Formation of Multilayered Sporadic E under an Influence of Atmospheric Gravity Waves (AGWs)

The formation of multilayered sporadic E by atmospheric gravity waves (AGWs), propagating in the mid-latitude lower thermosphere, is shown theoretically and numerically. AGWs with a vertical wavelength smaller than the width of the lower thermosphere lead to the appearance of vertical drift velocity nodes (regions where the ions’ vertical drift velocity, caused by these waves, is zero) of heavy metallic ions (Fe + ). The distance between the nearest nodes is close to the AGWs’ vertical wavelength. When the divergence of the ion vertical drift velocity at its nodes has a minimal negative value, then these charged particles can accumulate into Es-type thin layers and the formation of multilayered sporadic E is possible. We showed the importance of the ions’ ambipolar diffusion in the formation of Es layers and control of their densities. Oblique downward or upward propagation of AGWs causes downward or upward motion of the ion vertical drift velocity nodes by the vertical propagation phase velocity of these waves. In this case, the formed Es layers also descend or move upward with the same phase velocity. The condition, when the horizontal component of AGWs’ intrinsic phase velocity (phase velocity relative to the wind) and background wind velocity have same magnitudes but opposite directions, is favorable for the formation of the multilayered sporadic E at fixed heights of the sublayers. When the AGWs are absent, then horizontal homogeneous wind causes the formation of sporadic E but with a single peak. In the framework of the suggested theory, it is shown that, in the lower thermosphere, the wind direction, magnitude, and shear determine the development of the processes of ion/electron convergence into the Es-type layer, as well as their density divergence. Consideration of arbitrary height profiles of the meridional and zonal components of the horizontal wind velocity, in case of AGW propagation, should be important for the investigation of the distribution and behavior of heavy metallic ions on regional and global scales.


Introduction
The lower thermosphere is a weakly ionized medium and the behavior of the charged particles at the global and regional scales should be significantly determined by neutral wind. Neutral wind and its significant variations can be determined by atmospheric gravity waves (AGWs), planetary waves, and tidal motion [1,2]. The sporadic E is an ionospheric irregular structure observed in the lower thermosphere, which can be a manifestation of neutral atmosphere-ionosphere coupling in the presence of horizontal wind. The wind shear caused by tidal motion or atmospheric waves and instabilities, as well as wind and electric field directions, influence the formation and behavior of the Es layers [3][4][5][6][7].
In this study, the observed additional peak or sublayers (multilayered structures) in sporadic E [8][9][10] can be caused by vertical/oblique propagation of AGWs or tidal winds. Here, unlike the previous studies [11][12][13], homogeneous horizontal wind can cause single peak sporadic E (Es layer), but the 4 www.videleaf.com presence of AGWs with vertical wavelengths smaller than the width of the height region between about 90 and 150 km can cause additional sublayers (Es layers). In this case, the vertical changes in the wind direction, magnitude, and shear causes the appearance of nodes of ion vertical drift velocity-heights where the ion vertical drift velocity, caused by the AGWs, is zero. When in these regions of nodes, the drift velocity divergence has minimal negative values, then the ions/electrons converge in the Es-type thin layer. The distances between such nodes and the sporadic E sublayers are close to the AGWs' vertical wavelength. In this case, the Es layer densities are also controlled by ion ambipolar diffusion. When the ion vertical drift velocity divergence has maximal positive values at the region of its nodes, then the ion/electron density is depleted. The charged particles' density depletion is also an irregular/sporadic phenomenon, with increased interest of investigation [14].
According to the present theoretical study, the development of ion convergence processes and the formation of Es-type layers in a given mid-latitude lower thermosphere region, as well as their localization, are determined by horizontal wind direction, magnitude, and shear. In the framework of the presented theory, these processes are more predictable than by the windshear theory [15][16][17][18][19]. According to the windshear theory, the sporadic E is formed at heights, where ,e.g., for the northern hemisphere, due to wind velocity vertical shear, the southward wind changes to the northward one, or the eastward wind to the westward one [19]. Our study shows that the Es layers do not always form in the regions where horizontal wind changes direction to the opposite, which is an observed phenomenon [8,20,21]. We show analytically that the tendency of formation of the sporadic E can be caused by a certain direction, magnitude, and shear of arbitrary horizontal wind. In the presence of AGWs, the horizontal wind height profile is assumed as the sum of its homogeneous and variable parts caused by the AGWs (or tidal wind). This theoretical consideration will be followed by corresponding numerical simulations showing (1) the possibility of formation of multilayered sporadic E and its sublayers under influence of AGWs; (2)Es layers located at the ion vertical drift velocity nodes; (3) the importance of the effect of homogeneous www.videleaf.com horizontal wind on the ion/electron convergence processes; and (4) the possibility of formation of sporadic E by homogeneous winds. Considering the horizontal wind at about the 120 km height region, determined by the sum of the background wind and the AGW velocities, compared to the values of the Horizontal Wind Model 2014 (HWM14) and some measured data [1,22], we noted the importance of these waves in the formation of Es layers, at the region where they are frequently observed (below 120 km). The analytically obtained condition of formation and localization of the Es layers, which was applied to the linear AGWs, also includes the possibility of its development in case of various horizontal wind height profiles, which can be caused by both linear and nonlinear waves.
We demonstrate the applicability of the presented theory, describing heavy metallic ion behavior in the lower thermosphere, including the phenomena of sporadic E formation and ion depletion under influence of horizontal wind, taking into account the propagation of the AGWs. The applicability of this theory shows its importance for the investigation of regional and global peculiarities of heavy metallic ion distribution [23,24].

