We report an analysis of correlations between magnetic field and velocity fluctuations in the fast solar wind beyond 1 AU at high latitudes. We have found that on scales shorter than the microstream structures, there exists a well-defined dependence of the flow speed on the angle between the magnetic field vector and the radial direction. Solar wind is found to be slightly faster when the measured magnetic field vector is transverse to the velocity, while it is always slower when the magnetic field is parallel, or antiparallel, to the radial direction. We show that this correlation is a direct consequence of the high Alfvénicity of fast wind fluctuations and that it can be reasonably described by a simple model taking into account the main properties of the low-frequency antisunward Alfvén fluctuations as observed in the solar wind plasma. We also discuss how switchbacks, short periods of magnetic field reversals, naturally fit in this new observed correlation.
The bulk velocity of the quasi-stationary solar wind plasma displays large variations, ranging from 250–400 km/s in slow solar wind to 750 km/s and above in the fast wind. A more restricted spread of velocities is observed in the polar wind at high latitudes, where at solar minimum the Ulysses spacecraft measured very long periods of steady fast wind, with typical velocities between 700 and 850 km/s [e.g., McComas et al., 1998]. It is known that on a scale of a day or shorter, the velocity profile of fast solar wind is characterized by structures called microstreams, with typical extents of ~10 h [Neugebauer et al., 1995]. This finer structure of the solar wind velocity profile accounts for variations of the order of ±50 km/s in the bulk speed. On shorter scales, the polar wind is dominated by fluctuations which are Alfvénic [Smith et al., 1995], with typical transverse velocity fluctuations of ~20 km/s in the Ulysses data set considered here. The presence of Alfvénic fluctuations at various scales, from hours to minutes [Belcher and Davis Jr, 1971; Bruno et al., 1985], is one of the most well-established characteristics of the solar wind. In this framework, the observed correlation between velocity and magnetic fluctuations is such that they correspond to a flux of fluctuations that propagate outward from the Sun in the plasma frame. Gosling et al.  have pointed out that such a configuration often leads to one-sided velocity and magnetic field fluctuations; in this work we investigate further the connection between Alfvénic fluctuations and solar wind speed variations. In general, both advected structures of solar origin and large-amplitude, low-frequency Alfvénic fluctuations produce the velocity variations observed at scales smaller than the microstreams. The aim of this paper is to provide some insight into the relative importance of the two mechanisms that could be responsible for those variations.
2 Data Analysis
We analyze Ulysses observations from the Northern and Southern Hemispheres, during years 1994–1996 at Solar minimum. In this period the interplanetary magnetic field had a dipolar structure, with a single, opposite, and dominant polarity in each hemisphere [McComas et al., 1998]. Magnetic field data used here [Balogh et al., 1992] have 1 min resolution, and velocity measurements are obtained through the Solar Wind Observations Over the Poles of the Sun (SWOOPS) plasma experiment [Bame et al., 1992] and have 4 min resolution. Local microphysics properties, such as the proton core drift velocity, are obtained using a bi-Maxwellian model [Goldstein et al., 2010; Matteini et al., 2013].
2.1 Correlation Between V and θBR
Figure 1 shows the solar wind (proton) speed V as a function of the angle between the direction of the magnetic field and the radial direction. Note that the deviation of the solar wind flow from the radial direction is negligible in this data set, being always <5°, depending on the heliocentric distance, with a mean value of ~ 2°. This is because changes in the solar wind speed are mainly determined by changes in the radial component VR. Observations above 40° heliographic latitude are shown. Black dots represent North Hemisphere observations (outward magnetic field polarity, N), while red dots show the Southern Hemisphere (inward polarity, S). Figure 1 (bottom) shows the average value of data in bins of 5°. It can be seen that the solar wind bulk speed is correlated with θBR; it increases as θBR increases from 0° to 90° and decreases as θBR increases from 90° to 180°, with maximum values that are localized when the measured field is transverse to the flow. To our knowledge, such a correlation has not been identified before. We demonstrate here that this effect is connected to the properties of the solar wind Alfvénic fluctuations.
