Flux tube texture of the solar wind: Strands of the magnetic carpet at 1 AU?
Abstract
[1] It is argued here that the inner heliosphere is filled with a network of entangled magnetic flux tubes and that the flux tubes are fossil structures that originate at the solar surface. 65,860 flux tubes are collected from seven years of measurements with the ACE spacecraft at 1 AU by spotting the flux tube walls with large changes in the magnetic field direction and the vector flow velocity. The tube walls are associated with large changes in the ion entropy density and the alpha-to-proton ratio. The median size of the flux tubes at 1 AU is 4.4 × 105 km. The tubes are larger in slow wind than in fast wind. The tubes are on-average aligned with the Parker spiral, with a large spread in orientations. This large spread may be caused by slight misalignments of tubes in the corona. The flux tubes map to granule and supergranule sizes on the Sun. The amounts of magnetic flux in the tubes at 1 AU correspond to the amounts of magnetic flux in field concentrations in the magnetic carpet. It is argued that the flux tubes do not reconnect during the ∼100-h advection to 1 AU owing to the expansion of the solar wind. The flux tube texture impacts the flow properties of the solar wind, turbulence in the solar wind, energetic-particle propagation in the inner heliosphere, and the driving of the Earth's magnetosphere. A method for using measurements of the flux tube walls for the remote sensing of magnetic field dynamics in the magnetic carpet is suggested.
1. Introduction
[2] In this paper a picture is advanced wherein the solar wind plasma is composed of a network of individual magnetic flux tubes each containing a distinct plasma. The flux tubes are interwoven with distinctly different axial orientations from tube to tube. The flux tubes move independently, quantizing the solar wind flow. The MHD turbulence of the solar wind is confined to within the flux tubes and any turbulence-produced field-line wandering is confined to within a flux tube. A working hypothesis is that the flux tubes are seeded into the solar wind at the top of the magnetic carpet, wherein they survive intact the ∼100-h advection time to 1 AU. Hence the flux tubes seen in the solar wind are “fossil structures” and perhaps even “nonevolving fossil structures”. When mapped back to the Sun the solar wind flux tubes have sizes that correspond to granule and supergranule scale sizes. In this flux tube picture the observed temporal deviations of the IMF from the Parker-spiral direction are caused by small-angle misalignments in the flux tubes in the corona that become greatly exaggerated in the expanding solar wind.
[3] The flux tube texture of the solar wind is depicted in Figure 1 [see also Bartley et al., 1966, Figure 6; Bruno et al., 2001, Figure 4]. In the left panel the flux tubes are sketched as viewed looking down upon the ecliptic plane: they have a variety of diameters and are aligned on average along the Parker-spiral direction, with a spread in orientations. The different colors represent different plasma properties. In the right panel the flux tubes are sketched from an end-on view.
[4] Other solar wind researchers have put forth similar pictures about the underlying structure of the solar wind, notably the “magnetic field and plasma filaments” of Parker [1963], the “filamentary tubes” of early solar wind and cosmic ray observations [Ness et al., 1966; McCracken and Ness, 1966; Bartley et al., 1966], the “spaghetti” microstructure of Mariani et al. [1973], the “flow tubes” of Thieme et al. [1988, 1989], Tu and Marsch [1990, 1993], and Marsch [1991], and the “flux tube” picture of Bruno et al. [2001].
[5] In his investigation of the flow properties of the solar wind plasma, Borovsky [2006] found that large-scale flow patterns are often broken up into individually moving parcels of magnetized plasma. The parcels are separated by sharp interfaces characterized by strong jumps in the flow velocity, magnetic field orientation, plasma properties, and energetic-electron flux. The large-scale shear-flow patterns of CIRs were found to be fractured into distinct “slip zones” and the transverse flow patterns in CME sheaths were found to be broken up into independently flowing magnetized parcels.
[6] Researchers have used observations and theory to suggest that pressure-balance structures in the solar wind originate from filamentations in the solar corona or from polar plumes [Parker, 1963, 1964; Velli et al., 1994; Del Zanna et al., 1998; Casalbuoni et al., 1999; Reisenfeld et al., 1999; Del Zanna and Velli, 1999; Yamauchi et al., 2002, 2003]. Thieme et al [1988, 1989, 1990] mapped such solar wind structures to supergranule scale sizes on the Sun. [see also Neugebauer et al., 1995] for a discussion of solar wind “microstreams” mapping back to solar-surface structures). A fraction of the flux tubes considered here are pressure-balance structures, but not all: some neighboring flux tubes have very similar plasma properties and are characterized only by a localized magnetic field orientation and a localized plasma flow velocity.
[7] The flux tube texture of the solar wind plasma has many implications for heliospheric physics. The flux tube texture (1) affects the flow properties of the solar wind, (2) restricts the mixing action of MHD turbulence, (3) restricts turbulence-driven field-line wandering, (4) ducts particle transport in the heliosphere, (5) explains the mesoscale deviations of the magnetic field from the Parker spiral, and (6) imposes an underlying temporal pattern to the seeming random driving of magnetospheric activity.
[8] An understanding of the details of the flux tube network in the solar wind opens the possibility of a future technique to remotely sense the dynamic connections of open magnetic flux into the solar surface [cf. Fisk et al., 1999; Fisk, 2005; Mellor et al., 2005]. By examining the flux tube interfaces seen at 1 AU in energetic electrons versus plasma and magnetic fields, the rate of reconnection in the chromospheric magnetic carpet [cf. Schrijver et al., 1997; Handy and Schrijver, 2001; Close et al., 2004a] may be quantified.
[9] In this report we will use ACE plasma [McComas et al., 1998] and ACE magnetic field [Smith et al., 1998] measurements in the years 1998–2004 to analyze the flux tubes of the solar wind at 1 AU.
[10] This paper is organized as follows. In section 2 the methodology for locating the flux tube walls in the solar wind is explained. Section 3 deals with the orientations of the flux tubes. In section 4 the sizes of flux tubes at 1 AU are statistically examined and in section 5 the flux tubes observed in the solar wind are mapped back to the surface of the rotating Sun to obtain a statistical picture of the flux tube sizes where they originate in the solar corona. Section 6 deals with the MHD turbulence within the individual flux tubes. Section 7 discusses the possibility of using solar wind measurements to remotely sense the temporal dynamics of the connections of the open magnetic field lines into the solar surface. Section 8a discusses misconceptions about MHD turbulence in the solar wind, section 8b discusses reconnection of the flux tubes, section 8c discusses possible origins of the flux tubes away from the Sun, section 8d discusses the Alfvenic motions of the flux tubes, and section 8e discusses future work that is needed. Section 9a summarizes the findings of this report and section 9b summarizes the evidence that the flux tubes originate at the solar surface. Section 10 discusses the impact of the flux tube texture for heliospheric physics.
2. Flux Tube Boundaries in the Solar Wind
[11] The flux tubes of the solar wind are characterized by strong changes in the magnetic field direction from one tube to its neighbor. In crossing from one flux tube into the next, a spacecraft sees a rapid change in the magnetic field direction. In Figure 2 the angular change in the magnetic field direction at ACE every 128 s is binned for seven years of measurements. Between the magnetic field vector B1 at time 1 and the magnetic field vector B2 at time 2 which is 128 s later, the angular change Δθ is given by Δθ = invcos(B1•B2/∣B1∣∣B2∣). The occurrence distribution of Δθ in Figure 2 clearly has two components: a population of smaller changes that is well fit by exp(− Δθ/9.4°) and a population of larger changes that is well fit by exp(− Δθ/24.4°). The small-change population is interpreted to be angular fluctuations of the magnetic field caused by the MHD turbulence of the solar wind and the large-change population is interpreted to be crossings of the interfaces between flux tubes. (see also Bruno et al. [2001, Figure 4] for this interpretation.)
[12] The large changes in the magnetic field direction are temporally associated with other large changes in the solar wind plasma: large changes in the flow velocity, plasma density, ion temperature, electron temperature, alpha-to-proton density ratio, and field-aligned energetic-electron flux. When the temporal changes of these solar wind properties are examined, they also show dual populations. In the six panels of Figure 3, changes in the flow velocity, magnetic field strength, ion entropy density S = Ti/n2/3, electron temperature, and field-aligned 272-eV electron flux, are binned, each panel using one year of ACE measurements. For temporal data points i, i+1, i+2 … separated by 64 s, the differences in the plasma properties are calculated as follows. For panels a and b of Figure 3, the fractional change in the flow velocity ∣Δv∣/∣v∣ and the fractional change in the magnetic field strength ΔB2/B2 at time i are determined by comparing the differences in the i+1 and i-1 value (as were the Δθ values of Figure 2). For panels c, d, e, and f of Figure 3 the fractional changes in the ion entropy density, fractional change in the electron temperature, fractional change in the alpha-to-proton ratio, and the fractional changes in the field-aligned electron heat flux are determined by comparing the median of the three values i+1, i+2, and i+3 with the median of the values i-1, i-2, and i-3. As can be seen in the various panels of Figure 3, the distributions of change values separate into one population for small values and a second population for large values. The crossover points of the fits to the two distributions are indicated in each panel.
