SOME PROBLEMS OF IONOSPHERIC PHYSICS

I. ELECTRON DENSITY FLUCTUATIONS AND RADIO WAVE SCATTERING

Ya. L. Al’pert

CONTENTS

§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
§ 2. Effective scattering cross section of the ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
§ 3. Scattering in vertical sounding of the ionosphere; determination of $\overline{\left(\dfrac{\Delta N}{N}\right)^2}$ . . . . . . 434
§ 4. Further propagation of ultrashort waves; values of $\overline{\left(\dfrac{\Delta N}{N}\right)^2}$ and sizes of inhomogeneities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
§ 5. Further propagation of short waves caused by scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
§ 6. Some results of the analysis of experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
§ 7. Turbulence of the ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
Cited literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

§ 1. INTRODUCTION

Even a cursory but attentive acquaintance with the known data shows that, in essence, our knowledge of the physical nature of the ionosphere is very limited, and in literally every experiment we encounter phenomena that we are unable to explain in sufficient depth. Often this is due to fundamental difficulties characteristic of a number of problems arising in the study of the ionosphere and in certain adjacent areas of physics. Up to the present time, as is known, there are, for example, no reasonably complete theories of turbulence, of plasma or magnetohydrodynamic waves, of inelastic collisions in a plasma, etc., which are as necessary for the development of cosmic electrodynamics and for the study of plasmas of another type encountered by modern physics as they are for understanding the phenomena observed in the ionosphere.

Alongside this, since the object of study is a highly rarefied plasma remote from us and, moreover, subject to the influence of many forces that have been little studied, there are also specific difficulties arising, in particular, in the development of experimental investigations of the ionosphere that require the setting up of new experiments.

However, in some cases, especially in solving a number of practical questions and in analyzing various experimental results, we encounter the fact that many of the necessary data, long since fully accessible to measurement both methodically and technically, have until now

unfortunately, are absent simply because the corresponding measurements were not made; at the same time, there are many works that repeat previous studies and merely “detail” well-known facts.

In the present article, in the course of the exposition, some of the gaps in the study of the ionosphere and the problems that arise in this connection will be emphasized; new results will be presented, and it will be shown that in a number of cases it is possible to understand the observed phenomena more closely than can be inferred from the literature data.

Among the problems whose study is of interest for understanding the physical nature of the ionosphere, an important role is played by the question of the statistical nature1 of the ionosphere, connected with the investigation of the macrostructure, or, in other words, the macrophysics of the ionosphere. Certain aspects of this question are the subject of the present article.

Recently, in connection with new experimental results, it has become possible to approach this problem from a somewhat different side—namely, energetically. One of the most interesting and important experimental results of recent years, directly related to this problem, is the results of studying the propagation of ultrashort waves over long distances, caused by the scattering of radio waves in the ionosphere. On more careful examination of the data known from the literature, it is possible, as it seems to us, not only to explain them, but also to obtain from their analysis previously unknown data on the macrostructure of the lower part of the base of the ionosphere.

Along with this, an energetic approach to the results of the corresponding investigations relating to higher regions of the ionosphere makes it possible to show that the previously known experimental data obtained in the reflection of radio waves at frequencies lower than the critical frequency require a new interpretation, and that the question of the sizes of inhomogeneities in these regions of the ionosphere should be considered in a somewhat different light than was done earlier. As a result, apparently, it is possible to obtain an idea of how the fluctuations of electron density change with altitude, what the expected spectra of the sizes of inhomogeneities are, and, perhaps, to come closer to posing the problem of studying the mechanisms that determine the inhomogeneity and nonstationarity of the structure of the ionosphere.

Let us briefly discuss the initial data.

First of all, it should be emphasized that the range of phenomena considered here is analyzed mainly as applied to conditions when the ionosphere should be considered, as is commonly said, “quiet” or weakly disturbed. It is difficult to draw precise boundaries between the concepts of a quiet and a disturbed ionosphere; in the study of various physical quantities, or even when the experimental method is changed, these boundaries may vary greatly, especially since a characteristic feature of the properties of the ionosphere that interest us here is precisely its “unquiet”—variable—state, which is its natural feature. Earlier1, as a criterion of the quiet state of the ionosphere, as applied to the questions considered here, it was assumed that this corresponds to conditions under which, in vertical sounding, single signals are reflected from it as single, unsplit signals (Fig. 1, a) and when, as is said, diffuse reflections are not observed (Fig. 1, b). In the light of the data set forth below, one may speak of a more physical criterion, namely, assume that the quiet state of the ionosphere corresponds to the case when the angular spectrum of vertically reflected signals is narrow, which often means isotropic inhomogeneity of the medium.

Under the indicated conditions it was shown that the region forming the reflected signal, whose linear dimensions are of the order of the Fresnel zone, is inhomogeneous,

the scale of the inhomogeneity is sufficiently small, so that one may speak of a multitude of scattering centers contained in it. The basic experimental fact that led to these conclusions consists in the temporal and spatial variability of the amplitude and phase of the unit magnetically split signal reflected from the ionosphere. It was found that the field at the receiving point is described in the form

\[ E = a_0 \cos(\omega_0 t - \varphi_0) + \]

\[ + \sum_{(s)} a_s \cos \left[(\omega_0 + \Omega_s)t - \varphi_s\right], \tag{1} \]

i.e., it is formed by the specularly reflected wave and by a spectrum of waves with random amplitudes \(a_s\) and phases \((\Omega_s t - \varphi_s)\).

These results inevitably also led to another important conclusion, namely that the inhomogeneities, considered, say, as isolated formations—clouds—move chaotically, which leads to a Doppler frequency shift

\[ \Omega_s = 2\pi \frac{2v_s}{\lambda}, \]

where \(v_s\) is the irregular component of the velocity; at the same time horizontal drifts caused by winds in the ionosphere are also observed. The indicated properties of reflected radio waves showed that the ionosphere should be regarded as a medium in which the formation, disappearance, and re-formation of small-scale inhomogeneities in a state of continuous motion take place. This medium may be characterized as turbid, with degree of turbidity

\[ \beta^2 = \frac{a_0^2}{\sum u_s^2}, \tag{2} \]

Fig. 1. Doublets of magnetically split signals: a) in the case of a quiet ionosphere, b) in the case of diffuse reflections.

which is the ratio of the energy of the specularly reflected wave to the energy of the scattered waves received at the observation point. In the theory of oscillations the quantity \(\beta^2\) is often called the “signal-to-noise ratio.”

Impulse investigations of the ionosphere and their statistical analysis (see \(^{1,2,3,4}\)) made it possible to obtain an idea of the values of the indicated quantities mainly in the \(F\) region of the ionosphere (\(z \sim 250\text{–}400\) km); the results obtained are, in a certain sense, only averages for the \(F\) region as a whole, since there are still not enough data to determine whether these quantities vary with height within this region. At the same time, obtaining the height dependence is very important, in particular for the theoretical study of the mechanisms of these phenomena. Fewer results (with the exception of the value of the velocity \(u\) of horizontal drifts) are available for the \(E\) region (\(z \sim 100\text{–}120\) km), and only in one of the latest works \(^{5}\) have some data been obtained for the lower part of the ionosphere—the \(D\) region (\(z \sim 70\text{–}90\) km).

The results obtained are as follows.

From experiments on vertical sounding of the ionosphere, the linear dimensions \(\xi\) of small-scale irregularities have been determined. They vary within the limits from several tens to several hundreds of meters. The values most often encountered are

\[ \xi_0 \sim 200—300\ \text{m}, \tag{3} \]

and there are uncertain indications that the lower of the indicated values of \(\xi_0\) corresponds to region \(E\), and the upper one to region \(F\).

It should also be noted that, from an analysis of the scintillation of the intensity of Galactic radio sources and of solar radio emission at ultrahigh frequencies, values \(\xi_0 \sim 3—5\ \text{km}\) are obtained. However, from these data it cannot be established exactly at what altitude these irregularities are observed, and whether they do not correspond to the so-called sporadic layer \(F\), i.e., to conditions of an already disturbed ionosphere; therefore it is still difficult to draw any conclusions from these results. Further, recent experiments in the long-radio-wave range\(^5\) (down to frequencies of \(16\ \text{kHz}\)) show that at \(z \sim 100\ \text{km}\) values \(\xi \sim 300—500\ \text{m}\) are generally observed. Along with this, for \(z \sim 90\ \text{km}\) (at a frequency of \(71\ \text{kHz}\)) the same work gives irregularity sizes of \(5—6\ \text{km}\). The latter result is probably connected with the sensitivity of the experimental method, and should hardly be regarded as an indication that at lower ionospheric altitudes the values of \(\xi\) increase.

Data on chaotic velocities likewise do not make it possible to verify whether they differ in different layers. The root-mean-square values \(v_s\) generally vary within the limits from several tenths to ten or fifteen meters per second. There are indications that the values most often encountered are

\[ v_0 \sim (2—5)\ \text{m/sec}. \tag{4} \]

The situation is analogous with the quantity \(\beta^2\), which characterizes the degree of turbidity of the ionosphere. The values of \(\beta^2\) vary under different conditions within the limits from zero to thirty or forty and more; apparently the most probable value is

\[ \beta_0 \sim 2 \div 4, \tag{5} \]

and it has been established that values \(\beta > \beta_0\) occur more often. The distribution curve in \(\beta\) has approximately the form shown in Fig. 2.

Fig. 2: Schematic distribution curve of \(\beta\).

Along with the quantities indicated, from the experimental data one determines the angular spectrum \(\theta_0\) of the wave beam in vertical reflection, characterized by the root-mean-square value of the half-width of the wave beam. For both regions of the ionosphere, values of \(\theta_0\) are obtained which reach ten or more degrees in measurements at high frequencies (reflected from layers \(E\) and \(F\)); the most probable values are

\[ \theta_0 \sim (2 \div 5)^\circ, \tag{6} \]

where \(\theta_0\) is directly proportional to the wavelength; there are still too few data to determine the dependence of \(\theta_0\) on altitude.

Most results concern measurements of the velocities and horizontal drifts, especially in the lower part of the ionosphere\(^4\), where measurements were carried out by various methods (from the reflection of radio waves in the ionosphere, from meteor trails, from luminous clouds, etc.).

