Indications of the Impact of the Influence of Large-Scale Atmospheric Disturbances on Quasiperiodic ELF/VLF Emissions Inside the Plasmasphere

Abstract

The models of excitation of quasiperiodic ELF/VLF emissions with spectral shape repetition periods from 10 to 300 s are discussed. The primary cause of quasiperiodic (QP) emissions is cyclotron instability of electron radiation belts. Relatively slow processes of cyclotron instability evolution are well described within the framework of the plasma magnetospheric maser (PMM) theory based on the averaged self-consistent system of quasilinear equations for particles and waves. The presence of an eigen-frequency of oscillations of PMM parameters allows explaining many properties of QP 1 emissions, in which not very clear spectral bursts are hiss with resonant modulation mainly near the upper spectral boundary by geomagnetic pulsations of the Pc 3–4 range. The analysis of the general problem of equilibrium of radiation belts shows the possibility of its instability, which is caused by the difference in the pitch-angle dependences of the particle source power and the steady state distribution function. In the nonlinear mode of the specified instability, QP 2 emissions are formed, often with an increase in frequencies in individual spectral bursts. This paper mainly focuses on the study of QP 2 emissions with both a normal and an atypical time structure, as well as with large and fast dynamics of the frequency spectrum. Periodic large-scale atmospheric disturbances with a suitable frequency on the ionosphere can significantly affect the operating modes of the PMM and, as a consequence, the quasiperiodic VLF emissions in the magnetosphere. Infrasonic waves at the altitudes of the E region of the ionosphere can provide excitation of atypical quasiperiodic emissions due to a change in the reflection coefficient of whistler waves from the ionosphere from above. The obtained results are important for interpreting observational data on emissions associated with large-scale processes in the atmosphere. To analyze the magnetosphere response to earthquakes, observation data from the Van Allen Probe spacecraft were used. Also, specific examples of quasiperiodic emissions, probably associated with large-scale atmospheric processes, were obtained during the analysis of observational data.

1. Introduction

Near-Earth space is characterized by the presence of various electromagnetic radiations of the ELF/VLF ranges [1]. Many important properties of these magnetospheric radiations are considered in reviews and books [2,3,4].

In the electron radiation belts inside the plasmasphere, conditions typical for the operation of a plasma magnetospheric maser (PMM) often occur for electromagnetic VLF emissions. A relatively dense cold magnetized plasma and the ends of a magnetic trap form a resonator for electromagnetic waves. The active substance is energetic electrons with an energy of about 40 keV. The particle source acts as a pump, and the population inversion, associated with the natural transverse anisotropy of the energetic electron distribution function in the presence of a loss cone, provides cyclotron instability (CI) of electromagnetic waves [5]. In the morning and daytime subauroral magnetosphere, the PMM is a high-Q oscillatory system with an eigen-frequency. The presence of an eigen-frequency in the PMM, which corresponds to periodic processes of accumulation of energetic particles in radiation belts and their precipitation into the ionosphere during bursts of electromagnetic radiation, is the primary cause of the quasiperiodic (QP) electromagnetic VLF radiation with repetition periods of spectral forms of 10–300 s [5].

The mentioned QP emissions are usually observed in the morning and daytime sectors of the inner magnetosphere at frequencies of several kilohertz and have a spectral shape repetition period from ten seconds to several minutes (see, for example, [6,7,8,9,10]). They are recorded both by the DEMETER, CLUSTER, THEMIS, Van Allen Probe spacecraft (see, for example, [11,12]), during ground-based observations (see, for example, [13,14]), and from simultaneous space and ground-based data (see, for example, [15]).

In many cases, observations show the simultaneous appearance of modulated noise emissions, geomagnetic pulsations, and particle precipitations in conjugate regions of the ionosphere (see, for example, [16]). Properties of this type are characteristic of QP 1 emissions, probably caused by the influence of geomagnetic pulsations on the compression of the CI growth rate.

Along with those described above, QP 2 emissions with clearly repeating spectral forms are observed, usually not accompanied by geomagnetic pulsations [6,17,18]. The nature of such emissions is associated with the instability of the steady state of the radiation belts [19] and the development of a self-oscillatory process in them [20]. Many important questions of the theory of the formation of QP radiations have been investigated for conditions that guarantee small changes in the average frequency of electromagnetic radiation.

For the excitation of QP 2 emissions, no special waveguide propagation of electromagnetic waves is required, and therefore, within the framework of one event, they are observed in vast areas of the magnetosphere. At the same time, the dynamics of the PMM are sensitive to the conditions of reflection of electromagnetic waves from the ionosphere from above. Taking into account these two circumstances, one can expect an influence of large-scale atmospheric processes on the properties of VLF radiation in the magnetosphere, and in the case of QP emissions, this influence can be resonant. Due to the resonance effect on PMM, a correlation between the appearance of infrasonic waves in the atmosphere at the E layer of the ionosphere and atypical QP 2 emissions is possible. The influence of weak external influences associated with infrasonic waves on the ionosphere on the operation of PMM is considered.

In this paper, we study the excitation conditions of quasiperiodic VLF emissions with typical and atypical frequency dynamics inside the plasmasphere. We discuss probable channels of the influence of large-scale atmospheric processes on quasiperiodic emissions. Using the Van Allen Probe spacecraft observation data available online, we present different examples of QP emissions, including emissions with large and very fast dynamics of the frequency spectrum.

The formalism developed in this paper allows for the influence of large-scale atmospheric acoustic-gravity disturbances on the conditions for the excitation of VLF electromagnetic emissions within the plasmasphere to be taken into account. The possibility of such an influence is illustrated by observational data on atypical quasi-periodic emissions presented. Some of these observational data are associated with strong earthquakes.

Section 2 presents elements of the theory explaining the conditions for excitation of quasiperiodic VLF emissions inside the plasmasphere. Section 3 examines the channels of influence of large-scale atmospheric processes on the properties of quasiperiodic emissions. Section 4 discusses examples indicating the connection between large-scale atmospheric events and quasiperiodic VLF emissions based on observations by the Van Allen Probes spacecraft. Section 5 summarizes the results of the paper.

