Aleksander V. Kolesnichenko

A Synergetic Approach to the Description of Advanced Turbulence

An attempt is made to construct a phenomenological model of turbulence as a self-organization process in an open system. The representation of a turbulized continuum in the form of a thermodynamic complex consisting of two subsystems—the subsystem of averaged motion and the subsystem of turbulent chaos, which is considered, in turn, as a conglomerate of vortex structures of different space–time scales—made it possible to obtain, by methods of nonequilibrium thermodynamics, the defining relationships for the turbulent fluxes and forces that describe most comprehensively the transport and structurization processes in such a continuum. Using two interpretations of the Kolmogorov parameter (as a quantity that describes the rate of dissipation of energy into heat and as the rate of transfer of turbulent energy in the eddy cascade), the defining relationships were found for this quantity, thereby making the thermodynamic approach self-sufficient. An introduction into the model of internal parameters of the medium, which characterize the excitation of macroscopic degrees of freedom, made it possible to describe thermodynamically the Kolmogorov cascade process and to obtain a variety of kinetic equations (of the Fokker–Planck type in the configuration space) for the functions of distribution of small-scale turbulence characteristics, including the unsteady kinetic equation for the distribution of probability of dissipation of turbulent energy. As an example, a detailed derivation of such relationships is given for the case of stationary turbulence, when a tendency toward local isotropy is observed. In view of the wide occurrence of this phenomenon in nature, one might expect that the developed approach to the problem of modeling strong turbulence will find its use in astrophysical and geophysical applications.

Fundamentals of the mechanics of heterogeneous media in the circumsolar protoplanetary cloud: The effects of solid particles on disk turbulence

We formulate a complete system of equations of two-phase multicomponent mechanics including the relative motion of the phases, coagulation processes, phase transitions, chemical reactions, and radiation in terms of the problem of reconstructing the evolution of the protoplanetary gas-dust cloud that surrounded the proto-Sun at an early stage of its existence. These equations are intended for schematized formulations and numerical solutions of special model problems on mutually consistent modeling of the structure, dynamics, thermal regime, and chemical composition of the circumsolar disk at various stages of its evolution, in particular, the developed turbulent motions of a coagulating gas suspension that lead to the formation of a dust subdisk, its gravitational instability, and the subsequent formation and growth of planetesimals. To phenomenologically describe the turbulent flows of disk material, we perform a Favre probability-theoretical averaging of the stochastic equations of heterogeneous mechanics and derive defining relations for the turbulent flows of interphase diffusion and heat as well as for the “relative” and Reynolds stress tensors needed to close the equations of mean motion. Particular attention is given to studying the influence of the inertial effects of dust particles on the properties of turbulence in the disk, in particular, on the additional generation of turbulent energy by large particles near the equatorial plane of the proto-Sun. We develop a semiempirical method of modeling the coefficient of turbulent viscosity in a two-phase disk medium by taking into account the inverse effects of the transfer of a dispersed phase (or heat) on the growth of turbulence to model the vertically nonuniform thermohydrodynamic structure of the subdisk and its atmosphere. We analyze the possible “regime of limiting saturation” of the subdisk atmosphere by fine dust particles that is responsible for the intensification of various coagulation mechanisms in a turbulized medium. For steady motion when solid particles settle to the midplane of the disk under gravity, we analyze the parametric method of moments for solving the Smoluchowski integro-differential coagulation equation for the particle size distribution function. This method is based on the fact that the sought-for distribution function a priori belongs to a certain parametric class of distributions.

Modeling of the Turbulent Transport Coefficients in a Gas–Dust Accretion Disk

An heuristic way of modeling the turbulent exchange coefficients for Keplerian accretion disks surrounding solar-type stars is considered. The formulas for these coefficients, taking into account the inverse effects of dust transfer and potential temperature on the maintenance of shear turbulence, generalize to protoplanetary gas–dust clouds the expression for the turbulent viscosity coefficient in so-called α-disks which was obtained in a classic work by Shakura and Syunyaev (1973). The defining relationships are derived for turbulent diffusion and heat flows, which describe, for the two-phase mixture rotating differentially at an angular velocity Ω(r, z), the dust and heat transfer in the direction perpendicular to the central plane of the disk. The regime of limiting saturation by small dust particles of the layer of “cosmic fluid” located slightly above (or below) the dust subdisk is analyzed.