Note that despite the spread of the points due to the accumulation of the statistics over more than 2 years of observations at different heliographic distances and latitudes, the trend is quite robust and general. As an example, Figure 2 shows an interval of approximately 10 days of observations from each hemisphere. In each case subsets of data depending on the value of θBR have been selected. Observations when the local magnetic field was approximately aligned to the flow, θBR < 40° in N and θBR > 140° for S, are indicated by blue diamonds, while cases with large deviations from the radial direction, θBR > 80° in N and θBR < 100° for S, are marked by the red diamonds. It is apparent that observations at transverse B (red) always lie on the top part of the velocity profile and identify peaks of local maxima in the velocity profile, regardless to the dominant polarity. On the contrary, observations when the field is parallel (Figure 2, top) or antiparallel (Figure 2, bottom) correspond to local minima and track the lower part of the velocity profile. Such a trend is observed all the time in the data set in the polar fast wind, while it is, in general, weaker at lower latitudes and in the slow wind, where fluctuations are less Alfvénic. However, we have verified in other data sets (Helios, Wind) that also near the ecliptic the effect is present in high-speed streams and in slow wind when Alfvénic fluctuations are clearly present.
Figure 2 demonstrates that the global statistical trend shown in Figure 1 holds also at more restricted temporal scales. Such a conclusion is further confirmed in Figure 3, which focuses on a still shorter interval (~ 1 day). Figure 3 (top) shows the variation of θBR as a function of time (black line); in the same plot, the variation of the solar wind bulk speed is displayed in red. The magnetic field vector fluctuates significantly during the interval, with θBR ranging from 0° to 120°, and remarkably, it shows the same variability as observed in the bulk speed. Very clearly, the two quantities are highly correlated on the scale of the data resolution (4 min). This correlation holds in all the Ulysses fast wind measurements; it is then evident that the trend observed in Figure 1 really reproduces the distribution of this local effect over 2 years of observations.
It is interesting to note that occasionally, oscillations of θBR are large enough that a reversal of the radial magnetic field component is observed. These events are known as switchbacks; it has been proposed that they are generated either by large-scale Alfvénic turbulence [McComas et al., 1998] or by the interaction of the solar wind fluctuations with radial velocity shears [Landi et al., 2006]. During such events a consistent reversal of both macroscopic and microscopic main solar wind properties is observed, such as main direction of propagation of Alfvén fluctuations [Balogh et al., 1999], electron heat flux [Kahler et al., 1996], proton-alpha relative drift [Yamauchi et al., 2004], and proton beam drift [Neugebauer and Goldstein, 2012]. In particular Neugebauer and Goldstein  have shown that during switchbacks the core-beam structure of the solar wind proton distributions, aligned along Band usually antisunward, is reversed. Figure 3 (bottom) shows, for the same interval as in Figure 3 (top), the profile of the velocity drift of the core proton population with respect to the center of mass; in the plasma frame, this has a negative drift of about Δvc~−15/20 km/s and is always aligned with the local magnetic field (the secondary faster beam population drifts according to Δvb=−nc/nbΔvc, and the proton core-beam drift is typically of the order of the local Alfvén speed [Goldstein et al., 2000; Matteini et al., 2013]). At periods when θBR is larger than 90° (points above the dotted line in Figure 3 and highlighted by diamonds) the proton core-beam structure is reversed, and the core drift becomes positive, consistent with the analysis of Neugebauer and Goldstein . Note that the transition appears sharp because of the change in the sign of BR, but that |Δvc| (blue dashed line) remains approximately constant, suggesting that these events do not imply variations in the microphysics of the plasma. Also note that switchbacks are associated with local peaks in the proton speed.
The behavior discussed in previous sections can be qualitatively reproduced supposing a relatively simple model where it is taken into account that the angle θBR measured by the spacecraft is dynamically influenced by the magnetic field fluctuation δb and may then depart significantly from the orientation θ0 of the mean magnetic field B0. Figure 4 shows a schematic description of the magnetic field variations under typical fast solar wind conditions, assuming a positive underlying magnetic field polarity so that θ0<90°. We assume that velocity and magnetic field fluctuations are Alfvénic and that the magnetic field vector moves on a sphere of constant radius, i.e., B=constant. Under such conditions, the angle θBR observed by the spacecraft corresponds to a rotation, from B0 at θ0, of the instantaneous magnetic field B by a quantity proportional to δb. In particular, this results in an increase or decrease of the initial angle depending on the sign of the radial component δbR:
As shown in the figure, we expect the observed θBR angle to be closer to the radial for positive δbR (case on the left side), while negative δbR (case on the right) contributes to make B more transverse. Note that for sufficiently large fluctuations, the final θBR can be larger than 90° leading to a local reversal of the magnetic field with respect to the radial direction. At the same time, the anticorrelation between δb and δv makes the radial projection of the velocity fluctuations δvR opposite to δbR; as a consequence, when B is made more radial, this produces a negative δvR, (case on the left in the figure), while for more transverse magnetic field δvR is positive (case on the right).
