## 2 Kolmogorov Ocean Spectrum and Random Coaxial Double-Position Power Gaussian Vortex
### Kolmogorov Ocean Spectrum  
For a liner, isotropic, homogeneous, and clean absorbing seawater, the refractive index of seawater is [19; 20; 21]  
\[n_{oc}=n_{r}+in_{i} \tag{1}\]  
where \(n_{r}\) is the mean value of real part of refractive index of seawater, \(n_{i}\) is the mean value of imaginary part of refractive index of seawater that depends on the type of seawater, and \(i=\sqrt{-1}\) is the imaginary symbol. The turbulent characteristics of seawater are described by the following turbulent spectrum equation [24]  
\[ \begin{array}{l}\Phi_{n}(\kappa)=0.267\times 10^{-8\frac{\varepsilon^{-1/3} }{\omega^{2}}\kappa^{-11/3}}\Big{[}1+4.6(\kappa l_{0})^{2/3}\Big{]}\Big{[}1- \exp\left(-\frac{\kappa^{2}}{\kappa_{0}^{2}}\right)\Big{]}\\ \times\Big{[}\varpi^{2}\exp\left(-\frac{\kappa^{2}}{\kappa_{T}^{2}}\right)+1 \exp\left(-\frac{\kappa^{2}}{\kappa_{S}^{2}}\right)-2\varpi\exp\left(-\frac{ \kappa^{2}}{\kappa_{TS}^{2}}\right)\Big{]}\ \ \ 0<\kappa<\infty,\end{array}  \tag{2}\]  
where \(\kappa\) is the spatial frequency of turbulent fluctuations, \(\varepsilon\) is the rate of dissipation of kinetic energy per unit mass of fluid ranging from \(10^{-10}\) m\({}^{2}\)/s\({}^{3}\) to \(10^{-1}\) m\({}^{2}\)/s\({}^{3}\), \(\chi_{T}\) is the dissipation rate of the mean-squared temperature and has the range from \(10^{-10}\) K\({}^{2}\)/s to \(10^{-2}\) K\({}^{2}\)/s, \(l_{0}\) is the inner scale, \(\kappa_{0}=2\pi/L_{0}\), \(L_{0}\) is the outer scale of turbulence, \(\kappa_{T}=R_{T}/l\), \(\kappa_{S}=R_{S}/l_{0}\), and \(\kappa_{TS}=R_{TS}/l_{0}\). \(R_{j}=\sqrt{3}[W_{j}-1/3+1/(9W_{j})]^{3/2}Q^{-3/2}\), \((j=T,S,TS)\), \(W_{j}=\left\{\left[\frac{\text{Pr}_{T}^{2}}{(6\beta Q^{-2})^{2}}-\frac{\text{Pr} _{T}}{81\beta Q^{-2}}\right]^{1/2}-\left[\frac{1}{Z^{2}}-\frac{\text{Pr}_{T}}{ 6\beta Q^{-2}}\right]\right\}^{1/3}\), \(Q=2.35\), \(\text{Pr}_{T}\) and \(\text{Pr}_{S}\), respectively, represent the Prandtl number of the temperature and salinity, \(\text{Pr}_{TS}=2\text{Pr}_{T}\text{Pr}_{S}/(\text{Pr}_{T}+\text{Pr}_{S})\), \(\varpi\) defines the ratio of temperature and salinity contributions to the refractive index spectrum, which can vary in the interval [\(-5\),\(0\)] in the seawaters, \(\varpi=-5\) temperature is dominant in the induction of optical turbulence, while \(\varpi=0\) salinity is dominant in the induction of optical turbulence. Compared with the atmospheric turbulence spectrum [24], Equation (2) can be rewritten into the following ocean spectrum  
\[\Phi_{n}(\kappa)=\varphi_{n}(\kappa)f(\varpi,\kappa,\eta),\ \ 0<\kappa<\infty, \tag{3}\]  
with \"Kolmogorov ocean spectrum\"  
\[\varphi_{n}(\kappa)=0.033C_{\epsilon\chi_{T}}^{2}\kappa^{-11/3},\ \ \ 1/\eta<\kappa<1/L_{0}, \tag{4}\]  
Additionally,  
\[ \begin{array}{ll}f(\varpi,\kappa,\eta)&=\Big{[}1+4.6(\kappa l_{0})^{2/3} \Big{]}(1-\varpi)^{-2}\Big{[}1-\exp\left(-\frac{\kappa^{2}}{\kappa_{0}^{2}} \right)\Big{]}\\ &\times\left[\exp\left(-\frac{\kappa^{2}}{\kappa_{T}^{2}}\right)+\frac{1}{ \varpi^{2}}\exp\left(-\frac{\kappa^{2}}{\kappa_{S}^{2}}\right)-\frac{2}{ \varpi}\exp\left(-\frac{\kappa^{2}}{\kappa_{TS}^{2}}\right)\right].\end{array}  \tag{5}\]  
Here, \(C_{\epsilon\chi_{T}}^{2}=0.809\times 10^{-7}\varepsilon^{-1/3}\chi_{T}\varpi^{-2}(1- \varpi)^{2}\) defines the structural constants of the salinity-temperature fluctuation of oceanic turbulence. Consider that the physical mechanism behind the oceanic turbulence and atmospheric turbulence are same and in conjunction with the above discussion, using Equation (5), we can just write Rytov variance for a plane wave in a homogeneous isotropic weakly turbulent ocean as  
\[\sigma_{1x_{XT}}^{2}=1.23C_{x_{XT}}^{2}k^{7/6}z^{11/6} \tag{6}\]  
Formally, Equation (6) is the same as the Rytov variance of plane waves in the turbulent atmosphere, except that \(C_{x_{XT}}^{2}\) is replaced by the structure constant of atmospheric turbulence \(C_{n}^{2}\).