[13] Some of the large-change populations are associated with each other. In Figure 4 the fractional change in the flow velocity is plotted as a function of the angular change in the magnetic field direction. As can be seen, there is a population of points for small ∣Δv∣/∣v∣ and small Δθ (MHD turbulence) and then a population of large ∣Δv∣/∣v∣ and large Δθ that are related (flux tube wall crossings). As can be seen, a large change in Δθ corresponds also to a large change in ∣Δv∣/∣v∣.
[14] The large magnetic field direction changes Δθ are often accompanied by a strong change in the properties of the solar wind plasma. Examining the value of the entropy density S = PVγ = T/n2/3 in time series data is a powerful method to discern different plasmas [cf. Burlaga et al., 1990; Pagel et al., 2004] (see also Table 1 and Appendix of Borovsky et al. [1998]), since compressions and adiabatic transport will not change the S value of a plasma [cf. Goertz and Baumjohann, 1991]. Further, in a collisionless plasma S is preserved separately for the ions and the electrons. In the top panel of Figure 5 it is demonstrated that the large changes in the plasma (ion) entropy density S = Ti/n2/3 are associated with the large changes Δθ in the magnetic field direction. Here, the 128-s fractional change in S is binned separately for times when Δθ < 15° (black curve) and for times when Δθ > 50° (gray curve). Both distributions in Figure 5 are fit with exponentials over the range 0.6 < ΔS/S < 1.5 and the fits appear as the blue and red dashed curves: the functional form of the Δθ > 50° fit is exp(- (ΔS/S)/0.170) and the functional form of the Δθ < 15° fit is exp(- (ΔS/S)/0.168). Both are similar. As can be seen by extrapolating these two fits back toward ΔS/S in Figure 5, the Δθ < 15° distribution has a dual population (as in Figure 2), but the Δθ > 50° distribution does not. The Δθ < 15° population has a core population (fit by the green dashed curve) plus a tail population (fit by the red dashed curve), whereas the Δθ > 50° population is only the tail population (fit by the blue dashed curve). With the core population representing fluctuations in the plasma and the tail population representing boundaries in the plasmas, whenever Δθ > 50° (which is, whenever there is a flux tube wall as determined by the magnetic field) there is statistically a plasma boundary. Similarly in the bottom panel of Figure 5, the fractional change in the alpha-to-proton ratio (α/p) in the solar wind plasma is binned. The alpha-to-proton-ratio measurements are noisy, so instead of a 128-s change of the 64-s-resolution measurements, the 64-s measurements are smoothed with a running 5-min (320-s) average and 6-min changes of the 5-min averages are calculated and binned. The tails of both distributions are fit with exponential functions in the 0.6 < Δ(α/p)/(α/p) < 1.5 range: the exponential fit exp(−Δ(α/p)/(α/p)/0.154) for the Δθ < 15° distribution is plotted as the blue dashed line and the exponential fit exp(−Δ(α/p)/(α/p)/0.170) for the Δθ > 50° distribution is plotted as the red dashed line. Extrapolating the exponential fits back to Δ(α/p)/(α/p) < 0.6, one can see that the core population is much larger for the Δθ < 15° distribution than it is for the Δθ > 50° distribution: for Δθ > 50° the core is a factor of ∼2 above the tail distribution, whereas for Δθ < 15° the core is a factor of ∼10 above the tail distribution. This indicates the tendency to find large changes in the alpha-to-proton ratio across flux tube walls. As in the top panel, this indicates that plasma boundaries tend to occur with the large magnetic field directional changes.
| Property | Typical Value |
|---|---|
| Median tube diameter d | 5.6 × 105 km |
| Mean tube diameter | 1.4 × 106 km |
| Large eddy scalesize d/4 | 1.4 × 105 km |
| Age of flux tube | 100 h |
| Median tube-to-tube axial angle Δθ | 65° |
| Relative velocity between Neighboring tubes | 40.6 km/s |
| Tube sound crossing time | 3.4 h |
| Tube magnetosonic crossing time | 2.3 h |
| Turbulence amplitude at d/4 | 5 km/s |
| Eddy turnover time at d/4 | 19 h |
| Field-line-wandering scalesize | 2.9 × 105 km |
| Tube Alfven-speed crossing time | 3.05 h |
[15] In support of this, recent multispacecraft analysis have found that most of the discontinuities in the solar wind are tangential discontinuities [Horbury et al., 2001; Knetter et al., 2003, 2004; Riazantseva et al., 2005a, 2005b], i.e., they are plasma boundaries.
[16] The large changes in the magnetic field direction (which tend to be accompanied by large changes in the plasma properties) are interpreted to be the walls of flux tubes in the solar wind.
[17] To find the flux tube walls in the ACE data set, large changes in the magnetic field direction and large changes in the solar wind flow velocity are sought in solar wind measurements. In Figure 2 the Δθ occurrence distribution showed a breakpoint at Δθ ≈ 37° and in Figure 3a the ∣Δv∣/∣v∣ occurrence distribution showed a breakpoint at ∣Δv∣/∣v∣ ≈ 0.059. In both of these distributions it was argued that changes larger than the breakpoint values were owed to the spacecraft crossing from one flux tube into its neighbor. To locate flux tube walls in the 1998–2004 ACE data set, data points wherein the 128-s change in the magnetic field direction Δθ > 60° or the 128-s fractional change in the flow velocity ∣Δv∣/∣v∣ > 0.085 are marked as flux tube walls. For these two “trigger levels”, the fraction of triggers that were Δθ triggers only was 39.2%, the fraction that were ∣Δv∣/∣v∣ triggers only was 34.0%, and the fraction that were simultaneous Δθ and ∣Δv∣/∣v∣ triggers was 26.8%.
[18] In Figure 6 several plasma parameters measured by ACE for 8 h on 31 December 1998 are plotted. Times when the discontinuity algorithm (Δθ > 60° and/or ∣Δv∣/∣v∣ > 0.085) triggered are marked with the vertical dashed lines. In the top panel the three components of the magnetic field are plotted. The magnetic field orientation tends to undergo shifts in direction across the discontinuities. Note in the top panel that a prominent discontinuity at about 17 UT was missed by the algorithm. In the second panel of Figure 6 the local Alfven speed of the plasma B/(4πnmi)1/2 is plotted. Note the discontinuous changes in the Alfven speed of the plasma across the discontinuities. In the third panel the three components of the solar wind flow velocity are plotted (with 400 km/s subtracted off the radial component). Often there are distinct shifts in the flow velocity (including the radial velocity) across the discontinuities. In fact, the total speed of the solar wind (in the spacecraft frame) often jumps across the discontinuities. In the fourth of Figure 6 the fractional change ∣Δv∣/∣v∣ in the flow velocity over 128 s is plotted. Note the spikes in ∣Δv∣/∣v∣ at the trigger locations. Also, in the bottom panel of Figure 6 the angular change Δθ in the magnetic field direction over 128 s is plotted. Again, note the spikes in Δθ at the trigger locations. Similar plots to Figure 6 showing the solar wind plasma and flow broken up across discontinuities can be found in Figures 12–15 of Borovsky [2006].
3. Orientations of the Flux Tubes
[19] Figure 7 scatterplots of unit-vector directions of flux tube axes and of the normals to the flux tube walls are shown. The axial directions of the flux tubes are obtained from the mean vector magnetic field < B > within each flux tube (between one wall and the next). The normal directions to the flux tube walls are obtained using the cross-product method wherein the magnetic shear is taken to be aligned along the interface between adjacent flux tubes; in that case where the magnetic field changes from B1 to B2 across the wall, the normal to the wall is in the direction B1 × B2 [Burlaga and Ness, 1969; Knetter et al., 2004] When tested with multispacecraft measurements of discontinuities in the solar wind, Knetter et al. [2004] found that the cross-product method worked quite well for finding the normal directions of discontinuities, much better than minimum-variance methods did. For all of the flux tube boundaries found using the combined Δθ > 60° and/or ∣Δv∣/∣v∣ > 0.085 triggers, the normal directions are obtained from B1 × B2 (with B1 and B2 separated by 128 s). In the top two panels of Figure 7 the three-dimensional unit vector tips are plotted in r versus -t coordinates. The range of values of the expected parker spiral (2nd to 98th percentiles) for the measured values of vsw are indicated as the short curves labeled “P.S.”. As can be seen in the top-left panel, the flux tube axes tend to be aligned with the Parker spiral, with significant scatter. As can be seen in the top right panel, the flux tube-wall normals tend to be oriented perpendicular to the Parker spiral, with significant spread. In the bottom two panels of Figure 7 the unit-vector directions are viewed from along the 410-km/s Parker-spiral direction: all unit vectors are rotated about the n axis so that the r,t coordinates are converted into r′,t′ coordinates with t′ perpendicular to the 410-km/s Parker spiral direction. The Parker-spiral direction is in the center of each of the two plots at (t',n')= (0,0) in the bottom two panels of Figure 7. In the lower-left panel the direction vectors of the flux tube axes are plotted: as can be seen the axes tend to be aligned with the Parker spiral, with significant scatter. In the lower-right panel the normals to the flux tube walls are plotted: as can be seen the normals to the walls tend to be perpendicular to the Parker-spiral direction [see also Weimer et al., 2002, 2003; Bargatze et al., 2005]. Note also in the bottom-right panel that the distribution of wall normals are quasi-isotropic in the directions perpendicular to the Parker spiral.
