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The Astrophysical Journal, 710:456–461, 2010 February 10  C 2010.

doi:10.1088/0004-637X/710/1/456

The American Astronomical Society. All rights reserved. Printed in the U.S.A.

HELICAL LENGTHS OF MAGNETIC CLOUDS FROM THE MAGNETIC FLUX CONSERVATION Tetsuya T. Yamamoto1 , R. Kataoka2 , and S. Inoue3 1

Solar-Terrestrial Environment Laboratory, Nagoya University, Chikusa, Nagoya, 464-8601, Japan; [email protected] 2 Interactive Research Center of Science, Tokyo Institute of Technology, Meguro, 152-8550, Japan 3 National Institute of Information and Communications Technology, Koganei, 184-8795, Japan Received 2009 July 13; accepted 2009 December 21; published 2010 January 19

ABSTRACT We estimate axial lengths of helical parts in magnetic clouds (MCs) at 1 AU from the magnetic flux (magnetic helicity) conservation between solar active regions (ARs) and MCs with the event list of Leamon et al. Namely, considering poloidal magnetic flux (ΦP ) conservation between MCs and ARs, we estimated Lh in MCs, where Lh is the axial length of an MC where poloidal magnetic flux and magnetic twist exist. It is found that Lh is 0.01–1.25 AU in the MCs. If the cylinder flux rope picture is assumed, this result leads to a possible new picture of the cylinder model whose helical structure (namely, poloidal magnetic flux) localizes in a part of a MC. Key words: solar–terrestrial relations – Sun: corona – Sun: coronal mass ejections (CMEs) – Sun: magnetic topology estimated HR in MCs, and reported differences of HR between ARs and MCs. Luoni et al. (2005) showed that values of HR in an AR and in an MC are of the same order of magnitude, while toroidal magnetic flux (ΦT ) in the MC is about one tenth of that in the AR, based on the assumption that the axial length in the MC is 2.4 AU. Longcope et al. (2007) reported that HR in an MC is five times larger than that in an AR, while ΦT in the AR and in the MC are of the same order of magnitude, based on the assumption that the axial length in the MC is 2 AU. There is a possibility that they overestimated HR and poloidal magnetic flux (ΦP ) in the MCs owing to the assumed axial lengths (2–2.4 AU). Including the above papers, other papers (Lynch et al. 2005; Dasso et al. 2006) estimated HR in MCs with the assumed axial lengths (2–2.5 AU). On the other hand, Mandrini et al. (2005) and Rodriguez et al. (2008) estimated helical lengths of MCs with magnetic flux and magnetic helicity conservations. For example, Rodriguez et al. (2008) roughly guessed that the helical length of a CME that originated in AR 10720 was ∼1 AU around the Ulysses trajectory, ∼5 AU far from the Sun. They compared helicity density of the MC with the injected helicity content into AR 10720 (Zhang 2007). This helical length would indicate that the MC does not link to the solar atmosphere, or the helical length of the MC is shorter than the total axial length of the MC. Magnetic helicity is a conserved quantity in the interplanetary space, and is the sum of products of poloidal and toroidal fluxes (Equation (6) in Berger, 1999). This leads to the following equation in a simple flux tube configuration:

1. INTRODUCTION A magnetic cloud (MC) is a subset of an interplanetary coronal mass ejection (CME; e.g., MacQueen 1980), and is defined by nearly monotonic rotation of the magnetic field and larger field strength (Burlaga et al. 1981). The global picture of an MC is an enigma in solar-interplanetary science because of the limited in situ observations of the solar wind. In many studies, the reference global pictures of MCs are the cylinder model (e.g., Lepping et al. 1990), the torus model (e.g., Romashets & Vandas 2001), and the spheroidal model (e.g., Vandas et al. 1993) in order to explain the smooth rotation of the interplanetary magnetic field within MCs (see also Figure 5 of Vandas & Geranios 2001). These models have several ambiguities due to the limited data. Our paper is motivated by the results of Leamon et al. (2004). They compared magnetic fluxes and twist numbers of magnetic fields between active regions (ARs) and MCs in 12 events by analyzing solar and interplanetary magnetic field data. They estimated that twisting numbers in the ARs are 0.01–7.79, and those in the MCs are 21.6–124.8. Here the twisting number is defined as αL, where α is the force-free parameter, and L is the coronal loop length or the axial length of an MC. Leamon et al. (2004) assumed that the axial length of the MCs is “2.5 AU” under the cylinder model, while the other parameters are observed ones. In order to explain the marked difference of the twisting number, they suggested that some helical magnetic fields can provide twist to the MCs in the eruptive reconnection process. Another idea, however, naturally arises that the above assumption “2.5 AU” makes the differences of the twisting number between the ARs and the MCs. This is because a twisting field part, namely poloidal magnetic flux, possibly has a length shorter than 2.5 AU in an MC. Hereafter, we call the axial length where a poloidal field exists as the helical length Lh . Here we also mention the above assumption of “2.5 AU.” At present, as far as we know, only Larson et al. (1997) show the observational evidence of a magnetic field linking between an MC and the solar atmosphere. Referring to this paper, many authors, described in the following, assumed that the axial length of MCs is 2–2.5 AU, and estimated poloidal magnetic fluxes, total twists, and magnetic helicities. The relative magnetic helicity (HR ; Berger 1999) is another issue relating to the twisting number. Several papers recently

