Exploring the biases of a new method based on minimum variance for interplanetary magnetic clouds
AAstronomy & Astrophysics manuscript no. demoulin_MV_arXiv c (cid:13)
ESO 2018September 5, 2018
Exploring the biases of a new method based on minimum variancefor interplanetary magnetic clouds
P. Démoulin , S. Dasso , and M. Janvier LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place JulesJanssen, 92195 Meudon, France, e-mail:
[email protected] CONICET, Universidad de Buenos Aires, Instituto de Astronomía y Física del Espacio, CC. 67, Suc. 28, 1428 Buenos Aires,Argentina, e-mail: [email protected] Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Ciencias de la Atmósfera y los Océanosand Departamento de Física, 1428 Buenos Aires, Argentina, e-mail: [email protected] Institut d’Astrophysique Spatiale, UMR8617, Univ. Paris-Sud-CNRS, Université Paris-Saclay, Bâtiment 121, 91405 Orsay Cedex,France e-mail: [email protected]
September 5, 2018
ABSTRACT
Context.
Magnetic clouds (MCs) are twisted magnetic structures ejected from the Sun and probed by in situ instruments. They aretypically modeled as flux ropes (FRs).
Aims.
Magnetic field measurements are only available along the 1D spacecraft trajectory. The determination of the FR global charac-teristics requires the estimation of the FR axis orientation. Among the developed methods, the minimum variance (MV) is the mostflexible, and features only a few assumptions. However, as other methods, MV has biases. We aim to investigate the limits of themethod and extend it to a less biased method.
Methods.
We first identified the origin of the biases by testing the MV method on cylindrical and elliptical models with a temporalexpansion comparable to the one observed in MCs. Then, we developed an improved MV method to reduce these biases.
Results.
In contrast with many previous publications we find that the ratio of the MV eigenvalues is not a reliable indicator of theprecision of the derived FR axis direction. Next, we emphasize the importance of the FR boundaries selected since they stronglya ff ect the deduced axis orientation. We have improved the MV method by imposing that the same amount of azimuthal flux should bepresent before and after the time of closest approach to the FR axis. We emphasize the importance of finding simultaneously the FRaxis direction and the location of the boundaries corresponding to a balanced magnetic flux, so as to minimize the bias on the deducedFR axis orientation. This method can also define an inner flux-balanced sub-FR. We show that the MV results are much less biasedwhen a compromize in size of this sub-FR is achieved. Conclusions.
For weakly asymmetric field temporal profiles, the improved MV provides a very good determination of the FR axisorientation. The main remaining bias is moderate (lower than 6 ◦ ) and is present mostly on the angle between the flux rope axis andthe plane perpendicular to the Sun-Earth direction. Key words.
Physical data and processes: magnetic fields, Sun: coronal mass ejections (CMEs), Sun: heliosphere
1. Introduction
The Sun release mass and magnetic field in a permanent so-lar wind, and also as transients called coronal mass ejections(CMEs). Coronal remote white-light observations using corona-graphs have shown that the distribution of mass in some CMEspresent images consistent with twisted structures (e.g., Krall2007; Gopalswamy et al. 2013; Vourlidas et al. 2013; Wood et al.2017). When CMEs are observed in the interplanetary mediumthey are called interplanetary CMEs (ICMEs). The associationbetween CMEs and ICMEs was well established since severaldecades (e.g., Sheeley et al. 1985). In fact, twisted flux tubes,or flux ropes (FRs), are present also in several other systems inthe heliosphere, such as the Sun atmosphere, the solar wind, anddi ff erent locations of planetary magnetospheres (e.g., Fan 2009;Imber et al. 2011; Smith et al. 2017; Pevtsov et al. 2014; Kilpuaet al. 2017). Flux ropes can store and transport magnetic energyand, because their magnetic field lines can be strongly twisted, Send o ff print requests to : P. Démoulin FRs can also contain and transport important amounts of mag-netic helicity (e.g., Lynch et al. 2005; Dasso 2009; Sung et al.2009; Démoulin et al. 2016).When FRs are observed in situ by a crossing spacecraft, theypresent a large and coherent rotation of the magnetic field vec-tor. A particular set of events that present these FR characteris-tics corresponds to magnetic clouds (MCs). They are also char-acterized by a stronger magnetic field and a lower proton tem-perature than the typical solar wind (e.g., Burlaga et al. 1981;Gosling 1990). Thus, they have a low plasma beta, so that mag-netic forces are expected to be dominant.Interplanetary MCs have been systematically observed fromthe 80’s, and several models have been proposed to describetheir magnetic structure. The simplest one and generally usedto model the field in MCs is an axially symmetric cylindricalmagneto-static FR solution, with a relaxed linear force-free field,the Lundquist’s model (Lundquist 1950; Goldstein 1983). Thismodel describes relatively well the field distribution for a signif-icant number of observed MCs (e.g., Burlaga & Behannon 1982;
Article number, page 1 of 15 a r X i v : . [ a s t r o - ph . S R ] S e p & A proofs: manuscript no. demoulin_MV_arXiv
Lepping et al. 1990; Burlaga 1995; Burlaga et al. 1998; Lynchet al. 2003; Dasso et al. 2005; Lynch et al. 2005; Dasso et al.2006). Still, the Lundquist solution is known to have di ffi cultiesin fitting the magnetic field strength, in particular it was foundthat it frequently overestimates the axial component of the fieldnear the FR axis (e.g., Gulisano et al. 2005).Several other models have been developed. These modelsinclude more general FR properties, such as non-linear force-free field (Farrugia et al. 1999), non-force free magnetic config-urations (e.g., Mulligan et al. 1999; Hidalgo et al. 2000, 2002;Cid et al. 2002; Nieves-Chinchilla et al. 2016), and models withnon-cylindrical cross section (e.g., Vandas & Romashets 2003;Nieves-Chinchilla et al. 2009, 2018a).In particular, the elliptical model of Vandas & Romashets(2003) provides a better fit to some observed MCs having a fieldstrength more uniform than in the Lundquist solution. This in-dicates the existence of some flat FRs (Vandas et al. 2005). Onthe other hand, from a superposed epoch analysis, Masías-Mezaet al. (2016) showed that while slow MCs present a symmetricprofile of the magnetic field, fast ones present a magnetic inten-sity profile with a significantly stronger B near the front than atthe rear.The models presented above were typically compared andfitted to MC in situ observations, which are limited to the 1Dcut provided by the spacecraft trajectory inside the FR. This al-lows a local reconstruction of the FR cross section, and thenit is possible to make estimations of global magnetohydrody-namic (MHD) quantities, such as magnetic fluxes, twist, helicity,and energy (e.g., Dasso et al. 2003; Leamon et al. 2004; Dassoet al. 2005; Mandrini et al. 2005; Qiu et al. 2007; Démoulinet al. 2016; Wang et al. 2016). However, all these estimationsdepend directly on the FR orientation, and thus on the qualityof the method used to get it. Moreover, the orientation itself isan important property of a given FR in order to compare it withits solar origin (e.g., Nakwacki et al. 2011; Isavnin et al. 2013;Palmerio et al. 2017).Di ff erent methods to estimate the FR orientation from space-craft observations have been developed and applied to MCs.Some authors have used the so-called Grad-Shafranov methodto get the FR orientation, which consists in applying the Grad-Shafranov formalism, valid for describing general MHD mag-netostatic equilibria invariant in one direction (Sonnerup & Guo1996; Hau & Sonnerup 1999; Sonnerup et al. 2006; Isavnin et al.2011). This method use both the magnetostatic constraints andthe folding of the same magnetic field line when it is crossed bythe spacecraft during its in-bound and out-bound travel acrossthe FR. However, the method cannot recover the FR axis forthe simplest magnetic configurations, such as symmetric FRsor when the spacecraft crossed the FR close to its axis. Hu &Sonnerup (2002) argued that observed FRs are typically asym-metric and that this asymmetry removes the above di ffi culty. Infact a detailed study, Démoulin et al. , in preparation, show thatfor FRs typical of MCs the Grad-Shafranov method provides abiased orientation, which depends on the symmetry properties ofthe magnetic field components.Finally, the method that has less hypotheses, mainly becauseit does not introduce the details of a given model, is the Min-imum Variance method (MV). The MV method just requires awell ordered large scale variance of B in the three spatial direc-tions. MV has been extensively used to find the orientation ofstructures in the interplanetary medium (see e.g., Sonnerup &Cahill 1967; Burlaga & Behannon 1982; Hausman et al. 2004;Siu-Tapia et al. 2015). Several authors have shown that the MVmethod estimates quite well the orientation of the FR axis, when the distance between the axis and the spacecraft trajectory in theMC is small with respect to the FR radius (see e.g., Klein &Burlaga 1982; Bothmer & Schwenn 1998; Farrugia et al. 1999;Xiao et al. 2004; Gulisano et al. 2005, 2007; Ru ff enach et al.2012, 2015). Other authors use the MV method to get a first or-der approximation for the MC orientation. Then, they use thisestimation as a seed in a non-linear least squares fit of a mag-netic model to the data. This approach is expected to improve thecloud orientation (e.g., Lepping et al. 1990, 2003, 2006; Dassoet al. 2005).The most crucial information to introduce in any of the meth-ods to get the FR orientation is the start and end time of the FR.This is a major problem because di ff erent authors generally de-fine the FR boundaries at di ff erent places / times (e.g., Riley et al.2004; Russell & Shinde 2005; Dasso et al. 2006; Al-Haddadet al. 2013), with the consequent di ff erence on the determina-tion of both the axis orientation and the parameters for the pro-posed models. One of the reasons at the origin of these di ff erentboundaries is the partial erosion of the FR due to magnetic re-connection. If it happens in the FR front, this creates after theclosed FR a ’back’ which presents mixed signatures of a FR andstationary solar wind (Dasso et al. 2006). This erosion processalso can occur at the FR rear, producing a mixed region beforethe beginning of the real closed FR (Ru ff enach et al. 2015). Thislack of exact information on the FR boundary can have an im-portant influence on the proper determination of the FR axis di-rection. Indeed, comparing di ff erent methods show a large dis-persion of the estimated FR orientation (Al-Haddad et al. 2013;Janvier et al. 2015).In Sect. 2 we present a geometrical description of FRs in thesolar wind, proposing new angles to determine its orientation,and review the MV method and its application to get the orien-tation of MCs. In Sect. 3 we generate synthetic linear force-freeFRs with symmetry of translation along their axis, including thepossibility of expansion and cylindrical / elliptical cross-sections,and emulate the observed time-series observed by spacecraft.Then, in Sect. 4 the application of several variants of the MVtechnique are applied to the synthetic clouds, finding biases pro-duced by di ff erent MCs properties. In Sect. 5 a deeper analysisof the bias introduced due to a boundary selection is done, andwe proposed a new MV method to minimize this bias. Our con-clusions are given in Sect. 7, and in the Appendix we show theexpected coupling between field components when MV is ap-plied.
