Numerical study of L-mode waves with a magnetic field-aligned density duct in the polar ionosphere

In active ionospheric experiments, high-frequency pump waves can propagate to specific altitudes based on their polarization and local plasma conditions. Typically, ordinary mode (O-mode) pump waves reach altitudes where the local plasma frequency, $ωp$⁠, equals the pump wave frequency, $ω0$⁠. However, several studies have documented the propagation of O-mode pump waves beyond this expected reflection altitude, suggesting the involvement of mechanisms other than simple reflection.^1–9 This phenomenon can be explained by the conversion of O-mode to Z-mode waves. When O-mode pump waves are transmitted at the Spitze angle in a plane-stratified ionosphere, they can convert to the Z-mode at the plasma resonance, thereby effectively bypassing the O-mode reflection height through a “radio window.”^10,11

Theoretically, in a plain-stratified ionosphere, maximum Z-mode conversion occurs when the O-mode is directed along the Spitze angle within a narrow angular width of approximately one degree.¹² Pump waves transmitted within this angular range can efficiently pass through the plasma resonance and propagate into the overdense plasma region of the ionosphere. In contrast, waves outside this angular window are either absorbed or reflected at or below the plasma resonance. However, numerous observations have indicated that maximum plasma perturbation occurs when the wave's inclination angle aligns with the Earth's magnetic field, a phenomenon recognized as the magnetic zenith effect.^13–19 This deviation is believed to stem from two-dimensional inhomogeneities in the form of magnetic field-aligned irregularities and ducts, which enable the waves to propagate through regions beyond the standard radio windows.^13,14,20 These magnetic field-aligned irregularities occur in the ionosphere across a broad spectrum of transverse spatial scales, ranging from meters to hundreds of kilometers.^21–27 Cannon and co-workers explored the effects of various two-dimensional density profiles on O- to Z-mode conversion using a full-wave finite-difference time-domain (FDTD) code, reporting that 2D inhomogeneities alter the radio window.^28,29

Magnetic field-aligned irregularities in the ionosphere facilitate the propagation of O-mode waves beyond the height of plasma resonance and outside the standard radio window through two primary mechanisms. The first mechanism involves resonant scattering by small-scale, field-aligned density irregularities, which can transform an incident O-mode pump wave into a Z-mode wave.¹³ These small-scale irregularities can be rapidly generated, within seconds, due to the influence of the O-mode pump wave itself. Consequently, topside turbulence associated with Z-mode can be observed in a short timeframe. The second mechanism involves large-scale, field-aligned irregularities, which may be naturally occurring^30,31 or induced by pumping^32,33 in the ionosphere. These irregularities act as waveguides, compelling O-mode waves to propagate along the geomagnetic field lines as the L-mode.^34,35

It is important to note that a ground-based antenna can transmit pump waves as right-hand circularly polarized (RHCP) and left-hand circularly polarized (LHCP) waves, with the handedness defined relative to the direction of electron gyromotion in the ionosphere. An RHCP wave has an electric field that rotates in the same direction as electron gyromotion, whereas an LHCP wave rotates in the opposite direction. However, in ionospheric modification experiments, even before entering the ionosphere, an RHCP pump wave is referred to as the extraordinary (X)-mode pump wave, while an LHCP pump wave is the O-mode pump wave, regardless of the propagation angle. However, once inside the ionosphere, this classification changes depending on the wave's orientation with respect to the external magnetic field $B0$⁠. A parallel propagating X-mode wave is termed as the R-mode, while a perpendicular-propagating X-mode wave remains classified as the X-mode, with oblique angles in between as the R-X-mode. Similarly, a parallel propagating O-mode wave is designated as the L-mode, with its perpendicular counterpart as the O-mode and oblique angles as L-O-modes.³⁶ Conventionally, non-parallel propagating waves are typically simply labeled as the X-mode and O-mode for notational convenience. In this study, which focuses on O-mode pump waves, the L-mode denotes parallel propagating waves, while the O-mode refers to non-parallel propagation.

