Can Radioactive Emanations in a Seismically Active Region Affect Atmospheric Electricity and the Ionosphere?

Abstract—Anomalous variations in radon emissions from the soil are sometimes observed before and after strong earthquakes. In this paper, we theoretically analyze the known hypothesis that these variations in radioactive emanations in the surface layer of the atmosphere cause variations in the vertical background current in the atmosphere with subsequent changes in the electron density ${{n}_{e}}$ in the lower ionosphere. For this purpose, we will first assess the relation between radon emission variations and the vertical atmospheric current flowing into the ionosphere. Then we will solve a model problem concerning the spatial distribution of plasma density and electric field perturbations occurring in the E-layer of the ionosphere caused by an inflowing vertical current. It was believed that the current flowing into this layer contains only an electron component and is attenuated only by the recombination of electrons. The estimate of the maximum variation ${{n}_{e}}$ is at least 3–5 orders of magnitude smaller than the observed anomalous variations, which allows us to conclude that this hypothesis is not plausible.

INTRODUCTION: POSSIBLE MECHANISMS OF SEISMIC-IONOSPHERIC COUPLING

Ionospheric plasma dynamics are very sensitive even to relatively weak electric field perturbations and neutral gas motions. Therefore, ionosphere monitoring can be an effective tool for detecting large-scale perturbations, possibly related to different stages of the seismic process. The response of the ionosphere to seismic shock and seismic Rayleigh waves caused by acoustic-gravity waves (AGWs) is now a well-known effect and has become the physical basis of “ionospheric seismology” (see, for example, (Astafyeva, 2019)). At the same time, the physical interpretation of some anomalous ionospheric perturbations, which can be considered as precursors of earthquakes, is still an open question and an unsolved problem. Moreover, most papers consider possible mechanisms of lithospheric-ionospheric coupling only qualitatively, there are not even coarse models to quantify the possible effects. That is why we believe that the development of such estimation models is extremely important, since it will dramatically narrow the range of searches for possible directions of studying the seismoionospheric coupling. There are several proposed channels of how lithospheric and atmospheric processes affect the ionosphere (Surkov, 2000; Molchanov and Hayakawa, 2008; Surkov and Hayakawa, 2014).

– “Acoustic”—via AGWs in the neutral atmosphere (Gokhberg et al., 1994; Klimenko et al., 2011). Presumably, AGWs can be excited both during seismic events and before them as a result of unsteady release of gas emanations from the Earth’s crust or irregular heating of the surface layer of the atmosphere (Mareev et al., 2002). As the AGWs propagate to ionospheric altitudes, the amplitude of the gas velocity in the wave increases due to an exponential drop in the atmospheric density as the altitude increases. When penetrating into the ionosphere, the AGWs will cause variations in ionospheric plasma currents and densities. However, the possibility of pre-seismic AGWs has not yet found any direct observational evidence.

– “Electrostatic”—via a large-scale electrostatic field. Even in early studies of seismoionospheric coupling, it was suggested that ionospheric perturbations, possibly associated with impending earthquakes, may be caused by the effect of a large-scale seismogenic electric field on the ionosphere (Gokhberg et al., 1985). In this way, electrically charged thunderstorm clouds can affect the ionosphere (Park and Dejnakarintra, 1973). However, the weakening of the static electric field between the near-surface atmosphere and the ionosphere is too large to expect any ionospheric response to surface perturbations (Denisenko, 2015). Nevertheless, the possibility of this channel of lithosphere-ionosphere coupling cannot be completely excluded, because the nonstationarity of the source and the anisotropy of conductivity dramatically increase the efficiency of atmospheric field penetration into the ionosphere (Kim et al., 2017). Although reports on perturbations of the atmospheric electric field in seismically active regions appear from time to time (Rulenko et al., 1992; 2019), there is still no reliable observational confirmation that such global electric field perturbations occur before seismic events.

– “Current”—through the effects of currents flowing into the ionosphere. It has been suggested that the anomalous changes in the ionosphere are associated with increased radon (Rn) emanation from the soil. A several-fold increase in the Rn concentration at the ground surface compared to the background value was observed before some strong earthquakes (Virk and Singh, 1994; Inan et al., 2008; Giuliani and Fiorani, 2009; Yasuoka et al., 2009), although some researchers did not find any statistically significant changes in the Rn concentration before earthquakes (Pitari et al., 2014; Cigolini et al., 2015). α-Particles formed as a result of nuclear decay of Rn atoms make a significant contribution to air ionization only in the near-surface layer of the atmosphere, since the activity and concentration of Rn decrease rapidly as the altitude increases (Tverskoy, 1962; Zhang et al., 2011). The anomalous current above the seismic source may appear because radioactive emanations modulate the air conductivity and thus the atmospheric fair-weather current between the ionosphere and the Earth (Martynenko et al., 1994; Fuks et al., 1997; Harrison et al., 2013; Surkov, 2015). Technically, this mechanism is similar to the “orographic effect” in the physics of atmospheric electricity—changes in the electric field and current over mountains (Tzur et al., 1985).

