scholarly journals Propagation of Shock Waves in a Polytrope with a Toroidal Magnetic Field. II. Solution of Complete Differential Equations

1969 ◽  
Vol 22 (5) ◽  
pp. 605
Author(s):  
NK Sinha
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Shock Waves ◽  
Differential Equations ◽  
Toroidal Magnetic Field ◽  
Initial Values ◽  
Made In

The differential equations for the shock parameters along shock rays in the case of propagation of a spherically developed shock wave in a polytrope with a toroidal magnetic field, obtained in Part I, have been integrated numerically for a particular set of initial values. The results are compared with the corresponding results in Part I obtained by neglecting certain small terms and it is found that the effect of this omission is not significant. This substantiates the results and justifies the simplification made in Part 1.

scholarly journals Propagation of Shock Waves in a Polytrope with a Toroidal Magnetic Field. I. Simplified Solution of Differential Equations

1969 ◽  
Vol 22 (5) ◽  
pp. 589
Author(s):  
NK Sinha
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Approximate Solution ◽  
Partial Differential Equations ◽  
Shock Waves ◽  
Differential Equations ◽  
Toroidal Magnetic Field ◽  
Spherical Shock Wave ◽  
Simplified Solution ◽  
Spherical Shock

The propagation of an initially spherical shock wave in a polytrope with a magnetic field has been studied. The model chosen for the purpose was that of a poly trope with a toroidal magnetic field given previously by Sinha. Butler's method has been extended to transform the set of governing partial differential equations into a set of ordinary differential equations involving derivatives in the direction of propagation of the shock element at any point. An approximate solution is obtained and the effect of the toroidal magnetic field on the geometry of the front as well as on the effects brought about by the shock is discussed.

scholarly journals On Fast Magnetic Field Reconnection

1976 ◽  
Vol 71 ◽  
pp. 353-366 ◽  
Author(s):  
E. R. Priest ◽  
A. M. Soward
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Electrical Conductivity ◽  
Shock Waves ◽  
Similarity Solutions ◽  
Diffusion Region ◽  
Magnetohydrodynamic Equations ◽  
Reconnection Rate ◽  
Mathematical Basis ◽  
Rate Dependent

The first model for ‘fast’ magnetic field reconnection at speeds comparable with the Alfvén speed was put forward by Petschek (1964). It involves one shock wave in each quadrant radiating from a central diffusion region and leads to a maximum reconnection rate dependent on the electrical conductivity but typically of order 10-1 or 10-2 of the Alfvén speed. Sonnerup (1970) and Yeh and Axford (1970) then looked for similarity solutions of the magnetohydrodynamic equations, valid at large distances from the diffusion region; by contrast with Petschek's analysis, their models have two waves in each quadrant and produce no sub-Alfvénic limit on the reconnection rate.Our approach has been, like Yeh and Axford, to look for solutions valid far from the diffusion region, but we allow only one wave in each quadrant, since the second is externally generated and so unphysical for astrophysical applications. The result is a model which qualitatively supports Petschek's picture; in fact it can be regarded as putting Petschek's model on a firm mathematical basis. The differences are that the shock waves are curved rather than straight and the maximum reconnection rate is typically a half of what Petschek gave. The paper is a summary of a much larger one (Soward and Priest, 1976).

Magneto-gasdynamic flow over a wedge

1966 ◽  
Vol 25 (1) ◽  
pp. 165-178 ◽  
Author(s):  
D. C. Pack ◽  
G. W. Swan
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Shock Waves ◽  
Wave Pattern ◽  
Ionized Gas ◽  
Current Sheets ◽  
Gasdynamic Flow ◽  
The Stability ◽  
The Magnetic Field ◽  
Insight Into

