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

Abstract

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(κz), 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(κz) 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(κz)∝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 k7/6 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(κz).