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Solar system plasma Turbulence: Observations, inteRmittency and Multifractals

Final Report Summary - STORM (Solar system plasma Turbulence: Observations, inteRmittency and Multifractals)

Executive Summary:
The main objective of the FP7 project STORM was to make a systematic investigation of the in-situ space plasma data bases collected by ESA’s missions launched in the solar system, as well as of data from other relevant satellite data bases. We applied a systematic analysis of electromagnetic and plasma fluctuations in order to find evidence of turbulence and intermittency.
The analysis strategy adopted in STORM is built on the principle of increasing complexity, from lower order analyses (like, e.g. the Power Spectral Density - PSD) to higher order investigations (the Probability Distribution Functions – PDFs, Structure Functions - SFs, Fractals and Multifractals - MFs). The project made indeed a systematic survey, orbit by orbit, of data available from ESA data repositories and Principal Investigators. STORM produced catalogues of Power Spectral Density (PSD), Probability Distribution Functions (PDFs), partion functions and rank ordered multifractal spectra (MFs) at solar minimum (1997-1998, 2007-2008) and maximum (2000-2001) from data provided by Ulysses, Cluster, and Venus Express. Catalogues of the same type of analysis (PSD, PDFs, multifractal) were produced for magnetospheric data from Venus Express and Cluster, at solar minimum and maximum. Moreover the PSD, PDFs and multifractal analysis has been applied on geomagnetic data (nine global geomagnetic indices, AE, AL, AU, Dst, SYM-H, SYM-D, ASY-H, ASY-D, Dcm and data from individual Nordic observatories (Sodankylä, Nurmijärvi, and Belsk). Different types of turbulence models, like the p-model and the two-scale modified Cantor set models have been tested against solar wind and magnetospheric data. The catalogues of analyzed data are organized like a functional database and structured according to the type of targeted system (solar wind/magnetosphere/geomagnetic indices), solar cycle phase (minimum versus maximum), type of analysis (PSD, PDFs, multifractal). The catalogues, available online from include 4094 PSD spectra, 9566 PDFs and 15633 multifractal (partition function based and Rank Ordered - ROMA) spectra. On the scientific side this huge collection of analyses provides new insight on turbulent processes in solar system plasmas. In a series of recent studies published in international journals, the members of the STORM team showed that : (1) a systematic survey of Ulysses data at solar minimum and maximum reveals that the magnetic intermittency exhibit a tendency to decrease with the heliocentric distance; (2) statistically robust differences were found between the spectral properties of fast and slow wind magnetic turbulence at 0.72 astronomical units (in the vicinity of Venus) and solar minimum, (3) significant differences are found in the topology of the planetary magnetosheath turbulence between Venus and the Earth; Venus magnetosheath exhibit turbulent properties close to the “standard” picture (e.g. a robust detection of an inertial range); the terrestrial magnetosheath shows increased variability and many instances were found when the inertial range was not identified; (4) the critical behavior of geomagnetic fluctuations suggest a dependence of the Dst index intermittency on the solar cycle phase but virtual independence of the fractal properties of AE on the solar cycle, pointing towards an intrinsic nonlinear behavior of the magnetosphere. In addition to data analysis and scientific research STORM built an integrated library for non-linear analysis of time series that includes all the approaches adopted in STORM to investigate solar system plasma turbulence. This versatile analysis tool is prepared to offer the user a friendly environment tailored according to the STORM data analysis strategy, i.e. based on the principle of increasing complexity.

Project Context and Objectives:
The main objective of STORM is “to make a systematic investigation of the in-situ space plasma data bases collected by ESA’s missions launched in the solar system, Giotto, Ulysses, Rosetta, Cluster and Venus Express together with other satellite data bases, in particular NASA’s Cassini, Mars Global Surveyor and THEMIS. We use these data bases to perform a systematic analysis of electromagnetic and plasma fluctuations in order to find evidence of turbulence and intermittency. Our approach is meant to reveal new universal properties of intermittent and anisotropic turbulence and multifractals in solar system plasmas (solar wind; the planetary foreshock and magnetosheath, both for the quasi-parallel and quasi-perpendicular geometry, the terrestrial magnetospheric cusps, the Low Latitude Boundary Layers of magnetized planets) and how these properties vary within the solar cycle and with the distance from the Sun” (STORM Grant Agreement, EU contract 313038/2012).

1.1. Scientific Objectives of STORM

STORM targets three major categories of physical processes:

P1. Turbulent Energy cascade and dissipation;
P2. Intermittency, Multifractals, Scaling;
P3. Anisotropic and imbalanced turbulence;

Each of the topics listed above has its own associated scientific objectives, as indicated below:

P1. Energy cascade and dissipation:
i. To improve the physical insight on turbulence by investigating the topology of the energy transfer in the solar wind(in the ecliptic plane and at higher latitudes), as well as in planetary magnetospheres, and to search for quantitative measures for wave dispersion and/or coherent structures interaction;
ii. To improve the understanding of the dissipation mechanisms, in the solar wind and planetary magnetospheres and search evidence of coherent structure dissipation versus wave dumping
iii. To investigate the solar cycle effects on the energy cascade and dissipation mechanisms, in the solar wind and planetary magnetospheres and to analyse similarities and differences between the solar wind and the planetary magnetospheres
iv.To improve the current understanding of nonlinearities and nonstationarity of solar system plasma turbulence

P2. Intermittency, Multifractals, Scaling:
i. To compile a data base with intermittent events and their characteristics, and to discriminate between solar maximum and solar minimum, to compare fast and slow solar wind, closer and at larger distances from the Sun
ii. To use existing geomagnetic indices data bases as a test bed for investigating possible connections between multifractals and the concept of (Forced) and/or Self Organized Criticality (SFOC) and extract the SFOC parameters at solar minimum and solar maximum
iii. To advance the understanding of the coupling between the solar wind and the planetary plasmas by investigating the similarities and differences between solar wind, magnetospheric and ground based intermittency.
iv. To advance the understanding of the physical insight contained in the multifractal methods and their results for intermittent data in the solar wind, planetary plasma and ground based observations
v. To compile a data base with intermittent events and their characteristics, and to discriminate between solar maximum and solar minimum, to compare fast and slow solar wind, closer and at larger distances from the Sun

P3. Anisotropic and imbalanced turbulence:
i. To evaluate the anisotropy and intermittency of turbulence at solar maximum and minimum
ii. To estimate the importance of anisotropy and compressibility in the energy cascade;
iii. To establish an up-to-date turbulence theory describing best the observational results;

1.2. Operational/Technical Objectives of STORM

The operational/technical objectives defined to achieve the scientific objectives described above follows:
O1) To investigate the topology of the turbulent energy transfer and dissipation in solar system plasmas and to understand its variability in the heliosphere, where is sampled by ESA’s Giotto, Venus Express, Cluster, Rosetta and NASA’s THEMIS, Cassini and Mars Global Surveyor and at higher heliospheric latitudes (Ulysses) Work packages WP2, WP3, WP4, WP7 contributed to achieving this objective.
O2) To determine a relevant set of quantitative parameters for the description of the nonlinear state of the solar wind and planetary plasma environment. (e.g. PSD spectral exponent, statistical moments of incremental time-series in terms of scale parameter, structure function scaling exponent curve). Work packages WP2, WP3, WP4, WP7 contributed to achieving this objective.
O3) To evaluate the solar cycle effects on the intermittency of the turbulent transfer of energy in the solar wind and planetary plasmas; Work packages WP2, WP3, WP4, WP7 contributed to achieving this objective.
O4) To extract the multifractal properties of turbulent fluctuations in the solar wind and planetary magnetospheres using the partition function and the rank ordered (ROMA) approach; to compare the results of the two methods and better understand ROMA.
O5) To explore the effect of space weather events and to investigate the scaling and multifractal properties of the fluctuations of the geomagnetic indices at solar maximum versus solar minimum and to search for similarities and differences with scaling and multifractal properties of the solar wind.
O6) To use existing data bases as a test bed for investigating possible connections between multifractals and the concept of Forced and/or Self Organized Criticality (FSOC) and to extract the SFOC parameters at solar minimum and solar maximum, in the solar wind and planetary plasma environment; Work packageWP5 contributed to this objective.
O7) To-evaluate the anisotropy of turbulence in key magnetospheric regions (magnetosheath, magnetic cusps, boundary layers)and in the solar wind (at 1 AU), at solar maximum and minimum, using multi-spacecraft methods; Work package WP6 contributed to achieving this objective..
O8) To compile data bases including the nonlinear parameters determined for different regions of the solar wind and planetary magnetospheres system visited by the space missions as well as for terrestrial observations in terms of geomagnetic latitudes extended to global geomagnetic indices (AE, Dst, SYMH). Work packages WP2, WP3, WP4, WP7 contributed to achieving this objective.
O9) To create an integrated software library to include the full set of analysis methods devoted to the analysis of turbulence properties from time series provide by satellites; Work packageWP7 contributed to achieving this objective.
Project Results:
In this section we give a concise overview of the main scientific results and foreground provided by STORM. For each scientific objective we provide a summary of the progress towards the objectives and details for each task, and the scientific result.


