A NEW AND FAST WAY TO RECONSTRUCT A NONLINEAR FORCE-FREE FIELD IN THE SOLAR CORONA

Transcription

1 The Astrophysical Journal, 649: , 2006 October # The American Astronomical Society. All rights reserved. Printed in U.S.A. A A NEW AND FAST WAY TO RECONSTRUCT A NONLINEAR FORCE-FREE FIELD IN THE SOLAR CORONA M. T. Song, C. Fang, 2 Y. H. Tang, 2 S. T. Wu, and Y. A. Zhang Received 2006 March 6 accepted 2006 May 22 ABSTRACT We reexamine the method of upward integration of a nonlinear force-free field (NFFF), which is, as is well known, an ill-posed problem. It can be modified to a well-posed one by the following means: instead of using finite difference to express partial derivatives, we use smooth continuous functions to approach magnetic field values, write down three field components consisting of amplitude functions multiplying morphology functions, and reduce four basic NFFF equations to ordinary differential ones. They are then solved in an asymptotic manner (zeroth-order, first-order, etc.). Considering the physical meaning of, we found a self-consistent compatibility condition for the boundary values. Furthermore, a computation algorithm is proposed, similar to the usual time-dependent two-dimensional MHD simulation scheme. This algorithm is steady and robust against the noise in the magnetic field (in particular, the transverse field) measurement and is able to deal with concentrated photospheric currents. The algorithm runs very fast on an ordinary PC and lasts only 6 minutes for the (x y) mesh up to a height of 80 (= km 0: R ). So it provides a powerful tool for solar scientists to analyze the magnetic field properties of solar active regions and to make predictions of solar activity. Subject headinggs: MHD Sun: corona Sun: magnetic fields Online material: color figures. INTRODUCTION Since most solar eruptive phenomena, such as flares, coronal mass ejections, and eruptive prominences, take place in the lower corona, knowing the magnetic field configuration in the chromosphere and the lower corona plays a key role in understanding their physical origin. Unfortunately, a direct measurement of the coronal magnetic field is particularly difficult. Judge et al. (2002) pointed out that several forbidden lines (say, Fe xiii,hei,mgviii, and Si ix) are available, but the line-of-sight integration involved in the observations makes the data analysis a difficult inversion problem. So far there are no high-quality direct measurements of the coronal magnetic field. A complementary method of the field measurements is a precise extrapolation of the accurately observed photospheric magnetic field. Currently only potential field and linear force-free field (LFFF) approximations can be easily used, although it is well known that LFFF does not contain enough free energy to be released for an eruption in the framework of ideal or resistive magnetohydrodynamic (MHD) instabilities. However, during an eruptive MHD process a nonlinear force-free field (NFFF) can relax to a LFFF field with the same magnetic helicity ( Berger 984). Therefore, using the NFFF model to extrapolate the photospheric magnetic field is a promising good approximation. So far there are five method for computing NFFF (Wiegelmann 2004):. Upward integration method (Wu et al. 990 Amari et al. 997), which uses a kind of finite difference scheme to solve the height-dependent mixed elliptic-hyperbolic partial differential equations. However, it is an ill-posed problem, which involves Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 20008, China. 2 Department of Astronomy, Nanjing University, Nanjing 2009, China chengfang8@yahoo.com.cn. Center for Space Plasma and Aeronomic Research and Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville, Huntsville, AL (a) only the bottom boundary condition ( b) singularity points occurring at some heights where the magnetic field becomes infinite, even in the case of extrapolating upward a potential field ( ¼ 0), as pointed out by Low & Lou (990) and (c) in the numerical process, a growing mode that often erroneously increases the field. 2. Grad & Rubin (958) method (Sakurai 98 Aly 989). Starting with the potential field matching the observed vertical magnetic field B 0 z and selecting some field lines with a certain value 0, one can then solve a sequence of linear equations B n =: n ¼ 0and:<B nþ ¼ n B n,wheren is an iteration number.. MHD relaxation method (Chodura & Schluter 98 Roumeliotis 996 McClymont & Mikic 994), which solves three-dimensional MHD equations containing six variables, B and V. Starting with a potential field as an initial state, and imposing a vertical current (@B y x /@y) (or an electric field E) atthe bottom (z ¼ 0), the magnetic field B gradually approaches a state of NFFF due to the interaction between the friction force v and the Lorentz force J < B. This method has been widely accepted, since it matches the conservation laws in plasma physics, but it is time consuming. 