Theory of Sporadic E Formation under the Influence of AGWs and Background Horizontal Wind
In this section, we describe the theory of the formation of sporadic E in the nighttime mid-latitude lower thermosphere. For simplicity, we consider a case when the heavy metallic ions Fe + are dominant. Due to the quick quenching of molecular ions after sunset [10], the quasi neutrality in the region is mainly determined by the density of these metallic ions. The approximate analytical solution of the continuity equation for the ion/electron density height profile shows the possibility of formation of multilayered sporadic E in case of the presence of background horizontal wind and oblique propagation of the AGWs.
We study the mid-latitude lower thermosphere where the formation of sporadic E is mostly observed. The thermospheric 6 www.videleaf.com wind influences the development of ion convergence/divergence processes via combined action of the Lorentz forcing and ionneutral collision. Here, the ion ambipolar diffusion also influences the ion/electron motion and formation of a narrow and dense Es-type layer. We consider the horizontal neutral wind velocity is the unit imaginary number), and can be described by the following equations [25,26]:  (1) and (2), and its influence on the ion vertical drift velocity is negligibly small [5,19]. In Equations (3) and (4),  is the observed wave frequency at the given midlatitude region of the lower thermosphere, which, in the case of homogeneous background horizontal wind determined by the dispersion equation [27]: Here, g  is the AGWs intrinsic frequency (frequency relative to the background wind) and has the following form [25]:   where H is the atmospheric scale height;  (3) and (4)), and can lead to stationary ( 0   ) velocity profiles (Equations (1) and (2)) over the considered mid-latitude region. Such a stationary condition for propagation of the AGWs is similar to the case of tidal winds for which the horizontal wavelengths and period are expected to be much greater than those of the AGWs. 8 www.videleaf.com The dependence of ions' vertical drift velocity i w on the meridional x V and zonal y V components of the neutral particles horizontal wind velocity and ambipolar diffusion a D can be described by the following equations [6,13,28]: and where v w i is the ion vertical drift velocity caused by only horizontal wind via combined action of the Lorentz forcing and ion-neutral collisions [19]. In Equations (7) and (8),  (7) and (8)). These coefficients in v w i (Equation (8) To investigate the behavior of the height profile of the ion/electron density ) , In the continuity Equation (13), the production and loss rates of the long-lived metallic ion Fe + or the electrons for the characteristic time of sporadic E formation (e.g., about 1h) are www.videleaf.com neglected [30]. Here, the behavior of ) , ( t z N e under the influence of the meridional x V and zonal y V components of the horizontal wind velocity (V) and their vertical shears ( , Equation (14), and ) ( ' z C sh , Equation (15). The ions' ambipolar diffusion, Equation (11), is also taken into account. Here, for a given wind velocity magnitude and direction, the vertical changes ( (14)). On the basis of Equations (9) and (10) ( 2   2 ) and below its bottom of about z < 90 km( (ion gyrofrequency equal to the ion-neutral frequency) occurs at a height of about z=121 km. www.videleaf.com