2. Conditions of Excitation of Quasiperiodic VLF Radiations Inside the Plasmasphere

2.1. Cyclotron Instability of Electron Radiation Belts

Within the plasmasphere, conditions are met under which effective interaction is possible of whistler emissions whose propagation is determined by the well-known dispersion equation [1,21]

$${\left({\displaystyle \frac{kc}{\mathrm{\omega }}}\right)}^2=1+{\displaystyle \frac{{\mathrm{\omega }}_p^2}{\mathrm{\omega }\left({\mathrm{\omega }}_B\left|cos\mathrm{\theta }\right|-\mathrm{\omega }\right)}},$$

(1)

with energetic electrons at the cyclotron resonance

$$\mathrm{\omega }-k_{\parallel }{V}_{\parallel }={\mathrm{\omega }}_B,$$

(2)

where $\mathrm{\omega }$ and $k_{\parallel }$ are the frequency and longitudinal (along the magnetic field $\overrightarrow{B}$) component of the wave vector $\overrightarrow{k}$ of the electromagnetic wave, $V_{\parallel }$ and ${\mathrm{\omega }}_B$ are the longitudinal velocity and cyclotron frequency of the electron, ${\mathrm{\omega }}_p$ is the plasma frequency, and c is the speed of light in a free space. Calculations show that inside the plasmasphere at smaller angles $\mathrm{\theta }$ between the wave vector and the magnetic field, cyclotron resonance is most important. With strictly longitudinal propagation, there is no interaction at other resonances.

The fundamental reason for cyclotron instability is the transverse anisotropy $T_{\perp L}/T_{\parallel L}>1$ of the distribution function of energetic electrons with transverse temperature $T_{\perp L}$ and longitudinal temperature $T_{\parallel L}$, where the quantities with the index $L$ refer to the region of the magnetic equator. In reality, the cyclotron instability can be non-exponentially weak only if the parameter

$${\mathrm{\beta }}_{*}={\left({\displaystyle \frac{{\mathrm{\omega }}_{pL}}{{\mathrm{\omega }}_{BL}}}\right)}^2{\displaystyle \frac{<V_{\parallel L}^2>}{c^2}}>1,$$

(3)

where $<V_{\parallel L}^2>$ is the mid-square longitudinal velocity of energetic electrons with energy close to $40\hspace{0.33em}\mathrm{k}\mathrm{e}\mathrm{V}$. The condition (3) is rather stringent and can usually be satisfied in the plasmasphere or in clouds of the plasma separated from it (clouds of detached plasma), which are observed in the dayside magnetosphere after strong magnetic disturbances. It is clear that a convenient integral characteristic of the instability of the radiation belt is the magnitude of the amplification of whistler waves during one passage through a magnetic trap:

$$\mathrm{\Gamma }={\displaystyle \underset{-l/2}{\overset{l/2}{\int }}\left(2\gamma /V_g\right)dz}$$

(4)

where $2\gamma /V_g$ is the amplification factor included in the transfer equation, $\gamma$ is the cyclotron instability growth rate, $V_g$ is the whistler wave group velocity and $l$ is the magnetic trap length. The frequency corresponding to the highest amplification of electromagnetic radiations can be written as [5]

$${\mathrm{\omega }}_{\circ }={\mathrm{\omega }}_{BL}\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\displaystyle \frac{1}{{\mathrm{\beta }}_{*}}},\left(1-{\displaystyle \frac{T_{\parallel L}}{T_{\perp L}}}\right)\right\}$$

(5)

Typically, the amplification (4) is comparatively small, and the effect of wave energy accumulation is characterized by the reflection coefficient from the ionosphere. Electromagnetic emissions are actually recorded in observation, if the total amplification per one pass through the magnetospheric resonator is greater than the losses. Absolute instability is implemented when the CI threshold is exceeded and

$${\mathrm{\Gamma }}_{max}>\left|\mathrm{l}\mathrm{o}\mathrm{g}R\right|,$$

(6)

where $R$ is the reflection coefficient for the energy of the electromagnetic waves incident on the ionosphere from above. The last condition is satisfied in many cases inside the plasmasphere, where it is possible to excite various VLF noise emissions. The spectrum of electromagnetic emissions is often localized near the frequency

$$<\mathrm{\omega }>\simeq {\displaystyle \frac{{\mathrm{\omega }}_{BL}}{{\mathrm{\beta }}_{*}}}.$$

(7)

The average frequency of electromagnetic emissions does not have a notable drift and is quite stable if the parameter ${\mathrm{\beta }}_{*}>>1$.

2.2. Basic Equations of Plasma Magnetospheric Maser

To analyze relatively slow ($\Delta t>>T_g>T_b$) processes in PMM we can use a self-consistent system of quasilinear equations averaged over the oscillations of energetic particle ($T_b$ is the period of bounce oscillations) and waves ($T_g$ is the period of group propagation) between reflection points. If the spectrum of the electromagnetic waves is relatively narrow and localized in a region of frequencies (7) low in comparison with the cyclotron frequency ($\mathrm{\omega }<<{\mathrm{\omega }}_B$). Then the averaged system of quasilinear equations is written as [5]:

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }t}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D\in {\displaystyle \frac{\mathsf{\partial }F}{\mathsf{\partial }x}}\right)-{\displaystyle \frac{F}{T_0}}+J, \\ {\displaystyle \frac{\mathsf{\partial }\in }{\mathsf{\partial }t}}=\left({\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{x_c}{\overset{x_{max}}{\int }}K\frac{\mathsf{\partial }F}{\mathsf{\partial }x}dxdV}}\right)\in -\mathrm{\nu }\in . \end{gathered}$$

(8)

The first Equation of system (8) describes the diffusion of energetic electrons over pitch angles due to the interaction with waves, and the second Equation of system (8) is the equation of electromagnetic wave transport with an average growth rate of the CI. In system (8) $F\left(t,x,V\right)$ is the distribution function of energetic electrons, $J\left(x,V\right)$ is the particle’s source power, $T_0$ is the time of particle disappearance (due to collisions, for example), $x=V_{\perp L}/V$ is the sine of pitch angle at the center of the magnetic trap, $\in \left(t\right)$ is the energy density of the waves, and $\mathrm{\nu }$ is the rate of decay. In these equations $D\left(x,V\right)$, $K\left(x,V\right)$, and $x_{max}\left(V\right)$ are known positive functions and $x_c$ is the boundary of the loss cone in velocity space. The non-stationary modes of operation of the PMM that interest us are typical for the so-called weak pitch-angle diffusion [22], which occurs in the case of relatively low power sources of energetic electrons. For such a case, we can use a simple boundary condition for the distribution function

$$F=0$$

(9)

at the boundary of the loss cone at $x=x_c={\sigma }^{-1/2}$, $\sigma =B_{max}/B_L$ is the mirror ratio. The distribution function of energetic electrons in a magnetic field tube is normalized by the condition

$$2\mathrm{\pi }\sigma {\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{x_c}{\overset{1}{\int }}F{T}_b{V}^{3}xdxdV=N}}$$

(10)

where $N$ is the content of energetic electrons in a magnetic field tube with a unit cross-section at the level of the ionosphere.

