Radio Science

Generation of extremely low frequency radiation by ionospheric electrojet modulation using powerful high-frequency heating waves

Authors


Abstract

[1] Using amplitude-modulated high-frequency (HF) heating waves for electron heating, the conductivity of plasma and thus the embedded electrojet currents in high-latitude ionosphere can be modulated accordingly to set up the ionospheric antenna current for extremely low frequency (ELF) wave generation. Electron heating is hampered by inelastic collisions of electrons with neutral particles (mainly due to vibration excitation of N2), which cause the induced modulation current to remain at a low level. However, this inelastic collision loss rate drops rapidly to a low value in the energy regime from 3.5 to 6 eV. Thus, as the heating power exceeds a critical level, electron heating suddenly reaches an unexpected high level, resulting to a near step increase (of about 5 to 8 dB depending on the modulation waveform and frequency) in the spectral intensity of ELF radiation. The dependency of this critical HF heating-wave power on the modulation frequency is determined for three heating wave modulation forms: (1) rectangular wave, (2) sine wave, and (3) half-wave rectified wave.

1. Introduction

[2] Use of an amplitude-modulated powerful HF wave to heat electrons in the polar electrojet region results in the modulation of the electron conductivity in a similar fashion. Electrojet current, which is driven by a DC space charge field, begins to oscillate in time. The AC part of the current radiates electromagnetic waves at the modulation frequency and its harmonics. Experiments [Stubbe et al., 1981, 1982, 1985; Barr and Stubbe, 1984a, 1984b, 1991a, 1991b; Ferraro et al., 1982, 1984; James et al., 1984; Lee et al., 1990; McCarrick et al., 1990; Rietveld et al., 1986, 1989; Barr et al., 1991; Villasenor et al., 1996] and theoretical studies [Stubbe and Kopka, 1977; Stubbe et al., 1982; Papadopoulos and Chang, 1985; Papadopoulos et al., 1990; Kuo and Lee, 1983, 1993; Kuo, 1993; Kuo et al., 1998, 2000] on ELF/VLF wave generation by this approach in high-latitude ionosphere have been conducted for the past two decades. The generation efficiency and signal quality are critical to practical applications.

[3] Papadopoulos et al. [1990] suggested a beam-painting approach, which can enhance the ionospheric antenna gain by enlarging the modulated region of the electrojet. Kuo and Lee [1993] and Kuo [1993] showed that the increasing dependence of the elastic electron-neutral collision frequency νen on the electron temperature Te, i.e., νenTe5/6, provided a positive feedback channel to excite a stimulated thermal instability. This active process enhanced the electrojet modulation. The results of numerical and analytical analyses [Kuo et al., 1998, 2000] further showed that the signal quality and generation efficiency depended on the mode type (O or X) and frequency of the HF heating wave and the wave modulation scheme and frequency. The results further showed that the stimulated thermal instability was, in fact, the dominant process in the initial electron temperature modulation. However, this instability was quickly saturated at low levels by the nonlinear damping of inelastic collisions.

[4] The inelastic collision frequency of electrons (e.g., vibration excitation of N2 and O2) has a strong dependence on the electron temperature. It starts with a rapidly increasing dependence in the low electron temperature regime. This increasing dependence on the electron temperature slows down as the electron temperature further increases. The data curve is presented by Gurevich [1978, p. 57, Figure 5], who shows that the inelastic collision cross section of the electron, after reaching a peak at about 2.5 eV of the electron temperature (i.e., electron energy at about 3.5 eV), decreases rapidly with a further increase of the electron temperature. It stays at low values for 2.5 < Te < 4.5 eV before the optic excitation and ionization processes become significant. It suggests that the electron temperature modulation be enhanced drastically by increasing the heating power to a critical level [Kuo et al., 2002].

[5] In the present work, the electron thermal energy equation that governs the nonlinear evolution of electron heating by HF heating wave is analyzed numerically for three heating wave modulation forms: (1) rectangular wave, (2) sine wave, and (3) half-wave rectified wave. The spectral intensities of radiations are then evaluated. In section 2 the modal equations for the numerical analysis are derived, which include an equation relating the radiation magnetic field and electron temperature modulation in the electrojet current and an electron thermal energy equation governing the electron temperature modulation by the modulated heating wave. Numerical results are presented in section 3. Summary and conclusions are drawn in section 4.

2. Modal Equations

[6] A space charge field E0 = equation imageE0 drives the electron current density in the polar electrojet

equation image

where the geomagnetic field Bo = −equation imageB0 is assumed. In equation (1), νenTe5/6; hence the electron electrojet current can be modulated through the electron temperature Te.

