This point of view is the result of data analysis of scaling laws of turbulent fluctuations made by using wavelets filters (see Appendix C) instead of the usual Fourier transform. Unlike the Fourier basis, wavelets allow a decomposition both in time and frequency (or space and scale). In analyzing intermittent structures it is useful to introduce a measure of local intermittency, as for example the Local Intermittency Measure (LIM) introduced by Farge (1992) (see Appendix C).
The spatial structures generating intermittency have been investigated by Veltri and Mangeney (1999), using the Haar basis applied to time series of thirteen months of velocity and magnetic data from ISEE s/c. Analyzing intermittent events, they found that intermittent events occur on time scale of the order of few minutes and that they are one-dimensional structures (in agreement with Carbone et al., 1995b). In particular, they found different types of structures which can represent two different categories:
- Some of the structures are the well known one-dimensional current sheets, characterized by pressure balance and almost constant density and temperature. When a minimum variance analysis is made on the magnetic field near the structure, it can be seen that the most variable component of the magnetic field changes sign. This component is perpendicular to the average magnetic field, the third component being zero. An interesting property of these structures is that the correlation between velocity and magnetic field within them is opposite with respect to the rest of fluctuations. That is, when they occur during Alfvénic periods velocity and magnetic field correlation is low; on the contrary, during non-Alfvénic periods the correlation of structure increases.
- A different kind of structures looks like a shock wave. They can be parallel shocks or slow-mode shocks. In the first case they are observed on the radial component of the velocity field, but are also seen on the magnetic field intensity, proton temperature, and density. In the second case they are characterized by a very low value of the plasma parameter, constant pressure, anti-correlation between density and proton temperature, no magnetic fluctuations, and velocity fluctuations directed along the average magnetic field.
However, Salem et al. (2009), as already anticipated in Section 3.1.1, demonstrated that a monofractal can be recovered and intermittency eliminated simply by subtracting a small subset of the events at small scales.
Given a turbulent time series, as derived in the solar wind, a very interesting statistics can be made on the time separation between the occurrence of two consecutive structures. Let us consider a signal, for example or derived from solar wind, and let us define the wavelets set as the set which captures, at time , the occurrence of structures at the scale . Then define the waiting times , as that time between two consecutive structures at the scale , that is, between and . The PDFs of waiting times are reported in Figure 82. As it can be seen, waiting times are distributed according to a power law extended over at least two decades. This property is very interesting, because this means that the underlying process for the energy cascade is non-Poissonian. Waiting times occurring between isolated Poissonian events, must be distributed according to an exponential function. The power law for represents the asymptotic behavior of a Lévy function with characteristic exponent . This describes self-affine processes and are obtained from the central limit theorem by relaxing the hypothesis that the variance of variables is finite. The power law for waiting times we found is a clear evidence that long-range correlation (or in some sense “memory”) exists in the underlying cascade process.
On the other hand, Bruno et al. (2001), analyzing the statistics of the occurrence of waiting times of magnetic field intensity and wind speed intermittent events for a short time interval within the trailing edge of a high velocity stream, found a possible Poissonian-like behavior with a characteristic time around 30 min for both magnetic field and wind speed. These results are to be compared with previous estimates of the occurrence of interplanetary discontinuities performed by Tsurutani and Smith (1979), who found a waiting time around 14 min. In addition, Bruno et al. (2001), taking into account the wind speed and the orientation of the magnetic field vector at the site of the observation, in the hypothesis of spherical expansion, estimated the corresponding size at the Sun surface that resulted to be of the order of the photospheric structures estimated also by Thieme et al. (1989). Obviously, the Poissonian statistics found by these authors does not agree with the clear power law shown in Figure 82. However, Bruno et al. (2001) included intermittent events found at all scales while results shown in Figure 82 refer to waiting times between intermittent events extracted at the smallest scale, which results to be about an order of magnitude smaller than the time resolution used by Bruno et al. (2001). A detailed study on this topic would certainly clarify possible influences on the waiting time statistics due to the selection of intermittent events according to the corresponding scale.
