The properties of acoustic–gravity waves (AGWs) in the ionospheric D layer and their role in the D layer–lower thermosphere interaction are studied using the dispersion equation and the reflection coefficient. These analytical equations are an elegant tool for evaluating the contribution of upward-propagating acoustic and gravity waves to the dynamics of the lower thermosphere. It was found that infrasound waves with a frequency of ω>0.035 s−1, which propagate almost vertically, can reach the lower thermosphere. Also, gravity waves with a frequency of ω<0.0087 s−1, with a horizontal phase velocity in the range of 159 m s−1<vh< 222 m s−1 and a horizontal wavelength of 115 km <λp< 161 km, are important for the lower thermosphere dynamics. These waves can cause a temperature rise in the lower thermosphere and have the potential to generate middle-scale travelling ionospheric disturbances (TIDs). The reflection coefficient for AGWs is highly temperature-dependent. During maximum solar activity, the temperature of the lower thermosphere can rise several times. This is the situation where infrasound waves become a prime candidate for the ionospheric D layer–lower thermosphere interaction since strongly reflected gravity waves remain trapped in the D layer. Knowing the temperatures of the particular atmospheric layers, we can also know the characteristics of AGWs and vice versa.
Introduction
Acoustic–gravity waves (AGWs) are able to transport energy and momentum between different layers of the atmosphere. Understanding these waves is essential if we want to comprehend the atmosphere as a system where the layers are coupled. The ionosphere is a part of the Earth's atmosphere located between about 60 and 1000 km above the Earth's surface where the charged particles significantly influence its physical and chemical properties . Knowledge about typical AGW characteristics in the ionosphere is important for modelling of interaction between the ionized and the neutral atmosphere. Ionosphere is constantly exposed to various influences from outer space as well as from the terrestrial atmosphere and lithosphere. Non-periodic and sudden events, such as solar flares (), coronal mass ejection (), solar eclipses , supernova explosions followed by hard X and γ radiation , and lightnings , and some processes in the terrestrial lithosphere, such as volcanic eruptions and earthquakes (), induce space- and time-varying ionospheric perturbations. These disturbances cause numerous complex physical, chemical, and dynamical phenomena in ionosphere () and may directly affect human activities, especially in the telecommunications.
The atmospheric monitoring depends on the altitude of the considered atmospheric layer. The ionospheric D layer at an altitude of about 60 to 90 km, lies below the area being studied by satellite observations and above the region where balloon measurements find their application. Therefore, its monitoring is based on rocket and radar measurements and on the propagation of very low frequency and low-frequency (VLF/LF) radio waves . In this way, it is possible to observe a large part of the low ionosphere and detect local perturbations and sudden events.
The ionospheric D layer and lower thermosphere below 140 km, where AGWs with the specific frequencies and wavelengths are detected, are the focus of this article. We considered the conditions for propagation of AGWs in the D ionospheric layer and their reflection/transmission on the plane boundary between this layer and the lower thermosphere. This is a way to study the interaction between the ionosphere and the lower thermosphere and to analyse the influence of AGWs on thermospheric processes and characteristics.
The article is structured as follows: Sect. 2 contains the basic theory of AGWs and the derivation of their dispersion equation. Section 3 presents the analytical equation for the AGW reflection coefficient. In Sect. 4, the propagation of AGWs through the ionospheric D layer as well as their reflection/transmission properties are analysed. The discussion and conclusions are displayed in Sects. 5 and 6, respectively.
Basic equations
The D layer is a part of the ionosphere where typical atmosphere models give nn∼1021 m−3 for the neutral particle density and np∼108 m−3 for charged plasma particles and where electric and magnetic effects play a minor role in the local atmosphere dynamics. This is why hydrodynamic (HD) equations rather than magneto-hydrodynamic (MHD) equations can be used to analyse wave propagation. The standard set of HD equations describes the dynamics of adiabatic processes in a neutral atmosphere in the presence of gravity, g=-gez, with constant acceleration of g=9.81 m s−2.
The first equation is that of continuity and ideal gas equation:∂ρ∂t+∇⋅(ρv)=0,p=ρRT.
The second equation is the momentum equation:ρ∂v∂t+v⋅∇v=-∇p+ρg.
