(mhd) relaxation method with a flux-rope emergence model - IOPscience

The Astrophysical Journal, 768:119 (7pp), 2013 May 10 C 2013.

doi:10.1088/0004-637X/768/2/119

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

FORCED FIELD EXTRAPOLATION: TESTING A MAGNETOHYDRODYNAMIC (MHD) RELAXATION METHOD WITH A FLUX-ROPE EMERGENCE MODEL X. S. Zhu, H. N. Wang, Z. L. Du, and Y. L. Fan Key Laboratory of Solar Activity, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China; [email protected] Received 2012 December 5; accepted 2013 March 18; published 2013 April 22

ABSTRACT We undertake an attempt to reconstruct the Sun’s non-force-free magnetic field. The solar corona is often considered to be magnetohydrostatic. We solve the full MHD equations with a semi-realistic atmosphere model to attain this stationary state. Our method is tested with a Sun-like model which simulates the emergence of a magnetic flux rope passing from below the photosphere into the corona. Detailed diagnostics shows that our method can model the forced field more successfully than the optimization and potential method, but it still needs to be applied to real data. Key words: magnetic fields – magnetohydrodynamics (MHD) – methods: numerical – Sun: corona Online-only material: color figures

However, none of the evolutionary methods includes a high beta region at the lower boundary like the photosphere. The zero-beta methods neglect the pressure, which means the dynamic interactions between the plasma flow and magnetic field are ignored. The existing non-zero-beta methods only consider the low beta situation over the whole computational domain. For this reason, we present a new implementation of the MHD relaxation method to extrapolate the forced magnetic field. Since the semi-realistic atmosphere model is considered, the plasma beta at the lower boundary can vary from tens to a few hundreds. We choose a final state of a flux-rope emergence simulation produced by Manchester et al. (2004) as a Sun-like reference model. We pay attention to what degree our method is able to recover such a magnetic field. This paper is structured as follows. In Section 2, the Manchester model is introduced. In Section 3, our MHD relaxation method is described and the details of its implementation on testing with the Manchester model are presented. In Section 4, results are reported and analyzed. Finally, conclusions are drawn in Section 5.

1. INTRODUCTION The Sun’s magnetic field plays an important role in a variety of solar dynamical processes, such as extreme ultraviolet (EUV) and X-ray bright points, flares, and coronal mass ejections (CMEs). Studies of the topology and dynamics of the coronal magnetic field allow us to understand the mechanisms of energy release. Unfortunately, the coronal magnetic field is difficult to measure directly. In order to obtain a three-dimensional (3D) field in the corona, a force-free assumption is often employed for extrapolation from the photospheric magnetic measurement. Although this assumption is valid in the corona, the magnetic field at the photosphere is not force-free (Metcalf et al. 1995; Moon et al. 2002; Tiwari 2012). The inconsistency between force-free models and forced boundary conditions leads to poor solutions when using photospheric data as input (Metcalf et al. 2008). Wiegelmann et al. (2006) developed a preprocessing procedure to derive a suitable boundary for the extrapolation by eliminating the net force and torque. The revised boundary, however, does not exactly match observations. Actually, the forced boundary can be treated easily by the evolutionary method (McClymont & Mikic 1994; Mikic & McClymont 1994; Roumeliotis 1996; McClymont et al. 1997; Valori et al. 2005; Wu et al. 2006; Fan et al. 2011; Jiang et al. 2011). We divide the evolutionary method into the zero-beta method, which seeks the solution of momentum and induction equations, and the non-zero-beta method, which seeks the solution of full MHD equations. In the zero-beta method, the steady state of the forced field is attained by using the resistivity and viscosity. Jiao et al. (1997) show the forced boundary layer in their solution for active region AR 7220/7222 in which the Lorentz force is balanced by viscous drag. In the non-zero-beta method, the Lorentz force is balanced not only by viscosity but also by gas pressure and gravity. Wu et al. (2006) solved a set of full MHD equations with the lower boundary given by the projected characteristics (Nakagawa 1980, 1981a, 1981b). In Wu’s model, the initial atmosphere is a gravity stratification layer with a temperature of T = 1 × 105 K to mimic the solar environment. Such an isothermal layer simulating the chromosphere or low corona is often used to balance the forced magnetic field (Fan et al. 2011; Jiang et al. 2011; Fan et al. 2012).