Seismic events also supposedly cause upward flows of charged aerosols, which can carry the electric field and current from the surface atmosphere to the lower ionosphere (Pulinets and Boyarchuk, 2004; Klimenko et al., 2011; Sorokin and Hayakawa, 2014). It should be noted that there are no direct observations of such upward flows. The main problem of this hypothesis is that due to the large mass of aerosols, it is difficult to explain their export to altitudes of over 10 km. Therefore, to explain ionospheric anomalies, which may be associated with seismic events, we have to assume the presence of large “aerosol” currents and external electric fields which are many orders of magnitude higher than the observed values of these parameters in the lower atmosphere.

Early studies revealed cases of plasma density anomalies forming over the epicenter of impending earthquakes, with the help of vertical sounding of the ionosphere or data from low-orbit satellites (collected in the book by (Pulinets and Boyarchuk, 2004)). Since such observations rarely coincide with a seismically active region, these results remain an untested, though promising, speculation. Much more material for the study of seismoionospheric coupling is provided by modern means of continuous monitoring of the ionosphere.

Changes in the phase and amplitude of very low-frequency (VLF) waves (3–30 kHz) propagating in the Earth–ionosphere waveguide along traces above the earthquake preparation source were used to search for ionospheric earthquake precursors. Such phase and amplitude anomalies were indeed detected 3–6 days before some earthquakes (Rozhnoi et al. 2004; 2009; Hayakawa et al., 2010). The changes in amplitude and phase of the received VLF wave signal may be associated with local changes in the density of the lower layers of the ionosphere in the first Fresnel zone above the earthquake site. According to (Harrison et al., 2010), it is the anomalous vertical current above the radioactive emanation outlet that may be the cause of the displacement of the upper wall of the waveguide, leading to VLF wave anomalies.

Much greater opportunities to monitor almost the entire ionosphere appeared with the emergence of global navigation satellite systems (GNSS), such as GPS, GLONASS, etc. GNSS tools provide information on the variations of the integral parameter along a ray path—the total electron content (TEC). The ever-expanding TEC observations are becoming a global method for monitoring the ionospheric response to upward AGWs generated by seismic waves (Occhipinti et al., 2013) and tsunamis (Shalimov et al., 2019; Sorokin et al., 2019; Sorokin et al., 2019). The easy availability of huge amounts of TEC data triggered a surge of studies aimed at searching for ionospheric precursors of earthquakes. Based on the analysis of these data, there were reports of anomalous TEC variations observed several days before the earthquake (Pulinets and Ouzounov, 2011; Nenovski et al., 2015), although many researchers doubt the presence of earthquake precursors in the ionosphere and consider the stated effects dubious. As a possible mechanism of TEC perturbation, it was assumed that the local perturbation of the electric field in the E-layer of the ionosphere is swept into the F-layer and leads to changes in the plasma and TEC density in the upper ionosphere (Klimenko et al., 2011).

One of the possible mechanisms of lithospheric-ionospheric coupling is presumed to be a change in air conductivity in the lower atmosphere as a result of increased radon emanation from soil in seismically active regions before and after an earthquake. This assumption serves as the basis for the hypothesis that enhancement of radioactive emanations leads to changes in the background atmospheric current flowing in the global electrical circuit, which, in turn, may lead to variations in electron concentration in the ionosphere. This hypothesis was used to qualitatively interpret the precursory anomalies in the lower (Rapoport et al., 2004; Harrison et al., 2010; 2013) and upper (Ouzounov et al., 2011; Pulinets and Davidenko, 2014) ionosphere. However, so far there are no theoretical estimates that could confirm or refute this hypothesis. The main purpose of this study is to test this hypothesis and to assess whether the effect of radon on air conductivity in the lower atmosphere can cause anomalies in the ionosphere that can be associated with seismic events.

MEDIUM MODEL AND BASIC EQUATIONS

The main sources of air ionization in the lower atmosphere are cosmic rays and radioactive elements, mostly the heavy gas radon, which penetrates from the soil into the atmosphere. Air ionization caused by the radioactive decay of radon leads to the appearance of free electrons. The lifetime of free electrons is very short, since electrons quickly attach to neutral molecules in the order of 1 ns. Therefore, air conductivity in the lowest atmospheric layer mostly depends on the presence of light and cluster ions and partially— charged aerosols. The radon concentration drops rapidly as the altitude increases, so that its effect on the air ionization and conductivity becomes negligible at altitudes of over several kilometers.

Let us assume that the background atmospheric current is directed vertically upward along the $z$ axis. The density of this current ${{J}_{z}}$ is determined by Ohm’s law ${{J}_{z}} = - {{\sigma }_{a}}{{d\varphi } \mathord{\left/ {\vphantom {{d\varphi } {dz}}} \right. \kern-0em} {dz}}$, where ${{\sigma }_{a}}$ is the air conductivity, and $\varphi$ is the electric potential. Using this law, let us express the atmospheric current density through the potential difference ${{\varphi }_{a}}$ between the ionosphere and the ground, as well as through the total electrical resistance $R$ of the vertical atmospheric air column with a unit cross section and height $h$ equal to the distance between the ground and the bottom edge of the ionosphere:

$${{J}_{a}} = \left| {{{J}_{z}}} \right| = \frac{{{{\varphi }_{a}}}}{R},\,\,\,\,R = \int\limits_0^h {\frac{{dz}}{{{{\sigma }_{a}}\left( z \right)}}} .$$

(1)

Theoretical estimates show that the integral resistance of the atmosphere will decrease by 10–20% if there is a local increase in the radon concentration near the ground surface by 2–3 times (Harrison et al., 2010; Surkov, 2015). These are exactly the changes in radon concentrations that were observed, for example, two months before the Kobe earthquake with magnitude ${{M}_{w}} = 6.9$, which occurred on January 17, 1995 (Yasuoka et al., 2009).

