## 3 Discussion
be, and the smaller the disturbance to the signal vortex modes will be. As for the latter, when \(\varpi\) is close to 0, salinity dominates heavily, and the curve goes down dramatically, which means that the vortex mode is more sensitive to salinity fluctuations than to temperature fluctuations in turbulence. To further understand the impact of seawater on the received probability of vortex modes of the BGLW, we plot the dissipation rate of the mean-squared temperature \(\chi_{T}\) and the dissipation rate of kinetic energy per unit mass \(\varepsilon\) of fluid in Figure 5. Figure 5 reveals that the received probability of signal vortex modes increases with the increase of \(\varepsilon\) or the decrease of \(\chi_{T}\), which can be explained that the larger \(\varepsilon\) or the smaller \(\chi_{T}\) mean a weaker seawater turbulence, so beam has smaller wander. More significantly, Figure 5 shows that \(\chi_{T}\) affects vortex modes transmission much more than \(\varepsilon\) does. Also, to understand the influence of beam waist and receiving aperture size on the received probability of signal vortex modes, we show the evolution curve of the received probability of signal vortex modes with the change of beam waist \(w_{0}\) and receiving aperture diameter \(D\) in Figure 6. Figure 6 shows that with the increase of the beam waist \(w_{0}\), the received probability of signal vortex modes  
Figure 4: Received probability of the vortex modes of localized wave of Bessel–Gaussian amplitude envelope in seawater turbulence under different ratio of temperature and salinity \(\varpi\) and inner scale \(\eta\). Figure 5: Received probability of the vortex modes of localized wave of Bessel–Gaussian amplitude envelope in seawater turbulence under different dissipation rate of the mean-squared temperature \(\chi_{T}\) and dissipation rate of kinetic energy per unit mass of fluid \(\varepsilon\). keeps increasing until saturation, and a maximum effective waist \(w_{0b}\) exists before the saturated zone. Furthermore, the received probability of signal vortex modes increases with the decrease of the receiver aperture size \(D\). The rationality of each numerical curve in Figure 6 can be explained by that the optical path difference among rays passing through turbulence within the range of the received beam diameter varies with the increase or decrease of the size of the beam waist and the receiving aperture. For the waist effect, when the beam waist is smaller than the outer scale of turbulence, with the increase of the beam waist, the beam wander generated by turbulence decreases, the optical path difference between various rays in the receiving aperture decreases, and the wavefront distortion of the transmitted beam caused by turbulence disturbance is reduced, so as to improve the received probability of signal vortex modes. However, when the beam waist is larger than or equal to the outer scale of turbulence, with the increase of the beam waist, the beam wander generated by turbulence is still decreasing. However, rays in the receiving aperture pass through turbulence with different refractive index. Thus, the reduction of optical path difference caused by the decrease of beam wander and the increase of optical path difference caused by the turbulence with different refractive index will be \"complementary\". It is this kind of \"complementarity\" that makes the received probability appear at \"saturation\" or decrease with the further increase of waist. For the receiving aperture size effect, when the aperture is small, the received signal wave is traveling horizontally along the optical axis and passing through the same type of turbulence, and the optical path difference between the rays caused by turbulence is small. However, with the increase of receiving aperture, not only the difference of turbulence passing through but also the difference of geometric path of each ray increases. Thus, the wavefront fluctuation of received optical signal increases, and the received probability of signal vortex modes decreases.