We comment on the paper 'Saturation and Multifractality of Lagrangian and Eulerian Scaling Exponents in Three-Dimensional Turbulence' by D. Buaria and K.R. Sreenivasan. We point out some physical interpretation of the results discussed in the paper.

The conventional theory of small-scale magnetic field generation in a turbulent flow considers time-reversible random flows. However, real turbulent flows are known to be time irreversible: the presence of energy cascade is an intrinsic property of turbulence. We generalize the 'standard' model to account for the irreversibility. We show that even small time asymmetry leads to significant suppression of the dynamo effect at low magnetic Prandtl numbers, increases the generation threshold and may even make generation impossible for any magnetic Reynolds number. We calculate the magnetic energy growth rate as a function of the parameters of the flow.

A set of exact integrals of motion is found for systems driven by homogenous isotropic stochastic flow. The integrals of motion describe the evolution of (hyper-)surfaces of different dimensions transported by the flow, and can be expressed in terms of local surface densities. The expression for the integrals is universal: it represents general geometric properties and does not depend on the statistics of the specific flow.

The dynamics of a statistically homogeneous isotropic magnetic field generated by an incompressible turbulent plasma flow with a large, yet finite magnetic Prandtl number is discussed. It has been found that, on scales smaller than the Kolmogorov viscosity scale, the nonlinear feedback of a magnetic field to the fluid dynamics decreases exponentially, despite the fast exponential growth of the magnetic field. It is shown that the anisotropy of diffusion in a cosmic plasma leads to an additional decrease in the feedback. It is demonstrated that the feedback degeneracy leads to an energy paradox, which is resolved at a later stage of the development of initial disturbances when the spatial scale of magnetic fluctuations approaches the Kolmogorov scale. The possibility of the nonlinearity degeneracy in more complex systems is discussed: a similar phenomenon can occur in hydrodynamic turbulence, which makes it possible to find a key to the theoretical analysis of the latter.

We consider the line, surface, and volume elements of fluid in stationary isotropic incompressible stochastic flow in d-dimensional space and investigate the long-time evolution of their statistic properties. We report the discovery of a family of d! - 1 stochastical integrals of motion that are universal in the sense that their explicit form does not depend on the statistics of velocity. Only one of them has been discussed previously.

We consider a natural generalization of the Kazantsev-Kraichnan model for small-scale turbulent dynamo. This generalization takes account of statistical time asymmetry of a turbulent flow, and, thus, allows to describe velocity fields with energy cascade. For three-dimensional velocity field, generalized Kazantsev equation is derived, and evolution of the second order magnetic field correlator is investigated for large but finite magnetic Prandtl numbers. It is shown that as $Pr_m \to \infty$, the growth increment tends to the limit known from the T-exponential (Lagrangian deformation) method. Magnetic field generation is shown to be weaker than that in the Gaussian velocity field for any direction of the energy cascade, and depends essentially on the Prandtl number.

We consider the kinematic stage of evolution of magnetic field advected by turbulent hydrodynamic flow. We use a generalization of the Kazantsev–Kraichnan model to investigate time irreversible flows. In the viscous range of scales, the infinite-time limit of the spectrum is a power law, but its slope is more flat than that predicted by the Kazantsev model. This result agrees with numerical simulations. The rate of magnetic energy growth is slower than that in the time-symmetric case. We show that for high magnetic Prandtl turbulent plasma, the formation of the power-law spectrum shape takes very long time and may never happen because of the nonlinearity. We propose another ansatz to describe the spectrum shape at finite time.

We consider finite-dimensional systems of linear stochastic differential equations ∂_{t}x_{k}(t)=A_{kp}(t)x_{p}(t), A(t) being a stationary continuous statistically isotropic stochastic process with values in real d×d matrices. We suppose that the laws of A(t) satisfy the large-deviation principle. For these systems, we find exact expressions for the Lyapunov and generalized Lyapunov exponents and show that they are determined in a precise way only by the rate function of the diagonal elements of A.

The paper [Djenidi et al., Phys. Fluids 33(3), 031703 (2021)] considers a classical issue of an anomalous scaling of velocity structure functions in a high-Reynolds number turbulent flow. The paper offers a mathematical proof of the ground-breaking result: the intermittency is an artifact of the Reynolds number finiteness. However, the proof contains a technical error that makes this conclusion ungrounded.

Kinematic dynamo in incompressible isotropic turbulent flows with high magnetic Prandtl number is considered. The approach interpreting an arbitrary magnetic field distribution as a superposition of localized perturbations (blobs) is proposed. We derive a relation between stochastic properties of a blob and a stochastically homogenous distribution of magnetic field advected by the same stochastic flow. This relation allows to investigate the evolution of a localized blob at late stage when its size exceeds the viscous scale. It is shown that in 3-dimansional flows, the average magnetic field of the blob increases exponentially in the inertial range of turbulence, as opposed to the late-Batchelor stage when it decreases. Our approach reveals the mechanism of dynamo generation in the inertial range both for blobs and homogenous contributions. It explains the absence of dynamo in the two-dimensional case and its efficiency in three dimensions. We propose the way to observe the mechanism in numerical simulations.

We consider forced small-scale magnetic field advected by an isotropic turbulent flow. The random driving force is assumed to be distributed in a finite region with a scale smaller than the viscous scale of the flow. The two-point correlator is shown to have a stationary limit for any reasonable velocity statistics. Its spatial dependence is found to be a power law. The scaling exponent is found to be close to 3.

