## 1 Introduction  
The development of marine wireless optical communications is of great significance in remote submersibles, submarines, analysis of marine resources, and exploration of undersea gas and oil. Therefore, the construction of an underwater high-speed, secure and high-capacity optical communication system is important. Because the use of vortex modes can achieve secure optical communication with high information content [1], research on the propagation of the vortex mode in turbulent media has gotten ferocious attention in the field of wireless optical communication [1; 2; 3; 4]. However, when the vortex wave transits through turbulent seawater, it will suffer attenuation and wavefront distortion due to the various absorbing substance and the fluctuations of refractive index [5]. Considering that the vortex mode is determined by the wavefront structure, the wavefront deformation will generate mode distortion. It is noted that the wavefront deformation also includes phase distortion caused by optical systems [6], and \"branch point\" [7] contributed by the turbulent channel can also cause mode distortion. It is distortion of vortex mode that causes vortex crosstalk and vortex noise, which will reduce the quality of a wireless optical communication system with vortex modes. For this reason, a large number of studies have been reported about the propagation of vortex mode in turbulence of seawater [8; 9; 10; 11; 12; 13; 14]. For instance, [PERSON] et al. [8] studied the evolution of Gaussian Schell model vortex beam in seawater turbulence. [PERSON] et al. [9] analyzed the propagation of partially coherent flat-topped hollow beam carrying vortex mode in seawater turbulence. [PERSON] et al. [10] researched the evolution of four-fold vortex mode of green light (520 nm) in underwater channels of 2 m by experiments. [PERSON] et al. [11] investigated the evolution of partially coherent four-petal Gaussian beam with vortex modes in seawater turbulence and found that when either decreasing the coherence length, the relative strength of the temperature and salinity fluctuations or the dissipation rate of kinetic energy per unit mass of fluid, or increasing the dissipation rate of the mean square temperature, the beam will soon be converted into a Gauss-like beam. [PERSON] and [PERSON] [12] studied the propagation of a stochastic electromagnetic vortex beam in the turbulence of seawater, and found that a smaller topological charge corresponds to a smaller spectral degree of polarization. Besides, a localized wave is one that has no diffraction for a long distance in the absence of turbulence [15]. For this reason, [PERSON] et al. [16] researched the transiting of vector electromagnetic localized wave in the absence of turbulence, and [PERSON] et al. [17] explored the propagation law of a new class of non-diffracting optical pulses with vortex mode. Recently, we investigated the received probability of vortex mode carried by an X wave in the turbulence of oceans. However, because the effect of frequency distribution is ignored in the calculation of the phase structure function [18], and the received probability function is a multiple integral, it is difficult to use this model to analyze problems such as bit error rate, channel capacity and so on. Therefore, by the Gaussian spectrum of the frequency of pulse, we superimpose the monochromatic Bessel-Gaussian beams to constitute the localized wave of Bessel-Gaussian amplitude envelope (BGLW) carrying vortex modes, and establish a new received probability of vortex modes that can be used to analyze signal bit error rate and the channel capacity of the communication link. In this paper, for the BGLW with vortex modes in anisotropic seawater turbulence, we present a spectrum average mutual coherence function (SAMCF) by introducing a spectrum average coherence length of spherical wave. Using this SAMCF, we set up a more concise model of received probability of vortex modes carried by the BGLW in anisotropic seawater turbulence.