According to the cold plasma theory, the refractive index of the L-mode in a magnetized homogeneous plasma is given by $n||2=1−ωp2/ω(ω+ωc)$⁠, where $ωc$ represents the electron cyclotron frequency.³⁷ As such, the L-mode aligns with the O-mode dispersion surface for $ω>ωp$ (i.e., below the plasma resonance) and follows the Z-mode surface for $ω<ωp$ (i.e., above the plasma resonance), which is also referred to as the slow branch of the X-mode. Thus, while the O-mode waves can propagate only up to the plasma resonance, both the Z-mode and L-mode waves can travel through regions of higher plasma density regions, surpassing the limits of O-mode propagation. Conventionally, the term Z-mode encompasses waves with wave vectors ranging from perpendicular to parallel to the ambient magnetic field $B0$⁠. In contrast, the L-mode specifically refers to waves with wave vectors that are parallel or anti-parallel to the ambient magnetic field $B0$⁠.

Several experiments have been conducted in the ionosphere to investigate L-mode waves. One such ionospheric heating experiment at the European Incoherent Scatter Scientific Association (EISCAT) Heating facility involved directing pump waves toward the magnetic zenith, well beyond the standard radio window. Plasma parameters were subsequently measured with the EISCAT UHF (ultra high frequency) radar at various elevations around the magnetic zenith. The study involved modeling the temporal evolution of the electron temperature profile by integrating the electron energy equation and comparing the results with the measured plasma parameters. The findings indicated that the electron temperature enhancements and the associated heating rates were highest at the magnetic zenith, and the altitude range of heating was more extensive there than at other elevations.³⁸ In another EISCAT experiment, a pump wave directed toward the magnetic zenith resulted in observable enhancements at the altitude where the bottom-side and top-side high-frequency ion lines (BHFIL and THFIL) align with the reflection altitude where $ω0≈ωp$⁠.^39,40

An experiment at the EISCAT heating facility has provided evidence of L-mode wave propagation in the ionosphere. The experiment transmitted pump waves from the ground along the magnetic zenith, with a frequency lower than that of the daytime F-region of the ionosphere. The CASSIOPE (Cascade, Smallsat, and Ionospheric Polar Explorer) spacecraft detected these transmitted waves near the Earth's magnetic field in the topside of the ionosphere. However, the study was based on the dispersion characteristics of L-mode in a homogeneous plasma.⁴¹ Given that L-mode waves in the ionosphere are largely guided by large-scale magnetic field-aligned irregularities, a comprehensive understanding of these waves in inhomogeneous field-aligned density irregularities is crucial for advancing our knowledge of the L-mode wave propagation, especially beyond the O-mode reflection region and in the context of the magnetic zenith effect.

Ground-based and in situ ionospheric measurements can be affected by additional factors such as various nonlinear effects, including modulation instability, formation of solitons, acceleration of electrons, and formation of cavitons possibly occurring in the localized region near the reflection height.^42–44 For simplicity, we exclude these aspects in this work and focus on the effect of a two-dimension density duct within the limit of linear wave dynamics.

In this study, we employed numerical simulations to explore the characteristics of L-mode interactions with a field-aligned density duct. Our investigation covered two scenarios of field-aligned density ducts: one without and another with an L-mode reflection layer. The analysis utilized both L-mode and O-mode waves for a more comprehensive comparison, providing insights into the role of L-mode waves in wave propagation beyond the O-mode reflection height and the magnetic zenith effects. Our simulation approach uses the two-dimensional FDTD method, which applies finite-difference approximations to solve Maxwell's equations in both space and time, modeling the behavior of electromagnetic waves. This method is particularly useful for multidirectional inhomogeneous and anisotropic systems, as its grid-based structuring effectively incorporates medium properties at every point. The highly inhomogeneous and dynamic nature of plasma makes the FDTD method an ideal tool for simulating the behavior of plasma waves.

The structure of this paper is as follows. In Sec. II, we introduce the electron number density profiles to represent the field-aligned density duct, the absorbing boundary conditions, and the mathematical framework for the FDTD method. In Sec. III, we present the detailed results of our numerical simulations. We focus on magnetic field-aligned density ducts, both with and without L-mode reflection layers, including an examination of the spatial distribution of electric fields and the time-based evolution of electric field energies. Finally, in Sec. IV, we discuss our findings and conclude the paper.

