In the emerging ionosphere–space–weather paradigm, investigating the dynamical properties of ionospheric plasma irregularities using advanced computational nonlinear algorithms provide new insights into their turbulent-seeming nature, for instance, the evidence of energy distribution via a multiplicative cascade. In this study, we present a multifractal analysis of the equatorial F region in situ data obtained from two different experiments performed at Alcântara (2.4

Present ionospheric research is transiting towards ionospheric space weather that goes beyond the ground- and space-based communication interruptions to influence decision-making communities on social, economical, and physical infrastructural policies. The enhancements in ionospheric plasma irregularities driven by space weather conditions demand an accurate characterization of the dynamical properties of the electron density and its complex nonlinear variation

Various rocket experiments and numerical simulations have been performed and contributed to our understanding of the generation and development of ionospheric irregularities.

The spectral analysis, though widely used, falls short in characterizing nonstationary data as stationarity is assumed in the data, which is equivalent to presuming homogeneous turbulence; hence, a more robust method is necessary to analyze nonstationary data

Recent advances in the computational algorithms based on fractal formalism, supplemented with mathematical modeling derived from probabilistic measures, have conclusively substantiated the occurrence of the energy cascading process in turbulent sites in the solar and interplanetary environment as well as in the laboratory using Kolmogorov's formalism as the basis

Various different approaches had been explored to understand nonlinear characteristics and intermittency in ionospheric irregularities, like structure function analysis

Structure function analysis performed on ionospheric high-latitude in situ data have revealed the intermittent nature of ionospheric irregularities owing to the large deviations from the Kolmogorov's K41 universal power-law index proposed for neutral fluid turbulence

In all the abovementioned studies, the main feature which gets highlighted is that the power spectra point to large deviations from the homogeneous turbulence described by the Kolmogorov spectrum (

A detrended fluctuation analysis (DFA;

The MFDFA has wide applications in many branches of science, such as medicine

In this work, we explore the low-latitude equatorial F region in situ data obtained from two different experiments and performed from the same rocket launching station. In the first experiment, done on 18 December 1995, the rocket traversed through various medium- to large-scale plasma irregularities during its descent, which were associated with the generalized Rayleigh–Taylor instability

In the equatorial ionosphere, the evening PRE is considered as an important seeding mechanism for the post-sunset F region irregularities, as quick and acute uplift of the electric field escalates the rate of growth of the generalized Rayleigh–Taylor instability

The equatorial launching station of Brazil is located at Alcântara (2.24

Some of the key results from the aforementioned

From the same rocket launching station, Alcântara, a two-stage VS-30 Orion sounding rocket was launched at 19:00 LT, on 8 December 2012, under favorable conditions for strong spread F. During the

Multifractal detrended fluctuation analysis

To implement the MFDFA, a plasma density time series

Now we have a total of

Applying a linear fit to the fluctuation function profile on the log-log plot yields the generalized Hurst exponent,

The

Based on the generalized two-scale Cantor set, the

Comprehensive MFDFA for the first experiment: panel

Multifractal analysis measures for the first experiment: the time series at mean heights are listed in the first column, the second column shows the degree of multifractality (

Six time series of in situ observations of electric field fluctuations from the F region are selected from the first experiment performed on 18 December 1995, corresponding to the mean heights of 264.58, 270.22, 292.37, 324.00, 358.56, and 429.65 km in the downleg. Similarly, from the second experiment performed on 12 December 2012, we selected three time series of electron density fluctuations from the F region, corresponding to the mean heights of 339.94, 348.99, and 400.24 km in the downleg.
These time series are subjected to the multifractal analysis. Primarily, the profile is obtained by differencing the time series, i.e.,

Multifractal analysis measures for the second experiment: For the time series at mean heights listed in the first column, the second column shows degree of multifractality (

Comprehensive MFDFA for the second experiment: panel

In the MFDFA, fluctuation function

The multifractal spectrum illustrates how segments with small and large fluctuations deviate from the average fractal structure. The shape and width of the multifractal spectrum are also important measures to quantify the nature of multifractality present in the data.
For

MFDFA for the first experiment: the time series and its corresponding multifractal spectrum with the

A width of the spectrum can be quantified by

The multifractal spectrum reflects the characteristics of the

MFDFA for the second experiment: panels

Figure

Similar to Fig.

