stability and dynamics of a flux rope formed via flux ... - IOPscience

The Astrophysical Journal, 773:21 (9pp), 2013 August 10 C 2013.

doi:10.1088/0004-637X/773/1/21

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

STABILITY AND DYNAMICS OF A FLUX ROPE FORMED VIA FLUX EMERGENCE INTO THE SOLAR ATMOSPHERE 1

J. M. An1 and T. Magara2 School of Space Research, Kyung Hee University, 1 Seocheon-dong, Giheung-gu, Yongin, Gyeonggi-do 446-701, Korea; [email protected] 2 Department of Astronomy and Space Science, School of Space Research, Kyung Hee University, 1 Seocheon-dong, Giheung-gu, Yongin, Gyeonggi-do 446-701, Korea; [email protected] Received 2012 September 25; accepted 2013 May 24; published 2013 July 22

ABSTRACT We study the stability and dynamics of a flux rope formed through the emergence of a twisted magnetic flux tube into the solar atmosphere. A three-dimensional magnetohydrodynamic simulation has been performed to investigate several key factors affecting the dynamics of the flux rope. The stability of the flux rope is examined by deriving the decay index of the coronal magnetic field surrounding the flux rope. We investigate a transition between the quasi-static and dynamic states of the flux rope through an analysis of the curvature and scale height of emerging magnetic field. A practical application of this analysis for global eruptions is also considered. Key words: magnetohydrodynamics (MHD) – Sun: corona – Sun: coronal mass ejections (CMEs) – Sun: magnetic topology Online-only material: color figures

instabilities such as the kink instability (Sturrock et al. 2001; Fan & Gibson 2004) and torus instability (Kliem & Török 2006) could also drive global eruptions. The tether-cutting model proposed by Moore et al. (2001) explains that the weakening of the line-tying effect via magnetic reconnection produces global eruptions. Furthermore, in several multi-polar flux systems the magnetic reconnection between different flux domains leads to the global eruption (Antiochos et al. 1999; Chen & Shibata 2000; Kusano et al. 2004). These works mentioned above nicely explain the eruptive process of magnetic structures on the Sun, although they mostly prescribe particular magnetic configurations for pre-eruptive structures, so the origin of these pre-eruptive structures, that is, how they are self-consistently formed in the solar atmosphere before the global eruption, is not clear. In this respect, studies based on flux emergence have demonstrated the self-consistent formation of pre-eruptive structures on the Sun. Matsumoto et al. (1998) and Magara & Longcope (2001) explain the formation of sigmoidal structures on the Sun. Manchester et al. (2004) show how a flux rope is formed in the corona via flux emergence. Archontis & Török (2008) and Archontis & Hood (2012) discuss the evolution of a flux rope interacting with some particular preexisting coronal magnetic fields. The present paper aims at shedding some light on the stability and dynamics of a flux rope formed via flux emergence. Toward this end, we have performed a three-dimensional MHD simulation for the emergence of an isolated magnetic flux tube from a subsurface region into the solar atmosphere. Our particular interest is to investigate several key factors affecting the dynamics of a flux rope formed through the emergence of the flux tube. We discuss a transition between the quasi-static and dynamic states of the flux rope through an analysis of the curvature and scale height of emerging magnetic field. The stability of the flux rope is discussed especially from the viewpoint of the torus instability (Kliem & Török 2006). Kliem & Török (2006) and several following works (Török & Kliem 2007; Fan & Gibson 2007; Schrijver et al. 2008; Liu 2008; Guo et al. 2010; Xu et al. 2012; Kumar et al. 2012) focus on the decay index of the coronal magnetic field surrounding a flux

