Occurrence of fluctuations in the amplitude and phase of the radio signal in a turbulent atmosphere

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Introduction

Random changes in atmospheric parameters have a significant impact on the operation of radio lines [1]. The properties of radio lines in the troposphere, stratosphere, and ionosphere are determined by many parameters, including solar activity, thermal regimes [2; 3], air humidity [4; 5], and medium density. High-quality radio signal transmission is the main goal of the development of communication networks in the Russian Federation [6]. Random turbulent air fluctuations play an important role in high-quality radio signal transmission.

In this study, based on the model of interaction between the deterministic wave process of propagation of a radio wave transmitting an information signal in a turbulent atmosphere and the stochastic wave process of atmospheric turbulence [7], we consider the fluctuations (or pulsations) in the amplitude and phase of an electromagnetic wave that affect the quality of the received radio signal. In radiophysics, the term fluctuation is commonly used, whereas the theories of turbulence more often use the term pulsations. In this study, we assume that fluctuations in radio wave parameters arise from atmospheric turbulence.

1. Fluctuations in electromagnetic wave eikonal

In Ref. [7], the equation for the electric field in an electromagnetic wave is expressed in the following form:

$k^2\mathbf{D}=\text{curl\hspace{0.17em}curl\hspace{0.17em}}\mathbf{E},$ (1)

where $\mathbf{D}=\left(1+2n^{\text{'}}\right)\mathbf{E}$ is the electric field induction [7], E is the electric field strength, $n^{\text{'}}$ is the turbulent pulsations of the refractive index of the atmosphere, and k is the wave vector of the electromagnetic wave.

By substituting the quantity $\mathbf{D}=\left(1+2n^{\text{'}}\right)\mathbf{E}$ into Eq. (1) and using the well-known vector analysis equation, we obtain $k^2\mathbf{E}+2k^2{n}^{\text{'}}\mathbf{E}\approx$$-\Delta \mathbf{E}+\text{grad\hspace{0.17em}div\hspace{0.17em}}\mathbf{E}.$ By neglecting the second term on the right side, in accordance with the equation $\text{div}\mathbf{D}=\mathrm{0,}$ we derive the following expression:

$\Delta \mathbf{E}+k^2\mathbf{E}\approx -2k^2{n}^{\text{'}}\mathbf{E}.$ (2)

For any Cartesian component E of vector E, Eq. (2) can be presented in scalar form, as follows:

$\Delta E+k^{2}E\approx -2k^2{n}^{\text{'}}E.$ (3)

Similar to Ref. [7], we assume that, for some conditionally allocated volume V (Fig. 1) with turbulent pulsations of the atmosphere, a plane electromagnetic wave is incident to the electric field strength in the wave, which can be expressed as follows:

${\mathbf{E}}_0=\mathbf{p}{A}_0{e}^{i\mathbf{kX}},$ (4)

where X is the coordinate of propagation of the incident wave; p is the unit vector in the plane of oscillation of vector E₀ perpendicular to the direction of wave propagation, i.e., wave vector k; A₀ is the wave amplitude; and kX is the wave phase. Volume V is in the region X > 0, and plane X = 0 is the left boundary of this volume.

Because of the influence of turbulent pulsations in volume V, the scalar components of the electric field in this volume can be expressed as follows:

$E=A{e}^{iS}=e^{\phi },$ (5)

where A is the amplitude of the electromagnetic wave, S is its phase, and

$\phi =\ln A+iS$ (6)

is the so-called eikonal of an electromagnetic wave, which, for the incident wave $E_0,$ has the form ${\phi }_0=\ln A_0+i{S}_0.$

Eikonal fluctuations due to turbulent pulsations can be determined using the following equation:

${\phi }^{\text{'}}=\phi -{\phi }_0=\ln \left(\frac{A}{A_0}\right)+i\left(S-S_0\right)={\chi }^{\text{'}}+i{S}^{\text{'}},$ (7)

where the parameter ${\chi }^{\text{'}}=\ln \left(A/A_0\right)$ characterizes the amplitude fluctuations and the magnitude $S^{\text{'}}=S-S_0$ characterizes the phase fluctuations of the electromagnetic wave due to turbulent pulsations.

Considering $\phi ={\phi }_0+{\phi }^{\text{'}},$ and therefore, $E=$$e^{{\phi }_0+{\phi }^{\text{'}}}$ from the Eq. (2), we obtain the following expression:

$\Delta \left({\phi }_0+{\phi }^{\text{'}}\right)+{\nabla }^2\left({\phi }_0+{\phi }^{\text{'}}\right)+k^2=-2k^2{n}^{\text{'}}.$ (8)

In accordance with Eq. (4), the eikonal of the incident wave is ${\phi }_0=$$\ln A_0+ikX;$ therefore, $\nabla {\phi }_0=ik$ and $\Delta {\phi }_0=\mathrm{0,}$ and Eq. (8) is transformed into the following form:

$\Delta {\phi }^{\text{'}}+\left(2ik+\nabla {\phi }^{\text{'}}\right)\nabla {\phi }^{\text{'}}=-2k^2{n}^{\text{'}}.$ (9)

