Ion gyroharmonic structures in stimulated radiation during second electron gyroharmonic heating: 2. Simulations

Authors


Abstract

Characteristics of the Stimulated Electromagnetic Emission (SEE) spectrum recorded during ionospheric heating near the second electron gyroharmonic frequency, 2fce, have attracted attention due to their possible connection to artificially generated airglow and artificial ionospheric layers. Two newly discovered SEE spectral features within 1 kHz frequency shift relative to the pump frequency are (1) discrete narrowband structures ordered by the local ion gyrofrequency involving parametric decay of the pump field into upper hybrid/electron Bernstein (UH/EB) and ion Bernstein (IB) waves and (2) broadband structures that maximize around 500 Hz downshifted relative to the pump frequency involving parametric decay of the pump field into upper hybrid/electron Bernstein and oblique ion acoustic (IA) waves [Samimi et al., 2013]. In this paper, a two-dimensional particle-in-cell Monte Carlo Collision computational model is employed in order to consider nonlinear aspects such as (1) electron acceleration through wave-particle interaction, (2) more complex nonlinear wave-wave processes, and (3) temporal evolution of irregularities through nonlinear saturation. The simulation results show that the IB-associated parametric decay is primarily associated with electron acceleration perpendicular to the geomagnetic field. More gyroharmonic lines are typically associated with more electron acceleration. Electron acceleration is reduced when the pump frequency is sufficiently close to 2fce. The IA-associated parametric decay instability is primarily associated with electron tail heating along the magnetic field and electron acceleration is reduced when the pump frequency is sufficiently close to 2fce. Characteristics of caviton collapse behavior become prevalent in this case. Results are discussed within the context of some recent experimental observations.

1 Introduction

Stimulated Electromagnetic Emission (SEE) is a secondary electromagnetic (EM) wave that is generated in the ionosphere in response to the strong high-frequency electromagnetic pump wave. Spectral features of SEE can be used as an important diagnostic tool to obtain information about the ionosphere [Leyser, 2001]. For example, SEE may be used to estimate electron temperature [Bernhardt et al., 2009] in the interaction region, the local strength of the geomagnetic field [Leyser et al., 1992], and the density of minor ion species such as H+ during disturbed geomagnetic conditions [Bordikar et al., 2013, 2014]. SEE was predicted for the first time by Stenflo and Trulsen [1978] and later on was observed experimentally by Thidé et al. [1982]. Leyser [2001] has reviewed the experimental observations, theories, and simulation works of all the classic SEE spectral features.

Experimental observations report that during heating near the second electron gyroharmonic frequency, 2fce, field aligned irregularities (FAIs) are stronger [Fialer, 1974], artificial airglow is enhanced [Djuth et al., 2005; Kosch et al., 2007], and strong radar echoes from the E region are measured [Hysell and Nossa, 2009; Hysell et al., 2010]. These reports are in contrast to the observations during heating near the higher harmonics of the electron gyrofrequency i.e., nfce for n≥3 [Djuth et al., 2005] where all the aforementioned phenomena are strictly suppressed. On the other hand, during heating near 2fce SEE exhibits new strong spectral features within 1 kHz of the pump frequency [Bernhardt et al., 2011; Scales et al., 2011; Samimi et al., 2012, 2013; Mahmoudian et al., 2013; Bordikar et al., 2013, 2014; Fu et al., 2013] that may have links to the aforementioned observations.

Two distinct but physically related SEE spectral features are (1) discrete narrowband structures ordered by the local ion gyrofrequency called stimulated ion Bernstein scatter (SIBS) [Bernhardt et al., 2011; Samimi et al., 2013; Mahmoudian et al., 2013] and (2) broadband structures with maximum near 500 Hz downshifted relative to the pump frequency that will be called ion acoustic parametric decay (IAPD) [Samimi et al., 2013]. These spectral features are observed when the pump frequency is within approximately 10 kHz below 2fce [Mahmoudian et al., 2013; Fu et al., 2013]. The study of the generation mechanism of these new SEE characteristics can provide clues toward understanding different behavior of the ionosphere during heating near 2fce in comparison to higher electron gyroharmonics. Moreover, since artificial ionized layers in the ionosphere are produced during heating near harmonics of the electron gyrofrequency [Pedersen et al., 2010, 2011], these new SEE features may be able to play a fundamental role in their detection and tracking. The capability of employing the SEE spectral features to track the artificially generated ionospheric layer and also their possible connection to the artificial airglow are new applications that are currently under investigation. There is evidence that the IAPD feature is consistently associated with artificially generated ionospheric layers [Bernhardt et al., 2013]. Furthermore, the SIBS feature has been recently observed simultaneously with the artificial airglow [Mahmoudian et al., 2013].

The theory of the generation mechanism of these spectral characteristics in terms of parametric decay instabilities was discussed in the companion paper [Samimi et al., 2013] and will be very briefly reviewed in the next section. However, this model does not provide any information about important aspects of nonlinear evolution including electron acceleration due to wave-particle interaction, more strongly nonlinear wave-wave processes, and temporal evolution of irregularities. Such important aspects may be linked to experimental observations such as airglow and radar measurements. Particle-in-cell (PIC) plasma computational models [e.g., Birdsall and Langdon, 1991] are tools that can begin to at least qualitatively answer some of these unanswered questions and shed light on the nonlinear aspects of the generation mechanism of these newly discovered SEE features in their local source region. A 1.5-D PIC model was used previously to study the generation mechanism of a number of the classic SEE features such as downshifted maximum (DM), upshifted maximum (UM), downshifted peak (DP) [Scales et al., 1997; Hussein and Scales, 1997], and the broad upshifted maximum (BUM) [Hussein et al., 1998; Scales and Xi, 2000; Xi and Scales, 2001]. A 2.5-D PIC model was used to study heating experiments near 2fce and look at the generation process of the above mentioned spectral features of the SEE [Samimi et al., 2012]. Preliminary results of the simulation showed significant electron acceleration due to wave-particle interaction [Samimi and Scales, 2012]. The present work will provide a more comprehensive study of the nonlinear evolution using computational models. This paper is organized as follows. In the next section the experimental and theoretical context for the simulations to follow are provided. Then the PIC simulation model is discussed. Next, the results of the simulations are presented. Finally, discussion and conclusions are provided.