Note that the opposite effect is expected when the dominant polarity is negative and θ0 lies between 90° and 180° (not shown in the figure), as for the Southern Hemisphere in the Ulysses data set considered here: It is easy to see that in this case a positive δbR causes a larger departure of θBR from the radial, leading to a more transverse B, while negative δbR pushes the magnetic field toward an antiparallel alignment with the flow. Despite the fact that in this case magnetic fluctuations produce opposite rotations of B with respect to Figure 4, it is quite straightforward to show that this is not the case for the resulting velocity fluctuations δv, since antisunward Alfvén fluctuations imply now δb and δv to be positively correlated for an inward magnetic field.
Remarkably, this produces the same picture for both polarities: When the field is made more transverse or even reversed (switchbacks case), then δvR is always positive; the wind bulk velocity is thus locally increased. On the other hand, when B is more aligned with the flow, it always leads to a negative δvR, thus resulting in a decrease of V. Note that such a behavior is also consistent with the presence of one-sided fluctuations, as first discussed by Gosling et al. . The effect of the possible correlations are schematically summarized in Table 1. As a consequence, the empirical behavior observed in Figure 1 can be recovered.
Table 1. Bulk Velocity Variations for Outward Propagating Alfvén Fluctuations
Variation in V
more ∥ (⇒0)
more ∥ (⇒180)
Quantitatively, we expect magnetic and velocity fluctuations to be correlated through the Alfvén speed (, where B and ρ are the magnetic field intensity and plasma density, respectively) as δb/B = ±δv/vA. However, in the solar wind the Alfvénic ratio rA, the ratio between kinetic and magnetic energy of fluctuations, is known to be smaller than 1. Typically rA < 0.3 in the polar wind [Goldstein et al., 1995; Bruno and Carbone, 2005]. As a consequence we expect a velocity correlation. From equation (1), and assuming B ~ B0, we can then express the expected radial velocity variations:
where the upper (lower) sign refers to inward (outward) magnetic field, under the assumption of dominant antisunward fluctuations.
Moreover, equation (2) provides a prediction for δvR as a function of θBR that can be directly tested in the observations. Figure 5 shows the distribution of δvR which is obtained by subtracting a mean proton speed profile from the observations, for both (top) Northern and (bottom) Southern Hemispheres. Here an average speed based on 2 h has been used; however, the results are not significantly dependent on the length of the averaging interval, provided that it is sufficiently larger than the 4 min resolution of the data set but smaller than the typical microstream scale (~10 h).
In the figure the red line corresponds to the fit of the data with the formula
Assuming an average angle θ0 ~ 60°, as suggested by the data, we obtain vwave ~ 18 km/s for north, while, assuming θ0 ~ 120°, vwave ~ 14 km/s for south. The value of vwave is found to be well correlated with vA; selecting shorter intervals at different heliocentric distance or latitude, vwave varies consistently with the change observed in the Alfvén speed. At the same time, we find here a ratio between magnetic and velocity fluctuations significantly smaller than the measured Alfvén speed, as expected according to rA in the polar wind.
An independent estimate of vwave can be obtained from the direct analysis of the correlation between the normal component of the fluctuations δvN and δbN/B[see, for example, Goldstein et al., 1995]. Values obtained in this way for different time intervals confirm that the observed correlation is systematically smaller than the local Alfvén speed. At the same time, such estimates when compared to our values of vwave, suggest that our model underestimates its amplitude by approximately 30%. This quantitative discrepancy can be due to the simplified assumptions in our empirical model, which only aims to qualitatively reproduce the observed trend and does not include more complex details about the physics of the system, as well as to the limitation about the plasma composition, neglecting for example the possible contribution of alpha particles to the dynamics. More detailed work to improve the present model is planned.
Several interesting implications for the solar wind physics may be derived from the present findings. As already mentioned, apparent magnetic field reversals naturally arise from the dynamics proposed. We find that switchbacks show properties consistent with the rest of the background plasma in agreement with analysis of the local particle distributions [e.g., Kahler et al., 1996; Yamauchi et al., 2004]. We then confirm that such events in the Ulysses data do not constitute isolated structures embedded in the solar wind, but on the contrary, are related to the continuous oscillation of the magnetic field vector under the effect of Alfvénic plasma motion. Whether these oscillations are only the result of the broadband low-frequency large-amplitude Alfvén fluctuations activity or if structures like velocity shears also contribute to the deformation of the magnetic field, leading to switchbacks as extreme cases [Landi et al., 2006] remains an open question. Also, note that in the low-speed wind at low latitudes, magnetic field switchbacks may also be produced by interchange reconnection between open and closed field lines. Moreover, due to the θBR-Vcorrelation, these events also correspond to peaks in the solar wind speed profile, in agreement with the analysis presented by Neugebauer and Goldstein  and as it can also be inferred from Gosling et al., [2009, Figure 1].
