An Improved Approach to Non-Force-Free Coronal Magnetic Field Extrapolation

Solar Phys (2008) 247: 87–101DOI 10.1007/s11207-007-9090-7

An Improved Approach to Non-Force-Free CoronalMagnetic Field Extrapolation

Qiang Hu · Brahmananda Dasgupta

Received: 26 October 2006 / Accepted: 13 November 2007 / Published online: 1 December 2007© Springer Science+Business Media B.V. 2007

Abstract We develop an approach to deriving the three-dimensional non-force-free coronalmagnetic field from vector magnetograms. Based on the principle of minimum dissipationrate, a general non-force-free magnetic field is expressed as the superposition of one poten-tial field and two constant-α (linear) force-free fields. Each is extrapolated from its bottomboundary data, providing the normal component only. The constant-α parameters are dis-tinct and determined by minimizing the deviations between the numerically computed andmeasured transverse magnetic field at the bottom boundary. The boundary conditions re-quired are at least two layers of vector magnetograms, one at the photospheric level andthe other at the chromospheric level, presumably. We apply our approach to a few analytictest cases, especially to two nonlinear force-free cases examined by Schrijver et al. (SolarPhys. 235, 161, 2006). We find that for one case with small α parameters, the quantitativemeasures of the quality of our result are better than the median values of those from a setof nonlinear force-free methods. The reconstructed magnetic-field configuration is valid upto a vertical height of the transverse scale. For the other cases, the results remain valid to alower vertical height owing to the limitations of the linear force-free-field solver. Becauseour method is based on the fast-Fourier-transform algorithm, it is much faster and easy toimplement. We discuss the potential usefulness of our method and its limitations.

Keywords Magnetohydrodynamics · Magnetic fields: corona · Corona: models · Magneticfields: models

1. Introduction

A recent work by Hu and Dasgupta (2006) (hereafter Paper I) presented the first attemptto derive a three-dimensional (3D) non-force-free coronal magnetic field from photosphericmagnetograms based on the principle of minimum dissipation rate (MDR). MDR is closelyrelated to the principle of minimum entropy production rate of irreversible thermodynamics

Q. Hu (�) · B. DasguptaInstitute of Geophysics and Planetary Physics, University of California, Riverside, CA 92521, USAe-mail: [email protected]

88 Q. Hu, B. Dasgupta

first enunciated by Prigogine (1947). It states “The steady state of an irreversible processis characterized by a minimum value of the rate of entropy production.” For most cases,the entropy-production rate is equivalent to the energy-dissipation rate. The simplest exam-ples are the electric current distribution in a circuit of resistors and the velocity profile ofa viscous liquid flowing through a duct (Poiseuille flow), for which the corresponding totalohmic dissipation rate and the total viscous dissipation rate are minimum, respectively.

Analogous to the principle of minimum energy (e.g., Taylor, 1974), which yields aconstant-α (linear) force-free magnetic field (LFFF),

∇ × B = αB, (1)

the MDR yields a field that is more general and non-force-free (e.g., Dasgupta et al., 1998).It has been applied to magnetohydrodynamics (MHD) (Montgomery and Phillips, 1988)and has been successful in explaining the configurations of certain plasma-fusion deviceswith finite plasma-pressure gradient (Dasgupta et al., 1998, 2002; Bhattacharyya, Janaki,and Dasgupta, 2000, 2003). In recent work, Bhattacharyya and Janaki (2004) developed anMDR-based theory of dissipative relaxed states in a two-fluid plasma with external helicitydriving, which is more suitable for an open system with flow and further applied it to de-rive solar arcade structures (Bhattacharyya et al., 2007). This MDR theory originates fromtwo-fluid MHD and follows a variational approach. The basis is the set of generalized mo-mentum balance equations, and the system, with flow, always satisfies the MHD equations(Bhattacharyya and Janaki, 2004).

