnetic field dependence of the catastrophic relaxation that provide solid support for the theory of this phenomenon presented in our 2006 JETP Letter.

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

We employ the full FENE-P model of the hydrodynamics of a dilute polymer solution to derive a theoretical approach to drag reduction in wall-bounded turbulence. We recapture the results of a recent simplified theory which derived the universal maximum drag reduction (MDR) asymptote, and complement that theory with a discussion of the cross-over from the MDR to the Newtonian plug when the drag reduction saturates. The FENE-P model gives rise to a rather complex theory due to the interaction of the velocity field with the polymeric conformation tensor, making analytic estimates quite taxing. To overcome this we develop the theory in a computer-assisted manner, checking at each point the analytic estimates by direct numerical simulations (DNS) of viscoelastic turbulence in a channel.

Drag reduction by polymers in wall turbulence is bounded from above by a universal maximal drag reduction (MDR) velocity profile that is a log law, estimated experimentally by Virk as V+(y+) 11.7log y+-17. Here V+(y) and y+ are the mean streamwise velocity and the distance from the wall in "wall" units. In this Letter we propose that this MDR profile is an edge solution of the Navier-Stokes equations (with an effective viscosity profile) beyond which no turbulent solutions exist. This insight rationalizes the universality of the MDR and provides a maximum principle which allows an ab initio calculation of the parameters in this law without any viscoelastic experimental input.

We address the "additive equivalence" discovered by Virk and co-workers: drag reduction affected by flexible and rigid rodlike polymers added to turbulent wall-bounded flows is limited from above by a very similar maximum drag reduction (MDR) asymptote. Considering the equations of motion of rodlike polymers in wall-bounded turbulent ensembles, we show that although the microscopic mechanism of attaining the MDR is very different, the macroscopic theory is isomorphic, rationalizing the interesting experimental observations.

The drag reduction by polymers in turbulent flows was analyzed. The drag reduction, in a recent theory, was found to be consistent with the effective viscocity growing linearly with the distance from the wall. the reduction in the Reynolds stress overwhelmed the increase in the viscous drag by the self-consistent solution. By using the direct numerical simulations it was shown that a linear viscosity profile reduced the drag in agreement with the theory and in close correspondence with direct simulations of the FENE-P model at the same flow conditions.

The mechanism of drag reduction by polymers in turbulent wall-bounded flows was investigated. A new logarithmic law for the mean velocity was derived with a slope that fit existing data. The phenomenonology of the maximum drag reduction asymptote which is the maximum drag reduction attained by polymers was developed on the basis of the equations of fluid mechanics. It was observed that the mechanism of drag reduction is the suppression of the Reynolds stress in the elastic sublayer.

We suggested a multizone shell (MZS) model for wall-bounded flows accounting for the space inhomogeneity in a piecewise approximation, in which the cross-sectional area of the flow, [Formula presented] is subdivided into j zones. The area of the first zone, responsible for the core of the flow, [Formula presented] and the areas of the next j zones, [Formula presented] decrease toward the wall like [Formula presented] In each j zone the statistics of turbulence is assumed to be space homogeneous and is described by the set of shell velocities [Formula presented] for turbulent fluctuations of the scale proportional to [Formula presented] The MZS model includes a set of complex variables [Formula Presented] describing the amplitudes of the near-wall coherent structures of the scale [Formula presented] and responsible for the mean velocity profile. The suggested MZS equations of motion for [Formula presented] and [Formula presented] preserve the actual conservation laws (energy, mechanical, and angular momenta), respect the existing symmetries (including Galilean and scale invariance), and account for the type of nonlinearity in the Navier-Stokes equation, dimensional reasoning, etc. The MZS model qualitatively describes important characteristics of the wall-bounded turbulence, e.g., evolution of the mean velocity profile with increasing Reynolds number Re from the laminar profile toward the universal logarithmic profile near the flat-plane boundary layer as [Formula presented].

The weak version of universality in turbulence refers to the independence of the scaling exponents of the nth order structure functions from the statistics of the forcing. The strong version includes universality of the coefficients of the structure functions in the isotropic sector, once normalized by the mean energy flux. We demonstrate that shell models of turbulence exhibit strong universality for both forced and decaying turbulence. The exponents and the normalized coefficients are time independent in decaying turbulence, forcing independent in forced turbulence, and equal for decaying and forced turbulence. We conjecture that this is also the case for Navier-Stokes turbulence.