The efficiency of the CI weakly depends on the angle of the wave normal with respect to the magnetic field up to angles of the order of $\mathrm{\theta }\simeq {30}^{\circ }$. Therefore, waveguide propagation is not required to implement the non-stationary CI modes considered below. This conclusion of the theory is confirmed by the known data of simultaneous observations on several spacecraft (see, for example, [23]).

2.3. Eigen-Frequency of PMM and QP 1 Emissions Excitation

The system of Equation (8) has a steady state corresponding to the balance between the inflow of particles from the source and their precipitation into the ionosphere. The stable steady state $F_{\circ }$, ${\mathrm{\epsilon }}_{\circ }$ may correspond to the generation of VLF hiss. The study of the steady state stability led to the expressions [20]

$${\mathrm{\Omega }}_J={\left({\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{x_c}{\overset{1}{\int }}\widehat{K}Jdx}dV}\right)}^{1/2}$$

(11)

for oscillations near a steady state with frequency ${\mathrm{\Omega }}_J\simeq {\left(\mathrm{\nu }/T_l\right)}^{1/2}$ and growth rate

$${\gamma }_J={\displaystyle \frac{{\mathrm{\epsilon }}_{\circ }}{2{\mathrm{\Omega }}_J^2}}\left({\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{x_c}{\overset{x_{max}}{\int }}K\frac{{\mathsf{\partial }}^2}{\mathsf{\partial }{x}^2}\left(D\frac{\mathsf{\partial }J}{\mathsf{\partial }x}\right)dxdV}}\right),$$

(12)

where $\left|{\gamma }_J\right|\simeq 1/T_l$. Here $\mathrm{\nu }$ is the average rate of decay of whistler waves in the magnetospheric resonator in the second Equation of the system (8), and

$$T_l\simeq {\displaystyle \frac{N}{2S}}$$

(13)

is the mean life time of energetic electrons in the magnetic trap, taking into account all factors. In Equation (13), $N$ is the average content of energetic electrons in the magnetic field tube, and $S$ is the flux of energetic electrons precipitating into one of two conjugate regions of the ionosphere.

Periodic oscillations in the PMM with frequency (11) can be realized at certain values of the power of the source of energetic particles. For example, in the evening sector of the magnetosphere, the power of the sources may be insufficient to achieve the threshold of the CI. In the night sector of the magnetosphere, on the contrary, the power is excessively high. In such a case, the amplification of electromagnetic waves is high and the energy density of the waves very quickly adjusts to the steady state level, and the excitation of quasiperiodic emissions does not occur. For typical conditions in the pre- and post-noon local time sectors of the Earth’s magnetosphere, $T_J=2\mathrm{\pi }/{\mathrm{\Omega }}_J$ is between 10 and 300 s. and the $Q_J$-factor of the oscillations ($Q_J={\mathrm{\Omega }}_J/2{\mathrm{\nu }}_J$, if ${\gamma }_J=-{\mathrm{\nu }}_J<0$) is of the order of several tens. The PMM in the morning and dayside subauroral magnetosphere is an oscillatory system responsible for the excitation of QP electromagnetic emissions in the VLF range with repetition periods of spectral forms of 10–300 s. The existence in PMM weakly damped oscillations is possible 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 relatively low, the accumulation of particles continues, and their content exceeds the steady state 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. If the Q-factor is large enough, then the stationary generation mode is realized. Therefore, the conditions corresponding to the relatively small reflection coefficient of electromagnetic waves from the ionosphere can be very important. The presence of an eigen-frequency in the radiation belts, which corresponds to the periodic processes of accumulation of energetic particles in the radiation belts and their precipitation into the ionosphere during electromagnetic radiation bursts, is the root cause of quasiperiodic VLF radiation. The PMM is especially sensitive to external periodic actions at frequencies close to its eigen-frequency ${\mathrm{\Omega }}_J$. It should be noted that the comparison of the spectra of geomagnetic pulsations and the spectra of the envelope during the implementation of QP 1 VLF emissions in the PMM, carried out in the review [24], confirmed the conclusions about the presence of high-quality oscillations with a frequency of (11) during the implementation of a developed CI in the morning and daytime magnetosphere.

According to Equation (12) for certain angular dependence ${\gamma }_J>0$ and the build up of the parameters oscillations is possible. This is due to the anisotropy modulation of the distribution function. Let us explain the possibility of exciting such self-oscillations in the PMM in more detail.

2.4. Self-Oscillations in the PMM and Excitation of QP 2 Emission

To demonstrate the possibility of implementing self-oscillations in the PMM, we transform the system of Equation (8) to a simpler form. We introduce into the analysis the Sturm-Liouville problem [19,20,25]

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D\left(x,V_{\circ }\right){\displaystyle \frac{\mathsf{\partial }{Z}_q}{\mathsf{\partial }x}}\right)=-{\mathrm{\delta }}_q{Z}_q$$

(14)

in which the eigen-functions $Z_q$ satisfy the same boundary conditions as the distribution function. The eigen-functions of the self-adjoint differential operator can always be chosen as real, and the eigen-values are positive (${\mathrm{\delta }}_q>0$). We assume the power of the source of energetic electrons is characterized by a small spread in the modulus of the velocity near $V=V_{\circ }$

$$J={\displaystyle \sum _{q=1}^{\infty }{J}_q}{Z}_q\mathrm{\delta }\left(V-V_{\circ }\right).$$

(15)

Then the solution for the distribution function can naturally be sought in the form:

$$F={\displaystyle \sum _{q=1}^{\infty }{F}_q}{Z}_q\mathrm{\delta }\left(V-V_{\circ }\right)$$

(16)

Using the orthogonality of the system of eigen-functions, we obtain the following equations in total derivatives:

$$\begin{gathered} {\displaystyle \frac{d{F}_q}{dt}}=-{\mathrm{\delta }}_q\in F_q-{\displaystyle \frac{F_q}{T_{\circ }}}+J_q, \\ {\displaystyle \frac{d\in }{dt}}=\left({\displaystyle \sum _{q=1}^{\infty }{h}_q{F}_q-\mathrm{\nu }}\right)\in , \end{gathered}$$

(17)

where $h_q={\displaystyle \underset{x_c}{\overset{x_{max}}{\int }}K\frac{d{Z}_q}{dx}dx}$.