It follows from formula (1) that, for a given value of ${{\varphi }_{a}}$, small variations in the vertical air column resistance $\delta R$ and the atmospheric background current density $\delta {{J}_{a}}$ are related by the equation ${{\delta {{J}_{a}}} \mathord{\left/ {\vphantom {{\delta {{J}_{a}}} {{{J}_{a}}}}} \right. \kern-0em} {{{J}_{a}}}} = - {{\delta R} \mathord{\left/ {\vphantom {{\delta R} R}} \right. \kern-0em} R}$. Thus, a decrease in the resistance $\left( {\delta R < 0} \right)$ entails a corresponding increase in the background atmospheric current density.

The vertical atmospheric current coming through the bottom boundary of the ionosphere is partially diffused through the conductive E-layer of the ionosphere and partially penetrates into the F-region. Let us first investigate the distribution of fields and currents in the E-layer of the ionosphere. For this purpose, let us consider a simple flat-layered model of the ionosphere and magnetosphere, in which the geomagnetic field ${{{\mathbf{B}}}_{0}}$ is directed along the vertical $z$ axis, and the E-region of the ionosphere is located in the layer $0 < z < l$.

We will assume that an increase in radon activity from the soil causes a variation in the atmospheric current density $\delta {{J}_{a}}$. This current variation flows along the $z$ axis into the ionosphere through its bottom boundary $\left( {z = 0} \right)$. Let us consider a stationary problem in which currents and electric fields do not depend on time. The current density distribution at the bottom boundary of the ionosphere will be given as follows:

$$\delta {{J}_{a}}\left( {r,0} \right) = \delta {{J}_{{\max }}}\exp \left( { - {{{{r}^{2}}} \mathord{\left/ {\vphantom {{{{r}^{2}}} {r_{0}^{2}}}} \right. \kern-0em} {r_{0}^{2}}}} \right),$$

(2)

where $r$ is the polar radius; ${{r}_{0}}$ is the characteristic transverse size of the current $\delta {{J}_{a}}$, equal in order of magnitude to the size of the seismically active region or the width of the fault, i.e. the value of about hundreds of km.

Ohm’s law for the current density $\delta {\mathbf{J}}$ and electric field E in the anisotropic conductive E-layer of the ionosphere has the form:

$$\delta {\mathbf{J}} = {{\sigma }_{\parallel }}{{{\mathbf{E}}}_{\parallel }} + {{\sigma }_{P}}{{{\mathbf{E}}}_{ \bot }} + {{{{\sigma }_{H}}\left( {{{{\mathbf{B}}}_{0}} \times {{{\mathbf{E}}}_{ \bot }}} \right)} \mathord{\left/ {\vphantom {{{{\sigma }_{H}}\left( {{{{\mathbf{B}}}_{0}} \times {{{\mathbf{E}}}_{ \bot }}} \right)} {{{B}_{0}}}}} \right. \kern-0em} {{{B}_{0}}}}{\text{,}}$$

(3)

where ${{\sigma }_{\parallel }}$ is the longitudinal conductivity of the ionospheric plasma; ${{\sigma }_{P}}$ and ${{\sigma }_{H}}$ are the Pedersen and Hall conductivities, respectively; ${{{\mathbf{E}}}_{\parallel }}$ and ${{{\mathbf{E}}}_{ \bot }}$ denote the parallel and perpendicular components of the electric field with respect to the geomagnetic field ${{{\mathbf{B}}}_{0}}$. To make the problem simpler, it is assumed that the ionospheric plasma conductivity components are constant within the E-layer of the ionosphere and equal to their average layer thickness values. This problem is axisymmetric because the local geomagnetic field ${{{\mathbf{B}}}_{0}}$ is directed along the vertical $z$ axis, and the function $\delta {{J}_{a}}$ only depends on $r$. Therefore, we will further use a cylindrical coordinate system in which all functions do not depend on the azimuthal angle $\varphi$. Then the projections of the vector $\delta {\mathbf{J}}$ on the unit axes of the cylindrical coordinate system will look like this:

$$\delta {{J}_{z}} = {{\sigma }_{\parallel }}{{E}_{z}},\,\,\,\,\delta {{J}_{r}} = {{\sigma }_{P}}{{E}_{r}},\,\,\,\delta {{J}_{\varphi }} = {{\sigma }_{H}}{{E}_{r}},$$

(4)

where ${{E}_{z}}$ and ${{E}_{r}}$ are the vertical and horizontal projections of the electric field strength vector. By inserting the projections of the current density vector (4) into the current continuity equation $\nabla {\kern 1pt} \cdot {\kern 1pt} \delta {\mathbf{J}} = 0$, we will obtain:

$${{\sigma }_{\parallel }}\frac{{\partial {{E}_{z}}}}{{\partial z}} + {{\sigma }_{P}}\frac{1}{r}\frac{\partial }{{\partial r}}\left( {r{{E}_{r}}} \right) = 0.$$

(5)

Moreover, in the stationary case, the equation $\nabla \times {\mathbf{E}} = 0$ is satisfied, from which it follows that

$$\frac{{\partial {{E}_{r}}}}{{\partial z}} - \frac{{\partial {{E}_{z}}}}{{\partial r}} = 0.$$

(6)

It should be noted that in the case of vertical field ${{{\mathbf{B}}}_{0}}$ considered here, the Hall conductivity is not included in equations (5) and (6), and, therefore, does not affect the electric field. Nevertheless, this conductivity has an effect on the azimuthal current $\delta {{J}_{\varphi }}$.