And the final equation is an adiabatic law for a perfect gas:∂p∂t+v⋅∇p=γpρ∂ρ∂t+v⋅∇ρ.
Here, R=R0/M is the individual gas constant for molecules with molar mass M, R0=8.314 J mol−1 K−1 is the universal gas constant, and γ=cp/cv=(j+2)/j is the ratio of specific heats for gas particles with j=5 degrees of freedom.
Dispersion equation for AGWs
In what follows, we consider waves whose wavelengths are sufficiently small in comparison with the Earth's radius, RE=6371 km. Therefore, the plane-parallel geometry can be applied in a locally isothermal medium. Under these assumptions, the atmosphere is taken to be vertically stratified, initially in hydrostatic equilibrium, and then perturbed by harmonic waves of small amplitude. This means that Eqs. ()–() can be linearized by taking any physical quantity ψ(x,y,z,t) as a sum of its basic-state unperturbed value ψ0(z) and a small first-order perturbation δψ(x,y,z,t); i.e. ψ(x,y,z,t)=ψ0(z)+δψ(x,y,z,t), where δψ(x,y,z,t)=ψ′(z)ei(kxx+kyy-ωt) and |ψ′|≪|ψ0|. Equations ()–(), linearized with these perturbations, reduce to three equations: one for the unperturbed basic state and two for small perturbations. The unperturbed basic state is described by
ddzlnρ0(z)+1H=0,p0=ρ0RT0,withT0=const.,
the solution of which is
ρ0(z)=ρ0(0)e-z/Horp0(z)=p0(0)e-z/H,
where H=p0(0)/ρ0(0)=vs2/γg=const. is the characteristic scale height of the isothermal atmosphere.
The small perturbations are governed by the following equations :
dξz′dz=C1ξz′-C2p′,dp′dz-gdρ0dzξz′=C3ξz′-C1p′,
where ξz′=ivz′/ω is the z component (i.e. the vertical component) of the fluid displacement, while p′ is the pressure perturbation. The coefficients in Eq. () are
C1=gvs2,C2=ω2-kp2vs2ρ0(z)vs2ω2,C3=ρ0(z)ω2+g2vs2.
The density distribution, ρ0(z), is given by Eq. (), and kp2=kx2+ky2 designates the square of the horizontal wavenumber. Equations ()–() allow for the following solutions for the vertical displacement, ξz′, and the pressure perturbation, p′:
ξz′(z)=ξz′(0)ez2Heikzz,p′(z)=p′(0)e-z2Heikzz.
Equation () with solutions from Eq. () yields the following dispersion equation for AGWs:
kz2=ω2(ω2-ωco2)-kp2vs2(ω2-ωBV2)ω2vs2.
Here, kz is the vertical wavenumber, ωco2=γ2g2/4vs2=vs2/4H2 is the square of the acoustic-wave cutoff frequency, and ωBV2=(γ-1)g2/vs2 is the square of the Brunt–Väisälä frequency. This equation is quadratic in ω2, which indicates the existence of two wave modes in the considered stratified atmosphere: acoustic and gravity modes. Stratification in a vertical direction, caused by gravity and given by Eq. (), introduces cutoff frequencies – an acoustic cutoff frequency below which acoustic waves cannot propagate and a Brunt–Väisälä frequency above which gravity waves cannot propagate. Therefore, the branches of acoustic and gravity waves are present. Between them are evanescent waves that do not propagate (Fig. ).
Dispersion curves for AGWs. Two sets of curves are related to acoustic and gravity waves, which cannot propagate below the acoustic cutoff frequency, Ωco=ωcoH/vs and above the Brunt–Väisälä frequency, ΩBV=ωBVH/vs, respectively.
The dispersion equation, Eq. (), can be expressed in terms of wavelengths and wave frequency, ω, in the following way:
λz2(ω)=A2(ω)λp2λp2-A0(ω),
where
A0(ω)=4π2vs2(ω2-ωBV2)ω2(ω2-ωco2),A2(ω)=4π2vs2ω2-ωco2.
This equation will be useful for further analysis.