[20] In Figure 8 the axial orientations of the flux tubes are statistically compared with the velocity dependent Parker-spiral direction P, where Pr = 1, Pt = −405/vsw, and Pn = 0. The distribution of angles between the mean magnetic field Btube within the individual flux tubes and the expected Parker spiral P are shown as the gray curve in the top panel of Figure 8. As can be seen, the mean angle between Btube and P is ∼42.7°. In the top panel of Figure 8 the occurrence distribution of the angle between the wall normal B1 × B2 and the expected Parker spiral P is shown as the black curve. This distribution is peaked toward 90°, as expected for flux tubes approximately aligned with the Parker spiral. For comparison, the distribution of angles between randomly oriented vectors and the Parker spiral is shown as the dashed curve: this curve has a sinθ dependence according to the amount of phase-space area as a function of θ in polar coordinates. In the bottom panel of Figure 8 the density of occurrences is recalculated by dividing the curves of the top panel by sinθ. The flux tube axes are oriented about the Parker-spiral direction with a spread in angles of about 45°. The normal directions of the flux tube walls are oriented perpendicular to the Parker-spiral direction with a similar spread in angles.
4. Flux Tube Sizes at 1 AU
[22] In Figure 9 the distribution of wall-to-wall thicknesses d at 1 AU for the 65,860 flux tubes in the 1998–2004 ACE data set is shown as the black curve. The median value of d is 4.44 × 105 km = 70 RE and the mean value of d is 1.13 × 106 km = 178 RE. (Note that these thicknesses are not the “diameters” of the flux tube, if they are round, because all satellite crossings are not through the center of the tubes; for random crossings through cylindrical-shaped tubes, the mean wall-to-wall thickness is π/4 times the diameter of the tube.) These tube sizes agree with the structure scale sizes of 45–100 RE determined from two-spacecraft cross correlations in the solar wind [Richardson and Paularena, 2001].
[23] In Figure 9 two subsets of the ACE measurements are separated, slow wind (v < 350 km/s) and fast wind (v > 650 km/s), and the distributions of flux tube sizes d at 1 AU for the two types of wind are plotted. The slow category contains 6.2% of the total number of flux tubes and the fast category contains 10%. The median sizes of d are 98 RE for the slow wind and 67 RE for the fast wind. This systematic difference in flux tube scale size supports the notion of a solar-surface origin for the flux tubes, wherein the slow wind (from the edges of coronal holes) typically undergoes a greater expansion in the corona than does the fast wind (from the center of coronal holes) [e.g., Arge et al., 2003; Suzuki, 2006].
[24] It is argued here that flux tubes with large diameters d are associated with ejecta plasma [see also Joselyn and McIntosh, 1981; Wright and McNamara, 1983]. One attribute of ejecta solar wind is that the ion temperature Ti is depressed [Gosling et al., 1973; Richardson and Cane, 1995]; in particular the ion temperature is depressed relative to a Ti-versus-v curve. In Figure 10 the sizes of the flux tubes in the solar wind at 1 AU are plotted as a function of Ti/Ti medain (black points), where Ti is the average of the ion temperature across the flux tube and Ti median is the median value of Ti as a function of speed v in the OMNI2 data set. Ti medain (in keV) is given by Ti medain = 1.28 × 10−8 v3.324 for v < 372 km/s and Ti medain = 0.0572v − 16.79 for v > 372 km/s [Borovsky and Steinberg, 2006]. When Ti is substantially less than Ti median (i.e., when the solar wind ion temperature is anomalously low, ejecta is expected [e.g., Richardson and Cane, 1995; Elliott et al., 2004]. The red curve in Figure 10 is a 300-point running average of the black points; the red curve shows the trend underlying the scatter of the data. As can be seen, the largest flux tubes tend to occur for anomalously low-temperature solar wind. This supports a conclusion that the solar wind flux tubes have statistically larger diameters in ejecta plasma.
5. Mapping the Flux Tubes to the Sun
[26] In Figure 11 the distribution of Δt values observed by ACE for 65,860 flux tubes is converted into a distribution of distances Δx on the solar surface. The units of Δx are Mm, where 1 Mm = 103 km. As indicated by the arrows in the plot, the median value of the distribution is 2.27 Mm = 3.15 arcsec and the mean value is 6.44 Mm = 8.94 arcsec. The distribution of flux tube sizes spans a broad range of scalesizes. As noted in section 4, the large-Δx portion of the distribution is dominated by ejecta solar wind. Also plotted in Figure 11 are distributions of solar granules [after Roudier and Muller, 1986, Figure 6; Title et al., 1989, Figure 19] and supergranules [after Simon and Leighton, 1964, Figure 7d; Haganaar et al., 1997, Figure 7]. Also, indicated as horizontal bars are the supergranule sizes obtained with high-resolution measurements by Srikanth et al. [2000] and the spectrum of scalesizes of mesogranules as determined by Shine et al. [2000]. As can be seen in Figure 11, the broad distribution of flux tube sizes spans the granule and supergranule scales. It should be anticipated that a more-accurate mapping will shift the distribution of flux tube sizes on the solar surface toward smaller values (1) because of off-equatorial mapping and (2) because the mapping will connect onto concentrated regions of solar longitude.
[27] If seeded into the solar wind from near the solar surface, the flux tube sizes are in the range of granule and supergranule scales. This agrees with a picture wherein bundles of open magnetic flux from the photosphere poke through gaps in the magnetic carpet [cf. Close et al., 2004b]: these open-flux bundles will form flux tubes with a multitude of magnetic separatrices in the corona [e.g., Priest et al., 2002] (the separatrices being the flux tube walls), with about 5 separatrices per supergranule [Close et al., 2005].
[28] The large orientation changes from one flux tube to the next seen at 1 AU may be a consequence of slight misalignments of the flux tubes in the corona. As illustrated in Figure 12, flux tubes with small differences in axial orientation near the Sun map out into the expanding solar wind to have large differences in axial orientation at 1 AU. In the left panel of Figure 12 three flux tubes lying in the solar equatorial plane at 5 Rs are sketched: the black flux tube is aligned with the local Parker-spiral direction (for a 410-km/s wind) and the red and green flux tubes are oriented ±1.5° from the Parker spiral. In the right panel those three flux tubes are sketched at 1 AU after passively being mapped in the transversely expanding solar wind (passively means frozen into the radial plasma flow). In the right panel of Figure 12 the black tube is still oriented with the local Parker-spiral direction but the red and green flux tubes are now much more than ±1.5° away. Hence the large deviations in magnetic field orientation about the Parker spiral observed at 1 AU may be owed to the misalignment of flux tubes above the solar surface.
6. Solar Wind Turbulence and the Solar Wind Flux Tubes
[30] The discontinuous interfaces between flux tubes have a profound influence on the analysis of turbulence in the solar wind. To show this, the February 1998 to December 2004 ACE plasma [McComas et al., 1998] and magnetic field [Smith et al., 1998] data set is broken into 13,133 nonoverlapping 4.55-h-long subintervals and each subinterval is Fourier analyzed separately. A 4.55-h interval consists of 256 measurements separated by 64 s, which is the time resolution of the SWEPAM plasma instrument on ACE. Since 256 is a power of two, using data records that are 256 points long makes for efficient and accurate (no leakage) Fourier transforms. The subinterval length of 4.55 h is comparable to the correlation time (in the spacecraft frame) of the solar wind turbulence, which is 0.7−4 h [Matthaeus and Goldstein, 1982; Tu and Marsch, 1995; Matthaeus et al., 1999; Feynman et al., 1996] for the magnetic field fluctuations, and perhaps twice as long for the velocity fluctuations [Matthaeus and Goldstein, 1982]. Data dropouts in the ACE magnetic field and plasma measurements are replaced by linear interpolation across the data gap. A record is kept of which subintervals have magnetic field-data replacements and which subintervals have plasma-data replacements. If 20 or more data points in a 256-point subinterval need to be replaced by interpolation, that subinterval is removed from the analyses. Each variable f(t) in each data subinterval is detrended prior to analysis by subtracting a line from f(t) where the line goes through the endpoints f(tmin) and f(tmax) of the subinterval. The purpose of the detrending is to eliminate the jump in thedata when the ends of the subinterval are joined by the assumption of periodicity in the Fourier analysis. Discrete Fourier transforms of the detrended solar wind measurements are obtained using standard fast-Fourier-transform (FFT) methods [Cooley and Tukey, 1965; Otnes and Enochson, 1972]. The number of flux tube walls (discontinuities) Nd in each 4.5-h data subinterval is also recorded.