In particular, a special class of solutions of an MDR-based theory can be expressed asa linear superposition of several LFFFs with distinct α parameters (Dasgupta et al., 1998;Bhattacharyya et al., 2007; Paper I). In Paper I, we took advantage of this property anddeveloped an approach to extrapolate coronal magnetic field in non-force-free state by su-perposing two LFFFs from a single-level magnetogram. However, the original approach inPaper I utilized a somewhat simplified version of the MDR theory, as developed by Bhat-tacharyya et al. (2007) and it was applied to the solar corona magnetic-field structures. Thoseauthors showed a rigorous derivation of magnetic-field and plasma-flow equations resultedfrom MDR based on a two-fluid plasma description. Those equations led to magnetic-fieldarcade-structure solutions and certain flow patterns that resemble observations.

In this paper, we further improve our original approach (Paper I) to deriving a 3D non-force-free coronal magnetic field by adopting a complete implementation of the MDR the-ory, as to be described in Section 2. In Section 3, we illustrate our method by applying itto the same analytic non-force-free case (Low, 1992) examined in Paper I. In Section 4, wecarry out a comparison study of exactly the same two semi-analytic nonconstant α (non-linear) force free (NLFF) cases (Low and Lou, 1990) as Schrijver et al. (2006) (hereafterS06). They tested a set of NLFF field reconstruction methods against two semianalytic casesand made thorough evaluations by quantitative measures. Although it is not explicit whetherthe solution to our MDR-based system includes the NLFF field, it is useful to compareour results with theirs to explore the applicability and success of our method. There aresimply too few, closed-form analytic 3D solutions of non-force-free fields; Low’s active-region model (Case 0 in the following) is the only one that we knew of when we developedour method, although we have since learned that there are a few more (e.g., Low, 1982;Kaiser and Salat, 1997; Petrie, 2006). Therefore we choose the most popular and somewhatmore versatile Low and Lou’s NLFF solutions as “general” cases to test our method. More-over, we set out to do a side-by-side comparison study with S06. Therefore exactly the sametwo cases examined there are presented here in the same way as well. In the last section, weconclude and discuss the significance and limitations of our method.

Non-Force-Free Coronal Magnetic Field 89

2. An MDR-based Approach

2.1. Governing Equations and Exact Solution

Following Bhattacharyya and Janaki (2004) and Bhattacharyya et al. (2007), the plasma(consisting of ions and electrons) was described by two-fluid equations. They introduced agauge-invariant helicity in terms of the canonical momenta (combination of fluid momentaand the vector potential) and the canonical vorticity and formulated the variational principlefor MDR with the generalized helicity dissipation rate as constraints. Then they minimizedthe integral composed of total ohmic and viscous dissipation rates, subject to those con-straints. A variational approach yielded the following decoupled equations for the magneticfield (B) and the fluid vorticity (ω) (for details, see Bhattacharyya et al., 2007):

∇ × ∇ × B + a1∇ × B + b1B = ∇ψ, (2)

∇ × ∇ × ω + a2∇ × ω + b2ω = ∇χ. (3)

Here, a1, a2, b1, and b2 are constants involving the parameters of the system. The right-hand sides are gradients of arbitrary scalar functions. In Paper I, a version of Equation (2)is used by omitting the right-hand-side term, which is equivalent to subtracting a fractionof the potential field ∇ψ from B, then substituting such a modified field for the originalB, as demonstrated by Bhattacharyya et al. (2007). An exact solution was obtained by thesuperposition of two LFFFs. Although such an omission simplified the problem, especiallythe boundary conditions (as will be shown), it reduced the generality of the solution. Ineffect, such a solution generally needs correction by adding a potential field, as will beimplemented here.

To eliminate the ambiguity associated with the undetermined right-hand side of Equa-tion (2), we take an extra curl operation on both sides, which results in

∇ × ∇ × ∇ × B + a1∇ × ∇ × B + b1∇ × B = 0. (4)

One exact solution can be written as a superposition of three LFFFs,

B = B1 + B2 + B3, (5)

where each LFFF satisfies

∇ × Bi = αiBi , i = 1,2,3, (6)

with constant parameter αi . Therefore, it follows that

∇ × B = α1B1 + α2B2 + α3B3, (7)

∇ × ∇ × B = α21B1 + α2

2B2 + α23B3. (8)

It is clear from these equations that the solenoidal condition of the solution B is explicitlysatisfied, since each Bi is apparently divergence-free.