The results of these measurements show that, beginning at a height of approximately 80 km and in region \(E\), the most frequently encountered values of the velocity and of the velocity gradient with height are of the order

\[ u_0 \sim 70 \ \mathrm{m/sec}, \qquad \frac{du_0}{dz} \sim 3.6 \ \mathrm{m/sec\cdot km}. \tag{7} \]

There are grounds for concluding that in region \(F\) the velocities are somewhat greater, and the gradients smaller:

\[ u_0 \sim 100 \ \mathrm{m/sec}, \qquad \frac{du_0}{dz} \sim 1 \ \mathrm{m/sec\cdot km}, \tag{8} \]

and that at the top of region \(F\) \((z \sim 450\text{--}550 \ \mathrm{km})\) (according mainly to radio-astronomical observations) the most probable values are \(u \sim 200\text{--}300 \ \mathrm{m/sec}\). True, this last result, as well as the data cited earlier on the scales of inhomogeneities obtained from radio-astronomical observations, may correspond only to a disturbed state of the ionosphere accompanied by increased activity of the Earth’s magnetic field.

The ionospheric parameters considered above do not include, as is readily apparent, one of the principal physical characteristics of its inhomogeneity—namely, the magnitude of the fluctuations of the electrical conductivity, characterized by the relative value of the mean-square deviation of the electron concentration \(N\), that is, by the quantity

\[ \overline{\left(\frac{\Delta N}{N}\right)^2}. \]

This is explained by the fact that, up to the present time, the corresponding results have not been given in the literature. It is shown below how

\[ \overline{\left(\frac{\Delta N}{N}\right)^2} \]

can be determined from the data of various experiments, and the values obtained are presented. In particular, data are used on the long-distance propagation of ultrashort and short waves in the case when it occurs by scattering in the ionosphere. The corresponding analysis naturally requires, as far as possible, an accurate estimate of the energy scattered by inhomogeneities and received at the point of observation; for this, in turn, it is necessary to know the value \(\sigma\)—the effective scattering cross section of the ionosphere. The use, as is often done for calculating the scattering of ultrashort waves in the ionosphere (by analogy with calculations in acoustics\(^{7}\)), of the so-called Booker–Gordon formula\(^{6}\) for \(\sigma\), apparently results from a certain misunderstanding, since in its derivation the spatial correlation coefficient is approximated by the expression

\[ e^{-r/\xi_0}, \]

which is not consistent with the results of ionospheric studies, from which it follows that the approximation is closer to

\[ e^{-\left(r/\xi_0\right)^2} \]

(§ 2). Therefore, in some cases the Booker–Gordon formula leads not only to quantitative but also to fundamental contradictions (see §§ 2, 3). Similarly, in the analysis of experimental data, other quantities characterizing the ionosphere are also incorrectly used\(^{8,9}\), which likewise leads to apparent difficulties.

Another fundamental consideration, which until now has also not been taken into account in analyzing experimental data in the case when waves are reflected from the ionosphere (frequencies below the critical ones or the maximum applicable in oblique propagation), is as follows. It is usually assumed that the scattered waves returned by the ionosphere (the second term in formula (1)) are the result of backscattering by inhomogeneities. Meanwhile, such a view contradicts the physical picture for the following reasons. If only waves were accepted,

scattered backward, then the angular spectrum of these waves would be broad, since their amplitudes vary approximately as \(\dfrac{1}{R}\) (where \(R\) is the distance from the observation point to the scattering center), i.e., sufficiently slowly. Therefore, for sufficiently small values of \(\beta_0^2\), often observed experimentally, the reflected signal would be broadened in time, and its angular spectrum would be broad; at the same time, this does not agree with the results of experiments in a quiet or weakly disturbed ionosphere. If one proceeds from this picture, then an attempt at a quantitative determination of \(\overline{\left(\dfrac{\Delta N}{N}\right)^2}\) (see § 3) would lead to unreasonable values, or to the conclusion that \(\beta_0^2\) should predominantly have very large values \((\beta_0 \to \infty)\), and not the values (5). At the same time, even without special calculations it is easy to understand that in the case when \(\xi_0/\lambda \gg 1\), scattering occurs mainly forward. Corresponding calculations with various formulas for \(\sigma\) show that the ratio of the energy \(P_r(0)\) of waves scattered forward to the energy \(P_r(\pi)\) of backscattered waves, i.e., the quantity \(\dfrac{P_r(0)}{P_r(\pi)}\), must reach, under ordinary experimental conditions, values of order \(10^3\)–\(10^6\) and higher. This indicates that the bundle of scattered waves reaches the observation point in the following way: being scattered forward, the waves are reflected from the upper part of the ionosphere, which does not transmit them, i.e., in the same region where the specularly reflected wave is reflected, and then return to the observation point. If one proceeds from this picture, then the calculations immediately give a narrow angular spectrum (6), non-contradictory values of \(\overline{\left(\dfrac{\Delta N}{N}\right)^2}\), and remove a number of other difficulties.

From what has been said above it is clear that, for a consistent analysis of the questions considered here, it is first of all necessary to calculate and analyze the effective scattering cross section of the ionosphere.

§ 2. EFFECTIVE SCATTERING CROSS SECTION OF THE IONOSPHERE

If an isolated body is placed in the field of an electromagnetic wave, the electrical properties of which are characterized by a known spatial function \(\varepsilon\) of the complex dielectric constant, then, in the general case, in order to calculate the structure of the field varying under the influence of this body, it is necessary to solve the corresponding diffraction problem. Usually the solution of such problems, even for bodies of the simplest shape, involves great computational difficulties. However, in a number of cases it is possible, using various considerations, often geometrical-optical ones, to calculate approximately the secondary field (the scattering field) excited by this body, or to obtain semiempirical formulas characterizing the properties of this field.

The scattering field is conveniently characterized by the effective scattering cross section

\[ \sigma(\theta)=\frac{P_r}{p_0}, \tag{9} \]

equal to the ratio of the energy \(P_r\), scattered by the given body in the direction making an angle \(\theta\) with the direction of the wave incident on the body, to the energy \(p_0=\dfrac{c}{4\pi}[\mathbf{E}\mathbf{H}]\) of the latter. As is seen from (9), the effective cross section, by definition, depends on the electrical and geometrical properties of the body, on the angle \(\theta\), and has the dimension of area.

In the case, however, when the waves are scattered not by isolated particles, but by their aggregate, i.e., by the whole medium as a whole, the electrical properties of which

inhomogeneous, in particular, may vary from point to point, another approach to the concept of an effective cross section is necessary. A rigorous calculation of the wave field in such a medium often leads to still more complicated problems, since additional fundamental difficulties arise, due to the complexity of the physical processes occurring in such media. The physics of the processes responsible for scattering is included in the concept of the effective cross section, which, however, is already defined as a quantity characterizing the ratio of the energy scattered by a unit volume into a unit solid angle to \(p_0\).

Thus, in the present case \(\sigma\) has the dimension \(\dfrac{1}{\text{cm},\ \text{steradian}}\), and

\[ \frac{P_r}{p_0} = \sigma(\theta)\, dV \cdot d\Omega, \tag{10} \]

where \(d\Omega\) and \(dV\) are, respectively, the elements of solid angle and volume, and it is assumed that \(dV\) has linear dimensions small in comparison with the scale of variation of the electrical properties of the medium—the scale of the inhomogeneity \(\xi\)—and with the wavelength. It goes without saying that the whole volume \(V\) in which scattering is considered is assumed to be large in comparison with \(\xi^3\).

If the inhomogeneity of the medium is described, as in the ionosphere, by an irregular function, then in calculating the effective cross section one uses the statistical characteristics of the medium—the spatial correlation coefficient of the fluctuations of the electrical parameters of the inhomogeneities, i.e., of the deviations \(\Delta\varepsilon\) from the mean value of the dielectric constant—and until it becomes possible to approach the corresponding calculation from the physical side, i.e., on the basis of an analysis of the mechanisms responsible for the phenomena under consideration, the results of such a calculation are sufficiently general and adequate to the phenomena being studied. At the same time, however, it is important to know sufficiently well from some data (mainly from the results of measurements) the correlation function, equal to

\[ \overline{(\Delta\varepsilon)(\Delta\varepsilon')^{*}_{r}} = \int_{(V)} (\Delta\varepsilon)(\Delta\varepsilon')^{*}\, dV, \tag{11} \]

where \(\Delta\varepsilon\) and \(\Delta\varepsilon'\) are, respectively, the deviations of \(\varepsilon\) from its mean value \(\overline{\varepsilon}\) at two neighboring points mutually separated by a distance \(r\). Usually, in calculations of this kind, the correlation coefficient is used, defined by the formula (if \(\overline{\Delta\varepsilon}=0\))

\[ \rho(r)= \frac{\displaystyle \int_{(V)}(\Delta\varepsilon)(\Delta\varepsilon')^{*}\, dv} {V\cdot \overline{\Delta\varepsilon^{2}}}, \tag{12} \]

where

\[ V\cdot \overline{\Delta\varepsilon^{2}} = \int_{(V)} \Delta\varepsilon^{2}\, dv. \tag{13} \]

In formulas (11) and (12) the sign “\(*\)” denotes complex conjugation. Since for our purposes it is sufficient to take into account only the real part of the refractive index, this sign will be omitted in the final formulas.

Let now, at some point \(P\) of the volume \(V\) (Fig. 3), the field of the incident electromagnetic wave be equal to

\[ E_0 e^{i(\omega t-kR_0)}, \tag{14} \]

where \(R_0\) is the distance from the point \(P\) to the source of radiation, \(\omega\) is the angular frequency

waves, and \(k=k_0\sqrt{\bar{\varepsilon}}=\dfrac{\omega}{c}\sqrt{\bar{\varepsilon}}=\dfrac{2\pi}{\lambda}\) is the wave number. Written in this form, the expression for the field is not sufficiently general. Indeed, depending on whether the wave is a monochromatic or quasi-monochromatic group of waves—an impulse—it is necessary, taking into account the dependence of the mean value \(\bar{\varepsilon}\) on the coordinate of the point \(P\), to write, instead of \(kR_0\), the phase or group path of the wave in the form

\[ \frac{\omega}{c}\int_0^{R_0}\sqrt{\bar{\varepsilon}}\,dR \quad \text{or} \quad \frac{\omega}{c}\int_0^{R_0} \frac{d\left(\sqrt{\bar{\varepsilon}}\cdot\omega\right)}{d\omega}\,dR, \tag{15} \]

where

\[ \frac{c}{\dfrac{d\left(\sqrt{\bar{\varepsilon}}\cdot\omega\right)}{d\omega}} =\frac{d\omega}{dk} \]

is the group velocity. Since, however, the expression \(kR_0=\dfrac{2\pi R_0}{\lambda}\) is used here, it is thereby assumed that instead of (15) one may use a certain effectively introduced wavelength, which, with allowance for the propagation of the wave over the entire path from the point of radiation to the point of observation, is on the average close to the value \(\lambda\) for \(\bar{\varepsilon}=1\). Such an assumption, of course, is entirely legitimate in calculations in the ionosphere when the frequencies \(\omega\) are far from the values of the critical frequencies \(\omega_c\); it may lead, however, to discrepancies in the results of calculation if \(\omega\sim\omega_c\). In some cases the effective wavelength may differ from \(\lambda_0=\dfrac{2\pi c}{\omega}\) by several times. We do not consider here what consequences this leads to in various concrete cases.

