# Non-Classic Atmospheric Optical Turbulence: Review
## 3 Theoretical Modeling of Optical Turbulence
### Obukhov-Corrsin Theory  
We begin by discussing the Obukhov-Corrsin theory, suitable for characterization of the 3D atmospheric, boundary-free, homogeneous and isotropic turbulence, for which \(Re\) can reach values on the order \(10^{8}\)-\(10^{9}\). Additionally, the _Peclet number_ must be sufficiently high, and the _Mach number_ (both dimensionless) must be subsonic. The Peclet number (for temperature) is defined as a ratio of the rate of advection to the rate of diffusion as follows:  
\[Pe_{T}=RePr=\frac{uL}{\alpha_{T}}, \tag{15}\]  
and the Mach number is a characteristic of the fluid velocity \(u\) relative to the speed of sound \(u_{s}\) in the medium as follows:  
\[Ma=\frac{u}{u_{s}}. \tag{16}\]  
For \(Ma>1.35\) and \(Ma<0.65\) the flow is characterized as _supersonic_ and _subsonic_, respectively. The fluctuations of the refractive index are conventionally treated as a random process with stationary increments. It is a possibility of involving such a random process since the average value of the refractive index is a slowly-varying function of time. Then, it is sufficient to characterize the refractive-index by its two moments-- the average value and the autocovariance function as follows:  
\[\bar{n}(\mathbf{r})=\left\langle n(\mathbf{r})\right\rangle_{M}, \tag{17}\]  
\[B_{n}(\mathbf{r}_{1},\mathbf{r}_{2})=\left\langle[n(\mathbf{r}_{1})-\bar{n}( \mathbf{r}_{1})][n(\mathbf{r}_{2})-\bar{n}(\mathbf{r}_{2})]\right\rangle_{M}. \tag{18}\]  
Here, subscript \(M\) stands for the statistical average over the realizations of turbulence, and \(\mathbf{r}\) is the 3D position vector. If turbulence is homogeneous, using the Wiener-Khinchin theorem, it is also possible to characterize the classic fluctuations in the spatial frequency domain via the power spectrum:  
\[\Phi_{n}(\mathbf{\kappa})=\frac{1}{(2\pi)^{3}}\iiint B_{n}(\mathbf{r}_{d})\exp [-i\mathbf{\kappa}\cdot\mathbf{r}_{d}]d^{3}r_{d}, \tag{19}\]  
\(\mathbf{r}_{d}=\mathbf{r}_{1}-\mathbf{r}_{2}\) and \(\mathbf{\kappa}=\kappa_{x}\hat{\mathbf{x}}+\kappa_{y}\hat{\mathbf{y}}+\kappa_ {z}\hat{\mathbf{z}}\) being the difference vector and the 3D vector of spatial frequencies, respectively. Further, if turbulence is also isotropic, its 3D spectrum can also be written as \(\Phi_{n}(\mathbf{\kappa})\), where \(\mathbf{\kappa}=|\mathbf{\kappa}|\). Based on the Kolmogorov-Obukhov theory for the turbulent velocity field, the power spectrum for any advected scalar field can be determined by the Obukhov-Corrsin theory [49, 50], according to which the equilibrium turbulent regime is split into three regimes: _inertial-convective, viscous-convective_ and _viscous-diffusive_, depending on the participating scales. The inertial-convective regime corresponds to scales between the _outer scale_\(L_{0}\) (the largest eddy size) and the following:  
\[\eta_{K}=\sqrt[4]{v^{3}/\varepsilon},\quad\eta_{B}=\sqrt[4]{va_{d}^{2}/ \varepsilon}, \tag{20}\]  
known as the _Kolmogorov scale_ [m] and the _Batchelor scale_ [m], where \(\varepsilon\) [m\({}^{2}\) s\({}^{-3}\)] is the _turbulent kinetic energy dissipation rate_ (per unit mass) and \(\alpha_{d}\) is _diffusivity_ [m\({}^{2}\)/s]. The viscous-convective regime occurs for smaller scales, and for even smaller ones, the energyrapidly dissipates within the viscous-diffusive regime. In cases when the Prandtl number is sufficiently small, the inertial-convective regime can be directly followed by the inertial-diffusive regime. The general form of the 3D power spectrum temperature or for refractive index fluctuations is as follows:  
\[\Phi_{j}(\kappa)=\frac{C_{0}}{4\pi}\varepsilon^{-1/3}\chi_{j}\kappa^{-11/3}g( \kappa\eta_{K}), \tag{21}\]  
where \(C_{0}=0.72\) is the Obukhov-Corrsin constant, \(\chi_{j}\) is the _variance dissipation rate_, having units [K\({}^{2}\) s\({}^{-1}\)] for temperature, and [s\({}^{-1}\)] for the refractive index.