The key role of the family of hydromagnetic helicity invariants in connection with the generation and maintenance of magnetic fields in geophysical and astrophysical contexts is discussed. The influence of compressibility and rotation on the turbulent mass transport in helical hydromagnetic flows is investigated using a phenomenological approach at very high Reynolds numbers. The fluctuating effects entering into the averaged MHD equations through their correlation contributions and representing the hydromagnetic turbulent stress, turbulent electromotive force and a number of other correlation functions are modeled using linear closure relations (in the absence of reflective symmetry of small-scale motions) and differential equations for four helical chiral turbulence descriptors, which are: total turbulent plasma energy, turbulent transverse helicity, turbulent residual energy and turbulent residual helicity. It is believed that the model equations for these descriptors, combined with the compressible MHD mean field equations, allow the most complete construction of a self-consistent model of the turbulent dynamo. The ultimate goal of the undertaken research is the development of models of helical hydromagnetic turbulence capable of operating effectively in the hypersonic regime.

In the framework of entropic cosmology and Prigogine’s gravitational theory on the link between geometry and matter, providing the production of particles in the cosmological fluid, as well as under the assumption of exchange entropy at the event horizon, a one-liquid model of the evolution of a spatially flat, homogeneous, and isotropic Universe has been developed. To construct it, the energy conservation equation was derived from the first law of thermodynamics by accounting for the gravitationally-induced creation of matter and the exchange energy processes on the apparent horizon of the Universe. On the basis of this equation and the fundamental Friedmann equation describing the expansion of the Universe, as well as in the entropic formalism context, we constructed the modified Friedmann–Robertson–Walker equations that can be used to study various dynamical aspects of the evolution of the Universe with adiabatic creation of matter. When deriving them, we used several forms of exchangeable phenomenological entropies associated with the region of the apparent cosmological horizon.

The hydrodynamic instability of a magnetized, self-gravitating rotating anisotropic plasma is analyzed in the collisionless approximation and considering the heat flux vector based on the modified Chu–Goldberger–Low equations. A dispersion relation is obtained, on the basis of which simplified cases of propagation of low-amplitude disturbance waves and the derivation of modified criteria for hydrodynamic instability are discussed. Using the obtained dispersion relation, three simple cases are treated when the disturbance wave propagates across, along, and obliquely to the magnetic field vector. It is shown that the anisotropy of pressure and heat flow not only changes the classical criterion of the Jeans instability, but also leads to the appearance of new wave modes and causes the appearance of new unstable domains. It has been found that the presence of uniform rotation of the plasma reduces the critical wave number and has a stabilizing effect on the criterion of gravitational instability when the disturbance wave propagates transversely, without having an effect in the case of longitudinal propagation. These results are important for developing evolutionary magnetohydrodynamic models of astrophysical collisionless plasma.

The problem of self-gravitational instability of an astrophysical rotating plasma in a strong magnetic field with an anisotropic pressure tensor is studied on the basis of the Chew–Goldberger–Low (CGL) quasi-hydrodynamic equations modified by generalized polytropic laws. Using the general form of a dispersion relation obtained by the normal-mode perturbation method, a discussion is provided of the propagation of small-amplitude perturbation waves in an infinite homogeneous plasma medium for transverse, longitudinal, and oblique directions with respect to the magnetic field vector. It is shown that different polytropic indices and anisotropic pressures not only change the classical Jeans instability condition but also cause the appearance of new unstable regions. Modified Jeans instability criteria are obtained for isotropic MHD equations and anisotropic CGL equations owing to the influence of the polytropic indices on gravitational and firehose instabilities for astrophysical plasma. It is shown that in the case of a longitudinal mode of perturbation wave propagation, the Jeans instability criterion does not depend on uniform rotation. In the case of the transverse propagation regime, the presence of rotation reduces the critical wave number and exerts a stabilizing effect on the growth rate of the unstable regime.

