- 1Center for Space Plasma and Aeronomic Research, Department of Space Science, The University of Alabama in Huntsville, Huntsville, AL, United States
- 2Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, United States
Coronal mass ejections (CMEs) represent one type of the major eruption from the Sun. Their interplanetary counterparts, the interplanetary CMEs (ICMEs), are the direct manifestations of these structures when they propagate into the heliosphere and encounter one or more observing spacecraft. The ICMEs generally exhibit a set of distinctive signatures from the in-situ spacecraft measurements. A particular subset of ICMEs, the so-called Magnetic Clouds (MCs), is more uniquely defined and has been studied for decades, based on in-situ magnetic field and plasma measurements. By utilizing the latest multiple spacecraft measurements and analysis tools, we report a detailed study of the internal magnetic field configuration of an MC event observed by both the Solar Orbiter (SO) and Wind spacecraft in the solar wind near the Sun-Earth line. Both two-dimensional (2D) and three-dimensional (3D) models are applied to reveal the flux rope configurations of the MC. Various geometrical as well as physical parameters are derived and found to be similar within error estimates for the two methods. These results quantitatively characterize the coherent MC flux rope structure crossed by the two spacecraft along different paths. The implication for the radial evolution of this MC event is also discussed.
1 Introduction
Magnetic clouds (MCs) represent an important type of space plasma structures observed by in-situ spacecraft missions in the solar wind. They have been first identified in the in-situ spacecraft measurements of magnetic field and plasma parameters, and have been studied for decades, based on heliospheric mission datasets [1–4]. These include the earlier missions such as the Interplanetary Monitoring Platform (IMP), Helios, and Voyager missions. In later times, a number of NASA/ESA flagship missions, including Advanced Composition Explorer (ACE) [5], Wind [6], Ulysses [7], and Solar and TErrestrial RElations Observatory (STEREO) [8], have contributed greatly to the study of Solar-Terrestrial physics in general, and to the characterization of MC structures in particular. Generally speaking, the opportunities for one MC structure to be encountered by two or more spacecraft are rare, but when they do occur, it offers a unique opportunity for correlative and combined analysis between multiple spacecraft datasets (see references below).
A few such examples include an early study by [9] by using five spacecraft and the series of MC events in May 2007. During 19–23 May 2007, the newly launched twin STEREO spacecraft, Ahead and Behind, i.e., STEREO-A and B, respectively, were separated from Earth by
One commonly applied quantitative analysis method for MCs based on single-spacecraft in-situ data usually adopts the approach of an optimal fitting to an analytic solution, such as the well-known linear force-free field (LFFF) Lundquist solution [17], against the time series of magnetic field components within a selected interval. These solutions have limited one-dimensional (1D) spatial dependence, i.e., exhibit spatial variation in the radial dimension away from a central axis only. Recently we have improved the optimal fitting approach by extending the Lundquist solution to a quasi-three dimensional (3D) geometry [18, 19], based on the so-called Freidberg solution [20]. It represents a more general 3D configuration that can account for, to a greater degree, the significant variability in the in-situ measurements of MCs, such as the asymmetric magnetic field profile and sometimes the relatively large radial field component. An alternative two-dimensional (2D) method has also been applied to in-situ modeling of MCs, by employing the Grad-Shafranov (GS) equation, describing a two and a half dimensional (2–1/2D) configuration in quasi-static equilibrium [21–24]. This so-called GS reconstruction method is able to derive a 2D cross section of the structure traversed by a single spacecraft, yielding a complete quantitative characterization of the magnetic field configuration composed of nested cylindrical flux surfaces for a flux rope. Such a solution generally conforms to a cylindrical flux rope configuration with an arbitrary 2D cross section. The GS reconstruction method has been applied in a number of multi-spacecraft studies of MCs [see, e.g. [14, 25]], including the aforementioned MC events in May 2007 during the earlier stage of the STEREO mission. In addition, it has been widely applied to a variety of space plasma regimes with extended capability and additional improvement [26].
