Magnetohydrodynamic (MHD) waves in a stratified rotating plasma in a gravitational field are investigated in the Boussinesq approximation. A theory of flows on an
f
-plane, on a nontraditional
f
...-plane (with regard to the horizontal component of the Coriolis force), on a β-plane, and on a nontraditional β‑plane is developed. In each case, linear solutions to systems of three-dimensional MHD equations in the Boussinesq approximation are obtained that describe magnetic gravito-inertial waves, magnetostrophic waves, and magnetic Rossby waves. All existing types of three-wave interactions are found with the use of dispersion equations. In the case of magnetic Rossby waves in the β-plane approximation, it is shown that the low-frequency mode of a magnetic Rossby wave in the Boussinesq approximation is equivalent to that in the shallow-water MHD approximation. By the multiscale expansion method, a system of amplitude equations for interacting waves and the increments of two types of instability occurring in the system, decay and amplification, are obtained. For each type of three-wave interactions, it is shown that there is a difference in the coefficients and differential operators in the system of three-wave interactions.
The wave processes in a rotating layer of compressible astrophysical plasma with a stable stratification and a linear entropy profile are studied theoretically. The compressibility is taken into ...account in the anelastic approximation. In this approximation the acoustic waves are filtered out, the system contains terms with a potential temperature (entropy), and the continuity equation contains an initial stratified density profile. The Coriolis force in the magnetohydrodynamic equations for a compressible astrophysical plasma is considered in four different approximations: a traditional
f
-plane, a nontraditional
f
-plane (given the horizontal component of the Coriolis force), a traditional β-plane, and a nontraditional β-plane. Linear and nonlinear theories of wave processes have been developed for each Coriolis force approximation under consideration. New types of waves have been found, with the rotation, magnetic field, gravity, and compressibility serving as their restoring mechanisms. The compressibility effects are represented in the new dispersion equations by the Brunt–Väisälä frequency for compressible stratified flows dependent on both initial density and pressure profiles. All of the realizable types of three-wave interactions have been revealed through a qualitative analysis of the dispersion curves. A system of equations for the amplitudes of interacting waves and the growth rates of parametric instabilities have been derived by the multiscale expansion method.
Direct ESS has some disadvantages, which are seen even in the case of repeated games when the sequence of stage ESSs may not constitute the direct ESS in the repeated game. We present here the ...refinement of the ESS definition, which eliminates these disadvantages and represents the base for the definition of ESS in games in extensive form. The effectiveness of this approach for multistage n-person games is shown for metagame (this notion is used for the first time), in which under some relevant conditions, the existence of ESS is proved, and ESSs are constructed using threat strategies.
Conditions for an arbitrary jump occurrence in isentropic flow are studied. It is shown that the jump in gas-dynamic parameters arises as a result of the evolution of a self-similar flow. The concept ...of self-focusing Riemann waves is introduced. It is shown that an arbitrary jump is formed only by these waves and the conditions for its generation are found. It is shown that there exists a critical velocity, below which a discontinuity cannot be formed isentropically. The second critical value of velocity, exceeding which a discontinuity is formed only in the presence of a vacuum region is found. It is shown that there are only two classes of solitary shock waves: those that form in a medium containing a vacuum region and those that form in a continuous medium. It is shown that not every fall of the Riemann wave leads to the appearance of a shock wave. The results obtained are of a general physical nature, since they are based only on the properties of quasilinear hyperbolic systems.
The shallow water approximation is generalized for describing large-scale flow of a liquid in the gravity field with a free surface. The classical shallow water equations are an alternative to the ...solution of the complete system of hydrodynamics equations in the gravity force field; however, the classical approximation does not take into account the density nonuniformity of a liquid layer. We have analyzed the flow of a thin layer of a rotating liquid with a free surface with account for the compressibility effects. We have obtained a system of quasi-linear differential equations describing the flow of a compressible liquid in the shallow water approximation. The solutions to this system have been obtained in the form of linear Poincare waves on the
f
plane and the Rossby waves on the β plane in compressed flows. The nonlinear dynamics of the Rossby waves in compressible flows is analyzed using the method of multiscale expansions. The resulting three-wave equations for the amplitudes of interacting waves are analyzed for parametric instabilities, and their increments are determined.
