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).
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