2 Observations and Theory Context

In the companion paper [Samimi et al., 2013] experimental observations of the new SEE spectral features were discussed in detail. For reference with the simulations to follow Figure 1 shows example spectra of the two types of narrowband SEE (within 1 kHz of the pump frequency) observed during 2fce heating: (1) the discrete SIBS structures and (2) the IAPD feature that peaks at 436.7 Hz downshifted relative to the transmitter frequency. The IAPD structure has a weaker anti-Stokes sideband at 432 Hz upshifted relative to the pump frequency. These two SEE features are reported when the transmitter frequency is approximately (2fce−10kHz)≤f0≤2fce [Mahmoudian et al., 2013; Fu et al., 2013]. The SIBS structures are observed during the nighttime experiment. The uniform approximation method [Lundborg and Thidé, 1985, 1986] was used to evaluate the electric field strength and its direction relative to the geomagnetic field around the upper hybrid altitude [Samimi et al., 2013]. Since D and E layers do not exist during the nighttime and the ionospheric density is lower than the daytime, the electric field is close to perpendicular to the geomagnetic field at the upper hybrid altitude. This evaluation shows that the angle of the electric field relative to the geomagnetic field is more off perpendicular when the IAPD feature is observed.

Figure 1.

Experimental observations of narrowband SEE spectral features. (a) Stimulated ion Bernstein scatter (SIBS) and (b) the ion acoustic parametric decay (IAPD) recorded at HAARP facilities in July 2011.

A summary of the observational features of the SIBS structures for second gyroharmonic heating are as follows. The time evolution of the structures shows that the third and the fourth gyroharmonic structures typically have the fastest growth rate. All the spectral lines start to develop at the same time which implies simultaneous generation rather than a nonlinear cascading process. The bandwidth is typically less than the ion gyrofrequency fci, and the maximum spectral amplitude occurs for math formulawhere n is the harmonic number. The threshold excitation transmitter power was estimated to be 0.8 MW (63 MW effective radiated power (ERP)). Experimental observations show that the discrete SIBS structures start to appear above the noise level of the power spectrum after approximately 6 s, while the IAPD spectral feature develops much faster. This implies the growth rate of the ion acoustic parametric instability is higher than the ion Bernstein decay instability which has consistencies with theory. The SIBS structures have now been observed for third gyroharmonic heating, and also it has been shown that the IAPD observed for second gyroharmonic heating is equivalent to the classic downshifted peak (DP) SEE feature observed for third gyroharmonic heating [Mahmoudian et al., 2013].

Parametric decay of the pump wave into another upper hybrid/electron Bernstein (UH/EB) and either neutralized ion Bernstein (IB) waves or an oblique ion acoustic (IA) wave was proposed as a generation mechanism for the SIBS and IAPD SEE features, respectively [Scales et al., 2011; Samimi et al., 2012, 2013]. The general dispersion relation under weak coupling condition is described by [Porkolab, 1974]

display math(1)

where ω1=ω0ωs, ϵ(ω)=1+χe(ω)+χi(ω), and ϵe(ω)=1+χe(ω). Note that in equation (1) coupling terms corresponding to the other harmonics of ωn=ωs±nω0 are neglected because either those terms are off resonance or the coupling coefficient is very small and negligible.

The susceptibility of the jth species is given by

display math(2)

where math formula, k is the wave number, k(k) is the component of k perpendicular (parallel) to the magnetic field B, ρj is the gyroradius, ζjn=(ω+iνjnΩn)/kvtj, Ωn is the angular gyrofrequency (in radian/s), vtj is the thermal velocity, νj is the collision frequency, Γn(bj)=In(bj) exp(−bj), Z is the Fried Conte function, In is the modified Bessel function of the first kind of order n, λDj is the Debye length, and βe the coupling coefficient, proportional to the pump field E0, is given by

display math(3)

and math formula. It was shown that this model has many consistencies in line with experimental observations of the SIBS structures, the IAPD feature, and the IAPD feature with the embedded SIBS structures [Samimi et al., 2013]. The angle of the pump field relative to the background magnetic field and the pump field strength are the two critical parameters that determine the transition of the low-frequency decay mode from the neutralized IB modes to the oblique IA mode. As the off-perpendicular angle of the electric field relative to the magnetic field, θE, increases, instead of the discrete neutralized IB modes responsible for the SIBS structures, the oblique IA mode that generates the broadband structure is excited [Samimi et al., 2013]. The threshold excitation level of the neutralized IB modes is less than the oblique IA mode. The growth rate of the oblique IA mode is always larger than the growth rate of the neutralized IB modes that is in line with the experimental observations [Samimi et al., 2013]. The frequency shift of the pump field relative to 2Ωce(Ωce=2πfce) determines which neutralized IB mode has the fastest growth. As the frequency shift increases below 2Ωce, the higher modes such as third, fourth, and even eighth have the highest growth rate consistent with observations [Mahmoudian et al., 2013]. While above 2Ωce usually the first or the second mode has the highest growth rate.

