3.2 Turbulence studied via Elsässer variables
The Alfvénic character of solar wind fluctuations,especially within corotating high velocity streams,
suggests to use the Elsässer variables (Appendix 13.3) to separate the “outward” from the “inward”
contribution to turbulence. These variables, used in theoretical studies by Dobrowolny et al. (1980a),
Dobrowolny et al. (1980b
), Veltri et al. (1982), Marsch and Mangeney (1987), and Zhou and
Matthaeus (1989), were for the first time used in interplanetary data analysis by Grappin et al. (1990) and
Tu et al. (1989b). In the following, we will describe and discuss several differences between “outward”
and “inward” modes, but the most important one is about their origin. As a matter of fact,
the existence of the Alfvénic critical point implies that only “outward” propagating waves of
solar origin will be able to escape from the Sun. “Inward” waves, being faster than the wind
bulk speed, will precipitate back to the Sun if they are generated before this point. The most
important implication due to this scenario is that “inward” modes observed beyond the Alfvénic
point cannot have a solar origin but they must have been created locally by some physical
process. Obviously, for the other Alfvénic component, both solar and local origins are still
possible.
3.2.1 Ecliptic scenario
Early studies by Belcher and Davis Jr (1971), performed on magnetic field and velocity fluctuations
recorded by Mariner 5 during its trip to Venus in 1967, already suggested that the majority of the Alfvénic
fluctuations are characterized by an “outward” sense of propagation, and that the best regions where to
observe these fluctuations are the trailing edge of high velocity streams. Moreover, Helios spacecraft,
repeatedly orbiting around the Sun between 
to 
, gave the first and unique opportunity to
study the radial evolution of turbulence (Bavassano et al., 1982b; Denskat and Neubauer, 1983).
Successively, when Elsässer variables were introduced in the analysis (Grappin et al., 1989), it was finally
possible not only to evaluate the “inward” and “outward” Alfvénic contribution to turbulence but also to
study the behavior of these modes as a function of the wind speed and radial distance from the
Sun.
Figure 35 (Tu et al., 1990) clearly shows the behavior of 
(see Appendix 13.3) across a high speed
stream observed at 
. Within fast wind 
is much higher than 
and its spectral slope shows
a break. Lower frequencies have a flatter slope while the slope of higher frequencies is closer to a
Kolmogorov-like. 
has a similar break but the slope of lower frequencies follows the Kolmogorov slope,
while higher frequencies form a sort of plateau.
This configuration vanishes when we pass to the slow wind where both spectra have almost equivalent
power density and follow the Kolmogorov slope. This behavior, for the first time reported by Grappin
et al. (1990), is commonly found within corotating high velocity streams, although much more clearly
expressed at shorter heliocentric distances, as shown below.
Spectral power associated with outward (right panel) and inward (left panel) Alfvénic fluctuations,
based on Helios 2 observations in the inner heliosphere, are concisely reported in Figure 36. The
spectrum, if we exclude the high frequency range of the spectrum relative to fast wind at 
,
shows an average power law profile with a slope of 
, consistent with Kolmogorov’s scaling. The lack
of radial evolution of 
spectrum brought Tu and Marsch (1990a) to name it “the background spectrum”
of solar wind turbulence.
Quite different is the behavior of 
spectrum. Close to the Sun and within fast wind, this spectrum
appears to be flatter at low frequency and steeper at high frequency. The overall evolution is towards the
“background spectrum” by the time the wind reaches 
.
In particular, Figure 36 tells us that the radial evolution of the normalized cross-helicity has to be
ascribed mainly to the radial evolution of 
rather than to both Alfvénic fluctuations (Tu and
Marsch, 1990a). In addition, Figure 37, relative to the Elsässer ratio 
, shows that the hourly
frequency range, up to 
, is the most affected by this radial evolution.
As a matter of fact, this radial evolution can be inferred from Figure 38 where values of 
and
together with solar wind speed, magnetic field intensity, and magnetic field and particle density
compression are shown between 
and 
during the primary mission of Helios 2. It clearly appears
that enhancements of 
and depletion of 
are connected to compressive events, particularly within
slow wind. Within fast wind the average level of 
is rather constant during the radial excursion while
the level of 
dramatically decreases with a consequent increase of the Elsässer ratio (see
Appendix 13.3.1).
Further ecliptic observations (see Figure 39) do not indicate any clear radial trend for the
Elsässer ratio between 
and 
, and its value seems to fluctuate between 
and
.
