Dynamic spectral characteristics of high-resolution simulated equatorial plasma bubbles

Abstract

Manifestations of severe nighttime equatorial ionospheric disturbances have been observed for decades. It is generally accepted that the phenomena are caused by large depletions, referred to as equatorial plasma bubbles (EPBs), which are initiated on the rising unstable bottom side of the nighttime F layer. Physics-based simulations have enhanced our understanding of the EPB phenomenon. However, until very recently, stochastic structure smaller than ∼ 10 km was not well resolved. Recent high-resolution EPB simulations have extended the resolution to hundreds of meters, which provides a unique opportunity to characterize intermediate-scale EPB structure.This paper presents a summary analysis of simulated high-resolution intermediate-scale EPB structure. Estimation of altitude-dependent power law spectral density function parameters provides an altitude versus time history of the intermediate-scale structure development. Local structure onset is associated with successive bifurcation of rising EPBs. Developed structure characterized by a two-component power law spectral density function ultimately subtends several hundred kilometers in altitude.Two-component inverse power-law structure was first observed in early in situ rocket measurements. It has been observed in diagnostic measurements of beacon-satellite and GPS scintillation data as well as in situ measurements from Atmospheric Explorer and C/NOFS satellites. The EPB simulation data fully support the reported EPB diagnostics as well as a correlation between the turbulent strength and the large-scale spectral index parameter estimates. However, recent analyses have shown that the correlation is an intrinsic property of power-law parameter estimation.

Introduction

The terminology equatorial spread F (ESF), plumes, and equatorial plasma bubbles (EPBs) evolved, respectively, from ionospheric sounder, coherent radar backscatter, and diagnostic measurements. In situ and remote EPB radio-propagation diagnostics are formally time series generated by the motion of the probe or the interrogating propagation path. Interpreting such diagnostic measurements is challenging because altitude, magnetic field, and temporal structure variations are invariably intermingled. Moreover, time-to-space conversion depends on an unknown structure drift. The dependence of propagation diagnostics on path-integrated structure further complicates the interpretation of diagnostic measurements.

Physics-based simulations provide an exceptional opportunity to generate definitive structure development measurements. Although the underlying physics has been well established for decades, simulating the generation and dissipation of steep gradients that evolve in unstable regions has, until very recently, limited the resolution that could be achieved to kilometer scales. Recently, simulations that exploit advanced computational capabilities have resolved EPB structure to hundreds of meters.

Methods/Experimental

High-resolution simulations described in a survey paper by Yokoyama (2017) were made available for the EPB structure analysis presented in this paper. Yokoyama (2017) also reviewed the historical development of EPB simulations, which were introduced in the early 1980s. To make the EPB simulations as representative of real-world conditions as possible, multiple EPBs were initiated with an eastward E×B drift. The number of EPBs initiated depends on the initial conditions. However, the simulation analyzed in this paper shows that intermingling of structure from multiple EPBs populates extended structure regions.

The conditions under which EPBs develop and how the large-scale structure evolution can be reconciled with diagnostic measurements, particularly the coherent-backscatter radar echoes that delineate the plumes, have been studied extensively (Hysell 2000). Less attention has been given to characterizing intermediate-scale stochastic structure from tens of kilometers to hundreds of meters. Figure 1 is a perspective view of the developed EPB structure. The left frame shows the central meridian plane slice, which emphasizes smoothly varying field-aligned structure. The right frame shows vertical and horizontal slice planes, which emphasize stochastic cross-field structure.

Fig. 1

Perspective view of the EPB simulation environment. The left frame shows the central meridian plane. The right frame shows the structure in two orthogonal slice planes

Stochastic structure is definitively reproduced only in slice planes that cut across field lines. For this study, evolving structure from the equatorial slice plane and two offset slice planes, identified by the rays in Fig. 2, were extracted from the three-dimensional simulations. The electron density variation in Fig. 2 shows that field-aligned structure intercepts systematically varying background electron density. The offset slice planes allow exploration of the field-aligned structure translation.