A 2D geometry in the xz plane was set up to simulate the interaction between electromagnetic waves and plasma. In our simulation, we used an input source wave of frequency $ω0=2π×3.5×106 s−1$ (⁠ $f0=3.5 MHz$⁠) along with the corresponding vacuum wavelength $λ0=c/f0=85.7 m$⁠. Given the high-frequency nature of the radio waves, we assumed the absence of ion motion, incorporating only electron motion. The background electron plasma density gradient was modeled along the $+z$ direction, with an additional gradient along the x axis representing the density duct. The geomagnetic field $B0$ was assumed to be uniform and oriented along the $−z$ direction. While on a large scale the magnetic field and the density gradient may not be strictly anti-parallel in the polar ionosphere, it was reasonable to make such a local assumption.

In our model, the upper hybrid (UH) resonance frequency $ωUH$ is close to the pump wave frequency near the source region, but the initially pumped waves become insensitive to the UH resonances in the given density profiles in the sense that the pump waves remain mostly field-aligned in such regions.

An LHCP pump wave $E+=Ex+iEy=Eo(i+ij)$ with a magnitude of $Eo=1 mV/m$ was continuously driven at the bottom of the simulation box with the two different incident angles $θ=0$ and $θ=5°$ relative to the z axis, respectively. The former represents the L-mode, while the latter represents the O-mode wave or notational convenience. To prevent reflections and minimize their impact on the simulation results, the entire simulation area was encased in complex frequency-shifted perfectly matched layers (CFS-PMLs).⁴⁵ These layers were created by adding a material layer with an adjusted complex permittivity and permeability at the edges of the simulation domain. These complex values were chosen to make the reflection coefficient of the wave equal to zero at the boundary, thereby successfully absorbing the incident wave and dissipating its energy as it traverses the PML layer.

In Figs. 2 and 3, we present the characteristics of pump waves in a field-aligned density duct in the case where there is no L-mode reflection layer. The left column of Fig. 2 represents the case where a pump wave propagates parallel to the magnetic field within the density duct, indicating an L-mode configuration. On the contrary, the right column of Fig. 2 shows the case where a pump wave enters the simulation box at an angle of $θ=5°$ relative to the z axis, representing an O-mode configuration. In Fig. 3, we compare the transmission energies $Ux$ and $Uy$ (⁠ $Ui=∫S112ε0Ei2dτ$⁠, $i=x,y$⁠) of L-mode and O-mode for the perpendicular electric field components $Ex$ and $Ey$⁠, as recorded in the uppermost part (⁠ $S1$⁠) of the simulation box, which corresponds to the region of $z>3100$ above the plasma resonance layer in Fig. 1(a).

In Figs. 2(a)–2(d), it is evident that both the $Ex$ and $Ey$ electric field components propagate effectively along the ambient magnetic field above the plasma resonance layer. This observation is crucial, highlighting the role of a field-aligned density duct in channeling waves along the ambient magnetic field above the plasma resonance layer, applicable to both L- and O-mode waves. However, it is distinctly shown in Fig. 3 that L-mode waves exhibit higher energy transmission into the region above the plasma resonance compared to O-mode waves. Below the plasma resonance layer, Fig. 2 shows the spatial distribution of the perpendicular electric field components, $Ex$ and $Ey$⁠, become differentiated between the two types of pump waves. In this region, the magnetic field-aligned density duct guides the pump waves to propagate along the magnetic field direction. The L-mode waves, already aligned with the magnetic field, allow a more substantial portion of wave energy to pass through the plasma resonance layer.

Conversely, O-mode waves, characterized by their inclined entry angle, undergo multiple reflections within the plasma resonance layer of the duct and struggle to align with the direction of the ambient magnetic field. These reflections result in the stimulation of the $Ez$ electric field component within the duct. As illustrated in Fig. 2(f), $Ez$ creates a 2D standing wave pattern beneath the plasma resonance, leading to enhanced energy absorption at the plasma resonance and reduced transmitted energy through the plasma resonance layer for the O-mode wave. In the case of the L-mode, there is only an excitation of $Ez$ near the plasma resonance, as shown in Fig. 2(e). As a result, the L-mode absorbs less energy and can transmit more energy beyond the O-mode reflection region. This is consistent with the feature shown in Fig. 3, where the initial pump wave power is more efficiently delivered to the upper region beyond the plasma resonance layer in the case of the L-mode.