The physical quantities in the dispersion equation can be made dimensionless by appropriate scalings: Kp=kpH, Kz=kzH, Ω=ωH/vs, Ωco=ωcoH/vs=0.5, and ΩBV=ωBVH/vs=γ-1/γ=0.45. Now, the dispersion equation, Eq. (), for AGWs has the following dimensionless form:
Kz2=Ω2-Ωco2-Kp2(Ω2-ΩBV2)Ω2.
The acoustic waves with a frequency of Ω>Ωco propagate in the vertical direction if Kz2>0. This is fulfilled when
Kp2<Ω2(Ω2-Ωco2)Ω2-ΩBV2,
i.e. when the dimensionless horizontal phase velocity is
Vh2=Ω2Kp2>Ω2-ΩBV2Ω2-Ωco2.
Gravity waves with a frequency of Ω<ΩBV propagate in the vertical direction if Kz2>0, i.e. when Kp2 and Vh2 in Eqs. () and () have the opposite sign.
The AGWs become evanescent if Kz2<0, i.e. for the frequency of ΩBV<Ω<Ωco (Fig. ). The boundary between propagating and evanescent regions is given by Kz=0. Acoustic waves with frequencies close to the acoustic cutoff frequency, Ω≈Ωco=0.5, are more influenced by gravity than those with high frequencies when Ω≫Ωco. Hence, gravity–modified acoustic waves and pure acoustic waves coexist in the stratified atmosphere . Equation () shows that the vertical wavenumber Kz has a maximum value for Kp=0; i.e. the following applies:
Kzmax=K=Ω2-Ωco2.
This equation describes acoustic waves that propagate only in the vertical direction.
Gravity waves, in contrast to acoustic waves, are not able to travel vertically with Kp=0, which means there are no pure vertically propagating gravity waves . Therefore, they propagate obliquely through the stratified atmosphere in accordance with Eq. (). For the very low frequencies, when Ω≪ΩBV=0.45, gravity waves propagate with
Kz≈KpΩBVΩ,i.e.λpλz≈ωBVω.
Dimensionless equations are used because they are valid in each stratified medium, like the Earth, planets, or solar atmosphere. When we rewrite them using characteristic frequencies and temperatures, we obtain the equations for particular atmospheric layers as done in Sect. 4 for the ionospheric D layer.
Reflection coefficient of AGWs
The considered basic state in the stratified atmosphere is composed of two half-spaces with constant sound speeds separated by a horizontal plane boundary, z=0. The two regions are characterized by the corresponding neutral atmosphere densities, ρ01 and ρ02, adjacent to the lower and upper side of the boundary z=0. The unperturbed density profile can be expressed as follows:
ρ0(z)=ρ01e-z/H1,z<0,region 1,ρ0(z)=ρ02e-z/H2,z>0,region 2,
where H(n)=vsn2/γg, n=1,2. There is a density, pressure, and temperature jump across z=0. The boundary condition that has to be applied at z=0 in the basic state is the continuity of the unperturbed pressure, p0, at z=0, which yields the following:
ρ02ρ01=vs12vs22=T1T2=s=const.
The boundary conditions for perturbations are continuity of both the vertical fluid displacement ξz′ and the pressure perturbation p′-gρ0(z)ξz′ at the boundary z=0. Also, the energy density of the perturbations has to diminish to zero as |z| tends to infinity.
The harmonic wave, which propagates through regions (1) and (2), does not change its frequency and the horizontal wave vector component, Kp, parallel to the boundary z=0. However, the vertical wave vector component, Kz, has a discontinuity at the boundary z=0, where it changes from Kz1 to Kz2 according to the dispersion equation, Eq. (). We assume that a wave propagates from the lower region, region 1, upward towards the boundary z=0, and that the waves continuing past it are absorbed with no reflection in the upper region, region 2. In this case, in the lower region, the perturbations are the superposition of the incident and reflected waves, while in the upper region, there is only the transmitted wave. The reflection coefficient of AGWs is defined as the square of the absolute value of the reflection amplitude. Using dimensionless physical values for brevity, the reflection coefficient can be written as follows (see details in ):
R=1-γ21Vh2-1-s2sVh2-1+(s-1)Vh22+γ2Ω2Vv12Vv12Vv22⋅s2(sVh2-1)2-1(Vh2-1)21-γ21Vh2-1-s2sVh2-1+(s-1)Vh22+γ2Ω2Vv12Vv1Vv2⋅ssVh2-1+1Vh2-122+2γΩVv1(Vh2-1)1-γ21Vh2-1-s2sVh2-1+(s-1)Vh21-γ21Vh2-1-s2sVh2-1+(s-1)Vh22+γ2Ω2Vv12Vv1Vv2⋅ssVh2-1+1Vh2-122.