1. INTRODUCTION Emergence of magnetic field from the solar interior into the solar atmosphere, called flux emergence, is regarded as a key process for forming structures of magnetic field in the solar atmosphere where various kinds of active phenomena such as solar flares, jets, and coronal mass ejections (CMEs) are occasionally observed (Svestka & Cliver 1992; Low 1996; Golub & Pasachoff 1997; Priest & Forbes 2002; Aschwanden 2004; Forbes et al. 2006). The helioseismology suggests that magnetic fields producing these active phenomena are amplified in the tachocline between the radiative zone and convective zone (Miesch 2005). After traveling through the convective zone (Fan 2009), these magnetic fields eventually emerge to the solar surface via magnetic buoyancy (Parker 1955). They then expand rapidly due to the abrupt decrease of the surrounding gas pressure, thereby forming magnetic structures in the solar atmosphere where free magnetic energy is stored (Shibata & Magara 2011). Among those active phenomena mentioned above, solar flares are one of the typical phenomena caused by the dissipation of the coronal electric currents that supply the free energy. There is another group of phenomena known as global eruptions, which are caused by the imbalance among forces acting on magnetic structures. CMEs are one of these phenomena driven by the upward Lorentz force dominating other forces. An intensive effort has been made toward understanding the physical mechanism of those magnetically driven global eruptions. Low & Hundhausen (1995) explain the destabilizing effect of mass drain on a magnetic structure. Increasing the shear and/or twist of magnetic field at the solar surface is another possible driver of global eruptions (Mikic et al. 1988; Biskamp & Welter 1989; Chen 1989; Steinolfson 1991; Inhester et al. 1992; Mikic & Linker 1994; Choe & Lee 1996; Török et al. 2004). The cancellation of magnetic flux at the solar surface causes the imbalance between the magnetic pressure force and magnetic tension force, leading to the global eruption (van Ballegooijen & Martens 1989; Forbes & Isenberg 1991; Amari et al. 2000; Linker et al. 2003; Roussev et al. 2003; Lin 2004). Ideal MHD 1

The Astrophysical Journal, 773:21 (9pp), 2013 August 10

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Table 1 Units of Physical Quantities Physical Quantity

Unit 2 Λph a csph b 2 Λph /csph ρph c ρph cs2ph Tph d (ρph cs2ph )1/2

Length Velocity Time Gas density Gas pressure Temperature Magnetic field Notes. a Photospheric gas pressure scale height. b Photospheric adiabatic sound speed. c Photospheric gas density. d Photospheric temperature.

Figure 1. Initial distributions of gas density (dashed line), pressure (dotted line), temperature (dot-dashed line), and magnetic pressure (solid line) along z-axis.

and Δz increase toward 4 as |x|, |y|, and z increase. The total number of grids are Nx × Ny × Nz = 371 × 303 × 597. The initial state of the simulation is in mechanical equilibrium, consisting of an isolated magnetic flux tube and the background plasma atmosphere stratified under uniform gravity. The flux tube is placed below the surface horizontally along the y-axis, and the axis of the flux tube is placed at (0, y, −4). The flux tube is characterized by the so-called Gold–Hoyle profile: −b r θˆ + yˆ B = B0 , (6) 1 + b2 r 2

rope, which plays a crucial role in producing a global eruption via that instability. To see how flux emergence distributes the magnetic field in the corona, we use a method of potential-field extrapolation to derive the decay index from the simulation. In the next section we explain the details of the simulation. We show results on the flux rope reproduced by the simulation in Section 3. The decay index analysis of the flux rope is presented in this section. In Section 4 we discuss a transition between the quasi-static and dynamic states of the flux rope through the analysis of the curvature and scale height of emerging magnetic field. We also consider the possibility of a practical application of this analysis for global eruptions on the Sun.

where yˆ and θˆ represent the axial and azimuthal directions of the flux tube, while r, B0 , and b are the radial distance from the axis of the flux tube, field strength at the axis, and field-line twist, respectively. For the present study we assume that the radius of the flux tube is 2 while B0 = 17.3 and b = 1. Figure 1 shows the initial distributions of plasma and magnetic field along the z-axis. Since the goal of the present study is to investigate the dynamic evolution of a flux rope in the solar atmosphere, especially in the corona, we in this study adopt a larger simulation domain and distribute finer grids over a coronal region than in our previous work (Magara 2012). We examine a case where the gas pressure is reduced in the outer atmosphere, which is favorable for the continuous expansion of emerging magnetic field (Tajima & Shibata 1997). To do so, we broadened the range in the lower atmosphere along which the temperature remains constant and the gas pressure drops rapidly, so that a smaller coronal gas pressure than in the previous simulation is obtained (the location where T = 50Tph is shifted from z = 7.5 for the previous work to z = 10 for the present work). To trigger the emergence, we applied the following velocity perturbation to the flux tube during 0 < t < tr : ⎧ y π t v0 ⎪ for |y| λ2 ⎨ 2 cos 2π λ sin 2 tr

