Ionospheric modifications in high frequency heating experiments

Ionospheric heating and modification by powerful high-frequency (HF) waves transmitted from the ground has been a very active research area over the past four decades.^1–3 It is a platform for experimental and theoretical investigation of wave-wave and wave-particle interactions in magnetized plasma. Considerable advancement toward the understanding of nonlinear plasma processes has been recognized through the observations of various heating induced phenomena^3,4 at Arecibo, Puerto Rico, Tromso, Norway, Gakona, Alaska, and others. The HF transmitter of the High Frequency Active Auroral Research Program (HAARP)⁵ is the most updated facility, which was built in Gakona, Alaska for conducting ionospheric heating experiments. It has a rectangular planar array of 180 elements. Each element consists of a low band (2.8–7.6 MHz) and a high band (7.6–10 MHz) crossed dipole antennas. Each crossed dipole radiates circularly polarized wave up to 20 kW. Overall, the HF transmitter at Gakona, Alaska, radiates circularly polarized waves up to 3.6 MW in the frequency band from 2.8 MHz to 10 MHz. The antenna gain, which increases with the radiating frequency, varies from 15 dB to 30 dB. Thus an effective radiated power (ERP) up to 95 dBw is available in heating experiments, which explore modification effects on the bottom-side of the ionosphere.

HF heater wave interacts with the charged particles in the ionosphere and only electrons can effectively respond to the fast oscillation of the HF wave electric field. Neutral particles can also indirectly experience the presence of the HF heater wave through elastic and inelastic collisions with electrons. In the daytime, the bottomside of the ionosphere is low, where the electron-neutral collision frequencies are large; electrons can be heated by the HF heater wave directly. In the nighttime, electrons cannot be heated effectively by the HF heater wave directly in the F region of the ionosphere where the electron-ion collision frequency is dominant. Due to large mass ratio, only a small fraction of the electron quiver energy in the wave field is transferred to the neutral particles in the elastic collisions; thus the temperature elevation of the neutral particles is small. The inelastic collisions which increase the internal energy of the neutral particles involve mainly supra-thermal electrons with energies exceeding 2 eV.

The electron heating in the D and E regions could make an impact on the ionosphere. A long heating period can change the ionization balance, which involves the processes of (1) photo-ionization by the solar illumination, (2) electron-heating-caused reduction of the recombination coefficient, and (3) enhancement of the electron attachment coefficient caused by the heating of electrons and oxygen molecules. Among the three processes, the first one is crucial and thus the heating effect can be noticeable only in the daytime. Moreover, the electrojet current appeared in these regions can be modulated by intensity modulated HF heater wave to become a virtual antenna, which radiates extremely low frequency/very low frequency (ELF/VLF) waves.⁶ 

Most of the HF heating experiments were focused on the F region modification, where the wave energy delivered to the plasma is also mainly absorbed by the electrons. Because the Coulomb (electron-ion) collisions cannot efficiently damp the electromagnetic (EM) wave delivered to the F region of the ionosphere, the frequency of the HF heater wave is chosen to be less than the maximum cutoff frequency of the ionosphere to keep the wave energy in the bottom-side region. Moreover, O-mode (rather than X-mode) heater is transmitted so that parametric instabilities can be excited to achieve a fast conversion of the HF heater wave into electrostatic (ES) plasma waves,⁷ which are confined in plasma. Therefore, non-thermal processes prevail in the F region where the energy thermal and temperature equilibrium times are much longer. New phenomena attributed to wave-electron and wave-wave interactions were observed.

In this work, several featured experimental observations are discussed in Sec. II to explore the underlying physical mechanisms. In Sec. III, parametric coupling of high frequency and low frequency plasma waves in the presence of an HF heater wave is formulated and analyzed for instabilities excited near the O-mode HF reflection height and near the upper hybrid resonance layer. The threshold fields of the instabilities excited in the heating experiments conducted at the three heating sites: Arecibo, Puerto Rico, Tromso, Norway, and Gakona, Alaska are evaluated in Sec. IV. A summary is given in Sec. V.

It is intended to show that the experimental observations presented in the following all engage to the excitation of parametric instabilities. These observations include

Backscatter radars receive return signals from incoherent and coherent backscatters, wherein incoherent backscatter is mainly caused by the non-uniformity of the background plasma and coherent one is attributed to backward Bragg scattering by the plasma waves (electron plasma waves as well as ion plasma waves). In the presence of a plasma wave (ω, k), the radar signal (ω_R, k_R) is scattered to produce new signals (ω_Rs, k_Rs), whose frequencies and wavevectors are imposed by the Bragg scattering (matching) conditions: ω_Rs = ω_R ± ω and k_Rs = k_R ± k. Because ω_R ≫ ω, |k_Rs| ≅ |k_R|; in the backscattering, k_Rs ≅ −k_R and k = k_p ≅ ∓2 k_R. Thus, the wavelength of the plasma waves that set up coherent backscatter of radar signals has to be half of that of the radar signal and to propagate parallel or anti-parallel to the radar signal. The parallel and anti-parallel propagating plasma waves produce frequency down-shifted and up-shifted radar returns, respectively. Offset by the radar frequency, the frequency spectrum of backscatter-radar returns distributes on both sides of the central frequency at zero. The spectral lines on the negative side correspond to up-going plasma waves, and those on the positive side correspond to down-going plasma waves. In the unheated quiet ionosphere, the spectral intensity of plasma lines in the backscatter-radar returns is in the noise level.

When O-mode HF heater waves (ω₀, k₀ ≅ 0) were transmitted in heating experiments, both up-going and down-going plasma waves were recorded by backscatter radars. The frequency spectra of the electron plasma lines contain discrete spectral peaks located at HF heater wave frequency ω₀ and at frequencies downshifted from the HF heater wave frequency by Δω ≅ (2N + 1)2k_RC_S, N = 0, 1, and 2, where 2k_RC_S is an ion acoustic frequency and C_S is the ion acoustic speed. These lines were enhanced by the HF heater wave and thus named “HFPLs.” The spectral features suggest that HFPLs are correlated to the oscillating two stream instability (OTSI), parametric decay instability (PDI), and Langmuir cascade. First, these parametric instabilities are only excited by the O mode HF heater wave. OTSI excites Langmuir waves (ω, k) together with non-oscillatory purely growing modes (ω_s ≅ 0, k_s ≅ −k); thus the spectral peak of excited Langmuir waves is located at the heater frequency ω₀. PDI decays the HF heater wave to Langmuir waves (ω = ω₀ − ω_s, k) and ion acoustic waves (ω_s, k_s ≅ −k). The wavevector of the HFPLs is k_p ≅ ∓2 k_R, thus the frequency ω of the spectral peak is downshifted from the heater frequency ω₀ by ω_sp = k_spC_S = 2k_RC_S, i.e., ω = ω₀ − 2k_RC_S. The PDI excited Langmuir wave (ω, k) can cascade to a new Langmuir wave (ω′, k′ ≅ −k) and an ion acoustic wave (ω_s, k_s ≅ 2 k), and the cascade process may proceed further to generate more cascade lines. The frequency, ω′, of the first cascade plasma line is downshifted from ω by about 4k_RC_S, and thus from ω₀ by about 6k_RC_S.

