Advances in Mathematical Physics

Volume 2018, Article ID 8253210, 12 pages

https://doi.org/10.1155/2018/8253210

## Magnetoacoustic Heating in Nonisentropic Plasma Caused by Different Kinds of Heating-Cooling Function

Gdansk University of Technology, Faculty of Applied Physics and Mathematics, Ul. Narutowicza 11/12, 80-233 Gdansk, Poland

Correspondence should be addressed to Anna Perelomova; lp.ude.gp@avomolerep.anna

Received 25 July 2018; Accepted 2 September 2018; Published 24 September 2018

Academic Editor: Alexander Iomin

Copyright © 2018 Anna Perelomova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The nonlinear phenomena which associate with magnetoacoustic waves in a plasma are analytically studied. A plasma is an open system with external inflow of energy and radiation losses. A plasma’s flow may be isentropically stable or unstable. The nonlinear phenomena occur differently in dependence on stability or instability of a plasma’s flow. The nonlinear instantaneous equation which describes dynamics of nonwave entropy mode in the field of intense magnetoacoustic perturbations is the result of special projecting of the conservation equations in the differential form. It is analyzed in some physically meaningful cases; those are periodic magnetoacoustic perturbations and particular cases of heating-cooling function. A plasma is situated in the straight magnetic field with constant equilibrium magnetic strength which form constant angle with the direction of wave propagation. A plasma is initially uniform and equilibrium. The conclusions concern nonlinear effects of fast and slow magnetoacoustic perturbations and may be useful in direct and inverse problems.

#### 1. Introduction

The study of propagation of magnetohydrodynamic (MHD) perturbations has a key role in the astrophysical applications. It is of crucial importance in solar hydrodynamics, in particular in formation of coronal loops, fluid flows in thermonuclear reactors, interstellar gases, and planetary nebulae. Gaseous plasma is an open system with disturbed adiabaticity of wave perturbations. This may be modeled by some heating-cooling function which reflects the physical conditions of energy inflow and radiation losses [1–3]. Linear dynamics of the MHD waves, that is, waves of infinitely small magnitude in an open plasma which may be isentropically unstable, has been well studied in the last decade. The main conclusions are common in hydrodynamics of open systems [4, 5]. Flows of other fluids in thermodynamical nonequilibrium reveal analogous features which are described by the similar equations [6, 7]. Apart from complexity of the MHD system of equations which imposes coexistence of slow, fast sound modes, and Alfvén modes, along with the nonwave modes, other linear effects go into play, such as mechanical damping and thermal conduction of a plasma. The general nonlinear dynamics of perturbations in plasma is not far well understood. As for the peculiarities of nonlinear evolution of wave MHD perturbations without accounting for their interaction with other modes, they are well understood starting from some particular but important cases which concern one-dimensional flows along and across the magnetic field [8–10] and ending by various numerical investigations. Nakariakov and coauthors in [11] advanced in the analysis of one-dimensional propagation of the MHD wave perturbations at arbitrary angle between the straight magnetic field and direction of wave’s propagation. They have derived the generic weakly nonlinear evolutionary equation which describes slow and fast perturbations in active plasma. Authors of [12] concluded about possibility of self-organization of MHD waves and described theoretically slow magnetoacoustic shock autowaves with magnitude independent on the initial or boundary conditions. They discovered that the extension of a heating-cooling function in the Taylor series accounting for quadratic nonlinear terms “introduces new physics such as existence of solitary waves”. The conclusions concern nonlinear dynamics of individual slow and fast wave perturbations. The formation of the asymmetric autowave pulse in the similar geometry of a flow was described in [13].