Eq. (9) is a nonlinear equation. However, considering the smallness of the turbulent pulsations of the eikonal ${\phi }^{\text{'}},$ we assume that $2ik\gg \nabla {\phi }^{\text{'}}$ and the equation of the eikonal fluctuations of Eq. (9) can be expressed in linear form, as follows:

$\Delta {\phi }^{\text{'}}+2ik\nabla {\phi }^{\text{'}}=-2k^2{n}^{\text{'}}.$ (10)

Let us apply Eq. (10) to a specific geometry (Fig. 1). The direction along the X coordinate is preferred because an electromagnetic wave propagates along this direction. Fluctuations of the eikonal ${\phi }^{\text{'}}\left(Х\right)$ along the X-axis can be considered the result of the superposition of scattered waves in the dX layer of volume V on the electromagnetic wave (Fig. 1). Then, Eq. (10) can be rewritten as follows:

$\Delta {\phi }^{\text{'}}+2ik\frac{\partial {\phi }^{\text{'}}}{\partial Х}=-2k^2{n}^{\text{'}},$ (11)

where

$\Delta {\phi }^{\text{'}}=\frac{{\partial }^2{\phi }^{\text{'}}}{\partial Y^2}+\frac{{\partial }^2{\phi }^{\text{'}}}{\partial Z^2}.$ 

 

Fig. 1. Interaction of a plane electromagnetic wave in volume V with wave fluctuations arising due to turbulent pulsations

 

Discarding ${\partial }^2{\phi }^{\text{'}}/\partial X^2,$ we assume that the main influence on the electromagnetic wave is exerted by scattered waves ${Е}^{\text{'}}$ inside the flat layer dX (Fig. 1). At the front of a plane electromagnetic wave, partial scattering of this wave occurs because of turbulent fluctuations. The scattered waves immediately add up to a wave propagating along the X-axis. This addition occurs in a narrow dX layer at the wavefront.

Using Eq. (7) and accepting $n^{\text{'}}=n_1+i{n}_2,$ we obtain the equations for the real and imaginary parts of ${\phi }^{\text{'}}={\chi }^{\text{'}}+i{S}^{\text{'}}$ as follows:

$\Delta {\chi }^{\text{'}}-2k\frac{\partial S^{\text{'}}}{\partial Х}=-2k^2{n}_1,$ (12)

$\Delta S^{\text{'}}+2k\frac{\partial {\chi }^{\text{'}}}{\partial Х}=-2k^2{n}_2,$ (13)

where

$\Delta =\frac{{\partial }^2}{\partial Y^2}+\frac{{\partial }^2}{\partial Z^2}.$

Based on the system expressed in Eqs. (12) and (13), and by determining the Laplacian Δ on the left and right sides of these equations, we obtain separate equations for and as follows:

$\left({\Delta }^2+4k^2\frac{{\partial }^2}{\partial {Х}^2}\right){\chi }^{\text{'}}=-2k^2\left(\Delta n_1+2k\frac{\partial n_2}{\partial X}\right),$ (14)

$\left({\Delta }^2+4k^2\frac{{\partial }^2}{\partial {Х}^2}\right)S^{\text{'}}=-2k^2\left(\Delta n_2-2k\frac{\partial n_1}{\partial X}\right).$ (15)

We consider the turbulent pulsations of the refractive index to be a real value so that $n_2=0.$ Therefore, the previous equations are simplified as follows:

$\left({\Delta }^2+4k^2\frac{{\partial }^2}{\partial {Х}^2}\right){\chi }^{\text{'}}=-2k^2\Delta n_1,$ (16)

$\left({\Delta }^2+4k^2\frac{{\partial }^2}{\partial {Х}^2}\right)S^{\text{'}}=4k^3\frac{\partial n_1}{\partial X}.$ (17)

2. Two-point turbulent correlations

Let us consider the two-point correlation relationships between the amplitude pulsations of the electromagnetic wave and the turbulent pulsations of the refractive index of the atmosphere for points X₁ and X₂ along the X-axis (Fig. 1). For this purpose, we derive Eq. (16) for points X₁ and X₂ and multiply these equations, as follows:

$\left({\Delta }^2+4k^2\frac{{\partial }^2}{\partial {Х}_1^2}\right)\left({\Delta }^2+4k^2\frac{{\partial }^2}{\partial {Х}_2^2}\right)\times$ (18)

$\times ⟨{\chi }^{\text{'}}\left(X_1\right){\chi }^{\text{'}}\left(X_2\right)⟩=4k^4{\Delta }^2⟨n_1\left(X_1\right)n_1\left(X_2\right)⟩,$

where the angle brackets denote the spatial averaging.