[31] The amplitude of the “turbulence” in each 4.55-h-long data subinterval is determined as follows. First the time series data (vr, vt, vn, Br, Bt, Bn) is Fourier transformed (after detrending with a line). Then all Fourier coefficients pertaining to frequencies f < 1 × 10−3 Hz are zeroed out and an inverse Fourier transform is then performed. The RMS fluctuation levels δvr, δvt, δvn, δBr, δBt, δBn of the Fourier-filtered variables vr, vt, vn, Br, Bt, Bn in each data subinterval are then calculated. These RMS amplitudes are taken as measures of the turbulence amplitude. In Figure 14 the amplitudes δv and δb of the turbulence in each 4.55-h-long data subinterval are plotted as functions of the number of discontinuities Nd in each 4.55-h subinterval, where δv = (δvr2 + δvt2 + δvn2)1/2, and where δb = (δBr2 + δBt2 + δBn2)1/2/(4πmin)1/2 is the magnetic field fluctuation amplitude in Alfven units. The δv points are in red and the δb points are in blue. Random numbers between −0.5 and + 0.5 are added to the integer Nd values in the plot to spread the points horizontally so their vertical distributions can be clearly seen. As can be seen, the “turbulence” amplitudes are strongly correlated with Nd; the linear correlation coefficients with Nd are Rcorr = + 0.54 for δv, Rcorr = + 0.58 for δb, and Rcorr = + 0.56 for δB (not plotted). 100-point running averages of the amplitudes are also plotted to see the trends underlying the scatter of the data points; the average of δv is plotted in pink and the average of δb is plotted in black. Note the distinct increase the amplitudes going from Nd = 0 to Nd ≠ 0. For Nd ≠ 0, the amplitude of the “turbulence” is a strong function of the number of discontinuities Nd. This indicates that the Fourier power in the data intervals is dominated by the discontinuous flux tube walls.
[32] In Figure 15 the Alfven ratio rA of each 4.55-h-long data subinterval is plotted (black points) as a function of the number of discontinuities Nd in each 4.55-h subinterval. The Alfven ratio is rA = (δv/δb)2, where δv and δb are the velocity-fluctuation amplitude and magnetic field-fluctuation amplitude (in Alfven units) for f ≥ 1 × 10−3 Hz, as described earlier in this section. Random numbers between −0.5 and +0.5 have been added to the integer number of discontinuities per subinterval Nd to spread the points in the plot to make their distribution visible. To discern the trend in the spread of points, a 400-point logarithmic running average of rA is plotted in red in the figure. (A logarithmic average of rA is exp(〈loge(rA)〉), where 〈〉 denotes averaging.) As can be seen from the red curve in Figure 15, for Nd = 0 (which shows up at points between −0.5 and +0.5) the average Alfven ratio is above unity, and for Nd ≠ 0 the average Alfven ratio is below unity (see also Figure 14). This may indicate that the well-known rA < 1 property of the solar wind turbulence [cf. Matthaeus and Goldstein, 1982; Bruno et al., 1985; Tu et al., 1989; Roberts et al., 1990] may be related to the presence of flux tube walls in the solar wind, as suggested by Tu and Marsch [1992, 1993]. As can be seen by the red curve in Figure 15, for Nd > 0 there is only a slight variation of rA with Nd.
[33] From Figures 14 and 15 it can be concluded that the walls of the solar wind flux tubes profoundly ruin an analysis of solar wind turbulence. The analysis of discontinuity-free intervals (Nd = 0) was an analysis of the MHD turbulence of the solar wind; the intervals containing discontinuities were contaminated. In section 8b some resulting misconceptions about solar wind turbulence are elaborated upon, and in section 8e a new approach to analyzing turbulence in the solar wind is called for.
[34] In examining turbulent transport in the solar wind, Borovsky [2006] noted that there was an absence of turbulent mixing of plasmas across solar wind discontinuities; often there are sharp changes in the plasma density, ion temperature, electron temperature, plasma beta, and ionic composition across these boundaries. This led to the conclusion that the discontinuities are boundaries to the turbulence rather than structures imbedded within the turbulence. The findings are similar in the statistical analysis of section 2 here (see Figure 5). This raises a significant issue for the MHD turbulence of the solar wind: is the solar wind turbulence a homogeneous turbulence [e.g., Grappin et al., 1982; Grappin and Mangeney, 1996] or does its confinement within a flux tube change its properties? The effects of boundaries on MHD turbulence are not well studied, particularly for collisionless plasmas. For Navier-Stokes turbulence, some of the effects boundaries are (1)blocking of velocity components near walls [Uzkan and Reynolds, 1967; Perot and Moin, 1995], (2) producing non-Gaussian fluctuation statistics [Thomas and Hancock, 1977; Brumley and Jirka, 1987; Handler et al., 1993; Walker et al., 1996], and (3) producing changes in the spectral index of the turbulence [Townsend, 1976; Kreplin and Eckelmann, 1979]. It had been argued that the MHD turbulence of the Earth's plasma sheet is fundamentally different from the turbulence of the solar wind because the plasma sheet turbulence is a “turbulence-in-a-box” [Borovsky et al., 1997, Borovsky and Funsten, 2003]; however, owing to the flux tube texture of the solar wind, the two types of turbulence may not be so dissimilar.
[35] A number of turbulence-related quantities are calculated and entered into Table 1 for a typical flux tube in the solar wind at 1 AU. The median wall-to-wall distance through a flux tube (accounting for the orientation of the flux tube with respect to the solar wind flow) is 4.55 × 105 km (see Figure 9). Since for random cuts across a circle the average wall-to-wall distance is π/4 times the diameter d of the circle, the median diameter of a flux tube in Table 1 is taken to be 4/π times 4.55 × 105 km, which is d = 5.6 × 105 km. The mean diameter of a flux tube is taken to be 4/π times 1.13 × 106 km, which is d = 1.4 × 106 km. From analogy with Navier-Stokes turbulence confined between walls [e.g., Vaezi et al., 1997], we anticipate that the large-eddy scalesize in a flux tube to be 
of the diameter of the tube. Thus the large eddy scale in the typical flux tube is estimated to be 1.4 × 105 km in Table 1. The turbulence amplitude within the flux tube is estimated as follows. From Figure 14, the amplitude of turbulence in discontinuity-free data intervals is ∼5 km/s in the f > 1 × 10−3 Hz bandpass as seen by a spacecraft. For 410 km/s wind, this amplitude corresponds to fluctuations at wavelengths of 4 × 105 km and smaller, which have gradient scalesizes of approximately 1 × 105 km, which is close to the large-eddy scale.
[36] Since there appears to be an absence of turbulent mixing across the flux tube walls (section 2) [cf. Borovsky, 2006], it follows that any turbulence-driven magnetic field-line wandering will be confined to the insides of the flux tubes. For a typical flux tube in the solar wind, the field-line-wandering distance can be estimated as follows (see Table 1). The diffusive scalesize λ is given approximately by λ = (Dt)1/2, where D is the appropriate diffusion coefficient and t is the age of the flux tube. The appropriate diffusion coefficient is the eddy-diffusion coefficient for the action of eddies that are about 0.25 times the diameter d of the flux tube [Vaezi et al., 1997]. For a median-sized flux tube with d = 5.6 × 105 km, the eddy size d/4 is 1.4 × 105 km. According to Borovsky [2006, Figure 5] the eddy-diffusion coefficient for that scalesize in the solar wind is D = 6.85 × 1015 cm2/s. However, in that paper, the amplitude of the solar wind turbulence was overestimated since flux tube boundaries were not yet excluded from the solar wind data; the value of the turbulence amplitude used for that figure was 15 km/s at the correlation scale. A better estimate for the amplitude of the true turbulence of the solar wind is about 5 km/s at the correlation scale (see Table 1). That reduces the eddy-diffusion coefficient to D = 2.28 × 1015 cm2/s. Using this value for D and using t = 100 hr for the age of the flux tube yields the field-line-wandering scalesize λ = (Dt)1/2 = 2.9 × 105 km. This value is entered into Table 1. This scalesize λ is comparable to the radius of a tube. Depending on the turbulence amplitude and the tube size, the field-line-wandering distance can be smaller than the tube or larger than the tube; in cases where it is larger than the tube, confinement of the wandering by the tube walls modifies the field-line-wandering picture.
7. Remote Study of the Solar Surface
[37] An understanding of the detailed properties of the flux tube network in the solar wind could lay the groundwork for a future technique to remotely sense the dynamic connection of open magnetic flux through the chromospheric magnetic carpet into the photosphere.
[38] At 1 AU, the solar wind plasma is about 100 h old, which is the advection time of the plasma from the Sun to 1AU. The plasma boundaries and magnetic field boundaries seen in the solar wind at 1 AU reflect details of the magnetic field connections into the photosphere 100 h earlier. Suprathermal electrons travel from the Sun to 1 AU much faster, and boundaries seen in the suprathermal-electron (strahl) measurements at 1 AU reflect more-recent magnetic field connections into the photosphere. This is depicted in the two panels of Figure 16. In the left-hand panel a case where no reconnection is ongoing at the solar surface is sketched; here the magnetic field connection from the heliosphere into the photosphere is constant with time. Two bundles of field lines penetrating from the photosphere through the magnetic carpet are sketched: a bundle on the left and a bundle on the right. From the spot on the photosphere where the left bundle originates, green plasma is emitted (noted as the light-green shading) and green suprathermal electrons are emitted (noted as the dark green coloring of the field lines). From the spot on the photosphere where the right bundle originates, red plasma is emitted (noted as the light-red shading) and red suprathermal electrons are emitted (noted as the red coloring of the field lines). In a steady state configuration such as this, away from the Sun the boundary between the red and green plasma is the same as the boundary between the red and green suprathermal-electron strahl, as labeled in the left panel of Figure 16. Contrary to this case, if the open field lines are undergoing reconnection with the magnetic carpet, then the mapping of the open flux from the heliosphere into the photosphere changes with time. The sketch in the right-hand panel of Figure 16 depicts a case where initially no reconnection goes on (as in the left-hand panel) and then interchange reconnection commences. In the interchange reconnection near the solar surface, open magnetic field lines emanating from the right patch are reconnecting with a closed arch, reducing the number of open field lines in the right patch and increasing the number of open field lines in the left patch. The plasma boundary reflects the amount of open flux emanating from the right and left patches on the photosphere as a function of time: away from the Sun there was an equal amount of red and green plasma corresponding to the time before reconnection commenced, but nearer to the Sun the plasma distribution reflects the reduction of open flux from the right patch (red) and the increase to the left (green). The fast supratheramal electrons paint the field lines out from the Sun very rapidly. Their boundary away from the Sun reflects the recent magnetic field mapping into the photosphere. Hence as noted in the right-hand panel, the energetic-electron boundary differs in location from the plasma (and magnetic field) boundary. Thermal-electron boundaries could be used to mark the boundaries at intermediate timescales (∼10 h). (see Onsager et al. [1991], Lockwood [1995], and Lavraud et al. [2006] for similar methodologies used to remotely sense reconnection in the Earth's magnetosphere.)
