Fitting a toroidal forceâfree field to multispacecraft observations of a

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115, A10113, doi:10.1029/2010JA015539, 2010

Fitting a toroidal force‐free field to multispacecraft observations of a magnetic cloud Tomoko Nakagawa1 and Ayako Matsuoka2 Received 7 April 2010; revised 29 June 2010; accepted 23 July 2010; published 22 October 2010.

[1] A torus‐type flux rope model with an arbitrary aspect ratio was applied to an

interplanetary magnetic cloud observed by ACE and Nozomi on 16–18 April 1999, when Nozomi was 0.2 AU downstream of ACE in the solar wind within 3° of heliocentric longitude. The large and small radii of the torus, the direction of the symmetric axis, and the crossing points of the spacecraft were determined so that they would minimize the sum of the square of the difference between the model field and the hourly averages of the observed field. Self‐similar expansion of the flux rope was assumed in proportion with the heliocentric distance. The best fit model had large and small radii of 0.16 and 0.09 AU, respectively. Both spacecraft passed through the northern part of the torus. Difference in the magnetic field observed by the two spacecraft was explained by the difference in their paths through the magnetic cloud. The model fit was consistent with the direction of the vector normal to the preceding planar magnetic structures. The chirality of the flux rope was positive (left handed), suggesting that the solar source was on the Northern Hemisphere. Assuming a probable association with the filament disappearance observed on 13 April 1999 at N16 E00, it is inferred that the filament had traveled in interplanetary space across the ecliptic plane. It was also found that nearly the same fitting result was obtained using a single‐spacecraft observation in the case of a torus‐shaped magnetic cloud with a small aspect ratio. Citation: Nakagawa, T., and A. Matsuoka (2010), Fitting a toroidal force‐free field to multispacecraft observations of a magnetic cloud, J. Geophys. Res., 115, A10113, doi:10.1029/2010JA015539.

1. Introduction [2] A magnetic cloud is an interplanetary magnetic field structure characterized by a large smooth rotation of the intense magnetic field and a low‐beta plasma [Burlaga et al., 1981; Burlaga et al., 1990]. Magnetic clouds have been associated with solar phenomena such as filament eruptions [Marubashi, 1986] and coronal mass ejections (CMEs) [Burlaga et al., 1982], thus making them a key to connecting the Sun with interplanetary space. [3] During the passage of a magnetic cloud, a large southward magnetic field can last for more than several hours, causing serious effects in the planets’ magnetospheres. The duration of the southward magnetic field depends on the properties of the magnetic cloud such as the scale size, convection speed, inclination of the overall structure with respect to the ecliptic plane, direction of the core field, and the impact parameter of the planets with respect to the axis of the flux rope. To predict the possible geomagnetic storms that might be caused by the passage of the cloud, it is desirable to

1 Information and Communication Engineering, Tohoku Institute of Technology, Sendai, Japan. 2 ISAS, JAXA, Sagamihara, Japan.

Copyright 2010 by the American Geophysical Union. 0148‐0227/10/2010JA015539

determine the three‐dimensional configuration of the magnetic cloud in advance. [4] The scale size and the attitude of each magnetic cloud have been estimated from in situ observations by fitting a model field. Many authors have employed flux ropes of the force‐free magnetic fields, which satisfy r × B = aB to model the magnetic clouds. Most commonly used was a cylindrically symmetric, constant a force‐free field [Burlaga et al., 1981; Klein and Burlaga, 1982; Burlaga, 1988; Marubashi, 1986]. On the other hand, it has been assumed that the both ends of the magnetic clouds are connected to the Sun. To represent the configuration of the closed loops, a toroidal force‐free field was introduced by Ivanov et al. [1989]. Romashets [1993] and Ishibashi and Marubashi [2004] employed the toroidal force‐ free field solution by Miller and Turner [1981] assuming a large aspect ratio (that is, a thin toroid with a large global radius), which may not always be applicable to the magnetic clouds observed in interplanetary space. The model fitting was improved by Romashets and Vandas [2001, 2003], who introduced the toroidal force‐free field for arbitrary aspect ratio. [5] It has often been difficult to determine the overall curvature of the torus from a single spacecraft observation. For example, the magnetic field of the northern part of a torus‐ type magnetic cloud might be reproduced by the magnetic field of the southern part of another toroidal model with an opposite axial (toroidal) field. To determine the overall

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NAKAGAWA AND MATSUOKA: MULTISPACECRAFT FITTING MAGNETIC CLOUD

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fit the ACE observations: a cylindrical force‐free field and a torus‐shaped force‐free field with large aspect ratio [Miller and Turner, 1981], and obtained better fitting result from the torus‐shaped model. It resulted in reasonable agreement with the filament eruption of the solar source. The best fit model they obtained was a torus‐shaped force‐free field with a major radius of 0.3 AU and a cross‐sectional radius of 0.07 AU through which the spacecraft supposedly penetrated at the southern tip. [8] In this paper, a torus‐type flux rope model with an arbitrary aspect ratio applied to the interplanetary magnetic cloud was tested by observations carried out by the ACE and Nozomi spacecraft.