2. THE REFERENCE MODEL Manchester et al. (2004) designed a 3D ideal MHD simulation to model the emerging flux rope passing from the top layer of the convection zone into the corona. The adiabatical MHD equations (with the ratio of specific heats γ = 5/3) are non-dimensionalized using photospheric values of ρph = 2.7 × 10−7 g cm−3 (density), Hph = 150 km (pressure scale height), Tph = 5100 K (temperature), vth = 6.5 km s−1 (isothermal sound speed), and Bph = 1200 G (magnetic field). The simulation domain is (−60, −60, −20) (x, y, z) (60, 60, 95.2), which models the region about 18 × 18 × 17.3 Mm. A block adaptive mesh is employed in the simulation to efficiently resolve structures over many length scales. For the initial state, a horizontal, uniform twisted magnetic flux rope was embedded in a hydrostatic, plane-parallel atmosphere. Density in the middle of the rope is lower than in the background, so that this region becomes buoyant. As time goes on, this buoyant region rises to the photosphere and expands into the corona. Eventually, the flux rope in the model corona moves slowly, which indicates that the magnetic field is almost stationary. The model data used 1

The Astrophysical Journal, 768:119 (7pp), 2013 May 10

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Figure 1. Magnetic field lines of the Manchester reference model. Field lines that leave the computation region through the sides or top boundaries are blue; others are red. (A color version of this figure is available in the online journal.)

here are the final equilibrium state above z = 0 with the uniform Cartesian grids 300×300×246 (see Figure 1). The grid spacing is 58 km. Figure 2(a) shows the plasma β variation with height z (grid points). β ∼ 156 at z = 0, which seems much larger than the active region on the photosphere (Gary 2001). For 30 < z < 195, the magnetic field dominates the plasma since β is basically less than 0.1 and reaches a minimum of 0.05, which is comparable with Gary’s results of 10−1 β 10−3 near height 1 Mm. Then, β increases and peaks at 0.9 because of the finite size of the magnetic structure which also causes an increase in the Lorentz force ratio as we can see below. The Lorentz force can be written as the divergence of the Maxwell stress tensor: Mij = −

Bi Bj B2 δij + , 8π 4π

then the net force can be calculated as Bi Bj B2 δij ds + ds, Fnet = ∇Mij dV = − 8π 4π

and 1 Fp = 8π

2 Bz + Bx2 + By2 dxdy,

(6)

where Fp is the integrated magnetic pressure force. Figure 2(b) shows that the net force radio in 20 < z < 70 is smaller than 0.05, which compares closely to the value at a height of 600 km (Metcalf et al. 1995). Then, it begins to increase at z = 70. Examination of the mathematical expression for Fxyz reveals the reason for this change. The surface integral at the lower boundary can be written as Mxyz,z dxdy = − ∇Mij dV − Mij ds1 , (7)

(1)

where s1 represents the whole surface except for the lower boundary. When the lower boundary is selected below z = 70, the lower force-free domain dominates the two integrals on the right-hand side of Equation (7). This leads to the small Mxyz,z dxdy in 20 < z < 70 and then to the small net force radio in the same region. When the bottom boundary rises beyond z = 70, the effect of the high-beta region z > 200 can no longer be ignored since the scale height reaches 30000 km (106 K). This forced high beta region leads to an increase of net force radio above z > 70 although the plasma remains force-free in 70 < z < 200 (see Figure 2(c)). It is, however, not enough to assess the Lorentz force by using net force on the plasma by using net force which loses much local information. Since the full information of the 3D region is available, we can compute the local Lorentz force at each point (fx , fy , fz ) (Metcalf et al. 2008). Figure 2(b) shows f/fp 1 near z0 due to the high plasma β. In 30 < z < 180, f/fp has an average value of 0.2. Then, above z = 180, the ratio increases rapidly to 0.8 at the upper boundary. Such behavior of the local Lorentz force is consistent with plasma β variation. It is worth noting that a difference exists between the net and local Lorentz

