LENGTHS OF WANDERING MAGNETIC FIELD LINES IN THE TURBULENT SOLAR WIND

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1 The Astrophysical Journal, 653:1493Y1498, 006 December 0 # 006. The American Astronomical Society. All rights reserved. Printed in U.S.A. LENGTHS OF WANDERING MAGNETIC FIELD LINES IN THE TURBULENT SOLAR WIND B. R. Ragot Helio Research, P.O. Box 1414, Nashua, NH Received 006 August 10; accepted 006 August 31 ABSTRACT Charged energetic and suprathermal particles propagating in the solar wind (SW), as in most astrophysical environments, follow magnetic field-line irregularities down to very short scales of turbulence. Their minimal path length, or path length in the absence of scattering, is determined by the length of the traced magnetic field line resolved to those scales. The lengths of turbulent field lines are investigated here as functions of resolution scale and turbulence condition in the SW. In situ measurements on board Helios, turbulent field-line simulations, and theory all give excellent agreement for the dependence of the turbulent length on the resolution scale, as well as for the average field-line lengthening. Close to 1 AU, the length of a turbulent field segment is increased on average by close to 50%, with even longer field lines in some slow SW streams. While in fast SW most of the field-line lengthening is due to shortscale irregularities, wandering on a 0.1 AU scale can produce in slow SW as much lengthening as irregularities on shorter scales, explaining a strong stream-to-stream variability. In solar impulsive particle events, noticeable travel delays of the first arriving particles should result from the turbulent lengthening of the magnetic field lines, with significant variations of these travel delays from one event to another. Subject headinggs: cosmic rays interplanetary medium magnetic fields plasmas Sun: particle emission Sun: radio radiation turbulence waves 1. INTRODUCTION In most astrophysical environments, and in the solar wind (SW) in particular, suprathermal and energetic particles, both electrons and ions, are generally highly magnetied. Their gyroradii are much shorter than most turbulent scales. In the absence of Coulomb collisions these charged particles are forced to follow the magnetic field lines down to fluctuation scales corresponding to a few times their gyroradius, irregularities on even shorter scales being responsible for their scattering in pitch angle. When estimating the path length of unscattered particles, one therefore has to measure the length of the turbulent field lines resolved down to those scales of a few particle gyroradii. One should expect these path lengths to increase with decreasing particle energy and gyroradius. The effect of small-scale irregularities, however, is often omitted in the evaluation of the length of a field line. In the SW in particular, magnetic field lines are often assumed to have the length of a smooth Parker spiral, given by Z " r L spiral ¼ dr 1 þ (r r # s) 1= r 1 VSW " " # 1= (r r s ) VSW ¼ 1 þ V SW (r r s) # r þ V SW arcsinh (r r s ) V SW between radii r 1 and r,wherer s is the radius at the potential field source surface and :7 ; 10 6 rad s 1 is the solar rotation rate. For a SW speed V SW 350 km s 1, a smooth magnetic field line following a Parker spiral between the Sun and 1 AU has, according to equation (1), a length of about 1. AU. In the study of impulsive solar electron events, this value of 1. AU is often assumed in order to determine the injection time of the electrons (Haggerty & Roelof 00; Haggerty et al. 003). An r 1 ð1þ 1493 underestimated length of the field line may falsely lead to a conclusion of delayed electron injection relative to the metric type III burst generally accompanying the bursts of kev E 100 kev electrons. Intheirtestofthec/v onset plots, Kahler & Ragot (006) compared the electron path lengths L c =v obtained from the c/v plots with the Parker spiral lengths of equation (1). They found L c =v broadly distributed between 0.15 and.7 AU and generally <L spiral, which invalidated the c/v plot method and possibly its underlying assumptions of impulsive and energy-independent injection onsets and/or scatter-free propagation of the electrons. A comparison of L c =v to the actual lengths of the turbulent magnetic field lines resolved to the scale of the electron gyroradii could make their conclusion even stronger, if the small-scale irregularities of the turbulent field lines were to make the turbulent SW field lines noticeably longer. We are not aware of any better estimates for the lengths of magnetic field lines in the turbulent SW. We therefore propose to determine in this short paper the correction factor that should be applied, locally, to the length of an unperturbed magnetic field segment to obtain the real length of a turbulent field segment of the SW. To that end, turbulent magnetic field lines from independent SW streams are simulated on a broad range of length scales, under the assumption of quasilinearity, that is, under the assumption that the cross-field displacements can be neglected to first order in the estimate of those cross-field displacements (Ragot 006c, hereafter R06c). It has been previously shown through comparison of generalied quasilinear (GQL) results with in situ data (Ragot 006d, hereafter R06d) and with the results of fully nonlinear calculations (Ragot 006b, hereafter R06b) that, in the inner heliosphere and under quiet SW conditions, quasilinearity is a valid assumption for the calculation of individual magnetic field lines ( R06c; Ragot 006e, hereafter R06e) or of the mean cross-field displacements (R06b; R06c; R06d; R06e)uptoatleastafractionofanAU.Ashortsummaryof the GQL theory and its range of applicability in the SW can be found in x..1 of R06d, and a detailed summary of the

