**4. Excitation of whistler waves in a turbulent plasmapause boundary layer**

In this section we will discuss parametric interaction of quasi-electrostatic lower oblique resonance waves excited by electron and ion diamagnetic currents and hot anisotropic ion distributions [6–9, 23, 24] with ELF waves in the turbulent

plasmasphere boundary layer (TPBL). It is demonstrated below that this nonlinear mechanism can be responsible for generation of broadband, oblique Very Low Frequency (VLF) whistler (W) waves at frequencies much greater than the LH resonance frequency. It is important because the well known whistler generation mechanism by energetic electrons is unavailable in the TPB. Below we present the results of 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]. Due to instabilities the LOR and ELF waves are generated. Results of simulation confirm that due to nonlinear interaction electromagnetic whistler waves propagating in oblique direction are excited. The frequency of excited waves is well above the Lower Hybrid frequency what is in agreement with experimental results.

In general, parametric interaction of two waves, ωk1 and Ωk2 , produces sidebands at the combination frequencies, ω�, that satisfy the matching conditions (4) and (5). In the case in question, we have ω� ωk1 > > Ωk2 , that is, the high-frequency (VLF) and low-frequency (ELF) counterparts, with kj j � < <j j k1 j j k2 . A general approach for solution of this problem was developed to explain symmetric sidebands, observed during active experiments with injection of a high-power VLF pump whistler wave [1] and modulated electron beam [2] into the ionosphere. It was demonstrated that beat wave interaction between the artificially excited VLF wave and natural ELF emissions can produce observed VLF sidebands. This can be viewed as a first step in the process of a broad VLF spectrum formation because subsequent interaction produces secondary sideband waves and this process continues until the broad range of wavenumbers in k-space is excited. This leads to the requirement that analytical description of the problem should be capable to correctly capture nonlinear interaction in the broad range of wavenumbers and wave frequencies.

In [1, 2, 6, 25] equations written in the Fourier space were used and it was sufficient for obtaining the estimate for sideband amplitudes. To study a nonlinear stage of excited wave turbulence we will switch from the Fourier analysis to description in time and space. To do so Maxwell's equations together with equations of motion of magnetized electrons and unmagnetized ions in hydrodynamic approximation will be used. It is well known that to describe correctly nonlinear evolution of VLF turbulence it is necessary to use 3D description [26–28]. We will use two different systems of equations for description of ELF and VLF waves. They are connected through nonlinear terms containing vector nonlinearities. As a result of cumbersome but straightforward manipulations as in [1, 2, 25], we can obtain nonlinear set of equations for parametric interaction of the VLF and ELF waves, which can be found in [7]. The resulting system of nonlinear equations which describes evolution of VLF turbulence and appearance of electromagnetic whistler waves was solved numerically. Using the developed FORTRAN code which employs the predictor–corrector quasi-spectral numerical scheme detailed analysis of nonlinear mechanism of electromagnetic whistler wave generation from the quasielectrostatic LOR wave spectra was demonstrated. Main results of this analysis can be found in [25, 29].

Numerical analysis was carried out in a simulation box with the grid size 256 x 32 x 128. In the x direction it includes 16 VLF wavelengths, in the y direction 2 ELF wavelengths and in the z direction 1 ELF wavelength, what corresponds to 4 VLF wavelengths. In all directions periodic boundary conditions were applied.

An initial value problem was solved with VLF and ELF pump waves turn-on all the time. The adaptive time stepping was implemented with initial dimensionless time step <sup>Δ</sup><sup>t</sup> <sup>¼</sup> <sup>2</sup><sup>π</sup> � <sup>10</sup>�<sup>2</sup> <sup>Δ</sup>t<sup>≈</sup> <sup>10</sup>�<sup>5</sup> <sup>s</sup> . The computation takes a few days on a standard PC. The input conditions are taken close to the observed values in the plasmasphere [23]:

