G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Complexity & non-potentiality of the solar corona G. Aulanier ( Observatoire de.

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Complexity & non-potentiality of the solar corona G. Aulanier ( Observatoire de Meudon, LESIA )

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Overview of this tutorial 1. Basic physics 2. Learning from 3D MHD simulations 3. Magnetic field extrapolations 4. Toward SDO & other future missions 1.1 Introduction 1.2 Non-potentiality 1.3 Complexity 2.1 Current sheets & reconnection in quasi-separatrices 2.2 Breakout model of an observed eruption 3.1 Motivation 3.2 Potential & linear force-free fields 3.3 Non-linear force-free fields 1. Basic physics 1.1 Introduction 1.2 Non-potentiality 1.3 Complexity 2. Learning from 3D MHD simulations 2.1 Current sheets & reconnection in quasi-separatrices 2.2 Breakout model of an observed eruption 3. Magnetic field extrapolations 3.1 Motivation 3.2 Potential & linear force-free fields 3.3 Non-linear force-free fields 4. Toward SDO & other future missions

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Complexity & non-potentiality Yohkoh SXT, SXR 11:48 UT TRACE, FeXI 171A July 14 1998, 12:05 UT – 14:00 UT At the origin of all solar flares & eruptions Among the major goals of all upcoming solar instruments

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Magnetic energy : storage & release  Magnetically driven activity Corona :  ~ E Th / E B ~ 2  P / B² < 1  Long-duration energy storage phase a few days (flares) to a few weeks (prominence eruptions)  Sudden energy release & triggering of active phenomenon Alfvénic timescales ~ a few minutes

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Overview of this tutorial 1. Basic physics 1.1 Introduction 1.2 Non-potentiality 1.3 Complexity 2. Learning from 3D MHD simulations 2.1 Current sheets & reconnection in quasi-separatrices 2.2 Breakout model of an observed eruption 3. Magnetic field extrapolations 3.1 Motivation 3.2 Potential & linear force-free fields 3.3 Non-linear force-free fields 4. Toward SDO & other future missions

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Pre-eruptive B : force-free fields  Conservation of momentum : d t (  u )= 0  d t u = – (u.  ) u + (  ) –1 (  x B) x B +  P +  g t A ²/t² = u²/c A ² + 1 +  +  L / H P  Slow evolution : t ~ days >> t A ~ minutes Photospheric velocities : u ~ 0.1 km/s << c A ~ 1000 km/s « Cold » plasma :  = 0.0001 – 0.1 << 1 Loop sizes : L~ 10 – 100 Mm ~ H p ~ 50 Mm  J x B = 0 &  x B =  J  Field-aligned currents :  x B =  B

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Force-free fields : three classes  Potential fields :  = 0  Linear force-free fields :  = cst  Non-linear force-free fields :  = varying   x (  x B =  B )   ² B +  ²  B = 0  Helmoltz equation has analytical solutions   x B = 0  B =    B defined by a scalar potential    (  x B =  B )  ( B   )  = 0  A field line is defined by its  value

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Non potentiality : free magnetic energy  Potential field :  x B 0 = 0 ;   B 0 = 0 ; B 0 =    E B0 = III ½ B 0 ² dV  E B = III ½ B 0 ² dV + III ½ B 1 ² dV + III B 0  B 1 dV = E B0 + E B1 + III    B 1 dV = E B0 + E B1 + III   (  B 1 ) dV = E B0 + E B1 + II  B 1  dS = E B0 + E B1 > E B0  Same as Kelvin’s theorem for incompressible fluids  Potential field = lower bound of energy  Non potential field : B = B 0 + B 1 ;   B 1 = 0 ; II B 1  dS = 0

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA How to store energy in the corona  Wavelengths L of coronal waves with C = C A ~ cst :  Energy burst during dt : L ~ C A dt ~ 10 Mm (for C A = 200 km/s & dt = 50 s)  Slow & continuous motion of a footpoint : L ~ L coronal loop > 10 Mm  Corona / photosphere interface (assuming equal B) :  C A cor / C A phot ~ (  phot /  cor ) ½ ~ (10 17 cm -3 / 10 9 cm -3 ) ½ ~ 10 4  L wavelength / H P scale-height > 10 4 km / 10 2 km > 10 2  Paradigm :  The Sun has no experimental-like well-defined confining boundaries  But energy stored for  t >> t Alfvén