HR = I ΦT0 ΦP0 = I (ΦT1 + ΦT2 )ΦP0 ,

(1)

that shows a right- or left-handed twist where I is +1 or −1. We can interpret this equation in the solar-interplanetary environment as follows. The top panel of Figure 1 shows that an AR has toroidal magnetic flux ΦT0 , net poloidal magnetic flux ΦP0 , and magnetic helicity ΦT0 ΦP0 . If an MC is ejected from a part of the AR where toroidal magnetic flux is ΦT1 , then the MC should have magnetic helicity ΦT1 ΦP0 . In the other part of the AR, magnetic helicity is equal to ΦT2 ΦP0 as is shown in the bottom left panel of Figure 1. Equation (1) leads to the fact that the net poloidal flux in the MC is equal to the net poloidal flux in the AR. From this magnetic flux (magnetic helicity) conservation law, we can obtain helical lengths of MCs. 456

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Recent numerical simulations showed that poloidal and toroidal magnetic fields could alternate their positions between an AR and a CME in the torus or spheroidal field configuration (e.g., Gibson & Fan 2008). From this result, we could consider the following equation:

must define. Since Lepping et al. (1990), scientists fitted MCs with several force-free field models described above (cylinder, torus, and spheroidal models). Dasso et al. (2006) compared for cylinder force-free models, and reported that physical parameters estimated from different models show differences less than a factor of 2 in their Table 1. Recently the Grad– Shafranov equation has been applied in some papers (e.g., M¨ostl et al. 2009), and we can obtain pressure and magnetic field over the cross section of an MC (e.g., M¨ostl et al. 2009). As for start and end boundaries, Russell & Shinde (2005) show that different authors sometimes defined different boundaries for the same event, and event lists made by different authors include inconsistent events. It is natural that if differences of duration times are a few times (e.g., Figures 1 and 3 of Russell & Shinde 2005), estimated magnetic parameters could have some uncertainties, especially for ΦMC,T . Dasso et al. (2006) investigated magnetic parameters variation due to different end boundaries, and reported that helicity per unit length has an uncertainty of a factor of 2 for different boundaries. Later, we will discuss an effect of boundary selection, and will compare physical parameters of Leamon et al. (2004) with those investigated by Lepping and the Wind/MFI team. In this paper, our purposes are to obtain axial lengths of the helical part in MCs from the magnetic flux conservation with the data of Leamon et al. (2004), and to discuss magnetic field structures of MCs. Equation (1) is applied in this study, because Leamon et al. (2004) used the cylinder model. In Section 2, we compare toroidal and poloidal magnetic fluxes between the ARs and the MCs, and estimate helical lengths of the MCs from the magnetic flux conservation. In Section 3, the MC models and coronal magnetic flux variations are discussed. Conclusions are briefly summarized in Section 4.

ΦT0 ΦP0 = ΦT0 (ΦP1 + ΦP2 ).

2. HELICAL LENGTHS OF THE MAGNETIC CLOUDS

ΦP0

ΦT0

ΦP0

ΦP2

ΦP1

ΦT1

ΦT2 ΦT0

ΦP1 ΦP0 ΦT0

ΦT1

ΦP0

ΦP2

ΦT2 Equation (1)

ΦT0 Equation (2)

Figure 1. Schematic illustration of Equations (1) and (2). The dotted lines in the bottom left panel show the torus and spheroidal models.