The solution for the flow of a fully ionized gas over a wedge of finite angle is known for the case when the applied magnetic field is aligned with the incident stream. In this flow there are current sheets on the surfaces of the wedge. When the magnetic field is allowed to deviate slightly from the stream, the current sheets may move into the gas and become shock waves. The magnetic fields adjacent to the wedge above and below it have to be matched. A perturbation method is introduced by means of which expressions for the unknown quantities in the different regions may be determined when there are four shocks attached to the wedge. The results give insight into the manner in which the shock-wave pattern develops as the obliquity of the magnetic field to the stream increases. The question of the stability of the shock waves is also examined.

scholarly journals Magnetohydrodynamic waves within the medium separated by the plane shock wave or rotational discontinuity

2005 ◽  
Vol 23 (5) ◽  
pp. 1889-1908 ◽  
Author(s):  
A. A. Lubchich ◽  
I. V. Despirak
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Surface Wave ◽  
Shock Waves ◽  
Dispersion Equation ◽  
Reference Frame ◽  
Discontinuity Surface ◽  
Mhd Waves ◽  
Magnetoacoustic Waves ◽  
Rotational Discontinuity

Abstract. Characteristics of small amplitude plane waves within the medium separated by the plane discontinuity into two half spaces are analysed. The approximation of the ideal one-fluid magnetohydrodynamics (MHD) is used. The discontinuities with the nonzero mass flux across them are mainly examined. These are fast or slow shock waves and rotational discontinuities. The dispersion equation for MHD waves within each of half space is obtained in the reference frame connected with the discontinuity surface. The solution of this equation permits one to determine the wave vectors versus the parameter cp, which is the phase velocity of surface discontinuity oscillations. This value of cp is common for all MHD waves and determined by an incident wave or by spontaneous oscillations of the discontinuity surface. The main purpose of the study is a detailed analysis of the dispersion equation solution. This analysis let us draw the following conclusions. (I) For a given cp, ahead or behind a discontinuity at most, one diverging wave can transform to a surface wave damping when moving away from the discontinuity. The surface wave can be a fast one or, in rare cases, a slow, magnetoacoustic one. The entropy and Alfvén waves always remain in a usual homogeneous mode. (II) For certain values of cp and parameters of the discontinuity behind the front of the fast shock wave, there can be four slow magnetoacoustic waves, satisfying the dispersion equation, and none of the fast magnetoacoustic waves. In this case, one of the four slow magnetoacoustic waves is incident on the fast shock wave from the side of a compressed medium. It is shown that its existence does not contradict the conditions of the evolutionarity of MHD shock waves. The four slow magnetoacoustic waves, satisfying the dispersion equation, can also exist from either side of a slow shock wave or rotational discontinuity. (III) The expressions determining the polarisation of the MHD waves are derived in the reference frame connected with the discontinuity surface. This form of presentation is much more convenient in investigating the interaction of small perturbations with MHD discontinuities. It is shown that the perturbations of the velocity and magnetic field associated with the surface magnetoacoustic wave have the elliptic polarisation. Usually the planes of polarisation for the perturbations of the velocity and magnetic field are not coincident with each other. Keywords. Space plasma physics (Discontinuities; Shock waves) – Interplanetary physics (Discontinuities; Interplanetary shocks) – Magnetospheric physics (Solar windmagnetosphere interactions)

scholarly journals Converging Cylindrical Symmetric Shock Waves in a Real Medium with a Magnetic Field

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1177 ◽  
Author(s):  
Munesh Devi ◽  
Rajan Arora ◽  
Mustafa M. Rahman ◽  
Mohd Junaid Siddiqui
Keyword(s):  
Magnetic Field ◽  
Shock Waves ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Inertial Confinement Fusion ◽  
Matrix Form ◽  
Ideal Medium ◽  
Group Theoretic Method ◽  
Real Medium ◽  
Cylindrical Shock Waves