With a suitable value and angular dependence of the particle source power ($h_2{J}_2<0$), self-oscillations are established in the system. Such a conclusion is confirmed by an analysis of the phase plane. The auto-oscillations correspond to QP 2 VLF emissions [14].

2.5. Quantitative Characteristics of Quasiperiodic VLF Emissions

In the developed auto-oscillations the separate pulse duration [20] is

$$t_p~{\mathrm{\nu }}^{-1}~5-30\hspace{0.33em}s.$$

(18)

Pulse repetition period is

$$T_p~2\mathrm{\nu }{\mathrm{\Omega }}_J^{-2}~10-300\hspace{0.33em}s.$$

(19)

Maximum value of the whistler waves energy density is

$${\in }_{max}~2{\mathrm{\nu }}^2{\mathrm{\Omega }}_J^{-2}{\in }_{\circ }.$$

(20)

2.6. Atypical QP 2 Emissions

It was noted in papers [26] that an acoustic-gravity wave in the atmosphere can periodically change the reflection coefficient of whistler wave incident on the ionosphere from above. Advanced calculations show that in the morning and dayside magnetosphere, the corresponding modulation depth can reach 10% [27]. In model calculations based on the system of Equation (5), we can use

$$\mathrm{\nu }={\mathrm{\nu }}_{\circ }\left(1+\mathrm{\mu }\mathit{cos}\left(\mathrm{\Omega }t\right)\right),$$

(21)

where ${\mathrm{\nu }}_{\circ }$ is the rate of decay of the energy density of the whistler wave, $\mathrm{\mu }$ characterizes the modulation depth, and $\mathrm{\Omega }$ is the frequency of the acoustic-gravity wave at ionospheric altitudes. To substantiate the statements already made and to demonstrate the expected effects, it is sufficient to restrict ourselves to the case when the first two angular modes $J_1$, $J_2$, are included in the power of the particle source (15). The corresponding calculations can be fully specified in the important case noted in [5], when in the system (8), $D=D_{\circ }x$, $K=K_{\circ }{x}^2$. For such a case, detailed calculations are given in the article [28], where it is shown that in dimensionless variables the system of Equation (17) is written in the form

$$\begin{gathered} {\displaystyle \frac{d{n}_1}{d\mathrm{\tau }}}=-p_1^2\mathrm{\epsilon }{n}_1-\mathrm{\eta }{n}_1+j_1, \\ {\displaystyle \frac{d{n}_2}{d\mathrm{\tau }}}=-p_2^2\mathrm{\epsilon }{n}_2-\mathrm{\eta }{n}_2+j_2, \\ {\displaystyle \frac{d\mathrm{\epsilon }}{d\mathrm{\tau }}}=r_1\mathrm{\epsilon }{n}_1+r_2\mathrm{\epsilon }{n}_2-\left(1+\mathrm{\mu }\mathit{cos}\left(\Delta \mathrm{\tau }\right)\right)\mathrm{\epsilon }. \end{gathered}$$

(22)

Here $\mathrm{\tau }={\mathrm{\nu }}_{\circ }t$, $n_{1,2}\left(\mathrm{\tau }\right)={\displaystyle \frac{K_{\circ }{x}_{max}^2}{{\mathrm{\nu }}_{\circ }}}{F}_{1,2}$, $\mathrm{\epsilon }\left(\mathrm{\tau }\right)={\displaystyle \frac{D_{\circ }}{{\mathrm{\nu }}_{\circ }{x}_{max}}}\in$, $\mathrm{\eta }={\displaystyle \frac{1}{{\mathrm{\nu }}_{\circ }{T}_{\circ }}}$, $\Delta ={\displaystyle \frac{\mathrm{\Omega }}{{\mathrm{\nu }}_{\circ }}}$. If $x_c/x_{max}=0.01$, then $p_1=0.548$, $p_2=2.475$, $r_1=0.0566$, $r_2=-0.6864$. The system of Equation (22) allows modeling many important processes in the PMM, the implementation of which is determined by the power of energetic electron sources and external factors.

At $j_2=0$, $n_2=0$, $\mathrm{\mu }=0$, relaxation oscillations of the PMM parameters are possible with the phase portrait of the system and oscillograms of the PMM parameters shown in Figure 1. Recall that we noted relaxation oscillations when discussing QP 1 radiation. At $\mathrm{\mu }=0$, self-oscillations of the PMM parameters with the characteristics explained in Figure 2b are possible. We noted self-oscillations when discussing QP 2, an example of which is shown in Figure 2a. At suitable amplitudes and frequencies of atmospheric action, characterized by parameters $\mathrm{\mu }$, $\Delta$, atypical self-oscillations of the PMM parameters are recorded.

Note that according to the Wiener-Khinchin theorem [29], the Fourier transform of the autocorrelation function is equal to the square of its spectral density. Therefore, the Fourier transform of the sum of the autocorrelation functions of the three orthogonal components of the wave magnetic field is proportional to ${\mathrm{\epsilon }}_{\mathrm{\omega }}$, where ${\mathrm{\epsilon }}_{\mathrm{\omega }}$ is the spectral density of the energy of whistler waves.

Figure 3a shows an example of atypical QP 2 emissions, in which bursts of different intensity periodically alternate. Figure 3b shows the results of a model calculation with alternating bursts. Figure 4a shows an example of atypical QP 2 emissions, in which bursts of different intensity form a chaotic sequence. Figure 4b shows the results of a model calculation with a chaotic sequence of bursts.