vz = y− 2L− λ |y| ⎪ ⎩ v20 cos 2π [4L−2λ2 ] y sin π2 tt for |y| λ2 . r

2. MODEL To reproduce the formation of a pre-eruptive flux rope via flux emergence, we use the three-dimensional ideal MHD equations given by ∂ρ + ∇ · (ρv) = 0, (1) ∂t ∂v 1 (∇ × B) × B − ρg0 zˆ , (2) ρ + (v · ∇) v = −∇P + ∂t 4π ∂B = ∇ × (v × B) , ∂t

(3)

∂P + ∇ · (P v) = − (γ − 1) P ∇ · v, ∂t

(4)

and P =

ρT , μ

(5)

where ρ, v, B, P, g0 , γ , μ, , and T indicate the gas density, fluid velocity, magnetic field, gas pressure, gravitational acceleration, adiabatic index (γ = 5/3 is assumed), mean molecular weight (μ = 0.6 is assumed), gas constant, and temperature, respectively. The units of these physical quantities are listed in Table 1. By using the units, we derived dimensionless equations from the original equations, which were then solved by means of a modified Lax–Wendroff scheme (Magara 1998) in Cartesian coordinates where the z-axis is directed upward. The domain of the simulation is (−200, −200, −10) < (x, y, z) < (200, 200, 190), and z = 0 corresponds to the surface of the Sun. This domain is discretized into grids whose size is (Δx, Δy, Δz) = (0.1, 0.2, 0.1) for (−8, −12, −10) < (x, y, z) < (8, 12, 40), while Δx, Δy,

(7) where tr = 5, λ = 30, L = 200, and v0 = 0.31. This produces a single Ω-shaped emerging section of the flux tube. The boundary condition of the simulation is the same as the one used in our previous work mentioned above. At y = ±200 a periodic boundary condition is imposed, while open boundary conditions are imposed at x = ±200 and z = 190. A fixed, impermeable boundary condition is set at z = −10. In addition 2


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z

z

x

x

y

y

z

z

x

x y

y

Figure 2. Snapshots of an emerging flux tube taken at t = 16, t = 22, t = 41, and t = 50. Magnetic field lines are given in green, while the red line represents the axis of the flux tube. Gray-scale maps at z = 0 indicate photospheric magnetic flux. (A color version of this figure is available in the online journal.)

surface (t = 41; also see Magara et al. 2011). There are distinct evolutionary patterns of the axis of an emerging flux tube, and how the difference between these patterns affects the nature of the emerging magnetic field is discussed in Magara (2013). In the present case, we obtained the Lagrangian evolution of the axis of the flux tube by following tracer particles initially distributed over the simulation domain and found that the flux tube with the axis bodily emerges to form a flux rope in the corona, as shown in Magara (2013). A bodily emerging flux tube has also been reported in other works (Magara & Longcope 2003; Hood et al. 2009; MacTaggart & Hood 2009; Magara et al. 2011). Finally the flux rope seems to enter an eruptive phase at t ≈ 47.

to them, we place a wave damping zone near all the boundaries to reduce the influence of reflected waves on the simulation. We terminated the simulation before emerging magnetic field reaches the upper and side boundaries. 3. RESULTS 3.1. Overview of the Simulation The evolution of an emerging flux tube presented here is similar to what has been reported in many works, and the details of these works are given in a recent review by Shibata & Magara (2011). Let us consider Figure 2 to briefly discuss the evolution of an emerging flux tube. When the flux tube reaches the surface of the Sun (t = 16), a bipolar distribution of magnetic field appears at the surface, and positive and negative polarity regions show shear motions (Manchester 2007) as the inner part of the flux tube emerges (t = 22). This part forms a sheared arcade in the corona in the late phase of emergence during which a quadrupolar-like distribution of magnetic field is observed at the