It is seen from the above description that the multifractal spectrum is sufficient to assess the multifractal nature; henceforth we show the time series and the corresponding multifractal spectrum for the remaining chosen heights.

Variation of the mean density and the degree of multifractality with the mean height for the six selected time series from the first experiment in a 3-D plane. These variations are shown in a 2-D plane of the mean density (main image) and the degree of multifractality (inset).

Figure

For the time series corresponding to the mean height of 264.58 km, the multifractal spectrum is slightly right-skewed, which can be inferred from measure

For the time series corresponding to the mean height of 270.22 km, the multifractal spectrum is slightly left-skewed, which can be inferred from measure

For the time series corresponding to the mean height of 292.37 km, the multifractal spectrum is left-skewed, reflected in measure

For the time series corresponding to the mean height of 358.56 km, the multifractal spectrum is right-skewed, reflected in measure

For the time series corresponding to the mean height of 429.65 km, the multifractal spectrum is right-skewed, also reflected in measure

Figure

For the time series corresponding to the mean height of 348.99 km, the multifractal spectrum is left-skewed, reflected in measure

For the time series corresponding to the mean height of 400.24 km, the multifractal spectrum is almost symmetrical. This is reflected in measure

Figure

In this work, we investigate the in situ F region electric field and electron density measurements obtained from the two experiments carried out near the equatorial sites in Brazil using the MFDFA to understand the complexity in the data and to identify the signature of multiplicative energy cascades in irregularities.

In all the time series, we obtained

In the second experiment, we considered a total of six time series, out of which three time series exhibited a monofractal nature, and the remaining three showed weaker multifractality and are presented here.

Finally, we intend to test the potential of this algorithm in deciphering the morphology of the cascading phenomena. For this, we choose the first experiment where the rocket intercepted a plasma bubble.

The presence of a plasma bubble characterized by large-scale irregularities, which in turn is reflected in the low density, is observed around a mean height of 292.37 km. Contrarily, stronger multifractality is observed at this height. This inverse variation is in agreement with the turbulent-seeming multiplicative cascade process. On the other hand, as the rocket traversed higher altitudes, the mean density increased while the multifractality became weaker. This suggests that the cascading process resulted in smaller-scale irregularities by dissipating energy.

We conclude at this point where we have presented the schematic hypothesis based on the multifractal analysis of plasma irregularities in the ionospheric F region.

The data used in this analysis are available at the National
Institute for Space Research library archive through, available at:

All authors have contributed to the analysis and development of the manuscript.

The authors declare that they have no conflict of interest.

This article is part of the special issue “7th Brazilian meeting on space geophysics and aeronomy”. It is a result of the Brazilian meeting on Space Geophysics and Aeronomy, Santa Maria/RS, Brazil, 5–9 November 2018.

The authors are thankful to the Institute of Aeronautics and Space (IAE/DCTA) and
Alcântara Launch Center (CLA) for providing sounding rockets and for the launch
operation, respectively. Also, we are grateful to Abraham L. Chian for his constructive
comments and review on PSD-based ionospheric studies, which has been adopted in this
article, and to another anonymous reviewer for his or her comments that have increased
the clarity of the article. Neelakshi Joshi thanks CAPES for supporting her PhD and is grateful to Anna Wawrzaszek for the discussion and guidance obtained on the multifractal analysis and

This research was mostly supported through the CAPES scholarship for the PhD program in applied computing at INPE (grant no. 0028-15/2015). We hereby certify that this grant was active from March 2015 to February 2019.

This paper was edited by Igo Paulino and reviewed by Abraham C. L. Chian and one anonymous referee.