1.1. Definition of Solar Wind Data bases:

The following tasks are completed:
• define and construct solar wind databases at solar maximum, D1MAXSW (1999 Ulysses, 2000-2001, Ulysses, Cluster),
• define and construct solar wind databases at solar minimum D3MINSW (2007-2008, Ulysses, Cluster, Venus Express), D5MINSW (1997-1998, Ulysses)

1.1.1. Selection of Ulysses data
We used Ulysses magnetic field data from VHM-FGM magnetometer at 0.5 Hz and plasma data from Ulysses SWOOPS at 8 minute resolution. A main objective of Ulysses data survey and selection is to identify the “pure” states of the solar wind: slow and fast. We defined a set of criteria and thresholds to identify the solar wind type and origin, i.e. fast and slow wind originating in the polar coronal holes and respectively the streamer belt. The data selection is made based on the analysis of 5 solar wind parameters: (1) the radial velocity, (2) the Oxygen ion ratio O7+ /O6+, (3) the magnetic Compressibility factor, (4) the proton density np, (5) the proton temperature. One individual criterion and threshold was assigned to each of the 5 solar wind parameters. Interplanetary transients like CMEs and shocks were excluded from the analysis. We used 6-hour averages to select the threshold values for the five plasma and magnetic field parameters. Their associated scores were used to distinguish between the slow and fast streams. For each data sample we constructed a consolidated (final) score that specifies how many of the five individual scores are satisfied such that the sample can be classified as fast or slow wind. The consolidated score is formed by the sum of the 5 individual scores. Each individual score is equal to either 1 if the value of the corresponding parameter is larger than the threshold defined for fast wind or 0 otherwise (slow wind). If the consolidated/final score is equal to 4 or 5 we classify the sample as fast solar wind (FW). When the final score is equal to 0, 1 or 2 the sample is categorized as slow solar wind (SW). Cases when the score was 3 were disregarded from the analysis.

Based on the final score, we identified 46 time intervals of fast solar wind and 43 time intervals of slow solar wind. The information on these data is stored into an excel file, where, in addition to the information about final scores and basic characteristics, we have also included the information about the number of samples and data gaps for each interval. This file (Ulysses_data_selection.xls) is available in the ftp repository ( registration is required) together with the data catalogue. The final number of slow and fast solar wind intervals used for PSD calculation are the following:
In 1997-1998 (D5MINSW) 12 slow wind and 5 fast wind intervals.
In 1999-2001 (D1MAXSW) 27 slow solar wind and 32 fast wind intervals.
In 2007-2008 (D3MINSW) 4 slow wind and 9 fast wind intervals.

1.1.2. Selection of Cluster data included in D1MAXSW, D3MINSW

The selection of Cluster solar wind intervals is based on the simultaneous scanning of the following parameters: spacecraft position, magnetic field magnitude (Balogh et al., 1997), ion velocity, omni-directional ion energy flux (Reme et al., 1997) and wave energy density. The time intervals when Cluster was in the solar wind were selected close to the orbit apogee, between February and April 2001 and added to the D1MAXSW data base for solar maximum. Time intervals close to the Cluster apogee between January and April 2007 and respectively between January and April 2008 were added to the D3MINSW database for solar minimum.

We analyzed data from Cluster 1 and 3 since these two spacecraft offer the necessary data. In order to analyze only undisturbed solar wind data we excluded the time periods when Cluster encounter the ion and electron foreshocks . The ion foreshock effects are identified as time intervals when the flux (FE) of the highest energy channels (about 6 – 30 keV) of CIS-HIA instrument exceeds a given threshold (see the STORM deliverable Report D2.1). In order to exclude the electron foreshock we reject intervals in which the wave energy (E) measured by WHISPER exceeds a threshold value. These criteria for selecting the solar wind intervals “contaminated” by magnetospheric ions and electrons can only be applied when there are simultaneous measurements available in ESA Cluster Active Archive for the five parameters mentioned above.

The solar wind data intervals are further classified as fast (when plasma bulk velocity V > 450 km/s) and slow (V < 450 km/s) wind intervals. In 2001 (D1MAXSW) we find 21 slow wind and 1 fast wind intervals from Cluster 1 and 20 slow wind intervals from Cluster 3. In 2007-2008 (D3MINSW) we find 57 slow wind and 18 fast wind intervals from Cluster 1 and 47 slow wind and 7 fast wind intervals from Cluster 3.

1.1.3. Selection of Venus Express data included in D3MINSW

The selection of data from Venus Express is based on records from the VEX-MAG magnetometer and the ASPERA plasma analyser. Due to the highly eccentric orbit, Venus Express spends most of the orbital time in the solar wind and therefore provides valuable data for turbulence studies. Moreover, in 2007-2008, the time interval included in the D3MINSW database, there is an inferior conjunction between Venus and Earth (VEX and Cluster), between July and August 2007.
VEX solar wind data selection is based on the plasma bulk velocity measured by ASPERA at a resolution of 196 seconds. The contamination of solar wind by Venusian ion foreshock is reduced because, on the one hand, Venus has no intrinsic magnetic field, thus the induced magnetosphere is quite small and the foreshock size is reduced accordingly and, on the other hand, ASPERA is switched on for time intervals of the order of one hour only, close to the orbital apogee, more than 50000 kilometres away from Venus.

We used a selection threshold for fast solar wind consistent with the one defined for Cluster at 1 AU for the same time interval, 2007-2008, i.e. we consider the sample is fast solar wind if the velocity is larger than 450 km/s. We included only those time intervals larger than one hour that do not have data gaps longer than 30 seconds. VEX intersected a total number of 29 fast solar wind intervals in 2007 - 2008 (16 in 2007 and 13 in 2008).

1.2. Power spectral density (PSD) and Probability Distribution functions (PDFs) in the solar wind

The following tasks are completed:
• compute PSD and PDFs of magnetic field and velocity fluctuations from selected data bases;
• build a catalogue of PSD for magnetic field fluctuations for each of the databases D1MAXSW,D3MINSW,D5MINSW;
• build a catalogue of PDFs for magnetic field fluctuations for each of the databases D1MAXSW, D3MINSW,D5MINSW

1.2.1. Computation of Power Spectral Densities for solar wind at solar minimum and maximum

The Power Spectral Density (PSD) spectrum of magnetic field and velocity fluctuations (for limited time intervals) were computed using the Welch algorithm whose description is included in the STORM D2.1 report (Virtanen et al., 2014). The results are organized in data repositories/catalogs structured following the definitions of the STORM databases. The PSD catalog is organized as an ftp repository integrated into the website of the project at the password protected section of the site. The structure of the ftp repository is defined according to the guidelines stated in the Grant Agreement and its main pillars are the three databases:
• D5MINSW between 1997-1998 (includes Ulysses data and PSDs),
• D1MAXSW between 1999 - 2001 (includes Ulysses and Cluster data and PSDs) and
• D3MINSW between 2007-2008 (includes Ulysses, Cluster and Venus Express data).

The folder for each interval contains PSD and Data stored separately for each satellite. The database includes one graphical file (in PNG format) for each PSD computed for fast/slow solar wind intervals. The filename is defined such that it describes the type of analysis (PSD), the satellite name (Uly, C1, C3, VEX), the solar wind type (slow SW or fast SW), the analysed quantity (Bx, By, Bz, B2, Vsw ) and the start and end hour of the data interval (yy_mm_dd_hh). The PNG figure file contains a time series plot of the original satellite data and of the normalized (variance set to one) and the detrended (linear trend removed) satellite data, as well as a plot of the power spectral density. Each Figure includes information about the length of the data interval, data coverage and the number of segments, the window type and the amount of overlap used in Welch method of PSD calculation. PSD spectral data (frequency and power) are also stored in separate txt-files in the same folder with the same file name as PNG figure. The different missions contribute to the PSD database as follows:
• Ulysses: we have computed 27 PSDs included in D5MINSW (12 fast wind, 15 slow wind), 59 PSDs included in D1MAXSW (33 fast wind, 26 slow wind) and 49 PSDs included in D3MINSW (45 fast wind, 4 slow wind).
• Cluster: we have computed 22 PSDs from Cluster 1 data (1 fast wind, 21 slow wind) and 20 PSDs from Cluster 3 data (all slow wind) that were included in D1MAXSW. We have also computed 75 PSDs from Cluster 1 data (18 fast wind, 57 slow wind) and 54 PSDs from Cluster 3 data (7 fast wind, 47 slow wind) that were included In D3MINSW database.
• Venus Express: magnetic field data provide 374 PSD spectra (183 for 2007 and 191 for 2008) of which 110 PSD spectra correspond to fast wind streams (64 PSD spectra in 2007 and 46 PSD spectra in 2008).

1.2.2. Computation of Probability Density Functions for solar wind at solar minimum and maximum

We constructed an incremental measure of turbulent fluctuation based on differences over a range of scales, tau, defined as the difference between samples separated by a time interval tau - an integer multiple of the measurement resolution. Various values of tau identify/select various scales. When the Taylor hypothesis is satisfied the temporal scales may be transformed into Doppler shifted spatial scales. The scales selected for this study follow the incremental rule generally used in turbulence studies, i.e. they are distributed over several octaves, multiples of powers of 2. Technically the procedure is based on a sliding window of length tau across the time series. The results are collected in a statistical ensemble of differences, corresponding to the scale tau. Then we computed the histogram of the statistical ensemble of fluctuations against a number of bins, normalize it such that the resulting product is a density function. The procedure is repeated for each scale tau and the Probability Density Function is thus obtained for the range of considered scales.

