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:

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 );
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 (Roberts et al., 1990 );
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, 1992 ).

A fourth possibility was added by Tu and Marsch (1995a ), based on their model (Tu and Marsch, 1993 ). 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 $sc$ and $rA$ .

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., 2000a ; Malara et al., 2001 , 2002 ; Primavera et al., 2003 ).

The first results of Ulysses magnetic field and plasma measurements in the polar regions, i.e., above $± 30o$ latitude (left panel of Figure 43 ), revealed the presence of Alfvénic correlations in a frequency range from less than $1$ to more than $10 h$ (Balogh et al., 1995 ; Smith et al., 1995 ; Goldstein et al., 1995a ) in very good agreement with ecliptic observations (Bruno et al., 1985 ). 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., 1985 ) and strong velocity shears (Roberts et al., 1987a ). 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.

Figure 44:

Magnetic field and velocity hourly correlation vs. heliographic latitude (adopted from Smith et al., 1995

Figure 44

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 $o 80.2$ south, at the end of 1994. Not only the degree of $db - dv$ 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. (1996

) 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., 1982b

; Tu et al., 1989b

; Roberts et al., 1992

), 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 45

Figure 45:

Normalized magnetic field components and magnitude hourly variances plotted vs. heliographic latitude during a complete latitude survey by Ulysses (adopted from Forsyth et al., 1996

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 corotating 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. (1995c

) and Forsyth et al. (1996

) 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.

Figure 46:

Spectral indexes of magnetic fluctuations within three different time scales as indicated in the plot. The bottom panel shows heliographic latitude and heliocentric distance of Ulysses (adopted from Horbury et al., 1995c

Results in Figure 46

show the evolution of three different time scales. 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. (1996a

) 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 ), Burlaga (1992b ), Marsch and Tu (1993a

), Ruzmaikin et al. (1995

), and Horbury et al. (1996b )) since the spectral scaling $a$ is related to the scaling of the structure function $s$ by the following relation: $a = s + 1$ (Monin and Yaglom, 1975 ). Horbury et al. (1996a

), 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 $-2 10 Hz$ 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. (1995a

) 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., 1985

Horbury et al. (1995a ) 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 (1986 ) and Ruzmaikin et al. (1993 ), and is based on the estimate of the scaling of the length function $L(t )$ with the scale $t$ . This function is closely related to the first order structure function and, if statistical self-similar, has scaling properties $L(t ) ~ tl$ , where $l$ is the scaling exponent. It follows that $L(t )$ is an estimate of the amplitude of the fluctuations at scale $t$ , and the relation that binds $L(t)$ to the variance of the fluctuations $2 s(2) (dB) ~ t$ is:

L(t ) ~ N (t )[(dB)2]1/2 oc ts(2)/2-1,

where $N (t)$ represents the number of points at scale $t$ and scales like $t- 1$ . Since the power density spectrum $W (f )$ is related to $2 (dB)$ through the relation $2 fW (f ) ~ (dB)$ , if $-a W (f ) ~ f$ , then $s(2) = a - 1$ , and, as a consequence $a = 2l + 3$ (Marsch and Tu, 1996 ). 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.

Figure 47:

Spectral exponents for the $B z$ component estimated from the length function computed from Ulysses magnetic field data, when the s/c was at about $4 AU$ and $o ~ - 50$ latitude. Different symbols refer to different time intervals as reported in the graph (figure adopted from Horbury et al., 1995a

Results in Figure 47

show the existence of two different regimes, one with a spectral index around the Kolmogorov scaling extending from $1.5 10$ to $3 10 s$ 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 (1996

). These authors compared the spectral exponents, estimated using the same method of Horbury et al. (1995a

), 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 48

where it appears rather clear that the slope of the $B z$ 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, 1996

). 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.

Figure 48:

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. (1995a

Forsyth et al. (1996

) 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 15 to learn about the $RT N$ reference system). The variance along the radial direction was computed as $sR2 = < BR2 > - < BR >2$ 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 $-a r$ , but limiting the fit to the radial excursion between $1.5$ and $3 AU$ , these authors obtained $a = 3.39± 0.07$ for $2 sr$ , $a = 3.45 ± 0.09$ for $s2T$ , $a = 3.37 ± 0.09$ for $s2N$ , and $a = 2.48 ± 0.14$ for $s2B$ . 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 $a = 3$ . 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.

Figure 49:

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 (adopted from Forsyth et al., 1996

Horbury and Balogh (2001

) 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. (1996

). However, Horbury and Balogh (2001

), supposing a possible latitude dependence, performed a multiple linear regression of the type:

where $w$ is the power density integrated in a given spectral band, $r$ is the radial distance and $h$ is the heliolatitude ( $0o$ at the equator). Moreover, the same procedure was applied to spectral index estimates $a$ of the form $a = Aa + Ba log10 r + Ca sin h$ . Results obtained for $Bp,Cp, Ba, Ca$ are shown in Figure 50

Figure 50:

(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 $750 km s-1$ (adopted from Horbury and Balogh, 2001

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 50

. 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, 1990a

). 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 $k oc r1.1$ , in agreement with earlier estimates (Horbury et al., 1996a

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 totally unexpected. Horbury and Balogh (2001 ) and Horbury and Tsurutani (2001 ) estimated that the power observed at $80o$ is about $30%$ less than that observed at $30o$ . These authors proposed a possible 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 51 , 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.

Figure 51:

(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 (adopted from Horbury and Balogh, 2001

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 $26o$ 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 (2001

) 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 51 ), 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., 1982a ; Bruno and Bavassano, 1991 ). 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 (2001 ) found remarkable good agreement between observations by Ulysses at $2.5 AU$ and by Helios at $0.7 AU$ . In particular, Figure 52 shows Ulysses and Helios 1 spectra projected to $1 AU$ for comparison.

Figure 52:

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 $1 AU$ (adopted from Horbury and Balogh, 2001

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.