In the simulation model that will be described in the next section, because of the restrictions of the computational time, an artificial ion-electron mass ratio mi/me=400 is used instead of the real ion-electron mass ratio, i.e., mi/me=16×1836 for O+ions. This small ion-electron mass ratio provides acceptable separation of the electron and ion time scales and will not qualitatively alter the physics. Calculations of the decay instability from equation (1) with the scaled simulation parameters are provided here. These calculations are employed as a guidance for the simulations and provide a framework for comparing and justifying the simulation results. The calculations of the decay instability using reduced mass ratio show qualitatively similar behavior to what is obtained using the real ion-electron mass ratio.

Figure 2 is one example of the calculations that shows the effect of the angle of the pump electric field relative to the geomagnetic field on the low-frequency decay mode for vosc/vte=0.9, ω0=2Ωce−5Ωci, Te/Ti=3, θE=11.4° (Figure 2a) and θE=22.9° (Figure 2b) where vosc=qeE0/meω0. These parameters are in the same regime that was used for the calculations of the decay instability in the companion paper for the real ion-electron mass ratio [Samimi et al., 2013]. The left vertical axis shows the normalized frequency, the right vertical angle shows the corresponding normalized growth rate, and the bottom axis shows the normalized perpendicular wave number. The solid lines are the dispersion relation, and the dashed lines are the corresponding growth rate. These calculations show that as the pump electric field becomes more off perpendicular to the background magnetic field the oblique IA wave responsible for the IAPD structures is excited instead of the neutralized IB waves. It should be noted that for simplicity the calculations in Figure 2 are made with the fixed propagation angle θk=θE. Typically, for the IB parametric decay instability, the maximum growth occurs for θkθE. However, it should be noted that the IA parametric decay usually has substantial growth out to very oblique angles relative to the background magnetic field, i.e., θk≈40°. Increase in the ratio Te/Ti also allows the propagation to larger off-perpendicular angles due to reduction in ion Landau damping. This effect becomes particularly important during electron wave-particle interaction as will be seen in the simulations discussed in section 4. The elevation in electron-ion temperature ratio Te/Ti due to wave-particle interaction also has some bearing on the number of SIBS lines observed which is in line with equation (1) which will be discussed.

Figure 2.

The solid lines show the dispersion relation (i.e., frequency versus wave number), and the dashed lines show the corresponding growth rate of the low-frequency decay mode obtained from equation (1) for ω0=2Ωce−5.0Ωci, vosc/vte=0.9, mi/me=400, Te/Ti=3 (a) θE=11.4° and (b) θE=22.9°. Note for larger θE, ion acoustic decay instability is preferentially excited over ion Bernstein decay instability (which produces SIBS).

In the experiments described in the companion paper [Samimi et al., 2013] frequency sweeping was not available. The preponderance of the observations are such that f0≤2fce which implies the emphasis of the simulation investigation here. Although frequency sweeping near 2fce is considered to some degree here, a more thorough study is planned for the future as some past experimental work [Kosch et al., 2005, 2007] has specifically considered heating above the gyroharmonic frequencies (f0>2fce). Behavior of SIBS structures in SEE has been considered under frequency sweeping near 3fce [Mahmoudian et al., 2013], and connections with conclusions of this investigation will be discussed later.

3 Simulation Model

A periodic two space and three velocity dimensional particle-in-cell Monte Carlo Collision (PIC-MCC) computational model using standard techniques [Birdsall and Langdon, 1991] is employed in this study to consider the nonlinear aspects of the SEE generation mechanism in the upper hybrid source region. The boundary conditions are periodic. The background magnetic field is assumed to be in the z direction, and pump electric field is slightly off perpendicular relative to the background magnetic field. θE determines this off-perpendicular angle that is the angle of the E field relative to the y axis. In fact, it is the complementary angle to the angle of the electric field relative to the magnetic field. The simulations are conducted near the upper hybrid resonance altitude that is the source region of the SIBS and IAPD SEE spectral features according to the analytical model of section 2. Thus, the frequency of the pump field is ω0=ωuhΔωwhere ωuh=2Ωce and Δω is a variable pump wave frequency shift from the gyroharmonic frequency.

Figure 3 shows the schematic of the simulation domain for this local model. The simulation box is elongated in the z direction. The length in y is ly=512Δy and in z is lz=2048Δz where Δy and Δz are the grid cell sizes in y and z, respectively and for all cases to follow Δy=Δz. Also, Δy,z=λD or 2λD. The simulation grid is designed so that the wavelength of all of the low-frequency decay modes predicted by the analytical model of section 2 fit in the simulation domain. In the actual interaction region, ωpe≈2π(2.4×106) rad/s, Ωce≈2π(1.4×106) rad/s, νen≈2π(400) rad/s, λDe≈1 cm, vte≈105 m/s, Ωci≈2π(50) rad/s, and ρci≈2.5 m. This implies the typical simulation box size corresponds to spatial scales of tens of meters, and the typical simulation run time corresponds to a few hundred milliseconds which is consistent with the time period of initial early time observation of the SEE features under consideration in the spectrum. The pump field strengths here correspond to E0≈10 V/m as described in Samimi et al. [2013]. Particles are distributed uniformly in space, and 49 particles per species per grid cell are used in the simulation. The total electric charge at each grid point is calculated using the linear weighting technique. Two types of species are considered in the model: electrons and reduced mass ions. The initial velocity distribution of the electrons and ions is Maxwellian. It is assumed that at the beginning of the simulation both species are at the same temperature, i.e., Te=Ti unless otherwise noted. The ion-electron mass ratio mi/me=400 is large enough to separate time scales of the electrons and ions. Since in the interaction region, the wavelength of the pump field is assumed very long, the dipole approximation (k0≈0) is used to represent the pump electric field, i.e., math formula). The pump electric field is spatially uniform across the simulation domain. In each cycle, the superposition of the pump electric field and the electric field due to the particle displacements are calculated. Linear weighting is used to find the total electric field at the position of each particle. The current collisional model assumes elastic electron-neutral collisions that does not include the dependence of the cross section on electron energy [e.g., Vahedi and Surendra, 1995]. A more sophisticated collision model, although straightforward to implement, is beyond the scope and intent of the current study and is planned for future investigation.

