1 INTRODUCTION

In many cases, the plasma magnetospheric maser (PMM) in the morning and daytime subauroral magnetosphere is a high-Q oscillatory system responsible for the excitation of quasi-periodic electromagnetic emissions in the VLF range with repetition periods of spectral forms of 10–300 s. The relatively dense magnetized plasma and the magnetic trap foot regions form a resonator for VLF whistler waves; the active substance is high-energy electrons with energy of approximately 40 keV. The particle source plays the role of pumping and the population inversion associated with the transverse anisotropy of the distribution function of energetic electrons ensures the cyclotron instability of electromagnetic waves (Bespalov and Trakhtengerts, 1986). The presence of an eigen frequency in the radiation belts, which corresponds to periodic processes of accumulation of energetic particles in the radiation belts and their precipitation into the ionosphere during electromagnetic radiation pulses, is the root cause of quasi-periodic VLF radiation. These emissions (they are called QP emissions) with frequencies of several kilohertz are usually observed in the morning and daytime sectors of the inner magnetosphere and have a repetition period of spectral forms from several seconds to several minutes (see, for example, (Sato and Kokubun, 1980; Smith et al., 1998; Engebretson et al., 2004)). They are recorded both on the Cluster, Van Allen probes, and THEMIS spacecraft, for example, (Hayosh et al., 2013; Titova et al., 2015; Nemec et al., 2016a, 2016b) and ground-based observations, for example, (Manninen et al., 2012; Manninen et al., 2013). In many cases, observations show the simultaneous appearance of modulated noise emissions, geomagnetic pulsations, and particle precipitation in conjugated regions of the ionosphere, for example, (Raspopov and Kleimenova, 1977). This type of property is typical for QP 1 emissions, probably due to changes in the growth rate of cyclotron instability by geomagnetic compression pulsations.

Along with those described above, in some cases QP 2 radiation with a clearer repetition of spectral forms, not accompanied by geomagnetic pulsations occurs. The nature of such radiations is most likely associated with the instability of the stationary state of the radiation belts (Bespalov, 1981) and the development of a self-oscillating process in them (Bespalov, 1982).

The purpose of this work was to study the possibility of the resonant influence of weak external influences caused by atmospheric infrasonic disturbances on processes in the plasma magnetospheric maser. The effectiveness of the periodic impact on the quality factor of the magnetospheric resonator of electromagnetic waves in the VLF range is discussed. Within the framework of model calculations, the longitudinal dependence of the depth of modulation of the energy density of electromagnetic waves in a magnetic field tube is considered.

2 THE INFLUENCE OF RESONANT EXTERNAL FORCES ON PMM DYNAMICS

A plasma magnetospheric maser is a quasi-closed subsystem of the magnetosphere in which the interaction of electrons from radiation belts with electromagnetic waves of the whistler range occurs at the cyclotron resonance. In the morning and daytime magnetosphere in this subsystem, the existence of weakly damped oscillations is possible, which occur according to the following scenario: the accumulation of energetic particles under the action of the source ensures that the cyclotron instability threshold is reached; the energy density of the whistler waves then increases and, if the Q-factor of the magnetospheric resonator is not very high, the accumulation of particles continues and their content exceeds the stationary level, at which the action of the particle source is compensated by their precipitation into the ionosphere. Precipitation into the ionosphere then increases, the number of energetic particles decreases, electromagnetic waves are decay, and the system returns to a state close to the original one.

In the simplest cases, when the power of energetic electron sources is relatively small and coincides in terms of angular dependence with the first eigenfunction of the operator of quasilinear pitch-angle diffusion, the anisotropy of the distribution function and the average frequency of electromagnetic radiation practically do not change with time. Then, the dynamics of relatively slow processes in the PMM is described by the following system of balance-type equations (Bespalov and Trakhtengerts, 1986):

$$\frac{{\partial N}}{{\partial t}} + {{\Omega }_{d}}\frac{{\partial N}}{{\partial \psi }} = - \delta \left( \psi \right)\varepsilon N - \frac{N}{{{{T}_{l}}\left( \psi \right)}} + J\left( \psi \right){\text{,}}$$
(1)
$$\frac{{\partial \varepsilon }}{{\partial t}} = h\left( \psi \right)\varepsilon N - {{\nu }_{{{\text{eff}}}}}\left( \psi \right)\varepsilon + a\left( \psi \right).$$
(2)

