Coronal Magnetic Field Extrapolations

Transcription

1 3 rd SOLAIRE School Solar Observational Data Analysis (SODAS) Coronal Magnetic Field Extrapolations Stéphane RÉGNIER University of St Andrews

2 What I will focus on Magnetic field extrapolation of active regions Potential and nlff magnetic extrapolations Cartesian coordinates Full disc potential extrapolations How to analyse magnetic configurations What I will not talk about Quiet-Sun magnetic extrapolations Synoptic maps and cycle variations Ground-based observatories Linear force-free extrapolations (just briefly) Non force-free magnetic fields Rafael Nadal

3 Photospheric magnetic field as boundary conditions Line-of-sight magnetic fields Vector magnetic fields in Cartesian coordinates

4 Coronal equilibrium Three characteristic times are important: eq time of equilibrium, A Alfvén transit time, rec reconnection time. Typically, the Alfvén transit time is several minutes and the reconnection time is few seconds. The equilibrium can relax to a minimum energy state (Woltjer 1958) Time evolution can be describe by a series of linear force-free equilibria (Heyvaerts and Priest 1984) The equilibrium will more likely relax to a nonlinear force-free field We have shown that we can successfully describe the time evolution of an active region with a time series of nonlinear force-free equilibria with a time interval of 15 min (Régnier and Canfield 2006)

5 Why to study reconstruction methods? No 3D measurements of the magnetic field in the corona; Access only to the magnetic field components (line-of-sight and vector) on the photosphere and/or in the chromosphere; Understanding the magnetic field in the corona gives inputs for flare and CME models (e.g., storage of energy, site of magnetic reconnection)

6 Why to focus on nonlinear force-free fields? Next stage after potential and linear force-free fields, and which contain more information on the complex nature of solar magnetic fields; The corona (< 2.5 R ) has been thought to be force-free (Gold & Hoyle 1960); the scale-height of plasma pressure and gravity are large compared to the magnetic field (plasma <1) in the corona; Dissipation of Hall and Pedersen currents in the chromosphere; only parallel currents remain in the corona; Recent numerical simulations of flux emergence including partial ionization (Leake and Arber 2006) have shown that the final state of the coronal magnetic field is force-free.

7 Force-free fields We consider that the corona is dominated by the magnetic field (lowcoronal magnetic field is assumed to be force-free. plasma) and therefore the Magneto-hydrostatic equilibrium Force-free equilibrium j p B g 0 j B 0 j 0 j B B j ( r ) B 0 Potential Field Linear Forcefree Field Nonlinear Forcefree Field

8 Properties of force-free magnetic fields Necessary integral properties of a force-free field (not sufficient) from Molodensky (1969) and Aly (1989) No magnetic force No magnetic torque Molodensky (1969), Aly (1989)

9 Potential and lff magnetic fields 1. Equation and boundary conditions 2. Fourier transform: Gary s method 3. PFSS 2.1 Equations 2.2 Applications 4. Example of potential field study

10 1. Equation and boundary conditions Potential field model Boundary conditions B z from observations on the bottom boundary; On the sides and top: open, periodic, closed, mixed Potential magnetic energy - Minimum energy state - Total magnetic flux ( 2 ) - Distribution of charges

11 2. Fourier transform: Gary s method Fourier transform of magnetic components Boundary condition on the photosphere Fourier equations on the photosphere (same at z) Condition to get non trivial solutions Important limit 2 L where L is the characteristic length of the box

12 3. PFSS: Potential Field Source Surface Potential in spherical harmonics where Corresponding magnetic field components

13 3. PFSS: Potential Field Source Surface Boudary conditions Synoptic map of the radial component of the magnetic field on the photosphere (full sphere) The magnetic field is radial at the source surface

14 4. Example of potential field study

15 Nonlinear force-free fields 1. Equations and boundary conditions 2. Methods 2.1 Existing methods 2.2 Optimization vs Grad-Rubin 2.3 More comparison 3. Applications to the corona

16 1. Equations and boundary conditions Some analytical and semi-analytical solutions analytical solutions in cylindrical, spherical and toroidal coordinates have been derived (e.g. Chandrasekhar 1956, Gold-Hoyle 1960, Buck 1965, Low 1973, Titov and Démoulin 1999, Török et al. 2004); applications to thin twisted flux tubes in the corona and to magnetic clouds have been performed with the Gold & Hoyle solutions; well-known semi-analytical solutions were derived by Low & Lou (1990); these solutions have been used to test nonlinear force-free reconstruction techniques in Cartesian coordinates.