Figure 3.

Simulation geometry for the local SEE generation model at the upper hybrid layer for second electron gyroharmonic heating.

4 Simulation Results

According to the proposed analytical model of section 2, the IB and IA parametric decay instabilities are responsible for the generation of the SIBS and IAPD in the narrowband SEE spectrum, respectively. Several PIC simulations were conducted to study each of these parametric instabilities. In this section, simulation results of the IB parametric decay instability are discussed. The effect of the pump field strength and the pump field frequency proximity to 2Ωce on the wave coupling and the particle acceleration are studied. Next, PIC simulation of the IA parametric decay instability is presented. The possible diagnostic information during the development of this process is discussed. It should be noted that for the simulation runs presented in section 4.1, the electron-neutral collision frequency is νen=0, although a number of simulations were run including collisional effects within the framework of the model described. The latter simulations did not show significant qualitative changes other than slight reductions in electrostatic field energy growth and particle acceleration as would be expected by linear growth calculations. Due to the emphasis in this investigation, these results have not been described in detail for brevity.

4.1 IB Parametric Decay Instability

For the first simulation case, the pump field parameters are chosen based on the analytical calculations of Figure 2a, i.e., vosc/vte≈0.9, ω0=2Ωce−5Ωci, and θE=11.45°. Figure 4 shows evolution of the electrostatic field energy across the magnetic field, i.e., math formula, and the frequency power spectrum of the corresponding electric field component, i.e., Ey. The time intervals over which the power spectra are shown are chosen according to the electrostatic (ES) field energy evolution across the magnetic field. The field energy along the magnetic field (not shown) exhibits similar temporal behavior but is weaker by an order of magnitude. The field energy grows to the saturation state and then starts to slowly decay away. An estimation of the growth rate from the field energy is γ/Ωci≈0.3 which is in rough agreement to the analytical model of section 2 which predicts γ/Ωci≈0.12 in Figure 2a. Note that in the simulation, the electron temperature increases as the time evolves, while in the analytical model, the temperature is assumed constant which most likely accounts for slightly higher growth rate in the simulation in comparison to the analytical model. The frequency power spectrum exhibits three spectral lines ordered by ion gyrofrequency. This result is along the prediction of the analytical model for excitation of the IB waves. Each spectral line is associated with one of the IB modes. Figure 5 shows the electric potential wave number spectrum (|φ(kz,ky)|2) and the electron density (ne(z,y)) at the saturation time (Ωcit=13) for the simulation of Figure 4. The wave number spectrum of the earlier time corresponding to the beginning of the field energy growth (Ωcit=6) (not displayed) shows initially the first (n=1) IB mode line develops. As the simulation continues and the electron temperature increases, the wave number area over which the instability is developing becomes larger. Slightly shorter wavelength corresponding to the second (n=2) IB mode appears in the wave number spectrum. The first island in Figure 5 (0.4≤kyρci≤0.8) is due to the first two IB modes that were described. Then the second island (0.85≤kyρci≤1) that is obvious in Figure 5 develops. This island represents the third (n=3) IB mode. Modal analysis of the two islands in Figure 5 confirms the development of the different IB modes at different wavelength as discussed. The second island connects to the first island at later times. In other words the wavelength of the various IB modes overlaps. The wave number of the IB modes obtained from the wave number spectrum is approximately in the same range that is predicted by the analytical model in Figure 2 which is 0.5<kρci<1.5. Note that this process is not cascading [e.g., Zhou et al., 1994]. It will be clarified in the next simulation that all the ion Bernstein modes are destabilized at the same time (but grow at different rates). The enhancement of the electron temperature destabilizes a larger number of the IB modes [Samimi et al., 2012], and the results are consistent with the analytical model of section 2.

Figure 4.

Evolution of the simulated IB parametric decay instability, electrostatic field energy, and the frequency power spectrum of the component of the electric field across the background magnetic field for ω0=2Ωce−5.0Ωci, vosc/vte=0.9, and θE=11.4°. The frequency power spectrum shows harmonic spectral features associated with the IB parametric decay instability. Note that the electric field normalization is defined by math formula.

Figure 5.

Snapshots of the electric potential wave number spectrum and the electron density at the saturation time for the simulation of Figure 4. Electron density irregularities due to the IB parametric decay instability propagate approximately along the pump wave vector, i.e., θkθE. Note that initially the long wavelength mode corresponding to the first ion Bernstein (IB) mode grows and later as the electron temperature increases, the higher IB modes are destabilized. Also, note that the electron density is normalized to the initial uniform electron density distribution. The electric potential is normalized as math formula and is displayed logarithmically in the wave number spectrum.

The electron density shows four dominant wavefronts in the simulation domain with some additional subtle fine structure corresponding to the higher harmonics. The direction of the propagation of the wavefronts is roughly aligned with the pump electric field vector, i.e., θkθE, which is consistent with the equation (1) of section 2. This of course implies that the maximum growth of the waves is aligned with the pump field vector. It is also consistent with the location of the IB instability on the wave number spectrum.

