Observations of Alfvénic turbulence evolution
in the 3-D heliosphere
Bruno Bavassano
Istituto di Fisica dello Spazio Interplanetario (CNR), Roma, Italy
Abstract. The heliospheric plasma is an excellent laboratory to study the behaviour of collisionless Alfvénic
turbulence. This is a fundamental topic in both plasma physics and astrophysics. The impressive amount of
observations at different solar distances and latitudes collected in last three decades allows to get a good view
of the turbulence behaviour in the heliosphere inside 5 AU. In present review the focus will be on the evolution
of the Alfvénic turbulence in both low- and high-latitude solar wind, as derived from old observations on the
ecliptic and from recent observations by Ulysses at polar latitudes.
INTRODUCTION
The solar wind, a high Reynolds’ numbers collisionless plasma, is the most accessible medium in
which to study Alfvénic turbulence. This is a topic
of fundamental importance for both plasma physics
and astrophysics. Alfvénic turbulence in solar wind
1) contributes to plasma heating and acceleration,
2) accelerates particles to high energies, and 3) affects cosmic ray propagation. In the seventies and
the eighties impressive advances have been made in
the knowledge of Alfvénic turbulent phenomena in
solar wind. In those days spacecraft observations
were confined to a small latitudinal belt around the
solar equator. In the nineties, with the launch of
the Ulysses spacecraft, investigations have been extended to the high-latitude regions of the heliosphere.
This has allowed to study how Alfvénic turbulence
evolves in polar wind, a plasma flow in which the effects of large-scale inhomogeneities are considerably
less important than in low-latitude wind. With this
new laboratory relevant advances have been made.
In present review the Ulysses observations of turbulence evolution in polar wind will be discussed and
compared to those typical of near-equatorial solar
wind.
A short overview of the parameters that are generally used to describe Alfvénic fluctuations will be preliminarily given. Basic quantities are the Elsässer’s
variables (z± ). They are defined as z± = v±b, where
v and b are the velocity and magnetic field vectors, respectively,√and b is scaled to Alfvén units
(i.e., divided by 4πρ, with ρ the mass density).
Taking into account how the sign of the Alfvénic correlation depends on the propagation direction with
respect to the background magnetic field, it has become common to use the above definition in the
case of a background magnetic field pointing to the
Sun, while the equation z± = v∓b is taken for the
opposite polarity. With this choice we have that,
whatever the polarity is, z+ (z− ) fluctuations always
correspond to modes with an outward (inward) direction of propagation, with respect to Sun, in the
plasma frame. An exhaustive discussion on the use
of Elsässer variables in solar wind turbulence studies
has been performed by Tu and Marsch [1995]. The
relative weight of the energies (per unit mass) e+ and
e− associated to z+ and z− fluctuations, respectively,
is measured by the Elsässer ratio rE = e− /e+ . An
analogous measure for the energies eV and eB of
the v and b fluctuations is given by the Alfvén ratio rA = eV /eB . Other related parameters are the
normalized cross-helicity σC = (e+ −e− )/(e+ +e− ) =
(1−rE )/(1+rE ) and the normalized residual energy
σR = (eV −eB )/(eV +eB ) = (rA −1)/(rA +1).
Both outward and inward propagating fluctuations
are observed in the solar wind. Outward fluctuations
mainly have a solar origin (or, more precisely, inside
the Alfvén critical point). Conversely, inward propagating fluctuations can only be generated outside
such critical distance. As well known, the presence
of both kinds of fluctuation leads to the development
of nonlinear interactions.
The main features of the Alfvénic turbulence evo-
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
377
lution versus radial distance (i.e., versus transit time
to the observer) will be now discussed for ecliptic
and polar wind. Ecliptic results are mainly based on
Helios and Voyagers observations, while polar results
obviously rely on Ulysses measurements.
ECLIPTIC TURBULENCE
Since the first studies on Alfvénic fluctuations in
ecliptic wind it was clear that fast streams (or, more
precisely, their trailing edge) are the best places to
observe them. Helios spacecraft, with a systematic
covering of the inner heliosphere from 0.3 to 1 AU,
gave the first opportunity to study the turbulence
evolution with solar distance.
Figure 1 (from Marsch and Tu [1990]) shows how
e+ and e− power spectra (solid and dotted line, respectively) vary when solar distance increases from
0.3 to 0.9 AU. The e+ spectrum declines faster than
that of e− , with the result that the two spectra approach each other. At the same time the spectral
slopes evolve in such a way that an extended inertial
regime expands to low frequencies.
