To identify the type of the observed wave, its parameters should be compared to the theoretical predictions. As an example in Fig.  we plotted relative variations of densities of individual gases above the pole. The increase of the amplitude with the growth of the molecular mass (on the right) and the quasi-antiphase oscillations of He and heavy gases (on the left) are the characteristic signatures of AGW .

The AGW variation of the density, observed from a satellite, is the result of the superposition of several factors: (1) the longitudinal wave compression due to the pressure gradient; (2) the adiabatic expansion/contraction of the gas; (3) the change in the background density at the vertical displacement of the volume under the influence of gravity. The difference between oscillation profiles in gases is caused by the difference in the vertical distribution of their density above the turbopause. In the observed variations of heavy gases N2 (28 amu) and Ar (40 amu), the changes of the background density dominates. The wave variations of light gas He (4 amu) are caused mainly by the adiabatic expansion/contraction. During the maximal vertical displacement of the gas volume upwards, it expands adiabatically, so there is a minimum of He density. However, this extension is not enough to compensate the background gradient densities of heavy gases, and these gases demonstrate a maximum in the density. At the maximal vertical displacement of the gas downwards, there is a maximum of He density and a minimum of N2 and Ar densities. For gases O (16 amu) and N (14 amu) the adiabatic expansion/contraction is almost compensated by the change in background density and the elastic compression prevails in the resulting variations of the density. For this reason, the density variations of O and N are shifted in phase with respect to densities of N2 and Ar, and their resulting amplitudes are small. The details of these features are described earlier in Dudis and Reber (1976) and Fedorenko (2009).

To check if the wave propagates in space it is necessary to study the relation between the vertical velocity vz and the vertical displacement h of an elementary volume due to the wave. For a propagating monochromatic wave vz=iωh, and the variations of vz are ahead of h by π/2 in the direction of propagation . The vertical velocity vz was measured by the WATS experiment onboard DE2 . The vertical displacement h can be calculated by the relative variations of densities of two gases using a formula hH=mm1-m2δn1n1-δn2n2, where n1, n2 are the densities of the gases, δn1, δn2 are their absolute variations, H=kTn/mg is the height scale, k is the Boltzmann constant, Tn is the neutral temperature, m is the mean molecular mass, m1, m2 are the molecular masses of the gases, g is the gravitational acceleration.

Figure  shows a typical example of the variations of vz and h above the polar cap. It is easy to see that the wave propagates in reverse flight direction (leftwards in the plot). Almost all observed polar AGWs were propagating towards the dayside . Note that the systematic phase shift between the density variation of different gases in Fig.  also indicates that the wave is propagating in space. A positive phase shift between the variations of O and N densities and the densities of heavier gases like N2 and Ar indicates the direction of wave propagation .

Thus we conclude that the perturbations observed above the polar caps can be identified as propagating AGWs.

It is necessary to determine the AGW frequency from in situ measurements to compare them to the ground-based TID observations. The easiest way to do this is to use the relation (Eq. ) between the amplitudes of the variations of vz and h. In Fig.  they are plotted against each other for seven different orbits. We chose the orbital segments above the polar caps in quiet conditions within the 250–280km altitude range. Two of them are in the northern hemisphere, and five are in the southern one. All these orbits show approximately the same dependence, which means that the corresponding frequencies are very close to each other. A linear fit to these dependencies gives the mean frequency ω≈9.8×10-3 s-1, which corresponds to a period T≈640 s. The sound velocity in these conditions is about cs≈850 m s-1, and the isothermic Brunt–Väisälä frequency ωb=gγ-1/cs≈9.1×10-3 s-1, where γ is the ratio of specific heats. Within the estimation accuracy ω is close to ωb or slightly exceeds it. Note that ω is the intrinsic AGW frequency.

The horizontal phase velocity ux can be estimated from the measured horizontal wavelength λxs=2π/kxs, which is a projection of the actual horizontal wavelength λx onto the orbit, and the AGW period T. For the dominant values λxs=500–600km and T≈(650±50) s the estimated horizontal phase velocity equals uxs=λxs/T≈770-920 m s-1, which is close to the speed of sound at given conditions. Of course, AGWs can never become supersonic . In fact, it is always overestimated due to the projected wavelength being always larger than the actual one, λxs>λx. The actual coefficient depends on the angle between the wavefront and the orbit plane.

In the altitude range from 250 to 400km no altitude dependence of AGW amplitude was observed (see Fig. ). Each point on Fig.  corresponds to a maximum amplitude of a separate wave train. The classical AGW theory predicts an exponential growth of the AGW amplitude as it propagates upwards due to the conservation of the wave energy in the atmosphere with the exponentially decreasing background density. In the real atmosphere above about 200 km, the wave energy losses increase sharply due to the viscosity, especially when the mean free path of the particles becomes comparable with the wavelength. Therefore, in the real atmosphere, the increase in the amplitude of the AGW with height becomes slower. It is possible that the energy losses due to the molecular viscosity compensate the increase in the amplitude of the AGW. Another possible explanation is that AGWs propagate quasi-horizontally.

At altitudes between 250 and 400km polar AGWs create vertical flows of energy up to 0.1 erg cm-2 s-1 . This value is comparable to the energy brought by the precipitating particles in quiet geomagnetic conditions.

