Space-Charge Effects in the Current-Filamentation or Weibel Instability

We consider how an unmagnetized plasma responds to an incoming flux of energetic electrons. We assume a return current is present and allow for the incoming electrons to have a different transverse temperature than the return current. To analyze this configuration we present a nonrelativistic theory of the current-filamentation or Weibel instability for rigorously current-neutral and nonseparable distribution functions, $f_0(p_x,p_y,p_z)\ne f_x(p_x)f_y(p_y)f_z(p_z)$. We find that such distribution functions lead to lower growth rates because of space-charge forces that arise when the forward-going electrons pinch to a lesser degree than the colder, backward-flowing electrons. We verify the growth rate, range of unstable wave numbers, and the formation of the density filaments using particle-in-cell simulations.

Figure 1

(a) Electron distribution function showing that the electrons are stable to two-stream instability and (b) comparison between simulation and theory illustrating the reduction of the growth rate due to space-charge effects.

Figure 2

(a) Electron density isosurfaces for the hot beam ($p_{t2}=0.28mc$, red) and the cold beam ($p_{t1}=0.11mc$, blue) at ${\omega }_{p}t=248.5$ confirm the number of filaments predicted by our theory, (b) zoomed box shows $k_z=0$, (c) ion filaments (green) and hot beam filaments (red) are in complementary locations (zoomed box) and (d) ion filaments (green) and cold beam filaments (blue) overlap (zoomed box).