The moments of the PDFs and particularly the fourth order one (the kurtosis) provide a more quantitative description of intermittency. The flatness is equal to 3 for fluctuations that follow a normal distribution (Gaussian). Thus the departure of the flatness from this value is a quantitative measure of the “strength” of intermittency for the corresponding scale. For each scale we also compute the flatness. This procedure is applied on all solar wind data selected from Ulysses, Cluster and Venus Express observations. The PDFs are computed for each component of the magnetic field and for the plasma bulk speed (when the time interval allows reasonable statistics for the plasma moments).The results are stored as graphical files as well as ASCII data files for further analysis.

We produced 4317 Probability Density Functions (PDFs) for the three core missions (Ulysses, Cluster, Venus Express), an unprecedented, to our knowledge, effort to investigate the intermittency of the solar wind plasma turbulence. The details of the PDFs results obtained for each of the three missions are given below. A more detailed overview is included in the Deliverable report (Echim et al., 2014) Ulysses contribution to PDFs computed in the solar wind
The Probability Density Functions and the flatness were computed for all the Ulysses time intervals already analyzed and included in the databases of Power Spectral Densities (Virtanen et al., 2014). The PDFs and the Flatness were computed for each component of the magnetic field provided by the Ulysses magnetometer (Balogh et al., 1992) in the RTN system, BR, BN, BT, and also for the square of the magnetic field, a measure of the magnetic field energy.
We have analyzed 135 time intervals of Ulysses data, of which 45 for “pure” slow solar wind and 90 for “pure” fast solar wind. We have computed a total number of 540 PDFs for Ulysses, distributed as follows:
• 156 PDFs and respectively 116 PDFs for fast and respectively slow wind at solar max, 1999-2001,
• 24 PDFs and respectively 48 PDFs for fast and respectively slow solar wind at minimum, 1997-1998
• 180 PDFs and respectively 16 PDFs for fast and respectively slow solar wind at minimum, 2007-2008

All the PDFs (and the corresponding flatness values) are stored in the data base and have been also included in six Annexes of report on the deliverable D2.2 (Echim et al., 2014). The names of the files are self explanatory and indicate the type of analysis, the time period, the spacecraft and the type of wind. For each set of PDFs computed for Ulysses magnetic field data we include in the data base a picture file in PNG format that evidences the relevant scale behavior of the PDFs, separately for each component, together with information about the position of the spacecraft (radial distance and heliospheric latitude) and an illustration of the analyzed time series. For each scale defined for the PDFs we also compute the flatness. The results are organized following the same procedure as for the PDFs: one graphical file is produced to illustrate the flatness for all the components of the magnetic field measured by Ulysses and all scales. An ASCII file stores the values for further analysis. Express contribution to PDFs computed in the solar wind

We have computed the PDFs and the flatness for 575 time intervals of magnetic field records provided by the VEX MAG magnetometer (Zhang et al., 2006) in the solar wind at 0.72 AU, close to the spacecraft apogee, between 2007 and 2008.Venus Express contributes only to the D3MINSW. A full list of these time intervals is included as an Appendix of the Deliverable report. Since VEX sweeps the solar wind on a daily basis, its contribution to the solar wind package is consistent. Nevertheless the magnetic noise on board the spacecraft raises difficult calibration problems. Magnetic field data were available from ESA Planetary Science Archive database, ASPERA data are available from the French AMDA database (Automated Mutli-Dataset Analysis, We analyzed magnetic field data at a resolution of 1 Hz; MAG-VEX data is also available at 32 Hz.

The PDFs and the flatness are computed for each component of the magnetic field, but also for the magnitude and its square. The field is preprocessed by subtracting the average and by dividing by the standard deviation. We have computed a total number of 2935 PDFs for Venus Express (this amount cumulates the PDFs for all components of the magnetic field): 1500 PDFs for data recorded in 2007 of which 515 and respectively 985 for fast and respectively slow wind, and 1435 PDFs for data recorded in 2008 (of which 355 PDFs for fast wind). For each analyzed variable we produce a graphical that illustrates the PDFs for four relevant scales and the flatness for all the scales considered in the analysis. The PDFs for all scales and the flatness are also stored in a data file saved in ASCII ( Cluster contribution to PDFs computed in the solar wind

Due to the seasonal changes of the orbit compared to the Sun-Earth axis, Cluster spends longer time intervals in the solar wind during only several months per year (in spring). Nevertheless, due to the highly eccentric orbit the spacecraft probe the solar wind in regions where the planetary ion effect is minimum as discussed in the previous section of this report.

We have computed the PDFs and the flatness for a total of 176 time intervals, of which 101 time intervals of magnetic field data (Balogh et al., 1997) from Cluster 1, and 75 time intervals of magnetic field data from Cluster 3, in 2001 and respectively 2007-2008. Due to the high resolution and availability of plasma moments provided by the Cluster ion spectrometer CIS (Rème et al., 1997) we have been able to compute some preliminary PDFs of the plasma bulk velocity, although the limited number of samples reveals only the central part of these PDFs. Cluster is the only mission targeted by STORM that allowed this type of analysis.

We produced a total number of 842 PDFs for Cluster, of which 206 PDFs are included in the database D1MAXSW at solar maximum (2001) and 636 PDFs are included in the D3MINSW database at solar minimum (2007-2008), 104 PDFs for fast wind and 532 for slow solar wind. For each PDF computed with Cluster data we create four files:
• The graphical file in compressed format (PNG)
• The script file in native Matlab language to produce the graphical file(.fig)
• The PDF data file that includes the PDFs for all the scales considered in the analysis
• The Flatness data file that includes the values of F from (2) for all considered scales.
The graphical files are stored individually in the databases but are also included in the Annexes of the Report on the D2.2. deliverable. The file format and file name convention for Cluster PDFs and Flatness follows the same principles defined for Ulysses.

1.3. Quantitative assessment of intermittency in the solar wind

The following tasks are completed:
“compute the Local Intermittency measure from magnetic field measurement and wavelets representation” (STORM GA 313038)

A series of mother wavelets were tested on data from solar wind (mainly Ulysses and Cluster), like Haar, Morlet, Daubechies. This type of analysis provided feedback on the implementation of the corresponding module in the Interactive Nonlinear Analysis (INA) toolbox. A thorough analysis with wavelet based LIM was applied on magnetic field records from Venus Express and Cluster with particular focus on the time intervals corresponding to the impact of a Corotating Interaction Region (CIR) in January 2007. The analysis show similar behavior at the two planets, Venus and respectively the Earth, and the occurrence of several Local Intermittency Measure (LIM) events, i.e. regions with enhanced shears of magnetic field.

1.4. Partition Function Multifractal analysis of solar wind turbulence

The following tasks are completed:
“compute the multifractal spectrum for magnetic and velocity fluctuations fit turbulence models” (STORM GA 313038)

The multifractals are described by an infinite number of the generalized dimensions, D_q (Halsey et al., 1986) and by the multifractal spectrum f(alfa) (Ott, 1993). The generalized dimensions D_q, are calculated as a function of a continuous index q (Grassberger & Procaccia, 1983; Hentschel & Procaccia, 1983). An alternative description is provided by the singularity spectrum f(alfa), as a function of a singularity strength (e.g. Ott, 1993). This function describes singularities occurring in considered probability measure attributed to different regions of the phase space of a given dynamical system. Since the singularity multifractal spectrum is easier to interpret theoretically by comparing the experimental results with the models under study we provided this description of multifractality as final result of the analysis. Additional details on the multifractal methods are included in the STORM Deliverable Report D2.3.

In addition to the partition function analysis of data we also attempted to fit the multifractal spectra with turbulence models. Two such models were considered, the symmetric p-model (Meneveau & Sreenivasan, 1987) and the two-scale modified Cantor set model by Macek & Wawrzaszek (2008, 2009). The fitting procedure was systematically applied on all the data analysed with partition function multifractals. The fit by turbulence models proved to be useful when the data are scarce and the Dalpha value cannot be evaluated directly from observations. Instead we use the turbulence model fit to evaluate Dalpha. In general the two-scale model performs better than the p-model for observations of turbulence in the solar wind. In total we produced 2612 Partition Function multifractal (PFMS) spectra for the solar wind magnetic field observations distributed as follows:
289 PFMS spectra from Ulysses and Cluster for D1MAXSW database (solar maximum, 2000-2001) available from Multifractals/PFMS,
2261 PFMS spectra from Ulysses, Cluster and Venus Express, for D3MINSW database (solar minimum, 2007-2008) available from wind Multifractals/PFMS/
62 PFMS spectra from Ulysses, for D5MAXSW database (solar minimum, 1997-1998) available from and

1.5. Rank Ordered Multifractal analysis (ROMA) of solar wind turbulence

The following objectives are achieved:
“compute the ROMA spectrum for magnetic and velocity fluctuations, check possible crossovers, identify regime of turbulence based on the fractal exponents” (STORM GA 313038)

The Rank Ordered Multifractal Analysis is a novel technique invented by Chang and Wu (2008) in order to explore the singular nature of the subdominant fluctuations at all scales. A detailed description of the method and its theoretical background is included in the STORM Deliverable report D2.3. Let us mention here that the rank ordering procedure search for scaled-sized fluctuations whose incremental measure satisfies a monofractal scaling, although the monofractal “changes” with the rank. It implies solving a transcendental equation (see, for instance, Chang et al., 2015 or Chang, 2015). The first step is to isolate ranks of fluctuations and investigate their statistical properties. Then find the monofractal index for each range of scaled-size fluctutations, called Y and defined as Y= DQ/Ts. where DQ is the incremental measure computed with the procedure described above for the PDFs computation. T is the scale and s is the monofractal index. The procedure is repeatedly performed for all bins DY and provides the spectrum s(DY). The spectrum is discrete since the data are limited but it can be continuous (Chang and Wu, 2008). We applied two different strategies to find the ROMA spectra: one is based on the investigation of the monofractal scaling bin by bin and order by order, the second approach is based on a global minimization procedure. Theoretical and technical details are included in the report D2.3..

