# A Survey of Structure of Atmospheric Turbulence in Atmosphere and Related Turbulent Effects
## 3 Theoretical Studies of Turbulent Effect
### Theoretical Studies of Non-Kolmogorov Turbulent Effect  
Contrary to many experimental facts and some relevant theoretical results of Kolmogorov turbulence, the study of optical wave propagation in non-Kolmogorov atmospheric turbulence has attracted considerable attention, and experts and scholars at home and abroad have begun to study the topics related to optical wave propagation in non-Kolmogorov atmospheric turbulence. To our knowledge, a more influential study is summarized below. [PERSON] developed a refractive index fluctuation spatial power spectrum with a general power ratio. It has the following form:  
\[\Phi_{n}(\kappa,\alpha)=A(\alpha)\ \tilde{C}_{n}^{2}\ \kappa^{-\alpha},\kappa>0, \ 3<\alpha<4, \tag{8}\]  
where \(\alpha\) denotes the spectral power-law exponent that can assume all the values between the range of 3 to 4, \(\tilde{C}_{n}^{2}\) is a generalized refractive-index structure parameter with units \(m^{3-\alpha}\), and \(A(\alpha)\) is defined by the following:  
\[A(\alpha)=\frac{1}{4\pi^{2}}\ \Gamma(\alpha-1)\ \cos\Bigl{(}\frac{\alpha\pi}{2} \Bigr{)}, \tag{9}\]  
where the symbol \(\Gamma(x)\) represents the gamma function. Then, the Rytov smoothing perturbation approximation method was used to study theoretically the representative amplitude and phase effects of optical wave propagation in non-Kolmogorov turbulence, and the logarithmic amplitude fluctuation variances and coherence length of the plane wave and spherical wave under the condition of weak fluctuation were derived. The analytic expressions were obtained [10]. [PERSON] et al. studied optical wave propagation by using a non-classical power spectrum, which described media based on the report of statistical law of non-Kolmogorov atmospheric turbulence, and developed a spatial power spectrum of refractive index fluctuation with a generalized power law. Based on the Mellin transform, the structural functions of plane and spherical waves under the condition of weak fluctuations were derived, and then the Strehl ratio was calculated by using the structural functions, and the relative contributions of logarithmic amplitude structural functions and phase structural functions to the wave structural functionswere studied, which were often used to describe the statistical properties of an optical field [48]. [PERSON] et al. obtained an analytical expression for the forward difference of non-Kolmogorov turbulent waves by using Zernik polynomials [49]. [PERSON] et al. developed a theoretical method and then used this method to analyze the temporal and spatial characteristics of the phase fluctuation of optical wave propagation in non-Kolmogorov turbulence and deduced the spatial structure function, time structure function, and time frequency spectrum of the phase fluctuation. The cross-correlation functions of long exposure and short exposure and the Strehl ratio of imaging were calculated [50]. [PERSON] et al. also studied adaptive compensation of distributed beacons in non-Kolmogorov turbulence [51]. [PERSON] introduced a fractional Brownian motion representation of the turbulent wavefront phase and applied the method to the wavefront construction of non-Kolmogorov turbulence [52]. [PERSON] et al. developed a new stochastic phase screen time series simulation algorithm for dynamic atmospheric turbulence wavefront generator, which was applicable not only to Kolmogorov turbulence but also to non-Kolmogorov turbulence, and studied the spatial and temporal statistical laws of atmospheric turbulence by using this method [53]. [PERSON] et al. developed a spatial power spectrum with a general power index, then studied the angle-of-arrival fluctuation of optical wave propagation in non-Kolmogorov turbulence by using the geometric optical approximation method and deduced the analytical expressions of the variance of the angle-of-arrival fluctuation of plane wave and spherical wave under the condition of weak fluctuation.