The paper presents a closed system of three-dimensional hydrodynamic equations of averaged motion, intended for modeling spiral turbulence in a rotating astrophysical disk. Diffusion equations for the averaged vortex and an equation for the transport of the integral vortex helicity are derived. A general concept of the emergence of energy-intensive mesoscale coherent vortex structures in a thermodynamically open turbulent chaos subsystem, associated with the realization of a reverse cascade of kinetic energy in mirror-asymmetrical disk turbulence, is formulated. It is shown that negative viscosity in a rotating three-dimensional disk system is apparently a manifestation of cascade processes in helical turbulence, when reverse energy transfer from small vortices to larger ones takes place. It is also shown that the relatively long decay of turbulence in the disk is associated with the lack of mirror symmetry of the anisotropic field of turbulent velocities about its equatorial plane. The work comprises a review aimed at improving new models of astrophysical nonmagnetic disks, for which the effects of helical turbulence play a decisive role.

The paper analyzes one-dimensional simple waves and small-amplitude perturbations in radiating and scattering gray gas. The governing equation of radiation acoustics describing the dynamics of simple waves is derived. The conditions of radiation-thermal dissipation and radiation resistance force are introduced into this equation to describe the propagation with dissipation and attenuation of various radiation perturbation waves. The phenomenological approximate Whitham method is used to investigate non-equilibrium wave phenomena in radiative medium. This method is an effective way to analyze fundamental modes when more than one velocity appears in the governing equation. The use of this method is demonstrated in this paper by considering the evolution of one-dimensional harmonic waves caused by a short-wave initial perturbation of the equilibrium state of the radiating and scattering medium. Analytical solutions are obtained for all wave modes, which allow us to interpret their physical meaning. These solutions can be, in particular, an additional test for radiative hydrodynamic codes operating in the radiative acoustics regime. The presented approach may be useful in detailing higher-order numerical Godunov schemes for radiation acoustics problems.

The propagation of linear acoustic disturbances in an infinite, homogeneous, gray radiating plasma, initially in mechanical and radiation equilibrium, is considered. An exact governing equation for radiation acoustics in a radiating gray gas is derived, taking into account the influence of the transverse magnetic field. Radiation magnetohydrodynamics (MHD) is described by three hydrodynamic equations and two radiative momentum equations, making extensive use of the formalism of radiation thermodynamics. With the aim of more reliably describing the evolution of radiation magnetic–acoustic disturbance waves with scattering and attenuation, the conditions of radiation-thermal dissipation, the force of radiation resistance, as well as magnetic force and Joule heat are introduced into these equations. In this case, the Eddington approximation is used, which allows one to study the modes of radiation magnetohydrodynamic waves in two asymptotic cases—optically thin and thick gas. The exact control equation derived in the work made it possible, using the heuristic Whitham method, to obtain a set of approximate control equations of the lowest order, each of which is part of a reliable approximation to the exact equation in a certain region of the independent time variable. The relatively simple form of such equations made it possible to study the physical processes occurring in each radiation magnetic–acoustic wave without a formal solution to the full problem.

The aim of this article is to acquaint the reader to the rapidly developing stochastic-thermodynamic theory of irreversible processes using as an example the modeling of structured turbulence. Here, we consider a synergetic approach to the development of a phenomenological model of extremely developed turbulence in a compressible homogeneous fluid, taking into account the nonlinear cooperative processes in it. Inclusion in the model of a set of random variables as internal parameters of the turbulent chaos subsystem, associated with its microstructure, makes it possible in this case to derive, using thermodynamic methods, the Fokker–Planck–Kolmogorov (FPK) kinetic equations in configuration space. Stabilization of the chaos subsystem near the next stationary nonequilibrium state in configuration space corresponds to the transition of the turbulent system to a new state, adequate to the emergence of complex spatiotemporal CSs in a turbulent flow.

In this article, the possibility of influence of the hydrodynamic helicity appearing in a rotating disk on synergetic structuring of cosmic substance and on the emergence of the negative turbulent viscosity effect in it is investigated. It is shown that prolonged turbulence damping in a disk can be partly associated with the absence of reflection symmetry of the anisotropic field of turbulent velocities relative to its equatorial plane. It is shown that negative viscosity in the rotating disk system is apparently a manifestation of cascade processes in helical turbulence, when the inverse energy transfer from small to larger vortices occurs.