Fig. 2

Meridian slice plane with overlaid rays locating vertical equatorial slice plane and two offset slice planes excised for structure analysis

The left frame in Fig. 3 shows developed equatorial-plane structure detail 1 h after initiation. Seeded bottom-side perturbations initiated five EPBs that evolved at different rates depending on their zonal location at initiation. Slice planes were sampled uniformly from 300 to 800 km with 1120 zonal samples at 333.56 m and 1821 vertical samples at 700.83 m. The right frame shows the zonal average electron density (blue) with an overlaid smoothed profile (red). Figures 4 and 5 show the offset1 and offset2 structure summaries. The structure flux tubes intercept the offset planes at progressively lower altitudes. The structure in the offset2 slice plane is mapped below the 300 km lower limit.

Fig. 3

The left frame shows the equatorial plane structure at 1 h. The right frame shows the path-integrated density (blue) with a smoothed profile overlaid (red)

Fig. 4

The left frame shows the offset1 plane structure at 1 h. The right frame shows the path-integrated density (blue) with a smoothed profile overlaid (red)

Fig. 5

The left frame shows the offset2 plane structure at 1 h. The right frame shows the path-integrated density (blue) with a smoothed profile overlaid (red)

The zonal average electron densities shown in the right frames of Figs. 3, 4, and 5 are proportional to zonal path-integrated total electron content (TEC), which can be measured with navigation satellite transmissions received by low orbiting satellite occultations (Tsai et al. 2011). Although such measurements do not resolve the intermediate-scale structure directly, scintillation of the probing signals can be processed for structure diagnostics.

To summarize the structure evolution, the slice frames were cyclically shifted to compensate for the 120 mps eastward drift. Periodic simulation boundary conditions confined the zonal extent of the realizations. The residual formed by subtracting the smoothed average profiles provides a measure of the height-dependent structure development. The time resolution for the EPB simulations is 0.1 s but reported at 10 s intervals. Figure 6 summarizes the evolution of the structure residuals at 100 s intervals. Structure onset can be identified at a specific time and altitude, which is the point where the highest EPB penetrates the F-region peak electron density.

Fig. 6

Evolution of height-dependent residual structure at 100 s for equatorial, offset1, and offset 2 slices

To explore the structure onset detail, Fig. 7 shows four consecutive 10 s zoomed images of the most rapidly progressing central EPB. The steepening gradient at the head of the EPB generates local depletions flanked by enhancements, which are referred to as bifurcations. Each bifurcation initiates a secondary bifurcation. The process of successive bifurcation creates a fractal-like structure cascade. The progression in Fig. 4 shows that once initiated, successive bifurcation proceeds very rapidly. A more detailed discussion can be found in Yokoyama et al. (2014).

Fig. 7

Numbered frames show zoomed views at 10 s intervals of the central EPB that generated the structure shown in Fig. 3

Structure characterization

EPB electron density slice-plane realizations are formally two-dimensional scalar fields, N_e(y,z), where y and z represent cross-field and altitude, respectively. Assuming that N_e(y,z) is statistically homogeneous, the stochastic structure can be characterized by a two-dimensional spectral density function (SDF), which is formally the expectation of the intensity of two-dimensional Fourier decompositions of N_e(y,z) realizations.

Power-law models

Published in situ measurements and remote diagnostics imply an underlying two-component power-power law SDF. The following analytic representation is introduced to guide structure characterization:

$$ \Phi_{N_{e}}(q)=C_{s}\left\{ \begin{array}{c} q^{-p_{1}}\text{ for}~q\leq q_{0} \\ q_{0}^{p_{2}-p_{1}}q^{-p_{2}}\text{ for}~ q>q_{0} \end{array} \right., $$

(1)

where

$$ q=\sqrt{q_{y}^{2}+\beta q_{z}^{2}}, $$

(2)

is the magnitude of the spatial frequency vector [q_y,q_z] in radians per meter. The β coefficient accommodates projection of the radial variation of field-aligned structure. The defining parameters are turbulent strength, C_s; the break frequency, q₀; and the spectral indices, p_n corresponding to subranges of spatial frequencies smaller than (n=1) and larger than (n=2) q₀.