Comparison of the results of the simulation for vosc/vte=0.4 (not shown) and vosc/vte=0.9 shown on Figure 4 indicates that the IB parametric decay instability starts to develop for the weaker pump field, i.e., vosc/vte=0.4, later than for vosc/vte=0.9. The growth rate of the field energy γ/Ωci≈0.17 for vosc/vte=0.4 is approximately one half of the growth rate for vosc/vte=0.9. Therefore, the growth rate almost linearly scales with the pump field strength. The analytical model of section 2 predicts that for the lower pump field strength, a fewer number of the IB modes are destabilized. The frequency power spectrum for vosc/vte=0.4 (not shown) only shows one IB mode which is again consistent with the theory.

In the next two simulations, the pump frequency is offset further from the electron gyroharmonic frequency (2Ωce). The analytical model predicts that for heating below 2Ωce, as the offset frequency of the pump field relative to 2Ωce increases, higher IB modes are destabilized. The higher modes have shorter wavelength. Figure 6 shows the field energy across the magnetic field, the electron kinetic energy across the magnetic field, and the frequency power spectrum for vosc/vte=0.6 and θE=12.8°. For the simulation shown in the left ω0=2Ωce−7.5Ωci and for the one shown in the right ω0=2Ωce−10Ωci. It is important to note that although the pump field strength is vosc/vte=0.6, the field energy grows larger by an order of magnitude than the previous case for which vosc/vte=0.9. Again this is due to the fact the larger shift in pump frequency from 2Ωce is more impactful than the reduction in pump amplitude vosc/vte compared to the previous simulation parameters. A rough estimation of the growth rate from the field energy growth is γ/Ωci≈0.6 for ω0=2Ωce−7.5Ωci and γ/Ωci≈1 for ω0=2Ωce−10Ωci. The electron kinetic energy across the field increases at the same time that the field energy is growing, and the IB parametric decay instability is developing. Comparison of the evolution of the field energy and the electron kinetic energy indicates that the instability saturates in electron acceleration across the magnetic field. It should be noted that some parallel electron acceleration is observed (not shown) with significantly less amplitude than the perpendicular electron acceleration and occurs after the waves have saturated in amplitude. At the saturation time of the field energy, the electron kinetic energy across the magnetic field is almost at its maximum. After the time that the field energy saturates, the enhancement of the electron kinetic energy across the field is relatively small. Therefore, it is observed that the perpendicular electron acceleration due to the IB parametric decay instability is significantly reduced as the pump frequency approaches 2Ωce. A number of other simulations were performed at other offsets of the pump and this general trend holds. The frequency power spectrum also exhibits less number of the IB spectral lines for smaller offset frequencies relative to 2Ωce. The results of the simulations shown in Figure 6 exhibit up to 10 IB spectral lines for ω0=2Ωce−10Ωci in comparison to five IB spectral lines for ω0=2Ωce−7.5Ωci. These simulations also show that the excitation of a larger number of harmonics is associated with more electron acceleration across the magnetic field.

Figure 6.

The field energy across the magnetic field, the electron kinetic energy across the magnetic field, and the frequency power spectrum of the component of the electric field across the magnetic field for θE=12.8° and vosc/vte=0.6. For the simulation shown in the left ω0=2Ωce−7.5Ωci and for the ones in the right ω0=2Ωce−10Ωci. Note that the kinetic energy across the magnetic field grows significantly as the instability is developing and more IB modes are destabilized with a larger shift of the pump frequency below 2Ωce. Note the electric field normalization is given by math formula.

Figure 7 shows the electric potential wave number spectrum and the electron density at the saturation time for the simulation of Figure 6 (right). The wave number spectrum shows wave modes with shorter wavelength in comparison to Figure 5, i.e., up to kyρci≈10; kzρci≈2.3. The higher IB modes are destabilized at shorter wavelength in comparison to the first or the second mode. Note that all the wave modes appear in the wave number spectrum at the same time. It is consistent with the experimental observation and the analytical model of section 2 that predicts simultaneous generation of all of the IB decay modes rather than cascading processes. The electron density also shows long wavelength waves as well as fine structuring at shorter wavelength which is in line with the wave number spectrum. The perpendicular wave number of the nth harmonic of the IB mode is roughly kρcin[Samimi et al., 2013]. This is consistent with the frequency power spectrum of Figure 6 (right) that exhibits 10 IB lines and the wave number spectrum of Figure 7.

Figure 7.

Snapshots of the electric potential wave number spectrum and the electron density at the saturation time for the simulation of Figure 6(left). Note that electron density irregularities due to the IB parametric decay instability propagate approximately along the pump wave vector, i.e., θkθE. Finer structure compared to Figure 5 indicates higher harmonic generation. Also, note that the electron density is normalized to the initial uniform electron density distribution. The electric potential is normalized as math formula and is displayed logarithmically in the wave number spectrum.