Here, Vv1 and Vv2 are the vertical phase velocities of AGWs in regions 1 and 2, respectively, given by the following equations:
Vv1=ΩKz1=VhΩVh2(Ω2-Ωco2)-(Ω2-ΩBV2)
and
Vv2=ΩKz2=VhΩsVh2(Ω2-sΩco2)-(Ω2-sΩBV2),
while Vh is the horizontal phase velocity given by Eq. (). If Vv12 and Vv22 are positive, AGWs propagate through regions 1 and 2, respectively. If Vv12,Vv22<0, these waves are evanescent and not of interest to this study.
Results
In this section, the analytical equations derived in Sects. 2 and 3 are used to analyse the propagation of AGWs and their reflection/transmission properties in the ionospheric D layer.
AGWs in the ionospheric D layer
Acoustic–gravity waves which propagate in the lower ionosphere below 90 km can be generated from below, where hydrodynamic motions can be induced by atmospheric convective motions ; in the lithosphere (); and from above due to sunrise and sunset effects (). These perturbations may result in various patterns of either eigenmodes or driven linear waves in the atmosphere. The focus of this research is on the driven AGWs and their role in the ionosphere and the lower thermosphere interaction. Therefore, propagation of AGWs in the vertical direction is particularly important.
For the considered isothermal ionospheric D layer with a temperature of T=250 K and γ=1.4, the sound velocity is vs=γRT=317 m s−1, and H=7317 m. This is in accordance with . For a gravity–modified acoustic wave with a frequency near the acoustic cutoff frequency of ω=0.022 s−1≥ωco=0.021 s−1, Eq. () enables the calculation of λz≥λzmin≈460 km. For the pure acoustic wave, with a frequency much greater than acoustic cutoff frequency (ω≫ωco, i.e. ω=10⋅ωco=0.21 s−1), this value is λz≥λzmin≈9.2 km. It can be noticed that gravity–modified acoustic waves have much longer vertical wavelengths than pure acoustic waves. Therefore, acoustic waves with frequencies near the acoustic cutoff frequency, ωco, have the best chance for vertical propagation through the ionospheric D layer towards the lower thermosphere. Acoustic waves in Fig. are detected in the ionospheric D layer using VLF waves .
Vertical wavelength, λz, of acoustic waves from the dispersion equation, Eq. (), as a function of the horizontal wavelength λp, for a given frequency of ω>ωco=0.021 s−1.
Gravity waves with a high frequency, ω≈ωBV, and with a low frequency, ω≪ωBV, are presented in Figs. and , respectively. Equation () shows that low-frequency gravity waves have much longer horizontal than vertical wavelengths; i.e. they propagate more horizontally than vertically (Fig. ). In addition, for a given λp, the vertical wavelengths of low-frequency gravity waves are shorter than those of gravity waves with a frequency that is close to the Brunt–Väisälä frequency (Figs. ) and (). The vertical phase velocities of these waves are smaller than those of high-frequency gravity waves. Therefore, high-frequency gravity waves propagate faster upward through the ionospheric D layer towards the lower thermosphere. Figures and show gravity waves that were found in the ionospheric D layer . They can be induced in situ at sunrise and sunset due to motions of the solar terminator. Low-frequency gravity waves are observed near the OH layer at an altitude of about 87 km and near the O2 layer at an altitude of about 94 km by the mesospheric temperature mapper . Their frequencies are ω=0.0011 s−1 and ω=0.0014 s−1, respectively, and are even lower than the gravity wave frequencies in Fig. .
Vertical wavelength, λz, of gravity waves from the dispersion equation, Eq. (), as a function of the horizontal wavelength, λp, for a given frequency of ω≈ωBV=0.0195 s−1.