However, low values of the normalized cross-helicity can also be associated with a particular type of
uncompressive events, which Tu and Marsch (1991) called Magnetic Field Directional Turnings or MFDT.
These events, found within slow wind, were characterized by very low values of 
close to zero and low
values of the Alfvén ratio, around 
. Moreover, the spectral slope of 
, 
and the
power associated with the magnetic field fluctuations was close to the Kolmogorov slope. These
intervals were only scarcely compressive, and short period fluctuations, from a few minutes to
about 
, were nearly pressure balanced. Thus, differently from what had previously been
observed by Bruno et al. (1989), who found low values of cross-helicity often accompanied by
compressive events, these MFDTs were mainly uncompressive. In these structures most of the
fluctuating energy resides in the magnetic field rather than velocity as shown in Figure 40 taken
from Tu and Marsch (1991). It follows that the amplitudes of the fluctuating Alfvénic fields
result to be comparable and, consequently, the derived parameter 
. Moreover,
the presence of these structures would also be able to explain the fact that 
. Tu and
Marsch (1991) suggested that these fluctuations might derive from a special kind of magnetic structures,
which obey the MHD equations, for which 
, field magnitude, proton density, and
temperature are all constant. The same authors suggested the possibility of an interplanetary
turbulence mainly made of outwardly propagating Alfvén waves and convected structures
represented by MFDTs. In other words, this model assumed that the spectrum of 
would be
caused by MFDTs. The different radial evolution of the power associated with these two kind of
components would determine the radial evolution observed in both 
and 
. Although the
results were not quantitatively satisfactory, they did show a qualitative agreement with the
observations.
These convected structures are an important ingredient of the turbulent evolution of the fluctuations
and can be identified as the 2D incompressible turbulence suggested by Matthaeus et al. (1990) and Tu and
Marsch (1991).
3.2.2 On the nature of Alfvénic fluctuations
The Alfvénic nature of outward modes has been widely recognized through several frequency decades up
to periods of the order of several hours in the s/c rest frame (Bruno et al., 1985). Conversely, the nature of
those fluctuations identified by 
, called “inward Alfvén modes”, is still not completely clear. There
are many clues which would suggest that these fluctuations, especially in the hourly frequencies range, have
a non-Alfvénic nature. Several studies on this topic in the low frequency range have suggested that
structures convected by the wind could well mimic non-existent inward propagating modes (see the review
by Tu and Marsch, 1995a). However, other studies (Tu et al., 1989b) have also found, in the
high frequency range and within fast streams, a certain anisotropy in the components which
resembles the same anisotropy found for outward modes. So, these observations would suggest a
close link between inward modes at high frequency and outward modes, possibly the same
nature.
Figure 41 shows power density spectra for 
and 
during a high velocity stream observed at
(similar spectra can be also found in the paper by Grappin et al., 1990 and Tu et al., 1989b). The
observed spectral indices, reported on the plot, are typically found within high velocity streams encountered
at short heliocentric distances. Bruno et al. (1996) analyzed the power relative to 
and 
modes,
within five frequency bands, ranging from roughly 
to 
, delimited by the vertical solid lines
equally spaced in log-scale. The integrated power associated with 
and 
within the selected
frequency bands is shown in Figure 42. Passing from slow to fast wind 
grows much more
within the highest frequency bands. Moreover, there is a good correlation between the profiles
of 
and 
within the first two highest frequency bands, as already noticed by Grappin
et al. (1990) who looked at the correlation between daily averages of 
and 
in several
frequency bands, even widely separated in frequency. The above results stimulated these authors
to conclude that it was reminiscent of the non-local coupling in 
-space between opposite
modes found by Grappin et al. (1982) in homogeneous MHD. Expansion effects were also taken
into account by Velli et al. (1990) who modeled inward modes as that fraction of outward
modes back-scattered by the inhomogeneities of the medium due to expansion effects (Velli
et al., 1989). However, following this model we would often expect the two populations to be somehow
related to each other but, in situ observations do not favor this kind of forecast (Bavassano and
Bruno, 1992)
An alternative generation mechanism was proposed by Tu et al. (1989b) based on the parametric decay
of 
in high frequency range (Galeev and Oraevskii, 1963). This mechanism is such that large amplitude
Alfvén waves, unstable to perturbations of random field intensity and density fluctuations, would decay
into two secondary Alfvén modes propagating in opposite directions and a sound-like wave propagating in
the same direction of the pump wave. Most of the energy of the mother wave would go into the sound-like
fluctuation and the backward propagating Alfvén mode. On the other hand, the production of 
modes
by parametric instability is not particularly fast if the plasma 
, like in the case of solar wind
(Goldstein, 1978; Derby, 1978), since this condition slows down the growth rate of the instability. It is also
true that numerical simulations by Malara et al. (2000, 2001, 2002), and Primavera et al. (2003) have
shown that parametric decay can still be thought as a possible mechanism of local production of
turbulence within the polar wind (see Section 4). However, the strong correlation between 
and
profiles found only within the highest frequency bands would support this mechanism and
would suggest that 
modes within these frequency bands would have an Alfvénic nature.