In Arecibo HF heating experiments, the intensity and the originating height of the HFPLs showed overshoot^8,9 and down-shifting^10,11 in time, respectively. This is realized that the relevant parametric instabilities prefer to excite Langmuir waves along the geomagnetic field. On the other hand, the Langmuir waves ascribed to the HFPLs propagate vertically, i.e., having an oblique propagation angle of 40° with respect to the geomagnetic field at the Arecibo site. These waves are not the preferred ones and will be suppressed in time by those propagating with smaller oblique angles and having larger growth rates.⁹ The matching height of the Langmuir wave decreases with the increase of the effective electron temperature,¹¹ which is proportional to the total spectral intensity of the Langmuir waves.¹² 

Moreover, the spectral distribution of the HFPLs on the negative side (corresponding to up-going plasma waves) and on the positive side (down-going plasma waves) is asymmetric,¹³ the spectral intensity on the positive side appears to be stronger. Both up-going and down-going plasma waves are excited by the parametric instabilities. Because the plasma density in the bottomside of the ionosphere increases with the altitude, the wavenumber k of the down-going plasma wave is increasing during the downward propagation; on the other hand, the up-going plasma wave continues to decrease its wavenumber k until k = 0 at ω_p = ω, where the wave is reflected to become down-going plasma wave. Because parametric instabilities are excited very close to the reflection height of the plasma waves, the reflected plasma waves do not experience severe spatial attenuation before joining the down-going plasma waves excited directly by the parametric instabilities. These reflected waves also backscatter radar signals to produce frequency up-shifted radar returns, which enhance the spectral intensity of the HFPLs on the positive side.

HF heating enhanced airglow is ascribed to the emissions of atomic oxygen in the excited states. Atomic oxygen is collisionally excited by heating produced energetic electron fluxes. Supra-thermal electron flux in the energy range of ∼3–6 eV is generated by parametrically excited electron plasma waves;⁴ it collisionally excites atomic oxygen to produce airglow, which consists of emissions mainly at 630 nm and 557.7 nm. The minimum electron energies to cause airglow at 630 nm and 557.7 nm are 3.1 eV and 5.4 eV, respectively.¹⁴ Airglow enhancements have been observed in the O-mode HF heating experiments. The enhancement of airglow at 777.4 nm was also observed as the O-mode HF heating wave was transmitted at near the second and third harmonic of the electron gyro-frequency.^15,16 The minimum electron energy to excite 777.4 nm emissions is 10.7 eV,¹⁵ which is well beyond the supra-thermal level. The mechanism of producing such high energy level electron flux involves electron harmonic cyclotron resonance processes.

Electron gyrates, according to the right-hand rule, around the magnetic field at a cyclotron frequency f_c. When it interacts with a left-hand/right-hand (with respect to the magnetic field direction, set along the z axis) circularly polarized wave at a frequency f, the rotation frequency of the wave field seen by the electron is shifted to f₁ = f ± f_c. Thus, a right-hand circularly polarized wave field at a frequency f = f_c becomes a dc field as seen by the electron. This is the (fundamental) cyclotron resonance, the interaction leads to nonzero average in the electron speed.

If f = nf_c, where the integer n > 1 and the wave field is uniform in space, then the net effect on the electron velocity is zero because electron is interacting with a uniform ac field. Even though the wave field has a spatial variation, for example, in the x direction, but the Larmour radius of the electron gyration is neglected, i.e., assuming that the electron interacts with the wave field at a fixed location (at the guiding center) where the wave field is a constant ac field, the net effect of interaction on the electron velocity is still zero.

The finite Larmour radius of the electron gyration makes the electron to experience the spatial variation of the wave field; the field amplitudes in the acceleration phase and in the deceleration phase are different, it results to a nonzero net effect on the electron speed in one gyration period. This is called “finite Larmour radius effect.” As f – f_c = Nf_c, where the integer N ≥ 1, the net effect in each gyration period remains at the same phase so that the net effect on the electron motion will accumulate in time, similar to that in the fundamental cyclotron resonance. Moreover, the interaction changes the Larmour radius in time, setting up a positive feedback to the interaction. The harmonic cyclotron resonances, under the conditions f – f_c = Nf_c, rely on the finite Larmour radius effect,¹⁷ which is weighed in terms of k_⊥v_e⊥/Ω_e, the product of the wavenumber k_⊥ (measuring the spatial variation of the wave field) and the Larmour radius v_e⊥/Ω_e of the electron gyration, where v_e⊥ is the electron speed and Ω_e = 2πf_c. The finite Larmour radius effect works to shift down the wave frequency to the fundamental cyclotron resonance frequency as well as to provide a positive feedback to the interaction. Thus, heating at the harmonic cyclotron resonances directly by the HF heating wave is not effective because the wavenumber of the HF heating wave is very small. On the other hand, an O-mode HF heating wave can excite parametric instabilities to produce short-scale upper hybrid waves, which can interact effectively with electrons at cyclotron harmonic resonance. This is demonstrated in Fig. 1 showing the energize electrons that are generated through electron cyclotron harmonic resonance interaction with parametrically excited upper hybrid waves.^18,19 As shown, extra-thermal electrons are generated in second and third harmonic cyclotron resonance cases. These extra-thermal electrons enhance airglow and produce ionizations when collide with the background neutral particles.

Energetic electron fluxes in the energy range from 10 to 25 eV have been detected in situ by a probe in rocket²⁰ as well as on the ground by radar inferred by ultra up-shifted frequency band²¹ and by the enhancement of airglow at 777.4 nm during the O-mode HF heating.¹⁵ Plasma waves excited by the parametric instabilities are likely to be responsible for the electron acceleration to such high energy level. Langmuir waves can resonantly interact with electrons through the Doppler shift, and upper hybrid waves can introduce finite Larmour radius effect to resonantly interact with electrons through the cyclotron harmonic shift. A process involving electron cyclotron harmonic resonance interaction with parametrically excited upper hybrid waves has been formulated and analyzed^18,19 to explain the observation of artificial ionization layers (AILs) presented in the following.