Analytical Approach for Ion/Electron Density Height Profile Behavior and Condition of Sporadic E Formation under Influence of Horizontal Wind
To show the importance of the (14) and (15), we solve Equation (13)  , has the following form: At the initial time of o t t  , Equation (16) corresponds to the ion/electron Gaussian-type distribution layer with maximal density om N (peak density) at the corresponding om z z  height (peak height). In case of the absence of wind ( 0 According to Equation (16), when, at the peak height of the ion/electron density initial layer,( ). Here, it is important to note that, when the has its maximal values at certain www.videleaf.com heights, then the ion/electron convergence into a thin horizontal layer becomes possible. The assumed dominance of the longlived heavy metallic ions Fe + in the nighttime lower thermosphere also assumes the negligibly small changes in total electron content (TEC). If, in this case, the relatively big increase in ion/electron density exceeds their diffusive displacement, then the ion/electron convergence into a thin layer and the sporadic E may be formed. So, according to Equation (16), the condition necessary for the formation of sporadic E is: According to Equations (7)-(10), (14), and (15), when In this case, the condition (17) of the ion/electron convergence into a thin layer (when ) may happen for the given height region of the lower thermosphere. Hereafter, the www.videleaf.com coefficients ' v C , Equation (14), and ' sh C , Equation (15), will be referred to as the ion convergence ( ) rates, respectively, caused by the horizontal wind (V) and its vertical shear.

Vertical Motion of Es Layers and Their Localization during Presence of Homogeneous Wind and AGWs
According to Equation (16), the development of the ion/electron convergence processes ( ) and the formation of the Es layer is also possible during the drift of its peak height i . In this case, taking into account the vertical changes of the ion drift velocity v w i , the Es layers upward ( ) motion will continue until its localization either at the region of the ion drift velocity node or at the one where it vanishes. Here, under the ion vertical drift velocity v w i (Equation (8)) nodes, we assume the height region with 0 ) heights, where this drift velocity modulation by the AGWs is zero ( ). Later, in numerical simulations, we will consider the case where the v w i nodes are determined by the AGWs and the certain directed homogeneous wind can cause the Es-type layer localization at height regions In the lower thermosphere, at the height region 90-150 km, for the modeled [1] and observed [22] horizontal wind height profiles, the region with 0 ' v  C always occurs determined by the wind direction and magnitude (Equation (14)), and/or the one 14 www.videleaf.com , Equation (17), is the more precise criteria for the formation of the Es layer than the condition According to the height profile of the horizontal wind, given by Equations (1) and,therefore, the formation of a high density thin Estype layer by only the wind magnitude and direction effect is also possible. During propagation of AGWs, the wind direction, magnitude ( processes as well. Here, the character of changes in the wind direction and magnitude with height depends on the vertical wavelength z  . When  4) and (8) ; therefore, the formation of the multilayered sporadic E is possible.