[39] In the right-hand panel of Figure 16, the perpendicular distance between the plasma boundary and the suprathermal-electron boundary at 1 AU should be related to the number of magnetic field lines that reconnect in the ∼100h since the plasma left the Sun. By statistically examining the flux tube interfaces seen at 1 AU in suprathermal electrons versus plasma and magnetic fields, the time dependence of magnetic flux penetrating the magnetic carpet to form interfaces in the corona could be explored. It could be possible to test time-dependent open-flux models [Fisk et al., 1999; Priest et al., 2002; Wang and Sheeley, 2004; Fisk, 2005] using spacecraft measurements at 1 AU. With an understanding of the signatures, it might be possible to remotely study how reconnection works in the magnetic carpet, discerning interchange reconnection [e.g., Wang and Sheeley, 2004] from reconnection between pairs of open flux tubes [e.g., Parnell and Galsgaard, 2004; Mellor et al., 2005] and discerning reconnection at the base [e.g., Fisk, 2005] from reconnection in the canopy [Priest et al., 2005]. It might also be possible to measure timescales for reconnection [e.g., Schrijver et al., 1997; Handy and Schrijver, 2001]. Also, it might be possible to measure the timescales and spatial scales needed to calculate open-flux diffusion coefficients [cf. Fisk and Schwadron, 2001; Cohen et al., 2006].
[40] Remotely studying the dynamics of the magnetic connections into the photosphere, along with a coordinated study of the properties of the solar wind plasma and solar wind electrons, could provide information about the acceleration and heating of the solar wind [Fisk et al., 1999; Priest et al., 2002; Klimchuk, 2006], about the magnetic carpet [e.g., Parnell, 2001; Haganaar et al., 1997; Schrijver and Title, 2003; Wang and Sheeley, 2004], and about stellar winds in general [Landstreet, 1992; Schmelz, 2003; Pevtsov et al., 2003; Gudel, 2004; Holzwarth, 2005].
8. Discussion and Future Work
[41] In this section several topics dealing with the properties of the solar wind flux tubes are discussed and a call for future work is made.
8.1. Misconceptions in the Literature About the MHD Turbulence of the Solar Wind
[42] If the flux tube texture of the solar wind is ignored, there are a number of mistakes that can be made about the turbulence of the solar wind. These are the following.
[43] 1. It is likely that we have overestimated the amplitude of the turbulence in the solar wind. In Figure 14 it is shown that the amplitude of the solar wind “turbulence” is linearly proportional to the number of discontinuities in a data interval. The protocol for solar wind-turbulence data analysis has been to select large data intervals (days) to analyze [e.g., Matthaeus et al., 1986]. Selecting long data intervals guarantees the inclusion of discontinuities into the analysis, which can dominate the Fourier power and RMS fluctuation levels.
[44] 2. The measurements that are interpreted as the correlation length of the MHD turbulence of the solar wind may in fact be measurements of the flux tube sizes in the solar wind plasma. In Figure 17 a typical wave number spectrum for solar wind turbulence is sketched [cf. Goldstein et al., 1995, Figure 1]. As denoted in the sketch, the diameters of flux tubes are somewhat smaller than the 1 × 106 km to 4 × 106 km correlation length for solar wind fluctuations [Matthaeus and Goldstein, 1982; Feynman et al., 1996; Smith et al., 2001]: the mean wall-to-wall crossing distance 1.13 × 106 km is considerably less than a factor of 2π different from a typical correlation length of 2 × 106 km. In Table 1 it is predicted that the correlation length of the MHD turbulence within the tubes would be d/4 ∼ 1.4 × 106 km. Actual measurements of the turbulence correlation lengths are important to make (see section 8e).
[45] 3. Because our estimates of the turbulence amplitudes (point 1) and turbulence correlation lengths (point 2) may be wrong, our estimates of the scattering and transport coefficients for solar wind turbulence may be wrong. These transport coefficients pertain to the pitch angle scattering and diffusion of energetic particles in the heliosphere [Michalek, 2001; Zimbardo, 2005; Shalchi et al., 2006], turbulence-driven magnetic field-line wandering [Gray et al., 1996; Zimbardo et al., 2004; Ruffolo et al., 2006], and eddy viscosity of the solar wind flow [Korzhov et al., 1985; Verma, 1996; Borovsky, 2006].
[46] 4. Magnetic field-line wandering calculations might need to be rethought, not only because the coefficients of diffusion may be smaller than thought, but also because field-line wandering is confined to within the flux tubes. For a cylindrical tube, the diffusion of field lines would be solved in a circular cylinder with zero radial flux at the cylinder wall [e.g., Carslaw and Jaeger, 1959].
[47] 5. Because our values of the turbulence amplitudes may be overestimated, our calculations of the solar wind heating by the dissipation of the turbulence cascade may also be overestimated. A lower amplitude of turbulence means (1) less energy in the turbulence and (2) a longer eddy turnover time. These conspire to produce a reduced rate of spectral transfer of energy.
[48] 6. The measures we have of the intermittency of solar wind measurements may not be measurements of the intermittency of the turbulence. As pointed out by Bruno et al. [2001], the sources of intermittency of the solar wind data correspond to the interfaces between flux tubes. These interfaces are boundaries forced on the turbulence, they are not a property of the turbulence. To discern whether there is intermittency in the turbulence (i.e., coherent structures in the turbulence or a patchiness of the turbulence intensity), solar wind data without flux tube walls must be analyzed. Intermittency in the turbulence is important to determine since it can effect the physics of the turbulence cascade.
[49] 7. In wave number anisotropy measurements of the solar wind turbulence [cf. Matthaeus et al., 1990], the perpendicular-to-B spectrum of fluctuations may be produced by the walls of flux tubes (see Figure 7), not by turbulence with predominantly perpendicular wave numbers. This is particularly true for low-frequency (large-scalesize) fluctuations producing correlations at timescales of a fraction of an hour and longer. Further, when determining orientations perpendicular and parallel to the magnetic field, the mean field direction within each tube should be used; making a mean field direction that involves more than one flux tube (which is more than one plasma element, and more than one box of turbulence) does not make sense.
[50] 8. The interpretation of the very-low-frequency fluctuations (the so-called “energy subrange”) as a spectrum of Alfven waves [e.g., Saur and Bieber, 1999] may be inadequate. The “energy containing scales” correspond to frequencies of 2.7 × 10−6 to 8.0 × 10−6 Hz [Matthaeus and Goldstein, 1986; see also Goldstein and Roberts, 1999], which corresponds to periods of 3.4 h to 102 h. In the advecting solar wind, these correspond to wavelengths of 5 × 106 km to 1 AU. These wavelengths are much longer than a tube diameter and rather than consider Alfven waves one should consider the Alfvenic motion of flux tubes (see section 8d). If these very-low-frequency fluctuations as seen by a spacecraft are caused by the passage of one flux tube after another across the spacecraft with the flux tubes undergoing low-frequency Alfvenic motions, then locally in space in the solar wind there may not be a “spectrum” of fluctuations. This leads to the question of what drives the turbulence within the flux tubes of the solar wind plasma, since decay of a “spectrum” of low frequency Alfven waves [Tu et al., 1984; Tu and Marsch, 1995; Horbury and Tsurutani, 2001; Goldstein et al., 2005] may not be a viable mechanism.
[51] 9. The assumption that the turbulence of the solar wind is a homogeneous turbulence may be wrong (see also section 7). The flux tube walls which form boundaries to the turbulence may be important for the driving, dissipation, and dynamics of the MHD turbulence in the collisionless solar wind.
8.2. Reconnection of Flux Tubes
[52] If the flux tubes of the solar wind originate in the corona, the question arises as to why flux tube reconnection [cf. Linton et al., 2001; Linton, 2006] does not destroy the flux tube pattern in the 100 hours or so of advection time from the Sun to 1 AU.
