Hydrodynamic Aspects of Modeling of the Mass Transfer and Coagulation Processes in Turbulent Accretion Disks

This paper considers, in the context of modeling the evolution of a protoplanetary cloud, the hydrodynamic aspects of the theory of concurrent processes of mass transfer and coagulation in a two-phase medium in the presence of shear turbulence in a differentially rotating gas–dust disk and of polydisperse solid particles suspended in a carrying flow of solid particles. The defining relations are derived for diffuse fluxes of particles of different sizes in the equations of turbulent diffusion in the gravitational field, which describe the convective transfer, turbulent mixing, and sedimentation of disperse dust grains onto the central plane of the disk, as well as their coagulation growth. A semiempirical method is developed for calculating the coefficients of turbulent viscosity and turbulent diffusion for particles of different kinds. This method takes into account the inverse effects of dust transfer on the turbulence evolution in the disk and the inertial differences between disperse solid particles. To solve rigorously the problem of the mutual influence of the turbulent mixing and coagulation kinetics in forming the gas–dust subdisk, the possible mechanisms of gravitational, turbulent, and electric coagulation in a protoplanetary disk are explored and the parametric method of moments for solving the Smoluchowski integro-differential coagulation equation for the particles' size distribution function is considered. This method takes into account the fact that this distribution belongs to a definite 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 a-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 O(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.     

Temperature distribution in the solar nebula at successive stages of its evolution

We have constructed a model of the solar nebula that allows for the temperature and pressure distributions at various stages of its evolution to be calculated. The mass flux from the accretion envelope to the disk and from the disk to the Sun, the turbulent viscosity parameter α, the opacity of the disk material, and the initial angular momentum of the protosun are the input model parameters that are varied. We also take into account the changes in the luminosity and radius of the young Sun. The input model parameters are based mostly on data obtained from observations of young solar-type stars with disks. To correct the input parameters, we use the mass and chemical composition of Jupiter, as well as models of its internal structure and formation that allow constraints to be imposed on the temperature and surface density of the protoplanetary disk in Jupiter’s formation zone. Given the derived constraints on the input parameters, we have calculated models of the solar nebula at successive stages of its evolution: the formation inside the accretion envelope, the evolution around the young Sun going through the T Tauri stage, and the formation and compaction of a thin dust layer (subdisk) in the disk midplane. We have found the following evolutionary trend: an increase in the temperature of the disk at the stage of its formation, cooling at the T Tauri stage, and the subsequent internal heating of the dust subdisk by turbulence dissipation that causes a temperature rise in the formation zone of the terrestrial planets at the high subdisk density and the opacity in this zone. We have obtained the probable ranges of temperatures in the disk midplane, i.e., the temperatures of the protoplanetary material in the formation region of the terrestrial planets at the initial stage of their formation.     

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.     

Gravitational Instability in the Dust Layer of a Protoplanetary Disk with Interaction between the Layer and the Surrounding Gas

We consider gravitational instability of the dust layer in the midplane of a protoplanetary disk with turbulence and shear stresses between the gas in the disk and that in the dust layer. We solve a linearized system of hydrodynamic equations for perturbations of dust (monodisperse) and gas phases in the incompressible gas approximation. We take into account the gas drag of solid particles (dust aggregates), turbulent diffusion and the velocity dispersion of particles, and the perturbation of the azimuthal velocity of gas in the layer upon the transfer of angular momentum from solid particles to it and from this gas to the surrounding gas in the disk. We obtain and solve the dispersion equation for the layer with the ratio of surface densities of the dust phase and gas being well above unity. The following parameters of gravitational instability in the dust layer are calculated: the critical surface density of solid matter and the Stokes number of particles corresponding to the onset of instability, the wavelength range in which instability occurs, and the rate of its growth as a function of the perturbation wavelength in the circumsolar disk at radial distances of 1 and 10 AU. We show that at 10 AU, the maximum instability growth rate increases due to the transfer of angular momentum of gas in the layer to gas outside it, a new maximum emerges at a longer wavelength, a long-wavelength instability “tail” forms, and the critical surface density initiating instability decreases relative to that determined without the transfer of angular momentum to gas outside the layer. None of these effects are observed at 1 AU, since instability in this region probably develops faster than the transfer of angular momentum to the surrounding gаs of a protoplanetary disk occurs.     