Results and Discussion
By numerical solutions of Equation (13) (1) and (2), also can form the Es layer, in cases when there are no atmospheric waves. In the presented simulation, the neutral densities of the lower thermosphere are used from the NRLMSISE-00 model [35], for the midlatitude regions 45° ± 2° N, 45°± 2° E, and I=61° ±2°. The results can be extended for other mid-latitude regions of the northern or/and southern hemispheres, as well as for various directions and magnitudes of the horizontal wind. www.videleaf.com

Formation of Es Layers at the Fixed Heights by Stationary AGWs
At the beginning, we considered how the AGWs with observed frequency 0   (stationary AGWs), as per Equations (5) and (6), influence on the behavior of the ion/electron density and formation of sporadic E. In this case, the northward wind velocity 0  ox V is opposite and equal to the southward intrinsic phase velocity 0     Ion drift velocity also has nodes in the regions between the Es layers, where occur in the vicinity of heights 108 km, 120 km, and 133 km. When the wind direction, magnitude (with The initial phase of AGWs (Equations (3) and (4)) slightly influences the formation process and behavior of the Es layers in the lower thermosphere. Changes of the initial phase can shift the height of the ion drift velocity nodes. Figure 1a,b also show that ion ambipolar diffusion has a significant effect on the formed Es layers densities (Equations (11), (16), and (17) ) leads to a bigger decrease of the Es densities (Figure 1b).
Note that, according to the analytical solution (16), the location of the initial ion/electron layer peak height mo h close to the ion drift velocity node (caused by AGWs) is more favorable for formation of a relatively high density Es layer than its other locations. These various cases are not demonstrated for brevity. Figure 1c shows that the only homogeneous meridional ). Such propagation of the AGWs cause wave-like variations in the height of the ion drift velocity nodes and its convergence (or divergence) rates as well. ). In this case, the Es layers initial peak height is located at the same region as in Figure 1a. Here, the Es layers are descending (panel a and b) with the speed about equal to the wave vertical phase velocity (−0.2m/s). In this case the ion drift velocity nodes are also descending by the speed close to the AGWs' vertical phase velocity (panel e). Here the regions with maximal (or minimal) ion convergence (or divergence) rates shifting, similarly to the downward motion of the waves. This is expectable from Equations (1)-(10), (14), and (15). In this case, the descending regions of the ion/electron convergence (along with descending nodes) also cause accumulation of the charged particles into thin layers. These formed Es-type layers follow the nodes and convergence regions, so they move downward with the AGWs' vertical phase velocity (  (Figure 1a,b). In the case of the descending layers, the formation of one highdensity layer at about 120 km and below occurs, where the portion of homogeneous wind in the value of

Descending Es Layers Formed by Oblique Downward Propagating AGWs
sh is significant. In the case of absence of wind direction and value effect (with , the peak densities of the Es-type layers are significantly smaller (Figure 2b) than during both wind velocity and shear effects (Figure 2a). In this case, unlike the one when 0   , the sporadic E formed mostly with three sublayers is descending to the region with 0 w  i and 0 ) , sh . This causes the lower peak ( km h m 99 1  ) of the Es layer to vanish (panels a and b).

Upward Motion of Es Layers Formed by Oblique Upward Propagating AGWs
The formation of multilayered sporadic E by oblique upward propagating AGWs is also possible, similarly to the case of oblique downward propagating AGWs (Figure 3).  (Figure 1). Here, the effect of wind total convergence on the Es layers' density ( Figure 3a) is bigger than the wind shear effect only (Figure 3b). Figure 3 also shows the importance of ambipolar diffusion on the formation of the Es layers. In the case of upward motion of the Es layers, the density peaks are more noticeable at the lower heights (from 99 km) (Figure 3a,b). The upper peak is vanishing due to an increase in ion ambipolar diffusion. Here, the density in the upper peak is less than in case of the AGWs with an observed frequency 0   (Figure 1a,b).  (2) and (4);(e) ion drift velocity i w , Equation (7) . Here, the AGWs' parameters are the same as in Figure   1.

