Temperature variance dissipation equation and its relevance for optical turbulence modeling

The 3D spectrum Φ(κ) 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 Φ(κ)=0.033CT2κ(-11/3)h(κl(0)), where κ=|κ| is the wavenumber, CT2 is the temperature structure parameter, l(0) is the inner temperature scale, and h(κl(0) is a universal function that approaches 1 for wavenumbers in the inertial range and drops to zero for κl(0)≫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≈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: ∫0∞h(y)y(1/3)dy=(27/10)Γ(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)JMOPEW0950-034010.1080/09500349214551931], do not. The dissipation constraint can be used to "recalibrate" the coefficients in these models.