The corresponding variation in the relative weight
of outward and inward fluctuation energies is shown
by Figure 2 (from Marsch and Tu [1990]). The small
values of rE observed at 0.3 AU in the core of the
Alfvénic regime (frequencies around 10−4 − 10−3 Hz)
have disappeared at 0.9 AU. However, in spite of this
rE increase, the predominance of z+ fluctuations remains a clear feature of the Alfvénic turbulence observed by Helios inside 1 AU. Bavassano et al. [2001],
using Ulysses data from the ecliptic phase of the mission, have shown that this situation persists at least
up to 5 AU.
FIGURE 1. Power spectra of e+ and e− (solid and dotted
line, respectively) in the trailing edge of fast streams at
0.29 and 0.87 AU (adapted from Marsch and Tu [1990],
copyright 1990 American Geophysical Union, modified
by permission of American Geophysical Union).
FIGURE 3.
Radial variation of the Alfvén ratio
rA (=eV /eB ) as seen by Helios and Voyagers between
0.3 and 20 AU (from Roberts et al. [1990], copyright
1990 American Geophysical Union). The 9-hr curve (see
squares) shows that rA , after the fast decrease at small
distance, remains nearly unchanged.
FIGURE 2.
Power spectrum of the Elsässer ratio
rE (= e− /e+ ) in the trailing edge of fast streams at
0.29 and 0.87 AU (adapted from Marsch and Tu [1990],
copyright 1990 American Geophysical Union, modified
by permission of American Geophysical Union).
Another relevant feature of the solar wind Alfvénic
fluctuations is the emergence, for increasing solar distance, of a predominance of the magnetic fluctuation
energy. This decreasing trend for the Alfvén ratio
rA (=eV /eB ) clearly appears in Helios data [Bruno
et al., 1985; Marsch and Tu, 1990]. It has been shown
by Bavassano and Bruno [2000] that this change in
the relative amplitude of v and b fluctuations occurs
without any relevant variation of their correlation.
The decreasing trend of rA , however, is not with-
378
out limit, in other words a seemingly final stage is
reached in which the value of rA remains nearly unchanged. This is seen in Figure 3 (from Roberts et
al. [1990]), combining Helios and Voyagers observations to give an overall view of the radial variation
of rA between 0.3 and 20 AU. The 9-hr curve, that
of the two reported curves best describes the typical
MHD scales, shows that rA , after a fast and pronounced decrease, remains nearly unchanged.
POLAR TURBULENCE
Observations by Ulysses during its first out-ofecliptic orbit have shown that, at low solar activity,
the high-latitude solar wind is a fast and relatively
steady flow. A relevant feature of the polar wind is
the ubiquitous presence of an intense flow of Alfvénic
fluctuations (e.g., Goldstein et al. [1995]; Horbury
et al. [1995]; Smith et al. [1995]; Bavassano et al.
[1998] ). Similarly to previous ecliptic observations
in fast streams, a largely dominant fraction of these
fluctuations is outward propagating, with respect to
the Sun, in the solar wind frame. The relevance of
the Ulysses observations, besides their exploratory
character, is that the relative absence of structure
in polar flow offers the opportunity of studying the
evolution of Alfvénic turbulence under almost undisturbed conditions.
FIGURE 4. The plot shows power spectra of z+ and
z− (upper and lower curve, respectively) in polar wind at
(left panel) ∼ 2 AU and (right panel) ∼ 4 AU (adapted
from Goldstein et al. [1995], copyright 1995 American
Geophysical Union, modified by permission of American
Geophysical Union).
In Figure 4 power spectra of z+ and z− at about
2 and 4 AU in polar wind as obtained by Goldstein
et al. [1995]) are shown. The spectral evolution appears qualitatively similar to that observed in ecliptic wind, with the development of a turbulent cascade with increasing distance that moves to lower frequencies the breakpoint between the f −1 and f −5/3
regimes. However, in polar wind the breakpoint is at
higher frequency than at similar distances in ecliptic
wind (see Horbury et al. [1996]). Thus, spectral evolution in polar wind is slower than in ecliptic wind.
Further evidence on this is given in Figure 5. The
plot shows the power spectra of magnetic field components (solid circles) and magnitude (squares) as
estimated at 1 AU from trends in power scalings measured by Ulysses in polar wind between 1.4 and 4.1
AU (solid lines) and by Helios in ecliptic fast wind inside 1 AU (dashed lines) [Horbury and Balogh, 2001].