Discussion
According to the presented theoretical mechanism, the multilayered sporadic E, which is an observed phenomenon [8][9][10], can be formed by AGWs (Figures 1-3). We have showed that the formation of these layers can happen at the ion vertical drift velocity nodes, where vertical drift velocity divergence ( ) has a minimal negative value or the total is maximal. In this case, the distances between the Es layers and also the regions of ion/electron depletion (with are almost the same as the AGWs' vertical wavelength. The downward ( Figure 2) or upward (Figure 3) motions of the Es sublayers should correspond to the wave propagation in the lower thermosphere in case of the presence of homogeneous horizontal wind. Homogeneous background horizontal wind with a certain direction, in case of the absence of AGWs, can form a single-peak sporadic E (Figure 1c). Such behavior of ion/electron density (Figures 1-3) is predictable by Equation (16). In this case, the ion/electron total convergence rate  (14) and (15), can be described by the following equation: The first term ( (18) . This maximum in ion/electron convergence rate occurs at about the 121 km height where the ion gyrofrequency is equal to the ion-neutral collision frequency ( ). There can be four nodes for the vertical wavelength km z 15   of the AGWs in the height region of about 90-150 km. In this case, the maximal positive (or minimal negative) values in the total convergence rates ' ' v sh C C  at the ion drift velocity nodes also could be four. So, these nodes and maximal positive (or minimal negative) values of the ion/electron convergence rate should be localized by the vertical wavelength distances from each other, as was obtained in the numerical results of the formation of the Es layers (Figures 1-3).
Equations (18) Here, for the considered parameters of AGWs, the changes in the distances of the Es layer peak heights, due to the vertical changes of the phase ) (z  , as per Equation (21), are relatively small (Figures 1-3). Equations (18)-(21) also show the possibility of accumulation of ions/electrons into the Es layers at almost fixed heights (Figure 1), in case of a stationary wave ( 0   ).
During propagation of AGWs with greater values in the vertical phase velocity, the time of ion/electron density accumulation into the Es-type layers will be smaller than in case of stationary waves.
Planetary-scale tidal winds or atmospheric waves (with ) resemble the homogeneous horizontal wind. In this case, the condition of max can take place in the region of the greater horizontal wind velocity and the formation of single-peak sporadic E by homogeneous wind is possible (Figure 1c). Therefore, the formation of double-peak sporadic E by tidal wind only or by planetary waves should be rarer than during the propagation of AGWs. The similar horizontal wind profiles of Equations (3) and (4) also have been used for investigation of the formation of the Es layers under the influence of tidal wind [36].
We have used the non-dissipative linear AGW model with exponential growth of the amplitude, Equations (3)-(6) [25], to study the formation of multilayered sporadic E. According to the present theory, using Equations (16) and (17) (5)) is about twice greater than in case of the nonlinear effect ( , and were not taken into account. We assume that, at the region of about 120 km or above, the nonlinear decay of the considered AGWs is also important, especially for the relatively big time. In this case, the AGWs decay into the short scale linear waves increases the number of the ion vertical drift velocity nodes above 120 km. The newly formed convergence nodes in turn result in the ion/electron content redistribution into these nodes and the Es layers peak densities should be smaller above the 120 km height than during its formation in case of the absence of nonlinear processes. Taking into account the observed high values in the horizontal wind below 120 km [22], which, in addition to the tidal wind, can be caused by AGWs, shows the important role of AGWs in more frequent formation of Es layers below 120 km.
AGWs with short wavelengths ( ) also can be important for the formation of the quasi-periodic echo-like structures [37]. In this case, one can also take into account the influence of the vertical component of velocity perturbation, caused by short-period and small-scale AGWs, on the ion vertical drift velocity, as it was done in [13]. As we have shown, the oblique upward propagation of the AGWs can influence the upward motion of the heavy metallic ion layers ( Figure 3); therefore, the extension of the presented theory for equatorial regions should be important to study the formation of the bubblelike structures of plasma [38]. www.videleaf.com The obtained results stimulate the study of the observed coupling of Es layer formation with tectonic events [39] and thunderstorms [24], by generation and propagation of the atmospheric waves from the troposphere to the lower thermosphere. In this case, the presence of linear, nonlinear, and ducted-type AGWs [26] during the formation of the Es layers can be taken into account.
The above study shows the important influence of the AGWs and the background horizontal wind on the behavior of the midlatitude lower thermosphere heavy metallic ions [2]. Here, for the given height profiles of meridional and zonal wind velocity, the development of heavy metallic ion convergence (or divergence) processes is possible. The localization regions of the Es layers can be predicted theoretically. Development of this theory should be important for investigation of heavy metallic ion distribution at the global and regional scales. It is important to take into account the AGWs' propagation from the troposphere to the lower thermosphere for understanding of the regional peculiarities of sporadic E formation and, correspondingly, the heavy metallic ion distribution [23,24,40]. In this case, considering the background electric field, causing the ions' additional drift perpendicular to the geomagnetic field [36,41], also will be important for predictability of Es layers formation and Fe + ion distribution at the mid-latitude and equatorial regions.It is also important to consider ion horizontal transport caused by atmospheric waves and tidal motion, which is the goal of a future study.

