Structural characterization of the equatorial F region plasma irregularities in the multifractal context

In the emerging ionosphere-space-weather paradigm, investigating dynamical properties of ionospheric plasma irregularities using advanced computational non-linear algorithms is providing new insights into their turbulent-like nature, for instance, the evidence of energy distribution via multiplicative cascade. In this study, we present multifractal analysis of the equatorial F region in situ data obtained from two different experiments performed at Alcântara (2.4°S; 44.4°W), Brazil to explore their scaling structures. First experiment observed several large-medium scale plasma bubbles whereas second experiment 5 observed vertical uplift of the base of F region. Multifractal detrended fluctuation analysis and p-model fit is used to analyze the plasma density fluctuation time series. Result shows presence of multifractality with degree of multifractality 0.53−0.93 with 0.3≤ p≤ 0.4 cascading probability for first experiment. Another experimental data also exhibits multifractality with degree of multifractality 0.19− 0.27 with 0.42≤ p≤ 0.44 cascading probability in the ionospheric plasma irregularities. Our results confirm the nonhomogeneous nature of plasma irregularities and characterize the underlying nonhomogeneous multiplicative 10 cascade hypothesis in the ionospheric medium. Differences in terms of scaling and complexity in data belonging to different types of phenomena are also addressed.

data obtained from these two different in situ experiments. To corroborate our results, the singularity spectrum obtained from the MFDFA is fitted with a p-model (Meneveau and Sreenivasan, 1987) based on the generalized two-scale Cantor set. Details on the experiments are given briefly in section 2. Methods are described in section 3. Results of the analyses are discussed in section 4 followed by concluding remarks in section 5.
2 in situ experiments 5 The equatorial launching station of Brazil is located at Alcântara (2.24 • S, 44.4 • W, dip latitude 5.5 • S). SONDA III rocket was launched at 21:17 LT, on December 18, 1995 under favourable conditions for formation of plasma bubble. During the ∼ 11 min flight, plane of rocket trajectory was almost orthogonal to the geomagnetic field lines, spanned ∼ 589 km distance horizontally with an apogee at altitude ∼ 557 km. Rocket-born electric field double probe (EFP) measured electric field fluctuations related with ionospheric plasma irregularities. In the upleg profile (ascent of the rocket), the F region base is clearly observed around 10 300 km, but without any large scale depletion or bubble. On the other hand, several plasma bubbles of large-medium scale were observed in the downleg profile (descent of the rocket), around the base of F region and also topside of it, but without any sharp indication of the F region base from altitude above 240 km. Rocket traversed through regions of different altitudes separated by a few hundreds of kilometers during upleg and downleg so this might elucidate the large differences observed in ascent and Some of the key results from the analysis indicate -(1) initiation of cascade process, owing to Rayleigh-Taylor instability mechanism, near the base of F region that resulted in the development of the plasma bubbles or large scale irregularities, and (2) subsequently, advecting energy to higher altitudes, smaller scale irregularities were observed, owing to Cross-Field instability 20 mechanism (Muralikrishna et al., 2003;Muralikrishna P. and Abdu M. A., 2006;Muralikrishna P. and Vieira L. P., 2007).
From the same rocket launching station, Alcântara, a two-stage VS-30 Orion sounding rocket was launched at 19:00 LT, on December 8, 2012, under favorable conditions for strong spread-F. During the ∼ 11 min flight, the rocket trajectory was in the north-northeast direction towards magnetic equator, ranging ∼ 384 km horizontally with an apogee at ∼ 428 km.
conical Langmuir probe (CLP) onboard the rocket measured electron density fluctuations associated with ionospheric plasma 25 irregularities. In this experiment, the F region base was clearly observed in the downleg profile around 300 km, with some small scale fluctuations in the F region. Ground equipment, digisonde, near launching station, showed fast uplift of the F region base before sunset and its continuation for more than two hours, indicating possibility of pre-reversal enhancement of the vertical plasma drift (Savio et al., 2016). Further explanation of in situ experiment and data analysis is found in Savio et al.

Multifractal detrended fluctuation analysis
Implementation of the MFDFA consists of first, compute the profile by integrating the time series and then divide it into nonoverlapping and equidistant N s segments of length s, referred to as scales. These segments are then detrended using linear least squares. The variance is calculated over the all segments.
Then followed by averaging the root mean square, the q th order fluctuation function is computed.
When q = 0, logarithmic averaging should be used to calculate fluctuation function,

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Applying linear fit to the fluctuation function profile on log-log plot yields the generalized Hurst exponent, h(q), for each moment q as F q (s) ∝ s h(q) . The computed generalized Hurst exponent h(q) can be related to the classical multifractal scaling (or mass) exponent as τ (q) by τ (q) = qh(q) − 1. Singularity (or multifractal) spectrum is calculated using h(q) where α represents singularity strength and f (α) represents set of multifractal dimensions: Detailed implementation can be found in Kantelhardt et al. (2002).

p-model
The p-model is proposed by Meneveau and Sreenivasan (1987) to model energy cascading process in the inertial range of 20 fully developed turbulence for the dissipation field. The p-model starts with a coherent structure with an assumed specific energy flux per unit length which then undergoes a binary fragmentation at each cascading step, distributing the energy flux with probabilities p1&p2 among the fragments l1&l2. Based on the generalized two-scale Cantor set, the p-model consider equal scales (l1 = l2) and unequal weights (p1 = p2 and p1 + p2 ≤ 1). When p1 + p2 ≤ 1, loss in p parameter given by dp = model claims to display all multifractal properties of one-dimensional section of dissipation field for fully developed turbulence.
Multifractality ceases to exist for p = 0.5. Analytical formulation for the generalized two scale Cantor set is given by is useful to determine the generalized multifractal dimensions which represents the singularity spectrum. (Halsey et al., 1986).