Motivated by the large effect of turbulent drag reduction by minute concentrations of polymers, we study the effects of a weakly space-dependent viscosity on the stability of hydrodynamic flows. In a recent paper [Phys. Rev. Lett. 87, 174501, (2001)], we exposed the crucial role played by a localized region where the energy of fluctuations is produced by interactions with the mean flow (the \u201ccritical layer\u201d). We showed that a layer of a weakly space-dependent viscosity placed near the critical layer can have a very large stabilizing effect on hydrodynamic fluctuations, retarding significantly the onset of turbulence. In this paper we extend these observations in two directions: first we show that the strong stabilization of the primary instability is also obtained when the viscosity profile is realistic (inferred from simulations of turbulent flows with a small concentration of polymers). Second, we analyze the secondary instability (around the time-dependent primary instability) and find similar strong stabilization. Since the secondary instability develops around a time-dependent solution and is three dimensional, this brings us closer to the turbulent case. We reiterate that the large effect is not due to a modified dissipation (as is assumed in some theories of drag reduction), but due to reduced energy intake from the mean flow to the fluctuations. We propose that similar physics act in turbulent drag reduction.

A theory of clustering of inertial particles advected by a turbulent velocity field caused by an instability of their spatial distribution is suggested. The reason for the clustering instability is a combined effect of the particles inertia and a finite correlation time of the velocity field. The crucial parameter for the clustering instability is the size of the particles. The critical size is estimated for a strong clustering (with a finite fraction of particles in clusters) associated with the growth of the mean absolute value of the particles number density and for a weak clustering associated with the growth of the second and higher moments. A new concept of compressibility of the turbulent diffusion tensor caused by a finite correlation time of an incompressible velocity field is introduced. In this model of the velocity field, the field of Lagrangian trajectories is not divergence free. A mechanism of saturation of the clustering instability associated with the particles collisions in the clusters is suggested. Applications of the analyzed effects to the dynamics of droplets in the turbulent atmosphere are discussed. An estimated nonlinear level of the saturation of the droplets number density in clouds exceeds by the orders of magnitude their mean number density. The critical size of cloud droplets required for cluster formation is more than [formula presented].

Victor Iosifovich Belinicher, a prominent Russian theoretical physicist, was among the passengers on the airplane traveling from Tel Aviv, Israel, to Novosibirsk, Russia, that was accidentally hit by a Ukrainian antiaircraft missile fired during military exercises on 4 October 2001. This senseless tragedy ended the life of a highly respected member of the scientific community, a colleague, a friend, and a teacher.Born on 7 November 1945 in Sverdlovsk, Russia, Belinicher was one of the first students who graduated from the newly established Novosibirsk University in the Akademgorodok Scientific Center in Novosibirsk. In 1971, he received his PhD in physics and mathematics, under the supervision of Dmitrii Shirkov, for work on the Lagrangian formalism for particles of arbitrary spin. He earned his DSc degree in 1982 on the theory of the photogalvanic effect.In 1973, Belinicher joined the Institute of Automation and Electrometry of the Russian Academy of Sciences (RAS) as a research scientist. He particularly enjoyed the atmosphere of intense scientific interaction and intellectual challenge at Akademgorodok. He studied a variety of problems in condensed matter physics, semiconductor physics, and the physics of nonlinear phenomena. One of his most important contributions was a comprehensive theory of the photogalvanic effect in crystals that lack inversion symmetry. The systematic microscopic theory worked out by Belinicher and his colleagues addressed the most important mechanisms producing the effect, ultimately connecting the effects origin to the absence of a detailed balance condition in media that lack a center of symmetry. A review article by Belinicher and Boris I. Sturman published in 1980 in Soviet Physics Uspekhi on the photogalvanic effect remains one of the most cited works in the field.While at the institute, Belinicher began a long-lasting collaboration with one of us (Lvov). In 1984, they developed a new diagrammatic techniquewhich generalized the Keldysh formalism to the case of spin operatorsfor nonequilibrium processes in magnetism. In subsequent work in 1987, they formulated the scale-invariant theory of hydrodynamic turbulence. An earlier approach, advanced in the 1960s by Henry Wyld, suffered from divergences in each order of perturbation theory, and known regularizations were limited to the first order. Belinicher and Lvov found a transformation, now named after them, which removes divergences from all orders. This notable work created a basis for further intensive studies of hydrodynamic turbulence by means of modern methods of theoretical physics.Belinicher joined the RASs Institute of Semiconductor Physics in 1988 as a leading research scientist. His interests turned toward the newly discovered high-T c superconductors and to the general problem of strong electronic correlations associated with those materials. Belinicher worked intensely to develop a model that would describe the essential electronic properties of high-T c materials. The contributions he made, together with his graduate students and colleagues, to the understanding of the physics of the Hubbard model, spin liquids, and spinpolaron mechanisms of superconductivity significantly influenced progress in those areas. In recent years, he devoted much of his time to the derivation of a consistent field theory of the two-dimensional antiferromagnet and to yet another strongly correlated problemCoulomb blockade in a system of quantum dots. Undoubtedly, Belinichers contributions to physics will continue to serve as a basis for new findings, and his work will become part of textbooks for future generations of physicists.Belinicher also lectured and mentored many students at Novosibirsk University, where he was appointed as a professor of physics in 1995. He taught both undergraduate and advanced theoretical courses.In the 1990s, Belinicher was a visiting professor at several institutions in Europe and Israel, particularly at the University of Coimbra in Portugal, and the Weizmann Institute of Science in Rehovot, Israel, where he contributed invaluably to many studies. His outgoing, cooperative character together with an extraordinary dedication to physics brought him many friends and professional colleagues from all around the world.Belinicher was an intense and persistent man who believed deeply in maintaining the quality of his research. For him, science was the absolute priority in his life. He approached nonscientific problems with the same energetic attitude that he applied to his research, whether they involved an issue in turbulent Russian politics or a development of the local computer network. He taught students to treasure the time they devoted to research.Belinicher was open and optimistic, and generously shared his energy and vigor. He was a bright, talented theorist who pursued the deepest and most complicated problems in physics. He was full of plans for the future when the terrible accident ended his life. His family, friends, and colleagues have suffered a great loss