2. Geometry of Flux ropes
The coordinates generally used for the analysis of data providedby spacecraft located in the vicinity of Earth are defined in theGeocentric Solar Ecliptic (GSE) system of reference. It is de-fined with an orthogonal base of unit vectors ( ˆ x GSE , ˆ y GSE , ˆ z GSE ).ˆ x GSE points from the Earth toward the Sun, ˆ y GSE is also in theecliptic plane and in the direction opposite to the Earth rotationmotion around the Sun, and ˆ z GSE points to the north pole of theheliosphere (Fig. 1a). A similar coordinate system is the helio-graphic radial tangential normal (RTN) system of reference (e.g.,Fränz & Harper 2002) where the following could also be tran-scripted.The FR axis orientation is classically defined with respect tothe GSE system in spherical coordinates with the polar axis cho-sen as + ˆ z GSE , by two angles: the longitude ( ϕ axis ) and the latitude( θ axis ). However these angles are not the natural ones to describe Article number, page 2 of 15émoulin et al.: Improving minimum variance method for expanding symmetric flux ropes z FR,pr ! (a) $ y GSE ! z GSE ! x GSE ! i $ λ $ MC y FR ! x FR ! z FR ! z FR,pr ! z FR ! x FR ! x GSE ! λ ! λ ! (b) ! Fig. 1.
Definitions of the angles of the FR axis. (a)
Schema showingthe observation, here GSE, and FR frames. The FR frame is definedby the vectors ˆ x FR , ˆ y FR , ˆ z FR where ˆ z FR is along the FR axis and in thedirection of the axial magnetic field, ˆ x FR is orthogonal to ˆ z FR and inthe plane defined by ˆ x GSE and ˆ z FR , and ˆ y FR completes the orthonormaldirect basis. Two rotations are needed to pass from the GSE frame to theFR frame. They are defined by the angles i and λ which are respectivelythe inclination and the location angle (spherical coordinates with thepolar axis ˆ x GSE ). (b) Schema showing the meaning of the angle λ in theplane of the FR axis shown in panel (a) in light blue. ˆ z pr , FR , in green, isthe projection of ˆ z FR in the plane orthogonal to ˆ x GSE , see panel (a). the FR axis orientation with respect to its translation direction( ≈ − ˆ x GSE ). Indeed, the geometrical configuration of the space-craft crossing is defined both by the closest approach distanceand the angle between the spacecraft trajectory and the FR axisorientation. A rotation of the FR around the spacecraft trajectorydoes not change the geometry of the spacecraft crossing, whileit changes both ϕ axis and θ axis . This implies that the same cross-ing geometry is present along curves defined in the { ϕ axis , θ axis }space. Even worse, this choice of reference system has the dis-advantage to set the polar axis ( θ = ± ◦ ) along ˆ z GSE whichis both a possible and an un-particular axis direction for MCswhile the polar axis is singular as it corresponds to any values of ϕ axis . Therefore, the coordinates { ϕ axis , θ axis } are not appropriateto study the orientation of the FR.The motion of MCs is mainly radial away from the Sun, es-pecially away from the corona (where limited deflection couldoccur). Then, the radial direction is a particular direction. Wethen set a new spherical coordinate system having its polar axisalong ˆ x GSE (Fig. 1a). This direction could correspond to thespacecraft crossing a FR leg. However, such crossing does notallow to detect the rotation of the magnetic field as the FR is typ-ically only partially crossed on one side. It implies that FRs withan axis direction almost parallel to ˆ x GSE are mostly not present inMC data sets. Then, the direction ˆ x GSE can be used as the polaraxis of the new spherical coordinate system.The local axis direction of the FR is called ˆ z FR (with B z , FR > i ) and the location ( λ ) angles as shown in Fig. 1.More precisely, let us define the unit vector ˆ z pr , FR , in green inFig. 1a, along the projection of the FR axis on the plane definedby (ˆ y GSE , ˆ z GSE ), then i is the angle from ˆ y GSE to ˆ z pr , FR . Similarly, λ is the angle from ˆ z pr , FR to ˆ z FR .If the FR axis is located in a plane, i is the inclination of thisplane (in light blue) on the ecliptic as shown in Fig. 1a. And if we further suppose that the distance to the Sun is monotonouslydecreasing from the FR apex to any of the FR leg when followingthe FR axis, the angle λ is evolving monotonously along the FRfrom − ◦ in one of the FR leg, to λ = λ = ◦ in the other leg. It implies that λ can be used implicitlyas a proxy of the location where the spacecraft intercepts the FR(as shown in Fig.1c of Janvier et al. 2013). Finally, the aboveconclusions extend to other coordinate systems such as RTN.We next define the FR frame. ˆ z FR corresponds to the FR axis,as defined before. Since the speed of MCs is much higher thanthe spacecraft speed, we assume a rectilinear spacecraft trajec-tory defined by the unit vector ˆ d . We define ˆ y FR in the directionˆ z FR × ˆ d and ˆ x FR completes the right-handed orthonormal base( ˆ x FR , ˆ y FR , ˆ z FR ) which defines the FR frame (Fig. 1a). The MV method finds the directions in which the projection ofa series of N vectors has an extremum mean quadratic deviation(e.g., Sonnerup & Scheible 1998). This method can be appliedto the time series of the magnetic field B measured in situ acrossMCs, and it provides an estimation of the FR axis orientation asfollows.The mean quadratic deviation, or variance, of the magneticfield B in a given direction defined by the unit vector ˆ n is: σ n = N N (cid:88) k = (cid:16) ( B k − (cid:104) B (cid:105) ) · ˆ n (cid:17) = (cid:104) ( B n − (cid:104) B n (cid:105) ) (cid:105) (1)where the summation is done on the N data points of the timeseries and (cid:104) B (cid:105) is the mean magnetic field.The MV method finds the direction ˆ n where σ n is extremum.The constraint | ˆ n | = σ n n j = (cid:88) i = m i j n i = λ L n j (2)where λ L is the Lagrange multiplier and m i j = (cid:104) B i B j (cid:105) − (cid:104) B i (cid:105)(cid:104) B j (cid:105) (3)The matrix m i j is symmetric with real positive eigenvalues:0 (cid:54) ev min (cid:54) ev int (cid:54) ev max . Their corresponding orthogonaleigenvectors ( ˆ x MV , ˆ z MV , ˆ y MV ) are the directions of minimum,intermediate and maximum variation of the magnetic field, re-spectively.The magnetic configuration of MCs is assumed to be a FRwith some generic properties which have direct implications onthe eigenvalues and eigenvectors found by the MV applied toMC data as follows. First, the axial field, B z , FR , is stronger on theFR axis and decline to a low value near the boundary. Second, theazimuthal field vanishes on the axis (to have no singular electriccurrent) and typically grows to a magnitude comparable to theaxial field strength at the FR periphery.For trajectories passing close enough from the FR axis, theazimuthal field is mostly in the ˆ y FR direction. Then, the vari-ance of B y , FR is expected to be about twice the one of B z , FR .The variance of B x , FR is the lowest one. Then the eigenvectors( ˆ x MV , ˆ y MV , ˆ z MV ) provide an estimation of ( ˆ x FR , ˆ y FR , ˆ z FR ), takinginto account that the direction of the eigenvectors provided byMV could need a change of sign in order to satisfy the conven-tion defined before for the FR frame. Next, since ˆ y FR is perpen-dicular to ˆ d (defining the spacecraft trajectory), then the angle Article number, page 3 of 15 & A proofs: manuscript no. demoulin_MV_arXiv between ˆ y MV and ˆ d allows to test how close is the MV framefrom the FR frame. Finally, as the spacecraft trajectory is furtherfrom the FR axis, this estimation of the FR frame is expectedto be worse. This is quantitatively tested in Sects. 4-6 with themodels presented in Sect. 3.
