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"The Solar Wind as a Turbulence Laboratory"
Roberto Bruno and Vincenzo Carbone 
Abstract
1 Introduction
1.1 What does turbulence stand for?
1.2 Dynamics vs. statistics
2 Equations and Phenomenology
2.1 The Navier–Stokes equation and the Reynolds number
2.2 The coupling between a charged fluid and the magnetic field
2.3 Scaling features of the equations
2.4 The non-linear energy cascade
2.5 The inhomogeneous case
2.6 Dynamical system approach to turbulence
2.7 Shell models for turbulence cascade
2.8 The phenomenology of fully developed turbulence: Fluid-like case
2.9 The phenomenology of fully developed turbulence: Magnetically-dominated case
2.10 Some exact relationships
2.11 Yaglom’s law for MHD turbulence
2.12 Density-mediated Elsässer variables and Yaglom’s law
2.13 Yaglom’s law in the shell model for MHD turbulence
3 Early Observations of MHD Turbulence in the Ecliptic
3.1 Turbulence in the ecliptic
3.2 Turbulence studied via Elsässer variables
4 Observations of MHD Turbulence in the Polar Wind
4.1 Evolving turbulence in the polar wind
4.2 Polar turbulence studied via Elsässer variables
5 Numerical Simulations
5.1 Local production of Alfvénic turbulence in the ecliptic
5.2 Local production of Alfvénic turbulence at high latitude
6 Compressive Turbulence
6.1 On the nature of compressive turbulence
6.2 Compressive turbulence in the polar wind
6.3 The effect of compressive phenomena on Alfvénic correlations
7 A Natural Wind Tunnel
7.1 Scaling exponents of structure functions
7.2 Probability distribution functions and self-similarity of fluctuations
7.3 What is intermittent in the solar wind turbulence? The multifractal approach
7.4 Fragmentation models for the energy transfer rate
7.5 A model for the departure from self-similarity
7.6 Intermittency properties recovered via a shell model
8 Observations of Yaglom’s Law in Solar Wind Turbulence
9 Intermittency Properties in the 3D Heliosphere: Taking a Look at the Data
9.1 Structure functions
9.2 Probability distribution functions
10 Turbulent Structures
10.1 On the statistics of magnetic field directional fluctuations
10.2 Radial evolution of intermittency in the ecliptic
10.3 Radial evolution of intermittency at high latitude
11 Solar Wind Heating by the Turbulent Energy Cascade
11.1 Dissipative/dispersive range in the solar wind turbulence
12 The Origin of the High-Frequency Region
12.1 A dissipation range
12.2 A dispersive range
13 Two Further Questions About Small-Scale Turbulence
13.1 Whistler modes scenario
13.2 Kinetic Alfvén waves scenario
13.3 Where does the fluid-like behavior break down in solar wind turbulence?
13.4 What physical processes replace “dissipation” in a collisionless plasma?
14 Conclusions and Remarks
Acknowledgments
A Some Characteristic Solar Wind Parameters
B Tools to Analyze MHD Turbulence in Space Plasmas
B.1 Statistical description of MHD turbulence
B.2 Spectra of the invariants in homogeneous turbulence
B.3 Introducing the Elsässer variables
C Wavelets as a Tool to Study Intermittency
D Reference Systems
D.1 Minimum variance reference system
D.2 The mean field reference system
E On-board Plasma and Magnetic Field Instrumentation
E.1 Plasma instrument: The top-hat
E.2 Measuring the velocity distribution function
E.3 Computing the moments of the velocity distribution function
E.4 Field instrument: The flux-gate magnetometer
F Spacecraft and Datasets
References
Footnotes
Updates
Figures
Tables

4 Observations of MHD Turbulence in the Polar Wind

In 1994 – 1995, Ulysses gave us the opportunity to look at the solar wind out-of-the-ecliptic, providing us with new exciting observations. For the first time heliospheric instruments were sampling pure, fast solar wind, free of any dynamical interaction with slow wind. There is one figure that within our scientific community has become as popular as “La Gioconda” by Leonardo da Vinci within the world of art. This figure produced at LANL (McComas et al., 1998) is shown in the upper left panel of Figure 51View Image, which has been taken from a successive paper by (McComas et al., 2003Jump To The Next Citation Point), and summarizes the most important aspects of the large scale structure of the polar solar wind during the minimum of the solar activity phase, as indicated by the low value of the Wolf’s number reported in the lower panel. It shows speed profile, proton number density profile and magnetic field polarity vs. heliographic latitude during the first complete Ulysses’ polar orbit. Fast wind fills up north and south hemispheres of the Sun almost completely, except a narrow latitudinal belt around the equator, where the slow wind dominates. Flow velocity, which rapidly increases from the equator towards higher latitudes, quickly reaches a plateau and the wind escapes the polar regions with a rather uniform speed. Moreover, polar wind is characterized by a lower number density and shows rather uniform magnetic polarity of opposite sign, depending on the hemisphere. Thus, the main difference between ecliptic and polar wind is that this last one completely lacks of dynamical interactions with slower plasma and freely flows into the interplanetary space. The presence or not of this phenomenon, as we will see in the following pages, plays a major role in the development of MHD turbulence during the wind expansion.