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA  When an Alfvén waves reaches the photosphere  At the wave-front, over 1% only of the whole wavelength  Propagation speed  by a factor 10 4  Velocity amplitude  by a factor 10 8  This leads to a quasi-complete reflexion back into the corona - This is not only the result of strong  differences, it requires a sharp interface ! - Its is not always valid : e.g. steep waves & shocks, short loops, very short energy bursts Energy storage : line-tying  Line-tying = extreme assumption = full reflexion

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Origin of Energy : emergence & motions  Sub-photospheric emergence  Current carrying flux tube from convection zone  Flux tubes traveling the whole CZ  twist necessary  Slow photospheric motions  Twisting of 1 or 2 of the polarities  Shearing motions // inversion line  Energy stored in closed field lines only  Evacuation of E B at Alfvénic speeds in open fields

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Non-bipolar fields : complex topologies  2.5-D & 3D models :  Quadru-polar fields  Null point B=0  separatrix surfaces z x  In 3D : spine field line & fan surface Karpen et al. (1998) Aulanier et al. (2000)

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Complexity : current sheet formation  Quasi-spontaneous current sheet formation in 2.5-D : z x y x  Field line equation :  y =  B y d xz /B xz = B y  d xz /B xz  ( B xz   xz )  B y  = 0 since J x B = 0 & d/dy = 0  On each side of separatrix :  y equal & d xz /B xz different  Jump in B y y x

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Null point : magnetic reconnection  Basic principle in a current sheet :  dB/dt =   ² B & field line equation  reconnection  mass & energy conservations  u in /C A = Lu -½ (Sweet-Parker regime)  The Switch-on problem :  shearing separatrix  spontaneous J sheet  no flare, but heating  Advect stronger B, increasing , stronger driving, other physics (Petscheck, Hall…)  Or separatrix-less reconnection… (Aulanier, 2004, La Recherche)

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA no 3D null point Quasi-separatrices Generic four flux concentrations model  Topology / geometry :  Continuous field line mapping  Sharp connectivity gradients

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Log Q  = J / B J  =  x B pas de symétrie 2.5D Quasi-separatrices J (z=0) Gradual formation of current layers  Current layers & topology :  Along the pre-existing Quasi Separatrix Layer (QSL)  J sheet thinnest in Hyperbolic Flux Tube (HFT)  Thickness decreases with time in HFT Aulanier et al. (2005)

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Formation of current layers : where & how  Current sheets :  In pre-existing QSL  For any boundary motion  Thickness of J ~ thickness of QSL Aulanier et al. (2005)

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Slip-running reconnection in 3D  Field line dynamics :  Coronal reconnection  Alfvénic continuous footpoint slippage  Origin of apparent fast motion of particle impact along flare ribbons ?

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Yohkoh SXT, SXR 11:48 UT TRACE, FeXI 171A 12:05 UT – 14:00 UT A case study : The July 14, 1998 eruption

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Realistic model of B & line-tied motions  Coronal field :  B(Kitt Peak) modified to have |B z max |=2900 G  potential field extrapolation view from earth  Local photospheric twisting :  1 polarity in  -spot  satisfies dB z /dt = 0  u max = 2% C A (slow) top view

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA   -spot coronal configuration :  Null point aside of (not above) the sheared field lines at z = 3.9 Mm  Sheared fields beneath the fan surface Projection view of The full 3D MHD domain 2.5-D MHD breakout model (Antiochos et al. 1999) A generalized magnetic breakout ? Aulanier et al. (2000)

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Dynamics of sheared & complex coronal fields Projection view (field lines & B z [z=0] )2D cut of currents J Potential field

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Projection view (field lines & B z [z=0] )2D cut of currents J Eruption with continuous photospheric motions & null point reconnection t = 0000 s, n dump = 00 Dynamics of sheared & complex coronal fields

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Projection view (field lines & B z [z=0] )2D cut of currents J Eruption with continuous photospheric motions & null point reconnection t = 0810 s, n dump = 27 Null point reconnection Dynamics of sheared & complex coronal fields

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Projection view (field lines & B z [z=0] )2D cut of currents J Eruption with continuous photospheric motions & null point reconnection t = 0900 s, n dump = 30 Null point reconnection Dynamics of sheared & complex coronal fields

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Projection view (field lines & B z [z=0] )2D cut of currents J Eruption with continuous photospheric motions & null point reconnection t = 0990 s, n dump = 33 Null point reconnectionFlux tube acceleration Dynamics of sheared & complex coronal fields