In situ measurements are one-dimensional scans. If the structure volume were stochastic in all three dimensions, the measured one-dimensional SDF would be represented by a two-dimensional integration of the three-dimensional SDF. For field-aligned two-dimensional stochastic structures a slice plane containing the one-dimensional scan must be constructed. Configuration-space realizations populate arbitrarily oriented slice planes for extrapolation (Rino et al. 2018). For the EPB analysis, the cross-field orientation of the slice planes were selected for direct structure measurement. One-dimensional SDFs are related to (1) by the integration

$$ \Phi_{N_{e}}^{1}(q)=\int_{-\infty}^{\infty}\Phi_{N_{e}}(q_{y},q_{z}) \frac{dq_{z}}{2\pi }. $$

(3)

For (3) to be well defined, the power-law variation must be specified in more detail. In the EPB realizations, there is a transition from stochastic to trend-like variation at small spatial frequencies. At sufficiently high frequencies, the physics supporting the EPB simulations is incomplete. Furthermore, as already noted, the stochastic structure itself varies with altitude. To capture these details, the following height-dependent one-dimensional SDF hypothesized for EPB structure characterization:

$$ \Phi_{N_{e}}\left(q \right) =C_{s}\left\{ \begin{array}{c} q^{-\eta_{1}}\text{for}~q\leq q_{0} \\ q_{0}^{\eta_{2}-\eta_{1}}q^{-\eta_{2}}\text{for}~q>q_{0} \end{array} \right.. $$

(4)

The one-dimensional model captures a broad range of structure characteristics as defined by the turbulent strength C_s, the spectral indices η_n, and the break frequency q₀. For example, if η₁≃0, q₀ can be interpreted as an outer scale. If η₁≃η₂, the SDF is a single power law. Generally, η₁≤η₂. However, enhanced low-frequency structure might lead to the opposite ordering, η₁>η₂. In all cases, the variation of C_s provides a measure of overall structure intensity.

Establishing the relation between η_n and p_n, which is nominally η_n=p_n−1, is beyond the scope of this study. However, ionospheric structure models can be validated by comparing predicted height-dependent one-dimensional structure characteristics with the measured EPB structure.

Irregularity parameter estimation

Irregularity parameter estimation (IPE) systematically adjusts the defining parameters to minimize a measure of the disparity between an SDF estimate and the theoretical SDF. An IPE procedure for estimating scintillation intensity SDF parameters was introduced by Carrano and Rino (2016). The original IPE procedure was refined to maximize the likelihood that the periodogram was derived from a realization with the theoretical SDF (Carrano et al. 2017). For characterizing the EPB SDFs, the maximum likelihood estimation (MLE) procedure was adapted for power-law SDF estimation as described in Rino and Carrano (2018). Power-law parameter estimation is more challenging than intensity scintillation parameter estimation because of the singular behavior of unmodified power-law SDFs at zero frequency.

The MLE SDF estimate is the average of M periodograms, formally

$$ \widehat{\Phi}_{N_{e}}=\frac{1}{M}\sum\limits_{l=1}^{M}\widehat{\Phi }_{n}^{(l)}, $$

(5)

where the periodogram is defined as

$$ \widehat{\Phi }_{n}^{(l)}=\frac{\Delta y}{N}\left\vert \sum_{k=0}^{N-1}N_{e}^{(l)}\left(k\Delta y\right) \exp \{-ink/N\}\right\vert^{2}. $$

(6)

The index n corresponds to the spatial frequencies

$$ 2\pi/(N\Delta y)\leq n\Delta q \leq 2\pi /\Delta y, $$

(7)

where Δy is the y sample interval, and Δq=2π/(NΔy) is the spatial-frequency resolution. The index l identifies the altitude at which the zonal scan is extracted. One can show that

$$ \left\langle \widehat{\Phi}_{N_{e}}\right\rangle={\Phi}_{N_{e}} $$

(8)

MLE exploits the fact that the probability distribution function (PDF) of the periodogram is well approximated by a χ distribution with 2 degrees of freedom. The χ distribution with 2M degrees of freedom follows for the summation.

It is well known that periodogram estimates are contaminated by the sidelobes of end-point discontinuities. Moreover, efficient discrete Fourier transformation (DFT) evaluation requires N to be even with as many factors as possible, ideally a power of 2. The Welch method (Welch 1995) uses windowing and segmentation with averaging. Periodogram variants, such as maximum entropy estimates (Fougere 2009), provide additional variants. However, MLE relies on unbiased spectral estimates with χ distributions, whereby it is desirable to stay as close to (6) as possible. After some exploration, it was found that using the full 373.6 km y extent of the data zero extended to a nice FFT number gave the best results. Following (Rino and Carrano 2018), periodograms from two altitudes (M=2) were averaged.