4.2 IA Parametric Decay Instability

The IA decay instability has the maximum growth rate at propagation angles that are larger than the angle of the pump electric field vector, i.e., θkθE. The instability develops at higher electron temperatures more efficiently due to reduction in ion Landau damping. Thus, initial electron to ion temperature ratio Te/Ti=3 is used in the next set of simulations for enhanced computational efficiency. Also, although not essential, electron-neutral collisions are useful to clarify important aspects of the evolution of the instability and are included here. It is assumed the ratio of electron-neutral to electron-plasma frequency νen/ωpe=4×10−4throughout this section. Other parameters were chosen similar to the parameters of Figure 2b, i.e., vosc/vte=0.9, ω0=2Ωce−5Ωci and θE=22.9°. The corresponding electrostatic field energy along the magnetic field, the electron kinetic energy along the magnetic field, and the frequency power spectrum of the electric field along the magnetic field (Ez) are shown in Figure 8. The growth rate of the IA parametric decay instability roughly estimated from the field energy growth is γ/Ωci≈0.87. It is close to γ/Ωci≈0.95 predicted by the analytical model of section 2. Note that in the calculations of Figure 2b collision is not considered which results in a larger growth rate. Comparison of the field energy evolution and the electron kinetic energy indicates that the IA parametric instability saturates in the electron acceleration along the magnetic field. It should be noted that some perpendicular electron acceleration (not shown) small in comparison to the parallel electron acceleration also occurs. After the saturation time, the field energy and the electron kinetic energy exhibit two more phases of growth and saturation. This is likely due to the nonlinear caviton collapse [Weatherall et al., 1982] that will be described in more detail shortly. The frequency power spectrum exhibits the IAPD structure approximately 14Ωci downshifted from the pump frequency. A weak anti-Stokes line above the pump frequency can also be observed. Furthermore, a second structure at approximately 25Ωci downshifted relative to the pump frequency is recognizable. This has consistencies with the experimental observations [Samimi et al., 2013]. Figure 9 shows the electric potential wave number spectrum and the electron density at the saturation time of the IA parametric decay instability. The electron density irregularity wavefronts propagate at θk≈68°. Therefore, these waves propagate significantly off the direction of the pump electric field vector at the upper hybrid layer. Again this is in contrast to the IB decay instability irregularities that would propagate more in alignment with the pump electric field vector. The wave number of the IA mode is estimated kρci≈6.7. The destabilized IA mode maximizes at ω/Ωci=14.2. The sound speed cs/vte(0)≈0.061 is easily estimated from the well-known IA dispersion relation (ω=kcs). The electron temperature at the same time that the IA decay instability develops is Te/Te0=1.52. The corresponding analytic sound speed math formulawhich is in a very good agreement with the phase speed obtained from the frequency and wave number spectrum. This calculation validates that the instability observed in the simulation is the IA parametric decay instability as predicted by the theory in section 2. Note that the other strong wave mode at kyρci≤1 is due to the n=1 IB parametric decay instability.

Figure 8.

The field energy along the magnetic field, the electron kinetic energy along the magnetic field, and the frequency power spectrum of the component of the electric field along the background magnetic field for ω0=2Ωce−5Ωci, θE=22.9° and vosc/vte=0.9. The frequency power spectrum exhibits the IAPD feature. Note that the kinetic energy along the magnetic field grows significantly as the instability is developing. Note that the electric field normalization is math formula.

Figure 9.

Snapshots of the electric potential wave number spectrum and the electron density for the simulation of Figure 8. Note that fine structures indicate electron density irregularities due to the IA parametric decay instability. They propagate at a large oblique angle θk≈68° relative to the magnetic field where θk>>θE. Also, note that the electron density is normalized to the initial uniform electron density distribution. The electric potential is normalized as math formula and is displayed logarithmically in the wave number spectrum.

Figure 10 shows evolution of the electric potential wave number spectrum for the simulation of Figure 8. At the beginning of the IA parametric decay (Ωcit=2), the high-frequency decay mode develops at kzρci∼5.8, kyρci∼3.3. The wavelength of this wave mode reduces as the instability evolves. This high-frequency mode intensifies and then weakens, while its wavelength reduces. At Ωcit=10 this wave mode disappears from the wave number spectrum, and another wave mode at longer wavelength, i.e., kρci≈4.8, appears. This wave mode as shown in Figure 10 (bottom) intensifies, while its wavelength reduces. Again the wave mode weakens and ultimately disappears from the wave number spectrum at Ωcit=23. Another cycle of the wave mode formation, intensification, and suppression starts. Each of these cycles corresponds to one cycle of the field energy and the electron kinetic energy growth, saturation, and decay that is shown in Figure 8.

Figure 10.

Evolution of the electric potential wave number spectrum for the simulation of Figure 8. The electric potential is normalized as math formula and is displayed logarithmically in the wave number spectrum.

Figure 11 shows contours of the electric field magnitude |E| on a logarithmic scale right after the electrostatic field energy can be seen to maximize near Ωcit=15 in Figure 8. Contours are elongated in a direction nearly across the magnetic field due to the highly oblique propagation angle. In a very short period of time (from Ωcit = 16 to Ωcit = 17) the intensity of the electric fields inside the contours increases by approximately 20%. The intensified areas of the contours are at the same locations of electron density depletions (not shown). This implies trapping of the electric fields inside the density cavities. At the same time, the electric potential wave number spectrum shows intensification of the wave mode shown in Figure 10. The electron kinetic energy shown in Figure 8 reaches its maximum at approximately this time period which implies transfer of the electrostatic energy trapped in the density cavities to the electron kinetic energy. After the time snapshots of Figure 11, the electric field magnitude rapidly decays away.

Figure 11.

Contours of the electric field magnitude prior to a collapse for the simulation of Figure 8 showing rapid intensification of the field. The electric field is normalized as math formula and is shown logarithmically.

The nonlinear evolution of the wave number spectrum and the electric field intensity appears to be characteristic of caviton collapse considered in past works on Langmuir turbulence during heating experiments [e.g., Weatherall et al., 1982; DuBois et al., 1990]. Characteristics observed in the present simulations which support this hypothesis include the following: (1) After the new cavity is nucleated, its wavelength reduces which implies that the caviton becomes smaller [DuBois et al., 1990]. (2) The corresponding contours of the electric field strength (Figure 11) indicate that the electric field is intensified inside the caviton [Weatherall et al., 1982]. (3) If the wave number of the instability is averaged over a period of time, it is relatively broad [DuBois et al., 1990]. (4) As the wavelength of the new wave mode reduces, the phase speed becomes less than the ion sound speed that can produce caviton collapse [Nicholson and Goldman, 1978]. And (5) The electrostatic field energy is transferred to the electron kinetic energy along the magnetic field and generates tail heating in the electron velocity distribution function along the magnetic field [DuBois et al., 1993; Newman et al., 1990]. The low-frequency power spectrum (not shown) confirms that initially the IA parametric decay instability develops in the simulations as described. The IA parametric instability is expected to possibly play a role in the nucleation of the initial cavitons [Dubois et al., 1990]. Detailed theory of the process just described is out of the scope and intent of the current investigation. However, relevant components will be considered in the future.

