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Frontiers in Astronomy and Space Sciences ·

Volume 13 - 2026 | https://doi.org/10.3389/fspas.2026.1916524

Estimating the length scales of magnetic-field-line wandering in the solar wind at 1 AU: ACE–earth magnetic-connection events

  • Joseph E. Borovsky1*corresponding author
  • John T. Steinberg2
  1. 1Space Science Institute, Boulder, CO, United States
  2. 2Independent Researcher, Los Alamos, NM, United States
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Abstract

Open heliospheric magnetic-field lines wander along the Parker-spiral direction, and the length scale of the wandering provides information about the parallel wavelength of solar wind turbulence and affects scatter-free transport of energetic charged particles in the solar wind. While single-spacecraft measurement of length scales along the mean magnetic-field directions (e.g. along the Parker spiral) has long been known to be challenging, occasions when magnetic-field lines connect a solar-wind spacecraft to the Earth’s bow shock enable an estimation of the length scales. A large set of Advanced Composition Explorer (ACE)–Earth magnetic-connection events, identified by the ACE detection of backstreaming strahl electrons from the Earth’s bow shock, is used to estimate the length scales associated with the wandering heliospheric magnetic field at 1 AU. Our estimates indicate that the local radius of curvature of the magnetic-field lines is ∼0.016 AU and that the global length scales of the field-line variations along the Parker-spiral direction are ∼0.072 AU. These estimations are necessarily coarse due to the simplification of the geometry of magnetic-field lines to circular arcs or sine curves, and we assume that backstreaming electrons from the Earth’s bow shock mainly come from the bow shock nose. Nevertheless, here, we provide a novel perspective to make use of single-spacecraft measurement of particles and fields to probe structures parallel to the mean heliospheric magnetic field. The global length scales estimated here agree with estimations of field-aligned mean-free-paths for cosmic ray scattering, with braiding lengths associated with granule/supergranule motion on the photosphere/corona and lengths associated with the wandering of the open flux at the Sun produced by interchange reconnection in the low corona.

Keywords

1 Introduction

The wandering and braiding of the solar-wind magnetic-field lines is important for understanding the magnetic structure of the heliosphere and its turbulence therein. Additionally, the wandering of the open solar-wind magnetic-field lines along the Parker-spiral direction is a factor for the scatter-free transport of energetic charged particles in the solar wind (Mazur et al., 2000; Chollet and Giacalone, 2011; Trenchi et al., 2013). The wandering is also a factor in the lengthening of magnetic-field lines (Reames and Stone, 1986; Kahler and Ragot, 2006; Ragot, 2006). Field-line random-walk theoretical calculations for the solar wind have been performed to study particle transport (Jokipii and Parker, 1969; Giacalone et al., 2000; Pommois et al., 2001), field-line transport (Pommois et al., 2002; Weinhorst and Shalchi, 2010a; Ragot, 2011), field-line lengths (Ragot, 2006), and structure evolution (Zimbardo et al., 2004; Kaghashvili et al., 2006). The wandering length scale compared with perpendicular length scales may provide information about the nature of anisotropy in turbulence, which complements observations, such as anisotropy of power spectral density (Horbury et al., 2008) and anisotropy in wavevector space (He et al., 2013; Yan et al., 2016), and may be useful for testing and comparing various turbulence models such as the Maltese cross or superposition model (Matthaeus et al., 1990; Ruffolo et al., 2008; Weinhorst and Shalchi, 2010b; Zank et al., 2020), the Goldreich–Sridhar model (Goldreich and Sridhar, 1995; Goldreich-Sridhar, 1997; Lithwick et al., 2007), or the Boldyrev model (Boldyrev, 2005; Boldyrev, 2006; Perez and Boldyrev, 2010). Furthermore, the comparison might provide information about intermittency in the turbulence (Kaghashvili et al., 2006; Zimbardo et al., 2008; Ragot, 2010; Pei et al., 2016).

Measuring the braiding length scale (wiggle wavelength) of solar-wind magnetic-field lines is problematic. In effect, one needs two-point measurements of a single field line. A problem with such two-spacecraft measurements is knowing when the two spacecraft are on the same magnetic-field line. In the present investigation, this difficulty is overcome by using magnetic-connection events between a single spacecraft (ACE) and the Earth’s bow shock. A clear signal of the magnetic connection is made by observing on ACE the magnetic-field-aligned energetic electron strahl from the Sun backscattering off the bow shock.

