# Non-Classic Atmospheric Optical Turbulence: Review
## 3 Theoretical Modeling of Optical Turbulence
### Non-Kolmogorov Turbulence  
It was shown long time ago by [PERSON] that turbulence does not always follow the Kolmogorov power spectrum model, even in the inertial range [59]. This is due to the fact that the abstraction of turbulent energy by buoyancy forces leads to a sharp decrease in the rate of energy transfer with the wave number. As a consequence, the kinetic energy is transported across the spectrum with a non-uniform rate, while decreasing with increasing the wave number.  
Analytic models for the isotropic, non-Kolmogorov power spectrum were used since the 1990s (e.g., [60]). A comprehensive model developed in Reference [61], including the small- and the large-scale cut-offs, has the following form:  
\[\Phi_{n}(\kappa)=A(\alpha)\widetilde{C}_{n}^{2}\frac{\exp[-\kappa^{2}/\kappa_ {m}^{2}]}{(\kappa^{2}+\kappa_{0}^{2})^{\alpha/2}},\quad 3<\alpha<5, \tag{28}\]  
where \(\widetilde{C}_{n}^{2}(z)\) has units \(m^{3-\alpha}\) and  
\[A(\alpha)=\frac{\Gamma(\alpha-1)}{4\pi^{2}}\cos[\alpha\pi/2]. \tag{29}\]  
Here, the small and the large spatial frequency cut-offs are as follows:  
\[\kappa_{0}=2\pi/L_{0},\quad\kappa_{m}=c(\alpha)/l_{0}, \tag{30}\]  
with  
\[c(\alpha)=[2\pi\Gamma(5-\alpha/2)A(\alpha)/3]^{1/(\alpha-5)}. \tag{31}\]  
The major effect of varying \(\alpha\) on the turbulence structure is in the weight distribution among the participating turbulent scales: for smaller values of \(\alpha\), more energy is contained in smaller scales, and hence, the turbulence acquires a finer, more granular structure; for larger values of \(\alpha\), more energy is attributed to larger scales, and the turbulence fluctuations takes a cruder structure.  
It was suggested in Reference [62] that the non-Kolmogorov power spectrum has a dependence on height, such as \(z\), above the ground, i.e., the following:  
\[\Phi_{n}(\kappa,z)=A[\alpha(z)]\widetilde{C}_{n}^{2}(z)\kappa^{-\alpha(z)}, \tag{32}\]  
where  
\[A[\alpha(z)]=\frac{\Gamma(\alpha(z)-1)}{4\pi^{2}}\sin[\pi(\alpha(z)-3)/2], \tag{33}\]  
Figure 7: Function \(g_{l}\) depending on \(\kappa l_{0}\).  
and the dependence \(\alpha=\alpha(z)\) is obtained by combining the Kolmogorov power spectrum (\(\alpha_{1}=11/3\), \(A=0.033\)) in the boundary layer (up to \(\approx\)2 km above the ground), power law (\(\alpha_{2}=10/3\), \(A=0.015\)) for helical turbulence in the troposphere (\(\approx\)2-8 km) and the power law (\(\alpha_{3}=5\), \(A=0.0024\)) pertinent to the stratosphere (above \(\approx\)8 km) as follows:  
\[\alpha(z) =\frac{\alpha_{1}}{(1+z/H_{1})^{b_{1}}}+\frac{\alpha_{2}(z/H_{1} )^{b_{1}}}{1+(z/H_{1})^{b_{1}}}\cdot\frac{1}{1+(z/H_{2})^{b_{2}}} \tag{34}\] \[+\frac{\alpha_{3}+(z/H_{2})^{b_{2}}}{1+(z/H_{2})^{b_{2}}}.\]  
Here, \(H_{1}\) and \(H_{2}\) are the adjustable separation altitudes, and \(b_{1}\) and \(b_{2}\) are the suitable regime matching coefficients. The corresponding dependence of \(\widetilde{C}_{n}^{2}\) on height \(z\) was also discussed in [62] (see also work [63; 64] by the same authors). Figure 8 shows the \(\alpha\)-height dependence for two sets of parameters \(b_{1}\) and \(b_{2}\).  
Some other modeling aspects and representations relating to non-Kolmogorov turbulence were also discussed in [65; 66; 67].