In a more compact form, by denoting B̃ = (B1,B2,B3)T, we can write Equations (5), (7),

and (8) as

VB̃ = C̃, (9)


where the column vector C̃ = (B,∇ × B,∇ × ∇ × B)T, and the matrix V is a (transposed)Vandermonde matrix of order three, with element α

j−1i at the j th row and ith column. It

is guaranteed inversible provided that the parameters αi, i = 1,2,3 are distinct, so that itsdeterminant, |V| = ∏

j>i(αj − αi), is always nonzero. Therefore, it is always valid to solve

B̃ = V−1C̃. (10)

The equation for B resulting from such a system is

∇ × ∇ × ∇ × B − (α1 + α2 + α3)∇ × ∇ × B

+ (α1α2 + α2α3 + α3α1)∇ × B + α1α2α3B = 0. (11)

To satisfy Equation (4) and the requirement of distinctness of parameters αi , one such pa-rameter has to be set to zero; that is, the corresponding Bi becomes a potential field, sothat the term associated with B vanishes. Owing to the exchangeability of the subscript i,we arbitrarily choose α2 = 0 and α1 �= α3 �= 0. Consequently, the specific equation that themagnetic field B satisfies becomes

∇ × ∇ × ∇ × B − (α1 + α3)∇ × ∇ × B + α3α1∇ × B = 0. (12)

For the special case, −α1 = α3 = α, it is reduced to

∇ × ∇ × ∇ × B = α2∇ × B. (13)

This equation is equivalent to taking an extra curl on both sides of ∇ ×∇ × B = α2B, whichwas derived by Chandrasekhar and Woltjer (1958).

2.2. Application to Coronal Field Extrapolation

Similar to the situation of superposition of two LFFFs in Paper I, this system of Equa-tions (10) is utilized to provide the boundary conditions for each LFFF, Bi , i = 1,2,3 (onlythe normal components), including the potential field B2, at height 0 (z = 0), given vectormagnetograph measurement B on the transverse (x, y) plane within a certain field of view.One single-level vector magnetogram is sufficient for calculating (∇×B)z. However, at leasttwo layers of simultaneous vector magnetogram of the same region, presumably at both thephotospheric and the chromospheric level (see, e.g., Choudhary, Sakurai, and Venkatakrish-nan, 2001) are needed to evaluate (∇ × ∇ × B)z at z = 0 by various means. For instance, byrewriting ∇ × ∇ × B = −∇2B and using ∇ · B = 0, one obtains

(∇ × ∇ × B)z = −∇2Bz = −∇2t Bz + ∂

∂z

(∂Bx

∂x+ ∂By

∂y

)

, (14)

where the subscript “t” denotes operation in the transverse plane. The first term of this equa-tion can be evaluated at z = 0 by using only the photospheric vector magnetograph data, andthe second term has to be approximated by using two layers of vector magnetograms. Alter-natively, if more than one layer of observations above the photospheric level are available,only the normal components of two extra layers of greater than zero height are needed todirectly approximate the second-order derivative, ∂2Bz/∂z2.

The determination of the constant parameters α1 and α3 is carried out the same way bytrial and error as in Paper I, by minimizing the normalized vector error (Amari, Boulmeza-oud, and Aly, 2006), denoted En by S06, of the transverse magnetic field at z = 0. Namely,


let bt = b1t + b2t + b3t represent the numerical solution of the transverse field, for totalM = N2 grid points, with N being the number of grid points along the x or y-dimension.The normalized vector error between bt and the “exact” (measured) Bt is then

En =M∑

i=1

|Bt,i − bt,i |/M∑

i=1

|Bt,i |. (15)

The LFFF solver utilized is the one developed by Alissandrakis (1981), which employs thefast-Fourier-transform (FFT) algorithm and requires only the normal component. One pairof possibly optimal parameters α1 and α3 is determined when the corresponding En reachesa minimum. The final reconstruction result is simply the sum, b = b1 + b2 + b3, of twoLFFFs corresponding to two optimal constant α parameters, respectively, and one potentialfield within a certain volume in the z > 0 domain.