3. Test models
We describe in this section the geometrical aspects of the space-craft trajectory, and the simulation of measured magnetic fieldalong the trajectory. The magnetic models include key ingre-dients observed or consistent with observations of MC: typicalmagnetic profiles, expansion, flatness of the FR cross section andasymmetry due to magnetic reconnection. These models are de-scribed in the FR frame, with the origin of the reference systemset at the FR axis.
During the spacecraft crossing of the FR, its small accelerationhas a negligible e ff ect on the measured quantities (i.e., on thetime where they are measured, Démoulin et al. 2008). Then, weconsider a motion with a constant velocity V c during the space-craft crossing. Supposing that the FR moves along − ˆ x GSE , thespacecraft position, r S / C ( t ), in the FR frame is: x ( t ) = V c t cos λ y ( t ) = y p (4) z ( t ) = V c t sin λ where t is the time with t = | y p | , to the FR axis. Thus, t < t > B FR ( x , y , z , t ),is known in the FR frame (with an analytical or a numerical sim-ulation). This magnetic field is transformed to the GSE frameby two rotations (with angles i FR and λ FR ). Then, inserting thespacecraft trajectory r S / C ( t ), Eq. (4), in B GSE ( r ), limits the de-scription of B to the one observed along the spacecraft trajectory,noted B obs ( t ), which is only a function of time. After samplingthe time uniformly, this time series models the synthetic mag-netic field observed by a spacecraft crossing the magnetic fieldconfiguration comparable to in situ data obtained in MCs.The direct MV method, Sect. 2.2, and later on includingsome additional procedures, Sect. 5, are applied below to theabove synthetic field. The estimated angles, i MV and λ MV , arenext compared to i FR and λ FR allowing to test the performanceof the MV method on an ensemble of models by scanning theparameter space. We consider a general magnetic field configuration, B ( x , y , z ), associated to an element of fluid locatedat ( x , y , z ) and defined at time t . Below, to be coher-ent with Eq. (4), we simply set t = e ( t ) implies x = e ( t ) x , y = e ( t ) y , z = e ( t ) z where x , y , z are the newcoordinates of the same element of fluid, but at time t , with e (0) =
1. Only an isotropic expansion is considered here sinceit preserves the force balance of force-free configurations whilekeeping the magnetic field structure, while a non-isotropic ex-pansion introduces forces which deform the FR so that it wouldrequire a numerical simulation to determine the correspondingexpanded configuration.The expanded magnetic field, B , is function of space andtime: B ( x , y , z , t ) = B ( x / e ( t ) , y / e ( t ) , z / e ( t ) ) e ( t ) . (5)The term e ( t ) at the denominator is included to preserve themagnetic flux. Equation (5) describes the temporal evolution inthe 3D space. Introducing Eq. (4) into Eq. (5), this limits thedescription of B to the observed field along the spacecraft tra-jectory with B obs ( t ) only function of time.The expansion is driven by the decrease of total pressure inthe surrounding of the FR as it is moving away from the Sun(Démoulin & Dasso 2009a). This total pressure can be approxi-mated by a power law of the distance from the Sun to the FR cen-ter ( D ( t )). The force balance at the FR boundary implies that e ( t )is also a power law of D ( t ): e ( t ) = ( D ( t ) / D ) ζ where ζ is the nor-malized expansion factor (e.g., Démoulin et al. 2008) and D is areference distance taken here as the spacecraft distance from theSun. The velocity of the FR core, V c , is approximately constantduring the spacecraft crossing, as in Eq. (4), then D ( t ) = V c t + D with the spacecraft being located at D = D at t =
0. In thisframe work e ( t ) is: e ( t ) = (1 + τ ) ζ ≈ + ζ τ , (6)with τ = V c t / D , (7)which corresponds to the time di ff erence between the observedtime and the time when the axis reaches D , normalized by thetravel time from the Sun to the distance D at velocity V c . Thelinear approximation on the right side of Eq. (6) provides a goodapproximation for most MCs because ζ ≈ τ max << τ max being half of the FR crossing time (Démoulin et al.2008). It implies that the expansion typically introduces only asmall modification to the observed field profile compared to acase without expansion. We consider in this subsection a cylindrically symmetric mag-netic field configuration for the FR, with its axis along the z di-rection in the FR frame. Then, at a time t = B ( r ) is onlyfunction of the distance to the axis (cid:113) x + y , or equivalently ofthe relative position of an element of fluid from the FR axis, ρ ,that is the distance normalized by the FR radius, b . Consider-ing only expansion (e.g., no magnetic di ff usion) ρ is preservedduring the self-similar isotropic expansion: ρ = (cid:113) x + y b = (cid:112) x ( t ) + y ( t ) b ( t ) , (8)where b and b ( t ) are the radius of the FR at t = t re-spectively. As other spatial variables (Sect. 3.2), b ( t ) is simplyrelated to b as b ( t ) = e ( t ) b . ρ is a Lagrangian marker of therelative distance to the FR axis. It follows the magnetic field as Article number, page 4 of 15émoulin et al.: Improving minimum variance method for expanding symmetric flux ropes - Bx - - - - By - Bz - B - - - - Bx - - - - - - By - - Bz - - B (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:5)(cid:6)(cid:7)(cid:8) - Bx - - - - By - Bz - B (a)$$$b/a=1,$$$p$=$0$(b)$$$b/a=1,$$$p$=$,$0.5$(c)$$$b/a=2,$$$p$=$,$0.5$ ζ=0$ζ=1$ζ=0$ζ=0$ ζ=1$ζ=2$ t$ t$ t$ t$t$ t$ t$ t$t$ t$ t$ t$ ζ=2$ ζ=2$ζ=2$ζ=0$ Fig. 2.
Simulations of observed magnetic field components in the FR frame and | B | versus time (in hour) for synthetic expanding (a,b) circular and(c) elliptical linear force-free FR with positive magnetic helicity. The impact parameter is defined by p = y / b where b is the FR half-size in thedirection orthogonal to the simulated FR crossing. The FR boundaries are set at B z =
0. The results with three values of the normalized expansionfactor, ζ , are shown with di ff erent colors. A value of ζ ≈ V x , GSE ( t ) in MCsfrom 0.3 to 5 AU). The mean observed case, b = . B = t = V c =
400 km.s − are only scalingthe axis. the expansion occurs. In the FR frame, the magnetic field can bewritten simply as B ( ρ, t ) = B ( ρ ) / e ( t ) . (9)The spacecraft rectilinear trajectory satisfies y p = constant.We define the impact parameter as p = y p / b . Using Eq. (4),Eq. (8) is rewritten along the spacecraft trajectory as: ρ obs ( t ) = (cid:112) ( τ cos λ D / b ) + p e ( t ) (10)where the subscript "obs" has been added to specify that ρ isdetermined along the simulated spacecraft trajectory so ρ obs isfunction of time (and it corresponds to the di ff erent elements offluid observed by the spacecraft), while ρ , defined in Eq. (8),is a general Lagrangian marker, so independent of time. Sincethe simulated magnetic field along the spacecraft trajectory is B ( ρ obs ( t )) / e ( t ), the magnitudes of | τ | and ζ determine howstrongly the expansion a ff ects the modeled magnetic field (com-pared to a case without expansion).In the FR frame, B = ( − B θ sin θ, B θ cos θ, B z ) where B θ ( ρ )and B z ( ρ ) are the azimuthal and axial components respectively. Its components along the spacecraft trajectory are: B x , FR ( t ) = − B θ ( ρ obs ) e ( t ) p ρ obs e ( t ) B y , FR ( t ) = B θ ( ρ obs ) e ( t ) V c t cos λ b ρ obs e ( t ) (11) B z , FR ( t ) = B z ( ρ obs ) e ( t )For a linear force-free field (Lundquist’s FR): B θ, FR ( ρ obs ) = B J ( αρ obs ) B z , FR ( ρ obs ) = B J ( αρ obs ) (12)where B is the axial field strength of the FR axis. We define α as the first zero of the Bessel function J , α ∼ .