We have studied rotating magnetohydrodynamic flows of a thin layer of astrophysical plasma with a free boundary in the β-plane. Nonlinear interactions of the Rossby waves have been analyzed in the ...shallow-water approximation based on the averaging of the initial equations of the magnetic fluid dynamics of the plasma over the depth. The shallow-water magnetohydrodynamic equations have been generalized to the case of a plasma layer in an external vertical magnetic field. We have considered two types of the flow, viz., the flow in an external vertical magnetic field and the flow in the presence of a horizontal magnetic field. Qualitative analysis of the dispersion curves shows the presence of three-wave nonlinear interactions of the magnetic Rossby waves in both cases. In the particular case of zero external magnetic field, the wave dynamics in the layer of a plasma is analogous to the wave dynamics in a neutral fluid. The asymptotic method of multiscale expansions has been used for deriving the nonlinear equations of interaction in an external vertical magnetic field for slowly varying amplitudes, which describe three-wave interactions in a vertical external magnetic field as well as three-wave interactions of waves in a horizontal magnetic field. It is shown that decay instabilities and parametric wave amplification mechanisms exist in each case under investigation. The instability increments and the parametric gain coefficients have been determined for the relevant processes.
In this article, we demonstrate how an original effective "metal-free" and "chromatography-free" route for the synthesis of 3-thiocyanatopyrazolo1,5-
pyrimidines has been developed. It is based on ...electrooxidative (anodic) C-H thiocyanation of 5-aminopyrazoles by thiocyanate ion leading to 4-thiocyanato-5-aminopyrazoles (stage 1, yields up to 87%) following by their chemical condensation with 1,3-dicarbonyl compounds or their derivatives (stage 2, yields up to 96%). This method is equally effective for the synthesis of 3-thiocyanatopyrazolo1,5-
pyrimidines, both without substituents and with various donor (acceptor) substituents in the pyrimidine ring.
In the paper the formulas are provided for the derivatives of Cauchy-type integral
which are smooth on the skeleton of the polydisk of functions
. These formulas express the derivatives of the order
...of
through the derivatives of lower order (Theorem 2.1). They are used for estimating the smoothness of the derivatives of the Cauchy-type integral in terms of Hölder order scale (Theorem 3.1).
A theory of large-scale flows in a rotating astrophysical plasma under conditions of non-trivial properties of the physical medium, which are not described by the classical hydrodynamic theory of ...plasma, is developed. As a first step, the theory is developed within a neutral fluid model to describe astrophysical plasma, with a subsequent generalization in mind to take into account magnetic effects. Such a model is of independent importance for studying turbulent dynamo in star-forming regions in galaxies and hydrodynamic instabilities in poorly ionized disks, for describing meridional flows below convective zones in low-mass stars and on the Sun, as well as for studying oscillations of the Sun and stars. Therefore, the results obtained have a wider application, e.g., for describing geophysical currents. The theory is based on two key ideas developed in plasma astrophysics: the use of a shallow water model with large-scale compressibility and the use of a two-layer shallow water model. Equations for two-layer shallow water are derived taking into account rotation and the effect of flow sphericity on rotation, in which the effects of large-scale compressibility are taken into account in the upper layer. For a rotating system, dispersion relations are obtained for Poincaré waves in two-layer shallow water, taking into account large-scale compressibility; similar dispersion relations for Poincaré waves are obtained in the high-frequency limit taking into account the effect of sphericity on rotation; in the low-frequency limit, a dispersion relation is obtained for Rossby waves. It is shown that the dispersion relations for Poincaré waves, taking into account the sphericity of the flow, have a qualitatively different form, which leads to three-wave interactions of Poincaré waves and the interaction of two Poincaré waves with a Rossby wave, which are not observed in a single-layer flow of a compressible fluid. All types of three-wave interactions for the flows under consideration are studied using the method of multiscale expansions.
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