Here, \(N\) is the content of energetic electrons in a magnetic field tube with a unit cross section at the level of the ionosphere; \(J\left( \psi \right)\) is the power of particle sources in this tube, \(\psi \) is the azimuth angle; \(\varepsilon \) is the average energy density of whistler waves in a magnetic field tube; \({{T}_{l}}\) is the average lifetime of energetic electrons in a magnetic trap without taking into account the effect of cyclotron instability; \({{\nu }_{{{\text{eff}}}}} = 2\left| {\ln R} \right|T_{g}^{{ - 1}}\) is the damping decrement of the whistler wave, R is the coefficient of reflection from the ionosphere from above; \({{T}_{g}}\) is the group propagation time of whistler waves in the magnetospheric resonator; \({{\Omega }_{d}}\) is the average angular velocity of electron drift due to inhomogeneity and curvature of the magnetic field; \(a\) is the local power of other possible sources of whistling electromagnetic waves associated, for example, with lightning discharges in the atmosphere.

Inside the plasmasphere, the inequality \({{\beta }_{*}} = {{({{{{\omega }_{{pL}}}V} \mathord{\left/ {\vphantom {{{{\omega }_{{pL}}}V} {{{\omega }_{{BL}}}c}}} \right. \kern-0em} {{{\omega }_{{BL}}}c}})}^{2}} \gg 1,\) is usually satisfied; then, the values \(\delta \) and \(h\) can be estimated using the relations \(\delta = {{{{\omega }_{{BL}}}} \mathord{\left/ {\vphantom {{{{\omega }_{{BL}}}} {B_{L}^{2}}}} \right. \kern-0em} {B_{L}^{2}}},\) \(h = {{{{\omega }_{{BL}}}} \mathord{\left/ {\vphantom {{{{\omega }_{{BL}}}} {({{n}_{{pL}}}\sigma l)}}} \right. \kern-0em} {({{n}_{{pL}}}\sigma l)}},\) where \(L\) is the parameter of the magnetic shell; \({{\omega }_{B}}\) and \({{\omega }_{p}}\) are the cyclotron and plasma frequencies of electrons; \(V\) is the characteristic velocity of energetic electrons; \({{B}_{L}}\) is the magnetic field at the top of the magnetic tube; \({{n}_{p}}\) is the concentration of cold plasma; \(\sigma \) is a mirror relation; and \(l\) is the length of the magnetic tube between the conjugated regions of the ionosphere.

We confine ourselves to consideration of relatively slow and large-scale external influences, such that the inequalities \(\Omega {{T}_{b}} \ll 1,\) \(\Omega {{T}_{g}} \ll 1,\) \({{\Delta r} \mathord{\left/ {\vphantom {{\Delta r} {{{R}_{ \circ }}}}} \right. \kern-0em} {{{R}_{ \circ }}}} \leqslant 1,\) \(\Omega \sim {{\Omega }_{J}},\) where \(\Omega \) and \(\Delta r\) are the frequency and spatial scale of the external impact converted to the equatorial section; \({{T}_{b}}\) is the period of bounce oscillations; \({{T}_{g}}\) is the period of group propagation of whistler waves between conjugated regions of the ionosphere; and \({{R}_{ \circ }}\) is the radius of the Earth. An analysis of the equations of a plasma magnetospheric maser near the equilibrium state leads to formulas for the oscillation frequency (Bespalov and Trakhtengerts, 1986)

$${{\Omega }_{J}} = {{(hJ)}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}},$$
(3)

and their damping decrement

$${{\nu }_{J}} = {{hJ} \mathord{\left/ {\vphantom {{hJ} {2{{\nu }_{{{\text{eff}}}}}.}}} \right. \kern-0em} {2{{\nu }_{{{\text{eff}}}}}.}}$$
(4)

For the problem we are considering in the morning and daytime magnetosphere, the condition \({{\nu }_{J}} \ll {{\Omega }_{J}},\) is often satisfied-and characteristic decay time of relaxation oscillations is much longer than their period. For typical conditions in the daytime magnetosphere, the period of external influences that can count on a resonant response lies in the range from 10 to 300 seconds; these are the characteristic time scales of infrasonic waves that are observed at ionospheric heights.

