A comparison of methods for finding magnetic nulls in simulations and in situ observations of space plasmas
Vyacheslav Olshevsky, David Pontin, Benjamin Williams, Clare Parnell, Huishan Fu, Yangyang Liu, Shutao Yao, Yuri Khotyaintsev
AAstronomy & Astrophysics manuscript no. nulls © ESO 2021January 7, 2021
A comparison of methods for finding magnetic nulls in simulationsand in situ observations of space plasmas
V. Olshevsky , D. I. Pontin (cid:63) , B. Williams , C. E. Parnell , H. S. Fu , Y. Liu , S. Yao , and Y. V. Khotyaintsev KTH Royal Institute of Technology, Lindstedtsvagen 5, SE-10044 Stockholm, Sweden Main Astronomical Observatory of NAS, Akademika Zabolotnoho 27, 03680, Kyiv, Ukraine School of Mathematical and Physical Sciences, University of Newcastle, University Drive, Callaghan, NSW 2308, Australiae-mail:
[email protected] University of St Andrews, North Haugh, St Andrews, UK Beihang University, 37 Xueyuan Road, Haidian District, Beijing 100191, China Institute of Space Sciences, Shandong University, Weihai, China State Key Laboratory of Space Weather, National Space Science Center, Chinese Academy of Sciences, Beijing, China Swedish Institute of Space Physics, Box 537, SE-751 21 Uppsala, SwedenReceived September ??, 2020; accepted ????
ABSTRACT
Context.
Magnetic nulls are ubiquitous in space plasmas, and are of interest as sites of localised energy dissipation or magneticreconnection. As such, a number of methods have been proposed for detecting nulls in both simulation data and in situ spacecraft datafrom Earth’s magnetosphere. The same methods can be applied to detect stagnation points in flow fields.
Aims.
In this paper we describe a systematic comparison of di ff erent methods for finding magnetic nulls. The Poincaré index method,the first-order Taylor expansion (FOTE) method, and the trilinear method are considered. Methods.
We define a magnetic field containing fourteen magnetic nulls whose positions and types are known to arbitrary precision.Furthermore, we applied the selected techniques in order to find and classify those nulls. Two situations are considered: one in whichthe magnetic field is discretised on a rectangular grid, and the second in which the magnetic field is discretised along synthetic‘spacecraft trajectories’ within the domain.
Results.
At present, FOTE and trilinear are the most reliable methods for finding nulls in the spacecraft data and in numericalsimulations on Cartesian grids, respectively. The Poincaré index method is suitable for simulations on both tetrahedral and hexahedralmeshes.
Conclusions.
The proposed magnetic field configuration can be used for grading and benchmarking the new and existing tools forfinding magnetic nulls and flow stagnation points.
Key words. magnetic topology – Sun – space plasma
1. Introduction
Astrophysical plasmas are typically characterised by high mag-netic Reynolds numbers, and their magnetic fields are found toexhibit a complex structure on a range of scales. For example,observations from missions studying the Earth’s magnetosphere(Cluster (Escoubet et al. 2001) and the Magnetospheric Multi-scale (MMS) mission (Burch et al. 2016)) show highly fluctu-ating fields both in the magnetotail (Fu et al. 2017) and magne-tosheath (Retinò et al. 2007). Extrapolations of the solar coronalmagnetic field based on photospheric magnetograms similarlyshow enormous complexity in the magnetic connectivity be-tween photospheric flux fragments (e.g. Schrijver & Title 2002;Close et al. 2003). In order to understand the detailed dynamicsof such highly complex fields, we need to identify the featuresof the magnetic field at which localised energy conversion – typ-ically mediated by magnetic reconnection – takes place. Can- (cid:63)
Corresponding author didates for locations of magnetic reconnection in complex 3Dfields include magnetic nulls (points at which the magnetic fieldstrength, B , is zero), their associated separatrix surfaces, and theseparator lines that are formed by the intersections of these sep-aratrix surfaces (for a review, see e.g. Pontin 2012; Priest 2014).Such magnetic nulls have been detected in spacecraft data fromthe magnetotail, magnetopause, magnetosheath, and foreshock(e.g. Xiao et al. 2006; He et al. 2008; Deng et al. 2009; Wendel& Adrian 2013; Guo et al. 2016; Fu et al. 2019; Chen et al. 2017,2019). In extrapolations of the solar coronal field they are foundin abundance, with the number of nulls increasing exponentiallyas the photosphere is approached (Longcope & Parnell 2009).Moreover, magnetic reconnection at these nulls has been impli-cated in energy release in, for example, solar flares and jets (e.g.Masson et al. 2009; Yang et al. 2015; Kumar et al. 2019). Beingan isolated point, a magnetic null is not easy to detect in discretedata. This has motivated the development of methods which in- Article number, page 1 of 12 a r X i v : . [ phy s i c s . s p ace - ph ] D ec & A proofs: manuscript no. nulls fer the existence of magnetic null points, both in simulations andin spacecraft data.Methods for finding topological singularities and other spe-cial features are becoming increasingly important for researchersworking with huge amounts of simulation and observationaldata. A topological analysis is extremely useful for observers asspecific features (e.g. magnetic null points) are likely locationsfor energetic events in the Sun or the Earth’s magnetosphere (forappropriate external perturbations). The same analysis allowsus to distinguish the important subsets of the huge amounts ofdata collected from satellites or telescopes. A topological analy-sis serves two main purposes when applied to numerical sim-ulations: It allows the identification and classifications of thedata sets (or simulation sub-domains) of potential interest, andit could also be used for data compression. Finally, as discussedabove, in both observations and simulations, certain topologicalfeatures have very important physical implications and serve asa framework to understand the physical processes that drive theobserved dynamics.This paper addresses a subclass of topological analysis tech-niques, namely the identification of the stagnation points of 3Dvector fields. In the present discussion (around space and astro-physical plasmas), we mainly discuss the topology of the mag-netic field. The same analysis, however, is applicable to othervector fields, such as the flow velocity (Wang et al. 2020), solong as the field is divergence-free. We designed and undertook a‘challenge’ to compare the performance of di ff erent null findingapproaches to understand possible deficiencies and weaknessesof several pieces of software. Limited comparisons were previ-ously made between di ff erent methods (Fu et al. 2020), but theyeither did not include as many methods, or they did not include a‘ground truth’ in which the exact existence and positions of thenulls are known (Haynes & Parnell 2007; Eriksson et al. 2015).Our aim is to understand how the choice of method could pos-sibly influence the analysis of observations or simulations, andwhat are each method’s strengths and weaknesses. We comparethe most popular methods in the literature that can be automatedto quickly analyse many di ff erent instances of input data (seeSection 3).