Fig. 3. Toward the derivation of the formula for the effective scattering cross section.

Thus, if the field of the incident wave is equal to (14), then the dipole moment caused by the field at the point \(P\) is

\[ \Delta p=\frac{\Delta\varepsilon}{4\pi}\,E_0 e^{i(\omega t-kR_0)}\,dV, \tag{16} \]

respectively, the field vectors excited at the point of observation by each elementary volume dipole are equal to

\[ \begin{aligned} \Delta E&=-\,k_0^2\Delta p\,\frac{e^{-ikR}}{R}\sin\psi,\\ \Delta H&=\sqrt{\bar{\varepsilon}}\,\Delta E, \end{aligned} \tag{17} \]

the total field

\[ E=\frac{k_0^2}{4\pi}\int_{(V)} \frac{E_0\Delta\varepsilon}{R}\,\sin\psi\, e^{i[\omega t-k(R_0+R)]}\,dV \tag{18} \]

and the complex energy density at the point of observation is equal to

\[ \frac{c}{4\pi}EH^* = \frac{c\sqrt{\bar{\varepsilon}}}{4\pi}\, \frac{k_0^4}{(4\pi)^2} \iint (\Delta\varepsilon)(\Delta\varepsilon')^* \frac{EE_0'\sin\psi\sin\psi'}{RR'} \times \]

\[ \times e^{-ik[(R-R')+(R_0-R_0')]} \, dV\,dV', \tag{19} \]

where the index \((\,{}'\,)\) indicates that the values of the corresponding quantities are taken at the point \(P'\), separated from the point \(P\) by a distance \(r\). In expression (19) the integral can be rewritten in the form

\[ \begin{aligned} I &= \int_{(V)} \int_{(V')} (\Delta \varepsilon)(\Delta \varepsilon')^{*} \frac{E_{0}E_{0}'}{RR'} \sin \psi \sin \psi'\, e^{-ik[(R-R')+(R_{0}+R')]} \, dV\,dV' \\ &= \int_{(V)} \frac{E_{0}E_{0}'}{RR'} \sin \psi \sin \psi'\, e^{-ik[(R-R')+(R_{0}-R_{0}')]} \, dV \int (\Delta \varepsilon)(\Delta \varepsilon')^{*}\,dV \end{aligned} \tag{20} \]

or, using (12), we obtain:

\[ I = V\overline{\Delta \varepsilon^{2}} \int \rho(r)\, \frac{E_{0}E_{0}'}{RR'} \sin \psi \sin \psi'\, e^{-ik[(R_{0}-R_{0}')+(R-R')]} \, dV', \tag{21} \]

where \(\rho(r)\)—the correlation coefficient of the quantity \(\Delta \varepsilon\)—is assumed to depend only on the coordinate \(r\), measured relative to some point chosen arbitrarily in the volume \(V\); thus an isotropic irregularity of the medium is allowed.

In order to calculate the integral (21), it is necessary, first of all, to establish the form of the function \(\rho(r)\).

Naturally, so long as there is no theory of the mechanisms of the phenomena that give rise to fluctuations, the choice of the corresponding function can be made mainly on the basis of experimental data. The results of ionospheric investigations have shown, at any rate in the range of short and ultrashort waves of interest to us in the present paper, that the frequency and angular spectra of the scattered waves are narrow (see (4), (1a), and (6)) and, owing to the randomness of these quantities, the spectra are also symmetric. This means that the energy spectra of the oscillations received (1) are described by a Gaussian function, namely:

\[ \left. \begin{aligned} W(\omega_{0}+\Omega) &= \frac{W_{0}}{\sqrt{2\pi\sigma_{\Omega}}}\, e^{-\frac{\Omega^{2}}{2\sigma_{\Omega}^{2}}}, \\[6pt] W(\theta) &= \frac{W_{0}}{\sqrt{2\pi\theta_{0}}}\, e^{-\frac{\theta^{2}}{\theta_{0}^{2}}}, \end{aligned} \right\} \tag{22} \]

where \(\omega_{0}\), the carrier, is the central frequency of the spectrum and

\[ \left. \begin{aligned} W_{0} &= \sum a_{s}^{2}, \qquad \overline{\theta^{2}}=\theta_{0}^{2}, \\[6pt] \sigma_{\Omega}^{2} &= \overline{(\omega_{s}-\omega_{0})^{2}} = \overline{\Omega^{2}} = \Omega_{0}^{2} = 4\pi^{2}\frac{4v_{0}^{2}}{\lambda^{2}} . \end{aligned} \right\} \tag{23} \]

If one uses the indicated properties of the energy spectra and the known theorem expressing the correlation coefficient of random oscillations in terms of their energy spectra (see 10), it follows directly that the temporal and spatial correlation coefficients of the oscillations

\[ \sum a_{s}\cos[(\omega_{0}+\Omega_{s})t-\varphi_{s}] \]

(see (1)) have the form

\[ \rho(\tau)\sim e^{-\frac{\tau^{2}}{\tau_{0}^{2}}}, \qquad \rho(r)\sim e^{-\frac{r^{2}}{\xi_{0}^{2}}}, \tag{24} \]

where \(\tau_{0}\) and \(\xi_{0}\) correspond to the root-mean-square values of the “periods” of the frequency spectrum \(\left(\tau_{0}\sim \dfrac{2\pi}{\Omega_{0}}\right)\) and of the sizes of the inhomogeneities \(\left(\xi_{0}^{2}=\overline{\xi^{2}}\right)\)

(see, for example, \(^{1}\)). The experimental data known in the literature, based mainly on analysis of the field amplitude at a single point, show that formula (24) for \(\rho(\tau)\) is confirmed quite well over wide ranges. There are fewer data verifying \(\rho(r)\), since this is connected with the need for more complicated simultaneous observations at many points. However, from the available experimental results, and also from the similarity of the derivation of \(\rho(r)\) and \(\rho(\xi)\), based on the narrowness and symmetry of the energy spectrum, one may conclude with considerable justification that formula (24) rather well approximates the behavior of the spatial correlation coefficient in the ionosphere, if one starts from the fact (see \(^{10}\)) that the diffraction pattern at the earth’s surface is similar to the diffraction pattern near the rough “screen” (the ionosphere). Let us now substitute formula (24) for \(\rho(r)\) into the integral (21). In doing so, we note that, since \(\rho(r)\) is a rapidly decreasing function of \(r\), the integral may be extended to infinity and, moreover, the factor

\[ \frac{E_0 E_0' \sin\psi \sin\psi'}{RR'} \simeq \frac{E_0^2 \sin^2\psi}{R^2}, \tag{25} \]

which changes little over the distances at which the value of \(e^{-r^2/\xi^2}\) practically differs from zero, may be taken outside it, since under real conditions \(\xi_0 \ll R\). As a result, instead of (19) we obtain that the complex energy density at the point of observation is equal to

\[ \frac{c}{4\pi}EH^* = \frac{c\sqrt{\bar{\varepsilon}}\,E_0^2}{4\pi R^2} \cdot \left(\frac{k_0^2 \sin^2\psi}{4\pi}\right)^2 \overline{\Delta \varepsilon^2} \int_0^\infty e^{-r^2/\xi_0^2} \cdot e^{-ik\left[(R_n-R_0')+(R-R')\right]}\,dV. \tag{26} \]

Taking further into account that \(P_r=R^2\dfrac{c}{4\pi}\operatorname{Re}(EH^*)\) and \(p_0=\dfrac{c\sqrt{\bar{\varepsilon}}}{4\pi}E_0^2\), we obtain, after certain transformations of the real part of (26), using (10),

\[ \sigma = \left(\frac{k_0^2\sin\psi}{4\pi}\right)^2 \frac{4\pi\overline{\Delta\varepsilon^2}}{2k_0\sin\dfrac{\theta}{2}} \int_0^\infty e^{-\left(\dfrac{r}{\xi_0}\right)^2} \cdot r\sin\left(2k\sin\frac{\theta}{2}\,r\right)\,dr \tag{27} \]

or, taking for the ionosphere

\[ \varepsilon=1-\frac{4\pi Ne^2}{m\omega^2}=1-\frac{\omega_N^2}{\omega^2}, \]

\[ \sigma = \left(\frac{\Delta N}{N}\right)^2 \left(\frac{\omega_N}{\omega}\right)^4 \frac{\sqrt{\pi}}{8\lambda} \left(\frac{2\pi\xi_0}{\lambda}\right)^3 \sin^2\psi e^{-\left(\dfrac{2\pi\xi_0}{\lambda}\sin\dfrac{\theta}{2}\right)^2}. \tag{28} \]

In deriving (28), the influence of the earth’s magnetic field and the number of collisions in the expression for \(\varepsilon\) were not taken into account. It should also be noted that in formula (28) it would seem to be artificially written that \(\sigma\sim\omega_N^4\sim N^2\), since one could cancel \(N^2\) with the denominator of the term \(\dfrac{\overline{\Delta N^2}}{N^2}\). However, physically this particular way of writing is more correct; since \(\Delta N\) must be proportional to \(N\), it is more expedient to introduce its relative value.

From formula (28) for \(\sigma\) it is immediately evident that its most essential factor is the exponential term, which shows,

that the predominant part of the energy is scattered within the angular aperture

\[ \sin \frac{\theta}{2} \sim \frac{\lambda}{2\pi \xi_0}. \tag{29} \]

As \(\xi_0/\lambda\) increases, the value of \(\sigma\) decreases very rapidly, and the scattering characteristic acquires an extremely sharp forward directivity (at \(\theta \sim 0\)) already for \(\xi_0 \sim \lambda\). As an illustration, Fig. 4 gives two directivity diagrams for different values of \(\xi_0/\lambda\).

