## 3 Discussion  
Now, we begin to comprehensively analyze the influence of the beam parameters and turbulent parameters on the transmission of vortex modes carried by BGLW. In the following discussion, unless otherwise specified, we set the parameters: \(z=200\,\mathrm{m}\), \(m=1\), \(m_{0}=1\), \(w_{0}=0.1\,\mathrm{m}\), \(D=0.3\,\mathrm{m}\), \(\alpha=0.001\), \(\alpha=-4.5\), \(T_{0}=20\,\mathrm{fs}\), \(\zeta=2\), \(\epsilon=10^{-6}\,\mathrm{m}^{2}/\mathrm{s}^{3}\), \(\eta=0.001\,\mathrm{m}\), and \(\chi_{T}=10^{-8}\,\mathrm{K}^{2}/\mathrm{s}\). The evolution curve of the spectrum average coherence length of spherical wave has been given by Figure 1a,b. Next, Figure 2 will display the influence of beam parameters and channel parameters on signal vortex mode transmission; Figures 3-5 will display the influence of turbulence parameters on signal vortex mode transmission; Figures 6 and 7 will display the influence of beam parameters on signal vortex mode transmission. Figure 2a gives the received probability of signal vortex modes (\(|m-m_{0}=0|\)) under different co-moving coordinate \(\xi\) and topological charge number \(m_{0}\), and Figure 2b gives the received probability of signal vortex modes under different propagation distance \(z\) and topological charge number \(m_{0}\). Figure 2a shows that increasing the topological charge number \(m_{0}\) will reduce the received probability of signal vortex modes. But, what surprises us is when topological charge number of the vortex mode increases from 0 to 6, the received probability decreases less than 0.016 which is far less than the Laguerre-Gaussian beam [1]. And Figure 2b clearly shows that the received probability is still above 0.75 even the propagation distance \(z\) reaches 500 m. These results once again indicate that, in turbulent seawater, the transmission of the vortex mode carried by the BGLW is better than that of the vortex mode carried by Laguerre-Gaussian beam. To further research the effects of the temperature fluctuation strength of seawater turbulence, we plot the received probability of signal vortex modes under different \"temperature structure\" constant \(C_{m}^{2}\) and anisotropic factor \(\zeta\) in Figure 3. We can see that as \(C_{m}^{2}\) changes from \(10^{-18}\,\mathrm{m}^{-2/3}\) to \(10^{-14}\,\mathrm{m}^{-2/3}\), the received probability of the signal vortex modes is always decreasing. When \(C_{m}^{2}\) is closer to \(10^{-18}\,\mathrm{m}^{-2/3}\), i.e., weak turbulence, turbulence has little disturbance for the vortex mode. This is because that beam wander can be neglected in very weak turbulence. When \(C_{m}^{2}\) is closer to \(10^{-14}\,\mathrm{m}^{-2/3}\), i.e., strong turbulence, turbulence severely disturbing with signal propagation. Figure 3 also shows larger anisotropic factor \(\zeta\) is conducive to the transiting of signal vortex modes. To explore the effects of turbulent parameters on the received probability of vortex modes of the BGLW, in Figure 4, we plot the curve of the received probability as the function of the ratio of temperature and salinity \(\varpi\) and the inner scale \(\eta\). We can see that the received probability of signal vortex modes increases with the increase of inner scale \(\eta\) or the decrease of \(\varpi\). The former is consistent with the theory that the larger the inner scale of the turbulent eddy is, the smaller the scintillation will  
Figure 3: Received probability of the vortex modes of localized wave of Bessel–Gaussian amplitude envelope in seawater turbulence under different “temperature structure” constant \(C_{m}^{2}\) and anisotropic factor \(\zeta\). Figure 2: Received probability of the vortex modes of localized wave of Bessel–Gaussian amplitude envelope under different co-moving coordinate \(\xi\) and topological charge number \(m_{0}\) in (**a**) and Received probability of the vortex modes of localized wave of Bessel–Gaussian amplitude envelope under different propagation distance \(z\) and topological charge number \(m_{0}\) in (**b**).