Set Of Magnetohydrodynamic Equations Research Articles

The Dynamical Structure of the Outflows Driven by a Large-scale Magnetic Field

Abstract A large-scale magnetic field is crucial in launching and collimating jets/outflows. It is found that the magnetic flux can be efficiently transported inward by a fast-moving corona above a thin disk. In this work, we investigate the dynamical structure of the outflows driven by the large-scale magnetic field advected by a hot corona. With the derived large-scale magnetic field, the outflow solution along every field line is obtained by solving a set of magneto-hydrodynamic equations self-consistently with boundary conditions at the upper surface of the corona. We find that the terminal speeds of the outflows driven from the inner region of the disk are ∼0.01–0.1c. The temperatures of the outflows at a large distance from the black hole are still as high as several ten keV. The properties of the magnetic outflows derived in this work are roughly consistent with the fast outflows detected in some luminous quasars and X-ray binaries (XRBs). The total mass-loss rate in the outflows from the corona is about 7%–12% of the mass-accretion rate of the disk. The three-dimensional field geometry, the velocity, temperature, and density of the outflows derived in this work can be used for calculating the emergent spectra and their polarization of the accretion disk/corona/outflow systems. Our results may help understand the features of the observed spectra of XRBs and active galactic nuclei.

On the relation between DC current locations and an EUV bright point: A case study

Context. Motion of the photospheric plasma forces the footpoints of magnetic flux tubes to move. This can give rise to electric currents in the solar atmosphere. The dissipation of these electric currents and the consequent heating of the solar plasma may be responsible for the formation of Extreme-UltraViolet (EUV) and X-ray bright points. Earlier bright point models usually consider either the emergence or the canceling of photospheric magnetic features as being responsible for reconnection causing the bright point. Aims. We investigate the consequences of different patterns of horizontal photospheric plasma motion for the generation of electric currents in the solar atmosphere and locate them with respect to an observed EUV bright point. The goal is to find out whether these currents might be responsible for the heating of bright points. Methods. To perform this study we use a “data driven” three dimensional magnetohydrodynamic model. The model solves an appropriate set of magnetohydrodynamic equations and uses, as initial condition, the magnetic field extrapolated from the line-of-sight component of the photospheric magnetic field observed by MDI/SoHO and the height-stratified, equilibrium density and temperature of the solar corona. We apply different patterns of horizontal photospheric plasma motion, derived from the temporal evolution of the photospheric magnetic structures in the course of the bright point lifetime, as boundary conditions of the model. Results. All applied patterns of horizontal photospheric plasma motion (shearing, convergence and fragmentation) lead to the formation of electric currents in the chromosphere, transition region and corona. Currents do not develop everywhere in the region where the motion is applied but in specific places where the magnetic field connectivity changes significantly. An important result is that the position where the electric currents develop is independent of the motion pattern used as boundary condition of the model. A comparison with data obtained by TRACE in the 1550 A channel and by the EIT in the 195 A channel shows that the region where the strongest current concentrations are formed coincides with the region where the EUV bright point appears.

Self-gravitating rotating anisotropic pressure plasma in presence of Hall current and electrical resistivity using generalized polytrope laws