Fig. 4. Angular characteristics of the effective scattering cross section of the ionosphere for two different values of \(\xi_0/\lambda\).

Fig. 5. Dependence of the effective scattering cross section on \(\xi_0/\lambda\) for various values of \(\theta\).

For given values of the angle \(\theta\) (which is determined by the position of the observation point) and of the wavelength \(\lambda\), the optimal values of the dimensions of the inhomogeneities \(\xi_0\), which determine the maximum scattering, are found from the relation \(\frac{d\sigma}{d\xi_0}=0\), whence one obtains

\[ (\xi_0)_M = \frac{\sqrt{3}\,\lambda}{2\pi \sqrt{2}\,\sin \frac{\theta}{2}} . \tag{30} \]

From Fig. 5, in which the curves of the dependence of \(\sigma\) on \(\frac{\xi_0}{\lambda}\) are shown for several values of \(\theta\), it is seen that for \(\xi \gtrless (\xi_0)_M\) \(\sigma\) decreases rapidly, especially with increasing \(\theta\).

The indicated properties of \(\sigma\) lead to the following physical picture of scattering. If the medium consists of inhomogeneities of various sizes, then the greatest part of the energy arriving at the observation point, forming an angle \(\theta\) with the direction of the incident wave, is due to inhomogeneities whose sizes lie in the vicinity of \((\xi_0)_M\), provided that the waves scattered forward \((\theta \sim 0)\) cannot find other paths of arrival at the receiving point. If

the reception at the observation point is not limited by the value of the angle \(\theta\), as, for example, in the case of vertical sounding of the ionosphere at frequencies below the critical frequencies, when waves scattered forward \((\theta \sim 0)\) can be reflected from the upper regions of the ionosphere, the main role is then played by inhomogeneities of the largest size. In this case the aperture angle of the scattered waves (their angular spectrum), determined by relation (30), assumes the smallest possible value. Thus, in this case the reception of scattered waves is due not to scattering in the backward direction toward the observation point \((\theta \sim \pi)\), as is usually assumed, but to forward scattering \((\theta \sim 0)\).

Further, from formula (28) it is seen that, other conditions being equal, the scattering energy is proportional to

\[ \left(\frac{\omega_N}{\omega}\right)^4, \]

i.e. it increases rapidly with increasing electron concentration \(N\). Therefore, when irradiating a scattering medium in which the spectrum of the sizes of the inhomogeneities is approximately the same at all points, but \(N\) changes, the greatest part of the scattered energy must correspond to regions with maximum values of \(N\). This circumstance should be borne in mind in analyzing some results of measurements of radio-wave scattering by the ionosphere (see § 4), in which, beginning from the \(D\) region \((z \sim 80\ \text{km})\) up to the maximum of the \(F\) region \((z \sim 400\ \text{km})\), the value of \(N\) changes by approximately \(10^3\) times, so that \(\sigma\) must change by \(10^6\) times.

From what has been said it is clear that a detailed analysis of experimental data on the scattering of radio waves in the ionosphere, on the basis of the expression obtained above for \(\sigma\) and the physical picture of the phenomena occurring thereby that follows from its consideration, may make it possible to obtain data on the sizes of the inhomogeneities \(\xi_0\), the fluctuations of the electron density \(\overline{\left(\frac{\Delta N}{N}\right)^2}\), etc. In doing so, however, careful treatment of the various ionospheric data is required*).

In the following paragraphs the results of a corresponding analysis of various experimental data are given.

§ 3. SCATTERING IN VERTICAL SOUNDING OF THE IONOSPHERE; DETERMINATION OF \(\overline{\left(\frac{\Delta N}{N}\right)^2}\)

The results of the preceding paragraph show that, in vertical sounding of the ionosphere, up to frequencies \(\omega \leqslant \omega_c\), where \(\omega_c\) is the critical frequency of the corresponding region of the ionosphere, the field at the observation point (see (1)) is formed in the following way. Since usually

\[ \frac{2\pi \xi_0}{\lambda} \gg 1 \]

(see (3)), the scattering occurs mainly forward—in the direction of incidence of the wave; the scattered waves are then reflected in the region where the refractive index \(p\) is equal to zero

\[ \left(1-\frac{4\pi N e^2}{m\omega^2}\right), \]

and then arrive at the receiving point.

*) Let us note here that the authors of papers \(^{8}\) and \(^{11}\), in estimating the energy scattered by the ionosphere at ultrashort waves, in a number of cases use ionospheric data insufficiently carefully and consistently. At the same time, in \(^{8}\), as also in \(^{9}\), an expression for \(\sigma\) is adopted which was obtained in \(^{6}\) under the assumption that \(\rho \sim e^{-r/l}\). Such an approximation for \(\rho\) does not correspond to the experimental data on the ionosphere, quite apart from the fact that it is in principle unsuitable (see, for example, \(^{12}\)), since the derivative \(d\rho/d\xi\) is different from zero at \(\rho=0\), and this leads to a discontinuity of the refractive-index indicator at this point. Application of this formula and other inconsistencies in the indicated papers lead to disagreement with experiment by a factor of 1000 and more (see \(^{8,11}\)); (see the note on p. 442).

Let us calculate, using formula (28) for \(\sigma\), the ratio of the energy \(\sum_s a_s^2\) of the waves scattered by the entire thickness of the ionosphere and arriving at the observation point, to the energy \(a_0^2\) of the specularly reflected wave, i.e. the value \(\beta^{-2}\) (see (2)). For this purpose we assume that the electron concentration in the corresponding region of the ionosphere varies with height \(z\) according to a parabolic law (which, as is known, approximates the dependence \(N(z)\) quite well), i.e. that

\[ N(z)\sim \frac{\omega_N^2}{\omega^2} = \frac{\omega_c^2}{\omega^2} \left[ 1-\left(\frac{z_m+z_0-z}{z_m}\right)^2 \right], \tag{31} \]

where \(z_0\) is the height of the base of the parabolic layer, and \(z_m\) is its semi-thickness. For such a dependence \(N(z)\), the height \(z_s\) of reflection of a wave of frequency \(\omega \leqslant \omega_c\) is equal (for \(\varepsilon=0\)) to

\[ z_s=z_0+z_m(1-m),\qquad m=\sqrt{1-\left(\frac{\omega}{\omega_c}\right)^2}. \tag{32} \]

If the attenuation of the wave in the layer is neglected, i.e. if, as in the preceding paragraph, the number of collisions is taken to be zero—which in the problem under consideration does not lead to significant changes in the results of the calculations—and, owing to the narrowness of the angular spectrum \(\theta\) of the scattered waves (see (6)), the curvature of the Earth and of the ionosphere is also neglected, then one may write that

\[ d\left(\sum_s a_s^2\right) = \frac{P_i}{(4\pi z^2)^2}\,\sigma\, dV, \tag{33} \]

where \(P_i\) is the energy radiated by the antenna (it is assumed that the receiving and transmitting antennas are identical and nondirectional),

\[ \left. \begin{aligned} dV &\simeq \pi \left(\frac{z\theta_0}{2}\right)^2 dz,\\ a_0^2 &\simeq \frac{P_i}{16\pi z_s^2}. \end{aligned} \right\} \tag{34} \]

As a result, using (28), one obtains

\[ \frac{1}{\beta_0^2} = \frac{\sum a_s^2}{a_0^2} = \]

\[ = \frac{\pi\sqrt{\pi}}{4e} \frac{\lambda^2 \xi_0}{\lambda_c^4} (\alpha+1-m)^2 \int_{z_0}^{z_0+z_m(1-m)} \left(\frac{\Delta N}{N}\right)^2 \left[ 1-\left(\frac{z_m+z_0-z}{z_m}\right)^2 \right]^2 \frac{dz}{z^2}, \tag{35} \]

where

\[ \alpha=\frac{z_0}{z_m},\qquad \lambda_c=\frac{c}{\dfrac{\omega_c}{2\pi}},\qquad m=\sqrt{1-\left(\frac{\lambda_c}{\lambda}\right)^2} \tag{36} \]

and in the derivations it has been assumed that

\[ \theta_0=\frac{\lambda}{2\pi \xi_0}. \]

Taking outside the integral (35) the quantity \(\overline{\left(\dfrac{\Delta N}{N}\right)^2}\) and thereby assuming that as a result of the calculations there will be obtained a certain averaged value of it in the region of heights where the value of

Fig. 6. Dependence of \(\left\{\beta^2\left(\dfrac{\Delta N}{N}\right)^2\right\}\) on \(\dfrac{\lambda_c}{\lambda}\) for region \(E\) of the ionosphere.

Fig. 7. Dependence of \(\left\{\beta^2\left(\dfrac{\Delta N}{N}\right)^2\right\}\) on \(\dfrac{\lambda_c}{\lambda}\) for region \(F\) of the ionosphere.

integral (35), we ultimately obtain a formula determining \(\overline{\left(\dfrac{\Delta N}{N}\right)^2}\) as a function of the wavelength, the quantity \(\beta_0\), and other parameters of the ionosphere:

\[ \overline{\left(\frac{\Delta N}{N}\right)^2} = \frac{4e}{\pi\sqrt{\pi}}\, \frac{\lambda_c^4}{z_m\lambda^3(a+1-m)^2\beta_0^2\cdot M}, \tag{37} \]

where \(\left(\text{taking } \zeta=\dfrac{z}{z_m}\right)^{*}\)

\[ \begin{aligned} M &= \int_{a}^{a+1-m} \frac{d\zeta}{\zeta^2} \left[1-(1-a-\zeta)^2\right]^2 \\ &= \frac{1-m}{a+1-m} \left\{ 4a(a+1)(a+2) +2(1-m)(a+1)(a+2) -\frac{2}{3}(1-m)^2(a+3) +\frac{(1-m)^3}{3} \right\} \\ &\quad -4a(a+1)(a+2)\ln\left(1+\frac{1-m}{a}\right). \end{aligned} \tag{38} \]

In Figs. 6 and 7, for several values of the ionospheric parameters typical of the \(E\) and \(F\) regions, curves are given characterizing the dependence of

\[ \left\{\beta_0^2\,\overline{\left(\frac{\Delta N}{N}\right)^2}\right\} \]

on the ratio

\[ \frac{\omega}{\omega_c}=\frac{\lambda_c}{\lambda}, \]

calculated from formulas (37) and (38). If these curves are used and the value \(\beta_0^2\sim 10\) is adopted (see (5)), taking into account that the results of the known\({}^{1}\) measurements of \(\beta_0\) refer predominantly to the case \(\omega\sim(0.8—0.9)\omega_c\), when a doublet of magnetically split signals is usually observed and the necessary purity of the experiments is ensured, then it follows that in the \(E\) region \((z\sim 100—130\ \text{km})\)

\[ \sqrt{\overline{\left(\frac{\Delta N}{N}\right)^2}} \sim (1\div 4)\cdot 10^{-2} \tag{39} \]

and in the \(F\) region \((z\sim 250—400\ \text{km})\)

\[ \sqrt{\overline{\left(\frac{\Delta N}{N}\right)^2}} \sim (0.3\div 1)\cdot 10^{-2}. \tag{40} \]

The estimates made show that the relative values of the fluctuations of the electron concentration apparently do not change greatly, although a tendency toward their decrease is observed, on passing from one region of the ionosphere to another.

