TB, AR, APJ/ApJ461877/ART, 1/03/2013

The Astrophysical Journal, 766:1 (10pp), 2013 ??? doi:10.1088/0004-637X/766/1/1C© 2013. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

A GLOBAL MAGNETIC TOPOLOGY MODEL FOR MAGNETIC CLOUDS (II)

M. A. HidalgoDepartamento de Fısica, Universidad de Alcala, Apartado 20, E-28871 Alcala de Henares, Madrid, Spain; miguel.hidalgo@uah.es

Received 2012 December 20; accepted 2013 January 31; published 2013 ???

ABSTRACT

In the present work, we extensively used our analytical approach to the global magnetic field topology of magneticclouds (MCs), introduced in a previous paper, in order to show its potential and to study its physical consistency.The model assumes toroidal topology with a non-uniform (variable maximum radius) cross-section along them.Moreover, it has a non-force-free character and also includes the expansion of its cross-section. As it is shown, themodel allows us, first, to analyze MC magnetic structures—determining their physical parameters—with a varietyof magnetic field shapes, and second, to reconstruct their relative orientation in the interplanetary medium from theobservations obtained by several spacecraft. Therefore, multipoint spacecraft observations give the opportunity toinfer the structure of this large-scale magnetic flux rope structure in the solar wind. For these tasks we use data fromHelios (A and B), STEREO (A and B), and Advanced Composition Explorer. We show that the proposed analyticalmodel can explain quite well the topology of several MCs in the interplanetary medium and is a good starting pointfor understanding the physical mechanisms under these phenomena.

Key words: magnetic fields – solar–terrestrial relations – solar wind – Sun: coronal mass ejections (CMEs) –Sun: magnetic topology

Online-only material: color figures

1. INTRODUCTION

When coronal mass ejections (CMEs) drive through the in-

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terplanetary medium they are often referred to as interplanetaryCMEs and in general they are classified in magnetic clouds(MCs) and ejectas, which recently have been an active sub-ject of interest in solar physics above all because they havebeen revealed to be one of the most determinant phenomenain the relationship between Sun and Earth, mainly for theirimplications in geomagnetic storms.

More recently, the study of MCs has become more and moreimportant because of the possibility of being able to measurethose magnetic structures observed by several spacecraft, whichprovides a valuable scenario for reconstructing their globaltopology in the interplanetary medium. Besides, the multispace-craft observations at different Sun–Earth distances provide in-formation on their evolution (Kilpua et al. 2009; Mostl et al.2009; Nieves-Chinchilla et al. 2011), allowing us to improvethe analytical models used for their study and therefore the un-derstanding of the physical mechanisms happening in them.

Concerning the topology of those magnetic structures, one ofthe main questions is their flux rope or non-flux-rope character,as discussed in detail in Hidalgo et al. (2012, and referencestherein). To clarify this point, the development of some physicalmodel is necessary, either analytical or numerical. In particular,from our point of view analytical models provide us with anappropriate frame work to understand the physics within theevolution of this kind of structure and is also a good startingpoint for the optimization of numerical models.

Then fitting the MC models to the data, we can check thepresence of this topology in any particular event observedin the interplanetary medium and obtain physical informationon the phenomenon—like the behavior of the plasma currentdensity or the orientation of the corresponding magnetic struc-ture (i.e., the latitude, θ , with respect to the ecliptic plane andthe longitude with respect to the Sun–Earth line, φ).

One of the most important characteristics of any MC is itsstability during its propagation in the interplanetary medium(the MCs are even observed at distances larger than5 AU—several events have been seen by the Ulysses space-craft, and, additionally, there are traces of them at distances ofthe order of 100 AU appearing in the measurements of bothVoyager spacecraft) and of course its implications for the gen-eration of geomagnetic storms.