The solar-wind magnetic field exhibits fluctuations at all spatial scales and all time scales, as observed by a solar-wind spacecraft. The Fourier power spectral density of the time series of solar-wind magnetic-field measurements exhibits a power-law in frequency in an inertial range: converting that to a wavenumber k spectra using the Taylor frozen-in hypothesis yields a power-law wavenumber spectra in an inertial range from approximately 0.01 AU down to approximately 100 km (cf. Figure 1 of Goldstein and Roberts (1999) or Figure 25 of Bruno and Carbone (2013)). With k being the spatial wavenumber, at 1 AU, the statistical form of the magnetic spectrum is ∼k-1.56 in the fast solar wind (coronal-hole-origin plasma) and ∼k-1.70 in the slower streamer-belt solar wind (cf. Table 5 of Borovsky (2012) and see also Tessein et al. (2009) and Borovsky and Smith (2026)). Most of the magnetic Fourier power in the inertial range is contained in directional discontinuities (current sheets) in the solar wind (Siscoe et al., 1968; Sari and Ness, 1969; Borovsky, 2010a; Li et al., 2011; Borovsky and Podesta, 2015; Borovsky and Smith, 2026), leading to a depiction of the magnetic structure of the solar wind involving flux tubes (also called flux bundles or magnetic filaments) (e.g. McCracken and Ness, 1966; Michel, 1967; Bruno et al., 2001; Borovsky, 2008; Greco et al., 2009; Ruffolo et al., 2013), with the magnetic discontinuities forming the boundaries between flux tubes and with turbulence inside the tubes (Bruno et al., 2001; Borovsky, 2008; Qin and Li, 2008; Neugebauer and Giacalone, 2010). The magnetic-field lines in an individual flux tube are likely confined to the inside of that flux tube: some of the discontinuities are strahl-intensity boundaries indicating that field lines do not cross the discontinuity (Borovsky, 2020a), and some of the discontinuities at 1 AU are ion-composition boundaries indicating that they are fossils from the solar corona and that field lines do not cross the boundaries (Borovsky, 2020a). The flux tubes have a long-wavelength wiggle, and the magnetic-field lines inside the flux tubes exhibit this long-wavelength wiggle along with shorter-wavelength wiggles (cf. Figure 5 of Bruno et al, 2001). Certainly, owing to the large fraction of the Fourier power residing in the coherent directional discontinuities, the fluctuations at all spatial scales of the magnetic-field lines of the solar wind are not randomly phased variations (Borovsky and Mina, 2020). To estimate the wandering length scales of magnetic-field lines in the present study, an oversimplified picture of the solar wind will be used wherein only one spatial scale is analyzed and the multi-scale wiggling of the field lines is ignored.

Many studies indicate that gradients in the solar-wind plasma are much stronger in the direction perpendicular to the local magnetic-field direction than in the parallel direction (e.g., Weimer et al., 2003; Knetter et al., 2003, 2004; Weimer and King, 2008). Consistent with this, the local flux-tube texture of the solar-wind plasma is drawn with tubes that have diameters much smaller than their bending length scales (e.g. Figure 6 of Bartley et al., 1966; Figure 3 of McCracken and Ness, 1966; Figure 5 of Bruno et al., 2001; Figure 1 of Borovsky, 2008 Figure 22 of Borovsky, 2010b; and Figure 101 of Bruno and Carbone, 2013). When spatial scales of fluctuations along the mean field direction B o are inferred from single-spacecraft measurements of advecting solar wind, it is suspected that the results are dominated by the tipped current sheets of the flux tube walls and that the actual wandering length scales along B o may be much longer than indicated by those single-spacecraft time-series cuts through the advected solar-wind flux tubes.

At times, observation of backstreaming suprathermal electrons can indicate that a solar-wind spacecraft is magnetically connected to Earth’s bow shock (Feldman et al., 1982; Stansberry et al., 1988). Sunward-traveling field-aligned energetic-ion events also indicate these magnetic-connection events (Anderson, 1981; Sanderson et al., 1981; Haggerty et al., 2000), and often, the ion events coincide with backscattering electron events (Scholer et al., 1981; Mitchell et al., 1983; Haggerty et al., 2000). The origin of the sunward ions is thought to be either the energization of ions at the bow shock or the escape of energetic ions from the Earth’s magnetosphere (Sarris et al., 1978, Sarris et al., 1987; Scholer et al., 1981). Such magnetic connections provide unique opportunities to study the behavior of magnetic-field lines in the solar wind. ISEE-3 observations first established that the backstreaming electrons (Scholer et al., 1980; Feldman et al., 1982) and ions (Anderson, 1981; Mitchell et al., 1983) can be observed as far as 1.5 × 106 km upstream of Earth at the L1 region. Since anti-sunward field-aligned electron beams, also known as strahl electrons, are almost always present, the backstreaming field-aligned electrons from Earth’s bow shock gives rise to counterstreaming electron beams.

For the ACE spacecraft in a halo orbit around the L1 Lagrangian point (see Figure 1), this connection can be seen in the ACE suprathermal electron measurements. This backstreaming process is sketched in Figure 2, where a magnetic-field line (black) from the Sun intersects both ACE (green) and the Earth’s bow shock (blue). The flow of field-aligned strahl electrons along the field line is depicted in red. Outward-streaming strahl electrons from the Sun are depicted at the bottom of Figure 2, and backscattered electrons from the bow shock are depicted near the top of the figure. Counterstreaming strahl electrons (primary electrons from the Sun and backscattered electrons from the shock) are detected at ACE. Figure 3 shows an example of counter-streaming in electron pitch angle distributions measured by the ACE SWEPAM instrument (McComas et al., 1998) at 272-eV electron energy. Analysis of the strahl at 272 eV is a common procedure for the electron instrument on ACE (e.g. Steinberg et al., 2005; Skoug et al., 2006; Borovsky, 2018a, Borovsky, 2021; Borovsky and Runov, 2022). In this example of pitch-angle distributions, strahl electrons are continuously present at a 180° pitch angle in a toward-the-Sun magnetic-field sector, i.e., they are moving outward from the Sun anti-parallel to the local magnetic field, while Sunward backstreaming electrons appear in sporadic bursts at 0° pitch angle, which is opposite to the direction of Sun-generated Strahl electrons. The outward electron strahl is a beam of magnetic-field-aligned electrons originating from the solar corona that propagate away from the Sun (Feldman et al., 1976; Pilipp et al., 1987; Maksimovic et al., 2005; Boldyrev and Horaites, 2019) and is almost always seen in the solar wind at 1 AU (Borovsky, 2021).