2. Theoretical background
Magnetic nulls are locations in space at which the magnetic fieldis zero, and in the generic case this occurs at isolated points. Thestructure of the magnetic field in the vicinity of these points canbe characterised by linearising the field about the point. We notethat for any generic (stable) null this linearisation is non-zero,and the local topology of the field linearisation can be shownto be the same as the local topology of the field itself – seeHornig & Schindler (1996). The eigenvalues and eigenvectors ofthe magnetic field Jacobian ∇ B at the null determine the spine-fan structure of the field, as described in detail in Fukao et al.(1975); Parnell et al. (1996). Since ∇ · B =
0, the eigenvaluessum to zero. The eigenvectors associated with the two same-signeigenvalues locally define a plane in which magnetic field linesapproach or recede from the null, known as the fan surface (or Σ -surface). The remaining eigenvector defines the direction ofthe spine line (or γ -line), along which field lines recede fromor approach the null. If the same-sign eigenvalues have negativereal parts, the null has topological degree + A-type null or negative null.If the same-sign eigenvalues have positive real parts, the topo-
Fig. 1.
Visualisation of the test magnetic field showing the null pointsand associated field line structures, together with the simulated space-craft trajectories. Null points of topological degree + −
1) lie at thecentre of blue (red) spheres and their fan field lines are represented incyan (magenta). Spine lines are black. The red, blue, green and goldcurves are the simulated spacecraft trajectories. logical degree is −
1, and we have a
B-type null or positive null.One further pertinent distinction is between nulls for which allthree eigenvalues are real ( radial nulls ), and those for which twoeigenvalues are complex conjugates ( spiral nulls ). In the lattercase, the field lines form a spiral pattern in the fan surface, andnulls are sometimes denoted as being of A s - or B s -type (this oc-curs when a su ffi ciently strong component of electric current ispresent parallel to the spine line). The magnetic configuration used to test the null finding methodsis based on a triply-periodic field that has previously been usedto initiate turbulence simulations (Politano et al. 1995). To thisfield various perturbations are added in order to make the dispo-sition of null points less ‘regular’. Some of these perturbationstake the form of ‘flux rings’, that are inserted in such a way asto lead to a pitchfork bifurcation of one of the pre-existing nulls,leading to the creation of two additional nulls (as in e.g. Wyper& Pontin 2014). This is done in such a way that all nulls can beaccounted for based on theoretical arguments. Following the ad-dition of the perturbations as described, the exact null locationscan no longer be obtained analytically. Instead they are obtainedusing Newton’s method. Since the field itself is still known an-alytically, the null location can still be found to arbitrary preci-sion. Further, since the field is prescribed analytically, the Jaco-bian of B can also be calculated exactly at these points, and thusthe topological degree of the null and the local orientation of thespine line and fan surface can be determined as above. Details forconstructing the magnetic field are presented in the Appendix A.The null points and their spine and fan structures are representedwithin the volume of interest ( x , y , z ∈ [ − π/ , π/ Article number, page 2 of 12. Olshevsky et al.: Methods for finding magnetic nulls
The di ff erent null finding methods are described in the follow-ing section. These are designed to be used to find nulls on ei-ther hexa- or tetrahedral meshes of data points (obtained fromnumerical simulations), or on time-series of quadruplets of mea-surements (taken by the Cluster or MMS spacecraft). We there-fore generate two di ff erent types of data sets from the modelmagnetic field. In the first case, we evaluate the magnetic fieldcomponents on a rectilinear grid of points with various di ff erentresolutions. In the second, we define four trajectories throughthe domain, evaluating the magnetic field components at a dis-crete set of points along these trajectories (see Fig. 1). Thesepoints are chosen in each instance to lie at the corners of a regu-lar tetrahedron, to mimic typical spacecraft configurations. Thetrajectories are designed so as to pass close to some of the nullpoints, and further from others – see the Appendix A for details.We compare the results for three di ff erent sizes of tetrahedron(corresponding to di ff erent spacecraft separations).The magnetic field as defined in Appendix A is dimension-less. Particularly in the context of the trajectory data from hy-pothetical spacecraft tetrahedra, it is relevant to compare withphysical length scales. One possible dimensionalisation wouldbe to consider our domain to be equivalent to the largest fully-kinetic simulations a ff orded by present-day codes and supercom-puters, since these were recent data sets on which null finderswere applied. Those simulations consider domains extending fortens of ion inertial lengths d i (Pucci et al. 2017; Olshevsky et al.2018). If we suppose that the domain size is 20 d i in each di-mension, then our ‘small-scale’ and ‘medium-scale’ tetrahedrahave spacecraft separations of 0 . d i and 0 . d i , respectively.The inter-spacecraft separation of the MMS constellation canchange between 5 − km which corresponds to 0 . − . d i in the magnetotail, and 0 . − . d i in the magnetopause. Hence,our ‘small-scale’ tetrahedron aims at resembling the electron dif-fusion region scales covered by the MMS mission. The inter-spacecraft separation of the Cluster mission varies 200 − km ,resembling 0 . − . d i in the magnetotail, and 2 − d i in themagnetopause. This larger separation – of the order of the iondi ff usion region – dictates the choice of the ‘medium’ and ‘large’scales for our study.
3. Methods
This section describes the three methods that we have compared,their theoretical formulation and implementation.