3.2. Curvature and Scale Height Analysis of Emerging Magnetic Field In this subsection we explain two quantities characterizing the dynamic evolution of emerging magnetic field in the corona. We recall an evolution equation of emerging field lines given 3


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in Magara (2004), which describes how the apex height of emerging field lines temporally changes under the magnetic pressure force, tension force, and gravitational force. The gas pressure force is neglected because the corona is basically a low plasma-β region. The equation is considered at the symmetry axis (z-axis) of emerging field lines, which may be written as d 2Z ρ 2 = FM + FT + FG , dt

Figure 4 shows the temporal developments of the location (solid line) and velocity (dashed line) of the axis of the rising flux rope, which are measured at the z-axis. In the phase 47 t 50, the rising speed of the axis increases rapidly, indicating a transition to an eruptive phase. 3.3. Decay Index Analysis for the Torus Instability Kliem & Török (2006) and Török & Kliem (2007) are probably the first works pointing out the importance of the torus instability for flux rope eruptions (CMEs) on the Sun. The essence of their works is that the instability is controlled by the decay index of the coronal magnetic field surrounding a flux rope, namely, B ∝ z−n , (14)

(8)

where Z(t) is the apex height of a selected field line and ρ is the gas density measured at the apex, while FM , FT , and FG represent the vertical components of the magnetic pressure force, tension force, and gravitational force: ∂ B2 , (9) FM = − ∂ z 8π 1 B · ∇Bz 4π B2 = − κ, 4π

where n is the decay index. The torus instability occurs when a rising flux rope reaches a height at which n exceeds a critical value. Following the procedure for decay index analysis given in several previous works (Fan 2010; Aulanier et al. 2010), we use a method of potential-field extrapolation explained in Magara & Longcope (2003) to obtain a coronal magnetic field from the photospheric magnetic field provided by the simulation. The decay index is then calculated for this coronal field. Figure 5 shows the distributions of the decay index along the z-axis, obtained at t = 41 (solid line), 44 (dotted line), 47 (dashed line), and 50 (dot-dashed line), respectively. The diamonds indicate the locations of the axis of the rising flux rope at these times. From this figure it is found that the decay index at the location of the axis is well above 2 when the rapid acceleration occurs (at t ≈ 47; also see Figure 4), which is opposite to the previous result that the typical value of the decay index for eruption onset is smaller than 2 (Fan 2010; Aulanier et al. 2010). We discuss this issue in the next section.

FT =

and

FG = −ρg0 .

(10)

(11)

Here κ is the curvature of the field line, taking positive values when the field line has an Ω-shape. Note that the tension force is given by the curvature term alone at the symmetry axis. When we define H as the scale height of emerging magnetic field along the symmetry axis, namely, H −1 = −

1 ∂B , B ∂z

(12)

4. DISCUSSION

Equation (9) is given by FM =

In this section we discuss the dynamic evolution of the flux rope presented in the previous section. First let us consider the profiles of κ and 1/H . In Figure 3 we show that these quantities increase toward their common peak and then decrease with height. A similar profile is found in a potential field produced by a pair of monopoles placed within an interval of l on the surface (x, y, 0): ⎛ ⎛ ⎞ ⎞ ⎡ ⎤ x + 2l x − 2l ⎝ y ⎠ ⎝ y ⎠ ⎢ ⎥ ⎢ ⎥ ⎥ z z Φ⎢ ⎢ ⎥ Bpot = 3 − 3⎥ , ⎢ 2π ⎢ 2 2⎥ 2 l 2 l x − 2 + y 2 + z2 ⎦ ⎣ x + 2 + y 2 + z2

2

B 1 . 4π H

(13)