Figure 4 shows the spatial distribution of electric field components in a field-aligned density duct with an L-mode reflection layer. It is clear that the L-mode reflection layer redistributes the wave energies above the plasma resonance. In comparison to the previous case, it is found that the $Ex$ and $Ey$ components of the L-mode and O-mode waves are likely to propagate at various angles above the plasma resonance layer. This indicates that the non-parallel pump waves now propagate along the Z-mode branch, while parallel propagating waves are designated as the L-mode. Consequently, the non-parallel waves above the plasma resonance region should belong to the Z-mode. This transformation, resulting from the L-mode reflection, is significant because the Z-mode waves are strongly absorbed in the plasma resonance. Therefore, we observe strong $Ex$ and $Ez$ amplitudes near the plasma resonance.

To further examine the differences between the L-mode and O-mode waves, we analyzed the time series of the total electric field energies (⁠ $Ui=∫S212ε0Ei2dτ$⁠, $i=x,y,z$⁠), which are added up in the whole simulation box (⁠ $S2$⁠) of Fig. 4. Figure 5 shows how wave energy of each component $Ui$ evolves over time. When comparing the energy components $Ux$(blue) and $Uy$(red) for both modes (solid lines for L-mode and dashed lines for O-mode), it is evident that $Ux$ and $Uy$ in the L-mode are greater than in the O-mode. This supports the idea that the L-mode is better at transferring wave energy.

It should be noted in Fig. 5 that the $Uz$(green) component initially shows greater values for O-mode, but the $Uz$ of the L-mode overtakes as time progresses. Initially, the multiple reflections of the O-mode below the plasma resonance layer stimulate $Ez$⁠, as observed in the absence of the L-mode reflection layer. However, as time progresses, after reflection from the $ωL$ layer, the $Uz$ of the L-mode dominates over that of the O-mode. This occurs as the L-mode wave transfers more energy to the region above the plasma resonance. Then, upon reflecting off the $ωL$ layer, the L-mode wave has more energy available to impart to the plasma modes or $Uz$ than the O-mode. Additionally, it is shown that the peak amplitude of plasma resonance for $Ex$ is greater than $Ey$ in both Figs. 2 and 4. Due to the two-dimensional plasma inhomogeneity, the plasma resonance curve is no longer normal to the field-aligned direction, resulting in $Ux$ being greater than $Uy$ for both cases. This is attributed to the electron density nonuniformity along the x axis, leading to local resonance motion in $Ex$⁠.

In this work, we investigated the characteristics of wave dynamics when the field-aligned density ducts encompass a frequency range of the local plasma resonances (⁠ $ωp$⁠) and/or the L-mode reflection layer (⁠ $ωL$⁠). If the frequency becomes higher or the density becomes relatively low in our model, the UH resonance can be located in the region of field-aligned density ducts. Many studies have been conducted on the excitation of UH resonances when the inhomogeneity gradient lies perpendicular or slanted to the ambient magnetic field.^48–52 To remove complexities arising from the wide frequency range covering the L-mode cutoff, the plasma resonance, and the UH resonance, we utilized a two-dimensional FDTD model in this study, focusing on the investigation of interactions of pump waves with and without L-mode reflection layered field-aligned density ducts.^53,54

In the scenario without an L-mode reflection layer, our findings clearly show that the L-mode waves transmit through the plasma resonance layer with significantly large wave energy compared to O-mode waves, as illustrated in Figs. 2 and 3. This suggests that the L-mode waves are more effective than the O-mode waves in stimulating physical processes related to wave propagation beyond the O-mode reflection region. These observations strongly support the feature proposed by Leyser and Nordblad³⁴ and Nordblad and Leyser³⁵ that the L-mode can propagate deeper into the plasma.