During solar maximum (look at the upper right panel of Figure 51View Image) the situation dramatically changes and the equatorial wind extends to higher latitudes, to the extent that there is no longer difference between polar and equatorial wind.

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Figure 51: Large scale solar wind profile as a function of latitude during minimum (left panel) and maximum (right panel) solar cycle phases. The sunspot number is also shown at the bottom panels. Image reproduced by permission from McComas et al. (2003), copyright by AGU.

4.1 Evolving turbulence in the polar wind

Ulysses observations gave us the possibility to test whether or not we could forecast the turbulent evolution in the polar regions on the basis of what we had learned in the ecliptic. We knew that, in the ecliptic, velocity shear, parametric decay, and interaction of Alfvénic modes with convected structures (see Sections 3.2.1, 5.1) all play some role in the turbulent evolution and, before Ulysses reached the polar regions of the Sun, three possibilities were given:

  1. Alfvénic turbulence would have not relaxed towards standard turbulence because the large scale velocity shears would have been much less relevant (Grappin et al., 1991);
  2. since the magnetic field would be smaller far from the ecliptic, at large heliocentric distances, even small shears would lead to an isotropization of the fluctuations and produce a turbulent cascade faster than the one observed at low latitudes, and the subsequent evolution would take less time (Roberts et al., 1990Jump To The Next Citation Point);
  3. there would still be evolution due to interaction with convected plasma and field structures but it would be slower than in the ecliptic since the power associated with Alfvénic fluctuations would largely dominate over the inhomogeneities of the medium. Thus, Alfvénic correlations should last longer than in the ecliptic plane, with a consequent slower evolution of the normalized cross-helicity (Bruno, 1992Jump To The Next Citation Point).

A fourth possibility was added by Tu and Marsch (1995aJump To The Next Citation Point), based on their model (Tu and Marsch, 1993Jump To The Next Citation Point). Following this model they assumed that polar fluctuations were composed by outward Alfvénic fluctuations and MFDT. The spectra of these components would decrease with radial distance because of a WKB evolution and convective effects of the diverging flow. As the distance increases, the field becomes more transverse with respect to the radial direction, the s/c would sample more convective structures and, as a consequence, would observe a decrease of both σc and r A.

Today we know that polar Alfvénic turbulence evolves in the same way it does in the ecliptic plane, but much more slowly. Moreover, the absence of strong velocity shears and enhanced compressive phenomena suggests that also some other mechanism based on parametric decay instability might play some role in the local production of turbulence (Bavassano et al., 2000aJump To The Next Citation Point; Malara et al., 2001aJump To The Next Citation Point, 2002; Primavera et al., 2003Jump To The Next Citation Point).

The first results of Ulysses magnetic field and plasma measurements in the polar regions, i.e., above ±30 ∘ latitude (left panel of Figure 51View Image), revealed the presence of Alfvénic correlations in a frequency range from less than 1 to more than 10 h (Balogh et al., 1995Jump To The Next Citation Point; Smith et al., 1995Jump To The Next Citation Point; Goldstein et al., 1995aJump To The Next Citation Point) in very good agreement with ecliptic observations (Bruno et al., 1985Jump To The Next Citation Point). However, it is worth noticing that Helios observations referred to very short heliocentric distances around 0.3 AU while the above Ulysses observations were taken up to 4 AU. As a matter of fact, these long period Alfvén waves observed in the ecliptic, in the inner solar wind, become less prominent as the wind expands due to stream-stream dynamical interaction effects (Bruno et al., 1985Jump To The Next Citation Point) and strong velocity shears (Roberts et al., 1987aJump To The Next Citation Point). At high latitude, the relative absence of enhanced dynamical interaction between flows at different speed and, as a consequence, the absence of strong velocity shears favors the survival of these extremely low frequency Alfvénic fluctuations for larger heliocentric excursions.