To solve Eq. (18), we present the two-point correlation functions $B_{nn}=$$⟨n_1\left(X_1\right)n_1\left(X_2\right)⟩$ and $B_{\chi \chi }=$$⟨{\chi }^{\text{'}}\left(X_1\right){\chi }^{\text{'}}\left(X_2\right)⟩$ using the Fourier transform on the Fourier spectra of turbulent fluctuations $F_{nn}\left(\mathbf{\zeta },X_1,X_2\right)$ and fluctuations of the electromagnetic wave amplitude $F_{\chi \chi }\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right),$ as follows:

$B_{nn}=\int e^{-i\zeta \rho }{F}_{nn}\left(\zeta ,X_1,X_2\right)d\zeta ,$ (19)

$B_{\chi \chi }=\int e^{-i{\mathbf{k}}^{\mathbf{\text{'}}}\rho }{F}_{\chi \chi }\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right)d{\mathbf{k}}^{\mathbf{\text{'}}},$ (20)

where ${\mathbf{k}}^{\mathbf{\text{'}}}$ is the wave vector of electromagnetic fluctuations, $\zeta$ is the wave vector of turbulent pulsations, $\rho$ are the radii vectors with the beginning at points on the X-axis in the planes of (Y, Z), whose modules are calculated using the equation $\rho =\sqrt{Y^2+Z^2}$ (Fig. 1).

By substituting Eqs. (19) and (20) into Eq. (18), we obtain the following expression:

$\int \left({\Delta }^2+4k^2\frac{{\partial }^2}{\partial {Х}_1^2}\right)\left({\Delta }^2+4k^2\frac{{\partial }^2}{\partial {Х}_2^2}\right).$ (21)

Operators that do not depend on the integration parameters ${\mathbf{k}}^{\mathbf{\text{'}}}$ and $\zeta$ are introduced under the integration sign.

Using ${\Delta }^2{e}^{-i\zeta \rho }={\zeta }^4{e}^{-i\zeta \rho }$ on the right side of Eq. (21) and ${\Delta }^2{e}^{-i{\mathbf{k}}^{\mathbf{\text{'}}}\rho }={k^{/}}^4{e}^{-i{\mathbf{k}}^{\mathbf{\text{'}}}\rho }$ on the left side, we obtain the following expression:

$\int \left({k^{\text{'}}}^4+4k^2\frac{{\partial }^2}{\partial {Х}_1^2}\right)\left({k^{\text{'}}}^4+4k^2\frac{{\partial }^2}{\partial {Х}_2^2}\right)\times$ (22)

$\times e^{-i{\mathbf{k}}^{\mathbf{\text{'}}}\rho }{F}_{\chi \chi }\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right)d{\mathbf{k}}^{\mathbf{\text{'}}}=$

$=\int 4k^4{\Delta }^2{e}^{-i\zeta \rho }{F}_{nn}\left(\zeta ,X_1,X_2\right)d\zeta .$

At this stage of the analysis, the physical law of the influence of turbulent pulsations on the electromagnetic wave needs to be set. In further transformations, we assume that the correlations of turbulent pulsations are similar to those of electromagnetic wave parameters, which arise from turbulent pulsations. In particular,

$B_{nn}=\mu B_{\chi \chi },$ (23)

i.e.,

$\int e^{-i\zeta \rho }{F}_{nn}\left(\zeta ,X_1,X_2\right)d\zeta =$

$=\mu \int e^{-i{\mathbf{k}}^{\mathbf{\text{'}}}\rho }{F}_{\chi \chi }\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right)d{\mathbf{k}}^{\mathbf{\text{'}}},$

where $\mu$ is a constant scale proportionality factor.

Using Eq. (23), Eq. (22) can be rewritten as follows:

$\int \left({\left(\frac{k^{/2}}{2k}\right)}^2+\frac{{\partial }^2}{\partial {Х}_1^2}\right)\left({\left(\frac{k^{/2}}{2k}\right)}^2+\frac{{\partial }^2}{\partial {Х}_2^2}\right)\times$ (24)

$\times e^{-i{\mathbf{k}}^{\mathbf{\text{'}}}\rho }{F}_{\chi \chi }\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right)d{\mathbf{k}}^{\mathbf{\text{'}}}=$

$=\frac{\mu }{4}\int {\zeta }^4{e}^{-i{\mathbf{k}}^{\mathbf{\text{'}}}\rho }{F}_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right)d{\mathbf{k}}^{\mathbf{\text{'}}}.$

By equating the integrands, we obtain the following expression:

$\left({\left(\frac{{k^{\text{'}}}^2}{2k}\right)}^2+\frac{{\partial }^2}{\partial {Х}_1^2}\right)\left({\left(\frac{{k^{\text{'}}}^2}{2k}\right)}^2+\frac{{\partial }^2}{\partial {Х}_2^2}\right)\times$ (25)

$\times F_{\chi \chi }\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right)=\frac{\mu }{4}{\zeta }^4{F}_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right).$

By solving Eq. (25) under the initial condition $F_{\chi \chi }\left({\mathbf{k}}^{\mathbf{\text{'}}}\mathrm{,0,0}\right)=0$ (Fig. 1), we determine G using Green’s function, as follows:

$F_{\chi \chi }\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right)=$

$=\frac{\mu }{4}{\zeta }^4\underset{0}{\overset{X_1}{\int }}\underset{0}{\overset{X_2}{\int }}G\left(\mathbf{k},{\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2,\upsilon ,\xi \right)F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\upsilon ,\xi \right)d\upsilon d\xi .$ (26)

where X₁ and X₂ are the observation coordinates of the two-point correlations on the X-axis and $\upsilon$ and $\xi$ are the coordinates of the source of the influence of turbulence on the electromagnetic wave (Fig. 1) using the so-called influence function $G(\mathbf{k},{\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2,\upsilon ,\xi ).$

