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We investigate a stationary pair production cascade in the Crab pulsar magnetosphere. The charge depletion due to global flows of charged particles, causes a large electric field along the magnetic field lines. Migratory electrons and/or positrons are accelerated by this field to radiate gamma-rays via curvature and (resonant + non-resonant) inverse-Compton processes. Some of such gamma-rays collide with the X-rays and the magnetic field lines to materialize as pairs, which partially screen the original electric field. By solving the two-dimensional Poisson equation describing the electric field on the poloidal plane together with the Boltzmann equations for particles and gamma-rays, we demonstrate that (1) the gap is located just above the last-open field line and its trans-field thickness is much less than the longitudinal width, (2) the pair production within the gap is self-regulated slightly below the Goldreich-Julian rate, (3) the gap extends from the vicinity of the stellar surface towards the outer magnetosphere but does not reach the light cylinder, (4) space-time dragging effect becomes important close to the star (in the Poisson equation), (5) the created pairs outside of the gap are heavily concentrated in the equatorial region because of the perpendicularly thin gap geometry. We also discuss a unification scheme of polar-cap and outer-gap accelerator models.