Vertical wavelength, λz, of gravity waves from the dispersion equation, Eq. (), as a function of the horizontal wavelength, λp, for a given frequency of ω≪ωBV=0.0195 s−1.
Reflection coefficient of AGWs at the D layer–lower thermosphere boundary
We assume that z=0 (Eq. ) is the plane boundary between the ionospheric D layer at an altitude of 60–90 km, i.e. region 1, and lower thermosphere at an altitude of 90 to about 140 km, i.e. region 2. At this boundary, AGWs coming from below can be reflected in the D layer or transmitted into the lower thermosphere. The temperature of the D layer is T1=250 K, while the temperature of the lower thermosphere is T2=500 K. Therefore, Eq. () gives s=0.5. The reflection coefficients of acoustic and gravity waves will be analysed separately.
Reflection coefficient of the acoustic waves
Figure shows the reflection coefficient as a function of frequency, Ω>Ωco, for acoustic waves at the D layer–lower thermosphere plane boundary, z=0, when s=0.5. Acoustic waves in the frequency range Ωco<Ω<0.8 are reflected on this boundary to a somewhat greater extent. The reflection coefficient strongly decreases with increasing frequency, and acoustic waves with a frequency of Ω>0.8, i.e. ω=0.035 s−1, can easily propagate through the D layer–lower thermosphere boundary. These waves could affect the thermospheric temperature and dynamics by depositing their momentum and energy in the lower thermosphere. The value of the horizontal phase velocity, Vh, does not significantly affect the reflection coefficient, except in the case when Vh=1/s=1.41, i.e. for the horizontal phase velocity of the acoustic waves, vph=1.41vs=447 m s−1, when total internal reflection occurs. Waves with this horizontal velocity cannot penetrate the thermosphere. Acoustic waves with horizontal phase velocity Vh>1.41 can propagate through the D layer–lower thermosphere boundary and extend further into the thermosphere, especially if Ω>1. Their reflection coefficient slowly decreases with the increase in Vh for a given frequency Ω.
Reflection coefficient for acoustic waves at the D layer–lower thermosphere plane boundary, z=0, as a function of frequency, Ω, and parameter s=0.5. If Vh=1/s=1.41, the reflection coefficient is R=1, and total internal reflection occurs.
Reflection coefficient of the gravity waves
The reflection coefficient for gravity waves with Ω<ΩBV increases when the frequency Ω increases and decreases with increasing horizontal phase velocity, Vh, for a given frequency Ω (Fig. ). These waves can propagate in both regions, in the ionospheric D layer and in the lower thermosphere, if their frequencies are lower than the cutoff frequency Ω=sΩBV=0.32 or ω=0.014 s−1 and their horizontal phase velocities are lower than Vh=ΩBV/Ωco=0.9, i.e. vh=0.9vs=285 m s−1. Gravity waves with frequencies much lower than the Brunt–Väisälä frequency and with high horizontal phase velocities are candidates for crossing the D layer–lower thermosphere boundary. Contrary to this, gravity waves with frequencies near the cutoff frequency of Ω=sΩBV are strongly reflected at the D layer–lower thermosphere boundary. For the horizontal phase velocity of Vh=0.9, total internal reflection occurs, and the reflection coefficient is equal to unity.
Reflection coefficient for gravity waves at the D layer–lower thermosphere plane boundary, z=0, as a function of frequency, Ω, and parameter s=0.5. If Vh=ΩBV/Ωco=0.9, the reflection coefficient is R=1, and total internal reflection occurs.