The problem of locating a magnetic null is essentially a problemof finding a root of a continuous divergence-free vector field. ThePoincaré index or topological degree method for finding suchroots was introduced by Greene (1992). This technique has beenapplied to various kinetic simulations by Olshevsky et al. (2015,2016) and spacecraft observations by Eriksson et al. (2015);Xiao et al. (2006). The key assumption of the method consistsin the linearity of the field around a null, therefore a field in theneighbourhood of the null is given by B i = ( ∇ B ) i j (cid:16) x j − x j (cid:17) , (1)where the summation is implied over repeating indices, x j de-note the coordinates of the null, and ( ∇ B ) = ∂ B i /∂ x j | x j = x j is the magnetic field gradient, a 3 × D = (cid:88) k sign [ λ k λ k λ k ] , (2)where λ k , k , k are the eigenvalues of the ∇ B at the k -th null. Asmentioned above, in the generic case nulls do not degenerate (inreality they can be degenerate only at one instant of time during abifurcation process), and all three eigenvalues are non-zero. Theimplication of this fact is that magnetic nulls are isolated. As thetopological degree is strongly conserved, it provides a measureof the di ff erence between the number of positive and the numberof negative nulls in the given volume of space. If the volume ofspace is su ffi ciently small, one can assume it encloses exactlyone null if D (cid:44) π , gives the number of times the triangles cover aunit sphere in the magnetic field space. This is the sum of thesigns of all the nulls of the field inside the sphere (see Eq. (2)).We note that each area has a sign, and it is important to observethe order of vertices in the triple cross product B · B × B toget the sign correctly. The ‘plus’ sign corresponds to the outwarddirected flux, while the ‘minus’ refers to the inward field flux.For our implementation of the Poincaré index method weuse the formula for the solid angle subtended by three vectorsproposed by van Oosterom & Strackee (1983):tan (cid:32) Ω (cid:33) = B · B × B B B B + ( B · B ) B + ( B · B ) B + ( B · B ) B . (3)Evaluation of the solid angle this way is faster and more stablethan the conventional use of the Cosine theorem. In particular,there is no need for zero-denominator checks when using mod-ern programming languages, as errors are handled by the arctan2function.Once a cell which encloses a null is found, it is possible to geta more precise estimate of the null location inside this cell usingthe Secant theorem (Greene 1992). However, as noted by Greene(1992), this estimate may often be misleading, even giving loca-tions outside the cell. Our experiments confirmed this problem,therefore it is more practical to assume the null is located in thecentre of mass of the cell.The topological classification of a null is straightforward onhexahedral cells where finite di ff erences can be used to deducethe magnetic field Jacobian. A technique for ∇ B computation intetrahedral cells is given in Khurana et al. (1996). Article number, page 3 of 12 & A proofs: manuscript no. nulls
The null-finder based on the Poincaré index method combinesmagnetic field measurements into a set of either 4 or 8 mag-netic field vectors given in the vertices of a cell. It computes thetopological degree and returns either a very small number closeto zero (meaning no null is present inside the cell), or a num-ber close to 1 or −
1, meaning there is a null inside the cell. Inpractice, the thresholds of being ‘close to zero’ or ‘close to one’are regulated by the numerical precision. Similarly to the trilin-ear method described in Section 3.2, only those grid cells areselected for analysis, in which none of the components of themagnetic field have the same sign at all vertices (this being in-compatible with the existence of a null in the cell). If at leastone component of the magnetic field has the same sign in all thevertices, the field can’t go to zero inside this cell (in the linearapproximation). This pre-selection reduces the computation costdramatically in a typical simulation or observation, where only afraction of measurements comprise field nulls.
The trilinear method for finding the locations of null pointsin a numerical grid under the trilinear assumption was origi-nally formulated by Haynes & Parnell (2007). The algorithmdescribed below di ff ers from Haynes & Parnell (2007) by usinga deca-section method (like the bisection method) rather thanNewton-Raphson method for converging on the location of thenull points. There are three stages to the method: the reductionstage, bilinear stage and the sub-grid stage.The reduction stage just reduces the amount of work done bythe algorithm in the bilinear stage. Each grid cell is checkedin turn for a change in sign in the magnetic field. Essentially,a grid cell cannot contain a null point under the trilinear as-sumption if all 8 values of the grid cell vertices are of thesame sign (see above).The bilinear stage actually checks for the possibility of a nullpoint within a grid cell. The zero isosurfaces ( B i =
0) of thethree components of the magnetic field will intersect at a nullpoint if a null exists. This triple intersection is di ffi cult to finddirectly numerically. However, two of the three componentsof the magnetic field’s zero isosurfaces will intersect to forma line which the null point must lie on. This line must alsointersect with the grid cell faces. The magnetic field on thesegrid cell faces is now only bilinear and therefore the locationsof the intersection points of this line and the cell faces (say P and P ) can be found analytically. The values of the thirdcomponent of the magnetic field (unused to form the line)can be found at P and P : if a null point exists, then thisthird magnetic field component must be of opposite signs at P and P . By using this test, the algorithm can detect whichgrid cells may contain null points.The final, sub-grid stage is simply the first two stages whichare repeated at sub-grid cell resolution to identify the loca-tions of the null points at the required accuracy. Each null-containing grid cell is split into a new grid of 10 × × https: // bitbucket.org / volshevsky / magneticnullchallenge The methods used for detecting the sign of the null points usea convergence-style method. They are fully detailed in Williams(2018). A field line about a null point can be written as r ( s ) = a e λ s e + a e λ s e + a e λ s e (4)where λ i and e i are the corresponding eigenvalues and eigenvec-tors of M = ∇ B evaluated at the null point and a i are constants.By repeated multiplication of equation (4) by M (and assumingthat λ is the eigenvalue corresponding to the eigenvector asso-ciated with the spine line), we obtain M n · r ( s ) → a λ n e λ s e . (5)This allows the eigenvector associated with the spine line to beidentified. This convergence is used by the Sign Finder to clas-sify the signs of the null points. This also identifies the eigen-vectors associated with the fan plane. However, the Sign Finderdoes not find any of the values of eigenvalues of the null point oridentify if it is a spiral null point. If this information is desired,it must be found by an alternative method. The algorithm for finding null points using the trilinear methodis implemented in the Magnetic Skeleton Analysis Tools. It is aFortran based package for analysing the skeleton of divergence-free vector fields . The first-order Taylor expansion (FOTE) method is based onTaylor expansion of the magnetic field in the vicinity of a null(Fu et al. 2015): B ( r ) = ∇ B ( r − r ) , (6)where ∇ B is the Jacobian matrix derived from four-point mea-surements, r is the location of one of the four spacecraft, r isthe location of interest, and B ( r ) is the magnetic field at the lo-cation of interest. Requiring B ( r ) = r − r ) < d i , (2) the following dimensionless error parametersare both smaller than 0 . η = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ · B max( ∇ B ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (7) ζ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ + λ + λ max( | real ( λ ) | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (8)where where λ , λ , and λ are the eigenvalues of the Jacobianmatrix ∇ B . The quantitative criteria for qualifying FOTE are de-rived from the comprehensive tests of the simulation data (Fuet al. 2015). https: // github.com / benmatwil / msatArticle number, page 4 of 12. Olshevsky et al.: Methods for finding magnetic nulls The algorithm for finding null points and identifying their typeusing the FOTE method is implemented in Matlab. The time-series quadruple data are used in such test. At each samplingpoint, a null point position relative to the spacecraft is calculatedby Equation 6. Since the spacecraft trajectories generated artifi-cially are given, a null point location in the spatial domain canbe obtained.As we have introduced, owing to the linear assumption, theidentification of a remote null point is not reliable. Thus, we set athreshold distance. Only the magnetic nulls below such thresholddistance are further evaluated.