Burlaga et al. (1981) established the definition of thesephenomena from the magnetic field and solar wind plasmadata, and since then many models and techniques have beendeveloped in the literature with the goal of understanding the realtopology and evolution of the MCs in the interplanetary medium.However, most of them are local models that assume cylindricaltopology, either with a circular cross section (Lepping et al.1990; Hidalgo et al. 2002a) or including a distortion of theircross sections (as a first approach with an elliptical deformation;Mulligan & Russell 2002; Hidalgo et al. 2002b) and, on theother hand, introducing a local expansion of them (Farrugiaet al. 1995; Hidalgo 2003, 2005).

Some of the models mentioned above have a force-free char-acter; however, from the analysis of the data from the interplan-etary medium, it is found that the relaxation of the force-freecondition (and, as a consequence, including the plasma pressure,which is important in the study of the physical mechanisms atplay in the expansion of the MCs; Hidalgo 2003, 2005, andreferences therein) and the cylindrical approximation are bothnecessary in order to approach the global structure of the MCsin a more accurate frame (Marubashi 1997). Thus, to achievethis last purpose, analytical models with different topologieshave been developed (Farrugia et al. 1995; Romashets &Vandas 2001; Romashets et al. 2010), or more recently, a torusgeometry with a non-uniform cross-section (variable maximumradius) (Hidalgo & Nieves-Chinchilla, 2012).

With the aim of testing the potentiality and consistency ofour toroidal model (Hidalgo & Nieves-Chinchilla 2012), in

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the present paper we have analyzed, on one hand, MCs withdifferent apparent magnetic structures and compared on theother hand, the measurements made by several spacecraft ofthe same event at several distances from the Sun some of themcloser than 1 AU.

The outline of the paper is as follows. In Section 2 wesummarize the toroidal model and in Section 3 we describe theprocedure followed to compare the experimental measurements.(In any case, a more detailed description is available in Hidalgo& Nieves-Chinchilla 2012.) The data analyzed in this work aredescribed in Section 4, where we show the fits obtained foreight selected MC events, two of them observed by severalspacecraft (the 1978 January event observed by Helios A and B;and the 2007 May and November, events observed by AdvancedComposition Explorer (ACE) and the two STEREO spacecraft, Aand B). Finally, in Section 5, a summary and discussion sectionhas been added with an analysis of the results obtained from themodel for the selected MCs.

2. THE MODEL

In Hidalgo & Nieves-Chinchilla (2012), a global model for theMCs was introduced assuming a torus topology with a circularbut maximum variable radius cross-section along it.

Once that topology is established, we can obtain theanalytical solutions in the appropriate coordinate systemto describe such geometry, (ϕ, ψ , η), not only for themagnetic field vector, given by B = Bϕeϕ + Bψ eψ +Bηeη, but also for the plasma current density j =j e + jp = (je

ϕ + jpϕ )eϕ + (je

ψ + jp

ψ )eψ + (jeη + j

pη ) eη, where

j e and jp correspond to the electron and proton current densi-ties, respectively.

Hence, for the poloidal component of the magnetic field weget the expression

Bϕ = B0ϕ(ψ) cos(ϕ) − μ0jψr cosh(−ρ0η + f (ψ)), (1)

with B0ϕ(ψ) being an integration constant and ρ0 the mean radius

of the torus. For the axial component of the magnetic field wefind

Bψ = B0ψ (ψ) + μ0jϕrcosh(−ρ0η + f (ψ)), (2)

where B0ψ (ψ) is the axial magnetic field at the axis of the torus.

And, finally, for the third component of the magnetic field

Bη = −2cos(ϕ)S

{B0

0ψsin(ψ/2) +1

Cμ0α(t0 − t)

× r2cosh(−ρ0η + f (ψ))

}(3)

where S =√

sin2(ψ)/sin(ψ).In all the expressions above r the radial parameter related to

its cross-section, ϕ is the corresponding polar angle, and ψ isthe angular coordinate of the axis of the torus, the axial angle.(In Figure 1 of Hidalgo & Nieves-Chinchilla 2012, a sketchof this topology is shown.) f = f (ψ) is an auxiliary functiondepending on the coordinate ψ and responsible for the angulardependence of the maximum radius of the cross-section of theMC along the torus. This function will be a good starting pointto develop a model for more complex toroidal topologies. In thecases where this function is constant, this corresponds to a toruswith a uniform cross-section along it. Concerning this auxiliary

Figure 1. Data obtained by the Helios A spacecraft for the magnetic cloud from1975 March and the results from the toroidal model. The magnetic field strength,B, the Cartesian SSE-components (Bx, By, Bz), the proton plasma beta, and thebulk solar wind velocity are shown. The vertical dotted lines represent the timeboundaries of the cloud. Superimposed on the experimental data, the toroidalmodel predictions are shown with solid lines (see the text for details).