We suggested a one-fluid model of a turbulent dilute suspension which accounts for the "two-way'' fluid-particle interactions by k-dependent effective density of suspension and additional damping term in the Navier-Stokes equation. We presented analytical description of the particle modification of turbulence including scale invariant suppression of a small k part of turbulent spectrum (independent of the particle response time) and possible enhancement of large k region [up to the factor (1+ϕ)2/3]. Our results are in agreement with qualitative picture of isotropic homogeneous turbulence of dilute suspensions previously observed in laboratory and numerical experiments.

The major difficulty in developing theories for anomalous scaling in hydrodynamic turbulence is the lack of a small parameter. In this letter we introduce a shell model of turbulence that exhibits anomalous scaling with a tunable parameter ∈, 0 ≤ ∈ ≤ 1, representing the ratio between deterministic and random components in the coupling between N identical copies of the turbulent field. Our numerical experiments give strong evidence that in the limit N → ∞ anomalous scaling sets in proportional to ∈⁴. This result shows consistency with the nonperturbative closure proposed by the authors in Phys. Fluids, 12 (2000) 803. In this procedure closed equations of motion for the low-order correlation and response functions are obtained, keeping terms proportional to ∈⁰ and ∈⁴, discarding terms of orders ∈⁶ and higher. Moreover we give strong evidences that the birth of anomalous scaling appears at a finite critical ∈, being ∈_c ≈ 0.6.

The statistical objects characterizing turbulence in real turbulent flows differ from those of the ideal homogeneous isotropic model. They contain contributions from various two- and three-dimensional aspects, and from the superposition of inhomogeneous and anisotropic contributions. We employ the recently introduced decomposition of statistical tensor objects into irreducible representations of the SO(3) symmetry group (characterized by j and m indices, where [Formula Presented] to disentangle some of these contributions, separating the universal and the asymptotic from the specific aspects of the flow. The different j contributions transform differently under rotations, and so form a complete basis in which to represent the tensor objects under study. The experimental data are recorded with hot-wire probes placed at various heights in the atmospheric surface layer. Time series data from single probes and from pairs of probes are analyzed to compute the amplitudes and exponents of different contributions to the second order statistical objects characterized by [Formula Presented] 1, and 2. The analysis shows the need to make a careful distinction between long-lived quasi-two-dimensional turbulent motions (close to the ground) and relatively short-lived three-dimensional motions. We demonstrate that the leading scaling exponents in the three leading sectors [Formula Presented] 1, and 2) appear to be different but universal, independent of the positions of the probe, the tensorial component considered, and the large scale properties. The measured values of the scaling exponent are [Formula Presented] [Formula Presented] and [Formula Presented] We present theoretical arguments for the values of these exponents using the Clebsch representation of the Euler equations; neglecting anomalous corrections, the values obtained are 2/3, 1, and 4/3, respectively. Some enigmas and questions for the future are sketched.

