We use the magnetic field measurements from four spacecraft of the Cluster-II
mission (three events from 2005 to 2015) for the analysis of turbulent
processes in the Earth's magnetotail. For this study we conduct the spectral,
wavelet and statistical analysis. In the framework of statistical
examination, we determine the kurtosis for selected events and conduct
extended self-similarity evaluation (analysis of distribution function
moments of magnetic field fluctuations on different scales). We compare the
high-order structure function of magnetic fluctuations during dipolarization
with the isotropic Kolmogorov model and three-dimensional log-Poisson model
with She–Leveque parameters. We obtain power-law scaling of the generalized
diffusion coefficient (the power index that varies within the range of
0.2–0.7). The obtained results show the presence of super-diffusion
processes. We find the significant difference of the spectral indices for the
intervals before and during the dipolarization. Before dipolarization the
spectral index lies in the range from

The wavelet analysis shows the presence of both direct and inverse cascade processes, which indicates the possibility of self-organization processes, as well as the presence of Pc pulsations.

The physical process responsible for the onset of magnetospheric substorms
remains an unsolved mystery in spite of more than 5 decades of intense
research efforts after the discovery of this episodic disturbance in the
ionosphere and the magnetosphere. Many potential processes have been proposed
by e.g.

A four-satellite ESA mission, named Cluster-II, and a five-satellite NASA
mission, named Time History of Events and Macroscale Interactions during
Substorms (THEMIS), were launched to identify the location where the substorm
disturbances are initiated in the magnetotail

The strategy adopted by these missions is to have some satellites situated at
different downtail distances to identify the originating location of substorm
disturbance. This strategy turns out not to be foolproof as magnetic
reconnection was later recognized to be localized in the local time extent
and not a large-scale process as originally envisioned

On the other hand, both scenarios have common consequences such as impulsive
particle acceleration, dipolarization, and formation of a current wedge

Investigation of the magnetotail is significantly complicated by the presence
of turbulence due to instability resulting in a “catastrophic” alteration
of the flow and magnetic field structure

An analytical or numerical solution of the turbulent plasma dynamics (in
three-dimensional geometry) and determination of turbulence features at large
timescales are not currently possible. Therefore, statistical properties of
turbulence associated with large-scale invariance are determined
experimentally along with estimation of spectral indices in the assumption of
power laws for plasma parameters. This allows one to get an idea of the
physical properties of plasma turbulence and a description of the transport
processes in the turbulent regions in qualitative and quantitative terms

In this work, the spectral and statistical approach was carried out to
examine the features of the magnetic field dipolarization in the Earth's
magnetotail for three events (12 September 2015, 15 October 2005,
1 October 2005). The methods and approaches used in the work are described in
detail and tested in the works by

The data of the magnetic field for this analysis were obtained by the
spacecraft (SC) of the Cluster-II mission in the near-Earth tail for three
events (two events in 2005 and one event in 2015) during the dipolarization
of the magnetic field (see Fig.

The locations of the satellites.

The event of 2015 satisfies the most the conditions of dipolarization for the
CD model by

In the dipolarization region the fluctuations of the magnetic field greatly
differ from the region before dipolarization: in particular for the event on
1 October 2005 the magnetic field variations normalized to the current mean
value are

Features of the dipolarization fronts (DFs).

Estimated values of plasma characteristics in the dipolarization region.

Since the region of dipolarization is traced by four space vehicles, we were
able to estimate the speed and direction of the dipolarization front (DF)
motion, the thickness of the front (Table

Moreover, according to

The calculated values of the coefficients are also given in Table

Within the spectral analysis, the spectral power density (PSD) was built from
the frequency

The results of PSD analysis.

To find the break points and the slope of the spectrum, we used a piecewise
linear approximation of

The results of spectral analysis.

During the time before the dipolarization (interval 1), for all events and
spacecrafts, there is no sharp change in the PSD power law in the inertial
interval (the exponent varies in the range from

Within the framework of the wavelet analysis for a series of measurements

The continuous wavelet transform of the discrete signal

The results of the continuous wavelet transform of the magnetic field module
in the dipolarization region are shown in Figs. 4 to 6. The time range was
chosen to include the dipolarization interval (interval 2) with some margin
(

The results of wavelet analysis for event 12 September 2015. The cone of influence is shown by the shaded region.

The results of wavelet analysis for event 15 October 2005. The cone of influence is shown by the shaded region.

The results of wavelet analysis for event 1 October 2005. The cone of influence is shown by the shaded region.

In Fig.