(2)

As is shown in the bottom right panel of Figure 1, Equation (2) indicates that an MC has toroidal magnetic flux ΦP1 , poloidal magnetic flux ΦT0 , and magnetic helicity ΦT0 ΦP1 . This leads to the fact that the net poloidal flux in an MC is equal to the toroidal flux in an AR. This magnetic flux conservation could be applied for the torus and spheroidal models. Hereafter, the one-to-one relationship between an MC and an AR is assumed as the same as Leamon et al. (2004). There is, however, another possible relationship between an MC and its coronal source. Mandrini et al. (2007) investigated magnetic flux balance between an MC and large-scale coronal dimming regions around the X17 flare of AR 10486. Originally, Webb et al. (2000) suggested from a single event observation that coronal dimming regions in an AR are marks of footpoints of an MC. Mandrini et al. (2007) showed that magnetic fluxes do not match between the MC and the coronal dimming regions, and suggested an analysis of the global magnetic configuration with such large-scale dimming. Note that Mandrini et al. (2007) also assumed the axial length, 2 AU, in order to estimate poloidal magnetic flux in the MC. On the other hand, Qiu et al. (2007) showed that magnetic fluxes in MCs are correlated with photospheric magnetic fluxes in flare regions. Poloidal fluxes in MCs were estimated with the assumed axial lengths (1 AU). They concluded that magnetic fluxes in MCs are relevant to reconnected magnetic fluxes in the low corona. Uncertainties of magnetic parameters in an MC are also a subject we should pay attention to. There are two major origins for uncertainties. One is a model to fit in situ magnetic data, and another is start and end boundaries of an MC that we

In Leamon et al. (2004), they analyzed 12 solar-interplanetary events that occurred from 1995 to 2000. Columns 2–5 in Table 1 show observed properties of these events; the dates of the eruptive events in the solar atmosphere, the dates of the vector magnetograms, and the NOAA AR numbers. They analyzed solar magnetic field data observed at the Mees Solar observatory (Hawaii) and interplanetary magnetic field data observed with the Wind satellite. Magnetic parameters we adopt are taken from their Tables 1–3. For a precise description of their data, see Leamon et al. (2004). Hereafter, their magnetic values are indicated by italic subscripts. Note that their events 3, 4, 11, and 12 are excluded from this study. Leamon et al. (2004) reported that these events have the opposite signs of α between the ARs and the MCs. This is inconsistent with Equations (1) and (2). We show only event 3 in Table 1 in order to discuss errors of the observed magnetic fluxes in the later section. 2.1. Linear Force-free Fields In this study, for the MCs and the ARs, we assume a flux tube configuration in the linear force-free field (Lundquist 1950) as follows: αBP = −

αBT =

∂BT , ∂r

1 ∂ (rBP ), r ∂r

(3)

(4)

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Table 1 Physical Parameters of the Analyzed Events No.

Year

Eruption

Magnetogram

AR

ΦAR,T (TWb)

ΦAR,P (TWb)

ΦMC,T (TWb)

Φpol (TWb)

Lh (AU)

1 2 ... 3 ... 5 ... 6 7 8 9 ... ... 10

1995 ... ... ... ... 1998 ... 1999 ... ... 2000 ... ... ...

Feb 04 15:56 Feb 28 08:46 ... Dec 11 03:31 ... Apr 29 16:58 ... Feb 14 11:16 Aug 04 04:11 Sep 17 22:28 Jul 14 09:27 ... ... Jul 25 02:48

Feb 04 01:51 Feb 27 17:17 Feb 28 19:45 Dec 10 17:22 Dec 11 17:15 Apr 28 16:38 Apr 29 16:39 Feb 11 19:26 Aug 02 16:36 Sep 20 17:00 Jul 11 16:52 Jul 14 16:39 Jul 17 16:32 Jul 21 20:28

7834 7846 ... 7930 ... 8210 ... 8457 8651 8700 9077 ... ... 9097

16.5 24.9 34.3 9.40 10.5 30.7 30.9 43.3 169. 16.3 69.5 64.0 29.8 86.4

0.71 0.14 3.95 2.32 1.74 6.41 4.99 2.54 22.9 1.23 69.0 44.4 6.07 15.6

4.74 3.26 3.26 18.5 18.5 16.0 16.0 28.6 9.59 0.99 45.5 45.5 45.5 5.33

1.92 1.85 2.79 ... ... 2.76 1.36 2.83 4.10 3.19 4.66 2.98 1.59 2.63

0.05 0.01 0.37 ... ... 0.30 0.23 0.07 1.25 0.19 0.88 0.57 0.08 1.02

Note. The first column “No.” is the same as that of Leamon et al. (2004).