The ROMA spectra were collected in catalogues organized for each of the three databases D1MAXSW (2000-2001), D3MINSW (2007-2008) and D5MINSW (1997-1998). In total we produced 1364 ROMA spectra for the solar wind magnetic field observations distributed as follows:
240 ROMA spectra for D1MAXSW database (solar maximum, 2000-2001) available from Multifractals/ROMA,
1088 ROMA spectra for D3MINSW database (solar minimum, 2007-2008) available from wind Multifractals/ROMA/
36 ROMA spectra for D5MAXSW database (solar minimum, 1997-1998) available from and

In summary the activities devoted to solar wind turbulence and intermittency achieved the STORM objectives, and the main results are summarized below:
The project constructed a data base of solar wind Probability Density Functions with dominant non-Gaussian properties for solar maximum (2000-2001) and solar minimum (1997-1998, 2007-2008)
The project constructed a data base of solar wind partition function and ROMA multifractal spectra for solar maximum (2000-2001) and solar minimum (1997-1998, 2007-2008). Intermittent models of turbulence were tested and provided in general good results.
The project investigated for the first time the evolution of solar wind intermittency with the radial distance for more than one solar cycle for fast and slow solar wind (Wawrzaszek et al., 2015).
The project provides a comparative investigation the properties of intermittent turbulence for fast and slow solar wind in the inner heliosphere for different phases of the solar cycle and different radial distances (Bruno et al., 2014; Teodorescu et al., 2015, Bruno and Telloni, 2015; Telloni et al., 2015.
The project advanced the general understanding of turbulence and intermittency in the solar wind (Bruno et al., 2014, Zaqarashvili et al., 2014, Macek et al., 2014, Vörös, et al., 2015; Yordanova et al., 2015, Narita and Marsch, 2015).


This work package is devoted to the investigation of turbulence, intermittency and multifractals in the planetary plasmas, with emphasis on the Earth and Venus. The tasks defined for the period and how they were accomplished is explained in the following. We followed the same lines adopted for solar wind turbulence, aiming to provide an image of turbulence in the planetary environments that has the same features as the one obtained for solar wind.

2.1.Planetary data base definition

The following objectives are achieved:
• define and construct planetary plasma databases at solar maximum, D2MAXMSPH (2000-2001, Cluster)
• define and construct planetary plasma databases at solar minimum D4MINMSPH (2007-2008, Cluster, Venus Express)

2.1.1.Data selection

The detection of magnetosheath intervals from Cluster data was implemented for the periods Feb-April 2001 and January – April 2002, corresponding to solar maximum activity; and January-April 2007 and January – April 2008, corresponding to solar minimum activity. The selection procedure is based on the simultaneous scanning of the following parameters: spacecraft position, magnetic field magnitude, ion velocity, ion temperature and omni-directional ion energy flux. We used the flux-gate magnetometer (FGM) and the ion spectrometer (CIS) measurements. The parameters are downloaded from Cluster Active Archive (CAA). The minimal length of an interval is 35 minutes. In order to exclude magnetosheath mixing with ion foreshock in the data selection procedure we reject intervals in which the highest energy channels (16 – 30 keV) exceeds certain threshold - FE > (1.000.000 - 2.000.000) keV cm-2 s-1 keV-1. The thresholds assume different values due to a monthly variability in the parameters. It is worth noting that the selection criteria work only when data from all parameters are simultaneously available. Data selection for the terrestrial plasma turbulence
Data set is provided by Cluster fluxgate magnetometer (Balogh et al., 2001), averaged over 4-second spin, in the season February to April, year 2001 - 2002 for the regions of magnetosheath and LLBL/Cusp; in the season February to April, year 2007-2008 for the regions of magnetosheath; and August to September, year 2001 or 2002 for plasma sheet and lobe. Cluster formed a nearly regular tetrahedron at the size in the range between 100 and 1000 km, suitable for detailed wave analysis.
• Magnetosheath: The maximum time length is set to 35 minutes, a typical time length needed to resolve turbulent mirror mode structures in the Earth magnetosheath.
• LLBL/cusps: Interval list is obtained from Echim et al. (2007), Balan et al. (2006), Bogdanova et al. (2005) and Nykyri et al. (2011). Time interval is set to 35 minutes to keep the spectral estimate consistent with the Earth magnetosheath spectra.
• Lobe: the time intervals list is provided by the ECLAT FP7-project (running at the Space Research Institute, Graz) which already identified time intervals for different plasma domains in the Earth magnetotail using Cluster data and list of intervals is given in Appendix A3. Time interval is set to 35 minutes.
• Plasma sheet the Interval list is provided again by FP7 ECLAT project. Time interval is set to 35 minutes. Data selection for Venus plasma turbulence
We used Venus Express magnetometer data at 1-Hz resolution (Zhang et al., (2008)). Data are provided directly by the magnetometer PI at the Space Research Institute, Graz. Later on during the project the data VEX data selection was further refined by inclusion of ASPERA plasma data (Barabash et al., 2008) and the magnetosheath intervals were selected from a simultaneous scanning of magnetometer and plasma data (see Deliverable Report D3.3 for details). Saturn plasma turbulence
Cassini magnetometer data (Dougherty et al., 2004) are available in the Planetary Data System (PDS). Interval list is obtained from the Cassini trajectory plotting tool (provided by the Department of Physics at the University of Iowa) to avoid that the spacecraft is outside the magnetosphere. On some days in January, February, and October 2005, Cassini crosses the Saturn magnetopause at the stand-off distance between 22 Rs and 27 Rs (Achilleos et al., 2008). Time interval is set to 35 minutes. After these periods, no magnetosheath data are available due to the spacecraft maneuver. Mars plasma turbulence
Mars Global Surveyor magnetic field data are obtained from Magnetometer/Electron Reflectometer (MAG/ER) investigation (Acuña et al., 1992; Acuña et al., 1998), available in PDS. Spacecraft orbits during the aerobraking phase (September 1997 - November 1998) have a good coverage in magnetosheath (Espley et al., 2004); Apoapsis decreases from about 54000 km altitude down to about 450 km (cf. Mars planetary radius is about 3400 km). The dayside ionopause/magnetopause is located at about 1.3 planetary radii from the center (i.e. about 1000 km altitude from the surface); the bow shock distance is about 1.7 planetary radii (about 2000 km altitude). Mars Global Surveyor 3-second averaged fluxgate magnetometer data (Acuña et al., 1992) are used from the Planetary Data System (PDS). Interval list is obtained from Luhmann (2002) and the time interval is set to 25 minutes. No Mars magnetosheath data are available after the mapping phase (after 1999). Comet Halley Turbulence
8-second data from Giotto magnetometer (Neubauer et al., 1987) are used (data available in PDS). Interval list is obtained from Glassmeier (1987)) and time interval is set to 35 minutes during its Halley flyby trajectory.

2.2. Power spectral density and Probability Distribution functions in the planetary plasma

2.2.1. Power spectral density analysis
Prior to the computation of the Power Spectral Density (PSD) the magnetic field is transformed into the mean-field-aligned (MFA) coordinate system. The background or large-scale magnetic field is averaged over the entire time series. To estimate the time-dependent PSD, the Welch algorithm is applied on all selected data.

PSD Analysis for D2MAXMSPH database
In the catalogue D2MAXMSPH devoted to solar maximum (2001-2002) we introduced the list of time intervals and the PSD plots for the Earth magnetosheath. The complete list of intervals and PSD plots is available on-line at The catalogue includes 381 PSD spectra for the terrestrial magnetosphere.

PSD Analysis for D4MINMSPH database
This catalogue includes the complete list of time intervals and PSD plots in Earth magnetosheath for solar minimum (2007-2008) and in the Venus magnetosheath (2006-2009). We computed 337 PSDs for the terrestrial magnetosheath, 101 PSD spectra for Venus magnetosheath, 4 PSD spectra for Saturn and Mars turbulence, and 5 PSD spectra for comet Halley magnetosheath turbulence.

2.2.2. Probability Distribution functions analysis

Probability density functions (PDF) are computed for temporal increments of P(t) physical time-series recorded in the planetary plasma following the procedure discussed in 1.2.2. The lower limit of time-scales tau_imin is determined by the sampling time and represents a physical limit for the investigation of small-scale plasma fluctuations. The upper limit of the studied time-scales, on the other hand, was set with the constraint of ensuring statistically reliable number of samples for the PDF study, even for the case of the highest-scale increment time-series. Intermittency in the physical fluctuations appears in the scale-dependency of PDFs computed for different lag times. Scaling can also be investigated through the scale-dependent behaviour of the fourth statistical moment of the increment series, i.e. the flatness. PDFs and flatness vs. scale graphs were compiled for the mean field aligned vector components and absolute vector values of the original planetary plasma time-series. We also investigated the fluctuations in the transversal vector component. All the results were included in the STORM databases D2MAXMSPH (2000-2001) and D4MINMSPH (2007-2008).

