Within the spectral analysis, the spectral power density (PSD) was built from the frequency f, and the power-law dependence PSD(f)∼fa was determined. To determine the PSD of the signal for a series of N measurements Xn, a discrete Fourier transform was used: PSD=2Nfs1N∑n=0N-1Xnexp⁡2πinjN2, where n=0,1…N-1, j=0,1…N/2.

To find the break points and the slope of the spectrum, we used a piecewise linear approximation of log⁡PSD 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.  and Table .

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 ). 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 ωc/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 for event 1 October 2005, in the region of the magnetic field dipolarization, the percentage of oxygen ions in relation to protons (〈n(O+)〉/〈n(H+)〉) is 21.1±10.0% (SC C3) and 9.3±1.5% (SC C4), and the percentage of helium in relation to protons (〈n(He+)〉/〈n(H+)〉) ∼2.4±0.3% (SC C3) and ∼4.8±0.7% (SC C4); for event 15 October 2005 – 〈n(O+)〉/〈n(H+)〉∼11.1±1.0% (SC C4) and 〈n(He+)〉/〈n(H+)〉∼3.4±0.5% (SC C4); for the 12 September 2015 event – 〈n(O+)〉/〈n(H+)〉∼18.9±7.3% (SC C4) and 〈n(He+)〉/〈n(H+)〉∼15.8±5.4% (SC C4)). At the same time, the exponent lies in the range from -2.2 to -1.53 on large timescales of 0.01-ωc/2, and at smaller timescales ωc/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.

Within the framework of the wavelet analysis for a series of measurements Xn (n=0,1…N-1) with time step δt, a Morlet wavelet was used: Ψ0(η)=π-14eiω0ηe-η22, where ω0 is the dimensionless frequency, and η the dimensionless time.

The continuous wavelet transform of the discrete signal Xn is defined as the convolution of the mother wavelet whose argument is scaled and transmitted with a signal : Wn(s)=∑n′N-1xn′Ψ*(n′-n)δts, where (*) is the complex conjugate, |Wn(s)|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.

In Fig.  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  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  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 Bz 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 RE. The largest number of cascade processes is observed at a distance of 15–16 RE, and the largest number of inverse cascades is in the range 13–14 RE.

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 . 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 . The observation of intermittency implies that a non-linear, non-homogeneous energy transfer takes place in the system .

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).

The value of the kurtosis was determined by the moments of the second and fourth orders from the formula by : K(τ)=S4(τ)(S2(τ))2, where Sq(τ)=〈|B(t+τ)-B(τ)|q〉 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. . 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 ).

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.

The presence of intermittency is indicated by the analysis of the first-order structure function (Fig. ). For a self-affine signal, S(τ)≈τ-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 . In our case, the Hölder exponent is in the range h≈0.659±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: Sq(τ)=〈|B(t+τ)-B(t)|q〉∼τζ(q), 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 Sq(τ)∼Sp(τ)ζ(q)ζ(p) allows one to find ζ(q) and estimate the types of turbulent and diffusion processes . 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 ζ(q)=(1-Δ)q3+Δ1-β1-βq3, 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, Δ=β=2/3 (SL) .

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

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. . 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 . 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 . The coefficient of generalized diffusion is determined by the parameters of the structural function ζ(q) (intermittency and singularity) by the relations by and : D∝τR,R=R̃(-1),R̃(q)=q-ζ(3q),R=Δ(1/β-1). 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 , , and , 〈δx2〉∝Dτ∝τδ with an indicator δ∝1+R≈1.20–1.77>1, this dependence means the existence of super-diffusion.