2. Observations

Figure 1. Positions of Nozomi and ACE on 17 April 1999 in the solar ecliptic coordinate system. Nozomi was 0.2 AU downstream of ACE in the solar wind and within 3° of heliocentric longitude. (top) The orbits as seen by an observer in the ecliptic plane. (bottom) A view of the orbits looking down on the ecliptic plane.

structure, multispacecraft observations with separation of the order of the scale length of the magnetic cloud are desired. [6] A good opportunity occurred on 16–18 April 1999, when the Advanced Composition Explorer (ACE) and Nozomi spacecraft detected a magnetic cloud at radial distances of 1.0 and 1.2 AU, respectively, from the Sun within 3 degrees of heliocentric longitude. Nozomi is a Japanese Mars explorer that carried out observations of the interplanetary magnetic field in its cruise phase [Yamamoto and Matsuoka, 1998; Nakagawa et al., 2002]. Figure 1 shows the positions of ACE and Nozomi on 17 April 1999. The separation of 0.2 AU between the two spacecraft is of the order of the scale size of typical magnetic clouds and suitable for examining the structure of the magnetic cloud. [ 7 ] This is the same event analyzed by Ishibashi and Marubashi [2004]. They tried a couple of model fields to

[9] Figure 2 shows hourly averages of the magnetic field observed by (Figure 2a) ACE at 1.0 AU from the Sun and (Figure 2b) Nozomi at 1.2 AU from the Sun for 5 days around the time of the passage of the magnetic cloud in RTN coordinates. ACE observed the magnetic cloud at (−0.88, −0.46, 0.00) AU in ecliptic coordinates during the period from 20:00 UT on 16 April 1999 to 20:00 UT on 17 April 1999. Prior to the magnetic cloud, ACE detected a shock wave driven by the magnetic cloud at 10:30 UT on 16 April 1999. The magnetic cloud was also detected by Nozomi at (−1.08, −0.49, 0.06) AU during the period from 20:00 UT on 17 April 1999, to 18:00 UT on 18 April 1999. The delay of the arrival is consistent with the travel time of the magnetic cloud calculated from the typical solar wind speed of 391 km/s measured at ACE during the event. [10] The thin curves in Figure 2b show the magnetic field observed by ACE shifted for the travel time to the Nozomi position. Figure 2b shows that the magnetic field observed by the two spacecraft was nearly the same except for Alfvénic fluctuations until the arrival of the magnetic cloud and its preceding shock wave. Upon the arrival of the magnetic cloud, as recognized by the enhanced field magnitude B and depressed plasma beta (not shown here), the magnetic field began to rotate differently at each spacecraft. The most prominent instance of this was the BN component, which was positive at Nozomi while negative at ACE during the passage of the magnetic cloud. There was also a difference in the BT component. It would be natural to think that the magnetic field differences were due to the different paths through different parts of a single cloud, although there is a possibility that each spacecraft observed an entirely different cloud.

3. Model Fitting 3.1. Toroidal Force‐Free Magnetic Field With Arbitrary Aspect Ratio [11] In this paper, an attempt was made to fit a single toroidal force‐free model to the simultaneous observations of the magnetic cloud by ACE and Nozomi. [12] As a model flux rope, we employed an analytical solution of force‐free magnetic fields inside a toroid with an arbitrary aspect ratio [Romashets and Vandas, 2003]. The model field was given in radial, toroidal, and poloidal components, Bm, Bh, and B, in the toroidal coordinate system

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Figure 2. Hourly averages of the magnitude of the magnetic field B, the three components BR, BT, and BN in RTN coordinates, and the radial component of the plasma bulk velocity VR observed by (a) ACE at (−0.88, −0.46, 0.00) AU and (b) Nozomi at (−1.08, −0.49, 0.06) AU in ecliptic coordinates during the period from 15 April 1999 to 20 April 1999. The vertical lines indicate the start and end times of the magnetic cloud. The dashed line indicates the shock wave preceding the cloud, detected by ACE at 10:30 UT on 16 April 1999. The thin curves in Figure 2b are the magnetic field observed by ACE shifted for the travel time to the Nozomi position. No velocity observation was available at Nozomi. (m, h, ), which would asymptotically correspond to the Br, B, and Bz components in a cylindrical coordinate system (r, , z) in the limit of large aspect ratio. The toroidal coordinates are illustrated in Figure 3 of Romashets and Vandas [2001]. For simplicity, we take only the zeroth‐order terms of equations (18)–(20) in the work of Romashets and Vandas [2003, 2009]: B0 ¼ 0; a cosh Fð1 þ 0 ; 1 þ 0 ; 2; Þ; B0 ¼ hB0 2h sinh3 B0 ¼ B0

a Fð0 ; 0 ; 1; Þ; h sinh

where B0 is the constant that controls the magnitude of the magnetic field, h is the chirality (h = 1 for left‐hand twist or “anti‐parallel type” of Marubashi [1986], while h = −1 for right‐hand twist or “parallel‐type”), a is the parameter a¼