In AGW, in addition to the potential energy of acoustic compression, there is another kind of potential energy – thermobaric or gravitational potential energy associated with the vertical displacement of the gas volume under the action of the gravity. Polar AGWs also have a peculiar property: their acoustic and gravitational energy densities are almost equal when averaged over a period . Let us write the average potential energy of an AGW in the form Ep=14ρ0vx2ωkxcs2+vz2ωbω2, where vx and vz are the norms of the horizontal and the vertical components of the particles' velocities. Since the average over a period kinetic energy of an AGW is equal to Ek=14ρ0vx2+vz2, then, assuming Ep≈Ek, it follows from Eqs. () and () that vx2+vz2=vx2ωkxcs2+vz2ωbω2.

One can see from Eq. () that the extreme case vx=0 yields ω=ωb (Brunt–Väisälä oscillations), and the opposite case vz=0 yields ω=kxcs (Lamb mode).

DE2 data testify that both the horizontal and vertical components of velocities are non-zero and close to each other. Since ω≈ωb, ux must tend to cs, according to Eq. (). The classical AGW theory prohibits the propagation of AGWs with such parameters in an isothermal atmosphere, since they fall into a restricted area in the kx vs. ω plot.

Thus, DE2 data analysis tends towards the existence of a dominant AGW mode in the polar thermosphere with ω≈ωb and ux→cs, which contradicts the classical theory.

The wind pattern in the polar thermosphere is due to the superposition of solar heating and the magnetospheric convection projected along the field lines. The spatial structure of this pattern is quite complex and the wind velocity can reach 300–1000 m s-1 depending on the geomagnetic activity .

The AGWs observed from a satellite can occur directly in the upper atmosphere, but they may also be connected to sources in the lower atmosphere. Using an extended gravity wave scheme (Yiğit et al., 2008) that accounts for realistic wave propagation and dissipation showed that lower atmospheric gravity waves propagate into the thermosphere and affect the general circulation of the high-latitude thermosphere under various conditions (Yiğit et al., 2009, 2012, 2014). Also, numerical simulations suggest that gravity waves propagate from the surface to the thermosphere (Hickey et al., 2009, 2010). In the polar ionosphere, there are very powerful sources of AGWs associated with particle precipitation and the dissipation of currents. Therefore, the AGWs caused by low atmospheric sources in the polar region are very difficult to distinguish. Phase velocities of observed AGWs exceed the speed of sound in the lower atmosphere. This probably indicates their upper atmospheric origin.

The directions of AGW propagation follow a systematic behaviour suggesting that their propagation is controlled by the wind . The most evident feature is the predominant propagation of AGW from the nightside to the dayside, i.e. towards the wind. During the polar day or the polar night the directions of AGW propagation follow the seasonal variation of the wind pattern in both hemispheres .

To illustrate this point we plotted in Fig.  the dominant directions of AGW propagation in the northern hemisphere for different UT and KP values. Once can see that the AGW azimuths tend to follow the direction of Earth's rotation (counter-clockwise in the northern hemisphere). During the polar night (January–February) AGWs are tightly packed inside the polar cap and almost can not be detected outside the auroral oval. In this case the directions of AGW propagation are driven by the local polar wind pattern. During the polar day (June–July) an additional flow of AGWs towards the direction 14:00–15:00 LST can be seen. This flow is due to the wind caused by solar heating. On the nightside this flow escapes past the auroral oval, forming a tail-like structure. On the dayside the AGW activity is constrained within the polar cap.

In the southern hemisphere this picture looks similar to the northern one. The main differences are more widely spread AGW azimuths and systematically lower AGW amplitudes. Apparently, this can be explained by a greater distance between the magnetic and the geographical poles in the south as compared to the north. The wind-driven circulation in the polar thermosphere can be considered as a superposition of two wind systems . One of them is the global wind circulation arising due to the absorption of solar UV- and EUV-radiation (tied to the geographic coordinate system). The second one is the vortex circulation arising from the convective motion of ionospheric plasma (tied to the geomagnetic coordinate system). The geomagnetic input introduces a strong UT-variation in thermospheric dynamics due to the offset of the geomagnetic poles from the geographic poles. The spatial area, disturbed by the polar thermospheric circulation, is greater in the southern hemisphere due to the greater distance between the geographic and geomagnetic poles. Accordingly, the region of increased wave activity associated with the wind in the southern hemisphere is also more extended.

Another clear indicator of AGW-wind coupling is the dependence of the AGW amplitude on the wind velocity . During AGW propagation, the density, pressure, temperature, and velocity are periodically changed. The amplitudes of these variables are connected to each other by the wave polarization relations. They are proportional to each other . We have analysed the variations of density, primarily because they were measured with greater precision and higher time resolution than the velocity and the temperature. Figure  shows the dependence of the AGW amplitude on the wind velocity to be almost linear. For an explanation of the functional form of this dependence needs further theoretical investigations. It should be noted that the waves with amplitudes below 1 % are usually detected in mid-latitudes, and the waves with amplitudes over 2 % are mostly seen in high latitudes.

The influence of the conductive ionosphere on the propagation of AGWs is significant at thermospheric altitudes. For further theoretical analysis of the interaction between AGW and the wind one needs also to consider the influence of the geomagnetic field on AGW properties (Kaladze et al., 2008; Khantadze et al., 2010).