1. Introduction

Li et al. [1] use the Kolmogorov and non-Kolmogorov structure functions of the refractive index evaluated at the separation set to ${L_0}$ to calculate the variance of the index of refraction fluctuations, $\sigma _n^2$ (see Eq. (2) in this paper). This quantity is then kept fixed when comparing quantities resulting from predictions of non-Kolmogorov theories. The motivation for the approach of Ref. [1] is appealing, but the main deficiency of it is that it starts with the (implicit) assumption of infinite outer scale (Eq. (15) in Ref. [1]), only to introduce a finite outer scale in the subsequent equations. The approach requires starting from von Kármán model, where ${L_0}$ is explicitly present, in the first place. In this paper we rectify this inaccuracy.

2. Anchoring non-Kolmogorov turbulence on the outer-scale in the von Kármán spectrum

Let us focus on the following non-Kolmogorov power spectrum valid only for isotropic turbulence with zero inner scale [2,3]:

It is shown in Ref. [3] that for Kolmogorov power law the structure function of the fluctuations of the index of refraction associated with the von Kármán power spectrum, Eq. (1) with $\alpha = 11/3$, and at very large separation distance, $\rho \gg {L_0}$ takes the form

Similarly, for the spectrum with non-Kolmogorov power law shown in Eq. (1), the structure function at very large separation distance, $\rho \gg {L_0}$ can be expressed as

To solve integrals of the form as in Eq. (3), Sasiela uses the Mellin transform [3]

Using Eqs. (4) and (5), Eq. (3) can be expressed in closed form as follows

Note that Eq. (6) reduces to Eq. (2) when $\alpha = 11/3$ as it should be for the Kolmogorov turbulence case. Additionally observe that both structure functions in Eqs. (2) and (6), at large separation distances, $\rho \gg {L_0}$ saturate to double the value of the variance of the refractive index fluctuations, $2\sigma _n^2.$ The variance is by definition the covariance ${B_n}(\rho )$ calculated at zero separation distance, $\rho = 0$. For isotropic and homogeneous turbulence the variance or total power is $\sigma _n^2 = {B_n}(0 )= 4\pi \cdot \int\limits_0^\infty {{\kappa ^2}} {\Phi _n}({\kappa ,\alpha } )d\kappa$. To find the relation $\beta (\alpha )$ between the non-Kolmogorov and Kolmogorov refractive index structure constants, we impose the equality of the two structure functions, Eq. (2) and Eq. (6), and we deduce the main result of this paper

The refractive index structure constant $\tilde{C}_n^2$ can be obtained directly from $C_n^2$ once the outer scale is fixed and the power law exponent $\alpha$ is known. Note that the factor $\beta (\alpha )\cdot A(\alpha )$ represents the turbulence scaling due to non-Kolmogorov turbulence. In Fig. 1 we plot the 3D power spectrum, Eq. (1), multiplied by $4\pi \cdot {\kappa ^2}$ (to obtain the 1D version) for several power laws with outer scale fixed at 5 m and $C_n^2 = 1.7 \cdot {10^{ - 14}}{\textrm{m}^{ - 2/3}}$. It can be easily seen in Fig. 1 that the total power remains constant while the power law is changed because we defined $\beta (\alpha )$ to keep the total power of spectrum equal to the variance, $\sigma _n^2$ for any power law $\alpha$.

 

Fig. 1. 1D power spectrum (Eq. (1) multiplied by $4\pi \cdot {\kappa ^2}$) for different power laws $\alpha$ with fixed outer scale, ${L_0} = 5\,\textrm{m}$ and $C_n^2 = 1.7 \cdot {10^{ - 14}}{\textrm{m}^{ - 2/3}}$. The conservation of the total power for each plot is clearly visible.

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3. Summary

To summarize, in this paper we found a new relation between the structure constant in the von Kármán spectrum and the structure constant in the von Kármán spectrum with non-Kolmogorov power law. The approach we used is based on a measurable physical parameter, the outer scale, and it ensures that the total power of the fluctuations of the refractive index is conserved when the power law changes. Only normalized spectra allow for a meaningful comparison of their effects on various observables. Because a change in the power law brings with itself a change in how the power is distributed across spatial wavenumbers, different observables will react differently to different normalizations. Our approach provides natural and rigorous normalization. To our knowledge, all previous attempts to compare Kolmogorov and non-Kolmogorov turbulence, including Ref. [1] fail to conserve power and as such are not physical.