Differences in solar wind cross-helicity and residual energy during the last two solar minima



[1] The minimum of solar cycle 23 exhibited anomalous characteristics with respect to its predecessors. Other than the differences in the standard indicators of activity, such as sunspots, polar magnetic fields were also found to be different. These differences had a range of effects in the solar wind. In this work we study the Alfvénicity and the residual energy of magnetic field fluctuations in two intervals from the solar minima in cycles 22 and 23 using Ulysses data. We find that the differences between the minima are mainly related to the greater prominence of convected structures in the broader equatorial region in cycle 23. Thus the total power in the fluctuations was lower in cycle 23, mainly due to a reduction in Alfvénic fluctuations. We also show that the relationship between the normalized residual energy and cross helicity is scale dependent.

1. Introduction

[2] Both spacecraft and ground-based observations have revealed that the solar minimum in the last Solar Cycle (hereafter SC) 23 was quite different from the minimum in SC 22, as well as previous minima for which equivalent observations are available. In particular, it was longer and weaker than the previous one, with smaller sunspot numbers. The simple dipolar configuration of the corona, typical of solar minima, was not well recovered during the minimum in SC 23 [McComas et al., 2008]. The polar coronal holes were found to be ∼15% smaller in SC 23 than in SC 22, but the magnetic field strength in both polar regions was found to be ∼40% smaller in SC 23 [Wang et al., 2009]. As a related effect, the magnetic neutral line in the corona remained at higher latitudes than in the previous minima; this effect, coupled with the greater complexity of coronal structures, has led to a generally different heliospheric configuration [de Toma et al., 2010]. The magnetic flux density r2 BR, with r indicating the radial distance and BR representing the radial component of the magnetic field, underwent a reduction of about 33% [see Smith and Balogh, 2008]. McComas et al. [2008] showed that the wind was about 17% less dense, slower (∼14%), and with lower dynamic pressure (∼20%) in the last minimum.

[3] Smith and Balogh [2008] have also shown that the total variance of the magnetic field fluctuations exhibited a decrease in the minimum of SC 23 by about 25% with respect to SC22. Any possible link between the decrease in the background magnetic field and in the fluctuations was not investigated.

[4] In the present work we compare the normalized cross-helicity and the normalized residual energy [Bavassano et al., 1998] during the minima in SC 22 and SC 23. The former quantifies the degree of correlation between the velocity and the magnetic field fluctuations, namely it gives information about the Alfvénicity of the wind; the latter is a measure of the balance between the kinetic and the magnetic energy present in the fluctuations [see Bruno and Carbone, 2005, and references therein].

2. Observations

[5] A direct comparison of the properties of magnetic field fluctuations between SC 22 and 23 has been performed by analyzing Ulysses data in the interval during the solar minimum in SC 22 covering the two years 1994 and 1995 (hereafter M1) and in the interval during the solar minimum in SC 23 starting from 2006.5 and ending at the beginning of 2008 (hereafter M2). During both intervals, the spacecraft was first moving towards high southern heliolatitudes (max 80°S), then a fast latitude scan took place, during which the spacecraft moved from 80°S to 80°N. Afterwards, it again moved to lower northern heliolatitudes. For the following analysis, 60 sec resolution magnetic field data from the MAG instrument on board Ulysses [Balogh et al., 1992] have been analyzed in the RTN reference frame, where R indicates the radial Sun-spacecraft anti-sunward direction, T is the tangential direction coming from the cross-product between the solar rotation axis and R, and N is in the plane containing the R axis and the rotation axis of the Sun. Figure 1 shows in log-lin axis the variances, σi2 = 〈(Biequation imagei)2τ, (the index i runs over the magnetic field components) with τ = 1 day being the timescale of average computation, of both the magnetic field magnitude (Figure 1, top) and of the magnetic field components for the M1 (black lines) and for the M2 (red lines) intervals as a function of the Ulysses heliolatitude. While the variances of ∣B∣ do not differ significantly during the two minima, the variances of Bi exhibit marked differences: in M1 they are more than a factor 2 larger than in M2 at high latitudes, i.e., ∣λ∣ ≥ 30° (both north and south). The variances in the two fast latitude scans are characterized by a strong variability caused probably by different structures. It is worth remarking that the level of compression, indicated by the normalized variance of the magnetic field magnitude, is low during M1, as well as in M2 with the exception of a slight increase during the 2007 latitude scan. The panels in Figure 1 show that the components of the magnetic field transverse to the radial direction are characterized by greater variances, in agreement with previous observations [Forsyth et al., 1996]. In the work by Smith and Balogh [2008] a direct comparison between the two minima, showed that during the last solar cycle the total variance underwent a decrease of 25%.