FOTE method has shown great powers in automatic null detec-tion. However, the FOTE method requires the magnetic fieldsaround the spacecraft to be quasi-linear, so that its accuracy isreduced when dealing with strongly non-linear magnetic fields.Recently, a new method entitled ‘Second-Order Taylor Ex-pansion’ (SOTE) was proposed by Liu et al. (2019) to overcomethe linear limitation. This method is based on the second-orderTaylor expansion of the magnetic field B ( x , y , z , t ) = a x + b y + c z + d xy + e xz + f yz + l x + m y + n z + B , (9)where a , b , c , d , e , f , l , m , n , and B are vector coe ffi cients. Thefollowing constraints can be applied to these equations: ∇ · B = ∇ × B = u J , where the current density is derived fromparticle moments: J = ne ( V i − V e ). To completely determineall the coe ffi cients in Equation 9, the SOTE method utilises twosets of four-point measurements of magnetic field and particles,by assuming the structures to be quasi-stationary and solving theprecise trajectory of the spacecraft.The SOTE method is good at reconstructing non-linear struc-tures, for example, the null-point pairs in this study. Thus, itenables the analysis of null-point pairs in space plasmas. How-ever, for null point detection, the SOTE method should have thesame performance as FOTE method since the FOTE reconstruc-tion is essentially a local approximation of SOTE reconstruction.What’s more, the SOTE method cannot be applied to a time-varying structure while the FOTE method could reveal the tem-poral evolution of a magnetic structure.
4. Results
The methods considered here can be broken down into two cat-egories: Some methods are designed to take eight data points asinput (meaning 24 data values when the three component of B are included), motivated by the need to find magnetic nulls insimulation data utilising rectilinear grids of points. Both the Tri-linear and Poincaré methods have been applied previously in thisway. By contrast, other methods were developed to find nulls indata from the four-spacecraft missions, Cluster and MMS, andtherefore take as input the magnetic field components measuredat four points in space: the FOTE and Poincaré index methodswere used in this context. In the following sections we considerthese two cases separately. In Figure 2 we plot the locations of nulls found when the mag-netic field is evaluated on equally-spaced grids with resolutions
Fig. 2.
Locations of magnetic nulls found by the Poincaré index methodfor di ff erent grid resolutions. Cubic boxes outline the grid cells wherenulls are detected in the 20 grid (larger boxes), 30 grid (smallerboxes). Colour spheres show the nulls found in the 80 grid. Colourdenotes the topological type of the null as described in the legend. Fig. 3.
3D rendering of the magnetic field obtained using the trilinearmethod and associated Magnetic Skeleton Analysis Tools, for 80 reso-lution. Positive and negative null points are represented as red and bluespheres respectively, spine lines from positive and negative null pointsare represented as thick red and blue lines respectively and the fieldlines originating from the fan planes of positive and negative null pointsare drawn as thinner red and blue lines respectively. , 30 , and 80 . We see that the detection and location of someof the nulls is relatively stable between the di ff erent resolutions,while other null detections exhibit large di ff erences for the dif-ferent resolutions. The null points detected at 20 and 80 reso-lution are listed in Table 1. This is discussed further below. Article number, page 5 of 12 & A proofs: manuscript no. nulls
Fig. 4.
Illustration of the e ff ect of di ff erent grid resolution for the recon-structed magnetic field structure using the trilinear method. The mag-netic field in the vicinity of the same null point is reconstructed fromthe 20 grid data (left) and the 30 grid data (right). The reconstructedfield structure changes from a sink to a divergence-free null point whenthe resolution is increased. Red and orange field lines are traced in theforward and backward direction, respectively, of the magnetic field. Figure 3 illustrates a typical output of the magnetic skeletonanalysis and application of the trilinear method to a 80 grid.Table 1 shows the results by using the trilinear method to findthe locations of the null points in the test magnetic field on dif-ferent resolution grids. In the 20 grid, the trilinear method onlyfinds 12 null points and is only able to classify 11 of these. It can-not locate two of the 14 null points which exist in the analyticalfield.More closely analysing the unclassified null point revealsthat, under the trilinear assumption, this point is represented by asink with the field lines twisting into the point (Figure 4). How-ever, it turns out that when the magnetic field is evaluated ona grid with 30 resolution, this point is revealed to be a truedivergence-free null point (Figure 4). There is clearly just a res-olution issue and the vector field is not approximately trilinearlocally to this null point at 20 resolution.The two null points which cannot be located in the 20 gridare actually located within the same grid cell at 20 resolution.From analysis in the higher resolution grids (where these twonull points are now located in di ff erent grid cells), we find oneof these null points is positive and the other is negative. Sincethis pair of null points comprises both a positive and a negativenull point, together they have a topological degree of zero and sothey cannot be found when in the same grid cell.At 30 resolution, all 14 null points are now found using thetrilinear method. However there is still one null which the al-gorithm is unable to classify. The exact same change as aboveoccurs with null point 12 between 30 and 40 resolution. Thefield lines around null point 12 show that it appears to be a sourcein the trilinear approximation at 30 resolution and becomes anegative divergence free null point at 40 resolution. At 40 res-olution and higher, all 14 null points are found and classifiedcorrectly. The null point locations and types for the Poincaré index andtrilinear methods are compared to the true values in Table 1.Before comparing these results it is worth making some impor-tant notes. First, the trilinear method as currently implementeddoes not distinguish spiral nulls, since it does not make use ofthe Jacobian matrix eigenvalues to determine the null type (seeabove). In principle this could be done in the same way as for the Poincaré method (taking finite di ff erences over the grid to eval-uate the Jacobian matrix). Second, the Poincaré index method ascurrently implemented does not seek the nulls at sub-grid reso-lution, thus the centre of the cell is reported as the null location.By contrast, the trilinear method fits a field to the data, with thenull point of this fitted field within the cell being reported.With the above in mind we can compare the nulls found bythe two methods, in Table 1. First, we see that at 20 resolu-tion, both methods are imperfect. The trilinear method missestwo nulls (10 and 12 – though note that as mentioned abovethese exist in the same grid cell at this resolution), while onefails the classification process. The Poincaré index method per-forms a little worse: in addition to the two nulls missed by thetrilinear method, nulls 1 and 6 are also not found, while these isa false-positive null as well (bottom row of the table). The prob-lematic null 4 is again wrongly classified, while the spiral natureof null 14 is not picked up.At 80 resolution both methods show much better results,as expected. Both methods find all 14 nulls, with only null 4wrongly classified by the Poincaré index method (again as B rather than B s ). As a result of the fact that the field is fitted onthe grid allowing some sub-grid resolution to the null detection,the trilinear method generally gives a more accurate estimate ofthe null point location. Table 2 shows the results of the FOTE method testing on nullpoint location and identification. The types, coordinates andminimum distances to spacecraft of these null points are given.In the ‘medium-scale’ tetrahedron configuration, the space-craft separation is about 0 .
