  We use two-dimensional particle-in-cell simulations to investigate the
generation and evolution of the magnetic field associated with the propagation
of a jet for various initial conditions. We demonstrate that, in general, the
magnetic field is initially grown by the Weibel and Mushroom instabilities.
However, the field is saturated by the Alfv'en current limit. For initially
non-magnetized plasma, we show that the growth of the magnetic field is delayed
when the matter density of the jet environment is lower, which are in agreement
with simple analytical predictions. We show that the higher Lorentz factor
($\gtrsim 2$) prevents rapid growth of the magnetic fields. When the initial
field is troidal, the position of the magnetic filaments moves away from the
jet as the field strength increases. The axial initial field helps the jet
maintain its shape more effectively than the troidal initial field.
