Two Dimensional Configuration and Temporal Evolution of Sparking discharges in Pulsars

We have investigated the evolution of a system of sparking discharges in the inner acceleration region (IAR) above the pulsar polar cap. The surface of the polar cap is heated to temperatures around $10^6$ K and forms a partially screened gap (PSG) due to thermionic emission of positively charged ions from the stellar surface. The sparks lag behind the co-rotation speed during their lifetimes due to variable $E$x$B$ drift. In a PSG the sparking discharges arise in locations where the surface temperatures go below the critical level ($T_i$) for ions to freely flow from the surface. The sparking commences due to the large potential drop developing along the magnetic field lines in these lower temperature regions and subsequently the back streaming particles heat the surface to $T_i$. The temperature regulation requires the polar cap to be tightly filled with sparks and a continuous presence of sparks is required around its boundary since no heating is possible from the closed field line region. We have estimated the time evolution of the sparking system in the IAR which shows a gradual shift in the spark formation along two distinct directions resembling clockwise and anti-clockwise motion in two halves of the polar cap. Due to the differential shift of the sparking pattern in the two halves, a central spark develops representing the core emission. The temporal evolution of the sparking process was simulated for different orientations of the non-dipolar polar cap and reproduced the diverse observational features associated with subpulse drifting.