Based on the Rytov approximation of light propagation in weak turbulent atmosphere, the closed-form expressions of field and average irradiance of each one of the four fundamental families of Helmholtz-Gauss (HzG) beams: cosine-Gauss beams, stationary Mathieu-Gauss beams, stationary parabolic-Gauss beams, and Bessel-Gauss beams, which are propagating in weak turbulent atmosphere, are obtained. The results show that the field and average irradiance can be written as the product of four factors: complex amplitude depending on the z-coordinate only, a Gaussian beam, a factor of complex phase perturbation induced by atmospheric turbulence, and a complex scaled version of the transverse shape of the non-diffracting beam. The effect of weak atmospheric turbulence on irradiance distribution of the HzG beam can be ignored.