Another feature shown in Figure 42 that favors these conclusions is the fact that both 
and
keep the direction of their minimum variance axis aligned with the background magnetic
field only within the fast wind, and exclusively within the highest frequency bands. This would
not contradict the view suggested by Barnes (1981). Following this model, the majority of
Alfvénic fluctuations propagating in one direction have the tip of the magnetic field vector
randomly wandering on the surface of half a sphere of constant radius, and centered along the
ambient field 
. In this situation the minimum variance would be oriented along 
,
although this would not represent the propagation direction of each wave vector which could
propagate even at large angles from this direction. This situation can be seen in the right hand
panel of Figure 89 of Section 9, which refers to a typical Alfvénic interval within fast wind.
Moreover, 
fluctuations show a persistent anisotropy throughout the fast stream since the
minimum variance axis remains quite aligned to the background field direction. This situation
downgrades only at the very low frequencies where 
starts wandering between 
and 
.
On the contrary, in slow wind, since Alfvénic modes have a smaller amplitude, compressive
structures due to the dynamic interaction between slow and fast wind or, of solar origin, push the
minimum variance direction to larger angles with respect to 
, not depending on the frequency
range.
In a way, we can say that within the stream, both 
and 
show a similar behavior as we look at
lower and lower frequencies. The only difference is that 
reaches higher values at higher frequencies
than 
. This was interpreted (Bruno et al., 1996) as due to the fact that transverse fluctuations of
carry much less power than those of 
and, consequently, they are more easily influenced by
perturbations represented by the background, convected structure of the wind (e.g., TD’s and
PBS’s). As a consequence, at low frequency 
fluctuations may represent a signature of
the compressive component of the turbulence while, at high frequency, they might reflect the
presence of inward propagating Alfvén modes. Thus, while for periods of several hours 
fluctuations can still be considered as the product of Alfvén modes propagating outward (Bruno
et al., 1985), 
fluctuations are rather due to the underlying convected structure of the wind.
In other words, high frequency turbulence can be looked at mainly as a mixture of inward
and outward Alfvénic fluctuations plus, presumably, sound-like perturbations (Marsch and
Tu, 1993a). On the other hand, low frequency turbulence would be made of outward Alfvénic
fluctuations and static convected structures representing the inhomogeneities of the background
medium.
(Tu et al., 1990
computed from 
fluctuations for different time intervals
indicated by the arrows (adopted from 
and 
computed from 
and 
fluctuations.
Spectra have been computed within fast (H) and slow (L) streams around 
and 
as
indicated by different line styles. The thick line represents the average power spectrum obtained from
all the about 
spectra, regardless of distances and wind speed. The shaded area is the 
width related to the average (adopted from 
over 
within fast wind at 
and 
in the left and right
panels, respectively (adopted from 
. In the lower
panels the authors reported: 
, 
, 
, 
, magnetic compression, and number density
compression, respectively (adopted from 
over 
within fast wind between 
and 
as observed by Ulysses in
the ecliptic (adopted from 
and 
spectra (top) and 
(bottom) during a slow wind interval
at 
. Right column: kinetic 
and magnetic 
energy spectra (top) computed from
the trace of the relative spectral tensor, and spectrum of the Alfvén ratio 
(bottom) (adopted
from 
and 
during a high velocity stream observed at
. Best fit lines for different frequency intervals and related spectral indices are also shown.
Vertical lines fix the limits of five different frequency intervals analyzed by 
(thick line) and 
(thin line), within the 
frequency bands chosen, is shown in the lower
panels. Right panel: wind speed profile is shown in the top panel. Values of the angle between the
minimum variance direction of 
(thick line) and 
(thin line) and the direction of the
ambient magnetic field are shown in the lower panels, relatively to each frequency band (adopted
from