Formation of electrostatic shock waves in space plasma

Saito et al. (2003) note that even if there is charge neutrality and no magnetic field the plasma flows generate the intense electric and magnetic fields. This is referred to the counter-streaming instability that has the process similar to Weibel instability with anisotropic temperature (Weibel, 1959). Califano et al. (1997) investigated, both analytically and numerically, electron-electron counter-streaming instability in electron-ion plasma by using both two-fluid equations and Maxwell's equations. They derived the theoretical dispersion relation and applied it to beam-plasma instability in laser plasma. Kazimura et al. (1998) investigated electron-positron counter-streaming instability by using both linear theory and 'particle-in-cell' simulation based on Buneman (M1993). They derived the dispersion relation from four-fluid equations with Maxwell's equations, and compared the results derived from linear theory with the results of simulation. They noticed the linear stage of counter-streaming instability. In Haruki and Sakai (2003) there was investigated the non-linear stage of counter-streaming instability in pair plasma. They also derived the theoretical dispersion relation by using four-fluid equations with Maxwell equations, and compared the growth rate derived from the dispersion relation with that of simulation result. Furthermore in non-linear stage they showed the generation of electrostatic shock waves which are caused by electrostatic counter-streaming instability. The generated electrostatic waves create some high energy particles. In the papers of Califano et al. (1997), Kazimura et al. (1998), Haruki and Sakai (2003) they do not take into account the possible role of the background magnetic field Bo. Saito et al. (2003) developed this research and investigated the counter-streaming instability in pair plasma as well as particle acceleration taking into account the background magnetic field Bo parallel to the direction of counter-streaming.

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