Figure 1.

From top to bottom, daily averaged variances of the magnetic field magnitude and of the magnetic field components in the RTN reference frame as a function of the Ulysses latitude, relevant to M1 (black lines) and to M2 (red lines) (see text). Axis are log-lin.

[6] In this paper, the nature of the magnetic field fluctuations has been studied, both in SC 22 and SC 23, by analyzing their Alfvénicity via the cross-helicity, which is related to the degree of correlation between the velocity field and the magnetic field fluctuations [Roberts et al., 1987]. For the Alfvénic fluctuations the relationship between the fields is δv = ∓(δb/B0)VA [Barnes and Hollweg, 1974] where B0 is the mean magnetic field magnitude and VA = B0/equation image is the Alfvén velocity, being ρ0 the average proton mass density. The minus sign in the equation indicates that waves are propagating outwards from the Sun, while the plus sign refers to an inward propagation [Roberts et al., 1987]. Following the definition of cross-helicity, Hc, in the worky by Matthaeus and Goldstein [1982], we computed the normalized cross-helicity

equation image

over windows of time length τ; E is the total energy of the fluctuations. In equation (1) we put δv = V − 〈V〉 and δb = B − 〈B〉; in addition, the magnetic field fluctuations are expressed in Alfvén unit as b = (B/B0)VA. Along with the normalized cross-helicity, we also study the evolution of the normalized residual energy [Bavassano et al., 2000], namely

equation image

which quantifies the balance between magnetic and kinetic energy in the fluctuations. To compute the above quantities, we have used the 4 min resolution plasma data from the SWOOPS experiment [Bame et al., 1992] on board Ulysses and the magnetic field data, which have been degraded to the same resolution of the plasma data.

