Perpendicular Shock Waves Research Articles

On the structure of collisionless magneto—plasma shock waves at super—critical Alfvén—Mach numbers

Ions reflected from the sharp potential rise in a magneto-plasma, perpendicular shock wave result hi a broad, small-amplitude ‘foot’ in the magnetic field proffle. A theory is given for the structure of this foot, which shows its shape to be a rather flat parabolic curve commencing at about half a Larmor radius in front of the shock front proper. The theory also predicts the amplitude of the foot just in front of the sharp rise as a function of Alfvén—Mach number. Close agreement with experiment is found.

Structure of a Perpendicular Shock Wave in a Plasma

Based on the Boltzmann equations for fully ionized gases, analytic solutions of weak perpendicular shock waves are obtained. The result shows that all weak shock waves (in ionized gases as well as in nonionized gases) have similar profiles given by the hyperbolic-tangent function of the distance. The shock thickness is found to depend on four basic parameters, namely, the mean free path of the gas, the length of penetration, the Larmor radius based on thermal speed, and the mass ratio of electron and ion. From the explicit expression of the shock thickness, various significant cases are derived and discussed. The most interesting case occurs when the mean free path is larger than the electron length of penetration, which is in turn larger than the electron Larmor radius; it is found that both the scaled and the actual shock thickness can be smaller than the length of penetration—a result contrary to that obtained for collisionless plasmas.

Interactions of Hydromagnetic Waves

A generalized Rankine-Hugoniot relation for hydromagnetic shock waves propagating perpendicularly to the magnetic field is discussed. Then the interaction of the shock or rarefaction waves with the contact surface on which the magnetic field has a discontinuity is, analyzed by a graphical method. The motion of the high temperature part of a plasma surrounded by the strong magnetic field and the electric current is investigated applying the result of the above analysis. It is shown that under some initial condition the contracting motion of the plasma occurs succeeding its expansion. The results obtained by them show that the propagation of shocks is generally complicated for the arbitrary configuration of the magnetic field. However, it was shown by Taniuti G ) that the analysis of the fluid with infinite conductivity can be comparatively simplified, and that the propagation of hydromagnetic waves may be described in the way analogous to that of the ordinary hydrodynamics so far as the one-dimensional propagation of the hydromagnetic wave perpendicular to the magnetic field is concerned. In the present article, we deal with these perpendicular shock and rarefaction waves and their interaction. In § 2, the various shock relations, involving an equation which corresponds to the Rankine-Hugoniot equation in the hydrodynamics, will be presented. 4 ) The analysis of the interaction between the shock, rarefaction wave and the contact discontinuity will be given in § 3. In § 4 we shall discuss, on the basis of these results, an initial value problem in which the fluid is initially at rest and the high density part of the fluid associated with a weak magnetic field is surrounded by the low density part with a strong magnetic field. The subsequent b~haviour of the fluid and the magnetic field will be calculated in detail and it will be pointed out that under certain conditions the contact surface can oscillate due to its interaction with the rarefaction and shock waves. We hope that this result concerning the motion of the contact surface may be of some use for the qualitative explanations of some experimental facts, resulting from the pinched discharge in the cylindrical tube, although it may be difficult to find any actual phenomenon exactly corresponding with this kind of initial value problem.

Structure of a Magnetohydrodynamic Shock Wave in a Plasma of Infinite Conductivity

The structure of a magnetohydrodynamic perpendicular shock wave (magnetic field perpendicular to the direction of propagation) in a plasma of infinite conductivity has been analyzed. For the nonmagnetic case, the width of the shock front is found to be larger than in a non-ionized gas. Further, the width of the shock (in terms of mean free path within the shock) as a function of the shock strength as determined from the Stokes-Navier equations (which applies to weak shocks) is found to join smoothly at $M\ensuremath{\simeq}1.3$ with the same function as derived from the Mott-Smith analysis (which applies to strong shocks). The magnetic field tends to make the shock front narrower for shocks of moderate strength. It does not have appreciable effect for strong shocks.Application of the above results to solar radio noise and current theories of the origin of cosmic rays is indicated.