The effect of spirality on the evolution of turbulence in the solar protoplanetary cloud

We analyze the possible effect of hydrodynamic spirality that develops in a rotating disk on the synergetic structurization of cosmic matter and on the development of negative turbulent viscosity in cosmic matter within the framework of the problem of the reconstruction of the evolution of the protoplanetary cloud that surrounded the early Sun. We show that comparatively slow damping of turbulence in the disk can be partially due to the lack of reflective symmetry of the anisotropic field of turbulent velocities about its equatorial plane. We formulate the general concept of the development of energy-intensive coherent mesoscale vortex structures in the thermodynamically open system of turbulent chaos associated with the realization of inverse cascade of kinetic energy in mirror-nonsymmetrical disk turbulence. Because of energy release, the inverse cascade produces a hierarchical system of mass concentrations with a fractal density distribution, which ultimately initiate the mechanisms of triggered cluster formation. We use the methods of nonequilibrium thermodynamics to prove the possibility of the development of negative viscosity in the three-dimensional case in terms of the two-scale hydrodynamic description of maximally developed disk turbulence. Negative viscosity in a rotating disk system appears to be a manifestation of cascade processes in spiral turbulence where inverse energy transfer from small to larger vortices occurs. Within the framework of asymmetric mechanics of turbulized continua, we physically substantiated the phenomenological formula for the turbulent stress tensor of Wasiutynski, which is widely used in the astrophysical literature to explain the differential rotation of various cosmic objects by “anisotropic viscosity.” The aim of our study is, first and foremost, to improve a number of representative hydrodynamic models of cosmic natural turbulized media, including the birth of galaxies and galaxy clusters, birth of stars from the diffuse medium of gas and dust clouds, formation of accretion disks and subsequent accumulation of planetary systems, and also the formation of gaseous envelopes of planets, atmospheres, etc. This paper continues the application of stochastic and thermodynamic approach to the synergetic description of the structured turbulence of astrogeophysical systems, which we have been developing in a series of our papers (Kolesnichenko, 2004, 2005; Kolesnichenko and Marov, 2006; Marov and Kolesnichenko, 2002, 2006).

Synergetic Mechanism of the Development of Coherent Structures in the Continual Theory of Developed Turbulence

The aim of this paper is to further develop the continual theory of structurized turbulence in shear flows in a fluid modeled by a superposition of two mutually penetrating continua, where the first continuum refers to the averaged field of turbulent motion, and the second, to the turbulent spacetime chaos, which includes an ensemble of mesoscale coherent structures localized in space. It is believed that, in the case of an increase of the supercriticality level, mesoscale structures are generated by small-scale vortex formations, which, in the case of a two-level turbulence model, are described by additional internal parameters of chaos, e.g., generalized angular velocities characterizing vorticities of the pulsational hydrodynamic field. I discuss the possibility of synergetic formation of mesoscale coherent structures from turbulent chaos, which differs strongly from the complete chaos of thermodynamic equilibrium, due to phase synchronization of relatively large small-scale vortices (maximum oscillations within a certain spectral interval) in the presence of noise due to the “thermal” structure of the vortex continuum. I interpret such a mechanism of the formation and evolution of coherent structures in the thermodynamically open subsystem of turbulent chaos in terms of the theory of dynamical systems. The aim of this study is to develop a number of representative hydrodynamic models of natural space environments, including the evolution of the Solar System, turbulent transfer on planets and in their atmospheres, ecological problems, etc. It continues the stochastically thermodynamic approach to the synergetic description of structurized turbulence in astrogeophysical systems that I have been developing in a series of previous papers (Kolesnichenko, 2002, 2003, 2004).

Differential Models for the Closure of Turbulently Averaged Hydrodynamic Equations for a Chemically Active Continuous Medium

The problem of constructing semiempirical second-approximation turbulence models for a multicomponent chemically active gas mixture with a variable density and variable thermophysical properties is addressed. The set of closing differential transfer equations for the various one-point (one-time) second correlation moments of the fluctuating thermohydrodynamic parameters appearing in the averaged hydrodynamic equations of mean motion for a reacting mixture is derived. The closure problem for a chemically active medium is generally greatly complicated due to the necessity of averaging the nonlinear “source terms” of substance production in chemical reactions with an exponential behavior. Therefore, we propose an original procedure for averaging the rates of chemical reactions of any order and outline a scheme for semiempirical modeling of these additional correlations. Approximating expressions containing universal empirical coefficients that need not be chosen again for each new flow are used in modeling the third-order correlations in the transfer equations. It should be emphasized that although these additional equations are semiempirical, the invariant models of fully developed turbulence in chemically active gases based on them are fairly flexible. In particular, they allow the influence of the mechanisms of convection, diffusion, formation, redistribution, and dissipation of stochastic turbulent characteristics for the field of fluctuating thermohydrodynamic parameters on the spatiotemporal distribution of averaged thermohydrodynamic parameters for the medium to be taken into account. Basically, the approach we developed is widely used in numerical simulations of real reacting turbulized fluid flows with a significant influence of the flow prehistory on the turbulence characteristics at a point. On the other hand, it is used to derive more accurate algebraic relations for the turbulent transport coefficients in multicomponent shear mixture flows (and as applied to the specificity of modeling natural media), which is embodied in this chapter of the book.