Multi-parameter MLE used a MATLAB implementation of the Nelder-Mead simplex algorithm (Olsen and Nelsen 1975). The procedure is surprisingly robust in that fits were made to quasi-deterministic SDFs with no stochastic structure as well SDFs from realizations with fully developed stochastic structure. The two classes are readily distinguished by the reported IPE parameters.

Results

MLE-IPE with M=2 was performed over each set of slice-plane scans. Figures 8, 9, and 10 summarize the parameter estimates. The periodogram sample interval is twice the height sampling (1.4 km). The C_s>200 estimates shown in the upper frames of Figs. 8, 9, and 10 capture the unstructured regions identified in Fig. 6. From the second and third frames in Figs. 8, 9, and 10, we see that η₁<η₂ within the unstructured regions. Outside the structured regions, the pattern is reversed. For ease of interpretation, σ_b=2π/q₀ is reported rather than the break frequency.

Fig. 8

MLE-IPE SDF parameter estimates from equatorial slice plane realizations at 100 s, where σ_b=2π/q0

Fig. 9

MLE-IPE SDF parameter estimates from offset1 slice plane realizations at 100 s, where σ_b=2π/q0

Fig. 10

MLE-IPE SDF parameter estimates from offset2 slice plane realizations at 100 s, where σ_b=2π/q0

Figure 11 compares representative equatorial slice-plane SDFs extracted from the unstructured (upper frame) and structured (lower frame) altitude ranges. The smoothly varying structure generates enhanced structure at the lower spatial frequencies with sidelobes populating the higher frequencies. The MLE-IPE parameters capture the SDF envelope with η₁>η₂ and a break frequency scale near 5 km. The developed structure populates the lower frequencies more uniformly with a more rapid decrease at higher frequencies. MLE-IPE captures the structure with η₁<η₂ and larger C_s values. The transition from unstructured to structured SDFs necessarily includes SDFs with η₁≃η₂.

Fig. 11

SDFs (magenta) with overlaid theoretical SDFs from IPE parameters (red)

The break scale estimates within the structured region as summarized in the lower frames of Figs. 8, 9, and 10 show more variability. Movie presentations of slice plane realizations and the associated SDFs highlight the intermingling of the large-scale EPB structure with the background. The several kilometer size of the bifurcations shown in Fig. 7 appears to be a lower bound on the break frequency. However, the intermingling of the EPBs with the background evidently modulates the structure.

Discussion

The simulations reveal abrupt structure onset as a transition from quasi-deterministic structure with a steep low-frequency power-law index to the more representative two-component SDF structure with η₁<η₂. To the extent that field lines mapped from the equatorial plane are captured in the offset planes, the structure characteristics are nearly identical.

To summarize the stochastic structure characteristics, Figs. 12, 13, and 14 show probability distributions of the structure with CsdB>200 dB. The developed structure is fairly uniform. The η₁ and η₂ distributions show peaks just below 1.5 and just above 2.5. These values are consistent with the C/NOFS results reported by Rino et. al (2016). They are also consistent with the parameters Retterer (2010) used in the PBMOD ionospheric scintillation model, with allowance for the relation p_n=η_n+1. Moreover, EPB C_s range is comparable to the reported C/NOFS values when the unscaled C/NOFS values are translated to common electron density units.

Fig. 12

Probability distributions of equatorial plane stochastic structure

Fig. 13

Probability distributions of offset1 plane stochastic structure

Fig. 14

Probability distributions of offset2 plane stochastic structure

The C/NOFS break scales reported by Rino et. al (2016) are smaller than the break scales from the EPB analysis. One possibility is the resolution of wavelet scale spectra used for the C/NOFS analysis. Alternatively, as noted in the previous section, the EPB break scale is evolving and sensitive to the background structure. Precise measurement of the initiation and evolution of the break scale is a topic for targeted special study.

