3.1 Spectral analysis

Within the spectral analysis, the spectral power density (PSD) was built from the frequency f, and the power-law dependence PSD(f)∼f^a was determined. To determine the PSD of the signal for a series of N measurements X_n, a discrete Fourier transform (Daly and Paschmann, 2000) was used:

where $n=\mathrm{0},\mathrm{1}\mathrm{\dots }N-\mathrm{1}$, $j=\mathrm{0},\mathrm{1}\mathrm{\dots }N/\mathrm{2}$.

Table 3The results of PSD analysis.

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To find the break points and the slope of the spectrum, we used a piecewise linear approximation of logPSD from log(f) in the frequency range 0.005–∼1.0 Hz for the pre-dipolarization interval and 0.01–3.0 (events 1 October 2005 and 15 October 2005) and 0.01–1.0 (event 12 September 2015) Hz for dipolarization. The limitation of frequencies at a high level is due to the presence of instrumental noise, and at the low-frequency range due to the amount of data sampling and the edge effect of the smoothing procedure. The PSD results for the absolute value of the magnetic field are shown in Fig. 3 and Table 3.

Figure 3The results of spectral analysis.

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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 −2.08 to −1.68). During dipolarization (interval 2), the situation is significantly different. There is an increase in the “steepness” of PSDs for higher frequencies than the kink frequency, which means more efficient energy transfer from large to smaller scales. For practically all spectra of interval 2, the kink frequency is less than or close to the average value of the proton gyrofrequency (Table 2). The kink frequency determines the characteristic frequency of the type change (i.e. the energy transfer rate) of the turbulent cascade in the inertial range. In particular, for events 12 September 2015 and 15 October 2005 the break corresponds to about half of the proton frequency ω_c∕2. The fact that the break is observed at frequencies smaller than the proton gyrofrequency may indicate a significant effect of heavy ions at the distances considered (according to the measurements of the density by the CIS instrument (Rème et al., 2001) for event 1 October 2005, in the region of the magnetic field dipolarization, the percentage of oxygen ions in relation to protons ($\langle n\left({\mathrm{O}}^{+}\right)\rangle /\langle n\left({\mathrm{H}}^{+}\right)\rangle$) is 21.1±10.0 % (SC C3) and 9.3±1.5 % (SC C4), and the percentage of helium in relation to protons ($\langle n\left({\mathrm{He}}^{+}\right)\rangle /\langle n\left({\mathrm{H}}^{+}\right)\rangle$) $\sim \mathrm{2.4}\pm \mathrm{0.3}\mathit{\%}$ (SC C3) and $\sim \mathrm{4.8}\pm \mathrm{0.7}\mathit{\%}$ (SC C4); for event 15 October 2005 – $\langle n\left({\mathrm{O}}^{+}\right)\rangle /\langle n\left({\mathrm{H}}^{+}\right)\rangle \sim \mathrm{11.1}\pm \mathrm{1.0}\mathit{\%}$ (SC C4) and $\langle n\left({\mathrm{He}}^{+}\right)\rangle /\langle n\left({\mathrm{H}}^{+}\right)\rangle \sim \mathrm{3.4}\pm \mathrm{0.5}\mathit{\%}$ (SC C4); for the 12 September 2015 event – $\langle n\left({\mathrm{O}}^{+}\right)\rangle /\langle n\left({\mathrm{H}}^{+}\right)\rangle \sim \mathrm{18.9}\pm \mathrm{7.3}\mathit{\%}$ (SC C4) and $\langle n\left({\mathrm{He}}^{+}\right)\rangle /\langle n\left({\mathrm{H}}^{+}\right)\rangle \sim \mathrm{15.8}\pm \mathrm{5.4}\mathit{\%}$ (SC C4)). At the same time, the exponent lies in the range from −2.2 to −1.53 on large timescales of $\mathrm{0.01}-{\mathit{\omega }}_{\mathrm{c}}/\mathrm{2}$, and at smaller timescales ω_c∕2–3 Hz, the value lies in the range from −2.89 to −2.35. The greatest difference at different timescales is observed for the 2015 event.