2.3. Summary of Procedures

In brief summary, the procedure for reconstructing a non-force-free coronal magnetic fieldfrom at least two layers of vector magnetograms consists of the following two steps:

1. Set α2 = 0 and search through LFFF parameters α1 and α3 (α1 > α3, owing to theexchangeability of subscripts 1 and 3). Apply an LFFF solver to calculate the trans-verse components bit for each αi , utilizing the normal component Biz [given by Equa-tions (10)], i = 1,2,3, at z = 0 only. A pair of optimal αi, i = 1,3, is found, for whichthe En, given by Equation (15), is minimum.

2. Solve for B1 and B3 in the domain z > 0, for the optimal α1 and α3 found in Step 1,respectively. Calculate the potential field B2 in z > 0, and obtain B = B1 + B2 + B3. Ifcertain plasma information is available, evaluate plasma pressure, density, and tempera-ture profiles based on appropriate MHD equations, optionally with gravity.

It is not straightforward, though, to obtain the plasma state, since the full set of MHDequations has to be considered. Ideally one should solve both Equations (2) and (3) to obtainthe magnetic field and flow, then employ the full set of MHD equations to solve for theplasma states. For a slowly varying situation, the force balance can be interpreted in thetime-average sense (see, e.g., Montgomery and Phillips, 1988).

This approach has wide application to more general non-force-free coronal magnetic-field structure. Its performance, at the present stage, is plausible, as will be demonstrated inthe following sections with a few analytic test cases, including NLFF models. It requires twolayers of simultaneous vector magnetic field data on the bottom boundary, while periodicboundary conditions are assumed on side boundaries since each LFFF, by default, is solvedby an FFT-based algorithm (Alissandrakis, 1981). This is challenging for observations butmay not be entirely impossible. It looks desirable to further increase the number of LFFFs(Bi ) to attain a higher order system by taking extra curl operations. However, an attemptto increase to order four failed because the governing equation for B is not consistent withEquation (4).

3. An Analytic Non-Force-Free Test Case: Case 0

We first examine the same analytic test case as Paper I to illustrate the method. This is an an-alytic 3D non-force-free model for solar active regions developed by Low (1992). Figure 1a


Figure 1 3D view of magnetic field lines of Case 0: (a) analytic solution (Low, 1992); (b) numerical extrap-olation result on a 32 × 32 × 16 grid. Bottom image at vertical height 0 shows the normal field componentdistribution with gray scales from black to white indicating strongly negative to strongly positive field.

Figure 2 Contour plot of the Endistribution in (α1, α3) parameterspace for Case 0. The diagonaldashed line marks α1 = α3 whereno value is defined. Theoutermost contour is of value0.40. The inner ones are of levelsdecreased by 0.04 consecutively.The “+” sign denotes a globalminimum; the “×” sign denotesthe location of optimal choice.The dotted lines outline theapproximate boundary of the“small-scale” solution domain.See text for details.

shows one magnetic field line configuration with three flux ropes lying low horizontally overthe “inversion line” where the normal component changes sign. We use the vector magneticfield data from the bottom two layers as boundary conditions. A contour plot of the En

distribution in the (α1, α3) parameter space (normalized to be dimensionless) is shown inFigure 2. It shows multiple local minima with the overall minimum 0.28 at the locationmarked by the “+” sign. We search through the locations of local minima and choose theone marked by the “×” sign with En = 0.29, and α1 = 0.3789, α3 = −0.3891 as our optimalchoice. The solution obtained with this pair of α parameters, as shown in Figure 1b, has thebest resemblance with the exact solution as judged from the field line plot. For instance, themain feature of low-lying flux ropes is well recovered. All three flux ropes are present. Aview from the top as given in Figure 3 makes this more evident. In practical applications,when this situation arises, multiple observations, such as remote-sensing images implyingfield line configuration, are crucial to discern the optimal solution that best represents thetrue magnetic field structure. The corresponding transverse magnetic field vectors at z = 0are plotted in Figure 4. For the most part, the numerical solution agrees very well with theexact solution, but not near the upper left (−5,5) and lower right (5,−5) corners, nor near


Figure 3 2D (top) view of magnetic field lines of Case 0. Format is the same as Figure 1.