Let us use the following property of the Dirac δ-function:

$F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right)=$ (27)

$=\underset{0}{\overset{X_1}{\int }}\underset{0}{\overset{X_2}{\int }}\delta \left(X_1-\upsilon \right)\delta \left(X_2-\xi \right)F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\upsilon ,\xi \right)d\upsilon d\xi .$

Substituting Eqs. (26) and (27) into Eq. (25), we obtain the following expression:

$\underset{0}{\overset{X_1}{\int }}\underset{0}{\overset{X_2}{\int }}\left({\left(\frac{{k^{\text{'}}}^2}{2k}\right)}^2+\frac{{\partial }^2}{\partial {Х}_1^2}\right)\left({\left(\frac{{k^{\text{'}}}^2}{2k}\right)}^2+\frac{{\partial }^2}{\partial {Х}_2^2}\right)\times$ (28)

$\times G\left(\mathbf{k},{\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2,\upsilon ,\xi \right)F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\upsilon ,\xi \right)d\upsilon d\xi =$

$=\underset{0}{\overset{X_1}{\int }}\underset{0}{\overset{X_2}{\int }}\delta \left(X_1-\upsilon \right)\delta \left(X_2-\xi \right)F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\upsilon ,\xi \right)d\upsilon d\xi .$

From Eq. (28), we obtain the following auxiliary equation:

$\left({\left(\frac{{k^{\text{'}}}^2}{2k}\right)}^2+\frac{{\partial }^2}{\partial {Х}_1^2}\right)\left({\left(\frac{{k^{\text{'}}}^2}{2k}\right)}^2+\frac{{\partial }^2}{\partial {Х}_2^2}\right)\times$ (29)

$\times G\left(\mathbf{k},{\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2,\upsilon ,\xi \right)=\delta \left(X_1-\upsilon \right)\delta \left(X_2-\xi \right).$

If $X_1\ne \upsilon$ and $X_2\ne \xi ,$ i.e., outside the points of influence, and therefore, $\delta \left(X_1-\upsilon \right)$ $\delta \left(X_2-\xi \right)=\mathrm{0,}$ then a particular solution to Eq. (29) can be expressed as follows:

$G\left(\mathbf{k},{\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2,\upsilon ,\xi \right)=$ (30)

$=B\sin \frac{{k^{\text{'}}}^2\left(X_1-\upsilon \right)}{2k}\sin \frac{{k^{\text{'}}}^2\left(X_2-\xi \right)}{2k},$

where B is the integration constant.

Thus, the solution to Eq. (25) according to Eq. (26) has the following form:

$F_{\chi \chi }\left({\mathbf{k}}^{\mathbf{\text{'}}},X_1,X_2\right)=\frac{\mu }{4}B{\zeta }^4\underset{0}{\overset{X_1}{\int }}\underset{0}{\overset{X_2}{\int }}\sin \frac{{k^{\text{'}}}^2\left(X_1-\upsilon \right)}{2k}\times$ (31)

$\times \sin \frac{{k^{\text{'}}}^2\left(X_2-\xi \right)}{2k}{F}_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\upsilon ,\xi \right)d\upsilon d\xi .$

Considering that $\chi$ and n are dimensionless quantities, and $F_{\chi \chi }$ and $F_{nn}$ have the same dimension, we select the integration constant B from the condition $B=4.$

If at the wavefront $X_1=X_2=L$ (Fig. 1), then the solution to Eq. (31) has the following form:

$F_{\chi \chi }\left(k^{\text{'}}\right)=\mu {\zeta }^4\underset{0}{\overset{L}{\int }}\underset{0}{\overset{L}{\int }}\sin \frac{{k^{\text{'}}}^2\left(X_1-\upsilon \right)}{2k}\times$ (32)

$\times \sin \frac{{k^{\text{'}}}^2\left(X_2-\xi \right)}{2k}{F}_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\upsilon ,\xi \right)d\upsilon d\xi ,$

where $F_{\chi \chi }\left({\mathbf{k}}^{\mathbf{\text{'}}},L,L\right)=F_{\chi \chi }\left(k^{\text{'}}\right)$ is derived. Notably, a single point is considered along the X-axis to observe the effect of turbulent pulsations on an electromagnetic wave.

For a two-point correlation $B_{SS}=⟨S^{\text{'}}\left(X_1\right)S^{\text{'}}\left(X_2\right)⟩$ characterizing fluctuations in the phase of an electromagnetic wave, Eq. (17) has the derivative $\partial n_1/\partial X$ on the right side. Therefore, in Eq. (30) at the point $X_1=$$X_2=L,$ the sines for the Fourier spectrum of phase fluctuations are replaced by cosines, as follows:

$F_{SS}\left(k^{\text{'}}\right)=\mu {\zeta }^4\underset{0}{\overset{L}{\int }}\underset{0}{\overset{L}{\int }}\cos \frac{{k^{\text{'}}}^2\left(L-\upsilon \right)}{2k}\times$ (33)

$\times \cos \frac{{k^{\text{'}}}^2\left(L-\xi \right)}{2k}{F}_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\upsilon ,\xi \right)d\upsilon d\xi ,$

where $F_{SS}\left({\mathbf{k}}^{\mathbf{\text{'}}},L,L\right)=F_{ss}\left(k^{\text{'}}\right)$ is also derived.