In D2MAXMSPH we included 1985 PDFs from Cluster in the terrestrial magnetosphere. In D4MINMSPH we included 1685 PDFs from Cluster in the terrestrial magnetosphere, 1930 PDFs from Venus Express in the Venus magnetosheath, and 15 PDFs from Cassini.

We applied the The Local Intermittency Measure (LIM) analysis in different planetary contexts, mainly to investigate inermittency in the magnetosheaths, at the interface with the solar wind. The algorithm, as described in the previous section of this report devoted to LIM analysis of solar wind data, is also fully implemented in the Interactive Nonlinear Analysis (INA) toolbox such that the teams within STORM Consortium can easily use this analysis approach.

2.3. Multifractal structure of fluctuations in planetary magnetosheaths

The following activities have been performed:
• compute the partition function multifractal spectrum for magnetospheric magnetic and velocity fluctuations
• compute the ROMA spectrum for magnetospheric magnetic and velocity fluctuations

A structural and systematic multifractal analysis with both the partition function as well as the rank ordered (ROMA) formalisms was applied on data from planetary plasma environments. The targeted databases are magnetosheath observations at Earth from Cluster (during solar maximum, 2001-2002, and minimum, 2007-2008) and at Venus from Venus Express (during solar minimum, 2007-2008).

We followed the same steps defined for the multifractal analysis of the solar wind turbulence with some specific cautions adopted : (1) we used non-detrended data; (2) all data gaps have been identified, flagged by 9999.9999 and removed; (3) the data are transformed in The Mean Field Aligned (MFA) system with zMFA axis aligned with a background magnetic field obtained from averaging the magnetic field records within the analyzed time interval; (4) in general the multifractal spectra are computed for the total field, B, the parallel component, Bz_MFA , the perpendicular component Bperp_MFA and B2 (a measure of magnetic field energy). In some cases the analysis is applied on B and Bz_MFA only. All the multifractal spectra are collected into catalogues organized as a function of the targeted system, spacecraft and type of analysis.

The partition function multifractal analysis is applied, as in the case of solar wind data, following the steps described in previous chapter, devoted to Turbulence, Intermittency and Multifractals in the solar wind, at solar maximum and minimum The measure is defined for the scale of one second although data resolution is higher for Cluster. We analized all the data available from Cluster and Venus Express in the magnetosheath at solar maximum (Cluster) and solar minimum (Cluster and Venus Express). The same reference scale (1 sec) is used for Venus Express partition function multifractal analysis. Two turbulence models were tested for Cluster data that are considered to be less affected by the magnetic noise onboard the spacecraft than those from the fluxgate magnetometer on Venus Express.

The stationarity of data was tested and will be further checked for data included in scientific publications. Nevertheless we choose to include into the catalogues and databases the results of all ROMA analyses, although in some cases stationarity is not strictly satisfied.
We produced a total of 2716 partition function multifractal spectra for planetary turbulence distributed as follows :
• 856 partition function multifractal spectra for the terrestrial magnetosphere (mainly magnetosheath) for D2MAXMSPH data base (at solar maximum, 2000-2001) available from
• 316 partition function multifractal spectra for the terrestrial magnetosphere (mainly magnetosheath) for D4MINMSPH (at solar minimum, 2000-2001) available from
• 1544 partition function multifractal spectra for Venus magnetosheath for D4MINMSPH (at solar minimum, 2000-2001) available from

The range of scales for ROMA analysis is selected based on previous STORM results, namely the scale behaviour of the kurtosis, or flatness. The average behaviour of the flatness at minimum (2007-2008) for the Venus and Earth is shown in Figure 11 below, adapted the from STORM Deliverable report D3.2. Similar results are obtained for Cluster at solar maximum (2000-2001). Based on this average behaviour we have been able to identify different scaling ranges. Of utmost interest for intermittency studies is the range of scales where flatness increases with decreasing scales. Such a range was identified in the average behaviour of the flatness. Although the individual flatness profiles may show departure from this average trend it turns out that in general the range of scales [1.1 1.4 2.2 2.9 4.3 5.8 8.6] seconds bears similar features for a majority of data recorded in the magnetosheath at solar minimum. We applied ROMA on this unique range of scales for all the Cluster and Venus Express data included in this analysis.

The convergence of the ROMA analysis of planetary data towards meaningful results was a problem of an even increased complexity than for solar wind data analysis. Indeed, the limited amount of available data, even at full resolution, affected drastically the results. In some cases the ROMA spectra couldn’t be determined. For the planetary plasma analysis we choose to apply the Approach A tested for solar wind data, i.e. following the steps described originally by Chang and Wu (2008). Nevertheless attempts were made to apply the refined method (Wu et al., 2010) but this approach is less adaptable to a semi-automatic algorithm. We produced a total of 9429 ROMA spectra for planetary turbulence distributed as follows :
• 4479 ROMA spectra for the terrestrial magnetosphere (mainly magnetosheath) for D2MAXMSPH data base (at solar maximum, 2000-2001) available from
• 2616 ROMA spectra for the terrestrial magnetosphere (mainly magnetosheath) for D4MINMSPH (at solar minimum, 2000-2001) available from
• 2334 ROMA spectra for Venus magnetosheath for D4MINMSPH (at solar minimum, 2000-2001) available from


We analysed the intermittency of geomagnetic indices fluctuations and compared with quantitative measures of the intermittency in the magnetosphere and the solar wind measured simultaneously by satellites. Two categories of indices are considered: the auroral electrojet indices, AU, AL, AE and the high resolution global indices SYM-H, SYM-D, ASYM-H, ASYM-D.

3.1. Geomagnetic indices data

The geomagnetic indices AE, AL, AU, SYM-H, SYM-D, ASY-H and ASY-D were extracted from the World Data Center for Geomagnetism in Kyoto, Japan. The AE, AL and AU are the auroral zone electrojet indices derived from variations of the horizontal (H) component of the geomagnetic field observed at several (10-13) auroral zone magnetometer stations in the northern hemisphere. The AU and AL indices are, respectively, defined as the largest and smallest momentary values of all the stations. Thus they provide a quantitative measure of the intensity of the eastward and westward auroral electrojets, respectively. AE is the difference between the two, AE = AU - AL. The AE, AU and AL indices are defined at one-minute resolution.

The standard measure for ring current intensity is the Dst index and its one-minute version SYM-H index, which is calculated using measurements from six stations at low and mid-latitudes. SYM-H (SYM-D) describes the average disturbance in the H (declination, D) component. The range of variations of H and D disturbances measured at different stations are measured by ASY-H and ASY-D indices. The ASY-H and ASY-D indices are designed to measure the maximum local time asymmetry of the ring current.

We used one-minute magnetic field measurements from the four low-latitude magnetometer stations of Hermanus (HER), Honolulu (HON), Kakioka (KAK) and San Juan (SJG) (i.e. the Dst stations) to construct a new index analogous to the SYM-H index called the Dcm index. The one-minute Dcm index follows the idea of the hourly Dcx index introduced by Mursula and Karinen (2005). However, in order to extract the regular solar quiet variations (i.e. the QDC) of the geomagnetic field, the Kalman filter method was used in the derivation of the index. The details are given n the deliverable report D4.1. A local Dcm index is computed for each station. The global Dcm index is the average of the four local indices. The Dcm index is analogous to the SYM-H index, both measuring global activity as an average of the ring current intensity. As an analog to the ASY-H index, an asymmetric version of the Dcm index called the Asy-Dcm index is also constructed. It is defined as the maximum difference of any two local Dcm indices for each minute.

The new Am index is also calculated using the Kalman filter method. The index is defined as a sliding 3-minute range of the irregular variations of a station’s H component. First the quiet day trend is removed from the H component, then a sliding 3-minute window is moved point-by-point across the time-series. The index value at each point is the range (maximum-minimum value) within the 3-minute window, which consists of three adjacent points centered on the current point. The Am index is computed for stations in Belsk (BEL), Nurmijärvi (NUR) and Sodankylä (SOD). The above-mentioned three new geomagnetic indices Dcm, Asy-Dcm and Am use one-minute magnetometer measurements extracted from the World Data Center for Geomagnetism in Edinburgh. In addition to these indices, we also used the raw H component data of the same stations (BEL, NUR, SOD) without any further pre-processing in monthly sections to produce PSDs and PDFs.

3.2. Evaluation of “turbulence” and intermittency of geomagnetic indices fluctuations

The Power Spectral Densities (PSDs) and Probability Distribution Functions (PDFs) are computed for all the indices and individual observatory data described above. We focus on two time intervals at solar maximum, between 2000-2001, and respectively at solar minimum, between 2007-2008, when STORM provides analyses of turbulence in the solar wind (at 1 AU) and the terrestrial magnetosheath.
The PSDs were computed with the Welch approach, similar to the algorithm implemented in the STORM INA library. The PDFs are computed from the standard differential measure based on differences computed for a range of scales, tau.

