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 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.     

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.     

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).     

Diamagnetic pumping near the base of a stellar convection zone

The property of inhomogeneous turbulence in conducting fluids to expel large‐scale magnetic fields in the direction of decreasing turbulence intensity is shown as important for the magnetic field dynamics near the base of a stellar convection zone. The downward diamagnetic pumping confines a fossil internal magnetic field in the radiative core so that the field geometry is appropriate for formation of the solar tachocline. For the stars of solar age, the diamagnetic confinement is efficient only if the ratio of turbulent magnetic diffusivity η_T of the convection zone to the (microscopic or turbulent) diffusivity η_in of the radiative interior is η_T/η_in ≥ 10⁵. Confinement in younger stars requires larger η_T/η_in. The observation of persistent magnetic structures on young solar‐type stars can thus provide evidence for the nonexistence of tachoclines in stellar interiors and on the level of turbulence in radiative cores. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Properties of the negative effective magnetic pressure instability

As was demonstrated in earlier studies, turbulence can result in a negative contribution to the effective mean magnetic pressure, which, in turn, can cause a large‐scale instability. In this study, hydromagnetic mean‐field modelling is performed for an isothermally stratified layer in the presence of a horizontal magnetic field. The negative effective magnetic pressure instability (NEMPI) is comprehensively investigated. It is shown that, if the effect of turbulence on the mean magnetic tension force vanishes, which is consistent with results from direct numerical simulations of forced turbulence, the fastest growing eigenmodes of NEMPI are two‐dimensional. The growth rate is found to depend on a parameter β_* characterizing the turbulent contribution of the effective mean magnetic pressure for moderately strong mean magnetic fields. A fit formula is proposed that gives the growth rate as a function of turbulent kinematic viscosity, turbulent magnetic diffusivity, the density scale height, and the parameter β_*. The strength of the imposed magnetic field does not explicitly enter provided the location of the vertical boundaries are chosen such that the maximum of the eigenmode of NEMPI fits into the domain. The formation of sunspots and solar active regions is discussed as possible applications of NEMPI (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The turbulent destruction of interstellar clouds

The interaction of an astrophysical shock with a cloud typically occurs at high Reynolds number, and in such cases will be highly turbulent. However, the formation of fully developed turbulence is usually prevented by the artificial viscosity inherent in hydrodynamical simulations. Upstream structures mean that the flow behind the shock is also likely to be turbulent, as it sweeps over such inhomogeneities. We study the nature of adiabatic shock-cloud interactions using a subgrid compressible k–ε turbulence model.     

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.     

Spontaneous Formation of Magnetic Flux Concentrations in Stratified Turbulence

The negative effective magnetic pressure instability discovered recently in direct numerical simulations (DNSs) may play a crucial role in the formation of sunspots and active regions in the Sun and stars. This instability is caused by a negative contribution of turbulence to the effective mean Lorentz force (the sum of turbulent and non-turbulent contributions) and results in the formation of large-scale inhomogeneous magnetic structures from an initially uniform magnetic field. Earlier investigations of this instability in DNSs of stably stratified, externally forced, isothermal hydromagnetic turbulence in the regime of large plasma ?? are now extended into the regime of larger scale separation ratios where the number of turbulent eddies in the computational domain is about 30. Strong spontaneous formation of large-scale magnetic structures is seen even without performing any spatial averaging. These structures encompass many turbulent eddies. The characteristic time of the instability is comparable to the turbulent diffusion time, L ²/?? _t, where ?? _t is the turbulent diffusivity and L is the scale of the domain. DNSs are used to confirm that the effective magnetic pressure does indeed become negative for magnetic field strengths below the equipartition field. The dependence of the effective magnetic pressure on the field strength is characterized by fit parameters that seem to show convergence for larger values of the magnetic Reynolds number.     

Studying the influence of turbulent mixing on thermonuclear burning regimes during X-ray bursts of neutron stars using the <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">ε</Emphasis></Subscript> model

We study the influence of turbulent mixing on the development of thermonuclear flashes in the surface layers of neutron stars. A simple K _ε model that includes various physical processes is used to describe the turbulent processes. In contrast to the widespread mixing-length theory, the K _ε model does not require using additional dimensional parameters, traces the development of turbulence in dynamics, describes the various turbulence development scenarios (gravitational and shear instabilities, convection, semiconvection, etc.) in a unified way, and can be used in multidimensional numerical simulations. Empirical constants of the model are chosen on the basis of experimental data and direct numerical simulations of typical processes. We have used the Era and Tigr-3T software packages to numerically simulate thermonuclear flashes in the accretion-renewable atmospheres of neutron stars. Turbulence is shown to accelerate significantly the transport of released energy to the stellar surface. Mixing equalizes the concentrations of matter components throughout the burning layer and increases the amount of matter involved in the thermonuclear burning during a flash.     