Figure 4 The transverse magnetic field vectors at z = 0 for Case 0: (a) exact solution; (b) numerical solution.

the bottom (y = −5) and right (x = 5) boundaries. It is unusual that the transverse fieldvectors form a “sink” (“source”) at regions of positive (negative) normal field (as also seenin Low, 1992). We do not seek to provide the physical interpretation here, which is beyondthe scope of this paper. Figure 5 shows the transverse components at height z = 1.25. Onecan see larger deviations between numerical and exact solutions.

The quality of the reconstructed field-line configuration is significantly better than thatin Paper I, where only the remnant of the central flux rope was visible. However the over-all structure, especially the overlying arcade at greater vertical heights, shows deviationsfrom the actual configuration, resulting in altered connectivity for certain field lines. This ismainly due to the relatively large α values. As illustrated in Figure 2, the chosen, optimalvalues of α1 and α3 are all outside of the domain corresponding to well-behaved “small-scale” solutions corresponding to |α| ≤ αmax ≈ 2π/N (Alissandrakis, 1981; Gary, 1989).These small α values yield solutions exponentially decaying in height (Alissandrakis, 1981;Gary, 1989). However, the larger α values (|α| > αmax) result in “large-scale” solutions thatoscillate and do not vanish at infinite height. The larger the α value, the larger the spatialoscillating frequency becomes. The combined effect of the unphysical behavior owing tothe large α value, in addition to the finite boundary effect and nonuniqueness (e.g., Chiu


Figure 5 The transverse magnetic field vectors at z = 1.25 for Case 0. Format is the same as Figure 4.

and Hilton, 1977) of each LFFF solution, contributes to the distortion of the numerical so-lution at greater heights. In this case, the height up to which a quantitatively valid solutionis attainable is greatly limited, because of the limitations of the LFFF solver.

4. Quantitative Comparison with S06

In what follows, we carry out a comparison study with S06, by applying our method, justoutlined, to the same two semianalytic NLFF cases as S06. One can explicitly show that theNLFF equation with a varying α parameter is generally not compatible with the double-curlversion of the MDR equation employed in Paper I. For the triple-curl Equation (4), it is notclear. We intend to use these cases as “blind” test cases, without the need to know whetherthey are force-free or not, but rather concentrate on a comprehensive set of quantitativemeasures of the agreement between the exact and the numerical solutions. In turn, the qualityof these measures is also indicative of the satisfaction of the solenoidal condition and theforce-freeness if the solution is known to be force-free. The following two cases are of thesame notation and presented in the same order as S06 to facilitate comparison. The sameset of “figures of merit” proposed by those authors for all of their calculations is utilized aswell. The computations are performed in the same spatial domain: x, y ∈ [−1,1], z ∈ [0,2],but all on a 643 grid, and only the bottom two layers of vector magnetic field are utilized asboundary conditions for both cases. The inner volume is x, y ∈ [−0.5,0.5] and z ∈ [0,1] ona 323 grid.

4.1. Case I

This is the same Case I as S06, a semianalytic NLFF solution generated by Low and Lou(1990). A field-line plot is shown in Figure 6a for the inner volume of the 323 grid only. Thecontour plot of En is given in Figure 7. The optimal choice of α parameters are α1 = 0.01247and α3 = −0.01402 with En = 0.13, based on the quantitative measures developed by S06,as well as qualitative inspection of the agreement between the numerically reconstructedfield line configuration and the exact result. The corresponding optimal solution is shown inFigure 6b, and the set of “figures of merit” is given in Table 1.

The set of figures shows good agreement with the exact solution. Our results are generallybetter or comparable with the medians from the six NLFF methods presented in S06. Note


Figure 6 3D view of magnetic field lines of Case I. Format is the same as Figure 1.

Table 1 Figures of merit for Case I: (a) entire volume of the 643 grid; (b) inner volume of the 323 grid.