4, then ρ obs = ρ obs ≈
1, while some cases with ρ obs (cid:39) ζ ≈ . ± . ζ ≈ . ± . ζ ≈ . ± . ζ = , ,
2, which
Article number, page 5 of 15 & A proofs: manuscript no. demoulin_MV_arXiv are respectively the cases without, with typical and with twice aslarge of an expansion rate than observed. As ζ or / and the FR ra-dius increase, the simulated observed field is more asymmetric.Finally, the case ζ = B components for not extremely large FR radius (with aradius (cid:46) . B x , FR and B z , FR temporal profiles are nearly symmetricin time while B y , FR is nearly antisymmetric. A non-zero impactparameter p introduces a component B x , FR , whose magnitude in-creases with | p | , see Eq. (11). This property can be used in ob-servations to estimate | p | if a B θ ( ρ ) profile is assumed (see Sect.4.3 in Démoulin & Dasso 2009b). Based on di ff erent models andthe analysis of synthetic FRs, an empirical method to estimate | p | from the observed B x , FR profile was developed in Gulisano et al.(2007). Vandas & Romashets (2003) generalized the linear-force freefield of Lundquist to a FR with an elliptical cross-section. Alongand across the trajectory, more precisely in the x and y directionsof the FR frame, the maximum FR extension is 2 a and 2 b ,respectively. The Lundquist’s solution is recovered for b = a .Since the encountered or overtaking solar wind tends to com-press the FR in the radial direction away from the Sun, then therelative sizes are expected to satisfy a ≤ b .With invariance along the FR axis, the linear force-free fieldequations are reduced to the Helmholtz equation (cid:52) A + α A = B x , B y , B z ) = ( ∂ A /∂ y , − ∂ A /∂ x , α A ). Vandas & Romashets(2003) solved this equation with elliptic cylindrical coordinates,one of the few coordinate systems where the Helmholtz equationhas separable solutions. They set the magnetic field to be tangen-tial to the elliptical boundary with a vanishing axial component.For all b / a values, they found an analytical solution expressedwith the even Mathieu function of zero order.The above elliptical model can be set in self similar expan-sion as done above for the Lundquist’s field. The sizes become a = a e ( t ) and b = b e ( t ). The cartesian components of the el-liptical model solution ( B x , B y , B z ) are first expressed in functionof x , y . Then, the use of Eqs. (4) and (5) provide the simulatedcrossing of the FR. The impact parameter p is generalized to p = y p / b , with y p being the minimum signed distance to the FRaxis. Finally, Eq. (8) is generalized to ρ = (cid:113) ( x / a ) + ( y / b ) = (cid:113) ( x / a ) + ( y / b ) (13)with a , b defined at t = a , b at t . The selected boundarycondition implies B z , FR ( ρ = =
0. Finally, along the spacecrafttrajectory ρ obs ( t ) is still expressed by Eq. (10).An example of field component profiles of this ellipticalstructure is shown in Fig. 2c with b / a =
2. The main di ff erencewith the circular case, Fig. 2b, is a flatter B ( t ) profile. Indeed,since the magnetic tension is less important due to the FR elon-gation in the y direction, the outward gradient of the magneticpressure is lower then B is more uniform. In the absence of magnetic di ff usion and reconnection a givenvalue of ρ (Eq. (8)) traces a selected cylindrical shell within theFR. Along the spacecraft trajectory the front and rear boundariesare located at ρ front and ρ rear , respectively. Below, we considereroded FRs, with 0 < ρ front ≤ < ρ rear ≤
1, in order to include possible events where the peeling of the FR via mag-netic reconnection can be present (see, e.g., Dasso et al. 2006;Ru ff enach et al. 2012, 2015).The temporal series of B is computed in the time interval[ t F , t R ] from Eq. (11), or its equivalent for the elliptical case. Be-cause the closest approach is set at t =
0, Eq. (4), one has theordering: t F < < t R . t F and t R are found by solving Eq. (10),with ρ obs ( t ) = ρ front and ρ rear , respectively. We suppose belowthat p < ρ front and p < ρ rear , so that the FR is crossed by thespacecraft.The case of the front boundary and ζ ≥ ρ obs ( t ) in the in-bound branch is a decreasing function oftime (for t < ρ obs ( t F ) = ρ front has a unique solution when p < ρ front .The case of the rear boundary is more case-dependent. Thisis because both the numerator and the denominator of Eq. (10)are both increasing with time for ζ > t >
0. For ζ <
1, thenumerator is growing faster than the denominator so ρ obs ( t R ) = ρ rear has a unique solution (when p < ρ rear ). For larger ζ valuesthere are two t R solutions or even none for ζ (cid:38)
4. The physicalsolution in the first case is the one closer to the axis while inthe second case the expansion rate is so large that the simulatedspacecraft never crosses the rear boundary. This last case is notobserved as the maximum ζ values observed within unperturbedMCs is 1 . . .
4. Tests of minimum variance
A classical test of the quality of the MV method to distinguish di-rections is to analyze the separation between the variance (eigen-values) associated with the three eigenvalues ev min , ev int and ev max of the matrix m i j defined by Eq. (3) (see references inSect. 1). Indeed, if two directions have similar variances, so sim-ilar eigenvalues, say for example, x and z directions, a rotation ofthe frame around the third eigenvector ( y ) is expected to providealso similar variances in the two rotated directions ( x (cid:48) and z (cid:48) ).This implies that the x , z directions are not well defined.However, a large separation of the three eigenvalues is a nec-essary but not su ffi cient condition to have a precise determina-tion of the FR axis direction. A systematic bias could be presentdue to the mixing of the field components to achieve extremumvariances in the eigenvector directions. To understand this possi-ble bias, let us analyze the Lundquist’s FR with a non-negligibleimpact parameter p as an example. | B x , FR | has a shape compara-ble to B z , FR (Fig. 2b). Then, the MV combines B x , FR and B z , FR by a rotation of frame to produce the flattest possible B x , MV (sowith the lowest variance). This produces a bias in the MV frameorientation which increases with | p | , confirming the results ofGulisano et al. (2007). Indeed, in general, the MV method willmix components which are correlated, see Appendix A.More generally, this bias and other ones are present for theaxis direction obtained when the MV is applied to expandingFRs. Below, we first identify, then correct, these biases as muchas possible. They are measured by the positive and acute an-gles θ x , θ y , θ z between the x, y, z directions of the MV and FRframes, respectively (these angles can be defined as signed, butthe absolute value is su ffi cient for our purpose). Next, we com-pute ∆ i = i MV − i FR and ∆ λ = λ MV − λ FR which quantify the Article number, page 6 of 15émoulin et al.: Improving minimum variance method for expanding symmetric flux ropes ��������� p Θ x , Θ y , Θ z p - - - - - - - Δ i , Δ λ p ev min ev int , ev int ev max 0.2 0.4 0.6 0.8 p B z ,max ratio ��������� p Θ x , Θ y , Θ z p - - - Δ i , Δ λ p ev min ev int , ev int ev max 0.2 0.4 0.6 0.8 p B z ,max ratio (a) MV0(b) MV1 b/a=1, ξ=0 Fig. 3.
Test of the MV method with the Lunquist’s FR without expansion ( ζ = B z , FR ≥
0. (a) shows the results for MV0 (MV applied to B ) and (b) for MV1 (MV applied to ˆ B ). The FR axis is oriented with i = ◦ and λ = ◦ so that the simulated observed and FR frames are the same (Fig. 1). Left column: θ x , θ y , θ z are the angles in degrees between the x, y, z directions,respectively, of the FR and MV frames. Middle left column: ∆ i and ∆ λ quantify the di ff erence of axis orientations between the FR and the onefound by MV, so the orientation biases. Middle right column: ratios of the eigenvalues ev min , ev int and ev max . Right column: ratio of the maximumvalues of B z , MV to B z , FR (amplified by a factor of 100 for the first ratio). The horizontal dotted lines and the yellow regions are landmarks set at thesame locations in di ff erent figures for the graphs with the same quantities for a better comparison due to the di ff erent scales. (a)$(b)$ (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) - t B x - t - - B y - t B z - t B x / B - t - B y / B - t B z / B (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:9) - t B x - t - - B y - t B z - t B x / B - t - B y / B - t B z / B FR,$MV0,$MV1,$$b/a=1,$$ξ=0$$
Fig. 4.
Magnetic field components ver-sus time (in hour) in the FR frame(black), in the MV0 frame (orange, MVusing B ) and in the MV1 frame (green,MV using ˆ B ). The various compo-nents are compared in the same graphs.All panels are for a static ( ζ = b / a = p = − .
5, is se-lected to show the di ff erences ( p < B x > B and (b)shows ˆ B . The other parameters are B = t = V c =
400 km.s − , b = . di ff erence of axis orientations found by the MV and the correctone simulated with the FR model. Finally, we quantify the biason the axial field by the ratio B z , max , MV / B z , max , FR .From the tests made, we conclude that a large separation ofthe three eigenvalues is needed but this is far from being suf-ficient to have a precise axis determination. For example, for | p | (cid:46) .