At the bottom boundary of the ionospheric $E$‑layer, the normal component of the current density should be continuous. This leads to the boundary condition: $\delta {{J}_{z}}\left( {r,0} \right) = \delta {{J}_{a}}\left( {r,0} \right)$ where $\delta {{J}_{a}}\left( {r,0} \right)$ means the atmospheric current density flowing into the ionosphere, which is determined by equation (2). In order to estimate the maximum effect in the E-layer, we will first neglect the current flowing into the F-layer, i.e. we will assume that at the top boundary of the E-layer the following condition is satisfied: $\delta {{J}_{z}}\left( {r,l} \right) = 0$. See the Appendix for the solution to this problem obtained using the Hankel transform by the parameter $r$.

ASSESSMENT OF ELECTRIC FIELDS AND VARIATIONS OF ELECTRON CONCENTRATION IN THE IONOSPHERE

In this section, we will examine how changes in the atmospheric background current are related to the electron concentration in the E-layer of the ionosphere. We will solve the problem of electric field distribution in the E-ionosphere $\left( {0 < z < l} \right)$ by applying the inverse Hankel transform to equation (A8). The radial component of the electric field will be written as:

$$\begin{gathered} {{E}_{r}} = \frac{{\delta {{J}_{{\max }}}r_{0}^{2}}}{{2{{\sigma }_{\parallel }}}}\int\limits_0^\infty {\frac{{\cosh \left\{ {\lambda \left( {l - z} \right)} \right\}}}{{\lambda \sinh \left( {\lambda l} \right)}}} \\ \times \,\,\exp \left( { - \frac{{kr_{0}^{2}}}{4}} \right){{J}_{1}}\left( {kr} \right){{k}^{2}}dk, \\ \end{gathered} $$

(7)

where ${{J}_{1}}\left( {kr} \right)$ stands for the Bessel function of the first kind; $\lambda = k{{\left( {{{{{\sigma }_{P}}} \mathord{\left/ {\vphantom {{{{\sigma }_{P}}} {{{\sigma }_{\parallel }}}}} \right. \kern-0em} {{{\sigma }_{\parallel }}}}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}$. The integral sum in formula (7) is mainly obtained in the interval $0 \leqslant k < {{k}_{0}} = {2 \mathord{\left/ {\vphantom {2 {{{r}_{0}}}}} \right. \kern-0em} {{{r}_{0}}}}$, since, with $k \gg {{k}_{0}}$, the integrand decreases rapidly due to the exponentially decreasing multiplier $\exp \left( { - {{{{k}^{2}}r_{0}^{2}} \mathord{\left/ {\vphantom {{{{k}^{2}}r_{0}^{2}} 4}} \right. \kern-0em} 4}} \right)$. Now let us estimate the value of the arguments of the hyperbolic functions in equation (7) for the interval $0 < k < {{k}_{0}}$. By selecting the following numerical values of the parameters: ${{r}_{0}} = 100$ km, $l = 30$ km, ${{\sigma }_{\parallel }} = 0.1$ S/m, ${{\sigma }_{P}} = {{10}^{{ - 4}}}$ S/m (night conditions), we will obtain the inequality: $\lambda l < \left( {{{2l} \mathord{\left/ {\vphantom {{2l} {{{r}_{0}}}}} \right. \kern-0em} {{{r}_{0}}}}} \right){{\left( {{{{{\sigma }_{P}}} \mathord{\left/ {\vphantom {{{{\sigma }_{P}}} {{{\sigma }_{\parallel }}}}} \right. \kern-0em} {{{\sigma }_{\parallel }}}}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}} \approx 0.02$. This means that we can simplify integrand of (7) by assuming that $\cosh \left\{ {\lambda \left( {l - z} \right)} \right\} \approx 1$ and $\sinh \left( {\lambda l} \right) \approx \lambda l$. Then in the first approximation, we will get:

$${{E}_{r}} \approx \frac{{\delta {{J}_{{\max }}}r_{0}^{2}}}{{2{{\Sigma }_{P}}}}\int\limits_0^\infty {\exp \left( { - \frac{{{{k}^{2}}r_{0}^{2}}}{4}} \right){{J}_{1}}\left( {kr} \right)dk} ,$$

(8)

where ${{\Sigma }_{P}} = {{\sigma }_{P}}{{l}_{{}}}$ is the height-integrated Pedersen conductivity of the E-layer. It should be noted that in this approximation, ${{E}_{r}}$ only depends on the radius $r$, and the dependence on $z$ appears in the next approximation.