The topic “converging shock waves” is quite useful in Inertial Confinement Fusion (ICF). Most of the earlier studies have assumed that the medium of propagation is ideal. However, due to very high temperature at the axis of convergence, the effect of medium on shock waves should be taken in account. We have considered a problem of propagation of cylindrical shock waves in real medium. Magnetic field has been assumed in axial direction. It has been assumed that electrical resistance is zero. The problem can be represented by a system of hyperbolic Partial Differential Equations (PDEs) with jump conditions at the shock as the boundary conditions. The Lie group theoretic method has been used to find solutions to the problem. Lie’s symmetric method is quite useful as it reduces one-dimensional flow represented by a system of hyperbolic PDEs to a system of Ordinary Differential Equations (ODEs) by means of a similarity variable. Infinitesimal generators of Lie’s group transformation have been obtained by invariant conditions of the governing and boundary conditions. These generators involves arbitrary constants that give rise to different possible cases. One of the cases has been discussed in detail by writing reduced system of ODEs in matrix form. Cramer’s rule has been used to find the solution of system in matrix form. The results are presented in terms of figures for different values of parameters. The effect of non-ideal medium on the flow has been studied. Guderley’s rule is used to compute similarity exponents for cylindrical shock waves, in gasdynamics and in magnetogasdynamics (ideal medium), in order to set up a comparison with the published work. The computed values are very close to the values in published articles.

One-dimensional spherical shock waves in an interstellar dusty gas clouds

2021 ◽  
Vol 76 (5) ◽  
pp. 417-425
Author(s):  
Astha Chauhan ◽  
Kajal Sharma
Keyword(s):  
Shock Wave ◽  
Shock Waves ◽  
Differential Equations ◽  
Shock Propagation ◽  
Dusty Gas ◽  
Shock Strength ◽  
Efficient System ◽  
One Dimensional ◽  
Arbitrary Strength ◽  
Spherical Shock

Abstract A system of partial differential equations describing the one-dimensional motion of an inviscid self-gravitating and spherical symmetric dusty gas cloud, is considered. Using the method of the kinematics of one-dimensional motion of shock waves, the evolution equation for the spherical shock wave of arbitrary strength in interstellar dusty gas clouds is derived. By applying first order truncation approximation procedure, an efficient system of ordinary differential equations describing shock propagation, which can be regarded as a good approximation of infinite hierarchy of the system. The truncated equations, which describe the shock strength and the induced discontinuity, are used to analyze the behavior of the shock wave of arbitrary strength in a medium of dusty gas. The results are obtained for the exponents from the successive approximation and compared with the results obtained by Guderley’s exact similarity solution and characteristic rule (CCW approximation). The effects of the parameters of the dusty gas and cooling-heating function on the shock strength are depicted graphically.

Structure of small-amplitude quasiparallel magnetohydrodynamic shock waves in plasmas with anisotropic viscosity and thermal conductivity

1998 ◽  
Vol 60 (4) ◽  
pp. 695-710 ◽  
Author(s):  
M. S. RUDERMAN
Keyword(s):  
Magnetic Field ◽  
Thermal Conductivity ◽  
Shock Wave ◽  
Shock Waves ◽  
Small Amplitude ◽  
Critical Value ◽  
Stationary Points ◽  
Mhd Waves ◽  
Stable Node ◽  
Anisotropic Viscosity

Shock waves in plasmas with strongly anisotropic viscosity and thermal conductivity are considered. The analysis is restricted to the case where the plasma beta is less than unity. The set of two equations that governs propagation of small-amplitude MHD waves at small angles with respect to the unperturbed magnetic field in such plasmas is derived. A qualitative analysis of this set of equations is carried out. It is shown that the shock structure is described by a solution that is a separatrix connecting two stationary points: a stable node and a saddle. This solution describes the structure of a fast quasiparallel shock wave, and it only exists when the ratio of the magnetic field component, perpendicular to the direction of shock-wave propagation after and before the shock is smaller than a critical value. This critical value is a function of the plasma beta. The structures of shock waves are calculated numerically for different values of the shock amplitude and the ratio of the coefficients of viscosity and thermal conductivity.