(A color version of this figure is available in the online journal.)

function, our first approach is to consider f (ψ) = Csin(ψ/2),where C is an adjustable constant.

Moreover, in order to introduce the expansion of the cross-section of the MC, we assume a linear time dependence inthe components of the plasma current density, i.e., | j | =| j e + jp| ≈ (t0 − t), where t is the time variable (in our analysisit corresponds to the time of the passage of the spacecraftthrough the cloud, considering t = 0 at the entrance of it).t0 is a time parameter characterizing the expansion of the cloud.Now, one more consequence of the expansion of the cross-section when imposing magnetic field flux conservation is theappearance of a time evolution in the integration constants ofboth components of the magnetic field, the axial and poloidal,that we also assume them to be linear dependent, i.e., B0

ϕ ≈(t0 − t) and B0

ψ ≈ (t0 − t) (Hidalgo 2011).To have direct applicable analytical expressions for the

magnetic field components of the MC and then to be ableto fit them to the corresponding experimental data, we haveto create some additional hypotheses about the behavior ofthe different physical magnitudes appearing in them. Thus,we suppose the poloidal magnetic field integration constant

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Figure 2. Same as in Figure 1 but for the magnetic cloud from 1978 Aprilobtained by the Helios B spacecraft.

(A color version of this figure is available in the online journal.)

Table 1Results Obtained from the Toroidal Model for MCs of Several Topologies

Spacecraft Event MC Interval 〈vSW〉 θ t φt

(YY Month) (doy) (km s−1) (deg) (deg)

Helios A 75 March 63.686→64.098 475 114 173a

Helios B 78 April 114.531→114.788 385 62 148a

ACE 00 November 311.964→312.742 525 53 212b

ACE 03 May 150.082→150.436 650 118 99b

ACE 03 November 324.460→325.029 580 145 206b

Notes.a Referred to the vector Sun–S/C (A or B).b Referred to the vector Sun–Earth.

Table 2Results Obtained from the Toroidal Model for the Magnetic Cloud from

1978 January, Observed by the Two Helios Spacecraft

Spacecraft MC Interval 〈vSW〉 θ t φt

(doy) (km s−1) (deg) (deg)

Helios A 3.609→4.054 720 74 171a

Helios B 4.384→5.264 550 34 172a

Note. a Referred to the vector Sun–S/C (A or B).

Figure 3. Data obtained by the ACE spacecraft for the magnetic cloud from 2000November, and the result for the toroidal model. The magnetic field strength,B, the Cartesian GSE-components (Bx, By, Bz), the proton pressure, and thebulk solar wind velocity are shown. The vertical dotted lines represent the timeboundaries of the cloud. Superimposed onto the experimental data, the toroidalmodel predictions are shown with solid lines.

(A color version of this figure is available in the online journal.)

Table 3Results Obtained from the Toroidal Model for the Magnetic Cloud from

2007 May, Observed by both STEREOs and ACE Spacecraft

Spacecraft MC Interval 〈vSW〉 θ t φt

(doy) (km s−1) (deg) (deg)

STEREO A 141.623→142.052 475 89 180a

STEREO B 142.193→142.648 450 63 205a

ACE 141.982→142.539 450 68 190b

Notes.a Referred to the vector Sun–S/C (A or B).b Referred to the vector Sun–Earth.

to be B0ϕ(ψ) = B0

0ϕsin (ψ/2), where B00ϕ depends on r. Also,

for the axial magnetic field at the axis of the torus, we writeB0

ψ (ψ) = B00ψ |cos(ψ/2)|, imposing B0

0ψ as a constant. On theother hand, for the poloidal component of the current density weassume jϕ = α(t0 − t)r|cos(ψ/2)|, and for the axial componentjψ = λ(t0 − t)sin(ψ/2), where α and λ are constants, and theywill be parameters of the model. Therefore, we now have the

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Figure 4. Same as in Figure 3 but for the magnetic cloud from 2003 Mayobtained by the ACE spacecraft. The proton plasma beta is shown instead of theproton pressure for the sake of clarity in fitting the boundaries of the MC.

