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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. (1980bJump To The Next Citation Point), 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. (1990Jump To The Next Citation Point) and Tu et al. (1989bJump To The Next Citation Point). 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 0.3 to 1 AU, gave the first and unique opportunity to study the radial evolution of turbulence (Bavassano et al., 1982bJump To The Next Citation PointDenskat and Neubauer, 1983). Successively, when Elsässer variables were introduced in the analysis (Grappin et al., 1989Jump To The Next Citation Point), 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 35View Image (Tu et al., 1990Jump To The Next Citation Point) clearly shows the behavior of ± e (see Appendix 13.3) across a high speed stream observed at 0.3 AU. Within fast wind + e is much higher than - e 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. e - has a similar break but the slope of lower frequencies follows the Kolmogorov slope, while higher frequencies form a sort of plateau.

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Figure 35: Power density spectra ± e computed from ± dz fluctuations for different time intervals indicated by the arrows (adopted from Tu et al., 1990, © 1990 American Geophysical Union, reproduced by permission of American Geophysical Union).
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. (1990Jump To The Next Citation Point), is commonly found within corotating high velocity streams, although much more clearly expressed at shorter heliocentric distances, as shown below.
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Figure 36: Power density spectra e- and e+ computed from dz- and dz+ fluctuations. Spectra have been computed within fast (H) and slow (L) streams around 0.4 and 0.9 AU as indicated by different line styles. The thick line represents the average power spectrum obtained from all the about 50 - espectra, regardless of distances and wind speed. The shaded area is the 1s width related to the average (adopted from Tu and Marsch, 1990bJump To The Next Citation Point, © 1990 American Geophysical Union, reproduced by permission of American Geophysical Union).
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 36View Image. The e-spectrum, if we exclude the high frequency range of the spectrum relative to fast wind at 0.4 AU, shows an average power law profile with a slope of - 1.64, consistent with Kolmogorov’s scaling. The lack of radial evolution of e-spectrum brought Tu and Marsch (1990aJump To The Next Citation Point) to name it “the background spectrum” of solar wind turbulence.

Quite different is the behavior of + espectrum. 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 0.8 AU.

In particular, Figure 36View Image tells us that the radial evolution of the normalized cross-helicity has to be ascribed mainly to the radial evolution of + erather than to both Alfvénic fluctuations (Tu and Marsch, 1990aJump To The Next Citation Point). In addition, Figure 37View Image, relative to the Elsässer ratio rE, shows that the hourly frequency range, up to ~ 2 .10- 3 Hz, is the most affected by this radial evolution.

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Figure 37: Ratio of - eover + ewithin fast wind at 0.3 and 0.9 AU in the left and right panels, respectively (adopted from Marsch and Tu, 1990aJump To The Next Citation Point, © 1990 American Geophysical Union, reproduced by permission of American Geophysical Union).
As a matter of fact, this radial evolution can be inferred from Figure 38View Image where values of e-and + etogether with solar wind speed, magnetic field intensity, and magnetic field and particle density compression are shown between 0.3 and 1 AU during the primary mission of Helios 2. It clearly appears that enhancements of e-and depletion of e+are connected to compressive events, particularly within slow wind. Within fast wind the average level of e-is rather constant during the radial excursion while the level of + edramatically decreases with a consequent increase of the Elsässer ratio (see Appendix 13.3.1).
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Figure 38: Upper panel: solar wind speed and solar wind speed multiplied by sc. In the lower panels the authors reported: sc, rE, e-, e+, magnetic compression, and number density compression, respectively (adopted from Bruno and Bavassano, 1991Jump To The Next Citation Point, © 1991 American Geophysical Union, reproduced by permission of American Geophysical Union).
Further ecliptic observations (see Figure 39View Image) do not indicate any clear radial trend for the Elsässer ratio between 1 and 5 AU, and its value seems to fluctuate between 0.2 and 0.4.
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Figure 39: Ratio of e-over e+within fast wind between 1 and 5 AU as observed by Ulysses in the ecliptic (adopted from Bavassano et al., 2001Jump To The Next Citation Point, © 2001 American Geophysical Union, reproduced by permission of American Geophysical Union).
However, low values of the normalized cross-helicity can also be associated with a particular type of uncompressive events, which Tu and Marsch (1991Jump To The Next Citation Point) called Magnetic Field Directional Turnings or MFDT. These events, found within slow wind, were characterized by very low values of sc close to zero and low values of the Alfvén ratio, around 0.2. Moreover, the spectral slope of e+, e-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 40 min, 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 40View Image taken from Tu and Marsch (1991Jump To The Next Citation Point). It follows that the amplitudes of the fluctuating Alfvénic fields ± dz result to be comparable and, consequently, the derived parameter sc --> 0. Moreover, the presence of these structures would also be able to explain the fact that rA < 1. Tu and Marsch (1991Jump To The Next Citation Point) suggested that these fluctuations might derive from a special kind of magnetic structures, which obey the MHD equations, for which (B . \~/ )B = 0, 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 e-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 s c and r A. Although the results were not quantitatively satisfactory, they did show a qualitative agreement with the observations.
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Figure 40: Left column: e+and e-spectra (top) and sc (bottom) during a slow wind interval at 0.9 AU. Right column: kinetic e v and magnetic e B energy spectra (top) computed from the trace of the relative spectral tensor, and spectrum of the Alfvén ratio rA (bottom) (adopted from Tu and Marsch, 1991Jump To The Next Citation Point).
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 (1991Jump To The Next Citation Point).

