Two-stream instability, wave energy, and the energy principle

A generalized Poynting theorem for a system of uniform electron beams is obtained. Two examples of the two-stream instability with beams of equal density are used to discuss the relation between negative wave energy and negative potential energy, which arises in the energy principle of ideal magnetohydrodynamics. In the first example, v 10 > v 20 , while in the second example, v 20 = − v 10 , where v 10 , 20 are the equilibrium beam velocities. Both cases can be interpreted in terms of the energy density arising from the generalized Poynting theorem. The first instability is due to the coupling of negative and positive energy waves at a frequency k ( v 10 + v 20 ) ∕ 2 . The second instability is due to the coupling of the same two perturbations, but at zero frequency. In this case, there is no oscillatory (wave) energy, but the beam electrons still make a negative contribution to the total energy.