Observations of optical emissions by Pedersen et al.^22,23 indicated that new ionization layers were produced by the powerful O-mode HF heating wave with frequency set near double the electron gyro-frequency (∼2.85 MHz). The ionization energy of the atomic oxygen “O” is 13.6 eV. Experiments^24–26 were conducted during twilight and early evening hours in Alaska local time, when the photo-ionization was weak. The wave‐ionosphere interaction occurs in the region around 230 to 250 km, where the O⁺ ions are dominant. The experimental results showed that the enhanced optical emissions descended in the background F-region ionosphere²⁷ and the produced artificial ionization layer emerged from the base of the ambient F region as a relatively thin layer, seen directly in the ionograms of the digisonde.²⁸ 

AILs were also observed in later experiments with the heater frequencies set at 4.34 MHz and at 5.8 MHz, which are around the third and fourth harmonic of the electron gyro-frequency. Digisonde ionograms show that AILs also emerge from the base of the ambient F region as relatively thin layers, similar to those formed with the heater frequency set at second harmonic of the electron gyro-frequency. In the parametric excitation, upper hybrid waves are excited by the HF heater wave through lower hybrid decay waves. Thus, the upper hybrid wave frequency f_uk = f₀ – f_Lks, where f₀ is the heater frequency and the lower hybrid frequency f_Lks = f_LHξ^1/2 with f_LH ∼ (Ω_eΩ_i)^1/2/2π and ξ = 1 + (m_i/m_e)(k_z²/k_⊥²). In HAARP, the lower hybrid resonance frequency f_LH ∼ 8.3 kHz. For ξ = 1–10, f_Lks ∼ 8.3–26.3 kHz. Hence, f_uk has a bandwidth of 18 kHz, which covers the change of the 2nd harmonic electron gyro-frequency over an altitude range of about 12 km (because the geomagnetic field intensity increases slightly with a decrease of the altitude).It makes accessible for the match of cyclotron harmonic resonance over this altitude region where electrons can be accelerated continuously, while moving downward, through harmonic cyclotron resonance interaction with the spatially and frequency distributed upper hybrid waves along a slightly increasing geomagnetic field; the density of energized electrons which cause optical emissions increases with the decrease of the altitude and those energized to exceed the ionization energy, 13.6 eV, of the atomic oxygen “O” reach the maximum amount at the bottom of the resonance region.¹⁹ It results in a major ionization occurrence at the bottom of the F region. This explains¹⁹ the descending feature of the enhanced optical emissions²² in the development of an AIL and the emergence of the AIL as a relatively thin layer at the bottom of the F region.²³ 

Digisonde is HF radar probing the electron density distribution in the bottomside of the ionosphere. This radar transmits O-mode and X-mode sounding pulses with the carrier frequency f swept from 1 to 10 MHz and records sounding echoes in an ionogram. A sounding echo represents the backscatter of a corresponding sounding pulse from a layer of the ionosphere, where the electron plasma density N(h′) matches the cutoff density N_cO = (f/9000)²cm⁻³ or N_cX = f(f – f_e)/(9000)²cm⁻³ set by the carrier frequency f of the O-mode or X-mode sounding pulse, where h′ and f_e are the virtual height of the layer and the electron gyro-frequency. The virtual height h′(f) is determined by the time delay τ of the echo to be h′ = cτ/2, where c is the speed of light in free space. h′(f) can be converted to a true height profile N(h) by the plasma density dependent group velocity determined self-consistently. At HAARP site, a profile conversion (NHPC) algorithm,²⁹ which is a Fortran code inverting scaled ionogram trace data h′(f) into N(h) profiles with PC, is available in the software program Standard Archiving Output (SAO) Explorer.³⁰ 

The digisonde radiates at a large cone angle, each sounding pulse can be decomposed into many rays, which have different ray trajectories and only backscattered rays can return to the digisonde and are recorded as the ionogram echoes. In the unperturbed ionosphere, only a few rays, which are close to the vertical transmission, are backscattered. Thus, the virtual height traces of the sounding echoes in the ionogram have narrow virtual height spreads. However, when the ionosphere is perturbed, ray trajectories are also perturbed. In particular, in the presence of large-scale field-aligned density irregularities (FAIs with scale lengths of a few hundreds of meters to kilometers), ray trajectories can be significantly modified.³¹ Hence, in the presence of FAIs, the traces in the ionogram are not contributed by the returns of the vertically incident rays.

In order to achieve backscatter in the presence of FAI, the incident direction of the ray has to have an off vertical angle.³¹ There are multiple obliquely incident rays can be backscattered to the digisonde receiver to produce multiple sounding echoes at the same radar frequency but at different return times,³² resulting to the spread of the virtual height traces. In naturally perturbed ionosphere, the spread usually appears in the frequency band corresponding to the F region of the ionosphere, and thus termed “Spread-F.”

In HF heating experiments, spread-F induced by the O-mode HF heater wave has also been observed and is termed “artificial spread-F.” This is exemplified in Fig. 2 comparing a pair of ionograms without and with heating effect recorded during the heating experiment conducted on November 20, 2009 from 21:10 to 23:00 UTC (12:10 to 14:00 local time). As shown, the artificial spread-F extends from 2.1 to 4.5 MHz indicated by arrows; on the other hand, the parametric instabilities excited by the O-mode heater of 3.2 MHz occur only locally near the HF reflection height (at 3.2 MHz) and around upper hybrid resonance region (at 2.88 MHz).

Kuo and Schmidt³³ show that large scale FAIs can be generated directly by the HF heater wave through thermal filamentation instability. However, O-mode heater generates sheet-like FAIs parallel to the meridian plane,³⁴ which are not effective to enhance virtual height spread of the sounding echo. Another mechanism of large-scale FAI generation is thermal instability,³⁵ which is driven by the downward and upward heat flow from the localized anomalous heating of parametric instabilities excited near the HF reflection region and the upper hybrid resonance region. The generation occurs in an extended region along the geomagnetic field.

Artificial ionization enhancement was also observed in experiments, conducted on November 16 and November 20, 2009, observing artificial spread-F,³¹ even though the O-mode HF heater wave frequency was not close to an electron cyclotron harmonic resonance frequency. Experiments were conducted around local solar noon when the photo-ionization was strong and the wave-electron interaction occurred in the lower F region (<180 km) of the ionosphere, where the electron-ion effective recombination coefficient depends strongly on the electron temperature T_e.³⁶ Through parametric instabilities and thermal diffusion, anomalous electron heating reduced the recombination coefficient over a large region, which changed the balance between the photo-ionization and the recombination loss, giving rise to an electron density enhancement in the heated region below ∼180 km. This is demonstrated in Fig. 3, in which electron density distributions (a) at 21:42:00 UT, after the heater off within 10 s and (b) at 21:43:00 UT, after the heater off more than 60 s, are presented for comparison. As shown, the distribution at 21:42:00 UT has a higher density in the entire modified region. The percentage of the electron density increase exceeds 10% over a large altitude region (>30 km) from below to above the HF reflection height of ∼170 km.