[54] In Figure 18 the lifetime to reconnection of two flux tubes is calculated as a function of distance from the solar surface to 1 AU: one flux tube has a total magnetic flux Φ = 2 × 1017 Mx and the other has a total flux Φ = 5 × 1018 Mx (the median and mean values of the distribution of Figure 13). The plasma number density is taken to be n = 3.2 × 105 cm−3 at the solar surface (which is an underestimate at the surface [Guhathakurta et al., 1996; Antonucci et al., 2004]) and n = 7 cm−3 at 1 AU (1 AU = 215 Rsun) with an n ∝ r−2 variation in between; the magnetic field strength is taken to be B = 2.77 gauss at the solar surface [e.g., Schrijver et al., 1997] and B = 6 nT at 1 AU with a B ∝ r-2 variation in between. The θ value is taken to be the angle between radial and the local Parker-spiral direction (see Figure 12). The local lifetime to reconnection, given by expression (5), is plotted in Figure 18 as the two black curves for the two flux tubes. Also plotted as the dashed curve is the age of the solar wind τage = r/vsw for vsw = 410 km/s. As can be seen, smaller flux tubes have reconnection lifetimes that are shorter than the age of the solar wind plasma; hence, if reconnection were ongoing, the smaller flux tubes would have undergone reconnection. Larger flux tubes could survive without alteration by reconnection. Reconnection should not destroy the flux tubes nor change the magnetic topology from open to closed or open to disconnected, but it would produce layers of mixing with plasma from neighboring tubes becoming magnetically connected and with magnetic field orientations between the orientations in the two neighboring tubes. The relaxation of magnetic tension in the layer would also result in highly Alfvenic transverse flows.
[55] At 1 AU, the solar wind flux tubes appear not to be reconnecting; this lack of reconnection is indicated by the flux tube walls, which are much thicker than both the ion-inertial length and the ion gyroradius [Joachim Birn, private communication, 2007]. The wall thicknesses [Burlaga et al., 1977; Lepping and Behannon, 1986; Riazantseva et al., 2005a, 2005b] are consistent with Bohm diffusion acting for the ∼100-h age of the solar wind plasma [Borovsky, 2006].
[56] If the current sheet separating two plasmas is not thin enough, the two plasmas will not reconnect [Hesse et al., 2001; Pritchett, 2005] To get thin flux tube walls so that the flux tubes would reconnect may require that the flux tubes be driven together [e.g., Birn and Schindler, 2001]. However, in fact, in the expanding solar wind the flux tubes are being pulled apart (see Figure 12). In Figure 19 the expansion velocity of two flux tubes, one with total flux Φ = 2 × 1017 Mx and the other with total flux Φ = 5 × 1018 Mx, are plotted as functions of the distance from the Sun in a 410-km/s solar wind. Here the expansion velocity is the velocity at which the centers of two adjacent flux tubes move apart owing to the expansion of the solar wind plasma. Also plotted in Figure 19 is 0.1vA*, where vA* is the component Alfven speed for two adjacent flux tubes. As can be seen, for the larger flux tube the expansion velocity dominates over any reconnection inflow speed for r > 2 Rs. For the smaller flux tube, the expansion velocity is lower than the reconnection-inflow speed, but still the flux tubes are not driven together. It is suggested here that the expansion of the solar wind plasma is probably the chief reason why the flux tubes do not reconnect. In regions of the solar wind where compression might overcome expansion (such as CME sheaths or in some portions of CIRs), one might look for thin flux tube walls and reconnection caused by driving.
8.3. Origin Away From the Sun?
[57] The solar origin means that the flux tubes seen at 1 AU are “fossil structures” in the solar wind. An alternative is considered here: the formation of the flux tubes away from the Sun as part of the solar wind's evolution. Four possible alternate explanations of the origin of the flux tubes are suggested.
[58] First, as pointed out by Parker [1994, 2004], magnetic separatrices (current sheets) tend to form spontaneously as a plasma relaxes toward equilibrium (see also Priest et al. [2002] and Close et al. [2004c] for similar processes). If the solar wind plasma is born with some magnetic field “interlacing”, then discontinuities may arise as the solar wind plasma relaxes toward equilibrium. This relaxation may be modified by the transverse expansion of the solar wind as it advects away from the Sun, with the transverse expansion intensifying the magnetic shear in the solar wind (see Figure 12).
[59] Second, discontinuities may be produced by the action of MHD turbulence in the solar wind. A consequence of the cascade of energy in MHD turbulence from large-scale fluctuations to small scales is the formation of intense, coherent, thin current sheets [e.g., Matthaeus and Lamkin, 1986; Dmitruk et al., 2004]. If the current sheets formed in the turbulence can be intense enough, then some of the solar wind discontinuities could be structures within the solar wind turbulence rather than boundaries to the solar wind turbulence. (see also Giacalone et al. [2000], Zimbardo et al. [2004], and Chuychai et al. [2005] for discussions of flux concentrations in MHD turbulence.)
[60] Third, researchers have considered the possibility that tangential discontinuities and pressure-balance structures may originate from the nonlinear steepening of outward-traveling Alfven waves [e.g., Malara et al., 1996; Parenti et al., 1997; Vazquez and Hollweg, 1999; Kellogg and Horbury, 2005].
[61] A fourth possibility needs to be considered for the solar wind. Mesoscale coherent flows (“zonal flows”) are known to spontaneously form in unstable low-β plasmas owing to nonlinear energy transfer from drift-wave turbulence [Lin et al., 1998; Diamond et al., 2005; Itoh et al., 2006]. Zonal flows with velocities of up to 0.5vA can also be driven in higher-β plasmas by drift-Alfven turbulence [Naulin et al., 2005; Steinhauer and Guo, 2006]. When the zonal flows are planar, they can create transport barriers in the large-scale plasma [Lin et al., 1998; del-Castillo-Negrete, 2000].
[62] A strong argument against all four of these mechanisms as the explanation of the origin of the flux tubes in the solar wind is the fact that the properties of the plasma often change across the discontinuities of the solar wind (e.g., Figure 5). If the discontinuities were to form in the solar wind plasma as the plasma ages (i.e. form away from the Sun), then it is hard to see how the plasma can be different across the discontinuity. In particular, changes in the ionic composition (see Figure 5 ) [see Borovsky, 2006] are hard to explain in this manner.
8.4. Alfvenic Motions of the Flux Tubes
[63] The flux tubes in the solar wind are distinct magnetic entities that move coherently with distinct velocities relative to one another. When crossing from one flux tube to another, there is a change in the vector magnetic field (owed mostly to the orientation of one tube relative to its neighbor) and there is a change in the vector velocity (owing to a bulk motion of one tube relative to its neighbor). In the ACE data set the median value of the velocity difference between neighboring tubes is 40.6 km/s (see Table 1). The changes in the magnetic field and velocity are highly Alfvenic, i.e. Δv/vA ≈ ΔB/Bo (see Figure 4). This is consistent with the historical findings that the field changes and velocity shears across solar wind tangential discontinuities are highly Alfvenic [e.g., Denskat and Burlaga, 1977; Neugebauer, 1985; De Keyser et al., 1998]. This Alfvenicity of the tangential discontinuities has been somewhat of a mystery; explanations focusing on Alfven waves in the two plasmas across the discontinuity and on surface waves at the discontinuity have been put forth [e.g., Hollweg et al., 1982; Neugebauer, 1985; Vasquez, 2005].
[64] We interpret this Alfvenicity as the flux tubes undergoing Alvenic motion.
[65] Nonaxisymmetric (kink) disturbances in a thin flux tube propagate at the Alfven velocity and axisymmetric disturbances in a thin flux tube propagate at just below the Alfven velocity [Wilson, 1979; Edwin and Roberts, 1983]. These Alfvenic motions could be caused by a number of effects. (1) The temporal enhancement of the braiding of the flux tubes caused by the expansion of the solar wind as it advects away from the Sun (see Figure 12) could be putting the network of flux tubes out of equilibrium, and the dynamical relaxation toward equilibrium in the low-dissipation plasma could lead to Alfvenic motions. (2) Similarly, the network of tangled flux tubes could be put out of equilibrium by large-scale shears and compressions in the solar wind, leading to Alfvenic motions. (3) Temporal changes in the connection of the flux tubes in the corona (interchange reconnection) could be launching outward propagating thin-tube kink modes that persist out into the heliosphere [cf. Fisk et al., 1999; Cranmer and van Ballegooijen, 2005]. (4) Reconnection events between flux tubes in the corona and in the solar wind could be launching Alfvenic disturbances into the network of flux tubes. (5) Reconnection events between flux tubes in the solar wind could be creating layers of Alfvenically moving plasma as magnetic tension relaxes. (6) Differences in the radial plasma flow velocity in neighboring tubes near the Sun can result in shear motions between the tubes at 1 AU [cf. Siscoe, 1970], launching Alfvenic disturbances. Effects (1), (2), (3), and (4) have the Alfvenic motions of the flux tubes initiated out in the inner heliosphere whereas effects (3) and (4) have the Alfvenic motions of the flux tubes initiated at the Sun [cf. Casalbuoni et al., 1999].
[66] Clearly, work needs to be done to fully characterize the relative motions of the flux tubes to discern whether there are kink, sausage, or torsional body waves, to discern whether there are surface waves, and to gather evidence that would point to the cause of the Alfvenic disturbances.
8.5. Future Work
[67] To fully understand the nature of the solar wind in light of the network of flux tubes, six science tasks must be accomplished: (1) determine the properties of discontinuities in the solar wind, (2) determine the properties of the individual flux tubes, (3) determine the properties of the flux tube network and determine its evolution, (4) learn the origin of the flux tube texture, (5) reanalyze the solar wind turbulence excluding flux tube walls, and (6) assess the impacts of this flux tube texture for heliospheric physics.
