Characterization of the Initial Filamentation of a Relativistic Electron Beam Passing through a Plasma

The linear instability that induces a relativistic electron beam passing through a plasma with return current to filament transversely is often related to some filamentation mode with the wave vector normal to the beam or confused with Weibel modes. We show that these modes may not be relevant in this matter and identify the most unstable mode on the two-stream or filamentation branch as the main trigger for filamentation. This sets both the characteristic transverse and longitudinal filamentation scales in the nonresistive initial stage.

Figure 1

Weibel, two-stream, and filamentation modes.

Figure 2

Growth rates on the TSF branch in terms of $\mathbf{Z}=\mathbf{k}{V}_b/{\omega }_p$ with ${\mathbf{V}}_b\parallel \widehat{\mathbf{z}}$. (a) Cold beam, cold plasma (see also 23); (b) hot beam, cold plasma; and (c) hot beam, hot plasma. Parameters are $\alpha =0.05$ and ${\gamma }_b=4$ for (a)–(c), $V_{tb\perp }=V_b/10$ for (b),(c), and $V_{tp}=V_b/10$ for (c).

Figure 3

Right-hand side of Eq. (7) for $t=1/{\omega }_p$. Parameters are $V_{tp}=c/30$, $\alpha =0.1$, and ${\gamma }_b=7$ yielding ${\delta }_M\sim 0.16{\omega }_p$.