Spectral Characteristics of Atmospheric Turbulence
The wavenumber power spectrum of refractive index fluctuations in the atmosphere has important consequences on a number of problems involved in the propagation and scattering of optical waves. On the basis of the above 2/3 power-law expression, it can be deduced, and the associated power spectral density for refractive-index fluctuations can be described, by the following expression [8,46]:
This is the well-known Kolmogorov spectrum, which was calculated in normalized form, Ф(к) = Ф(к)/0.033СП in Reference 9 for inertial and dissipation ranges. Here к is the spatial frequency or wavenumber, к = (2n/l) (in units rad • m-1). We present in Figure 1.2 the corresponding computations following Reference 8.
As shown by Tatarskii [19,45], the Kolmogorov spectrum is theoretically valid only in the inertial subrange. The use of this spectrum is justified only within that subrange or over all wave numbers к = 2n/l, if the outer scale is assumed to be infinite and the inner scale negligibly small. We note that the three-dimensional (3D) classical power spectrum (1.40) with к-11/3 power law is related to 1D spectrum with к-5/3 power law due to the following relation [8,17-20].
where Ф^° and Ф3-0(к) are the 1D and 3D power spectrum, respectively.
The spectrum measures the distribution of the variance of a variable over scale sizes or periods. If the variable is a velocity component, the spectrum also describes the distribution of kinetic energy over spatial periods.
Using the relation between the structure function and the covariance, and the Wiener-Khinchin theorem, the relation between the structure function and the power spectrum is given by References 8, 17-20.
Figure 1.2 Kolmogorov normalized spectrum, shown for the inertial and dissipation ranges; the dashed vertical line indicates к = 2n//o.
We should notice that in practice of atmospheric optical communication, the velocity power spectrum is not of interest, when the optical properties of turbulence need to be characterized. In this case, Obukhov has adopted a Kolmogorov’s 3D law to the 1D law by use of the concept of “passive scalar” [45-47], which states that fluctuations in passive scalar quantities (e.g., temperature field, refractive index, etc.) associated with turbulent structure function, has a similar form, as (1.36), under assumption that the turbulent field is locally homogeneous and isotropic. Therefore, he introduced the 1D spectral power function, Ф5(к), which describes the power spectrum of a given passive scalar S 
where CS = Const • ?T • ?-1/3 is the structure constant derived by Obukhov . Here ?T is the heat flux intensity over the spectrum (thermal dissipation rate).
Finally, based on Obukhov’s law (1.43) and on the 2/3 power law (1.36) for the structure function, the associated 3D spectrum for the refractive index fluctuations over the inertial subrange has the same form, as that obtained by Kolmogorov and expressed by (1.40). Therefore, the Equation 1.40 is known in the literature as the Obukhov—Kolmogorov power spectrum [44—47]. The model
is valid only for the inertial subrange, although it is often extended over all wave numbers by assuming the inner scale is zero and the outer scale is infinite. It is usually assumed that the inertial subrange determines the optical properties of the turbulent atmosphere.
Other spectrum models have been proposed for making calculations when inner scale and/or outer scale effects cannot be ignored. The following isotropic forms of spectra, which take into account deviations from a power law in the region of turbulence outer and inner scale, can be used in calculations [8,17-21].
Thus, for mathematical convenience, by von Karman was assumed that the turbulence spectrum is statistically homogeneous and isotropic over all wave numbers. A spectral model that he used in this case is that combines the three regions defined by (1.1), called in literature, as the von Karman spectrum [8,20]:
Here Km = (5.92/l0) and k0 = (C0/L0), where 1 < C0 < 8n is the scaling constant for the outer scale parameter [8,20]. Note that even though the last equation describes the entire spectrum, its value in the input range must be considered only approximate, because it is generally anisotropic and depends on how the energy is introduced into the turbulence. This model, unlike the models shown above, does not have a singularity at к = 0. The von Karman spectrum (1.44) was computed in Reference 8, and we present it here in Figure 1.3.
Figure 1.3 von Karman normalized spectrum, shown for all ranges; the dashed vertical line indicates к = 2n//o-
As shown in Reference 8, although the Tatraskii spectrum “explodes” near the origin, the von Karman spectrum inclination is suppressed in this region. For other regions, the two spectra are almost identical. Therefore, we can emphasize that the von Karman spectrum is almost identical to the Tatarskii spectrum except for a difference in small values of wave number.
However, we should notice that the last two spectra have the correct behavior only in the inertial range: that is, the mathematical form that permits the use of these models outside the inertial range is based on mathematical convenience, and not because of any physical meaning. The Tatarskii spectrum has been shown to be inaccurate in about 50% for predicting the irradiance variance for the strong- turbulence regime in optical propagation experiments and in more than 40% for weak turbulences [19,20,47].
Hill (see details in References 8, 20) developed a numerical spectral model with a high wave-number rise that accurately fits the experimental data. He has performed a hydrodynamic analysis that led to a numerical spectral model with the small rise (or bump) at high wave numbers near Km ^ 1/l0 that fit the experimental data of temperature spectrum [8,20]. The bump in temperature spectrum also induces a corresponding spectral bump in the spectrum of the refractive index fluctuations that can have important consequences on a number of statistical quantities important in problems involving optical wave propagation.
However, because Hill’s model is described in terms of a second-order differential equation that must be solved numerically, the corresponding spectrum cannot be used in analytic developments. An analytic approximation to the Hill spectrum, which offers the same tractability as the von Karman model (1.44), was developed by Andrews and colleagues [8,20].
This approximation, commonly called the modified atmospheric spectrum (or just modified spectrum), is given by References 22—25, and it is valid for wave numbers in the range of 0 < к < ж.
where кп = 3.3/l0. We compare the modified model (1.45) with that proposed by von Karman and present this comparison in Figure 1.4, following computations made in Reference 8. Numerical comparison of results based on the above equation and the Hill spectrum reveal differences no larger than 6%, but generally within 1%—2% of each other. Note that these forms of the turbulence spectrum are used for computational reasons only and are not based on physical models. The modified model, which is obviously based on Hill’s numerical spectral model, is closed to von Karman model for к < 600 m-1 and what is important, it provides good agreement with experimental results [8,20].
Figure 1.4 Comparison between the modified and von Karman normalized spectra; the dashed vertical line indicates к = 2n/V
We also should notice that expression (1.45) is similar to the von Karman functional form (1.44), except for the terms within the brackets that characterize the high wave number spectral bump.
We can summarize that in the simple Kolmogorov model of the turbulence, the atmosphere is usually described as a single turbulent layer in which the variations of the refractive index with temperature and pressure induce both phase and amplitude fluctuations of the propagating wavefront. In addition, it is usually assumed that the time scale of temporal changes in the atmospheric layer of wave propagation is much smaller than the time it takes the wind to blow the turbulence over the receiver aperture (Taylor’s hypothesis of frozen turbulence, see Reference 8). The spatial and temporal properties of this single layer are thus linked by the wind speed Vof the layer as w = к ? V, where w is the frequency in rad/s.
For elevations around the boundary layer (2—3 km) and above, the exponent 11/3 in Equation 1.40, and Cl change as functions as elevation. This does not conform to the Kolmogorov empirical assumptions and is called non-Kolmogorov turbulence [48—51]. It affects wireless communication reliability and image quality. Some special cases of how the non-Kolmogorov processes can occur around the bounded irregular turbulent atmospheric layers are also discussed here in Chapter 3.