Kraichnans model of passive scalar advection in which the driving (Gaussian) velocity field has fast temporal decorrelation is studied as a case model for understanding the anomalous scaling behavior in the anisotropic sectors of turbulent fields. We show here that the solutions of the Kraichnan equation for the n-order correlation functions foliate into sectors that are classified by the irreducible representations of the [Formula Presented] symmetry group. We find a discrete spectrum of universal anomalous exponents, with a different exponent characterizing the scaling behavior in every sector. Generically the correlation functions and structure functions appear as sums over all these contributions, with nonuniversal amplitudes that are determined by the anisotropic boundary conditions. The isotropic sector is always characterized by the smallest exponent, and therefore for sufficiently small scales local isotropy is always restored. The calculation of the anomalous exponents is done in two complementary ways. In the first they are obtained from the analysis of the correlation functions of gradient fields. The theory of these functions involves the control of logarithmic divergences that translate into anomalous scaling with the ratio of the inner and the outer scales appearing in the final result. In the second method we compute the exponents from the zero modes of the Kraichnan equation for the correlation functions of the scalar field itself. In this case the renormalization scale is the outer scale. The two approaches lead to the same scaling exponents for the same statistical objects, illuminating the relative role of the outer and inner scales as renormalization scales. In addition we derive exact fusion rules, which govern the small scale asymptotics of the correlation functions in all the sectors of the symmetry group and in all dimensions.

We show that the Sabra shell model of turbulence, which was introduced recently, displays a Hamiltonian structure for given values of the parameters. The requirement of scale independence of the flux of this Hamiltonian allows us to compute exactly a one-parameter family of anomalous scaling exponents associated with 4th-order correlation functions.

We develop a consistent closure procedure for the calculation of the scaling exponents ζ_n of the nth-order correlation functions in fully developed hydrodynamic turbulence, starting from first principles. The closure procedure is constructed to respect the fundamental rescaling symmetry of the Euler equation. The starting point of the procedure is an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents ζ_n. This hierarchy was discussed in detail in a recent publication by V. S. L'vov and I. Procaccia. The scaling exponents in this set of equations cannot be found from power counting. In this paper we present in detail the lowest nontrivial closure of this infinite set of equations, and prove that this closure leads to the determination of the scaling exponents from solvability conditions. The equations under consideration after this closure are nonlinear integro-differential equations, reflecting the nonlinearity of the original Navier-Stokes equations. Nevertheless they have a very special structure such that the determination of the scaling exponents requires a procedure that is very similar to the solution of linear homogeneous equations, in which amplitudes are determined by fitting to the boundary conditions in the space of scales. The renormalization scale that is necessary for any anomalous scaling appears at this point. The Hölder inequalities on the scaling exponents select the renormalization scale as the outer scale of turbulence L. We demonstrate that the solvability condition of our equations leads to non-Kolmogorov values of the scaling exponents ζ_n. Finally, we show that this solutions is a first approximation in a systematic series of improving approximations for the calculation of the anomalous exponents in turbulence.

We propose a scheme for the calculation from the Navier-Stokes equations of the scaling exponents in of the nth order correlation functions in fully developed hydrodynamic turbulence. The scheme is nonperturbative and constructed to respect the fundamental rescaling symmetry of the Euler equation. It constitutes an infinite hierarchy of coupled equations that are obeyed identically with respect to scaling for any set of scaling exponents zeta(n). As a consequence the scaling exponents are determined by solvability conditions and not from power counting. It is argued that in order to achieve such a formulation one must recognize that the many-point spacetime correlation functions are not scale invariant in their time arguments, The assumption of full scale invariance leads unavoidably to Kolmogorov exponents. It is argued that the determination of all the scaling exponents in requires equations for infinitely many renormalized objects. One can however proceed in controlled successive approximations by successive truncations of the infinite hierarchy of equations. Clues as to how to truncate without reintroducing power counting can be obtained from renormalized perturbation theory. To this aim we show that the fully resummed perturbation theory is equivalent in its contents to the exact hierarchy of equations obeyed by the nth order correlation functions and Green's function. In light of this important result we can safely use finite resummations to construct successive closures of the infinite hierarchy of equations. This paper presents the conceptual and technical details of the scheme. The analysis of the high-order closure procedures which do not destroy the rescaling symmetry and the actual calculations for truncated models will be presented in a forthcoming paper in collaboration with V. Belinicher. (C) 1998 Elsevier Science B.V. All rights reserved.

