Three-dimensional magnetohydrodynamic simulations of the Crab nebula

In this paper we give a detailed account of the first 3D relativistic magnetohydrodynamic (MHD) simulations of Pulsar Wind Nebulae (PWN), with parameters most suitable for the Crab Nebula. In order to clarify the new features specific to 3D models, reference 2D simulations have been carried out as well. Compared to the previous 2D simulations, we considered pulsar winds with much stronger magnetisation, up to \sigma=3, and accounted more accurately for the anticipated magnetic dissipation in the striped zone of these winds. While the 3D models preserve the separation of the post termination shock flow into the equatorial and polar components, their relative strength and significance differ. Whereas the highly magnetised 2D models produce highly coherent and well collimated polar jets capable of efficient "drilling" through the supernova shell, in the corresponding 3D models the jets are disrupted by the kink mode current driven instability and "dissolve" into the main body of PWN after propagation of several shock radii. Our results show that contrary to the expectations based on 1D analytical and semi-analytical models, our numerical solutions with highly magnetized pulsar winds still produce termination shocks with radii comparable to those deduced from the observations. We present polarized synchrotron maps constructed from the 3D simulations, showing that the wealth of morphological features observed with Hubble and Chandra can well be reproduced with 3D-MHD models.