A new era has begun for solar and heliospheric physics with the launch of the Parker Solar Probe (PSP) [27] and the Solar Orbiter (SO) [28] missions. They will not only yield unprecedented new discoveries of never-before explored territories, but also provide two additional sets of in-situ measurements at different locations in the heliosphere. PSP will plunge closer to the Sun and reach a heliocentric distance below 0.1 au, and SO will provide highly anticipated measurements over a range of heliocentric distances and beyond the ecliptic plane. In this study, we examine one MC event detected during the month of April 2020 by both SO and Wind spacecraft when they were approximately aligned radially from the Sun, but separated by a radial distance of
2 Event Overview
The SO mission observed its first ICME event on April 19, 2020 (day of year, DOY 110) at a heliocentric distance
FIGURE 1. (A) Time series data from Wind spacecraft (from top to bottom panels) the magnetic field components in GSE-X (blue), Y (red), and Z (gold) coordinates and the magnitude (black), the solar wind speed, the proton number density (blue; left axis) and temperature (black; right axis), the proton β, and the proton plasma pressure and the axial magnetic pressure (red). Two sets of vertical lines mark the intervals for the GS reconstruction (green) and the optimal fitting to the Freidberg solution (red), respectively, and are denoted beneath the last panel. (B) The multiple spacecraft and planets locations around April 20, 2020 in the ecliptic plane (courtesy of the STEREO Science Center). (C) The corresponding SO magnetic field measurements in the RTN coordinates (see legend).
In Figure 1A, two intervals are marked for the subsequent analysis of the ICME/MC flux rope structure via the GS reconstruction method (between 11:36 and 22:28 UT) and the optimal 3D Freidberg solution fitting approach (between 12:41 and 23:15 UT) on April 20, 2020. The in-situ measurements enclosed by the vertical lines indicate clear signatures for an MC: 1) elevated magnetic field magnitude, 2) relatively smooth rotation in field components (i.e., mainly the GSE-Z component varying from negative to positive values), and 3) depressed proton temperature and β value. The corresponding measurements of magnetic field components at SO show similar features with slightly enhanced magnetic field magnitude. The plasma measurements were not available during these earlier time periods of the mission [16]. In particular, the rotation in the N component of the magnetic field at SO corresponds well to the rotation in the GSE-Z component at Wind, while the East-West components (along T and the GSE-Y directions) are approximately reversed. For a typical cylindrical flux rope configuration crossed by a single spacecraft, the magnetic field component with a uni-polar pattern usually corresponds to the field component along the axis of the flux rope, while the change in the north-south or east-west component usually indicates the rotation of the transverse field about the axis. Therefore these signatures, for this particular MC event, hint at a flux rope configuration lying near the ecliptic with the axial direction pointing eastward (positive GSE-Y component, aligned with the thumb of the left hand) with respect to the Sun and with a left-handed chirality (the handedness; GSE-Z component rotating from southward to northward direction, aligned with the other four fingers). Given the difference in the magnetic field magnitude and a 1-day time delay consistent with the radial separation distance between SO and Wind [16], it is plausible to consider an evolution between the two spacecraft as well as the spatial variation, assuming that the two spacecraft crossed the same structure along different paths mainly due to their longitudinal separation. In what follows, we present our analysis results and discuss the interpretations.