In the GSE X–Y plane, the green points indicate the locations of ACE around L1 at the times when magnetic connection to the bow shock occurred. The approximate shape of the bow shock is indicated in blue using the Fairfield–Greenstadt model (Fairfield, 1971; Greenstadt et al., 1990).
Figure 1. In the GSE X–Y plane, the green points indicate the locations of ACE around L1 at the times when magnetic connection to the bow shock occurred. The approximate shape of the bow shock is indicated in blue using the Fairfield–Greenstadt model (Fairfield, 1971; Greenstadt et al., 1990).
A sketch of the backscattering of the electron strahl from the Sun off of the Earth’s bow shock creating a bi-directional (Sunward and anti-Sunward) double strahl at the ACE spacecraft.
Figure 2. A sketch of the backscattering of the electron strahl from the Sun off of the Earth’s bow shock creating a bi-directional (Sunward and anti-Sunward) double strahl at the ACE spacecraft.
Example of an ACE electron pitch angle distribution plot for 30 October 1998. The color bar is in s<sup>3</sup>cm<sup>-3</sup> units, and the scale is logarithmic.
Figure 3. Example of an ACE electron pitch angle distribution plot for 30 October 1998. The color bar is in s3cm-3 units, and the scale is logarithmic.

The occurrence of Sunward counterstreaming electrons does not exclusively imply magnetic connections to Earth’s bow shock. It is well-known that counterstreaming often appears upstream and downstream of interplanetary shocks (Bame et al., 1981; Zwickl et al., 1983), in coronal mass ejections that have two magnetic footpoints on the Sun (Gosling et al., 1987), and downstream of corotating interaction regions (Gosling et al., 1993; Steinberg et al., 2005). In addition, counterstreaming can appear due to the depletion of halo particles around a 90° pitch angle (Gosling et al., 2001, Gosling et al., 2002; Skoug et al., 2006). The authors also observe that counterstreaming electrons are commonly present near the sector reversal regions where interplanetary magnetic field reverses its polarity (the “strahl-confusion zones” of Xu and Borovsky, 2015): this could also represent plasma with two magnetic footpoints on the Sun.

This study identified counterstreaming electrons not associated with any of the plasma conditions listed above, wherein the counterstreaming is attributed to terrestrial origin. A large set of ACE–Earth magnetic connection events is collected and used to uniquely estimate (1) the local radius of curvature of solar-wind magnetic-field lines and (2) the more global wandering length scale (i.e. braiding length scale) along the Parker spiral.

There are some analogies between this study utilizing these ACE–Earth magnetic connections and proposals to inject artificial field-aligned electron beams from spacecraft in the Earth’s nightside magnetosphere, wherein, like the strahl, the electron beam travels 5 to 10 RE along the magnetospheric magnetic field to hit the Earth’s ionosphere and illuminate the atmosphere, thereby establishing unambiguous accurate magnetic connections between the magnetosphere and its physical processes and the ionosphere and its physical processes (Borovsky et al., 2020). In the experiment, some of the energetic beam electrons from the spacecraft will backscatter off the Earth’s atmosphere (Banks et al., 1974), but it is unlikely that the backscattered electrons will directly hit the beam spacecraft owing to strong gradient-and-curvature drifts in the magnetosphere that will shift the electrons off of the spacecraft’s magnetic-field line (Borovsky et al., 2022).

This manuscript is organized as follows. In Section 2, we discuss the ACE–Earth-connection event selection. Section 3 presents estimates of the radius of curvature of field lines that connect from ACE at L1 to the Earth’s bow shock. Section 4 presents estimates of the wandering wavelengths of magnetic-field lines along the Parker spiral using sine-wave analysis. Section 5 contains discussions about the obtained measures of the field lines with expectations and about the backscattering from the bow shock.

2 Event selection

The event selections were made by visually inspecting counterstreaming electrons observed by ACE in 16 months of measurements, namely, May through November of 1998, October through December of 2006, and January through June of 2007. To remove counterstreaming associated with other sources, broad time intervals when ACE observed interplanetary coronal mass ejections (Jian et al., 2006; Richardson and Cane, 2010), CIRs (Jian et al., 2011), and sector reversals (sector boundaries) (Crooker et al., 1998) were excluded from the study and not inspected for events, and 12-h periods before entering and after leaving the above plasma regions were excluded. In addition, 24-h periods before and after the interplanetary shocks in the CFA Interplanetary Shock Database (www.cfa.harvard.edu/shocks/) were excluded from the study. To pick out counterstreaming associated with backstreaming electrons from Earth’s bow shock, the following three selection criteria were applied:

  • Strahl electrons are present continuously 12 h before and after events.
  • Enhancement of electron flux at 0° or 180° by backscattering electrons is clearly identifiable visually in ACE 272-eV electron pitch angle distribution plots.
  • Similar enhancements are seen in ACE electron pitch angle distributions at energies between 100 and 500 eV.

The time resolution of the ACE SWEPAM electron instrument is 64 s. By defining an “event” to be a 64-s interval in the ACE measurements where the proper counterstreaming electrons appear, 12,625 events were collected that also had magnetic-field and plasma-velocity data. This covers approximately 0.02% of all of the 64-s intervals that were inspected. Almost all of the 64-s events were part of longer events that were typically less than 10 min in duration. Note that the events are not screened for strahl-intensity discontinuities or for magnetic-field directional discontinuities.

It is well-known that when spacecraft–Earth connection events happen, Ψ, the angle between instantaneous vector B and the spacecraft–Earth line is often small (Feldman et al., 1982; Stansberry et al., 1988). Figure 4 shows the relative frequency for both ambient solar wind (blue) and ACE–Earth connection events (red). The magnetic-field direction was measured at ACE using the ACE Magnetic Fields Experiment (Smith et al., 1998). All ACE observations were used from all of 1998, 2006, and 2007 to obtain the statistics for ambient solar wind, and the magnetic field was averaged over 64 s to reduce fluctuations. The ambient solar wind has small probability for small Ψ angles and large probability for large oblique angles. However, 70% of samples in connection events have Ψ < 30°. The statistics of the Ψ values of our connection events are consistent with the earlier study of connection events with the ISEE-3 spacecraft at L1 (Stansberry et al., 1988). It is worthwhile pointing out that Ψ in the ambient solar wind peaks at approximately 60ο, which can be explained by a combined effect of turbulently random fluctuations of B and a tendency to approach Parker-spiral directions (Stansberry et al., 1988; Borovsky, 2010b).