Formation of Planetesimals in the Trans-Neptunian Region of the Protoplanetary Disk

We consider the formation of cometlike and larger bodies in the trans-Neptunian region of the protoplanetary gas–dust disk. Once the particles have reached 1–10 cm in size through mutual collisions, they compact and concentrate toward the midplane of the disk to form a dust subdisk there. We show that after the subdisk has reached a critical density, its inner, equatorial layer that, in contrast to the two subsurface layers, contains no shear turbulence can be gravitationally unstable. The layer breaks up into 10¹²-cm clumps whose small fragments (10⁹ cm) can rapidly contract to form bodies 10 km in size. We consider the sunward drift of dust particles at a velocity that decreases with decreasing radial distance as the mechanism of radial contraction and compaction of the layer that contributes to its gravitational instability and the formation of larger (100 km) planetesimals. Given all of the above processes, it takes 10⁶ yr for planetesimals to form, which is an order of magnitude shorter than the lifetime of the gas–dust protoplanetary disk. We discuss peculiarities of the structure of planetesimals.     

Long-wavelength MHD instability in the prefront of collisionless shocks with accelerated particles

Collisionless shocks in turbulent space plasmas accelerate particles by the Fermi mechanism to ultrarelativistic energies. The interaction of accelerated particles with the plasma inflow produces extended supersonic MHD flows of multicomponent plasma. We investigate the instabilities of a flow of three-component turbulent plasma with relativistic particles against long-wavelength perturbations with scales larger than the accelerated particle transport mean free path and the initial turbulence scales. The presence of turbulence allows us to formulate the system of single-fluid equations, the equation of motion for the medium as a whole, and the induction equation for the magnetic field with turbulent magnetic and kinematic viscosities. The current of accelerated particles enters into the induction equation with an effective magnetic diffusion coefficient. We have calculated the local growth rates of the perturbations related to the nonresonant long-wavelength instability of the current of accelerated particles for MHD perturbations in the WKB approximation. The amplification of long-wavelength magnetic field perturbations in the flow upstream of the shock front can affect significantly the maximum energies of the particles accelerated by a collisionless shock and can lead to the observed peculiarities of the synchrotron X-ray radiation in supernova remnants.     

Gravitational instability in the dust layer of a protoplanetary disk: Interaction of solid particles with turbulent gas in the layer

Gravitational instability of the dust layer formed after the aggregates of dust particles settle toward the midplane of a protoplanetary disk under turbulence is considered. A linearized system of hydrodynamic equations for perturbations of dust (monodisperse) and gas phases in the incompressible gas approximation is solved. Turbulent diffusion and the velocity dispersion of solid particles and the perturbation of gas azimuthal velocity in the layer upon the transfer of angular momentum from the dust phase due to gas drag are taken into account. Such an interaction of the particles and the gas establishes upper and lower bounds on the perturbation wavelength that renders the instability possible. The dispersion equation for the layer in the case when the ratio of surface densities of the dust phase and the gas in the layer is well above unity is obtained and solved. An approximate gravitational instability criterion, which takes the size-dependent stopping time of a particle (aggregate) in the gas into account, is derived. The following parameters of the layer instability are calculated: the wavelength range of its subsistence and the dependence of the perturbation growth rate on the perturbation wavelength in the circumsolar disk at a radial distance of 1 and 10 AU. It is demonstrated that at a distance of 1 AU, the gas–dust disk should be enriched with solids by a factor of 5–10 relative to the initial abundance as well as the particle aggregates should grow to the sizes higher than about 0.3 m in order for the instability to emerge in the layer in the available turbulence models. Such high disk enrichment and aggregate growth is not needed at a distance of 10 AU. The conditions under which this gravitational instability in the layer may be examined with no allowance made for the transfer of angular momentum from the gas in the layer to the gas in a protoplanetary disk outside the layer are discussed.     