Case I Cveca CCS

b E′n

c E′m

d εe εpf

(a) Entire volume

Our results 0.99 0.93 0.73 0.44 1.01 1.31

Mediansg (S06) 0.98 0.76 0.68 0.38 0.99 1.26

LFF (S06) 0.88 0.90 0.50 0.42 0.77 1.00

Potential (S06) 0.85 0.82 0.45 0.35 0.77 1

Exact (S06) 1 1 1 1 1 1.29

(b) Inner volume

Our results 0.99 0.99 0.85 0.79 0.98 1.23

Mediansg (S06) 1.00 0.94 0.82 0.71 0.96 1.18

LFF (S06) 0.88 0.91 0.54 0.49 0.80 1.00

Potential (S06) 0.86 0.87 0.50 0.44 0.81 1

Exact (S06) 1 1 1 1 1 1.24

aVector correlation, Equation (28) in S06.

bCauchy – Schwarz, Equation (29) in S06.cE′

n = 1 − En, complement of En, Equation (15), but for a volume.

dE′m = 1 − Em, complement of Em, Equation (31) of S06.

eTotal magnetic energy normalized to the exact value.

fRatio of magnetic energy over that of the potential field of the same normal magnetic field distribution,obtained by the Fourier method (Gary, 1989) with α ≡ 0.gMedians obtained from relevant rows “(b)” to “(g)” of Table I and II in S06.

that both optimal α values are well within the “small-scale” solution domain (|α| < αmax ≈0.1). Together with a well-determined transverse field at z = 0 (shown in Figure 8), a goodnumerical result is warranted, especially in the inner volume. Although larger deviations ofthe transverse field are seen at greater height (Figure 9), the reconstructed solution remainsvalid up to a height of the transverse scale.


Figure 7 Contour plot of the Endistribution for Case I. Format isthe same as Figure 2. Within thelevel 0.14, all contours are oflevel 0.13.

Figure 8 Transverse magnetic field vectors at z = 0 for Case I. Format is the same as Figure 4.

4.2. Case II

This is the same Case II as S06, and it is also a solution from Low and Lou (1990). Theexact solution in the inner volume is shown in Figure 10a. For this case, since large α valuesbeyond the domain of “small-scale” solutions yield solutions of dramatically distorted fieldstarting at small heights, we limit our allowed α parameters to be within the “small-scale”domain (i.e., |α1|, |α3| < 2π/64 ≈ 0.1). The result for the En distribution is shown in Fig-ure 11. We choose the pair of α values α1 = 0.003117 and α3 = 0.001558 that correspondsto a global minimum, En = 0.35, as our optimal choice to obtain our numerical solutionshown in Figure 10b. Compared with the exact solution in panel a, the low-lying field linesseem to agree well, but the high rising field lines show apparent deviations. The transversefield at z = 0 shown in Figure 12 seems to agree well mostly in the center region, and the


Figure 9 Transverse magnetic field vectors at z = 0.5 for Case I. Format is the same as Figure 4.

Figure 10 3D view of magnetic field lines of Case II. Format is the same as Figure 1.

corresponding value En is significantly larger than that of Case I. The agreement of thetransverse field at height z = 0.5 (shown in Figure 13) seems to be worse than Case I.

The set of “figures of merit” is given in Table 2. For this case, our results are generallyworse than the medians from the six NLFF methods in S06, although they perform better forthe inner volume. They are better than the results from the LFF method and are compara-ble with those from the potential field calculation. Nonetheless, as anticipated, the numbers(not given here) are much improved when they are calculated for a volume of lower height.This indicates that our result is valid to a limited vertical height. Cautions have to be exer-cised especially when considering field line behaviors at greater heights. These argumentsare largely supported by visual inspection of both panels of Figure 10. The reason for rela-tively poor performance of our method, especially in contrast to Case I, is probably that thiscase is closer to a potential-field configuration as hinted by the magnetic energy, which isjust slightly larger than the potential field, and the field is more compact and more nearlypotential in a larger volume surrounding the center (S06). Our theoretical basis may breakdown, as exemplified by the nearly zero optimal α values and a relatively poorly determinedminimum En.


Figure 11 Contour plot of theEn distribution for Case II.Format is the same as Figure 2.

Figure 12 Transverse magnetic field vectors at z = 0 for Case II. Format is the same as Figure 4.

4.3. Order of the Method

Our method, based on an FFT algorithm, has run times that scale as N3 log2 N for a cubic N3

grid, whereas the methods employed in S06 scale at least as N5. Therefore our method runsapproximately N2/ log2 N times faster than most NLFF reconstruction methods reportedin S06. For instance, in the two test cases here with a 643 grid, our results are obtained inseconds in IDL on a desktop Pentium 2.8-GHz PC.

