1 2 3 4 K 0.05 0.10 0.15 0.20 0.25 0.30 0.35$^{Δ}$ FIG. 2: Growth rate δ as a function of K for H = 0 . 6 and θ = π/ 6. rescaled growth rate of the instability δ in terms of the rescaled wave vector K is plotted on Figure 1 without any approximation (top), and according to the longitudinal approximation (bottom). As expected, the exact calculation displays unstable filamentation modes for K$_{z}$ = 0 which are bypassed by the approximation. Apart for the quasi normal wave vectors, the approximation is very good, and even exact for K$_{x}$ = 0. The most striking features here is the connection of the two two-stream branches along the flow axis K$_{z}$ . When restricting the calculation to this axis, these two branches seem disconnected. After extending the analysis for arbitrarily oriented wave vectors, we now see that there is a continuum of unstable modes bridging between them. An analytical analysis of the dispersion function for ( K$_{x}$, K$_{z}$ ) = (0 , 0) is possible and allows for an analytical characterization of the bridge between the two-stream branches. For the sake of the discussion, we switch to polar coordinates, and consider θ = π/ 6. Figure 2 pictures the growth rate as a function of K in this direction. Figures 3 now display the behavior of the dispersion function for K = 2 (stable), K = 3 (unstable) and K = 4 (stable). The stability of the system can be explained from the respective position of the singularities of the dispersion function. According to Eq. (8), III. CONNECTION BETWEEN THE TWO-STREAM MODES AND UNSTABLE DOMAIN K$_{x}$ ≡ K sin θ, K$_{z}$ ≡ K cos θ, (10) 5