Nonlinear force-free modeling of the solar coronal magnetic field

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    Nonlinear force-free modeling of the solar coronal magnetic fieldT. WiegelmannMax-Planck-Institut fur Sonnensystemforschung, Max-Planck-Strasse 2, 37191 Katlenburg-Lindau, Germany

    Abstract. The coronal magnetic field is an important quantity because the magneticfield dominates the structure of the solar corona. Unfortunately direct measurements ofcoronal magnetic fields are usually not available. The photospheric magnetic field is mea-sured routinely with vector magnetographs. These photospheric measurements are ex-trapolated into the solar corona. The extrapolated coronal magnetic field depends on as-sumptions regarding the coronal plasma, e.g. force-freeness. Force-free means that all non-magnetic forces like pressure gradients and gravity are neglected. This approach is welljustified in the solar corona due to the low plasma beta. One has to take care, however,about ambiguities, noise and non-magnetic forces in the photosphere, where the mag-netic field vector is measured. Here we review different numerical methods for a nonlin-ear force-free coronal magnetic field extrapolation: Grad-Rubin codes, upward integra-tion method, MHD-relaxation, optimization and the boundary element approach. We brieflydiscuss the main features of the different methods and concentrate mainly on recentlydeveloped new codes.


    1. Introduction

    1.1. Why do we need nonlinear force-free fields?

    2. Nonlinear force-free codes

    2.1. Grad-Rubin method

    2.2. Upward integration method

    2.3. MHD relaxation

    2.4. Optimization approach

    2.5. Boundary element or Greens function likemethod

    3. How to deal with non-force-free boundaries andnoise ?

    3.1. Consistency check of vector magnetograms

    3.2. Preprocessing

    4. Code testing and code comparisons

    4.1. The NLFFF-consortium

    5. Conclusions and Outlook

    1. IntroductionInformation regarding the coronal magnetic field are im-portant for space weather application like the onset offlares and coronal mass ejections (CMEs). Unfortunatelywe usually cannot measure the coronal magnetic field di-rectly, although recently some progress has been made[see e.g., Judge, 1998; Solanki et al., 2003; Lin et al.,2004]. Due to the optically thin coronal plasma directmeasurements of the coronal magnetic field have a line-of-sight integrated character and to derive the accurate3D structure of the coronal magnetic field a vector to-mographic inversion is required. Corresponding feasibil-ity studies based on coronal Zeeman and Hanle effect

    Copyright 2008 by the American Geophysical Union.0148-0227/08/$9.00






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    measurements have recently been done by Kramar et al.[2006] and Kramar and Inhester [2006]. These directmeasurements are only available for a few individual casesand usually one has to extrapolate the coronal magneticfield from photospheric magnetic measurements. To doso, one has to make assumptions regarding the coronalplasma. It is helpful that the low solar corona is stronglydominated by the coronal magnetic field and the mag-netic pressure is orders of magnitudes higher than theplasma pressure. The quotient of plasma pressure p andmagnetic pressure, B2/(20) is small compared to unity( = 20p/B

    2 1). In lowest order non-magnetic forceslike pressure gradient and gravity can be neglected whichleads to the force-free assumption. Force free fields arecharacterized by the equations:

    jB = 0 (1)B = 0j, (2) B = 0, (3)

    where B is the magnetic field, j the electric current den-sity and 0 the permeability of vacuum. Equation (1)implies that for force-free fields the current density andthe magnetic field are parallel, i.e.

    0j = B, (4)

    or by replacing j with Eq. (2)

    B = B, (5)

    where is called the force-free function. To get someinsights in the structure of the space dependent function, we take the divergence of Eq. (4) and make use ofEqs. (2) and (3):

    B = 0, (6)

    which tells us that the force-free function is con-stant on every field line, but will usually change from onefield line to another. This generic case is called nonlinearforce-free approach.

    Popular simplifications are = 0 (current free poten-tial fields, see e.g., Schmidt [1964]; Semel [1967]; Schat-ten et al. [1969]; Sakurai [1982]) and = constant (linearforce-free approach, see e.g., Nakagawa and Raadu [1972];Chiu and Hilton [1977]; Seehafer [1978]; Alissandrakis[1981]; Seehafer [1982]; Semel [1988]). These simplifiedmodels have been in particular popular due to their rel-ative mathematical simplicity and because only line-of-sight photospheric magnetic field measurements are re-quired. Linear force-free fields still contain one free globalparameter , which can be derived by comparing coro-nal images with projections of magnetic field lines (e.g.,Carcedo et al. [2003]). It is also possible to derive an av-eraged value of from transverse photospheric magneticfield measurements (e.g. Pevtsov et al. [1994]; Wheatland[1999]; Leka and Skumanich [1999]). Despite the popu-larity and frequent use of these simplified models in thepast, there are several limitations in these models (seebelow) which ask for considering the more sophisticatednonlinear force-free approach.