2 1494 RAGOT Vol. 653 theory/observation comparison of R06d in x of R06e. In our simulations, we apply the SRND, or Sums of Random Numbers Distribution, method introduced by R06c for the generation of projected magnetic field lines and generalied by R06e for the generation of three-dimensional magnetic field lines. In x, the lengths of the simulated field lines versus resolution scale, and their statistics, are studied for various SW conditions. In x 3, a theoretical estimate of the average length is made and compared to the averaged simulation results. The lengths of SW wandering field lines are also computed from Helios in situ magnetic field data in x 4 and compared to the simulation and theoretical results.. LENGTHS OF MAGNETIC FIELD LINES IN INDIVIDUAL STREAMS: SIMULATION RESULTS Our goal is to estimate the correction factor to be applied to the length of a segment of smooth Parker spiral field line in order to obtain the real length of a turbulent field segment in the inner heliosphere. It has been shown that the GQL theory and its quasilinear approximation are valid in the inner heliosphere under quiet turbulence conditions (R06b; R06d), and realistic turbulent field lines can now be simulated with a powerful new method, the SRND method (Ragot 006a, hereafter R06a; R06c). Here we apply the generalied SRND method of R06e to generate threedimensional field lines from the magnetic power spectra measured in slow and fast SWat 0.3 and 1 AU during quiet SW periods near solar minimum. The SRND method is based on the generation of sums of random numbers from the analytically derived distributions of these sums as a shortcut to the direct summation of an astronomically large number of random numbers; typically are needed to generate a realistic turbulent field over four decades of length scales from a three-dimensional spectrum. We generate 50 magnetic field lines under each of the following SW conditions, slow (V SW 350 km s 1 )andfast(v SW 650 km s 1 ) at 0.3 and 1 AU during quiet periods. Each of the field lines is 4 ; 10 1 cm long in projection along the main magnetic field B 0, i.e., of the order of 1 day long. The projections of the three-dimensional spectra used for these simulations can be found in R06e (or R06d for slow SW and R06a for fast SW). They were determined from Helios data on the same time intervalsasusedinx4todetermine the observational lengths. We then compute the length of each simulated field line as a function of the resolution scale along the main field. The results are shown in Figure 1, rescaled by the projected length ¼ 4 ; 10 1 cm, for resolution scales between 10 9 and 10 1 cm. Four major points should be noted about Figure 1. First, the ratios L/, orl/l smooth, are significantly larger than 1, implying that a turbulent field line can be significantly longer than its projection along the main field, by as much as 50% or more. This should result in travel delays for the particles propagating along the turbulent field lines, even in the absence of pitchangle scattering, which has been studied earlier (Lintunen & Vainio 004; Sái et al. 005) as a possible explanation of travel delays. For instance, a 10 MeV proton traveling between 0.7 and 1 AU from the Sun with a gyroradius cm and a parallel speed v 3 ; 10 9 cm s 1 should be delayed in slow SW by ½L spiral (0:7 AU; 1AU)/v Š(L/) 15 1 minutes (delay times broadly distributed between 3 and 7 minutes with a few more extreme values) and in fast SW by 8:5 minutes. The delay time in the slow SW could actually be even more broadly distributed if the large-scale wandering of the traced field line significantly decreases or increases the radial transport of the field line (see x 5). Between 0. and 0.4 AU from the Sun, a 10 MeV proton has a gyroradius 10 9 cm and should incur a travel delay of 3 1 minutes in slow SW and 4 1 minutes in fast SW. To Fig. 1. Ratio L/ as function of the resolution scale for SW conditions as indicated in each panel. The ratio L/ is the correction factor that should be applied to the length L smooth of a field-line segment s smooth approximation to obtain the length of the real turbulent field-line segment seen at the resolution. Here the smooth approximation of a field-line segment is just its projection along the main magnetic field B 0. The lengths L are computed from field lines simulated in independent SW streams from the equivalent of about turbulent modes. Each of the 50 simulated field lines in each panel is about 4 ; 10 1 cm long in projection and, therefore, the statistics become relatively poor when the resolution approaches 10 1 cm.