A virtual height bump usually occurs at frequencies near foF1 in the presence of an F1 layer, the true height profile indicates that there is a density ledge (cusp) at foF1, which retards the propagation of sounding signals with frequencies f close to foF1. Spread-F normally extends over a frequency band in the ionogram and is recognized to be caused by the large-scale FAIs in the plasma. However, localized anomalous echo spread appearing as a bump in the ionogram trace has also been observed in the O-mode HF heating experiments. This is exemplified in Fig. 4, showing a combined ionogram from a pair of heater off ionograms acquired at 21:42 UT and 21:43 UT after the O mode heater wave of 3.2 MHz turned on at 21:40 UT for 2 min.³⁷ 

Both ionograms show spread of echoes, except the spread in 21:42 UT is considerably larger and contains a noticeable bump located next to the plasma frequency (∼2.88 MHz) of the upper hybrid resonance layer of the HF heater wave. This heating-induced bump in the ionogram trace at 2.88 MHz is similar in its appearance to the virtual height bump at foF1 in the presence of an F1 layer; the similarity suggests that there is a heater-stimulated ionization ledge (cusp) appears in the upper hybrid resonance region, which explains the appearance of the bump.³⁷ This ledge can be created by the thermal pressure force of the local heating by the HF heater wave excited parametric instabilities in the upper hybrid resonance region.

Heat diffuses from the upper hybrid resonance region downward and upward to generate large scale FAIs,³⁵ which enhance Spread-F. It also increases the electron temperature and reduces the electron-ion effective recombination rate, to cause ionization enhancement over a large region. Due to the field-aligned nature of the upper hybrid waves, these waves could not be detected directly by the UHF radar. The distinctive virtual height bump in Fig. 4 suggests that the digisonde can be a key diagnostic instrument to explore the upper hybrid waves excited in the HF heating experiment.

On November 20, 2009 from 21:00 to 23:04 UT, an experiment was conducted with 2 min on and 2 min off. In the on period, the polarization of the heater wave was switched alternately with O mode and X mode. Echo spread with bump(s) occurs only after O mode heating. The spread was fading away in the subsequent off period and eliminated further by the X-mode heater. In essence, it was 2 min on and 6 min off. The Sun was above the HAARP horizon for the entire experiment period. Therefore, there was no precondition on the background plasma for each O mode heating period. The time development of bumps in the ionogram trace was also observed.³⁷ This is demonstrated in a sequence of six ionograms, presented in Figs. 5(a)–5(f). The echoes in each of the ionograms, were acquired beginning the moment the O mode heater wave turns off. The results show the time change of the virtual height spread as well as the development of the bumps next to the HF reflection height and next to the upper hybrid resonance layer. In Fig. 5(a), two bumps indicated by arrows appear in the regions below 3.2 MHz (i.e., HF reflection height) and 2.88 MHz (i.e., upper hybrid resonance height). Based on the locations of these speculative bumps, one may infer that the Langmuir PDI and upper hybrid PDI were excited concurrently by the HF heater wave. As the experiment proceeded, the Langmuir PDI bump shifted down slightly in frequency and was gradually weakened as indicated in Figs. 5(b) and 5(c); on the other hand, the upper hybrid PDI bump shifted up slightly in frequency and became stronger. This upper hybrid PDI bump evolved to a steady state level as shown in the sequence of Figs. 5(d)–5(f), and the Langmuir PDI bump was disappeared completely in Figs. 5(e) and 5(f). This development infers that the PDI was suppressed by instabilities draining heater energy in the upper hybrid resonance region.

Anomalous attenuation of HF radar echoes over a wide-band was observed in Boulder heating experiments; it was attributed to Bragg scattering of HF radar signals by HF-induced (via parametric instabilities) field-aligned density irregularities of a few tens of meters; the scattered signals did not return to the radar receiver to cause a loss in echoes' intensities.

Shown in Fig. 6 is the first observation of wide-band absorption in HAARP heating experiments³⁷ conducted on July 27, 2011. Ionograms presented in columns A and D were recorded during the heater on periods and in columns B and C were recorded during the heater off periods. The 11 heater-on ionograms were recorded after turning on the heater for 30 s, and the 12 heater-off ionograms were recorded right after turning off the heater. Heating leads to the decrease of the ionogram echo amplitude and only the strongest echoes are recorded. Thus, the heating effect can be manifested by the disappearance of ionogram echoes in the heater-on and heater-off ionograms. In fact, the heater-on ionograms recorded 15 min after the start of the experiment (i.e., at 01:53:31 and later) already show wideband disappearance of echoes.

Moving down along each column of Fig. 6, one observes clearly a decrease in the number of ionogram echoes in both on and off ionograms. The ionogram echoes disappear almost completely in the on ionograms of the last three rows (i.e., from 01:56 to 02:06 UT). This represents wideband absorption on the HF signals. Though wideband attenuation of ionosonde signals is generally attributed to the presence of medium- to large-scale density irregularities (i.e., a few tens of meters to a few kilometers) in the background ionosphere,⁴ nonlinear thermal instability^38,39 also produces periodic density irregularities in the D and lower E regions of the ionosphere. Both elastic and inelastic electron-neutral collision frequencies are electron temperature dependent and modified by the HF heating. Such temperature dependence provides a feedback channel to the heating, leading to the excitation of a thermal instability and the subsequent nonlinear evolution of the produced density perturbation after exceeding a threshold. The spatial distribution of the periodic density irregularities is parallel to the geomagnetic field, which for Gakona, AK is 14° off the zenith. The combination of the increasing D region absorption to weaken the ionosonde signals and the scattering of the ionosonde signals by the produced periodic density irregularities causes the disappearance of ionogram echoes over a wideband.