2.7. Broadband QP 2 Emissions

In the previous sections we assumed the condition ${\mathrm{\beta }}_{*}>>1$ to be satisfied and ignored the dynamics of the frequency spectrum in Equation (7) of electromagnetic radiation. However, when the given inequality becomes less strict, the dynamics of the frequency spectrum of waves take place. For a quantitative discussion of this issue, it is necessary to use a more general system of equations compared to (8). The generalizing Equation (8) includes the integral of the frequency spectrum of waves and the change in this equation is not very significant. Equation (8) changes fundamentally, and takes the form

$${\displaystyle \frac{\mathsf{\partial }{\mathrm{\epsilon }}_{\mathrm{\omega }}}{\mathsf{\partial }t}}={\displaystyle \frac{2}{T_g}}\left(\mathrm{\Gamma }-\left|\mathrm{l}\mathrm{n}R\right|\right){\mathrm{\epsilon }}_{\mathrm{\omega }}=\left({\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{x_c}{\overset{1}{\int }}\widehat{K}FdxdV}}\right){\mathrm{\epsilon }}_{\mathrm{\omega }}-\mathrm{\nu }{\mathrm{\epsilon }}_{\mathrm{\omega }}$$

(23)

Here ${\mathrm{\epsilon }}_{\mathrm{\omega }}\left(t\right)$ is the spectral density of the energy of whistler waves, $\mathrm{\Gamma }\left(t,\mathrm{\omega }\right)$ is the average gain of whistler waves during a single passage through the radiation belt, $\mathrm{\nu }\left(\mathrm{\omega }\right)=2\left|\mathrm{l}\mathrm{n}R\right|/T_g$ is the decay rate due to all factors, $R\left(\mathrm{\omega }\right)$ is the effective energy reflection coefficient from the ionosphere from above, and $\widehat{K}\left(x,V,\mathrm{\omega }\right)$ is the known operator in the expression for the gain of whistler waves during a single passage through the radiation belt. The differential operator $\widehat{K}\left(x,V,\mathrm{\omega }\right)$ has a rather complicated form (see, for example, [4,5])

$$\widehat{K}=G\left(\mathrm{\omega },x,V\right)\left[\left(Vx\right){\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }V}}+\left({\displaystyle \frac{{\mathrm{\omega }}_{BL}}{\mathrm{\omega }}}-x^2\right){\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\right],$$

(24)

where $G\left(\mathrm{\omega },x,V\right)$ is some known nonnegative function.

Let us first consider qualitatively the dynamics of the frequency spectrum under the conditions of the quasiperiodic mode QP 2, that is not associated with geomagnetic pulsations. It turns out to be very useful to analyze the properties of the averaged coefficient of cyclotron amplification of whistler waves during a single passage of the radiation belt $\mathrm{\Gamma }\left(\mathrm{\omega },F\right)$, which is included in Equation (23). The most important are two frequency dependences $\mathrm{\Gamma }\left(\mathrm{\omega },J\right)$ and $\mathrm{\Gamma }\left(\mathrm{\omega },F_{\circ }\right)$. The mutual position of the maxima of these dependencies determines the direction of the frequency change within the electromagnetic burst. To explain the physical meaning of this statement, let us consider the initial stage of accumulation of energetic electrons in a magnetic trap under the initial conditions, when the spectral energy density is at the noise level ${\mathrm{\epsilon }}_{\mathrm{\omega }}\left(t=0\right)={\mathrm{\epsilon }}_{*}$, and there are practically no particles in the magnetic trap $F\left(t=0\right)=0$. In this case, in accordance with the initial Equation (8), the distribution function will increase according to the linear law

$$F=J\cdot t.$$

(25)

Therefore, at this stage the gain has the following form $\mathrm{\Gamma }\left(\mathrm{\omega },F=J\cdot t\right)=t\mathrm{\Gamma }\left(\mathrm{\omega },J\right)$ and, in accordance with Equation (23), the maximum of the spectral energy density as particles accumulate will be determined by the frequency maximum of the function $\mathrm{\Gamma }\left(\mathrm{\omega },J\right)/T_g\left(\mathrm{\omega }\right)$.

With further accumulation of particles, diffusion in the space of adiabatic invariants is connected, and the maximum of the wave spectrum shifts closer to the maximum of the spectrum in the steady state. The steady state of the system of waves and particles itself is determined by averaged quasilinear equations, which at ${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}=0$ have a solution for the distribution function $F=F_0\left(x,V\right)$ and the spectral density of the wave energy ${\mathrm{\epsilon }}_{\mathrm{\omega }}={\mathrm{\epsilon }}_{0\mathrm{\omega }}$, corresponding to the gain factor $\mathrm{\Gamma }\left(\mathrm{\omega },F_0\right)$. When the quasiperiodic mode is implemented, the system does not reach a steady state, particles precipitate in the ionosphere from the magnetic trap and the process of particle accumulation begins again. The frequencies corresponding to the maximum spectral density of wave energy are largely determined by the anisotropy of the pitch-angle distribution function. The higher the anisotropy, the higher the frequency. Therefore, at the initial stage, the wave spectrum is determined by the angular dependence of the particle source power, and closer to the maximum of the electromagnetic radiation burst. It is determined by the distribution function $F_0$ in the steady state. As a result, the frequency of electromagnetic radiation increases within a separate burst if the dependence of the particle source power on the pitch-angle is less pronounced than that of the steady state distribution.

Usually, when QP 2 type emission is realized, a change in the average radiation frequency does occur (see Figure 5), from which one can draw a conclusion about the angular dependence of the particle source power. If the radiation frequency increases within the electromagnetic burst, this indicates that the particle source power is more isotropic than it could be in a steady state; if the frequency decreases, the particle source has high anisotropy.

In addition to those already noted, another atypical QP 2 emissions are encountered, for which the spectral forms (the logarithm of the spectral density of the wave energy) can be represented as a function with separable variables

$$\mathrm{l}\mathrm{o}\mathrm{g}\left({\mathrm{\epsilon }}_{\mathrm{\omega }}\left(t\right)\right)=\mathrm{l}\mathrm{o}\mathrm{g}\left(\mathrm{\epsilon }\left(t\right)\right)\Delta \left(\mathrm{\omega }\right)$$

(26)

where the non-negative function $\Delta \left(\mathrm{\omega }\right)$ characterizes the frequency dependence. Figure 6 shows a typical example of the spectrogram of quasiperiodic emissions, which show elements with a large and almost synchronous change at different frequencies with a period of about one minute. Synchronicity is understood here in the sense that the logarithm of the spectral energy density at different frequencies has the same time dependence. The emissions were observed by the Van Allen Probe spacecraft during local daytime. Panels Figure 6b,c show the onboard magnetometer data and the plasma density data, indicating observations inside the plasmasphere.