    Our aim is to review recent developments of the ex-trapolation of nonlinear force-free fields (NLFFF). Forearlier reviews on force-free fields we refer to [Sakurai ,


    1989; Aly , 1989; Amari et al., 1997; McClymont et al.,1997] and chapter 5 of Aschwanden [2005]. Here wewill concentrate mainly on new developments which tookplace after these earlier reviews. Our main emphasis is tostudy methods which extrapolate the coronal magneticfield from photospheric vector magnetograms. Severalvector magnetographs are currently operating or planedfor the nearest future, e.g., ground based: the solarflare telescope/NAOJ [Sakurai et al., 1995], the imagingvector magnetograph/MEES Observatory [Mickey et al.,1996], Big Bear Solar Observatory, Infrared PolarimeterVTT, SOLIS/NSO [Henney et al., 2006] and space born:Hinode/SOT [Shimizu, 2004], SDO/HMI [Borrero et al.,2006]. Measurements from these vector magnetogramswill provide us eventually with the magnetic field vec-tor on the photosphere, say Bz0 for the normal and Bx0and By0 for the transverse field. Deriving these quanti-ties from the measurements is an involved physical pro-cess, which includes measurements based on the Zeemanand Hanle effect, the inversion of Stokes profiles (e.g.,LaBonte et al. [1999]) and removing the 180 ambiguity(e.g., Metcalf [1994]; Metcalf et al. [2006]) of the hori-zontal magnetic field component. Special care has to betaken for vector magnetograph measurements which arenot close to the solar disk, when the line-of-sight and nor-mal magnetic field component are far apart (e.g. Garyand Hagyard [1990]). For the purpose of this paper wedo not address the observational methods and recent de-velopments and problems related to deriving the photo-spheric magnetic field vector. We rather will concentrateon how to use the photospheric Bx0, By0 and Bz0 to de-rive the coronal magnetic field.

    The transverse photospheric magnetic field (Bx0, By0)can be used to approximate the normal electric currentdistribution by

    0jz0 =By0x



    and from this one gets the distribution of on thephotosphere by

    (x, y) = 0jz0Bz0


    By using Eq. (8) one has to keep in mind that ratherlarge uncertainties in the transverse field component andnumerical derivations used in (7) can cumulate in signifi-cant errors for the current density. The problem becomeseven more severe by using (8) to compute in regionswith a low normal magnetic field strength Bz0. Specialcare has to be taken at photospheric polarity inversionlines, i.e. lines along which Bz0 = 0 [see e.g., Cupermanet al., 1991]. The nonlinear force-free coronal magneticfield extrapolation is a boundary value problem. As wewill see later some of the NLFFF-codes make use of (8)to specify the boundary conditions while other methodsuse the photospheric magnetic field vector more directlyto extrapolate the field into the corona.

    Pure mathematical investigations of the nonlinearforce-free equations [see e.g. Aly , 1984; Boulmezaoud andAmari , 2000; Aly , 2005] and modelling approaches notbased on vector magnetograms are important and occa-sionally mentioned in this paper. A detailed review ofthese topics is well outside the scope of this paper, how-ever. Some of the model-approaches not based on vectormagnetograms are occasionally used to test the nonlinearforce-free extrapolation codes described here.


    1.1. Why do we need nonlinear force-free fields?

    A comparison of global potential magnetic fieldmodels with TRACE-images by Schrijver et al. [2005] re-vealed that significant nonpotentially occurs regulary inactive regions, in particular when new flux has emergedin or close to the regions.

    Usually changes in space, even inside one activeregion. This can be seen, if we try to fit for the op-timal linear force-free parameter by comparing fieldlines with coronal plasma structures. An example can beseen in Wiegelmann and Neukirch [2002] where stereo-scopic reconstructed loops by Aschwanden et al. [1999]have been compared with a linear force-free field model.The optimal value of changes even sign within the in-vestigated active regions, which is a contradiction to the = constant linear force-free approach (see Fig. 1).

    Photospheric distributions derived from vectormagnetic field measurements