Resonant influence on the dynamics of a plasma magnetospheric maser can be exerted by relatively small external influences that modify the quantities \(h\) and \({{\nu }_{{{\text{eff}}}}}\) in the average transport equation (2):

$$\begin{gathered} \frac{{\partial \varepsilon }}{{\partial t}} = h\left( {1 + {{\mu }_{1}}\frac{{b(t,\psi )}}{{{{B}_{L}}}}} \right)\varepsilon N \\ - \,\,{{\nu }_{{{\text{eff}}}}}\left( {1 - {{\mu }_{2}}\frac{{r(t,\psi )}}{R}} \right)\varepsilon + a(\psi ). \\ \end{gathered} $$
(5)

Here, in the first order of perturbation theory, it is taken into account that \(h\) and \({{\nu }_{{{\text{eff}}}}}\) depend on the magnetic field and the reflection coefficient from the ionosphere from above, \(b\left( {t,\psi } \right)\) is the perturbation of the total magnetic field \({{B}_{L}} + b\) at the top of the magnetic field tube; \({{\mu }_{{1,2}}} \sim 1\) are numerical coefficients; and \(r\left( {t,\psi } \right)\) is the perturbation due to external influence of the reflection coefficient \(R + r\) whistler waves from the ionosphere from above. Issues of modulation of the cyclotron instability growth rate with a change in \(h\) were carefully studied both experimentally and theoretically, for example, (Bespalov and Kleimenova, 1989) when considering the formation of radiation QP 1 due to hydromagnetic compression waves in the magnetosphere. In accordance with equation (5), a change in the decay rate \({{\nu }_{{{\text{eff}}}}},\) can have an equally effective effect on the PMM dynamics which we will consider below. We note that the cavity Q-switching is widely used to control the operating modes of laboratory laser systems (Khanin, 1999).

3 Q-SWITCHING OF A PLASMA MAGNETOSPHERIC MASER BY INFRASONIC WAVES

We consider the main directions for improving the step model influence of atmospheric infrasonic waves on the conditions for excitation of whistler electromagnetic radiation in the magnetosphere, which was formulated in (Bespalov et al., 2003; Bespalov and Savina, 2012).

3.1 Infrasonic waves in the atmosphere

At present, on the basis of numerous experimental and theoretical studies, it has been established that at the heights of the ionosphere there are infrasonic perturbations of time and space scales of interest to us, for example, (Hines, 1972; Lay and Shao, 2011; Nishioka et al., 2013; Pilger et al., 2013). Published data on disturbances of the total electron density of the ionosphere with periods of less than 5 minutes, correlating with ground sources caused by earthquakes, volcanoes and ground explosions (Dautermann et al., 2009; Liu et al., 2011).

When solving the problem of the propagation of acoustic-gravity waves with periods from 10 to 300 s, a number of difficulties arise. The main ones seem to be taking into account realistic altitude profiles of the atmospheric parameters of the indicated scales and the nonlinearity associated with the exponential growth of the velocity perturbation with height. In (Savina and Bespalov, 2014; Bespalov and Savina, 2015), the authors showed the possibility of filtering acoustic-gravity waves due to the inhomogeneity of the temperature profile. The effect manifests itself when the wave reaches a level at which for infrasound with frequent \(\Omega \) and horizontal wave vector \({{k}_{ \bot }}\) condition is met: \({{k}_{ \bot }} = {\Omega \mathord{\left/ {\vphantom {\Omega {{{c}_{s}},}}} \right. \kern-0em} {{{c}_{s}},}}\) where \({{c}_{s}}\) is the speed of sound. This effect confirms the possibility of accumulating the energy of infrasonic waves at the heights of the region D and E of the ionosphere and the formation of waveguide channels. Using the approach that was used in these works, we carried out a numerical analysis, choosing altitude profiles of atmospheric temperature and viscosity according to the MSIS-E-90 model (Hendin, 1991). In Figure 1a the dependences of the characteristic parameters of acoustic-gravity waves on height are plotted in accordance with this model (1500 UT on August 10, 2012 in the region of 65° N, 45° E). The results of calculating the dependence of the vertical velocity perturbation amplitude \(\left| W \right|\) and pressure \(\left| P \right|\) from the vertical coordinate \(z\) are shown in Fig. 1b (in the caption to the figure, the zero index marks the value of the quantity on the surface of the Earth and \({{\rho }_{0}}\) is the density of the undisturbed atmosphere on the Earth’s surface). The calculations used a formal change of variables for the perturbation of the vertical velocity \(w = \left| {W\left( z \right)} \right|\exp \left( {\int {{{dz} \mathord{\left/ {\vphantom {{dz} {2H}}} \right. \kern-0em} {2H}} + i{{k}_{ \bot }}x} - i\Omega t} \right)\) and pressure disturbances \(p = \left| {P\left( z \right)} \right|\exp \left( { - \int {{{dz} \mathord{\left/ {\vphantom {{dz} {2H}}} \right. \kern-0em} {2H}} + i{{k}_{ \bot }}x} - i\Omega t} \right),\) where \(H\) is the height of the homogeneous atmosphere and \(x\) is the horizontal coordinate. The calculation taking the viscosity into account was carried out according to the method proposed in (Savina and Bespalov, 2014). It can be seen from the figure that for a wave with a period of 150 s (\({\Omega \mathord{\left/ {\vphantom {\Omega {{{\Omega }_{{g0}}}}}} \right. \kern-0em} {{{\Omega }_{{g0}}}}} \simeq 2.1\)) and a horizontal scale of approximately 100 km, filtration can be expected at an altitude of approximately 160 km, where \({{\Omega }_{{g0}}}\) is the value of the Brunt–Väisälä frequency at the Earth’s surface calculated for an isothermal atmosphere. In this case, the source of infrasonic disturbances located on the Earth forms a standing structure up to the heights of the region E. We note that the height of the level at which filtration occurs is determined by the temperature profile of the atmosphere and depends on the horizontal scale and infrasound frequency.

