The properties of relativistic jets, their interaction with the ambient environment, and particle acceleration due to kinetic instabilities are studied self-consistently with Particle-in-Cell simulations. An important key issue is how a toroidal magnetic field affects the evolution of an e$^{pm}$ and an e$^{-}$ - p$^{+}$ jet, how kinetic instabilities such as the Weibel instability (WI), the mushroom instability (MI) and the kinetic Kelvin-Helmholtz instability (kKHI) are excited, and how such instabilities contribute to particle acceleration. We show that WI, MI and kKHI excited at the linear stage, generate a quasi-steady $x$-component of electric field which accelerates and decelerates electrons. In this work, we use a new jet injection scheme where an electric current is self-consistently generated at the jet orifice by the jet particles. We inject both e$^{pm}$ and e$^{-}$ - p$^{+}$ jets with a toroidal magnetic field (with a top-hat jet density profile) and for a sufficiently long time in order to examine the non-linear effects of the jet evolution. Despite the weakness of the initial magnetic field, we observe significant differences in the structure of the strong electromagnetic fields that are driven by the kinetic instabilities. We find that different jet compositions present different strongly excited instability modes. The magnetic field in the non-linear stage generated by different instabilities becomes dissipated and reorganized into a new topology. The 3-dimensional magnetic field topology indicates possible reconnection sites and the accelerated particles are significantly accelerated in the non-linear stage by the dissipation of the magnetic field and/or reconnection. This study will shed further light on the nature of astrophysical relativistic magnetized jet phenomena.
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