 Full paper
 Open Access
 Published:
A threedimensional stochastic structure model derived from highresolution isolated equatorial plasma bubble simulations
Earth, Planets and Space volume 75, Article number: 64 (2023)
Abstract
Ionospheric structure is characterized by the space–time variation of electron density. However, our understanding of the physical processes that initiate and sustain intermediatescale structure development does not relate directly to statistical measures that characterize the structure. Consequently, highresolution physicsbased equatorial plasma bubble simulations are essential for identifying systematic relations between statistical structure measures and the underlying physics that initiates and sustains the structure evolution. An earlier paper summarized the analysis of simulated equatorial plasma bubble (EPB) structure initiated with a quasiperiodic bottomside perturbation that generated five plasma bubbles. The results are representative of real environments. However, the association of the structure development with individual EPBs was difficult to ascertain. This paper summarizes the analysis of new results from single isolated EPB realizations with varying parameters that affect the structure development. The evolution of the single isolated EPB realizations reveal what we have identified as a canonical structure evolution pattern manifest in the space–time development of four quantitative spectral parameters. The onset of structure occurs when the plasma bubble penetrates the Fregion peak. The parameter evolution from the initiation point have a fishlike appearance. The threedimensional structure model can be used to interpret in situ and remote diagnostic measurements as well as predicting the deleterious effects of propagation disturbances on satellite communication, navigation, and surveillance systems.
Graphical Abstract
Introduction
Although the ionosphere is an extremely complex system, electron density, \(N_e(\textbf{r},t)\), is a sufficient observable for structure characterization separate from the state of the ionosphere and underlying physics. We consider a region of interest (ROI) centered on a GPS coordinate, \(\textbf{r}_{0}\), with temporal evolution initiated at universal time, \(t_0\). We let
where the background electron density, \(\overline{N}_{e}(\textbf{r} _{0},t_{0})\), is derived from a global ionospheric model and \(DN_{e}(\Delta \textbf{r},\Delta t)\) represents a class of structure that supports statistical measures. Formally,
The term \(\Delta N_{e}(\textbf{v}_{\text {eff}}\Delta t)\) represents intermediatescale structure that evolves slowly enough to support spatially invariant configurations over measurement intervals. The term \(\delta N_{e}\) represents diffusive process that removes structure at small scales.
In situ probes measure electron density directly. Electromagnetic waves that propagate through the ionosphere can be processes to measure pathintegrated electron density. Either way, an effective scan velocity, \(\textbf{v}_{\text {eff}}\), translates intercepted spatial structure to a time varying form that can be manipulated to generate representative diagnostic time series.
The structure model represented by (2) was developed to characterize stochastic structure. The remainder of this paper will address structure associated with the phenomenon know as Equatorial Spread F, which is an historical reference to anomalous ionogram signatures. Electron density depletions, commonly called equatorial plasma bubbles (EPBs), are the principal manifestations of spread F. Highresolution physicsbased plasma simulations, as described in Yokoyama (2017), provide an ideal testbed for quantitative stochastic structure characterization.
Homogeneous stochastic structure can be characterized by a spectral density function (SDF), which is formally the average intensity of a spatial Fourier decomposition of the structure. In an earlier study (Rino et al. 1918) published in situ and remote structure measurements led us to postulate a onedimensional SDF of the form
The variable \(q_{s}\) is spatial frequency measured in radians per meter. Four parameters; turbulent strength, \(C_{s}^{1}\), the spectral indices, \(\eta _{1}\), and \(\eta _{2}\), and the break frequency, \(q_{0}\) define the SDF. An iterative parameter estimation (IPE) procedure, which is described in Rino et al. (1918), was used to generate maximum likelihood estimates of the structure parameters.
The original study initiated an EPB structure realization with a bottomside perturbation representative of a traveling ionospheric disturbance, which spawned five initially separated bubbles. The purpose of this paper is to present new results from single isolated EPB realizations. The new results characterize a canonical EPB structure evolution.
A threedimensional structure model was developed to accommodate arbitrary orientations of the ROI relative to the magnetic field direction. Applications of the model for propagation diagnostics and performance analysis of satellite beacon, navigation, communication, and surveillance applications will be discussed in the discussion section. The next section summarizes the singlebubble results.
Results
Simulations were performed in an asymmetrically sampled dipole magneticfieldaligned coordinate system. A bottomside Flayer perturbation initiates 60min EPB realizations. Uniformly spaced electron density samples were extracted from twodimensional oblique slice planes. Each slice plane spans an altitude range from 300 to 800 km with a crossfield extent of 373.4 km. Altitude was sampled at 300.19m intervals. The crossfield sample interval was 222.4 m. Periodograms were computed for each crossfield scan after trend removal. Trend removal is a necessary initial step that determines the largest resolved spatial scale within the ROI. In the original study 10 periodogram estimates were averaged prior to IPE parameter estimation. Slice planes were processed at 100s intervals.