Relative occurrence frequency of the angle between instantaneous magnetic field direction (64s average) and Earth–ACE line for both the ambient solar wind (blue) and ACE–Earth connection events (red).
Figure 4. Relative occurrence frequency of the angle between instantaneous magnetic field direction (64s average) and Earth–ACE line for both the ambient solar wind (blue) and ACE–Earth connection events (red).

The Xu and Borovsky (2015) algebraic scheme is used to temporally categorize the solar-wind plasma that passes the Earth into four major types based on their origins from solar-surface features: (1) coronal-hole-origin plasma, (2) streamer-belt-origin plasma, (3) sector-reversal-region plasma, and (4) ejecta. The scheme makes the categorization based on the proton specific entropy, the Alfven speed, and the velocity-to-proton-temperature ratio of the solar-wind plasma at Earth, the properties of which are determined from the OMNI solar-wind plasma data set (King and Papitashvili, 2005). Of the total connection times between ACE and the Earth, 52.9% occurred when ACE was in streamer-belt-origin plasma, 38.95% occurred when ACE was in coronal-hole-origin plasma (high-speed streams), 6.6% occurred when ACE was in sector-reversal-region plasma, and 1.5% occurred when ACE was in ejecta. Examining the 16 months of data used to collect the events, in that data set, the occurrence rate for the four types of solar-wind plasma were 38.5%, 26.8%, 22.5%, and 12.2%, respectively. There is a relative rarity of connection events in ejecta plasma and in sector-reversal-region plasma: the reasons are probably that (1) ejecta often has a counterstreaming Sunward strahl and backscattering may not be noticeable in the presence of this counterstreaming strahl (or may even produce a reduction in the Sunward strahl) and (2) the main strahl is often very weak or absent in sector-reversal-region plasma (cf. Figure 1 of Borovsky, 2021). As a result of the events mostly occurring in coronal-hole and streamer-belt plasma, the magnetic-field perturbations that create the connections between ACE and the Earth are dominantly in Alfvénic magnetic-field fluctuations passing the ACE–Earth system (cf. Figure 8 and Figure 11b of Borovsky et al., 2019, and cf. Borovsky, 2020b). Non-Alfvénic solar-wind plasma also has a magnetic-field structure similar to that in the Alfvenic solar wind (Borovsky, 2008).

3 Estimation of local radius of curvature of field lines

To estimate the local curvature of the interplanetary magnetic field in each ACE–Earth connection event, the magnetic-field-line shape is approximated as a circular arc that connects from the ACE spacecraft position to the Earth, with the tangent to the circle at ACE being the measured magnetic-field direction vector at ACE. The circle lies in the plane defined by the two vectors d (the connection from ACE to the Earth) and B (the magnetic-field direction measured at ACE using the ACE Magnetic Fields Experiment; Smith et al., 1998. 64-s averages of the measured field are used to obtain the direction of B.). With two points (ACE and Earth) on the circle and the tangent direction at one of the points (ACE), the circle can be derived uniquely, and the radius of curvature (the radius of the circle) can be uniquely obtained. If α = arccos(B⋅d/(|B| |d|)) is the angle between B and d, then the radius of the circle is given by Equation 1

r = d / 2 ⁡ sin α . (1)

Admittedly, there is no a priori reason that field lines should take circular shapes, but a circle connecting ACE to Earth yields the weakest estimate of the local radius of curvature of the magnetic-field line that ACE sees. In fact, wandering spaghetti-shaped field lines are probably a better model for the geometric complexity of the heliospheric magnetic field, but the circle solution is obtainable. Without further assumptions, the information a magnetic connection event offers is the vector distance d between two points (ACE and Earth) threaded by the same field line and the vector direction of B at one of those points (ACE). In this and the next section, the source region of the observed backstreaming electrons (i.e., the location where the field line intersects the Earth’s bow shock) will be simplified to a single point target. This simplification is discussed in detail in Section 5.1. Without knowing the exact location on the terrestrial bow shock that connects with the spacecraft, it will be assumed that the location can be approximated by the center of the Earth for simplicity and clarity. It is shown in Section 5.1 that our models are not very sensitive to the exact choice of this location, providing it is not in the bow shock tail region. However, the bow shock in the magnetotail region has a weak compression ratio (cf. Section 5.1) and is unlikely to reflect out-from-the-Sun propagating strahl electrons.

A sketch of a circle fit is shown in Figure 5. A hypothetical magnetic-field line (red) that connects ACE (green) to the Earth’s bow shock (blue) is shown, and the circle that is fit to the locations of ACE and the bow shock, accounting for the direction of the magnetic field at ACE, is shown as the purple dashed curve. Notably, the radius of curvature of the circle is an approximation to the radius of curvature of the actual magnetic-field line.

Sketch of a circle fit to the magnetic-field information at ACE and the bow shock during a magnetic-connection event.
Figure 5. Sketch of a circle fit to the magnetic-field information at ACE and the bow shock during a magnetic-connection event.

With the above assumptions, circles are fit to all connection events to obtain the local curvature radii. Figure 6 shows the frequency histogram of the fitted radius. As can be seen, there is a long tail to the occurrence distribution associated with measured magnetic fields at ACE that point close to the Earth, yielding estimates of the radius of curvature that are very large. For the 12,625 ACE–Earth magnetic connection events, the median and mean values of the radius of curvature of the field lines are 370 RE (0.016 AU) and 513 RE (0.022 AU), respectively.