(2)

where s represents six boundary surfaces. For an isolated magnetic structure like the reference model (see Figure 3), the components of the net Lorentz force in the half-space z > 0 can be computed by the following surface integrals (Low 1985), provided that the magnetic field falls off enough as x, y, or z approaches infinity 1 Fx ≈ − (3) Bx Bz dxdy, 4π 1 Fy ≈ − (4) By Bz dxdy, 4π 2 1 Fz ≈ − (5) Bz − Bx2 − By2 dxdy, 8π 2


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(a)

(b)

(c)

Figure 2. (a) Horizontal averages of the plasma β with height; z is the vertical grid number. (b) Net Lorentz force, F = (Fx2 + Fy2 + Fz2 )1/2 , measured relative to 2 2 2 1/2 the magnetic pressure force Fp . (c) Horizontal averages of the total Lorentz force, f = (fx + fy + fz ) , measured relative to the horizontally averaged magnetic pressure gradient force fp = ∇B 2 /2.

Figure 4. Profiles of density (solid), pressure (dotted), and temperature (dashed) of the initial equilibrium. Figure 3. Magnetogram of vertical magnetic field of the reference model. White/black regions represent the direction of the magnetic field outward/ inward of the lower boundary.

where the dependent variables are the mass density ρ, the velocity v, the magnetic field B, the total pressure Pt = Pg + B 2 /2 (with Pg the gas pressure), and the total energy density E = Pg /(γ − 1) + ρv 2 /2 + B 2 /2 (with γ = 5/3 the ratio of specific heats for a fully ionized hydrogen plasma). We apply a total variation diminishing (TVD) Lax–Friedrichs scheme (Harten 1983; Tóth & Odstrcil 1996) to solve the above equations. TVD schemes have the ability to capture sharp shock without any misleading oscillations. The spatially and temporally second-order accuracy is achieved by a linear approximation of primitive variables and Hancock’s predictor scheme, respectively. Powell’s source terms −∇ · B(0, B, 0, v)T are used to eliminate the instability of divergence B (Powell et al. 1999). These terms are proportional to ∇ · B, thus analytically they always remain zero. Several other effective divergencecontrolling techniques, including diffusive approach (Marder 1987; Dedner et al. 2002) and the Poisson projection method, are often used in evolution methods (Valori et al. 2007, 2010; Jiang et al. 2011). The initial state comprises a plane-parallel multilayered hydrostatic model (Fan 2001), embedded with a potential magnetic field (Sakurai 1982) determined by the normal component of the vector magnetogram. The thermal quantities of the atmosphere are plotted as functions of z in Figure 4. The outflow condition is applied for both side and upper boundaries. At the lower boundary the density, pressure, and the normal component of the magnetic field are fixed, and the velocity is set to 0. As the system evolves, the transverse field is slowly changed from the initial condition to the observed field. This is called the “stress and relax” approach (Roumeliotis

force in assessing the magnetic field, and the local Lorentz force f/fp is more accurate. We summarize the comparison between the reference model and the Sun below. The plasma beta at z = 0 is much larger than that in the active region of the photosphere. The plasma beta in 30 z 200 is comparable to that in the real corona. Due to the finite size of the main magnetic structure (smaller and lower than the typical active region), β and the two Lorentz force ratios become too large above z = 200 compared with the Sun. Therefore, this region is not considered in our relaxation and comparison. 3. THE MHD RELAXATION METHOD In this study, we solve the ideal, compressible MHD equations, which can be written in conservative form as: ∂ρ + ∇ · (ρv) = 0, ∂t