4. Optimization approach method ( Wheatland et al Wiegelmann 2004), which introduces an integration where the squares of the Lorentz force and the field divergence are contained and then makes it attain a minimum state by a time-dependent evolution method. The result gives an NFFF field. 5. Other methods (see Wiegelmann et al Yan & Li 2006). The common drawback of the above methods (), (2), and () is that they need a long computation time to obtain a useful result. In view of these, examination of method () is renewed in this paper. We try to transform the ill-posed problem into a well-posed one, by the following means: (a) For each height step, instead of using a finite-difference scheme, an analytical asymptotic solution is taken as accurately as possible. (b) We give definite boundary values at the bottom, top, and four lateral surfaces. The bottom

2 NONLINEAR FORCE-FREE FIELD IN SOLAR CORONA 085 boundary is, of course, the data from photospheric vector magnetograms. (c) We distinguish the boundary compatibility conditions, which involve B 0 x B0 y B0 z,and0 at the bottom z ¼ 0, for the case of an analytical solution from that of using observed magnetic fields. The key point of our idea is that the NFFF of solar active regions must be a similarity solution consisting of several smooth and continuous functions, i.e., analogous to the magnetic field given by usual MHD simulations. The magnetic configurations in every adjacent layers are similar and do not suffer from abrupt variations. This is just what we see from the observations of the chromosphere and the photosphere. This paper is arranged as follows: The basic equations and our asymptotic solution are presented in x 2. In x we investigate the peculiarity of and the compatibility condition. Our algorithm to get NFFF is described in x 4. Two examples of NFFF computations are given in x 5. One is for the analytical solution given by Low & Lou (990), and the other is based on the vector magnetogram of the active region NOAA 965. The conclusion and discussion are given in x BASIC EQUATIONS AND SOLUTIONS Under Cartesian coordinates the general NFFF equations, :< B ¼ B and :=B ¼ 0, can be written z z þ B z z B z y B z x where is a slowly varying function to be determined later. Now we suppose that a set of similarity solutions with secondorder continuous partial derivatives exists in a certain height range, 0 < z < H, and can be written in the form B x ¼ (x y z)f (x y z) B y ¼ 2 (x y z)f 2 (x y z) B z ¼ (x y z)f (x y z) where, 2,and mainly depend on z andslowlyvarywith (x y), while F F 2,andF mainly rely on x and y but weakly vary with z. Equation (5) is the mathematical representation of the similarity solutions. The similarity property is embodied by the slow varying of six /:::@F / in the z-direction. Due to a great variety of magnetic fields in active regions, it is impossible to seek an analytical solution for the magnetic field and in a volume within an active region (say, ¼ km km). However, we can construct analytical asymptotic solutions within a thin layer: and z k < z < z kþ. The expression is the extension of the active region: x < x < x 2, y < y < y 2,andz k ¼ kz,wherez ¼ H/K and k ¼ 2 :::K K. At first, we construct the solution in the bottom layer [0 < z < z, ztx and y x ¼ (x 2 x )/N, y ¼ ( y 2 y )/M ]. For instance, given N ¼ 80, M ¼ 60, x ¼, then z is taken to be The values at the bottom boundary (z ¼ 0) are given ðþ ð2þ ðþ ð4þ ð5þ by the vector magnetograph at N M sampling points: (B 0 x B0 y B 0 z ) i j, where i ¼ 0 : : :N and j ¼ 0 2: : :M. The values at the lateral surface boundary (x ¼ x x 2, and y ¼ y y 2 ) are artificially set to be tiny (e.g., 0 0 G, assuming the field maximum 500 G), since it does not disturb the main results. As for the values at the top boundary (z ¼ z), we do not need to set them exactly, because in an active region the natural tendency of the magnetic field extending upward into the corona is the decrease of its strength and its magnetic energy density. Although jb x j jb y j, and jb z j sometimes fall or rise at a single point, the total energy RR B 2 dxdy z for a layer surely decreases with height. The boundary values of and its limits are discussed in x. Here the boundary values of are simply fixed by solar observations. Substituting equation (5) into equations () (4) and considering that the amplitude functions,, 2, and, depend