Fig. 1.
figure 1

The height dependence of the normalized: (a) limiting acoustic frequency \({{{{\Omega }_{A}}} \mathord{\left/ {\vphantom {{{{\Omega }_{A}}} {{{\Omega }_{{g0}}}}}} \right. \kern-0em} {{{\Omega }_{{g0}}}}}\) (dashed curve), Brandt Väisälä frequency \({{{{\Omega }_{g}}} \mathord{\left/ {\vphantom {{{{\Omega }_{g}}} {{{\Omega }_{{g0}}}}}} \right. \kern-0em} {{{\Omega }_{{g0}}}}}\) (solid curve), values \({{{{c}_{s}}{{k}_{ \bot }}} \mathord{\left/ {\vphantom {{{{c}_{s}}{{k}_{ \bot }}} {{{\Omega }_{{g0}}}}}} \right. \kern-0em} {{{\Omega }_{{g0}}}}}\) (dash-dotted curve) and \({\Omega \mathord{\left/ {\vphantom {\Omega {{{\Omega }_{{g0}}}}}} \right. \kern-0em} {{{\Omega }_{{g0}}}}}\) (dotted line); (b) the amplitudes of wave disturbances for vertical velocity \(\left| {{{W}_{*}}} \right| = {{\left| {W\left( z \right)} \right|} \mathord{\left/ {\vphantom {{\left| {W\left( z \right)} \right|} {\left| {{{W}_{0}}} \right|}}} \right. \kern-0em} {\left| {{{W}_{0}}} \right|}}\) (solid curve) and pressure \(\left| {{{P}_{*}}} \right| = {{\left| {P\left( z \right)} \right|} \mathord{\left/ {\vphantom {{\left| {P\left( z \right)} \right|} {{{\rho }_{0}}c_{{s0}}^{2}}}} \right. \kern-0em} {{{\rho }_{0}}c_{{s0}}^{2}}}\) (dashed curve).

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Another possibility of the occurrence of atmospheric disturbances at ionospheric heights with a frequency close to the frequency \({{\Omega }_{J}}\) is that the instability of the atmosphere due to high temperature gradients under conditions where the Brunt–Väisälä frequency becomes higher than the limiting acoustic frequency (Savina, 2001).

3.2 Variations in the Electron Density in the Ionosphere and Q-Switching of the PMM

Acoustic-gravitaty waves that arise as a result of natural processes can provide perturbations of the ionospheric plasma. To estimate the perturbations of the electron density under the action of infrasound, we will use the approach developed in the works of B.N. Gershman (Gershman, 1974), assuming that the ionospheric plasma at altitudes of approximately 110 km is a small admixture in the atmospheric medium. For electrons and ions, the usual equations of two-fluid quasi-hydrodynamics are written. The infrasonic wave enters these equations through collisions of neutrals with electrons and ions. When estimating the perturbation of the electron concentration \(n\) one can use the equation (Bespalov and Savina, 2012)

$$\frac{{\partial n}}{{\partial t}} = - \frac{\partial }{{\partial z}}\left\{ {\frac{{\nu _{{in}}^{2} + \Omega _{B}^{2}{{{\cos }}^{2}}\chi }}{{\nu _{{in}}^{2} + \Omega _{B}^{2}}}\left( {nw - {{D}_{a}}\frac{{\partial n}}{{\partial z}}} \right)} \right\}.$$
(6)

