The closed-form expressions for the mean-squared beam width and the far-field divergence angle of partially coherent Hermite-Gaussian (H-G) array beams propagating in atmospheric turbulence are derived. The far-field divergence angle is chosen as the characteristic parameter of beam directionality,and the directionality of partially coherent H-G array beams propagating in atmospheric turbulence is studied. It is shown that under certain conditions,partially coherent H-G array beams may generate the same directionality as a Gaussian laser beam in free space and also in turbulence. Furthermore,it is found that,the directionality of H-G array beams in terms of the far-field divergence angle is not consistent with that in terms of the normalized far-field average intensity distribution in free space,but both may be consistent in atmospheric turbulence. The result is some different from the behavior of GSM array beams. For the incoherent combination case,GSM array beams with the same far-field divergence angle as a Gaussian beam will have the same normalized far-field intensity distribution as the Gaussian beam in free space.