In a series of recent works it was proposed that shell models of turbulence exhibit inertial range scaling exponents that depend on the nature of the dissipative mechanism. If true, and if one could imply a similar phenomenon to Navier-Stokes turbulence, this finding would cast strong doubts on the universality of scaling in turbulence. In this Letter we propose that these \u201cnonuniversalities\u201d are just corrections to scaling that disappear when the Reynolds number goes to infinity.

The phenomenology of the scaling behavior of higher order structure functions of velocity differences across a scale R in turbulence should be built around the irreducible representations of the rotation symmetry group. Every irreducible representation is associated with a scalar function of R which may exhibit different scaling exponents. The common practice of using moments of longitudinal and transverse fluctuations mixes different scalar functions and therefore may mix different scaling exponents. It is shown explicitly how to extract pure scaling exponents for correlation functions of arbitrary orders.

We develop expressions for the nonlinear wave damping and frequency correction of a field of random, spatially homogeneous, acoustic waves. The implications for the nature of the equilibrium spectral energy distribution are discussed.

The anomalous scaling behavior of the nth-order correlation functions F_n of the Kraichnan model of turbulent passive scalar advection is believed to be dominated by the homogeneous solutions (zero modes) of the Kraichnan equation B_nF_n = 0. In this paper we present an extensive analysis of the simplest (nontrivial) case of n = 3 in the isotropic sector. The main parameter of the model, denoted as ζ_h, characterizes the eddy diffusivity and can take values in the interval 0≤ζ_h≤2. After choosing appropriate variables we can present nonperturbative numerical calculations of the zero modes in a projective two dimensional circle. In this presentation it is also very easy to perform perturbative calculations of the scaling exponent ζ₃ of the zero modes in the limit ζ_h→0, and we display quantitative agreement with the nonperturbative calculations in this limit. Another interesting limit is ζ_h→2. This second limit is singular, and calls for a study of a boundary layer using techniques of singular perturbation theory. Our analysis of this limit shows that the scaling exponent ζ₃ vanishes as √ζ₂/|Inζ₂, where ζ₂ is the scaling exponent of the second-order correlation function. In this limit as well, perturbative calculations are consistent with the nonperturbative calculations.

The theoretical calculation of the scaling exponents that characterize the statistics of fully developed turbulence is one of the major open problems of statistical physics. We review the subject, explain some of the recent developments, and point out the road ahead.

Elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described. We focus here on the structure functions of velocity differences that satisfy inertial range scaling laws S-n(R)similar to R(zeta n), and the correlation of energy dissipation K-epsilon epsilon(R)similar to R(-mu). The goal is to understand from first principles what is the mechanism that is responsible for changing the exponents zeta(n) and mu from their classical Kolmogorov values. In paper II of this series [V. S. L'vov and I. Procaccia, Phys. Rev. E 52, 3858 (1995)] it was shown that the existence of an ultraviolet scale (the dissipation scale eta) is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields. The leading scaling exponent in this family was denoted Delta. The exact resummation of ladder diagrams resulted in a ''bridging relation,'' which determined Delta in terms of zeta(2): Delta = 2-zeta(2). In this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling (i.e., zeta(n) not linear in n) with the renormalization scale being the infrared outer scale of turbulence L. It is shown that deviations from the classical Kolmogorov 1941 theory scaling of S-n(R) (zeta(n) not equal n/3) must appear if the correlation of dissipation is mixing (i.e., mu>0). We suggest possible scenarios for multiscaling, and discuss the implication of these scenarios on the values of the scaling exponents zeta(n) and their ''bridge'' with mu.

Fusion rules in turbulence address the asymptotic properties of many-point correlation functions when some of the coordinates are very close to each other. Here we put to the experimental test some nontrivial consequences of the fusion rules for scalar correlations in turbulence. To this aim we examine passive turbulent advection as well as convective turbulence. Adding one assumption to the fusion rules, one obtains a prediction for universal conditional statistics of gradient fields. We examine the conditional average of the scalar dissipation field [Formula Presented] for [Formula Presented] in the inertial range and find that it is linear in [Formula Presented] with a fully determined proportionality constant. The implications of these findings for the general scaling theory of scalar turbulence are discussed.