Figure

Figure

Thus, during the dipolarization, the magnetometers of all spacecraft recorded
powerful signals with periods of 50, 100, 125, 166, and 200

In the presence of intermittency in magnetic field fluctuations, the energy cascade is characterized by non-homogeneous non-linear transfer of energy among smaller and smaller structures, with the result of concentrating the energy on limited regions of space.

This effect becomes more and more intense at smaller and smaller scales. More
properly, intermittency corresponds to scale-dependent, non-Gaussian,
heavy-tailed probability distribution functions (PDFs) of the field
fluctuations

In order to determine the presence of intermittence, an analysis of the value
of the excess for all the SCs of the considered events has been performed,
and the Hölder parameter

The results of kurtosis.

The value of the kurtosis was determined by the moments of the second and
fourth orders from the formula by

For SC C3 and C4, changes in the value of kurtosis are very similar. The
largest jump is observed for C1, 12 September 2015. A sharp drop in the kurtosis is
observed on the scales to the ion-cyclotron frequency (Table

The “gap” of values for interval 2 for very small

Thus, for a region of dipolarization at small timescales, we have a distribution with a sharper vertex and broad wings (the excess value is greater than 3) than for a normal distribution.

The example of Hölder exponents.

The presence of intermittency is indicated by the analysis of the first-order
structure function (Fig.

Also, for the interval prior to dipolarization, the variations “caused” by the presence of spacecraft spin effects in the data are clearly visible.

To compare the type of turbulent processes with the available models of turbulent processes, an analysis of the high-order structural function was done, allowing one to characterize the properties of heterogeneity at small timescales.

In this case, the structural function is determined by the ratio:

The existence of the criterion of generalized self-similarity for an
arbitrary pair of structural functions

Dependence of the order of the structure function for different timescales during dipolarization (event 15 October 2005).

The results of ESS analysis (during DP). Ratio of the power of the

ESS-analysis parameters and diffusion coefficients.

The power law of the type

The results of scaling the moments of the probability density function for
different orders of

The resulting values of

As a result of the analysis, it can be concluded that the relative variations of the magnetic field during the dipolarization exceed the value before dipolarization by more than 5 times. The distribution functions of magnetic field fluctuations during the disruption of the current layer indicate the non-Gaussian statistics of processes, as well as the excess of large-scale perturbations generated by the source.

Comparing the structure functions of the magnetic field fluctuations during
dipolarization with the Kolmogorov model, it is impossible to describe
turbulent processes on small timescales using a homogeneous model. Using the
coefficients of intermittency and singularity of turbulent processes found in
the ESS analysis, the power law of the generalized diffusion coefficient on
the scale was obtained (the power index varies within the range from

One of the important results is the significant difference of the spectral
indices for the intervals before and during the dipolarization. Before
dipolarization the spectral index lies in the range from

Also, within the framework of the research the following results were obtained:

the higher the PSD value, the greater the value of the height of the excess;

the log-Poisson model of turbulent processes with She–Leveque parameters corresponds to variations in the value of

the spectral indices correlate with the values of the diffusion coefficient.

Thus, during dipolarization the large-scale and multi-fractal disturbances of the magnetic field are observed and the presence of inverse cascade processes also indicates the possibility of self-organization processes.

In this paper we only used open-access data. The Cluster data were downloaded from the Cluster
Science Archive version 2.0 at

LVK formulated the goal and tasks of the investigations, carried out the statistical and spectral analysis of features of the magnetic field fluctuations in the Earth's magnetosphere tail, and also carried out the analysis of obtained results and calculated characteristics of the turbulent processes. BAP selected the considered work events and conducted the spectral and wavelet analysis of the data of the Cluster-II mission. ATYL carried out analysis of the obtained results. EAK and EEG selected and analyzed the dipolarization events used for the subsequent analysis of turbulence properties. ASP took part in the wavelet analysis of the data. All authors took part in the discussion about obtained results and preparation the article for publication.

The authors declare that they have no conflict of interest.

The work was conducted in the framework of a complex programme of the National Academy of Science of Ukraine in Plasma Physics, with support of the the education programme of Ministry of Education and Science of Ukraine no. 2201250 “Education, Training of students, PhD students, scientific and pedagogical staff abroad”, grant Az. 90 312 from the Volkswagen Foundation (“VW-Stiftung”) and the International Institution of Space Research (ISSI-BJ).

We also thank the Principal Investigators and teams of FGM and CIS instruments of the Cluster mission. Edited by: Elias Roussos Reviewed by: Gaetano Zimbardo and one anonymous referee