only on z in the vicinity of the sampling points (x i y j ), we can then from equations () (4) get d dz (x i y j z) (x i y j z) ¼ þ (x i y j z) 2 F 2 (x i y j z) d 2 dz (x i y j z) 2(x i y j z) ¼ (x i y j z) F (x i y j z) d dz (x i y j 2 (x i y j z) (x i y j z) ¼ 2 (x i y j (x i y j z) (x i y j z) F (x i y j z) ¼ 2 for 0 z z ð6þ where we /@ zj x¼xi y¼y j 2 /@ zj x¼xi y¼y j zj x¼xi y¼y j ¼ 0, because it is the requirement of the definition of amplitude functions. Now we start to seek the solutions of zerothorder, first-order, and second-order approaches progressively.. Zeroth-order approach. Due to the similarity, we suppose in the zeroth-order approximation that all the wave functions, F, F 2,andF, are fully independent of z, i.e., the six /:::@F /, only are functions of (x y). Thus, equation (6) can be reduced to a solvable ordinary differential equation with constant coefficients as d dz (x i y j 0) (x i y j 0) ¼ þ (x i y j 0) 2 F 2 (x i y j 0) d 2 dz (x i y j 0) 2(x i y j 0) ¼ (x i y j 0) F (x i y j 0) d dz (x i y j 2 (x i y j 0) (x i y j 0) ¼ 2 (x i y j (x i y j 0) (x i y j 0) F (x i y j 0) ¼ 2 for 0 z z: ð7þ Since (x i,y /(x i y j 0):::, etc. are known, we can immediately find the analytical solution of the coupled linear ordinary differential-equation system (7) (see, e.g., Miller & Michel 982). Note that the six /@F / :::@F / j xi y j 0, are computed accurately by three smooth continuous functions (see the Appendix). 2. First-order approach. In the first-order approximation we consider the slow variations of magnetic configuration at z ¼ z, i.e., the weak dependence on z of the six / :::@F /. Multiplying F F 2 F (x i y j 0) by the

3 086 SONG ET AL. Vol. 649 TABLE Zeroth- to Fourth-Order Approach Results Approach 2 (deg) Angle < B z > (G) B x (G) B y (G) B z (G) 0th-order st-order nd-order rd-order th-order solutions, 2, of equation (7), we obtain NM field values at the plane z ¼ z, then use double quadratic functions to approach these values (see the Appendix), and calculate the partial derivatives to get the new six wave / :::@F / and new. For new z / (x i y j z)/ :::, etc. Thus, we can write down the first-order approximations for these quantities () (x i y j z) (x i y j 0) " þ (z)(z) (x i y j (x i y j 0) () (x i y j z) ¼ (x i y j 0) þ (z)(z) () (x i y j z) (x i y j 0) for 0 z z ð8þ where seven () () () 2 () 2 () () /,and(), are expressed by linear functions of z. SincezT xy, such linear expressions are accurate enough. We then construct the first-order differential equations as follows: d dz () (x i y j 0) ¼ (x i y j z) þ () (x i y j z) 2 F 2 (x i y j 0) d 2 dz () 2(x i y j 0) ¼ (x i y j z) () (x i y j z) F (x i y j 0) d dz () (x i y j 0) ¼ (x i y j () 2 (x i y j z) 2 : ð9þ Equation (9) is a differential equation with varying coefficients whose solution can be found in the usual textbooks, or equivalently can be got numerically using the Runge-Kutta method with double-precision variables. Thus, we can obtain the firstorder approach,, 2,and, and multiplying by F, F 2,andF, respectively, yields the first-order approximation of the magnetic field at the plane z ¼ z.. Second-order approach. Using double quadratic functions to approach the magnetic fields given by the above first-order procedure and taking their derivatives, we obtain new wave functions and new. By writing analogous equations (8) and (9) again and solving these equations, one gets the second-order approximations for the magnetic field and at the plane z ¼ z. 4. Third-order approach: similar to the second-order one. Obviously, if we should repeat the above asymptotic procedure dozens or 00 times to get a better field approach, then this method has a little meaning. In fact, we found that the second-order approach can already yield a fine result:, 2,, and are almost unchanged. The same method can be used in other layers (k )z z kz, wherek 6¼ 0. Note that when z 6:0 (40,000 km), we should set the lateral and the top (z ¼ H ) boundary conditions to be a potential field configuration. Table gives the zeroth- to fourth-order approach results, computed in the layer 0 z z, corresponding to an arbitrarily selected point (i ¼ 8, j ¼ 40) located in the active region NOAA 965 on 2000 September 5. In the table, Angle means the angle in degrees between :<B and B,andB x, B y,andb z are in gauss.. PECULIARITY OF AND COMPATIBILITY CONDITION The first property of is that it characterizes the scale of magnetic field variation, since from equation (4), nh i o (B y ) iþ j (B y ) i j =2x nh i o (B x ) i jþ (B x ) i j =2y (B z ) i j 0: 0:5 L ¼ x y: ð0þ L That is, is of the order of L, the reciprocal of L,whenx is taken to be the scale length L. Therefore, the