In the same study by Bruno et al. (2001), these authors analyzed in detail an event characterized by a strong intermittent signature in the magnetic field intensity. A comparative study was performed choosing a close-by time interval which, although intermittent in velocity, was not characterized by strong magnetic intermittency. This time interval was located a few hours apart from the previous one. These two intervals are indicated in Figure 96 by the two vertical boxes labeled 1 and 2, respectively. Wind speed profile and magnetic field magnitude are shown in the first two panels. In the third panel, the blue line refers to the logarithmic value of the magnetic pressure , here indicated by ; the red line refers to the logarithmic value of the thermal pressure , here indicated by and the black line refers to the logarithmic value of the total pressure , here indicated by , including an average estimate of the electrons and s contributions. Magnetic field intensity residuals, obtained from the LIM technique, are shown in the bottom panel. The first interval is characterized by strong magnetic field intermittency while the second one is not. In particular, the first event corresponds to a relatively strong field discontinuity which separates two regions characterized by a different bulk velocity and different level of total pressure. While kinetic pressure (red trace) does not show any major jump across the discontinuity but only a light trend, magnetic pressure (blue trace) clearly shows two distinct levels.
A minimum variance analysis further reveals the intrinsic different nature of these two intervals as shown in Figure 97 where original data have been rotated into the field minimum variance reference system (see Appendix D.1) where maximum, intermediate and minimum variance components are identified by , , and , respectively. Moreover, at the bottom of the column we show the hodogram on the maximum variance plane , as a function of time on the vertical axis.
The good correlation existing between magnetic and velocity variations for both time intervals highlights the presence of Alfvénic fluctuations. However, only within the first interval the magnetic field vector describes an arc-like structure larger than on the maximum variance plane (see rotation from A to B on the 3D graph at the bottom of the left column in Figure 97) in correspondence with the time interval identified, in the profile of the magnetic field components, by the green color. At this location, the magnetic field intensity shows a clear discontinuity, changes sign, shows a hump whose maximum is located where the previous component changes sign and, finally, keeps its value close to zero across the discontinuity. Velocity fluctuations are well correlated with magnetic field fluctuations and, in particular, the minimum variance component has the same value on both sides of the discontinuity, approximately 350 km s–1, indicating that there is no mass flux through the discontinuity. During this interval, which lasts about 26 min, the minimum variance direction lies close to the background magnetic field direction at so that the arc is essentially described on a plane perpendicular to the average background magnetic field vector. However, additional although smaller and less regular arc-like structures can be recognized on the maximum variance plane , and they tend to cover the whole interval.
Within the second interval, magnetic field intensity is rather constant and the three components do not show any particular fluctuation, which could resemble any sort of rotation. In other words, the projection on the maximum variance plane does not show any coherent path. Even in this case, these fluctuations happen to be in a plane almost perpendicular to the average field direction since the angle between this direction and the minimum variance direction is about .
Further insights about differences between these two intervals can be obtained when we plot the trajectory followed by the tip of the magnetic field vector in the minimum variance reference system, as shown in Figure 98. The main difference between these two plots is that the one relative to the first interval shows a rather patchy trajectory with respect to the second interval. As a matter of fact, if we follow the displacements of the tip of the vector as the time goes by, we observe that the two intervals have a completely different behavior.
Within the first time interval, the magnetic field vector experiences for some time small displacements around a given direction in space and then it suddenly performs a much larger displacement towards another direction in space, about which it starts to wander again. This process keeps on going several times within this time interval. In particular, the thick green line extending from label A to label B refers to the arc-like discontinuity shown in Figure 97, which is also the largest directional variation within this time interval. Within the second interval, the vector randomly fluctuates in all direction and, as a consequence, both the 3D trajectory and its projection on the maximum variance plane do not show any large empty spot. In practice, the second time interval, although longer, is similar to any sub-interval corresponding to one of the trajectory patches recognizable in the left hand side panel. As a matter of fact, selecting a single patch from the first interval and performing a minimum variance analysis, the maximum variance plane would result to be perpendicular to the local average magnetic field direction and the tip of the vector would randomly fluctuate in all directions. The first interval can be seen as a collection of several sub-intervals similar to interval #2 characterized by different field orientations and, possibly, intensities. Thus, magnetic field intermittent events mark the border between adjacent intervals populated by stochastic Alfvénic fluctuations.