(A color version of this figure is available in the online journal.)

Table 4Results Obtained from the Toroidal Model for the Magnetic Cloud from

2007 November, Observed by Both STEREOs and ACE Spacecraft

Spacecraft MC Interval 〈vSW〉 θ t φt

(doy) (km s−1) (deg) (deg)

STEREO A 324.363→324.776 425 20 100a

STEREO B 323.945→324.295 440 16 75a

ACE 324.012→324.378 480 21 175b

Notes.a Referred to the vector Sun–S/C (A or B).b Referred to the vector Sun–Earth.

analytical expressions for the three components of the magneticfield.

3. PROCEDURE

The procedure to compare the model to the experimen-tal observations is also explained in detail in Hidalgo &Nieves-Chinchilla (2012). In this section again, we only sum-marize the main steps to be applied in the implementation of thefitting procedure.

Figure 5. Same as in Figure 1 but for the magnetic cloud from 2003 Novemberobtained by the ACE spacecraft.

(A color version of this figure is available in the online journal.)

For all the MCs analyzed in the present work wehave considered that the propagation velocity of the MCsin the interplanetary medium is mainly determined by the highvalue of the x-GSE component of the solar wind velocity at thecorresponding MC interval. Then, we assumed that the path ofthe spacecraft is almost parallel to the x-GSE direction. Addi-tionally, taking the mean solar wind velocity in the cloud intervalconsidered, 〈vsw〉, we determine the position of the spacecraftat any time t.

With all the assumptions above, the theoretical local Cartesianmagnetic field components can be determined as:

Bx = − sin(ϕ) cos(ψ)Bϕ − sin(ψ)Bψ − cos(ϕ) cos (ψ)Bη

By = − sin(ϕ) sin(ψ)Bϕ + cos(ψ)Bψ − cos(ϕ) sin(ψ)Bη

Bz = cos(ϕ) Bϕ − sin(ϕ)Bη, (4)

where Bϕ , Bψ , Bη are given by Equations (1)–(3), respectively.From these equations the theoretical local magnetic field compo-nents in the GSE system are easily obtained. On the other hand,taking into account the parameters related to the orientation ofthe cloud (the orientation of the axis of the MC—the latitude, θ ,and the longitude, φ—and the minimum distance between thespacecraft path and its axis, y0), we can deduce the theoreticallocal magnetic field components in the GSE system. Moreover,the coordinates rspc(t), ϕspc(t), and ψ spc(t) of the spacecraft at

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(a) (b)

Figure 6. Data obtained by both Helios spacecraft for the magnetic cloud from 1978 January, and the corresponding fits for the toroidal model, (a) Helios A and(b) Helios B. The magnetic field strength, B, the Cartesian SSE-components (Bx, By, Bz), the proton pressure, and the bulk solar wind velocity are shown in bothfigures. The vertical dotted lines represent the time boundaries of the cloud. Superimposed on the experimental data, the toroidal model predictions are representedwith solid lines.

(A color version of this figure is available in the online journal.)

any time t inside the MC also have to be expressed as a functionof the orientation of the cloud before fitting.

There is no significant difference between expression (21)appearing in Hidalgo & Nieves-Chinchilla (2012) and thisexpression (4), where the term (r∂f (ψ)/ρ0∂ψ) sin(ϕ) Bψ hasbeen neglected due to its dependence on ρ0 and the conditionsassumed to obtain Equations (1)–(3), i.e., (ρ0>).