Parametrically excited high frequency plasma waves set up a ponderomotive force acting on electron plasma. This force modifies the plasma density distribution in which ions follow electrons via the induced self-consistent electric field. The induced density perturbation tends to develop in the field-aligned nature because the diffusion loss along the geomagnetic field is large. The density perturbation redistributes the high frequency plasma waves, which are cumulative in the density depletion regions where the wave phase speed is lower. This in turn reinforces the ponderomotive force to further increase the density perturbation. Such a positive feedback excites an electrostatic filamentation instability,⁴⁰ which generates meter-scale FAIs and filaments the intensity of the high frequency plasma waves. The short-scale FAIs can also be generated directly by the HF heater wave via upper hybrid OTSI,^41,42 which decays the HF heater wave to two upper hybrid sidebands and a short-scale FAI. It was demonstrated that short-scale density irregularities could scatter ground transmitted VHF signals (wavelengths of 1–10 m) back to the ground at a different site, over the horizon distance away from the transmitter site,⁴³ a possible communications link.

Short-scale FAIs (0, k_I) can also scatter (convert) high frequency plasma waves (ω, k) into EM radiation (ω₀, k₀), where the frequency ω₀ and wavevector k₀ of the EM radiation are imposed by the frequency and wavevector matching conditions ω₀ = ω and k₀ = k ± k_I. Because the wavenumber k₀ of the EM radiation is much smaller than that of electron plasma wave, i.e., k₀ ≪ k, the wavevector matching condition requires that k ≅ ∓k_I. Thus, the scattered high frequency plasma wave has to be a near field-aligned mode having a wavenumber close by the wavenumber of a FAI. Emissions with frequency downshifted spectra have been detected on the ground by HF receivers, mainly at high latitude heating sites. The spectra resembled those of upper hybrid/electron Bernstein waves excited by parametric instabilities.⁴⁴ These parametrically excited electrostatic waves, which could not be detected by backscatter radar due to their propagation direction nearly perpendicular to the geomagnetic field, were converted, via scattering by short-scale FAIs, into EM emissions,^45–47 which propagate downward to the ground. These emissions were generated through instabilities, thus termed “SEEs.”⁴⁸ SEEs require the presence of short-scale density irregularities, which are sensitive to the heater frequency in reference to the electron cyclotron harmonic resonance frequencies. Therefore, various spectral features of SEEs have been observed by fine-tuning of the heater frequency around each electron cyclotron harmonic resonance frequency.⁴⁹ 

We next show that parametric instabilities can indeed be excited by the O-mode HF heater waves transmitted from all three HF heating facilities at Arecibo, EISCAT, and HAARP.

Plasma is a nonlinear dielectric medium; high, and low frequency plasma modes oscillate in plasma as thermal fluctuations in the absence of external sources and couple to each other parametrically. When a large amplitude high frequency wave E_p(ω₀, k_p) (either EM or ES) appears in plasma, the parametric coupling makes it to act as a pump wave which excites high and low frequency plasma waves concurrently. The electric field of the high frequency pump wave sets up a quiver motion v_eq(t) in the electron plasma. In the low frequency plasma wave field, electrons and ions oscillate together to maintain quasi-neutrality, i.e., n_es(ω_s, k_s) ≅ n_is(ω_s, k_s) = n_s(ω_s, k_s), facilitates low frequency plasma wave to buildup charge density perturbation n_s(ω_s, k_s). Thus, a space charge current of the density -en_sv_eq is produced in the electron plasma. This current drives beat waves E(ω, k) and E′(ω′, k′) with their wavevectors and frequencies governed by the matching conditions

These beat waves, in turn, also couple with the pump wave to exert on electrons a low frequency nonlinear force, which has the same frequency and wavevector as n_s(ω_s, k_s). This is a feedback loop in the parametric coupling. The strength of the coupling depends on the nature of the induced beat waves. The coupling is strong as the beat waves are plasma modes. When the feedback is positive and large enough to overcome linear losses of the coupling waves, the coupling becomes unstable and the coupling waves grow exponentially in the expense of pump wave energy. This is called “parametric instability,” by which the pump wave E_p(ω₀, k_p) decays to two sidebands E(ω, k) and E′(ω′, k′) through a low frequency decay mode n_s(ω_s, k_s). The parametric coupling is imposed by the frequency and wavevector matching conditions (1) as well as a threshold condition on the pump electric field intensity. When the decay mode n_s(ω_s, k_s) has a finite oscillation frequency, two sidebands cannot satisfy the same dispersion relation concurrently. The frequency-upshifted sideband E′(ω′, k′) is off resonant with the plasma and can be disregard to reduce the coupling to three-wave interaction.

The most effective parametric instabilities excited directly by the HF heater wave are (1) PDI and (2) OTSI, in both mid-latitude and high-latitude regions.^4,50,51 The sidebands are Langmuir waves and upper hybrid waves; however, the instabilities involving Langmuir waves as sidebands have to compete with those excited in the upper hybrid resonance region, where the upper hybrid waves are the sidebands of the instabilities. The wavenumber k₀ of the heater is much smaller than the wavenumbers of the electrostatic sidebands and decay modes, thus a dipole pump, i.e., k₀ = 0, is generally assumed.

Parametric excitation of Langmuir/upper hybrid waves ϕ(ω, k) and low-frequency plasma waves n_s(ω_s, k_s) by electromagnetic or Langmuir/upper hybrid pump waves E_p(ω₀, k_p) are explored in the following, where E_p, ϕ, and n_s denote electric field of a pump wave, electrostatic potential of the Langmuir/upper hybrid sideband, and density perturbation of the low frequency decay mode, respectively. Langmuir waves can have large oblique propagation angles (with respect to the geomagnetic field B₀ = −$ẑ$ B₀), upper hybrid waves have near 90° propagation angles, and low-frequency plasma waves include ion acoustic/lower hybrid waves as well as purely growing modes. When the theory is used to explain the observations, however, it is noted that HF enhanced plasma and ion lines are mainly monitored by UHF and VHF backscatter radars, thus upper hybrid waves, lower hybrid waves, and field-aligned purely growing modes cannot be detected and the observed HFPLs and high frequency enhanced ion lines (HFILs) are contributed by waves propagating oblique to the geomagnetic field at an angle conjugate to the magnetic dip angle.

The coupled mode equation for the Langmuir/upper hybrid sideband is derived from the electron continuity and momentum equations, and Poisson's equation

where n_e = n₀ + δn_e + n_s; n₀ and δn_e are the unperturbed plasma density and electron density perturbation associated with Langmuir/upper hybrid waves, respectively; m is the electron mass, Ω_e = eB₀/m is the electron cyclotron frequency and v_te = (T_e/m)^1/2 is the electron thermal speed; E = E_P+ E_L and E_L = −∇ϕ; and the adiabatic relationship ∇P_e = 3T_e∇δn_e is used; ν_et = ν_en + ν_ei + ν_eL = ν_e + ν_eL is the effective electron collision frequency, where ν_e = ν_en+ ν_ei, ν_en is the electron-neutral elastic collision frequency, ν_ei is the electron-ion Coulomb collision frequency [ν_ei = 2.632 (n₀/T_e^3/2) lnΛ ≅ 39.5(n₀/T_e^3/2) ≅ 4.87 × 10⁻⁷(f_p²/T_e^3/2), here, lnΛ ≅ 15 is assumed; n₀ is in cm⁻³, T_e is in K, and f_p is the electron plasma frequency] and ν_eL is a phenomenological collision frequency to incorporate the electron Landau damping effect [ν_eL = (π/2)^1/2(ω_a²ω_p²/k_zk²v_te³)exp(−ω_a²/2k_z²v_te²); where ω_a is the plasma wave frequency, i.e., ω_a = ω and ν_et = ν_eh for the high frequency sideband and ω_a = ω_s and ν_et = ν_es for the low frequency decay mode].