where BP is the poloidal field component, BT is the toroidal field component, and r is the radius. Then the toroidal magnetic flux, ΦT , is obtained from Equation (4) and the following equation: 

R

ΦT =

2π rBT dr, 0

2π R BP (R), α

=

(5)

where R is the tube radius. The poloidal magnetic flux, ΦP , is obtained from Equation (3) and the following equation:  ΦP =

L



BP drdL, 0

=

R

0

L (BT (0) − BT (R)), α

(6)

where L is the axial length of the flux tube. The following relation is obtained from Equations (5) and (6): ΦP 

αL ΦT . 2.5π

(7)

Here BT and BP , respectively, are given by the zeroth- and firstorder Bessel functions (Lundquist 1950; Lepping et al. 1990). 2.2. Magnetic Fluxes in the ARs and in the MCs Here we compare the toroidal magnetic fluxes of the MCs (ΦMC,T ) with the toroidal and poloidal magnetic fluxes of the ARs (ΦAR,T and ΦAR,P ). In Leamon et al. (2004), ΦAR and ΦMC are the toroidal magnetic fluxes in the ARs and in the MCs, respectively. In this paper, these are shown as ΦAR,T and ΦMC,T , respectively. Leamon et al. (2004) showed multiple values of ΦAR for each event, and we applied the average value of ΦAR as ΦAR,T . Dispersions of ΦAR are, at most, about 50% of ΦAR,T , excluding event 8 whose dispersion is about 80%. We can estimate ΦAR,P from Equation (7) with unsigned (αL)AR and ΦAR,T . Table 1 shows these magnetic fluxes. From Table 1, it is found that most of ΦAR,T are larger than ΦAR,P , and event 3 shows larger values of ΦMC,T than ΦAR,T and ΦAR,P . Including event 3, Leamon et al. (2004) discussed a possibility that large-scale dipole magnetic fields reconnected with the

magnetic fields in the ARs. From vectormagnetograms obtained by Hinode, Tsuneta et al. (2008) showed that net magnetic flux in the south polar region is 1.8 × 1014 Wb. Benevolenskaya (2004) reported similar values ((1.5–2.5) × 1014 Wb) of unsigned magnetic fluxes around the Northern and Southern poles from SOHO/MDI magnetograms observed during 1996–2003. Considering 1.8 × 1014 Wb as the polar magnetic flux, we estimated contribution of the polar magnetic flux to the ARs as follows: l Φpol = 1.8 × 1014 , (8) 2π Rsun where Φpol is the contribution of the polar magnetic flux, Rsun is the solar radius, and l is the characteristic lengths of the ARs shown in Table 2 of Leamon et al. (2004). Column 9 in Table 1 shows Φpol . From these values, it is found that the values of Φpol are smaller than or equal to those of ΦAR,T and ΦAR,P , excluding events 1 and 2. In these events, Φpol is 10 times larger than ΦAR,P . These values of Φpol , however, have little effect on our results. Hereafter we ignore Φpol . Here we also mention an error of ΦMC,T . Figure 2 shows a magnetogram observed at the Kitt Peak National Solar Observatory on 1995 December 11. The circle in this figure shows AR 7930 where the event 3 occurred. As shown in this figure, there is no AR, namely no strong magnetic flux source, other than AR 7930. And as is shown in Table 1, ΦAR,P of AR 7930 are less than ΦAR,T , and ΦAR,T of AR 7930 are about half of ΦMC,T . Therefore, we concluded that ΦMC,T have an error, at least, of the order of a factor of 2. Note that, from Dasso et al. (2006), an uncertainty of ΦMC,T due to different cylinder model fittings is less than 40% in their Table 1. 2.3. Helical Lengths Helical lengths of the MCs (Lh ) are estimated from the following equation: Lh =

2.5π ΦMC,P , αMC ΦMC,T

(9)

where αMC is the force-free parameter of the MCs listed in Leamon et al. (2004). This equation is derived from Equation (7). From the one-to-one relationship between an MC and an AR we assumed, ΦAR,P is adopted as ΦMC,P . Namely, no reconnection

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Table 2 Physical Parameters of the Events Analyzed by Leppinga and the Wind MFI Team No.