According to Equations (10) and (13), κ and 1/H represent the magnetic tension and pressure forces normalized by the common factor (4π )−1 B 2 . Equations (8), (10), (11), and (13) indicate that 1/H > κ is required for any Ω-loop being accelerated upward because the gravitational force always works downward. We then examine the spatial distributions of κ and 1/H in the corona. Figure 3 shows the distributions of κ, 1/H , x-component of emerging magnetic field (Bx ), and magnetic pressure (Pm ) along the z-axis, obtained at t = 41 (a), 44 (b), 47 (c), and 50 (d). Bx (the middle panel at each time) is mostly negative when z is small, while it has a jump to taking positive values as z increases. Pm (the bottom panel at each time) shows a relatively uniform distribution below the location of the jump in Bx , and above that location it decreases with z. On the other hand, both κ and 1/H (the top panel at each time) tend to increase with z in a similar way around the location of the jump, indicating that the magnetic pressure force and tension force are almost balanced. After their common peak, κ tends to decrease more rapidly than 1/H although both quantities have local maxima and minima. At t = 41 the axis of the rising flux rope is below the location of the peak (Figure 3(a)), and then the axis is shifted to the range where κ and 1/H decrease with height as explained above (Figures 3(b)–(d)).

(15) where Φ represents the total flux of the potential field. Along the z-axis (symmetry axis) where y- and z-components of the field are zero, the magnetic field curvature is given by

1 ∂ Bz (0, 0, z, t) 3z κ=− = 2 , Bx (0, 0, z, t) ∂x x=y=0 z2 + 2l (16) while the scale height is given by

1 ∂ Bx (0, 0, z, t) 1 3z =− = 2 . H Bx (0, 0, z, t) ∂z z2 + 2l x=y=0 (17) 4


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Figure 3. (a) Distributions along z-axis of κ (solid line) and 1/H (dashed line) in the top panel, Bx in the middle panel, and log10 Pm in the bottom panel are shown. Time is 41. The vertical line in the top panel indicates the location of the axis of the rising flux rope. (b) Same as (a) but for t = 44. (c) Same as (a) but for t = 47. (d) Same as (a) but for t = 50.

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(a) Q

3

κ, 1/H

P

2 R

1

0

1

2

z

3

4

(b) Figure 4. Temporal developments of the location (solid line) and velocity (dashed line) of the axis of the rising flux rope, measured at z-axis.

R

Q P

Figure 6. (a) Distributions of κ and 1/H along z-axis are presented for a bipolar potential field. The normalization unit of length is given by a distance between positive and negative poles placed at z = 0. P, Q, and R represent the field lines indicated in Figure 6(b). (b) Configuration of the potential field is presented. (A color version of this figure is available in the online journal.)

Figure 5. (a) Distributions of the decay index along z-axis are presented for extrapolated potential field at t = 41 (solid line), 44 (dotted line), 47 (dashed line), and 50 (dot-dashed line). The diamonds indicate the locations of the axis of the rising flux rope at these times.

Figure 8 presents the temporal development of this ratio during the transition between the quasi-static and dynamic states of the flux rope. This figure shows that the aspect ratio is 1.14 at t = 47 when the rapid upward acceleration of the flux rope seems to occur, which is significantly larger than the values suggested by previous works on the onset height of eruptions. Vrsnak et al. (1991) use an arc model for prominences to report that the aspect ratio of eruptive prominences is about 0.4 (in their derivation of the aspect ratio they divide height by half footpoint distance, so their value corresponds to twice our value, that is, 0.8), while Chen & Krall (2003) made a theoretical model of CMEs to show that the rapid acceleration of an erupting flux rope should occur shortly after the rope axis has evolved beyond a semi-circular shape (the aspect ratio is 0.5, same as the potential-field case mentioned above). Currently we do not have any reliable explanation of the difference between our result and these previous results. We will investigate the cause of this difference in our future study. We now consider the possibility of a practical application of the described κ–H analysis for observed global eruptions (CMEs). Since, at present, it is not possible to fully capture the structure of the coronal magnetic field only by observations, the extrapolation of the coronal field from the photospheric magnetic field is frequently used. We again use the potential field explained in Section 3.3 to derive the distributions of κ (and 1/H , they are identical) along the z-axis at the different times. The result is presented in Figure 9 where the solid and dotted vertical lines in each panel represent the location of the axis of the rising flux rope and the location of maximum κ,