With the aid of (4), the three orthogonal components of (3) are combined into a single scalar equation

where ω_p = (n₀e²/mε₀)^1/2 is the electron plasma frequency.

The right hand side (RHS) of (5) is assembled into four groups. The first two groups contain linear response terms and the last two contain coupling terms. The terms in the fourth group can be neglected because these terms are much smaller than those in the third group in providing parametric coupling. Equations (2), (4), and (5) are then combined to a coupled mode equation⁴⁰ for the high frequency (Langmuir/upper hybrid) sideband ϕ(ω, k)

where 〈 〉 stands for a filter, which keeps only terms having the same phase function as that of the function ϕ on the left hand side. Although (6) is derived from the fluid equations, the kinetic effect of electron Landau damping has been included phenomenologically in the collision damping rate.

Both electrons and ions respond effectively to the low frequency wave fields, hence the formulation for the coupled mode equation of the low frequency decay mode n_s(ω_s, k_s) involve electron and ion fluid equations. Because electrons and ions tend to move together, the formulation is simplified by introducing quasi-neutral condition: n_si ≅ n_se = n_s. The ion fluid equations are similar to (2) and (3), except that the subscript e is changed to i, and the charge −e changed to e. Moreover, the collision terms ν_etv_e and ν_itv_i in the electron and ion fluid equations are replaced by ν_ei(v_e − v_i) + (ν_{en +} ν_eL)v_e and ν_ie(v_i – v_e) + ν_iv_i, respectively, where ν_i = (ν_in + ν_iL), ν_in is the ion-neutral collision frequency, ν_iL ≅ (π/2)^1/2(ω_s²/k_zV_s)(T_e/T_i)^3/2exp(−ω_s²/2k_z²v_ti²) accounts for the ion Landau damping effect on the low frequency decay mode, and V_s = (T_e/M)^1/2 and M is the ion mass.

With the aid of ν_ie = (m/M)ν_ei and mΩ_e = MΩ_i, and neglect the electron inertial term and the ion convective term in the momentum equations, these two momentum equations are combined to be

where Ω_i is the ion cyclotron frequency, C_s = [(T_e + 3T_i)/M]^1/2 is the ion acoustic speed, and M is the ion (O⁺) mass; (m/M)ν_e ≪ ν_i is applied. With the aid of ∇ · v_is = −∂_t n_s/n₀ from the ion continuity equation, (7) becomes

where the second term on the RHS of (8) can be expressed in terms of (n_s/n₀) from the two equations derived by taking the operations “$×ẑ$” and “∇ ·” on the electron and ion momentum equations.

The coupled mode equation of the low frequency (ion acoustic/lower hybrid) decay mode is then derived to be⁵² 

where v_ti = (T_i/M)^1/2 is the ion thermal speed; the coupling terms a_p = 〈v_e·∇v_e〉 and J_B = 〈n_ev_e〉 arise from plasma nonlinearities and can be expressed explicitly by taking the linear part of the electron velocity response and the electron density response to the total high frequency wave fields.

Decays of an O-mode EM dipole pump E_p(ω₀, k_p = 0) into a Langmuir/upper hybrid sideband ϕ_±(ω_±, k_±) and a purely growing or ion acoustic/FAI or lower hybrid decay mode n_s(ω_s, k_s) in the spatial region below the HF reflection height are studied in the following; ϕ_± and n_s denote the sideband's electrostatic potentials and purely growing or ion acoustic/FAI or lower hybrid mode's density perturbation, respectively; ϕ₊ = ϕ(ω, k₁) and ϕ₋ = ϕ′(ω, k₁′ = − k₁), i.e., ω_± = ω and k₊ = k = −k₋; k = $ẑ$ k_z + $x̂$ k_⊥; frequency and wavevector matching conditions lead to ω = ω₀ – ω_s* and k_s = −k. In this spatial region the O-mode heater field is given to be

where E_p⊥,z = E_p⊥,z exp(−iω₀t), E_p⊥,z = E_0⊥,z/2, and c.c. represents complex conjugate. It is noted that E_0⊥,z varies with location (i.e., altitude); for instance, near the HF reflection height E_0z ≅ E₀ and thus E_p ≅ $ẑ$ E_pz + c.c. ≅ $ẑ$ E₀cosω₀t; and in the upper hybrid resonance region E_0⊥ ≅ E₀ and thus E_p ≅ (⁠$x̂$ + i $ŷ$⁠)E_p⊥ + c.c. ≅ E₀(⁠$x̂$ cos ω₀t + $ŷ$ sin ω₀t).

With the aid of (10), Eqs. (6) and (9) become

and

where n_s+* = n_s(ω_s, k_s = − k₁) = n_s−.

Equations (11) and (12) are analyzed in the k-ω domain, where the spatial and temporal variation of physical functions in (11) and (12) are set to have the form of p = p exp[i(κ·r – ϖt)], where κ and ϖ = ϖ_r + ir are the appropriate wavevector and complex frequency of each physical quantity. Thus, (11) and (12) are converted to the coupled algebraic equations to be

and

where sin²θ = k_⊥²/k², ω_k²= ω_p²+ 3k²v_te² and ω_u²= ω_p²+ Ω_e²; (13) is simplified with the condition (Ω_e/ω)⁴≪ 1.

In the absence of the pump field (HF heater wave), (13) and (14) reduce to the dispersion equations of the high frequency and low frequency plasma modes engaged in the considered parametric couplings. The eigen-frequencies ω_r and ω_sr of the high frequency and low frequency plasma modes are derived to be

and

where ω_Lks = (Ω_eΩ_iξ + k²C_s²/ξ)^1/2 and ξ = 1 + (M/m)cot² θ; ω_r is the eigen-frequency of the electron plasma mode (Langmuir wave) for (k_z/k_⊥)² ≫ m/M, and of the upper hybrid mode for (k_z/k_⊥)² ≪ 1. In the regions near the HF reflection layer and near the upper hybrid resonance layer, the HF heater wave parametrically excites Langmuir waves and purely growing mode, and upper hybrid waves and field-aligned density irregularities, respectively. In the region between the HF reflection layer and upper hybrid resonance layer, the HF heater wave excites Langmuir waves and ion acoustic waves parametrically, and excites upper hybrid waves and lower hybrid waves parametrically in the region below the upper hybrid resonance layer.