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Projection view (field lines & B z [z=0] )2D cut of currents J Eruption with continuous photospheric motions & null point reconnection t = 1080 s, n dump = 36 Flux tube acceleration Dynamics of sheared & complex coronal fields

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA TRACE 171A View from Earth ( field lines & J z [z=0] ) Eruption with photospheric motions supressed TRACE observations vs. MHD model

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA  Physical & observational ingredients :  Photospheric twisting & slow expansion 0 < t < 750  Magnetic energy: E B free = 9.5 % E B potential field t = 990  Null point reconnection & leaning sideward of overlaying fields rooted in the  -spot750 < t < 1080  Fast eruption of sheared fields & 2-ribbon flare990 < t < 1440 ( idem if photospheric driving supressed )  Moving brightenings observed in EUV = d t J z (z=0) during reconnection A generalized magnetic breakout ?  So far difficulties to calculate full eruption :  numerical instabilities in current sheets  calculation halts  work in progress

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA To link models & observations 2.5-D MHD breakout model SoHO/EIT 195 filament eruption  Magnetic field extrapolations :  Model of B coronal using observed B phot as boundary conditions  To better analyze observed events knowing B corona  To test models & provide B initial for 3D MHD simulations (Antiochos et al. 1999)

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA 1. Basic physics 1.1 Introduction 1.2 Non-potentiality 1.3 Complexity 2. Learning from 3D MHD simulations 2.1 Current sheets & reconnection in quasi-separatrices 2.2 Breakout model of an observed eruption 3. Magnetic field extrapolations 3.1 Motivation 3.2 Potential & linear force-free fields 3.3 Non-linear force-free fields 4. Toward SDO & other future missions Overview of this tutorial

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Potential & linear force-free field extrapolations  Assumption :  = cst (  =0 for potential)   ² B +  ²  B = 0 : Helmoltz equation : analytical solutions  Fourier, Bessel functions, spherical harmonics (Nakagawa & Raadu 1972, Alissandrakis 1981, Démoulin et al. 1997, Chiu and Hilton 1977, Semel 1988, Altschuler & Newkirk 1969, Schrijver & DeRosa 2003 …)  Advantages & limits : + Fast computation  low computer memory & power + Based on analytical formulas  low dependance on algorithm + Do not require full B phot  vector magnetograms rare & noisy + Overall topology  most topological regimes are stable – Lower bounds on E B & H B  poor estimation of free energy & helicity – Small-scale shear  largest field lines most affected by  –  limits  cannot treat highly stressed fields –  = cst  no mixed sheared & potential fields & no return currents

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Setting force-free parameter  cst Yohkoh/SXT lfff extrapolation Yohkoh/SXT H  (DPSM / Pic du Midi)   chosen to best match - large SXR loops - transverse B phot if available  small connectivities weakly depend on 

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Yohkoh/SXT H  (DPSM / Pic du Midi) SXR loops Arch Filament System Démoulin et al. (1997), Schmieder et al. (1997) Flare ribbons : footpoints of QSLs = gradients of connectivites Confined flare topologies in active regions

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Pre-CME topologies in active regions Aulanier et al. (2000) Eruption precursor : shear Alfvén wave along null spine because of reconnection

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Pre-CME topologies between hemispheres Delannée et al. (1999) Large-scale dimmings : footpoints of TIL’s pushed from below during eruption

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Pre-eruptive topologies : potential & lfff sufficient  Major results :  Topology (skeletons & QSL’s) of overlaying weakly stressed fields  Location of associated current sheets reconnection particle acceleration sites particle impacts confined flare ribbons dimmings during CMEs  Some codes already « available » to the community :  Potential Field Source Surface (PFSS) on SSWIDL Schrijver & DeRosa (2003)  FRench Online MAGnetic Extrapolations (FROMAGE) on the WWW Démoulin et al. (1997), Aulanier et al. (1999) Such extrapolations well addess the issue of global connectivity

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Aulanier & Schmieder (2002) Aulanier et al. (2000) Topology of filaments with constant-  extrapolations Aulanier et al. (1999) Full field lines magnetic dips H  / EUV filament bodies (feet) : magnetic dips within (aside) a flux tube of low twist  <1.5 

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA 9-hour evolution on Sep 25, 1996 VTT/MSDP 08:43 UT 12:14 UT17:04 UT 15:57 UT SoHO/MDI 07:40 UT 15:59 UT 17:35 UT 12:53 UT Aulanier et al. (1999) Evolving filaments with constant-  extrapolations