[68] 1. Determine the properties of discontinuities in the solar wind. A statistical analysis of the discontinuities in the solar wind is called for. It is very important to obtain an understanding of what is in the solar wind and where it comes from. A large survey of the properties of discontinuities using high-time-resolution plasma and magnetic field measurements would yield much information. One question to answer is: Are there different types of discontinuities in the solar wind? (besides the older classification of tangential versus rotational). Some discontinuities may be plasma boundaries (flux tube walls), some may be current sheets produced with MHD turbulence, or fractures in large-scale velocity shears, or current sheets formed by global magnetic relaxation, or steepened of Alfven waves. Another issue to answer is: How thick are the various types of discontinuities and what controls their thicknesses? In different situations, the thicknesses could be governed by Bohm diffusion, reconnection, current-sheet thinning, stabilizing effects to thinning, Landau damping, etc. Also, Why are the discontinuities are so Alfvenic? is a longstanding question that needs to be settled. Finally, a specific questions is: Which discontinuities are flux tube walls created at the Sun and which ones are structures that evolved in the solar wind.
[69] 2. Determine the properties of the individual flux tubes. To attain this, flux tube measurements must be collected and statistically analyzed. The following properties of the solar wind flux tubes must be discerned: (1) flux tube diameters, (2) flux tube total magnetic flux, (3) the magnetic twist within the flux tubes, (4) the currents carried in the flux tube interiors and within the flux tube walls, (5) the axial plasma flows within the flux tubes, and (6) the flow vorticity within the flux tubes.
[70] 3. Determine the properties of the flux tube network and determine its evolution. It is very important to use single-spacecraft statistics and multispacecraft comparisons to attain a realistic depiction of the pattern of flux tubes in the solar wind emanating from the various types of regions on the solar surface. The flux tube-network information will include (1) the distribution of flux tube orientations about the Parker-spiral direction, (2) the flux tube cross-sectional shapes, (3) the braiding lengthscale, (4) the distribution of relative velocities between neighboring flux tubes, and (5)the sizes of the plasma-property jumps between neighboring flux tubes. This will provide rudimentary information about the braiding, dynamics, and evolution of the flux tubes as the solar wind advects away from the different types of regions on the Sun.
[71] 4. Learn the origin of the flux tube texture. The flux tubes in the solar wind may have their origin above the solar surface or they may form in the solar wind far from the Sun. Learning their origin means discerning whether their properties are consistent with (1) fossil structures from the magnetic carpet, (2) in situ formation during relaxation of the solar wind plasma, or (3) structures created by the action of MHD turbulence in the solar wind.
[72] 5. Reanalyze the solar wind turbulence excluding flux tube walls. The analysis of the turbulence of the solar wind needs to be redone with care to exclude the walls of the flux tubes from the data selected. This means that a statistical survey must be done to discern flux tube walls (plasma boundaries) from current sheets that could be part of the turbulence. Analyzing the turbulence within individual tubes separately and treating the individual analyses as an ensemble is the way to proceed. With the boundary discontinuities out of the analyses, the amplitude of the turbulence, the spectra of the turbulence, the correlation lengths of the turbulence, and the intermittency of the turbulence need to be determined. Orienting the analysis with respect to the magnetic field direction in each tube, the perpendicular-parallel anisotropy of the turbulent fluctuations need to be determined.
[73] 6. Assess the impacts of this texture for heliospheric physics. (see also section 9b for more-detailed discussions.) Almost certainly the fact that the solar wind plasma is separated into distinct tubes of plasma has an impact on solar wind and heliospheric physics. It has been observed that the flux tube texture affects the nature of the large-scale flows of the solar wind; large numbers of shear zones in the solar wind need to be examined to gain a more-detailed understanding of these flow effects. The effects of the flux tube walls on the turbulence within the flux tubes needs to be assessed and the mechanisms that drive MHD turbulence within the flux tubes need to be determined. The effect on the transport of energetic particles in the inner heliosphere by the “magnetic ductwork” formed by the tangled flux tubes needs to be investigated, including the scatter of lower-energy energetic particles by the MHD turbulence within the tubes and the scatter of higher-energy energetic particles by the tangling of the tubes themselves. Assessing how shocks propagate in the inhomogeneous flux tube solar wind is important, as is assessing how the cellularization of the solar wind affects the acceleration of particles by shocks. Finally, looking for signatures of the flux tube texture of the solar wind in the various geomagnetic indices will provide information about (1) the effects of the tube structure on the magnetosphere (e.g., the triggering of substorms by the passage of a flux tube wall) and (2) response timescales for solar wind changes to affect the various phenomena in the magnetosphere (e.g., the polar-cap currents, dayside aurora, nightside aurora, and the ring current).
9. Summary
[74] In section 9.1 the findings of this paper are summarized. In section 9.2 a summary of the evidence that the solar wind flux tubes originate at the Sun is given.
9.1. Summary of the Findings
[75] 1. At 1 AU in the solar wind, discontinuities can be found by noting temporal changes in the magnetic field field direction, the magnetic field strength, the ion properties, the electron temperature, or the field-aligned electron heat flux. These discontinuities are interpreted to be the walls of individual magnetic flux tubes containing distinct plasmas.
[76] 2. The flux tube walls are most easily found via large temporal changes in the magnetic field direction and the vector plasma-flow velocity.
[77] 3. Large changes in the ion entropy density and the alpha-to-proton ratio of the plasma are associated with flux tube walls.
[78] 4. The sizes of the tubes are measured by assuming they are frozen into the solar wind flow and measuring the wall-to-wall distance. There is a wide distribution of sizes, with the median tube “cross section” at 1 AU being about 4.4 × 105 km.
[79] 5. At 1 AU the tubes sizes are larger in slow wind than they are in fast wind.
[80] 6. At 1 AU, the largest tubes are found in anomalously cool solar wind plasma. This is interpreted as fewer flux tubes occurring in ejecta.
[81] 7. The tubes are tilted strongly with respect to each other. This suggests a braiding.
[82] 8. The tubes are on-average aligned with the Parker-spiral direction, with a large spread in orientations.
[83] 9. The large changes seen in the magnetic field direction and the plasma flow velocity from one tube to the next are very Alfvenic. This suggests that the flux tubes are undergoing Alfvenic motions.
[84] 10. Because of the lack of plasma mixing from one tube to the next, it is argued that the flux tube walls form boundaries to the MHD turbulence of the solar wind. Hence the turbulence resides within the individual flux tubes.
[85] 11. The measured amplitude of solar wind fluctuations in a data interval is linearly proportional to the number of flux tube walls in that interval. This argues that the walls of flux tubes dominate the Fourier power of the solar wind turbulence within the flux tubes.
[86] 12. Because the flux tube walls are thick at 1 AU, it is assumed that the flux tubes are not reconnecting with each other at 1 AU.
[87] 13. It is hypothesized that the flux tubes are created above the solar surface and that they survive the ∼100-h advection to 1 AU. Hence the flux tubes at 1 AU are “fossil structures”.
[88] 14. When mapped from 1 AU to the rotating solar surface, the diameters of the flux tubes correspond to granule and supergranule scales.
[89] 15. The amount of magnetic flux Φ in the flux tubes at 1 AU has the same distribution as the amount of magnetic flux in the magnetic field concentrations of the magnetic carpet.
[90] 16. The large-angle changes in orientation between adjacent flux tubes at 1 AU could be caused by slight misalignments of neighboring flux tubes above the solar surface, whereupon advection in a frozen-in fashion in the expanding solar wind greatly amplifies the misalignments.
[91] 17. Calculations showed that if reconnection of flux tubes were to have occurred during the advection of the solar wind from the corona to 1 AU, the smaller tubes (Φ less than about 2 × 1017 Mx) would have undergone reconnection and the larger tubes would not. It is argued that reconnection of flux tubes does not occur because solar wind expands so tubes are not driven together.
[92] 18. Studying the flux tubes in the solar wind at 1 AU opens the possibility to gather information about the changing connections of interplanetary magnetic field lines into the magnetic carpet.
9.2. Evidence That the Solar Wind Flux Tubes Originate at the Sun
[93] The findings (3), (5), (12), (14), and (15) above support the hypothesis that the flux tubes observed in the solar wind at 1 AU by ACE were created at the solar surface.
[94] First, changes in the properties of the plasma (ion entropy density and alpha-to-proton ratio) occur across the boundaries between flux tubes (Figure 5). This indicates that the flux tubes and the plasma are created together. Forming the flux tubes inside of a large plasma cannot produce these changes.
[95] Second, when mapped to the Sun, the sizes of the flux tubes seen at 1 AU by ACE correspond to subsupergranule scale sizes. This agrees with expectations that several open-flux bundles per supergranule [e.g., Close et al., 2005] connect from the photosphere into the solar wind.
[96] Third, the flux tubes are larger in slow wind than they are in fast wind (Figure 9). The slow wind undergoes a larger expansion factor in the corona, so if the flux tubes are created below the corona then the flux tubes should also undergo greater expansion and so it makes sense that they are larger.
[97] Fourth, the distribution of the total amount of magnetic flux Φ in the tubes as measured by ACE at 1 AU is very similar to the distribution of the total magnetic flux Φ of flux bundles in the magnetic carpet (Figure 13). If the flux tubes from the magnetic carpet are advected out in the solar wind, Φ should be conserved and can be used to identify the bundles. This appears to be the case.
