4. Simulation

4.1. The Calculation Results

[30]  Thus equations system (2) with the initial and boundary conditions was solved, using the computer program MADRL created by us. The profiles of the background density and temperature of the atmosphere from the MSISE 90 model (ftp://nssdcftp.gsfc.nasa.gov/models/atmospheric/hwm93) were used in the simulation. We used various versions of the input data as parameters of the disturbance source. Here we present the results calculated for one case. Choosing the values of these parameters we took into account the fact that in practice it is not simple to determine precise values of the disturbances pulse parameters and they depend on the particular launching. However, as far as our main goal is to study general properties of the disturbances from such sources, we took approximate values of the parameters of the acoustic pulse: u_m = 200 m s^-1, P = 23 s, z_m = 110 km, D_z = 10 km [Nagorsky, 1998].

Figure 4

[31]  It is widely known that the temperature stratification and the zonal wind influence the AGW propagation in the atmosphere. In order to separate these influences from each other, we considered in our calculations step-by-step different versions of the atmosphere model. At first we simulated the problem in the isothermal and quiet atmosphere where the zonal wind is absent. The results of this simulation show that after the disturbance arrival to the considered region, there are generated AW and IGW. Disturbances with scales of hundreds of kilometers are observed at considerable horizontal distances. By their spatial and time characteristics these disturbances belong mainly to the IGW class. These waves exceed considerably AW by their intensity. Figure 4 shows the part of the calculated region, where they are clearly manifested 5000 s after the passing of the disturbance through the model region boundaries. The results of the calculations show that the main component of the velocity is the horizontal one. The period of these waves varies from 6 min to 14 min and increases linearly in the process of horizontal propagation. One can see from Figure 4 that the wave fronts are almost vertical. Moreover, a nonlinear deformation is observed, that is steepening of the wave profile caused by a large value of the Mach number [Landau and Lifshits, 1988]. It was found that disturbances in the isothermal atmosphere mainly contain a continuous spectrum of IGW. One can see in Figure 4 that the saturation point (i.e., the height of the maximum amplitude of the wave) is situated at a height of h 150 km. All the above mentioned properties of IGW found by us agree with the results of many authors [Francis, 1975]. The latter results have been obtained using analytical and numerical calculations with more simple models describing the IGW propagation. This shows that our numerical method has a high enough accuracy.

Figure 5

[32]  As it was expected, taking into account of the real profile of the atmosphere temperature leads to results quite different from the first case. Main difference is the appearance of a discrete wave spectrum which arises as a result of the ducting of the wave in the atmosphere waveguides. These waveguides are formed due to the wave reflection from the temperature gradients and the Earth surface. Many studies were performed using spectral models to investigate properties of these waves in different modes [Francis, 1975; Gavrilov, 1985]. However, our model allows us to find a common picture of AGW propagation in the real atmosphere where the picture of disturbances is formed by a superposition of all modes of the ducted waves and continuum and also by a nonlinear interaction between different harmonics. Figure 5 shows the horizontal component of the atmosphere particles velocity versus the coordinates in the time moment t = 5000 s for the January profiles of the atmosphere temperature and density. It is evident that the complicated shapes of wave surfaces differ considerably from the previous case. In Figure 5 ducted waves are observed at heights of 100-150 km. This wave spectrum belongs to the thermospheric modes G₀' arising as a result of the AGW ducting into MTW [Francis, 1975]. Comparing Figures 5 and 4, one can see that the wave amplitudes in the case of the real stratification of the temperature are much less than in the isothermal atmosphere, this fact once more proving a ducting of waves. The values of the vertical component of motion velocity of the particles in the wave, the variations in the atmosphere temperature and density demonstrate similar dependencies on time and space.

Figure 6

[33]  Simulating the ionosphere disturbances caused by propagation of AGW, we found that characteristics of these irregularities depend strongly on the direction of the geomagnetic field. We demonstrate this fact for some cases. Figure 6 shows variation in the electron concentration versus the coordinate at the time moment t = 5000 s for the model with the real stratification of the density and temperature. Here Figure 6a presents the case when the magnetic field is not taken into account, that is the variation in the electron concentration is calculated using formula (8); Figure 6b corresponds to the cases when the magnetic field is directed horizontally, i.e., in the vicinity of the geomagnetic equator; Figure 6c presents the case when the magnetic field is directed vertically, i.e., in the vicinity of the magnetic pole; and Figure 6d corresponds to the situation near the rocket site KSC (28.5^o N, 279.3^o E) (the OX axis is directed eastward). It is evident that if the magnetic field is perpendicular to the particles velocity ( b_x = 0.0, b_z =0.0 ), then disturbances are absent. Figure 6 demonstrates how the magnetic field changes strongly the picture of the ionosphere wave-like irregularities. Because of the vertical component the shape of the disturbances becomes oblique. Our studies show that orientation of the magnetic field influences not only the spatial picture of the ionosphere disturbances, but their time characteristics.

[34]  One can see from the results that the horizontal lengths of wave-like disturbances are hundreds of kilometers, the propagation velocity is about 300 m s ^-1, and the amplitude is of the order of a few percents of the background electron concentration. These are well-known medium-scale traveling ionosphere disturbances (TID) [Francis, 1975].