3 No., 006 TURBULENT LENGTHENING 1495 obtain the total travel delay between the Sun and 1 AU would require the knowledge of L/ at the very least at one additional intermediate distance from the Sun, to estimate the travel delay incurred in the range 0.4 to 0.7 AU. Here we can set a lower limit for the average travel delay between 0.4 and 0.7 AU of 1:68 ; 3minutes 5 minutes and 1:56 ; 4minutes 6 minutes in slow and fast SW, respectively, but actual values, closer to those near 1 AU, are more likely. The factors 1.68 and 1.56 are the values of the ratio L spiral (0:4 AU; 0:7 AU)/L spiral (0: AU; 0:4 AU).The resulting travel delay for an unscattered 10 MeV proton traveling from a few solar radii to 1 AU could exceed 3 14 minutes in slow SW and 19 4 minutes in fast SW. The travel delays of unscattered 5 kevelectrons with gyroradii afew10 7 cm at 1 AU should be similar to those of the 10 MeV protons in slow SW, perhaps even longer in fast SW. Of course the travel delays are shorter for faster particles, and longer for slower particles, since they are proportional to the travel time and the correction factors L/ are decreasing functions of the particle gyroradius. Second, as already noted in the previous point, the absolute values of L/,orL/L smooth, are relatively broadly spread, by as much as 35%, 17%, 13%, and 6% of the average values at 10 9 cm resolution in the four panels of Figure 1, and third, for each turbulence spectrum, all curves of L/ versus display the same variations with up to at least cm. The spread of the absolute values at a given resolution reflects the variability of x and y from one field line to another on the longest scales. The parallel evolution of the curves relative to each other toward the shorter resolution scales is due to the independence of the scales and the higher statistics at the shorter resolution scales. The jumps x and y on the scale are fairly independent of the jumps on another scale 0. Also, whereas the value of L/ on the longest scale, 10 1 cm, is given in Figure 1 by only four field segments, it results at ¼ 10 9 cm from as many as 4000 segments. Fourth, the length ratios L/ increase with decreasing resolution, making the path lengths of the particles following the self-similar turbulent field lines down to scales ¼ (E ) dependent on their energy E. Unscattered particles of lower energies have longer path lengths. This dependence is relatively weak, however, especially in the slow SW, where the turbulence spectra steepen at relatively low frequency already. Even in the fast SW, the difference in the path lengths of particles with gyroradii cm (e.g., low pitch-angle, 10 MeV protons at 1 AU) and cm (e.g., 1 GeV protons at 1 AU) is only of the order of 7% or 8% close to 1 AU and 9% or 10% close to 0.3 AU. More comments on the shape of the curves in Figure 1 and the differences between SW conditions follow in xx 3 and 5. The histograms of the ratios L/ at resolution 10 9 cm are presented in Figure. They show again the spread in the values of L/ for our sets of 50 independent field lines. All 50 field lines of a given SW condition are generated from the same magnetic power spectrum. They are independent because they are computed from independent realiations of the turbulence with independent phases. They represent segments of magnetic field lines in independent SW streams. The histograms of Figure reflect the variability of the field-line lengths from one SW stream to another, not including possible variations in the turbulence spectra themselves. The spectra used in our computations were obtained by Fourier transform with a single time window of several days at all frequencies. The high spectral variability demonstrated in Figure 4 of R06d is not accounted for. The increase in the ratios L/ at short resolution scales may therefore be overestimated by a factor of roughly the inverse of the fraction of time during which the high-frequency short-time fluctuations are actually Fig.. Histograms of the ratios L/ of Fig. 1 at 10 9 cm. The lengths of wandering magnetic field lines vary from one SW stream to another, even for a fixed spectrum of turbulence. Variability is stronger in the slow SW where, unlike in the fast SW, wandering on a 0.1 AU (10 1 cm) scale can produce as much lengthening as irregularities on the shorter scales. close to the high-frequency long-time fluctuations used in our computation, perhaps as high as a factor of or 3 in some streams. Since the increase of L/ with decreasing at short is moderate, however, especially in slow SWand at 1 AU, the effect of the short-time higher frequency spectral variability should not significantly alter the conclusions of this paper. Also, we chose periods of very quiet SW turbulence, where the highfrequency variability may be reduced.

