The solar wind can be regarded as a network of entangled magnetic flux tubes and Alfvénic fluctuations propagating within each flux tube . Flux tubes can emerge locally in the solar wind as a consequence of the magnetohydrodynamic turbulent cascade . An alternative view describes coherent structures as “fossile” structures that emanate from the solar surface and are advected by the solar wind .

The probability distribution functions (PDFs) of turbulent space plasmas display sharp peaks and fat tails on small scales within the inertial subrange , as well as departures from self-similarity and monofractality . These features are due to the presence of rare, large-amplitude coherent structures which dominate the statistics of fluctuations on small scales and can be quantified by the computation of statistical moments.

A robust parabolic dependence between the normalized third-order moment (skewness) and the normalized fourth-order moment (kurtosis) has been found in local concentrations of contaminants in atmospheric turbulence as found by . also found a similar skewness–kurtosis parabolic relation using global data of sea-surface temperature fluctuations. reported a similar skewness–kurtosis dependence in electron density fluctuations in plasma confinement experiments. obtained a skewness–kurtosis parabolic relation for datasets of human reaction times for visual stimuli. Since then, the presence of a skewness–kurtosis relation in different physical scenarios has attracted much attention and has been associated with the presence of non-Gaussian fluctuations due to coherent structures . The skewness–kurtosis parabolic relation was also found in time series of two-point differences of the modulus of the magnetic field by . They demonstrated that the parabolic relation is due to nonlocal interaction between large-scale structures and small-scale intermittency.

In this paper we investigate the skewness–kurtosis relation during a triple interplanetary magnetic flux rope (IMFR) event detected by Cluster-1 in the solar wind. This event was recently characterized by . They demonstrated the occurrence of magnetic reconnection at the interface region of two IMFRs and that this reconnection can be the origin of interplanetary intermittent turbulence. Our results show that the skewness–kurtosis parabolic relation is enhanced during the reconnection between flux ropes, and that is a natural consequence of the interaction between flux ropes.

This paper is organized as follows. Section  presents the statistical tools employed for the data analysis, including the equations to compute the skewness and the kurtosis. Section  describes the triple-IMFR event. The skewness–kurtosis relation is analyzed in detail in Sect. . The interpretations of these results are presented in Sect. . Finally, we conclude in Sect. .

Let θi, i=1,…,N be the time series of a quantity of interest (e.g., the modulus of the magnetic field |B|). The skewness of θi can be computed as follows: S=1N∑i=1Nθi-θiσ3, where θi represents the average of θi, N represents the number of data points, and σ is the SD of θi. The flatness of θi is given by F=1N∑i=1Nθi-θiσ4, from which the kurtosis can be obtained by K=F-3.

For a Gaussian function S=K=0. The skewness quantifies the degree of asymmetry of the PDF of θi, whereas the kurtosis quantifies the departure of the flatness of the PDF of θi from the flatness of a Gaussian distribution which is equal to 3. The definition of kurtosis in Eq. () is sometimes called “excess kurtosis” .

A common way to characterize asymmetry and non-Gaussianity of θi as a function of scale τ is through the time series of two-point differences: δθi(τ)=θi+τ-θi. The skewness of θi on scale τ is then S(τ)=1N∑i=1Nδθi-δθiστ3, and the flatness is F(τ)=1N∑i=1Nδθi-δθiστ4, where στ is the SD of δθi(τ). From Eq. () the kurtosis as a function of scale is obtained by K(τ)=F(τ)-3.

A functional relation between the skewness and the kurtosis of θi as defined by Eqs. ()–() has been observed in a variety of scenarios e.g.,. This relation is given by K=αS2+β, where α and β are the coefficients that characterize a parabolic curve.

We compute the α and β coefficients by applying a least-square fit between (S,K) values obtained from the observational data and Eq. () following the Levenberg–Marquardt algorithm , which is a popular method to fit a dataset into nonlinear equations. In order to quantify how well the computed (S,K) values are fitted into Eq. () we employ the correlation index r which measures the correlation between two datasets Xi and Yi, i=1,…,N: r=1σXσY∑i=1N(Xi-Xi)(Yi-Yi)N, where σX and σY represent the SD of Xi and Yi, respectively. The correlation index r∈[-1,1]. If r=1 there is complete correlation between Xi and Yi, whereas r=-1 indicates anticorrelation. The value r=0 represents absence of correlation.

In summary, the analysis is described by the following steps:

Compute S and K from the modulus of magnetic field |B| using Eqs. ()–().