Because it is assumed that $X_1=$$X_2=L,$ a single coordinate of the source of the influence of turbulence on the electromagnetic wave with the influence function G is also logically assumed, i.e., $\upsilon =\xi .$ Consequently, $F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\upsilon ,\xi \right)=$$F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right).$

Eqs. (32) and (33) characterize the relationship between turbulent pulsations of the refractive index of the atmosphere and fluctuations in the amplitude and phase of the electromagnetic wave.

Let us perform some transformations in Eq. (32). We determine the integral of as follows:

$F_{\chi \chi }\left(k\right)=-\mu {\zeta }^4\frac{2k}{k^{/2}}\underset{0}{\overset{L}{\int }}\underset{0}{\overset{L}{\int }}\sin \frac{{k^{\text{'}}}^2\left(L-\varsigma \right)}{2k}d\frac{{k^{\text{'}}}^2\left(L-\varsigma \right)}{2k}\times$ (34)

$\times \sin \frac{{k^{\text{'}}}^2\left(L-\xi \right)}{2k}{F}_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right)d\xi =$

$=2\mu \frac{{\zeta }^{4}k}{{k^{\text{'}}}^2}\underset{0}{\overset{L}{\int }}\left(\sin \frac{{k^{\text{'}}}^2\left(L-\xi \right)}{2k}-\cos \frac{{k^{\text{'}}}^{2}L}{2k}\sin \frac{{k^{\text{'}}}^2\left(L-\xi \right)}{2k}\right)\times$

$\times F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right)d\xi =2\mu \frac{{\zeta }^{4}k}{{k^{\text{'}}}^2}\underset{0}{\overset{L}{\int }}\left(\sin \frac{{k^{\text{'}}}^2\left(L-\xi \right)}{2k}-\right.$

$-\left.\frac{1}{2}\left(-\sin \frac{{k^{\text{'}}}^2\xi }{2k}+\sin \frac{{k^{\text{'}}}^2\left(2L-\xi \right)}{2k}\right)\right)F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right)d\xi .$

Given that L is close to $\xi$ but not equal to $\xi ,$ we can replace

$\sin \frac{{k^{\text{'}}}^2\left(L-\xi \right)}{2k}\approx \frac{{k^{\text{'}}}^2\left(L-\xi \right)}{2k}.$

Notably, the main influence on the electromagnetic wave occurs immediately after it arrives in the turbulent region of volume V (Fig. 1). We use the condition $\xi \to \mathrm{0,}$ therefore,

$\sin \frac{{k^{\text{'}}}^2\xi }{2k}\approx \frac{{k^{\text{'}}}^2\xi }{2k}$ and $\cos \frac{{k^{\text{'}}}^2\xi }{2k}\approx 1.$

Thus, Eq. (34) is transformed into the following form:

$F_{\chi \chi }\left(k^{\text{'}}\right)=2\mu \frac{{\zeta }^{4}k}{{k^{\text{'}}}^2}\underset{0}{\overset{L}{\int }}\left(\frac{{k^{\text{'}}}^2\left(L-\xi \right)}{2k}-\right.$ (35)

$\left.-\frac{1}{2}\left(-\frac{{k^{\text{'}}}^2\xi }{2k}+\sin \frac{{k^{\text{'}}}^2\left(2L-\xi \right)}{2k}\right)\right)F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right)d\xi =$

$=\mu \frac{{\zeta }^{4}k}{{k^{\text{'}}}^2}\underset{0}{\overset{L}{\int }}\left(\frac{{k^{\text{'}}}^2}{k}\left(L-\xi \right)+\frac{{k^{\text{'}}}^2\xi }{2k}-\sin \frac{{k^{\text{'}}}^2}{k}L+\right.$

$+\left.\frac{{k^{\text{'}}}^2\xi }{2k}\cos \frac{{k^{\text{'}}}^2}{k}L\right)F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right)d\xi =$

$=\mu {\zeta }^{4}L\underset{0}{\overset{L}{\int }}\left(1-\frac{\sin \frac{{k^{\text{'}}}^2}{k}L}{\frac{{k^{\text{'}}}^2}{k}L}-\frac{\xi }{2L}\left(1-\cos \frac{{k^{\text{'}}}^2}{k}L\right)\right)\times$

$\times F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right)d\xi .$

Similarly, from the Fourier spectrum of phase fluctuations expressed in Eq. (33), we obtain the following equation:

$F_{SS}\left(k^{\text{'}}\right)=$ (36)

$=\mu {\zeta }^{4}L\underset{0}{\overset{L}{\int }}\left(1+\frac{\sin \frac{{k^{\text{'}}}^2}{k}L}{\frac{{k^{\text{'}}}^2}{k}L}-\frac{\xi }{2L}\left(3+\cos \frac{{k^{\text{'}}}^2}{k}L\right)\right)\times$