4 1496 RAGOT Vol. 653 Finally, while the peaks of the histograms in Figure give most likely correction factors for the field-line length of the order of 1.45 and 1.33 in slow and fast wind around 1 AU, and 1.16 and 1.3 around 0.3 AU, correction factors as small as 1. and as large as 1.8 are also possible in some streams at 1 AU. From the most likely values, one can estimate that most field lines in the SW from the Sun to 1 AU have lengths in excess of 1.5AUbothinquiet slow and quiet fast SW. This is a rough estimate. A more precise determination of the lengths of magnetic field lines between the Sun and 1 AU would require extending our study to a greater number of locations in the inner heliosphere. 3. AVERAGE LENGTHS: THEORETICAL ESTIMATE The length L ; of a field line of total projected length and resolution is the sum P N n¼1 l ;n of the lengths l ;n ¼ (x) ;n þ (y) ;n þ ()1= for each adjacent field-line segment n, considered as a straight segment, with a projected length on the main field. The total number of segments constituting the field line is N ¼ /(). The average length of such a field line is given by D L ; ¼ () þ (r) 1= E ; ðþ where (r) ¼ (x) þ (y) is distributed according to the distribution (see x 4.1 in R06e) Pf(r) ¼ X g¼ X e X = ; ð3þ with ¼ (r). Provided that / exceeds the correlation wavenumber k cor of the turbulent phases (R06c) and for a system sufficiently large in all directions, it can be shown that the cross-field displacements x and y are random numbers drawn from the Gaussian distribution f X (x) ¼ 1= 1 e (x) = (see R06c). Also, x and y are the real part and imaginary part of a complex random number with a phase uniformly distributed between 0 and. It follows that the modulus r of this complex random number is distributed according to equation (3). One can easily check that x ¼ r cos and y ¼ r sin with r distributed according to equation (3) and uniformly distributed between 0and have Gaussian distributions f X. Estimating the average in equation () with the probability distribution of equation (3), we obtain L ; ¼ 1 þ 1= 1 e () = ; ð4þ where : x! R 1= x 0 d e is the error function, also known as the probability integral. We compare in Figure 3 the theoretical estimate of equation (4) (dash-dotted lines) to the average of the 50 curves shown in each panel of Figure 1 (solid lines). From our estimate (eq. [4]) of the average length L ; of a wandering field line, it is clear that the dependence of the length L ; on the resolution scale is related to the magnetic field lines transport exponent defined by dðlogh(r) iþ/d ð log Þ or, more roughly, by (r) / () (Ragot 1999, 001; R06c). The moresupradiffusive the field lines, the weaker this dependence of L ; on is. If the value of the transport exponent approaches, the dependence on of equation (4) nearly vanishes. When the spectrum steepens toward higher wavenumbers, as happens in the SW, the transport exponent increases toward Fig. 3. Ratio L/ as a function of the resolution scale showing excellent agreement between simulation, theory, and in situ observations. Solid lines: Averages of the 50 curves obtained from simulated field lines in Fig. 1. Dash-dotted lines: Theoretical estimates of eq. (4). Crosses with error bars: Ratioscomputed from Helios data in the same streams where the turbulence spectra used for the simulations were computed. shorter length scales ( Ragot 1999, R06c), and the dependence of L ; becomes weaker toward the shorter. In Figures 1 and 3, this is most noticeable in the slow SW at 1 AU, where the turbulence has had the longest time to evolve. In the fast SW, the flatter spectra and resulting lower transport exponent still produce a relatively strong increase toward shorter resolutions of the ratio L/. Flatter spectra imply more short-scale irregularities

5 No., 006 TURBULENT LENGTHENING 1497 along the field lines, as illustrated in Figure 3 of R06c and R06e by examples of magnetic field lines computed in the same SW conditions as here. In the fast SW, most of the field-line turbulent lengthening is due to the short-scale irregularities. By contrast, in the slow SW, wanderingona0.1au10 1 cm scale can produce as much turbulent lengthening as the shorter scale irregularities, resulting in the stronger variability of the slow SW case in Figure 1 and its broader distributions in Figure. Although very close, simulation results and theory prediction candifferbyasmuchas1%y% in Figure 3. The discrepancy is due to the difficulty of deprojecting the one-dimensional spectra. The three-dimensional turbulence spectra used for the simulations do not give by projection onto the direction of k the exact same spectra as used as input in the theory. They can differ by a few percent in some range of wavenumbers. 