3.2 Wavelet analysis

Within the framework of the wavelet analysis for a series of measurements X_n ($n=\mathrm{0},\mathrm{1}\mathrm{\dots }N-\mathrm{1}$) with time step δt, a Morlet wavelet (Torrence and Compo, 1998) was used:

where ω₀ is the dimensionless frequency, and η the dimensionless time.

The continuous wavelet transform of the discrete signal X_n is defined as the convolution of the mother wavelet whose argument is scaled and transmitted with a signal (Farge, 1992; Grinsted et al., 2004; Jevrejeva et al., 2003):

where (^*) is the complex conjugate, $\left|W_n\right(s){|}^{\mathrm{2}}$ is the wavelet power spectrum, and s is the wavelet scale. Index 0 in Ψ₀ denotes that the function is normalized.

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 (±5 min) to exclude the influence of the edge effects of the wavelet transform on the explored intervals. The upper limit of the wavelet transform is limited by the Nyquist frequency. The sampling frequency of the measurements makes it possible to analyse the presence of high-frequency fluctuations in addition to the low-frequency components.

Figure 4The results of wavelet analysis for event 12 September 2015. The cone of influence is shown by the shaded region.

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Figure 5The results of wavelet analysis for event 15 October 2005. The cone of influence is shown by the shaded region.

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Figure 6The results of wavelet analysis for event 1 October 2005. The cone of influence is shown by the shaded region.

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In Fig. 4 the wavelet analysis of the magnetic field magnitude for the event on 12 September 2015 is presented. In this case C3 and C4 were located ahead of C1 and C2, with C2 being the furthest in the magnetotail. Inverse and direct cascades are present in wavelet analysis at multiple times: 13:47:30 (dipolarization onset) and 13:53:00, both spanning 0.02–0.2 Hz in the frequency domain. This signal broadens for C1 and C2 wavelets and breaks up into smaller time-frequency forms: e.g. the signal on 13:53:00 UT becomes stronger in time and frequency domains of 1 min and 0.01 Hz correspondingly. Wavelet transform for C1 is characterized by a prevalence of intensity enhancements in the wide frequency range at an earlier stage of the turbulent phase of dipolarization at 13:47:30–13:50:30 as compared to transforms for C2, C3, and C4. Also for C1, it is interesting to note the fact of coexistence of the inverse and direct cascades simultaneously starting at 13:47:30, and wherein the first one lasts for 2 min with frequency decrease from 0.015 to 0.008 Hz and the more intense second one lasts for 2.5 min with slight frequency increase from 0.015 to 0.02 Hz.

Figure 5 presents the wavelet transform for event 15 October 2005. Taking into account the cone of influence (COI), there are no strong enhancements presented for C3 and C4. For both these satellites, only high-frequency short signals are present. Transformations for C1 and C2 have a much richer frequency content. The component 0.008–0.01 Hz is present on all wavelet analysis, with the maximum amplitude shown at the C2 satellite. For C1 this signal has a broader structure, starting from 06:56 until 07:18, with frequency spanning from 0.005 to 0.02 Hz. Both of the satellites hold the same structure of short-term high-frequency signals up to 1 Hz.