In the framework of the basic problem of cosmogony, which is associated with the reconstruction of the protoplanetary solar disk at the very early stages of its existence, a closed system of MHD equations on the mean flow scale is formulated. This system is intended for numerical solution of problems on interconsistent simulation of the structure and evolution of the accretion protoplanetary disk and its corona. The model of a thin (but optically thick) disk is considered, in which turbulence dissipation due to kinematic and magnetic viscosity, opaqueness of the medium, accretion from the surrounding space, and the action of turbulent αω dynamo on the generation of magnetic field, as well as the magnetic force and energy interaction between the disk and its corona are taken into account.

—In this article, a closed system of magnetohydrodynamic (MHD) equations on the mean flow scales, which is intended for simulating turbulent flows of electroconductive media in the presence of an electromagnetic field, is derived in the single-fluid MHD approximation.

Proceeding from the concept of the fractal nature of dust clusters, we develop an evolutionary hydrodynamic model of the formation and growth of loose dust aggregates in an aerodisperse medium of a laminar disk that was originally composed of gas and solid (sub)micron particles. In contrast to a number of classical studies, in which the aggregation process was modeled within an “ordinary” continuous medium and, consequently, the multifractional structure of the dust component of a protoplanetary cloud and the fractal nature of dust clusters being formed during the evolution were often ignored, this paper proposes to consider an ensemble of loose dust aggregates as a special type of a continuous medium, namely, a fractal medium, in which there are points and domains not occupied by dust particles.

In contrast to several classical studies, in which gravitational instability criteria for astrophysical disks are derived within traditional hydrodynamics or kinetics, it is proposed to consider the set of loose gas–dust clusters of an accretion protoplanetary disk as a special type of continuous medium, i.e., a fractal medium whose phase velocity space contains points and areas not filled by its components. Within the Tsallis formalism of nonadditive statistics (thermodynamics), intended to describe the behavior of anomalous systems, i.e., systems with a strong gravitational interaction of its individual parts and the fractal nature of the phase space, linearized equations are obtained for oscillations of a solid-state rotating disk on the basis of modified Navier–Stokes hydrodynamic equations (the so-called q -hydrodynamics equations) and in view of dissipative effects, and a derivation is given of the dispersion equation in the WKB approximation. An analysis is conducted of axisymmetric oscillations of a differentially rotating astrophysical gas–dust space object to obtain modified Jeans and Toomre gravitational instability criteria for disks with a fractal structure.

—In this paper, in relation to the problem of reconstructing the evolution of a preplanetary gas and dust cloud, an attempt is made to develop a model of a turbulent heterogeneous medium and to construct on this basis a new class of mathematical models of space media that take into how the character and development of turbulence are influenced by the inertial properties of a polydisperse mixture of dust particles, processes of heat and mass transfer and coagulation, phase transitions, chemical reactions, and radiation. This can significantly expand the capabilities of numerical modeling of various physical phenomena in complex space media such as accretion gas and dust disks formed in stars of various classes during their differential rotation around the center of gravity, to study their structure, physicochemical and hydrodynamic properties, and time evolution.

Using the Verlinde formalism, several scenarios for the evolution of the Friedmann–Robertson–Walker universe are considered, which are possible within the entropic cosmology based on a new modification of the Sharma–Mittal entropic measure. The study within the nonextensive statistical theory uses several entropic measures associated with the cosmological horizon due to the holographic data stored in it. Several versions of the generalized Friedmann equations have been constructed, which can serve as an effective theoretical basis for describing the dynamic evolution of a flat, homogeneous, and isotropic universe, generating diverse forms of matter contained in it. The approach proposed, which involves the use of probabilistic non-extensive aspects of the cosmological horizon, meets the well-known basic requirements for thermodynamic modeling of the dynamic behavior of outer space without involving the concept of hypothetical dark energy.