Figure 12 shows the distribution function of the electron velocity along the magnetic field. It shows strong tail heating in the electron distribution function. The curve of the electron distribution function at Ωcit=5 shows enhancement of the number of electrons with vz≈6vte0at the tail of the distribution function. Recent experimental studies show that the IAPD structure is typically observed with the artificially generated ionospheric layers for 2fce heating [Bernhardt et al., 2013]. The significant enhancement of the electron velocity increases the collision frequency that results in more ionization of the neutral particles. Enhancement of the ionization ratio generates the new ionospheric layers.

Figure 12.

Time evolution of the distribution of the electron velocity along the magnetic field for the simulation of Figure 8. Note that the IA decay instability generates strong electron tail heating.

The results here indicate significant acceleration of the electrons along the magnetic field at the upper hybrid layer associated with the IA parametric decay instability which would appear to affirm the possible connection with the generation or triggering mechanism of the artificial ionospheric layers. Figure 13 compares the electron kinetic energy along the magnetic field for vosc/vte=0.9 and θE=22.9 for three pump frequencies: ω0=2Ωce−5Ωci, ω0=2Ωce−10Ωci, and ω0=2Ωce−15Ωci. For all the three cases the IA parametric decay instability develops. For the ω0=2Ωce−10Ωci, and ω0=2Ωce−15Ωci, the IB parametric decay also develops which increases electron kinetic energy across the magnetic field as well. As the pump frequency approaches 2Ωceelectron acceleration reduces. This general trend has been reported in experimental observations for heating near higher harmonics of the electron gyrofrequency [Honary et al., 1995].

Figure 13.

Evolution of the electron kinetic energy along the magnetic field for vosc/vte=0.9 and θE=22.9° for three simulations with −5Ωci, −10Ωci and −15Ωci pump frequency offset relative to 2Ωce. Note that as the pump frequency approaches 2Ωce, electron acceleration reduces.

It should be mentioned that for the offset frequencies −10Ωci and −15Ωci, although the electron temperature enhancement is higher, the nonlinear caviton collapse process is not as strong as for the case with −5Ωcioffset frequency. The caviton collapse process is dominant in the regimes where the ion Landau damping is high [DuBois et al., 1990]. Enhancement of the electron temperature reduces ion Landau damping. Since enhancement of electron kinetic energy (electron temperature) reduces for the pump frequencies closer to 2Ωce, the caviton collapse process is more dominant.

Note that in the absence of collisions, as expected, electron acceleration along the magnetic field is more significant. However, electron acceleration across the magnetic field is more efficient when collisions are included since electron-neutral collisions demagnetizes electrons and allows more efficient electron acceleration across the magnetic field. Therefore, in collisional simulations of the IA decay instability, the IB decay instability typically develops at the same time for sufficiently large shifts of the pump frequency from 2ωce.

It should be mentioned that the IA decay instability can be preferentially excited over the IB decay instability for stronger pump electric fields, i.e., vosc/vte≥1.5 as well. As it was stated before, the analytical model also predicts that the stronger pump field excites the IA mode instead of the IB modes since the ions become unmagnetized due to large growth rate.

5 Discussion and Conclusion

Results of computational modeling of heating experiments near 2fce are in line with the experimental observations of new narrowband SEE spectral features within 1 kHz of the pump frequency and the analytical description of their generation mechanism. According to the analytical model, the discrete structures ordered by ion gyrofrequency so-called stimulated ion Bernstein scatter (SIBS) are generated through parametric decay of the pump field into an UH/EB wave and neutralized IB modes. The results of the simulation of the IB parametric instability in the source upper hybrid layer shows all the IB spectral lines are generated at the same time rather than cascading which is consistent with the theory and experimental observation [Samimi et al., 2013]. The IB decay instability saturates in electron acceleration across the magnetic field. The electron kinetic energy efficiently increases while the IB instability is growing. As the pump field frequency approaches sufficiently close to 2fce, electron acceleration across the magnetic field reduces and a fewer number of the IB modes are destabilized. Therefore, excitation of a larger number of the IB modes is associated with more electron acceleration across the magnetic field.

All the experimental observations of the SIBS spectral lines are reported during nighttime or near sunset [Bernhardt et al., 2011; Samimi et al., 2012, 2013; Mahmoudian et al., 2013]. The D region is absent during this time period. Thus, electron temperature is enhanced more efficiently by the heater. Recent experimental investigation confirms simultaneous observation of artificial airglow and the SIBS structures in the SEE spectrum [Mahmoudian et al., 2013]. SIBS in the SEE spectrum and the artificial airglow are observed when the transmitter frequency is below 2Ωce[Mahmoudian et al., 2013]. Previous experimental investigations indicate that above 2fce the artificial airglow is enhanced [Kosch et al., 2005, 2007]. This enhancement is attributed to the development of the parametric decay instability and the oscillating two stream instability (OTSI) at the same time [Kosch et al., 2007]. The model of section 2 also predicts that above 2fce, the growth rate of the IB parametric instability is larger. Computational study of the heating above 2fce will be the subject of future investigation.