6, we show in Fig. 3b a case where the two lowest eigen-values are more separated when | p | increases, up to | p | ≤ . θ x , θ z , ∆ λ in theleft panels). B or ˆ B The MV technique, described in Sect. 2.2, can be applied to thetime series of B or ˆ B = B / | B | defined within the modeled FR(and later on with in situ data). These approaches are called MV0and MV1 and were used, for example, by Siscoe & Suey (1972)and Gulisano et al. (2007), respectively.MV0, applied to the Lundquist’s model has an orientationbias increasing linearly with p (Fig. 3a, see θ x , θ z and ∆ λ varia-tions). This bias is reduced with MV1 (Fig. 3b). Next, θ y = ∆ i = B x , FR , B z , FR are symmetric and B y , FR is antisymmetric with time so that they cannot be mixedby MV (see Appendix A). These tests agree with the results ofGulisano et al. (2007) and precise the origin of the bias since the Article number, page 7 of 15 & A proofs: manuscript no. demoulin_MV_arXiv ��������� p Θ x , Θ y , Θ z p - - - - - - - Δ i , Δ λ p ev min ev int , ev int ev max 0.2 0.4 0.6 0.8 p B z ,max ratio ��������� p Θ x , Θ y , Θ z p - - - Δ i , Δ λ p ev min ev int , ev int ev max 0.2 0.4 0.6 0.8 p B z ,max ratio (a) b/a=1(b) b/a=2 MV1, ξ=1
Fig. 5.
Test of the MV1 method with FRs expanding with a typically observed rate ( ζ = B z , FR ≥
0. (a) shows the results for a circular FR and (b) for an elliptical one. The FR axis is oriented with i = ◦ and λ = ◦ so that thesimulated observed and FR frames are the same (Fig. 1). Left column: θ x , θ y , θ z are the angles in degrees between the x, y, z directions, respectively,of the FR and MV frames. Middle left column: ∆ i and ∆ λ quantify the di ff erence of axis orientations between the FR and the one found by MV, sothe orientation biases. Middle right column: ratios of the eigenvalues ev min , ev int and ev max . Right column: ratio of the maximum values of B z , MV to B z , FR (amplified by a factor of 100 for the first ratio). The horizontal dotted lines and the yellow regions are landmarks to compare the graphs. ��������� - t B x - t - - - B y - t B z - t B x / B - t - - B y / B - t B z / B (a) b/a=1(b) b/a=2 ��������� - t B x - t - - B y - t B z - t B x / B - t - B y / B - t B z / B FR, MV0, MV1, ξ=1
Fig. 6.
Magnetic field components ver-sus time (in hour) in the FR frame(black), in the MV0 frame (orange, MVusing B ) and in the MV1 frame (green,MV using ˆ B ) for FRs expanding with atypically observed rate ( ζ = p = − . ff erences. (a)shows the results for a circular FR and(b) for an elliptical one. The other pa-rameters are B = t = V c = − , b = . use of i and λ is more adapted to separate the relevant magneticcomponents than the longitude and latitude of the FR axis. Thisfurther justifies the choice of i and λ angles because, when usingthem, the bias of the orientation remains mainly on λ .Moreover, the origin of the biases is illustrated for ζ = B components of the Lundquist’s FR drawn in the FRframe (black lines) and MV0 frame (orange lines) in Fig. 4a. B x , FR and B z , FR are combined to provide a flat B x , MV (Fig. 4a,left panel). More generally, for a cylindrically symmetric FR, B x = B θ ( ρ obs ) p /ρ obs in the FR frame from Eq. (11), then B x = constant if the azimuthal component B θ is linear with the ra-dius ρ obs . In this case, the variance of B x vanishes and the MV0method would exactly associate this minimum variance directionto ˆ x FR . In brief, there would be no orientation biases. However, ingeneral this linear dependence is only approximately true close to the FR center for cylindrically symmetric FR (using a Taylorexpansion of B θ ).The above bias decreases when normalizing B to unity since | B | decreases away from the FR axis, compensating partly thedecrease of B θ . It implies that B θ / | B | is more linear with ρ obs than B θ . This e ff ect is directly seen on the B y component for p = B θ = | B y | and it is still well present for p = − . B x / | B | ismore uniform than B x and the MV1 frame is closer to the FRframe than the MV0 one (Fig. 3). Then, the B components in theMV1 frame (green lines in Fig. 4) are closer to the ones in theFR frame (black lines) than when MV0 is applied to B of theLundquist’s field (orange lines). Article number, page 8 of 15émoulin et al.: Improving minimum variance method for expanding symmetric flux ropes ��������� p Θ x , Θ y , Θ z p - - - Δ i , Δ λ p ev min ev int , ev int ev max 0.2 0.4 0.6 0.8 p B z ,max ratio ��������� p Θ x , Θ y , Θ z p Δ i , Δ λ p ev min ev int , ev int ev max 0.2 0.4 0.6 0.8 p B z ,max ratio (a) ρ rear = 0.9(b) ρ rear = 0.5 b/a=1, MV1, ξ=0 Fig. 7. E ff ect of the flux unbalance on the results of MV1 applied to the Lundquist’s FR without expansion ( ζ = B z , FR = ρ front =
1) and the rear limit is set at afraction ρ rear of the FR radius. For comparison the case ρ rear = θ x , θ y , θ z are the angles in degrees between thex, y, z directions, respectively, of the FR and MV frames. Middle left column: ∆ i and ∆ λ quantify the di ff erence of axis orientations between theFR and the one found by MV, so the orientation biases. Middle right column: ratios of the eigenvalues ev min , ev int and ev max . Right column: ratioof the maximum values of B z , MV to B z , FR (amplified by a factor of 100 for the first ratio). The horizontal dotted lines and the yellow regions arelandmarks. The MV bias identified above is due to a mix between B x , FR and B z , FR . By definition of the orientation of the z axis, B z , FR ≥ B x , FR has the sign of − p B θ . Then,the bias ∆ λ has also the sign of − p B θ . For example, Fig. 3 showsa case with positive helicity, B θ B z ≥
0, and p ≥ ∆ λ ≤
0. Other cases are simply obtained by changing the sign of ∆ λ accordingly to the signs of p and magnetic helicity.The e ff ect of expansion for a typical ζ value, ≈
1, introducesonly a small extra bias on i , | ∆ i | < ◦ , as seen by comparingFig. 5a to Fig. 3b. The expansion induces an asymmetry in thefield components (Fig. 2), then the MV method combines a partof B y , FR with B z , FR , Fig. 6a, introducing a bias for θ y and ∆ i values. Other quantities ( θ x , θ z , ∆ λ , the ratio of eigenvalues and B z , max , MV ) are not significantly a ff ected by the expansion. Thissmall e ff ect of expansion is a generic result for all the modelstested.Increasing the aspect ratio b / a enhances more significantlythe bias (Fig. 5b, we note the change of vertical scale on the leftpanels). In fact, since the B ( t ) profile gets flatter with increasing b / a value (Fig. 2c, right panel) MV1 is converging to MV0 as b / a is larger, so the bias increases (as shown above with Fig. 3comparing MV0 to MV1). In contrast, as b / a increases, the ra-tio ev min / ev int decreases (Fig. 5, compare middle right panels),so this ratio is again not a reliable indicator of the quality ofthe axis direction determination. Furthermore, as b / a increases, B x , FR ( t ) behaves closer to B z , FR ( t ) then both MV0 and MV1 ro-tate the frame combining B z , FR and B x , FR , trying to get B x , MV asflat as possible (compare the two left panels of Fig. 6). This fur-ther increases | ∆ λ | so the bias in the determined axis orientation(Fig. 5b).The bias on the axis orientation introduced by the MV im-plies a systematic overestimation of the axial component B z , asshown in Fig. 4, which increases with | p | (Fig. 3, right panels). This e ff ect is stronger for MV0 than MV1 as expected from theorientation bias results. This bias on B z also increases slightlywith b / a value (Fig. 5, right panels). This is illustrated with anexample shown in the right panels of Fig. 6. The MV method ro-tates the MV frame so that the bump shape of B x , FR is removedin B x , MV . This strengthens B z , MV independently of the sign of p and of magnetic helicity.
