## 2 Theoretical Model  
### Seawater Turbulence and Localized Wave of Bessel-Gaussian Amplitude Envelope  
In order to make the conclusions of this study closer to the real seawater environment, we research the propagation of BGLW with vortex modes in anisotropic turbulence. For the anisotropy of turbulent seawater that exists only along the \(z\) direction of propagation, we have the power spectrum of the refractive index fluctuations [13]  
\[\phi^{\prime}(\kappa^{\prime},\zeta)=0.388C_{m}^{2}\zeta^{2}\kappa^{\prime-11/3} \left[1+2.35(\kappa^{\prime}\eta)^{2/3}\right]f(\kappa^{\prime},\varpi), \tag{1}\]  
where \(\kappa^{\prime}=\sqrt{\kappa_{z}^{2}+\zeta^{2}\kappa_{\rho}^{2}}\) is the spatial frequency of the refractive index fluctuations, \(\kappa_{\rho}=\sqrt{\kappa_{x}^{2}+\kappa_{y}^{2}}\), under the Markov approximation, \(\kappa^{\prime}\) can be further simplified as \(\kappa^{\prime}=\zeta\kappa_{\rho}\), \(\zeta\) is the turbulence anisotropy index; \(\varpi\) is the ratio of temperature and salinity contributions to the refractive index spectrum, which can vary in \([-5;0]\) in the seawater with \(-5\) and \(0\) being heavily temperature and salinity dominating respectively, \(\varepsilon\) is the dissipation rate of kinetic energy per unit mass of fluid that ranges from \(10^{-10}\,\text{m}^{2}/\text{s}^{3}\) to \(10^{-1}\,\text{m}^{2}/\text{s}^{3}\), \(\chi_{T}\) is the dissipation rate of the mean-squared temperature and has the range from \(10^{-10}\,\text{K}^{2}/\text{s}\) to \(10^{-4}\,\text{K}^{2}/\text{s}\); \(f(\kappa^{\prime},\varpi)=\exp(-A_{T}\delta)+\varpi^{-2}\exp(-A_{S}\delta)-2 \varpi^{-1}\exp(-A_{TS}\delta)\), \(\eta\) is the inner scale, \(C_{m}^{2}=10^{-8}\varepsilon^{-1/3}\chi_{T}\) is the \"temperature structure\" constant and denotes the temperature fluctuation strength of seawater turbulence for given \(\varpi\)[19], \(\delta=8.284(\kappa^{\prime}\eta)^{4/3}+12.978(\kappa^{\prime}\eta)^{2}\), \(A_{T}=1.863\times 10^{-2}\), \(A_{S}=1.9\times 10^{-4}\), and \(A_{TS}=9.41\times 10^{-3}\). Using the Rytov approximation, we can express the complex amplitude of a Bessel Gaussian (BG) monochromatic vortex wave in turbulent seawater at the receiver plane as [20]  
\[u(\rho,\theta,z,\omega)=u_{m_{0}}(\rho,\theta,z,\omega)\exp\left[\mathrm{i}S( \rho,\theta,z,\omega)\right], \tag{2}\]  
where \((\rho,\theta,z)\) means cylindrical coordinates, \(\omega\) is the angular frequency of the beam, \(S(\rho,\theta,z,\omega)\) represents the phase perturbation of spectrum arising from seawater turbulence, \(u_{m_{0}}(\rho,\theta,z,\omega)\) denotes the complex amplitude of the BG monochromatic vortex wave in the absence of turbulence and is expressed as [18; 21]  
\[ \begin{split} u_{m_{0}}(\rho,\theta,z,\omega)=\frac{(-\mathrm{i} )^{m_{0}+1}E_{0}\omega w_{0}^{2}}{2 zc\sqrt{1+z_{\xi}^{2}}}\exp\left[-\frac{ \omega^{2}w_{0}^{2}}{4c^{2}(1-\mathrm{i}z_{\xi})}\left(\sin^{2}\alpha+\frac{ \rho^{2}}{z^{2}}\right)\right]\\ \times I_{m_{0}}\left(\frac{\omega^{2}w_{0}^{2}\rho\sin\alpha}{2 zc ^{2}(1-\mathrm{i}z_{\xi})}\right)\exp\left[\mathrm{i}\frac{\omega}{c}\left( z+\frac{\rho^{2}}{2z}\right)+\mathrm{i}m_{0}\theta+\mathrm{i}\arctan\left(z_{\xi} \right)\right],\end{split}  \tag{3}\]  
where \(w_{0}\) is the beam waist, \(I_{m_{0}}(...)\) represents the modified Bessel function of the first kind, \(m_{0}\) is the topological charge number of vortex mode, \(\alpha\) is the Bessel cone angle, \(E_{0}\) is a constant, \(z_{\xi}=\omega w_{0}^{2}/(2 zc)\) is the non-dimensional coordinate.