Naturally, these data are preliminary in character. However, it is evident that detailed use of the method described here makes it possible to determine not only the quantity \(\overline{\left(\dfrac{\Delta N}{N}\right)^2}\), but also to investigate its dependence on height and on various experimental conditions. In this connection, since the factor \([1-(1-a+\zeta)^2]^2\) in integral (38) is always less than unity and rapidly tends to unity as one approaches the upper limit of the integral, corresponding to the reflection height \(z_s\) of the wave (see (30)), where \(\varepsilon=0\), the value

\({}^{*}\) Let us note here that for ionospheric data observed experimentally, formula (38) is usually the difference of two large quantities of equal order, and therefore careful handling of it is required in numerical calculations. It should also be borne in mind that formula (37) is unsuitable when \(\beta_0=0\), i.e., when there is no mirror-reflected wave and it is no longer possible to speak simply of fluctuations of the electron density.

the integral depends mainly on its upper limit. This is physically explained by the fact that \(\sigma \sim \omega_N^4 \sim N^2\), and only a small region \(\Delta z\) of the layer in the neighborhood of \(z=z_{\theta}\), where \(\omega_N^4\) reaches its maximum value, effectively participates in the scattering. Estimates show that at some frequencies \(\Delta z\) constitutes only a small fraction of the half-thickness of the layer \(z_m\), and therefore the values

\[ \overline{\left(\frac{\Delta N}{N}\right)^2}, \]

determined by formulas (37) and (38), are mean values for a small range of heights \(z\). This shows that, with an appropriate arrangement of experiments, it will be possible to study in sufficient detail the altitude dependence of

\[ \overline{\left(\frac{\Delta N}{N}\right)^2} \]

within a single layer.

§ 4. FURTHER PROPAGATION OF ULTRASHORT WAVES; VALUES OF \(\overline{\left(\frac{\Delta N}{N}\right)^2}\) AND SIZES OF INHOMOGENEITIES

In the last five or six years, much attention has been attracted by the discovery of a new, fairly stable type of radio transmission of ultrashort radio waves in the range from \(2\)–\(3\) to \(10\)–\(12\) m beyond the limits of the visible horizon, over distances reaching one and a half to two thousand and more kilometers. It was established many years ago that, in a number of cases, waves \(7\)–\(10\) m long, considerably shorter than the minimum usable wavelengths for which the ionosphere becomes transparent, cover very great distances: thus, for example, cases were known when television transmissions in England and Germany were received in America and Africa, etc. However, these phenomena attracted due attention only recently\(^8\), after which their systematic study began, as well as their use for practical purposes\(^9\).

It became clear that the principal mechanism responsible for the transfer of appreciable energy of ultrashort radio waves over large distances is scattering by inhomogeneities of the ionosphere\(^8\), meteor trails\(^13\), and, on a number of paths, by ionized formations of the aurora borealis\(^14\). The participation of meteor ionization, along with the ionosphere, in the long-distance propagation of ultrashort waves (u.s.w.) manifests itself in two of its main features.

Fig. 8. Daily variation of the intensity of ultrashort waves at a frequency of \(49.8\) MHz and a distance of \(1243\) km (April 1951\(^8\)).

First, in the diurnal variation of intensity there is observed a midday maximum, corresponding to the maximum ionization of the ionosphere, and a morning maximum, coinciding with the time of maximum meteor activity (Fig. 8).

Second, the amplitude of the received waves, always representing an irregularly varying function of time (the sum of random oscillations—see (1)), has a smoother form around midday (Fig. 9, a), when the field is due chiefly to the ionosphere, and becomes more irregular, with finer periods, when the “meteor component” of the field is comparable with, and at individual moments predominates over, the “ionospheric” component of the field (Fig. 9, b), and bursts—flares—of field intensity are observed.

Up to the present time there have been no data that would make it possible to estimate the ratio of the intensities of the two components at different times of day. There is, however, sufficient basis for concluding that in po

loudened hours the ionospheric component of the field considerably exceeds the meteoric one. Therefore, in analyzing the role of the ionosphere in the propagation of ultra-short waves, one should first of all consider the results of measurements at this time of day.

In the light of the questions of interest to us here, if one uses the results of analyses of the time variation of the amplitudes of the ionospheric component, given in various works (see, for example, \(^{15}\)) and showing that variations in the frequency spectrum of the received oscillations change within the limits from tenths of a hertz to several hertz, then with the aid of (1a) one obtains velocities of chaotic motions

\[ \left. \begin{aligned} v_s &\sim 0.2 \div 1.2\ \text{m/sec},\\ v_0 &\sim 2\ \text{m/sec}, \end{aligned} \right\} \tag{41} \]

i.e., close to the values given above (see (4)).

Further, from the series of experiments described in the literature it follows that the height \(z\) of scattering of ultra-short waves in the daytime is predominantly equal to \(75\text{--}80\ \text{km}\), and at night to \(85\text{--}90\ \text{km}\) \(^{16}\).

And, finally, for the subsequent analysis, the results of measurements of the energy \(P_{\mathrm{p}}\) of the received waves at different frequencies and at different distances from the transmitter in the afternoon hours are important. The data obtained, borrowed by the author from the corresponding graphs of work \(^{9}\), are summarized in Tables I, II, and III.

Fig. 9. Recording curves of the amplitude of the ultra-short-wave field \(^{15}\):
a) at local noon; b) in the morning hours.

For the analysis of the experimental data it is necessary to calculate the energy \(P_{\mathrm{p}}\) received at the observation point. Since the corresponding measurements are usually carried out with directional (so-called rhombic) antennas, we shall proceed from the fact that the receiving and transmitting antennas are of the same type, so that the region of the ionosphere illuminated by their directional diagrams is situated symmetrically with respect to the observation and radiation points (Fig. 10, a). The gain coefficient of the antennas, as is known, is equal to

\[ g(\alpha,\gamma)=g_0 |f(\alpha,\gamma)|^2,\qquad g_0=\frac{4\pi A_0}{\lambda^2}, \tag{42} \]

where

\[ g_0=4\pi:\left\{\int_{4\pi}|f(\alpha,\gamma)|^2\,d\Omega\right\} \]

is the maximum gain, and \(f(\alpha,\gamma)\) is the complex directional characteristic of the antenna; \(\alpha\) and \(\gamma\) are angles measured from the horizon in the vertical and horizontal planes; \(d\Omega\) is an element of solid angle; and \(A_0\) is the aperture or, in other words, the effective cross section of the antenna. From the definition of \(g_0\) it is evident that \(\lambda^2/A_0\) is equal to \(\Omega_0\), the effective solid angle of radiation of the antenna.

Table I

Results of measurements at frequency \(f=49.8\) MHz and various distances \(d\)
(Fig. 16 of paper \(^{9}\))

\(d_1 = 491\) km	\(\theta_1/2 = 19^\circ\)	\((P_{\mathrm{п}})_3 : (P_{\mathrm{п}})_1 = 4.8\) and \(7.6\)
\(d_2 = 592\) km	\(\theta_2/2 = 16.4^\circ\)	\((P_{\mathrm{п}})_3 : (P_{\mathrm{п}})_2 = 5.3\)
\(d_3 = 811\) km	\(\theta_3/3 = 13^\circ\)	—

Table II

Results of measurements at distance \(d=1243\) km \((\theta/2=11.8^\circ)\) and at various
frequencies (Fig. 19 of paper \(^{9}\))

\(f_1 = 27.775\) MHz	\(\lambda_1 = 10.8\) m	\((P_{\mathrm{п}})_1 : (P_{\mathrm{п}})_2 = 69\ (50)\)
\(f_2 = 49.8\) MHz	\(\lambda_2 = 6\) m	\((P_{\mathrm{п}})_2 : (P_{\mathrm{п}})_3 = 1580\ (2240)\)
\(f_3 = 107.8\) MHz	\(\lambda_3 = 2.78\) m	—

Table III

Averaged results of measurements at distance \(d=1243\) km \((\theta/2=11.8^\circ)\)
and at various frequencies (Fig. 19 and Fig. 8 of paper \(^{9}\))

\(f_1 = 27.775\) MHz	\(E = 28\) dB \((34.5\ \text{dB})\)	\((P_{\mathrm{п}})_1 = 2.7 \cdot 10^{-13}\) W
\(f_2 = 49.8\) MHz	\(E = 17\) dB \((16\ \text{dB})\)	\((P_{\mathrm{п}})_2 = 2.1 \cdot 10^{-14}\) W
\(f_3 = 107.8\) MHz	\(E = -7\) dB \((-12.15\ \text{dB})\)	\((P_{\mathrm{п}})_3 = 2.1 \cdot 10^{-16}\) W

(\(\text{In dB above }1\ \mu\text{V};\ 1\ \mu\text{V corresponds to }4.2 \cdot 10^{-16}\ \text{W or } -154\ \text{dB}\))

Using (10), it may now be written that the received energy scattered by an elementary volume is equal to

\[ \Delta P_{\mathrm{п}} = \sigma p_0 \Delta \Omega' dV = \sigma \left\{ \frac{P_{\mathrm{и}} g_0 |f(\alpha,\gamma)|^2}{4\pi R_0^2} \right\} \left\{ \frac{A_0'}{R^3} \right\} dV, \tag{43} \]

where \(P_{\mathrm{и}}\) is the radiated power and \(\dfrac{A_0'}{R^3}=\Delta\Omega'\) is the solid angle of reception of the antenna. Making use of expression (42), we obtain that

\[ \frac{P_{\mathrm{п}}}{P_{\mathrm{и}}} = \frac{g_0^2 \lambda^2}{(4\pi)^2} \iiint_{\varepsilon,\tau,R} \frac{\sigma |f(\alpha,\gamma)|^2 |f(\alpha',\gamma')|^2}{R_0^2 R^2} \, dV, \tag{44} \]

where \(\alpha'\) and \(\gamma'\) are the values of the angles measured at the receiving point.

