In this paper, we describe results of new high-resolution axisymmetric relativistic magnetohydrodynamic (MHD) simulations of pulsar wind nebulae. The simulations reveal strong breakdown of the equatorial symmetry and highly variable structure of the pulsar wind-termination shock. The synthetic synchrotron maps, constructed using a new more accurate approach, show striking similarity with the well-known images of the Crab Nebula obtained by Chandra and the Hubble Space Telescope. In addition to the jet–torus structure, these maps reproduce the Crab's famous moving wisps whose speed and rate of production agree with the observations. The variability is then analysed using various statistical methods, including the method of structure function and wavelet transform. The results point towards the quasi-periodic behaviour with the periods of 1.5–3 years and MHD turbulence on scales below 1 year. The full account of this study will be presented in a follow-up paper.