# Bit Error Rate of Multi-Gaussian Correlated Asymmetric Bessel Beam Through Turbulent Ocean
## 3 Numerical Simulations and Discussion
The temperature and salinity ranges selected according to the spectrum of OTOPS cover all the natural water conditions [32]. As the temperature and salinity increase, the BER keeps rising because the lower reception probability of OAM signal mode at this time brings the reduction in SNR. Further observation reveals that there exists significant differences in the impacts of temperature and salinity changes on the level of BER. Under the same transmission distance, the change in temperature has a greater impact on BER, while the impact of salinity is relatively gentle. This indicates that the propagation of the MGCAB beam is more sensitive to the variation in seawater temperature. Since the surface temperature is higher than the deep temperature in actual oceanic environments, the selection of a deep cold ocean channel will result in better signal transmission quality. In addition to temperature and salinity concentration, the temperature-salinity gradient ratio \(H\) also contributes to turbulence effects. As shown in Figure 5, the BER of the  
Figure 4: Effects of (**a**) average temperature \(\left\langle T\right\rangle\) and (**b**) average salinity concentration \(\left\langle S\right\rangle\) on the BER of the MGCAB beam under different transmission distances \(z\). The three-dimensional diagrams are given in (**c,d**) corresponding to \(z=50\) m and \(z=150\) m, respectively. Figure 3: Effects of average salinity concentration \(\left\langle S\right\rangle\) on (**a**) the received mode probability distribution of the MGCAB beam and (**b**) the signal reception probability under different transmission distances \(z\). MGCAB beam slowly increases with \(H\) at first. When \(H\) reaches about \(-5\)\({}^{\circ}\)C/ppt, the BER decreases but later experiences a rapid rise with further elevation of \(H\). This can be attributed to the variation in beam spatial coherence radius \(\rho_{0}\). An increase in \(H\) at small values causes the diminution in turbulence eddy diffusivity. As a result, the beam spatial coherence radius reduces, implying the occurrence of more wavefront distortion and modal crosstalk. Nevertheless, as \(H\) increases to about \(-5\)\({}^{\circ}\)C/ppt, the water expansion coefficient becomes negative [32]. The spatial coherence radius increases first but later decreases with the further rise of \(H\), causing a \"groove\" in the curve of BER within the [\(-5\),0) interval. The influences of other turbulence parameters such as the temperature dissipation rate \(\chi_{T}\) and kinetic energy dissipation rate \(\varepsilon\) on BER are also reflected in Figure 5. A lower level of BER can be obtained in conditions of smaller \(\chi_{T}\) and larger \(\varepsilon\) due to the weaker oceanic turbulence. Compared with \(\varepsilon\), \(\chi_{T}\) has a more significant impact on beam propagation performance. The results are consistent with the conclusions about \(\chi_{T}\) and \(\varepsilon\) in previous reports based on the Nikishov oceanic turbulence spectrum [37]. To investigate the impact of modulation system on the BER of the MGCAB beam, Figure 6a,c illustrate the relationship between BER and modulation order \(M\) under various transmission distances \(z\). During the transmission process, the BER of high-order modulation is higher than that of low-order modulation. The reason is that, when the modulation order increases, the constellation point number of QAM increases and their mutual distances become smaller, which makes the modal interference more severe in oceanic turbulence. Compared with the traditional OOK modulation [18], the low-order modulation of QAM not only has a lower BER but also carries a larger amount of information per symbol, which can significantly enhance the signal transmission efficiency. To further investigate the BER characteristics of the MGCAB beam under different turbulent conditions, we reduce the value of temperature dissipation rate \(\chi_{T}\) in Figure 6c to construct a weaker turbulence condition than that in Figure 6a.