The expression (44), as is easy to see, is a generalized radar formula.

In general form, the evaluation of the integral entering into (44) is a very difficult problem. However, in various special cases it can be substantially simplified.

Fig. 10. Toward the derivation of the formula for the energy \(P_{\mathrm{p}}\).

In the case of reception of ultrashort waves, when the effective solid angle of radiation of the antenna is small and the linear dimensions \(V\) (Fig. 10, b) are small in comparison with the distances \(R_0\) and \(R\), one may assume that within the volume \(V\) the values \(\sigma \sim \mathrm{const}\) and \(R_0 \sim R\), take outside the integral \(\{|f(\alpha,\gamma)|^2\}_{\mathrm{av}}\), \(\{|f(\alpha',\gamma')|^2\}_{\mathrm{av}}\), and write

\[ V=\frac{R^2\Omega_0}{\sin \frac{\theta_0}{2}}\Delta z =\frac{R_0^2\lambda^2\Delta z}{A_0\{|f(\alpha,\gamma)|^2\}_{\mathrm{av}}\sin \frac{\theta_0}{2}} \tag{45} \]

(see Fig. 10, a). As a result we obtain, using (42),

\[ \frac{P_{\mathrm{p}}}{P_{\mathrm{i}}} =\sigma \frac{A_0\Delta z}{R_0^2\sin \frac{\theta_0}{2}} \{|f(\alpha,\gamma)|^2\}_{\mathrm{av}} . \tag{45'} \]

If both antennas operate in the directions of their principal maxima, then in formula (45′) the factor \(\{|f(\alpha,\gamma)|^2\}\) is omitted. Thus, in the final

form, using (28) and assuming that the incident waves are horizontally polarized \(\left(\psi=\dfrac{\pi}{2}\right)\), and also bearing in mind that the value of the antenna cross section \(A_0\) is calculated with account of its orientation relative to the scattering region, we obtain:

\[ \frac{P_{\mathrm{p}}}{P_{\mathrm{i}}} = \frac{\sqrt{\pi}}{8\lambda} \left(\frac{\Delta N}{N}\right)^2 \left(\frac{\omega_N}{\omega}\right)^4 \left(\frac{2\pi \xi_0}{\lambda}\right)^3 \frac{A\Delta z}{R_0^2\sin\dfrac{\theta_0}{2}} e^{-\left(\frac{2\pi \xi_0}{\lambda}\right)^2\sin^2\dfrac{\theta_0}{2}} . \tag{46} \]

On the basis of the considerations set forth above (§ 2), we shall first of all determine from the experimental data, with the aid of formula (46), the dimensions of the inhomogeneities \(\xi\).

If one uses the measurement results given in Table I at one frequency and at different distances, and assumes that under the experimental conditions

\[ \left\{\left(\frac{\Delta N}{N}\right)^2 \left(\frac{\omega_N}{\omega}\right)^4\right\} \]

and \(\xi\) remained unchanged, then the value of \(\xi\) can be determined from the relation

\[ \frac{(P_{\mathrm{p}})_3}{(P_{\mathrm{p}})_1} = \frac{R_1^2\sin\dfrac{\theta_1}{2}} {R_3^2\sin\dfrac{\theta_3}{2}} e^{-k^2\xi^2\left(\sin^2\dfrac{\theta_3}{2}-\sin^2\dfrac{\theta_1}{2}\right)} \tag{47} \]

and the analogous relation for \((P_{\mathrm{p}})_3:(P_{\mathrm{p}})_2\). Substituting the corresponding numbers from Table I, we obtain:

\[ \xi = 6.2\ \text{m};\quad 6.9\ \text{m};\quad 8.0\ \text{m}. \]

In the same way, from the measurement data at one distance and different frequencies (Table II), it follows that

\[ \xi \sim 4.8\ \text{m}\quad \text{and}\quad 5.6\ \text{m}. \]

The optimum values of \(\xi\), estimated theoretically with the aid of formula (30), are obtained of approximately the same order. As a result one may accept that in the experiments indicated the ultrashort waves were scattered by inhomogeneities whose linear dimensions*)

\[ \xi_0 \sim 6\ \text{m}. \tag{48} \]

Choosing for what follows \(\xi_0=6\ \text{m}\), we calculate from the data of Table III, with the aid of formula (46), the quantity

\[ \left(\frac{\Delta N}{N}\right)^2 . \]

For this it is necessary to choose the values of \(N\) and \(\Delta z\) at the height where scattering occurs. Unfortunately, up to the present time there are no sufficiently accurate measurement results for \(z\sim 80\ \text{km}\). It may be assumed, according to various data, that in this altitude region \(N\sim 2\cdot10^2 \div 10^3\)**). Choosing \(N\sim 5\cdot10^2\) and \(\Delta z=10\ \text{km}\), we obtain from the data of Table III:

\[ \sqrt{\left(\frac{\Delta N}{N}\right)^2} \simeq (0.1 \div 1.3)\cdot 10^{-2}. \tag{49} \]

Comparing (49) with (39) and (40), calculated for higher regions of the ionosphere, we see that, within the accuracy of the estimates made, the values of the electron-density fluctuation differ little from one another in

*) In the work cited above\({}^{8}\), without further analysis, values \(\xi_0=200\ \text{m}\) are used (see (3)). The attempt by the author of the present article to determine, by the method described here, from the data of Tables I and II, the value of \(\xi\) with the aid of the Booker–Gordon formula\({}^{6,8}\) for \(\sigma\), led to imaginary values of \(\xi\).

**) It is surprising that in works\({}^{8}\) and\({}^{11}\) values \(N\sim(2 \div 5)\cdot10^4\), corresponding to the electron concentration of \(E\) or \(E_{\mathrm{spor}}\), are taken, although from the data of the same series of experiments it follows that the scattering heights are substantially smaller than the heights of \(E\) and \(E_{\mathrm{spor}}\).

SOME PROBLEMS IN THE PHYSICS OF THE IONOSPHERE

in order of magnitude. Naturally, these estimates, like the calculations of $\xi_0$, are preliminary in character. It is clear, however, that by the method described a more detailed and complete investigation of these quantities is possible.

In concluding this paragraph it is useful to draw attention to one more circumstance. From the results of the measurements cited above it is clear that further propagation of ultrashort waves occurs by scattering in the lower part of the ionosphere. This fact is not surprising if one proceeds from the point of view (§ 2) that the principal factor determining their scattering is the size of the inhomogeneities—the wave “selects” that region of the medium in which inhomogeneities of optimal dimensions are contained. However, if one departs from this point of view and assumes, as, for example, was done in work $^{8}$, that the scattering of ultrashort waves is caused by inhomogeneities of size $\sim 200\ \text{m}$, then it is necessary to understand why the waves were not scattered in the experiments described $^{8,9}$ predominantly at altitudes of $100$–$110\ \text{km}$, where the maxima of the principal antenna lobes were directed, or else in the $F$ region. In these regions, as is known, $\xi_0 \sim 200\ \text{m}$ (see (3)), and together with this the electron concentration increases, respectively, by $10^2$ and by $10^3$ times, so that the energy of the scattered waves should have increased approximately by $10^4$ and $10^6$ times. The answer to the question why scattering did not occur in the $F$ region may, to a certain extent, be the circumstance that the principal maxima

Fig. 11. Radiation pattern of the antenna used in experiments with ultrashort waves: a) projection of $|f(\alpha,\gamma)|$ onto the vertical plane; b) projection of $|f(\alpha,\gamma)|$ onto the horizontal plane.

of the antennas used in the experiments $^{8,9}$ were directed at lower altitudes. Indeed, if one calculates the radiation patterns of these antennas from the data given in the cited works (Fig. 11), it turns out that the energy radiated by the antenna in the direction of the $F2$ layer decreases by approximately $400$–$500$ times, so that the received energy should decrease by $2\cdot 10^5$ times, while the ionization factor gives an increase of the same order. However, the fact that scattering occurred at an altitude of $80\ \text{km}$ gives some grounds for assuming that the most favorable conditions for scattering of ultrashort waves existed in this part of the ionosphere, because here there is a sufficient number of inhomogeneities of small size. To verify this important conclusion it is necessary to carry out experiments with antennas whose radiation patterns vary with height and, in particular, experiments on waves of greater length, say on short waves. The corresponding experiments would also make it possible to investigate the spectrum of inhomogeneity sizes as a function of altitude.

§ 5. FURTHER PROPAGATION OF SHORT WAVES CAUSED BY SCATTERING

Unfortunately, despite many years of studies of short waves, there are no data that would make it possible to analyze them analogously to how this was done in the preceding section; one can find only indications of facts of distant propagation of short waves in the case when the ionosphere became transparent for them. At the same time, it is difficult to overestimate the importance of such investigations. With a proper and sufficiently complete formulation of the corresponding experiments, their results may be very fruitful for the study of the structure of the ionosphere. In this connection we shall dwell on this question and, as an example, analyze the results of one experiment.

First of all, let us calculate more fully the ratio \(\dfrac{P_{\mathrm{p}}}{P_{\mathrm{i}}}\) of the received and emitted energies. Because the directivity of antennas decreases with increasing wavelength, the region \(V\) illuminated by them increases. Therefore, without knowing the effective dimensions of \(V\), one cannot, a priori, disregard in the calculations the dependence of \(\sigma\), \(\theta\), \(R_0\), and \(R\) on the coordinates of the point. Naturally, it is most essential here to take into account the dependence of these quantities on the height \(z\). Since a complete calculation proves to be rather cumbersome and complicated, one has to introduce certain simplifications which, however, preserve the dependence on \(z\), which plays the main role here.