[98] We have argued that the flux tubes are not destroyed by reconnection because the solar wind expands, pulling the flux tubes apart rather than driving them together. Indeed, when examined at 1 AU the walls of the flux tubes are thicker than the characteristic scale sizes of thin current sheets undergoing reconnection. It also seems to be the case that the solar wind turbulence does not destroy the identities of the flux tubes. Perhaps the strong magnetic shear across the interfaces between adjacent tubes makes the flux tube walls impervious to the action of the turbulence.
10. Impact of the Flux Tube Texture on Heliospheric Phenomena
[99] The cellularization of the solar wind into distinct magnetized tubes of plasma will have several impacts on solar wind and heliospheric physics. Some of those impacts are discussed briefly below.
10.1. Solar Wind Flow
[100] The network of tangled, resilient flux tubes changes the flow properties of the solar wind. It has been observed previously [Borovsky, 2006] that large-scale shear flows (corotating interaction regions and coronal-mass-ejection sheaths) “fracture” along the boundaries between the flux tubes. This fractured flow differs from the flow of a homogeneous plasma. Normally, fluids are treated as a continuum: continuous and smoothly varying [cf. Shepherd, 1965; Hughes and Young, 1966; Batchelor, 1967]. These are so-called “simple fluids”. The solar wind with its resilient flux tubes is more of a “complex fluid” with underlying structure. Owing to interactions and entanglements of underlying structures, complex fluids can have properties such as elasticity (memory) and non-Newtonian viscosity [e.g., Cheremisinoff, 1988; Yoshizawa and Israelachvili, 1993; Tirrell, 1994; Ganesan et al., 2006].
10.2. Solar Wind Turbulence
[101] If the turbulence of the solar wind is indeed confined to the insides of the flux tubes, then boundary effects may be important for the turbulence. Boundaries can produce anisotropies in the turbulence [cf. Uzkan and Reynolds, 1967; Anthony and Wilmarth, 1992] and boundaries can introduce additional mechanisms to drive turbulence [cf. Hussain and Reynolds, 1975; Townsend, 1976]. Questions arise about the interpretation of the very-low-frequency fluctuations of the solar wind magnetic field and velocity; with wavelengths longer than the tube diameters, the interpretation of these fluctuations as a spectrum of Alfven waves needs to be rethought. The mechanisms by which the turbulence within the flux tubes is driven need to be explored.
10.3. Energetic-Particle Transport
[102] If magnetic field lines are confined to the insides of the individual flux tubes, then the tangled network of flux tubes forms a “magnetic ductwork” for the transport of energetic particles in the inner heliosphere [cf. Bartley et al., 1966; Mazur et al., 2000]. The flux tubes approximately follow the Parker spiral, but the details of energetic-particle transport will depend upon the manner in which flux tubes wander in the network. The individual flux tubes duct the electron heat flux from the Sun, suprathermal electron bursts, and the lower-energy solar energetic particle (SEP) events. Inside the tubes that duct the particles, MHD turbulence will act to scatter the particles, and the scattering coefficients will depend on the properties of the turbulence inside the tubes. For higher-energy particles with gyroradii comparable to or larger than the tube diameters, the tangled-tube pattern (see Figure 1) affects particle transport by acting as a scatterer [see also Fisk and Sari, 1973]. The scattering coefficients for these high-energy particles will depend on the braiding pattern of the tube network, which may depend on Leighton-type diffusion owed to fluid motions on the photosphere [Leighton, 1964] or to Fisk-type diffusion of the flux tube connections into the magnetic carpet [Fisk and Schwadron, 2001]. For a typical magnetic field strength of 6 nT at 1 AU, 100-MeV protons or 500-MeV electrons have gyroradii comparable to tube radii. Hence cosmic rays and the higher-energy protons of SEPs will be scattered by the tangled-tube pattern.
10.4. Interplanetary Shocks
[103] The propagation of interplanetary shock waves through the solar wind plasma will be affected by the mesoscale inhomogeneous texture of the plasma. The differences in the magnetic field orientation from tube to tube change the shock normal angle from tube to tube and the differences in plasma properties from tube to tube change the shock Mach number. These changes can lead to localized velocity changes and localized refractions of the shock front, altering the global propagation of a shock through the inner heliosphere [cf. Heinemann, 2002; Williams and Dyson, 2002; Poludnenko et al., 2002; Fry et al., 2003]. Pertinent to the acceleration of particles by interplanetary shocks, (1) the changes in the shock normal angle from one flux tube to the next may place limits on shock-drift mechanisms and (2) the confinement of shock-energized particles in a flux tube that can cross shock in two locations can intensify the energization of particles by the shock [cf. Sandroos and Vainio, 2006].
10.5. Solar Wind/Magnetosphere Coupling
[104] The flux tube texture of the solar wind will lead to (at least) three effects at Earth: (1) intervals of quasi-steady driving of the magnetosphere, (2) the association of the occurrence of substorms with flux tube walls, and (3) the association of hot flow anomalies and magnetospheric transients with flux tube walls. Geomagnetic activity in the Earth's magnetosphere is controlled chiefly by the orientation of the solar wind magnetic field and by the magnitudes of v, B, and n of the solar wind Borovsky [2007]. In the flux tube network, these quantities each vary with time in a bimodal fashion (see Figures 2 and 3): there are small fluctuations inside the tubes (owing to turbulence) and large changes when tube walls are crossed. Hence the Earth is driven in quasi-steady intervals punctuated by sudden changes in the driving level. In Figure 20 the time dependence of one very simple solar wind driver function F = vBz [cf. Burton et al., 1975; Pudovkin et al., 1985] is analyzed, where v is the solar wind speed and Bz is the component of the solar wind magnetic field that is perpendicular to the Sun-Earth line and parallel to the projection of the Earth's magnetic-dipole orientation as viewed from the Sun. Using ACE spacecraft measurements for the 5 years 1996–2000, the driver function F is calculated every 64 s, then a 5-min running average of F is made, and the change ΔF in the 5-min-averaged F over every 5-min difference is calculated. The absolute value of ΔF is binned and the resulting occurrence distribution is plotted in Figure 20, similar to the quantities in Figures 2 and 3. As can be seen, a dual population is seen in the changes ΔF: a population of small changes and a population of large changes. For solar wind/magnetosphere coupling this is interpreted as quasi-steady driving (small changes) associated with the Earth being within a flux tube and large changes in the driving when the Earth crosses a flux tube wall. The quasi-steady driving intervals should have typical durations of a fraction of an hour. It remains to be seen whether this dual population of temporal changes is seen in the geomagnetic-activity indices that measure the coupling of the solar wind to the Earth's magnetosphere-ionosphere system. It is well known that magnetospheric substorms can be triggered by sudden changes in the direction of the solar wind magnetic field [e.g., Rostoker and Falthammer, 1967; Rostoker, 1983; Lyons et al., 1996]. Hence it is suggested herein that substorm occurrences may be associated with the passages of flux tube walls over the Earth. However, not all wall passages will produce substorms; about 5 substorms occur per day [Borovsky et al., 1993] whereas about 50 flux tubes pass the Earth per day. Also, not all substorms will have an association with walls; substorms can occur spontaneously, without identifiable triggers in the solar wind [McPherron et al., 1986; Hsu and McPherron, 2003]. The changes in the solar wind magnetic field associated with the passage of a flux tube wall should result in rapid changes in the properties of dayside reconnection, leading to sudden changes in magnetospheric flow and sudden changes in magnetosphere-ionosphere currents. Hot flow anomalies [Schwartz, 1995; Safrankova et al., 2000; Omidi and Sibeck, 2007] are caused by the interaction of tangential discontinuities with the Earth's bow shock. These anomalies produce substantial local perturbations to the properties of the solar wind plasma and its flow, producing an enhanced perturbation on the Earth's magnetosphere and ionosphere [Sibeck et al., 1999; Kataoka et al., 2002] that includes sudden large changes in the size of the magnetosphere, magnetic impulse events in the magnetosphere, sudden intensifications of aurora, and traveling convection vortices in the ionosphere.
10.6. Extrapolation to Astrophysical Plasmas
[105] Since the solar wind plasma is filled with magnetic flux tubes from the Sun, it is not a large, homogeneous plasmas. Does that make the solar wind different from astrophysical plasmas of interest, such as stellar winds, the interstellar medium (ISM), accretion disks, and the intergalactic medium (IGM)? The in situ observations of the solar wind have made the solar wind a useful laboratory to study MHD turbulence [cf. Terasawa and Scholer, 1989; Goldstein et al., 1995]; how appropriate is the extrapolation of those measurements to astrophysical plasmas? On the contrary, could the ISM be seeded with flux tubes from stellar winds and as a result could the flux tube texture of the solar wind provide important information about the behavior of that unreachable plasma?
Acknowledgments
[106] The author wishes to thank Joachim Birn, Roberto Bruno, Benoit Lavraud, Mark Linton, Bill Matthaeus, Eckart Marsch, Michael Ruderman, and John Steinberg for helpful conversations and Randy Bremmer, Ruth Skoug, and Chuck Smith for ACE data. This work was supported by the NASA Research in the Structure of the Solar Wind at 1 AU Program, by the NASA Heliospheric Guest Investigator Program, and by the Los Alamos National Laboratory Laboratory Directed Research and Development Program.
[107] Amitava Bhattacharjee thanks Eric Priest and Robert J. Leamon for their assistance in evaluating this paper.