Code No.b

Year

Start Time (UT)

End Time (UT)

B0 (nT)

R0 (Gm)

ΦMC (TWb)

αMC (Gm−1 )

δST c (hr)

δET d (hr)

δL e

1 2 6 7 8 9 10

1 2 39 41 42 46 47

1995 1995 1999 1999 1999 2000 2000

Feb 08 05:48 Mar 04 10:48 Feb 18 14:18 Aug 09 10:48 Sep 21 21:06 Jul 15 21:06 Jul 28 21:06

Feb 09 00:48 Mar 05 03:47 Feb 19 12:18 Aug 10 15:48 Sep 22 05:05 Jul 16 09:53 Jul 29 10:06

15.2 14.9 13.1 12.1 18.0 45.2 17.8

16.2 12.4 38.0 20.3 6.8 24.0 17.2

5.4 3.1 25.7 6.8 1.1 35.3 7.1

0.148 0.194 0.063 0.118 0.356 0.100 0.140

2.8 −0.2 0.3 0.8 0.1 2.1 0.1

2.8 −0.2 0.3 −0.2 0.1 0.9 −0.9

0.12 0.08 −0.23 −0.20 0.15 −0.16 0.20

Notes. The first column “No.” is the same as that of Leamon et al. (2004). a See http://lepmfi.gsfc.nasa.gov/mfi/mag_cloud_pub1.html. b Event number shown in MFI/Wind magnetic cloud Web site. c Differences of start times between Leamon et al. and Lepping. d Differences of end times between Leamon et al. and Lepping. e Differences of L between Leamon et al. and Lepping. h

1000

11-Dec-1995 15:51:16.736

Y (arcsecs)

500

SU N AR 7930

0

Figure 3. Illustration of the cylinder MC model having a localized helical part.

-500 -1000 -1000

-500

0 500 X (arcsecs)

1000

Figure 2. NSO Kitt Peak magnetogram. The full gray scale spans Bz = ±300 G, where Bz is the longitudinal field strength.

with other ARs is assumed during the erupting process. With ΦAR,P , we can obtain Lh shown in Table 1. It is found that Lh is 0.01–1.25 AU. These lengths are clearly shorter than 2.5 AU. We will discuss magnetic structures of the MCs with these lengths. 3. DISCUSSION 3.1. About the Cylinder MC Model In the previous section, we obtained the helical lengths (Lh ) of the MCs. Note that we do not consider these lengths as total axial lengths of the MCs. From Lh , we could not decide whether or not the MCs root on the solar atmosphere, and whether or not magnetic reconnection occurs in the interplanetary space. These issues are beyond the scope of this paper. From Table 1, the values of Lh are shorter than 2.5 AU. If we consider the error of a factor of 2, events 7 and 10 may show Lh longer than 2 AU, and the other events show Lh shorter than 2 AU. From this result, we propose a possible new picture of the cylinder MC model that only a part in the cylinder has the helical structure. Figure 3 shows this picture. Solid lines in this figure show magnetic field lines. Around the apex of the cylinder MC model, solid lines show helical configuration, while dashed lines show non-helical configuration for comparison. In this figure, one difficulty is identifying a force to make poloidal fields localize in a part of the MC. At present we consider a candidate resolving this difficulty. The candidate is the solar wind blowing in the cylinder field. From rough estimation, it is found that the solar wind faster by

70 km s−1 than the velocity of an MC can support poloidal fields in a part of the cylinder. Here we considered that the pressure of the magnetic field is equal to the dynamic pressure of the solar wind in an MC. The following equation is used: np mp 2 BP2 = v , 8π 2

(10)

where np is the proton number density, mp is the proton mass (1.67 × 10−24 g), and v is the velocity for the dynamic pressure. For this estimation, it is assumed that BP is 10 nT (= 10−4 G), and np is 10 cc−1 . One of our next subjects is to construct a theoretical MC model consistent with our obtained description (e.g., Figure 3), in addition to the previous models (Kumar & Rust 1996; D´emoulin & Dasso 2009). 3.2. About the Torus and Spheroidal MC Models Some magnetic field data of MCs could be fitted not only by the cylinder MC model but also by the torus (Romashets & Vandas 2001) and spheroidal (Vandas et al. 1993) MC models. The torus and spheroidal MC models are expected to be closed in the interplanetary space, and should have consistent magnetic fluxes with those of an AR, as is shown in Figure 1. In these models, the major radius, Rh , is defined as follows: Rh =

Lh . 2π

(11)

Using this Rh , we could decide either the torus model or the spheroidal model is appropriate for an observed MC as follows. We checked ratios k = r/Rh , where r is the minor radius of the torus model. The torus MC model, from its geometrical definition, should have k less than unity, while the spheroidal MC model should have k more than unity. In order to obtain r, Rh , and k of the torus model for the events of Leamon et al.

