Evgenii Aleksandrovich Mikhailov

The magnetic field generation studies in astronomy lead to a number of interesting problems in mathematical physics. In the dynamo theory, the problem is reduced to a system of parabolic equations for the field components. Assuming that the field grows exponentially, we obtain an eigenvalue problem for the corresponding elliptic operator. The possibility of the field generation and behaviour of the system is characterized by the spectra of the operator. If all eigenvalues lie in the left half of the complex plane, the perturbations will decay. On the other hand, if some of the eigenvalues have positive real parts, the large-scale structures of the field can be generated. From the astrophysical point of view, galactic magnetic fields are very important to study. One of the main problems is connected with the peripheral regions, where the properties of the medium complicate the operator structure. We can use the perturbation theory to find the eigenvalues. However, the problem can be solved analytically by considering some specific approximations. We can find the spectra using numerical approaches in the case of the conditions that are close to the real ones. In this paper, we solve eigenvalue problems for different operators which are connected with magnetohydrodynamic processes in outer parts of galaxies.

Magnetic fields play an important role in the structure and evolution of accretion disks. There are strong arguments in favor of the fact that they are related to the action of the dynamo mechanism. The structure of the magnetic field that can be obtained using various approximations is discussed. Particular attention is paid to the possibility of the emergence of essentially non-axisymmetric solutions.

A large number of galaxies have regular magnetic fields of several microgauss. Their evolution is described usnig the dynamo mechanism. It is based on helicity of the turbulent motions in the interstellar medium and differential rotation of the object. The process is described by mean field dynamo equation which is quite difficult to be solve both analytically and numerically. So different models taking into account geometrical shape of the object are used. One of the most popular approaches is connected with thin disc approximation which replaces the z-derivatives by algebraic expressions. However, as for discs with large half-thickness it does not give proper results. So we consider the thick disc model which can be useful for the objects with quite large half-thickness. Here we give the estimates of the magnetic field growth rate and check them numerically. It has been shown that for thick disc it is much more difficult to obtain the growing field. One of the most interesting questions (which has not been studied before) is connected with phase diagrams for the magnetic field. We have found a stable point and describe the typical phase trajectories leading to it

The possibility of generating a magnetic field at a large distance (10–15 kpc) from the center of the galaxy has been investigated. The magnetic field can be generated both by the dynamo mechanism, considering nonlinear effects, and by magnetorotational instability. Presumably, the nonlinear effects of the dynamo mechanism play a more significant role at this distance.

The existence of magnetic fields in spiral galaxies is beyond doubt and is confirmed by both observational data and theoretical models. Their generation occurs due to the dynamo mechanism action associated with the properties of turbulence. Most studies consider magnetic fields at moderate distances to the center of the disk, since the dynamo number is small in the marginal regions, and the field growth should be suppressed. At the same time, the computational results demonstrate the possibility of magnetic field penetration into the marginal regions of galaxies. In addition to the action of the dynamo, magnetorotational instability (MRI) can serve as one of the mechanisms of the field occurrence. This research is devoted to the investigation of MRI impact on galactic magnetic field generation and solving the occurring eigenvalue problems. The problems are formulated assuming that the perturbations may possibly increase. In the present work, we consider the eigenvalue problem, picturing the main field characteristics in the case of MRI occurrence, where the eigenvalues are firmly connected with the average vertical scale of the galaxy, to find out whether MRI takes place in the outer regions of the galaxy. The eigenvalue problem cannot be solved exactly; thus, it is solved using the methods of the perturbation theory for self-adjoint operators, where the eigenvalues are found using the series with elements including parameters characterizing the properties of the interstellar medium. We obtain linear and, as this is not enough, quadratic approximations and compare them with the numerical results. It is shown that they give a proper precision. We have compared the approximation results with those from numerical calculations and they were relatively close for the biggest eigenvalue.

At present, there is little doubt that accretion discs surrounding compact astrophysical objects such as black holes, white dwarfs, and neutron stars may have magnetic field structures. Thus, they explain the transfer of angular momentum between different parts of the disc and some other processes. There are various ways to explain the occurrence of these magnetic fields. In this paper, we study the possibility of generation of magnetic fields due to the Biermann battery mechanism. It is associated with radial flows of protons and electrons. Due to their different masses, they interact differently with the rotating medium, producing circular currents that generate magnetic fields. Previously, a similar process was studied for galactic discs and it was shown that the battery mechanism can generate initial magnetic fields in such objects. Here, we discuss the action of the Biermann battery for accretion disks. This requires solving an integral equation of the second kind, which arises if we take into account the self-interaction of the magnetic field. It is shown that co-rresponding fields are quite significant and can play an important role in the evolution of magnetic fields in discs.

