Tosiya Taniuti

Perturbation Method for a Nonlinear Wave Modulation. II

A perturbation method given in a previous paper of this series is applied to two physical examples, the electron plasma wave and a nonlinear Klein‐Gordon equation. In these systems, and probably in most physical systems, an assumed condition for a mode of l = 0 is not valid. Consequently, the direct application of the method is impossible. In the present paper, we shall illustrate by these examples how this difficulty can be overcome to allow us to use the method. As a result we shall find that, in either case, the original equation can be reduced to the nonlinear Schrodinger equation.

Precursor of Ion‐Acoustic Quasishock Wave in Collisionless Plasma

A weak, ion‐acoustic quasishock wave which propagates into a uniform state is considered. A relevant initial value problem of the Vlasov equation is solved to give the long time behavior of a precursor which is a stream of ions reflected by the electrostatic potential. An asymptotic expansion in terms of the small amplitude is used. The ratio of the ion temperature to the electron temperature is assumed small so that effects of the reflected ions appear as a correction to the Korteweg‐de Vries equation which governs the wave propagation due to the nonresonant ions. The connection with the stationary solution is examined.

Deflagration Waves in Laser Compression. I

The structure of a stationarily propagating deflagration wave, driven by the absorption of laser energy, is investigated by use of a two temperature hydrodynamic model. It is shown that nonlinear electron thermal conduction and electron-ion temperature relaxation are responsible for most of the structure. In a steady compression model, plasma motion of the whole system is determined with the aid of energy conservation of the system.

Nonlinear Drift Waves

Turbulence and solitary vortex structures of electrostatic drift waves are presented using a simple model equation based on two-dimensional cold ions and Boltzmann electrons.

Steady Axisymmetric Toroidal Equilibrium in Ideal Magnetohydrodynamics

A self‐consistent theory of steady axisymmetric toroidal equilibrium in ideal magnetohydrodynamics with poloidal rotation about the magnetic axis and toroidal flow around the torus is described. An application of the characteristic theory of hyperbolic partial differential equations to the steady hydromagnetic equations yields critical rotational velocities (poloidal) for the formation of standing hydromagnetic waves, which depend on the direction of the wave normal. For low β and large aspect ratio, the critical velocities for the slow (magnetosonic) mode are, for any direction, nearly equal to a velocity found by Stringer for the onset of anomalous diffusion. Flows with poloidal motion about the critical slow velocities are investigated by analogy with nozzle flow in conventional gasdynamics, the distance along the axis of nozzle corresponding to the poloidal angle. The periodicity of entropy with respect to the poloidal angle excludes shock transitions and, as a result, trans‐slow‐magnetosonic flows. A class of continuous, self‐consistent solutions in the standard tokamak ordering is obtained in terms of the expansion of the square root of the inverse aspect ratio, in which the poloidal flow approaches, at a point corresponding to the throat of nozzle, the critical slow velocity in the poloidal direction; the solutions are found to be approximated by continuous solutions in the usual low‐β approximation which does not make use of Amperes law.

Higher Order Corrections to the Soliton-Velocity and the Linear Dispersion Relation

Higher-order corrections to the velocities of solitons in the weakly dispersive system, which are obtained so as to eliminate the secular terms appearing in the higher-order approximations, are derived from the linear dispersion relation.

Oblique Hydromagnetic Waves in a Cold Plasma

Steady, one‐dimensional, oblique, hydromagnetic waves of small but finite amplitudes in a cold plasma are considered. Analysis is based on an equation derived by Saffman and is restricted to a class of the waves with a large critical Mach number and high‐frequency oscillations. Solutions are given in an asymptotic expansion. In the lowest order of perturbation the equation is integrated to give the solutions in closed form, which exhibit compressional solitary waves.

Envelope solitons in nonlinear acoustics

Abstract A quasi-monochromatic acoustic wave of small but finite amplitude propagating in a cylindrical wave-guide is studied by means of the reductive perturbation method incorporated with the matched expansion method. Envelope solitons (holes) are shown to propagate in the axial direction and damp slowly due to the presence of a viscous boundary layer.

Nonlinear Dynamics of Low-β Plasma and Drift-Wave Studies

The nonlinear dynamics of low-β plasma is discussed on the basis of a model equation derived from the two-fluid model with a small but finite electrical resistivity in the direction of the magnetic field. Attention is focused on the deviation from the Boltzmann distribution for electrons and the validity of a model equation for the drift-wave turbulence proposed by Hasegawa and Mima. The anomalous diffusion due to convective cells and the evolution of the energy spectrum of the latter equation are examined by means of the modified quasi-normal approximation which has been found to be useful for the turbulence of the Navier–Stokes equation.

EXISTENCE CONDITIONS FOR COLLISIONLESS HYDROMAGNETIC SHOCK WAVES ALONG THE MAGNETIC FIELD.

This paper is concerned with existence conditions for steady hydromagnetic shock waves propagating in a collisionless plasma along an applied magnetic field. The electrostatic waves are excluded. The conditions are based on the requirement that solutions of the Vlasov-Maxwell equations deviate from a uniform state ahead of a wave. They are given as the conditions on the upstream flow velocity in the wave frame (i.e. in the form of inequalities among the upstream flow velocity and some critical velocities). The conditions crucially depend on the pressure anisotropy, and demonstrate possibilities of exacting collisionless shock waves for high β plasmas.