Rino et al. Progress in Earth and Planetary Science
https://doi.org/10.1186/s40645-018-0243-0(2018) 5:83

Progress in Earth andPlanetary Science

RESEARCH ARTICLE

Open Access

Dynamic spectral characteristics ofhigh-resolution simulated equatorial plasma
bubblesCharles Rino1*, Tatsuhiro Yokoyama2 and Charles Carrano1

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 haveenhanced 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 ofmeters, 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 ofaltitude-dependent power law spectral density function parameters provides an altitude versus time history of theintermediate-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 measurementsfrom 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.
Keywords: Equatorial spread F, Power-law ionospheric structure, Convective plasma instability, Structure
characterization

Introduction
The terminology equatorial spread F (ESF), plumes, andequatorial 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 generationand dissipation of steep gradients that evolve in unstableregions 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*Correspondence: charles.rino@bc.edu
1Institute for Scientific Research, Boston College, 140 Commonwealth Ave.,
Chestnut Hill, MA 02467, USA
Full list of author information is available at the end of the article

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)© The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to theCreative Commons license, and indicate if changes were made.

Rino et al. Progress in Earth and Planetary Science(2018) 5:83

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 withan eastward E × B drift. The number of EPBs initiateddepends 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 reconciledwith diagnostic measurements, particularly the coherentbackscatter radar echoes that delineate the plumes, havebeen 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.
Stochastic structure is definitively reproduced only inslice 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, wereextracted from the three-dimensional simulations. The
electron density variation in Fig. 2 shows that field-aligned
structure intercepts systematically varying backgroundelectron density. The offset slice planes allow explorationof the field-aligned structure translation.

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The left frame in Fig. 3 shows developed equatorialplane structure detail 1 h after initiation. Seeded bottomside 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 thezonal average electron density (blue) with an overlaidsmoothed 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.
The zonal average electron densities shown in theright frames of Figs. 3, 4, and 5 are proportional tozonal 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 theintermediate-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 structuredevelopment. 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

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

Rino et al. Progress in Earth and Planetary Science(2018) 5:83

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Fig. 2 Meridian slice plane with overlaid rays locating vertical equatorial slice plane and two offset slice planes excised for structure analysis

intervals. Structure onset can be identified at a specifictime and altitude, which is the point where the highest
EPB penetrates the F-region peak electron density.
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 thehead of the EPB generates local depletions flanked byenhancements, which are referred to as bifurcations. Eachbifurcation 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. 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)Rino et al. Progress in Earth and Planetary Science(2018) 5:83

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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)Structure characterization
EPB electron density slice-plane realizations are formally two-dimensional scalar fields, Ne (y, z), where
y and z represent cross-field and altitude, respectively. Assuming that Ne (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 intensityof two-dimensional Fourier decompositions of Ne (y, z)
realizations.
Power-law models

Published in situ measurements and remote diagnostics
imply an underlying two-component power-power law

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)(2018) 5:83

Rino et al. Progress in Earth and Planetary Science

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Fig. 6 Evolution of height-dependent residual structure at 100 s for equatorial, offset1, and offset 2 slices

SDF. The following analytic representation is introduced
to guide structure characterization:Ne (q) = Cs

q−p1 for q ≤ q0
p2 −p1 −p2q0q
for q > q0

,

(1)

where
q=



qy2 + βqz2 ,

(2)



is the magnitude of the spatial frequency vector qy , qz
in radians per meter. The β coefficient accommodates

Fig. 7 Numbered frames show zoomed views at 10 s intervals of the central EPB that generated the structure shown in Fig. 3

