On the thermodynamic derivation of differential equations for turbulent flow transfer in a compressible heat-conducting fluid

This paper considers the modern approach to the thermodynamic modeling of developed turbulent flows of a compressible fluid based on the systematic application of the formalism of extended irreversible thermodynamics (EIT) that goes beyond the local equilibrium hypothesis, which is an inseparable attribute of classical nonequilibrium thermodynamics (CNT). In addition to the classical thermodynamic variables, EIT introduces new state parameters—dissipative flows and the means to obtain the respective evolutionary equations consistent with the second law of thermodynamics. The paper presents a detailed discussion of a number of physical and mathematical postulates and assumptions used to build a thermodynamic model of turbulence. A turbulized liquid is treated as an indiscrete continuum consisting of two thermodynamic sub-systems: an averaged motion subsystem and a turbulent chaos subsystem, where turbulent chaos is understood as a conglomerate of small-scale vortex bodies. Under the above formalism, this representation enables the construction of new models of continual mechanics to derive cause-and-effect differential equations for turbulent heat and impulse transfer, which describe, together with the averaged conservations laws, turbulent flows with transverse shear. Unlike gradient (noncausal) relationships for turbulent flows, these differential equations can be used to investigate both hereditary phenomena, i.e., phenomena with history or memory, and nonlocal and nonlinear effects. Thus, within EIT, the second-order turbulence models underlying the so-called invariant modeling of developed turbulence get a thermodynamic explanation. Since shear turbulent flows are widespread in nature, one can expect the given modification of the earlier developed thermodynamic approach to developed turbulence modeling (see Kolesnichenko, 1980; 1998; 2002–2004; Kolesnichenko and Marov, 1985; Kolesnichenko and Marov, 2009) to be used in research on a broad class of dissipative phenomena in various astro- and geophysical applications. In particular, a major application of the proposed approach is the reconstruction of the processes in the preplanetary circumsolar disk, which might help solve the fundamental problems of stellar-planetary cosmogony.     

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.     

Thermodynamic Modeling of Developed Structural Turbulence Taking into Account Fluctuations of Energy Dissipation

A thermodynamic approach to the construction of a phenomenological macroscopic model of developed turbulence in a compressible fluid is considered with regard for the formation of space–time dissipative structures. A set of random variables were introduced into the model as internal parameters of the turbulent–chaos subsystem. This allowed us to obtain, by methods of nonequilibrium thermodynamics, the kinetic Fokker–Planck equation in the configuration space. This equation serves to determine the temporary evolution of the conditional probability distribution function of structural parameters pertaining to the cascade process of fragmentation of large-scale eddies and temperature inhomogeneities and to analyze Markovian stochastic processes of transition from one nonequilibrium stationary turbulent-motion state to another as a result of successive loss of stability caused by a change in the governing parameters. An alternative method for investigating the mechanisms of such transitions, based on the stochastic Langevin-type equation intimately related to the derived kinetic equation, is also considered. Some postulates and physical and mathematical assumptions used in the thermodynamic model of structurized turbulence are discussed in detail. In particular, we considered, using the deterministic transport equation for conditional means, the cardinal problem of the developed approach—the possibility of the existence of asymptotically stable stationary states of the turbulent-chaos subsystem. Also proposed is the nonequilibrium thermodynamic potential for internal coordinates, which extends the well-known Boltzmann–Planck relationship for equilibrium states to the nonequilibrium stationary states of the representing ensemble. This potential is shown to be the Lyapunov function for such states. The relation is also explored between the internal intermittence in the inertial interval of scales and the fluctuations of the energy of dissipation. This study is aimed at constructing representative models of natural space environments. It develops a synergetic approach to modeling the structurized turbulence of astrophysical and geophysical systems, which was proposed by the author in previous papers (Kolesnichenko, 2002, 2003).     

On the simulation of helical turbulence in an astrophysical nonmagnetic disk

The problem of the influence of vortex helicity on the synergic structuring of cosmic matter in it, as well as the appearance of the effect of negative viscosity in three-dimensional gyrotropic turbulence, were studied in the framework of the fundamental problem of simulating the evolution of a rotating astrophysical nonmagnetic disk—in particular, the accretion disk—surrounding the Sun at the early stage of its existence. The evolution equations for averaged vorticity and vortex helicity, as well as rheological relations for the turbulent flow of heat and asymmetrical tensor of the turbulent stress in helical turbulence, were obtained. The demonstrative dependence of helicity on the rotation velocity, density (temperature) gradients, and turbulent energy of the disk gas was established. The role of helicity in the appearance of the inverse Richardson-Kolmogorov energy cascade from small vortices to larger ones and the related process of the generation of the power-consuming macroscale coherent vortex formations appearing in gyrotropic turbulence at high Reynolds number were discussed. The results of the numerical experiments confirming the real existence of the inverse energy cascade in helical turbulence were analyzed. It was assumed that the relatively long-term decay of turbulence in the solar protoplanet cloud can be due to the absence of the reflective symmetry of the anisotropic field of the turbulent velocities with respect to its equatorial plane. As the concept of the inverse energy cascade in three-dimensional helical turbulence is more and more reliably confirmed in numerical experiments, accounting for this effect affecting the structure and dynamics of the astrophysical nonmagnetic disk becomes important during its simulation.     

