MHD of the solar corona : Non-potentiality, complex topologies & line-tying Guillaume Aulanier ( Observatoire de Meudon, LESIA ) n° 1

Complexity & non-potentiality of coronal B At the origin of all solar activity TRACE, FeXI 171A July 14 1998, 12:05 UT – 14:00 UT n° 3 Yohkoh SXT, SXR 11:48 UT Among the major goals of all upcoming solar instruments But not easily measurable…

Activity : storage & release of magnetic energy Magnetically driven activity Corona : b ~ ETh / EB ~ 2mP / 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 n° 4

Overview 1.) Physics of the solar corona 2.) 3D MHD simulations 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion 1.) Physics of the solar corona 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion 1.) Physics of the solar corona 2.) 3D MHD simulations 3.) 3D MHS filament models 5.) Conclusion n° 2

Pre-eruptive B : field-aligned currents Conservation of momentum : dt ( r u )= 0 h dt u = – (u .s) u + (mr)–1 (sx B) x B + sP + rg tA²/t² = u²/cA² + 1 + b + b L / HP In the solar corona : h Slow evolution : t ~ days >> tA ~ minutes Photospheric velocities : u ~ 0.1 km/s << cA ~ 1000 km/s « Cold » plasma : b = 0.0001 – 0.1 << 1 Loop sizes : L~ 10 – 100 Mm ~ Hp ~ 50 Mm n° 5 h J x B = 0 & sx B = mJ  sx B = aB

Force-free fields : three classes h sx B = 0  B = sY h B defined by a scalar potential Potential fields : a = 0 Linear force-free fields : a = cst h sx (sx B = a B )  s² B + a² B = 0 h Helmoltz equation has analytical solutions h sh(sx B = a B )  ( B hs)a = 0 h A field line is defined by its a value n° 6 Non-linear force-free fields : a = varying

Overview 1.) Physics of the solar corona 2.) 3D MHD simulations 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion does not count

Non-potentiality |a| > 0 : free magnetic energy Potential field : sx B0 = 0 ; sh B0 = 0 h EB0 = III ½ B0² dV ; B0 = sF ; II B1h dS = 0 Non potential field : B = B0+ B1 ; sh B1 = 0 h EB = III ½ B0² dV + III ½ B1² dV + III B0hB1 dV = EB0 + EB1 + III (sF) hB1 dV = EB0 + EB1 + III [ sh(F B1) – F (shB1) ] dV = EB0 + EB1 + III sh(F B1) dV = EB0 + EB1 + II F B1 h dS n° 7 = EB0 + EB1 > EB0 h Same as Kelvin’s theorem for incompressible fluids h Potential field = lower bound of energy for a given Bzphot

How to store energy in the corona Paradigm : h The Sun has no experimental-like well-defined confining boundaries h But energy stored for Dt >> tAlfvén Wavelengths L of coronal waves with C = CA ~ cst : h Energy burst during dt : L ~ CA dt ~ 10 Mm (for CA= 200 km/s & dt = 50 s) h Slow & continuous motion of a footpoint : L ~ Lcoronal loop > 10 Mm n° 8 Corona / photosphere interface (assuming equal B) : h CAcor / CAphot ~ (rphot / rcor)½ ~ (1017 cm-3 / 109 cm-3 )½ ~ 104 h Lwavelength / HP scale-height > 104 km / 102 km > 102

Energy storage : line-tying When an Alfvén waves reaches the photosphere h At the wave-front, over 1% only of the whole wavelength h Propagation speed m by a factor 104 h Velocity amplitude m by a factor 108 h This leads to a quasi-complete reflexion back into the corona - Not only the result of strong r differences, it requires a sharp interface ! - Not always valid : e.g. steep waves & shocks, short loops, very short energy bursts Line-tying = extreme assumption = full reflexion n° 9

Origin of Energy : emergence & motions Sub-photospheric emergence h Current carrying flux tube from convection zone h Flux tubes traveling the whole CZ  twist necessary Slow photospheric motions h Twisting of 1 or 2 of the polarities h Shearing motions // inversion line n° 10 Energy stored in closed field lines only h Evacuation of EB at Alfvénic speeds in open fields