ð1Þ

ð2Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R20 r02 ;

ð4Þ

determined from the large and small radii of the toroid (R0 and r0, respectively), is a positive constant, hm is the Lamé constant

ð3Þ

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h ¼

a ; ðcosh cos Þ

ð5Þ

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and F is the hypergeometric function. The arguments a0, b0, and x of the hypergeometric function F are given as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 42 ; 4

ð6Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 42 ; 0 ¼ 4

ð7Þ

0 ¼

1þ

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and ¼

1 : sinh2

ð8Þ

Romashets and Vandas [2003] assumed that the axial field component B would vanish at the boundary of the flux rope. To satisfy this assumption, the parameter was set as 2:41

a : r0

ð9Þ

Details of the selection of were given by Romashets and Vandas [2001, 2003]. We employ the assumption and the value of according to Romashets and Vandas [2003]. [13] Once the parameters R0, r0, B0, and h are given, we can calculate the force‐free field (Bm, Bh, B) at any point (m, h, ) in the toroidal coordinate system. Figure 3 illustrates an example of a toroidal force free field with an aspect ratio of R0/r0 = 3.0 and a chirality h = 1. It has a circular cross section like the toroidal model with a large aspect ratio. In contrast to the large aspect ratio models, the position of the maximum field magnitude is not on the center of the circle of the cross section in the arbitrary aspect ratio model; rather it is shifted to the inner hole of the torus [Romashets and Vandas, 2003], because the poloidal magnetic field lines become closer to each other on the inner side of the torus than on the outer side. To balance the pinch effect produced by the poloidal field, the axial field also becomes stronger on the inner side. Figure 3 also demonstrates that a small displacement in the measurement position can cause a large difference in the magnetic field. 3.2. Model Assumptions [14] In addition to the model field structure, some other assumptions were made to reproduce the observations. The magnetic cloud was assumed to be launched radially outward from the Sun at a constant speed of 391 km/s, as measured by ACE in such a direction that ACE would traverse the cloud. The precise launch direction was specified with the point P of intersection of the spacecraft trajectory with the plane including the main core axis. Figure 4 illustrates a model torus and the crossing point P. The location of the crossing point P was indicated by the toroidal angle p measured from the northern top of the torus (jpj < p/2 for northern crossing, and jpj > p/2 for southern crossing), and the distance rp of the point P from the central toroid Figure 3. An example of the model field with an aspect ratio R0/r0 = 3.0. (a–e) The magnetic field at several cuts in the torus (illustrated at the top), in which the bars indicate the magnetic field vectors projected onto the cross section and the gray scale indicates the magnitude of the magnetic field with darker colors representing a more intense field. The bottom two plots show the axial and poloidal components, By and Bz at y = 0, which correspond to B and Bh, respectively. Figure 3a shows the magnetic field is poloidal (that is, no axial field) at the boundary of the flux rope as assumed. Figure 3e shows the most intense field is shifted toward the symmetric axis of the torus. Comparison of these plots shows that a small difference in measurement location can cause a large difference in the magnetic field. 4 of 10

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magnetic field to be observed at the ACE and Nozomi positions each hour of the day of detection of the magnetic cloud. [18] The direction of the toroidal field (the sign of B0) was determined so that it would reproduce the radial component BR of the magnetic field observed in the magnetic cloud (BR > 0 for the present case). The chirality h was determined by testing the two possible candidates, then choosing the one that better reproduced the observations. [19] The parameters R0, r0, A, A, p, rp, and Dt were determined so that they would minimize the difference between the model field and the observed field S ¼ SA þ SN ;

ð10Þ

summed over the two spacecraft denoted by the subscripts A for ACE and N for Nozomi. The sum of the squares of the differences SA ¼