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Figure 52: Magnetic field and velocity hourly correlation vs. heliographic latitude. Image reproduced by permission from Smith et al. (1995Jump To The Next Citation Point), copyright by AGU.

Figure 52View Image shows the hourly correlation coefficient for the transverse components of magnetic and velocity fields as Ulysses climbs to the south pole and during the fast latitude scanning that brought the s/c from the south to the north pole of the Sun in just half a year. While the equatorial phase of Ulysses journey is characterized by low values of the correlation coefficients, a gradual increase can be noticed starting at half of year 1993 when the s/c starts to increase its heliographic latitude from the ecliptic plane up to ∘ 80.2 south, at the end of 1994. Not only the degree of δb − δv correlation resembled Helios observations but also the spectra of these fluctuations showed characteristics which were very similar to those observed in the ecliptic within fast wind like the spectral index of the components, that was found to be flat at low frequency and more Kolmogorov-like at higher frequencies (Smith et al., 1995). Balogh et al. (1995) and Forsyth et al. (1996Jump To The Next Citation Point) discussed magnetic fluctuations in terms of latitudinal and radial dependence of their variances. Similarly to what had been found within fast wind in the ecliptic (Mariani et al., 1978; Bavassano et al., 1982bJump To The Next Citation Point; Tu et al., 1989bJump To The Next Citation Point; Roberts et al., 1992Jump To The Next Citation Point), variance of magnetic magnitude was much less than the variance associated with the components. Moreover, transverse variances had consistently higher values than the one along the radial direction and were also much more sensitive to latitude excursion, as shown in Figure 53View Image.

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Figure 53: Normalized magnetic field components and magnitude hourly variances plotted vs. heliographic latitude during a complete latitude survey by Ulysses. Image reproduced by permission from Forsyth et al. (1996Jump To The Next Citation Point), copyright by AGU.
In addition, the level of the normalized hourly variances of the transverse components observed during the ecliptic phase, right after the compressive region ahead of co-rotating interacting regions, was maintained at the same level once the s/c entered the pure polar wind. Again, these observations showed that the fast wind observed in the ecliptic was coming from the equatorward extension of polar coronal holes.

Horbury et al. (1995cJump To The Next Citation Point) and Forsyth et al. (1996Jump To The Next Citation Point) showed that the interplanetary magnetic field fluctuations observed by Ulysses continuously evolve within the fast polar wind, at least out to 4 AU. Since this evolution was observed within the polar wind, rather free of co-rotating and transient events like those characterizing low latitudes, they concluded that some other mechanism was at work and this evolution was an intrinsic property of turbulence.

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Figure 54: Spectral indexes of magnetic fluctuations within three different time scale intervals as indicated in the plot. The bottom panel shows heliographic latitude and heliocentric distance of Ulysses. Image reproduced by permission from Horbury et al. (1995cJump To The Next Citation Point), copyright by AGU.

Results in Figure 54View Image show the evolution of the spectral slope computed across three different time scale intervals. The smallest time scales show a clear evolution that keeps on going past the highest latitude on day 256, strongly suggesting that this evolution is radial rather than latitudinal effect. Horbury et al. (1996aJump To The Next Citation Point) worked on determining the rate of turbulent evolution for the polar wind.

They calculated the spectral index at different frequencies from the scaling of the second order structure function (see Section 7 and papers by Burlaga, 1992a,b; Marsch and Tu, 1993aJump To The Next Citation Point; Ruzmaikin et al., 1995Jump To The Next Citation Point; and Horbury et al., 1996b) since the spectral scaling α is related to the scaling of the structure function s by the following relation: α = s + 1 (Monin and Yaglom, 1975). Horbury et al. (1996aJump To The Next Citation Point), studying variations of the spectral index with frequency for polar turbulence, found that there are two frequency ranges where the spectral index is rather steady. The first range is around 10−2 Hz with a spectral index around − 5∕3, while the second range is at very low frequencies with a spectral index around − 1. This last range is the one where Goldstein et al. (1995aJump To The Next Citation Point) found the best example of Alfvénic fluctuations. Similarly, ecliptic studies found that the best Alfvénic correlations belonged to the hourly, low frequency regime (Bruno et al., 1985Jump To The Next Citation Point).