These differences in the dynamics of the orientation of the field vector can be appreciated running the two animations behind Figures 99 and 100. Although the data used for these movies do not exactly correspond to the same time intervals analyzed in Figure 96, they show the same dynamics that the field vector has within intervals #1 and #2. In particular, the animation corresponding to Figure 99 represents interval #2 while, Figure 100 represents interval #1.
The observations reported above suggested these authors to draw the sketch shown in Figure 101 that shows a simple visualization of hypothetical flux tubes, convected by the wind, which tangle up in space. Each flux tube is characterized by a local field direction and intensity, and within each flux tube the presence of Alfvénic fluctuations makes the magnetic field vector randomly wander bout this direction. Moreover, the large scale is characterized by an average background field direction aligned with the local interplanetary magnetic field. This view, based on the idea that solar wind fluctuations are a superposition of propagating Alfvén waves and convected structures (Bavassano and Bruno, 1989), strongly recalls the work by Tu and Marsch (1990a, 1993) who suggested the solar wind fluctuations being a uperposition of pressure balance structure (PBS) type flux tubes and Alfvén waves. In the inner heliosphere these PBS-type flux tubes are embedded in the large structure of fast solar wind streams and would form a kind of spaghetti-like sub-structure, which probably has its origin t the base of the solar atmosphere.
The border between these flux tubes can be a tangential discontinuity where the total pressure on both sides of the discontinuity is in equilibrium or, as in the case of interval #1, the discontinuity is located between two regions not in pressure equilibrium. If the observer moves across these tubes he will record the patchy configuration shown in Figure 100 relative to interval #1. Within each flux tube he will observe a local average field direction and the magnetic field vector would mainly fluctuate on a plane perpendicular to this direction. Moving to the next tube, the average field direction would rapidly change and magnetic vector fluctuations would cluster around this new direction. Moreover, if we imagine a situation with many flux tubes, each one characterized by a different magnetic field intensity, moving across them would possibly increase the intermittent level of the fluctuations. On the contrary, moving along a single flux tube, the same observer would constantly be in the situation typical of interval #2, which is mostly characterized by a rather constant magnetic field intensity and directional stochastic fluctuations mainly on a plane quasi perpendicular to the average magnetic field direction. In such a situation, magnetic field intensity fluctuations would not increase their intermittency.
A recent theoretical effort by Chang et al. (2004), Chang (2003), and Chang and Wu (2002) models MHD turbulence in a way that recalls the interpretation of the interplanetary observations given by Bruno et al. (2001) and, at the same time, reminds also the point of view expressed by Farge (1992) in this section. These authors stress the fact that propagating modes and coherent, convected structures share a common origin within the general view described by the physics of complexity. Propagating modes experience resonances which generate coherent structures, possibly flux tubes, which, in turn, will migrate, interact, and, eventually, generate new modes. This process, schematically represented in Figure 102, which favors the local generation of coherent structures in the solar wind, fully complement the possible solar origin of the convected component of interplanetary MHD turbulence.
Interesting enough is to look at the statistics of the angular jumps relative to the orientation of the magnetic field vector. Studies of this kind can help to infer the relevance of modes and advected structures within MHD turbulent fluctuations. Bruno et al. (2004) found that PDFs of interplanetary magnetic field vector angular displacements within high velocity streams can be reasonably fitted by a double log-normal distribution, reminiscent of multiplicative processes following turbulence evolution. As a matter of fact, the multiplicative cascade notion was introduced by Kolmogorov into his statistical theory (Kolmogorov, 1941, 1991, 1962) of turbulence as a phenomenological framework to accommodate extreme behavior observed in real turbulent fluids.
The same authors, studying the radial behavior of the two lognormal components of this distribution concluded that they could be associated with Alfvénic fluctuations and advected structures, respectively. In particular, it was also suggested that the nature of these advected structures could be intimately connected to tangential discontinuities separating two contiguous flux tubes (Bruno et al., 2001). Whether or not these fluctuations should be identified with the 2D turbulence was uncertain since their relative PDF, differently from the one associated with Alfvénic fluctuations, did not show a clear radial evolution. As a matter of fact, since 2D turbulence is characterized by having its k vectors perpendicular to the local field it should experience a remarkable evolution given that the turbulent cascade acts preferably on wave numbers perpendicular to the ambient magnetic field direction, as suggested by the three-wave resonant interaction (Shebalin et al., 1983). Obviously, an alternative solution would be the solar origin of these fluctuations. However, it is still unclear whether these structures come directly from the Sun or are locally generated by some mechanism. Some theoretical results (Primavera et al., 2003) would indicate that coherent structures causing intermittency in the solar wind (Bruno et al., 2003a), might be locally created by parametric decay of Alfvén waves. As a matter of fact, coherent structures like current sheets are continuously created when the instability is active (Primavera et al., 2003).