17 2.1 NLFFF methods Vertical integration method (Wu et al. 1990, Démoulin et al. 1992, Song et al. 2006) Grad & Rubin method (Grad and Rubin 1958, Sakurai 1981, Aly 1988, Amari et al. 1997, 1999, Wheatland 2004, Amari et al. 2006, Inhester and Wiegelmann 2006, ) Optimization method (Pridmore-Brown et al. 1981, Wheatland et al. 2000, Wiegelmann et al. 2003) Evolutionary techniques (Mikic and McClymont 1994) Stress-and-Relax technique (Roumeliotis 1996) Boundary element method (Yan and Sakurai 2000, Li et al. 2004) Magneto-frictional method (Yang et al. 1986, van Ballegooijen et al. 2000, Mackay et al. 2000, 2001, Valori et al. 2005) Finite element method (Amari et al. 2006) ( )

18 2.1 NLFFF methods

19 2.2 Optimization vs Grad-Rubin Problem To find the magnetic configuration the closest to a nonlinear force-free and divergence free magnetic field which matches all components of the magnetic field at the boundaries. This is an ill-posed boundary value problem Minimization of a functional L: Following Wheatland et al. (2000): where and The iterative process starts from a potential field, then solves the above equation, and updates the new magnetic field configuration following: and increases t until the functional L is minimized.

20 2.2 Optimization vs Grad-Rubin Problem To find the nonlinear force-free field associated with those boundary conditions corresponding to a well-posed boundary value problem The Grad-Rubin (1958) scheme separates the nonlinear equations into two linear systems of equations (Aly 1988, Amari et al. 1999): Transport of along field lines (hyperbolic): Updating the 3d magnetic field (elliptic): Properties Existence and uniqueness of solution for small (Bineau 1972) in simply connected domains; extended to multiple connected domains (Boulmezaoud and Amari 2000); Sakurai (1981) and Wheatland (2004) schemes use the magnetic field (on the left) Amari et al. (1997, 1999), Inhester and Wiegelmann (2006) and Amari et al. (2006) use a scheme based on the vector potential to preserve div.b;

21 2.3 More comparison In Schrijver et al. (2006) and Amari et al. (2006), a quantitative comparison of different numerical nonlinear force-free fields was performed, based on the Low & Lou solutions (1990); The different methods performed well in strong field regions; The methods are compared in terms of the speed of computation, and of the accuracy with respect to the semianalytical model; The authors gave a ranking of methods as shown in the image on the right.

22 2.3 More comparison

23 2.3 More comparison DeRosa et al. (2009) Comparison of the integrated current density along the vertical dimension for all nlfff models

24 3. Applications to the corona Twisted flux bundles like sigmoids and filaments (Yan et al. 2000, Liu et al. 2002, Régnier et al. 2002, Régnier et al. 2004) Nature of filaments associated with eruptions (Yan et al. 2000) Evidence of twisted flux bundles supporting dense plasma, and not in quadrupolar-like magnetic dips (Régnier & Amari, 2004) Coronal loops (Yan et al. 2000, Bleybel et al. 2002, Liu et al. 2002, Wiegelmann et al. 2005, Régnier and Canfield 2006) black: observed loop, white: potential loop, orange: linear force-free field, yellow: nonlinear force-free field Wiegelmann et al. (2005)

25 3. Applications to the corona H features during a flare From Mikic & McClymont (1994), the field lines computed using the nonlinear forcefree field match the H features during a M-class flares, while the potential field lines do not match. Magnetic field and radio sources From Lee et al. (1998), good correlation between the reconstructed field (evolutionary method) and the sources of radio (microwaves) gyro-resonant frequencies at 4.9 GHz and at 8.4 GHz; Radio peaks are C1-C2 at 4.9 GHz, and X1-X2 at 8.4 GHz (squares: 430 G, crosses: 750 G); Sunspots are marked with a S letter.