Under conditions of developed self-oscillations at the stage of particle accumulation, the distribution function grows linearly with time (25), a rapid precipitation of particles occurs during a relatively short burst of electromagnetic radiation, and all details of the formation of the frequency spectrum of waves are controlled by Equation (23). Substituting the spectral energy density (26) and the distribution function in the form (25), we obtain that at the main stage of particle accumulation, the equation is satisfied

$${\displaystyle \frac{\mathsf{\partial }\mathrm{l}\mathrm{o}\mathrm{g}\left({\mathrm{\epsilon }}_{\mathrm{\omega }}\right)}{\mathsf{\partial }t}}={\displaystyle \frac{\mathsf{\partial }\mathrm{l}\mathrm{o}\mathrm{g}\left(\mathrm{\epsilon }\right)}{\mathsf{\partial }t}}\Delta \left(\mathrm{\omega }\right)={\displaystyle \frac{2}{T_g\left(\mathrm{\omega }\right)}}\left(t\mathrm{\Gamma }\left(\mathrm{\omega },J\left(x,V\right)\right)-\left|\mathrm{l}\mathrm{o}\mathrm{g}R\left(\mathrm{\omega }\right)\right|\right)$$

(27)

from which it follows that for the existence of broadband QP 2 radiations, the functions $\mathrm{\Gamma }\left(\mathrm{\omega },J\left(x,V\right)\right)$ and $\left|\mathrm{l}\mathrm{o}\mathrm{g}R\left(\mathrm{\omega }\right)\right|$ must have the same frequency dependence. It is important that large-scale atmospheric processes can modify the reflection coefficient from the ionosphere from above and ensure that the specified condition is met.

Note that the obtained result can be substantiated more formally. Let us assume that a developed periodic regime is realized. Then, with a suitable start of time counting, the distribution function in the interval between electromagnetic pulses grows proportionally to time, repeating the angular dependence of the particle source power $F=J\left(\kappa ,V\right)t_{\circ }$. The spectral density of electromagnetic waves satisfies the averaged transfer Equation (23), which can be written as

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }t}}\left({\displaystyle \frac{T_g}{2}}\mathrm{l}\mathrm{o}\mathrm{g}\left({\mathrm{\epsilon }}_{\mathrm{\omega }}\left(t\right)\right)\right)=\mathrm{\Gamma }-\left|\mathrm{l}\mathrm{o}\mathrm{g}R\right|$$

(28)

and carry out integration in Equation (28) over the period $T\left(\mathrm{\omega }\right)$, which conceivably can depend on the frequency

$$T\left(\mathrm{\omega }\right)={\displaystyle \frac{2\left|\mathrm{l}\mathrm{o}\mathrm{g}R\left(\mathrm{\omega }\right)\right|}{\mathrm{\Gamma }\left(\mathrm{\omega },J\right)}}$$

(29)

For QP 2 emissions, the spectral forms of which can be represented as a product of functions that depend on time and frequency. The period (29) does not depend on frequency. This is possible with the same frequency dependence of the reflection coefficient of whistler waves from the ionosphere from above $R\left(\mathrm{\omega }\right)$ and the averaged coefficient $\mathrm{\Gamma }$ of cyclotron amplification of whistler waves during a single passage of the radiation belt, in which the power of energetic electron sources in the magnetic field tube is substituted instead of the distribution function.

3. Some Channels of Influence of Natural Atmospheric Processes on the Operating Modes of PMM and Quasiperiodic VLF Radiations

Let us list the atmospheric processes that can, conceivably, influence the operation of the PMM. These are sources of whistler and relativistic particles associated with thunderstorm activity; changes in the current system caused by modification of the equatorial current jet; a response of the reflection coefficient of electromagnetic waves from the ionosphere from above to acoustic-gravity waves (see, for example, [27,30,31]). We will discuss the last case in more detail.

Periodic action of atmospheric disturbances with a suitable frequency on the ionosphere can significantly affect the operating modes of the PMM and, as a consequence, the quasiperiodic VLF radiation in the magnetosphere. Such a specific, effective reason for controlling the operating modes of the PMM is the periodic action on the quality factor of the magnetospheric resonator by modulating the rate of decay of the electromagnetic waves (21).

Estimates show that the eigen-frequency of the PMM ${\mathrm{\Omega }}_J$ is higher than the Brunt-Vaisla frequency at ionospheric altitudes ${\mathrm{\Omega }}_g$ and the limiting acoustic frequency ${\mathrm{\Omega }}_A$ and corresponds to the infrasonic frequency range. At present, based on numerous theoretical and experimental studies, there is no doubt about the existence of infrasonic disturbances at ionospheric altitudes with the spatial and temporal scales of interest to us (see, for example, [32,33,34]). Data have been published on disturbances of the total electron density of the ionosphere with periods of less than five minutes, correlating with ground sources (earthquakes, volcanoes, ground explosions [35,36,37]). For example, vertical movement of the Earth’s surface can become a source of infrasound waves. It is generally accepted that these waves propagate almost vertically and can reach ionospheric altitudes. The period of such infrasonic waves is greater than 10 s; waves with smaller periods attenuate due to viscosity and do not reach ionospheric altitudes (see, for example, [38] and references therein). Note that periods greater than 10 s are also typical for the PMM eigen-frequency.

To make rough preliminary estimates of the impact of earthquakes on the state of the ionosphere, it is natural to use the isothermal atmosphere model and use the dispersion relation for the case of almost vertical propagation of infrasound waves [39]

$${\mathrm{\Omega }}^2\simeq c_s^2\left({\displaystyle \frac{4{\mathrm{\pi }}^2}{{\mathrm{\Lambda }}_z^2}}\right)+{\mathrm{\Omega }}_A^2,$$

(30)

where $c_s$ is the speed of sound, and ${\mathrm{\Lambda }}_z$ is the vertical length of the infrasonic wave. Let the area captured by the earthquake be about $d=100$ km and the period of the oscillation process be about 2 min. With such periods, the impact on the PMM will be optimal. Then, considering the propagation of infrasound to be practically vertical and assuming the speed of sound to be about 300 m/s, we obtain a rough estimate of the vertical length of the infrasonic wave ${\mathrm{\Lambda }}_z\approx 18\hspace{0.33em}$ km, the width of the directivity diagram $\Delta \Theta \approx {20}^0$, and the area of illumination at an altitude of about 100 km is about 80 km.