The other SEE spectral feature observed during second electron gyroharmonic heating is the broadband structure that maximizes around 500 Hz. This feature is called ion acoustic parametric decay (IAPD). It is generated through the parametric decay of the pump wave into another UH/EB and an oblique IA mode. The IA parametric decay saturates in electron acceleration along the magnetic field. The electron kinetic energy along the magnetic field grows significantly, while the IA parametric decay is developing and causes tail heating in the electron velocity distribution along the magnetic field.

The electron kinetic energy and the electrostatic field energy show characteristic cyclical behavior of growth and saturation. The corresponding wave number spectrum shows that as the instability continues to evolve, the wavelength of the destabilized high-frequency mode reduces. While the wavelength is reducing, this wave mode intensifies, then weakens and ultimately disappears from the wave number spectrum. Next, while the destabilized wave mode in the wave number spectrum weakens, the electrostatic field energy also decays; however, the electron kinetic energy reaches to its maximum. Contours of the electric field intensity right after the field energy is maximized shows the electric field intensifies inside the electron density depletions. It implies that the electric field is trapped inside the electron density cavities. As it was mentioned before, this implies that the energy of the destabilized wave mode is transferred to the electrons. Another long wavelength wave mode is generated, and this cycle repeats.

Comparison of these results and the previous studies [Nicholson and Goldman, 1978; Weatherall et al., 1982; DuBois et al., 1990, 1993; Newman et al., 1990] provide evidence that this process is caviton collapse. The high-frequency wave mode in this process is the UH/EB wave that is generated by the IA parametric decay instability. This nonlinear process enhances the tail heating and accelerates electrons along the magnetic field significantly. This may have connection with generation of the artificial ionospheric layers. Recent experiments show simultaneous observation of the IAPD structure and the artificial ionospheric layer [Bernhardt et al., 2013]. In summary, the results of this study indicate that the IAPD would be consistent as a superior diagnostic for efficient electron acceleration along the magnetic field as opposed to SIBS.

The simulation shows that the destabilized IA waves are propagating nearly along the magnetic field. This instability develops when the electron temperature is high, i.e., Te/Ti≥3 which reduces the ion Landau damping. Comparing the results of the simulations of the IA parametric decay indicate that similar to the IB decay instability, at small offset frequencies of the pump field relative to 2fceelectron acceleration is more efficient. As the transmitter frequency approaches sufficiently close to 2fce the electron acceleration along the magnetic field reduces. Therefore, in general the simulation results show that sufficiently close to 2fcefor the both types of the parametric instabilities electron acceleration reduces. It is consistent with the measurements of the electron temperature during heating near third harmonic of the electron gyrofrequency, (3fce) [Honary et al., 1995]. Note that the SIBS and IAPD SEE features are observed for the transmitter frequencies approximately (2fce−10kHz)≤f0≤2fce[Mahmoudian et al., 2013; Fu et al., 2013]. There is no direct measurement of the electron temperature during heating near 2fce. As was mentioned before, Kosch et al. [2005, 2007] report enhancement of the artificial airglow when the transmitter frequency is slightly above 2fcewhich will be the subject of future computational modeling. Another investigation that confirms more efficient electron acceleration when the heater frequency has a small offset relative to 2fceis radar measurements of the field aligned irregularities (FAIs) [Hysell et al., 2010]. The field aligned irregularities (FAIs) are slightly suppressed very close to 2fce, while small offset of the transmitter frequency relative to 2fcegenerates strong FAIs [Hysell and Nossa, 2009; Hysell et al., 2010]. The suppression of the FAIs very close to 2fce is not comparable with the suppression of the FAIs during heating near the third or higher harmonics of electron gyrofrequency [Hysell et al., 2010]. PIC simulation of the OTSI instability that is responsible for the conversion of the EM wave into an ES UH/EB wave will be the subject of future study since it is easily adaptable to the model framework used here. The FAIs are generated by the ES UH/EB waves [Huang and Kuo, 1994].

The current simulation model considers parametric decay instability in the source upper hybrid layer and is a local model. It is the first step in providing a global model for the generation of the SEE. The SEE received on the ground is an EM wave generated in the source region, while the high-frequency decay modes considered with the local model of this study are ES UH/EB waves. These ES waves must of course convert back to the downward propagating EM waves. The nonlinear beat electric current that forms in the interaction region by the high-frequency UH/EB decay mode math formula may work as an antenna as the source of the EM wave [Zhou et al., 1994]. This nonlinear beat current can be obtained from the local PIC computational model. The vector potential and the electric and magnetic field of the down propagating EM wave can be easily calculated from the beat electric current [Zhou et al., 1994].

In the current model, a simple elastic electron-neutral collision is included. In order to study the association of the SEE features, the artificial airglow, and the generation of the artificial ionized layers, it is important in the future investigation to incorporate the electron-neutral excitation collision and the ionization collision in the model [Birdsall, 1991]. The PIC modeling of heating near higher harmonics of the electron gyrofrequency will provide a better picture of the parametric instability near electron gyroharmonic frequencies. There is of course potential for assessing the connection of other SEE features with airglow and artificial ionization layer formation at the higher electron gyroharmonics. Furthermore, consideration of the multi-ion species in the model is appealing due to the recent experimental observations of the H+spectral lines in the SEE spectrum [Bordikar et al., 2013, 2014]. Consideration of the density gradient in the model allows looking at the development of the parametric decay instability over an altitude range. It will provide a more accurate picture of the source region of the various SEE features. Two-dimensional modeling including density gradient and more realistic collisions is the next step that would lead us toward having a global model of the generation of SEE.

Acknowledgments

This work was supported by the National Science Foundation.

Masaki Fujimoto thanks Eliana Nossa for her assistance in evaluating this paper.