Here, \({{\nu }_{{in}}}\) is the frequency of collisions of ions with neutrals; \({{\Omega }_{B}}\) is the ion cyclotron frequency; \(\chi \) is the angle between the magnetic field and the vertical; \({{D}_{a}}\) is the coefficient of ambipolar diffusion; and \(w\) is the vertical velocity of neutral particles in an infrasonic wave. If we assume that there is an inhomogeneous infrasonic wave \(w(x,z,t) = A(z)\sin (\Omega t - {{k}_{ \bot }}x)\) with amplitude A (z) running along the horizontal coordinate \(x\) then relatively small variations in the electron concentration are determined by the expression

$$\delta n = \frac{\partial }{{\partial z}}\left( {\frac{{\nu _{{in}}^{2} + \Omega _{B}^{2}{{{\cos }}^{2}}\chi }}{{\nu _{{in}}^{2} + \Omega _{B}^{2}}}{{n}_{0}}A(z)} \right)\cos \left( {\Omega t - {{k}_{ \bot }}x} \right).$$
(7)

The modulation will be more noticeable in the morning ionosphere, where the stationary electron density gradients are greater than in the daytime. We note that in a relatively long infrasonic wave, not only the local concentration but also the total electron content in the ionosphere is modulated.

3.3 Modulation of the Reflection Coefficient of Whistler Waves from the Ionosphere by Infrasonic Waves

Let us return to the system of equations (1) and (2) that describe the dynamics of relatively slow processes in the PMM, which include average decrement of whistler wave damping \({{\nu }_{{{\text{eff}}}}},\) which is defined by the expression

$${{\nu }_{{{\text{eff}}}}} = 2\left| {\ln R} \right|T_{g}^{{ - 1}},$$
(8)

where R is the reflection coefficient from the ionosphere from above; \({{T}_{g}}\) is the group propagation time of whistler waves in the magnetospheric resonator.

It is well known that under real conditions whistler waves decay in a magnetospheric resonator is determined by many factors, for example, wave refraction and absorption in a magnetic tube. However, one stably important reason is the attenuation of waves in the ionosphere, which determines the reflection coefficient. In the case of the presence of weak infrasonic disturbances in the ionosphere, to estimate the reflection coefficient \(R + r\) of electromagnetic waves incident from the magnetosphere along the normal, one can use the relations (Bespalov et al., 2003)

$$R \simeq 1 - \frac{{2{{\omega }_{p}}{{\omega }^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}H{{\nu }_{{en}}}}}{{c{{{\left( {{{\omega }_{B}}\left| {\cos \chi } \right|} \right)}}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}}},$$
(9)
$$r \simeq - \frac{{{{\omega }_{p}}{{\omega }^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}}}}{{\nu }_{{en}}}}}{{\Omega c{{{\left( {{{\omega }_{B}}\left| {\cos \chi } \right|} \right)}}^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}}}}}}A\cos \Omega t = - {{r}_{0}}\cos \Omega t,$$
(10)

from which it follows that the reflection coefficient experiences modulation at the frequency of the infrasonic wave. Analysis and numerical calculations have shown that whistler waves decay more efficiently in the daytime ionosphere. This is due to the fact that the daytime lower boundary of the ionosphere is less sharp than the nighttime one, and whistler waves can penetrate into the region of strong absorption, where \({{\nu }_{{en}}} \sim {{\omega }_{{Be}}}.\) For the case of oblique incidence, the results of calculating the reflection coefficient were given in (Bespalov et al., 2018; Mizonova and Bespalov, 2021), on the basis of which, at reasonable amplitudes of the infrasonic wave, the depth of modulation of the logarithm of the reflection coefficient from the morning ionosphere can vary by approximately 10–15%.