[7] Figure 2 displays the normalized cross-helicity (Figure 2, top), the normalized residual energy (Figure 2, middle) as a function of time and the scatter plots of the cross-helicity versus the residual energy (Figure 2, bottom) during M1 (Figure 2, left) and M2 (Figure 2, right), computed at daily scale. The green lines indicate the values of σC during the fast latitude scans. While in M1 the passage from the southern latitudes to the northern ones is quite sharp and the change in the cross-helicity is very rapid, during M2 a more extended transition between σC ∼ −1 to σC ∼ +1 occurs. The change in sign of cross-helicity is due to the change in the polarity of the magnetic field, so that, according to the relationship δv ∼ ∓δb and to equation (1), the waves are outward propagating in both hemispheres, independently of the polarity of the magnetic field. The middle panels in Figure 2 show that the level of σR does not change from the minimum in SC 22 to the one in SC 23. At daily scales, it is around an average value of −0.5 which indicates the presence of an imbalance in favor of magnetic energy in the fluctuations. Magnetic fluctuations with high cross-helicity are routinely observed in the solar wind, especially at high latitudes [Bavassano et al., 1998, 2000]. The scatter plots of σc versus σR in Figure 2, at high latitudes (black dots) and during the equatorial scans (green crosses) in M1 (Figure 2, left) and M2 (Figure 2, right), indicate that the largest differences between M1 and M2 occur at low latitudes, while the wind from high latitudes exhibit the same level of Alfvénicity. This is in agreement with the results shown by Bavassano et al. [2009] for the velocity fluctuations analyzed in the last minima: the statistical properties of the polar wind remain unchanged. In M1 the dominant feature is the presence of fluctuations characterized by ∣σC∣ ≥ 0.5 and an excess of magnetic energy, i.e., σR ∼ −0.5, along with a small population of magnetic structures having σC ∼ 0 detected at low latitudes (green crosses in the scatter plot). In M2 both those populations are roughly equally present. Indeed, during the last minimum a shrinking of the polar holes was observed probably due to the weakening of the Sun's polar fields [Wang et al., 2009]; therefore the observed solar wind can be more affected by equatorial structures, as the equatorial scan during M2 is broader than in M1 (see Figure 2, top). Note in both the data sets the low latitude high σC events, which are due to the presence of high speed streams close to the equator; this is a typical feature of the solar minimum. The same scatter plots are shown in Figure 3 but at a timescale of one hour. The top panels display data during time periods at high latitudes, while the bottom panels during the transitions from one solar pole to the other. At such a scale both σC and σR exhibit a high level of variability within intervals of few hours (not shown); indeed, the plots in Figure 3 show the presence of another population in both M1 and M2 periods along with the Alfvénic one (namely, σR ∼ 0, ∣σC∣ ≥ 0.5). That population is characterized by close to zero cross-helicity and close to −1 residual energy. Such structures have been observed at 1 AU and beyond and they have been found to be non-compressive with constant density and close to zero velocity fluctuations [Tu and Marsch, 1991; Bruno et al., 2007]. As suggested by Tu and Marsch [1991] they can be caused by spatial changes of the magnetic field directions and they were named “Magnetic Field Directional Turnings” (MFDTs). These dominantly transverse structures are likely to correspond to the relative growth of transverse fluctuations at high latitudes, originally proposed by Jokipii and Kóta [1989]. These structures (which are convected by the solar wind) were recognized in the early analysis of the Ulysses high latitude observations [Goldstein et al., 1995]. Although non-Alfvénic (and therefore non-propagating), these structures are transverse and represent an important population in the spectrum of magnetic fluctuations. We have also looked at plasma parameters, in order to identify MFDTs, and we have found that they occur, indeed, at high latitudes. The presence of structures with σR ∼ −1 and σC ∼ 0 at both high and low latitudes might also be caused by the presence of boundaries separating adjacent structures. To quantify the amount of structures present in the two periods, we have calculated the number of points satisfying some threshold values in both σR and σC and, then, we have normalized to the total number of points in each data set. Thus,Alfvénic fluctuations represent the dominant population in M1, ∼36%, while they are only 18.5% of the M2 period; MFDTs cover about 24% of the M1 interval and ∼29% of the M2 interval. The population characterized by high cross-helicity and σR ∼ −0.5 (which is dominant at daily scale) is roughly 6.7% in M1 and 3.6% in M2. It is worth noticing that the remaining population, having close to zero cross-helicity (low correlation between the velocity and the magnetic field fluctuations) and close to zero residual energy (i.e., equipartition), is dominant in M2 (∼49%), while is about 33% in M1. These results indicate that M1 tends to be characterized by fluctuations with σC ≠ 0, especially at high latitudes (see the top left panel), and that M2 shows a wider variety of structures and waves, as the semi-plane σR ≤ 0 is quasi-homogeneously filled with observed values in Figure 3. At this scale also high latitude flows exhibit differences between the minima.

Figure 2.

(left) Normalized cross-helicity during M1, (top) the green line indicates the cross-helicity values during the fast latitude scan; (middle) normalized residual energy; (bottom) scatter plot of σC versus σR. Green crosses are relevant to the fast latitude scan period. The red circle indicates the limit σC2 + σR2 = 1. (right) Same as Figure 2 (left) but during M2. All the quantities are daily averages.

Figure 3.

Scatter plots of one hour resolution normalized cross-helicity versus normalized residual energy (left) for the period M1 and (right) for the period M2. (top) The scatter plots of high latitude data and (bottom) low latitude data. The linear relation σc = σR + 1 is also displayed (red dashed line).

[8] Under the assumption made by Tu and Marsch [1991], in which fluctuations are a mixture of an Alfvénic population and MFDTs, it is easy to find the linear relationship σc = σR + 1 [Bavassano et al., 1998; Bruno et al., 2007]. However, this approximation seems not to reproduce very well the observations. Indeed, if we consider the scatter plots shown in Figure 3, the majority of the values are broadly distributed around this line (red dashed line in the plots) and they do not seem to follow a linear relationship.

