**5. Parametric excitation of whistler waves: LSP simulation results**

A well-developed particle-in-cell plasma simulation code called Large Scale Plasma (LSP) [10] was used to perform 3D simulations of VLF field excitation. We have used the Large Scale Plasma (LSP) simulation code to force the VLF and ELF modes in a cold, magnetized plasma. One of the built in LSP models is the Lagrangian fluid model for both the ion and the electron species. This model was used to obtain presented results. Fluid particles in the model carry the fluid velocity. It is updated every time step with the help of the momentum equation. The fluid particles characteristics such as velocity and position are weighted on the simulation grid. This allows to involve the source terms which define excited electromagnetic fields through supplied density and current density. In addition, using the known density and fluid velocity an equation for temperature can also be solved on the grid. In this model plasma pressure can also be found on the grid assuming an ideal gas approximation. Next, the pressure gradient on the grid can be used in the momentum equation to update the particle velocities and fields. To allow for larger spatial grids and simulation time steps in LSP an implicit energy conservation scheme is used. In this way the scheme provides the Lorentz force push. Electric and magnetic fields are solved self-consistently. This approach allows to substantially reduce simulation time in comparison with implementation of the explicit field solvers. The initial distribution function is assumed to be Maxwellian to allow plasma to behave as an ideal gas.

Simulations to compare the Lagrangian model with the fully PIC results with application to the nonlinear interaction of VLF and ELF waves restricted by 2D approximation were carried out. Obtained results were very close and this was the reason the Lagrangian approach was used. This approach due to the dramatic reduction of simulation time allowed to perform 3D simulations what is necessary to obtain correct nonlinear description of parametric interaction. There are several other advantages in using a Lagrangian fluid approach. Fewer particles per cell are needed and simulations are much quieter. In presented simulations only eight particles per cell were used. To obtain similar quality result using a fully kinetic

approach it was needed to use 200 particles per cell. To carry out simulations a 3D Cartesian geometry was used with imposed externally magnetic field directed along a z axis. The amplitude of the magnetic field was chosen to be 0.3 G and plasma density � <sup>10</sup><sup>5</sup> cm�<sup>3</sup> what corresponds to the ionospheric parameters. In simulations hydrogen ions with a mass ratio of 1836:1 were used. An outlet boundary conditions were used and the wave was allowed to propagate out of a simulation box minimizing reflections and wave return back into the simulation box. In a cold plasma used in simulations no thermal expansion was observed and no particles were leaving a simulation box.

In LSP we can impose a traveling plane wave inside a simulation box. Both VLF and ELF waves are excited simultaneously. We choose *kx*, *ky* and *ω*. We then use the VLF and ELF dispersion to solve for *k*<sup>z</sup> for these waves. Therefore, unlike the waveguide approach which excites a seemingly random set of *k*-vectors consistent with the dispersion relation, we can target any mode we desire. Because we are directly exciting specific modes, we call this method the "Direct Excitation" (DE) method. The boundary conditions are also much simpler using the DE method. We have chosen to use outlet boundaries. The waves are free to leave the simulation domain and there is little reflection of the waves at the boundaries. Finally, the simulation domain is much smaller, which allows is to make the plasma region larger and therefore resolve smaller *k*-vectors. Lastly, because we can target specific *k*-vectors, using the DE method we can test the theory and compare results from LSP with the direct solution to the equations which describe the parametric interaction. The following *k*-vectors and frequencies are used in the simulation results shown below: kVLF ¼ ð Þ 3*:*5, 0, 1*:*1 ωpe*=*c and kELF ¼ ð Þ 0*:*1, 0*:*1, 0*:*11 ωpe*=*c. The ELF and VLF frequencies are 0*:*8ωLH and 12ωLH. This set of parameters leads to δω<sup>þ</sup> ≈0*:*013ωLH and δω� ≈0*:*05ωLH. Therefore, we expect the positive sideband to be a larger amplitude since δω<sup>þ</sup> is smaller than δω� and therefore closer to resonance. Indeed, we have seen that the positive sideband is often a larger amplitude. The simulation was run for �11 ELF periods. The VLF and ELF waves are exciting by specifying the *y*-component of the electric field only. This leads to the excitation of the other electric field components and the magnetic field. Note that we do not specifically excite *B*y. Therefore all field components are excited self-consistently when only one field component is excited. The black curves in **Figure 6** represent the solutions to the VLF and ELF dispersion equations. The most prominent modes occur at the wavenumbers which are driven externally. The VLF dispersion curve

**Figure 6.** *VLF/ELF wave power spectra of magnetic field component by at t = 0.*

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

### **Figure 7.**

*VLF/ELF wave power spectra of the magnetic field component By at time* t ¼ 4*:*5 � TELF*. Wave power spectra presented at* t ¼ 4*:*5 � TELF *are consistent with the VLF dispersion relation presented in Figure 1. This simulation result confirms nonlinear transformation of a quasi-electrostatic LOR to an electromagnetic VLF whistler waves.*

crosses the externally driven VLF *k*-vector and ELF dispersion crosses the externally driven ELF *k*-vector. Therefore, we are confident that we are driving the correct modes (**Figure 7**).

## **6. Conclusion**

In this Chapter using analytical methods and PIC simulation we analyzed efficiency of excitation of electromagnetic VLF whistler waves due to parametric interaction of quasi-electrostatic LOR and ELF waves in the ionospheric plasma.

δω� < < Ωks and values of the sideband amplitudes in agreement with experimental results. It is also possible to satisfy the condition for resonance excitation of VLF waves. If we take into account resonance broadening Δω due to finite collisions then Δω � <sup>ω</sup> ωce ν , where ν is the collision frequency. This means that for nonresonant excitation of sidebands to occur δω� > Δω must be satisfied. In the opposite case when δω� < Δω resonant excitation mechanism takes place.

A numerical model describing nonlinear parametric coupling of LOR with ELF waves in cold collisionless plasma has been developed in order to explain the generation of electromagnetic VLF whistler waves in the TPBL in the absence of energetic electrons. These electrons are usually viewed as a source for generation of electromagnetic VLF whistler waves and absence of them in the satellite data was an unanswered question for understanding of a generation mechanism. The results of the 3D LSP simulation confirm that nonlinearly excited waves exhibit spectral features consistent with the observed electromagnetic VLF whistler waves.

Using PIC simulations we have directly tested the nonlinear mechanism suggested in [6] by forcing a quasi-electrostatic whistler wave (i.e., a LOR wave) and an ELF mode to allow parametric interaction. Obtained simulation results confirm that this generation mechanism is capable to explain observed electromagnetic VLF modes. Simulation results clearly show that the LOR mode has cascaded to lower wave number electromagnetic VLF whistler modes. Therefore, the model proposed in [6] that the observed whistler waves are due to a parametric interaction between the LOR and ELF waves is consistent with the findings from the simulation results.