Horbury et al. (1995aJump To The Next Citation Point) presented an analysis of the high latitude magnetic field using a fractal method. Within the solar wind context, this method has been described for the first time by Burlaga and Klein (1986Jump To The Next Citation Point) and Ruzmaikin et al. (1993), and is based on the estimate of the scaling of the length function L (τ ) with the scale τ. This function is closely related to the first order structure function and, if statistical self-similar, has scaling properties ℓ L (τ ) ∼ τ, where ℓ is the scaling exponent. It follows that L (τ) is an estimate of the amplitude of the fluctuations at scale τ, and the relation that binds L(τ) to the variance of the fluctuations (δB )2 ∼ τs(2) is:

21∕2 s(2)∕2− 1 L(τ) ∼ N (τ)[(δB )] ∝ τ ,

where N (τ) represents the number of points at scale τ and scales like − 1 τ. Since the power density spectrum W (f ) is related to (δB )2 through the relation fW (f ) ∼ (δB )2, if W (f ) ∼ f −α, then s(2) = α − 1, and, as a consequence α = 2ℓ + 3 (Marsch and Tu, 1996Jump To The Next Citation Point). Thus, it results very easy to estimate the spectral index at a given scale or frequency, without using spectral methods but simply computing the length function.

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Figure 55: Spectral exponents for the Bz component estimated from the length function computed from Ulysses magnetic field data, when the s/c was at about 4 AU and ∘ ∼ − 50 latitude. Different symbols refer to different time intervals as reported in the graph. Image reproduced by permission from (from Horbury et al., 1995aJump To The Next Citation Point).

Results in Figure 55View Image show the existence of two different regimes, one with a spectral index around the Kolmogorov scaling extending from 101.5 to 103 s and, separated by a clear break-point at scales of 103 s, a flatter and flatter spectral exponent for larger and larger scales. These observations were quite similar to what had been observed by Helios 2 in the ecliptic, although the turbulence state recorded by Ulysses resulted to be more evolved than the situation seen at 0.3 AU and, perhaps, more similar to the turbulence state observed around 1 AU, as shown by Marsch and Tu (1996Jump To The Next Citation Point). These authors compared the spectral exponents, estimated using the same method of Horbury et al. (1995aJump To The Next Citation Point), from Helios 2 magnetic field observations at two different heliocentric distances: 0.3 and 1.0 AU. The comparison with Ulysses results is shown in Figure 56View Image where it appears rather clear that the slope of the Bz spectrum experiences a remarkable evolution during the wind expansion between 0.3 and 4 AU. Obviously, this comparison is meaningful in the reasonable hypothesis that fluctuations observed by Helios 2 at 0.3 AU are representative of out-of-the-ecliptic solar wind (Marsch and Tu, 1996Jump To The Next Citation Point). This figure also shows that the degree of spectral evolution experienced by the fluctuations when observed at 4 AU at high latitude, is comparable to Helios observations at 1 AU in the ecliptic. Thus, the spectral evolution at high latitude is present although quite slower with respect to the ecliptic.

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Figure 56: Spectral exponents for the Bz component estimated from the length function computed from Helios and Ulysses magnetic field data. Ulysses length function (dotted line) is the same shown in the paper by Horbury et al. (1995aJump To The Next Citation Point) when the s/c was at about 4 AU and ∼ − 50 ∘ latitude. Image reproduced by permission from Marsch and Tu (1996), copyright by AGU.

Forsyth et al. (1996Jump To The Next Citation Point) studied the radial dependence of the normalized hourly variances of the components BR, BT and BN and the magnitude |B | of the magnetic field (see Appendix D to learn about the RT N reference system). The variance along the radial direction was computed as σR2 = ⟨BR2 > − < BR ⟩2 and successively normalized to |B |2 to remove the field strength dependence. Moreover, variances along the other two directions T and N were similarly defined. Fitting the radial dependence with a power law of the form − α r, but limiting the fit to the radial excursion between 1.5 and 3 AU, these authors obtained α = 3.39 ± 0.07 for σ2r, α = 3.45 ± 0.09 for σ2T, α = 3.37 ± 0.09 for σ2N, and α = 2.48 ± 0.14 for σ2 B. Thus, for hourly variances, the power associated with the components showed a radial dependence stronger than the one predicted by the WKB approximation, which would provide α = 3. These authors also showed that including data between 3 and 4 AU, corresponding to intervals characterized by compressional features mainly due to high latitude CMEs, they would obtain less steep radial gradients, much closer to a WKB type. These results suggested that compressive effects can feed energy at the smallest scales, counteracting dissipative phenomena and mimicking a WKB-like behavior of the fluctuations. However, they concluded that for lower frequencies, below the frequency break point, fluctuations do follow the WKB radial evolution.

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Figure 57: Hourly variances of the components and the magnitude of the magnetic field vs. radial distance from the Sun. The meaning of the different symbols is also indicated in the upper right corner. Image reproduced by permission from Forsyth et al. (1996Jump To The Next Citation Point), copyright by AGU.

