  We have investigated the influence of velocity shear and a radial density
profile on the spatial development of the current driven kink instability along
helically magnetized relativistic jets via three-dimensional relativistic
magnetohydrodynamic simulations. In this study, we use a non-periodic
computational box, the jet flow is initially established across the
computational grid, and a precessional perturbation at the inlet triggers
growth of the kink instability. If the velocity shear radius is located inside
the characteristic radius of the helical magnetic field, a static
non-propagating current driven kink is excited as the perturbation propagates
down the jet. Temporal growth disrupts the initial flow across the
computational grid not too far from the inlet. On the other hand, if the
velocity shear radius is outside the characteristic radius of the helical
magnetic field, the kink is advected with the flow and grows spatially down the
jet. In this case flow is maintained to much larger distances from the inlet.
The effect of different radial density profiles is more subtle. When the
density increases with radius, the kink appears to saturate by the end of the
simulation without apparent disruption of the helical twist. This behavior
suggests that relativistic jets consisting of a tenuous spine surrounded by a
denser medium with a velocity shear radius outside the radius of maximum
toroidal magnetic field have a relatively stable configuration.
