We report nonlinear studies of the Weibel instability of a relativistic electron beam in a plasma. If «&«w^, the beam splits into self-pinched filaments at density w^. These filaments then recombine into a single dense beam, from which the return current is expelled. The ratio of final magnetic to final streaming energy is 0{v/y), and significant plasma heating occurs.

The propagation of a high current relativistic beam in a cold magnetized plasma is investigated using a model developed by Hammer and Rostoker for the unmagnetized case. In this model, the beam electrons are assumed to be undeflected from their zero-order orbits and the fields associated with the beam are switched on at time t = 0. The return current induced in the plasma is calculated as a function of beam and plasma parameters. It is demonstrated that the return current does not extend indefinitely but dies away inversely as the distance from the head of the beam with a characteristic length Ln = v0τa2/λE2, where v0 and a are the beam velocity and radius, τ is a phenomenological momentum relaxation time for the plasma electrons, λE = c/ωp, and ωp is the plasma frequency of the plasma electrons. When the beam is injected either parallel or perpendicular to a static magnetic field B0, it is found to be magnetically neutralized by a return current over a length of order Ln if a2/[λE2(1+Ω2/ωp2)]≫1, where Ω = eB0/m0c.

the net frequency shift observed in the laboratory is a combination of this red shift plus a blue shift due to the overall expansion of the upstream plasma. The increased red shift may be sufficient to yield a net red shift in the frequency of the reflected light.For simplicity we have discussed backscatter, but similar results should also obtain for sidescatter (the momentum transfer is less by a factor of W2). It should be noted that 2D effects could further reduce the time over which a large ref lectivity can occur. For example, the incident light may form narrow filaments, expelling the plasma laterally. '-' %e are grateful for recent discussions of light reflection with L. , Phys. Rev. Lett. 33, 634 (1974).

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