Discussion
It is known that high-frequency acoustic waves are strongly absorbed by the atmosphere . The rate of absorption is proportional to the wave frequency squared. Therefore, only acoustic waves with low frequencies (infrasound) may propagate through the ionospheric D layer and eventually through the lower thermosphere. Indeed, it was found that only acoustic waves with periods of less than 4 min, i.e. Ω>0.6, or ω>0.026 s−1 propagating almost vertically are able to reach the lower thermosphere (). In Fig. , the reflection coefficient for infrasound waves is presented since the dimensionless frequency Ω=1.6 corresponds to the frequency of ω=0.069 s−1, i.e. ν=ω/2π=0.01 Hz. These waves, with a horizontal phase velocity of vh>447 m s−1 and with a minimum vertical phase velocity of vvmin>317 m s−1 have the best chance of reaching the thermosphere if they propagate almost vertically with an infrasound frequency of ω>0.035 s−1. Although infrasound waves dissipate their energy in the lower thermosphere, they are not the first option for raising its temperature. Namely, the influence of acoustic wave energetics into the ionosphere/lower thermosphere is weak . It appears that the temperature in the thermosphere is increased by low-frequency gravity waves coming from below . Their reflection coefficient for the ionospheric D layer–lower thermosphere boundary is shown in Fig. . Gravity waves with a horizontal phase velocity of Vh<0.5 are easily reflected from the boundary between the D layer and lower thermosphere and will likely remain trapped at lower altitudes. Only waves with a horizontal phase velocity within 0.5<Vh<0.7, i.e. 159 m s−1<vh< 222 m s−1, and with a low frequency of Ω<0.2 or ω<0.0087 s−1 are important for the dynamics of the middle atmosphere. Horizontal wavelengths for these waves are in the range of 115 km <λp< 161 km. This is consistent with the results known from the scientific literature (), which emphasize that gravity waves with periods as short as 10 min (i.e. Ω<0.24 or ω<0.01 s−1) can carry significant momentum flux vertically. These waves with a wavelength of λp≈ 100–200 km play an important role in the interaction between the ionospheric D layer and the lower thermosphere. They are responsible for the generation of middle-scale travelling ionospheric disturbances (TIDs) with periods from 15 min to 3 h, velocities from 100 to 250 m s−1, and horizontal wavelength of approximately a few hundred kilometres (). It seems that they are causing a rise in temperature in the lower thermosphere through the process of gravity wave breaking and dissipation due to kinematic viscosity and thermal diffusivity (). A similar situation can be found in the solar atmosphere and on the photosphere–chromosphere boundary (), with the parameter s=0.6.
Gravity waves dissipate their energy contributing to local heating of the thermosphere at higher altitudes during extreme solar minimum since the kinematic viscosity is much smaller in warmer than in colder thermosphere at the same altitude . During extreme solar minimum, the lower thermosphere is relatively cold of T≈500 K, while during active solar conditions, the temperature in thermosphere can be T≈2000 K . The current, 25th solar cycle, which began in December 2019, is expected to have maximum activity in July 2025. This solar activity could increase the temperature in the lower thermosphere several times. The reflection coefficient for acoustic waves in active solar conditions varies with the frequency, Ω, and the parameter s= 250 K/2000 K = 0.125 as depicted in Fig. . The reflection coefficient decreases in the frequency range Ωco<Ω<1.5. Acoustic waves with Vh≥1/s≈2.83 are the best candidates to pass through the D layer–lower thermosphere boundary and propagate further into the thermosphere. Acoustic waves with Vh≫1/s are the most susceptible to reflection. This is the opposite situation compared to the reflection coefficient for acoustic waves with s=0.5 (Fig. ), where the waves with Vh≈1/s are the ones that are most prone to reflection. For a frequency of Ω>1.5, the reflection coefficient decreases very slowly and remains almost constant.
Reflection coefficient for acoustic waves at the D layer–lower thermosphere plane boundary, z=0, as a function of frequency, Ω, and parameter s=0.125.
The reflection coefficient for gravity waves in active solar conditions as a function of frequency, Ω, and the parameter s=0.125, is shown in Fig. . It has very high values for all gravity waves propagating with the allowed Vh<0.9. These waves can hardly pass the boundary between the ionospheric D layer and the lower thermosphere. It seems that they are trapped in the ionospheric D layer and cannot propagate through the thermosphere. Therefore, infrasound can play a significant role in the interaction between the ionospheric D layer and lower thermosphere during solar maximum activity.
Reflection coefficient for gravity waves at the D layer–lower thermosphere plane boundary, z=0, as a function of frequency, Ω, and parameter s=0.125.
