**1. Introduction**

The generation of VLF sideband emissions due to parametric interaction of LOR and ELF waves was first suggested in [1, 2] in an attempt to explain an experimental results observed in the ionosphere by the Aureol 3 satellite [3, 4] and during the CHARGE 2B ionospheric rocket experiment [5]. Sideband VLF wave emissions can be explained as secondary peaks above and below the primary peak. They results from parametric interaction of excited VLF and ELF waves. Next, nonlinear parametric interactions between quasi-electrostatic LOR and ELF waves was proposed as possible generation mechanisms of VLF whistler waves in the Turbulent Plasmosphere Boundary Layer (TPBL). Excitation of these waves was analyzed through an assessment of observations from the Cluster spacecraft and Van Allen Probes [6]. To further validate a model developed in [1, 2] and adapted in [6] to explain the observations of whistler waves in the plasmasphere. In [7] a numerical solution of a system of nonlinear equations describing parametric interactions between LOR and ELF pump waves excited in the TPBL by the diamagnetic ion currents and hot ion ring instabilities [8, 9] was analyzed. Obtained results show that nonlinear coalescence of the LOR and ELF waves leads to oblique electromagnetic VLF (whistler) emissions at frequencies much greater than the LH resonance frequency, in agreement with the observations. Finally Particle-In-Cell (PIC) simulation of parametric generation of electromagnetic whistler waves will be discussed. This simulation will be initiated by excitation of the forced wave electric field at VLF and ELF frequencies. Such initiation is possible due to the ability of the PIC code known as Large Scale Plasma (LSP) code [10] to excite traveling plane waves in a simulation box. Wave vectors and frequencies of excited in this way

modes are chosen to satisfy the ELF and VLF dispersion relations. It was demonstrated that quasi-electrostatic VLF and electromagnetic ELF waves in the process of nonlinear interaction were able to excite electromagnetic whistler waves. These results was obtained implementing a Lagrangian fluid model – part of the LSP package. Simulation results also reveal generation of multiple sideband emissions around the pump VLF wave. These simulation results strongly support analytical model presented in [1, 2] and used in [6, 7] to explain the observations of whistler waves in the plasmasphere boundary layer [11].

## **2. VLF waves in the ionosphere**

We analyze excitation of waves with frequencies *ω* several times above the lower hybrid resonance frequency, but below the one half of electron cyclotron frequency *i.e.*:

$$
\alpha \alpha\_{LH} < \alpha < \frac{1}{2} \alpha\_{\text{ct}},
\tag{1}
$$

where the lower hybrid frequency ωLH in the case when ω<sup>2</sup> ce < <ω<sup>2</sup> pe is given by:

$$
\alpha\_{LH}^2 = \frac{\alpha\_{pi}^2}{1 + \alpha\_{pe}^2/\alpha\_{ce}^2}.\tag{2}
$$

and ωpe is an electron plasma frequency. It is well known that in this case in a cold plasma only one mode can be excited. The character features of excited wave field at large distances from the source region can be explained using a plot presented at **Figure 1**. This plot is similar to the commonly used wave refractive index surface plot. The plot at **Figure 1** was obtained using the dispersion relation of VLF waves presented below:

**Figure 1.** *Wave number surface for a constant* ωLH <ω< <sup>1</sup> <sup>2</sup> ωce *with three critical points.*

*Parametric Interaction of VLF and ELF Waves in the Ionosphere DOI: http://dx.doi.org/10.5772/intechopen.100009*

$$\text{co}^2 = \frac{\text{m}\_{\text{i}}}{\text{m}\_{\text{e}}} \frac{\text{k}\_{\text{z}}^2}{\text{k}^2} \frac{\text{o}\_{\text{LH}}^2}{\left(1 + \frac{\text{o}\_{\text{ps}}^2}{\text{k}^2 \text{c}^2}\right)^2} \tag{3}$$