reasonable range of should be between L and L. Generally we do not restrict the values of, because if x decreases by 5 times, will then rise by 5 times. So, if there exists a strong current region with violent magnetic field variation, we then take more sampling points, making the field variation smoother. As a result, x falls while increases by the same factor. Therefore, we are able to deal with the current concentrated regions observed in the photosphere, as suggested by Démoulin et al. (997). Next, we study the effect of nonhomogeneous, i.e., when 9(x y z) 6¼ 0, on the magnetic field. The propagation equation of (x y z ¼ z 0 ) can be deduced from equations () (4): (B = :) ¼ 0. Thus, we are able to investigate the functions, B x B y B z,and, propagating x /@ z ¼ :::,@B y /@ z ¼ z /@ z ¼ :::,and@/@ z ¼ ::: z is analogous to t. Introducing a monochromatic p wave E ¼ E 0 exp I k x x þ k y y kz,where I ¼ ffiffiffiffiffiffi, kx and k y are the wavenumbers, k is the wave growth (or decay) rate along the z-axis, and E is one of the four quantities (B x, B y, B z, ). Inserting them into the wave equations (only keeping the first order of small quantities), :<B ¼ 0 B þ B 0 ðb =: Þ 0 þ ðb 0 =: Þ ¼ 0

4 No. 2, 2006 NONLINEAR FORCE-FREE FIELD IN SOLAR CORONA 087 where B 0 and 0 are initial quantities, and B and are small disturbed ones. We can then obtain a dispersion relation for the growth rate k: k B 0 2 z þ k k x B 0 x B 0 z þ k y B 0y B 0 z þ IB 2 0 z 0 z þ k 2 H 2 þ kh þ H 0 ¼ 0 where H 2, H,andH 0 are complicated functions of B 0, 9 0, k x, and k y.when9 0 0, it has the simplest form of the dispersion equation : 0 2 k x 2 k y 2 k 2 ¼ 0 (the case of Alissandrakis [98]), which leads at once to the growth rate k ¼Iðkx 2 þ ky 2 0 2Þ/2.If9 0 6¼ 0, there are four complex number roots for k: two growth modes and two decay modes. From our field computations, we often see that, 2,and vary between 0.97 and.02 (z ¼ 0:02, x ¼ ), where 0.97 means a decay mode and.02 means a growth mode. Generally,, 2,and cannot all be growth modes, but one is growth while the other two are decay modes, or one is decay while the other two are growth modes. Only in the area with strong magnetic field are these all decay modes. The expression 0 characterizes the action of plasma on the magnetic field. It plays the role of a medium, like nonuniform gas. The variability of 0 means that the magnetic field B 0 propagates through a nonuniform medium 0 : there is either absorption (decay) or amplification (growth). Of course, this is not a real amplification or absorption but the reorganization and adjustment of the magnetic field. However, the magnetic energies in the layers [ RR k z<z<(kþ) z (B2 /2) dx dy dz] are decreasing progressively upward from the bottom (z ¼ 0). From the properties of mentioned above, it is apparent that finding from the photospheric magnetograms should obey the following rules:. The expression j z¼z0 should be a slowly varying function of (x y). The inaccuracy of measured transverse fields leads to the uncertainty of the vertical y / x /,so that it is necessary to average these currents, say, h i < (B y ) iþ j (B y ) i j =2x > i j ¼ < (B z ) i j > h i < (B x ) i jþ (B x ) i j =2y > ðþ < (B z ) i j > where the symbol< > means the average of the quantities over the nearby nine sampling points (i, j ), (i, j), (i, j þ ), (i, j ), (i, j), (i, j þ ), (i þ, j ), (i þ, j), and (i þ, j þ ). 2. Suppose that we have taken enough sampling points NM to make B x, B y, and B z to be smooth continuous functions. Then the limitation of (L) <<þ(l) is physically a good choice for, so that for every -value greater than /L, we must set it to be /L or /L. In this way, the -distribution becomes physically and mathematically reasonable.. When (B z ) i j or < (B z ) i j > approaches zero, i.e., when crossing the magnetic polarity inversion line (B jj ¼ 0), there is a problem of how can be computed. It is suggested that we omit the contribution from the points with too low a B z value, as has been initiated by Démoulin et al. (992), i.e., admitting that is continuous when crossing the polarity inversion line (B jj ¼ 0). Next we study the compatibility conditions for the boundary values of B x, B y, B z, and at the bottom, z ¼ 0. Here we pose two categories of compatibility conditions:. One is for the boundary values provided by an analytical solutions, such as given by Low & Lou (990) or Chiu & Hilton (977). In this case the compatibility condition is x = z¼0 ð2þ B z j z¼0 which is just the same as equation (4). It can be seen that equation (2) is a very stringent condition: even in the case of B z! 0, it must be kept correctly. When the boundary values provided by Low & Lou (990) are used as a NFFF test computation example, we notice the fact that the values of B 0 x B0 y B0 z,and0 are selfconsistent. If the values are not self-consistent (such as observed magnetic fields), equation (2) must be invalid. 