Results and Interpretation
Six time series of in situ observations of electric field fluctuations from the F region are selected from the first experiment These time series are subjected to multifractal analysis. Primarily, the profile is obtained by differencing the time series i.e.
y = x(i + 1) − x(i), using the criterion based on the power exponent obtained in the DFA method, prescribed by (Ihlen E. (2012), Table 2) for biomedical time series to yield best results from the MFDFA method. We found the criterion to hold for the degree of multifractality and complexity of data. Degree of multifractality, ∆α, is the difference between maximum and minimum dimension, represent the deviation from average fractal structure, and directly relate to the parameters corresponding to multiplicative cascade process. Larger (smaller) value of ∆α infers stronger (weaker) multifractality in data.
Alongside ∆α, measure of asymmetry, A, is given by where the singularity strength, α 0 , corresponds to the maximum value of the singularity spectrum i.e., for f (α 0 ) = 1. When A = 1, singularity spectrum is symmetric in a sense that the time series is influenced with both the coarser as well as fine structures. When A > 1, the spectrum is left-skewed which implies that the time series is more influenced with the coarser structures (large fluctuations). When A < 1, the spectrum is right-skewed which implies that the time series is more influenced   Table 1. Lastly, the singularity spectrum is fitted with a p-model (shown with a continuous line), where the scales are equal i.e., l1 = l2 = 0.5 20 and the weights, p1 and p2, are varied such that p1 + p2 ≤ 1. Nevertheless, loss in p parameter had to be accounted to obtain an optimal fit. The loss factor, dp, signifies non-conservative energy distribution i.e., a dissipative energy cascading process in the inertial range. We have obtained a dissipative factor of 0.090, with p1 = 0.315. The p-model fit parameters are listed in Table   1.
Similar to Figure 1, Figure 3 shows detailed multifractal analysis of a time series from second experiment, corresponding to  Table 2.
It is seen from above discussion that the singularity spectrum is sufficient to assess the multifractal nature, henceforth we show time series and the corresponding singularity spectrum for the remaining chosen heights. Figure 2 shows time series  The optimal p-model fit obtained with parameters p1 = 0.399 and dp = 0.0355. Wawrzaszek A. and Macek W., 2010;Wawrzaszek et al., 2019).
In this work, we investigate the in situ F region electron density and electric field measurements obtained from past two 5 experiments carried near equatorial sites in Brazil using MFDFA to understand the complexity of the data and to identify the signature of multiplicative energy cascade in the irregularities present there. We selected altogether nine time series at altitudes supposedly in or near the irregularities. In all the time series, we obtained 1.5 > h(q) > 0.9, which indicate a long range correlation with persistent temporal fluctuations. In addition, we note that the h(q) profile monotonically decreases with respect to q, and that τ (q) shows deviation from linearity indicating the presence of multifractality in all time series. Measures 10 of the singularity spectra, A have shown presence of structures (both smaller or larger) in the fluctuations; and ∆α have shown weaker to stronger multifractality. The singularity spectra is fitted with p-model and we found weight parameter p1 to be different than 0.5 which confirms multifractality present in data. Accounting non-zero dissipation factor suggests that energy distribution across the eddies to be non-uniform.
We now focus attention on the second experiment during which base of F region was moving upwards, thereby indicating 15 possibility of PRE. We expected to find some different characteristics of this data owing to the fact PRE is the seeding mechanism for Rayleigh-Taylor instability. Of the six time series, three exhibited monofractal nature and remaining three showed weaker multifractality (considered in this study). ∆α and skewness are found to be smaller compared to the first experiment.
Singularity spectra for two time series are found to be left skewed with right truncated which shows time series has large structures but the spectrum is insensitive to smaller local fluctuations. But at altitude of 400.24 km, the spectrum is found 20 to be almost symmetrical, so may be this region has both large and small structures. Results for mean height of 348.99 km are different than other two heights and evident for some different kind of physical mechanism which can be described by multiplicative cascade process. Though time series are characterized by weaker multifractality, these data has fractal behavior with long range correlation. However, we argue that more detailed study is required to reach any definite conclusion on the turbulent-like mechanism driving the ionospheric irregular structures.

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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 rocket intercepted a plasma bubble. Muralikrishna et al. (2003) reported presence of predominant sharp peaks in the power spectra over a wide range of heights, and they attribute these to a developing plasma bubble that subsequently dissipated energy, reaching an equilibrium which is evidenced by the absence of peaks. Our multifractal analysis has captured this sequence of events. Figure 5 show the variation of mean density and ∆α with mean heights for the 30 selected six time series on a 3-dimensional plane. Presence of a plasma bubble characterized by large scale irregularities, that 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. These inverse variation is in line with the turbulent like 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. Two dimensional plots showing 35