Oscillating pulsar polar gaps
  An analytical model for oscillating pair creation above the pulsar polar cap is presented in which the parallel electric field is treated as a large amplitude, superluminal, electrostatic wave. An exact formalism for such wave is derived in one-dimension and applied to both the low-density regime in which the pair plasma density is much lower than the corotating charge density and the high-density regime in which the pair plasma density is much higher than the corotating charge density. In the low-density regime, which is relevant during the phase leading to a pair cascade, a parallel electric field develops resulting in rapid acceleration of particles. The rapid acceleration leads to bursts of pair production and the system switches to the oscillatory phase, corresponding to the high density regime, in which pairs oscillate with net drift motion in the direction of wave propagation. Oscillating pairs lead to a current that oscillates with large amplitude about the Goldreich-Julian current. The drift motion can be highly relativistic if the phase speed of large amplitude waves is moderately higher than the speed of light. Thus, the model predicts a relativistic outflow of pairs, a feature that is required for avoiding overheating of the pulsar polar cap and is also needed for the pulsar wind. 
Q. Luo and D. B. Melrose