Previous studies have demonstrated the L-mode propagation of the EISCAT Heating beam as transionospheric propagation for $f0<f0F2<f0+fe/2$⁠. In this frequency range, an L-mode wave is not reflected but rather passes through the ionospheric plasma density peak. This phenomenon was directly measured by the CASSIOPE spacecraft, as reported by Leyser et al.,⁴¹ and was also indirectly observed through EISCAT UHF radar, which detected ion acoustic lines in the topside ionosphere, as noted in the work of Rexer et al.^39,40 Another experiment for trans-ionospheric propagation of HF pump waves has been done at Sura facility. The HF pump waves were transmitted from the ground under the condition $f0<fOF2$⁠. The electric field of the pump waves was detected by the DEMETER (Detection of Electro-Magnetic Emissions Transmitted from Earthquake Regions) satellite at 670 km altitude, in the topside ionosphere where the local plasma frequency was lower than the pump wave frequency near the magnetic zenith, suggesting limitations of Z-to-O mode conversion and pointing to the need for alternative explanations. Our findings in the absence of the L-mode reflection layer show that the L-mode waves transmit efficiently through the plasma resonance layer. This behavior of the L-mode waves is consistent with the feature that pump waves made it through the topside plasma resonance area near the magnetic zenith.

When there is an L-mode reflection layer, a pronounced excitation of the $Ex$ and $Ez$ is observed near the plasma resonance, more so in the case of L-mode waves than O-mode, as shown in Figs. 4 and 5. These results are consistent with the observations at the EISCAT Heating facility focusing on L-mode waves, where it was found that electron heating is much stronger at the magnetic zenith than off-zenith. However, their explanation attributed the effectiveness of the L-mode to its perpendicular polarization, which efficiently energizes the ionosphere by stimulating UH oscillations. It should be noted that ducted waves possess wave vectors that are not completely parallel to $B0$⁠. Our analysis leads us to propose that the principal role of the L-mode waves is the energy transmission beyond the O-mode reflection height in the ionosphere. When the L-mode waves reflect off the inclined $ωL$ layer, the L-mode can alter its direction and propagate in a non-parallel direction as a Z-mode and significantly enhance the excitation of plasma modes near the plasma resonance since the Z-mode has strong absorption characteristics at these frequencies. This can be understood clearly by comparing the electric field magnitudes with and without the L-mode reflection layer. In the former case, the L-mode keeps propagating along the ambient magnetic field, resulting in no strong excitation of localized oscillations. On the contrary, strong excitation is observed in the latter case due to the change in the L-mode waves from the reflection layer. Thus, this change from the L-mode to the Z-mode seems to be a significant factor contributing to the magnetic zenith effect.

The conclusion of our numerical study on the L-mode characteristics is that L-mode waves can effectively transmit energy through the plasma resonance of a magnetic field-aligned density duct. When the plasma is dense enough to have an $ωL$ reflecting layer, which can change the direction of the L-mode propagation into non-parallel directions as the Z-mode, it is likely to cause a noticeable increase in localized oscillations of the electric fields at the plasma resonance. These insights clarify our understanding of the high-frequency wave dynamics beyond the O-mode reflection height and the magnetic zenith effects observed in the Earth's ionosphere when pump waves are transmitted along the geomagnetic field.

While this study provides valuable insights into wave behavior beyond the plasma resonance and magnetic zenith effects, it has certain limitations. First, it focuses exclusively on the bottom-side plasma resonance, which occurs below the ionospheric plasma density peak. A broader investigation is needed to explore how pump waves interact with the topside plasma resonance within a field-aligned density duct. Second, the study considers a pump wave frequency that is not a multiple of the electron cyclotron frequency. When the pump wave frequency approaches the integer multiples of the electron cyclotron frequency in the presence of irregularities, wave dynamics can undergo significant modifications.^8,39,40 Both aspects are left for future research.

This work was supported by KHU-20130752 and in part through the Institute of Information & Communications Technology Planning & Evaluation (IITP) Grant funded by the Korea Government (MSIT) (RS-2023-00235751). K. Kim was supported through the National Research Foundation of Korea Grants (NRF-2022R1F1A1074463 and NRF-2021R1A6A1A10044950) funded by the Korean Government.

The authors have no conflicts to disclose.

Danish Naeem: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Software (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Dong-Hun Lee: Conceptualization (supporting); Funding acquisition (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (equal). Kihong Kim: Conceptualization (supporting); Funding acquisition (equal); Project administration (equal); Writing – review & editing (equal).

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