Horbury and Balogh (2001Jump To The Next Citation Point) presented a detailed comparison between Ulysses and Helios observations about the evolution of magnetic field fluctuations in high-speed solar wind. Ulysses results, between 1.4 and 4.1 AU, were presented as wave number dependence of radial and latitudinal power scaling. The first results of this analysis showed (Figure 3 of their work) a general decrease of the power levels with solar distance, in both magnetic field components and magnitude fluctuations. In addition, the power associated with the radial component was always less than that of the transverse components, as already found by Forsyth et al. (1996Jump To The Next Citation Point). However, Horbury and Balogh (2001Jump To The Next Citation Point), supposing a possible latitude dependence, performed a multiple linear regression of the type:

log w = A + B log r + C sin𝜃, (59 ) 10 p p 10 p
where w is the power density integrated in a given spectral band, r is the radial distance and 𝜃 is the heliolatitude (0∘ at the equator). Moreover, the same procedure was applied to spectral index estimates α of the form α = A + B log r + C sin 𝜃 α α 10 α. Results obtained for B ,C ,B ,C p p α α are shown in Figure 58View Image.
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Figure 58: (a) Scale dependence of radial power, (b) latitudinal power, (c) radial spectral index, (d) latitudinal spectral index, and (e) spectral index computed at 2.5 AU. Solid circles refer to the trace of the spectral matrix of the components, open squares refer to field magnitude. Correspondence between wave number scale and time scale is based on a wind velocity of 750 km s–1. Image reproduced by permission from Horbury and Balogh (2001Jump To The Next Citation Point), copyright by AGU.

On the basis of variations of spectral index and radial and latitudinal dependencies, these authors were able to identify four wave number ranges as indicated by the circled numbers in the top panel of Figure 58View Image. Range 1 was characterized by a radial power decrease weaker than WKB (− 3), positive latitudinal trend for components (more power at higher latitude) and negative for magnitude (less compressive events at higher latitudes). Range 2 showed a more rapid radial decrease of power for both magnitude and components and a negative latitudinal power trend, which implies less power at higher latitudes. Moreover, the spectral index of the components (bottom panel) is around 0.5 and tends to 0 at larger scales. Within range 3 the power of the components follows a WKB radial trend and the spectral index is around − 1 for both magnitude and components. This hourly range has been identified as the most Alfvénic at low latitudes and its radial evolution has been recognized to be consistent with WKB radial index (Roberts, 1989; Marsch and Tu, 1990aJump To The Next Citation Point). Even within this range, and also within the next one, the latitude power trend is slightly negative for both components and magnitude. Finally, range 4 is clearly indicative of turbulent cascade with a radial power trend of the components much faster than WKB expectation and becoming even stronger at higher wave numbers. Moreover, the radial spectral index reveals that steepening is at work only for the previous wave number ranges as expected since the breakpoint moves to smaller wave number during spectrum evolution. The spectral index of the components tends to − 5∕3 with increasing wave number while that of the magnitude is constantly flatter. The same authors gave an estimate of the radial scale-shift of the breakpoint during the wind expansion around k ∝ r1.1, in agreement with earlier estimates (Horbury et al., 1996aJump To The Next Citation Point).

Although most of these results support previous conclusions obtained for the ecliptic turbulence, the negative value of the latitudinal power trend that starts within the second range, is unexpected. As a matter of fact, moving towards more Alfénic regions like the polar regions, one would perhaps expect a positive latitudinal trend similarly to what happens in the ecliptic when moving from slow to fast wind.

Horbury and Balogh (2001Jump To The Next Citation Point) and Horbury and Tsurutani (2001Jump To The Next Citation Point) estimated that the power observed at 80 ∘ is about 30% less than that observed at 30∘. These authors proposed a possible effect due to the over-expansion of the polar coronal hole at higher latitudes. In addition, within the fourth range, field magnitude fluctuations radially decrease less rapidly than the fluctuations of the components, but do not show significant latitudinal variations. Finally, the smaller spectral index reveals that the high frequency range of the field magnitude spectrum shows a flattening.

The same authors investigated the anisotropy of these fluctuations as a function of radial and latitudinal excursion. Their results, reported in Figure 59View Image, show that, at 2.5 AU, the lowest compressibility is recorded within the hourly frequency band (third and part of the fourth band), which has been recognized as the most Alfvénic frequency range. The anisotropy of the components confirms that the power associated with the transverse components is larger than that associated with the radial one, and this difference slightly tends to decrease at higher wave numbers.