5. Biases due to boundary selection
The boundaries of the time interval selected in the in situ datato find the FR orientation are typically set where abrupt changesof the magnetic field and plasma parameters are detected. How-ever, there are frequently some ambiguities in setting them sothat the selected boundaries depend on the author’s specific cri-teria. For example, the MC observed on October 18-20 1995 bythe Wind spacecraft was studied by several groups of authorswhich set di ff erent MC boundaries (Lepping et al. 1997; Larsonet al. 1997; Janoo et al. 1998; Collier et al. 2001; Hidalgo et al.2002; Dasso et al. 2006).The main origin of these discrepancies is that the measuredmagnetic field components and the plasma data, such as pro-ton temperature, plasma composition and properties of high en-ergy particles, do not always agree on the extension of the MC,and more generally of the magnetic ejecta (see, e.g., Russell &Shinde 2005). A part of this ambiguity is due to the partial mag-netic reconnection of the ejected solar FR with the magnetic fieldof the encountered solar wind or of an overtaking stream. Thisreconnection does not a ff ect the field and plasma on the sametime scale. For example a change of magnetic connectivity is af-fecting on short time scales (on the order of few tens of minutes)the propagation of high energy particles while the mix up of MCand solar wind protons from the distribution core takes days to Article number, page 9 of 15 & A proofs: manuscript no. demoulin_MV_arXiv be mixed after reconnection occurred. The above processes im-ply that di ff erent boundaries are often defined in di ff erent studiesof the same MC (e.g., Riley et al. 2004; Al-Haddad et al. 2013).The selection of boundaries have large implications on thederived FR orientation (e.g., Dasso et al. 2006). For example,the axis orientation derived from MV and a least square fit toa Lundquist’s model can be significantly di ff erent and withoutcoherence along the axis of a MC observed by four spacecraft(ACE, STEREO A and B, Wind) for a set of boundaries (Far-rugia et al. 2011), while consistent results are obtained betweenboth methods and along the FR axis when refined time inter-vals are used (Ru ff enach et al. 2012). The important e ff ect of theboundaries is further shown in Janvier et al. (2015) by the dif-ference in orientations found with the same set of MCs analyzedby Lynch et al. (2005) and Lepping & Wu (2010), while the FRaxes were determined in both cases by the same method (fit of B data to Lundquist’s model). The e ff ect of the boundary selection on the determined axis isillustrated in Fig. 7 using MV1 on a Lundquist’s FR without ex-pansion (in order to focus the analysis only on the boundary ef-fects). We set the front at ρ front = B z , FR =
0) and allows ρ rear <
1. The FR is crossed only for | p | < ρ rear so the large valuesof ∆ i , ∆ λ and B z , max , MV / B z , max , FR obtained for large | p | values inFig. 3b are absent in figures with ρ rear < p value.Already with ρ rear = . ρ rear = ff erent parame-ter). θ y , which was null in Fig. 3b, is comparable to θ x and θ z (leftpanel of Fig. 7a). This introduces mainly a rotation of the esti-mated FR axis by changing its inclination on the ecliptic plane( ∆ i > B y , FR with the other componentsin order to increase the variance of B y , MV . This is possible as thecomponents are no longer antisymmetric or symmetric with re-spect to the closest approach time, then they are partly correlated(see Appendix A).As ρ rear is further decreased the bias on i increases, and domi-nates the bias on λ for all p values. For example, with ρ rear = . i with a rotation of the determined FR axis on the or-der of 24 − ◦ compared to the known axis direction (Fig. 7b,middle left panel). Even worse, this bias on i is also present forlow impact parameter cases (bias ≈ ◦ ). This increasing bias isassociated with an increasing separation of the two lowest eigen-values (Fig. 7b, middle right panel). It shows once more that theeigenvalue separation is not a good criteria for estimating theprecision of the axis direction defined by MV.The location of the FR boundaries also introduces biaseson other parameters of the FR such as its radius (by chang-ing the geometry of the crossing) or its axial field strength(by mixing the field components). Still, it has only a moder-ate e ff ect on B z , max , MV / B z , max , FR since MV1 mixes B y , FR with B z , FR and B y , FR ≈ B z , FR is maximum (Fig. 6), so that B z , max , MV / B z , max , FR is comparable to the case ρ rear = p value (compare the left panel of Fig. 3 to those of Fig. 7). Since ∇ . B =
0, the same amount of azimuthal magnetic fluxshould be present in the FR front and rear (e.g., Dasso et al. 2006). This implies that in the FR frame the same amount offlux crosses the y-z plane before and after the time of the closestspacecraft approach to the FR axis, then for a locally straight FRaxis: (cid:34) FR B y d x d z = . (14)However, the FR axis of observed MC is expected to be bendtoward the Sun. Therefore, following field lines from the front tothe rear part, the same amount of flux is located in a slightlylonger region along the axis in the front region than at the rear.Nevertheless, the curvature radius of the axis, R c , is large, typ-ically several AUs (e.g., Janvier et al. 2015), compared to thedistance r to the FR axis ( r (cid:46) . r / R c , is dominant near the border of the FR, but it is still small.Moreover, the magnetic torque balance is expected to distributeequally the twist along the FR, at least locally. Then, the hy-pothesis of local invariance by translation along the FR axis isexpected to be a good approximation, at least away from the FRlegs (see Owens et al. 2012).Within the above approximations, the accumulated magneticflux between t F and t R , in the FR frame and per unit length alongthe axial direction, isd F y ( t F , t R )d z = (cid:90) t R t F B y ( t (cid:48) ) V x ( t (cid:48) ) d t (cid:48) . (15)Since d F y / d z includes the observed velocity component V x thisflux computation includes automatically the details of the localFR expansion. It is also valid for any FR cross-section shape.Next, for a given boundary time, either t F or t R , d F y ( t F , t R ) / d z = | t F − t R | so selecting only one FR.Another equivalent formulation is to remark that ∇ . B = z : B = ( ∂ A /∂ y , − ∂ A /∂ x , B z ( x , y )) , (16)where A ( x , y ) is the z component of vector potential. Integrat-ing any field line implies that it is located on a surface A ( x , y ) = constant. This property relate any location in the FR front to itscorresponding rear position. Along the spacecraft trajectory, so y = y p (defined in Eq. (4)), A ( x , y p ) can be derived by integra-tion of B y . Next, taking the origin of A at the front boundarythis implies A ( t , y p ) = − d F y ( t F , t ) / d z where the x coordinate hasbeen replaced by time for marking the position. Then, Eq. (15)can also be expressed with the vector potential component A , asdone, for example, in the Grad-Shrafranov equation for magne-tostatic field.The above property of flux conservation was used to definethe back region of MCs (the region which was part of the FR be-fore reconnection at the front occurred, Dasso et al. 2006, 2007)and, more generally, the amount of magnetic flux reconnectedwith the background field (Ru ff enach et al. 2015). Below, wefurther propose that Eq. (15) can also be used in a broader con-text to define coherent sub-FRs within the core of the physicalFR. The aim is to test the stability and to limit the bias of the FRaxis determined by the MV variance. Article number, page 10 of 15émoulin et al.: Improving minimum variance method for expanding symmetric flux ropes ρ b Θ x , Θ y , Θ z ρ b - - - - Δ i , Δ λ ρ b ev min ev int , ev int ev max 0.2 0.4 0.6 0.8 1.0 ρ b B z ,max ratio ��������� ρ b Θ x , Θ y , Θ z ρ b - - - - - - Δ i , Δ λ ρ b ev min ev int , ev int ev max 0.2 0.4 0.6 0.8 1.0 ρ b B z ,max ratio ��������� ρ b Θ x , Θ y , Θ z ρ b - - - - - - - Δ i , Δ λ ρ b ev min ev int , ev int ev max 0.2 0.4 0.6 0.8 1.0 ρ b B z ,max ratio (a) b/a=1, ξ=0(b) b/a=1, ξ=1(c) b/a=2, ξ=1 Fig. 8.
Test of MV1 applied only to the FR core. The results are shown in function of ρ b = ρ front = ρ rear , so with flux balance between the front andrear. (a,b) b / a =
1, (c) b / a =
2. (a) ζ = ζ = ρ b > p (otherwisethe FR core is not crossed) where p = .