At time scales smaller than few hours (see scale on
top) the general agreement appears good, in both
shape and magnitude. However, it is clearly seen
that the spectrum of magnetic components is steeper
in ecliptic wind, as expected for a faster evolution.
FIGURE 5. The plot shows the power spectra of magnetic field components (solid circles) and magnitude
(squares) as estimated at 1 AU from trends in power
scalings at Ulysses in polar wind between 1.4 and 4.1
AU (solid lines) and Helios in ecliptic fast wind inside
1 AU (dashed lines), from Horbury and Balogh [2001]
(copyright 2001 American Geophysical Union).
A general agreement exists on the fact that the
slower evolution for polar turbulence has to be ascribed to the lack of a large-scale stream structure.
The role of such a structure in accelerating turbulence evolution has been recently stressed by Bavassano et al. [1998] using Ulysses data at mid latitudes (in the first out-of-ecliptic orbit), where strong
gradients were dominant in the velocity pattern. It
should also be mentioned that the turbulence ap-
379
pears younger in polar flow since fluctuations at a
given distance have had less time to evolve due to the
higher wind speed (e.g., see Matthaeus et al. [1999]).
The radial variation of the Elsässer ratio rE and
the Alfvén ratio rA in polar wind is shown in Figure 6 (see also Bavassano et al. [2000a]). Both the
decline of the e+ predominance and the increase of
the eB predominance do not go beyond some limits,
in agreement with ecliptic observations.
on the ecliptic. As regards polar wind, the e+ values (hourly variances) exhibit the same radial gradient over all the investigated range of distances. In
contrast, for e− a change of slope around 2.5 AU is
clearly apparent. Another remarkable feature is the
good agreement of the Ulysses gradients with Helios
data.
0.6
0.5
rE
0.4
0.3
0.2
0.5
0.4
rA
0.3
0.2
0.1
1
2
3
4
5
R (AU)
FIGURE 6. Radial variation of (top) Elsässer ratio
rE (=e− /e+ ) and (bottom) Alfvén ratio rA (=eV /eB )
in polar wind, as obtained from hourly total variances of
the fluctuating vectors.
Agreement between polar and ecliptic observations
is found also for the radial variation of magnetic field
fluctuations. Figure 7 shows the decline with solar
distance for hourly variances of magnetic field components and magnitude in polar wind [Forsyth et al.
1996]. Best fit power laws (R−n ) for data in the
range from 1.5 to 3 AU (data at larger distances have
not been included to avoid effects related to compressive features) indicate a radial slope n of about 3.4
for the magnetic components. This value is in good
agreement with that found in ecliptic wind, at the
same scale and for a similar range of distances, by
Bavassano and Smith [1986] with Pioneer 10 and 11
data.
Alfvénic turbulence evolution, however, is better
described by the variation of z+ and z− , rather than
by magnetic field. Figure 8 is a composite plot combining the Ulysses observations in polar wind with
those by Helios in the trailing edge of fast streams
FIGURE 7. Radial variation of hourly variances of
magnetic field components and (lower curve) magnitude
(from Forsyth et al. [1996], copyright 1996 American
Geophysical Union).
The e− behaviour highlighted by Figure 8 is probably related to the presence, inside ∼2.5 AU, of local
generation effects. Turbulence generation in polar
wind is a very relevant point. Velocity shear probably is an important factor to generate turbulence
and drive its evolution in ecliptic solar wind, but
certainly this does not hold for polar wind, where
large-scale velocity gradients are almost absent. It
has been proposed (e.g., see Malara et al. [2000] and
Del Zanna [2001]) that in polar wind a role might be
played by the parametric decay. Simulations of the
non-linear development of the parametric decay for
large-amplitude non-monochromatic Alfvénic fluctuations [Malara et al., 2000] have shown that the final
state strongly depends on the value of β (thermal to
magnetic pressure ratio). For β < 1 the normalized
cross-helicity σC decreases, from an initial value of
1, to values close to 0. Thus, the instability appears
able to completely destroy the initial Alfvénic correlation. In sharp contrast, for β = 1 (a value closer to
real solar wind conditions) σC is around 0.5 in the
380
final state. In this case the parametric instability is
not able to go beyond some limit in the disruption
of the initial correlation between velocity and magnetic field fluctuations. This solution surely is qualitatively reminiscent of the Ulysses data behaviour
shown in Figure 8 (see also the rE variation in Figure 6). Analogous results are reached by Del Zanna
[2001] for arc-polarized Alfvén waves. It should be
stressed, however, that these models have many limitations (e.g., see discussion by Malara et al. [2001]).
