Masashi Kako

Stability of oblique modulation on an ion‐acoustic wave

Modulation instability is shown to be a general property of a wave in a nonlinear and dispersive medium when the modulation is allowed in the direction oblique to that of the wave phase velocity. As an example a modulation on an ion‐acoustic wave is shown to be unstable even if this wave is modulationally stable in the case of parallel or perpendicular modulation.

B. Nonlinear Modulation of Plasma Waves

The self-trapping and instabilities of modulated whistler waves propagating along a uniform magnetic field were first studied by Taniuti arid Washimi.l) Using the fluid model for the cold plasma, they derived a dispersive nonlinear equation (the nonlinear Schrodinger equation) to describe a slow modulation of a carrier wave of small but finite amplitude. This equation takes the same form as the equation appearing in the study of self-focusing in nonlinear optics. They then showed that the solitary wave, that is, envelope soliton exists while the plane wave is unstable against a modulation of amplitude and phase. Mizutani and Taniuti2) extended the theory to oblique propagation of stationary waves to obtain the envelope soliton, and later Mizutani3) derived the nonlinear Schrodinger equation for nonstationary oblique propagation and showed the modula~ional instability. These problems were also investigated by Pataraya.4) However, these works are concerned with the propagation of the ·carrier wave at those frequencies for which the group velocity is equal to the phase velocity. The extension to propagation at an arbitrary frequency is straightforward, and this was carried out, using the method of Taniuti and Yajima,5) by Hasegawa6) for the case of propagation along the magnetic field for all the frequencies of whistler waves and also ion-cyclotron waves. Further, Kako7) has investigated the modulation of carrier waves of arbitrary frequencies for all angles of propagation, so that the results obtained earlier are special cases. In this paper, we present an investigation of nonlinear wave

Laminar electrostatic shock waves generated by an ion beam

Strong laminar electrostatic shock waves have been experimentally observed when an ion beam is injected into a collisionless plasma. The structure of the shock is qualitatively different from one with a trailing wave train. A density depression follows behind the shock front, and no trailing wave train due to wave dispersion is found. A significant amount of ions reflected from and transmitted through the shock front form a precursor. The critical Mach number above which no shock is formed is found to be 1.5. Numerical simulations reported here reproduce the experimental observations very well. An analysis based on the water‐bag model accounts for the observed value of the critical ion‐beam velocity which gives the critical Mach number. It also points out that the reflected ions play an essential role in the persistence of the shock.

Equilibria of Field-Reversed Configuration with Subsidiary Coils

Two-dimensional equilibria of field-reversed configuration with subsidiary coils are computed by integrating the ideal magnetohydrodynamic equilibrium equation. The mirror coils have an effect to compress the plasma appreciably in the axial direction and hence sustain the compact torus. The cusp coils enlarge the plasma and consequently make the gradient of plasma density relaxed in the radial direction.

Nonlinear Wave Modulation in a Magnetized Collisionless Plasma

Nonlinear modulation of quasi-monochromatic electromagnetic waves propagating parallel to an external magnetic field is investigated with particular attention to contributions of resonant particles at the group velocity. The contributions of these resonant particles give rise to a nonlocal-nonlinear tram and modify the local-nonlinear term of the nonlinear schrodinger equation. An explicit expression of a coefficient of the nonlocal-nonlinear term is given for a plasma characterized by isotropic Maxwellian velocity distribution functions. In a vanishing temperature limit this coefficient becomes zero and the nonlinear Schrodinger equation agrees with that obtained by using a fluid model.