We repeat these steps for S(τ) and K(τ) of two-point differences using Eqs. () and () in the first step. There are several computational programs for data analysis that implement the Levenberg–Marquardt algorithm. Here we use the implementation available in the GNU Octave program .

We note that several papers regarding the relation between skewness and kurtosis have employed the definition of what we refer to as flatness (Eq. ). Throughout this paper we will focus on the kurtosis defined by Eq. ().

Figure a shows the time series of the modulus of magnetic field |B| obtained by the FGM instrument onboard Cluster-1 from 00:00 to 12:00 UT on 2 February 2002. During this interval Cluster-1 was in the solar wind upstream of the Earth's bow shock . The magnetic field data are collected by Cluster-1 at a resolution of 22 Hz . Figure also presents an overview of other in situ plasma parameters for the selected interval, namely, the three components of B in the GSE coordinates, the angles ΦB and ΘB of the solar wind magnetic field B relative to the Sun–Earth x axis in the ecliptic plane, and out of the ecliptic, respectively, in the polar GSE coordinates; the modulus of the ion bulk flow velocity |Vi|, the ion number density ni, the ion temperature perpendicular to the magnetic field Ti and the ion plasma βi, which is the ratio between plasma kinetic pressure and magnetic pressure. The Cluster-1 plasma measurements are given by the ion spectrometry experiment CIS .

This event is characterized by the presence of three interplanetary magnetic flux ropes. Magnetic flux ropes are magnetic structures described as bundles of twisted, current-carrying magnetic field lines bent into a tube-like shape, spiralling around a common axis . During this event three IMFRs were identified by using a combination of criteria for large-scale magnetic cloud boundary layers and small-scale IMFRs . The interval of each IMFR is indicated by horizontal arrows in Fig. a, and their timings are shown in Table .

A theoretical explanation of the parabolic relation between the skewness and kurtosis of turbulent fluids and plasmas is still an open question. proposed a nonlinear Langevin equation with external forcing that can account for the parabolic relation between S and K. extended this model to include self-generated internal instabilities in plasmas. argued that a parabolic relation can be obtained as a natural consequence of a number of constraints expected to be met for most physical systems. proposed a simplified model of a synthetic intermittent time series, constructed from a random number of coherent structures with random amplitudes embedded in a background Gaussian noise, and demonstrated that their model can predict a S–K parabolic relation. A similar study was performed by using a model of coherent plasma flux events.

Although a theoretical explanation of the S–K relation is still unclear, there is a consensus that the parabolic shape is due to non-Gaussianity related to coherent structures, whereas points near (S,K)=(0,0) correspond to Gaussian fluctuations. This is confirmed by models of synthetic time series. For example, proposed a model of intermittent time series which consists of a superposition of Gaussian and non-Gaussian random fluctuations. Their model includes a parameter that measures the deviation from Gaussianity. The resulting PDF derived from their model displays asymmetric long tails that reproduce measured distributions of plasma density fluctuations in plasma magnetic confinement devices as well as distributions of X-ray emissions detected from accretion disks . Their model also leads to a parabolic relation between S and K. observed a transition from a parabolic shape to the (S,K)=(0,0) point by increasing the intensity of the Gaussian noise in their model of synthetic time series, constructed by adding deterministic fluctuations and Gaussian noise. However, a quantification of the parabolic shape is needed for an objective comparison between different datasets. We have found that the computation of the correlation index r allows time series dominated by either Gaussian and non-Gaussian fluctuations to be clearly distinguished. Despite the simplicity of this approach, it represents an alternative way to compare the degree of non-Gaussianity due to asymmetry and fat tails in the PDFs of different datasets, and can be applied to observational data and results from numerical simulations.

The stochastic model of a time series proposed by assumes that the non-Gaussian fluctuations arise from a quadratic nonlinear term. By increasing the degree of non-Gaussianity the skewness and the kurtosis converge to extreme values: S=±22 and K=12. This means that experimental data governed by nonlinear processes of quadratic order should lead to S–K scatter plots with S∈[-22,22] and K<12. The scatter plots shown in Fig. a, c and e seem to agree with these limits; however, in Fig. d there are some points in which S<-22. also propose that processes described by higher-order nonlinearities can result in S–K parabolic shapes with S outside the interval [-22,22], which can explain the behavior of |B| during the magnetic reconnection occurring in the I23 interval.