Occurrence histogram of the measured radius of curvature assuming that the magnetic-field lines connecting ACE and Earth have the form of circular arcs.
Figure 6. Occurrence histogram of the measured radius of curvature assuming that the magnetic-field lines connecting ACE and Earth have the form of circular arcs.

4 Wandering length scale along the Parker spiral

To estimate the large-scale global wandering behavior of the interplanetary magnetic-field lines at 1 AU, a mean field is taken that is aligned with the Parker spiral but with a sinusoidally shaped perturbation that is propagating away from the Sun in the Parker-spiral direction while simultaneously being advected radially outward from the Sun by the solar-wind proton flow (Borovsky, 2020b; Nemecek et al., 2020). There is no reason to believe that the magnetic-field wandering would take the form of a periodic sine wave. Rather, as discussed in Section 5.2, if the field-line displacement were to be caused by an interchange reconnection in the solar corona, then a hyperbolic-tangent-like shift in the field might be a more appropriate approximation. However, here, for simplicity, a sine-wave description is exploited. The amplitude of the sinusoidal structures must be sufficiently large so that both ACE and Earth can fall on the same field line, leading to a magnetic connection. The wavelengths of the sine curves can be determined from the ACE-measured magnetic-field direction and the ACE spacecraft location relative to Earth.

In the vicinity of Earth (including the L1 Lagrangian-point region), the Parker-spiral direction is given by p ≡ −x + (405/|vx|)y, where x and y are the unit basis vectors for the GSE coordinate system in the X and Y directions and vx is the solar-wind proton radial velocity in km/s, as measured by ACE SWEPAM (McComas et al., 1998). The corotation speed of heliospheric structure from rotating the Sun is 405 km/s; 405 km/s = 2π (1 AU)/(27 days). The displacement vector from ACE to Earth is denoted as d. To describe the sine curve conveniently, the following coordinate system is introduced. Let n ≡ p × d be the normal to the plane defined by p and d. We define the unit vectors n* = n/|n|, p* = p/|p|, and q* = n × p. Then, the unit vectors p*, q*, and n* together form a right-hand coordinate system. It is easily seen in this coordinate system that a sine curve passing through ACE and Earth, lying in a plane defined by p and d, and having its axis parallel to the Parker-spiral direction p, would take the form of Equation 2:where p and q denote displacement along p* and q*. In equation 2, the variable A denotes the amplitude of the sine wave and should be related to the wavelength λ. δ and C give the phase shift and the offset, respectively, of the sine curve. Therefore, there are four free parameters in the sine-curve: A, λ, δ, and C. The independently known or measured parameters are d, B, and p. In the context of this simple model, the known independent parameters are reduced to three scalars, namely, (1) d = |d|, (2) α = arccos(B⋅p/|B||p|), the angle between B projected into the p–d plane and p, and (3) γ = arccos(d⋅p/|d||p|), the angle between d and p. In order to solve for a length scale (λ), one degree of freedom is eliminated by fixing the amplitude as follows: A = 0.6 λ. The choice A = 0.6 λ is taken from statistics of the distribution of angles of measured magnetic-field vectors about the Parker-spiral direction at 1 AU (Borovsky, 2010b): the standard deviation of the angles is approximately 45° in both the north–south and east–west directions. If the field lines were planar sine waves aligned along the Parker-spiral direction and the planes of the sine waves were randomly oriented about the Parker-spiral direction, then the statistics of sine-wave field lines with A = 0.6 λ would have 45° spreads about the Parker-spiral direction in the east–west and north–south directions. This choice of A makes the amplitude of the wave comparable to its wavelength and effectively constrains |α| and |γ| to both be ≤ arctan(2) ∼ 60°. Fitting a sine curve is obtained by solving Equations 3a–c: where d = |d|. Equation 3a is obtained from placing the ACE spacecraft at the origin of the p–q–n coordinate system at p = 0 and q = 0 in equation 2, which assures that ACE lies on the sine curve. Equation 3b enforces the requirement that the projection of the magnetic-field vector at ACE is tangent to the sinusoidal projection of the field line by setting dp/dq = tan(α) at p = 0 and q = 0. Equation 3c utilizes the fact that the Earth is also on the sinusoidal field line and that its distance from ACE is d. Knowing α, γ, and d, these three expressions can be solved for the wavelength λ of the sine curve, providing that both |α| and |γ| are < arctan(2), which is consistent with the choice A = 0.6 λ.

q = A ⁡ sin 2 π p / λ + δ + C , (2)

0 = A ⁡ sin δ + C , (3a)

tan α = 2 π A / λ cos δ , (3b)

d ⁡ sin γ = A ⁡ sin 2 π d ⁡ cos γ / λ + C , (3c)

In Figure 7, an example sine wave fit to the ACE–bow-shock configuration is shown in red, with the sine wave aligned along the Parker-spiral direction (black dashed line). ACE is shown in green, and the bow shock is shown in blue.

Sketch of a sine-wave fit to the ACE magnetic connection to the Earth’s bow shock.
Figure 7. Sketch of a sine-wave fit to the ACE magnetic connection to the Earth’s bow shock.

In Figure 8, the histogram of λ values is plotted for the assumption A = 0.6 λ: 998 solutions are attained. If A > 0.6 λ is taken, more solutions are obtained, but the amplitude of the field fluctuations about the Parker-spiral direction would not agree with the statistics of Borovsky (2010a). Two points can be made about the amplitude choice. (1) The ACE–Earth connection events are probably outside of the average statistics of the magnetic-field direction (cf. Figure 4), and (2) the statistics of the λ values obtained are not very sensitive to the value of A/λ used in the calculations. The histogram of Figure 8 is skewed with a long tail. The median value of the sine wavelength λ is 1,700 RE (0.072 AU). The mean (2,300 RE = 0.097 AU) is larger than the median, and it is biased toward the tail of the distribution. It is suggested, here, that the median value of λ better describes the bulk statistics.

