Resonant excitation of whistler wave via Cherenkov emission by sweeping radio frequency heated hot spot across the equatorial electrojet

Whistlers are an important mode of circularly polarized electromagnetic waves propagated in magnetized plasmas and were discovered about one century ago.^1–3 Whistlers play a very important role in the Earth's radiation belts,^4–6 and their excitation in laboratory plasmas^7,8 can provide a more controlled environment for understanding the underlying physical mechanisms of runway driven whistler instabilities in spherical tokamak and ITER.^9–11 Whistlers can propagate through overdense plasmas, having plasma frequency far exceeding the wave frequency $(ωp≫ω)$⁠, have very little attenuation (attenuation of a whistler wave is defined as the gradual decrease in the amplitude or energy of a whistler wave as it propagates through a plasma), and can also be used as a diagnostic tool for such plasmas.³ Naturally occurring whistlers, produced in thunderstorms and lightening discharges, indeed have been widely employed for learning about the ionosphere and magnetosphere. Whistlers can also be generated artificially by power high-frequency (HF) radio waves propagation in ionosphere.¹² Low-frequency whistlers produced near the North Pole can travel to the antipodal point near the South Pole along the magnetic lines of force of the Earth through the ionosphere and return back to the point of their origin with significant power level.¹³ One interesting characteristic of the whistler-mode is that they can be guided by inhomogeneities of the background plasma density; the physics of this process is not fully understood and the regions of higher or lower plasma density can confine the whistler wave in a “duct” along a magnetic field line in magnetosphere^3,14 and geometrical optics methods also used to study whistler wave propagation in magnetized plasmas.^15–17 

An important issue related to whistlers is their generation in the magnetosphere or in a laboratory. In addition to the thunderstorms and discharges, whistlers can be generated by electron beams or may be driven unstable by electron temperature anisotropy in plasmas.¹⁸ Modulated electron beams are capable candidates to excite whistlers through resonant interaction in magnetized plasmas.^19–24 Periodic modification of the ionosphere electrojet current by a ground-based radio transmitter is a potential scheme for the generation of extremely low-frequency (ELF) and very low-frequency (VLF) whistler waves.^25,26 These low-frequency waves subsequently couple to the earth-ionosphere waveguide (known as a duct) and propagate over long distances, facilitating underwater communication and underground exploration, while a fraction of their power goes upward toward the magnetosphere.

A most fascinating and important property of the active ionosphere is its potential to act as a frequency transformer that converts HF power injected from a high-power HF transmitter into the ionosphere into coherent lower frequency VLF/ELF waves.^27,28 The conversion principle relies on modulating the electrojet currents in the ionospheric D and E regions using amplitude-modulated HF heating. The low-frequency whistler's fields subsequently couple to the earth-ionosphere waveguide and propagate to long distances, facilitating underwater communication and underground exploration, while a fraction of their power goes upward toward the magnetosphere. While there are two major and accessible electrojets, the auroral and equatorial, most of the theoretical and experimental work focused on the auroral region.

With respect to very low-frequency/extremely low-frequency (VLF/ELF) generation, the major difference between the two electrojet regions is the magnetic geometry. In the auroral electrojet, the magnetic field is essentially vertical to the ionospheric stratification, while in the dip equator, it is parallel. Current schemes of ELF/VLF generation rely on periodic heating of the focal spot of the ionosphere, carrying electrojet current, by a ground-based high-frequency, high–power, high-gain antenna array.²⁹ 

In this paper, we proposed an efficient scheme to optimally excite a VLF whistler wave by sweeping the HF heated hot spot across the electrojet current. The equatorial ionosphere has an electron density maximum around a height of 100 km. Thus, there exists a natural duct to guide VLF whistler waves along the earth's magnetic field. By suitably choosing the HF heater frequency around the maximum plasma frequency in the duct, one can modify the Pedersen conductivity, hence the electrojet current. When one moves the hot spot across the electrojet current but along the magnetic field with the phase velocity of a whistler wave, one can successively reinforce the amplitude of the whistler. By repeating the hot spot with the frequency of the whistler, one can achieve high VLF power. The scheme has a resemblance to coherent Cherenkov emission.

In Sec. II, we study the electrojet current modulation, which is the source of whistler wave excitation. In Sec. III, we study the whistler wave excitation in a plasma duct by sweeping a hot spot via Cherenkov emission, and the results are discussed in Sec. IV.