nA X k¼1

2A;k

ð11Þ

over nA = 24 data points for ACE and Figure 4. Schematic illustration of a torus model with a large radius R0 and a small radius r0 that has a symmetric z axis in the direction (A, A) in the GSE coordinate system. The crossing point P of the ACE trajectory with the plane z = 0, including the main core axis, is given by the angle p measured from the x axis and the distance rp from the main core axis (x, y, and z are Cartesian coordinates with respect to the torus). (∣rp∣ < r0, rp > 0 for crossing the outer side of the central toroid, and rp < 0 for crossing the inner side near the central hole of the torus). [15] The attitude of the toroid, i.e., the inclination of the overall structure with respect to the ecliptic plane, given by the direction of the symmetric axis of the toroid (A, A) in the geocentric solar ecliptic (GSE) coordinate system at the ACE position, was assumed to be unchanged during the passage of the cloud from ACE to Nozomi. [16] Self‐similar expansion of the flux rope was assumed to be proportional to the heliocentric distance of the center of the torus. The large and small radii R0 and r0 of the torus were given at the ACE position. Those at the Nozomi position were assumed to be proportional to the radial distance of the center of the torus from the Sun. The magnitude of the magnetic field at Nozomi was reduced due to conservation of the magnetic flux. The plasma velocity of each part of the toroidal magnetic cloud was given by the sum of the constant velocity of the center of the torus and the expansion velocity of each part. Due to the self‐similar expansion, the velocity component of the plasma radially outward from the Sun becomes larger at the front side of the torus and smaller at the rear side. For simplicity, compression through interactions with the surrounding solar wind plasma (as suggested by Lepping et al. [2007]) and oblateness (as suggested by Vandas et al. [2006]) were not taken into account. 3.3. Fitting Procedure [17] By setting the parameters R0, r0, B0, h, A, A, p, rp, and the start time adjustment Dt, we can calculate the model

SN ¼

nN X k¼1

2N ;k

ð12Þ

for nN = 22 data points for Nozomi was calculated on the basis of the difference between the model field BM k and the hourly averages of the observed field BO k ormalized by their magnitudes as

2A;k

2N ;k

¼ ¼

X

BO A;k;j

j

jBO A;k j

X

BO N ;k;j

j

jBO N ;k j

BM A;k;j

!2

jBM A;k j BM N ;k;j

jBM N ;k j

; ð13Þ

!2 ;

following Lepping et al. [1990], where the summations are taken for the three components of the RTN coordinates. [20] Normalization by the magnitude of the magnetic field was employed to ensure that all parts of the magnetic cloud were given equal importance. A least square fitting without normalization is equivalent to giving extra importance to the central core part, which has an intense magnetic field. After fitting all the other parameters, the magnitude of B0 was calculated in one‐parameter least squares fit to the data. [21] As the scale size of the magnetic cloud was finite and no model field was defined outside the torus, there was a case where no model field was defined at the spacecraft position when the spacecraft was observing the magnetic cloud. In such a case, the square of the differences s2k was set to the maximum value (s2k = 22). Also, there was an alternative case in which the spacecraft was out of the observed magnetic cloud while it was calculated to be within a model flux rope. In those cases, the maximum value of s2k was added to the sum SA or SN for each additional 1‐h data point obtained within the model cloud.

4. Results 4.1. Northern and Southern Crossings [22] Table 1 lists the best fit parameters of the toroidal force free field model, together with the sum of the squares

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Table 1. Comparison of Fitting Results Present Study Northern Passage

Southern Passage

Isibashi and Marubashi [2004]

chirality left hand direction of core field − 0.163 R0 (AU) 0.089 r0 (AU) 1.8 R0 /r0 21.0 B0 (nT) 54.1 A (°) in GSE −11.1 A (°) in GSE position of spacecraft crossing −0.46 rp /r0 −40.6 p (°) adjustment of start hour 0.99 5.14 ACE deviation SA 9.48 Nozomi deviation SN total deviation S 14.63

left hand + 0.384 0.094 4.1 23.6 23.6 −18.4

left hand

−0.22 168.5 −0.06 15.83 28.89 44.72

0.3 0.07 4.3 23 33 −18

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of the deviation of the model field from the observations. The model and observations were compared for jpj < p/2 (northern crossing) and jpj > p/2 (southern crossing). The difference between the model and the observations was much smaller for the northern passage (SA = 5.14, SN = 9.48, and S = 14.63) than the southern passage (SA = 15.83, SN = 28.89, and S = 44.72). The results show that the ACE and Nozomi spacecraft passed through the northern part of a torus‐shaped flux rope whose large and small radii were 0.163 AU and 0.089 AU, respectively. The aspect ratio was 1.8. The direction of the symmetric axis of the torus was (A, A) = (54°, −11°). The chirality was positive, i.e., the helicity was left handed (antiparallel type), suggesting that the solar origin of the magnetic cloud was a filament on the northern hemisphere [Rust and Kumar, 1994]. [23] Figure 5 compares the best fit toroidal force‐free field model (thick curves) for the northern passage of ACE and Nozomi through the magnetic cloud with the observations (thin curves). The trajectories of ACE and Nozomi through

Figure 5. Comparison of the best fit toroidal force‐free field model (thick curves) and the magnetic field observed by ACE and Nozomi (thin curves) within the magnetic cloud. A scale of the maximum possible error for each measurement is indicated on the BR component. The magnetic field outside the magnetic cloud is not displayed. (bottom) The radial component of the plasma bulk velocity VR observed by ACE compared with that estimated from the model under the assumption of the self‐similar expansion of the toroidal model. To the right side of the panels is an illustration that shows the trajectories of Nozomi and ACE through the toroidal field model. Both spacecraft passed through the torus on the northern part. The small dots are trajectories outside the torus, the large dots are in the near half of the torus, and the intermediate‐sized dots are in the far half of the torus. 6 of 10