Turbulence generation in a flow with a low level of
large-scale velocity shear is still an open issue.
Helios
104
Ulysses
e+
e+, e- e
2 2
-
CONCLUDING REMARKS
_1.39
r
(km /s )
103
_0.42
r
_1.46
r
102
0.3
tween gradients observed in high-latitude wind and
in fast streams on the ecliptic. This has been seen to
hold both for magnetic field fluctuations (see above)
and for outward and inward Alfvénic fluctuations (as
shown by Bavassano et al. [2000b] and [2001]). However, recent results of Horbury and Balogh [2001]
have indicated that fluctuations in magnetic field
components at time scales shorter than about 1 day
(in the spacecraft frame) exhibit a non negligible dependence on latitude. Bavassano et al. [2002] have
reexamined this point applying the same method of
Horbury and Balogh [2001] to the Elsässer variables
and carefully removing data intervals with compressive features (as those seen in Figure 7). Their conclusion is that, at least for the core of the Alfvénic
regime, latitude does not appear to have an appreciable influence on the Alfvénic turbulence evolution
in polar wind.
0.5
1
r
3
5
(AU)
FIGURE 8. A composite plot combining Ulysses observations in polar wind with those by Helios 1 and 2 inside
1 AU on the ecliptic plane (adapted from Bavassano et
al. [2000b], copyright 2000 American Geophysical Union,
modified by permission of American Geophysical Union).
The values of e+ (e− ) are shown as squares (diamonds),
small for Ulysses and large for Helios. Best fit lines and
radial power laws for Ulysses data are given.
A last comment is about the nature of the Alfvénic
turbulence variations observed in polar wind (i.e., radial, or latitudinal, or both). Both distance and latitude concurrently vary along the Ulysses trajectory,
thus this point needs to be carefully examined. Several analyses of Ulysses data have indicated that the
variation of turbulence properties in high-latitude
wind is essentially radial, rather than latitudinal,
in nature (e.g., Goldstein et al. [1995]; Horbury et
al. [1995]; Forsyth et al. [1996]). A further robust argument in favor of the radial character of the
turbulence variation comes from the agreement be-
Ulysses observations, combined with previous results in the ecliptic, allow to get a quite complete
view of the Alfvénic turbulence evolution in the heliosphere from 1 to 5 AU.
Polar wind observations, when compared to previous ecliptic results, do not appear as a dramatic
break. Polar evolution is similar to that in the ecliptic, although slower. This well agrees with the view
proposed by Bruno [1992], a middle course between
a non-relaxing turbulence (due to the lack of velocity
shear, Grappin et al. [1991]) and a quick evolution
(due to the large amplitude of fluctuations relative
to the mean magnetic field, Roberts [1990]).
A still open question is about the driving mechanism(s) for polar turbulence evolution, given the low
level of large-scale velocity shear in the polar flow.
Parametric decay could be a good candidate, but
models still need relevant improvements.
It has been seen that in all the examined region the
Alfvénic fluctuations appear characterized by 1) a
predominance of outward fluctuations (i.e., a positive
cross-helicity) and 2) a predominance of magnetic
fluctuations (i.e., a negative residual energy).
As regards outward fluctuations, in high-latitude
wind their dominant character probably extends well
far away from the Sun. At low solar activity the
polar wind fills a large fraction of the heliospheric
cavity. For such conditions the outward fluctuations
may play a leading role in acceleration and diffusion of high-energy particles. In other words, models
based on the assumption of a negligible cross-helicity
should not be able to give a good description of these
381
phenomena.
As regards the imbalance in favour of magnetic energy, it does not appear to go beyond a given limit.
Several ways to get a different from zero residual energy have been proposed, for instance, 2-D processes
(e.g., Oughton et al. [1994]; Matthaeus et al. [1996])
or propagation in a non uniform medium (e.g., Hollweg and Lee [1989]). However, convincing arguments
to account for the existence of such limit have not
been given yet.
In conclusion, though non negligible advances have
still to be made, a satisfactory understanding of the
solar wind Alfvénic turbulence does not seem too far
away.
ACKNOWLEDGMENTS
Useful discussions with R. Bruno are gratefully acknowledged. The present work has been supported
by the Italian Space Agency (ASI).
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