In the last decade, the relativistic magnetohydrodynamic (MHD) modelling ofpulsar wind nebulae, and of the Crab nebula in particular, has been highlysuccessful, with many of the observed dynamical and emission propertiesreproduced down to the finest detail. Here, we critically discuss the resultsof some of the most recent studies: namely the investigation of the origin ofthe radio emitting particles and the quest for the acceleration sites ofparticles of different energies along the termination shock, by using wispsmotion as a diagnostic tool; the study of the magnetic dissipation process inhigh magnetization nebulae by means of new long-term three-dimensionalsimulations of the pulsar wind nebula evolution; the investigation of therelativistic tearing instability in thinning current sheets, leading to fastreconnection events that might be at the origin of the Crab nebula gamma-rayflares.
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