| Summary: | In this letter, we conducted a detailed investigation of the statistical properties, such as spectral density, spectral degree of coherence (SDOC), orbital angular momentum (OAM) flux density, and propagation factor M<sup>2</sup>, of a twisted Gaussian Schell-model (TGSM) beam carrying the cross phase in a turbulent atmosphere. Our findings revealed that atmospheric turbulence induces degeneration of the intensity distribution and spectral degree of coherence of a Gaussian Schell-model beam with the cross phase during propagation, while the twist phase acts as an antidote to degradation. Furthermore, we observed that the z-component of the time-averaged angular momentum flux is determined by the twist phase, whereas the cross phase influences the distribution of the OAM flux density in the beam. Additionally, we explored the variations in the propagation factor M<sup>2</sup> of a TGSM beam with the cross phase in a turbulent atmosphere. Notably, we discovered that the deleterious effects of the atmospheric conditions can be mitigated by modulating both the twist and the cross phases. This work contributes valuable insights for information transfer and optical manipulations.
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