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., 1985Jump To The Next Citation Point). Conversely, the nature of those fluctuations identified by dz-, 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, 1995aJump To The Next Citation Point). However, other studies (Tu et al., 1989bJump To The Next Citation Point) 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.

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Figure 41: Power density spectra for + eand - eduring a high velocity stream observed at 0.3 AU. 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 Bruno et al. (1996Jump To The Next Citation Point) (adopted from Bruno et al., 1996Jump To The Next Citation Point).
Figure 41View Image shows power density spectra for e+and e-during a high velocity stream observed at 0.3 AU (similar spectra can be also found in the paper by Grappin et al.Jump To The Next Citation Point, 1990Jump To The Next Citation Point and Tu et al.Jump To The Next Citation Point, 1989bJump To The Next Citation Point). The observed spectral indices, reported on the plot, are typically found within high velocity streams encountered at short heliocentric distances. Bruno et al. (1996Jump To The Next Citation Point) analyzed the power relative to e+and e-modes, within five frequency bands, ranging from roughly 12 h to 3 min, delimited by the vertical solid lines equally spaced in log-scale. The integrated power associated with + eand - ewithin the selected frequency bands is shown in Figure 42View Image. Passing from slow to fast wind + egrows much more within the highest frequency bands. Moreover, there is a good correlation between the profiles of e-and e+within the first two highest frequency bands, as already noticed by Grappin et al. (1990Jump To The Next Citation Point) who looked at the correlation between daily averages of e-and e+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 k-space between opposite modes found by Grappin et al. (1982) in homogeneous MHD. Expansion effects were also taken into account by Velli et al. (1990Jump To The Next Citation Point) 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., 1989Jump To The Next Citation Point). 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, 1992Jump To The Next Citation Point)
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Figure 42: Left panel: wind speed profile is shown in the top panel. Power density associated with e+(thick line) and e-(thin line), within the 5 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 + dz (thick line) and - dz (thin line) and the direction of the ambient magnetic field are shown in the lower panels, relatively to each frequency band (adopted from Bruno et al., 1996Jump To The Next Citation Point).
An alternative generation mechanism was proposed by Tu et al. (1989bJump To The Next Citation Point) based on the parametric decay of e+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 e-modes by parametric instability is not particularly fast if the plasma b ~ 1, like in the case of solar wind (Goldstein, 1978Jump To The Next Citation PointDerby, 1978), since this condition slows down the growth rate of the instability. It is also true that numerical simulations by Malara et al. (2000Jump To The Next Citation Point2001Jump To The Next Citation Point2002Jump To The Next Citation Point), and Primavera et al. (2003Jump To The Next Citation Point) 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 e+and e-profiles found only within the highest frequency bands would support this mechanism and would suggest that e-modes within these frequency bands would have an Alfvénic nature. Another feature shown in Figure 42View Image that favors these conclusions is the fact that both + dz and dz- 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 Bo. In this situation the minimum variance would be oriented along Bo, 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 89View Image of Section 9, which refers to a typical Alfvénic interval within fast wind. Moreover, + dz 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 h+ starts wandering between 0o and 90o. 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 Bo, not depending on the frequency range.

In a way, we can say that within the stream, both + h and - h show a similar behavior as we look at lower and lower frequencies. The only difference is that h- reaches higher values at higher frequencies than h+. This was interpreted (Bruno et al., 1996Jump To The Next Citation Point) as due to the fact that transverse fluctuations of dz- carry much less power than those of dz+ 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 - dz 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 dz+ fluctuations can still be considered as the product of Alfvén modes propagating outward (Bruno et al., 1985Jump To The Next Citation Point), - dz 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, 1993aJump To The Next Citation Point). 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.

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