Rapid structure development was observed after rising EPBs penetrated the Flayer peak density. Once initiated, the structure development expands and progresses upward. An altitude subrange that captures the developed structure is readily identified. Twocomponent SDFs with \(\eta _{1}\sim 1.5\) and \(\eta _{2}\sim 2.5\) were measured within the developed structure region. A single powerlaw with \(\eta \sim 2\) was measured below the developed structure region. In the original study the structure break scale (\(\sigma _{0}=2\pi /q_{0}\)) varied from less than 1 km to more than 6 km with no clear pattern. The breakscale variability was attributed to commingling of the expanding EPB plumes. The same patterns were observed in sliceplanes offset from the equatorial plane.
The singlebubble EPB simulations were processed with 10s time resolution. PSD averaging is a compromise between uncertainty and altitude resolution. Based on the original results, the new results were processed with non overlapping averages of two PSD estimates to increase the altitude resolution. An applied electric field induces a 120 mps westward drift. Each frame was shifted to center the EPB in the frame. The left frame in Fig. 1 shows the new equatorialplane EPB structure at 10min intervals starting at 30 min, which captures the structure onset. The right frame shows zoomed views at 40s intervals, which show the rapid structure onset. The developed left summary frames show enhanced structure on the westward EPB wall at altitudes below the developed structure. This structure is attributed to secondary instability generated by the \(E\times B\) drift of the wall gradient.
At structure onset quasiperiodic kilometerscale voids penetrate the bubble cap followed by fractallike finer scale structure development. The process is often described as successive bifurcation, which implies a doubling of the number of branches with half the parent structure dimension. The concentration of structure in the center of the sliceplane scans is ideally suited for spectral analysis. Each scan contains 1680 samples, from which the linear component connecting the end points has been removed. Discrete Fourier transforms (DFTs) are computed with zero padding to 1728 samples for efficient DFT computation. IPE parameter estimation was applied to averaged PSDs to generate the four structure parameter estimates.
The turbulentstrength parameter is used to identify structured sliceplane regions. The IPE parameters are well defined in the altitude range where CsdB is greater than 190 dB. Figure 2 shows the CsdB progression. Variance is determined by the integral of the SDF, which can be strongly influenced by the low spatial frequency contributions. However, as can be seen from the spectralindex parameters summarized in the upper frame of Fig. 3, the lowspatial frequency index is nearly invariant, whereby CsdB is a quantitative measure of the structure variance.
Figure 4 shows the progression of the breakscale estimates. Comparing, Figs. 2, 3, and 4 to the four frames in Rino et al. (1918). Figure 8 shows the improved definition of the structure evolution parameters, particularly the break scale. The singlebubble realization shows that the breakscale varies inversely with turbulent strength in the high CsdB region. Below the lower boundary edge of the enhanced turbulent strength, the SDF transitions to a single power law.
Figure 5 shows probability distribution functions (PDFs) of the IPE parameters. The SDFs with \(CsdB<190\) dB are distorted by sidelobes of undeveloped EPB structure, as discussed in Rino et al. (1918). The \(\sigma _{0}\) PDF shows two distinct peaks corresponding to the smaller and larger break scale populations. A definitive relation between the break scale and the scale associated with the structure onset remains to be demonstrated. Figure 6 shows the progression of the break scale in a 20 km altitude range centered at 550 km. The blue line is a leastsquares secondorder polynomial fit to the data. The break scale at initiation achieves is smallest value. As the structure spreads and decays the break scale increases.
Additional singlebubble simulations were generated with the ion content doubled and with the electric field reversed after initiation. Increased ion density slows the development but does not change the the canonical pattern. This can be seen qualitatively by comparing Fig. 7 to 1. The parameter summary plots shows the same canonical fish patterns, but delayed in time. Reversing the direction of the dynamo field suppressed the development, but did not change the overall fish patterns.
Figures 4 and 5 illustrate a canonical EPB structure development. Structure onset is initiated when the bubble cap penetrates the Fregion peak. Enhanced turbulent strength spreads to higher altitudes defining the fish structure. The twocomponent inversepowerlaw SDF starts with a shallow index, which transitions to a steeper index at a break scale that progressively decreases. Below the enhanced turbulence region a single powerlaw SDF characterizes the structure. The varying twocomponent powerlaw is confined to the structured region. A direct connection of defining SDF parameters to physical parameters that determine instability and initial growth rates has yet to be established.