A more recent analysis (Borovsky, 2008) on changes in the field direction experienced by the solar wind magnetic field vector reproposed the picture that the inner heliosphere is filled with a network of entangled magnetic flux tubes (Bruno et al., 2001) and interpreted these flux tubes like fossil structures that originate at the solar surface. These tubes are characterized by strong changes in the magnetic field direction as shown by the distribution illustrated in Figure 104 that refers to the occurrence of changes in the magnetic field direction observed by ACE for about 7 years for a time scale of roughly 2 minutes. Two exponential curves have been used to fit the distribution, one for the small angular change population and one for the large angular change population. The small angular-change population is associated with fluctuations active within the flux tube while, the second population would be due to large directional jumps identifying the crossing of the border between adjacent flux tubes. The same authors performed similar analyses on several plasma and magnetic field parameters like velocity fluctuations, alpha to proton ratio, proton and electron entropies, and found that also for these parameters small/large changes of these parameters are associated with small/large angular changes confirming the different nature of these two populations. Larger flux tubes, originating at the Sun, thanks to wind expansion which would inhibit reconnection, would eventually reach 1 AU.
In another recent paper, Li (2008) developed a genuine data analysis method to localize individual current sheets from a turbulent solar wind magnetic field sample. He noticed that, in the presence of a current sheet, a scaling law appears for the cumulative distribution function of the angle between two magnetic field vectors separated by some time lags. In other words, if we define the function to represent the frequency of having the measured angle between magnetic vectors separated by a time lag larger than we expect to have the following scaling relation:
As a matter of fact, if the distribution function above a certain critical angle is dominated by current-sheet crossing separating two adjacent flux tubes, we expect to find the scaling represented by relation 69. On the contrary, if we are observing these fluctuations within the same side of the current sheet is dominated by small angular fluctuations and we do not expect to find any scaling.
Using the same methodology, Li et al. (2008) also studied fluctuations in the Earth’s magnetotail to highlight the absence of similar structures and to conclude that most of those advected structures observed in the solar wind must be of solar origin.
Marsch and Liu (1993) investigated for the first time solar wind scaling properties in the inner heliosphere. These authors provided some insights on the different intermittent character of slow and fast wind, on the radial evolution of intermittency, and on the different scaling characterizing the three components of velocity. In particular, they found that fast streams were less intermittent than slow streams and the observed intermittency showed a weak tendency to increase with heliocentric distance. They also concluded that the Alfvénic turbulence observed in fast streams starts from the Sun as self-similar but then, during the expansion, decorrelates becoming more multifractal. This evolution was not seen in the slow wind, supporting the idea that turbulence in fast wind is mainly made of Alfvén waves and convected structures (Tu and Marsch, 1993), as already inferred by looking at the radial evolution of the level of cross-helicity in the solar wind (Bruno and Bavassano, 1991).
Bruno et al. (2003a) investigated the radial evolution of intermittency in the inner heliosphere, using the behavior of the flatness of the PDF of magnetic field and velocity fluctuations as a function of scale. As a matter of fact, probability distribution functions of fluctuating fields affected by intermittency become more and more peaked at smaller and smaller scales. Since the peakedness of a distribution is measured by its flatness factor, they studied the behavior of this parameter at different scales to estimate the degree of intermittency of their time series, as suggested by Frisch (1995).
In order to study intermittency they computed the following estimator of the flatness factor :
In particular, vector field, like velocity and magnetic field, encompasses two distinct contributions, a compressive one due to intensity fluctuations that can be expressed as , and a directional one due to changes in the vector orientation . Obviously, relation takes into account also compressive contributions, and the expression is always true.
- Magnetic field fluctuations are more intermittent than velocity fluctuations.
