Abstract

The 3D spectrum $\mathrm{\Phi }(\mathit{\kappa })$ of the turbulent air temperature fluctuations is a key quantity for the physics of optical propagation through the turbulent atmosphere. The standard model, which was derived in the 1950s by Tatarskii from the Obukhov–Corrsin theory of homogeneous and isotropic turbulence, is $\mathrm{\Phi }(\kappa )=0.033C_T^2{\kappa }^{-11/3}h(\kappa l_0)$, where $\kappa =|\mathit{\kappa }|$ is the wavenumber, $C_T^2$ is the temperature structure parameter, $l_0$ is the inner temperature scale, and $h(\kappa l_0)$ is a universal function that approaches 1 for wavenumbers in the inertial range and drops to zero for $\kappa l_0\gg 1$. Certain performance characteristics of optical systems, such as the scintillation index for small receiving apertures, depend sensitively on the functional form of $h(y)$ at $y\approx 1$. During the last 70 years, the optical-turbulence community has developed and applied various heuristic $h(y)$ models. There is a constraint that any valid $h(y)$ model has to fulfill: ${\int }_0^{\infty }h(y)y^{1/3}\mathrm{d}y=(27/10)\mathrm{\Gamma }(1/3)=7.233$. This constraint is a dimensionless form of the spectral temperature variance dissipation equation, which follows directly from first-principle fluid mechanics. We show that Tatarskii’s cutoff (1961) and Gaussian (1971) models fulfill this constraint, while three more recent models, including the widely used Andrews model [J. Mod. Opt. 39, 1849 (1992) [CrossRef]  ], do not. The dissipation constraint can be used to “recalibrate” the coefficients in these models.

© 2015 Optical Society of America

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