4.2 Polar turbulence studied via Elsasser variables

4.2 Polar turbulence studied via Elsässer variables

Goldstein et al. (1995a

) 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 $- 40o$ , and at $2 AU$ and around the maximum southern pass, as shown in Figure 53

. 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 54

, 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 corotating streams which resemble much better solar wind conditions at high latitude.

Figure 53:

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 $0.3 AU$ (adopted from Goldstein et al., 1995a

Anyhow, results relative to the normalized cross-helicity $s c$ (see Figure 54

) clearly show high values of $sc$ , around $0.8$ , which normally we observe in the ecliptic at much shorter heliocentric distances (Tu and Marsch, 1995a

). A possible radial effect would be responsible for the depleted level of $sc$ at $4 AU$ . Moreover, a strong anisotropy can also be seen for frequencies between $10-6 -10- 5 Hz$ with the transverse $sc$ much larger than the radial one. This anisotropy is somewhat lost during the expansion to $4 AU$ .

Figure 54:

Normalized cross-helicity and Alfvén ratio at $2$ and $4 AU$ , as observed by Ulysses at $o - 80$ and $o -40$ latitude, respectively (adopted from Goldstein et al., 1995a , © 1995 American Geophysical Union, reproduced by permission of American Geophysical Union).

The Alfvén ratio (bottom panels of Figure 54

) has values around $0.5$ for frequencies higher than roughly $-5 10 Hz$ , with no much evolution between $2$ and $4 AU$ . A result similar to what was for the first time obtained by Bruno et al. (1985

), Marsch and Tu (1990a

), and Roberts et al. (1990 ) in the ecliptic at about $1 AU$ . The low frequency extension of $rA _L$ together with $sc _L$ is probably due to the fact that Ulysses was longitudinally sampling Alfvénic fluctuations and has been considered by these authors not really indicative of the existence of such low frequency Alfvénic fluctuations. However, by the time Ulysses reaches to $4 AU$ , $sc _L$ has strongly decreased as expected while $rA _L$ gets closer to $1$ , making the situation even 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. (2000a ) studied in detail the evolution of the power $e+$ and $e-$ associated with outward $dz+$ and inward $dz -$ 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 $± dz$ and found two different regimes, as shown in Figure 55 . 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 corotating fast streams (Bruno and Bavassano, 1991 ; Tu and Marsch, 1990b ; Grappin et al., 1989 ). 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 $-1.39 r$ and $- 1.48 r$ , reported by Bavassano et al. (2000a ), 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.

Figure 55:

Left panel: values of hourly variance of $dz±$ (i.e., $e±$ ) vs. heliocentric distance, as observed by Ulysses. Helios observations are shown for comparison and appear to be in good agreement. Right column: Elsässer ratio (top panel) and Alfvén ratio (bottom panel) are plotted vs. radial distance while Ulysses is embedded in the polar wind (adopted from Bavassano et al., 2000a

This different radial behavior is readily seen in the radial plot of the Elsässer ratio $r E$ shown in the top panel of the left column of Figure 55

. 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.

Another interesting feature observed in polar turbulence is unraveled by Figure 56 from Bavassano et al. (1998 , 2000b ). The plot shows 2D histograms of normalized cross-helicity and normalized residual energy (see Appendix 13.3.1 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 $sc$ ) and of magnetic fluctuations (negative values of $sr$ ) 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 $s - ~ 0 c$ and $s -~ - 1 r$ , which might well be due to magnetic structures like the MFDT found by Tu and Marsch (1991 ) in the ecliptic.

Figure 56:

2D histograms of normalized cross-helicity $sc$ (here indicated by $sC$ ) and normalized residual energy $sr$ (here indicated by $sR$ ) for different heliospheric regions (ecliptic wind, mid-latitude wind with strong velocity gradients, polar wind) (adopted from Bavassano et al., 1998 , © 1998 American Geophysical Union, reproduced by permission of American Geophysical Union).

In a successive paper, Bavassano et al. (2002a

) 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 (2001 ), 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 (37 ). 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. (2000a ). As a matter of fact, the use of Elsässer variables (see Appendix 13.3.1) instead of magnetic field, and of selected data samples, leads to very small values of the latitudinal coefficient as shown in Figure 57 , where different contributions are plotted with different colors and where the top panel refers to the same dataset used by Horbury and Balogh (2001 ), while the bottom panel refers to a dataset omni-comprehensive of south and north passages free of strong compressive events (Bavassano et al., 2000a ). Moreover, the latitudinal effect appears to be very weak also for the data sample used by Horbury and Balogh (2001 ), although this is the sample with the largest value of the “ $C$ ” coefficient.

Figure 57:

Results from the multiple regression analysis showing radial and latitudinal dependence of the power $e+$ associated with outward modes (see Appendix 13.3.1). 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 $o e+$ 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 (adopted from Bavassano et al., 2002a

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 58

, 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 $e+$ .

Figure 58:

$e+$ (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 (adopted from Bavassano et al., 2001 , © 2001 American Geophysical Union, reproduced by permission of American Geophysical Union).