3. Left column: θ x , θ y , θ z are the angles in degrees between the x, y, z directions, respectively, of theFR and MV frames. Middle left column: ∆ i and ∆ λ quantify the di ff erence of axis orientations between the FR and the one found by MV, so theorientation biases. Middle right column: ratios of the eigenvalues ev min , ev int and ev max . Right column: ratio of the maximum values of B z , MV to B z , FR (amplified by a factor of 100 for the first ratio). The horizontal dashed dotted lines and the yellow regions are landmarks. We have shown in Sect. 4.2, for FRs with cylindrical symmetry,that the MV bias decreases as the azimuthal field component B θ or B θ / | B | (MV0 or MV1) has a more linear dependance withthe distance to the FR axis. For non singular distributions of theelectric current and cylindrically symmetric FRs, B θ vanishes onthe axis. A Taylor expansion of B θ ( r ) implies that B θ ( r ) is linearfor su ffi ciently small r values, so the bias of MV is expected tobe lower when MV is applied to the core of the FR. However,restricting the core has the disadvantage of considering less datapoints, and for real FRs it implies including relatively more smallstructures and noise / fluctuations that can compete with the FRstructure and produce an additional undesired bias. Then, we useEq. (15) to define sub-regions corresponding to inner FRs in thesimulated data. For the models considered, Eq. (15) is simplysolved by setting ρ front = ρ rear = ρ b . When real observationsare analyzed, getting the boundaries for this flux-balanced innersub-FR requires to know its orientation (i.e., B y in the FR frameis needed to compute F y ). Thus, it is needed to use a methodthat feedbacks on the best estimations of the orientation whilechecking that the flux balance criterion is met.Next, we describe how to impose the flux balance with data.The following method was tested with the above models wherethe flux balance is rigorously defined with ρ front = ρ rear . Themethod has two main steps: fixing either the front or rear bound-ary and scanning the other boundary. More precisely, the method first tests if there is more azimuthal flux at the rear than at thefront of the MC, then, if it is the case it defines the FR rear time.If no flux balance is achieved in the previous step, the end timeis then used as the rear boundary and the method defines the FRfront time when the azimuthal flux is conserved.We next describe this method with more details. Let us sup-pose first that the front boundary is fixed (e.g., it can be definedby a sharp jump in the magnetic and plasma data, or it can be alsolocated inside the MC to study the FR core). The MV methodis recurrently applied to the data limited in between t F and thescanned values of t R , then the flux balance is computed fromEq. (15), in the MV frame (function of t R ). The lowest valueof t R which satisfies the flux balance is selected as the FR rearboundary. If no flux balance is achieved, the same method is ap-plied by fixing t R at a sharp jump in the magnetic and plasmadata (at the latest time compatible with MC properties), or ear-lier on, again to study only the FR core. Then, t F is scanned.Finally, the largest value of t F which satisfies the flux balance isretained.In summary, present method defines the FR, or its core,which is present at the spacecraft crossing by imposing the az-imuthal flux balance. Since the MV eigenvectors are computedwith flux balance, the bias on the deduced FR axis orientation isminimized. Finally, this method of scanning boundaries to im-pose flux balance can be implemented in other methods, such asfitting a flux rope model. The limitation is the computation time, Article number, page 11 of 15 & A proofs: manuscript no. demoulin_MV_arXiv as a non-linear least-square fit is required for each tested bound-ary. Still, the computation time can be limited by including in themethod a more e ffi cient search of the zero of a function than asimple scan (the simplest being bisection or Newton’s root find-ing). As expected, the results of Fig. 8a show that the MV1 bias is sig-nificantly reduced as a smaller central part of the Lundquist’s FRis considered up to the limit where the FR is not crossed ( ρ b = p ).This contrast with the eigenvalues being closer as ρ b decreases,further showing another example where the ratios of eigenvaluesare not good indicators of the quality of the deduced FR axis.Finally, as a natural consequence of MV1 and FR frames be-coming closer with a lower ρ b value, the axial field is also betterrecovered when ρ b is decreased (right panel of Fig. 8a).The above trends with ρ b are also obtained with MV0 butstill with a larger bias than with MV1. Next, similar results areobtained when the expansion is included with a magnitude astypically observed (Fig. 8b). However, this improved axis deter-mination is less e ff ective as the FR is flatter, so as b / a is larger(Fig. 8c). Of course in application to observations, perturbationsof the FR field and the limitation to too few data points will con-strain the decrease of ρ b so close to the impact parameter p as inFig. 8.
6. Biases due to fluctuations
The e ff ects of fluctuations can separate two groups:- a change of position for the boundary determined by the fluxbalance,- a change of the variances in the x,y,z directions (with fixedboundaries).Let us call b y the fluctuation in the y direction. The flux bal-ance writes:d F y ( t F , t R )d z = (cid:90) t R t F ( B y ( t (cid:48) ) + b y ( t (cid:48) )) V x ( t (cid:48) ) d t (cid:48) = , (17)which is simpler to write in function of x : (cid:90) x R x F ( B y + b y ) d x = . (18)Let us fix t F . t R is changing by ∆ t , corresponding to the size ∆ x ,to satisfy the flux balance with fluctuations. The change of fluxdue to the main field B y is about B y , a ∆ x where B y , a is the am-plitude of the field near the FR boundary. The change of fluxdue to a sinusoidal fluctuation of wavelength l is at most b y , a /π ,where b y , a is the amplitude of the fluctuation, which correspondsto the contribution of half wavelength (the others cancel). Insert-ing these orders of magnitude in Eq. (18), and diving by the FRradius b to normalize, the magnitude of the boundary change ∆ x satisfies: ∆ xb (cid:46) π b y , a B y , a lb . (19)Both the last two fractions are typically <<
1, say on the order of0 .
1, so ∆ x / b < .
01, which will give a very small bias on the axis(Fig. 7a is for ∆ x / b = .
1, so the e ff ect would be much smaller,likely a factor of ten smaller). In conclusion, fluctuations would change significantly the computed boundary, so the axis deter-mined, unless they are both large in amplitude and with a largewavelength. As an example, to get the e ff ect shown in Fig. 7awe need ∆ x / b = .
1, then with b y , a / B y , a ≈ l / b both ratio needsto be ≈ .
4, and this is the most favorable case where half thewavelength perturbation is not compensated (odd number of halfwavelength across the FR).For the second e ff ect, the variance of Eq. (1) is rewritten withfluctuations as: σ n = (cid:104) ( B n + b n − (cid:104) B n + b n (cid:105) ) (cid:105)≈ (cid:104) ( B n + b n − (cid:104) B n (cid:105) ) (cid:105)≈ (cid:104) ( B n − (cid:104) B n (cid:105) ) (cid:105) + (cid:104) ( B n − (cid:104) B n (cid:105) ) b n (cid:105) + (cid:104) b n (cid:105)≈ (cid:104) ( B n − (cid:104) B n (cid:105) ) (cid:105) + (cid:104) b n (cid:105) (20)where the linear terms in b n are neglected because they con-tribute only by at most half wavelength (same as in previousparagraph). Then, the contributions of the fluctuations is mostlyto increase the variances of all B components. If isotropic fluc-tuations are introduced they have no e ff ect in the determinationof the variance extremums so on the eigenvectors. This is notthe case for fluctuations orthogonal to B (Alfvenic fluctuationsare expected in a low β plasma). Still, could the eigenvectors besignificantly rotated? Let suppose a rotation of x , y directions to x (cid:48) , y (cid:48) in the FR frame. Reducing (cid:104) b n (cid:105) in the x (cid:48) direction wouldneed a significant correlation of the perturbation in the x , y di-rections (see Appendix A). Moreover, this rotation would signif-icantly increase (cid:104) ( B n − (cid:104) B n (cid:105) ) (cid:105) by mixing B y with B x counteract-ing the above hypothetical decrease of variance due to fluctua-tions. The same consideration can be done with a rotation from x , z directions to x (cid:48) , z (cid:48) , with the di ff erence that it is less costlyto mix x and z component for the variance as they have moresimilar behavior than compared with the y component.In conclusion, fluctuations cannot change significantly the i angle. The e ff ect is expected to be larger on λ angle but still weakunless the fluctuations have a long wavelength, comparable tothe FR radius and are of large amplitude. This is very di ffi cultto achieve as we already know that the e ff ect of the expansion,which also can be thought as a kind of large-scale perturbation,is small.