The effects of uniform rotation, finite electrical resistivity, electron inertia, and Hall current on the self-gravitational instability of anisotropic pressure plasma with generalized polytrope laws have been studied. A general dispersion relation is obtained with the help of the relevant linearized perturbed magnetohydrodynamic (MHD) equations incorporating the relevant contributions of various effects of the problem using the method of normal mode analysis. The general dispersion relation is further reduced for the special cases of rotation; i.e., parallel and perpendicular to the direction of the magnetic field. The longitudinal and transverse modes of propagation are discussed separately for investigation of condition of instability. The effects of rotation, Hall current, finite electron inertia, and polytropic indices are discussed on the gravitational, “firehose,” and “mirror” instabilities. The numerical calculations have been performed to obtain the dependence of the growth rate of the gravitational unstable mode on the various physical parameters involved. The finite electrical resistivity, rotation, and Hall current have a stabilizing influence on the growth rate of the unstable mode of wave propagation. The finite electrical resistivity removes the effect of magnetic field and polytropic index from the condition of instability in the transverse mode of propagation for both the cases of rotation. It is also found that the Jeans criterion of gravitational instability depends upon rotation, electron inertia, and polytropic indices. In the case of transverse mode of propagation with the axis of rotation parallel to the magnetic field, it is observed that the region of instability and the value of the critical Jeans wavenumber are larger for the Chew–Goldberger–Low set of equations in comparison with the MHD set of equations. The stability of the system is discussed by applying Routh–Hurwitz criterion. The inclusion of rotation or Hall current or both together depresses the growth rate of mirror instability. We also note that the condition of mirror instability depends upon polytropic indices.

Using plasma physics to weigh the photon

The currently accepted value for the upper bound for the photon mass, mph, is 22 orders of magnitude less than the electron mass. As the mass mph is so incredibly small, it has essentially no effect on atomic and nuclear physics; and it is very difficult to improve this estimate by laboratory experiments. However, even a very small mass may have a significant effect on astrophysical phenomena occurring on a scale exceeding the photon Compton length (where for the currently accepted mass). A set of magnetohydrodynamic equations (assuming a finite photon mass) are used to analyze properties of the solar wind at Pluto's orbit. This yields an improved (reduced) by a factor of 70 estimate of the photon mass. Possible opportunities and challenges for the further reduction of the upper limit for mph based on the properties of larger-scale astrophysical objects are discussed.

Structure of intermediate shocks and slow shocks in a magnetized plasma with heat conduction

The structure of slow shocks and intermediate shocks in the presence of a heat conduction parallel to the local magnetic field is simulated from the set of magnetohydrodynamic equations. This study is an extension of an earlier work [C. L. Tsai, R. H. Tsai, B. H. Wu, and L. C. Lee, Phys. Plasmas 9, 1185 (2002)], in which the effects of heat conduction are examined for the case that the tangential magnetic fields on the two side of initial current sheet are exactly antiparallel (By=0). For the By=0 case, a pair of slow shocks is formed as the result of evolution of the initial current sheet, and each slow shock consists of two parts: the isothermal main shock and the foreshock. In the present paper, cases with By≠0 are also considered, in which the evolution process leads to the presence of an additional pair of time-dependent intermediate shocks (TDISs). Across the main shock of the slow shock, jumps in plasma density, velocity, and magnetic field are significant, but the temperature is continuous. The plasma density downstream of the main shock decreases with time, while the downstream temperature increases with time, keeping the downstream pressure constant. The foreshock is featured by a smooth temperature variation and is formed due to the heat flow from downstream to upstream region. In contrast to the earlier study, the foreshock is found to reach a steady state with a constant width in the slow shock frame. In cases with By≠0, the plasma density and pressure increase and the magnetic field decreases across TDIS. The TDIS initially can be embedded in the slow shock’s foreshock structure, and then moves out of the foreshock region. With an increasing By, the propagation speed of foreshock leading edge tends to decrease and the foreshock reaches its steady state at an earlier time. Both the pressure and temperature downstreams of the main shock decrease with increasing By. The results can be applied to the shock heating in the solar corona and solar wind.

Structure of slow shocks in a magnetized plasma with heat conduction

The structure of slow shocks in the presence of a heat conduction parallel to the local magnetic field is simulated from the set of magnetohydrodynamic equations. In this study, a pair of slow shocks is formed through the evolution of a current sheet initiated by the presence of a normal magnetic field. It is found that the slow shock consists of two parts: The isothermal main shock and foreshock. Significant jumps in plasma density, velocity and magnetic field occur across the main shock, but the temperature is found to be continuous across the main shock. The foreshock is featured by a smooth temperature variation and is formed due to the heat flow from downstream to upstream region. The plasma density downstream of the main shock decreases with time, while the downstream temperature increases with time, keeping the downstream pressure constant. It is shown that the jumps in plasma density, pressure, velocity, and magnetic field across the main shock are determined by the set of modified isothermal Rankine–Hugoniot conditions. It is also found that a jump in the temperature gradient is present across the main shock in order to satisfy the energy conservation. The present results can be applied to the heating in the solar corona and solar wind.