The catalog built for solar maximum, D2MAXMSPH (2000-2001), includes 360 PSDs computed for nine global geomagnetic indices AE, AL, AU, Dst, SYM-H, SYM-D, ASY-H, ASY-D, Dcm, and also for data from three individual observatories and the corresponding local Am indices. One PSD was computed for one month time interval. Similarly for the solar minimum interval D4MINMSPH in 2007-2008, we have included PSDs for 24 intervals for each index and station, amounting to a total of 360 PSDs. The same structure and data analysis strategy is adopted for the Probability Distribution Functions (PDFs). Thus we included a similar number of PDFs at solar minimum and maximum, for the same set of parameters. One PDF is computed for one month of data.

3.3. Multifractal analysis of geomagnetic indices fluctuations

The multifractal analysis of geomagnetic fluctuations was performed with the Partition Function and ROMA approaches along the lines described already in section "1. Turbulence, Intermittency and Multifractals in the solar wind, at solar maximum and minimum." The dependence of the multifractal character of two geomagnetic indices, AE and Sym-H, on the solar activity cycle has been further investigated . We evaluated how the multifractality changes over the solar cycle, and checked its time evolution for AE and Sym-H and compared it with the solar F10.7 radio flux, a measure of solar variability.

The catalogue of partition function multifractal and ROMA spectra is organized as an FTP repository integrated into the website of the project at ( The structure of the FTP repository is defined according to the guidelines stated in the Grant Agreement and its main pillars are the two databases: the solar maximum between 2000-2001 and the solar minimum between 2007-2008. The folder for each database contains partition function multifractal (MFS) and ROMA folders for the geomagnetic indices. Each index has its own folder containing the corresponding set of monthly MFS and ROMA spectra. The catalog contains a total amount of 672 PNG images of ROMA and multifractal and 336 text files for the MFS spectra data.


The data sets treated in STORM are used to advance our understanding on the relationship between turbulence, self organized criticality and multifractals, including the newly developed multifractal technique based on rank ordering (ROMA, Chang and Wu, 2008). The work was organized as a series of tasks as outlined below.

4.1. Theoretical understanding driven by data analysis

The first data base defined as test bed is the one that includes the geomagnetic indices AE, AL and AU geomagnetic indices. The first step of our endeavor is to search for self-organized criticality behavior possibly revealed by finite size scaling effects in the dynamics of the AE indices. In order to understand better the universality of the critical behavior of geomagnetic indices we performed tailored simulations with a stochastic sandpile model (Manna, 1991). The probability distribution functions of the sand-pile model "avalanches" show data collapsing upon finite size rescaling (FSS). This rescaling of PDFs resulting from simulations on lattices of different size L supports the critical nature of the Manna sand-pile model. With this insight from a numerical critical model we checked if FSS hypothesis is valid or not for the AE-index critical behavior on the different phases (solar maxima and minima) of the last three solar cycles, from January, 01, 1978 to December, 31, 2013.

The results of the FSS analysis on AE indices suggests that:
i) the FSS hypothesis is not fully satisfied by AE-index burst distribution, suggesting a more complex nature of the related physical event;
2) the sunspot number is a relevant active parameter in controlling the inner magnetospheric critical dynamics as monitored by the AE index during magnetic substorms.
In particular, the emerging picture seems to point towards a multiscaling nature of the data collapsing and FSS hypothesis. A possible approach to solve this problem is to use the ROMA technique introduced by Chang and Wu (2008) to find the scaling indices spectrum as a function of the rescaled variable.

This analysis was further expanded by investigating the F10.7 solar radio flux as a relevant active parameter and tested FSS effects for AE, AL and SYM-H geomagnetic indices. We obtained a lack of data collapsing and the absence of a master (invariant) distribution for AE and AL indices is well in agreement with the previous results on moment scaling, suggesting that the observed near-criticality dynamics cannot be related with a simple mono-scaling behaviour.

The application of this insight on in-situ data from solar system plasma turbulence was not straight forward. We tried various combination of parameters in order to find the one linked to turbulent dissipation. Several difficulties were encountered: (1) data in the dissipation regime, at small scales, were affected by instrumental limitations; (2) various attempted measures based on in-situ data (like, e.g. magnetic shear) did not show power-law behavior over multi-scale decades, thus was not possible to define “avalanches” and check self-organized criticality effects. This is certainly an investigation to be continued, in fact several new hints emerged during the last phase of STORM, to be tested in the near future.

Due to the gap/limitations in space plasma observational results for testing FSS and SOC with in-situ data we invesigated the critical bahavior based on simulations results of a shell model turbulence. The latter is a simplified simulation of turbulence that captures however some of the basic characteristics, like a robust dissipation estimation. The shell model is discussed in the Deliverable report D5.1 and will not be detailed here. Although the shell model is devoted to neutral turbulence it provides a direct and relevant measure for dissipation, estimated from instanteneous velocities and viscosity values. On this dissipative measure we applied the FSS analysis along the lines described above an found from moment analysis the scaling values. However the data collapsing was not satisfactory, the work on this topic requires more efforts.


5.1. Scaling properties of magnetospheric fluctuations

A study of kinetic instabilities in the magnetosheath turbulence at and bellow proton scales was performed, related to the the STORM's objectives “1. Determine degree of anisotropy (fluctuation anisotropy and wave vector anisotropy) and compressibility in the energy cascade of the respective ranges" and "2. Evaluate the existence and type of wave modes in energy cascade and dissipation regimes”. The analysis of several magnetosheath passes by Cluster spacecraft showed that (Breuillard et al., ApJ, submitted, 2016):
1) The break of power spectra of magnetic field fluctuations occurs at the largest characteristic ion scale, regardless of its nature;
2) Ion instabilities that superpose to the background turbulence can, depending on the plasma parameters, modify the spectra up to a frequency corresponding to the smallest ion scale.
3) In the small-scale range, when no waves are observed, the background turbulence is quasi-isotropic (Bx~By~Bz) with an index of -2.8 consistent with KAW, whistler and compressive Hall-MHD scenarios;
4) The observed non-gyrotropy of kinetic amplitude fluctuations can be due to a 2D turbulence developing in the presence of mirror modes, whereas gyrotropic amplitude fluctuations result from possible development of slab turbulence in the presence of AIC waves;
5) When strong waves dominate, the absolute value of spectral indices of the corresponding components are decreased (|α| ~ 2-2.7) except for the case where both instabilities are developed.

In another study (Voeroes et al., 2016) based on the analysis of high-resolution magnetosheath data from Cluster we attempted to identify thin currents produced by turbulence. The current sheets were detected by a combination of methods/parameters: 1.) the Partial Variance of Increments (PVI) - the normalized variance of the absolute value of magnetic field spatial increments between two spacecraft (Greco et al., 2009); 2.) the angle between magnetic field vectors - estimating the rotation of magnetic field across a discontinuity (e.g. current sheet): and 3.) the partial magnetic field derivatives. The joint occurrence of strong magnetic shear, high PVI indicates and non-zero partial derivatives determines that the corresponding discontinuity is a current sheet. Applying additional conditioning hundreds of proton scale (~0.5 s) currents sheets have been detected in the one-hour interval under investigation with this technique. It was also demonstrated that the smallest PDF values are normally distributed while the strongest discontinuities populate the tails of the distribution as expected from numerical simulations (Greco et al., 2009; Servidio et al., 2009; Matthaeus et al., 2015).

5.2. Investigation of weak turbulence

In STORM we used multi-spacecraft methods to quantify effects due to weak turbulence. The wave telescope method is an application of the minimum variance estimator (also called the least square estimator or the Capon estimator) to the four-spacecraft Cluster measurements in space. The method is a projection of the vectorial quantity (e.g. magnetic field) sampled at four spatial points from the spatial coordinates onto the 3-D wave vector domain (Glassmeier et al., 2001). The method plays a role of the Fourier transform to the wave vector domain. In contrast to the Fourier transform, the wave telescope method estimates the fluctuation amplitudes in the wave vector domain by fitting with a set of plane waves under the constraint of minimizing the isotropic noise in the data.

An advanced version of the wave telescope method is applied in STORM, i.e. the MSR method (Multipoint Signal Resonator, Narita et al., 2011). The effect of the finite noise is eliminated by employing an extended form of the MUSIC (Multiple Signal Classification, Schmitt, 1986) algorithm and coupling it to the minimum variance projection. The MSR method was developed particularly for studying waves and turbulence using four-point magnetic field data, and makes extensive use of the 12-by-12 covariance matrix (three components of the magnetic field measured at four spacecraft) by combining the minimum variance projection and the eigenvalue analysis of the covariant matrix. The MSR technique is based on the assumption that the measured fluctuations represent a set of plane waves and that the fluctuations contain small amplitude isotropic noise. The energy spectrum is given as a function of spacecraft-frame frequencies and wave vectors.

A fitting procedure is applied to the measured spectrum in two distinct domains. One is the plane spanned by the streamwise wavenumbers and the frequencies and the other is the 3-D wave vector domain after integrating the 4-D spectrum over the frequencies. The fitting procedure in the former domain determines the Doppler shift U and the broadening V, while the fitting in the latter domain determines the anisotropy coefficients.

