Linear growth rate of the electrostatic ion-ion instability for the symmetrical beam system of electron-ion plasma with the drift velocity of and the electron-to-ion temperature ratio of 9. The horizontal and the vertical axes are the wave vector in the -direction, , and that in the -direction , respectively. The gray-scale shows the linear growth rate normalized to the electron plasma frequency.
Linear growth rate of the electromagnetic beam-Weibel instability for the same beam system as in Fig. 1.
Time evolution of the -component of the electric field , averaged over the -direction (gray-scale). The horizontal and vertical axes are the -axis and the time, respectively. The electrostatic shock forms and propagates during . However, after that time, the shock slows down and finally fades away.
Number density of the ions at , (b) 3000, and (c) 5000. The horizontal and vertical axes are the -axis and the -axis in the electron skin depth, respectively. The electrostatic ion-ion instability, which generates the filamentary structure, develops with time.
(a) The charge density at and (b) its power spectrum. The horizontal and vertical axes in (b) are and in , respectively. The portion where the electrostatic mode develops shown in the panel (b) agrees well with those obtained from the linear analysis shown in Fig. 1.
Phase-space plots of the ions and electrons at , (b) 3000, and (c) 5000: from top to bottom, the and distributions of the ions and those of the electrons are shown. In the upstream of the shock, the ion heating is evident while the electron heating is small.
Velocity distributions of the ions in the space within at , (b) 3000, and (c) 5000. The heating in the -direction is remarkable.
Time development of the temperature in each direction immediately in front of the electrostatic shock normalized to the initial ion temperature : (a) the electrons, (b) the incoming ions, and (c) the reflected ions. The solid curves, the dashed curves, and the dotted curves show the , , and components, respectively. It is remarkable that the components of the temperatures of both ion populations rapidly increase within the period . Panel (d) shows the time development of the number density ratio of the reflected ions to the incoming ions .
Time evolutions of the -component of the electric field , as in Fig. 3 obtained from quasi one-dimensional simulations (see text): (a) for the anisotropic ion temperature and and (b) for the isotropic ion temperature with . The electrostatic shocks form and propagate without decaying in both cases.
Close-ups of (a) the electrostatic potentials, (b) the number density of the incoming ions , and (c) that of the reflected ions at for the two-dimensional simulation. In (a), the electrostatic potential in front of the shock, which is generated by the electrostatic ion-ion instability, reaches even about a half of the shock potential.
The phase space distribution of the ions within at (see Fig. 10). The feature of the ion trapping due to the nonlinear evolution of the electrostatic ion-ion instability is clear. This trapping results in the strong fluctuation in the incoming ions as is shown in Fig. 10(b) to destroy the shock structure.
(a) The phase space distribution of the ions within at . (b) The close-up of the distribution of the reflected ions around . The “secondary” electrostatic shock exists in the reflected ion beam around .
The linear growth rates of the two Weibel-type instabilities in the anisotropic Maxwellian beam system ( and ): the beam-Weibel instability with the wave vector in the -direction (solid curve) and the (ordinary) Weibel instability with the wave vector in the -direction (dashed curve). The dot-dashed curve shows the growth rate of the beam-Weibel instability in the isotropic Maxwellian beam system as in Sec. II (i.e., and ) as a reference. The growth rate of the beam-Weibel instability is reduced due to the temperature anisotropy in the ions. The Weibel instability in the -direction is stable in this case.
The -component of the normalized magnetic field, , at , (b) 3000, and (c) 5000 for the two-dimensional simulation, where . The magnetic field is generated by the beam-Weibel instability in front of the shock and it grows with time during the simulation.
PIC simulation of the Weibel-mediated shock for showing the formation of the shock qualitatively with parameters of and . The left panel shows the ion number density normalized by the upstream value , and the right panel shows the -component of the normalized magnetic field , from top to bottom at , 1000, and 2000, respectively. Due to the magnetic field generated by the beam-Weibel instability, a kind of collisionless shock, which is called the Weibel-mediated shock, is formed at .
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