2. The other is for the boundary values given by photospheric field measurements. In this case, we suggest that the compatibility condition could be taken as the average of equation (4) over a small range around a sampling point (x i y j ): (x i y j z 0 ) < (B z )j z¼z0 > ði jþ y j z¼z0 > ði jþ x j z¼z0 > ði jþ ðþ where RR the symbol< > means the average, e.g., < (B z )j z¼z0 > ði j i j B z (x y z 0 ) dxdy. Thesmallarea i j is taken to be 2 x x i 2x, 2y y y j 2y. In equation () we admit that is a slowly varying function that marks the scale of field change in the area i j. It is clear that without the validity of the compatibility equation (), no NFFF solution exists. Finally, if the bottom boundary values are given by an analytical solution, we suggest that the working basic equations are equations (), (2), (), and (2). Or if the bottom boundary values are given by observed vector magnetograms, then we can use equations(),(2),(),and(). 4. ALGORITHM In comparison with our case, we first review the algorithm adopted in usual MHD simulations. In a two-dimensional timedependent MHD simulation, for instance, the case of Harned & Kerner (985) dealing with the propagation of magnetoacoustic waves, one starts with a set of self-consistent initial quantities, v x (z i, x j ), B x (z i, x j ), and B z (z i x j ), that satisfy the basic x /@t ¼ B x /@ z /@t ¼ B x x /@t ¼ B z0 ð@b x z /Þ, and B z0 ¼ const. There are three quantities in the initial state (t ¼ 0). Then one fixes the time step t according to the criterion of numerical stability and starts the time-advance computation going from the nth time step to the (n þ )th time step. At the first stage, predictor advance computation, vx nþ, Bx nþ, and Bz nþ are obtained in an explicit formula from vx n, B x n,andb z n. At the second stage, semi-implicit advance computation, vx nþ, Bx nþ,andbz nþ are found out in a mixed scheme including both (v x B x B z ) n and (v x B x B z ) nþ by using the tridiagonal linear algebraic method. And then, using artificial viscosity, we can correct the velocity v x (with larger viscosity) and magnetic fields B x and B z (with smaller viscosity). Thus, the (n þ )th time-step computation ends, and then the (n þ 2)th time step starts and so on. In this way we can obtain numerical solutions with time advancing. Backing to our height-advance NFFF computation, we start work with a set of self-consistent initial values (B 0 x B0 y B 0 z 0 ) i j that satisfy the basic equations (), (2), (), and () (or [2]). Þ ¼

5 088 SONG ET AL. Vol. 649 Then we fix z ¼ 0:02x and begin z-advance computation: going from the kz to the (k þ )z height step. At the first stage, explicit form computation, we finish the zeroth-order approach given in x 2. At the second stage, similar to the semiimplicit procedure, we accomplish the first- and second-order approximations given in x 2, until (B x B y B z ) kþ become stable. And then by using artificial viscosity, we correct (B x B y ) kþ (with small viscosity), Bz kþ (with large viscosity), and kþ. Thus, the (k þ )th height step is completed, and then we can go to the (k þ 2)th height step and so on. In this way the NFFF solutions can be obtained within a large height range. We now write down the algorithm as follows:. First, we get the self-consistent boundary values in the plane z ¼ 0. From the field boundary values, Bx 0, By 0,andBz 0,givenby vector magnetograms or by an analytical solution, using equation () (for the observed magnetic field) or equation (2) (for the analytical solution) we calculate the 0 values consistent with such field values. 2. Then we take z ¼ 0:02x and use the Runge-Kutta method (with double-precision variables) described in x 2 to get the zeroth-, first-, and second-order approaches.. Next, we correct the calculated field B(x i y j ) by using the following artificial viscosity formulae: ½(B x ) i j Š correct ¼ (B x ) i j (! ) þ! 4 (B x) i j þ (B x ) iþ j þ (B x ) i j þ (B x ) i jþ ½(B z ) i j Š correct ¼ (B z ) i j (! 2 ) þ! 2 4 (B z) i j þ (B z ) iþ j þ (B z ) i j þ (B z ) i jþ where! ¼ 0:,! 2 ¼ 0:2, and ½(B y ) i j Š correct is similar to ½(B x ) i jš correct above. 4. Then we repeat (), (2), and () for the height step 2z. 5. Finally, we finish the height step z, 4z, and so on in the same way. Note that, in solving equation (7) one demands that F, F 2, and F are not close to zero. In dealing with very low field values, see the Appendix. 