The conditions for AGW propagation, as well as their reflection coefficient, strongly depend on the temperature through vs and parameter s (Eqs. and ). Therefore, any change in temperature can affect the propagation of AGWs and their reflection and transmission features. This means that the detection of these waves depends on the current temperature in the region being observed. A similar situation exists with the detection of AGWs by lidar or any other instrument because their positions relative to the wave source region will determine which AGW characteristic can be observed .
One of the important effects of AGWs and especially gravity waves is their influence on the concentration of charged particles in the ionospheric E layer embedded in the lower thermosphere at an altitude of 90–140 km. Namely, the concentration of charged particles becomes time-dependent in the presence of waves. The changed characteristics of this layer affect the reflection of radio waves and telecommunication connections . A similar situation is seen in strong natural hazards when earthquakes of magnitude Mv=5.5+ are studied by VLF/LF radio waves. A physical interpretation is based on atmospheric gravity waves which could alter the ionospheric E layer and modulate the height of the VLF/LF wave reflection .
An interesting approach to the study of linear AGWs has been made by . They suggest that AGWs in an isothermal atmosphere can be considered a superposition of oscillations that occur simultaneously at two natural frequencies – acoustic and gravitational – for a fixed wavelength.
AGWs driven from the Earth's surface or troposphere are typically characterized as primary or higher order (e.g. secondary) depending on how they propagate to thermospheric altitudes . Primary AGWs propagate directly through the thermosphere and can be modelled using linear theory. proposed a method for recognizing the types of linear AGWs in the atmosphere from satellite measurements. Higher-order AGWs are created when primary AGWs break in the upper atmosphere; nonlinear propagation theory is required to simulate them (). Considerable attention has recently been paid to the study of so-called secondary AGWs that arise as a result of instability and nonlinear interactions of primary wave modes propagating from atmospheric sources, among themselves, and with the mean flow. separated the horizontal spatial spectra of primary and secondary AGWs at fixed altitude levels in the middle and upper atmosphere at different time moments using a three-dimensional nonlinear high-resolution model AtmoSym. This separation of the spectra of primary and secondary AGWs makes it possible to estimate the relative contribution of secondary AGWs at different altitudes, at different times, and with a different stability of background temperature and wind profiles in the atmosphere. These issues are important for future research, and numerical models could be a good tool for them.
Conclusions
In this article, analytical equations are used to study AGWs propagation through the ionospheric D layer and the D layer–lower thermosphere interaction. The dispersion equation and the reflection coefficient show that infrasound waves with a frequency of ω>0.035 s−1 that propagate almost vertically can reach the lower thermosphere. Gravity waves propagate in both regions – the ionospheric D layer and the lower thermosphere – if their frequency is ω<0.014 s−1 and their horizontal phase velocity is vh<285 m s−1. Gravity waves with frequencies much lower than the Brunt–Väisälä frequency, ωBV=0.0195 s−1, propagate more horizontally than vertically because λp≪λz. These waves have lower vertical phase velocities than high-frequency gravity waves with ω≈ωBV, which travel faster through the ionospheric D layer towards the lower thermosphere. The reflection coefficient is the smallest for the gravity waves with the frequency of ω<0.0087 s−1, horizontal phase velocity of 159 m s−1<vh< 222 m s−1, and horizontal wavelength of 115 km <λp< 161 km, which is in accordance with the results known in the scientific literature (). These waves can generate the middle-scale TIDs and cause temperature rise in the lower ionosphere.
The reflection coefficient is highly temperature-dependent. It changes significantly during the pronounced solar maximum when the temperature in the lower thermosphere can rise several times. A strong increase in the reflection coefficient for gravity waves indicates that they cannot pass the D layer–lower thermosphere boundary. Therefore, infrasound waves are better interaction instruments.
There is broad scientific interest in the future study of AGWs. This is particularly attributed to the study of natural hazards, telecommunications and navigation, and space weather. Due to the complex nature of this process, differences between model results and observations are expected .
Data availability
Research data can be accessed via the references in the text, Sect. 4.
Competing interests
The author has declared that there are no competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.
Acknowledgements
The research and writing of this work was supported by the Montenegrin national project “Physics of Ionized Gases and Ionized Radiation”.
Review statement
This paper was edited by Dalia Buresova and reviewed by two anonymous referees.
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