(8)

∂ρv + ∇ · (ρvv − BB + Pt ) = ρg, ∂t

(9)

∂E + ∇ · [(E + Pt )v − B(B · v)] = 0, ∂t

(10)

∂B − ∇ × (v × B) = 0, ∂t

(11) 3


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Figure 5. Evolution of the six metrics (left) and plasma velocity (right) during the relaxation. The two field line divergence metrics, FLD_a and FLD_f, represent the fractions of area and flux, respectively. Table 1 Metrics for the MHD Relax Method Applied to Different Lower Boundaries Model ks16 ks17 ks18 ks19 ks20 ks21 ks22 ks23 ks24 ks25 ks26 ks27 ks28 ks29

Cvec

Ccs

En

Em

1-FLD(Area/Flux)

Beta

0.938(0.933, 0.729) 0.943(0.946, 0.732) 0.949(0.955, 0.735) 0.952(0.958, 0.739) 0.958(0.961, 0.742) 0.960(0.966, 0.746) 0.962(0.963, 0.749) 0.964(0.957, 0.753) 0.966(0.968, 0.757) 0.968(0.971, 0.761) 0.971(0.973, 0.765) 0.981(0.976, 0.769) 0.983(0.972, 0.773) 0.984(0.974, 0.776)

0.924(0.878, 0.828) 0.930(0.897, 0.835) 0.936(0.931, 0.841) 0.943(0.915, 0.847) 0.948(0.919, 0.853) 0.953(0.924, 0.857) 0.959(0.920, 0.862) 0.963(0.915, 0.865) 0.968(0.927, 0.869) 0.967(0.931, 0.873) 0.969(0.933, 0.877) 0.972(0.937, 0.880) 0.967(0.935, 0.884) 0.961(0.938, 0.886)

0.610(0.535, 0.364) 0.627(0.576, 0.373) 0.644(0.611, 0.382) 0.660(0.629, 0.391) 0.674(0.644, 0.400) 0.690(0.667, 0.407) 0.703(0.659, 0.414) 0.714(0.646, 0.421) 0.729(0.686, 0.428) 0.732(0.704, 0.435) 0.747(0.714, 0.442) 0.798(0.728, 0.449) 0.796(0.711, 0.455) 0.789(0.717, 0.454)

0.540(0.391, 0.345) 0.555(0.425, 0.359) 0.569(0.457, 0.372) 0.590(0.476, 0.384) 0.598(0.495, 0.396) 0.623(0.519, 0.406) 0.641(0.518, 0.416) 0.658(0.514, 0.425) 0.678(0.550, 0.433) 0.674(0.568, 0.441) 0.688(0.578, 0.448) 0.732(0.593, 0.445) 0.713(0.581, 0.463) 0.695(0.590, 0.464)

0.398/0.478(0.309/0.312) 0.400/0.496(0.350/0.352) 0.427/0.530(0.382/0.390) 0.439/0.536(0.399/0.398) 0.481/0.568(0.450/0.426) 0.545/0.569(0.561/0.449) 0.581/0.552(0.523/0.406) 0.653/0.529(0.510/0.362) 0.688/0.530(0.564/0.416) 0.673/0.553(0.575/0.428) 0.704/0.565(0.603/0.447) 0.768/0.666(0.638/0.473) 0.786/0.734(0.595/0.421) 0.805/0.787(0.584/0.414)

4.8 3.7 2.8 2.2 1.7 1.3 1.0 0.8 0.6 0.5 0.4 0.3 0.3 0.2

Notes. The first and second terms in the bracket represent the Wiegelmann and potential results (the FLD metrics of the potential results are not included). The plasma beta of the different lower boundaries is represented by the “Beta” parameter in this table.