[7] The electrojet is modulated in a region with dimensions much smaller than the wavelength of the radiation; thus the modulation current acts as a Hertz dipole placed at r′ = 0. The magnetic field of the radiation at the receiver site at r′ = r is obtained to be

equation image

where V is the volume of the current source, νA is averaged wave velocity over the propagation path from the source to the receiver, passing plasma region, and free space; and ∂tJe = (5n0e2E0/6m){[equation imagee2 − νen2) + equation imageeνen]/(νen2 + Ωe2)2}(νen/Te)∂tTe, which links the ELF/VLF radiation field to the electron temperature modulation in the background plasma of the electrojet.

[8] The electron thermal energy equation [Gurevich, 1978] is

equation image

where ve is the electron fluid velocity, δ(Te) is the average relative energy fraction lost in each collision, νe(Te) is the effective collision frequency of electrons with neutral particles, Tnis the temperature of the neutral particles (the explicit expression of the ionization loss term on the left-hand side (LHS) of equation (3) will be given later), Q is the total Ohmic heating power density contributed by the electrojet current and the HF heater wave, and m is the electron mass. Only temporal modulation is considered; two terms involving spatial derivatives on each side of equation (3) have been set to zero.

[9] Consider the electrojet modulation by amplitude-modulated x-mode heating wave, Ep = (equation imageiequation image)(εp/2)exp[i(k0z − ω0t)] + complex conjugate, where εp2 = εp02M(t) and M(t) is a power modulation function. The total Ohmic heating power density is given by

equation image

where ue = eE0/men2 + Ωe2)1/2, 〈∣νpe2〉 = νq2M(t), and νq2 = (eεp0/m)2/[(ω0 − Ωe)2 + νen2]; angle brackets indicate the time average over the HF wave period.

[10] The main processes involved in the inelastic collisions are the rotational and vibrational excitation of N2 and O2; the optical excitation process is neglected. The fractional electron heat loss rate through elastic and inelastic collisions with neutral particles can be written as [Gurevich, 1978]

equation image

where χ = Te/Te0 and Te0 = 1500 K are assumed; νen0 = νen(Te0); μI(χ) = χ−5/2 [14.73e6.98(1−1/χ) + 0.1e13.88(1−1/χ) + 3.81 × 10−4e19.34(1−1/χ) + 1.43 × 10−4e23.67(1−1/χ) + 2.7 × 10−5e26.1(1−1/χ) + 2.59 × 10−6e28.05(1−1/χ) + 3.54 × 10−7e30.87(1−1/χ)].

[11] The ionization loss rate on the LHS of equation (3) is given by [Gurevich, 1978]

equation image

where εi(O2) and εi(N2) are the ionization energies of O2 and N2, which are 12.1 eV ≅ 93 Te0 and 15.6 eV ≅ 120 Te0, respectively; νion(O2) = 4.3 × 102D1χ1/2e−94.3/χ and νion(N2) = 1.57 × 103D2χ1/2e−121.7/χ. Here D1 = (1 + 0.032χ + 5 × 10−4χ2 + 3.6 × 10−6χ3) and D2 = (1 + 0.025χ + 3 × 10−4χ2 + 1.7 × 10−6χ3).

3. Numerical Analysis and Results

[12] Equations (2) and (3) are the modal equations used in the following numerical analyses, which will be carried out for electrojet modulation in the region near 100 km altitudes. The heating wave frequency of ω0/2π = 4 MHz is used, and adopted E region parameters are TnTi ≅ 300 K, Te0 ≅ 1500 K, νen0 = 5 × 104 s−1, E0 = 50 mV/m, Mn/m = 5.52 × 104, Ωe/2π = 1.35 MHz, and νt0 = (Te0/m)1/2 = 1.5 × 105 m/s. In the strong heating power regime to be considered in the following, the numerical results are insensitive to the initial value of the electron temperature.

[13] In the numerical analysis, dimensionless variables and parameters are introduced: χ = Te/Te0 as that defined early, νenen0 = χ5/6, τ = νe0t/100, R = νe0r/100νA, B(R, t) = ∣cB/E0∣, ω10 = 100ω1en0, Tn/Te0 = 0.2, η = (eE0/mΩeνt0)2, b = (νen0e)2, and q = αM(t), where α = (νqt0)2, νq = 1.04 × 104εp0 m/s, εp0 is in V/m, and c is the speed of light in a vacuum. Choosing ε0 = 2.5 V/m as a reference heating wave field amplitude leads to νq0 = 2.6 × 104 m/s, α0 = α(νq = νq0) = 0.03, and α = 0.03p, where p = (εp00)2 is the heating wave power normalized to the reference power.

[14] Thus equations (3) and (2) are normalized to the dimensionless forms

equation image

and

equation image

where G0 = (5Vωpe2Ωeb3/2/24πc2νAR) × 10−4 and χ = χ(τ − R); the spatially dependent terms (i.e., the divergent terms) in equation (3) have been neglected.