5. RECONSTRUCTION OF CORONAL MAGNETIC FIELD AND TWO EXAMPLES OF NFFF Our first example is to compute a well-known NFFF analytical solution given by Low & Lou (990). The magnetic field is actually a two-dimensional force-free field (independent of azimuth ) that is shifted under two steps of Cartesian coordinate system transformation, rotating by an angle around the y-axis and moving the origin to a distance l along the z-axis. The NFFF solution is given as " B ¼ B 0 (r=r 0 ) dp d ˆr þ p ˆ þ (a # )p 2 ˆ ( 2 ) =2 ( 2 =2 ) where ¼ cos, r 0 ¼ 2, and B 0 ¼ 5. In the original global coordinate system, X ¼ r sin cos, Y ¼ r sin sin, Z ¼ r cos, and ¼ 2a r 0 p (r=r 0 ) ð4þ Fig.. Height variations of the NFFF configurations computed for the model given by Low & Lou (990). The component B z is shown by contours, while B x /BandB y /B are shown by vectors. (a) z ¼ 2:00, (b) z ¼ 0:00, (c) z ¼ 62:00. For comparison, the analytical solutions are also shown for (d ) z ¼ 2:00, (e) z ¼ 0:00, and ( f ) z ¼ 62:00. Solid and dashed contours in all figures refer to positive and negative B z in the photosphere at z ¼ 0, respectively. where p is a solution of the equation d 2 2 P() d 2 þ 2P() þ 2(a ) 2 ½P()Š ¼ 0 and ða Þ 2 ¼ 0:425. Then B r, B,andB are changed to B X, B Y, and B Z by the equations B X ¼ (B r sin þ B cos )cos B sin B Y ¼ (B r sin þ B cos )sin B cos B Z ¼ B r cos þ B sin and B X, B Y,andB Z are then transformed to B x, B y,andb z by X ¼ x cos ðz þ l Þsin Y ¼ y Z ¼ x sin ðz þ lþcos B x ¼ B X cos þ B Z sin B y ¼ B Y B z ¼ B X sin þ B Z cos ð5þ where ¼ 0:45 and l ¼ 0:r 0. The field given by equation (5) looks like that in a solar -type sunspot whose magnetic fluxes are concentrated in the central region and rapidly decay to zero at the rim of the domain (see Fig. ). Dividing the x or y effective domain into 64 intervals: x ¼ y ¼, 2 x 2, 2 y 2, it is easy to prove that, in the plane z ¼ 0,

6 No. 2, 2006 NONLINEAR FORCE-FREE FIELD IN SOLAR CORONA 089 TABLE 2 Comparison of the Extrapolation Quality between Different Models Models C vec C CS E n E m Low & Lou (990)... Wiegelmann (2004) Ours In order to rate the quality of our extrapolation, here we present six quantitative measures (Schrijver et al. 2006): P iðb i = b i Þ C vec ¼ hp P i =2 ij B ij 2 ij b ij 2 L d ¼ V C CS ¼ X (B i = b i ) (NM ) jb i i jjb i j ¼ X cos i (NM ) i P i E n ¼ jb i B i j P i jb ij E m ¼ X jb i B i j (NM ) B i i P ij J ¼ J i < B i j=b P i Z V i J i " # j:=bj max j@b x =j@b y = j =@ zj Fig. 2. Configurations of NFFF field lines computed for the model given by Low & Lou (990). The top panel is viewed from the viewpoint of ¼ 0, ¼ 225, l ¼ The bottom panel is viewed from the viewpoint of ¼ 0, ¼ 270, l ¼ [See the electronic edition of the Journal for a color version of this figure.] 0:289 0:27276, 622:595 G B x 76:8 G, 2296:65 G B y 099:889G, 80:96 G B z 45:6 G. Note that we take B 0 ¼ 5 in order to make the field more realistic, like that in solar active regions. The computation starts from the layer 0 z z z ¼ 0:02. At first the analytical functions, F (x, y), F 2 (x, y), and F (x y), and their partial derivatives are constructed at each network knot (x i y j ), according to the Appendix. The expression is taken directly from the compatibility equation (2) (not that coming from the analytical eq. [4]). The lateral and top boundary conditions are taken directly from the analytical solution itself instead of using a potential field (see Schrijver et al. 2006). Then the algorithm begins. About 200 time steps later we can reach the height z ¼ 64 (72,800 km above the photosphere). The results are given in Figures and 2. It can be seen that the computed configurations are very similar to those of the analytical solution (eq. [5]) in the box. where b i is the analytic magnetic field given by equation (5), J i is the current density, and N and M are the point numbers of the mesh used in our computation. For the perfect case, the first two quantities should be, while the others should be 0. Note that our definition of L d differs little from that of Schrijver et al. (2006), since numerically j:=bj is often not close to zero, while using j:=bj/ jbj to judge the divergence-free condition (:=B ¼ 0) is physically not correct. For comparison of the extrapolation quality with other methods (Schrijver et al. 2006), see Table 2. The measures J and L d are 2:9 0 and :56 0, respectively. From Table 2, we believe that our extrapolation can be compared with the computation given by Wiegelmann (2004). The second test example is to compute NFFF based on three components Bx 0, B y 0, and B z 0 given by the vector magnetogram of the active region NOAA 965 on 2000 