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Figure 6. The best fit toroidal force‐free field model based on the assumption of a southern passage (thick curves) compared with the observations by ACE and Nozomi (thin curves) within the magnetic cloud. The format is the same as Figure 5. The model field failed to reproduce the Nozomi observations. the toroidal field model are illustrated to the right of the panels. Nozomi traversed a more outer part of the torus than ACE did. The difference of the trajectories caused the difference in the value of the BN component of the magnetic field observed by the two spacecraft. [24] The model magnetic field agreed fairly well with the observations by ACE. The deviation was larger for Nozomi, but the direction of the magnetic field was approximately reproduced. It should be noted that the best fit model can simultaneously account for both the negative BN at ACE and the positive BN at Nozomi. The duration of the magnetic cloud was well reproduced for both ACE and Nozomi. [25] Figure 5 (bottom) shows the radial component of the plasma bulk velocity VR. The estimation of VR assuming the self‐similar expansion of the model (thick line) agreed with the ACE observations (thin line). Thus, Figure 5 shows that the northern passage best fit model agrees well with the observations of the magnetic cloud made by the two spacecraft. [26] Figure 6 shows the results of the fitting based on the assumption that the spacecraft traversed the southern part of the toroidal model. The resulting torus had a large radius of 0.384 AU, a small radius 0.094 AU, and a symmetric axis (A, A) = (23.6°, −18.4°). The chilarity was again positive (left‐handed helicity). The agreement of the model field with the ACE observations was good, but the start time of

the cloud detection did not agree with the observation, and the difference was larger at Nozomi. In addition, the positive BN as observed by Nozomi was not reproduced. Thus, Figure 6 shows that the southern passage best fit model did not agree as well with the observations as the northern passage best fit model did. [27] Table 1 also shows that the best fit parameters (R0 = 0.384, r0 = 0.094, R0 /r0 = 4.1, and B0 = 23.6 nT) calculated from the southern passage model were similar to those (R0 = 0.3, r0 = 0.07, R0/r0 = 4.3, and B0 = 23 nT) obtained by Ishibashi and Marubashi [2004]. It shows that the toroidal force‐free field with aspect ratio of 4.1 is well represented by the model with large aspect ratio. [28] Figure 7 is a three‐dimensional expression of the best fit models to illustrate the direction of the overall configuration of the torus, which might not be easily recognized from the direction of the symmetric axis (A, A) given in Table 1. The core axes of both models run northeast to southwest, consistent with the supposed origin of the filament disappearance between 20:36 UT on 12 April 1999 and 10:58 UT on 13 April 1999 [Ishibashi and Marubashi, 2004]. It is also recognized in the northern passage case that Nozomi trajectory was outer side of the axial core field while ACE trajectory was inner side of the core field; thus, each spacecraft observed different directions of the magnetic

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Table 2. Single Spacecraft Fitting Using ACE Observation

chirality direction of core field R0 (AU) r0 (AU) R0 /r0 B0 (nT) A (°) in GSE A (°) in GSE position of spacecraft crossing rp /r0 p (°) adjustment of start hour ACE deviation SA Nozomi deviation SN total deviation S

Northern Passage

Southern Passage

left hand − 0.199 0.099 2.0 22.6 55.1 −7.8

left hand + 0.491 0.092 5.3 24.6 23.6 −20.4

−0.25 −46.2 0.99 3.74 19.90 23.64

−0.22 165.3 −0.06 15.22 51.13 66.35

field. On the other hand, in the southern crossing case, both spacecraft passed through the torus at the inner side of the torus near the inner hole; thus, the southern crossing model cannot account for the difference of the magnetic fields observed by ACE and Nozomi. 4.2. Comparison With Single‐Spacecraft Fitting [29] We made another trial of a single‐spacecraft fitting by using only ACE observations, and compared the results with the multi‐spacecraft fitting. The results from the single‐ spacecraft northern crossing and southern crossing models are listed in Table 2. Again the northern crossing model gave a better fitting result. It should be noted that the results are quite similar to those obtained from the multi‐spacecraft fitting listed in Table 1. [30] It suggests that for a torus‐shaped magnetic cloud with a small aspect ratio like the present example, fitting to single‐spacecraft observations gives a good estimation of the global structure of the magnetic cloud.

Figure 7. Schematic illustrations of the best fit toroidal force‐free field models of (a and b) the northern passage and (c) the southern passage. Blue and green bars indicate the hourly averaged magnetic field vectors observed at ACE and Nozomi, respectively. In this schematic illustration, the effect of self‐similar expansion is not precisely reproduced. The field vectors at ACE (at 1 AU) are presented as they were observed, not being converted into their predicted values at the Nozomi position (1.2 AU).