Discussion
Although our analysis used structure snapshots, the progression of the structure development is slow enough to support the frozen structure hypothesis. An effective scan velocity allows conversion of the spatial structure to observable time series. A spatial structure model can be constructed by exploiting the fact that intermediatescale structure is field aligned. Consequently, spatial stochastic structure is defined by the crossfield structure that intercepts the equatorial plane. For typical ROI dimensions the variation of the intercepting fieldline direction can be neglected. The invariance of the simulation fieldaligned structure was confirmed by comparing structure summaries in offset oblique slice planes in the original analysis.
Air glow, artificial barium releases, and the aurora support a structure model comprising striations, which are formally fixed radial ionization distributions that follow individual magnetic field lines. Each striation has a characteristic scale size and peak intensity. The defining crossfield structure of a randomly distributed configurations of striations is a twodimensional isotropic SDF, \(\Phi _{N_e}(q_y,q_z)\). It is readily shown that the measured SDF from a horizontal scan is defined by the integration
A commonly used method of constructing threedimensional structure models starts with the isotropic spatial coherence function introduced by Shkarofsky (1968). Incorporating the quadratic form introduced by Singleton (1970), contours of constant correlation are transform to ellipsoidal surfaces. Striations are highly elongated ellipsoids. However, the model only accommodates a single power law. Moreover, the Shkarofosky model is acutely sensitive to largescale and smallscale cutoff parameters that suppress powerlawindexdependent singular behavior at small scales and the decay at large scales, respectively.
As an alternative, we introduced a configurationspace model comprising randomly distributed striations originating in a central slice plane (Rino et al. 2018). If the striations intercept the surface at an oblique angle, a rotation transforms the isotropic structure. For ellipsoidal striations, threedimensional and twodimensional spatial wave numbers are related as follows:
The quadratic form defined by \(q_s\) is the spectraldomain form of Singleton’s transformation. Although striations vary continuously, using a constant magnetic field direction approximates parallel structure within the ROI.
To accommodate a twocomponent inverse power law, we found that the size and intensity of the striations could be constructed to generate a twocomponent power law SDF (Rino et al. 2018). It is well known that powerlaw structure functions are associated with inverse powerlaw SDFs. The same relation connects the striation size and intensity to the powerlaw SDF wavelength and intensity. A realization of a canonical ESF structure is shown in Fig. 8. The blue curve is an average of onedimensional PDFs. The two overlaid curves are the analytic (magenta) and the target (red) SDFs. The paper describes procedures for generating configurations with target SDFs and subsequent algorithmic computation of onedimensional in situ and pathintegrated SDFs. The defining relation is Equation (18) in Rino et al. (2018). The defining twodimensional SDF, the threedimensional SDF, and diagnostic onedimensional SDFs all support twocomponent inverse powerlaw forms. The model reconciles the defining and measured inverse powerlaw parameters. For example, if \(p_n\) is the defining twodimensional index, \(\eta _n=p_n1\).
For pathintegrated diagnostics structure, correlation along the integration path influences the result. The defining relation is
where L is the path length. The approximation assumes negligible decorrelation over the integration path. This assumption is used in scintillation models to relate the weakscatter intensity powerlaw index to an in situ index (Carrano and Rino 2016), which is the same as the twodimensional to onedimensional relation for direct electron density sampling.
To summarize, the IPE analysis procedure for estimating parameters that characterize a twocomponent power law can be applied to any in situ or TEC measurements. Early spectral analyses of C/NOFS satellite data (Rino et al. 2016) led to the application of IPE for the simulation studies. Although intermediatescale structure associated with EPBs is responsible for for scintillation, the GNSS LBand operating frequencies were chosen to eliminate the deleterious effects of scintillation. The diffraction contribution to signal GNSS signal phase is most often negligibly small. Moreover, when significant scintillation is present, the largescale stochastic scintillation can be extracted and used to predict scintillation at other frequencies (Carrano et al. 2014).
Where definitive twocomponent signatures are observed, they can be interpreted against the canonical structure evolution. Where multiple structures are involved, canonical segments might still be identifiable. Nonuniform configuration structure realizations can be used to evaluate more complicated structures. Possibly nonuniform configurations could be incorporated in a model fitting procedure. For highly elongated structure striations are formally truncated at the ROI boundaries. Model results can be extracted for fieldaligned propagation. However, the interpretation of such results is problematic unless curvature is taken into account.
The very large data base of GNSS TEC data is ideally suited for IPE parameter estimation. This was demonstrated with auroral zone GPS data, which had very weak scintillation but significant stochastic TEC variation (Rino et al. 2019). The term stochastic TEC refers to TEC structure unaffected by diffraction. A documented library of Matlab utilities for constructing and manipulating twocomponent inverse powerlaw models is available on request to the corresponding author.
