## 1 Introduction  
Orbital angular momentum (OAM) communication has broad application prospects in FSO communications [1; 2; 3; 4; 5]. The OAM modes carried by vortex beams have theoretically infinite eigenmodes and can form an infinite-dimensional Hilbert space, which can be used in FSO communications to increase capacity and spectral efficiency via the mode-division multiplexing technique. The vortex beam carrying the OAM mode inevitably encounters atmospheric turbulence when it propagates in free space. The initial OAM state will be degraded by atmospheric turbulence, followed by the OAM mode's crosstalk generation for the power of it spreading to the neighboring OAM states [6; 7; 8; 9; 10; 11]. Therefore, the phase of the vortex beams is perturbed by atmospheric turbulence, the detection probability of the signal OAM state decreases [12], the signal-to-noise ratio (SNR) decreases, [13] and the bit error rate (BER) [14; 15; 16; 17; 18] increases. These are important parameters for evaluating the performance of FSO communication systems. The performance of communication channels in a turbulent atmosphere can be improved by employing partially coherent beams [19; 20; 21]. In the past few decades, the propagation properties of the many types of partially coherent beams, for example, the Gaussian-Schell beam [22], partially coherent rectangular array beams [23], the Gaussian-Schell model vortex beam [24], partially coherent radially polarized beams [25], Airy-Schell beams [26], partially coherent four-petal elliptic Gaussian vortex beams [27], partially coherent beams carrying a cross phase [28], and partially coherent twisted Gaussian beams [29]have been researched and used to mitigate the effects of a turbulent atmosphere on the signal quality of FSO communications. Non-uniformly correlated beams with spatially varying coherence are different from the Schell-model beams with uniform coherence distributions [30; 31; 32; 33; 34]. The characteristics of non-uniformly correlated beams can be used to measure the refractive indices of a uniaxial crystal based on its independent self-focusing property along the propagation directions [35]. Two high-index particles were trapped simultaneously and manipulated in the longitudinal direction via focused structured optical fields with a spatially non-uniform correlation [36]. Non-uniformly correlated beams can also be used to resist the negative effects induced by atmospheric turbulence [37; 38; 39; 40; 41; 42]. The research results show that the scintillation of Gaussian non-uniformly correlated beams and Hermite non-uniformly correlated beams was lower than that of the Gaussian-Schell model beams [37; 38]. The power-in-the-bucket value can be enhanced under conditions of weakly anisotropic turbulence by controlling the coherence parameter for non-uniformly correlated beams [39]. In one study, convex partially coherent flat-topped beams were used to improve the mean SNR and reduce the mean BER [40]. Laguerre non-uniformly correlated beams with a suitable beam order and coherence length were applied to reduce the detrimental effect caused by atmospheric turbulence [41]. The fiber-coupling efficiency of the Laguerre non-uniformly correlated beam was better than that of the conventional Gaussian-Schell-model beam in FSO communications [42]. Therefore, non-uniformly correlated high-order Bessel-Gaussian beams carrying the OAM state can be applied to improve the performance of OAM communications in FSO communications. In this study, we theoretically investigated the OAM state carried by non-uniformly correlated high-order Bessel-Gaussian beams disturbed by a turbulent atmosphere. The detection probability, SNR, and BER equation of the signal OAM states for non-uniformly correlated high-order Bessel-Gaussian beam propagation in a turbulent atmosphere were derived according to its intensity distribution at the receiver plane. The results show that the performance of OAM communications can be enhanced by non-uniformly correlated high-order Bessel-Gaussian beams.