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Figure 59: (a) Scale dependence of power anisotropy at 2.5 AU plotted as the log10 of the ratio of BR (solid circles), BT (triangles), BN (diamonds), and |B| (squares) to the trace of the spectral matrix; (b) the radial, and (c) latitudinal behavior of the same values, respectively. Image reproduced by permission from Horbury and Balogh (2001Jump To The Next Citation Point), copyright by AGU.

As already shown by Horbury et al. (1995b), around the 5 min range, magnetic field fluctuations are transverse to the mean field direction the majority of the time. The minimum variance direction lies mainly within an angle of about 26∘ from the average background field direction and fluctuations are highly anisotropic, such that the ratio between perpendicular to parallel power is about 30. Since during the observations reported in Horbury and Balogh (2001Jump To The Next Citation Point) and Horbury and Tsurutani (2001) the mean field resulted to be radially oriented most of the time, the radial minimum variance direction at short time scales is an effect induced by larger scales behavior.

Anyhow, radial and latitudinal anisotropy trends tend to disappear for higher frequencies. In the mean time, interesting enough, there is a strong radial increase of magnetic field compression (top panel of Figure 59View Image), defined as the ratio between the power density associated with magnetic field intensity fluctuations and that associated with the fluctuations of the three components (Bavassano et al., 1982aJump To The Next Citation Point; Bruno and Bavassano, 1991Jump To The Next Citation Point). The attempt to attribute this phenomenon to parametric decay of large amplitude Alfvén waves or dynamical interactions between adjacent flux tubes or interstellar pick-up ions was not satisfactory in all cases.

Comparing high latitude with low latitude results for high speed streams, Horbury and Balogh (2001Jump To The Next Citation Point) found remarkable good agreement between observations by Ulysses at 2.5 AU and by Helios at 0.7 AU. In particular, Figure 60View Image shows Ulysses and Helios 1 spectra projected to 1 AU for comparison.

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Figure 60: Power spectra of magnetic field components (solid circles) and magnitude (open squares) from Ulysses (solid line) and Helios 1 (dashed line). Spectra have been extrapolated to 1 AU using radial trends in power scalings estimated from Ulysses between 1.4 and 4.1 AU and Helios between 0.3 and 1 AU. Image reproduced by permission from Horbury and Balogh (2001Jump To The Next Citation Point), copyright by AGU.

It is interesting to notice that the spectral slope of the spectrum of the components for Helios 1 is slightly higher than that of Ulysses, suggesting a slower radial evolution of turbulence in the polar wind (Bruno, 1992; Bruno and Bavassano, 1992). However, the faster spectral evolution at low latitudes does not lead to strong differences between the spectra.

4.2 Polar turbulence studied via Elsässer variables

Goldstein et al. (1995aJump To The Next Citation Point) for the first time showed a spectral analysis of Ulysses observations based on Elsässer variables during two different time intervals, at 4 AU and close to − 40∘, and at 2 AU and around the maximum southern pass, as shown in Figure 61View Image. Comparing the two Ulysses observations it clearly appears that the spectrum closer to the Sun is less evolved than the spectrum measured farther out, as will be confirmed by the next Figure 62View Image, where these authors reported the normalized cross-helicity and the Alfvén ratio for the two intervals. Moreover, following these authors, the comparison between Helios spectra at 0.3 AU and Ulysses at 2 and 4 AU suggests that the radial scaling of e+ at the low frequency end of the spectrum follows the WKB prediction of 1∕r decrease (Heinemann and Olbert, 1980). However, the selected time interval for Helios s/c was characterized by rather slow wind taken during the rising phase the solar cycle, two conditions which greatly differ from those referring to Ulysses data. As a consequence, comparing Helios results with Ulysses results obtained within the fast polar wind might be misleading. It would be better to choose Helios observations within high speed co-rotating streams which resemble much better solar wind conditions at high latitude.

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Figure 61: Trace of e+ (solid line) and e (dash-dotted line) power spectra. The central and right panels refer to Ulysses observations at 2 and 4 AU, respectively, when Ulysses was embedded in the fast southern polar wind during 1993 – 1994. The leftmost panel refers to Helios observations during 1978 at 0.3 AU. Image reproduced by permission from Goldstein et al. (1995aJump To The Next Citation Point), copyright by AGU.

