Counterstreaming ion instability for arbitrary angles

The fully electromagnetic dispersion relation for two equidense counterstreaming ion beams flowing almost perpendicular to a magnetic field has been derived. The beams flow through a warm stationary electron background treated using the Vlasov equation. The wavelengths must therefore be much smaller than the ion Larmor radius, but are assumed to be much greater than the electron Larmor radius. There is then an instability that occurs when [β/(1+2β)]≦(m−/m+)(υ¯2/V02) (υ¯ and V0 are the electron and ion velocities). It is found that in the region of fastest growth and for small β the instability growth drops sharply when the angular deviation of the wave vector from the direction perpendicular to the B field is greater than (m−/8m+)1/2 for charged ions of mass m+ or ∼12° for protons. The transition to the zero field electrostatic case is also obtained by going to the β≫1 limit.