Taylor's hypothesis (Taylor, 1938) assumes that the fluctuating fields such as the flow velocity, the density, and the magnetic field of a flowing plasma are "frozen-in" into the flow such that the time series data of fluctuations are interpreted as spatial structures passing by the observer or the sensor standing in the flow. Fluctuating fields should not evolve during the measurement time. Therefore, Taylor's hypothesis is believed to be valid when the flow speed is high enough and when the measurement time is short enough. Taylor's hypothesis is formulated as the use of the Doppler relation when interpreting the time series data: w = k.U_flow
where w is the wave frequency and k is the wave number, U_flow is the plasma flow velocity. Taylor's hypothesis breaks down when this condition is violated. The study of turbulence spectrum from the wave telescope suggests two particular situations in which Taylor's hypothesis breaks down: (1) the finite-speed wave propagation and (2) the Doppler broadening. Out studies suggest that in the first case the effect of waves (mainly of Alfven type) waves needs to be included for a quantitative estimate of Taylor’s hypothesis in the solar wind.


The Interactive Nonlinear Analysis (INA) library is a tool designed to analyze measurements from: Cluster, Ulysses and Venus Express spacecraft and Geomagnetic Indices (e.g. AE, Dst). For other file types we have implemented general CDF, ASCII, and MAT (Matlab generated files) reading routines. There is also the possibility of generating synthetic signals which can be used to test the analysis methods. The library is defined to be versatile such that the user can use it to make a complete statistical analysis of a time series and compute key-parameters for turbulence like: (i) the Power Spectral Densities, (ii) the Probability Density Functions (PDF), (iii) the wavelet analysis and the local intermittency measures, (iv) higher order moments of PDFs (flatness), (v) the multifractal analysis. Although INA is written in Matlab in order to facilitate the usage of INA outside this programming environment, an executable standalone version was compiled and delivered ( Several analysis modules are available as described below.

6.1. Import Data

Although INA will not interrogate itself the databases, the IMPORT functions are adapted to the precise format of data available from the ESA and NASA databases. INA is also able to ingest general CDF, ASCII and .MAT files. In addition to external data, INA can also generate custom synthetic signals. There is also a section devoted to importing data files that have been previously exported from INA, which can be used to easily exchange and visualize specific analysis results. For other file types or formats, the user can check the section labeled OTHER, where one can also make requests for specific file formats to be implemented in future versions. The data import functionality is divided into four levels giving the user the chance to choose the Data Type (from spacecraft, from file, synthetic, other), the space mission (Cluster, ACE, Ulysses, Venus Express), Experiment (e.g. magnetometer), Data Source (e.g. Ulysses Final Archive).

6.2. Select variable and preprocess

INA adopts a strategy that drives the user to follow a linear progression from importing the data to their analysis:
This sequence is achieved by using the buttons from an “active” control panel displayed in all INA windows at right. There are seven main preprocessing options:
a) square, computes the square of the time series;
b) standardize, subtract the average and divides by the variance;
c) subtract mean;
d) square + standardize
e) square + subtract mean
f) wavelet denoising
g) handling data gaps.
Wavelet denoising performs an automatic denoising using wavelets. The denoising level is controlled by the user. The “handling data gaps” functionality gives the user the possibility to choose between two methods to handle data gaps. By default, INA automatically detects and linearly interpolates all the data gaps. Another option is to fill-in the data gaps with NaN values.

6.3. Spectral Analysis Module

The Analysis layer of INA can be viewed as a “hub” that connects all the analysis methods implemented in the library. The spectral analysis is one method, structured as two sub-modules described below:

6.3.1. Descriptive Statistics
The "Descriptive Statistics" analysis class of INA is designed as a “first look” to the data and includes two subclasses of methods:
a) Periodogram, and
b) Histogram.
The two subclasses accept a number of adjustable settings (in fact, only one per subclasses), and can be used as a “first degree” estimate of some basic statistical properties of the time series. The methods are based on built-in functions like periodogram (which gives a simple nonparametric estimate of the PSD of the input signal using a given window selected by the user), and histogram (which creates a histogram bar plot of the elements of the input signal, y, sorted into a user defined number of equally spaced bins along the x-axis between the minimum and maximum values of y).

6.3.2. Power Spectral Density module
The Power Spectral Analysis class contains two subclasses of methods:
a) Power Spectral Density (PSD) analysis with the Welch algorithm (PSD-Welch analysis) and
b) the analysis of the evolution of the PSD in time – the Spectrogram analysis.

This functionality is devoted to computing the Power Spectral Density (PSD) of the selected variable using the Welch algorithm. The user controls the analysis and can modify a list of options organized in five categories: i) parameters, ii) “logmean”, iii) “frequency zoom”, iv) display parameters and v) slope analysis. There are four adjustable display parameters that control the graphical illustration of the spectrum. The Welch algorithm is widely used in the community, for a reference in the field the user may check an ESA document available online at: The Welch approach is based on splinting the signal in N possibly overlapping sub-segments of data of length L each. The PSD is computed with a standard periodogram routine for each segment and the resulting PSDs for all segments are averaged to produce the final PSD result assigned to the analyzed time series. Prior to computing the PSD one applies a window in order to reduce the lobe effects. The user can modify the default analysis parameters for the PSD-Welch method by choosing a different window type, a different individual segment length and/or a different overlap between adjacent segments. INA offers 16 different types of windowing.

In addition to computing one PSD spectrum for a selected time interval using the Welch algorithm, INA also offers the possibility to compute a series of PSD spectra and display their evolution in time on a spectrogram. The spectrogram is a three dimensional plot (time, frequency, PSD) where the third dimension (corresponding to PSD values) is color coded. The user can choose the type of windowing applied prior to the Fourier analysis. The default type is the modified Bartlett-Hann window. It is also possible to choose the number of individual PSD spectra to be computed for the spectrogram. The user can also specify the degree of overlapping between two consecutive windows

6.4. Probability Distribution Functions

The Probability Density module includes four analysis tools, all related to the Probability Distribution Functions (PDFs):
1. Computing the probability distribution functions for a range of scales;
2. Computing the flatness (fourth order moment, or kurtosis) of the PDFs;
3. Computing a parameter to be used in an attempt to rescale all the computed PDFs;
4. Effectively apply the rescaling procedure based on the One Parameter Rescaling (OPR).

The PDFs subclass computes by default the unscaled Probability Density Functions (PDFs) for a number of N scales; each scale comprises a number of points equal to 2^tau, where tau takes values between tau_min, in general equal to 0, and tau_max-1, where tau_max is the smallest power of 2 for which 2^tau_max is still larger than the total length of the time series. The PDFs are obtained by moving a sliding window of length 2^tau over the entire time series to be analyzed and taking the difference between the right and left edge points of the window. The window is displaced by one point at each step, thus consecutive windows overlap. The normalized histogram of the differences/increments gives the PDF at that scale. The computation of PDFs has four adjustable parameters: the smallest scale, the increment between considered scales, the maximal scale, and the number of bins used for estimating he normalized histograms.

The Flatness routine computes by default the Flatness factor by normalizing the fourth order moment of the PDFs by the square of the second order moment. The flatness is computed for the same scales as those used for computing the PDFs. The Flatness computation has the same adjustable parameters as the PDFs method described above.

The one parameter rescaling factor is computed based on the assumption that the probability distributions can be described by a stable, symmetric shape (like a Gaussian or Levy) PDF. The scaling exponent is determined from the slope of the peak value of the PDFs, P (0, tau), versus the scale. The OPR function is used to obtain P (0, tau) as a function of tau and the slope is computed automatically.

6.5. Wavelet Analysis

This INA module is based on two subclasses of methods: a) Scalogram, and b) LIM (Local Intermittency Measure). The wavelet scalogram returns a three-dimensional color representation of the wavelet coefficients, where the horizontal axis represents the time t, the vertical axis represents the scale a, and the color scale (z axis) represents the logarithm of the squared modulus of the wavelet coefficients The Local Intermittency Measure (LIM) is defined from the wavelet representation as a normalized distribution, the normalization coefficient is given by the time average for the corresponding scale. LIM gives thus the ratio between a local wavelet coefficient associated to a time t and scale a and the time average of the coefficients belonging to the same scale a.

6.6. The structure function analysis

The structure function (SF) of order q, for the time series y(t) is defined as:
SF(q,tau)= <|y(t+tau)-y(t)|^q >
where <> means ensemble averaging, tau is the scale and q gives the order of the SF analysis. In the conventional structure function analysis one searches for a power law variation of SF with the scale tau. The case when the scaling exponent is a linear function of the order q is of special interest as it can mean topological self-similarity. The default set of SF analysis parameters can be modified by the user through dedicated "action" buttons of a control panel. From a set of scales and orders chosen by the user the program also computes the SF exponent as a function of q. The superposed linear fit of the SF exponent as a function of q, is one of the main results of the conventional structure function analysis, giving a direct visual representation of the linearity (mono-fractal) or non-linearity (multi-fractal) character of the structure function scaling.

6.7. Rank Ordered Multifractal Analysis (ROMA)

ROMA is a complex analysis method, even a brief and technical report needs an introduction on the specific characteristics of the method itself. We defered the reader to the brief introduction given in chapter 1.5 as well as in the tutorial available on-line and specifically dedicated to this novel type of analysis ( ROMA is method for the statistical description of fluctuations based on multifractals. Thus the output, the ROMA spectrum, must be supplemented by preliminary tests and analyses to understand and validate the results. The final aim of ROMA is to fully collapse the PDFs at all scales onto a single master curve. In practice the one parameter rescaling procedure, based on a single parameter does not collapse all the data. Therefore ROMA seach for those values that can achieve partially collapsing, for "chuncks" of PDFs. The final series of rescaling, fluctuations dependent, numbers form the ROMA spectrum. Details on the technical procedure on how this spectrum is obtained are given in the STORM Deliverable reports D2.3 and D3.3 as well as in the tutorial document cited above.