Figure 7

[35]  After that we can find the disturbances in TEC along the satellite-receiver ray. We assume that this satellite is located on the XOZ plane. Figure 7 presents the variation in TEC in the vertical direction when the GPS receiver is located in the point x = 400 km and the elevation angle of the satellite is q = 90^o (see Figure 2). This result is for the real atmosphere and the direction of the magnetic field ( b_x =0.51, b_z = -0.86 ), i.e., in the vicinity of KSC. One can see from Figure 7 that the profile of the variation in TEC is rough and asymmetric. Here two wave packets are presented: the first packet is low-frequency IGWs with a period of about 20 min and the second wave packet is ducted IGWs with a shorter period. As it has been noted above, there are many modes of ducted AGW. Taking into account that all heights take part in the profile formation, one can understand the cause of the picture complicity. Moreover, the dispersion and nonlinear effects also lead to deformation of the wave profile [Rudenko and Soluyan, 1975]. In order to demonstrate how the picture of TEC variation changes strongly depending on the elevation angle of the satellite and the receiver location, we consider the case when the receiver is located in the point x = 1000 km and q =170^o.

Figure 8

Figure 9

It is evident in Figure 8 that in this case quite a different picture is obtained, which differs from the previous one not only quantitatively but also qualitatively. Here AW with periods of a few hundreds seconds and also captured AW are seen.

[36]  To take into account possible influence of the wind, we assume that its direction coincides with the direction of OX (see Figure 2). Inclusion of the zonal wind into model does not considerably influence the final result. It is convincingly shown in Figure 9 where three curves show the TEC variations in the same registration conditions (the receiver location x = 600 km and the elevation angle q =90^o ) and at the same direction of the magnetic field. The first, second and third curves correspond to the January temperature profile in windless atmosphere, the January temperature profile and allowance for the zonal wind, and the July temperature profile without allowance for the wind, respectively. The more complicated structure of the secondary waves in the case of taking into account the wind can be explained by the fact that part of the waves is ducted by the wind [Gavrilov, 1985] and is superimposed on the other ducted waves. One can see from Figure 9 that the properties of the ionosphere response also insignificantly depend on the season. The higher amplitude of the waves in July may be explained by the higher temperature of the upper atmosphere in summer than in winter. Comparing Figures 9 and 7, we see that the period of the first N form wave increases with distance, because the distances of GPS receivers from the source are different in the first and second cases. Moreover, the profile of the variation in TEC at large distances from the source becomes smoother. It can be caused by the fact that acoustic waves attenuate quickly [Francis, 1975] and only IGW are observed at large distances from the source.

4.2. Comparison to Observations Data

[37]  The preliminary comparison of the obtained results to the experimental data shows that, first our model predicts an appearance of TIDs very often observed in the atmosphere from various sources including rocket launchings. The horizontal propagation and oblique forms of TID predicted by us are observed in many experiments [Adushkin et al., 2000]. If we consider the second half plane which is located to the left of the Mach cone where the angle between the magnetic field and the velocity is different, and also if we take into account variations in the magnetic field with horizontal distance, we conclude that actually the general picture of the ionosphere disturbances can have no symmetry relative the rocket trajectory. This is confirmed by the observation results carried out by the radiotomography method during the rocket launching from Plesetsk rocket site [Kunitsyn and Tereshchenko, 2003]. One can see from the results obtained that our numerical calculations provide the entire spectrum of the disturbances observed from rockets (acoustic waves, internal gravity waves, and secondary ducted waves). As it has been noted in section 1, all these waves are observed in the experiments conducted by different methods. The importance of our results is confirmed by the fact that our model (unlike the previous ones) is able to predict the appearance of long-period IGWs from a high-frequency disturbance. This work makes it possible to explain generation of the secondary waves in the disturbances spectra. The results confirm the experimental fact that the main ionosphere response for all rocket launchings has a form of N wave both for AW and IGW.

[38]  As for the data obtained by the transionosphere sounding of the upper atmosphere by signals of the satellite radio navigation system GPS, it is obvious from our results that the characteristic of these signals depends strongly on the receivers location, the satellite elevation angle, etc. Actually these disturbances are detected by the observers with the help of filtering the TEC time series within certain short intervals (3-5 min) in order to exclude the TEC variations due to the diurnal solar cycle [Calais and Minster, 1998]. However, our results show that the frequency range of these disturbances can be very wide. We see that the curve in Figure 8 qualitatively well describes the disturbances detected by Calais and Minster [1998]. The wave periods, delay time of the second packet relative the first packet, and horizontal phase velocities of the wave propagation almost coincide in our model and the experiment.

[39]  The IGW predicted by us were registered in many experiments. Comparing Figure 1 to Figures 7 and 9 we see that our model predicts an appearance of the first wave with large amplitude and period and the secondary waves following after it with shorter periods and smaller amplitudes. Thus the developed numerical method well describes generation of all types of waves during rocket launchings.

[40]  The disturbances amplitudes obtained are equal to 0.01-0.1 in the TECU units. It is confirmed in many observations [Afraimovich et al., 2001; Calais and Minster, 1998]. However, as it has been noted above, our main goal was to study the common properties of the atmosphere and ionosphere disturbances generated by rocket flights at large distances.