In this short note we present a brief overview of our recent progress in understanding the universal statistics of fully developed turbulence, with a stress on anomalous scaling.

It is shown that the idea that scaling behavior in turbulence is limited by one outer length L and one inner length η is untenable. Every nth order correlation function of velocity differences F_n\(R₁, R₂.,\) exhibits its own crossover length η_n to dissipative behavior as a function of R₁. This depends on n and on the remaining separations R₂, R₃., One result is that when separations are of the same order R, this scales as η_n\(R\)∼η\(R/L\)^x_n with x_n = \(ζ_m ζ_n+1 + ζ₃ ζ₂\)/\(2 - ζ₂), ζ_n the scaling exponent of the nth order structure function. We derive an infinite set of scaling relations that bridge the exponents of correlations of gradient fields to the exponents ζ_n, including the \u201cbridge relation\u201d for the scaling exponent of dissipation fluctuations μ = 2-ζ₆.

The lectures presented by one of us (IP) at the Les Houches summer school dealt with the scaling properties of high Reynolds number turbulence in fluid flows. The results presented are available in the literature and there is no real need to reproduce them here. Quite on the contrary, some of the basic tools of the field and theoretical techniques are not available in a pedagogical format, and it seems worthwhile to present them here for the benefit of the interested student. We begin with a detailed exposition of the naive perturbation theory for the ensemble averages of hydrodynamic observables (the mean velocity, the response functions and the correlation functions). The effective expansion parameter in such a theory is the Reynolds number (Re); one needs therefore to perform infinite resummations to change the effective expansion parameter. We present in detail the Dyson-Wyld line resummation which allows one to dress the propagators, and to change the effective expansion parameter from Re to O(1). Next we develop the ``dressed vertex" representation of the diagrammatic series. Lastly we discuss in full detail the path-integral formulation of the statistical theory of turbulence, and show that it is equivalent order by order to the Dyson-Wyld theory. On the basis of the material presented here one can proceed smoothly to read the recent developments in this field.

We show that the Kolmogorov-1941 picture of fully developed hydrodynamic turbulence (with the scaling of the structure functions S_n(R) α R^n/3) necessarily leads to an anomalous scaling for correlation functions which include the rate of energy dissipation e(t, r), these correlation functions being described by an independent index. The mechanism for anomalous scaling, suggested on the basis of the Navier-Stokes equation, is the multi-step interaction of eddies from the inertial interval with eddies at the viscous scale via a set of eddies of intermediate scales.

This paper is the first in a series of papers that aim at understanding the scaling behavior of hydrodynamic turbulence. We present in this paper a perturbative theory for the structure functions and the response functions of the hydrodynamic velocity field in real space and time. Starting from the Navier-Stokes equations (at high Reynolds number Re) we show that the standard perturbative expansions that suffer from infrared divergences can be exactly resummed using the Belinicher-L'vov transformation. After this exact (partial) resummation it is proven that the resulting perturbation theory is free of divergences, both in large and in small spatial separations. The hydrodynamic response and the correlations have contributions that arise from mediated interactions which take place at some space-time coordinates. It is shown that the main contribution arises when these coordinates lie within a shell of a ''ball of locality'' that is defined and discussed. We argue that the real space-time formalism that is developed here offers a clear and intuitive understanding of every diagram in the theory, and of every element in the diagrams. One major consequence of this theory is that none of the familiar perturbative mechanisms may ruin the classical 1941 Kolmogorov (K41) scaling solution for the structure functions. Accordingly, corrections to the K41 solutions should be sought in nonperturbative effects. These effects are the subjects of paper II (the following paper) and a future paper in this series that will propose a mechanism for anomalous scaling in turbulence, which in particular allows a multiscaling of the structure functions.

An exact relation between the Green's function and the dressed third-order vertex Γ was found for the Kardar-Parisi-Zhang (KPZ) model of surface roughening in (1+d) dimensions. This relation, of the Ward-identity type, follows from a hidden symmetry of the problem, which generalizes in some sense the Galilean invariance of the KPZ equation. This relation allows one to conclude that in the region of strong coupling, Γ-Γ0∼0.1Γ0, where Γ0 is the bare value of the vertex Γ. The identity is generalized for higher-order vertices, enabling us to predict some relations between observable correlation functions.