12 in dimensionless units. Consider-ing the linear assumption, the null points with the distance (fromthe tetrahedron centre) less than 1 are reserved. In total, we de-tected 14 null points, in which the null points 1, 2, 5, 6, 8, 9, 11,and 14 are included in Table 2 and thus are real null points. Theothers are misidentifications, generally with large distances (seenull points 4, 7, 12, 13). This is consistent with the properties ofFOTE method. Six real null points (null points 1, 3, 7, 12, 13,and 14 in Table A.2) are missed in the test. Among the missednull points, nulls 1 and 7 (see Table A.2) are located rather farfrom the spacecraft trajectory, and thus cannot be detected. Nullpoints 12 and 13 are close to each other (relative to the spacecraftseparation), possibly breaking the linearity of the field in theirvicinity. This explains why these null points are not detected byFOTE method.In the ‘small-scale’ tetrahedron configuration, the spacecraftseparation is about 0 . . .
25, while the misidentifications are with thedistances larger than 0 .
5. This means we could conveniently im-prove the credibility of FOTE method by decreasing the thresh-old distance.The null point detections for the two di ff erent tetrahedronsizes are shown graphically in Fig. 5, where the exact answeris also plotted in the top panel. In the fourth panel of the Fig-ure we also show the result of applying FOTE to an even largertetrahedron, for comparison with the Poincaré index method (seebelow). Article number, page 6 of 12. Olshevsky et al.: Methods for finding magnetic nulls
Table 1.
Comparison of nulls locations and types obtained using the trilinear and Poincaré index methods to the true locations and types, for twodi ff erent grid resolutions. Null x y z
Type x y z
Type x y z Trilinear Poincaré Exact (2 d.p.)1 B 0.30 -0.25 -1.49 - - - - B s . − . − .
482 A 0.04 -0.00 -0.01 A s -0.08 -0.08 -0.08 A s .
05 0 . − .
013 A 2.90 -0.16 -0.57 A A . − . − .
604 0 2.95 -0.17 0.17 A s B s . − .
15 0 .
225 A 3.29 0.10 0.40 A s A s .
23 0 .
06 0 .
256 B -0.20 3.31 0.42 - - - - B s − .
21 3 .
32 0 .
447 B -0.28 0.25 1.26 B s -0.41 0.25 1.24 B s − .
27 0 .
24 1 .
258 A 3.14 3.14 -0.04 A A .
14 3 . − .
049 A 0.02 -0.02 3.22 A s -0.08 -0.08 3.22 A s . − .
02 3 . B s .
32 0 .
15 2 . A s -0.08 3.22 3.22 A s − .
00 3 .
14 3 . A s .
31 0 .
15 3 . B s B s . − .
10 3 . B B s .
02 3 .
02 3 .
39- - - - - B s Trilinear Poincaré Exact (2 d.p.)1 B 0.30 -0.25 -1.48 B s B s . − . − .
482 A 0.05 -0.00 -0.01 A s A s .
05 0 . − .
013 A 2.86 -0.19 -0.60 A A . − . − .
604 B 2.98 -0.15 0.23 B B s . − .
15 0 .
225 A 3.23 0.06 0.25 A s A s .
23 0 .
06 0 .
256 B -0.21 3.32 0.44 B s -0.18 3.32 0.46 B s − .
21 3 .
32 0 .
447 B -0.27 0.24 1.25 B s -0.26 0.22 1.25 B s − .
27 0 .
24 1 .
258 A 3.14 3.13 -0.04 A A .
14 3 . − .
049 A 0.02 -0.02 3.21 A s -0.02 -0.02 3.24 A s . − .
02 3 . B s B s .
32 0 .
15 2 . A s -0.02 3.16 3.16 A s − .
00 3 .
14 3 . A s A s .
31 0 .
15 3 . B s B s . − .
10 3 . B s B s .
02 3 .
02 3 . With the ‘small-scale’ and ‘medium-scale’ tetrahedron config-urations described above, the nulls never pass exactly through the spacecraft tetrahedron. Therefore, to test the Poincaré indexmethod we have created trajectories with a ‘large-scale’ tetrahe-dron (see the Appendix), to ensure the possibility of (true) pos-itive results. The results of applying the Poincaré index methodto this data set are shown in the bottom panel of Figure 5. Weobserve that all nulls that happen to be enclosed by the artifi-cial spacecraft constellation, namely the nulls 8, 2, and 11, havebeen correctly identified. We have found that, in accordance withGreene (1992), the Secant method provides a bad estimate of theenclosed null location, often outside the tetrahedron. Therefore,the best practice is to provide the centre of the tetrahedron as thelocation of the null point.
As expected, the FOTE method is able to detect the nulls whenthey are some distance away from the tetrahedron. Moreover,FOTE detects a null feature in all cases where the null pointcomes within a distance of 1 from the tetrahedron centre, for alltetrahedron sizes tested. The accuracy of the distance and nulltype assessment tends to degrade for larger tetrahedron sizes, asexpected. The Poincaré index method also detects all nulls thatcould be expected (those that pass through the tetrahedron). Thetwo methods for assessing the type of the null perform similarly,with success rate around 50%. When multiple nulls are located
Article number, page 7 of 12 & A proofs: manuscript no. nulls
Index
R= 0.4 | R | O X
Index
MMS-type Trajectory | R | O X
Index
Cluster-type Trajectory | R | O X
Index | R | m i n O X
Index W i t h i n o r N o t { E x a c t " s m a ll - s c a l e "" m e d i u m - s c a l e "" l a r g e - s c a l e "" l a r g e - s c a l e " ; P o i n c a r é Fig. 5.
Null points detected based on the simulated spacecraft trajectories as described in Appendix A. In the top panel is the exact distance fromthe centre of the spacecraft tetrahedron to the nearest null ( | R | ), its type, and its number based on Table A.2. The second, third and fourth panelsshow the results from the FOTE method with a tetrahedron with S = . S = .
12, and S = .
4, respectively. The bottom panel shows theresults of applying the Poincaré index method for the large-scale tetrahedron with S = . close together (e.g. nulls 4 and 5), or too far from the spacecraft(in case of FOTE), both methods detect the presence of nulls, butshow noisy results in detection / distance / typing. Tetrahedron trajectory considerations
The results discussed above are based upon data measured alongtrajectories that traverse a circular path in the xy -plane of ourdomain. Clearly these trajectories do not mimic the behaviourof spacecraft constellations such as Cluster or MMS, which forexample in the magnetotail move only slowly as magnetic struc-tures are convected backwards and forwards past them. How- ever, we do not expect the nature of these trajectories to influencethe results. The shape of the trajectories is chosen to bring themclose to as many of the null points in the domain as possible, inorder to test the field reconstruction around each of those nulls,and thus make our analysis more robust. The null point identifi-cation is not a ff ected locally by the shape of the trajectory (sinceonly the data at a single time – or two adjacent times for SOTE –is used), but rather by the separation of the ‘spacecraft’. In orderto mimic the e ff ects of small-scale fluctuations in the fields andnoise, we added the small-scale fluctuations to the trajectories inEquations (A.3–A.5). Article number, page 8 of 12. Olshevsky et al.: Methods for finding magnetic nulls
Table 2.