We can similarly simplify the equation for the vertical component of the electric field. In the first approximation, we will get:

$${{E}_{z}} \approx \frac{{\delta {{J}_{{\max }}}r_{0}^{2}}}{{2{{\sigma }_{\parallel }}}}\left( {1 - \frac{z}{l}} \right)\int\limits_0^\infty {\exp \left( { - \frac{{{{k}^{2}}r_{0}^{2}}}{4}} \right){{J}_{0}}\left( {kr} \right)kdk} .$$

(9)

Integrals (8) and (9) are reduced to the tabular ones (Gradshteyn and Ryzhik, 2007). The result will be:

$$\begin{gathered} {{E}_{r}} \approx \frac{{\delta {{J}_{{\max }}}r_{0}^{2}}}{{2r{{\Sigma }_{P}}}}\left\{ {1 - \exp \left( { - \frac{{{{r}^{2}}}}{{r_{0}^{2}}}} \right)} \right\}, \\ {{E}_{z}} \approx \frac{{\delta {{J}_{{\max }}}}}{{{{\sigma }_{\parallel }}}}\left( {1 - \frac{z}{l}} \right)\exp \left( { - \frac{{{{r}^{2}}}}{{r_{0}^{2}}}} \right). \\ \end{gathered} $$

(10)

By direct substitution, we can see that approximate solutions (10) satisfy initial equations (5), (6).

Analysis of formulas (10) shows that the maximum values ${{E}_{r}}$ and radial current densities $\delta {{J}_{r}}$ are reached at $r \approx 1.2{{r}_{0}}$. The largest values of these quantities linearly depend on the amplitude of the inflowing atmospheric current $\delta {{J}_{{\max }}}$ and the characteristic size ${{r}_{0}}$ of the perturbed region: ${{E}_{{r\max }}} \approx {{0.32\delta {{J}_{{\max }}}{{r}_{0}}} \mathord{\left/ {\vphantom {{0.32\delta {{J}_{{\max }}}{{r}_{0}}} {{{\Sigma }_{P}}}}} \right. \kern-0em} {{{\Sigma }_{P}}}}$ and $\delta {{J}_{{r\max }}} \approx {{0.32\delta {{J}_{{\max }}}{{r}_{0}}} \mathord{\left/ {\vphantom {{0.32\delta {{J}_{{\max }}}{{r}_{0}}} l}} \right. \kern-0em} l}$. The longitudinal field ${{E}_{z}}$ practically disappears in the ionosphere due to the high conductivity of the plasma along the geomagnetic field (${{\sigma }_{\parallel }} \to \infty$).

The atmospheric current near the bottom boundary of the ionosphere is mostly carried by free electrons, since they are much more mobile than ions. In the E-layer of the ionosphere, the drift motion of electrons and ions and their contributions to the total current are different because electrons are magnetized and ions are not. Below, we will estimate the maximum values of electric field variations and electron concentrations in the lower ionosphere associated with this effect.

Let us consider a simple model of ionospheric plasma consisting of electrons and singly charged positive and negative ions and neutral molecules of the same type. The change in the electron concentration in the ionosphere ${{n}_{e}}$ is described by the balance equation:

$$\frac{{\partial {{n}_{e}}}}{{\partial t}} = {{\gamma }_{c}} - \alpha {{n}_{e}}{{n}_{ + }} - {{\nu }_{a}}{{n}_{e}} + {{e}^{{ - 1}}}\nabla {\kern 1pt} \cdot {\kern 1pt} {{{\mathbf{j}}}_{e}} + \nabla {\kern 1pt} \cdot {\kern 1pt} \left( {{{D}_{e}}\nabla {{n}_{e}}} \right).$$

(11)

Here ${{\gamma }_{c}}$ is the formation rate of electron-ion pairs as a result of short-wave radiation and cosmic rays; ${{n}_{ + }}$ is the density of positive ions; $\alpha$ is the electron-ion recombination coefficient; ${{\nu }_{a}}$ is the frequency of electrons attaching to neutral molecules; $e$ is the elementary charge; ${{{\mathbf{j}}}_{e}}$ is the electron current density; ${{D}_{e}}$ is the electron diffusion coefficient. All of the above coefficients depend on the altitude $z$. At the altitudes of the E-layer, the loss of electrons is mainly determined by the recombination of electrons with ions (Kelley, 1989). Therefore, in equation (11), we will neglect the attachment of electrons to air molecules and diffusion of electrons, i.e., the terms ${{\nu }_{a}}{{n}_{e}}$ and $\nabla {\kern 1pt} \cdot {\kern 1pt} \left( {{{D}_{e}}\nabla {{n}_{e}}} \right)$. In this case, it follows from the condition of quasi-neutrality of the plasma that ${{n}_{ + }} \approx {{n}_{e}}$.