In (3) the wave vector k is defined as k2 <sup>¼</sup> k2 <sup>⊥</sup> <sup>þ</sup> k2 <sup>z</sup> where k<sup>⊥</sup> and kz are the wave vector components perpendicular and along an external magnetic field. In **Figure 1** the wave vector component kz along the magnetic field is plotted versus k<sup>⊥</sup> assuming a constant ω. Most of the radiated by an antenna power can be found in a region in k space occupied by the quasi-electrostatic whistler waves with the parameter ω2 pe*=*k<sup>2</sup> c<sup>2</sup> ≤1. Another part of the wave spectrum in k space which satisfies the condition ω<sup>2</sup> pe*=*k2 c<sup>2</sup> > >1 belongs to the electromagnetic whistler waves and is radiated up to an angle 19.5° in oblique direction. This is the shadow boundary determined by the long wavelength inflection point and radiated power of these waves is small compared to the power radiated into the quasi-electrostatic part of the wave spectrum.

In [12] it was shown that most of the wave power is radiated perpendicular to the curve presented in **Figure 1** and depends from the distance as R�<sup>1</sup> except three critical points which define three directions. Two of them are inflection points d2 kz*=*dk<sup>2</sup> <sup>⊥</sup>. In these points a wave field dependence is given as R�5*<sup>=</sup>*6. The third critical point is defined from the equation d<sup>2</sup> kz*=*dk<sup>2</sup> <sup>⊥</sup> ¼ 0 and provides wave field dependence in the form R�1*=*<sup>2</sup> and corresponds to the wave power radiated along the direction of magnetic field.

## **3. Parametric excitation of VLF waves in the ionosphere**

Nearly monochromatic signals injected from ground-based VLF transmitters are known to experience bandwidth expansion as they traverse the ionosphere [13–17] and magnetosphere [18]. Several mechanisms have been proposed to explain this phenomenon based upon linear and nonlinear scattering assuming existence of magnetic-field-aligned plasma density irregularities. In the absence of ionosphere irregularities a mechanism based on a parametric instability was proposed in [19–21].

Reports on sideband signals associated with VLF transmitter signals are rather scarce. Spectral peaks have been identified near the magnetic equatorial plane on the ISEE satellite at approximately �55 Hz of the carrier frequency (13.1 and 13.6 kHz) of Omega pulses [22]. Similar peaks seem to be observed on the COS-MOS 1809 satellite and generated in the ionosphere by the carrier frequency 19 kHz [17]. Sidebands at approximately �500 Hz of the carrier frequency (11.9 and 12.65 kHz) of Alpha pulses have been observed in the ionosphere by the AUREOL 3 satellite [3, 4].

At first sight, the 50-Hz sidebands observed on AUREOL 3 seem to correspond the Riggin and Kelly [19] prediction in which the transmitted wave decays into a lower hybrid wave and an ion-acoustic type of oscillation. To account for the existence of two symmetric spectral peaks, one may replace the three-wave parametric instability considered by these authors by a four-wave parametric instability (or modulation instability) as suggested in [21]. According to this scheme, the ELF branch is due to a purely growing electrostatic mode with wave vector *k* large enough to provide sidebands � kVs j j off the transmitter frequency.

(Vs is the satellite velocity). This mode is excited in course of a four-wave process by the incident VLF transmitter wave. In our case, the ELF wave branch is

**Figure 2.**

*Averaged power spectral density of electric field. ELF natural emission at 500 Hz, VLF transmitted emission at 12.65 kHz, sidebands at frequencies (12.65 + 0.5) kHz and (12.65–0.5) kHz. This data were observed in ionosphere on AUREOL 3 satellite during experiments in framework of ARCAD project [3].*

clearly electromagnetic and as such is of natural origin. Therefore, another explanation in accord with this experimental data has to be found.

Sotnikov et al. [1] proposed another mechanism for the production of 500-Hz sidebands. It is based on nonlinear coupling between the transmitted wave and the ELF emission above the local proton gyrofrequency. The sidebands are shown to be forced oscillations, excited only where the coupling take place. We consider the nonlinear coupling model described in the articles [1, 2].