$\times F_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right)d\xi .$

The condition $\xi \to 0$ enables us to further simplify Eqs. (35) and (36), as follows:

$F_{\chi \chi }\left(k^{\text{'}}\right)=\mu {\zeta }^{4}L\left(1-\frac{\sin \frac{{k^{\text{'}}}^2}{k}L}{\frac{{k^{\text{'}}}^2}{k}L}\right)\underset{0}{\overset{L}{\int }}{F}_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right)d\xi ,$ (37)

$F_{SS}\left(k^{\text{'}}\right)=\mu {\zeta }^{4}L\left(1+\frac{\sin \frac{{k^{\text{'}}}^2}{k}L}{\frac{{k^{\text{'}}}^2}{k}L}\right)\underset{0}{\overset{L}{\int }}{F}_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right)d\xi .$ (38)

The analysis performed in Ref. [8] enables us to conclude that

$\underset{0}{\overset{L}{\int }}{F}_{nn}\left({\mathbf{k}}^{\mathbf{\text{'}}},\xi \right)d\xi =\pi F_{nn}\left(k^{\text{'}}\right)$

Hence,

$F_{\chi \chi }\left(k^{\text{'}}\right)=\pi \mu {\zeta }^{4}L\left(1-\frac{\sin \frac{{k^{\text{'}}}^2}{k}L}{\frac{{k^{\text{'}}}^2}{k}L}\right)F_{nn}\left(k^{\text{'}}\right),$ (39)

$F_{SS}\left(k^{\text{'}}\right)=\pi \mu {\zeta }^{4}L\left(1+\frac{\sin \frac{{k^{\text{'}}}^2}{k}L}{\frac{{k^{\text{'}}}^2}{k}L}\right)F_{nn}\left(k^{\text{'}}\right).$ (40)

If $\frac{{k^{\text{'}}}^2}{k}L\ll \mathrm{1,}$ , then, using the Taylor series expansion

$\sin \frac{{k^{\text{'}}}^2}{k}L\approx \frac{{k^{\text{'}}}^2}{k}L-{\left(\frac{{k^{\text{'}}}^2}{k}\right)}^3\frac{L^3}{6},$

we obtain the following expressions:

$F_{\chi \chi }\left(k^{\text{'}}\right)=\pi \mu {\zeta }^{4}L\left(1-\frac{1}{\frac{{k^{\text{'}}}^2}{k}L}\left(\frac{{k^{\text{'}}}^2}{k}L-{\left(\frac{{k^{\text{'}}}^2}{k}\right)}^3\frac{L^3}{6}\right)\right)\times$ (41)

$\times F_{nn}\left(k^{\text{'}}\right)=\frac{1}{6}\pi \mu L^3{\left(\frac{{k^{\text{'}}}^2}{k}\right)}^2{\zeta }^4{F}_{nn}\left(k^{\text{'}}\right),$

$F_{SS}\left(k^{\text{'}}\right)=\pi \mu {\zeta }^{4}L\left(1+\frac{1}{\frac{{k^{\text{'}}}^2}{k}L}\left(\frac{{k^{\text{'}}}^2}{k}L-{\left(\frac{{k^{\text{'}}}^2}{k}\right)}^3\frac{L^3}{6}\right)\right)\times$ (42)

$\times F_{nn}\left(k^{\text{'}}\right)=2\pi \mu L{\zeta }^4{F}_{nn}\left(k^{\text{'}}\right).$

Thus, the Fourier spectrum of fluctuations in the amplitude of an electromagnetic wave is proportional to the cube of the distance traveled by the wave in a turbulent medium, proportional to the fourth power of the wave number of wave fluctuations, and inversely proportional to the square of the wave number of the wave. The Fourier spectrum of fluctuations in the phase of an electromagnetic wave is proportional to the first power of the distance traveled by the wave. The dependence of the fluctuation parameters of an electromagnetic wave on the wave number of the turbulence is clarified in the subsequent section.

3. Influence of the energy spectrum of turbulence on the radio signal

The shorter the electromagnetic waves used for a radio signal, the more they are affected by turbulence, which is logical because, at shorter waves, their size approaches the size of turbulent fluctuations that distort the signal.

Eqs. (41) and (42) include the spectral function of pulsations of the wave number of the electromagnetic wave $F_{nn}\left(k^{\text{'}}\right)~F\left(\zeta \right).$ Initially, in Eq. (19), the spectral function $F_{nn}$ used the dependence on the wave number of turbulence. We assume that the wave numbers of turbulent pulsations are similar to the fluctuations in the wave number of an electromagnetic wave $\zeta ~{\mathbf{k}}^{\mathbf{\text{'}}}$ that arise from turbulent pulsations.