September 5. Figure shows the observed vector magnetogram and the boundary values of magnetic field on the (x y) mesh that we used in our computation. First we construct the analytical functions F, F 2, and F and their derivatives in the same way as in the above example, but the difference is that we should deal carefully with the analytical representation of the vertical current density (:< B) z y x /, since equation (4) connects B z to (B x B y ), the longitudinal and transverse components of magnetic field, which gives self-consistent boundary conditions for, B x, B y, and B z. Then the algorithm begins. As before, layer by layer, computing until 4000 time steps later, we reach the height z ¼ 80:0(26,000 km above the photosphere). The results are shown in Figure 4. Since the field configurations of adjacent layers are similar, the features only vary gradually. So we have confidence that they represent a real NFFF. Figure 5 depicts the configuration of field lines of the computed NFFF result. Note that some field lines from a negative polarity seem to end at one of the same

7 090 SONG ET AL. Vol. 649 Fig.. Vector magnetogram (left) and the boundary values (right) of the magnetic field on the (x y) mesh. Solid and dashed contours in all figures refer to positive and negative B z, respectively, in the photosphere at z ¼ 0. polarity. However, they connect the positive polarity regions. As a comparison, in Figure 6 we also give the field line configurations of the LFFF ( ¼ 0:, x ¼ ) given the integration of Chiu & Hilton (977). One can see the difference and similarity between the NFFF and LFFF. From Figure 5 it can be seen that, when z 2, there exists high shearing along the main inversion line and spiral configuration above the negative polarity sunspot, and a new magnetic region has been created where some solar flares took place later. 6. CONCLUSION AND DISCUSSION We propose a new and fast way to extrapolate and reconstruct a NFFF in the solar corona. We use smooth continuous functions to approach the magnetic fields, write down three field components consisting of amplitude functions multiplying morphology Fig. 4. Height variations of the NFFF configurations computed by use of the vector magnetogram of the active region NOAA 965 on 2000 September 5. The component B z is shown by contours, while B x /B and B y /B are shown by vectors. (a) z ¼ 0:0, (b) z ¼ 4:0, (c) z ¼ 8:0, (d) z ¼ 20:0, (e) z ¼ 58:0, and ( f ) z ¼ 88:0. Fig. 5. Configurations of the NFFF lines computed by use of the vector magnetogram of NOAA 965. The top panel is viewed from the viewpoint of ¼ 0, ¼ 225, l ¼ The bottom panel is viewed from the viewpoint of ¼ 0, ¼ 270, l ¼ The footpoints of the magnetic field lines are all the same as in Fig. 5.

8 No. 2, 2006 NONLINEAR FORCE-FREE FIELD IN SOLAR CORONA 09 Fig. 6. Configurations of LFFF lines computed by the integration method of Chiu & Hilton (977) for the vector magnetogram of NOAA 965. The top panel is viewed from the viewpoint of ¼ 0, ¼ 225, l ¼ The bottom panel is viewed from the viewpoint of ¼ 0, ¼ 270, l ¼ functions, and reduce four basic NFFF equations to ordinary differential ones. They are then solved in an asymptotic manner (zeroth-order, first-order, etc.). Considering the physical meaning of, we found a self-consistent compatibility condition for field boundary values. Furthermore, a computation algorithm is constructed, similar to the usual time-dependent two-dimensional MHD simulation scheme. Our computations of the NFFF configuration for two examples indicate that this algorithm is steady and robust against the noise in the magnetic field (in particular the transverse field) measurements and is able to deal with concentrated photospheric currents. Démoulin et al. (997) suggested that the computation of a solar NFFF should satisfy five criteria: () observed magnetic fields as the boundary condition (2) the ability to handle the complex topology and being versatile from one configuration to another () use of the observed concentrated current (4) insensitivity to the noise in the data, in particular, for the transverse field and (5) fast enough in order to study the time evolution of an active region, as well as a large number of cases. Our algorithm for NFFF basically satisfies all the criteria, is simple, and runs very fast. In an ordinary PC, for the (x y) mesh, the computation up to a height of 80 (=26,000 km) lasts only about 6 minutes. Due to the short CPU time, our algorithm can be used to investigate the time variation of the magnetic fields in active regions. As for the previous upward integration method, Wu et al. (990) indicated that () the accuracy deteriorates when the