Rino et al. Progress in Earth and Planetary Science(2018) 5:83

projection of the radial variation of field-aligned structure. The defining parameters are turbulent strength, Cs ;the break frequency, q0 ; and the spectral indices, pn corresponding to subranges of spatial frequencies smaller than(n = 1) and larger than (n = 2) q0 .
In situ measurements are one-dimensional scans. If thestructure volume were stochastic in all three dimensions,the measured one-dimensional SDF would be represented by a two-dimensional integration of the threedimensional SDF. For field-aligned two-dimensional
stochastic structures a slice plane containing the onedimensional scan must be constructed. Configurationspace 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. Onedimensional SDFs are related to (1) by the integration
 ∞
dqz
1
Ne (q) =.
(3)
Ne (qy , qz )
2π−∞
For (3) to be well defined, the power-law variationmust be specified in more detail. In the EPB realizations, there is a transition from stochastic to trend-likevariation at small spatial frequencies. At sufficiently highfrequencies, the physics supporting the EPB simulations
is incomplete. Furthermore, as already noted, the stochastic structure itself varies with altitude. To capture thesedetails, the following height-dependent one-dimensional
SDF hypothesized for EPB structure characterization:
q−η1 for q ≤ q0
Ne (q) = Cs.
(4)
η2 −η1 −η2q0
q for q > q0
The one-dimensional model captures a broad rangeof structure characteristics as defined by the turbulent
strength Cs , the spectral indices ηn , 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 Cs provides a measure of overall
structure intensity.
Establishing the relation between ηn and pn , which isnominally ηn = pn − 1, is beyond the scope of thisstudy. However, ionospheric structure models can bevalidated by comparing predicted height-dependent onedimensional structure characteristics with the measured
EPB structure.
Irregularity parameter estimation

Irregularity parameter estimation (IPE) systematicallyadjusts 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

Page 6 of 13(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 andCarrano (2018). Power-law parameter estimation is morechallenging 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 Ne = 1

M

M


(l)

n ,

(5)

l=1

where the periodogram is defined as
y
(l)

n =
N

N−1


2

Ne(l) (ky) exp{−ink/N} .

(6)

k=0

The index n corresponds to the spatial frequencies
2π/(Ny) ≤ nq ≤ 2π/y,(7)where y is the y sample interval, and q = 2π/(Ny)is the spatial-frequency resolution. The index l identifies
the altitude at which the zonal scan is extracted. One canshow that N e = N e
(8)

MLE exploits the fact that the probability distributionfunction (PDF) of the periodogram is well approximatedby a χ distribution with 2 degrees of freedom. The χ
distribution with 2M degrees of freedom follows for thesummation.
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 thatusing 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 inthat fits were made to quasi-deterministic SDFs with nostochastic structure as well SDFs from realizations with

Rino et al. Progress in Earth and Planetary Science(2018) 5:83

fully developed stochastic structure. The two classes arereadily distinguished by the reported IPE parameters.

Results
MLE-IPE with M = 2 was performed over each setof slice-plane scans. Figures 8, 9, and 10 summarize theparameter estimates. The periodogram sample interval istwice the height sampling (1.4 km). The Cs > 200 estimates shown in the upper frames of Figs. 8, 9, and 10
capture the unstructured regions identified in Fig. 6. Fromthe second and third frames in Figs. 8, 9, and 10, we seethat η1 < η2 within the unstructured regions. Outside
the structured regions, the pattern is reversed. For easeof interpretation, σb = 2π/q0 is reported rather than the
break frequency.
Figure 11 compares representative equatorial sliceplane 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 breakfrequency 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 Cs values. The transition from unstructured to structured SDFs necessarily
includes SDFs with η1  η2 .
The break scale estimates within the structured region
as summarized in the lower frames of Figs. 8, 9, and 10

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show more variability. Movie presentations of slice plane
realizations and the associated SDFs highlight the intermingling of the large-scale EPB structure with thebackground. The several kilometer size of the bifurcations shown in Fig. 7 appears to be a lower boundon 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 lowfrequency 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 arenearly 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 theparameters Retterer (2010) used in the PBMOD ionospheric scintillation model, with allowance for the relationpn = ηn + 1. Moreover, EPB Cs range is comparable to the reported C/NOFS values when the unscaled
C/NOFS values are translated to common electron
density units.