Our analysis is also helpful in the identification of possible structures of coronal origin from the background of solar wind fluctuations. We have shown that sufficiently large bending of the magnetic field lines may produce short intervals of significant enhancement of the bulk speed. This suggests that peaks often observed within the profile of the fast streams, and looking like distinct features at first glance, can be interpreted as part of the whole fluctuating solar wind velocity profile. This is, for example, consistent with the findings of Gosling et al.  who have associated the presence of analogous, but isolated, structures in the slow wind with pulsed Alfvén waves. As the amplitude of these enhancements—which are fairly frequent in the fast wind due to its high level of Alfvénicity—is related to the Alfvén speed and the change in angle, their relative importance in modifying the solar wind velocity profile is expected to vary as a function of heliocentric distance, playing a more important role closer to the Sun, where the ratio vA/V is larger. An improved description of the scaling of such radial fluctuations is needed, in order to correctly infer the underlying solar wind profile from observations [see, for example, Thieme et al., 1989, 1990].
The empirical correlation between V and the direction of the local magnetic field outlined in this work may also influence the statistical analysis of the solar wind fluctuations and their properties. In particular, properties of turbulent spectra obtained by looking at different directions with respect to the local field and applying the Taylor hypothesis assuming the velocity is uncorrelated with the magnetic field direction [e.g., Horbury et al., 2008; Wicks et al., 2010] should include this effect. We have shown that the local field direction is correlated to the flow speed, so that electromagnetic fluctuations within a given interval are advected on average with a larger flow speed when the local magnetic field is perpendicular than when it is radial. Preliminary estimations indicate that this introduces changes of the order of 10% in the estimated power levels. This effect will be larger closer to the Sun, however.
We have presented an analysis of velocity and magnetic field fluctuations in the fast solar wind using Ulysses observations at high latitude. We have focused on the proton speed fluctuations at scales shorter than microstreams (less than few hours) and their correlation with the direction of the local magnetic field vector B. As shown in Figure 1, a strong correlation exists in the solar wind between V and θBR, the angle between Band R, which is a direct consequence of the Alfvénic nature of fluctuations in the plasma.
The properties of the solar wind turbulence, in particular the constant amplitude of the total magnetic field and the presence of a dominant flux of antisunward propagating Alfvén fluctuations, introduce peculiar correlations in the geometry of the fluctuations. The resulting picture is summarized in Table 1. As a consequence, the solar wind is found to be slightly faster when the measured magnetic field vector is transverse to the velocity, while it is always slower when B is more parallel, or antiparallel, to the radial direction. Moreover, this also explains the presence of one-sided Alfvénic fluctuations (see discussion of Gosling et al., ). Using a simple model describing the geometry of the fluctuations as a function of θBR, we were able to reproduce the trend observed over 2 years of Ulysses measurements; moreover, we have found that the variation of δvRcan be reasonably well described by equation (3).
The amplitude of such variations is of the order of ~20 km/s in the polar wind beyond 1 AU, and it is related to the value of the local Alfvén speed. We expect this effect to increase approaching to the Sun where the Alfvénic contribution to the modulation of V might even exceed the variations associated with larger scale structures (microstreams) and possibly become dominant in the shaping of the solar wind bulk speed profile [e.g., Thieme et al., 1989].
We have shown that the existence of this correlation is also consistent with the presence of magnetic field reversals. Although our model describes switchbacks as the natural evolution of Alfvénic fluctuations, it is worth underlining that this does not constitute a complete explanation for the observed complex dynamics, since the physical mechanism driving and supporting solar wind fluctuations still remains a challenging open question. Nevertheless, we think that our present findings will be helpful to shed light on the origin of the solar wind low-frequency spectrum of fluctuations and its evolution [e.g., Verdini et al., 2012, Wicks et al., 2013]. More detailed analysis on the effects discussed in this work and on the possible implications for the preparation of the forthcoming space missions as Solar Orbiter and Solar Probe+ are planned.
We thank Marco Velli, Simone Landi, and Petr Hellinger for useful discussions. The research described in this paper was supported by the UK Science and Technology Facilities Council grant ST/K001051/1. Part of the research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
The Editor thanks Jack Gosling and an anonymous reviewer for their assistance evaluating this manuscript.