In the following, the coupled mode algebraic equations (13) and (14) are employed to analyze parametric instabilities relevant to the experimental observations presented in Sec. II.

As the RH circularly polarized HF heater wave propagates to the region near the reflection height, it converts to the O-mode with electric field E_p ≅ $ẑ$ E₀ cos ω₀t.

(1) OTSI—Excitation of Langmuir waves together with purely growing density striations by the HF heater wave.

This process involves two Langmuir sidebands ϕ_±(ω, ±k), and their coupled mode equations are deduced from (13) to be

The parallel (to the magnetic field) component k_sz of the wavevector of the short scale purely growing mode (ω_s = iγ_s, k_s) is not negligibly small, thus the short scale purely growing mode is mainly driven by the parallel component of the ponderomotive force induced by the high frequency wave fields. Use the relation Ω_eΩ_iξ ∼ Ω_e² k_z²/k_⊥² ≫ k²C_s², the coupled mode equation for the purely growing mode is deduced from (14) to be

Set ω_s = iγ_s and ω = ω₀ + iγ_s, (17) and (18) are combined to be

where Δω²= ω_p²+ 3k²v_te²+ Ω_e² sin²θ – ω₀².

Set γ_s = 0 in (19), the threshold field is obtained to be

Equation (20) shows that the threshold field of OTSI varies with the propagation angle θ and wavelength λ of the Langmuir sidebands as well as the location of excitation (i.e., Δω²). There is a preferential height layer at altitude h(k,θ) to excite (k,θ) lines, where ω_p²(h) = ω_p²(k,θ)= ω₀(ω₀ + ν_e) − 3k²v_te²− Ω_e²sin²θ, Δω²(k,θ) = ω₀ν_eh, and the threshold field (20) has the minimum

(2) PDI—Decay of HF heater wave to Langmuir sideband ϕ(ω, k) and ion acoustic wave n_s(ω_s, k_s).

The coupled mode equations (13) and (14) reduce to

and

Equations (22) and (23) are combined to obtain the dispersion relation

We now set ω = ω_r and ω_s = ω_sr in (24), i.e., the growth rate γ_k = 0, to evaluate the threshold field ε_pth = E_pth(k, θ) of the instability excited at an arbitrary height h₁, where ω_p²(h₁) = ω_r²− 3k₁²v_te²− Ω_e²sin²θ₁ is the matching height of the (k₁, θ₁) Langmuir wave. Thus in the general case that when the sideband and decay wave of the instability are driven waves, rather than eigen modes of plasma, the threshold field of the instability is obtained to be

where Δω₁²= ω_kθ² − ω_r²= 3(k²−k₁²)v_te²+ Ω_e²(sin²θ − sin²θ₁); ω_sr²= k²C_s² − ω_srν_iΔω₁²/ω_rν_eh.

It is shown in (25) that the threshold field varies with the propagation angle θ and wavelength λ of the Langmuir sideband as well as the location of excitation. When the instability is excited at the matching height h of its Langmuir sideband (k, θ), i.e., ω_r = ω_kθ and thus Δω₁ = 0, the threshold field is the minimum given by

Equations (20) and (25) show that as the oblique propagation angle θ of OTSI and PDI lines increases, their preferential excitation layers move downward and the threshold fields (21) and (26) increase. When the heater field E₀ is large, the spectral lines of the Langmuir sidebands excited by OTSI and PDI are expected to establish angular (θ) and spectral (k) distributions, as well as spatial (h) distributions in a finite altitude region. The altitude region of the OTSI is higher and narrower than that of the PDI.

1. OTSI—Excitation of upper hybrid waves together with field-aligned density irregularities by the HF heater wave.

The upper hybrid resonance layer is located below the O-mode HF reflection height and is accessible with the O-mode HF heater wave. In high latitude regions, such as at Tromso, Norway and Gakona, Alaska, RH circularly polarized heater can be transmitted along the geomagnetic field. The wave fields in the region below the upper hybrid resonance layer still remain at RH circular polarization, i.e., E_p = ($x̂$ + i$ŷ$)(E₀/2)exp(−iω₀t) + c.c., which decays into two upper hybrid sidebands ϕ_±(ω_±, k_± = ±$x̂$ k) and a field-aligned purely growing mode n_s(ω_s = iγ_s, k_s = −$x̂$ k), where ω_± = ω = ω₀ + iγ_s.

From (13), the coupled mode equations for upper hybrid sidebands are

where the notations n_s+* = n_s = n_s− are used again; ν_eL = 0.

The coupled mode equation for the field-aligned purely growing mode is deduced from (14) to be

where ν_iL = 0 and ν_i = ν_in.

Equations (27) and (28) are analyzed in the same way as that for (18) and (19). The dispersion relation of upper hybrid OTSI is then derived to be

where γ₁ = γ_s/ν_e, k_D = ω_p/v_te and Γ = ω_p²+ 3k²v_te²+ Ω_e²+ ν_eh² − ω₀².

Set γ₁ = 0 in (29), the threshold field ε_pth = |E_p|_th of the instability is obtained to be

The right hand side of (30) has to be positive, it leads to the condition that

The threshold field for Γ = Γ₀ = a + [a²+ ν_e²(ω₀²+ Ω_e²)²/ω₀²]^1/2 has the minimum value

1. PDI—Decay of HF heater wave to upper hybrid sideband ϕ(ω, k) and lower hybrid wave n_s(ω_s, k_s):

   For the upper hybrid sideband, the coupled mode equation (13) is reduced to

   $[−Γ+ iνeω]ϕ= iωp2[(1−Ωe/ω)/2k]E0(ns*/n0),$

   (33)

where Γ = ω_uk² −ω², ω_uk²= ω_k²+ Ω_e²+ ν_e², ω_k²= ω_p²+ 3 k_⊥²v_te², and ω_u²= ω_p²+ Ω_e².

For the lower hybrid decay mode, Ω_i² ≪ |ω_s|² ≪ Ω_e², and |∂_t²∇_⊥²| and |Ω_e²∇_z²| are in the same order of magnitude. Thus the coupled mode equation (14) is reduced to

where ω_Lks²= ω_LH²ξ + k²C_s²/ξ, ξ = 1 + (M/m)(k_z²/k_⊥²), and ω_LH²= ω_pi²/(1 + ω_p²/Ω_e²) ≅ Ω_eΩ_i. k_sx ≅ −k is substituted; Ω_e² ≪ ω_p² has been assumed.