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(2004), it is assumed that αMC ΦMC,T and r of the torus model is about one-third of αMC ΦMC,T and radius of the cylinder model. This is because Marubashi & Lepping (2007), who obtained fitting parameters of 17 MCs by adopting both the cylinder and torus MC models, showed such a tendency. As a result, events 5, 7, 8, and 10 showed k less than unity, while events 1, 2, 6, and 9 showed k more than unity. Therefore, events 1, 2, 6, and 9 are not appropriate for the torus model fitting, and the spheroidal MC fitting might be appropriate for these events. We will investigate consistencies of the torus and the spheroidal MC models in our future works. As a straightforward approach, for example, Kataoka et al. (2009) adopted a CME model with a spheroidal magnetic field in their three-dimensional magnetohydrodynamic simulation of the solar wind to quantitatively reproduce the in situ observation of the MC at 1 AU.

following two points: (1) obtained helicity injection rates have some uncertainties (Welsch et al. 2007); and (2) in order to increase poloidal fluxes in MCs by magnetic helicity injections, cylinder MCs must root on ARs. If MCs root on regions other than ARs, strong helicity injection rates would not be expected.

3.3. About the Magnetic Helicities and Magnetic Fluxes

δST = STLep − STLea ,

(14)

One of the major problems in Leamon et al. (2004) is different signs of α in events 3, 4, 11, and 12. There is one possibility of resolving the problems of different signs of α and of shorter Lh than 2.5 AU, as follows. In contrast to Equations (1) and (2), the following configuration could be considered:

δET = ETLep − ETLea ,

(15)

ΦT0 ΦP0 = ΦT1 ΦP1 + ΦT2 ΦP2 .

(12)

In this equation, ΦP1 and ΦP2 could take opposite signs and larger unsigned values. However, this magnetic configuration leads to the fact that helicity of the opposite sign is left in coronal loops after a CME. As a result, it may be viewed that helicity markedly increases in coronal loops. From our impression, the coronal helicity sign hardly changes after a CME occurrence. Figure 4 of Lim et al. (2007) shows that the coronal helicity takes the negative sign before and after the CMEs. They obtained the coronal helicity by fitting coronal loops with the linear force-free field. Table 4 of D`emoulin et al. (2002) showed the continuous positive signs of the coronal helicity on AR 7978 during five months. The cadence of their data is about one month. We need more quantitative analysis of coronal helicity variations. Helicity injection into MCs has a possibility of elongating Lh to 2.5 AU in the cylinder model. However, from the following estimation, analyzed helicity injection rates may not be enough. Adopting typical values of the MCs (αMC = 10−10 m−1 , Lh = 2.5 AU, and ΦMC,T = 1013 Wb), and assuming that the travel time of MCs to the Earth is 4 days, we obtained the increasing ˙ MC,P = 1.7 × 108 Wb s−1 , in an MC. rate of the poloidal flux, Φ ˙ AR,P ) We can also obtain injection rates of poloidal fluxes (Φ ˙ in ARs from magnetic helicity injection rates (HR ) with the following equation: ˙ ˙ AR,P = HR . Φ ΦAR,T

(13)

Here ΦAR,T is assumed to be constant. Recently, Pariat et al. (2006) show that H˙ R and ΦAR,T of AR 8210 are 1022 Wb2 s−1 and 1.6×1014 Wb, respectively. From these values, we obtained ˙ AR,P = 6.3 × 107 Wb s−1 . This value is one-third of the above Φ ˙ ΦMC,P . Recently, Yamamoto & Sakurai (2009) analyzed helicity injection rates statistically. From their results, we found that ˙ AR,P is 3.1 × 105 –2.1 × 107 Wb s−1 . the unsigned values of Φ ˙ AR,P is not enough for From these values, it is found that Φ Lh of 2.5 AU. In this scenario, we should pay attention to the