Angular fluctuations of relict radiation produced by cosmological turbulence

On the basis of a theory for the evolution of cosmological turbulence developed earlier by the present authors, the temperature fluctuations of the relict black-body radiation produced by turbulent motions are calculated. We take into consideration (i) the temperature fluctuations that appear at the moment of recombination (redshiftz10³) with the account of their subsequent diminishing as a result of the optical depth produced in the course of re-ionization of metagalactic gas by young galaxies at their bright phase, and (ii) the temperature fluctuations that emerge on the surface of the last scattering (1) atz10 from vortexes and from potential velocities generated by the vortexes. A comparison of these calculations with the available measurements (upper limits) for temperature fluctuations of relict radiation makes it possible to obtain important upper and lower bounds for the initial velocity of the vortex motions.     

Masers, Lasers and the Interstellar Medium

This paper discusses recent results obtained by myself and my colleagues in three domains of astrophysics: interstellar supersonic turbulence, circumstellar disks, and natural masers and lasers. S.A. Kaplan, S.B. Pikelner, and I.S. Shklovskii were among those who, 30-40 years ago, laid the foundation of these domains.H2O masers become an effective probe of supersonic turbulence associated with mass outflow from very young stars. They demonstrate a very low (1) fractal dimension of the spatial set on which turbulence dissipates its kinetic energy, and, thereby, a strong intermittency of the turbulence. They also indicate that supersonic turbulence, like incompressible turbulence, has an inner scale, on which the bulk of turbulent energy dissipates in low-Mach, random shocks. H2O masers themselves find thereby a new pumping source in these random shocks.Masers in hydrogen recombination lines, discovered 8 years ago, originate in a circumstellar disk surrounding a massive star MWC 349A. They give a possibility to investigate kinematics and structure of the disk. Far-infrared nydrogen recombination lines, recently detected in MWC349A from the Kuiper Airborn Observatory, proved to be amplified as well. They are the first known natural amplifiers of electromagnetic waves in the laser wavelength domain. Analysis of their radiation, along with the radiation in other recombination lines, gives a possible key to understanding the lack of optical lasers in the Universe.     

Spectrum of the galactic magnetic fields

The magnetic fields observed in the galactic disc are generated by the differential rotation and the helical turbulent motions of interstellar gas. On the scalesl=2k ^–1 which lie in the intervall ₀<l<l _e (l ₀100 pc is the energy-range scale of the galactic turbulence), the spectral density of the kinetic energy of turbulence (k ^–5/3) exceeds the magnetic energy spectral density (k ^–1). The equipartition between magnetic and kinetic energies is reached atl=l _e 6 pc in the intercloud medium and is maintained down to the scalel=l _d 0.03 pc. In dense interstellar cloudsl _e is determined by the individual cloud size andl _d 0.1 pc.The internal turbulent velocities in Hi clouds (cloud size is assumed to be 10 pc) lie in the range from 1.8 to 5.6km s^–1, fitting well within the observed range of internal rms velocities. Dissipation of the interstellar MHD turbulence leads to creation of temperature fluctuations with amplitudes of 150 K and 65 K in dense clouds and intercloud medium, respectively. The small-scale fluctuations observed in the interstellar medium may arise from such perturbations due to the thermal instability, for instance. Dissipation of the MHD turbulence energy provides nearly half of the energy supply needed to maintain the thermal balance of the interstellar medium.     

Analysis of nonradial pulsations of a homogeneous neutron star

Nonradial pulsations of an isolated neutron star were studied in two continuum models — a self-gravitating spherical mass of an inviscid, incompressible fluid and an elastic Fermi continuum. A detailed analytical derivation of natural spheroidal and torsional oscillation frequencies of a neutron star of mass 1.4M based on notions about neutron matter as an elastic Fermi continuum is given. A comparison of numerical estimates of natural frequencies shows that both models predict the same order of pulsations 10⁴ Hz; however, the frequencies of spheroidal modes calculated in the elastic Fermi continuum model are about 2–2.5 times higher than the frequencies calculated in the framework of the Kelvin hydrodynamic theory.Translated from Astrofizika, Vol. 38, No. 1, pp. 121–139, January–March, 1995.     