4. RESULTS FROM IN SITU DATA ANALYSIS We now turn to the determination of the wandering field-line lengths from in situ measurements. We use Helios magnetic field data with 6 s time resolution. From the Helios data, we select one slow and one fast stream of quiet SWat each distance, 0.3 and 1 AU, near solar minimum. The selected periods are 1976 January 6Y9 (slow wind at 0.97 AU 1 AU), 1977 April 19Y (slow wind at 0.3 AU), 1976 January 1Y6 (fast wind at 0.98 AU 1 AU), and 1976 April 13Y18 (fast wind at 0.3 AU). They are the same as the ones used to compute the turbulence spectra of x. We determine the displacements of the magnetic field lines by integrating the magnetic field fluctuations measured on board the spacecraft. The variations (x;y) obtained by integration of the magnetic field data are not, strictly speaking, cross-field displacements of particular magnetic field lines. The point of measurement, instead of following a particular field line, remains along a fixed direction and the magnetic field lines, froen in the SW flow, are passing by the spacecraft (R06d). If the GQL approximation holds, however, as was shown to be the case up to a fraction of an AU in the quiet SW (R06b; R06d; R06e), the mean displacements obtained by evaluating the field perturbation along a real field line is the same as the one obtained by evaluating the field perturbation along the main field, whose angle with the local field can be large (R06e). As long as the quantity or physical dependence we are trying to determine involves averaging over a sufficiently large number of intervals ( 3 ), we can thus expect accurate results from the integration of the measured fluctuations. This should at the very least guarantee an accurate dependence of L on the resolution scale for all T. We compute the lengths of the field lines for resolution scales between 10 9 and 10 1 cm. At the largest resolution scale ¼ 10 1 cm, we have a doen subintervals in each of the slow SW cases and over 30 in the fast SW cases. This guarantees sufficient statistics up to the largest resolution in the fast SW. In the slow SW, we should also be measuring values of L close to their average on. The results of our data analysis are shown in Figure 3 (crosses with error bars) in comparison to the average lengths of the simulated field lines (solid lines) and the theoretical estimates of x 3(dash-dotted lines). The variations with of L/ are again almost parallel to those of the simulated field-line lengths. Although slightly different in the slow SW from the average values found for the simulated field lines and from the theory, the values of L/ computed here from the Helios data are, given the distributions of Figure, very close to the average. Because each of the data intervals selected for analysis is 4Y5 timeslongerthan one simulated field line, the results from the in situ analysis are much closer to the average values than would be expected from the distributions of Figure. Note that a small error in the determination of the angle between the radial direction and the direction of the main magnetic field can induce significant variations in the computed length ratios. We checked that an error of is already sufficient to produce a variation in the length ratios comparable to the quarter width of the distributions in Figure. Such an error is likely to occur, for instance, when the fluctuations of the SW speed are significant in the time interval chosen for analysis. 5. CONCLUSION Charged energetic and suprathermal particles in turbulent magnetied plasmas are forced to follow magnetic field-line irregularities down to length scales of the order of a few times their gyroradius. Their path length and travel time therefore depend on the length of the field lines resolved at these length scales of a few gyroradii. Given the field-line self-similar structure demonstrated by R06c, it is clear that the length of a magnetic field line should depend on the resolution scale. Such a dependence, although predicted (Burlaga 1995, R06c), had not been estimated so far. Applying a new computational method (R06a; R06c; R06e) to simulate magnetic field lines from the equivalent of turbulent modes, we computed under various SW conditions the lengths L of a number of turbulent magnetic field-line segments with projected lengths ¼ 4 ; 10 1 cm along the main field. We computed these lengths L for resolution scales varying from 10 9 to 10 1 cm. We deduced from the length L ¼ L the correction factor L/ or L /L smooth that should be applied to the length L smooth of a smooth approximation of a field segment, its projection along the main field, to obtain the real length L of the turbulent field segment seen at the resolution. Our results are confirmed both by theoretical estimates of the average length ratio (eq. [4]) and by the length ratios obtained from Helios data under similar SW conditions (see Fig. 3). All three methods are independent from each other. The common inputs into simulation and theoretical estimates are the onedimensional turbulence spectra, deduced from the in situ data and corrected in the slow SW for the cross-flow effect (see xx.1,.., and 4 in R06d or x. in R06e). These one-dimensional spectra provide, through the GQL calculations (R06c), the values of needed for the theoretical estimates (eq. [4]) of the average lengths. The same one-dimensional spectra are deprojected to obtain the three-dimensional turbulence spectra needed for the simulations by the SNRD method (R06c; R06e). In our simulations, we used isotropic spectra, knowing that the particular geometry of the three-dimensional wavevector distribution has no effect on the mean field-line displacements h(x;y) i 1/ as long as the nonlinear regime has not been reached. The fact that the nonlinear regime is not reached on the scales less than a few 0.1 AU and for the quiet turbulence levels considered in this paper has been demonstrated earlier (R06b; R06d; R06e). The correction factor L / can be very significant. It varies from about 1:16 0:06 and 1:3 0:03 in slow and fast SW at0.3auto1:45 0:5 and 1:33 0:06 in slow and fast SW at 1 AU (with a few more extreme values, see Figs. 1 and ). The variations from one SW stream to another SW stream with the same turbulence spectrum but different turbulent phases are mainly due to the large-scale wandering of the field lines. As the resolution scale becomes finer and more short-scale irregularities are accounted for in the computation of the length L,the curves of the correction factors in different realiations of the

6 1498 RAGOT turbulence increase nearly parallel to each other, due to the higher statistics on the shorter turbulent scales (see Fig. 1). The correction L / increases with distance from the Sun because the decrease in r 4 of the squared magnetic field intensity B 0 is faster than that of the magnetic turbulent energy, implying a stronger field-line wandering at larger distance from the Sun in the inner heliosphere. Also, the dependence of this correction on the resolution is stronger in the fast SW, where the turbulence has less time to evolve and the turbulence spectra are flatter, implying more short-scale irregularities along the field lines (see Fig. 3 of R06c; R06e). In the fast SW, the spread of L/ between field realiations is less than the variation of L/ with the resolution.onecan therefore be confident that the ratio L/L smooth on a longer segment of a magnetic field line will be little modified by the curvature of the spiral field and the possibility for a field line to cut short outside the spiral. The small-scale irregularities are the main cause for the enhancement of the length L, not the larger scale wandering of the field lines. Even if the large-scale wandering produced a straight field line, radial out to 1 AU, the length enhancement due to the short-scale irregularities would still guarantee that L > L spiral as soon as cm. In the slow SW, similar extreme conditions would give L > 1:1AUfor cm. Fortheseestimates,weassumeafieldlinemadeofthreesegments: one straight radial segment from the Sun out to 0.3 AU, a second radial segment from 0.3 to 0.8 AU to which we apply the minimal lengthening factor at 0.3 AU of 1.1 in slow SW (1.14infastSW)at10 10 cm resolution (see Fig. 1), and a third radial segment from 0.8 to 1 AU to which we apply the less than minimal lengthening factor 1.5 (1.3) found at 1 AU. The lowest limit estimated here should only rarely be observed, if ever, because the three-dimensionality of the field-line wandering makes a straight shortcut along the radial and in the ecliptic plane for a distance of 1 AU very unlikely. Pei et al. (006) find that the field lines can be shortened by the field-line wandering when the random walk causes the field line to go straight along the radial instead of along the Parker spiral. This illustrates well the