As before, the factor \(\{|f(a,\gamma)|^2\}\times \{|f(\alpha',\gamma')|^2\}\) is taken outside the integral (44), and it is thereby meant that, correspondingly, the values of the gain and the effective area of the antennas will be calculated. We further assume that \(\sigma\), \(\theta\), \(R_0\), and \(R\) depend only on \(z\), i.e., their values along the line \(OO'\) (see Fig. 10, a)—the axis of symmetry of the scattering volume—are substituted under the integral; here it is assumed that the values of these quantities change little on the surface \(z=\text{const}\). As a result one may write, taking \(\theta_0/2\sim\alpha\),

\[ \left. \begin{aligned} dV&=\left\{\frac{R_0\,d\alpha}{\sin \dfrac{\theta_0}{2}}\right\} (R\cos\alpha\,d\gamma)\frac{dz}{2},\\[4pt] R_0&=R=\frac{z}{\sin \dfrac{\theta}{2}} . \end{aligned} \right\} \tag{50} \]

Therefore (44) takes the form

\[ \frac{P_{\mathrm{p}}}{P_{\mathrm{i}}} = \frac{g_0^2\lambda^3}{(4\pi)^2} \int_{\alpha_1}^{\alpha_2} \int_{\gamma_1}^{\gamma_2} \int_{z_1}^{z_2} \frac{\sigma(z,\theta_0)}{z^2} \sin\frac{\theta_0}{2}\cos\frac{\theta_0}{2}\, d\alpha\,d\gamma\,dz \tag{51} \]

or

\[ \frac{P_{\mathrm{p}}}{P_{\mathrm{i}}} = \frac{g_0^2\lambda^2}{2(4\pi)^2}(\gamma_2-\gamma_1) \int_{\alpha_1}^{\alpha_2} e^{-\left(\dfrac{2\pi\xi_0}{\lambda}\right)^2\sin^2\alpha}\, d(\sin^2\alpha) \int_{z_1}^{z_2} \frac{\sigma(z)}{z^2}\,dz, \tag{52} \]

where \(\gamma_1,\gamma_2,\alpha_1\), and \(\alpha_2\) characterize the angular aperture of the antenna lobes illuminating the scattering region, and

\[ \sigma(z)= \left(\frac{\overline{\Delta N}}{N}\right)^2 \left(\frac{\omega_N}{\omega}\right)^4 \frac{\sqrt{\pi}}{\delta\lambda} \left(\frac{2\pi\xi}{\lambda}\right)^3 . \]

The integral over \(z\) in expression (52), as is easily seen, is expressed through

the integral \(M\) (see (38)). As a result we find:

\[ \frac{P_{\mathrm{p}}}{P_{\mathrm{i}}} = \frac{g_0^2}{2\sqrt{\pi}} \left(\frac{\omega_c}{\omega}\right)^4 \frac{\xi_0(\gamma_2-\gamma_1)}{z_m} \left(\frac{\Delta N}{N}\right)^2 M \left\{ e^{-\left(\frac{2\pi \xi}{\lambda}\sin\alpha_1\right)^2} - e^{-\left(\frac{2\pi \xi}{\lambda}\sin\alpha_2\right)^2} \right\}. \tag{53} \]

It is not difficult to show that formula (53), like formula (46), obtained for the v.k.v., gives, for \(\xi_0 \sim 200\text{–}300\ \mathrm{m}\) and for the values of the other parameters encountered in short-wave experiments, negligibly small values of the scattered energy (\(\sim 10^{-30}\div 10^{-40}\ \mathrm{W}\) and less). At the same time, for example, on a path of length \(d\sim 2800\ \mathrm{km}\), with antennas of a known type, at \(\lambda\sim 20\ \mathrm{m}\), under conditions in which it exceeded \(\lambda_{\mathrm{mp}}\) by a factor of \(2\text{–}2.5\), the received energy reached \(10^{-15}\text{–}10^{-13}\ \mathrm{W}\). Here the heights \(z_1\) and \(z_2\) of the region irradiated by the main lobes of the antennas were of the order of \(300\div 500\ \mathrm{km}\). If one estimates \(\frac{P_{\mathrm{p}}}{P_{\mathrm{i}}}\) on the assumption that \(\xi\) had an optimum value (see (30)), i.e. was \(\sim 20\text{–}30\ \mathrm{m}\), then formulas (53) or (46) give, in order of magnitude, energy values close to those observed. However, the assumption that at heights of \(300\text{–}500\ \mathrm{km}\) there are a sufficient number of irregularities of the corresponding dimensions requires careful verification. (Indeed, the mean free paths here reach many hundreds of meters, so that at first glance the minimum dimensions of the irregularities may be at least of the same order.) Therefore one should also keep in mind another possibility for the propagation of short waves, namely by scattering at lower heights. In the experiment mentioned, for example, taking the sphericity of the Earth into account, it is easy to find that the minimum heights which were still illuminated by the antennas were of the order of \(170\text{–}180\ \mathrm{km}\), and it is possible that scattering occurred in this region. Other paths of wave propagation due to scattering at still lower heights are, however, not excluded. From this analysis it becomes clear that corresponding short-wave experiments at various distances from the transmitter, reaching the greatest possible propagation distances in the case of a single hop, and also at various frequencies (with \(\omega>\omega_{\mathrm{mp}}\)), may provide very valuable data needed for the study of the structure of the ionosphere. Naturally, it is important to accompany such experiments with measurements of scattering heights (see \(^{16}\)) and with height sounding of the ionosphere in a range including low frequencies, so as to ensure the acquisition of more detailed information on the height dependence \(N(z)\). Similar measurements are also important to carry out in experiments with ultrashort waves, which has not yet been done.

§ 6. SOME RESULTS OF THE ANALYSIS OF EXPERIMENTAL DATA

Let us briefly dwell on some conclusions that follow from the data presented above.

Although all the parameters considered, characterizing the inhomogeneity of the structure and motions in the ionosphere, have as yet been little studied and their further comprehensive investigation is required, it nevertheless seems possible to note a number of features already reliably revealed in their behavior, which apparently have substantial significance for the study of the mechanisms of the phenomena under consideration.

The altitude region of \(80\div 300\ \mathrm{km}\) and above, which is of interest to us here, is characterized by very strong changes both in the electron density \(N\) and in the density of neutral particles \(N_M\), and consequently also in the mean free paths. Table IV gives the corresponding data for noon and middle latitudes, the mean free paths being

\[ \Lambda = 1:\sqrt{2}\,\sigma_M N_M \]

calculated for values \(\sigma_M = 3 \cdot 10^{-15} \div 4.3 \cdot 10^{-16}\), given in various works.

Table IV

Ionospheric data

\(z\), km	\(T\), deg	\(N_M\), \(1/\mathrm{cm}^3\)	\(N\), el/\(\mathrm{cm}^3\)	\(\Lambda\), m
80	200	\(5 \cdot 10^{14}\)	\(5 \cdot 10^2 \div 10^3\)	\((0.4 \div 3)\,10^{-2}\)
100	220	\(3 \cdot 10^{13}\)	\((0.5 \div 2)\,10^5\)	\((0.8 \div 5)\,10^{-1}\)
200	800	\(10^{11}\)	\((1.5 \div 4)\,10^5\)	\((0.3 \div 2)\,10^2\)
300	1500	\(3 \cdot 10^9\)	\((5 \div 20)\,10^5\)	\((0.7 \div 6)\,10^3\)

It is seen from Table IV that the density of neutral particles and, correspondingly, the mean free paths change at the heights of interest to us by approximately \(10^5\) times, while the electron density changes by \(10^3\)—\(10^4\) times. At the same time, the dimensions of the inhomogeneities \(\xi_0\)—(3), the fluctuations of electron density—(39), (40) and (49), and the velocity of their chaotic motions—(4) and (41)—apparently change little with height, in any case considerably more slowly than \(N\) and \(N_m\). Thus, above all, the fact that there is no substantial change of \(\xi_0\), \(v_0\), and \(\left(\dfrac{\Delta N}{N}\right)^2\) with height attracts attention.

Further, the following circumstance draws attention. In the case of vertical sounding of the ionosphere, when the largest role in the scattering of radio waves should be played by inhomogeneities of the greatest size, for heights of 100—300 km and higher one obtains values \(\xi \sim 200\)—300 m\(^*\). It may therefore be assumed that these values correspond to the most frequently encountered maximum dimensions of inhomogeneities. At the same time, estimates of the sizes of vortices (see § 7), the lengths of longitudinal plasma waves or other types of inhomogeneous formations, as well as the long mean free paths, lead to the conclusion that even the minimum linear dimensions of inhomogeneities of 200—300 m would seem already to be prohibited above 200—250 km, whereas in reality they are nevertheless observed\(^{{**}}\). The noted circumstance is, at first glance, unexpected and difficult to explain. At the same time, indications of the presence of inhomogeneities of very small size (\(\xi_0 \sim 6\) m, see (48)) in the lower part of the ionosphere, on the contrary, agree with various estimates and qualitative considerations (see § 7).

Finally, data from studies of horizontal drifts caused by winds in the ionosphere show that wind velocities change little with height. Meanwhile, the still insufficiently verified data on the altitude gradient of winds (see (7)) show that the velocities should increase rapidly with height. Thus, on passing from region \(E\), where \(u_0 \sim 70\ \mathrm{m/sec}\), to region \(F\), the velocity should reach at a height of 200 km the value \(u \sim 400\ \mathrm{m/sec}\); at the same time, at these heights it is equal to 100 m/sec (see (8)). This apparently indicates that the velocity gradient has large values only in limited regions. It is possible that there exist local, sufficiently narrow regions of active development of winds with large—

\(^*\) It should be noted that in determining the dimensions of inhomogeneities, the corresponding formulas usually do not take into account the dependence of the wavelength on height, as was done above in determining \(\left(\dfrac{\Delta N}{N}\right)^2\) (§ 3).

\(^{{**}}\) See § 1 concerning inhomogeneities whose linear dimensions are of the order of several km.

large gradients and relatively windless extended regions. This leads to the idea that in the ionosphere there are peculiar boundary surfaces—“walls”—between which rapid streams of particles move, being braked at these walls, which may lead to turbulence of the particle flow. Whether this is so, and what the physical nature of these “walls” is, is unknown. It may be pointed out, however, that such a peculiar wall is, for example, the temperature minimum at an altitude \(z \sim 80\) km, which causes the occurrence of the so-called lunar oscillations of pressure in the atmosphere \(^{17}\).