Macro-and micro-turbulent filter functions for weak lines in stellar atmospheres

Following a similar discussion given earlier for the solar case (De Jager, 1972) we compute in this paper spectral line profiles for the spatial wavelengths in which a stellar motion field can be decomposed, and thereafter the macro-and micro-turbulent filter functionsf _M(k) andf (k), where is the optical scale height andf ²(k) dk the fraction of the energy of the turbulent motions between wavenumbersk andk+dk of the spectrum of turbulence that contributes to either kind of turbulence. If micro-and macro-turbulent velocity components are known for a certain star, and if the spectrum of turbulence is sharp enough, the ratiof _M/f would enable one to derive the average size of the turbulent elements in the star's atmosphere. The computations apply to weak lines in idealized stellar atmospheres, and refer to two cases: isotropic turbulence, and radial pulsations. These filters can be suitably used in a diagnostic method for the analysis of the motion field in the solar and stellar atmospheres. Some examples of applications to stars of very different kinds are given.     

Turbulent velocity in undisturbed and active photosphere

Profiles of weak Fraunhofer lines have been recorded photoelectrically in both the faculae and the undisturbed photosphere. The entrance slit of the double-pass spectrograph corresponded to 1 × 20 on the solar disk. Turbulent velocities were found by comparing half-widths of the observed profiles with those of model calculations. These latter were carried out with aid of an electronic computer for the Bilderberg Continuum Atmosphere (BCA) under the assumption of true absorption and depth independent turbulent velocities. Line-formation levels were derived from the contribution curves computed by the method of weighting functions. For the undisturbed photosphere a turbulent velocity of about 2.6 km/s was found with no appreciable increase with the depth. In facular filaments all lines formed above = 0.1 showed smaller turbulence velocities, the velocity differences being in the range between 0.2 and 0.6 km/s (see Table II).     

Stochastic-thermodynamic modeling of turbulent chaos with nonzero memory using fractional Fokker—Planck—Kolmogorov equations

A stochastic-thermodynamic approach to the derivation of the generalized fractional Fokker—Planck—Kolmogorov (FFPK) equations is considered. The equations describe turbulent transfer processes in a subsystem of turbulent chaos on the basis of fractional dynamics, which takes into account the structure and metric of fractal time. The actual turbulent motion of a fluid is known to be intermittent, since it demonstrates the properties that are intermediate between the properties of regular and chaotic motions. On the other hand, the process of the flow turbulization may be non-Markovian because of the multidimensional spatiotemporal correlations of pulsating parameters; in a physical language, this means that the process has a memory. The introduction of fractional time derivatives into the FFPK kinetic equations, used to find the probability distribution functions for different statistical characteristics of structured turbulence, makes it possible to use an unified mathematical formalism in considering the effects of memory, nonlocality, and time intermittence, with which we usually associate the presence of turbulent bursts against the background of less intense low-frequency oscillations in the background turbulence. This study is aimed at creating representative models of space and natural media. It is a development of the synergetic approach to the modeling of structured turbulence in astrogeophysical systems, which has been developed by the author in a series of papers (Kolesnichenko, 2002–2005).     

Simulation for electron acceleration by DC electric field in the presence of ion sound waves and associated hard X-ray emission

Time-dependent Fokker-Planck equation was numerically solved to demonstrate the dynamics of electrons in a uniform coronal loop with an applied axial DC electric field in the presence of ion-sound waves. This electric field is attributed to an anomalous resistivity due to the ion-sound turbulence caused by an initially given critical current density.The electron momentum distribution becomes a steady state in the whole turbulent region in a short time for which some electrons can be accelerated to the maximum electric potential K _c. The steady energy distribution of electrons flowing out the end of the turbulent region has a very hard power-law-like spectrum with an index of about 0.75. The associated hard X-rays from a thick target also show a hard spectrum with a photon spectral index of 1.3. In order for to be much greater as observed in impulsive X-ray bursts, it is required that the source is a sum of many elementary loops with a power-law-like distribution in K _c with an index = – + 2.5.     