Figure 6 demonstrates the wavelet analysis for the event on 1 October 2005. The third and fourth SCs were located relatively close, with the first and second slightly behind in the magnetotail. Although the onset of dipolarization begins at 04:49, where the B_z component becomes comparable with magnetic field magnitude B, the signal up to this moment is not devoid of high-amplitude changes. Transforms for C3 and C4 show strong signals in the frequencies ranging from 0.002 to 0.004 Hz, which span 10 min, with different times for intensity maxima: 04:39 for the third SC and 04:42 for the fourth. In both cases, a structure of inverse cascade can be traced before dipolarization onset: the frequency decreases from 0.005 to 0.002 Hz. The second inverse cascade lasts for 10 min starting from 04:57 in the time domain with a gradual decrease in the frequency range from 0.015 to 0.005 Hz, while at higher frequency it breaks up into smaller wave forms. Wavelet decomposition for C2 differs from the others, primarily by the absence of any cascade during the turbulent phase of dipolarization with a distinct component at 0.0035 Hz which spans for 8 min. It is interesting to note that just for this SC the spectral slope of the PSD spectrum is less in absolute value in comparison with other SCs: −2.5 against −2.8. The relatively short components, with durations of less than 2 min, extend in the frequency range from 0.02 up to 0.2 Hz. For C1 there is an enhancement with long duration at 0.004 Hz, which spans more than 20 min, and one with a gradual increase in frequency up to 0.007 Hz, i.e. direct cascade. Such prolonged intensity enhancements are observed with a wide frequency coverage beginning with 0.002 up to 0.01 Hz. A large number of high-frequency components appears in measurement signals from both satellites.

Thus, during the dipolarization, the magnetometers of all spacecraft recorded powerful signals with periods of 50, 100, 125, 166, and 200 s, corresponding to Pc4 (45–150 s) and Pc5 (150–600 s) pulsations, as well as direct and inverse cascade processes. The presence of inverse cascade processes indicates that together with the decay of the vortex structures, self-organization also takes place, i.e. smaller vortices are grouped into larger vortices. In the analysed events Pc pulsations were observed by all satellites – in the spatial range of 11–17 R_E. The largest number of cascade processes is observed at a distance of 15–16 R_E, and the largest number of inverse cascades is in the range 13–14 R_E.

3.3 Statistical analysis

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 (Frisch, 1995). Non-Gaussianity of the PDFs, which increases as the spatial scale decreases, is indeed due to the presence of the intense, phase-correlated fluctuations, due to the transfer of energy between contiguous eddies. It should be pointed out that spectral properties of the field are not essentially affected by intermittency. This is normally studied through the scaling properties of PDFs, or through their high-order moments (the structure functions), for which models and theoretical results exist (Frisch, 1995). The observation of intermittency implies that a non-linear, non-homogeneous energy transfer takes place in the system (Zimbardo et al., 2010).

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 h for spacecraft C1 has been determined. In this case, the statistical properties of the magnitude value of the magnetic field fluctuations at different timescales were analysed. The use of the Taylor hypothesis for various regions of the magnetospheric tail is detailed in Borovsky and Funsten (2003).

Figure 7The results of kurtosis.

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The value of the kurtosis was determined by the moments of the second and fourth orders from the formula by Zacks (1971):

where $S_q\left(\mathit{\tau }\right)=\langle \left|B\right(t+\mathit{\tau })-B(\mathit{\tau }){|}^q\rangle$ is the structure function of qth order, 〈…〉 is the time average of the data, τ is the timescale (time shift), multiple of measurements discretization 0.0445 s. When determining the excess value of the magnetic field fluctuations, the dependence of the functions K(τ) from the scale parameter τ was constructed. The significance of excesses for different mission SC and different events is shown in Fig. 7. It is clearly seen from the graphs that for the interval 1 (dotted line) for almost all satellites the value of K(τ) varies about 3 (in the range from 2 to 5), which is close to the normal distribution. The only exception is the measurement on the C1 spacecraft for 1 October 2005. Also, for interval 1, the spin tone of SC rotation is clearly observed by sheer accident near the gyrofrequency. For the dipolarization region (interval 2, solid line), the function K(τ) on small scales varies from 100 (C1, 12 September 2015) to 8 (C3, C4, 15 October 2005).

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 2).

The “gap” of values for interval 2 for very small τ can be explained by the instrumental error of observations.

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.

Figure 8The example of Hölder exponents.