Streaming Instability in the Gas-Dust Medium of the Protoplanetary Disc and the Formation of Fractal Dust Clusters

The sequence of evolution of the protoplanetary gas-and-dust disk around the parent star includes, according to modern concepts, its compression in the central plane and decay into separate dust condensations (clusters) due to the occurrence of various types of instabilities. The interaction of dust clusters of a fractal structure during their collisions is considered as a key mechanism for the formation and growth of primary solids, which serve as the basis for the subsequent formation of planetesimals and embryos of planets. Among the mechanisms contributing to the formation of planetesimals, an important place belongs, along with gravitational instability, hydrodynamic instabilities, in particular, the socalled streaming instability of the two-phase gas-dust layer due to its ability to concentrate dispersed particles in dense clots. In contrast to a number of existing models of streaming instability, in which dust particles are considered structurally compact and monodisperse, this paper proposes a more realistic model of polydisperse particles of fractal nature, forming dust clusters as a result of coagulation. The instability of the dust layer in the central plane of the protoplanetary disk under linear axisymmetric perturbations of its parameters is considered. A preliminary conclusion can be drawn that the proposed model of dust fractal aggregates of different scales increases the efficiency of linear growth of hydrodynamic instabilities, including the streaming instabilities associated with the difference between the velocities of the dust and gas phases.

     

Refractory Metal Nuggets – Formation of the first condensates in the Solar Nebula

As gas flowed from the solar accretion disk or Solar Nebula onto the proto-Sun, magnetic pressure gradients in the solar magnetosphere and the inner Solar Nebula provided an environment where some of this infalling flow was diverted to produce a low pressure, high temperature, gaseous, “infall” atmosphere around the inner Solar Nebula. The pressure in this inner disk atmosphere was mainly dependant on the accretion flow rate onto the star. High flow rates implied relatively high pressures, which decreased over time as the accretion rate decreased.In the first hundred thousand years after the formation of the Solar Nebula, accretional flow gas pressures were high enough to create submicron-sized Refractory Metal Nuggets (RMNs) – the precursors to Calcium Aluminum Inclusions (CAIs). Optimal temperatures and pressures for RMN formation may have occurred between 20,000 and 100,000 years after the formation of the Solar Nebula. It is possible that conditions were conducive to RMN/CAI formation over an 80,000 year timescale. The “infall” atmosphere and the condensation of refractory particles within this atmosphere may be observable around the inner disks of other protostellar systems.The interaction of forces from magnetic fields with the radiation pressure from the proto-Sun and the inner solar accretion disk potentially produced an optical-magnetic trap above and below the inner Solar Nebula, which provided a relatively stable environment in which the RMNs/proto-CAIs could form and grow. These RMN formation sites only existed during accretion events from the proto-solar disk onto the proto-Sun. As such, the formation and growth time of a particular RMN was dependent on the timescale of its nascent accretion event.Observational evidence suggests that RMNs were the nucleation particles for CAIs. As a consequence, the observed bimodal distribution of ²⁶Al in CAIs, where some CAIs have ²⁶Al while others do not, is probably due to the injection ²⁶Al during the short CAI formation period, where ²⁶Al was not present when the first CAIs were formed.     

Growth of planetesimals and clearance of disks around the T-tauri stars

A growth and contraction of dust condensations formed as a result of development of gravitation instability in a dust subdisk is discussed within the framework of the generally accepted scheme of evolution of a circumsolar pre-planetary disk. The time of evolution of condensations necessary for clearing the hypothetical disks around young stars of the Sun type is estimated.     

Interplanetary dust particles and impact erosion

Motion of the interplanetary dust particle under the action of collisions with much smaller interplanetary dust particles is investigated. The equation of motion is derived. Perturbation equations of celestial mechanics are also discussed. The results are compared with the Poynting-Robertson effect and the effect of solar wind on the motion of the interplanetary dust particles, from the point of view of observational data.     

Evolution of grains in a turbulent solar nebula

In accretion disk models of the solar nebula, turbulence is driven by convective instability. This mechanism requires high opacity, which must be provided by solid grains. Evolution of the grain size distribution in a turbulent disk is computed numerically, using realistic collisional outcomes and strengths of grain aggregates, rather than an arbitrary “sticking efficiency.” The presence of turbulence greatly increases the rate of grain collisions; the coagulation rate is initially much greater than in a nonturbulent disk. Aggregates quickly reach sizes ～0.1–1 cm, but erosion and breakup in collisions prevent growth of larger bodies for plausible aggregate impact strengths. These aggregates are too small to settle to the plane of the disk, and planetesimal formation is impossible as long as the turbulence persists. However, the opacity of the disk is reduced by aggregate formation; some combinations of opacity law and surface density produce an optically thin disk, cutting off turbulent convection. The disk may experience alternating periods of turbulence and quiescence, as grains are depleted by coagulation and replenished by infall from the presolar cloud. Planetesimals can form only during the quiescent intervals; it is argued that such episodes were rare during the lifetime of the accretion disk.     