4.3. Planar Magnetic Structures [31] Often a planar magnetic structure is formed in front of a coronal mass ejection. Jones et al. [2002] have shown that the direction of the vector normal to a sheath planar magnetic structure provides a good consistency check of the orientation of the structure of the driving coronal mass ejecta. [32] Table 3 lists the planar magnetic structures detected by ACE and Nozomi prior to their encounter with the magnetic cloud. The selection criteria for a planar magnetic structure are the small magnetic field component perpendicular to the plane of the structure, Bn /jBj < 0.1, and a high variability of the magnetic field, sB /jBj > 0.7, where sB is the standard deviation of the magnetic field variation [Nakagawa et al., 2000]. In 16 s magnetic field data obtained by ACE, a planar magnetic structure was found during the period from 12:00 to 15:00 on 16 April 1999. The vector normal to the plane was calculated to be (0.91, −0.35, 0.21) using minimum variance analysis [Sonnerup and Cahill, 1967]. The minimum variance lmin = 2.7 which is very small with respect to

Table 3. Planar Magnetic Structures Preceding the Magnetic Cloud Spacecraft

Time

Bn jBj

B jBj

lmin

lmed

lmax

e na

ACE Nozomi

12–15 UT, 16 April 1999 18–20 UT, 17 April 1999

0.06 0.07

1.00 0.94

2.7 0.94

12 13

31 41

(0.91, −0.35, 0.21) (0.83, −0.44, 0.34)

a

en, vector normal to the plane of the planar magnetic structure.

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the solar origin of the magnetic cloud was a filament on the northern hemisphere [Rust and Kumar, 1994]. Assuming that the solar origin was the filament disappearance observed on 13 April 1999 at N16 E00 as suggested by Ishibashi and Marubashi [2004], we can say that the filament traversed the ecliptic plane in interplanetary space toward the heliospheric current sheet, which was on the southern hemisphere at the longitude (∼240°) according to the coronal field map computed by the Wilcox Solar Observatory.

Figure 8. Planes of the planar magnetic structures that precede the magnetic cloud (red crosses) and the 20 min averages of the magnetic field vectors within the planar magnetic structures (light‐colored bars in front of the red crosses). Dark‐colored bars behind the red crosses are the magnetic field vectors within the magnetic cloud, with green for Nozomi and blue for ACE. The field vectors at ACE (at 1 AU) were converted into their predicted values at the Nozomi position (1.2 AU) considering the radial dependence of each component of the magnetic field (BR, BT, BN). Compared with Figure 7, the directions of the planes of the planar magnetic structures seem to be consistent with the northern passage model of the toroidal magnetic cloud. the medium and maximum variances, lmed = 12 and lmax = 31, indicates that an accurate normal vector was obtained [Lepping and Behannon, 1980]. In the 8 s data obtained by Nozomi there was also a planar magnetic structure starting from 18:00 and lasting for 2 h on 17 April 1999. The vector normal to the plane was (0.83, −0.44, 0.34). The vector normal to the Nozomi planar magnetic structure was directed rather northward than that of ACE. The difference of the direction is consistent with the northern crossing of the best fit model, because the vector normal to the surface of the model torus is supposed to be directed more northward at the Nozomi position than the ACE position. [33] Figure 8 is a three‐dimensional presentation of the planes of the planar magnetic structures (red crosses parallel to each plane of the structure), together with the magnetic field vectors within the planar magnetic structures (light colored bars) and within the magnetic cloud (dark colored bars). The top left red cross is the plane of the planar magnetic structure detected by Nozomi, while the lower‐right red cross is that observed by ACE. As seen in Table 3, the planar magnetic structure at NOZOMI was directed rather northward than at ACE. Note that the two planes are nearly parallel to the front surface of the torus‐shaped model of the northern passage, as displayed in Figure 7a, thus corroborating the view that our northern passage model is a correct description.

5. Discussion 5.1. Relationship With the Solar Origin [34] The chirality of the best fit force‐free field was positive, i.e., the helicity was left handed, suggesting that

5.2. Speed of Propagation [35] Figure 9 shows the timeline of the events starting from a halo CME at 0330 UT on 13 April 1999, which is assumed to be the origin of the magnetic cloud discussed in the present study [Ishibashi and Marubashi, 2004], followed by the interplanetary disturbance detected by the observation of interplanetary scintillation by the Solar‐Terrestrial Environment (STE) Laboratory, Nagoya University (M. Tokumaru, private communication), and the magnetic cloud observations by ACE and Nozomi. It seems that the assumption of constant speed of the center of the toroidal magnetic cloud was appropriate. 5.3. Deviation From the Model [36] For the best fit model, the difference between the model field and the observed field was larger for Nozomi (SN = 9.48) than for ACE (SA = 5.14), as listed in Table 1. The reason is not known. It seems that the difference was larger in the front half of the magnetic cloud (the earlier part of the observation) as recognized from Figure 5. It might be due to a distortion caused by the interaction between the cloud and the ambient solar wind, as suggested by Lepping et al. [2007], a rotation of the magnetic cloud, the assumption of the axial field vanishing at the boundary of the cloud, or the oversimplified model calculated with only the zeroth‐ order terms. It would require some numerical simulations to