Thus, at the present time the picture of the phenomena occurring in the ionosphere appears complex and obscure. There are no completed results of theoretical calculations that would make it possible to analyze them in sufficient depth in the light of one mechanism or another; there is a known tendency, when considering phenomena of this type (see, for example, \(^{11,18}\)), to ascribe them to turbulence, the present state of whose theory permits only certain estimates, based chiefly on similarity considerations \(^{19}\). In the following paragraph it will be shown what follows from the corresponding estimates.

§ 7. TURBULENCE OF THE IONOSPHERE

One of the most substantial arguments compelling us to seek in turbulence an explanation of the phenomena of interest here is the statistical character of a turbulent flow, in which, by its very nature, the formation, disappearance, and re-formation of vortices take place, and this process occurs spontaneously.

The scattering of electromagnetic waves by such a turbulent flow takes place as follows. In a homogeneous and isotropic turbulent flow, the pressure oscillations \(\Delta p_s\) of the inhomogeneities of this flow are described by the dimensional relation \(^{19}\)

\[ \Delta p_s \sim \rho(\Delta v_s)^2, \tag{54} \]

where \(\rho\) is the density of the medium, and \(\Delta v_s\) is the velocity difference of the vortices. These pressure oscillations lead to density oscillations and, consequently, to fluctuations of the electron density. Owing to the statistical nature of this process, the distribution of amplitudes of the field scattered by density fluctuations must obey Rayleigh’s law (see § 1).

Let us examine more closely what the relations of the theory of turbulence \(^{19}\) give for the ionosphere.

As is known, turbulent motions are characterized by Reynolds numbers

\[ \mathrm{Re}=\frac{\Delta U_0\cdot L_0}{\nu} =3\,\frac{\Delta U_0}{v_M}\frac{L_0}{\Lambda}, \tag{55} \]

where \(\Delta U_0\) is the difference in the laminar velocity of the flow and \(L_0\) is its thickness, \(\bar v_M=\sqrt{\dfrac{8kT}{\pi M}}\) is the thermal velocity of gas particles (\(k\) is Boltzmann’s constant, \(M\) is the particle mass), and \(\Lambda\) is the length of their free path.

The quantities \(L_0\) and \(\Delta U_0\) entering formula (55) are the largest of the linear scales and velocity differences possible in a turbulent flow, which is characterized by a spectrum of vortices of various sizes \(L_s\) (the microscale of turbulence) with velocity difference \(\Delta u_s\); the smallest of them are respectively equal to:

\[ L_s=\frac{L_0}{(\mathrm{Re})^{3/4}},\qquad \Delta u_s=\frac{\Delta U_0}{(\mathrm{Re})^{1/4}}. \tag{56} \]

To formulas (56) one may add the relation characterizing the fluctuations of electron density, which follows from (54). Indeed, putting

\[ \frac{\Delta p_s}{p}=\frac{\Delta \rho_s}{\rho}\sim \frac{\Delta N_s}{N}, \]

we obtain

\[ \frac{\overline{\Delta N_s^2}}{N^2}\sim \frac{u_s^4}{v_M^2}\sim \frac{u_s^4}{v_M^4}. \tag{57} \]

In order to estimate \(L_s\), \(\Delta u_s\), and \(\sqrt{\overline{\left(\dfrac{\Delta N_s}{N}\right)^2}}\), it is necessary to know the values of \(L_0\) and \(\Delta U_0\), which are difficult to choose from ionospheric data.

One may, for example, suppose that the characteristic thicknesses \(L_0\) of the flow are dimensions commensurate with the thicknesses of the so-called layers of the ionosphere, and that the velocity jump \(\Delta U_0\) corresponds to the velocity of the winds. If the velocity \(\Delta U_0\) is chosen in this way, it is thereby assumed that at the edges of the flow—at its “walls”—the velocity is equal to zero. From the corresponding values of the indicated parameters one obtains the data presented in Table V.

Table V

Values of \(v_s\), \(L_s\), and \(\sqrt{\overline{\left(\dfrac{\Delta N_s}{N}\right)^2}}\)

\(z,\) km	\(\Delta z,\) km	\(\overline{v},\) m/sec	\(U_0,\) m/sec	Re	\(\Delta u_s,\) m/sec	\(L_s,\) m	\(\sqrt{\overline{\left(\dfrac{\Delta N}{N}\right)^2}}\)
80	5	400	70	\((9—70)\,10^4\)	2.5—4	0.4—1	\(6\cdot 10^{-4}\)
100	5	400	70	\((5—40)\,10^3\)	5—8	2—8	\(3\cdot 10^{-4}\)
200	30	1000	100	50—400	80—40	160—1600	\(3\cdot 10^{-4}\)
300	100	1300	200	10—80	70—100	3.8—18	\(4\cdot 10^{-3}\)

Table V shows, first of all, that the Reynolds numbers are everywhere sufficiently large, which is usually a criterion for the occurrence of turbulence. However, whether this criterion remains valid for the ionosphere, and what “large” means in this case, is unknown; therefore it is hardly possible to draw any conclusions from this.

Second, it is seen that up to \(z\sim 200\) km the values of \(L_s\) and \(\Delta u_s\) do not contradict the experimental values of \(v_0\) and \(\xi_0\) given above; at an altitude of 300 km, however, \(\Delta u_s\), \(L_s \gg v_0, \xi_0\).

Third, the fluctuations of electron density \(\overline{\left(\dfrac{\Delta N_s}{N}\right)^2}\) are approximately 100 times smaller than the values obtained from experimental data.

If, further, one assumes that \(\Delta U_0\sim \left(\dfrac{dU_0}{dz}\right)\cdot \Delta z\) (see (7) and (8)), then, in contrast to the data of Table V, one obtains an increase in the discrepancy between \(L_s\) and \(\xi_0\) (see Table VI).

These estimates, as well as other variants of the calculations, show that in the lower regions of the ionosphere, at \(z \lesssim 200\) km, the minimal (!) linear dimensions and velocities of the interscale inhomogeneities are, in general, close to the experimental values \(\xi_0\) and \(v_0\). The values substantially diverge from them ...

Table VI

\(z,\ \mathrm{km}\)	\(\Delta z,\ \mathrm{km}\)	\(\Delta U_0,\ \mathrm{m/sec}\)	\(\mathrm{Re}\)	\(\Delta u_s,\ \mathrm{m/sec}\)	\(L_s,\ \mathrm{m}\)	\(\sqrt{\overline{\left(\dfrac{\Delta N_s}{N}\right)^2}}\)
80	5	18	2000	2.6	15	\(0.8\cdot 10^{-4}\)
100	5	18	100	5	140	\(4\cdot 10^{-4}\)
200	30	30	15	15	\(4\cdot 10^3\)	\(2\cdot 10^{-4}\)
300	100	100	5	70	\(30\cdot 10^3\)	\(9\cdot 10^{-3}\)

fluctuations of the electron density. Remaining within the framework of the notions of turbulence in the ionosphere, let us estimate the values of the expected fluctuations of the electron density \(N\), taking into account its gradient \(\dfrac{dN}{dz}\) and the velocity gradient \(\dfrac{dU_0}{dz}\).

Let us suppose that over distances of the order of the length \(L_s\) the jump in electron density is \(\Delta N_s \sim \dfrac{dN}{dz} L_s\). Then

\[ \sqrt{\frac{\overline{\Delta N_s^2}}{N}} \sim \frac{L_s}{N}\frac{dN}{dz} \tag{58} \]

and if we choose

\[ \begin{aligned} z &= 80\ \mathrm{km}, & \frac{dN}{dz} &= 1\ \frac{\mathrm{el}}{\mathrm{cm}^3\cdot \mathrm{m}}, & N &\sim 5\cdot 10^2, & L_s &= 6\ \mathrm{m},\\ z &= 100\ \mathrm{km}, & \frac{dN}{dz} &\sim 1.2\ \frac{\mathrm{el}}{\mathrm{cm}^3\cdot \mathrm{m}}, & N &\sim 10^5, & L_s &\cong 200\ \mathrm{m},\\ z &= 200\ \mathrm{km}, & \frac{dN}{dz} &\sim 2\ \frac{\mathrm{el}}{\mathrm{cm}^3\cdot \mathrm{m}}, & N &\sim 5\cdot 10^5, & L_s &\sim 300\ \mathrm{m}, \end{aligned} \tag{59} \]

then for all heights one obtains

\[ \sqrt{\overline{\left(\frac{\Delta N_s}{N}\right)^2}} \sim (1 \div 3)\cdot 10^{-3}. \tag{60} \]

Similarly, if we write

\[ \rho u \frac{du}{dz} \sim -\operatorname{grad} p \sim -\frac{dp}{dz}, \tag{61} \]

then, putting \(\dfrac{\Delta N_s}{N} \sim \dfrac{\Delta p_s}{p}\), we obtain

\[ \sqrt{\frac{\overline{\Delta N_s^2}}{N}} \sim \frac{u_s L_s}{v_M^2}\frac{du}{dz}. \tag{62} \]

The numerical values of the corresponding parameters give:

\[ \sqrt{\frac{\overline{\Delta N_s^2}}{N^2}} \sim 10^{-7}. \tag{63} \]

Thus, it is seen that the gradient \(\dfrac{dN}{dz}\) leads to values of the electron-density fluctuations closest to those obtained above from the experimental results.

The data considered, taken as a whole, show that the estimates of ionospheric parameters currently possible, characterizing its statistical inhomogeneity, and based on concepts of ionospheric turbulence, in a number of cases lead to noncontradictory results. Therefore the corresponding theoretical calculations, in particular for plasma, become especially important. In this connection we note that the estimates made above are, to a certain extent, not fully legitimate, since they are based on formulas which, strictly speaking, are applicable to media consisting of neutral particles. In the denser parts of the ionosphere the properties of the electron inhomogeneous formations that are of interest to us here apparently do not differ from the properties of inhomogeneous formations of neutral particles. However, it is risky to make the corresponding assumptions for a highly rarefied plasma, and to obtain an answer to such a question it is necessary to consider the problem of the motion of electrons in a turbulent flow, taking into account the external magnetic field and, as a first step, to obtain for electron formations relations analogous to those given above.

In conclusion it should once more be emphasized that the study of the mechanisms responsible for the statistical character of the structure of the ionosphere is of great interest and is one of the most interesting and important problems of modern physics.

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1. In the Russian original, the words here and below are letter-spaced for emphasis. ↩↩