5. Conclusions and Discussion

In conclusion, we improve upon our approach to deriving the 3D non-force-free coronalmagnetic field from vector magnetograms. Based on the MDR theory, the general non-force-free field is obtained as a superposition of one potential field and two LFFFs. Each


Figure 13 Transverse magnetic field vectors at z = 0.5 for Case II. Format is the same as Figure 4.

Table 2 Figures of merit for Case II: (a) entire volume of the 643 grid; (b) inner volume of the 323 grid. SeeTable 1 for notes.

Case II Cveca CCS

b E′n

c E′m

d εe εpf

(a) Entire volume

Our results 0.98 0.23 0.13 −5.21 1.06 1.23

Mediansg (S06) 0.99 0.53 0.72 −0.32 1.01 1.10

LFF (S06) 0.93 0.08 −0.80 −37. 1.04 1.15

Potential (S06) 0.92 0.35 0.47 −0.63 0.91 1

Exact (S06) 1 1 1 1 1 1.10

(b) Inner volume

Our results 0.99 0.62 0.68 −0.61 1.04 1.21

Mediansg (S06) 0.99 0.85 0.80 0.50 1.00 1.10

LFF (S06) 0.94 0.53 0.39 −3.1 1.01 1.10

Potential (S06) 0.92 0.66 0.57 0.30 0.91 1

Exact (S06) 1 1 1 1 1 1.10

is obtained by LFFF extrapolation that only utilizes their normal components, which aredetermined from at least two layers of vector magnetograms. The constant-α parameters ofeach LFFF are determined by optimizing the agreement between the numerical and mea-sured transverse magnetic field at the photospheric level. We apply this approach to a fewanalytic test cases, including two NLFF models examined by S06. Study of these analyticcases offers us insight into the characteristics of the method and the following conclusionscan be drawn:

1. For small, optimal αi parameters (both |α1| and |α3| < αmax), and well-determined trans-verse field at z = 0 (indicated by better uniquely determined and relatively smaller min-imum En), more reliable solutions can be anticipated, up to a certain vertical height ofapproximately the transverse scale.

2. For small, optimal αi, i = 1,3, parameters, but somewhat poorly determined transversemagnetic field at z = 0 (implied by relatively larger minimum En or a wider range of


multiple choices of possible optimal pairs of αi ), the solution may be valid to limitedlower heights.

3. For large, optimal αi parameters (either or both |α1|, |α3| > αmax), the valid solution isgreatly limited to lower height, mainly because of the unphysical behavior of the corre-sponding LFFF extrapolation.

In practical applications, we may start from the original approach of Paper I, using thesuperposition of two LFFFs and only the photospheric vector magnetogram. If observationspermit, we will use the fully implemented approach developed in the present work to refinethe analysis and follow the guidelines listed here to reconstruct and interpret our results.Multi-instrument and multi-wavelength observations must be sought to determine the op-timal solution, especially when the situation of multiple choices arises. We anticipate thatmany issues, including ones discussed in the following, have yet to be addressed by appli-cation to actual magnetograph measurements.

Our method imposes higher requirements for magnetograph observations. In practice,we need simultaneous vector magnetograph data at both photospheric and chromosphericlevels. In a study by Choudhary, Sakurai, and Venkatakrishnan (2001), those authors alreadyutilized nearly simultaneous photospheric and chromospheric magnetograms, although onlythe line-of-sight components. Those two layers were separated by about one grid size of thetransverse plane, which is ideal for our method to employ them as bottom boundary planesonce the vector-magnetic-field measurements are available for both planes. At the expenseof more observational input, the extra benefit is a low-coronal field in a non-force-free state.

In principle, our approach is also applicable to deriving the flow field [Equation (3)].By replacing magnetic field B by flow vorticity ω, exactly the same procedures can becarried out. However, the boundary conditions are harder to implement since they involveevaluations of double and triple curl of flow field. Nonetheless, Bhattacharyya et al. (2007)were able to derive an analytic model of the coronal flow, based on the MDR theory.