The electrojet current in the equatorial ionosphere lies at a height of 90–110 km, with a static electric field in the vertical direction. At this height, the electron density profile has a mild maximum, and the plasma is dominated by electron–neutral collisions. As shown in Fig. 1, the electric field induced by the closure of the field aligned currents in the electrojet region in the horizontal direction and drives a mixture of Hall and Pedersen currents.^30–32 The modification of the ionospheric conductivity by the heater affects predominately the Hall current and drives two whistler waves, one upward toward the magnetosphere and one downward toward the ground. The upward moving whistler has often been detected by overflying satellites and once by a shipboard detector in the conjugate region. When the downward moving whistler enters the ionosphere region where the electron–neutral collision frequency exceeds the electron cyclotron frequency $(νen>Ωc)$⁠, the whistler-driven current diffuses till the bottom of the waveguide, where the conduction current equals the displacement current $(εω≈σ)$⁠. It subsequently couples to the earth-ionosphere waveguide in a manner similar to the ground-based antennas. Normally, since the low-frequency modes correspond to the transverse electromagnetic mode (TEM), the coupling is due to the two vertical ant-parallel currents and is proportional to $I2(L/λ)2$⁠, where $I$ is the current driven near the bottom of the waveguide, and L is the indicative of the coupling efficiency of horizontal dipole (HD) sources.

This modulated nonlinear current density is the source of whistler wave excitation. As a result of current modulation in equatorial electrojet, a number of advantages follows: (i) in addition to the horizontal dipole antenna (HD) source due to modulation of the Cowling current, similar to the auroral electrojet, it generates a vertical dipole moment. This generates VLF/ELF whistler waves to the earth-ionosphere waveguide with efficiency by a factor $(L/λ)2$ over the HD generated in the auroral electrojet; (ii) the power injected upward in the whistler mode in the auroral electrojet is injected in the whistler guide located in the upper D and E regions and subsequently leak into the earth ionosphere wave; and (iii) since the index of refraction for the whistler waves lies between 50 and 100, resulting in a phase velocity on the order of 0.01 c, it is possible to generate the whistler waves by sweeping the heater along the magnetic field direction in proper phase with the particular mode we would like to excite.

Figure 2 shows the ratio of whistler wave amplitude to dc electrojet field $|A|/E0$ for the following parameters: $L//≈30 km,$ $ν0≈105 s−1$⁠, $ωp0/ωc=3$⁠, $ω/ωc=1/10,1/100,1/1000$⁠, $Teω/T0=0.5$⁠, the maximum ratio of whistler wave amplitude to dc electrojet field is nearly 1.6. This figure shows that the ratio of whistler wave amplitude to dc electrojet increases monotonically with $ω/ωc$ and saturates at a higher value of $ω/ωc$⁠. In Fig. 3, we have plotted the attenuation length of a whistler wave (the attenuation length of a whistler wave is defined as the distance over which the amplitude of a whistler wave decreases by a factor of e due to interactions with the plasma) for the following parameters: $ν0≈105 s−1$⁠, $ωp0/ωc=3$⁠, $ω/ωc=1/10,1/100,1/1000$⁠, the maximum attenuation length is about $ki−1≈60 km$⁠, and the attenuation length is longer than the $L//$ and the expression [Eq. (17)] for $|A|/E0$ is valid. The attenuation length decreases monotonically with increasing $ωp/ωc$⁠.

In conclusion, for CW generation of whistler wave of frequency $ω$⁠, one would need $L///λ//$ number of antenna arrays where $λ//=2π/kz=2π(c/ω)(ωωc)1/2/ωp$⁠. Once the hot spot created by array number 1 moves a distance $λ//$⁠, the second array must turn on to heat the previous spot and this spot must follow the preceding one with the same velocity $vp=ω/kz$⁠. Thus, a succession of hot spots would produce a continuous VLF wave. About 10 or 12 antenna array sets should suffice at $ω/2π≈1 kHz$⁠. Each antenna array would comprise a large number of phased antennae whose relative phases can be controlled electronically. By varying the phases, one can move the hot spot at the desired speed. All the antennae are operating simultaneously, but they heat the ionosphere at different locations separated by a whistler wavelength and these locations move with the phase velocity of the whistler. Each antenna array heats a local region at one instant for a duration $r0/vp$⁠, where $r0$ is the width of a hot spot. For any array antenna gain G, $r0≈2h/G1/2$⁠, where h is the height of the ionospheric duct. For efficient electrojet current modulation, $r0/vp$ should be of the order of temperature saturation time $τH$⁠, i.e., a fraction of a millisecond. This condition is met for whistlers in a frequency range of kilohertz. The ducted whistler could have a weak evanescent field on the ground $|E|ground≈|A|ducte−|kz|h$⁠, where $kz$ is the wavenumber in the duct. However, as the whistler travels toward higher latitude, it bends toward the ground and may have larger values of ground amplitude. One may also create a volume grating in the ionospheric duct using a different RF source and reflecting the whistler toward the ground.

The author has no conflicts to disclose.

Pawan Kumar: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.