3.4. Boundaries of an MC Event For some MC events, one open issue is how to define proper boundaries of an MC in in situ data as is described in the introduction. In order to estimate uncertainties of the magnetic parameters, we checked the start and end times and fitted magnetic parameters investigated by Lepping.4 Table 2 shows these parameters. Here event 5 is excluded, because in Lepping’s list, this event has the opposite sign of α between the AR and the MC. δST and δET in Table 2 are differences of start and end times between Leamon et al. and Lepping as follows:

where ST is the start time, ET is the end time, and subscripts “Lea” and “Lep” indicates times of Leamon and Lepping, respectively. As a result, time differences are less than 3 hr. We estimated an uncertainty of Lh as follows: δL =

L h − L , Lh

(16)

where L is the helical lengths obtained with Equation (9) and the magnetic parameters in Table 2. The right-end column in Table 2 shows that δL is less than 23%. Therefore, we could conclude that boundary selections have minor effect on our results. 4. SUMMARY In this paper, the magnetic flux (magnetic helicity) conservation law is applied to investigate the magnetic structure of MCs with the data listed in Leamon et al. (2004) under the main assumption of the one-to-one relationship between an MC and an AR. Our conclusions are as follows. 1. The helical lengths of the MCs (Lh ) are estimated to be 0.01–1.25 AU shown in Table 1 from the magnetic flux (magnetic helicity) conservation. 2. These helical lengths would lead to the new picture of the cylinder model. The helical part localizes in an MC, as is shown in Figure 3. 3. From the recent polar observations (Benevolenskaya 2004; Tsuneta et al. 2008), it is concluded that the polar magnetic field can supply magnetic fluxes on the order of 1012 Wb to the ARs. If a helical length of the cylinder MC model is 2–2.5 AU, these magnetic fluxes would not be enough. We acknowledge fruitful discussion with K. Kusano and K. Shibata. NSO/KPVT data used here were produced cooperatively by AURA/NSO, NASA’s GSFC, and NOAA/SEC. The authors are grateful to R. P. Lepping and the Wind MFI team for making their data readily available. This work was partially 4 MFI/Wind magnetic cloud Web site: http://lepmfi.gsfc.nasa.gov/mfi/mag_cloud_pub1.html.

No. 1, 2010

HELICAL LENGTHS OF MCS FROM THE MAGNETIC FLUX CONSERVATION

supported by the Grant-in-Aid for Creative Scientific Research “The Basic Study of Space Weather Prediction” (17GS0208, Head Investigator: K. Shibata) from the Ministry of Education, Science, Sports, Technology, and Culture of Japan. The work by R.K. was supported by a research fellowship of Special Postdoctoral Research Program at RIKEN. This work was partially supported by the Grant-in-Aid for Creative Scientific Research (20740286). REFERENCES Benevolenskaya, E. E. 2004, A&A, 428, L5 Berger, M. A. 1999, in Geophysical Monogr. 111, Magnetic Helicity in Space and Laboratory Plasmas, ed. M. R. Brown, R. C. Canfield, & A. A. Pevtsov (Washington, DC: AGU), 1 Burlaga, L. F., Sittler, E. C., Mariani, F., & Schwenn, R. 1981, J. Geophys. Res., 86, 6673 Dasso, S., Mandrini, C. H., D´emoulin, P., & Luoni, M. L. 2006, A&A, 455, 349 D´emoulin, P., & Dasso, S. 2009, A&A, 498, 551 D´emoulin, P., Mandrini, C. H., van Driel-Gesztelyi, L., Thompson, B. J., Plunkett, S., Kov´ari, Z., Aulanier, G., & Young, A. 2002, A&A, 382, 650 Gibson, K. G., & Fan, A. F. 2008, J. Geophys. Res., 113, A09103 Kataoka, R. T., Ebisuzaki, K., Kusano, D., Shiota, S., Inoue, T. T., & Yamamoto, M. Tokumaru 2009, J. Geophys. Res., 114, A10102 Kumar, A., & Rust, D. M. 1996, J. Geophys. Res., 101, 15667 Larson, D. E., et al. 1997, Geophys. Res. Lett., 24, 1911 Leamon, R. J., Canfield, R. C., Jones, S. L., Lambkin, K., Lundberg, B. J., & Pevtsov, A. A. 2004, J. Geophys. Res., 109, A05106 Lepping, R. P., Jones, J. A., & Burlaga, L. F. 1990, J. Geophys. Res., 95, 11957

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