Acoustic-gravity waves, having reached ionospheric altitudes, can cause modulation of the electron density and the associated change in the reflection coefficient of whistler waves from the ionosphere from above. The efficiency of the electron density modulation can be estimated by applying the usual equations of two-fluid quasi-hydrodynamics for electrons and ions. Assuming that the interaction between neutral particles and ionospheric plasma occurs through collisions of neutrals with electrons and ions [26], it is possible to obtain a relationship linking the electron density disturbance with the velocity of neutral particles. Electron density disturbances are determined by the following equation [26]

$${\displaystyle \frac{\mathsf{\partial }n}{\mathsf{\partial }t}}=-{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left\{{\displaystyle \frac{{\mathrm{\nu }}_{in}^2+{\mathrm{\omega }}_{Bi}^2{\cos }^2\mathsf{\chi }}{{\mathrm{\nu }}_{in}^2+{\mathrm{\omega }}_{Bi}^2}}\left(nw-D_a{\displaystyle \frac{\mathsf{\partial }n}{\mathsf{\partial }z}}\right)\right\}.$$

(31)

Here, ${\mathrm{\nu }}_{in}$ is the frequency of collisions of neutrals with ions, ${\mathrm{\omega }}_{Bi}$ is the ion cyclotron frequency, $\mathsf{\chi }$ is the angle between the magnetic field and the vertical, $D_a$ is the ambipolar diffusion coefficient, $w$ is the vertical velocity of neutral particles in an infrasonic wave, and $z$ is the vertical coordinate. As estimates show, at an altitude of about 100 km, the ambipolar diffusion coefficient $D_a\sim {10}^{-4}-{10}^{-2}\mathrm{k}{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and the characteristic diffusion time for the inhomogeneities of interest to us on the scale of tens of kilometers are of the order of several seconds, which is much less than the period of resonant infrasonic waves. In this case, the ionospheric plasma can be considered a passive admixture with respect to the neutral gas, i.e., spatial and temporal disturbances of the ionospheric plasma density follow disturbances of the atmospheric gas. If we assume that the vertical velocity in an inhomogeneous infrasonic wave is $w(z,t)=A(z)\sin (\mathrm{\Omega }t)$ with an amplitude $A\left(z\right)$, then relatively small variations in the electron density are determined by the expression

$$\mathrm{\delta }n={\displaystyle \frac{1}{\mathrm{\Omega }}}{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left({\displaystyle \frac{{\mathrm{\nu }}_{in}^2+{\mathrm{\omega }}_{Hi}^2{\mathrm{c}\mathrm{o}\mathrm{s}}^2\mathsf{\chi }}{{\mathrm{\nu }}_{in}^2+\hspace{0.17em}{\mathrm{\omega }}_{Hi}^2}}{n}_{0}A(z)\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\Omega }t\right).$$

(32)

The modulation will be more noticeable in the morning ionosphere, where the gradients of the stationary electron density are greater than in the daytime ionosphere. The disturbance of the electron density synchronous with the infrasonic wave can affect the operation of the PMM, for example, through the modulation of the reflection coefficient. The value of the reflection coefficient from the ionosphere largely determines the quality factor of the magnetospheric resonator for whistler electromagnetic waves and this affects magnetospheric processes. The problem of the influence of plasma density disturbances caused by infrasonic waves on the reflection coefficient of whistler-range electromagnetic waves incident on the morning ionosphere from above was considered in detail in the paper [27]. The expression for the averaged rate of decay of whistler waves included in Equation (8) has the form

$$\mathrm{\nu }=2{\displaystyle \frac{\left|\ln R\right|}{T_g}}$$

(33)

where $R$ is the reflection coefficient from the ionosphere and $T_g$ is the period of group propagation of whistler waves in the magnetospheric resonator. It was noted in papers [26] that an acoustic-gravity wave in the atmosphere can periodically change the reflection coefficient of the whistler wave incident on the ionosphere from above. For small harmonic variations in neutral and plasma density, Equation (33), the expression for $\mathrm{\nu }\left(t\right)$ we can write in a form identical to Equation (21)

$$\mathrm{\nu }\left(t\right)={\mathrm{\nu }}_0\left(1+\mathrm{\mu }\cos \mathrm{\Omega }t\right),$$

(34)

where $\mathrm{\mu }\simeq {\displaystyle \frac{2{\mathrm{\omega }}_{p0}{\mathrm{\omega }}^{1/2}H{\mathrm{\nu }}_{en0}}{R_0\left|\mathrm{l}\mathrm{o}\mathrm{g}{R}_0\right|c{\left({\mathrm{\omega }}_{He}\left|\cos \mathrm{\theta }\right|\right)}^{3/2}}}\left[A\left(z*\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\Omega }t\right)\right]$, $R_0$ is undisturbed reflection coefficient, and $H$ is characteristic scale of change in height of a quantity in the reflection region at $z=z*$ (see [26]).

Note that the problem of the influence of infrasonic wave-induced disturbances of plasma density on the propagation and reflection of electromagnetic waves of the whistler range propagating in the morning ionosphere from above was solved in the paper [27]. The calculations showed that the strongest modulation of the reflection coefficient of whistler waves from the morning ionosphere is associated with density disturbances at altitudes of about 80–100 km, where the rate of decay of propagating radiation modes increases by more than an order of magnitude within a fairly local region in altitude with an extent of less than 10–15 km. Changes in the electron density at altitudes below 60 km and above 110 km (outside the zone of strong wave attenuation) have no effect on the reflection coefficient and the magnitude of the wave magnetic field on the Earth’s surface. It was found that the efficiency of the impact of plasma density oscillations caused by an infrasonic wave on the reflection coefficient of electromagnetic waves is especially high if the horizontal scales of the whistler and infrasonic waves are close in magnitude.

Calculations of the reflection coefficient given in the papers [40] showed that at reasonable amplitudes of the infrasound wave, the modulation depth of the logarithm of the reflection coefficient from the morning ionosphere can change by 10 percent or even more, and an important parameter $\mathrm{\mu }\simeq 0.1$ included in Equation (21).

4. Signs of Connection Between Large-Scale Atmospheric Events and Atypical QP 2 Emissions from the Van Allen Probes Spacecraft

One of the promising areas of research into the nature of QP emissions in the inner magnetosphere is the study of their possible connection with large-scale atmospheric phenomena, such as earthquakes. Analysis of data from the EMFISIS instruments on board the Van Allen Probes revealed several atypical events whose morphology differs from QP 1 and QP 2 and at the same time demonstrates a spatio-temporal correlation with earthquakes.