Transition populations with η₁≃η₂ are very small. However, the simulated background structure is idealized and might not represent real-world background structure. Bhattacharyya et al. (2003) showed that the latitudinal dependence of EPB structure can be explained by invoking a single power-law structure in the lower F-region, with attendant smaller scintillation levels.

The offset1 and offset2 PDFs shown in the lower frames of Figs. 13 and 14 have peaks between 4 and 5 km, which is consistent with the dimensions of the initial bifurcations shown in Fig. 7. The distribution of larger break scales is associated with the aforementioned intermingling of background structure. The offset2 structure is fully contained in the enhanced background, which would explain the enhanced distribution of larger break scales.

Figure 15 shows a scatter diagram of the measured EPB parameters CsdB versus η₁. The correlation is identical to the correlation reported in Rino et al. (2016) from an analysis of 4 years of C/NOFS data. The overlaid log-linear dependence has the reported slope of 0.02 nepers per dB of C_s change. The tendency for the correlation to appear in narrow bands was also noted in the C/NOFS data. However, we now know from the analysis reported by Rino and Carrano (2018) that the correlation can be completely explained as a intrinsic property of power-law irregularity parameter estimation. The correlation occurs because the χ distribution with small degrees of freedom generates a significant population of errors larger than the mean. This is reflected in the C_s distributions shown in the upper frames of Figs. 12, 13, and 14, which favor larger C_s values.

Fig. 15

Scatter diagram of equatorial stochastic structure CsdB values versus η₁ (red), with log-linear overlay (blue)

As a consequence of the correlation, the true values of C_s and η₁ are likely to be closer to the central values in Fig. 15. As a test of this effect, the equatorial MLE-IPE was recomputed with M=10, which reduces the altitude resolution to 7 km. Figure 16 shows the probability distributions. The M=10 equatorial Cs distribution is more concentrated between 210 and 215 dB. The η₁ and η₂ peaks are sharpened somewhat, while larger scales dominate the break frequency scale. Because resolution requires larger segments, which reduces the number of segments that can be averaged, trades between resolution and statistical uncertainty are unavoidable. However, the variability of the break scale appears to convey information about the underlying structure. These are clearly topics for further study.

Fig. 16

Probability distributions of equatorial plane stochastic structure computed with M=10

Conclusions

The analysis of high-resolution EPB simulations presented in this paper supports that generally accepted hypothesis that developed EPB structure can be characterized by a one-dimensional two-component power-law SDF with η₁ somewhat smaller than 1.5 and η₂ somewhat larger than 2.5. The scale associated with the break frequency varies from the 4 to 5 km bifurcation scale to much larger values reflecting intermingling of the EPB structure with the F-region background. In units of electrons per cubic centimeter, the decibel turbulent strength parameter falls between 210 and 215 dB. A persistent correlation between the measured turbulent strength and the large-scale spectral index is an intrinsic property of power-law parameter estimation that researchers need to be aware of. Structure evolving in the equatorial plane maps along field lines with no significant structure variation. However, structure mapped below F-layer was not investigated.

The transition from smoothly varying background structure to stochastic structure is manifest by an SDF transition initially reflecting large-scale structure with η₁>η₂ to developed structure with η₁<η₂. The transition through a single power law with η₂≃η₂ is not a prominent feature, but we have argued that a smooth background is idealized and not representative of real ionospheric structure. Within the structured region, the structure is uniform over the 30 min interval with developed structure and over an altitude range of several hundred kilometers. The result favors the standard interpretation that tow-dimensional SDF characterizing the cross-field structure has the two-dimensional form (1) with p_n=η_n+1.

We conclude by recalling that the 1971 PLUMEX campaign successfully launched a powerful rocket into an EPB being tracked by coherent-scatter radar. The rocket carried a radio beacon and a Langmuir probe. Analysis of the PLUMEX radio-beacon and Langmuir probe data summarized by Rino et. al (1981) showed the first evidence of a two-component power-law structure. Numerical simulations being developed by the U. S. Naval Research Laboratory and reviewed by Yokoyama (2017) were being used to interpret the PLUMEX results. To quote the final sentence in the PLUMEX paper: Rapid progress is being made in such [numerical simulations], and we believe that the simulations have the potential to verify the results presented in [the PLUMEX] paper.