In the inertial interval of turbulence one asserts that the velocity structure functions Sn(r) scale like rnnζ. Recent experiments indicate that Sn(r) has a more general universal form [rf(r/η)]nnζ, where η is the Kolmogorov viscous scale. This form seems to be obeyed on a range of scales that is larger than power law scaling. It is shown here that this extended universality stems from the structure of the Navier-Stokes equations and from the property of the locality of interactions. The approach discussed here allows us to estimate the range of validity of the universal form. In addition, we examine the possibility that the observed deviations from the classical values of ζn=1/3 are due to the finite values of the Reynolds numbers and the anisotropy of the excitation of turbulence.

A mechanism for anomalous scaling in turbulent advection of passive scalars is identified as being similar to a recently discovered mechanism in Navier-Stokes dynamics [V. V. Lebedev and V. S. Lvov, JETP Lett. 59, 577 (1994)]. This mechanism is demonstrated in the context of a passive scalar field that is driven by a rapidly varying velocity field. The mechanism is not perturbative, and its demonstration within renormalized perturbation theory calls for a resummation of infinite sets of diagrams. For the example studied here we make use of a small parameter, the ratio of the typical time scales of the passive scalar vs that of the velocity field, to classify the diagrams of renormalized perturbation theory such that the relevant ones can be resummed exactly. The main observation here, as in the Navier-Stokes counterpart, is that the dissipative terms lead to logarithmic divergences in the diagrammatic expansion, and these are resummed to an anomalous exponent. The anomalous exponent can be measured directly in the scaling behavior of the dissipation two-point correlation function, and it also affects the scaling laws of the structure functions. It is shown that when the structure functions exhibit anomalous scaling, the dissipation correlation function does not decay on length scales that are in the scaling range. The implication of our findings is that the concept of an inertial range in which the dissipative terms can be ignored is untenable. The consequences of this mechanism for other cases of possible anomalous scaling in turbulence are discussed.

A review of magnon properties of yttrium-iron garnet (YIG), a classical object for experimental studies in magnetism, is presented. Both experimental and theoretical results concerned with thermodynamics and kinetics of YIG are described. The main purposes of the review are to introduce a new method of approximate calculation of the magnon spectra in magnets with large unit cell and to obtain by means of this method some basic properties of YIG. In particular, it is shown that the problem of calculating the frequencies of all the 20 magnon branches over the entire Brillouin zone contains two small parameters. First, because of the large number of magnetic atoms in the unit cell the distance between the nearest interacting magnetic atoms is small in comparison with the lattice constant and, accordingly, with the wavelength of a spin wave. An effective long-wavelength character thus arises in the problem. Second, there are a large number of wave-vector directions along which many elements of the Hamiltonian matrix vanish by symmetry in the basis which diagonalizes this matrix for k = 0. These matrix elements thus have an additional, angular smallness for arbitrary directions of k. These matrix elements can be taken into account using perturbation theory. As a result, the large elements of the Hamiltonian matrix are few in number, and they can be eliminated by several two-dimensional rotations. Approximate expressions, differing from the computer calculations by {less-than or approximate} 10%, are thus obtained for the frequencies. Neutron scattering data are used to find the values of the exchange integrals in YIG and to obtain the magnon spectra. It is shown that in the energy range T {less-than or approximate} 260 K only magnons of the lower branch are excited; the spectrum of these "ferromagnons" is quadratic in the wave vector only up to 40 K and becomes linear in the region ω_k {greater-than or approximate} 40 K. For temperatures up to 400 K the temperature dependence of the magnetization is calculated in the spin-wave approximation and good agreement with experimental data is found. A brief review of experimental data on magnon relaxation in YIG is presented. The magnon-magnon interactions which cause the magnon relaxation are described. The amplitude of the four-magnon exchange interaction is determined, and the temperature correction to the frequency is evaluated. This temperature correction is positive, in contrast to the case of simple cubic ferromagnet with nearest-neighbour interaction. The exchange relaxation rate is calculated for normal and umklapp processes. It is shown that the magnetic dipole interaction is important only for the ferromagnons; the amplitude of this interaction and the corresponding relaxation rate are determined. Three-magnon scattering processes are allowed only for wave vectors larger than a certain k_/; at k = k_/ there is a discontinuity in the wave-vector dependence of the damping. A calculation is given for the nonvanishing contribution to the relaxation at k = 0 on account of scattering processes involving optical magnons; this contribution is due to the local uniaxial anisotropy. The relative role of each of the investigated relaxation mechanisms is discussed, and the correspondence of the present results with the experimental data is examined.