The theory of inoizing shock waves in a magnetic field. Part 1. Skew and oblique shock waves, boundary conditions and ionization stability

1981 ◽  
Vol 26 (1) ◽  
pp. 29-53 ◽  
Author(s):  
M. A. Liberman ◽  
A. L. Velikovich
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Shock Waves ◽  
Electric Field ◽  
Boundary Conditions ◽  
Wave Front ◽  
Shock Wave Front ◽  
Neutral Gas ◽  
Supersonic Shock ◽  
Photo Ionization

The general theory of ionizing shock waves in a magnetic field has been constructed. The theory takes into account precursor ionization of a neutral gas ahead of the shock wave front, caused by photo-ionization, as well as by the impact ionization with electrons accelerated by a transverse electric field induced by the shock front in the incident flow of a neutral gas. The concept of shock wave ionization stability, being basic in the theory of ionizing shock waves in a magnetic field, is introduced. An additional equation for the electric field in the shock wave is obtained. This equation, together with the investigation of the singular point in the downstream flow behind the shock wave front, provides all the information required for solving the problem. For example, this provides two additional boundary conditions for the shock waves of type 2, determining the value and direction of the electric field in the incident flow. One additional boundary condition determines a relation between the value and direction of the electric field for supersonic shock waves of type 3. There are no additional boundary conditions for supersonic shock waves of type 4. The electric field ahead of the shock front has two degrees of freedom. As well as for shocks of other types, its value is less than that of the transverse electric field at which an ionization wave could be emitted by the shock wave front (the ionization stability condition). The additional relationship for supersonic waves of type 4 determines the onset of an isomagnetic (viscous) jump in the structure of the shock wave front. The boundary conditions and ionizing shock wave structures, considered earlier by the authors of the present paper in the ‘limit of electrostatic breakdown’, as well as the structural determination of the electric field, considered earlier by Leonard, are limiting cases in the theory developed here. The ionizing shock wave structures are shown to transform from the GD regime at a low shock velocity to the MHD regime at an enhanced intensity of the shock wave. The abruptness of such a transition (e.g. the transition width on the Mach number scale) is determined by precursor photo-ionization.

scholarly journals Magnetohydrodynamic Ionizing Shock Waves in a Conducting Medium

1965 ◽  
Vol 18 (4) ◽  
pp. 363 ◽  
Author(s):  
B Green ◽  
RM May
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Shock Waves ◽  
Field Strength ◽  
Shock Front ◽  
Ionization Energy ◽  
Conducting Medium ◽  
Flow Parameters ◽  
Magnetohydrodynamic Shock ◽  
Initial Magnetic Field

We present numerical calculations for the flow parameters (velocity, density, pressure, etc.) in a magnetohydrodynamic shock wave propagating in a conducting medium. The effect of ionization in the shock front is included. The results are presented graphically for a complete range of the initial magnetic field strength and direction, and for several arbitrary values of the ionization energy of the downstream fluid.

scholarly journals Turbulent Magnetic Field Amplification behind Strong Shock Waves in GRB and SNR

2011 ◽  
Vol 7 (S279) ◽  
pp. 335-336
Author(s):  
Tsuyoshi Inoue
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Shock Waves ◽  
Supernova Remnants ◽  
Gamma Ray ◽  
Three Dimensional ◽  
Strong Shock ◽  
Field Amplification ◽  
Field Lines ◽  
Strong Shock Waves

AbstractUsing three-dimensional (special relativistic) magnetohydrodynamics simulations, the amplification of magnetic field behind strong shock wave is studied. In supernova remnants and gamma-ray bursts, strong shock waves propagate through an inhomogeneous density field. When the shock wave hit a density bump or density dent, the Richtmyer-Meshkov instability is induced that cause a deformation of the shock front. The deformed shock leaves vorticity behind the shock wave that amplifies the magnetic field due to the stretching of field lines.

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