Determining the spectral function of turbulent pulsations is a complex and ambiguous task. First, pulsations of the refractive index $n^{\text{'}}$ are determined by pulsations of air pressure $p^{\text{'}}=\rho {u^{\text{'}}}^2$ [9], where $\rho$ is the average air density and ${u^{\text{'}}}^2~E$ is the square of pulsations of the longitudinal air velocity, which is proportional to the turbulence energy [9]. Consequently, $B_{nn}=$$⟨n_1\left(X_1\right)n_1\left(X_2\right)⟩~E\left(\zeta \right).$ The relationship between the functions $F\left(\zeta \right)$ and $E\left(\zeta \right)$ was investigated by many well-known scientists, including Heisenberg [10], Karman [11], Kovazhny [12], and Obukhov [9]. These scientists proposed various communication equations. Stewart and Townsend [13] showed that the function

$\underset{0}{\overset{\zeta }{\int }}F\left(\zeta \right)d\zeta$

represents the product of two factors, the first of which is the integral from $\zeta$ to $\infty$ and the other one is the integral from 0 to $\zeta ,$ irrespective of the exact equation of these factors.

For isotropic turbulence, we limit ourselves to the relatively simple Kovazhny equation:

$\underset{0}{\overset{\zeta }{\int }}F\left(\zeta \right)d\zeta =-2\alpha \left(\underset{\zeta }{\overset{\infty }{\int }}{E}^{\frac{3}{2}}\left(\zeta \right)d\zeta \right)\frac{2}{3}{\zeta }^{\frac{3}{2}},$ (43)

where $\alpha$ is a constant.

In the turbulence spectrum (Fig. 2), as the wave number of turbulence $\zeta$ increases, the size of the turbulent vortices decreases. At the maximum of the turbulence spectrum, there is a region of the so-called energy-containing turbulent vortices. Further along the spectrum of turbulent wave number $\zeta ,$ i.e., smaller vortices, there is the so-called inertial region of the spectrum. For the turbulence energy in the inertial region of the turbulence spectrum (Fig. 2), we employ the equations independently obtained by Kolmogorov [14], Onsager [15], and Weizsäcker [16] to obtain the following expression:

$E\left(\zeta \right)=C{\zeta }^{-\frac{5}{3}},$ (44)

where C is a quantity independent of the wave number of the turbulence $\zeta .$

 

Fig. 2. Spectrum of turbulence. Dependence of the turbulence energy on the wave number of turbulent pulsations

 

The inertial region of wave numbers is characterized by the fact that turbulence in this region is in statistical equilibrium, where the flow of energy from larger to smaller turbulent vortices is determined by the viscous dissipation of the smallest vortices. The smallest vortices are already beyond the inertial region, i.e., in the region described by Heisenberg’s law $~{\zeta }^{-7}.$ The viscous dissipation of turbulent vortices in the inertial region itself is insignificant. Turbulence in the inertial region does not depend on external conditions.

If we substitute Eq. (44) into Eq. (43), then we obtain

$\underset{0}{\overset{\zeta }{\int }}F\left(\zeta \right)d\zeta =\text{const,}$

and therefore, $F\left(\zeta \right)=\mathrm{0,}$ which, naturally, does not reflect the general properties of the spectrum for any turbulence wave number.

To obtain a more complete description of the spectral function of turbulence than that expressed in Eq. (44), we use the result obtained by Kovazhny:

$E\left(\zeta \right)=C_1{\zeta }^{-\frac{5}{3}}{\left(1-{\eta }_1{\zeta }^{\frac{4}{3}}\right)}^2,$ (45)

where C₁ and $\eta$ are quantities independent of the wave number of turbulent pulsations.

By substituting Eq. (45) into Eq. (43), we obtain the following expression:

$\underset{0}{\overset{\zeta }{\int }}F\left(\zeta \right)d\zeta =-2\alpha \left(\underset{\zeta }{\overset{\infty }{\int }}{E}^{\frac{3}{2}}\left(\zeta \right)d\zeta \right)\frac{2}{3}{\zeta }^{\frac{3}{2}}=$ (46)

$=-2\alpha \left(\underset{\zeta }{\overset{\infty }{\int }}{C}_1{\zeta }^{-\frac{5}{2}}{\left(1-2{\eta }_1{\zeta }^{\frac{4}{3}}\right)}^{3}d\zeta \right)\frac{2}{3}{\zeta }^{\frac{3}{2}}\approx$

$\approx -2\alpha \left(\underset{\zeta }{\overset{\infty }{\int }}{C}_1{\zeta }^{-\frac{5}{2}}\left(1-6{\eta }_1{\zeta }^{\frac{4}{3}}\right)d\zeta \right)\frac{2}{3}{\zeta }^{\frac{3}{2}}=$

$=-2\alpha {С}_1\left(\left(\frac{2}{3}\right){\zeta }^{-\frac{3}{2}}-36{\eta }_1{\zeta }^{-\frac{1}{6}}\right)\frac{2}{3}{\zeta }^{\frac{3}{2}}=$

$=-\frac{8}{9}\alpha {С}_1+48\alpha {С}_1{\eta }_1{\zeta }^{\frac{4}{3}}.$

Hence,

$F\left(\zeta \right)\approx 64\alpha {С}_1{\eta }_1{\zeta }^{\frac{1}{3}}.$ (47)

We conclude that, in Eqs. (41) and (42), the dependence on the turbulence wave number has the form ${\zeta }^{4+1/3}=$${\zeta }^{13/3}.$ A detailed comparison of four different spectral equations for turbulence energy is performed in Ref. [17]. A rather strong influence on the electromagnetic wave of the turbulent atmosphere, up to more than the fourth power of the wave number of the turbulence, is noteworthy.