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4. DATA

For the present study we have used data from severalspacecraft: Helios (A and B), ACE, and STEREO s (A and B).In fact, there are three events that were each observed byseveral spacecraft that we analyze in light of the model. Hence,to compare the MC orientations obtained for every impliedspacecraft, and in order to reconstruct it in the interplanetarymedium, we have to consider their relative positions andrespective coordinate systems used for representing the data.Therefore, we have to bear in mind that in the GSE coordinatesystem (ACE) the x–y plane corresponds to the Earth meanecliptic of the date and the + x-axis is the vector Earth–Sun of

the date; in RTN coordinates (STEREO), + x-axis is the vector(Sun-S/C) and + y-axis is the cross-product of the heliographicpolar axis and + x-axis; and, finally, and in SSE (Helios)coordinates the x–y plane corresponds to the Earth mean eclipticof the date, with the + x-axis being the projection of the vectorS/C-Sun on the XY-plane and + z-axis the ecliptic south pole.

To determine the time interval of the considered MC events,we have followed the physical criteria defined by Burlaga et al.(1981) to establish their boundaries and, hence, address thebehaviors of the corresponding magnetic field components,strength, and the plasma data (in this last case selecting thetime interval where a low proton pressure, proton temperature,or plasma beta is clearly defined). Thus, in Tables 1–4 the dayof year (doy) for the boundaries of every event analyzed aredetailed, in correspondence with the entrance/exit of the satellitethat observed each event. The mean velocity at the passage ofthe spacecraft, which is necessary for our analysis procedure, isalso given.

In Figures 1–6, 8, and 10 we represent data for every analyzedevent and present the fitting results obtained from the model. Thegraphs shown in all those figures are the magnetic field strength,

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(a) (b)

(c)

Figure 7. Relative positions of Helios A (a) and B (b) at the arrival time of the MC from 1978 January (http://nssdc.gsfc.nasa.gov/space/helios/heli.html). The locationsof each one are highlighted with a red hexagon. (c) A snapshot of the projection over the ecliptic plane of the MC in its evolution when it is closed to both spacecraft,as it is deduced and consistent with the results obtained from the fitting of the model.

(A color version of this figure is available in the online journal.)

B, the corresponding Cartesian GSE, RTN, or SSE-components(Bx, By, Bz), some of the proton thermodynamic magnitudes(temperature, pressure, or beta), and the proton bulk solar windvelocity. The vertical dotted lines represent the time boundariesof the cloud following, Burlaga’s criteria. For the plasma, inmost of the cases we consider the proton beta, although in someother ones (Figures 3, 6(b), and 10(c)), we choose the protonpressure because the boundaries of the MCs appear to be betterdefined; and for the MC of 2007 November (Figure 10(b)), theproton temperature is shown.

The MCs analyzed in the present study are divided in two

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groups:

1. On one hand, MCs whose magnetic field presents a cleartoroidal topology: 1975 March (Helios A, ρ0 ≈ 0.188 AU,Figure 1); 1978 April (Helios B, ρ0 ≈ 0.158 AU, Figure 2);2000 November (ACE, ρ0 ≈ 0.5 AU, Figure 3). 2. Onthe other hand, more non-conventional complex structures:2003 May (ACE, ρ0 ≈ 0.5 AU, Figure 4); 2003 November(ACE, ρ0 ≈ 0.5 AU, Figure 5). (2ρ0 corresponds to theSun/spacecraft distance.)

In the events observed by both Helios spacecraft, whenthey were close to the Sun, the mean value of the strengthof the magnetic field is significantly higher. However, ouraim in considering data from Helios at these distances

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(a) (b)

Figure 8. Data obtained by both STEREOs and the ACE spacecraft for the magnetic cloud from 2007 May, and the corresponding fits for the toroidal model,(a) STEREO A, (b) STEREO B, and (c) ACE. The magnetic field strength, B; the Cartesian RTN-components (Bx, By, Bz), (a) and (b), and the CartesianGSE-components; the proton plasma beta and the bulk solar wind velocity are shown. The vertical dotted lines represent the time boundaries of the cloud.Superimposed on the experimental data, the toroidal model predictions are represented with solid lines (see the text for details).