Numerical studies of reflection process on a field-reversed configuration plasma

A two-dimensional magnetohydrodynamic (MHD) simulation of a reflection on a field-reversed configuration (FRC) plasma is performed in the parameter range of the FRC injection experiment (FIX). The full set of MHD equations are solved on a rezoned Lagrangian mesh which employs an adaptive algorithm to concentrate the grid in regions of sharp plasma pressure gradients. It is shown from the simulation results that the FRC plasma is reflected by downstream magnetic mirror field at the end of the confinement region in the FIX machine without destroying the closed magnetic flux surfaces. The effects of this field on FRC plasma are discussed.

Surface structure formation in ion deposition

In the present paper, the possibility of surface structure formation in thin film ion deposition accompanied by undamped plasma oscillations is studied theoretically. A magnetic field parallel to the substrate surface and perpendicular to the ion flow direction is applied to the plasma region. A model of ion deposition from such a structure is developed for the purpose of calculation of the non-uniform deposition followed by the surface structure formation. The use of environments such as plasma density, magnetic field and ion energy required are examined. Ion flow transit through the oscillating plasma area was studied using a set of magneto-hydrodynamic equations. It was shown that with plasma density of 10 19 m −3, magnetic field value of 0.05 T and ion energy of 1000 eV, surface relief with a trench width of 10–1000 μm can be created. The maximum value of non-dimensional ridge height is found to be about 0.8 and the maximum value of non-dimensional ridge width of about 0.2 with the plasma–surface distance to the wavelength ratio being 1. This proves the possibility of formation of sharp enough profiles.

Consequences of nongyrotropy in magnetohydrodynamics

It is assumed in the magnetohydrodynamic set of equations that a plasma is both isotropic and gyrotropic at any instant of time. However, this cannot be always justified for applications in space plasmas which are essentially collisionless. By introducing two relaxation time scales, one for isotropization and another for gyrotropization, we propose a model which describes evolution of the field and the plasma variables, including all the pressure tensor elements. We discuss briefly the linear wave dispersion relation of the system, and then attempt to apply the model to the magnetic reconnection.

MHD ANALYSIS OF PETSCHEK-TYPE RECONNECTION IN NON-UNIFORM FIELD AND FLOW GEOMETRIES

Analytical studies of reconnection have, for the most part, been confined to steady and uniform current sheet geometries. In contrast to these implifications, natural phenomena associated with the presence of current sheets indicate highly non-uniform structure and time-varying behaviour. Examples include the violent outbursts of energy on the Sun known as solar flares, and magnetospheric phenomena such as flux transfer events, plasmoids, and auroral activity. Unlike the theoretical models, reconnection therefore occurs in a highly dynamic and structured plasma environment. In this article we review the mathematical tools and techniques which are available to formulate models capable of describing the effects of reconnection in such situations. We confine attention to variants of the reconnection model first discussed by Petschek in the 1960s, in view of its successful application in predicting and interpreting phenomena in the terrestrial magnetosphere. The analysis of Petschek-type reconnection is based on the equations of ideal magnetohydrodynamics (MHD), which describe the large-scale behaviour of the magnetic field and plasma flow outside the diffusion region, which we determine as a localised part of the current sheet in which reconnection is initiated. The approach we adopt here is to transform the MHD equations into a Lagrangian or so-called 'frozen-in' coordinate system. In this coordinate system, the equation of motion transforms into a set of coupled nonlinear equations, in which the presence of inhomogeneous magnetic fields and/or plasma flows gives rise to a term similar to that which appears in the study of the ordinary string equation in a non-homogeneous medium. As demonstrated here, this approach not only clarifies and highlights the effects of such non-uniformities, it also simplifies the solution of the original set of MHD equations. In particular, this is true for those types of problem in which the total pressure can be considered as a known quantity from the outset. To illustrate the method, we solve several 2D problems involving magnetic field and flow non-uniformities: reconnection in a stagnation-point flow geometry with antiparallel magnetic fields; reconnection in a Y-type magnetic field geometry with and without velocity shear across the current sheet; and reconnection in a force-free magnetic field geometry with field lines of the form xy = const. These case examples, chosen for their tractability, each incorporate some aspects of the field and flow geomtries encountered in solar-terrestrial applications, and they provide a starting point for further analytical as well as numerical studies of reconnection.