Let us consider a stationary case when the time derivative in equation (11) can be neglected. In this approximation, it follows from equation (11) that small variations in the electron current density $\delta {{{\mathbf{j}}}_{e}}$ and electron concentration $\delta {{n}_{e}}$ are related by the equation:

$$\delta {{n}_{e}} \approx \frac{1}{{2\alpha {{n}_{e}}e}}\nabla \cdot \delta {{{\mathbf{j}}}_{e}}.$$

(12)

This equation describes the perturbation of the ionospheric plasma density that occurs when the stationary current flows into the E-layer, spreads out through the layer, and attenuates due to recombination. The components of the vector $\delta {{{\mathbf{j}}}_{e}}$ can be found from the formulas: $\delta {{j}_{{ez}}} = {{\sigma }_{{e\parallel }}}{{E}_{z}}$ and $\delta {{j}_{{er}}} = {{\sigma }_{{eP}}}{{E}_{r}}$, where ${{\sigma }_{{e\parallel }}}$ and ${{\sigma }_{{eP}}}$ denote the components of the longitudinal and Pedersen conductivity of the plasma, conditioned by the motion of electrons only. It should be noted that the conductivity components included in the current in formula (10) are determined by the contribution of both electrons and ions, i.e., ${{\sigma }_{\parallel }} = {{\sigma }_{{e\parallel }}} + {{\sigma }_{{i\parallel }}}$ and ${{\sigma }_{P}} = {{\sigma }_{{eP}}} + {{\sigma }_{{iP}}}$, where the indices $e$ and $i$ refer to electrons and ions, respectively. Provided that $\delta {{n}_{e}} \ll {{n}_{e}}$, variations in the electron concentration have little effect on the conductivity of the ionospheric plasma. Therefore, according to the model we have adopted, the components of electronic and ionic conductivity have constant values within the $E$-layer of the ionosphere.

Let us express the electron current density components $\delta {{j}_{{ez}}}$ and $\delta {{j}_{{er}}}$ through electric field components (10) and plasma conductivity components, and then apply them to equation (12). Then after a number of transformations, we will get:

$$\frac{{\delta {{n}_{e}}}}{{{{n}_{e}}}} = \frac{{\delta {{J}_{{\max }}}\left( {{{\sigma }_{{e\parallel }}}{{\sigma }_{{iP}}} - {{\sigma }_{{i\parallel }}}{{\sigma }_{{eP}}}} \right)}}{{2\alpha e{{\sigma }_{\parallel }}n_{e}^{2}{{\Sigma }_{P}}}}\exp \left( { - \frac{{{{r}^{2}}}}{{r_{0}^{2}}}} \right).$$

(13)

We can simplify this formula by taking into account that ${{\sigma }_{{e\parallel }}} \gg {{\sigma }_{{i\parallel }}}$, and hence ${{\sigma }_{\parallel }} \approx {{\sigma }_{{e\parallel }}}$. The result will be:

$$\frac{{\delta {{n}_{e}}}}{{{{n}_{e}}}} \approx \frac{{\delta {{J}_{{\max }}}{{\sigma }_{{iP}}}}}{{2\alpha en_{e}^{2}{{\Sigma }_{P}}}}\exp \left( { - \frac{{{{r}^{2}}}}{{r_{0}^{2}}}} \right).$$

(14)

Thus, the spatial structure of the perturbation of the electron concentration in the E-layer of the ionosphere depends on $r$ approximately the same as the atmospheric current variations $\delta {{J}_{a}}$: these variations decrease in the transverse direction with the characteristic scale ${{r}_{0}}$ and have the maximum values at $r = 0$. Formula (14) leads to an estimation equation for the perturbation of the ionospheric plasma density:

$${{\left( {\frac{{\delta {{n}_{e}}}}{{{{n}_{e}}}}} \right)}_{{\max }}} \simeq \frac{{\delta {{J}_{{\max }}}}}{{2\alpha eln_{e}^{2}}}\frac{{{{\sigma }_{{iP}}}}}{{{{\sigma }_{P}}}},$$

(15)

where ${{\sigma }_{P}} = {{\sigma }_{{eP}}} + {{\sigma }_{{iP}}}$. For numerical estimates, we will use characteristic values of the parameters for the E-layer of the night ionosphere: $\alpha = {{10}^{{ - 13}}}$ m³/s, ${{n}_{e}} = 3.1 \times {{10}^{9}}$ m⁻³, $l = 50$ km, ${{\sigma }_{{iP}}} = 3 \times {{10}^{{ - 6}}}$ S/m, ${{\sigma }_{P}} = 5 \times {{10}^{{ - 6}}}$ S/m (Ivanov-Kholodnii, 1990; Kelley, 1989). The typical value of fair weather current density is in the order of ${{J}_{a}}$ = 2 pA/m². As an anomalous variation of this current, let us take a noticeable value of 50% of the background current, i.e., $\delta {{J}_{{\max }}} =$ 1 pA/m². From estimate (10), it follows that such a current produces a horizontal electric field in the night-time E‑layer (${{\Sigma }_{P}} = 0.3$ S) with the value $\sim {\kern 1pt} {{10}^{{ - 4}}}$ mV/m in the region with the scale ${{r}_{0}} = 100$ km. By inserting the current and ionospheric parameters into estimation equation (15), we will find the maximum value of the electron concentration variation: ${{\left( {{{\delta {{n}_{e}}} \mathord{\left/ {\vphantom {{\delta {{n}_{e}}} {{{n}_{e}}}}} \right. \kern-0em} {{{n}_{e}}}}} \right)}_{{\max }}} \approx 4 \times {{10}^{{ - 5}}}$. For daytime ionospheric parameters, the relative perturbation will be even smaller.