[9] Looking at Figure 3 (top) we note that, as also shown by Bruno et al. [2007], the two dominant populations are the Alfvén waves (i.e., σR ∼ 0, ∣σC∣ ∼ 1), and the MFDTs (i.e., σR ∼ −1, ∣σC∣ ∼ 0). This is especially true in the M1 interval and away from low latitudes. We repeated the calculation made by Tu and Marsch [1991] without making their assumption that there is no correlation between outward propagating waves and MFDTs. Assuming that MFDTs are convected static structures [Tu and Marsch, 1991], we consider that their characteristic speed is given by the solar wind proton bulk flow. It is instructive to consider what happens if the earlier assumption about the (lack of) correlation between structures and waves is abandoned. After some algebra, we obtain again the linear equation σC = σR + 1, which does not fit the observations. This means that other phenomena have to be taken into account to draw a scenario closer to the values observed.

3. Discussions

[10] This work represents an attempt to study the properties of solar wind turbulence during the last two solar minima. The computation of the normalized cross-helicity has revealed a difference between the two latitude scans during SC 22 and SC 23. While in M1 the sign of σC changed abruptly because of the change in sign of the radial magnetic field, passing from the southern pole to the northern one, during M2 the change in cross-helicity from one pole to the other was much more gradual because of the broader region of transition. This can be related to the shrinking of the polar holes observed in SC 23 [Wang et al., 2009]. However, no differences have been detected between the minima in the normalized residual energy, i.e., in the difference between the kinetic energy and the magnetic energy stored in the fluctuations, and an average value 〈σR〉 ∼ −0.5 at one day scale has been observed, thus indicating the presence of magnetic structures. In particular, a predominance of high cross-helicity and negative residual energy structures was observed in M1, while in the last minimum, along with such structures, another population of σC ∼ 0 and σR ∼ −0.5 is present and detected during the crossing of the equatorial region, possibly due to boundaries between adjacent structures [Borovsky, 2008]. In addition, it has been observed that the degree of compression (not shown) is very low both in M1 and in M2, however the latter shows an increase of compressibility in the magnetic field fluctuations during the latitude scan; correspondingly a close to zero σR is observed (see Figure 2 (middle right) around 2007.6).

[11] From the analysis of the magnetic field fluctuations at one hour scales a very clear difference between the two periods can be seen (see Figure 3): the M1 interval shows the predominance of Alfvén waves (i.e., σR ∼ 0, ∣σC∣ ∼ 1) which usually dominates the dynamics of the polar wind. However, the non-compressive magnetic structures observed at one day scale still survive, although they represent only a small fraction of the period. In addition, another population of non-compressive structures appears, characterized by σC ∼ 0 and σR ∼ −1, that is the MFDTs [Tu and Marsch, 1991; Bruno et al., 2007], related to abrupt changes in the direction of the magnetic field vector. They also represent a large fraction of the structures present in M2, during which the percentage of Alfvénic fluctuations decreases when compared to the previous minimum. The non-Alfvénic, but transverse, non-compressive and convected structures are likely to arise as a consequence of the relatively slower decay in the transverse components of B (i.e., BT, BN ∼ 1/r) [Jokipii and Kóta, 1989].

[12] In M2 the dominant population (∼50%) is characterized by low cross-helicity and low residual energy, which can be related to the presence of compressive structures during the last minimum (note that in the latitude scan in SC 23 the equatorial region is broader and characterized by strong variations in the variance of the magnetic field magnitude, see Figure 1 (top)). Such a wide variety of structures is clearly indicated in Figure 3 (right) where the parameter space σR ≤ 0 is quasi totally filled. In SC 22 minimum at high latitudes (see Figure 3, top left) there is a relative absence of observations close to σC = 0.

[13] We remark how a straightforward relationship between σR and σC is not easy to obtain and is not seen in the observations. The data tend to deviate from a linear relationship, being broadly distributed in the σRσC plane. It is hard to draw a consistent model which takes into account such a large variety of structures present in the wind streams, which are also highly scale dependent.