7. Conclusion
An accurate determination of the flux rope (FR) axis directionfrom the interplanetary data of B is important for statistical stud-ies determining the global axis shape, for the determination ofthe FR physical properties (e.g., the internal distribution of themagnetic twist, the amount of magnetic flux and helicity) as wellas for comparing the MC global properties and its orientation it-self to the ones determined from its solar source region. MV isone method to estimate the direction of the FR axis. However, asany present method, MV is known to have biases. Our main aimwas to identify them and to correct them as much as possible.The orientation of the axis is better defined by the locationangle λ and the inclination i , as defined in Fig. 1, rather thanthe traditional latitude and longitude of the axis, using ˆ z GSE asthe polar angle. For example, the biases of the MV method areof di ff erent origins and magnitudes for λ and i while the biasorigins are mixed for latitude and longitude. The angles λ and i are then important to identify, then to correct, the MV biases.For that, we test MV results with several methods and models.The eigenvalues ratio provided from applying MV to the FRobservations quantifies how di ff erent / similar are the variances Article number, page 12 of 15émoulin et al.: Improving minimum variance method for expanding symmetric flux ropes in the three di ff erent directions. The eigenvalues should be suf-ficiently separated to identify accurately the eigenvectors foundby MV. However, this eigenvalues ratio is not a proxy for quanti-fying the accuracy of the estimation of the real FR axis direction,in contrast to the claim of several previous publications.MV was first applied to the simulated B time series com-puted with a Lundquist’s model and its generalisation to an ellip-tical cross-section. This introduces an increasing bias on the axisdetermination as the simulated spacecraft crosses the FR furtheraway from its axis. This bias is reduced when the MV is appliedto B / B time series, confirming the previous results of Gulisanoet al. (2007). By analyzing cylindrical models we identify theorigin of this bias as due to the departure of a linear dependenceof the azimuthal field components with the distance to the FRaxis. Including expansion in the modeled FR, with an expansionrate as it is typically observed, does not have a significant e ff ecton the orientation obtained from MV.Another bias is due to the selection of the time interval usedto apply MV. In MC observations there are frequently severalsets of plausible boundaries depending on which magnetic andplasma data are used to define abrupt change of temporal be-havior. We verify on modeled synthetic FR that the selectionof the time interval boundaries is crucial to minimize the biasin the determined FR orientation. We next solve this di ffi cultyby imposing that the same amount of azimuthal magnetic fluxshould be measured before and after the closest approach of theFR axis. The estimation of the flux balance is computed in theMV frame scanning alternately the rear and the front boundary.We find, as expected, that the MV bias for the estimation of theFR axis direction is minimized when the azimuthal flux balanceis satisfied. This flux balance technique can be applied to othermethods, such as FR model fitting. We also anticipate a decreasein the axis orientation bias since the FR models have an intrin-sic flux balance, so that the selected part of the data should alsosatisfy this constraint.The above constraint of flux conservation also allows us toexplore a variable part of the FR core as input to MV. Indeed,the test made with cylindrical models shows that the bias in theaxis orientation is lower when an inner part of the FR is selected.This e ff ect is weaker for FRs with flatter cross sections. In appli-cations to observations, the limitation to the FR core is furtherjustified by the expected presence of larger perturbations at theFR periphery, due to interactions with the external ambient solarwind.However in observations, the presence of B fluctuations im-plies that the analyzed time interval should be su ffi ciently largeand does not a ff ect the axis determination too much. Then, acompromise should be taken between too small and too largetime intervals. This compromise is MC case dependent since itis linked to the intensity and distribution of B perturbations. Thiscan be realized in studied MCs by analyzing the evolution of i and λ in function of the position of one boundary (the other beingcomputed by flux balance).In the present study we limit the application of the MV toMCs with approximately symmetric B ( t ) observed profiles. Thiscorresponds to about 70% the MCs observed at 1 AU with anasymmetry parameter below 15% (Figure 3h of Lepping et al.2018) or with a distortion parameter (DiP) of 0 . ± .
07 (Table 4of Nieves-Chinchilla et al. 2018b). Next, on top of the expansion,which introduces only a weak asymmetry, an intrinsic asymme-try exist especially for the faster MCs (Masías-Meza et al. 2016).This spatial intrinsic asymmetry introduces another bias in theaxis determined by the MV. Its correction needs both the devel-opment of the MV method, and also the development of asym- metric models to test the amount of remaining bias. This will bethe object of another study.Still, the present results are important for applications tomostly slow MCs. For FR close to cylindrical symmetry and inexpansion, MV applied to B / B time series combined with theflux conservation gives results where the remaining biases aremostly present on the location angle λ with a magnitude typi-cally proportional to the impact parameter. It implies that, witha known axis shape, there is a bias on how far the spacecraftcrossing occurred from the axis apex, see Janvier et al. (2015).The sign of the bias depends on the sign of the product betweenthe impact parameter and of the magnetic helicity.At the opposite, the inclination i is well recovered (with abias less than 4 ◦ for an impact parameter below 0 . B / B time series, called MV1,is recommended for studying the amount of global rotation ofthe FR during its transit from the Sun to the in situ observationposition. Acknowledgements.
We thank the referee for her / his comments which improvedthe manuscript. S.D. acknowledges partial support from the Argentinian grantsUBACyT (UBA), PICT-2013-1462 (FONCyT-ANPCyT), and PIDDEF 2014 / References
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Appendix A: Minimum Variance and Symmetries
The mean quadratic deviation, or variance, of the magnetic field B in a given direction defined by the unit vector ˆ n is:Var n = σ n = N N (cid:88) k = (cid:16) ( B k − (cid:104) B (cid:105) ) · ˆ n (cid:17) (A.1)where the summation is done on the N data points of the timeseries and (cid:104) B (cid:105) is the mean magnetic field.Let us apply Eq. (A.1) to the FR magnetic components ob-served (or simulated) by a spacecraft, with B x ( t ) , B y ( t ) , B z ( t ) thecomponents in the FR frame (defined in Sect. 2.1). This definesa series of N data points B ki ( i = x , y , z ). We call the variancesof these field components Var i . We define also the correlationsbetween two specific components, asCor i , j = N N (cid:88) k = ( B k i − (cid:104) B i (cid:105) ) ( B k j − (cid:104) B j (cid:105) ) (A.2) Appendix A.1: Is the FR y -direction well recovered by MV? Let us suppose we are in the FR frame, and let us consider a 2Drotation by an angle δ around a fixed x axis, so supposing ˆ x MV = ˆ x FR , in order to analyze if MV will mix the components B y with B z when this rotation is permitted. Thus, the new component B y (cid:48) in the rotated frame ( y (cid:48) ) will be B y (cid:48) = B y cos δ − B z sin δ (A.3)Then, the variance in the new rotated y (cid:48) axis, Var y (cid:48) ( δ ), can bewritten asVar y (cid:48) ( δ ) = Var y (cos δ ) + Var z (sin δ ) − δ cos δ Cor y , z = Var y − Var z δ − Cor y , z sin 2 δ + Var y + Var z a cos 2( δ − δ b ) + b with a , b two constants with a > y > Var z . δ b is the bias angle of the MV: the rotationaround the axis ˆ x MV = ˆ x FR needed to maximize Var y (cid:48) ( δ ). Moreprecisely: a = (cid:115)(cid:32) Var y − Var z (cid:33) + Cor , z tan 2 δ b = − y , z Var y − Var z (A.4)A typical order of magnitude of δ b can be estimated with thefield B y = B sin t , B z = B cos t with t in the interval [ − π/ , π/ B y and B z profiles, in partic-ular their symmetry, and it is su ffi cient for an order of magni-tude estimation). This implies Var y ≈ B /
2, Var z ≈ B /
10 andCor y , z =
0. Indeed, in practice we have often in MCs the order-ing: | Cor y , z | << Var z << Var y (A.5)It implies δ b ≈ Cor y , z / Var y << x replacing z , so consid-ering a rotation by an angle δ (cid:48) around the z axis, so supposing ˆ z MV = ˆ z FR , and thus mixing B x with B y . As above, we have oftenin MCs the ordering: | Cor x , y | << Var x << Var y (A.7)This implies δ (cid:48) b ≈ − Cor x , y / Var y << B x islower than B z , so for a low impact parameter.We conclude that the y -direction, both orthogonal to the FRaxis and spacecraft trajectory, is expected to be well recoveredby MV when flux balance is achieved. This is summarized by alow expected bias for the i angle. Appendix A.2: Is the FR axis direction well recovered by MV?
The same analysis done in the previous sub-section can be ap-plied to the mix between B x with B z . Here it is clearer to lookat the bias of the x axis, as a minimum of variance is expectednearby while a saddle point is expected around the z axis (vari-ance decreases when changing in some directions, but increasesin other directions). This is not a problem as the variance matrixis symmetric, then it has orthogonal eigenvectors, so when thebias on both x - and y -axis are known, the bias on the z -axis canbe deduced.Let us consider a rotation by an angle δ y around the y axis,so supposing ˆ y MV = ˆ y FR . B x (cid:48) = B x cos δ y − B z sin δ y (A.9)Then, Var x (cid:48) ( δ y ) is written asVar x (cid:48) ( δ y ) = Var x (cos δ y ) + Var z (sin δ y ) − δ y cos δ y Cor x , z = − Var z − Var x δ y − Cor x , z sin 2 δ y + Var x + Var z y changed to x and extract-ing a negative sign in front of cos 2 δ y to write Var x (cid:48) ( δ y ) as − a cos 2( δ y − δ y , b ) + b so that a minimum of Var x (cid:48) ( δ y ) is presentclose to the x -axis for small δ y , b values (with a , b two constantswith a > z > Var x ). δ y , b is the bias angle of the MV forthis rotation: the rotation around the axis ˆ y MV = ˆ y FR needs tominimize Var x (cid:48) ( δ y ). More precisely: a = (cid:115)(cid:32) Var z − Var x (cid:33) + Cor , z tan 2 δ y , b = x , z Var z − Var x (A.10)Unlike previously, Eqs. (A.5) and (A.7), the ordering | Cor x , z | << Var x << Var z (A.11)is typically much less satisfied in MCs, so the bias of the rotation δ y , b around the y -axis, as provided by the minimum of variance,could be important, and it increases with Cor x , z so with the im-pact parameter. Since ˆ x MV and ˆ z MV are orthogonal this conclu-sion applies also to the axis direction estimated by MV.We conclude that the FR axis, or z , direction is expected tohave more bias than the y direction, except when the impact pa-rameter is small (which implies a small B x component, then asmall Cor x , z ). This is summarized by a larger bias for the λ anglethan for i except for small p values.values.