6.8. Exporting Analysis Results

The EXPORT functionality allows the user to save the results of the analysis either as a graphical or binary object or both. The functionality is designed to be easily accessible at any point during the analysis. For this reason, INA offers a wizard than can be called from the EXPORT button, available in any module of the program. INA automatically names each file by defining a root name in the form: “INA4v2_yymmddHHMMSS_layer_type”, where the first part gives the software version, the second part is a time stamp recording the exact time of the export, the third part contains the name of the layer and the fourth part differentiates between various exported file types. “Name” is an editable text field, and the user can modify it and choose his own root name. There is also the option of choosing an extension for the exported figures: the default option is .png (with a 600dpi resolution), but the user can also export .eps files and even .fig files (which can be used only inside MATLAB).
Potential Impact:
STORM promoted an innovative approach for the analysis of satellite and ground based data bases. It built a prototype scheme for turbulence analysis that goes beyond the low order analysis (like, e.g. power spectral densities) to assess the topology of fluctuations and the statistical properties of fluctuations at all available scales.

The project stimulated close scientific cooperation between groups operating with different methods and pertaining to different schools of academic thought. Many studies involved at least two teams and we report a relatively large number of mutual visits as well as joint participation and/or organization at/of relevant sessions at major international conferences.

There is a strong component of capacity building and innovation achieved in STORM. New collaborations were actively formed and performed very well and productively. Virtually each STORM team enlarged their field of research by targeting new methodologies and/or type of data and/or physical systems of the solar system. An effort of such scale is unprecedented at European level and could not have been possible without a dedicated funding tool like the SPACE Call of the European FP7 Programme.

The number of joint papers involving at least two STORM teams has steadily increased since the beginning of the project, a sign of an increased mutual collaboration. In average 30% of the published papers and 65% of the communications to conferences were co-authored by scientists from at least two different STORM teams.

STORM promoted the principle of fair societal return by making sure that STORM data and methods are available freely for the academic and scientific community, through various vectors, like the web page of the project and meetings organized on several occasions all over the world.

STORM adopted a consistent policy for the dissemination of the scientific results through publications in international scientific journals with very good scientific reputation and visibility like Journal of Geophysical Research, Geophysical Research Letters, Astrophysical Journal and Astrophysical Journal Letters, Annales Geophysicae.

The members of STORM presented the results of the project at all major scientific events of the field between 2013-2015 either as speaker and/or main organisers of scientific sessions (European and American Geophysical Unions, the International Association for Geomagnetism and Aeronomy). This activity will continue in the future as a number of on-going studies will end after the nominal closure of the project. Indeed, we can report a scientific "latency" of STORM that goes beyond the nominal life of the project. We will continue to acknowledge the EU support in all publications/conference materials resulting/initated from/by STORM. As an example, the results of STORM will be presented in the frame of the session devoted to European Funded Projects at the General Assembly of the European Geophysical Union in Vienna, April 2016.

The international workshop « Solar system plasma turbulence, intermittency and multifractals » organized in Mamaia, between 6-13 September was the ideal occasion to promote STORM at an international level and to share the expertise aquired during the project. STORM participants were seminal in lecturing during the summer school organized on the occasion. This activity is further detailed below.

Participation in STORM triggered increased national support for some of the local teams. At least eight young scientists from seven of the eight STORM teams received support from the project during 2013-2015. The teams also reinforced their position at national level and consolidated their international profile.

STORM is acknowledged as a precursor activity relevant for future European space missions, notably for THOR, an M4 class mission currently in Phase A development by the European Space Agency. Indeed THOR targets solar wind and magnetosheath turbulence, precisely the main fields of scientific interest for STORM. THOR proposal makes explicit reference to STORM as relevant repository of turbulence data analysis methodology,

Another future European mission, the L-class Solar Orbiter ( is also relevant when discussing the impact of STORM. Indeed the STORM data analysis and studies devoted to the radial evolution of turbulence and intermittency are of interest for this mission that will approach the sun at unprecedented short distace, probing perhaps the "craddle" of solar wind turbulence and intermitency.

The impact of STORM can be measured by the publications, communications, outreach activities resulting from the project. Indeed, STORM produced 44 published papers, 7 submitted manuscripts, 34 invited talks at international workshops and conferences,
68 contributed oral presentations at international workshops and conferences, 56 posters at international workshops and conferences
4 special sessions at international Conferences (EGU Vienna, 2014, AOGS Sapporo 2014, AGU Fall San Francisco 2014, IAGA Prague 2015)
International Workshop and School on solar system plasma turbulence and intermittency, Mamaia, Romania, 2015
Outreach activities:
- STORM Splinter Organized during the European Geosciences Union General Assembly Vienna, Austria, 27 April – 02 May 2014.
- STORM project presented in the Open Doors Day, MFGI, Budapest, Hungary, 23 April, 2014.
- STORM presented in the Open-door day in science "Lange Nacht der Forschung", Space Research Institute, Graz, Austria, 4 April, 2014.
- STORM poster presented in the Researchers Night in Bucharest, 27 Sep, 2013.
- The First Romanian Space Weather Day organized with STORM support and participation in Magurele, Bucharest, Romania, 25 June, 2013.
- STORM project introduced at the Open Doors Day, Space Pole Uccle, Brussels, Belgium, 25 May, 2013.
- STORM project presented in the Open Doors Day, MFGI, Budapest, Hungary, 9 May, 2013.
- Data analysis techniques for turbulence and intermittency, Echim M., Kovács P. and Munteanu C., summer school lecture in STORM2015, Workshop and School, Mamaia, Romania, 06-13 September, 2015
- Grazer Weltraumtag 2015 (Graz Space day 2015) 2 October 2015 Y. Narita and P. Bourdin have presented STORM turbulence research in the solar system at a public outreach program on the space research activities in the city Graz
- Macek W.M. Multifractal Turbulent Structures at the Heliospheric Boundaries, talk at International Space Science Institute, Meeting 13-27 April 2015, Bern, Switzerland.
- Macek W.M. Turbulence at the Heliospheric Boundaries and Beyond, talk at International Space Science Institute, Meeting 4-9 October 2015, Bern, Switzerland.
- STORM project presented in the open doors event, Földtudományos Forgatag, Hungarian Natural History Museum, 7-8 November 2015, Budapest, Hungary
- Wawrzaszek A., Analysis of the Solar System plasma turbulence in the frame of STORM project, talk at Space Research Centre, Polish Academy of Sciences, 21 May 2015, Warsaw, Poland.
- Wawrzaszek A., Multifractal description of the turbulence in the heliosphere, talk at University of Natural Sciences and Humanities, 21 October 2015 Siedlce, Poland.
- M. Echim, STORM – an FP7 project to study plasma turbulence, European Space Weather Week, 23-27 November 2015, Ostende, Belgium
- Grazer Weltraumtag 2014 (Graz space day 2014) 19 September 2014 Y. Narita, N. Dwivedi, and P. Bourdin have presented STORM turbulence research in the solar system at a public outreach program on the space research activities in the city Graz together with the current and upcoming spacecraft missions such as Venus Express, Cluster, and Rosetta
- STORM poster presented in the public event Researchers Night, Bucharest, 26 September, 2014.
- Turbulence in Space and on Earth, M. Echim, Open Doors Day, Space Pole, Uccle, Belgium, 6-7 October 2014
- STORM – an FP7 European project coordinated by IASB to study turbulence in the solar system, M. Echim, Open Doors Day, Space Pole, Uccle, Belgium, 6-7 October 2014

STORM Workshop and School 2015

The event called “International Workshop and School on solar system plasma turbulence, intermittency and multifractals (STORM 2015)” was organized in Mamaia, Romania from 6 to 13 September 2015 and joined 38 participants (of which 18 members of STORM) from 17 countries, a total of 37 presentations of which 9 tutorials, 9 invited talks, 7 contributed oral talks and 12 posters. The members of STORM presented 7 tutorials, 5 invited talks, 5 contributed talks and 2 posters. One afternoon was devoted to a hand-on session with STORM Interactive Nonlinear Analysis tool (INA). The workshop and school was attended by 13 international young scientists (11 from Europe, 3 members of STORM) whose participation was sponsored by international institutions like the European Geophysical Union (EGU), the European Space Agency (ESA), the International Association for Geomagnetism and Aeronomy (IAGA). The US Office for Naval Research Global (ONRG) sponsored the travel and accommodation of some international scientists.

The programme was organised as series of tutorial lectures before noon and contributed and invited papers in the afternoon. The lectures and talks were very well attended, the poster session gave the occasion for lively discussions and open exchange of ideas between generations. The hands-on session with the STORM Integrated Nonlinear Analysis (INA) library built within the FP7 project STORM was another opportunity for the audience to learn about and test live the functionalities of this tool adapted for the analysis of non-linear features of time series. It was an ideal occasion to make publicity to this analysis tool as virtually all the participants attended the hands-on session. Each participant received an USB key with a pre-installed virtual machine to be run on any type of operating system and on which a pre-compiled INA toolbox was installed such that the user has no need to access a MATLAB license.

The full details on the even are published in the webpage of the workshop:
List of Websites:
Dr. Marius Echim
Belgian Institute for Space Aeronomy
Avenue Circulaire 3
1180 Bruxelles