  With a Yang-Mills field, stratified shear flow initial state and a high
resolution (3+1)D Particle-in-Cell Relativistic (PICR) hydrodynamic model, we
calculate the $\Lambda$ polarization for peripheral Au+Au collisions at RHIC
energy of $\sqrt{S_{NN}}=200$ GeV. The obtained longitudinal polarization in
our model agrees with the experimental signature and the quadrupole structure
on transverse momentum plane. It is found that the relativistic correction (2nd
term), arising from expansion and from the time component of the thermal
vorticity, plays a crucial role in our results. This term is changing the
signature and exceeds the first term, arising from the classical vorticity.
Finally, the global polarization in our model shows no significant dependence
on rapidity, which agrees with the experimental data. It is also found that the
second term flattens the sharp peak arising from the classical vorticity (1st
term).
