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 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 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 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 74. 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
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 . 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 , 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 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 , , 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 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, 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 , indicating that there is
no mass flux through the discontinuity. During this interval, which lasts about , 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
.
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 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.
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.
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 , 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 . 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 ,
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 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.