An investigation on the compressibility effects arising into the near field of turbulent jets operated at high Reynolds number at Mach numbers M=0.3, 0.9 and 1.1 (under-expanded regime) is carried out with two-components planar PIV experiments with high resolution cameras. The arrangement of the PIV system allows for mapping out the flow field with a vector pitch equal to 0.17mm along the streamwise direction. The flow field is characterized by single-point statistics. For the under-expanded jet, the spatial evolution of the flow field within the potential core region is characterized by the presence of shock-cell structures that extend up to 4.5 downstream from the nozzle exit. Two-point cross-correlation technique is used to investigate the large scale coherent structures in the potential core and in the shear layer. The distribution of the longitudinal spatial correlations valuated in the potential core region of the jet exhibits a phase shifted pattern in sign that extends periodically along the jet axis. This spatial phase shift is approximately 0.5 for M=0.3 and 0.25 for M=1.1. Downstream from =2 and =0, the intensity of the peaks of the spatial correlations in the underexpanded jet is higher than that calculated at Mach 0.3 jet because of the presence of the shock-cell structures in the potential core.