(A color version of this figure is available in the online journal.)

near the Sun is due to the fact that the curvatures ofthe corresponding MCs are expected to be important (seebelow). However, even at 1 AU, near the ACE location, it ispossible to observe some MCs with significant curvatures,as was the case in the 2000 November event.

Then, for the MCs of 1975 March, 1978 April, and2000 November the strength of their magnetic fieldshave shapes associated with a toroidal topology; as seenin the corresponding figures, the strengths increase dur-ing the passage of the spacecraft at the MC time interval.This shape can only be explained by assuming that thereis a curvature in the MC topology. In fact, bearing in mindfor example an elliptic cross-section distortion, that shapecannot be related to some deformation of the cross-sectionof the MC because it would imply that the higher elonga-tion associated with the distortion would be higher in thedirection of the propagation of the MC, something that isphysically improbable.

In this first group of MCs, we have added two more eventswith more complex magnetic structures:

1. 2003 May, which in light of the model we can identifyas an MC; and 2003 November with a high strength andimportant consequences in the geomagnetic field.

2. MCs seen by two or more spacecraft: 1978 January (HeliosA and Helios B, ρ0 ≈ 0.47 AU, Figures 6(a) and (b)); 2007May (ACE, STEREO A, and STEREO B, ρ0 ≈ 0.5 AU,Figures 8(a)–(c)); and 2007 November (ACE, STEREO A,and STEREO B, ρ0 ≈ 0.5 AU, Figures 10(a)–(c)).

From the comparison of Figures 7(a) and (b), where the posi-tions of both Helios are represented at the time of the observa-tion, 1978 January (http://nssdc.gsfc.nasa.gov/space/helios/heli.html), the relative angular difference of both spacecraft wasaround 35◦. (In both figures each spacecraft position is high-lighted in red.)

On the other hand, in Figures 9 and 11 the relative positionsof the three other spacecraft used in the present multipoint study,STEREO (A and B) and ACE, are shown, each one at the momentof the passage of the corresponding MC through them, the 2007May and November events (the pictures were obtained from theWeb site, http://stereo-ssc.nascom.nasa.gov/where/). The threespacecraft were in the ecliptic plane and separated around 10o

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(c)

Figure 8. (Continued)

of longitude in the case of the 2007 May event, and around 30◦in the 2007 November event.

5. SUMMARY AND COMMENTS

We have achieved two kinds of analysis: first, of several MCswith different magnetic field shapes, and second, the same eventbut seen by several spacecraft in different locations. The mainpurpose of this last study is to test the toroidal topology modelthat we previously proposed. In Tables 1–4 we provide theorientation of every cloud as deduced from the fitting of themodel.

In Figures 1–6, 8, and 10 the toroidal model fitting results aresuperimposed with solid lines onto the experimental data for thethree magnetic field components and the strength of it.

1. MCs with different magnetic field structures. The MCs ofFigures 1–3 (clouds of 1975 March, 1978 April, and 2000November) show shapes in the strength of the magnetic fieldat different distances from the Sun that can unambiguouslybe associated with a toroidal topology. The only topologyexplaining this kind of shape, with an increased strengthalong the passage of the spacecraft inside of them, isassuming an intrinsic curvature of the MC.

On the other hand, in Figures 4 and 5 (MCs of 2003May and November, respectively, both seen at 1 AU), weshow two more complex magnetic field structures. Even

Figure 9. Relative positions of STEREO A, B, and ACE at the arrival time of theMC from 2007 May (http://stereo-ssc.nascom.nasa.gov/where/). A snapshot ofthe projection is superimposed over the ecliptic plane of the MC in its evolutionwhen it is closed to the three spacecraft, as it is deduced and consistent with theresults obtained from the fitting of the model. Because the latitude obtained bythe model is sufficiently high, it is represented here by the cross-section of themagnetic cloud

(A color version of this figure is available in the online journal.)

with a complex magnetic field topology, as in Figure 4, themodel can be fitted, allowing us to infer the MC topologyof this event and then showing its potentiality in the study

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of the topologies of these phenomena in the interplanetarymedium.