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The presence of intermittency is indicated by the analysis of the first-order structure function (Fig. 8). For a self-affine signal, $S\left(\mathit{\tau }\right)\approx {\mathit{\tau }}^{-h}$, where h is the Hölder exponent (note that the Hölder exponent is the Hurst exponent of first order, h=0.5 for Brownian motion). The higher value of h afterward indicates a persistent signal with a longer correlation than a random noise and may imply the occurrence of reorganization during dipolarization (Chang, 1992; Consolini and Lui, 2000). In our case, the Hölder exponent is in the range $h\approx \mathrm{0.659}\pm \mathrm{0.005}$ at the time of dipolarization.

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:

where 〈…〉 is the time average of the data, and τ is the time step.

The existence of the criterion of generalized self-similarity for an arbitrary pair of structural functions $S_q\left(\mathit{\tau }\right)\sim S_p(\mathit{\tau }{)}^{\frac{\mathit{\zeta }\left(q\right)}{\mathit{\zeta }\left(p\right)}}$ allows one to find ζ(q) and estimate the types of turbulent and diffusion processes (Dubrulle, 1994). In this case, the non-linear functional dependence ζ(q) from the order of the moment q for experimental data is a consequence of the intermittency of processes. For the interpretation of the non-linear spectrum ζ(q), the log-Poisson model of turbulence is used, in which the power index of the structural function is determined by the relation (Dubrulle, 1994; Kozak et al., 2011; She and Leveque, 1994)

where β and Δ are parameters that characterize intermittency and singularity of dissipative processes, respectively. It is important to note that within the framework of this model a stochastic multiplicative cascade is considered, and the logarithm of dissipation energy is described by the Poisson distribution. For isotropic three-dimensional turbulence, $\mathrm{\Delta }=\mathit{\beta }=\mathrm{2}/\mathrm{3}$ (SL) (She and Leveque, 1994).

Figure 9Dependence of the order of the structure function for different timescales during dipolarization (event 15 October 2005).

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Figure 10The results of ESS analysis (during DP). Ratio of the power of the qth-order structural function to the third-order function power. The experimental data for the magnetic field are marked with the symbol; the solid line corresponds to the value calculated using the formula in the log-Poisson cascade model for $\mathrm{\Delta }=\mathit{\beta }=\mathrm{2}/\mathrm{3}$ (SL), and the dotted line corresponds to the q=3 (K41).

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Table 4ESS-analysis parameters and diffusion coefficients.

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The power law of the type S_q(τ)∼τ^ζ(q) (i.e. self-similarity – linear dependence) is observed on limited timescale intervals (Fig. 9). For the considered satellite measurements, this interval is close to the value of the ion-cyclotron frequency during dipolarization (Table 2).

The results of scaling the moments of the probability density function for different orders of q in the analysis of small-scale turbulence and comparing them with the Kolmogorov model are shown in Fig. 10. The results of the ESS analysis of the satellite measurements indicate the heterogeneity of turbulent processes during the dipolarization to describe what can be a log-Poisson cascade model with fitting parameters. The obtained values of the parameters β and Δ are given in Table 4. In addition, the obtained values can be used to determine the characteristics of the diffusion transfer of plasma. In this case, the properties of diffusion are considered within the concept of a multi-fractal multiplicative cascade (Lovejoy et al., 1998). The coefficient of generalized diffusion is determined by the parameters of the structural function ζ(q) (intermittency and singularity) by the relations by Lovejoy et al. (1998) and Prokhorenkov et al. (2015):

This approach is used to estimate the transfer in a statistically inhomogeneous medium, and the index R, in general, is determined by the fractal properties of the medium and characterizes (on average) the topological properties (connection properties that determine the transfer) of a stochastic structure of turbulence.

The resulting values of R lie within the range from 0.20 to 0.77 (Table 4). Given that the law of particle displacement over time is given by the formula by Treumann et al. (1990), Chechkin et al. (2008), and Zaburdaev et al. (2015), $\langle \mathit{\delta }{x}^{\mathrm{2}}\rangle \propto D\mathit{\tau }\propto {\mathit{\tau }}^{\mathit{\delta }}$ with an indicator $\mathit{\delta }\propto \mathrm{1}+R\approx \mathrm{1.20}$–1.77>1, this dependence means the existence of super-diffusion.