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

B0 <sup>¼</sup> <sup>0</sup>*:*003 G, n0 <sup>¼</sup> <sup>10</sup><sup>2</sup> cm�3, <sup>ω</sup>ce <sup>¼</sup> <sup>5</sup>*:*<sup>3</sup> � <sup>10</sup><sup>4</sup> <sup>s</sup>�1, <sup>ω</sup>pe <sup>¼</sup> <sup>5</sup>*:*<sup>6</sup> � <sup>10</sup><sup>5</sup> <sup>s</sup>�<sup>1</sup> and <sup>ω</sup>LH <sup>≈</sup> <sup>1</sup>*:*<sup>2</sup> � <sup>10</sup><sup>3</sup> <sup>s</sup>�1. The input VLF pump wave is a monochromatic quasielectrostatic LOR wave at <sup>ω</sup><sup>1</sup> <sup>≈</sup>5ωLH, 3-D wavevector k1 <sup>¼</sup> <sup>ω</sup>pe <sup>c</sup> 8, 0, 0*:*94μ�1*=*<sup>2</sup> � �, and the amplitude E1 ¼ 2 mV*=*m . The input ELF wave is a monochromatic MS wave wit hΩ<sup>2</sup> <sup>≈</sup>0*:*<sup>77</sup> <sup>ω</sup>LH, k2 <sup>¼</sup> <sup>ω</sup>pe <sup>c</sup> 0, 0*:*05, 0*:*13μ�1*=*<sup>2</sup> � �, and E2 <sup>¼</sup> 2 mV*=*m. The values of ω<sup>1</sup> and k1, as well as Ω<sup>2</sup> and k2, satisfy the dispersion equation for Fast Magnetosonic (FMS) waves:

$$\alpha\_{\mathbf{k}}^{2} = \frac{\alpha\_{\mathbf{L}\mathbf{H}}^{2}}{\mathbf{1} + \frac{\alpha\_{\mathbf{p}\mathbf{s}}^{2}}{\mathbf{k}^{2}\mathbf{c}^{2}}} \left[ \mathbf{1} + \frac{\mathbf{M}}{\mathbf{m}} \frac{\mathbf{k}\_{\mathbf{z}}^{2}}{\mathbf{k}^{2}} \frac{\mathbf{1}}{\mathbf{1} + \frac{\alpha\_{\mathbf{p}\mathbf{s}}^{2}}{\mathbf{k}^{2}\mathbf{c}^{2}}} \right]$$

Note that the pump wave parameters are chosen specifically so that they are close to but not exactly satisfy the resonance conditions (Eq. (3)) required to get the maximal efficiency of parametric interaction, as described in [1, 2]. However, in the resonance case, the collisionless system of nonlinear equations crashes after only a few time steps because of singularities that cannot be avoided, unless collisional terms are included. Spatial spectra in 2D of the electrostatic potential δΦ of nonlinearly excited VLF waves (frames b, c, and d) and a pump wave Φ<sup>0</sup> (frame a) are presented in **Figure 4**. They were taken in the middle of the computational box (y = 16) at 10�<sup>3</sup> , 0.1, and 0.18 sec from the beginning of the computational run. Presented in **Figure 4** results clearly demonstrate that the spectral density of electromagnetic modes with kc< <ωpe absent initially starts to grow with time due to the wave cascade towards smaller wavenumbers. Eventually we find that electromagnetic VLF whistlers with frequencies from the range 3 <ω*=*ωLH <7*:*5 produce noticeable part of the excited wave spectrum. The wavenumbers of these waves are inside the rectangles in **Figure 4b**–**d**. These waves represent the long wavelength part of the dispersion relation for the FMS waves, which corresponds to an electromagnetic VLF whistler wave. Calculations were initiated with k<sup>⊥</sup> ≈kx > > ky. Time evolution of oblique electromagnetic VLF whistler waves with the wavenumbers

**Figure 4.**

*(a) – 2D representation of the VLF pump wave in Fourier space. (b) – 2D Fourier spectra of VLF density perturbations at time* T1*. (c) - 2D Fourier spectra of VLF density perturbations at time* 96 � T1*. (d) -2D Fourier spectra of VLF density perturbations at time* 171 � T1*.*