Overview 1.) Physics of the solar corona 2.) 3D MHD simulations 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion does not count

Definition of magnetic shear angle for a = constant Bz = B° sin(kx x) exp[-(kx² – a²)1/2 z] Single Fourier mode By = -a/kx B° cos(kx x) exp[-(kx² – a²)1/2 z] Bx = -(kx² – a²)1/2/kx B° cos(kx x) exp[-(kx² – a²)1/2 z] Potential vs. linear force-free field (at x=z=0) : h Bx = -(kx² – a²)1/2/kx B° & By = -a/kx B°  By = 0 for potential field only h q = atan( By / Bx ) = atan( a / (kx² – a²)1/2 )  Magnetic shear x y z x y z q Bx Bx By Bz>0 Bz<0 Bz>0 Bz<0

Localized line-tied shear : direct & return currents y a = 0 potential fields a = 0 potential fields a > 0 direct currents a < 0 return currents - to confine current-induced B so as to keep external B potential - Biot-Savart  Sj = 0 x z y x Bz<0 Bz>0

Overview 1.) Physics of the solar corona 2.) 3D MHD simulations 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion does not count

Non-bipolar fields : complex topologies 2.5-D & 3D models : h Quadru-polar fields h Null point B=0  separatrix surfaces h In 3D : spine field line & fan surface z x n° 11 Karpen et al. (1998) Aulanier et al. (2000)

Complexity : current sheet formation Quasi-spontaneous current sheet formation in 2.5-D : h Field line equation : Dy = S By dxz/Bxz = By S dxz/Bxz h ( Bxzhsxz) By = 0 since J x B = 0 & d/dy = 0 h On each side of separatrix : Dy equal & dxz /Bxz different  Jump in By z x y x y x n° 12

Null point : magnetic reconnection Basic principle in a current sheet : h dB/dt = hs² B & field line equation  reconnection (Aulanier, 2004, La Recherche) h mass & energy conservations  uin /CA = Lu -½ (Sweet-Parker regime) The Switch-on problem : h shearing separatrix  spontaneous J sheet  no flare, but heating n° 13 h Advect stronger B, increasing h , stronger driving, other physics (Petscheck, Hall…) h Or separatrix-less reconnection…

Overview 1.) Physics of the solar corona 2.) 3D MHD simulations 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion does not count

MHD equations in the LESIA code : zero-b i Continuity dt r = – (u .s) r – r (s.u) [ advection ; compressible term ] i Momentum dt u = – (u .s) u + (mr)–1 (sx b) x b + n ru [ advection ; Lorentz force ; pseudo-viscous term ] i Induction dt b = – (u .s) b – b (s.u) + (b .s) u + h rb + z s(s.b) [ ideal term ; resistive diffusion ; divb cleaner ] ~ ~

Numerical scheme Finite differences : 4th order (5 point derivatives) Predictor-Corrector : 3-step dams-Bashorth & 2-step Adams-Moulton explicit time-step with CFL condition Numerical mesh : non-uniform, structured, fixed stronger concentration where b stronger Explicit diffusion terms Laplacian for b Filter adapted to the mesh of u

Boundary conditions : line-tied & open bx, by, bz, ux, uy z r uz ghost cells domain | | | | | | r, bx, by, bz z ghost cells domain ux, uy, uz | | | | | | z y x z y x

Overview 1.) Physics of the solar corona 2.) 3D MHD simulations 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion

Twisting flux tubes : rotating sunspots

Twisting flux tubes : sub-Alfvénic line-tied motions 3D projection view Top view uxy/cA = 0.02 Aulanier, Démoulin & Grappin (2005)

Dynamics & equilibrium properties 0.2 0.4 0.6 uz/cA Equilibria z F /2p Dynamic phase Quiet phase Dynamic phase : even with sub-alfvénic boundary driving Equilibrium curves : ln(z/z0) = cst F² Why is there no kink ? Expansion  L k  F/L m  stabilizing effect

Electric currents : photosphere & corona Jz (z=0) S (J2 dz) potential return direct