FOTE method applied to measurement along trajectories. Thefinal column ( | R | ) gives the closest approach of the centre of the tetra-hedron to the null. Index Type x y z | R | medium-scale: S = .
12 Nulls found: 141 X 2.9372 -0.1778 0.1631 0.12942 A 3.1385 3.1349 -0.0326 0.12823 B 3.3098 3.5301 2.8745 0.61954 B 3.3493 2.9556 2.9583 0.94805 O -0.1163 3.1190 0.1774 0.52186 O 0.0454 -0.0002 -0.0109 0.02767 O 4.1302 0.8054 1.7989 1.73188 O -0.0011 3.1369 3.1722 0.07269 O 0.0184 -0.0149 3.2124 0.157410 As 3.1649 3.1176 -0.0065 0.410211 As 3.2514 0.0774 0.2758 0.266912 Bs 4.0884 0.7715 1.7469 1.689113 Bs -1.0344 1.1210 1.3510 1.663514 Bs 3.3374 0.1237 2.6662 0.3175small-scale: S = .
025 Nulls found: 101 X 2.9668 -0.1497 0.1298 0.14082 A 3.1386 3.1344 -0.0320 0.19743 B 3.4131 3.5137 2.6837 0.75734 B 3.0098 3.0167 3.4307 0.58735 O -0.1291 3.1253 0.2019 0.60166 O 0.0457 -0.0005 -0.0104 0.08987 O -0.0006 3.1376 3.1733 0.15568 O 0.0197 -0.0166 3.2153 0.23129 As 3.1832 3.1089 0.0113 0.568010 Bs 3.2822 0.1041 2.8751 0.2343
5. Conclusion
This work intends to help researchers who want to analyse nullpoints (stagnation points) of divergence-free vector fields in theirsimulations or observations. There are two situations that arecommonly encountered in practice: (i) numerical simulations onhexahedral or tetrahedral meshes, and (ii) data from tetrahedraof spacecraft (MMS and Cluster). Each of these cases was as-sessed independently using the same test magnetic field. This isthe first time that such methods have been tested and comparedagainst a ‘ground truth’ situation where null numbers, positions,and types were known to arbitrary precision based on an analyt-ical expression for the magnetic field. The main results of ourstudy relevant to ‘8-point methods’ used for rectangular meshesfrom numerical simulations are the following. – When the field is moderately resolved (1,2, or fewer gridpoints between nulls), both the Poincaré index and trilinearmethods give errors, but the trilinear method has no falsepositives (PI method has 1), fewer false negatives (2 vs 4),and the performance on null type is the same (one incorrectA vs B identification each). This suggests that the trilinearmethod is more robust when the field is quite ‘rough’ on thegrid. – When the field is well resolved, ( > – Both methods can be e ffi ciently implemented to run in lessthan 1 second on an 80 grid for the present test field, andshow no substantial di ff erence in scaling with resolution.Concerning 4-point methods typically applied to spacecraftdata, we have considered three di ff erent sizes for the spacecrafttetrahedron, the smallest two of which can be considered as‘Cluster-scale’ and ‘MMS-scale’ on the basis of a physically-motivated dimensionsalisation of the field – see Section 2.3. Weconclude that FOTE performs well in finding the nulls whenthey are not close together – roughly speaking, when the nullseparation is larger than the null-spacecraft distance. (The maindiscrepancies in Figure 5 are the use of X and O for nulls thatare close to 2D). On the other hand, for null pairs that are closetogether the detection method fails (interestingly it still detectsa null, but the inferred type jumps around a lot). Probably thiscould be used to indicate multiple adjacent nulls.The practical advice is to use the FOTE method for locat-ing the nulls in the fields measured by probes or spacecraft. Inthe numerical simulations on the rectilinear grids the trilinearmethod gives more accurate null location. On the meshes oneshould use a variation of the PI on either hexahedral or tetra-hedral cells. The null location produced by the latter should betaken in the middle of the cell as the secant method of the loca-tion estimation could produce unreliable results.
6. Discussion
We propose that the fields defined in the appendix and used herecould be used to test / benchmark future null finders. For examplea method based on an expansion in spherical harmonics used by(He et al. 2008; Li 2019). The original intention of this methodis to reconstruct magnetotail magnetic structure around magneticnull observed by local satellite (He et al. 2008; Guo et al. 2016).Based on satellite measurements, it reconstructs the magneticfield by taking advantage of a fitting function approach. To matchthe 12 observed magnetic field components, 10 fitting parame-ters are presented in 10 spherical harmonic functions, and theother two are in the Harris current sheet model (Harris 1962).Thus, by fitting the simultaneous magnetic field vectors, one canreconstruct the local magnetic field. The calculations in He et al.(2008) confirmed the existence of a magnetic field null in recon-nection event, and present a magnetic structure around a 3-D nullin the magnetotail. For convenience, we provide the theoreticalformulation of this method in Appendix B.In order to apply this method, four point measurements arerequired in the data cube. Any four points that form a tetrahe-dron in the data box introduced in Section 2 can be used forreconstruction experiments. For example, in a 80 size data box,one can first choose a 2 data box, and separate it as five indepen-dent tetrahedrons. Then five reconstructions can be done basedon the tetrahedrons. The following is to check for magnetic nullsin these reconstructed magnetic structures. The advantage is thatit can be reconstructed to get multiple nulls, while other methodscan’t judge the existence of multiple zeros in the area surroundedby multiple satellites. Multiple tetrahedrons can be randomly se-lected for reconstruction, and the results obtained together withall data points can be compared. However, it would be less e ffi -cient if it is used as an ergodic calculation similar to the Poincaréindex. Also the results of the magnetic nulls need to check thereconstructed magnetic structures manually, which need to befurther improved in the future. Once the automated procedure Article number, page 9 of 12 & A proofs: manuscript no. nulls for inspecting and classifying the reconstruction results is devel-oped, this method can be tested against the proposed challenge.
Acknowledgements.