The initial sample was formed based on USGS (United States Geological Survey) data for earthquakes with the magnitude $\mathrm{M}>4$, the epicenters of which are located on magnetic shells $L\ge 3$. Geomagnetic coordinates and magnetic shell values $L$ were calculated using geomagnetic field models taking into account the current configuration of the magnetosphere. The coincidence criterion was the location of one of the Van Allen Probe A or the Van Allen Probe B spacecraft at the time of the event in the range $\left|L_{sat}-L_{eq}\right|\le 0.2-0.3$, where $L_{sat}$ and $L_{eq}$ are the magnetic shells of the spacecraft and the earthquake. This approach allowed us to identify those cases in which it was possible to assume a potential impact of seismic activity on magnetospheric processes.

For the selected cases, the analysis of the spectrogram of electromagnetic radiation, cold plasma density, local magnetic field, as well as $Kp$ index and solar wind parameters were performed. Atypical QP emissions manifested themselves as sequences of clear bursts of radiation in the frequency range of 1–3 kHz with irregular amplitude modulation with repetition periods of 30–300 s. For the event of 21 August 2017 ($L\simeq 4.2,LT\simeq 13:00,Kp\simeq 2$), groups of signals alternating in amplitude were observed, coinciding in time with an earthquake of magnitude $\mathrm{M}=4.6\hspace{0.33em}$ (UT 20:52:23) in the Bering Strait area. Another example is 20 December 2017 ($L\simeq 4,\hspace{0.33em}\hspace{0.33em}Kp\simeq 1,\hspace{0.33em}{n}_p\simeq 900\hspace{0.33em}{\mathrm{s}\mathrm{m}}^{-3}$), where a similar structure correlated with an earthquake of magnitude $\mathrm{M}=4.5\hspace{0.33em}$ off the coast of Iceland (05:29:10 UT) in the absence of significant disturbances of the solar wind and IMF.

To assess the significance of the obtained results, control spectra were used in similar geophysical conditions, but without seismic events in the corresponding shells. For example, Figure 7 shows the spectrogram of 9 August 2017, with magnetic disturbance $Kp=1$, during the daytime, near the magnetic shell $L=4,2$, which approximately correspond to the conditions that took place when Figure 8 and Figure 9 were obtained. However, in this case, no pronounced amplitude modulation of electromagnetic radiation was noted, which confirms the connection of the identified features with seismic activity.

A possible mechanism of connection between seismic processes and magnetospheric radiation suggests the generation of infrasonic waves during an earthquake, their propagation into the upper layers of the atmosphere and the ionosphere, where the modulation of the reflection coefficient of VLF radiation occurs. As shown in the PMM model with external periodic action, even weak modulation is capable of initiating a significant modification of self-oscillatory modes of electromagnetic radiation generation.

Thus, the registered atypical QP emissions with an irregular structure and stable time correlation with seismic events may represent a response of the magnetospheric system to atmospheric-ionospheric disturbances initiated by earthquakes. The analysis of atypical QP 2 emissions, likely associated with strong earthquakes, carried out in this paper indicates the potential of the relevant studies, but does not claim to formulate a complete set of geophysical evidence.

5. Conclusions

The work on studying the signs of the influence of large-scale atmospheric disturbances on quasiperiodic ELF/VLF emissions inside the plasmasphere was implemented within the framework of a comprehensive structural analysis of modern theoretical models and observational data on electromagnetic waves in the inner magnetosphere. The study included mathematical modeling based on the theory of a plasma magnetospheric maser, statistical processing of observational data, and spectral analysis.

Conditions for the excitation of quasiperiodic VLF radiations inside the plasmasphere with spectral shape repetition periods are formulated. It is noted that excitation of quasiperiodic radiations does not require waveguide propagation of whistler waves and can be observed in large spatial regions. Many results are based on the fact that the interaction of electromagnetic waves with energetic electrons in a plasma magnetospheric maser at a suitable value of the energetic electron source power has an eigen-frequency corresponding to periodic stages of particle accumulation and their precipitation during bursts of electromagnetic radiation. The presence of an eigen-frequency makes the magnetospheric system sensitive to external periodic effects. Therefore, excitation of QP 1 emissions caused by the resonant effect of geomagnetic pulsations of the Pc 3–4 range on VLF radiation is possible. If the power of the source of energetic electrons has a suitable value and angular dependence, then self-oscillations of the PMM parameters are utilized, which determine the properties of QP 2 radiation not directly associated with geomagnetic pulsations.

Special attention is paid to atypical quasiperiodic emissions. Within the framework of the mathematical model of the PMM, it is shown that if the reflection coefficient from the ionosphere changes periodically, then, unlike QP 2 emissions, bursts of electromagnetic radiation can have alternating intensity. Generation of chaotic sequences of bursts is also possible. An internal reason for the frequency dynamics of VLF bursts of emissions is noted. Conditions for the frequency dependence of the reflection coefficient from the ionosphere from above, under which atypical broadband quasiperiodic VLF emissions are excited, are determined.

The calculations showed that infrasonic waves at the altitudes of the E region of the ionosphere can affect the PMM and provide excitation of atypical quasiperiodic emissions due to a change in the reflection coefficient of whistler waves from the ionosphere from above. It is noted that the strongest changes in the reflection coefficient associated with density disturbances occur in the morning magnetosphere at altitudes of about 80–100 km, within a region that is quite local in altitude (less than 10–15 km).

To analyze the magnetosphere response to earthquakes, observation data from the Van Allen Probe spacecraft were used. The results show that there are QP emissions with an irregular structure, the excitation of which is probably associated with seismic activity. The comparison of the observed events with the geophysical environment (solar wind speed and density, Kp index, Bz parameter, etc.) allowed us to assume the influence of external factors on the PMM operation modes, probably caused by infrasonic disturbances and seismic activity. In several cases, a correlation was noted between earthquakes and the occurrence of atypical QP emissions. This was additionally confirmed by comparing the orbits of the spacecraft and the epicenters of seismic events.

Further study of the details of excitation of quasiperiodic VLF emissions and emissions with significant frequency dynamics inside the plasmasphere has promising prospects and important diagnostic potential.