# Non-Classic Atmospheric Optical Turbulence: Review
## 2 Measurements and Synthesis
### Anisotropic Turbulence  
The classic [PERSON] theory, based on [PERSON]'s energy cascade, implies that in the inertial sub-range of scales, all turbulence eddies are isotropic. However, it appears possible, in free atmosphere, within the stably-layered stratosphere, for the turbulence to become anisotropic (direction-dependent) at large spatial scales (e.g., (26; 27)). A similar situation may appear in the close proximity of a hard boundary, i.e., ground surface, a building, an airplane, etc., i.e., within a boundary layer (28; 29; 30; 31). In their 1970 experimental paper, [PERSON], [PERSON] and [PERSON] illustrated in the laboratory (over 130 m path, at 1 m height) that in the close proximity of a boundary (ground), a pair of laser beams, co-aligned and set to propagate along the horizontal direction, would lead to short-exposure relative beam wander (\"relative dancing\"), depending on the placement of the two sources (32). In particular, the statistics were shown to qualitatively and quantitatively differ for horizontal and vertical orientations, implying the presence of turbulent anisotropy. This measurement procedure suggested a very simple, optics-based method for imprinting the signature of air turbulence onto the intensity statistics of propagating light. Since the fluctuating temperature of the turbulent air is the dominating random process affecting the fluctuating refractive index, the air velocity turbulence can be directly related to the refractive index turbulence, also termed _optical turbulence_. In particular, because temperature fluctuations are largely responsible for optical turbulence of the air, the anisotropic behavior of the temperature field carries over that in the fluctuating refractive index field. In order to develop the theory, a 3D Gaussian (ellipsoidal) correlation function model of the refractive index fluctuations of the turbulent air was implemented:  
\[B_{n}(\Delta x,\Delta y,\Delta z)=\langle n^{2}\rangle\exp\left[-\left(\frac{ \Delta x^{2}}{\Delta x_{0}^{2}}+\frac{\Delta y^{2}}{\Delta y_{0}^{2}}+\frac{ \Delta z^{2}}{\Delta z_{0}^{2}}\right)\right], \tag{14}\]  
where \(\langle n^{2}\rangle\) is the mean-square value of the refractive index about its mean value, and \(\Delta j_{0}\) is a typical deviation in direction \(j\), (\(j=x,y,z\)). Physically, the anisotropic turbulent medium can be viewed as the random collection of ellipsoidal lenses with average semi-axes lengths and orientations corresponding to geometrical directions \(x\), \(y\) (horizontal) and \(z\) (vertical). In the vicinity of a horizontal boundary, for example, ground, \(\Delta z_{0}<\Delta x_{0},\Delta y_{0}\). This discrepancy leads to qualitatively different refraction scenarios along the horizontal and vertical axes. The dancing of the two beams is characterized by the _wave structure function_, being a variance of the difference of the optical field measured at several separation distances. The structure functions \(D_{y}\) and \(D_{z}\), measured for horizontal and vertical sources' placement, are shown in Figure 2A,B, respectively. The solid lines denote the theoretical joint structure functions, while dots denote the experimental data. The discrepancy in \(D_{y}\) and \(D_{z}\) is related to the shape of \(B_{n}\) along directions \(x\) and \(y\). Since the 1990s, high-altitude atmospheric temperature field measurements have become available. A strong anisotropy in temperature was first found by [PERSON] et al. (33) from experimental observations of starlight scintillation at the intermediate altitudes. Another experimental campaign by [PERSON] et al. (34) revealed the presence in the atmosphere, at altitudes of up to 25 km, of very strong positive temperature gradients within very thin layers, or sheets, with vertical distortions up to 10 m and of horizontal extensions larger than 100 m. The anisotropy in the experimentally measured optical wavefront tilt's statistics was also revealed by [PERSON]: the outer scale in horizontal direction was found to be smaller than that in vertical (35).