This work has resulted from ISSI / ISSI-BJ team activity 416“Magnetic Topology E ff ects on Energy Dissipation in Turbulent Plasma". Article number, page 10 of 12. Olshevsky et al.: Methods for finding magnetic nulls
Appendix A: Details of the test magnetic field
The magnetic field model used in this paper is B = ( − y ) + sin z ) e x + (2 sin x + sin z ) e y + (sin x + sin y ) e z + . − y ) + x cos z + sin( y ) + . e x + . x − z ) + sin( x + z ) + √ / e y + . − x sin z + sin( y ) + y ) − . e z + (cid:80) i = ∇ × (cid:18) a i k i exp (cid:18) − ( x − X i ) + ( y − Y i ) a i − ( z − Z i ) l i (cid:19) e z (cid:19) + (cid:80) i = ∇ × (cid:18) a i k i exp (cid:18) − ( x − X i ) + ( z − Z i ) a i − ( y − Y i ) l i (cid:19) e y (cid:19) + (cid:80) i = ∇ × (cid:18) a i k i exp (cid:18) − ( y − Y i ) + ( z − Z i ) a i − ( x − X i ) l i (cid:19) e x (cid:19) (A.1)where the values of a i , k i , l i , X i , Y i and Z i are given in Table A.1. Table A.1.
Parameter values for the magnetic field in Equation (A.1). i a i k i l i X i Y i Z i − . − . − . π + .
191 0.0117 π π − .
037 0.24 − .
22 1 0.9 π + . π + . π − .
22 2 0.5 π/ π/ π/
26 1 1 1 0.073 0.0198 − . − .
22 2 0.5 π/ π/ π/ r ( s ) = (cid:32) π √ s ) + π + f ( s ) + . (cid:33) e x + (cid:32) π √ s ) + π + f ( s ) (cid:33) e y + (cid:18) (1 − . s ) π s ) + π + f ( s ) (cid:19) e z (A.2)where f ( s ) = .
01 sin (20 s + + .
004 sin (23 s + + .
011 sin (13 s + + .
007 sin (37 s +
13) (A.3) f ( s ) = .
01 sin (19 s + + .
005 sin (25 s + + .
009 sin (17 s + + .
013 sin (33 s +
13) (A.4) f ( s ) = .
007 sin (22 s + + .
006 sin (24 s + + .
01 sin (13 s + + .
003 sin (39 s +
13) (A.5)Next, four constant vectors are added to this this expression todetermine four neighbouring trajectories for the four spacecraft: V = S √ . e x + . e y + . e z ) , V = S √ − . e x − . e y + . e z ) , V = S √ − . e x + . e y − . e z ) , V = S √ . e x − . e y − . e z ) . These vectors lie at the corners of a regular tetrahedron, witheach point lying a distance S from the centre of the tetrahedron (which has side-length, or spacecraft separation, S √ / ) .Here we consider three di ff erent tetrahedron sizes, with S = . , S = . and S = . , which we refer to as ‘small-scale’, ‘medium-scale’ and ‘large-scale’, respectively .The null points within the domain together with the eigen-values of the associated Jacobian matrix are given in Table A.2. Appendix B: Spherical expansion methodformulation
The fitting model is designed based on a total of 12 functions,including a constant background field, a function taken from theHarris current sheet model (Harris 1962), and 10 spherical har-monic functions. For the convenience of describing the potentialfield, the spherical harmonic functions are adopted as a part ofthe fitting model. Such fitting can be expressed as B γ B θ B φ = (cid:101) B r (cid:102) B θ (cid:102) B φ + T xyz → γθφ · B tanh z − z L z + B (cid:102) B θ (cid:102) B φ , (B.1)where (cid:16) B γ , B θ , B φ (cid:17) are the three spherical coordinate systemmagnetic field components at a spatial position ( γ, θ, φ ). Thefirst term on the right-hand side describes a potential field fromthe spherical harmonic series shown below. The transform ma-trix T xyz → γθφ . converts vector field from a common spacecraftCartesian coordinate system to a spherical coordinate system.The x-direction background magnetic field together with themagnetic field in a Harris current model is shown in this equa-tion. Expression for (cid:101) B r , (cid:102) B θ , (cid:102) B φ is shown as (cid:101) B r = (cid:88) n (cid:88) m − ( n + (cid:18) R e r (cid:19) n + · (cid:0) q mn cos ( m ϕ ) + h mn sin ( m ϕ ) (cid:1) · P mn (cos θ ) (cid:102) B θ = R e r (cid:88) n (cid:88) m (cid:18) R e r (cid:19) n + · (cid:0) q mn cos ( m ϕ ) + h mn sin ( m ϕ ) (cid:1) · ( − sin θ ) · ∂∂θ (cid:0) P mn (cos θ ) (cid:1)(cid:102) B φ = R e r sin θ (cid:88) n (cid:88) m (cid:18) R e r (cid:19) n + · (cid:0) q mn ( − m ) · sin ( m ϕ ) + h mn · m · cos ( m ϕ ) (cid:1) · P mn (cos θ ) , where P mn is the associated Legendre function with degree n andorder m of [ n , m ] = [1 , , [2 , , [2 , , [3 , , [3 , q mn and h mn are the coe ffi cients in the spherical harmonic series. Appendix C: Null-finder validation tools
Data files and source codes for test-ing null-finders are available online athttps: // bitbucket.org / volshevsky / magneticnullchallenge. References
Burch, J. L., Moore, T. E., Torbert, R. B., & Giles, B. L. 2016, Space ScienceReviews, 199, 5Chen, X. H., Fu, H. S., Liu, C. M., et al. 2017, The Astrophysical Journal, 852,17Chen, Z. Z., Fu, H. S., Wang, Z., Liu, C. M., & Xu, Y. 2019, Geophysical Re-search Letters, 46, 10209Close, R. M., Parnell, C. E., MacKay, D., & Priest, E. R. 2003, Solar Phys., 212,251
Article number, page 11 of 12 & A proofs: manuscript no. nulls
Table A.2.
Null points in the domain and the associated eigenvalues - exact values.