This work has not considered the effect of currents exiting from the E-region into the F-region of the ionosphere and closing on the magnetically conjugate ionosphere. Estimates under the simplest models show that taking into account this effect leads to replacing ${{\Sigma }_{P}}$ in formula (14) by the sum ${{\Sigma }_{P}} + {{\bar {\Sigma }}_{P}}$, where ${{\bar {\Sigma }}_{P}}$ is the integral Pedersen conductivity of the conjugate ionosphere. Consequently, the relative variation of the electron concentration in the E-layer is still very small.

In conclusion, let us make a few remarks about the “radon” effect in the F-layer of the ionosphere. If we take into account that the current in the F-layer must be finite, we can obtain a rough estimate of the variation of the electron concentration in this region by using formula (12) with the F-layer parameters: $\alpha = {{10}^{{ - 16}}}{\kern 1pt} - {\kern 1pt} 3 \times {{10}^{{ - 14}}}$ m³/s and ${{n}_{e}} = {{10}^{{10}}} - {{10}^{{11}}}$ m⁻³ (for night conditions (Ivanov-Kholodnii, 1990)). The longitudinal conductivity of the plasma in this region of the ionosphere is much larger than the transverse conductivity, so the longitudinal electron current $\delta {{j}_{{ez}}}$ is carried here almost unchanged, i.e., $\nabla {\kern 1pt} \cdot {\kern 1pt} \delta {{{\mathbf{j}}}_{e}} \approx 0$ and, therefore, in the first approximation $\delta {{n}_{e}} \approx 0$. If we assume that the longitudinal current in the F-region attenuates for some reason and use a rough estimate: $\nabla {\kern 1pt} \cdot {\kern 1pt} \delta {{{\mathbf{j}}}_{e}} \approx {{\delta {{J}_{{\max }}}} \mathord{\left/ {\vphantom {{\delta {{J}_{{\max }}}} L}} \right. \kern-0em} L}$, where $L \approx 300 - 500$ km is the characteristic size of the F-region, we will obtain from formula (12): ${{\left( {{{\delta {{n}_{e}}} \mathord{\left/ {\vphantom {{\delta {{n}_{e}}} {{{n}_{e}}}}} \right. \kern-0em} {{{n}_{e}}}}} \right)}_{{\max }}} \approx$ $2 \times {{10}^{{ - 8}}}{\kern 1pt} - {\kern 1pt} {{10}^{{ - 3}}}$. Although this estimate varies over such a wide range, the relative variation in the electron concentration is very small. Consequently, variations in radon emanation from the soil and the associated changes in conductivity of the lower atmosphere have almost no effect on the electron concentration and TEC in either the E- or the F‑layers of the ionosphere.

DISCUSSION AND CONCLUSIONS

This work has examined the plausibility of the hypothesis that variations in radon emission could affect the vertical atmospheric background current, which, in turn, would lead to variations in ionospheric plasma density. To test this hypothesis, we have solved a model problem concerning the spreading and recombination of the vertical atmospheric current in the E-layer of the ionosphere and have given a theoretical estimate of the change in the electron concentration associated with this effect. The theoretical analysis has shown that local variations in the background current associated with changes in the radon concentration in the lower atmosphere have almost no effect on the distribution of electrons in the ionosphere. For example, atmospheric current variations of 50% associated with radon emission can cause a relative change in electron concentration ${{\left( {{{\delta {{n}_{e}}} \mathord{\left/ {\vphantom {{\delta {{n}_{e}}} {{{n}_{e}}}}} \right. \kern-0em} {{{n}_{e}}}}} \right)}_{{\max }}}\sim {{10}^{{ - 3}}}{\kern 1pt} - {\kern 1pt} {{10}^{{ - 5}}}$ in the ionosphere. Although the ionospheric parameters are subject to strong variations depending on the time of day, solar activity, etc., estimates of variations of ionospheric parameters give such small values, which makes the hypothesis under study hardly plausible. Only uniquely large changes in the integral conductivity of the atmosphere, for example, as a result of high-altitude nuclear explosions, volcanic eruptions, and over high mountain ranges, can cause changes in the fields and the plasma in the ionosphere (Ponomarev et al., 2011).

The conclusion about the implausibility of this hypothesis can be supported by the experimental results of the work by (Schekotov et al., 2021), which studied the possible connection between anomalous extremely low frequency (ELF) variations of the Earth’s electromagnetic field before earthquakes and pre-seismic air temperature changes, which could be associated with an increase in air ionization due to the growth of Rn activity in the surface layer of the atmosphere. Analysis of the observational data showed the absence of any correlation between these phenomena, since neither the position of the projection of the electromagnetic perturbation source nor the time of its appearance coincided with the similar parameters of the temperature anomalies. Moreover, even the more powerful ionizing radiation caused by the Fukushima nuclear power plant disaster did not lead to electromagnetic field anomalies in the surface atmosphere (Schekotov et al., 2021). This result is natural, since ionization itself does not lead to the formation of space charges in the absence of external forces. A possible mechanism of the effect of anomalously strong ionization of the surface atmosphere by radioactive emissions on the ionosphere during accidents at nuclear power plants needs to be considered separately (Boyarchuk et al., 2013).