Figure 5 (November 2003) corresponds to another exam-ple obtained for a very intense MC.

2. MC seen by several spacecraft. An increasingly impor-tant method in the study of MCs is multispacecraft mea-surements, because it allows the reconstruction of the realMC by comparing the results obtained using an analyticalmodel. It is also useful for testing the consistency of themodel.

We compare the results obtained by using our model toanalyze three MCs observed by different spacecraft. There is oneMC seen by the two Helios spacecraft A and B, that correspondsto the MC observed in 1978 January and two more observed bythe two STEREO spacecraft (A and B), and ACE that correspondto 2007 May and November.

(a) Event of 1978 January (measured by Helios A and B).This corresponds to the MC of 1978 January, an MC observed

by both Helios spacecraft (Figures 6(a) and (b)). In Figures 7(a)and (b) we show the position of Helios A (a) and Helios B (b) atthe time of the MC observation. In the case of Helios A, thereis a gap in the data that conditions the rare boundary of the MCand of course the time interval used in the fit. The boundariesused for the study are shown with dotted lines. The starting timeof the passage of Helios A, which first observed the MC, wasat 3.609 and the exit of the spacecraft was at 4.054; for HeliosB, the entrance was at 4.3084 and the exit at 5.264. So there isa delay between both observations consistent with the relativepositions of both spacecraft.

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(a)

(b)

Figure 10. Data obtained by both STEREOs and the ACE spacecraft for the magnetic cloud from 2007 November, and the corresponding fits for the toroidal model:(a) STEREO A, (b) STEREO B, and (c) ACE. The magnetic field strength, B, the Cartesian RTN-components (Bx, By, Bz), (a) and (b), and the CartesianGSE-components; the proton plasma beta and the bulk solar wind velocity are shown. The vertical dotted lines represent the time boundaries of the cloud.Superimposed on the experimental data, the toroidal model predictions are shown with solid lines (see the text for details).

(A color version of this figure is available in the online journal.)

Considering the results obtained from the fitting of the model,we can reconstruct the projection over the ecliptic plane ofthe MC, such as was deduced from the orientation parametersobtained (Table 2 and Figure 7(c)).

(b) Event of 2007 May (observed by ACE, STEREO Aand B).

This event has been studied in detail in other references (Liu

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et al. 2008; Kilpua et al. 2009; Mostl et al. 2009). The timeinterval selected by us and other authors is quite similar. Themean velocities measured by the different spacecraft are alsopractically the same.

Figures 8(a)–(c) show data that correspond to this MCfrom the three spacecraft used for our analysis, STEREO A,STEREO B, and ACE, respectively. Looking at the positionof every spacecraft, the selected arrival time for the event ineach spacecraft is consistent with the relative position of them(Figure 9 and Table 3).

In Figure 9, we also show the picture of the relative position ofevery spacecraft and the projection of the MC over the eclipticplane as deduced from the results of the fitting of the model;

in Table 3, the corresponding orientation obtained for everyspacecraft is consistent with each other.

In particular, because the time duration of the event inSTEREO A is less than in the two other spacecraft, we canestablish that the encounter and passage of this spacecraftthrough the event has to take place near the flank of the MC, asit is represented in Figure 9.

(c) Event of 2007 November (observed by ACE, STEREO Aand B).

This event was recently studied in detail by Ruffenach et al.(2012). In Figures 10(a)–(c), our fitting results are representedwith solid lines. The velocities measured by the three spacecraftat the respective MC time interval are consistent with each otherand with the position of the spacecraft, as is their duration andarrival times (Figure 11). In this figure, a picture of the relativepositions of the spacecraft and the projection of the MC over theecliptic plane are shown, as deduced from the fitting (Table 4).