Within the problem of modeling the evolution of circumstellar gas-and-dust clouds, the impact is discussed of black radiation on the Jeans magnetogravitational instability for a self-gravitating magnetized plasma disk, taking into account the effects of rotation, large-scale magnetic field, dissipative processes due to magnetic viscosity, and radiative heat transfer. A general dispersion relation is derived using the normal-mode analysis. This relation is used to obtain modified Jeans instability criteria for several special cases associated with different relative orientations of the magnetic field, the rotation axis, and the propagation vector of the perturbation wave. The gravitational stability condition for the system is discussed using the Routh–Hurwitz criterion. It is shown that in the case of both longitudinal and transverse propagation of the perturbation wave, the Jeans instability criteria are considerably modified due to radiative heat losses and the Stokes correction. The results obtained make it possible, inter alia, to better understand the problem associated with the evolution of circumstellar plasma disks.

A generalization of Chandrasekhar’s integral theorems on the equilibrium for matter and blackbody radiation in a protostellar gravitating spherically symmetric cloud has been obtained in the context of Kaniadakis nonextensive kappa statistics. The elements of deformed thermodynamics for an ideal gas, the deformed canonical Gibbs distribution, and the effective gravitational constant calculated in the Verlinde formalism are used for this purpose. The deformation parameter $$\kappa $$ measures the so-called degree of nonextensiveness of the cloud system. In addition, modified thermodynamic properties of blackbody radiation are discussed in the context of Kaniadakis statistics, in particular, the κ-analog of Stefan’s law for radiation energy and generalized expressions for the entropy, thermal capacity, and radiation pressure. The presented method of combining these anomalous physical processes provides an alternative to the well-known Chandrasekhar’s integral theorems for spherically symmetric gas configurations in a state of hydrostatic equilibrium, and restores all standard expressions in the limit κ → 0. The approach developed in this study can be used to construct new models of the evolution of nonextensive protostellar objects and stars.

Within the problem of modeling the evolution of a protostellar disk, a discussion is presented on the effect of radiation on the Jeans gravitational instability for a self-gravitating optically thick (for intrinsic infrared radiation) gas-and-dust medium, taking into account the influence of radiation pressure perturbations and radiative diffusion transfer on the critical wavelength. Two radiative diffusion approximations are considered: the case of perfect thermal equilibrium with the same temperature of matter and radiation and the case of the time dependence of the radiation field with an energy separation between radiation and matter. An analysis of the normal regime of modes is used to derive dispersion relations, which enable the derivation of modifications of the classical Jeans instability criterion under the influence of radiation pressure and radiation diffusion. In particular, it is shown that, in contrast to the system’s local thermodynamic equilibrium, where the acoustic velocity of perturbed gas propagates with the isothermal speed of sound, in the case of different temperatures of radiation and gas, the perturbing wave propagates with the adiabatic speed of sound in gas. The results obtained are aimed at solving the problem of gravitational instability of individual massive protostellar disks or self-gravitating radiative media characterized by large optical depths for their dust-transformed intrinsic infrared radiation.

Within the framework of Tsallis nonextensive statistics, the criteria for the Jeans gravitational instability are derived for a self-gravitating protoplanetary disk, whose substance consists of a mixture of a conducting ideal q-gas and modified radiation of a photon gas. The instability criteria are derived from the corresponding dispersion relations written for both neutral disk matter and magnetized plasma with modified blackbody radiation. The thermodynamics of a photon gas are constructed based on the nonextensive Tsallis quantum entropy, which depends on the deformation parameter. It is shown that blackbody q-radiation can stabilize the state of a nonextensive medium for a purely gaseous disk, and for an electrically conducting disk, the Jeans instability criterion is modified by the magnetic field and radiation pressure only in the transverse propagation mode of the disturbance wave.

A dynamic theory of the linear reaction from nonextensive quasi-equilibrium multibody systems to an external time-dependent perturbation is developed in quantum statistical mechanics based on the Tsallis parametric nonadditive entropy associated with the density matrix. For nonextensive quantum systems, a modification of the Kubo theory developed in quantum mechanics is proposed. The linear reaction theory is constructed based on a generalized canonical form of the density matrix obtained by maximizing the Tsallis quantum entropy by averaging the observed values over the escort distribution. Generalized expressions for admittance and response functions are presented that describe the linear response of the system to a weak external mechanical impact. The paper discusses the symmetry property for the relaxation function under time reversal and the Onsager reciprocity relation for generalized susceptibility. It is shown that these properties known in classical quantum statistics remain valid for anomalous systems.