Anyhow, results relative to the normalized cross-helicity σc (see Figure 62View Image) clearly show high values of σc, around 0.8, which normally we observe in the ecliptic at much shorter heliocentric distances (Tu and Marsch, 1995aJump To The Next Citation Point). A possible radial effect would be responsible for the depleted level of σc at 4 AU. Moreover, a strong anisotropy can also be seen for frequencies between 10−6 –10− 5 Hz with the transverse σc much larger than the radial one. This anisotropy is somewhat lost during the expansion to 4 AU.

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Figure 62: Normalized cross-helicity and Alfvén ratio at 2 and 4 AU, as observed by Ulysses at ∘ − 80 and ∘ − 40 latitude, respectively. Image reproduced by permission from Goldstein et al. (1995a), copyright by AGU.

The Alfvén ratio (bottom panels of Figure 62View Image) has values around 0.5 for frequencies higher than roughly 10 −5 Hz, with no much evolution between 2 and 4 AU. A result similar to what was originally obtained in the ecliptic at about 1 AU (Martin et al., 1973; Belcher and Solodyna, 1975; Solodyna et al., 1977; Neugebauer et al., 1984; Bruno et al., 1985Jump To The Next Citation Point; Marsch and Tu, 1990aJump To The Next Citation Point; Roberts et al., 1990). The low frequency extension of rA⊥ together with σc⊥, where the subscript ⊥ indicates that these quantities are calculated from the transverse components only, was interpreted by the authors as due to the sampling of Alfvénic features in longitude rather than to a real presence of Alfvénic fluctuations. However, by the time Ulysses reaches to 4 AU, σc⊥ has strongly decreased as expected while r A⊥ gets closer to 1, making the situation less clear. Anyhow, these results suggest that the situation at 2 AU and, even more at 4 AU, can be considered as an evolution of what Helios 2 recorded in the ecliptic at shorter heliocentric distance. Ulysses observations at 2 AU resemble more the turbulence conditions observed by Helios at 0.9 AU rather than at 0.3 AU.

Bavassano et al. (2000aJump To The Next Citation Point) studied in detail the evolution of the power e+ and e associated with outward + δz and inward − δz Alfvénic fluctuations, respectively. The study referred to the polar regions, during the wind expansion between 1.4 and 4.3 AU. These authors analyzed 1 h variances of ± δz and found two different regimes, as shown in Figure 63View Image. Inside 2.5 AU outward modes e+ decrease faster than inward modes e, in agreement with previous ecliptic observations performed within the trailing edge of co-rotating fast streams (Bruno and Bavassano, 1991Jump To The Next Citation Point; Tu and Marsch, 1990b; Grappin et al., 1989Jump To The Next Citation Point). Beyond this distance, the radial gradient of e becomes steeper and steeper while that of e+ remains approximately unchanged. This change in e is rather fast and both species keep declining with the same rate beyond 2.5 AU. The radial dependence of e+ between r−1.39 and r− 1.48, reported by Bavassano et al. (2000aJump To The Next Citation Point), indicate a radial decay faster than r− 1 predicted by WKB approximation. This is in agreement with the analysis performed by Forsyth et al. (1996) using magnetic field observations only.

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Figure 63: Left panel: values of hourly variance of ± δz (i.e., ± e) vs. heliocentric distance, as observed by Ulysses. Helios observations are shown for comparison and appear to be in good agreement. Right panel: Elsässer ratio (top) and Alfvén ratio (bottom) are plotted vs. radial distance while Ulysses is embedded in the polar wind. Image reproduced by permission from Bavassano et al. (2000aJump To The Next Citation Point), copyright by AGU.

This different radial behavior is readily seen in the radial plot of the Elsässer ratio rE shown in the top panel of the right column of Figure 63View Image. Before 2.5 AU this ratio continuously grows to about 0.5 near 2.5 AU. Beyond this region, since the radial gradient of the inward and outward components is approximately the same, rE stabilizes around 0.5.

On the other hand, also the Alfvén ratio rA shows a clear radial dependence that stops at about the same limit distance of 2.5 AU. In this case, rA constantly decreases from ∼ 0.4 at 1.4 AU to ∼ 0.25 at 2.5 AU, slightly fluctuating around this value for larger distances. A different interpretation of these results was offered by Grappin (2002). For this author, since Ulysses has not explored the whole three-dimensional heliosphere, solar wind parameters experience different dependencies on latitude and distance which would result in the same radial distance variation along Ulysses trajectory as claimed in Bavassano’s works. Another interesting feature observed in polar turbulence is unraveled by Figure 64View Image from Bavassano et al. (1998Jump To The Next Citation Point, 2000b). The plot shows 2D histograms of normalized cross-helicity and normalized residual energy (see Appendix B.3.1 for definition) for different heliospheric regions (ecliptic wind, mid-latitude wind with strong velocity gradients, polar wind). A predominance of outward fluctuations (positive values of σc) and of magnetic fluctuations (negative values of σr) seems to be a general feature. It results that the most Alfvénic region is the one at high latitude and at shorter heliocentric distances. However, in all the panels there is always a relative peak at σc ≃ 0 and σr ≃ − 1, which might well be due to magnetic structures like the MFDT found by Tu and Marsch (1991) in the ecliptic.