The results for the magnetic field components of the MCsseen by several spacecraft presented in this work lead us toinfer that their internal structures seem to be slightly writhe, as

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(c)

Figure 10. (Continued)

suggested by Jacobs et al. (2009) and Al-Haddad et al. (2011),in particular for the case of the MC of 1978 January.

Q6In light of these results, we think that the model presented

here can be considered as a useful tool for understanding thephysical mechanisms that affect the evolution of MC structuresevolving in the interplanetary medium. This model allows usto analyze the behavior of the plasma associated with suchmagnetic topologies, and not only the total plasma currentdensity, which is easily deduced from the set of equationswritten above, but also the plasma pressure (and, of course,the temperature).

Of course there are cases where the model does not workas well. A systematic study of many magnetic structures inthe interplanetary medium, most of them concerning flux-ropestructures, is discussed in detail in Hidalgo et al. (2012).

However, the high quality of fittings is clearly noticed in all thepresented cases, even for the events seen by different spacecraft,which support the consistency of the toroidal model and showits potentiality. Our next step will involve incorporating the

Figure 11. Relative positions of STEREO A, B and ACE at the arrival timeof the MC of 2007 November (http://stereo-ssc.nascom.nasa.gov/where/). Asnapshot of the projection is superimposed over the ecliptic plane of the MCin its evolution when it is closed to the three spacecraft, as it is deduced andconsistent with the results obtained from the fitting of the model.

(A color version of this figure is available in the online journal.)

behavior of the plasma inside the MC into the model and, as aconsequence, into the fitting procedure.

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This work was supported by the Comision Interministerialde Ciencia y Tecnologıa (CICYT) of Spain. Project References:AYA2010-12439-E and AYA2011-29727-C02-01. The authorsthank the teams of Helios (A and B), STEREOs (A and B),and ACE, for permission to use data and information from therespective Web sites.

REFERENCESQ8

Al-Haddad, N., Roussev, I. I., Mostl, C., et al. 2011, ApJL, 738, L18Burlaga, L. F., Sittler, E., Mariani, F., & Schwenn, R. 1981, JGR, 86, 6673Farrugia, C. J., Osherovich, V. A., & Burlaga, L. F. 1995, JGR, 100, 12293Hidalgo, M. A. 2003, JGR, 108, 1320

Q9Hidalgo, M. A. 2005, JGR, 108, 1320Hidalgo, M. A. 2011, JGR, 116, A02101Hidalgo, M. A., Cid, C., Vinas, A. F., & Sequeiros, J. 2002a, JGR, 107, 1002Hidalgo, M. A., & Nieves-Chinchilla, T. 2012, ApJ, 748, 109Hidalgo, M. A., Nieves-Chinchilla, T., & Cid, C. 2002b, GeoRL, 29, 1637Hidalgo, M. A., et al. 2012, SoPh, 10.1007/s11207-012-0191-6

Q10Jacobs, C., Roussev, I. I., Lugaz, N., & Poedts, S. 2009, ApJL, 695, L171Kilpua, E. K. J., Liewer, P. C., Farrugia, C., et al. 2009, SoPh, 254, 325

Q11Lepping, R. P., Burlaga, L. F., & Jones, J. A. 1990, JGR, 95, 11957Lepping, R. P., et al. 1997, JGR, 102, 1404

Q12Liu, Y., Luhmann, J. G., Huttunen, K. E. J, et al. 2008, ApJL, 677, L133Marubashi, K. 1997, in Coronal Mass Ejections, ed. N. Crooker, J. A. Joselyn,

& J. Feynman (Geophysical Monograph Series, Vol. 99; Washington, DC:AGU), 147

Mostl, C., Farrugia, C. J., Miklenic, C., et al. 2009, JGR, 114, A04102Nieves-Chinchilla, T., Gomez-Herrero, R., Vinas, A. F., et al. 2011, JASTP,

73, 1348Romashets, E. M., Vandas, M., & Poedts, S. 2010, SoPh, 261, 271Romashets, E. P., & Vandas, M. 2001, JGR, 106, 10615Ruffenach, A., Lavraud, B., Owens, M. J., et al. 2012, JGR, 117, A09101

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