Figure 9. Timeline of the events starting from a partial halo CME at 0330 UT on 13 April 1999 (adapted from the SOHO/LASCO CME catalogue [Yashiro et al., 2004], followed by the interplanetary disturbance detected by the STE Laboratory, Nagoya University (using the observation of interplanetary scintillation), and ending with the magnetic cloud observations by ACE and Nozomi.

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assess the effect of the interaction with the ambient solar wind.

6. Conclusion [37] A torus‐type flux rope model with an arbitrary aspect ratio applied to an interplanetary magnetic cloud was tested by observations made by the ACE and Nozomi spacecraft on 16–18 April 1999, when Nozomi was 0.2 AU downstream of ACE in the solar wind and within 3 degrees of heliocentric longitude. The results show that the ACE and Nozomi spacecraft passed through the northern part of a torus‐shaped flux rope with a large radius of 0.16 AU and a small radius of 0.09 AU resulting in an aspect ratio of 1.8. The direction of the symmetric axis of the torus was (A, A) = (54°, −11°). Nozomi traversed outer part of the torus than ACE did, detecting northward magnetic field, while ACE detected southward magnetic field during the passage through the magnetic cloud. The front surface configuration of the best fit model was consistent with the direction of the planar magnetic structures preceding the magnetic cloud. The chirality was positive, that is, the helicity was left handed, suggesting that the solar origin of the magnetic cloud was a filament on the northern hemisphere [Rust and Kumar, 1994]. Assuming that the solar origin was the filament disappearance observed on 13 April 1999 at N16 E00 as suggested by Ishibashi and Marubashi [2004], we infer that the filament had traveled in interplanetary space across the ecliptic plane toward the heliospheric current sheet which was on the southern hemisphere at the longitude. [38] It should also be noted that the use of a single‐ spacecraft observation resulted in a fairly good estimation of the torus‐shaped model magnetic cloud. It tells us that, for magnetic clouds that are not well reproduced by a cylindrical model but are well reproduced by a torus‐shaped model, it is possible to determine the global configuration of the magnetic cloud from observations made by a single spacecraft. [39] Acknowledgments. The authors thank Nozomi/MGF team for the Nozomi magnetic field observations. We thank the PLANET‐B project team for making a special effort to realize a magnetically clean spacecraft, which enabled magnetic field observations under conditions where the expansion of the MGF sensor mast was prohibited. The authors are grateful to Charles W. Smith for the use of ACE/MAG data and to David J. McComas for ACE/SWEPAM data. The authors are indebted to the ACE Science Data center for providing the ACE data. Interplanetary scintillation measurements of the solar wind have been carried out under the joint research program of the Solar‐Terrestrial Environment Laboratory of the Nagoya University. Wilcox Solar Observatory data used in this study were obtained via the web site http://wso.stanford.edu at 2010:06:23_0 0:16:36 PDT courtesy of J. T. Hoeksema. The Wilcox Solar Observatory is currently supported by NASA. Figure 7 was produced by K. Jinbo and M. Suzuki. This study was supported by the JSPS grant‐in‐aid for scientific research project 14540416. [40] Philippa Browning thanks the reviewers for their assistance in evaluating this paper.

References Burlaga, L. F. (1988), Magnetic clouds and force‐free‐fields with constant alpha, J. Geophys. Res., 93(A7), 7217–7224, doi:10.1029/JA093iA07p07217. Burlaga, L. F., E. Sittler, F. Mariani, and R. Schwenn (1981), Magnetic loop behind an interplanetary shock: Voyager, Helios, and IMP 8 observations, J. Geophys. Res., 86(A8), 6673–6684, doi:10.1029/JA086iA08p06673.

