9 Turbulent Structures

The non-linear energy cascade towards smaller scales accumulates fluctuations only in relatively small regions of space, where gradients become singular. As a rather different point of view (see Farge, 1992

) these regions can be viewed as localized zones of fluid where phase correlation exists, in some sense coherent structures. These structures, which dominate the statistics of small scales, occur as isolated events with a typical lifetime greater than that of stochastic fluctuations surrounding them. The idea of a turbulence in the solar wind made by a mixture of structures convected by the wind and stochastic fluctuations is not particularly new (see, e.g., Tu and Marsch, 1995a

). However, these large-scale structures cannot be considered as intermittent structures at all scales. Structures continuously appear and disappear apparently in a random fashion, at some random location of fluid, and carry a great quantity of energy of the flow. In this framework intermittency can be considered as the result of the occurrence of coherent (non-Gaussian) structures at all scales, within the sea of stochastic Gaussian fluctuations.

This point of view is the result of data analysis of scaling laws of turbulent fluctuations made by using wavelets filters (see Appendix 14) 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 14).

Looking at the field $fs(x)$ one can investigate the spatial behavior of structures generating intermittency. Relatively to the solar wind, the Haar basis has been applied to time series of thirteen months of velocity and magnetic data from ISEE space experiment for the first time by Veltri and Mangeney (1999a ). 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 $b$ parameter, constant pressure, anti-correlation between density and proton temperature, no magnetic fluctuations, and velocity fluctuations directed along the average magnetic field.

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 $u(t)$ or $b(t)$ derived from solar wind, and let us define the wavelets set $ws(r, t)$ as the set which captures, at time $t$ , the occurrence of structures at the scale $r$ . Then define the waiting times $dt$ , as that time between two consecutive structures at the scale $r$ , that is, between $ws(r, t)$ and $w (r,t + dt) s$ . The PDFs of waiting times $P(dt)$ are reported in Figure 74 . As it can be seen, waiting times are distributed according to a power law $-b P (dt) ~ dt$ 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 $P (dt)$ represents the asymptotic behavior of a Lévy function with characteristic exponent $a = b - 1$ . 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 recalled 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 74 . However, Bruno et al. (2001 ) included intermittent events found at all scales while results shown in Figure 74 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 that, 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 87 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 $Pm$ , here indicated by $PB$ ; the red line refers to the logarithmic value of the thermal pressure $Pk$ , here indicated by $PK$ and the black line refers to the logarithmic value of the total pressure $Ptot$ , here indicated by $PT = PB + PK$ , including an average estimate of the electrons and $a$ 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.

Figure 87:

From top to bottom, we show $81 s$ averages of velocity wind profile in $km s-1$ , magnetic field intensity in nT, the logarithmic value of magnetic (blue line), thermal(red line), and total pressure (black line) in $dyne/ cm2$ and field intensity residuals in $nT$ . The two vertical boxes delimit the two time intervals # 1 and #2 which were chosen for comparison. While the first interval shows strong magnetic intermittency, the second one does not (adopted from Bruno et al., 2001

A minimum variance analysis further reveals the intrinsic different nature of these two intervals as shown in Figure 88

where original data have been rotated into the field minimum variance reference system (see Appendix 15.1) where maximum, intermediate and minimum variance components are identified by $c3$ , $c2$ , and $c1$ , respectively. Moreover, at the bottom of the column we show the hodogram on the maximum variance plane $c3 - c2$ , 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 $90o$ on the maximum variance plane (see rotation from A to B on the 3D graph at the bottom of the left column in Figure 88 ) 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, $B[c3]$ changes sign, $B[c2]$ shows a hump whose maximum is located where the previous component changes sign and, finally, $B[c1]$ 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 $V [c ] 1$ has the same value on both sides of the discontinuity, approximately $-1 350 km s$ , 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 $11.9o$ 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 $c2- c3$ , and they tend to cover the whole $2p$ .

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 $9.3o$ .

Figure 88:

Left column, from top to bottom: we show magnetic field intensity, maximum $c3$ , intermediate $c 2$ and minimum $c 1$ variance components for magnetic field (blue color) and wind velocity relative to the time interval #1 shown in Figure 87

. Right below, we show the hodogram on the maximum variance plane $c3 - c2$ , as a function of time (blue color line). The red lines are the projection of the blue line. The large arc, from $A$ to $B$ , corresponds to the green segment in the profile of the magnetic field components shown in the upper panel. The same parameters are shown for interval # 2 (Figure 87

), in the same format, on the right hand side of the figure. The time resolution of the data is $81 s$ (adopted from Bruno et al., 2001

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 89

. 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 88 , 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.

Figure 89:

Trajectory followed by the tip of the magnetic field vector (blue color line) in the minimum variance reference system for interval # 1 (left) and # 2 (right). Projections on the three planes (red color lines) formed by the three eigenvectors $c1,c2,c3$ , and the average magnetic field vector, with its projections on the same planes, are also shown. The green line extending from label A to label B refers to the arc-like discontinuity shown in Figure 88

. The time resolution of the magnetic field averages is $6 s$ (adopted from Bruno et al., 2001

). (To see animations relative to similar time intervals click on Figures 90

for a time series affected by the intermittency phenomenon or at 91

for non-intermittent and intermittent samples.

Figure 90:

Trajectory followed by the tip of the magnetic field vector in the minimum variance reference system during a time interval not characterized by intermittency. The duration of the interval is $2000 × 6 s$ but the magnetic field vector moves only for $100 × 6 s$ in order to make a smaller file (movie kindly provided by A. Vecchio).

These differences in the dynamics of the orientation of the field vector can be appreciated running the two animations behind Figures 90

and 91

. Although the data used for these movies do not exactly correspond to the same time intervals analyzed in Figure 87

, they show the same dynamics that the field vector has within intervals # 1 and # 2. In particular, the animation corresponding to Figure 90

represents interval # 2 while, Figure 91

represents interval # 1.

Figure 91:

Trajectory followed by the tip of the magnetic field vector in the minimum variance reference system during a time interval characterized by intermittent events. The duration of the interval is $2000 × 6 s$ but the magnetic field vector moves only for $100 × 6 s$ in order to make a smaller file (movie kindly provided by A. Vecchio).

The observations reported above suggested these authors to draw the sketch shown in Figure 92

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 about 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 superposition 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 at 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 91 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.

Figure 92:

Simple visualization of hypothetical flux tubes which tangle up in space. Each flux tube is characterized by a local field direction, and within each flux tube the presence of Alfvénic fluctuations makes the magnetic field vector randomly wander about this direction. Moreover, the large scale is characterized by an average background field direction aligned with the local interplanetary magnetic field. Moving across different flux-tubes, characterized by a different values of $|B |$ , enhances the intermittency level of the magnetic field intensity time series (adopted from Bruno et al., 2001

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 93

, 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.

Figure 93:

Composite figure made adapting original figures from the paper by Chang et al. (2004 ). The first element on the upper left corner represents field-aligned spatio-temporal coherent structures. A cross-section of two of these structures of the same polarity is shown in the upper right corner. Magnetic flux iso-contours and field polarity are also shown. The darkened area represents intense current sheet during strong magnetic shear. The bottom element of the figure is the result of 2D MHD simulations of interacting coherent structures, and shows intermittent spatial distribution of intense current sheets. In this scenario, new fluctuations are produced which can provide new resonance sites, possibly nucleating new coherent structures

9.1 Radial evolution of intermittency in the ecliptic
9.2 Radial evolution of intermittency at high latitude