1996). The calculation is stopped when the plasma flow is dissipated sufficiently by the viscosity. In a series of tests, magnetic boundaries at different heights are used as extrapolations in the domain (−60, −60, 0) (x, y, z) (60, 60, 70) which is resolved by 150 × 150 × 90 grids. The starting heights of the atmosphere model are chosen as the corresponding heights of the magnetic data. The solutions are presented in the following section.

the mean vector error metric, Em =

We use six metrics (Schrijver et al. 2006) to quantify the degree of agreement between the reference field B and the extrapolated field b. They are: the vector correlation metric, 1/2 2 2 Cvec = Bi · bi |Bi | |bi | , (12) i

i

the Cauchy–Schwarz metric, Ccs =

1 Bi · bi , M i |Bi ||bi |

the normalized vector error metric, |bi − Bi | |Bi |, En = i

(13)

En = 1 − En ,

Em = 1 − Em ,

(15)

and the field line divergence (FLD) metric (Barnes et al. 2006; Metcalf et al. 2008), where the sum runs over all points i, M is the total number of grid points in the considered volume. If B and b are identical, then all the metrics are equal to 1. It is notable that Cvec and CCS are only sensitive to angle difference, while En and Em are sensitive to differences both in angle and vector norm. The FLD metric measures how well the topology of the field is reproduced by each method. For the same initiated point of reference and extrapolated field line, a score can be given to this point by the distance between the endpoints divided by the length of field line. A single score can be assigned by the fraction of the area at the lower boundary that has an FLD less than 10%. Another score considering the fraction of the flux has an FLD less than 10% is also meaningful. All the metrics are computed in the central region 100 × 100 × 70. The evolution of the these metrics (see Figure 5, left) shows the convergence property of the MHD relaxation method. This extrapolation uses the z = 27 data in the Manchester model as the lower boundary, which we mark as “ks27.” As also found in Jiang et al. (2011), all the metrics evolve fast at the “stress” stage. After that, the system begins to “relax” and finally becomes an equilibrium state where all the metrics converge. Figure 5

4. RESULTS

i

1 |bi − Bi | , M i |Bi |

(14)

i

4


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Figure 6. Plots of the six metrics as given in Table 1. Solid, dotted, and dashed lines indicate the MHD, Wiegelmann, and potential methods, respectively.

Figure 7. Magnetic fields for the Manchester reference model, MHD, and pot represent the reference field, the MHD relaxation field, and the potential field, respectively. (A color version of this figure is available in the online journal.)

six metrics. It is apparent that, compared with the potential extrapolation, the MHD method can much better recover the expanded and sheared flux rope in which non-potential energy is stored. However, in spite of the similarity of most lines, some details are quite different when inspecting this figure carefully. For a forced finite β plasma, the structures of the density, pressure, and temperature are important. Table 2 shows correlations of the three quantities between the reference model and relaxation results. (1) The correlations depend mainly on the initial state. The lower correlation of the higher magnetogram relaxation is caused by the lower correlation of the initial state. (2) The density and pressure correlations seem improved by the relaxations when using the high magnetogram as input. The reason for this is that, for β < 1 plasma, the Lorentz force at the bottom boundary can affect the structure of density and pressure. (3) The relaxations make the temperature correlation worse. The temperature response functions differ greatly to

(right) shows the average velocity, as time goes on, dissipated by the viscosity to about 7 × 10−3 Vf (Vf the fast speed), which also means the system is more and more slowly evolving and converges to a solution at the end. Generally speaking, the three extrapolations presented here do better when using the higher magnetogram as the lower boundary. Table 1 and Figure 6 show the results of the metrics, applied to boundary data at 16 z 29, for the MHD relaxation, optimization (Wiegelmann 2004), and potential methods, respectively. In this range, the plasma β at the lower boundary varies from 0.2 to 4.8 which is consistent with photospheric β in the active region (Gary 2001). It is notable that the MHD relaxation method performs better than the optimization and potential method in most metrics. The reference model and the solutions of the MHD and potential methods of ks22 and ks29 are shown in Figure 7. Overall, ks29 is better. This is consistent with the scores of 5


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Figure 8. Comparison of the density (solid), pressure (dotted), and temperature (dashed) profiles.