In the previous sections we showed and discussed the value of the correlation index measuring how well the S–K scatter plots fit with a parabola. As mentioned before, there is no agreement on the interpretation of the coefficients α and β in Eq. (). argues that the coefficients are not likely to offer relevant information about the underlying process. However, discussed an interpretation of the α and β coefficients based on their model of a synthetic time series. The value of the α coefficient depends on the statistics of the fluctuations due to coherent structures and is not necessarily constant in time. For the β coefficient, if the number of coherent structures in a time series can be represented as random independent variables that follow a Poisson distribution function (which models the occurrence of rare events), then β=3. Deviations from this value can be interpreted as a departure from the independence assumption, which means that there is interaction among coherent structures . Since we define kurtosis to be the flatness minus three, the previous statement is equivalent to say that deviations from β=0 are due to interacting coherent structures. From Table , we note that all intervals have nonzero values of β. Recall that this event is characterized by a rope–rope magnetic reconnection involving IMFR-2 and IMFR-3, with formation of a bifurcated current sheet acting as a source of intermittent turbulence . The interaction between the small-scale IMFR-2 and the medium-scale IMFR-3 occurring during this event gives support for the interpretation of the β parameter by .

demonstrated that the S–K parabolic relation is also observed for time series of two-point differences of |B| in the solar wind. They showed that this relation is enhanced in the presence of large-scale events such as interplanetary shocks, whereas for nonshock intervals, the parabolic relation is not observed. In this case the S–K parabolic relation represents a signature of direct coupling between large-scale structures (interplanetary shocks) and small-scale intermittency. Our results indicate that the S–K parabolic relation is present during reconnection between a small-scale IMFR with a duration of ∼60 min and a medium-scale IMFR with a duration of ∼7 h (see Table 1). The only region in which the parabolic relation is not observed is in the interior of IMFR-2. This region was found to have a low degree of intermittency and nearly monofractal scaling. Therefore, our results are in accordance with cross-scale coupling between IMFR scales and scales within the inertial subrange.

In this paper we investigated the relation between the skewness and the kurtosis during a triple-IMFR event on 2 February 2002. This event was divided into five regions, namely, the interior of IMFR-1, the interface of IMFR-1 and IMFR-2, the interior of IMFR-2, the interface of IMFR-2 and IMFR-3, and the interior of IMFR-3. We then computed the skewness S and the kurtosis K of |B| using a sliding window, and showed that the scatter plots of K as a function of S display a parabolic shape for all regions. The highest value of the correlation index computed by a least-square fit between the (S,K) values and Eq. () occurs at the interface of IMFR-2 and IMFR-3. This region was found to be the source of intermittent turbulence due to a magnetic reconnection between the small-size IMFR-2 and the medium-size IMFR-3 . Therefore, the enhanced S–K parabolic relation is related to non-Gaussian fluctuations due to coherent structures emerging from intermittent turbulence generated via magnetic reconnection. The lowest value of the correlation index was obtained at the interior of IMFR-2, in agreement with the results of , who found that this region is characterized by a low degree of non-Gaussianity and phase synchronization, and nearly monofractal scaling.

We also analyzed the S–K relation using two-point differences of |B| on two different scales within the inertial subrange. By computing the compensated PSD we selected an interval of frequencies in which all regions exhibit -5/3 scaling corresponding to the inertial subrange and selected two timescales representing the largest scale (τ=100 s) and the smallest scale (τ=10 s) within the inertial subrange. We found that the scatter plot of IMFR-2 on the largest scale (τ=100 s) and on the smallest scale (τ=10 s) accumulate around the (S,K)=(0,0) point. The least-square fit with Eq. () results in a low correlation index, which confirms that magnetic field fluctuations in this region are nearly Gaussian. All other regions displayed parabolic shapes. At τ=100 s, the correlation index is high for the interface of IMFR-2 and IMFR-3, indicating that the magnetic reconnection that occurs in this region can generate non-Gaussian fluctuations on the largest scale. On the smallest scale, the correlation index is higher for two regions, namely, the interior of IMFR-1 and the interface of IMFR-2 and IMFR-3. This result can be due to non-Gaussian fluctuations resulting from small-scale current sheets detected within these regions . Our analysis indicates that the S–K parabolic relation observed in interplanetary magnetic field turbulence is enhanced on small scales within the inertial subrange.

Our findings give support to the conclusion by that rope–rope magnetic reconnection acts as a source of interplanetary intermittent turbulence and suggest that magnetic reconnection is responsible for non-Gaussian PDFs with asymmetric shapes and fat tails. The results are also in agreement with the results of in that the S–K parabolic relation is a signature of direct coupling between IMFR scales and small-scale intermittency.