Non-linear evolution of the diocotron instability in a pulsar electrosphere: two-dimensional particle-in-cell simulations

Context. The physics of the pulsar magnetosphere near the neutron star surface remains poorly constrained by observations. Indeed, little is known about its emission mechanism, from radio to high-energy X-ray and gamma-rays. Nevertheless, it is believed that large vacuum gaps exist in this magnetosphere, and a non-neutral plasma partially fills the neutron star surroundings to form an electrosphere in differential rotation. Aims. According to several of our previous works, the equatorial disk in this electrosphere is diocotron and magnetron unstable, at least in the linear regime. To better assess the long term evolution of these instabilities, we study the behavior of the non-neutral plasma using particle simulations. Methods. We designed a two-dimensional electrostatic particle-in-cell (PIC) code in cylindrical coordinates, solving Poisson equation for the electric potential. In the diocotron regime, the equation of motion for particles obeys the electric drift approximation. As in the linear study, the plasma is confined between two conducting walls. Moreover, in order to simulate a pair cascade in the gaps, we add a source term feeding the plasma with charged particles having the same sign as those already present in the electrosphere. Results. First we checked our code by looking for the linear development of the diocotron instability in the same regime as the one used in our previous work, for a plasma annulus and for a typical electrosphere with differential rotation. To very good accuracy, we retrieve the same growth rates, supporting the correctness of our PIC code. Next, we consider the long term non-linear evolution of the diocotron instability. We found that particles tend to cluster together to form a small vortex of high charge density rotating around the axis of the cylinder with only little radial excursion of the particles. This grouping of particles generates new low density or even vacuum gaps in the plasma column. Finally, in more general initial configurations, we show that particle injection into the plasma can drastically increase the diffusion of particles across the magnetic field lines. The newly formed vacuum gaps cannot be replenished by simply invoking diocotron instability. Conclusions. Diocotron instability offers a new possibility to solve the current closure problem in a pulsar magnetosphere. It is a promising mechanism leading to highly unstable flows in the pulsar inner magnetosphere. When flowing towards the light cylinder, some relativistic and particle inertia effects (not included in this study) appear. Nevertheless, the system should remain unstable because of the relativistic diocotron or the magnetron instability. Therefore, we expect an electric current to circulate in the closed magnetosphere and to feed the base of the wind with charged particles.