26 How to analyse magnetic fields? 1. Characteristic parameters 2. Advanced parameters 1.1 PIL, Magnetic energy 1.2 Current density, CIL 3. 3D magnetic configurations 2.1 Magnetic Energy 2.2 Magnetic Helicity 2.3 Geometry of field lines 2.4 Magnetic Topology 2.5 Time series

27 2.1 Magnetic Energy Several quantities are of interest to understand the storage and release of energy: the total magnetic energy of a magnetic configuration: the free magnetic energy budget: the magnetic energy density along the vertical axis (see graphs on the left) the time evolution of those quantities Change of the magnetic energy and helicity in linear force-free fields AR8151: decaying active region with high current density exhibiting highly twisted flux bundles

28 2.2 Magnetic Helicity Relative magnetic helicity (Finn & Antonsen 1985, Berger & Field 1984): where A is the vector potential associated to B, and the index 0 corresponds to a reference field (usually taken to be the potential field). If then For this particular active region, the mutual helicity dominates due to the complex topology of the field and no evidence of twisted flux bundles (Régnier et al. 2005)

29 2.2 Magnetic Helicity The minimum energy state of a nlff fields with helicity Hm is the linear force-free field with the same helicity (Taylor s relaxation theory) Δx AR8151 AR 8151: old decaying active region with highly twisted bundles AR 8210: newly emerged active region with complex topology (C-flares) AR 9077: Bastille day flare 2000, post-flare phase AR 10486: Halloween event 2003 on 28th OCt., before X17 flare erg erg Δ x Δ x erg Δ x 32 erg AR8210 AR9077 AR0486 Free Magnetic Energy in Solar Active Regions above the Minimum-Energy Relaxed State Régnier, S., Priest, E. R. 2007, ApJ, 669, L53 Δ: Nonlinear force-free energy x: Linear force-free energy

30 2.3 Geometry of field lines Potential vs. NLFFF changing connectivity shear (aligned with PIL) twist (e.g., highly twisted bundles) E Potential Field Linear Force-free Field = Mm -1 Linear Force-free Field = Mm -1 Nonlinear Forcefree Field E E E E E

31 2.4 Magnetic Topology Magnetic topology: discontinuity of the mapping of magnetic field configurations In 2D (Parnell et al. 1996, Phys. Plas.): Topological elements: null points, separatricies Types of null points: X-points, O-points In 3D (Parnell et al. 1996): Topological elements: null points, separatrix surfaces, separators Types of null points: plenty of, e.g., prone and upright photospheric nulls Skeleton Definition: to find the skeleton corresponds to find all the topological elements in a given magnetic configuration, including null points, separatrix surfaces, spine field lines, separators. Basics of reconnection: In 2D, magnetic reconnection occurs at X-points. In 3D, magnetic reconnection can occur at null points, along separatrix surfaces and along separators where strong electric currents and current sheets can be created; and in absence of null points (e.g. at quasi separatrix layers)

32 2.4 Magnetic Topology We compare the topology of a magnetic configuration with a coronal null points using the same boundary conditions (left) potential field (centre) linear force-free fields with values between -1 and 1 Mm -1 (right) nonlinear force-free fields for a ring distribution of J z

33 2.5 Time series Method describing the time evolution of an active region as successive nonlinear force-free equilibria assuming that the evolution is slow enough (t > n.t A ) Moving Magnetic Features Emergence of a parasitic polarity in a pre-existing magnetic configuration having a broken fan-like topology. Flare scenario: reconnection process along separatrix surface, due to (apparent) horizontal motions of a parasitic polarity leading to small-scale re-organization of the magnetic field (Régnier & Canfield 2006) Pre-existing magnetic configuration NE Emerging and fast moving parasitic polarity SW Separatrix