It is important to choose a reasonable starting height of the atmosphere model to adapt the magnetogram at different heights. If the starting height is too low, the plasma is too heavy to be driven by the magnetic field, which leads to a long relaxation time. If it is too high, the plasma is too light to stabilize the forced magnetogram. Note the relaxation results in Figure 8: if the coronal plasma state is chosen over the whole region (Wu et al. 2006; Jiang et al. 2011; Fan et al. 2011, 2012), the plasma β employed in our computation at the lower boundary becomes very low, which means a force-free modeling in nature. In the present implementation, the ideal MHD is modeled. The change in topology during relaxation is realized by numerical resistivity. A resistivity term, however, can be specified easily in our extrapolation for a more complex magnetic topology in the corona. An application of our method to observational data available on the Solar Dynamics Observatory/Heliosesimic and Magnetic Imager will be presented in the future.

Table 2 Correlation Coefficients between the Reference Model and Relaxation Applied to Different Lower Boundaries Corr. Coeff ks16 ks17 ks18 ks19 ks20 ks21 ks22 ks23 ks24 ks25 ks26 ks27 ks28 ks29

Density

Pressure

Temperature

Beta

0.905(0.907) 0.908(0.910) 0.908(0.909) 0.904(0.905) 0.896(0.896) 0.882(0.881) 0.862(0.860) 0.836(0.834) 0.805(0.798) 0.775(0.770) 0.750(0.733) 0.723(0.672) 0.703(0.690) 0.692(0.681)

0.878(0.879) 0.864(0.864) 0.842(0.840) 0.813(0.809) 0.775(0.768) 0.735(0.724) 0.696(0.678) 0.663(0.642) 0.627(0.600) 0.602(0.565) 0.584(0.540) 0.566(0.519) 0.562(0.507) 0.559(0.494)

0.384(0.391) 0.376(0.389) 0.321(0.385) 0.353(0.383) 0.348(0.377) 0.329(0.375) 0.295(0.366) 0.256(0.365) 0.218(0.359) 0.223(0.351) 0.210(0.348) 0.148(0.333) 0.136(0.338) 0.157(0.335)

4.8 3.7 2.8 2.2 1.7 1.3 1.0 0.8 0.6 0.5 0.4 0.3 0.3 0.2

Note. The term in parentheses represents the correlation of the initial state.

We thank the referee for constructive comments. This work is jointly supported by the National Basic Research Program of China (973 Program) through grant 2011CB811406 and the National Natural Science Foundation of China (NSFC) through grants 11273031 and 10921303.

different wavebands in solar observations like coronal EUV images. For example, 131 Å and 171 Å are sensitive to the hot (∼11 MK) and cooler (∼0.6 MK) plasma (O’Dwyer et al. 2010; Lemen et al. 2012), respectively. The temperature reconstruction is not encouraging; because the temperature is not solved directly by the MHD equations, the density and pressure errors can be amplified by the division operation T = P /ρ. Figure 8 shows the comparison of these three quantities between the reference model and relaxation results of ks27.

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5. CONCLUSION By using the “stress-and-relax” approach, we solve the full MHD equations with a high-beta region at the lower boundary like the photosphere, and then we have a new implementation of the MHD relaxation method, which is applied to reconstruct a Sun-like forced field generated by Manchester et al. (2004). Metrics of the solutions carried out by the optimization (Wiegelmann 2004) and potential methods are also reported. A series of tests show that, when the value of beta at the lower boundary decreases, the difference between the reconstructed and reference magnetic fields becomes smaller. The results also show that our new implementation benefits from discarding the force-free assumption when extrapolating a forced magnetic field. The recovery of density and pressure structure can be improved by the relaxation when the plasma beta is not very large (β < 1). The relaxed temperature distribution, however, is not reliable. 6


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