**Figure 5.** *The whistler generation efficiency,* <sup>κ</sup><sup>w</sup> <sup>≈</sup> ð Þ <sup>ω</sup>ce*=*2ω<sup>w</sup> <sup>1</sup>*=*<sup>2</sup> Ew*=*ELH *in logarithmic scale versus time.*

from the rectangles in **Figure 4** is presented in more details in **Figure 5**. It shows the change in time of efficiency of wave transformation from quasi-electrostatic to an electromagnetic part of the wave spectra <sup>κ</sup><sup>w</sup> <sup>≈</sup> ð Þ <sup>ω</sup>ce*=*2ω<sup>w</sup> <sup>1</sup>*=*<sup>2</sup> Ew*=*ELH. The root of the mean-square amplitude of the quasi-electrostatic VLF wave field energy density is denoted as ELH whereas Ew represents an electromagnetic wave energy density inside the rectangles. Numerical results show that the amplitude of electromagnetic whistler waves increases with time and eventually reaches the value � 0*:*1ELH. It is worth mentioning that the analytical estimate presented in [6] provides similar values and is also consistent with the experimental data. Presented numerical results allow to implement the following scenario of electromagnetic VLF whistler waves generation based on beat wave excitation mechanism. A VLF pump wave in the process of nonlinear interaction with an ELF pump wave generates sideband waves, which in turn generate another sidebands and spread of the wave spectrum in k-space. This in turn leads to appearance of long wavelength waves corresponding to an electromagnetic whistlers. Amplitudes of these waves rapidly grow and can achieve very large values, up to the 30% of the quasi-electrostatic pump wave amplitudes.

In (a) – (d) electromagnetic VLF density perturbations with frequencies from the interval 3<ω*=*ωLH < 7*:*5 are placed inside a rectangle. To the right of each panel one can find color codes in logarithmic scale representing normalized wave amplitudes. Numerical setup with constant in time pump waves amplitudes was used. Chosen pump waves did not obey resonance conditions for sideband excitation and this resulted in relatively small sideband amplitudes. This is the reason it takes so much time to form a broad VLF wave spectrum presented in **Figure 4**. This in turn leads to relatively slow growth of VLF type perturbations. Numerical results also show that nonlinearly excited ELF perturbations does not contribute much to VLF perturbations.

This conclusion follows from the comparison with the results of simulations with exactly the same input parameters but without taking account of the ELF disturbance.

These waves have also been detected in the TPBL, which is devoid of substorminjected kiloelectronvolt electrons [11, 22]. These emissions represent a distinctive subset of the substorm/storm-related VLF whistler activity and provide the rate of pitch angle diffusion of the radiation belt (RB) electrons that can explain the plasmapause-radiation belt boundary correlation [11]. As the "standard" whistler generation mechanism by energetic electrons is unavailable in the TPBL, [6]

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

suggested nonlinear interactions between quasi-electrostatic LH oblique resonance (LOR) and ELF waves to be the source.

Free energy for enhanced waves comes from electron diamagnetic currents in the entry layer near the TPBL's outer boundary [6, 22], while diamagnetic ion currents and anisotropic (nearly ring like) hot ion distributions are the main contributors near the inner boundary [8, 9]. It is worth mentioning that electromagnetic VLF whistler waves with frequencies far exceeding the Lower Hybrid frequency were produced as a result of numerical solution of nonlinear equations describing interaction of quasi-electrostatic lower oblique resonance (LOR) waves and externally excited ELF waves. This result supports the suggestion that experimentally detected in the TPBL electromagnetic VLF whistler waves with frequencies well above the Lower Hybrid frequency can be produced in the process of nonlinear interaction between the LOR and ELF waves. Experimental results also show that one of the possible mechanisms for changes in the outer radiation belt boundary is connected with the presence of electromagnetic VLF whistler waves. Taking into account that due to the absence of substorm-injected kiloelectronvolt electrons the well known whistler generation mechanism is not applicable to the plasma sheet inner boundary, we can conclude that described above nonlinear generation mechanism can play an important role in this region. This statement is also supported by observations.