- Compressive fluctuations are more intermittent than directional fluctuations.
- Slow wind intermittency does not show appreciable radial dependence.
- Fast wind intermittency, for both magnetic field and velocity, clearly increases with distance.
- Magnetic and velocity fluctuations have a rather Gaussian behavior at large scales, as expected, regardless of type of wind or heliocentric distance.
Moreover, they also found that the intermittency of the components rotated into the mean field reference system (see Appendix D.1) showed that the most intermittent component of the magnetic field is the one along the mean field, while the other two show a similar level of intermittency within the associated uncertainties. Finally, with increasing the radial distance, the component along the mean field becomes more and more intermittent with respect to the transverse components. These results agree with conclusions drawn by Marsch and Tu (1994) who, analyzing fast and slow wind at 0.3 AU in Solar Ecliptic (SE hereafter) coordinate system, found that the PDFs of the fluctuations of transverse components of both velocity and magnetic fields, constructed for different time scales, were appreciably more Gaussian-like than fluctuations observed for the radial component, which resulted to be more and more spiky for smaller and smaller scales.
However, at odds with results by Bruno et al. (2003a), Tu et al. (1996) could not establish any radial dependence due to the fact that their analysis was performed in the SE reference system instead of the mean field reference system as in the analysis of Bruno et al. (2003a). As a matter of fact, the mean field reference system is a more natural reference system where to study magnetic field fluctuations.
The reason is that components normal to the mean field direction are more influenced by Alfvénic fluctuations and, as a consequence, their fluctuations are more stochastic and less intermittent. This effect largely reduces during the radial excursion mainly because in the SE reference system cross-talking between different components is artificially introduced. As a matter of fact, the presence of the large scale spiral magnetic field breaks the spatial symmetry introducing a preferential direction parallel to the mean field. The same Bruno et al. (2003b) showed that it was not possible to find a clear radial trend unless magnetic field data were rotated into this more natural reference system.
On the other hand, it looks more difficult to reconcile the radial evolution of intermittency found by Bruno et al. (2003b) and Marsch and Liu (1993) in fast wind with conclusions drawn by Tu et al. (1996), who stated that “Neither a clear radial evolution nor a clear anisotropy can be established. The value of P1 in high-speed and low-speed wind are not prominent different.”. However, it is very likely that the conclusions given above are related with how to deal with the flat slope of the spectrum in fast wind near 0.3 AU. Tu et al. (1996) concluded, indeed: “It should be pointed out that the extended model cannot be used to analyze the intermittency of such fluctuations which have a flat spectrum. If the index of the power spectrum is near or less than unity …P1 would be 0.5. However, this does not mean there is no intermittency. The model simply cannot be used in this case, because the structure function(1) does not represent the effects of intermittency adequately for those fluctuations which have a flat spectrum and reveal no clear scaling behavior”.
Bruno et al. (2003a) suggested that, depending on the type of solar wind sample and on the heliocentric distance, the observed scaling properties would change accordingly. In particular, as the radial distance increases, convected, coherent structures of the wind assume a more relevant role since the Alfvénic component of the fluctuations is depleted. This would be reflected in the increased intermittent character of the fluctuations. The coherent nature of the convected structures would contribute to increase intermittency while the stochastic character of the Alfvénic fluctuations would contribute to decrease it. This interpretation would also justify why compressive fluctuations are always more intermittent than directional fluctuations. As a matter of fact, coherent structures would contribute to the intermittency of compressive fluctuations and, at the same time, would also produce intermittency in directional fluctuations. However, since directional fluctuations are greatly influenced by Alfvénic stochastic fluctuations, their intermittency will be more or less reduced depending on the amplitude of the Alfvén waves with respect to the amplitude of compressive fluctuations.