[15] We now study the dependence of the radiation intensity on the heating power. A modulation frequency of f1 = 10 Hz is first considered. Thus ω10 = 0.04π, and p is the only variable parameter left in equation (7). For a given p, equation (7) is solved numerically subject to the initial condition χ(0) = 1. The result is then substituted into equation (8) to obtain the time-dependent radiation field B(R, t) with p as a variable parameter. The dependency of the spectral intensity I1 of the fundamental line of radiation on the normalized heating power p, varied from 1 to 7, is presented in Figure 1 for three heating wave modulation forms: (1) rectangular wave, (2) sine wave, and (3) half-wave rectified wave. It is shown that this functional dependence in each case has a narrow transition region of width Δp. As p passes this transition region to exceed a critical value pc = 5.84 for rectangular wave modulation, 4.4 for sine wave modulation, and 2.94 for half-wave rectified wave modulation, the spectral intensity of the signal is suddenly increased by near 8.3, 5.74, and 5.5 dB, respectively. The critical value pcalso varies with the modulation frequency f1. Similar calculations have been carried out for different modulation frequencies in the range from 10 to 100 Hz for the three modulation cases. This dependency, showing monotonic increase of the critical value pcwith the modulation frequency f1 in all three cases, is presented in Figure 2. Numerical results also show that the width Δp of the transition region increases and the change ΔI1 of I1 across the transition region decreases as the modulation frequency f1 increases. The dependencies of ΔI1 and Δp on f1 for the considered three cases are presented in Figure 3. The slopes in the transition region in all three cases are much larger than that at p = 1 (as well as that for p < pc), representing considerable increase in the generation efficiency. The improvement on the generation efficiency, however, goes down as the modulation frequency increases.

Figure 1.

The dependency of the spectral intensity I1 of radiation's fundamental line on the normalized heating power p in the three cases of different modulation waveforms. A (near) step increase in the spectral intensity occurs in each case as the heating power exceeds a respective critical level pc = 5.84 for rectangular wave modulation (labeled by R), 4.4 for sine wave modulation (labeled by S), and 2.94 for half-wave rectified wave modulation (labeled by H); the modulation frequency f1 = 10 Hz in all three cases.

Figure 2.

The dependency of the critical power pc on the modulation frequency f1 in the three cases of considered modulation waveforms.

Figure 3.

The dependencies of ΔI1, the change of I1 across the major enhancement region, and Δp, the width of that region, on the modulation frequency f1.

4. Summary and Conclusions

[16] Inelastic collisions of electrons with neutral particles (mainly due to vibration excitation of N2) introduce nonlinear damping to stabilize the electron heating by amplitude-modulated HF heating wave. However, the nonlinear damping rate μI on the LHS of the electron thermal energy equation (7) decreases rapidly after reaching a peak at χ = Te/Te0 ≅ 10. Therefore, as the HF heating power exceeds a critical level, significant electron heating causes a steep drop in the electron inelastic collision rate, resulting to a (near) step increase of the electron temperature and the generation efficiency of ELF radiation (as shown in Figure 1). This critical heating power varies with the modulation waveform and frequency. For example, the critical field amplitude, for using 4 MHz X-mode heating wave modulating at 10 Hz, is about 6.1 V/m in the case of rectangular wave modulation. It is reduced to about 5.22 and 4.27 V/m in the cases of sine wave and half-wave rectified-wave modulation, respectively. The half-wave rectified-wave modulation scheme has the lowest critical power requirement, but the rectangular wave modulation scheme advantageously increases the (near) step jump of the radiation intensity by 8.3 dB, comparing with 5.5 dB for the half-wave rectified-wave modulation and 5.74 dB for the sine wave modulation.

[17] The minimum critical power, evaluated from the E region parameters (∼100 km altitude) for 10 Hz modulation, requires the effective radiated power ERP ≥ 90 dBW. However, D region absorption during daytime or under disturbed conditions could severely reduce the HF power transmitted to altitudes near 100 km. This further increases the ERP requirement. Nevertheless, this extraordinary physical phenomenon could be explored in future heating experiments by the European Incoherent Scatter (EISCAT) facility's superheater [Stubbe, 1996] in Tromso, Norway, under favorable ionospheric conditions, or by the High Frequency Active Auroral Research Program (HAARP) heating facility [Kossey et al., 1999] in Gakona, Alaska, when its effective radiated power reaches its planned level.

Acknowledgments

[18] We are grateful to Paul Kossey, John Heckscher, and Lee Snyder, Air Force Research Laboratory at Hanscom Air Force Base, to Edward Kennedy, Naval Research Laboratory, and to M. C. Lee, MIT, for helpful discussions. This work was supported in part by the High Frequency Active Auroral Research Program (HAARP), Air Force Research Laboratory at Hanscom Air Force Base, and in part by the Office of Naval Research, grant ONR-N00014-03-1-0166.

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