We consider fluctuations of a magnetic field excited by an external force and advected by isotropic turbulent flow. It appears that non-Gaussian velocity gradient statistics and a finite region of pumping force provide the existence of a stationary solution. The mean-square magnetic field is calculated for arbitrary velocity gradient statistics. An estimate for possible feedback of the magnetic field on velocity shows that, for a wide range of parameters, stationarity without feedback would take place even in the case of intensive pumping of the magnetic field.

Evolution of stochastically homogeneous magnetic field advected by incompressible turbulent flow with large magnetic Prandtl numbers is considered at the scales less than Kolmogorov viscous scale. It is shown that, despite unlimited growth of the magnetic field, its feedback on the fluid's dynamics remains negligibly small.

On the basis of Kolmogorov’s 4/5 law (Kolmogorov, 1941; Landau, Lifschitz 1975) analytical relations for triple two-point correlations of velocity and velocity gradients in homogeneous isotropic incompressible turbulence are derived (Kopyev, Zybin, 2018). The corresponding correlation tensor can be expressed in terms of the dissipation rate, the second-order correlation function for the longitudinal velocity increment, and the new scalar function of distance between the points. However, some components of the tensor do not depend on the new function. The derived analytical results are in agreement with the data obtained from direct numerical simulations. The function can be well approximated in the inertial range by a constant value that depends on the dissipation only. The application of the obtained correlators in the turbulent transport theory is discussed (Il’yn, Sirota, Zybin, 2016; Kazantsev, 1968).The study was performed in Central Aerohydrodynamic Institute (TsAGI) with the funding of Russian Science Foundation (project No. 17-11-01271).

Statistical properties of a statistically homogeneous random magnetic field in a viscous diffusive fluid are derived from the evolution of a single blob of the magnetic field. It is shown that, although the magnetic field of a single blob decreases in time, the volume occupied by the magnetic field and its energy increase; this is the cause of the magnetic field growth in a homogeneous medium. We also get an exact expression for the increment of the magnetic field in the generalized (not time-inversible) Kraichnan model.

The impact of turbulent advection in reaction-diffusion systems is investigated for the viscous range of scales. We show that the population size can increase exponentially even in systems with density saturation, at the expense of exponential propagation of the reaction front. Exact expressions for scaling exponents of the density and population size are calculated in different intermediate asymptotics of the process. The system appears to demonstrate high intermittency.

(FIAN), an outstanding scientist, corresponding member of the Russian Academy of Sciences (RAS), professor, doctor of physico-mathematical sciences, principal research fellow of the Sector of Plasma Phenomena Theory at FIAN, professor at the Moscow Engineering Physics Institute (MEPhI), Viktor Pavlovich Silin, died on 12 January 2019. V P Silin graduated from the Physical Faculty of Lomonosov Moscow State University (MSU). His diploma thesis advisor was D I Blokhintsev. V P Silin's whole life in science was associated with the Lebedev Physical Institute, where he began working in 1949 immediately after he graduated from MSU and where he moved from junior research worker to head of the Department of Solid State Physics (1989±1995). During that time, he published over 700 scientific papers in various fields of physics. He is the author of four monographs fairly popular among specialists in plasma physics and the physics of condensed matter. The brilliant talent of Viktor Pavlovich became apparent as far back as the early 1950s, when he worked in the Department of Theoretical Physics at FIAN. Work on the development of the Tamm±Dankov method that had provided deeper insight into the nature of nuclear interactions appeared in that period. Simultaneously, he became engaged in the quantum many-body theory. Together with Yu L Klimontovich, he derived the kinetic self-consistent field equation describing a weakly ideal quantum Fermi gas. Using this equation, V P Silin predicted the existence of undamped oscillations in Fermi gas near the temperature of absolute zero. After L D Landau formulated the general theory of a Fermi liquid, these undamped oscillations were called zero sound. V P Silin also applied the Fermi-liquid theory to describe the properties of metals. The equation derived by him and known as the Landau±Silin equation became the basis of the description of collective effects in metals and allowed the prediction of cyclotron and spin waves in normal metals, a description of sound wave absorption in normal metals and conducting magnets, and the prediction of quantum spin waves. In 1970, V P Silin was awarded the USSR State Prize for the formulation of the theory of an electron Fermi liquid. Viktor Pavlovich remained interested in condensed matter physics in subsequent years. In the mid-1960s, he and P S Zyryanov conducted a large series of studies on the theory of waves in quantizing magnetic fields. These studies are well known to specialists in the field of semiconductor physics. A little later, V P Silin, together with younger disciples, developed a new approach to the description of magnetic and elastic properties of invar alloys, predicted surface quantum waves, and investigated collective excitations in ferroand antiferromagnets. One of the last areas of study in solid-state physics, to which Viktor Pavlovich had given attention till his last days, was working out the nonlocal electrodynamics of Josephson junctions and layered Josephson systems. He also obtained results of paramount importance in this area. The possibility of the existence of a whole range of new nonlinear vortex Josephson structures was predicted, the effect of quantization of Josephson vortex velocities was discovered, fast vortices were predicted, and the theory of Cherenkov radiation by vortices and vortex chains was formulated. In the late 1950s±early 1960s, V P Silin's scientific interests involved the rapidly developing plasma physics. The remarkable monograph by V P Silin and A A Rukhadze devoted to the problems of the electrodynamics of plasma and plasmalike media appeared in 1961. The monograph became a handbook for many generations of physicists and has recently been reissued. Almost immediately, a large series of papers appeared with NN Bogoliubov's ideas carried over to plasma physics. A number of new collision integrals in quantizing magnetic fields and in strong high-frequency Uspekhi Fizicheskikh Nauk 189 (5) 559 ± 560 (2019) DOI: https://doi.org/10.3367/UFNr.2019.03.038541 Translated by M V Tsaplina PERSONALIA PACS number: 01.60.+q