Large-scale zonal flow and magnetic field generation due to drift-Alfven turbulence in ionosphere plasma

In the present work, the generation of large-scale zonal flows and magnetic field by short-scale collision-less electron skin depth order drift-Alfven turbulence in the ionosphere is investigated. The self-consistent system of two model nonlinear equations, describing the dynamics of wave structures with characteristic scales till to the skin value, is obtained. Evolution equations for the shear flows and the magnetic field is obtained by means of the averaging of model equations for the fast-high-frequency and small-scale fluctuations. It is shown that the large-scale disturbances of plasma motion and magnetic field are spontaneously generated by small-scale drift-Alfven wave turbulence through the nonlinear action of the stresses of Reynolds and Maxwell. Positive feedback in the system is achieved via modulation of the skin size drift-Alfven waves by the large-scale zonal flow and/or by the excited large-scale magnetic field. As a result, the propagation of small-scale wave packets in the ionospheric medium is accompanied by low-frequency, long-wave disturbances generated by parametric instability. Two regimes of this instability, resonance kinetic and hydrodynamic ones, are studied. The increments of the corresponding instabilities are also found. The conditions for the instability development and possibility of the generation of large-scale structures are determined. The nonlinear increment of this interaction substantially depends on the wave vector of Alfven pumping and on the characteristic scale of the generated zonal structures. This means that the instability pumps the energy of primarily small-scale Alfven waves into that of the large-scale zonal structures which is typical for an inverse turbulent cascade. The increment of energy pumping into the large-scale region noticeably depends also on the width of the pumping wave spectrum and with an increase of the width of the initial wave spectrum the instability can be suppressed. It is assumed that the investigated mechanism can refer directly to the generation of mean flow in the atmosphere of the rotating planets and the magnetized plasma.     

Swirling hydrodynamic jets in viscous accretion flows

A study is made of axisymmetric, low sonic-Mach-number flows of a viscous fluid with angular momentum outside of a black-hole. The viscosity is an eddy viscosity due to turbulence in the sheared flows. Self-similar solutions arise naturally, reducing the Navier-Stokes equations to a set of nonlinear ordinary differential equations. These equations are solved analytically for flows of constant specific angular momentum and numerically for more general flows. For flows with non-constant specific angular momentum, the momentum flux density includes a planar discontinuity which is interpreted as an accretion disc. In general, two flow regions appear on each side of the disk, corresponding to accretion onto the disk and jet-like outflows along the ±z-axes. Physical interpretations of the solutions show that these flows arise in response to point sources of axial momentum at the origin directed in the ±z-directions. The power needed to maintain this momentum input is assumed to come from the mass accretion onto the black hole.The hydrodynamic flows are generalized to include a magnetic field. In the limit of infinite electrical conductivity, the possible types of flow patterns are the same as in hydrodynamic case. The magnetic field alters the relative amounts of reversible and irreversible momentum and angular momentum transport by the flow. For a flow with turbulent viscosity, the magnetic field acts to reduce the level of the turbulence and the effective value of the eddy viscosity.     

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 spectra of polarized Galactic foregrounds

It is believed that magnetic field lines are twisted and bend by turbulent motions in the Galaxy. Therefore, both Galactic synchrotron emission and thermal emission from dust reflects statistics of Galactic turbulence. Our simple model of Galactic turbulence, motivated by results of our simulations, predicts that Galactic disk and halo exhibit different angular power spectra. We show that observed angular spectra of synchrotron emission are compatible with our model. We also show that our model is compatible with the angular spectra of star-light polarization for the Galactic disk. Finally, we discuss how one can estimate polarized microwave emission from dust in the Galactic halo using star-light polarimetry.