# Non-Classic Atmospheric Optical Turbulence: Review
## 3 Theoretical Modeling of Optical Turbulence
### Anisotropic Turbulence  
In the most general case, the turbulent anisotropy can be modeled with the ellipsoid-like correlation function (or the power spectrum) discussed in Section 2.1. As was illustrated above with the optical measurements, such an ellipsoid can be arbitrarily oriented with respect to the ground. However, for analytical modeling, the ellipsoid with the semi-axis describing a vertical and two mutually orthogonal horizontal directions is typically used, in which the vertical semi-axis is shorter than two equal horizontal semi-axes. In this case, a typical turbulent eddy can be viewed as a horizontally oriented \"crepe\". Such a model was applied to two light propagation cases: (A) vertical path (see Figure 9) and (B) horizontal path (see Figure 10). See also [68] for qualitative theory of anisotropic irregularities.  
Figure 8: Three-layer model of non-Kolmogorov turbulence spectrum \(\alpha\) versus height \(z\) [m] above the ground. Solid curve: \(b_{1}=8\), \(b_{2}=10\); Dashed curve: \(b_{1}=15\), \(b_{2}=20\). Reprinted with permission from Reference [62].  
In case (A), the power spectrum model accounts for the turbulent eddy symmetry axis placed along the optical axis, for example, \(z\)[69]. In addition, since the deviation of the power law from the Kolmogorov value of \(11/3\) is also regarded to be a consequence of stratification and anisotropy [35], these two phenomena can be expressed together as follows:  
\[\Phi_{n}(\mathbf{\kappa},\alpha,\mu_{z})=\frac{\Gamma(\alpha-1)\cos(\alpha\pi/2) \widetilde{C}_{n}^{2}\mu_{z}^{2}}{4\pi^{2}[\mu_{z}^{2}\kappa_{z}^{2}+(\kappa_{ z}^{2}+\kappa_{y}^{2})]^{\alpha/2}}, \tag{35}\]  
where \(\Gamma\) stands for the Gamma function and \(\widetilde{C}_{n}^{2}\) is the generalized refractive index structure parameter with units \(m^{3-\alpha}\). The anizotropic factor \(\mu_{z}\) is related to the ellipse eccentricity in the \(z\) (vertical) direction.  
In case (B), the power spectrum describes the axis of eddy symmetry being orthogonal to the optical axis \(z\)[70; 71]:  
\[\Phi_{n}(\mathbf{x},\alpha,\mu_{x},\mu_{y})=\frac{\Gamma(\alpha-1)\cos(\alpha \pi/2)\widetilde{C}_{n}^{2}\mu_{x}\mu_{y}}{4\pi^{2}[\kappa_{z}^{2}+\mu_{x}^{2} \kappa_{x}^{2}+\mu_{y}^{2}\kappa_{y}^{2}]^{\alpha/2}}. \tag{36}\]  
Here, anisotropic factors \(\mu_{x}\) and \(\mu_{y}\) define the eccentricity of the ellipse in the \(x\)-\(y\) planes, i.e., the planes orthogonal to the direction of propagation. In these spectra, the inner scale and the bump at high spatial frequencies are not included but can also be modeled in.  
The power spectrum model of Reference [71] assumes the same degree of anisotropy for all turbulent scales in the inertial sub-range, according to Kolmogorov theory [13]. [PERSON] introduced an extension of this model to the situations when anisotropy may be scale-dependent [72]. In this model, the eddies on the scale comparable to the inner scale are spherically symmetric and gradually increase in the anisotropy in the vertical direction as the outer scale is approached. Other elaborate models for treating anisotropy were also suggested, e.g., in [73]).