Conclusions
Our results showed that fitting a twocomponent powerlaw SDF to physicsbased realizations of evolving EPB structure reveal a canonical pattern. A structure model was introduced that can be applied to estimate defining crossfield structure parameters. Detailed descriptions and references can be found in Yokoyama (2017) and Rino et al. (1918). The next step is to apply the analysis to real data. Using models to remove geometric and propagation effects has a long history. Weakscatter scintillation theory was used to derive probability of occurrence models of ionospheric structure (Vasylyev et al. 2022). More refined procedures have evolved which relate more directly to underlying physical processes Carrano et al. (2019), Carrano et al. (2018) Costa et al. (2011) and Bhattacharyya et al. (2016).
As noted in the discussion, we envision the IPE procedures as a complement to TEC structure analysis, which goes beyond EPBinitiated structure driven by processes that have their own canonical structure patterns. An example of IPE applied to auroral zone structure was presented in the discussion. Turbulence theory will most likely be needed to make direct connections to structure characteristics.
Abbreviations
 EPB:

Equatorial plasma bubble
 SDF:

Spectral density function
 PSD:

Power spectral density
References
Bhattacharyya A, Kakad B, Gurram P, Sripathi S, Sunda S (2016) Development of intermediate scale structure at different altitudes within an equatorial plasma bubble: implications for Lband scintillations. J Geophys Res 122:1015
Carrano CS, Groves KM, Delay SH, Doherty PH (2014) An inverse diffraction technique for scaling measurements of ionospheric scintillations on the gps l1, l2, and l5 carriers to other frequencies. Proceedings of the 2014 Institute of Navigation ION ITM meeting, San Diego, California, January 27029, 2014
Carrano CS, Rino CL (2016) A theory of scintillation for twocomponent power law irregularity spectra: overview and numerical results. Radio Sci 51:789–813. https://doi.org/10.1002/2015RS005903
Carrano CS, Rino CL, Groves KM, Doherty PH (2018) A technique for inferring zonal irregularity drift from singlestation GNSS measurements of intensity (s4) and phase \(( \phi )\) scintillations. Radio Sci. https://doi.org/10.1002/2015RS005864
Carrano CS, Rino CL, Groves KM (2019) On the relationship between the rate of change of total electron content index (roti), irregularity strength (ckl), and the scintillation index (s4). J Geophys Res Space Phys 124:2099–2112. https://doi.org/10.1029/2018JA02635
Costa E, de Paula ER, de Rezende LFC, Groves KM, Roddy PA (2011) Equatorial scintillation predictions from c/nofs planar Langmuir probe electron density fluctuation data. URSI Gen Assembly Sci Sympos 2011:1–4. https://doi.org/10.1109/URSIGASS.2011.6050901
Rino C, Yokoyama T, Carrano C (1918) Dynamic spectral characteristics of equatorial plasma bubbles. Progr Earth Planet Sci. https://doi.org/10.1186/s40645018082430
Rino CL, Groves KM, Carrano CS, Roddy PA (2016) A characterization of intermediatescale spread f structure from four years of highresolution C/NOFS satellite data. Radio Science 51:19. https://doi.org/10.1002/2015RS005841
Rino C, Carrano C, Groves K, Yokoyama T (2018) A configuration space model for intermediate scale ionospheric structure. Radio Sci 53:1472. https://doi.org/10.1029/2018RS00667
Rino C, Morton Y, Breitsch B, Carrano C (2019) Stochastic tec structure characterzation. J Geophys Res Space Phys 124. https://doi.org/10.1029/2019JA026958
Shkarofsky IP (1968) Generalized turbulence spacecorrelation and wavenumber spectrumfunction pairs. Can J Phys 46:2133–2153
Singleton DG (1970) Saturation and focusing effects in radiostar and satellite scintillations. J Atmos Terr Phys 32:187–208
Vasylyev D, Béniguel Y, Volker W, Kriegel1 M, Berdermann J (2022) Modeling ionospheric scintillation. J Space Weather Space Clim 12 (22). https://doi.org/10.1051/swsc/2022016
Yokoyama T (2017) A review on the numerical simulation of equatorial plasma bubbles toward scintillation evaluation and forecasting. Prog Earth Planet Sci 4:37. https://doi.org/10.1186/s4064501701536
Acknowledgements
The computation was performed on the FX100 supercomputer system at the Information 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 Institute for SpaceEarth Environmental Research (ISEE), Nagoya University, Japan. Support for CR and CC was provided by internal research funding.
Author information
Authors and Affiliations
Contributions
All of the simulations analyzed in this paper were performed by TY and generously reformatted and made available to CR who performed the analysis. The new results were initiated following analysis of the original simulations 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.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Rino, C., Yokoyama, T. & Carrano, C. A threedimensional stochastic structure model derived from highresolution isolated equatorial plasma bubble simulations. Earth Planets Space 75, 64 (2023). https://doi.org/10.1186/s40623023018236
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s40623023018236
Keywords
 Equatorial Spread F
 Powerlaw ionospheric structure
 Convective plasma instability
 Structure characterization