solution is close to the boundary and (2) the uniqueness of the solution has yet to be established. In our method, by using smooth functions and transforming the original equations to a set of ordinary differential ones, the uniqueness of the problem seems to be solved in a local small region (x i x < x < x i þ x, y j y < y < y j þ y, z 0 < z < z 0 þ z), so that, in a thin box, all six boundary conditions can determine the solution uniquely. The first problem indicated by Wu et al. does not exist in our present code. Moreover, it is noted that Démoulin et al. (992) indicated that no further improvement has been obtained with other types of smoothing functions F s. While we may point out that the usual discrete formula for describing partial differential derivatives must lead to some growing modes in a NFFF computation, so that it is impossible to obtain a good function F s.to avoid this problem we have transformed the partial differential equations into some ordinary differential ones. If anybody is interested in our code, please contact C. F. This work was supported by the National Natural Science Foundation of China ( NSFC) under the grant numbers 000, 02702, 02200, 0040, and 04000, as well as by the National Basic Research Priorities Project (2006CB0600) of China and the Chinese Academy of Sciences. This work was also supported by NASA grant NAG5-284 and NSF grant ATM We thank the anonymous referee very much for the valuable comments and suggestions. We use the double quadratic function defined in the region : APPENDIX : x i x x x i þ x to approach field components F, F 2,andF in the form y j y y y j þ y 8 4 ( þ M)( þ M ) M ¼ 5 7 >< L M ( ) ¼ 2 ( 2 )( þ M ) M ¼ ( þ M)( 2 ) M ¼ 4 8 >: ( 2 )( 2 ) M ¼ 9

9 092 SONG ET AL. where P ¼ (x x i )/x and ¼ ( y y j )/y. Then for any continuous point (x, y), B x, B y,andb z canbeexpressedbyasum 9 M¼ L M( M M )(B M ), where ( M M ) includes nine sampling points, i.e., ( )j M¼,(0 )j M¼2,( )j M¼,( 0)j M¼4, ( )j M¼5,(0 )j M¼6,( )j M¼7,( 0)j M¼8,and(0 0)j M¼9, corresponding to the points (x i, y j ), (x i, y j ), (x iþ, y j ), (x iþ, y j ), (x iþ, y jþ ), (x i, y jþ ), (x i, y jþ ), (x i, y j ), and (x i y j ), respectively. The expression (B M ) is the field value at one of the nine points. While their partial derivatives (@F /@F / :::) can be calculated analytically or computed numerically, / ½F (x þ x y) F (x x y) Š/x, withx ¼ y ¼ z ¼ 0 to 0,wheny x ¼. It is demanded in x 2thatF, F 2,andF are not very close to zero. For dealing with F and F 2, it is easy to avoid this defect by using the formulae ( F ¼ B x(x i y j ) B x (x i y j ) Bxmin B xmin sgn B x (x i y j ) Bx (x i y j ) < Bxmin ( F 2 ¼ B y(x i y j ) B y (x i y j ) Bymin B ymin sgn B y (x i y j ) By (x i y j ) < Bymin where B xmin and B ymin are taken to be B limit ¼½ðB 2 x þ B 2 y þ B 2 z ÞmaxŠ /2 /6600. For instance, supposing the maximum field strength be 500 G in the active regions, one can take B limit ¼ 2orGatz 0: when going to z 75:0, B limit decreases to 0 4 G. For dealing with F, we follow the suggestion given by Démoulin et al. (997), i.e., the contribution of such a point ( x i ȳ j ), where F ( x i ȳ j ) 0, is omitted. That is, considering the continuity of B z, we take the average of four adjacent field values [say, B z ( x i þ x/2 ȳ j ) B z ( x i x/2 ȳ j ) B z ( x i ȳ j þ y/2) and B z ( x i ȳ j y/2)]. Alissandrakis, C. E. 98, A&A, 00, 97 Aly, J. J. 989, Sol. Phys., 20, 9 Amari, T., Aly, J. J., Luciani, J. F., Boulmezaoud, T. Z., & Mikic, Z. 997, Sol. Phys., 74, 29 Berger, M. A. 984, Geophys. Astrophys. Fluid Dyn., 0, 79 Chiu, Y. T., & Hilton, H. A. 977, ApJ, 22, 87 Chodura, R., & Schluter, A. 98, J. Comp. Phys., 4, 68 Démoulin, P., Cuperman, S., & Semel, M. 992, A&A, 26, 5 Démoulin, P., Henoux, J.-C., Mandrini, C. H., & Priest, E. R. 997, Sol. Phys., 74, 7 Grad, H., & Rubin, H. 958, in Proc. 2nd Int. Conf. On Peaceful Uses of Atomic Energy,, 90 Harned, D. S., & Kerner, W. 985, J. Comp. Phys., 60, 62 Judge, P. G., Tomczyk, S., Livingston, W. C., Keller, C. U., & Penn, M. J. 2002, ApJ, 576, L57 REFERENCES Low, B. C., & Lou, Y. Q. 990, ApJ, 52, 4 McClymont, A. N., & Mikic, Z. 994, ApJ, 422, 899 Miller, R. K., & Michel, A. N. 982, Ordinary Differential Equations ( New York: Academic Press) Roumeliotis, G. 996, ApJ, 47, 095 Sakurai, T. 98, Sol. Phys., 69, 4 Schrijver, C. J., et al. 2006, Sol. Phys., 25, 6 Wheatland, M. S., Sturrock, P. A., & Roumeliotis, G. 2000, ApJ, 540, 50 Wiegelmann, T. 2004, Sol. Phys., 29, 87 Wiegelmann, T., Inhester, B., & Sakurai, T. 2006, Sol. Phys., 2, 25 Wu, S. T., Sun, M. T., Chang, H. M., Hagyard, M. J., & Gary, G. A. 990, ApJ, 62, 698 Yan, Y. H., & Li, Z. H. 2006, ApJ, 68, 62