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA 9-hour evolution on Sep 25, 1996 12:14 UT17:04 UT VTT/MSDP 08:43 UT15:57 UT Aulanier et al. (1999) Moving parasitic polarities : destruction / formation of dips & evolution of barbs Evolving filaments with constant-  extrapolations

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Performing non-linear force-free extrapolations  Properties :  = varying (as observed in vector magnetograms & in MHD models)   x B =  B & ( B   )  = 0 must be computed numerically in general  Common feature in most algorithms :  Stress imposed at 1 photospheric footpoint or on the whole photospheric plane or on all faces via increments of  bdry or B bdry or u bdry  Relax toward force-free state by imposing some transport method (e.g. physical or numerical) & conditions (e.g. E min or |JxB| min )  Great care with mathematical ill-posed methods ! i.e. redundant BC’s at boundary(ies)  discontinuities in domain  Simple vertical integration doomed to failure no side/upper BC’s  existing k modes growing as ~exp(k.z)

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Results : sheared loops & sigmoids Bleybel et al. (2002) B z PHOT  z PHOT Yohkoh/SXT Jiao, McClymont & Mikic (1997) B z PHOT  z PHOT Yohkoh/SXT Pre-post eruption E B & H B ; non-homogeneous shear & return currents

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Results : flux ropes & filaments Régnier & Amari (2004) van Ballegooijen (2004) courtsesy van Ballegooijen Filament bodies : magnetic dips within a flux tube of medium twist  ~ or > 2 

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Some results, but difficulties at every level  PDE’s  Growth (& instability) of non-physical spatial oscillations  Multiple solutions can exist for 1 same boundary condition  MHD-unstable solutions can exist  Numerical : for locally strong gradients d i B j  Physical : for regions of strong  values  Observational : spectro-polarimetry & magnetography  Inversion of Stokes profiles I,Q,U,V & solving 180 o ambiguity  B  Weak Q,U  noise in B xy & weak I  errors in B sunspots  Many non force-free regions where  >1 & in (quasi-) separatrices  Limited field of view  artificial flux imbalance

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Finding the best method(s)  Some input models :  Low & Lou (1990)  (   )/  < (  B z )/B z &  weak & no return currents  analytical solution Valori (2005), Schrijver et al. (2006), Inhester & Wiegelmann (2006), Amari et al. (2006)  Best methods :  Optmization method better for analytical models (with BC’s on 6 faces imposed)  Grad-Rubin (& magneto-frictional) better on symmetric twisted flux tubes  Grad-Rubin a priori best (well posed, no redundant BC’s), but  J x B  not imposed  convergence not ensured  losses of convergence found for strong  3D MHD [Török&Kliem03] or  Photospheric bipolar B z &  3D nlfff [Titov&Démoulin01] or  symmetries 2D arbitrary [Inhester&Wiegelmann06]  no analytical solution for most models

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA More realistic & demanding input models  (   ) /  > (  B z ) / B z   phot strong & varies faster than B z phot  narrow non force-free layers with strong  at the fan/spine footpoints   0 &  0 in one polarity  return currents from MHD simulation of the July 14, 1998 pre-eruptive B  = J z /B z (z=0)   ~ 0 on all 5 coronal faces  «open» field lines are potential B z (z=0) B z (z=0) & field lines

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Toward SDO & other future missions Non-linear fff extrapolations Take B PhotObs from vector magnetogram Calculate potential field B POT Prescribe Stress on bounary(ies) and tend toward B corona ~ EUV loops Super-computer resources (high resol. needed) Then analyze observations & analyze stability with MHD simulations Forward modeling : 3D MHD Take B z PhotObs from LOS magnetogram Calculate potential field B POT Prescribe u xy phot or E phot and tend toward B corona ~ EUV loops and/or B PhotMHD ~ B PhotObs Super-computer resources Then analyze observations & pursue MHD simulation to model coronal dynamics  Beyond potential and linear force-free fields analyses of observations :  MHD & nlfff algorithms to be validated & tested against observations  Observing full coronal field lines (multi- : SDO/AIA)  Measuring photospheric full vector B (SDO/HMI & inversion techniques)

G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Complexity & non-potentiality of the solar corona G. Aulanier ( Observatoire de.

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G. Aulanier – AIA HMI Team Meeting – Feb 13-17, 2006 – Monterey, CA, USA Complexity & non-potentiality of the solar corona G. Aulanier ( Observatoire de.

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