The work by (Pulinets and Ouzounov, 2011) suggests that an increase in air ionization by radioactive radon leads to an accumulation of charges on the so-called large ionic clusters (LIC), which become centers of water vapor condensation in the surface atmosphere. According to their estimates, the condensation of steam on the LIC will be accompanied by the release of latent heat of vaporization and intense heating of the air in the lower atmosphere, which will lead to an increase in the average air temperature by 2–3 K. This hypothesis could explain the increased level of upward infrared radiation flux, which has sometimes been observed on satellites over seismically active regions (Tronin, 1996; Tramutoli et al., 2001; Surkov et al., 2006). Nevertheless, the theoretical estimate shows that this hypothesis can explain this effect only if the LIC concentration is unrealistically high (Surkov, 2015), and the estimate of the real heat flux associated with the “radon effect” is 10–12 orders of magnitude lower than the value given in the work by (Pulinets and Ouzounov, 2011).

Thus, the only significant “radon effect” that can be explained and supported by theoretical analysis is a change in the air electrical conductivity in the lower atmosphere and a 10–15% decrease in the vertical air column resistance. The anomalous increase in radon concentration, which is sometimes observed in seismically active regions, cannot explain, at least according to the above hypothesis, the observed TEC changes in the ionosphere.

Additional ionospheric effects may be associated with an inhomogeneous distribution of the real unperturbed ionospheric plasma and the presence of external ionospheric fields. Small plasma perturbations due to “radon” or some other effect will lead to polarization of plasma inhomogeneities in the external electric field and transfer of the polarization field along magnetic induction lines to the upper ionosphere. In the collisionless plasma of the upper ionosphere, even a weak field can cause an altitude shift of the F-layer maximum. However, a numerical estimation of this effect needs to be considered separately.

APPENDIX

By applying the Hankel transform by $r$ to equations (5) and (6), we obtain:

$${{\sigma }_{\parallel }}e_{z}^{'} + k{{\sigma }_{P}}{{e}_{r}} = 0,$$

(A1)

$$e_{r}^{'} + k{{e}_{z}} = 0,$$

(A2)

where $k$ is the Hankel transform parameter, and the prime marks represent the derivatives of $z$. In addition, the following symbols are used here:

$$\begin{gathered} {{e}_{z}}\left( {k,z} \right) = \int\limits_0^\infty {{{E}_{z}}\left( {r,z} \right){{J}_{0}}\left( {kr} \right)rdr} , \\ {{e}_{r}}\left( {k,z} \right) = \int\limits_0^\infty {{{E}_{r}}\left( {r,z} \right){{J}_{1}}\left( {kr} \right)rdr} , \\ \end{gathered} $$

(A3)

where ${{J}_{0}}\left( x \right)$ and ${{J}_{1}}\left( x \right)$ are the Bessel functions of the first kind of zero order and first order. From equations (A1) and (A2), we obtain:

$$e_{r}^{{''}} - {{\lambda }^{2}}{{e}_{r}} = 0,$$

(A4)

where $\lambda = k{{\left( {{{{{\sigma }_{P}}} \mathord{\left/ {\vphantom {{{{\sigma }_{P}}} {{{\sigma }_{\parallel }}}}} \right. \kern-0em} {{{\sigma }_{\parallel }}}}} \right)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}$. The general solution of equation (A4) has the form:

$${{e}_{r}} = {{c}_{1}}\exp \left( {\lambda z} \right) + {{c}_{2}}\exp \left( { - \lambda z} \right),$$

(A5)

where ${{c}_{1}}$, ${{c}_{2}}$ are undefined coefficients.

At the bottom boundary of the E-layer at $z = 0$, the normal component of the current density should be continuous. By applying the Hankel transform to equation (2), we will write this boundary condition as follows:

$$e_{r}^{'}\left( {k,0} \right) = - \frac{{\delta {{j}_{{\max }}}kr_{0}^{2}}}{{2{{\sigma }_{\parallel }}}}\exp \left( { - \frac{{{{k}^{2}}r_{0}^{2}}}{4}} \right).$$

(A6)

The variation of the vertical current density $\delta {{j}_{z}}$ becomes zero at the top boundary of the E-layer $z = l$. This leads to the boundary condition $e_{r}^{'}\left( {k,l} \right) = 0$. By inserting solution (A5) for ${{e}_{r}}$ into the above boundary conditions, we arrive at a system of algebraic equations for the coefficients ${{c}_{1}}$ and ${{c}_{2}}$. The solution to this system is as follows:

$${{c}_{{1,2}}} = \frac{{\delta {{J}_{{\max }}}kr_{0}^{2}}}{{4{{\sigma }_{\parallel }}\lambda \sinh \left( {\lambda l} \right)}}\exp \left( { \mp \lambda l - \frac{{kr_{0}^{2}}}{4}} \right).$$

(A7)

By inserting these coefficients into formula (A5), we obtain the solution to the problem:

$${{e}_{r}} = \frac{{\delta {{J}_{{\max }}}kr_{0}^{2}\cosh \left\{ {\lambda \left( {l - z} \right)} \right\}}}{{2{{\sigma }_{\parallel }}\lambda \sinh \left( {\lambda l} \right)}}\exp \left( { - \frac{{kr_{0}^{2}}}{4}} \right).$$

(A8)

The vertical component of the field is found by means of the equation ${{e}_{z}} = - {{e_{r}^{'}} \mathord{\left/ {\vphantom {{e_{r}^{'}} k}} \right. \kern-0em} k}$. The final solution to the problem is found by the inverse Hankel transform.