3 Methods and Results
We have developed and applied both 2D and 3D flux rope models to in-situ spacecraft measurements of MCs. The 2D model is based on the Grad-Shafranov (GS) equation and is able to derive a 2D cylindrical configuration with nested flux surfaces of arbitrary cross section shape [see, e.g. [26]]. The 3D model is based on a more general LFFF formulation, the so-called Freidberg solution [20], and accounts for a greater deal of variability in the in-situ data through a rigorous
3.1 Grad-Shafranov Reconstruction Results
The GS reconstruction utilizes the spacecraft measurements of magnetic field
FIGURE 2. (A) The
This is a typical rendering of the GS reconstruction result as viewed down the z axis such that the flux surfaces (contours of A) are projected onto the cross-section plane as closed loops surrounding the center for a flux rope configuration. The axial magnetic field usually reaches the maximum at the center and decreases monotonically toward the outer boundary. Along the spacecraft path at
FIGURE 3. The Walén plot for the MC interval at Wind for which the GS reconstruction is applied. The HT analysis yields an HT frame velocity [−346.52, −10.81, 21.10] km/s, in the GSE coordinates. The slope of the regression line shown is denoted the Walén slope.
It is also informative to illustrate the magnetic field configuration in the perspective view toward the Sun with both Wind and SO spacecraft locations marked in Figure 4. This provides a direct 3D view toward the Sun (located at the same position as Wind in this view but at a distance 1 au away) along the Sun-Earth line. It is seen that the reconstructed flux rope structure based on the Wind in-situ data along its path shows selected spiral field lines with arbitrary colors winding around a central axis represented by the red straight field line, along the z axis direction, pointing approximately horizontally to the East with both Wind and SO spacecraft passing beneath the center of the flux rope, and separated mostly in the East-West direction. With the 2D reconstruction result from the Wind spacecraft, it enables a direct comparison between the derived magnetic field components along the SO spacecraft path, as shown in Figure 2B, and the actual measured ones returned by the spacecraft. Figure 5 shows such comparison of the three magnetic field components in the SO centered RTN coordinates. Figure 5A shows the component-wise time series within the MC interval at SO, while Figure 5B shows the corresponding one-to-one correlation plot, yielding a correlation coefficient
FIGURE 4. The 3D view toward the Sun (along the +GSE-X direction) of the selected field lines (with arbitrary colors) from the GS reconstruction result corresponding to Figure 2. The unit vectors of the GSE-Y and Z coordinates are denoted in green and black arrows, respectively. The cross section as seen in Figure 2B is shown to the right where the field lines originate and spiral along the z axis. The red straight field line originates from the center of the flux rope as marked in Figure 2B. The spacecraft locations of Wind and SO are marked by the blue and green dots, respectively. The z axis direction is denoted on top in the GSE coordinates.
FIGURE 5. (A) The comparison between the derived magnetic field components (solid curves) based on the GS reconstruction result from Wind in Figure 2, and the actual measurements (circles and error bars), along the SO spacecraft path. The field magnitude is in black. (B) The corresponding component-wise one-to-one scatter plot with the correlation coefficients between the two sets for all three components,
One main discrepancy is the underestimated magnitude of the
3.2 A Quasi-3D Configuration Based on the Freidberg Solution
We also apply an optimal fitting approach based on the quasi-3D Freidberg solution to the MC interval denoted in Figure 1A. For this interval, an HT frame velocity is obtained
Here the solution involves the Bessel’s functions of the first kind,
An optimal fitting approach based on
FIGURE 6. The optimal fitting result to the Freidberg solution for the MC interval marked in Figure 1A. The Wind spacecraft measurements of the magnetic field with uncertainty estimates are shown as error bars, while the corresponding analytic solution is given by solid curves (see legend). The horizontal axis is the integral index of the data points.
When compared with the GS reconstruction result, the significant distinction of this configuration represented by the Freidberg solution is the 3D nature, not present in any 2D configurations. There no longer exists distinctive 2D flux surfaces, and the field lines exhibit more general 3D features, not lying on discernable individual flux surfaces. Figure 7A demonstrates one cross section perpendicular to the z axis. The transverse field vectors are not tangential to the contours of
FIGURE 7. (A) One cross section of the optimal Freidberg solution where the colored contours represent
FIGURE 8. The 3D view of the field lines or the flux bundle of the Freidberg solution given in Figure 7B, from the same viewpoint and in the same format as Figure 4. The z axis orientation and the locations of Wind (blue dot) and SO (green dot) spacecraft are also marked.