The radial dependence of the intermittency behavior of solar wind fluctuations stimulated Bruno et al. (1999b) to reconsider previous investigations on fluctuations anisotropy reported in Section 3.1.4. These authors studied magnetic field and velocity fluctuations anisotropy for the same co-rotating, high velocity stream observed by Bavassano et al. (1982a) within the framework of the dynamics of non-linear systems. Using the Local Intermittency Measure (Farge et al., 1990; Farge, 1992; Bruno et al., 1999b) were able to justify the controversy between results by Klein et al. (1991) in the outer heliosphere and Bavassano et al. (1982a) in the inner heliosphere. Exploiting the possibility offered by this technique to locate in space and time those events which produce intermittency, these authors were able to remove intermittent events and perform again the anisotropy analysis. They found that intermittency strongly affected the radial dependence of magnetic fluctuations while it was less effective on velocity fluctuations. In particular, after intermittency removal, the average level of anisotropy decreased for both magnetic and velocity field at all distances. Although magnetic fluctuations remained more anisotropic than their kinetic counterpart, the radial dependence was eliminated. On the other hand, the velocity field anisotropy showed that intermittency, although altering the anisotropic level of the fluctuations, does not markedly change its radial trend.
Recently, Pagel and Balogh (2003) studied intermittency in the outer heliosphere using Ulysses observations at high heliographic latitude, well within high speed solar wind. In particular, these authors used Castaing distribution Castaing et al. (2001) to study the Probability Distribution Functions (PDF) of the fluctuations of magnetic field components (see Section 9.2 for description of Castaing distribution and related governing parameters definition and ). They found that intermittency of small scales fluctuations, within the inertial range, increased with increasing the radial distance from the Sun as a consequence of the growth to larger scales of the inertial range.
As a matter of fact, using the scaling found by Horbury et al. (1996a) between the transition scale (the inverse of the frequency corresponding to the break-point in the magnetic field spectrum) , Pagel and Balogh (2003) quantitatively evaluated how the top of the inertial range in their data should shift to larger time scales with increasing heliocentric distance. Moreover, taking into account that inside the inertial range and that the proposed scaling from Castaing et al. (2001) would be , we should expect that for the parameter . Thus, these authors calculated and at different heliocentric distances and made the hypothesis of a similar scaling for and , although this is not assured by the model. Figure 108 reports values of and vs. distance calculated for the top of the inertial range at that distance using the above procedure. The radial behavior shown in this figure suggests that there is no radial dependence for these parameters for all the three components (indicated by different symbols), as expected if the observed radial increase of intermittency in the inertial range is due to a broadening of the inertial range itself.
They also found that, in the RTN reference system, transverse magnetic field components exhibit less Gaussian behavior with respect to the radial component. This result should be compared with results from similar studies by Marsch and Tu (1994) and Bruno et al. (2003b) who, studying the radial evolution of intermittency in the ecliptic, found that the components transverse to the local magnetic field direction, are the most Gaussian ones. Probably, the above discrepancy depends totally on the reference system adopted in these different studies and it would be desirable to perform a new comparison between high and low latitude intermittency in the mean-field reference system.
Pagel and Balogh (2002) focused also on the different intermittent level of magnetic field fluctuations during two fast latitudinal scans which happened to be during solar minimum the first one, and during solar maximum the second one. Their results showed a strong latitudinal dependence but were probably not, or just slightly, affected by radial dependence given the short heliocentric radial variations during these time intervals. They analyzed the anomalous scaling of the third order magnetic field structure functions looking at the value of the parameter obtained from the best fit performed using the p-model (see Section 7.4). In a previous analysis of the same kind, but focalized on the first latitudinal scan, the same authors tested three intermittency models, namely: “lognormal”, “p” and “G-infinity” models. In particular, this last model was an empirical model introduced by Pierrehumbert (1999) and Cho et al. (2000) and was not intended for turbulent systems. Anyhow, the best fits were obtained with the lognormal and Kolmogorov-p model. These authors concluded that magnetic field components display a very high level of intermittency throughout minimum and maximum phases of solar cycle, and slow wind shows a lower level of intermittency compared with the Alfvénic polar flows. These results do not seem to agree with ecliptic observations (Marsch and Liu, 1993; Bruno et al., 2003a) which showed that fast wind is generally less intermittent than slow wind not only for wind speed and magnetic field magnitude, but also for the components. At this point, since it has been widely recognized that low latitude fast wind collected within co-rotating streams and fast polar wind share many common turbulence features, they should be expected to have many similarities also as regards intermittency. Thus, it is possible that also in this case the reference system in which the analysis is performed plays some role in determining some of the results regarding the behavior of the components. In any case, further analyses should clarify the reasons for this discrepancy.