  We investigate the dependence of the GRB jet structure and its evolution on
the properties of the accreting torus in the central engine. Our models
numerically evolve the accretion disk around a Kerr black hole using 3D general
relativistic magnetohydrodynamic simulations. We use two different analytical
hydrodynamical models of the accretion disk, based on the Fishbone-Moncrief and
Chakrabarti solutions, as our initial states for the structure of the collapsar
disk and the remnant after a binary neutron star merger, respectively. We
impose poloidal magnetic fields of two different geometries upon the initial
stable solutions. We study the formation and evolution of the magnetically
arrested disk state and its effect on the properties of the emitted jet. The
jets produced in our models are structured and have a relatively hollow core
and reach higher Lorentz factors at an angle $\gtrsim 9{^\circ}$ from the axis.
The jet in our short GRB model has an opening angle of up to $\sim 25^{\circ}$
while our long GRB engine produces a narrower jet, of up to $\sim 11^{\circ}$.
We also study the time variability of the jets and provide an estimate of the
minimum variability timescale in our models. The application of our models to
the GRB jets in the binary neutron star post-merger system and to the
ultra-relativistic jets launched from collapsing stars are briefly discussed.
