# Non-Classic Atmospheric Optical Turbulence: Review
###### Abstract  
Theoretical models and results of experimental campaigns relating to non-classic regimes occurring in atmospheric optical turbulence are overviewed. Non-classic turbulence may manifest itself through such phenomena as a varying power law of the refractive-index power spectrum, anisotropy, the presence of constant-temperature gradients and coherent structures. A brief historical introduction to the theories of optical turbulence, both classic and non-classic, is first presented. The effects of non-classic atmospheric turbulence on propagating light beams are then discussed, followed by the summary of results on measuring the non-classic turbulence, on its computer and in-lab simulations and its controlled synthesis. The general theory based on the extended Huygens-Fresnel method, capable of quantifying various effects of non-classic turbulence on propagating optical fields, including the increased light diffraction, beam profile deformations, etc., is then outlined. The review concludes by a summary of optical engineering applications that can be influenced by atmospheric non-classic turbulence, e.g., remote sensing, imaging and wireless optical communication systems. The review makes an accent on the results developed by the authors for the recent AFOSR MURI project on deep turbulence. non-classic turbulence; anisotropic turbulence; non-Kolmogorov turbulence +
Footnote †: journal: Applied  
19 July 20  
By the middle of the 19 th century, the interest in turbulence resurfaced. On studying instabilities in the pendulum's motion, [PERSON] introduced a parameter that characterizes fluid flows by the amount of their mixing [3], currently known as the _Reynolds number_ (after [PERSON] who was the first to make practical use of the concept [4]). The Reynolds number characterizes the ratio of inertial to viscous forces in the fluid:  
\[\text{Re}=uL/\
u, \tag{1}\]  
where \(u\) [m/s] is the speed, \(L\) [m] is the characteristic linear dimension, and \(\
u\) [m\({}^{2}\)/s] is the kinematic viscosity of the fluid. The transition from laminar (mixing-free) to turbulent flows depends on geometry--from \(2.3\times 10^{3}\) to \(5.0\times 10^{4}\) for pipe flow to \(10^{6}\) for boundary layers. In atmospheric cyclones, the number can reach values on the order of \(10^{12}\). On developing the tools for weather forecasting, [PERSON] came up in the early 1920s with the construction that is now known as the direct energy cascade or _Richardson energy cascade_[5]. In a classic 3D turbulent flow, the energy is mechanically injected into a volume filled with a fluid by means of a velocity field, such as wind in a gas or a current in a liquid. If the initial distribution of some scalar field (temperature, pressure, concentration of a chemical compound) within the volume is not uniform, the injected energy creates a few sizable eddies, which later break down into a larger number of smaller ones. The process repeats until a sufficiently small scale is reached and the mechanical energy starts dissipating into heat. A turbulent eddy is intuitively understood as a region of space in which the scalar field remains fairly correlated. In 1925, [PERSON] introduced a parameter, currently known as the _Prandtl number_, which characterizes the ratio of momentum diffusivity to diffusivity of an advected scalar. It is defined by the following expression: [6]  
\[Pr=\
u/\alpha_{j}, \tag{2}\]  
where \(\alpha_{j}\) [m/s\({}^{2}\)] is the diffusivity relating to the nature of the advected scalar: thermal for temperature advection, mass for molecular advection, etc. This parameter plays a crucial part in characterizing non-classic turbulent regimes. For atmospheric turbulence, the value of Prandtl number is about 0.72. Having been greatly influenced by the work of [PERSON], in 1935, Sir [PERSON] introduced the concept of locally isotropic 3D turbulence [7], the isotropy (or invariance, in the limit) being understood as the weak dependence of the flow's characteristics on orientations in space.