Occurrence histogram of fitted magnetic-field-line wavelengths assuming that the magnetic field lines are large sinusoidal structures in the solar wind connecting ACE with the Earth oriented along the Parker spiral.
Figure 8. Occurrence histogram of fitted magnetic-field-line wavelengths assuming that the magnetic field lines are large sinusoidal structures in the solar wind connecting ACE with the Earth oriented along the Parker spiral.

The sine-wavelength results are in agreement with the radius-of-curvature results of Section 3. The curvature κ(p) of the sine wave given by expression (2) q = Asin(2πp/λ) is κ(p) = [A2 (4π2/λ2) |sin (2πp/λ)|]/[1 + A (4π2/λ2) cos2 (2πp/λ)]3/2: if A = 0.6 λ (as taken in this sine calculation), then the median value of κ(p) over a period of the sine wave is κ = 0.259 λ, which yields a radius of curvature R = 1/κ = 0.39 λ. With the median value of R being 370 RE in Section 3 and the median value of the sine λ being 1,700 RE in this section, the ratio of R/λ is 0.22, which is less than a factor of 2 from the expected value.

5 Discussion

In this section, we (1) discuss the spatial extent of the Earth’s bow shock and where on the shock the observed backstreaming electrons may originate and (2) compare the estimated length scales from this study with expected length scales (summarized in Table 1).

TABLE 1. Comparison of estimated magnetic length scales.

Length scalekmREAU
Median calculated radius of curvature2.4 × 1063700.016
Median sinusoidal wandering length scale1.1 × 1071,7000.072
Cosmic-ray scattering length scale1.2 × 107–5 × 1071,900–7,0000.08–0.3
Magnetic-field correlation length1 × 1061600.007
Solar footpoint braiding length scale2 × 106–6 × 107300–9,0000.013–0.38

5.1 Strahl backscattering from the bow shock

To estimate the local radii of curvature and the wandering length scales, the angle between the measured magnetic field at ACE and the ACE–Earth line was used with the assumption that the magnetic-field line would bend to connect to the Earth at a distance d from ACE. These calculations did not account for the fact that the bow shock is not a localized target for the field line. Examining the sketch in Figure 1, the Fairfield–Greenstadt (Fairfield, 1971; Greenstadt et al., 1990) nominal bow shock has a radius of approximately 25 RE at X = 0 and a radius of approximately 120 RE at X = −250 RE. As viewed from L1 (approximately 235 RE), the X = 0 disk of the bow shock is approximately 6ο in radius and the X = −250 RE the disc of the bow shock is approximately 14ο in radius. (Notably, at low Mach number, the bow shock is broader but weaker.) Hence, an ACE magnetic-field line with < 14ο hits the bow shock; therefore, estimates of the radii of curvature in Section 3 and estimates of the wandering length scales in Section 4 may be underestimates.

An important question to ask is: Where on the global bow shock do the backstreaming-event electrons originate? One might speculate that regions of the bow shock that have higher compression ratios might be more likely to produce reflected electrons, with the compression ratio being an indicator of the amount of energy dissipated at the shock. In Figure 9, the measured density compression ratio of the bow shock in a global MHD simulation of the solar-wind-driven magnetosphere of Earth is plotted as a function of the GSE X location on the shock. The simulation (run number “Joseph Borovsky 021312 1″ at the CCMC) was run using the LFM (CMIT LFM-MIX) simulation code (Lyon et al., 2004; Merkin and Lyon, 2010) at the Community Coordinated Modeling Center (Rastatter et al., 2012). The code was run with the solar-wind flow vector coming in at an angle of 9.5ο the center of the cylindrical simulation grid where the gridpoint resolution is the highest. For the solar-wind magnetosonic Mach number of Mms = 6.6 in the simulation, the density compression ratio (the plasma number density downstream of the shock divided by the plasma number density upstream) of the bow shock at the nose (X = +14.5 RE) is 3.95. As seen in Figure 9, the bow-shock compression ratio drops strongly, going downtail from the dawn–dusk terminator (X = 0 RE): at X = −75 RE, the compression ratio is 2.0, and at X = −200 RE, the ratio is only 1.6. Certainly, the production of foreshock energetic electrons will be reduced for magnetic-field lines contacting the bow shock much beyond the Earth’s terminator. However, without a survey of the production of foreshock electrons from the actual bow shock, the spatial extent of the electron production region will remain unknown.

Density compression ratio of the Earth’s high-Mach-number bow shock as a function of the GSE X coordinate, as determined using a global MHD simulation. The simulation is available at the CCMC as run “Joseph Borovsky 021312 1.”
Figure 9. Density compression ratio of the Earth’s high-Mach-number bow shock as a function of the GSE X coordinate, as determined using a global MHD simulation. The simulation is available at the CCMC as run “Joseph Borovsky 021312 1.”

During an interval of connection to the bow shock, the magnetic-field direction at ACE will undergo temporal variations that will cause variations in the calculated values of the field-line curvature and variations in the calculated values of the field-line wandering length scale. In the picture of solar-wind flux tubes confining magnetic-field lines (cf. Figure 5 of Bruno et al., 2001 or Figure 101 of Bruno and Carbone, 2013), connection intervals are produced by the connection of magnetic flux tubes between the bow shock and ACE. The individual field lines within the flux tubes have variations in their vector directions, but they are still connected if the flux tube is connected. In the Bruno et al. (2001) picture, the major temporal variations in the direction of the magnetic field are owed to the variations in the orientations of the flux tubes, and minor temporal variations in the field direction are associated with the individual field lines within the flux tubes; a recent study by Borovsky and Smith (2026) agrees with this. If the flux-tube diameters were small, then the temporal boundaries of bow-shock-connection intervals at ACE would be associated with the tubes becoming connected and disconnected from either the bow shock or from ACE; the picture is more complicated by the fact that flux-tube diameters are comparable to the size of the Earth’s magnetosphere (e.g. Borovsky, 2008 or Figure 2 of Borovsky, 2018b). The statistics of connection intervals and the substructure within connection intervals is a topic for future study.