Equations (33) and (34) are combined to obtain the dispersion relation

where ν_i ≪ ν_es is assumed.

We now set ω = ω_uk and ω_s = ω_Lks in (35), i.e., the growth rate γ_k = 0, the minimum threshold field |E_updi|_m of the instability excited at the matching height of its upper hybrid sideband ϕ(k, θ) is obtained to be

where ν_e = ν_ei + ν_en, ν_es = ν_e + ν_eLs, and ν_eLs = (π/2)^1/2(M/m)^3/2[ω_LH⁴ξ/k³v_te³(ξ−1)^1/2]exp[−Mω_LH²ξ/2mk²v_te²(ξ−1)]. The plasma frequency at the matching height is ω_p = (ω₀² − Ω_e² − 3k²v_te²)^1/2< (ω₀² − Ω_e²)^1/2. It indicates that the upper hybrid PDI prefers to be excited below the upper hybrid resonance layer.

The definitions of the symbols used in (21), (26), (32), and (36), the results of the minimum threshold fields, are summarized in the following: C_s = [(T_e + 3T_i)/M]^1/2 is the ion acoustic speed and v_te = (T_e/m)^1/2 is the electron thermal speed; ω₀, ω_sr, ω_LH = (Ω_eΩ_i)^1/2, ω_p, and Ω_e,i are the heater wave, ion acoustic, lower hybrid, electron plasma, and electron and ion cyclotron radian frequencies; ν_eh = ν_en + ν_ei + ν_eL = ν_e + ν_eL, ν_en is electron-neutral elastic collision frequency, ν_ei is the electron-ion Coulomb collision frequency, ν_e = ν_en + ν_ei, and ν_eL = (π/2)^1/2(ω₀²ω_p²/k_zk²v_te³)exp(−ω₀²/2k_z²v_te²) is twice the electron Landau damping rate; ν_es = ν_e + ν_eLs, and ν_eLs= (π/2)^1/2(M/m)^3/2[ω_LH⁴ξ/k³v_te³(ξ - 1)^1/2]exp[−Mω_LH²ξ/2mk²v_te²(ξ − 1)] is twice the electron Landau rate on lower hybrid wave, where ξ = 1 + (M/m)(k_z²/k_⊥²); ν_i = (ν_in + ν_iL), ν_in is the ion-neutral collision frequency, ν_iL ≅ (π/2)^1/2(ω_s²/k_zV_s)(T_e/T_i)^3/2exp(−ω_s²/2k_z²v_ti²) is twice the ion Landau damping rate on ion acoustic wave; θ is the oblique angle of the wavevector with respect to the magnetic field; Γ₀ = a+ [a²+ ν_e²(ω₀²+ Ω_e²)²/ω₀²]^1/2 accounts for frequency mismatch, where a = (1 + Ω_e²/ω₀²) [ν_e²+ (3/2)Ω_eΩ_i]/(1 − Ω_e/ω₀).

The results of analyses presented in Sec. III are applied to understand observations presented in Sec. II. The relevant parametric values of the HF heating experiments conducted at Arecibo, Puerto Rico, Tromso, Norway, and Gakona, Alaska, are given in the following:

In Arecibo heating experiments, the parameters are: ω₀/2π = 5.1 MHz, Ω_e/2π = 1.06 MHz, T_e = T_i = 1000 K, v_te ∼ 1.23 × 10⁵m/s, v_ti = 7.17 × 10²m/s, C_s ∼ 1.43 × 10³m/s, and k_||0 ≅ 4.377π (i.e., λ_||0 = 0.457 m = λ_R/2 sin θ_m, where λ_R = 0.7 m is the wavelength of the 430 MHz radar signal and θ_m = 50° is the magnetic dip angle); ν_in = 0.5 s⁻¹ and ν_iL ∼ 1.64 × 10³(k_s/k_R) s⁻¹, where k_s and k_R are the wavenumbers of ion acoustic wave and radar signal; ν_eL≪ ν_en < ν_ei, and ν_e ≅ ν_en + ν_ei ≅ 500 s⁻¹.

In Tromso heating experiments, the parameters are: Ω_e/2π = 1.35 MHz, T_e = 1500 K, T_i = 1000 K, ν_in = 0.8/0.5 s⁻¹, v_te ∼ 1.5 × 10⁵m/s, v_ti = 7.17 × 10²m/s, C_s ∼ 1.52 × 10³m/s. In the case of 933 MHz radar, ω₀/2π = 5.423, k_||1 = 12.17π (i.e., λ_||1 = 0.1644 m), ν_iL1 ∼ 1.19 × 10⁴(k_s1/k_R1) s⁻¹; in the case of 224 MHz radar, ω₀/2π = 6.77 MHz, k_||2 = 2.92π (i.e., λ_||2 = 0.685 m), ν_iL2 ∼ 2.87 × 10³(k_s2/k_R2) s⁻¹; electron Landau damping rate can be neglected in both cases of heater frequencies, thus ν_e = ν_en + ν_ei ∼ 600 s⁻¹.

In HAARP heating experiments, the parameters are: ω₀/2π = 5 MHz, Ω_e/2π = 1.4 MHz, T_e = 1500 K, T_i = 1000 K, v_te ∼ 1.5 × 10⁵m/s, v_ti = 7.17 × 10²m/s, C_s ∼ 1.52 × 10³m/s, and k_||0 ≅ 5.77π (i.e., λ_||0 = 0.347 m = λ_R/2 sin θ_m, where λ_R = 0.67 m is the wavelength of the 450 MHz radar signal and θ_m = 75.6° is the magnetic dip angle); ν_in = 0.5 s⁻¹ and ν_iL ∼ 5.55 × 10³(k_s/k_R) s⁻¹. Again, ν_eL ≪ ν_en < ν_ei, and ν_e ≅ ν_en + ν_ei ≅ 600 s⁻¹.

We now use (21) and (26) to evaluate the threshold fields of OTSI and PDI which contribute to HFPLs; thus k_s = k = 2k_R and θ = θ₀ = 90° − θ_m, where 2k_R = 5.71π, 12.44π/2.99π, and 6π for Arecibo's 430 MHz radar, EISCAT's 933/224 MHz radars, and HAARP's 450 MHz radar; θ₀ = 40°, 12°, and 14.6°, at Arecibo, Tromso, and Gakona, respectively.

(1) OTSI

(2) PDI