Stochastic-Thermodynamic Modeling of Developed Structured Turbulence

The context is devoted to the development of a phenomenological model for the developed turbulence in a compressible homogeneous medium by taking into account nonlinear cooperative processes. The representation of a turbulized fluid motion as a thermodynamic system consisting of two continua, the subsystem of averaged motion and the subsystem of turbulent chaos, which, in turn, is considered as a conglomerate of vortex structures of various spatiotemporal scales, serves as the primary concept. We develop the ideas of a stationary-nonequilibrium state of the dissipatively active subsystem of turbulent chaos emerging due to the influx of negentropy from the external medium (the subsystem of averaged motion) and the appearance of relatively stable coherent vortex structures in the system when varying the flow control parameters. This allows some of the turbulent field rearrangement processes to be considered as self-organization processes in an open system. The methods of the stochastic theory of irreversible processes and extended irreversible thermodynamics are used to derive the defining relations for the turbulent fluxes and forces that close the system of averaged hydrodynamic equations and describe the transport and self-organization processes in the stationary-nonequilibrium case with completeness sufficient for practice.

Turbulent Chaos and Self-Organization in Cosmic Natural Media

Natural objects evolve from initial chaotic motions to order through a fascinating internal self-organization, which is embedded in their structure. The dynamics of this process is the focus of this work. They can be subjected to temporal and spatial variations or retain their stability for a long time. Ordered structures surround us ubiquitously on Earth; numerous examples of self-organization are observed in space. Turbulent flows characterized by a great variety of dynamical processes are widespread in the surrounding world. We mainly focus on the problems of macroscopic modeling these natural flows.

Influence of Hydrodynamic Helicity on the Evolution of Turbulence in an Accretion Disk

In this chapter we analyze the influence of hydrodynamic spirality that develops in a rotating disk on the synergetic structurization of disc matter. It is shown that comparatively slow damping of turbulence in the disk is caused by the lack of reflective symmetry of the anisotropic field of turbulent velocities relative its equatorial plane. The general concept of the development of energy-intensive coherent mesoscale eddy structures in the thermodynamically open system of turbulent chaos associated with the realization of inverse cascade of kinetic energy in mirror-nonsymmetrical disk turbulence is formulated. Because of energy release, the inverse cascade produces a hierarchical system of mass concentrations with a fractal density distribution, which ultimately initiate the mechanisms of triggered cluster formation. In turn, eddy coherent formations are responsible for intensification of mechanical and physical-chemical interactions of particles in the flow, which results in spontaneous setup and growth of dust clusters. Condensation, phase transition, and the processes of mass and heat transfer between different regions of the heterogeneous medium as well as significant modification of oscillation patterns (density wave spectrum) are encouraged. It is also shown that negative viscosity in a rotating disk system appears to be a manifestation of cascade processes in spiral turbulence where inverse energy transfer from small to larger vortices occurs.

Self-Organization of Developed Turbulence and Formation Mechanisms of Coherent Structures

Here the so-called H-theorem for the Kullback entropy is proved, from which it follows that any initial probability distribution for the internal coordinates of the subsystem of turbulent chaos under known assumptions asymptotically approaches a certain stationary state after a sufficiently long time. Here, we demonstrate that self-organization (i.e., the emergence of ordered dissipative structures with a lower symmetry than that of the initial state) is possible in principle in the thermodynamically open subsystem of turbulent chaos when the generation of coherent structures associated with the effect of multiplicative-noise-induced nonequilibrium phase transitions in the subsystem of chaos is possible in due course of temporal evolution of the quasi-equilibrium vortex subsystem. We show that if the multiplicative noise of chaos is intense enough, then the extrema of the probability density describing the stationary behavior of a stochastic vortex system differ significantly in both number and position from the stationary states corresponding to a deterministic system. Moreover, multiplicative noise can give rise to new stationary states, thereby changing the properties (in particular, the bifurcation diagrams) of the local stability of chaos themselves: the transition points can be displaced under the influence of intense noise in a turbulent fluid.