5.2 Comparison with expected length scales

The novel technique applied in this study is limited to giving information about magnetic-field line curvature measured on the scale size of the L1–Earth distance. The wandering length scales along the Parker-spiral found herein correspond to magnetic structures that could be categorized as having spatial scales in the “energy-containing range” of the solar-wind magnetic power spectral density, which is Fourier power at lower frequencies than the low-frequency breakpoint in the magnetic Fourier spectrum (cf. Figure 1 of Goldstein et al., 1995, or Figure 25 of Bruno and Carbone, 2013). At 1 AU, the low-frequency breakpoint timescales (1/f, where f is the frequency in Hz of the Fourier breakpoint) of the solar-wind magnetic spectrum vary considerably in the literature. In the fast wind, breakpoint timescales of 630 s (Figure 3 of Horbury et al., 1996) and 1,300 s (Figure 13 of Bruno and Carbone, 2013) have been reported. Advecting these timescales with solar-wind speeds of 600 km/s yields scale lengths of the Fourier breakpoint of 4 × 105–8 × 105 km, which is 60 RE–120 RE. The difficulty in finding a low-frequency breakpoint in the non-Alfvénic slow wind has been investigated by Bruno et al. (2019) and Dorseth et al. (2024). Concerning the low-frequency magnetic power spectral density of the solar wind, it must be realized that small-scale structures (∼15 min) in the magnetic-field time series at 1 AU, which are comprised of magnetic-field directional discontinuities and the stepped field change across them, add Fourier power at all frequencies, including into the “energy containing” range (Siscoe et al., 1968; Borovsky, 2010a). Hence, not all of the Fourier power in the energy-containing range of wavenumbers in the magnetic power spectral density comes from large-scale structures. Two-point correlations in the slow wind yield 80 RE (5 × 105 km) (Chang and Nishida, 1973). Correlations in general yield 1 × 106 km (160 RE) at 1 AU (Figures 2, 3b of Ruiz et al., 2014). These correlation lengths undoubtedly correspond to the crossing of tilted flux tubes in the flux-tube spaghetti with typical flux-tube diameters of ∼70 RE (Borovsky, 2008). The median value of the wandering length scales determined via the sine fits in Section 4 is 1,700 RE (0.072 AU): these wandering length scales correspond to sinusoidal advected timescales of fractions of a day and additionally in the energy-containing scales of the plasma. This is noted in Table 1.

The sinusoidal field-line-wandering length scales along the Parker spiral obtained in Section 4 (0.072 AU–0.097 AU, cf. Figure 8) are on the order of the well-accepted “Palmer consensus” values for the field-aligned mean-free-path for cosmic-ray scattering, which are 0.08 AU–0.3 AU at 1 AU (Palmer, 1982; Kunow et al., 1991; Droege et al., 1993). This is noted in Table 1.

In a simplistic picture, footpoint motions on the Sun will lead to braiding of open magnetic-field lines in the heliosphere. A rough estimate of the length scales of this braiding along the Parker-spiral direction can be obtained by estimating the timescales for footpoint motions at the Sun and then advecting those timescales outward along open field lines at the Alfvén speed plus solar-wind speed (c.f. Jokipii and Parker, 1969; Giacalone, 1999). Advecting past an observer in the solar wind, the temporal frequencies of the motions transverse to the mean field will be preserved; for the high-Mach-number solar wind at 1 AU, the Alfvén speed can be neglected compared with the solar-wind speed in estimating the length scale associated with the footpoint motions. For characteristic timescales for the convective motions of granules and supergranules in the photosphere, the Leighton (1964) values 400 s for granules and 7 × 104 s for supergranules (which is approximately the lifetime of a supergranule (Wang and Zirin, 1989; Roudier et al., 2014)) could be used. Advecting these timescales at a solar-wind speed of 400 km/s yields parallel scale lengths of 1.6 × 105 km (25 RE ∼ 0.0011 AU) and 2.8 × 107 km (0.19 AU) for granule and supergranule braiding of the solar-wind magnetic-field lines. The granular value is certainly lower than the range of values obtained in Section 4, but the supergranule braiding value is in the range. A more realistic picture of the solar braiding of solar-wind magnetic-field lines is obtained from the interchange reconnection of open flux tubes with closed flux tubes in the corona or upper chromosphere (Fisk and Schwadron, 2001; Crooker et al., 2002; Wang and Sheeley, 2004). Estimates for the timescale for the reconnection-enabled wandering of the open flux at the Sun can be obtained from flux-replacement (flux-recycling) timescale estimates in the chromosphere: these estimates vary from 1.4 h–3 h (Close et al., 2004, 2005) to ∼40 h (Schrijver et al., 1997; Schrijver, 2007). Advection outward at 400 km/s of the 1.4 h, 3 h, and 40 h footpoint timescales yields 2.0 × 106 km, 4.3 × 106 km, and 5.8 × 107 km for the length scales along the Parker spiral, which are 0.013 AU, 0.029 AU, and 0.38 AU, respectively. These expected braiding length scales are in the range of the values found in Section 4.