Conclusions
We showed the formation of multilayered sporadic E by atmospheric gravity waves (AGWs) propagating in the midlatitude lower thermosphere at about a 90-150 km height. We have used the non-dissipative linear AGW model with exponential growth of the amplitude. In this study, we have not taken into account the possible collapse of these waves during their interaction and propagation in the atmosphere. Such AGWs with a vertical wavelength smaller than the width of this region lead to appearance of the heavy metallic ion (Fe + ) vertical drift www.videleaf.com velocity nodes (height regions with 0   gy y gx x V C V C ). When the ion vertical drift velocity divergence at its nodes has a minimal negative value, then these charged particles can accumulate into the Es-type thin layers and the formation of multilayered sporadic E becomes possible. Here, the importance of the ions' ambipolar diffusion in the formation of Es layers and the control of their densities were shown. When at regions of ion drift velocity nodes, its divergence is positive and have maximal values, then the opposite phenomenon-density depletionoccurs. The distance between the nearest ion/electron convergence nodes is close to the AGWs' vertical wavelength. In case of horizontal homogeneous wind, the Es-type layer is localized at the ion drift velocity nodes or at the region where it vanishes.
It was demonstrated that the AGWs with a vertical wavelength of about a quarter of the width of the lower thermosphere (e.g., Here, significant depletion in ion/electron density between the two nearest Es-type layers was noted. According to the suggested theory and corresponding numerical results, when the horizontal component of the AGWs' intrinsic phase velocity (phase velocity relative to wind) and background wind velocity have the same magnitudes and opposite directions, it is favorable for the formation of multilayered sporadic E with sub-layers at fixed heights. During upward propagation of the AGWs, the Es layer densities are smaller than during its downward propagation. This case corresponds to the stationary height profiles of wind velocity. We have shown that the oblique downward or upward propagation of the AGWs causes the downward or upward motion of the ion vertical drift velocity nodes by the vertical propagation phase velocity of these waves. A horizontal homogeneous wind can form only a single-peak sporadic E.
Considering the horizontal wind at about the 120 km height region, determined by the sum of the background and AGW velocities, comparable to the values of the HWM14 and some measured data [1,22], we noted the importance of these waves in www.videleaf.com the formation of Es layers, at the region where they are frequently observed (below 120 km).
Within the suggested theory, the development of the processes of ion/electron convergence into an Es-type layer and its localization, determined by the wind direction, magnitude, and shear, is more predictable than in case of a windshear effect only. The applicability of this theory to horizontal wind considering propagation of AGWs, which was shown in this study, indicates its importance for investigation of the peculiarities of heavy metallic ion distribution on regional and global scales.