A model of the supergranulation network and of active-region plages

Analysis of magnetograph recordings made simultaneously in different spectral lines have shown that the quiet-region network and active-region plages with average field strengths less than about 100 G are made up by the same type of elementary structures, each having the same physical properties. Magnetograph data are used together with continuum, line profile, and EUV data to derive a model of these subarcsec, spatially unresolved elementary structures. The field strength at the center of each basic element is about 2 kG. The temperature enhancement starts at a height of about 180 km (above the level ₀ = 1 in HSRA), and increases rapidly with height. The brightness structures are coarser than the magnetic-field structures.The magnetic field cannot be contained by either gas pressure or dynamic pressure. The magnetic pressure must be balanced by the constricting force of strong electric currents along the magnetic filaments (pinch effect). A mechanism is proposed for the amplification of the field, involving vortex motions around the downdrafts in the network and plages. Efficient heating by hydromagnetic waves builds up an excess gas pressure inside the twisted fluxropes. The excess pressure is released by the ejection of spicules, which have to move out along the helically shaped field lines and thereby will acquire a spinning motion.The continuum emission in the fluxropes, which are located in the intergranular lanes, washes out the contrast between cell interiors and cell boundaries and creates an abnormal granulation pattern. When more and more magnetic flux is brought into a given area, the interaction between the fluxropes and the granulation starts to change the physical structure of the fluxropes. This begins at an average field B _obs 100 G, with a gradual transition to pores and sunspots as b _obs is increased.     

Two-dimensional stochastic motions and the problem of differential rotation

This paper deals with the spatial dependence of the angular velocity in a rotating turbulent fluid sphere. The original turbulence unaffected by the global rotation is assumed to be two-dimensional where the stochastic force field producing the turbulence does not possess a radial component. By using results of earlier papers we proceed to the treatment of a rotational rate, , no longer small compared to _c (frequency of turbulent mode). It is shown that for _c the angular velocity increases with increasing radius but no latitudinal dependence exists. Contrary to this, for 2 _c an equatorial acceleration is possible and related to negativity of the two-dimensional eddy viscosity. Furthermore, the outer layers rotate faster than the inner ones. These findings coincide with Gilman's numerical results. Ward's observations, as well as the characteristic scales of supergranulation and giant cells, suggest the presence of negative two-dimensional eddy viscosity on the Sun.     

Infrared Spectroscopy of V4334 Sgr (Sakurai's Object)

IR spectroscopy and photometry in the 0.8–2.4 and 3–14 mregions are reported for seven dates between March 21 1998 and July20 2000 UT. The shorter wavelength region displays a smooth continuum increasing to longerwavelengths that is indicative of the Wien tail of a Planck function. Only theHe I 1.0830 line is present early and it shows a P-Cygni profile which laterdisappears. The long wave spectra show a smooth continuum between 3 and 13m that was well fit by a gray body at 1000 K. A weak, unidentifiedemission feature appears between 8 and 10 m.     

Seismic Diagnostics on Stellar Convection Treatment from Oscillation Amplitudes of p-modes

The excitation rate P of solar p-modes is computed with a model of stochastic excitation which involves constraints on the averaged properties of the solar turbulence. These constraints are obtained from a 3D simulation. Resulting values for P are found 4.5 times larger than when the calculation assumes properties of turbulent convection which are derived from an 1D solar model based on Gough (1977)'s formulation of the mixing-length theory (GMLT). This difference is mainly due to the assumed values for the mean anisotropy of the velocity field in each case.Calculations based on 3D constraints bring the P maximum closer to the observational one.We also compute P for several models of intermediate mass stars (1 M 2 M).Differences in the values of P _max between models computed with the classical mixing-length theory and GMLT models are found large enough for main sequence stars to suggest that measurements of P in this mass range will be able to discriminate between different models of turbulent convection.     

Low frequency flickering of TT arietis: Hard and soft X-ray emission region

Using archival ASCA observations of TT Arietis, X-ray energy spectra and power spectra of the intensity time series are presented for the first time. The energy spectra are well-fitted by a two continuum plasma emission model with temperatures 1 keV and 10 keV. A coherent feature at 0.643 mHz appeared in the power spectra during the observation.     

The cosmic rays of superhigh energies

The paper considers the generation mechanism of the relativistic particles of superhigh energies (10¹⁸ eV) in a plasma where the supersonic turbulence and the hydrodynamic shock waves occur. It is found that the conditions necessary for the formation of this turbulence are realized in supernovae shells during the period of the outburst. The estimations of the energy gain rate of the charged particles and comparison with their energy loss rate conditioned by synchrotron radiation and collisions with photons and nuclei show that in the actually determined conditions of shells in Crab and Cassiopeia nebulae, at the early stages of their expansion, acceleration surpasses deceleration. And finally, the estimations of the total number of superhigh energy particles generated during the flare are in agreement with the observed data.