This footpoint-motion view is complicated by the fact that the open field lines of the heliosphere belong to open flux funnels at the Sun that have their solar magnetic footpoints trapped in the downflow lanes at the edges of supergranules (Dowdy et al., 1986; Gabriel, 1976; Peter, 2007) rather than having footpoints in the sea of granules where granule motions can shake the field lines The motions of open field lines may be because of interchange reconnection events with newly emerged closed arches that are advected into these downflow lanes wherein the arches are pressed against the bases of the open flux funnels and can open up via reconnection and feed plasma and magnetic-field perturbations into the open flux funnel (Tu et al., 2005; He et al., 2008; Wilhelm et al., 2011; Yang et al., 2013). As noted in Section 4, an interchange-reconnection event would probably provide a hyperbolic-tangent-like perpendicular displacement to open field lines, a shift in one direction that is not periodic. The appearance of such a propagating shift at 1 AU can be estimated from solar-corona simulations of interchange reconnection between newly emerged closed flux tubes and open flux funnels associated with supergranule conviction. Based on Figure 2 of Yang et al. (2013) and the analysis in that study, a field kink would probably propagate outward from the corona at approximately the local Alfvén speed. At the reconnection site at 2.5 Mm altitude in the figure, the authors quote the local Alfven speed in the simulation to be vA ∼ 30 km/s. If the reconnection was of the Petschek (collision-less) form, then the reconnection inflow speed would be taken to be ∼0.1 vA, but in the corona, the reconnection is resistive and much slower (e.g. Birn et al., 2001; Krista et al., 2011), so the reconnection rate will be taken to be the approximately 1 km/s flow speed of supergranule advection into the downflow lanes where the open flux footpoints are trapped. These two numbers give a local perpendicular-to-parallel kink scale of 1/30, which is like the A/λ scale ratio discussed in Section 4 for the “amplitude” over the “wavelength.” The local field strength in Figure 2 of Yang et al. (2013) is ∼2 G at the reconnection site at 2.5 Mm altitude. The typical solar wind magnetic field is ∼5 nT = 5 × 10−5 G at 1 AU, which means that a flux tube area increases by a factor of 2/5 × 10−5 = 400 in going from the corona to 1 AU, which is a flux-tube radius increase of 20. So the perpendicular jump of the field line increases by a factor of 20 going from the corona to 1 AU. The parallel scale of the kink is vAΔt, where Δt is the duration of reconnection and vA is the local Alfvén speed. The timescale Δt is preserved as the kink propagates. In the corona, vA equals 30 km/s, and at 1 AU, vA is typically 50 km/s. Hence, the perpendicular-to-parallel scale (which we will call A over λ) in the corona was 1/30, and at 1 AU, this would become 20/50 = 0.4. The value A/λ that was taken to be 0.6 in Section 4 (agreeing with the magnetic-field direction vectors about the Parker spiral) appears to be consistent with expectations for interchange reconnection events in the corona. These footpoint braiding scales of the two previous paragraphs are noted in Table 1.

The A ∼ 0.6 λ choice with the median λ of 1,700 RE (cf. Figure 8) yields a wandering amplitude A ∼ 1,000 RE about the Parker-spiral direction at 1 AU. At the 405 km/s corotation speed of structure at 1 AU, this gives a perpendicular amplitude of the field-line wandering of 1,000 RE/405 km/s = 4.4 h. Borovsky (2016) found time intervals of Alfvénicity in the coronal-hole-origin plasma at 1 AU (cf. Figure 14 of that study) that appeared to correspond to the sizes of open flux funnels at 1 AU, with approximately four funnels per day passing the Earth. This A ∼ 0.6 λ with λ ∼ 1700 RE is consistent with the sizes of these Alfvénicity intervals, and this could be interpreted as the wandering of field lines (and flux tubes) within open flux funnels.

6 Summary

A study of the behaviors of solar-wind magnetic-field lines was performed during 12,625 64-s data intervals of ACE–Earth connection. The quasi-local radius of curvature of the magnetic-field lines was calculated, yielding median values of ∼0.016 AU. The quasi-global wandering scale of the magnetic-field lines along the Parker-spiral direction was estimated, yielding median values of ∼0.072 AU. In Table 1, these estimates are compared with other estimated length scales in the solar wind.

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Footnotes

Edited by: Nithin Sivadas, National Aeronautics and Space Administration, United States

Reviewed by: Gilbert Pi, Charles University, Czechia

Harsha Gurram, University of Maryland, United States

Statements

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Funding

The author(s) declared that financial support was received for this work and/or its publication. JEB was supported at the Space Science Institute by the NASA LWS Program via grant 80NSSC23K0904, by the NSF Magnetospheric Program via grant AGS-2149822, and by the NASA HERMES Interdisciplinary Science Program via grant 80NSSC21K1406. Both authors were supported in part by the Los Alamos Space Weather Summer School.

Data availability

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found in the article/Supplementary Material.

Author contributions

JB: Supervision, Methodology, Conceptualization, Investigation, Data curation, Writing – original draft, Formal Analysis, Software. JS: Writing – review and editing, Investigation, Methodology, Conceptualization, Validation.

Acknowledgements

The authors thank Stuart Bale, Ed Roelof, David Sibeck, Ruth Skoug, and Michelle Thomsen for helpful conversations. The authors also thank Yuguang Tong, who helped with the identification of ACE–Earth connection events and performed original fits to field lines as a project at the Los Alamos Space Weather Summer School.

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Citation

Borovsky, J. E. & Steinberg, J. T. (2026). Estimating the length scales of magnetic-field-line wandering in the solar wind at 1 AU: ACE–earth magnetic-connection events. Frontiers in Astronomy and Space Sciences, Volume 13. https://doi.org/
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Copyright © 2026 Borovsky and Steinberg.

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*Correspondence: Joseph E. Borovsky

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