Under the assumption that the Kardar-Parisi-Zhang model (KPZ) possesses scale invariant solutions, there exists an exact calculation of the dynamic scaling exponent z=3/2. The authors prove that both KPZ and the related Kuramoto-Sivashinsky model indeed possess scale invariant solutions in 1+1 dimensions which are in fact the same for both models. The proof entails an examination of the higher order diagrams in the perturbation theory in terms of the dressed Green function and the correlator. Although each higher order diagram contains logarithmic divergences, endangering the existence of the scale invariant solution, the authors show that these divergences cancel in each order. The proof uses a fluctuation-dissipation theorem (FDT), which is an exact result for KPZ in 1+1 dimensions.

The scaling ranges of temperature and velocity fluctuations in thermally driven turbulence are studied by analyzing the various contributions to the equations of motion. The crossover wave number kB between Bolgiano-Obukhov and Kolmogorov-Obukhov scaling is estimated in terms of the forcings. By evaluating the thermal and buoyant stirrings and dissipations of Rayleigh-Bénard convection experiments we find kB much larger than L-1, the energy-containing scale, but smaller than (10η)-1, the viscous scale. For computer simulation of randomly thermal driven turbulence we find kB of the order of L-1. This might explain why the Bolgiano-Obukhov scaling was observed in laboratory experiments whereas Kolmogorov-Obukhov scaling was found in computer simulation of thermally driven turbulence.

The stationary spectrum of hydrodynamic convective turbulence is shown to be defined by influxes of two independent motion integrals: entropy and mechanical energy. A careful analysis of the conservation laws is performed. It is shown that in the inertial range of scales kinetic energy converts into potential energy due to presence of temperature fluctuations independently of the type of long-scale stratification (stable or unstable one). Under a purely entropic excitation (for example, by horizontal temperature gradient) the spectrum with constant entropy flux, F_vv ∼ k ^{-21 5}, fills the whole of the inertial interval and crossover to the Kolmogorov-Obukhov spectrum with constant energy flux, F_vv ∼ k ^{-11 3}, is absent. An estimate for crossover scale is obtained for a mixed method of excitation with both nonzero energy pumping and nonzero entropy extraction caused by an environment. A simple but consistent differential model is suggested for the description of the fluxes of energy and entropy in k-space. Two-flux universal spectraof the velocity and temperature fluctuations are obtained.

The long-wavelength properties of the Kuramoto-Sivashinsky equation are studied in 2+1 dimensions using numerical and analytic techniques. It is shown that this equation is not in the universality class of the Kardar-Parisi-Zhang model. Its roughening exponents are (up to logarithmic corrections) like those of the free-field theory, with dimension 2 being the marginal dimension for roughening. Assuming that the solution has logarithmic corrections, we derive a scaling relation for the exponents of the logarithmic terms. This solution is consistent order by order with the Dyson-Wyld diagrams. We explain why previous renormalization-group treatments failed.

The interaction of sound with hydrodynamic turbulence has been studied in detail. The sound absorption decrement, the correlation time and length and the frequency diffusion coefficient for the acoustic wave packet are calculated. The spectral composition of the sound radiated by a unit, turbulent volume and the spectral energy density of sound in equilibrium with the turbulence are studied. The region of applicability of the kinetic equation for sound with a linear dispersion low is found.

For a sequence of growing Reynolds numbers the azimuthal velocity component was measured by a laser Doppler velocimeter using data acquisition system. The power spectra and temporal behaviour of narrow-band filtered velocity flucuations for the spatial states of 28 and 30 Taylor vortices were studied to clarify the question how the Landau conception and stochastic attractor (SA) idea concern the laminar-turbulent transition. It is shown that the transition combines both the Landau and SA hypotheses features. (A)

The aim of the present paper is to construct a consistent nonlinear theory of ferromagnetic resonance which takes into account both the spin wave- uniform precession and spin wave- spin wave interaction. See also English translation in Sov Phys-Solid State v 13 n 2 Aug 1971 p 418-25