# Non-Classic Atmospheric Optical Turbulence: Review
### Kolmogorov Theory (1941-1942)  
After establishing the rigorous, mathematical theory of random functions in 1930s [9; 10] [PERSON] and his students attempted its application to isotropic turbulence. Greatly influenced by the results of [PERSON] and [PERSON], [PERSON] applied the concept of [PERSON]'s ensemble averages to a homogeneous and isotropic turbulent flow [11]. He also developed the system of equations governing the velocity correlation and implemented _quasi-normal_ approximation to obtain the closed system of the second- and third-order correlations [12]. Between 1941 and 1942, [PERSON] established the foundations of the classic turbulence theory by formulating three hypotheses regarding the equilibrium regime in a homogeneous turbulent flow at sufficiently high Reynolds numbers and for sufficiently small scales [13; 14; 15; 16; 17] (see also important parallel work by [PERSON] [18; 19; 20]). This theory largely elaborated on the Richardson turbulent energy cascade by assigning to a turbulent eddy a scale \(l\), a velocity vector \(\mathbf{u}(l)\) and a time \(\tau(l)\). It was then conjectured that the cascade is formed by the largest eddies, with a scale \(L_{0}\) (integral or outer scale), \(L_{0}\leq L\), \(L\) being the scale of entire region of mixing. It can be estimated as follows:  
\[L_{0}\sim K^{3/2}/\varepsilon, \tag{3}\]  
where \(K\) is the turbulent kinetic energy (reduced to unit mass) as follows:  
\[K=\frac{1}{2}\mathbf{u}\cdot\mathbf{u} \tag{4}\]  
and \(\varepsilon\,[\mathrm{m}^{2}\,\mathrm{s}^{-3}]\) is the energy dissipation rate. Then, by defining the _turbulent Reynolds number_ as  
\[Rt=u^{2}(l)l/\
u, \tag{5}\]  
and applying it to the integral scale, one obtains the following:  
\[Rt_{L}=\sqrt{K}L_{0}/\
u=K^{2}/(\varepsilon\
u), \tag{6}\]  
attaining very large values. The first [PERSON] hypothesis states: _(i) the turbulent motion of eddies with scales \(l<<L_{0}\) is statistically isotropic._ It appears possible to estimate a threshold scale \(L_{I}\) that separates the regimes with isotropic (\(L_{I}<l<L_{0}\)) and anisotropic (\(l<L_{I}\)) eddy scales, as \(L_{I}\sim L_{0}/6\). The second hypothesis postulates that _(ii) the statistics of the small-scale (\(l<L_{I}\)) eddy motion have a universal form that is uniquely determined by \(\varepsilon\) and \(\
u\)_. This range of scales is also known as the _universal equilibrium range_. It follows from (ii) that a set of parameters solely depending on \(\varepsilon\) and \(\
u\) can be introduced:  
\[l_{K}=\sqrt[4]{\
u^{3}/\varepsilon},\quad u_{K}=\sqrt[4]{\varepsilon\
u}, \quad\tau_{K}=\sqrt{\
u/\varepsilon},\quad Rt_{K}=l_{K}u_{K}/\
u=1. \tag{7}\]  
These are the characteristics of the smallest eddies before dissipation, carrying [PERSON]'s name. Note that \(Rt\) starts at a very large number when energy is injected at scale \(L_{0}\) and decreases to unity at scale \(l_{K}\). The third hypothesis states that _(iii) the statistics of motion of eddies with scales in the range \(l_{K}<<l<<L_{0}\) have a universal form that is uniquely determined by \(\varepsilon\) and are independent of \(\
u\)_. There is another scale, \(l_{D}\approx 60l_{K}\), that splits the universal dissipation range (\(l<l_{D}\)) in which motion is the result of viscous forces. The [PERSON] theory established the regime of classic turbulence conceptually, and also led to specific mathematical law governing the structure of turbulent velocity correlation function or, equivalently, its energy spectrum. On introducing spatial frequency as \(\kappa=2\pi/l\) (or \(kl_{K}\) for dimensionless version) and energy spectrum \(E_{K}(\kappa)\) such that\[K=\int\limits_{0}^{\infty}E_{K}(\kappa)dx, \tag{8}\]  
one finds at once that, in order for \(E_{K}(\kappa)\) to depend on \(\kappa\) and \(\varepsilon\), i.e., to be in the form  
\[E_{K}(\kappa)\sim\varepsilon^{p}\kappa^{q}, \tag{9}\]  
it suffices to perform dimensional analysis.