Next, analysis of the parametrically generated VLF turbulence has been developed in the articles by Sotnikov et al., [1, 2] as attempt to explain appearance of symmetric sidebands in frequency. Such parametrically generated waves were observed in multiple ionospheric experiments. In these experiments two types of waves were present, waves excited by a VLF transmitter and ELF waves excited due to natural processes in the ionosphere. It was demonstrated that beat wave excitation mechanism can be responsible for appearance of observed sidebands with comparable wave amplitudes. Sidebands were excited at combination frequencies given by:

$$o\_{\pm} = o\_{\mathbb{k}\_1} \pm o\_{\mathbb{k}\_2},\tag{4}$$

which can result in sideband emissions. The sideband wave numbers are matched according to

$$k\_{\pm} = k\_1 \pm k\_2 \tag{5}$$

Note that sidebands are not plasma eigenmodes but forced oscillations excited only where VLF to ELF wave coupling take place (**Figures 2** and **3**).

Using the cold plasma approximation the equations for the perpendicular to magnetic field sideband electric field components can be derived in the form [2]:

$$\begin{aligned} E\_{\perp k\_{+}} &= \frac{e}{2m} \frac{k\_{+}}{\Omega\_{e} \delta \alpha\_{+}} [E\_{k\_{1}} \times E\_{k\_{2}}]\_{x} \\\\ E\_{\perp k\_{-}} &= \frac{e}{2m} \frac{k\_{-}}{\Omega\_{e} \delta \alpha\_{-}} [E\_{k\_{1}} \times E\_{k\_{2}}]\_{x} \end{aligned} \tag{6}$$

*Parametric Interaction of VLF and ELF Waves in the Ionosphere DOI: http://dx.doi.org/10.5772/intechopen.100009*

**Figure 3.**

*ELF natural emission at 2 kHz, VLF transmitted emission at 17.95 kHz, sidebands at frequencies (17.95 + 2) kHz and (17.95–2) kHz. This data were observed during cooperative high-altitude rocket gun experiment (CHARGE 2B) carried out in march 1992.*

where

$$\frac{\delta\alpha\_{\pm}}{\Omega\_{\epsilon}} = \frac{k\_{\text{x1}}/k\_{1}}{1 + \alpha\_{p\epsilon}^{2}/\left(k\_{1}^{2}c^{2}\right)} \pm \frac{k\_{\text{x2}}/k\_{2}}{1 + \alpha\_{p\epsilon}^{2}/\left(k\_{2}^{2}c^{2}\right)} - \frac{(k\_{\text{x1}} \pm k\_{\text{x2}})/k\_{3\pm}}{1 + \alpha\_{p\epsilon}^{2}/\left(k\_{3\pm}^{2}c^{2}\right)},\tag{7}$$

k2 <sup>3</sup>� <sup>¼</sup> k2 <sup>1</sup> <sup>þ</sup> k2 <sup>2</sup> � 2k1k2 cosð Þθ and θ is the angle between **k** vectors.

Sidebands may be a result of nonlinear coupling of the VLF transmitter wave and the natural ELF emission above the local proton gyrofrequency. The VLF wave propagate through the ionosphere as a whistler mode.

$$
\alpha\_{k\_1} = \Omega\_\epsilon \frac{k\_{1\varepsilon}/k\_1}{1 + \alpha\_{p\epsilon}^2/\left(k\_1^2 c^2\right)}\tag{8}
$$

For the transmitted frequency ω*=*2π ¼ 12 kHZ, the corresponding wave number is k1 <sup>≈</sup> <sup>2</sup> � <sup>10</sup>�4cm�1. For the known parameters whistler propagates with ω2 pe*=*k<sup>2</sup> 1c<sup>2</sup> <sup>¼</sup> 1 at large angle to the magnetic field.

The characteristic frequency ωkz of the ELF wave is slightly above the ion gyrofrequency and k2 <sup>≈</sup><sup>3</sup> � <sup>10</sup>�<sup>5</sup> cm�1. These waves generally propagate at large angle to the geomagnetic field. As ω<sup>2</sup> pe*=*k2 2c2 > >1, it is described by

$$
\alpha\_{k\_2} = \Omega\_\epsilon \frac{k\_{2x}}{k\_2} \frac{k\_1^2 \varepsilon^2}{\alpha\_{pe}^2},
\tag{9}
$$

where Ω<sup>e</sup> is the electron cyclotron frequency.