effects of curvature and large-scale random walk. Due to the inclusion of shorter scale turbulence in our description of the field-line random walk, however, we find a different result. Namely, the short-scale irregularities guarantee field-line lengths in excess of 1.1 AU in slow SW and L spiral in fast SW for resolutions < cm, in spite of, as shown by Pei et al. (006), the large-scale wandering possibly shortening the large-resolution field lines by making them more radial in some cases. The minimal local lengthening factors are due to short-scale irregularities, not any random walk visible to the naked eye on the 1 AU scale. The field lines giving the minimal lengths look perfectly straight on the long scale. We estimate these lower limits to confirm the conclusions of Kahler & Ragot (006), namely, that most near-relativistic electron path lengths L c/v derived from the c/v onsetplotmethodfor impulsive solar electron events are shorter than the minimal length of a magnetic field line between the assumed electron injection at a few solar radii and their detection at 1 AU. In most cases, the length of the magnetic field lines traced by the electrons with gyroradii P afew10 7 cm actually exceeds 1.5 AU, making all but a few of the L c=v values obtained from the c/v plots shorter than the minimal path length. This result invalidates the c/v onset plot method, whether the unphysical path lengths derived from this method are due to experimental errors or incorrect underlying assumptions of the method. From the turbulent lengthening of the SW magnetic field lines demonstrated in this paper, one should expect significant travel delays of the impulsive solar energetic particles. These travel delays can be estimated to exceed, on the average, (1:5 AU L spiral;1au )/v, or about 5% of the travel time along a straight line of 1 AU in slow SW (closer to 50% in fast SW), with maxima as high as ( AU L spiral;1au )/v (or about 70%) and minima of the order of (1: AU L spiral;1au )/v (or a few percent). The variability from solar event to solar event of these travel delays should be high in slow SW. The existence of a travel delay due to the increased length of the wandering field lines in the SW is consistent with the conclusions of Cane (003) who, for 79 Electron, Proton, and Alpha Monitor (EPAM) near-relativistic solar electron events, correlated the electron delay times (times between the arrival of the first electrons and the detection of the type III radio burst at the Sun) with the times required for the associated type III radio bursts to drift to the lowest frequencies. Field-line wandering and travel delay was also inferred from the direct observation of some type III burst profiles (Reames & Stone 1986). While the travel time of energetic and suprathermal particles may be noticeably affected by the turbulent lengthening of the SW magnetic field lines, the variations with the particle energy of the path lengths may be too subtle (10% at most between particles with and cm gyroradii traveling from the Sun out to 1 AU) to have observable effects. We cannot exclude, however, that an enhancement of the turbulence in the medium- to high-frequency range (10 4 to 10 1 the ion gyrofrequency) could make the energy dependence of the path length much more significant. In general, moderate enhancements of the turbulence relative to the quiet SW levels considered in this paper could rapidly produce very strong effects through the increased length of the self-similar turbulent field lines. This material is based on work supported by the National Aeronautics and Space Administration under grant NNG04GB15G. The author thanks S. W. Kahler for useful discussions and the Air Force Research Laboratory for computer resources as a visitor of S. K. during the course of this research. Helios data were downloaded from the COHO Web site at REFERENCES Burlaga, L. F. 1995, Interplanetary Magnetohydrodynamics ( New York: Oxford Ragot, B. R. 001, Proc. of the 7th Int. Cosmic Ray Conf. ( Hamburg: Copernicus Gesellschaft), 393 Univ. Press) Cane, H. V. 003, ApJ, 598, a, ApJ, 64, 1163 (R06a) Haggerty, D. K., & Roelof, E. C. 00, ApJ, 579, b, ApJ, 644, 6 (R06b) Haggerty, D. K., Roelof, E. C., & Simnett, G. M. 003, Adv. Space Res., 3,. 006c, ApJ, 645, 1169 (R06c) d, ApJ, 647, 630 (R06d) Kahler, S. W., & Ragot, B. R. 006, ApJ, 646, e, ApJ, 651, 109 (R06e) Lintunen, J., & Vainio, R. 004, A&A, 40, 343 Reames, D. V., & Stone, R. G. 1986, ApJ, 308, 90 Pei, C., Jokipii, J. 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