Fig. 8 MLE-IPE SDF parameter estimates from equatorial slice plane realizations at 100 s, where σb = 2π/q0

Rino et al. Progress in Earth and Planetary Science(2018) 5:83

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Fig. 9 MLE-IPE SDF parameter estimates from offset1 slice plane realizations at 100 s, where σb = 2π/q0

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 notedin 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

Fig. 10 MLE-IPE SDF parameter estimates from offset2 slice plane realizations at 100 s, where σb = 2π/q0

Rino et al. Progress in Earth and Planetary Science(2018) 5:83

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Fig. 11 SDFs (magenta) with overlaid theoretical SDFs from IPE parameters (red)dependence of EPB structure can be explained by invoking
a single power-law structure in the lower F-region, withattendant 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 theinitial bifurcations shown in Fig. 7. The distribution oflarger break scales is associated with the aforementioned

Fig. 12 Probability distributions of equatorial plane stochastic structureintermingling 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 tothe correlation reported in Rino et al. (2016) from an analysis of 4 years of C/NOFS data. The overlaid log-linear

Rino et al. Progress in Earth and Planetary Science(2018) 5:83

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Fig. 13 Probability distributions of offset1 plane stochastic structure

dependence has the reported slope of 0.02 nepers per dBof Cs 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 theχ distribution with small degrees of freedom generates a

Fig. 14 Probability distributions of offset2 plane stochastic structure

significant population of errors larger than the mean. Thisis reflected in the Cs distributions shown in the upper
frames of Figs. 12, 13, and 14, which favor larger Cs values.
As a consequence of the correlation, the true valuesof Cs 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Rino et al. Progress in Earth and Planetary Science(2018) 5:83

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Fig. 15 Scatter diagram of equatorial stochastic structure CsdB values versus η1 (red), with log-linear overlay (blue)

probability distributions. The M = 10 equatorial Cs distribution is more concentrated between 210 and 215 dB.
The η1 and η2 peaks are sharpened somewhat, whilelarger scales dominate the break frequency scale. Because
resolution requires larger segments, which reduces thenumber of segments that can be averaged, trades between
resolution and statistical uncertainty are unavoidable.

However, the variability of the break scale appears toconvey information about the underlying structure. These
are clearly topics for further study.

Conclusions
The analysis of high-resolution EPB simulations presented
in this paper supports that generally accepted hypothesis

Fig. 16 Probability distributions of equatorial plane stochastic structure computed with M = 10

Rino et al. Progress in Earth and Planetary Science(2018) 5:83

that developed EPB structure can be characterized bya 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 muchlarger values reflecting intermingling of the EPB structurewith 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 andthe 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 planemaps along field lines with no significant structure variation. However, structure mapped below F-layer was notinvestigated.
The transition from smoothly varying backgroundstructure 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 asmooth background is idealized and not representative ofreal 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 pn = ηn + 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 evidenceof a two-component power-law structure. Numerical simulations being developed by the U. S. Naval ResearchLaboratory 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 thatthe simulations have the potential to verify the resultspresented in [the PLUMEX] paper.
Abbreviations
EPB: Equatorial plasma bubble; PSD: Power spectral density; SDF: Spectral
density function
Acknowledgements
The computation was performed on the FX100 supercomputer system at theInformation Technology Center, Nagoya University, and Hitachi SR16000/M1
system at NICT, Japan.
Funding
This work was supported by JSPS KAKENHI Grant Number JP16K17814. This
work was also supported by the computational joint research program of the

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Institute for Space-Earth Environmental Research (ISEE), Nagoya University,
Japan.
Support for CR and CC was provided under Advanced Data Driven
Specification and Forecast Models for the Ionosphere-Thermosphere System,
Air Force Contract FA9453-12-C-0205
Availability of data and materials
The simulation data are stored on the FX100 supercomputer system at theInformation Technology Center, Nagoya University, and Hitachi SR16000/M1
system at NICT, Japan. TY (tyoko@nict.go.jp) can provide data upon request.
Authors’ contributions
All of the simulations analyzed in this paper were performed by TY andgenerously reformatted and made available to CR who performed the analysis.
The analysis was conceived by CR and TY at the December 2016 AGU meeting
following a presentation by TY. CC has worked extensively to improve
scintillation diagnostics, particularly definitive SDF parameter estimation,
which was central to this study. All authors read and approved the final
manuscript.
Competing interests
The authors declare that they have no competing interest.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.
Author details
1 Institute for Scientific Research, Boston College, 140 Commonwealth Ave.,
Chestnut Hill, MA 02467, USA. 2 National Institute of Information andCommunications Technology, 4-2-1 Nukui-Kitamachi, Koganei, Tokyo
184-8795, Japan.
Received: 27 August 2018 Accepted: 16 November 2018

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