Index Type x y z λ λ λ B s . − . − . − . . + . j . − . j A s . − . − . . − . + . j − . − . j A . − . − . . − . − . B s . − . . − . . + . j . − . j A s . . . . − . + . j − . − . j B s − . . . − . . + . j . − . j B s − . . . − . . + . j . − . j A . . − . . − . − . A s . − . . . − . + . j − . − . j B s . . . − . . + . j . − . j A s − . . . . − . + . j − . − . j A s . . . . − . + . j − . − . j B s . − . . − . . + . j . − . j B s . . . − . . + . j . − . j Deng, X. H., Zhou, M., Li, S. Y., et al. 2009, J. Geophys. Res., 114, A07216Eriksson, E., Vaivads, A., Khotyaintsev, Y. V., Khotyayintsev, V. M., & André,M. 2015, Geophys. Res. Lett., 42, 6883Escoubet, C. P., Fehringer, M., & Goldstein, M. 2001, Annales Geophysicae, 19,1197Fu, H. S., Cao, J. B., Cao, D., et al. 2019, Geophysical Research Letters, 46, 48Fu, H. S., Vaivads, A., Khotyaintsev, Y. V., et al. 2017, Geophysical ResearchLetters, 44, 37Fu, H. S., Vaivads, A., Khotyaintsev, Y. V., et al. 2015, Journal of GeophysicalResearch (Space Physics), 120, 3758Fu, H. S., Wang, Z., Zong, Q., et al. 2020, Methods for Finding Magnetic Nullsand Reconstructing Field Topology (American Geophysical Union (AGU)),153–172Fukao, S., Ugai, M., & Tsuda, T. 1975, Rep. Ion. Sp. Res. Japan, 29, 133Greene, J. M. 1992, Journal of Computational Physics, 98, 194Guo, R., Pu, Z., Chen, L.-J., et al. 2016, Physics of Plasmas, 23, 052901Harris, E. G. 1962, Il Nuovo Cimento, 23, 115Haynes, A. L. & Parnell, C. E. 2007, Physics of Plasmas, 14, 082107He, J.-S., Tu, C.-Y., Tian, H., et al. 2008, Jour-nal of Geophysical Research: Space Physics, 113[ https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2007JA012609 ]Hornig, G. & Schindler, K. 1996, Phys. Plasmas, 3, 781Khurana, K., Kepko, E., Kivelson, M., & Elphic, R. 1996, Magnetics, IEEETransactions on, 32, 5193Kumar, P., Karpen, J. T., Antiochos, S. K., et al. 2019, Astrophys. J., 873, 93Li, S. 2019, International Journal of Geosciences, 10, 967Liu, Y. Y., Fu, H. S., Olshevsky, V., et al. 2019, The Astrophysical Journal Sup-plement Series, 244, 31Longcope, D. W. & Parnell, C. E. 2009, Solar Phys., 254, 51Masson, S., Pariat, E., Aulanier, G., & Schrijver, C. J. 2009, Astrophys. J., 700,559Olshevsky, V., Deca, J., Divin, A., et al. 2016, Astrophys. J., 819, 52Olshevsky, V., Divin, A., Eriksson, E., Markidis, S., & Lapenta, G. 2015, Astro-phys. J., 807, 155Olshevsky, V., Servidio, S., Pucci, F., Primavera, L., & Lapenta, G. 2018, Astro-phys. J., 860, 11Parnell, C. E., Smith, J. M., Neukirch, T., & Priest, E. R. 1996, Phys. Plasmas,3, 759Politano, H., Pouquet, A., & Sulem, P. L. 1995, Physics of Plasmas, 2, 2931Pontin, D. I. 2012, Phil. Trans. R. Soc. A, 370, 3169Priest, E. R. 2014, Magnetohydrodynamics of the Sun (Cambridge UniversityPress, Cambridge.)Pucci, F., Servidio, S., Sorriso-Valvo, L., et al. 2017, Astrophys. J., 841, 60Retinò, A., Sundkvist, D., Vaivads, A., et al. 2007, Nature Physics, 3, 236Schrijver, C. J. & Title, A. M. 2002, Solar Phys., 207, 223van Oosterom, A. & Strackee, J. 1983, Biomedical Engineering, IEEE Transac-tions on, 30, 125Wang, Z., Fu, H. S., Olshevsky, V., et al. 2020, The Astrophysical Journal Sup-plement Series, 249, 10Wendel, D. E. & Adrian, M. L. 2013, Journal of Geophysical Research (SpacePhysics), 118, 1571Williams, B. M. 2018, PhD thesis, University of St AndrewsWyper, P. F. & Pontin, D. I. 2014, Phys. Plasmas, 21, 082114Xiao, C. J., Wang, X. G., Pu, Z. Y., et al. 2006, Nature Physics, 2, 478Yang, K., Guo, Y., & Ding, M. D. 2015, Astrophys. J., 806, 171]Hornig, G. & Schindler, K. 1996, Phys. Plasmas, 3, 781Khurana, K., Kepko, E., Kivelson, M., & Elphic, R. 1996, Magnetics, IEEETransactions on, 32, 5193Kumar, P., Karpen, J. T., Antiochos, S. K., et al. 2019, Astrophys. J., 873, 93Li, S. 2019, International Journal of Geosciences, 10, 967Liu, Y. Y., Fu, H. S., Olshevsky, V., et al. 2019, The Astrophysical Journal Sup-plement Series, 244, 31Longcope, D. W. & Parnell, C. E. 2009, Solar Phys., 254, 51Masson, S., Pariat, E., Aulanier, G., & Schrijver, C. J. 2009, Astrophys. J., 700,559Olshevsky, V., Deca, J., Divin, A., et al. 2016, Astrophys. J., 819, 52Olshevsky, V., Divin, A., Eriksson, E., Markidis, S., & Lapenta, G. 2015, Astro-phys. J., 807, 155Olshevsky, V., Servidio, S., Pucci, F., Primavera, L., & Lapenta, G. 2018, Astro-phys. J., 860, 11Parnell, C. E., Smith, J. M., Neukirch, T., & Priest, E. R. 1996, Phys. Plasmas,3, 759Politano, H., Pouquet, A., & Sulem, P. L. 1995, Physics of Plasmas, 2, 2931Pontin, D. I. 2012, Phil. Trans. R. Soc. A, 370, 3169Priest, E. R. 2014, Magnetohydrodynamics of the Sun (Cambridge UniversityPress, Cambridge.)Pucci, F., Servidio, S., Sorriso-Valvo, L., et al. 2017, Astrophys. J., 841, 60Retinò, A., Sundkvist, D., Vaivads, A., et al. 2007, Nature Physics, 3, 236Schrijver, C. J. & Title, A. M. 2002, Solar Phys., 207, 223van Oosterom, A. & Strackee, J. 1983, Biomedical Engineering, IEEE Transac-tions on, 30, 125Wang, Z., Fu, H. S., Olshevsky, V., et al. 2020, The Astrophysical Journal Sup-plement Series, 249, 10Wendel, D. E. & Adrian, M. L. 2013, Journal of Geophysical Research (SpacePhysics), 118, 1571Williams, B. M. 2018, PhD thesis, University of St AndrewsWyper, P. F. & Pontin, D. I. 2014, Phys. Plasmas, 21, 082114Xiao, C. J., Wang, X. G., Pu, Z. Y., et al. 2006, Nature Physics, 2, 478Yang, K., Guo, Y., & Ding, M. D. 2015, Astrophys. J., 806, 171