Hydrodynamic and turbulent motions in the galactic disk

Statistics in absorption 21-cm data show two main types of clouds at low galactic latitudes: dense small clouds, many of them with molecular cores, with dispersions σ≈1.5 km s⁻¹ and large clouds forming the fine features of the spiral arms (the shingle like features) with a dispersion range α≈3–4 km s⁻¹. Sizes and dispersions of both types of clouds are compatible with the Kolmogorov law of turbulence: σ∞d ^1/3. The large clouds forming the shingle-like features can be considered as the largest clouds of a Kolmogorov spectrum (the initial vortices), or as the hydrodynamic features with minimum sizes in the Galaxy. In order to define hydrodynamic motions in the same sense as given by Ogrodnikov (1965) we use here the tensorial form of the Helmholtz theorem to obtain an approximation for the hydrodynamic motions depending on distances and seen from the local standard of rest:V _r ∞r. The intermediate range of sizes between turbulent motions and hydrodynamic motions is 100<d<300 pc which is also the range of sizes of the large clouds forming the fine features of the spiral arms. A classification on of motions in the Galaxy is postulated: (a) a basic rotation motion given by an smooth unperturbed curveΘ _b (R) associated to the old disk population. (b) Systematic motions of the spiral arms. (c) Systematic motions in the fine structure of the arms. For scale sizes smaller than these fine features one has turbulent motions according to the Kolmogorov law. The densities and sizes of the turbulent clouds behave asn _H ∝d ⁻² in a range of sizes 7 pc<d<300 pc. The obtained gas densities of the clouds are confirmed with the dust densities from photometric studies. The conditions for gravitational binding of the clouds are analyzed. Factors as the geometry and the magnetic field within the clouds increases the critic densities for gravitational binding. When we consider these factors we find that the wide component clouds have densities below such a critical value. The narrow component clouds have densities similar or above the critical value; but the real fraction of collapsing clouds remains unknown as far as the factor of geometry and the inner magnetic field of each cloud are not determinated.     

Diffussion of a scalar field in a compressible turbulent medium

The diffusion of scalar fields (temperature, density number of some admixture) in a compressible medium showing an isotropic, homogeneous and stationary turbulence is considered. The derived formulae for turbulent diffusivity χ_T(ξ) hold up to ξ ≈ 1, where ξ = u₀ τ₀/R₀ (u₀, τ₀, and R₀ are characteristic velocity, life-time, and correlation length of turbulent pulsations, respectively. The velocity field of turbulent motions u(r, t) is assumed to be known and the influence of the scalar field onto u(r, t) is neglected. It is shown that the velocity correlators, which change their signs in dependence on the space corrdinates, may give negative values for ξ_T(ξ) when ξ ≠ 0.     

A theory of differntial rotation based on the discussion of turbulent transport of angular momentum

This paper deals with the theory of the solar rotational law. We assume the turbulence to be of the largest influence compared with the momentum flux caused by molecular viscosity and meridional circulation. Firstly we use heuristical forms for the needed cross correlations Qrφ (turbulent radial momentum flux) and Qνφ (turbulent latitudinal momentum flux): Qrφ = −α₀r ϑΩ/ϑr · sin ν + Q₀ sin ν + Q₂ sin³ ν, Qνφ = −δ₀ ϑΩ/ϑν· sin ν + P₂ sin²θ cos θ. It is shown that a radial dependence of the angular velocity Ω is given by Q₀. Furthermore, the observed equatorial acceleration occurs in the case of non-negativity of Q₂ and/or P₂. Because of the spatial dependence of the solar angular velocity the coefficients of Q and P are unfortunately not to be measured. Secondly, we determine the coefficients with a theory founded upon the hypothesis that a rotating stochastical force field — independent from Ω — maintains an anisotropic turbulence. The global fast rotation produces, indeed, finite cross correlations Q₂ and P₂. It is suggested that horizontally directed turbulent motions with not too small radial correlations lengths and time scales of about 2 weeks could be responsible for the solar differential rotation. Finally, we show that also short-living turbulent horizontal modes provide the observed equatorial acceleration if they occur preferably at the equatorial region.     

A Consistent one-Dimensional Model for the Turbulent Tachocline

The first consistent model for the turbulent tachocline is presented, with the turbulent diffusivity computed within the model instead of being specified arbitrarily. For the origin of the 3D turbulence a new mechanism is proposed. Owing to the strongly stable stratification, the mean radial shear is stable, while the horizontal shear is expected to drive predominantly horizontal, quasi-2D motions in thin slabs. Here I suggest that a major source of 3D overturning turbulent motions in the tachocline is the secondary shear instability due to the strong, random vertical shear arising between the uncorrelated horizontal flows in neighboring slabs. A formula for the vertical diffusivity due to this turbulence, Equation (9), is derived and applied in a simplified 1D model of the tachocline. It is found that Maxwell stresses due to an oscillatory poloidal magnetic field of a few hundred gauss are able to confine the tachocline to a thickness less than 5 Mm. The integral scale of the 3D overturning turbulence is the buoyancy scale, on the order of 10 km, and its velocity amplitude is a few m s^–1, yielding a vertical turbulent diffusivity on the order of 10⁸ cm² s^–1.     

The Influence of a Uniform Magnetic Field of Arbitrary Strength on Turbulence

The spectral tensor of turbulent motion in an infinite conductive incompressible medium is given in the case of a uniform magnetic field of any strenght affecting a homogeneous turbulence. With the help of BOCHNER 's theorem we make sure that the trace u_i(x, t) u_i(x, t) is non-negative. The presence of a weak magnetic field causes a damping of the turbulence, in some cases a strengthening. For strong magnetic fields the norms of the velocity vectors parallel and perpendicular to B approach one and the same value. Compared with the correlation length measured perpendicular to the magnetic field the correlation length measured along the magnetic field increases. Furthermore, our formulas have allowed to calculate the dependence of the α-effect on the magnetic field.