While perturbations of infinitely small magnitude develop independently, nonlinearity makes different types of a fluid’s motion to interact in physical reality which deals with finite-magnitude perturbations. The wave modes may excite nonwave modes which form inhomogeneities in turn impacting on wave propagation; waves may be scattered by other waves, and so on [14–16]. Apart from nonlinearity, the reasons for interaction are irreversible losses in energy and momentum or/and the disturbance of adiabaticity. In description of nonlinear interaction of modes, we face with mathematical difficulties much serious as compared to the case of nonlinear dynamics of individual mode. The nonlinear excitation of the entropy mode in the field of slow and fast magnetosonic perturbations for different kinds of external heating-cooling function is the subject of this study. Following Nakariakov and coauthors, we consider weakly nonlinear flow of completely ionized gas affected by some generic heating-cooling function. Weak nonlinearity means the smallness of the Mach number in a flow: . Also, the impact of nonlinearity may be obtained as corrections to the linear results. This concerns the links between specific perturbations inside every mode and dynamic equations. The method which has been applied by the author in studies of nonlinear interaction of wave and nonwave modes in a number of different fluid flows is useful in investigation of magnetic flows as well. It allows subdividing nonlinear dynamic equations for perturbations in a specific mode from the system of conservation equations in a differential form. The procedure is in fact some linear combination of equations which eliminates all liner terms containing nonspecific perturbations. The dynamic coupling equations include the first-order derivatives with respect to time. The quadratic nonlinear terms which are products of nonspecific perturbations and their derivatives form the “foreign forces”. In the context of weak nonlinearity, the quadratic terms including magnetosonic perturbations are of importance. Nonlinear corrections in the dynamic equations for fast and slow magnetosonic perturbations may be also considered as a result of self-interaction of corresponding wave mode. Nonlinearity, even weak, accumulates in time and is responsible for formation of shock fronts, enrichment of perturbations’ spectrum, and nonlinear transfer of energy and momentum. In regard to magnetoacoustic heating, nonlinear transfer of the wave energy into the energy of the entropy mode leads to forming of the thermal lenses which in turn scatter sound. The effects of plasma’s boundaries are not considered. Nonlinear dynamic equations are valid with the accepted accuracy up to quadratic nonlinear terms, that is, up to terms proportional to the squared Mach number, . That concerns also terms originating from the heating-cooling function. The perturbations which specify the entropy mode enhance with time. In the frames of a model, their magnitudes should not exceed magnitudes of corresponding wave perturbations. When these magnitudes equalize, the entropy mode may have impact on the wave process. Hence, the equations are valid over some temporal and spatial domains, where wave perturbations remain dominant.

One may expect that not only the wave processes, but the magnetoacoustic heating reveals anomalous behavior in the isentropically unstable active plasma, as it happens to all acoustically active media independently on the physical reason of acoustical activity [6, 7]. The subsequent analysis confirms this conclusion. We do not consider mechanical and thermal losses in a plasma and its finite electrical conductivity which are well studied. They introduce additional attenuation and dispersion. The magnetoacoustic heating due to propagation of the MHD perturbations across the straight magnetic field has been studied by the author in [17, 18]. In this study, we analyze the instantaneous dynamic equation of excitation of the entropy mode by fast and slow MHD sound modes in the field of planar waves which form an arbitrary angle with the magnetic strength and discuss it in some physically meaningful cases of wave perturbations and the heating-cooling function. That concerns also the case of nonadiabatic instability. The first results in a special case of heating-cooling function which depends exclusively on temperature may be found in the recent author’s paper [19]. In this study, we make conclusions about entropy perturbations due to magnetosonic heating in general case.

#### 2. Perturbations of Infinitely Small Magnitude in the MHD Flow

The system of PDE equations consists of continuity equation, momentum equation for ionized gas, energy balance equation, and electrodynamic equations: where , , , and are the velocity of a plasma, the magnetic field strength, plasma’s pressure, and density, respectively, and is the magnetic permeability of free space. is the heating-cooling function which disturbs adiabaticity of wave perturbations in a plasma. The choice of the heating-cooling function which depends on pressure and density makes using these basic thermodynamic variables and expressing all other thermodynamic quantities such as internal energy as functions of pressure and density convenient. Following [11], it is assumed to be dependent on pressure and density. The third equation in the set (1) relates to an ideal gas with the caloric equation of state where denotes the internal energy of a gas and is the ratio of specific heats under constant pressure and constant density. We consider the same geometry of a flow as in [11]. The equilibrium magnetic strength lies in the plane. Its equilibrium magnitude is constant, and it forms constant angle with the positive direction of axis , so denote projections of onto axis and . Axis points to the direction of wave propagation. Figure 1 recalls the geometry of a planar flow.