As well known, magnetic fields in space are distributed very inhomogeneously. Some-times field distributions have forms of filaments with high magnetic field values. As many ob-servations show, such a filamentation takes place in convective cells in the Sun and other astro-physical objects. This effect is associated with the frozenness of the magnetic field into a medium with high conductivity that leads to compression of magnetic field lines and forming magnetic filaments. We show analytically, based on the general analysis, that the magnetic field intensifies in the regions of downward flows in both two-dimensional and three-dimensional convective cells. These regions of the hyperbolic type for magnetic fields play a role of a specific attractor. This analysis was confirmed by numerical simulations for 2D convective cells of the roll-type. Without dissipation the magnetic field grows exponentially in time and does not depend on the aspect ratio between horizontal and vertical scale of the cell. An increase due to compression in the magnetic field in the high conductive plasma is saturated due to the natural limitation associated with dissipative effects when the maximum magnitude of the magnetic field is of the order of the root of the magnetic Reynolds number Rem. For the solar convective zone the mean kinetic energy density exceeds mean magnetic energy density at least for two orders of magnitude that allows one to use the kinematic approximation for the MHD induction equation. In this paper based on the stability analysis we explain why downward flows influence magnetic filaments from making them more flat with orientation along interfaces between convective cells.

As is well known, magnetic fields in space are distributed very inhomogeneously. Sometimes, field distributions have forms of filaments with high magnetic field values. As many observations show, such a filamentation takes place in convective cells in the Sun and other astrophysical objects. This effect is associated with the frozenness of the magnetic field into a medium with high conductivity that leads to the compression of magnetic field lines and formation of magnetic filaments. We analytically show, based on the general analysis, that the magnetic field intensifies in the regions of downward flows in both two-dimensional and three-dimensional convective cells. These regions of the hyperbolic type in magnetic fields play the role of a specific attractor. This analysis was confirmed by numerical simulations of 2D roll-type convective cells. Without dissipation, the magnetic field grows exponentially in time and does not depend on the aspect ratio between the horizontal and vertical scales of the cell. An increase due to compression in the magnetic field of highly conductive plasma is saturated due to the natural limitation associated with dissipative effects when the maximum magnitude of a magnetic field is of the order of the root of the magnetic Reynolds number Re_m. For the solar convective zone, the mean kinetic energy density exceeds the mean magnetic energy density for at least two orders of magnitude, which allows one to use the kinematic approximation of the MHD induction equation. In this paper, based on the stability analysis, we explain why downward flows influence magnetic filaments, making them flatter with orientation along the interfaces between convective cells.

Nowadays it is strongly believed that in a large number of spiral galaxies there are regular magnetic fields. Their existence is proved by measurements of the Faraday rotation of polarization plane of electromagnetic waves which are received on modern radio telescopes. The theoretical description of magnetic fields generation is connected with the mean field dynamo mechanism. It is based on joint action of the ?-effect and differential rotation. At the first stage of evolution, magnetic fields enlarge exponentially with the growth rate described by the largest eigenvalue of the corresponding differential operator. If the field becomes comparable with the equipartition value, the nonlinear terms become more important, and they are connected with saturation of the field growth. The non- linear system of equations has several stationary points, and some of them are stable. According to the contrast structures theory for a quite small turbulent viscosity and some initial conditions, in different regions there will be magnetic fields of opposite directions, and they will be divided by narrow transition layers. Such features, for example, characterize the magnetic field of the Milky Way. Most of the existing works describe reversals for quite moderate distances from the galaxy center. However, it was shown previously that the magnetic field can principally be generated also in outer parts of galaxies, which are situated at distances of more than 10 kpc from the rotation axis. It is interesting to describe the appearance of reversals in such parts of galaxies. In this work we have studied the dynamo action in outer parts of the Milky-Way-like galaxies. It is shown that it is possible to generate opposite magnetic fields there. We have used random initial conditions to obtain spatial reversals. Because of a large volume of computations, they were carried out by using graphics processing units (GPUs).