Since the potential field is static, κ = 1/H is satisfied for all z. Figure 6(a) shows the common profile of κ and 1/H given by Equations (16) and (17), while Figure 6(b) shows the configuration of the potential field. P, Q, and R represent the field lines whose apices are below, at, and above the location of the peak of κ and 1/H . Figure 7(a) schematically illustrates the profiles of κ and 1/H for the emerging field reproduced by the simulation. Their profiles are similar to the one for the potential field, although there is a significant difference between the emerging and potential fields. The structure of the emerging field is divided into two parts separated at the maximum of κ and 1/H : the quasi-static part where the upward magnetic pressure force represented by 1/H is almost balanced by the downward tension force represented by κ, and the expansion part where the magnetic pressure force exceeds the tension force. These two parts are characterized by the distinct geometric shapes of emerging field lines: limited lateral expansion shape for the quasi-static part and fan expansion shape for the expansion part (Magara 2004; Isenberg & Forbes 2007), which are shown in Figure 7(b). An important issue for flux-rope dynamics is the location of the peak of κ and 1/H . In the case of the potential field explained above, the height of this location is equal to the half footpoint distance (see the field line Q in Figure 6). To compare with this, we derive the aspect ratio for the axis of the rising flux rope, which is defined as height divided by footpoint distance. 6


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Figure 7. (a) Spatial distributions of κ (solid line) and 1/H (dotted line) are schematically illustrated for emerging field. S and S represent the field lines outlined by thick red lines. The dotted blue lines indicate the field lines at the maximum of κ. (b) Evolution of emerging field lines in a limited lateral expansion shape (left) and fan expansion shape (right). Adapted from Magara (2004). (A color version of this figure is available in the online journal.)

and potential fields, so we need more work to verify whether the κ–H analysis is practically applicable to global eruptions. The decay index analysis with the extrapolated potential field also gives an inconsistent critical value for eruption onset compared to the value suggested by previous works (Fan 2010; Aulanier et al. 2010). The potential-field extrapolation used in this study is a rather simple approximation for the coronal magnetic field because it only uses the vertical component of the photospheric magnetic field. In a future study we will use the more advanced method of nonlinear force-free extrapolation (e.g., Inoue et al. 2012) to improve our decay index analysis. Finally, we like to point out that there are various parameters that can be expected to affect the dynamic and morphological evolution of flux ropes formed by flux emergence, as well as the resulting κ and H distributions. Some of these parameters could be the twist, radius, and undulation wavelength of an emerging flux tube. We will make a detailed investigation into these possible parameters and report results in our forthcoming papers.

Figure 8. Temporal development of aspect ratio for the axis of the rising flux rope.

both of which are for the emerging field reproduced by the simulation. The diamond and dashed vertical line in each panel show the locations of maximum κ and n = 1.5 for that potential field. The eruptive phase of the flux rope starts at t ≈ 47. The location of maximum κ is quite different between the emerging

The authors deeply appreciate many productive comments by anonymous referees. The authors also wish to thank the Kyung 7


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Figure 9. Distributions of κ along z-axis are presented for the potential field explained in Section 3.3. Times are t = 41 (top), 44 (second from the top), 47 (third from the top), and 50 (bottom). The solid and dotted vertical lines in each panel represent the location of the axis of the rising flux rope and the location of maximum κ for the emerging field reproduced by the simulation. The diamond and dashed vertical line in each panel show the locations of maximum κ and n = 1.5 for that potential field.

Hee University and Korea Meteorological Administration/ National Meteorological Satellite Center for their general support. This research was financially supported by Basic Science Research Program (2010-0009258, PI: T. Magara) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology, as well as the World Class University (WCU) program through the NRF (R31-10016).

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