Optical propagation through non-overturning, undulating temperature sheets in the atmosphere

It is a standard assumption in the theory of optical propagation through the turbulent atmosphere that the refractive-index fluctuations n₁(x) are statistically isotropic. It is well known, however, that n₁(x) in the free atmosphere and in the nocturnal boundary layer is often strongly anisotropic, even at very small scales. Here we present and discuss a model atmosphere characterized by randomly undulating, non-turbulent and non-overturning, quasi-horizontal refractive-index interfaces, or "sheets." We assume n₁(x)=v[z-h(x,y)], where v(z) is a random function that has a 1D spectrum V(κ_z), and where h(x,y) is a vertical displacement that varies randomly as a function of the horizontal coordinates x and y. We derive a closed-form expression for the 3D spectrum Φ(κ) and show that the horizontal 1D spectra have the same power law as V(κ_z) if the structure function of h(x,y) is quadratic. Moreover, we evaluate the scintillation index σI2 for a plane wave propagating horizontally through the undulating sheets, and we compare σI2 predicted for undulating sheets with Tatarskii's classical predictions of σI2 for fully developed, isotropic turbulence. For Phillips-type sheets, where V(κ_z)∝κz-2, in the diffraction limit we find σI2∝k (where k=2π/λ is the optical wavenumber), which is only slightly different from Tatarskii's famous k^7/6 law for propagation through fully developed, Obukhov-Corrsin-type, isotropic turbulence where Φ(κ)∝κ^-11/3. Our model predicts that σI2 is inversely proportional to the sheet tilt angle standard deviation 〈θx2〉, regardless of whether or not diffraction plays a role and regardless of the value of the power-law exponent of V(κ_z).