3.4 Some Results of Numerical Calculations

Let us write the system of equations (1) and (2) for a plasma magnetospheric maser in dimensionless variables, taking into account the atmospheric infrasonic wave propagating along the longitude

$$\frac{{\partial \tilde {N}}}{{\partial \tau }} = - \tilde {\varepsilon }\tilde {N} - \frac{{\tilde {N}}}{{{{\tau }_{0}}}} + j - {{\omega }_{d}}\frac{{\partial \tilde {N}}}{{\partial \psi }},$$
(11)
$$\frac{{\partial \tilde {\varepsilon }}}{{\partial \tau }} = \tilde {\varepsilon }\tilde {N} - {{\nu }_{*}}[1 + \mu \cos (\tau - k\psi )]\tilde {\varepsilon } + \alpha ,$$
(12)

where \(\tau = \Omega t,\) \({{\tau }_{0}} = \Omega {{T}_{l}},\) \({{\omega }_{d}} = \frac{{{{\Omega }_{d}}}}{\Omega },\) \({{\nu }_{*}} = \frac{{{{\nu }_{{{\text{eff}}}}}}}{\Omega },\) \(j = \frac{{hJ}}{{{{\Omega }^{2}}}},\) \(\alpha = \frac{{\delta a}}{{{{\Omega }^{2}}}},\) \(\tilde {N} = \frac{{hN}}{\Omega },\) \(\tilde {\varepsilon } = \frac{\varepsilon }{{{{\varepsilon }_{*}}}},\) \({{\varepsilon }_{*}} = \frac{\Omega }{\delta },\) \(\mu \simeq \frac{{{{r}_{0}}}}{R}.\)

When performing a numerical analysis of equations (11), (12), it was taken into account that \({{\tau }_{0}} \gg 1,\) \(j \sim 1,\) \({{\nu }_{*}} \geqslant 1,\) in steady state \(\tau \gg 1,\) in dimensional terms the horizontal length of the infrasonic wave was taken equal to 100 km, and the local power of other possible sources of whistling electromagnetic waves was chosen as being small (\(\alpha = 0.005\)). As a result, the dependence of the energy density of whistler waves in a magnetic field tube on local time was obtained and analyzed. The model dependences of the power of the sources of energetic particles in the magnetic field tube and the damping decrement of the whistler wave used in the calculations are shown in Figs. 2a, 2b. For the infrasonic wave, two possibilities were considered separately: the maximum amplitude of the infrasonic wave at noon (dotted line in Fig. 2c) and at midnight (dashed line in Fig. 2c). The time dependence of the energy density of whistler waves in a magnetic field tube is shown in Figs. 3 and 4. From the graphs shown in Fig. 3 it is seen that even a small infrasonic perturbation (\(\mu < 0.1\)) in the daytime ionosphere can cause a strong modulation of the energy density of whistler waves in the magnetic field tube in the daytime and morning magnetosphere, where the source power is not significant, and electromagnetic waves decay in the magnetospheric resonator is significant. For similar infrasonic disturbances at night, where the attenuation of electromagnetic waves in the magnetospheric resonator is much less than during the day, the level of the stationary value of the whistler wave energy density is much higher, but there is practically no modulation (Fig. 4). In the evening sector of the magnetosphere, the power of energetic electron sources is not sufficient to reach the cyclotron instability threshold.

Fig. 2.
figure 2

The local time dependence of the power of particle sources in the magnetic field tube (a), decrement of whistler wave damping from local time (b) and the amplitude of the infrasonic wave (c).

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Fig. 3.
figure 3

The dependence of the energy density of whistler waves in a magnetic field tube on time (UT) at different moments of local time (LT), if the maximum amplitude of infrasonic disturbances occurs at noon.

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Fig. 4.
figure 4

The dependence of the energy density of whistler waves in a magnetic field tube on time (UT) at different moments of local time (LT), if the maximum amplitude of infrasonic disturbances occurs at midnight.

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The modulation of the average energy density of whistler waves caused by infrasonic waves probably explains the nature of some types of QP emissions that are observed in the absence of hydromagnetic compression waves with a repetition period of spectral forms that lies in the range of atmospheric disturbances with periods of 10–300 s.

4 CONCLUSIONS

Thus, atmospheric infrasonic waves can influence the processes in the electron radiation belts and cause the formation of quasi-periodic VLF radiation. The following conditions are most favorable for such processes:

— the infrasonic wave should have a period from 10 to 300 s;

— the horizontal scale of the infrasonic wave should be approximately 100 km;

— the process can occur in the morning and daytime magnetosphere at subauroral latitudes.

Various natural phenomena excite intense infrasonic waves: volcanic eruptions, earthquakes, and severe thunderstorms. Waves with periods of 10–300 s and horizontal scales of the order of 100 km are observed in the ionosphere. Due to the possibility of a resonance effect on the operation of a plasma magnetospheric maser, one can expect a correlation between the appearance of infrasonic waves in the atmosphere and the quasi-periodic VLF radiation observed in the magnetosphere.