Finally, our method, based on an FFT algorithm, is much faster than most NLFF solversreported by S06. For instance, for a computation on a 643 grid, our method is about 600times faster. This makes it potentially useful for analyzing future high-resolution magne-tograms. However, when the number of grid points (N ) increases, the maximum allowableα parameter (αmax) decreases. This effectively reduces our domain of desirable α parame-ters corresponding to “small-scale” solutions. One well-known limitation of the LFFF ex-trapolation is the ill-posedness of the problem (see, e.g., Low and Lou, 1990). We are notaware of any means to completely overcome this, except for a few attempts to mitigate theproblem (Amari, Boulmezaoud, and Maday, 1998; Wheatland, 1999). Improvement of anLFFF solver that is less vulnerable to those weaknesses, but still maintains the virtue ofeasy implementation and fast speed, is highly desirable. The ultimate test and judgment areapplications to real data.

Acknowledgements The authors thank Prof. Debi Choudhary, Dr. Manolis Georgoulis, and Dr. Yang Liufor providing their codes and helpful discussions. We are grateful to Drs. Carolus Schrijver and Marc Derosafor providing their codes for generating the semianalytic solution of Case II. H.Q. acknowledges NASAGrant Nos. NNG04GF47G and NNG06GD41G for partial support. B.D. acknowledges NASA Grant No.NNG04GF83G for partial support.

References

Alissandrakis, C.E.: 1981, Astron. Astrophys. 100, 197.Amari, T., Boulmezaoud, T.Z., Aly, J.J.: 2006, Astron. Astrophys. 446, 691.


Amari, T., Boulmezaoud, T.Z., Maday, Y.: 1998, Astron. Astrophys. 339, 252.Bhattacharyya, R., Janaki, M.S.: 2004, Phys. Plasmas 11, 5615.Bhattacharyya, R., Janaki, M.S., Dasgupta, B.: 2000, Phys. Plasmas 7, 4801.Bhattacharyya, R., Janaki, M.S., Dasgupta, B.: 2003, Plasma Phys. Control. Fus. 45, 63.Bhattacharyya, R., Janaki, M.S., Dasgupta, B., Zank, G.: 2007, Solar Phys. 240, 63.Chandrasekhar, S., Woltjer, L.: 1958, Proc. Nat. Acad. Sci. 44, 285.Chiu, Y.T., Hilton, H.H.: 1977, Astrophys. J. 212, 873.Choudhary, D.P., Sakurai, T., Venkatakrishnan, P.: 2001, Astrophys. J. 560, 439.Dasgupta, B., Dasgupta, P., Janaki, M.S., Watanabe, T., Sato, T.: 1998, Phys. Rev. Lett. 81, 3144.Dasgupta, B., Janaki, M.S., Bhattacharyya, R., Dasgupta, P., Watanabe, T., Sato, T.: 2002, Phys. Rev. E 65,

046405.Gary, G.A.: 1989, Astrophys. J. Suppl. 69, 323.Hu, Q., Dasgupta, B.: 2006, Geophys. Res. Lett. 33, L15106.Kaiser, R., Salat, A.: 1997, J. Plasma Phys. 57, 425.Low, B.C.: 1982, Astrophys. J. 263, 952.Low, B.C.: 1992, Astrophys. J. 399, 300.Low, B.C., Lou, Y.Q.: 1990, Astrophys. J. 352, 343.Montgomery, D., Phillips, L.: 1988, Phys. Rev. A 38, 2953.Petrie, G.J.D.: 2006, Astrophys. J. Suppl. 166, 378.Prigogine, I.: 1947, Etude thermodynamique des phénomènes irrèversibles, Desoer, Liège.Schrijver, C.J., De Rosa, M., Metcalf, T., Liu, Y., McTiernan, J., Régnier, S., Valori, G., Wheatland, M.S.,

Wiegelmann, T.: 2006, Solar Phys. 235, 161.Taylor, J.B.: 1974, Phys. Rev. Lett. 33, 1139.Wheatland, M.S.: 1999, Astrophys. J. 518, 948.

An Improved Approach to Non-Force-Free Coronal Magnetic Field Extrapolation

Documents

Transcript of An Improved Approach to Non-Force-Free Coronal Magnetic Field Extrapolation