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Figure 64: 2D histograms of normalized cross-helicity σc (here indicated by σC) and normalized residual energy σr (here indicated by σR) for different heliospheric regions (ecliptic wind, mid-latitude wind with strong velocity gradients, polar wind). Image reproduced by permission from Bavassano et al. (1998), copyright by AGU.

In a successive paper, Bavassano et al. (2002aJump To The Next Citation Point) tested whether or not the radial dependence observed in e± was to be completely ascribed to the radial expansion of the wind or possible latitudinal dependencies also contributed to the turbulence evolution in the polar wind.

As already discussed in the previous section, Horbury and Balogh (2001Jump To The Next Citation Point), using Ulysses data from the northern polar pass, evaluated the dependence of magnetic field power levels on solar distance and latitude using a multiple regression analysis based on Equation (59View Equation). In the Alfvénic range, the latitudinal coefficient “C” for power in field components was appreciably different from 0 (around 0.3). However, this analysis was limited to magnetic field fluctuations alone and cannot be transferred sic et simpliciter to Alfvénic turbulence. In their analysis, Bavassano et al. (2002b) used the first southern and northern polar passes and removed from their dataset all intervals with large gradients in plasma velocity, and/or plasma density, and/or magnetic field magnitude, as already done in Bavassano et al. (2000aJump To The Next Citation Point). As a matter of fact, the use of Elsässer variables (see Appendix B.3.1) instead of magnetic field, and of selected data samples, leads to very small values of the latitudinal coefficient as shown in Figure 65View Image, where different contributions are plotted with different colors and where the top panel refers to the same dataset used by Horbury and Balogh (2001Jump To The Next Citation Point), while the bottom panel refers to a dataset omni-comprehensive of south and north passages free of strong compressive events (Bavassano et al., 2000aJump To The Next Citation Point). Moreover, the latitudinal effect appears to be very weak also for the data sample used by Horbury and Balogh (2001Jump To The Next Citation Point), although this is the sample with the largest value of the “C” coefficient.

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Figure 65: Results from the multiple regression analysis showing radial and latitudinal dependence of the power e+ associated with outward modes (see Appendix B.3.1). The top panel refers to the same dataset used by Horbury and Balogh (2001). The bottom panel refers to a dataset omni-comprehensive of south and north passages free of strong compressive events (Bavassano et al., 2000a). Values of e+ have been normalized to the value ∘ e+ assumed by this parameter at 1.4 AU, closest approach to the Sun. The black line is the total regression, the blue line is the latitudinal contribution and the red line is the radial contribution. Image reproduced by permission from Bavassano et al. (2002aJump To The Next Citation Point), copyright by AGU.

A further argument in favor of radial vs. latitudinal dependence is represented by the comparison of the radial gradient of e+ in different regions, in the ecliptic and in the polar wind. These results, shown in Figure 66View Image, provide the radial slopes for e+ (red squares) and e (blue diamonds) in different regions. The first three columns (labeled EQ) summarize ecliptic results obtained with different values of an upper limit (TBN) for relative fluctuations of density and magnetic intensity. The last two columns (labeled POL) refer to the results for polar turbulence (north and south passes) outside and inside 2.6 AU, respectively. A general agreement exists between slopes in ecliptic and in polar wind with no significant role left for latitude, the only exception being e in the region inside 2.6 AU. The behavior of the inward component cannot be explained by a simple power law over the range of distances explored by Ulysses. Moreover, a possible latitudinal effect has been clearly rejected by the results from a multiple regression analysis performed by Bavassano et al. (2002a) similar to that reported above for e+.

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Figure 66: e+ (red) and e (blue) radial gradient for different latitudinal regions of the solar wind. The first three columns, labeled EQ, refer to ecliptic observations obtained with different values of the upper limit of TBN defined as the relative fluctuations of density and magnetic intensity. The last two columns, labeled POL, refer to observations of polar turbulence outside and inside 2.6 AU, respectively. Image reproduced by permission from Bavassano et al. (2001), copyright by AGU.


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