Radial expansion of the magnetohydrodynamic equations for axially symmetric configurations

Abstract We introduce a general expansion approach to obtain a fully consistent closed set of magnetohydrodynamic equations in two independent variables, which is particularly useful to describe axially symmetric, time-dependent problems with weak variation of all quantities in the radial direction. This is done by considering the hierarchy of expanded magnetofluid equations in cylindrical coordinates and equating terms with equal powers in the radial coordinate r. From geometrical considerations it is shown that the radial expansions of the pertaining physical quantities are either even series or odd series in r; this introduces a significant reduction in the number of variables and equations. The closure of the system is provided by appropriate boundary conditions. Among other possible applications, the method is relevant for the analysis of structure and dynamics of magnetic field concentrations in stellar atmospheres.

Effect of Hall Current on the Magnetogravitational Instability of Anisotropic Plasma with Generalized Polytrope Law

AbstractThe effect of Hall current on the propagation of small perturbations through self gravitating anisotropic collisionless pressure plasma with generalized polytrope law is investigated. The poly‐trope law for pressure components parallel and perpendicular to the direction of magnetic field is utilized in the analysis. The effect of Hall current and finite conductivity is introduced in the generalized Ohm's law. Using the polytrope law and Ohm's law dispersion relations are obtained from linearized perturbation equations for wave propagation along and perpendicular to the direction of magnetic field. The dispersion relations incorporating polytrope indices are able to represent the Chew, Goldberger and Low approximation with double adiabatic equation of state for the anisotropic pressure and the magnetohydrodynamic set of equations with isothermal equation of state for the isotropic pressure. The effect of Hall current, finite conductivity and polytrope indices is discussed on the well known hose and gravitational instability. It is found that Jeans' criterion depends on polytrope indices and the condition of gravitational instability is determined for different special cases of interest.

Skinning process stability of the magnetic field in the solar active regions

Skinning process stability of the magnetic field in homogeneous plasma is studied. A set of magnetohydrodynamic equations is used. Dependence of electrical conductivity on the plasma parameters and radiation intensity in grey-body approximation are taken into account. The investigation is carried out on the model problems in linear approximation and by means of numerical solution of MHD equations. Threshold of stability and critical gradient of magnetic field in skin-layer are obtained. The model of the phenomenon proposed in the paper indicates on overheating instability of plasma with electric current in large gradient magnetic field zones as a possible trigger mechanism of solar flare origin.

The Normal Modes of Linearized Magnetohydrodynamics

The central purpose of this paper is to derive a general set of magnetohydrodynamic equations for a two component plasma in an external magnetic field and to find the eigenmodes of the linearized equations. The magnetohydrodynamic equations are derived from nonequilibrium thermodynamic principles. It is pointed out that a minimal set of phenomenological coefficients are found in this manner. The magnetohydrodynamic equations are linearized and then solved for the magnetohydrodynamic eigenmodes in the two special cases of the wave vector k parallel and perpendicular to the external magnetic field.