FIGURE 9. The comparison between the derived magnetic field components based on the optimal Freidberg solution fit to Wind spacecraft data, and the actual measurements, along the SO spacecraft path across the solution domain. Format is the same as Figure 5.
4 Discussion
We lay out, briefly, a consideration for the radial evolution of the MC, given the difference in the average magnetic field magnitude between SO and Wind during the MC interval, which can be partially accounted for by the spatial variations [see, also [16]]. Because the solar wind flow speed at Wind shows little variation, the expansion in the radial direction may be negligible for this event (also justified by the small Walén slope as shown in Figure 3). Therefore by assuming conservation of axial magnetic flux content and a constant angular extent of the MC flux rope cross section,
Here the average axial field
TABLE 1. Summary of geometrical and physical parameters for the MC based on Wind spacecraft measurements.
This study represents one step forward in the direction of quantifying how realistic MC model outputs are, based on one event study with available two-spacecraft in-situ observations. Future work would involve additional measurements and analysis based on remote-sensing observations, which will provide characterizations of solar source region (magnetic) properties of certain MC events to help further assess the fidelity of each model. The present implementations represent the best effort we have made in accounting for the variability in the in-situ measurements of MCs and proper error/uncertainty estimates of output parameters. Two models employed are deemed complementary and both are worth applying for individual event studies, as judged by the metrics, mainly, the combined correlation coefficients obtained from this two-spacecraft study with
5 Summary
In summary, we have examined one MC event in the solar wind by using the in-situ spacecraft measurements from both the Wind and SO missions located at heliocentric distances
It is worth noting that as multi-spacecraft measurements become increasingly more available, as partially illustrated in Figure 1B, new and exciting multi-messenger science will be enabled by using multiple analysis tools. It is highly anticipated that the constellations of current and future missions will usher in new frontiers in heliophysics research.
Data Availability Statement
Publicly available datasets were analyzed in this study. This data can be found here: NASA CDAWeb: https://cdaweb.gsfc.nasa.gov/index.html/.
Author Contributions
QH carried out the analysis and wrote the draft of the manuscript. WH helped with the visualization of the analysis results. LZ obtained the time-series data from SO and participated in the interpretation of the results. EL helped with the analytic verification of the Freidberg solution.
Funding
Funding is provided, in part, by NASA Grant Nos. 80NSSC21K0003, 80NSSC19K0276, 80NSSC18K0622, 80NSSC17K0016, and NSF Grant Nos. AGS-1650854 and AGS-1954503, to The University of Alabama in Huntsville.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Acknowledgments
WH and QH acknowledge NSO/NSF DKIST Ambassador program for support. The authors wish to thank Ms. Constance Hu for proofreading the manuscript. We thank the reviewers for useful comments that have helped improve the presentation of this manuscript.
Footnotes
1The polar angle δ is from the ecliptic north, and the azimuthal angle ϕ is measured from GSE-X towards GSE-Y axes, all in degrees.
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Keywords: magnetic clouds, magnetic flux ropes, coronal mass ejections, grad-shafranov equation, force-free field, solar orbiter, wind
Citation: Hu Q, He W, Zhao L and Lu E (2021) Configuration of a Magnetic Cloud From Solar Orbiter and Wind Spacecraft In-situ Measurements. Front. Phys. 9:706056. doi: 10.3389/fphy.2021.706056
Received: 06 May 2021; Accepted: 07 July 2021;
Published: 22 July 2021.
Edited by:
Daniel Verscharen, University College London, United KingdomReviewed by:
Nandita Srivastava, Physical Research Laboratory, IndiaYeimy Rivera, Harvard University, United States
Copyright © 2021 Hu, He, Zhao and Lu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Qiang Hu, cWgwMDAxQHVhaC5lZHU=