Conclusion

Based on the model of interaction between the deterministic wave process of propagation of electromagnetic waves in the atmosphere and the random wave process of turbulent pulsations of the atmosphere, the occurrence of fluctuations in the eikonal of an electromagnetic wave, which depends on the amplitude and phase of the wave and affects the quality of the transmitted radio signal, has been investigated. The influence of turbulent pressure pulsations in the atmosphere leads to pulsations of the refractive index, which in turn leads to pulsations of the electromagnetic wave parameters, particularly eikonal pulsations.

The resulting differential equation for eikonal fluctuations is split into two interrelated equations. Eq. (1) is for amplitude fluctuations, and Eq. (2) is for phase fluctuations.

Differential equations were derived for two-point turbulent correlations of the refractive index, with parameters depending on amplitude and phase fluctuations.

Using the Fourier transform and Green’s function method, a detailed solution to the differential equation for the parameter associated with the two-point correlation of turbulent fluctuations of the electromagnetic wave amplitude is obtained. A solution to the equation for the parameter associated with the two-point correlation of turbulent fluctuations in the phase of an electromagnetic wave is also presented. In this case, a physical assumption regarding the similarity of two-point correlations of turbulent pulsations and two-point correlations of fluctuations of electromagnetic wave parameters that arise from turbulent pulsations was used.

The Fourier spectrum of the two-point correlation of pulsations of the amplitude of an electromagnetic wave is proportional to the cube of the distance traveled by the wave in a turbulent medium, proportional to the fourth power of the wave number of wave fluctuations, and inversely proportional to the square of the wave number of the wave. The Fourier spectrum of the two-point correlation of phase fluctuations of an electromagnetic wave is proportional to the first power of the distance traveled by the wave.

The analysis performed, as well as the use of the turbulence energy spectrum, revealed that the dependence of the Fourier spectra of the two-point correlations of turbulent pulsations of the amplitude and phase of the electromagnetic wave is proportional to the wave number of turbulence to the power of 13/3.

In conclusion, a turbulent atmosphere has a rather strong influence on the quality of radio wave transmission.

About the authors

Dmitriy S. Klyuev

Povolzhskiy State University of Telecommunications and Informatics

Email: klyuevd@yandex.ru
ORCID iD: 0000-0002-9125-7076

Doctor of Physical and Mathematical Sciences, head of the Department of Radioelectronic Systems, Povolzhskiy State University of Telecommunications and Informatics, Samara, Russia. Author of over 250 scientific papers.

Research interests: electrodynamics, microwave devices, antennas, metamaterials.

Russian Federation, 23, L. Tolstoy Street, Samara, 443010, Russia

Andrey N. Volobuev

Samara State Medical University

Email: volobuev47@yandex.ru
ORCID iD: 0000-0001-8624-6981

Doctor of Technical Sciences, professor of the Department of Medical Physics, Mathematics and Informatics, Samara State Medical University, Samara, Russia. Author of over 400 scientific papers.

Research interests: biophysics, radiophysics.

Russian Federation, 89, Chapayevskaya Street, Samara, 443099, Russia

Sergei V. Krasnov

Samara State Medical University

Email: s.v.krasnov@samsmu.ru
ORCID iD: 0000-0001-5437-3062

Doctor of Technical Sciences, professor, chief of the Department of Medical Physics, Mathematics and Informatics, Samara State Medical University, Samara, Russia. Author of over 100 scientific papers.

Research interests: biophysics, information technologies in medicine, theory of artificial intellect.

Russian Federation, 89, Chapayevskaya Street, Samara, 443099, Russia

Kaira A. Adyshirin-Zade

Samara State Medical University

Email: adysirinzade67@gmail.com
ORCID iD: 0000-0003-3641-3678

Candidate of Pedagogical Sciences, associate professor of the Department of Medical Physics, Mathematics and Informatics, Samara State Medical University, Samara, Russia. Author of over 50 scientific papers.

Research interests: biophysics, radiophysics.

Russian Federation, 89, Chapayevskaya Street, Samara, 443099, Russia

Tatyana A. Antipova

Samara State Medical University

Email: antipovata81@gmail.com
ORCID iD: 0000-0001-5499-2170

Candidate of Physics and Mathematics Sciences, associate professor of the Department of Medical Physics, Mathematics and Informatics, Samara State Medical University, Samara, Russia. Author of over 50 scientific papers.

Research interests: physics, radiophysics.

Russian Federation, 89, Chapayevskaya Street, Samara, 443099, Russia

Natalia N. Aleksandrova

Samara State Medical University

Author for correspondence.
Email: grecova71@mail.ru
ORCID iD: 0000-0001-5958-3851

senior lecturer of the Department of Medical Physics Mathematics and Informatics, Samara State Medical University, Samara, Russia. Author of over 15 scientific papers.

Research interests: biophysics, radiophysics.

Russian Federation, 89, Chapayevskaya Street, Samara, 443099, Russia

References

Supplementary files