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Burlaga, L. F., L. Klein, N. R. Sheeley Jr., D. J. Michels, R. A. Howard, M. J. Koomen, R. Schwenn, and H. Rosenbauer (1982), A magnetic cloud and a coronal mass ejection, Geophys. Res. Lett., 9(12), 1317–1320, doi:10.1029/GL009i012p01317. Burlaga, L. F., R. P. Lepping, and J. A. Jones (1990), Global configuration of a magnetic cloud, in Physics of Magnetic Flux Ropes, Geophys. Monogr. Ser., vol. 58, edited by C. T. Russell et al., pp. 373–377, AGU, Washington D. C. Ishibashi, H., and K. Marubashi (2004), Structure of interplanetary magnetic cloud on April 16, 1999 and its origin estimated by fitting the torus‐shaped flux rope model, Geophys. Res. Lett., 31, L21807, doi:10.1029/2004GL020702. Ivanov, K. G., A. F. Harshiladze, E. G. Eroshenko, and V. A. Stiazhkin (1989), Configuration, structure, and dynamics of magnetic clouds from solar flares in light of measurements on board Vega 1 and Vega 2 in January–February 1986, Sol. Phys., 120, 407–419. Jones, G. H., A. Rees, A. Balogh, and R. J. Forsyth (2002), The draping of heliospheric magnetic fields upstream of coronal mass ejecta, Geophys. Res. Lett., 29(11), 1520, doi:10.1029/2001GL014110. Klein, L. W., and L. F. Burlaga (1982), Interplanetary magnetic clouds at 1 AU, J. Geophys. Res., 87(A2), 613–624, doi:10.1029/JA087iA02p00613. Lepping, R. P., and W. Behannon (1980), Magnetic field directional discontinuities: 1. Minimum variance errors, J. Geophys. Res., 85(A9), 4695–4703, doi:10.1029/JA085iA09p04695. Lepping, R. P., J. A. Jones, and L. F. Burlaga (1990), Magnetic field structure of interplanetary magnetic clouds at 1 AU, J. Geophys. Res., 95(A8), 11,957–11,965, doi:10.1029/JA095iA08p11957. Lepping, R. P., T. W. Narock, and H. Chen (2007), Comparison of magnetic field observations of an average magnetic cloud with a simple force free model: The importance of field compression and expansion, Ann. Geophys., 25, 2641–2648. Marubashi, K. (1986), Structure of the interplanetary magnetic clouds and their solar origins, Adv. Space Res., 6(6), 335–338. Miller, G., and L. Turner (1981), Force‐free equilibria in toroidal geometry, Phys. Fluids, 24, 363–365. Nakagawa, T., S. Kokubun, and T. Mukai (2000), Plasma velocity in interplanetary planar magnetic structures, Adv. Space Res., 26(5), 811–814. Nakagawa, T., et al. (2002), NOZOMI observation of the interplanetary magnetic field in 1998, Adv. Space Res., 29(3), 427–432. Romashets, E. P. (1993), Force‐free magnetic field in nonsymmetrical interplanetary cloud, Geomagn. Aeron, 31, 365–372. Romashets, E. P., and M. Vandas (2001), Dynamics of a toroidal magnetic cloud in the solar wind, J. Geophys. Res., 106(A6), 10,615–10,624, doi:10.1029/2000JA900174. Romashets, E. P., and M. Vandas (2003), Force‐free field inside a toroidal magnetic cloud, Geophys. Res. Lett., 30(20), 2065, doi:10.1029/ 2003GL017692. Romashets, E. P., and M. Vandas (2009), Correction to “Force‐free field inside a toroidal magnetic cloud”, Geophys. Res. Lett., 36, L12107, doi:10.1029/2009GL038342. Rust, D. M., and A. Kumar (1994), Helicity charging and eruption of magnetic flux from the sun, Solar Dynamic Phenomena and Solar Wind Consequences, in Proceedings of the Third SOHO Workshop held 26–29 September, 1994 in Estes Park, Colorado, edited by J. J. Hunt, Eur. Space Agency, ESA SP‐373, p. 39. Sonnerup, B. U. Ö, and L. J. Cahill Jr. (1967), Magnetopause structure and attitude from Explorer 12 Observations, J. Geophys. Res., 72(1), 171–183, doi:10.1029/JZ072i001p00171. Vandas, M., E. Romashets, S. Watari, A. Geranios, E. Antoniadou, and O. Zacharopoulou (2006), Comparison of force‐free flux rope models with observations of magnetic clouds, Adv. Space Res., 38(3), 441–446, doi:10.1016/j.asr.2004.11.026. Yamamoto, T., and A. Matsuoka (1998), PLANET‐B magnetic fields investigation, Earth Planets Space, 50, 189–194. Yashiro, S., N. Gopalswamy, G. Michałek, O. C. St. Cyr, S. P. Plunkett, N. B. Rich, and R. A. Howard (2004), A catalogue of white light coronal mass ejections observed by the SOHO spacecraft, J. Geophys. Res., 109, A07105, doi:10.1029/2003JA010282. A. Matsuoka, ISAS, JAXA, Yoshinodai, Chuo‐ku, Sagamihara, Kanagawa, 252‐5210, Japan. ([email protected]) T. Nakagawa, Information and Communication Engineering, Tohoku Institute of Technology, Yagiyamakasumi‐cho, Taihaku‐ku, Sendai, Miyagi, 982‐8577, Japan. ([email protected])

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Fitting a toroidal forceâfree field to multispacecraft observations of a

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