Sigmoïd topology : sheared lines vs. twisted flux rope Twisted B lines around vortex centers Whirled B lines on sigmoïd ends Sheared B lines on brightest parts

Twisted sunspots : Non-radial magnetic fields MHD model Aulanier, Démoulin & Grappin (2005) THEMIS / MTR courtesy G. Molodij

Overview 1.) Physics of the solar corona 2.) 3D MHD simulations 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion

Confined flare topologies from cst-a extrapolations Yohkoh/SXT lfff extrapolation Yohkoh/SXT H (DPSM / Pic du Midi) h a chosen to best match - large SXR loops - transverse Bphot if available h small connectivities weakly depend on a

Sharp mappings : quasi-separatrix layers (QSL’s) Arch Filament System H (DPSM / Pic du Midi) SXR loops Yohkoh/SXT Démoulin et al. (1997), Schmieder et al. (1997)

QSL’s in four flux concentrations model Quasi-separatrices no 3D null point Quasi-separatrices Topology / geometry : Continuous field line mapping Sharp connectivity gradients

Current sheet formation in QSL’s Log Q a = J / B J = sx B Aulanier, Pariat & Démoulin (2005) Current layers & topology : J (z=0) Along the pre-existing Quasi Separatrix Layer (QSL) J sheet thinnest in Hyperbolic Flux Tube (HFT) Thickness decreases with time in HFT pas de symétrie 2.5D Quasi-separatrices

Formation of current sheets : where & how In pre-existing QSL For any boundary motion Thickness of J ~ thickness of QSL Aulanier, Pariat & Démoulin. (2005)

Slip-running reconnection in 3D Aulanier, Pariat, Démoulin & DeVore (2006) Field line dynamics : Coronal reconnection Alfvénic continuous footpoint slippage Origin of apparent fast motion of particle impact along flare ribbons ?

Overview 1.) Physics of the solar corona 2.) 3D MHD simulations 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion

Potential & linear force-free field extrapolations Semi-analytical solutions h s² B + a² B = 0 : Helmoltz equation  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 Bphot  vector magnetograms rare & noisy + Overall topology  most topological regimes are stable – Lower bounds on EB & HB  poor estimation of free energy & helicity – Small-scale shear  largest field lines most affected by a – a limits  cannot treat highly stressed fields – a = cst  no mixed sheared & potential fields & no return currents

Extrapolation of filament channels 08:12 UT 07:52 UT

Plasma support in magnetic dips h Levitation of dense plasma H (prominence) ~ 50 - 500 Hg  no hydrostatic support e (prominence) ~ 0.01L(Bphot)  Alfvén wave support requires to be localized : why ? u (plasma) ~ 0.1 cA  field-aligned motions : role debated (B . s) B > 0 Bz = 0 h Full field line altitude (z) Filling of the dip : h dHa = Hg = 300 km Simulation of filament observations = 3D ensemble of all dips h

Topology of filaments with constant-a extrapolations Aulanier et al. (1999) Aulanier et al. (2000) Full field lines magnetic dips Aulanier & Schmieder (2002)

Constant-a extrapolations vs. MHD models Magnetic dips (IP & NP) Full field lines Full field lines magnetic dips Aulanier & Schmieder (2002) DeVore & Antiochos (2000) Full field lines Amari et al. (1999) Aulanier, DeVore & Antiochos (2006)

Overview 1.) Physics of the solar corona 2.) 3D MHD simulations 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion

Evolving filament barbs : observations 9-hour evolution on Sep 25, 1996 VTT/MSDP 08:43 UT 12:14 UT 17:04 UT 15:57 UT SoHO/MDI 07:40 UT 15:59 UT 17:35 UT 12:53 UT Aulanier et al. (1999)

Evolving filament barbs : constant-a extrapolations 9-hour evolution on Sep 25, 1996 VTT/MSDP 08:43 UT 15:57 UT 12:14 UT 17:04 UT Aulanier et al. (1999)

Comparisons with B measured in prominences Normal Inverse Aulanier & Démoulin (2003) |B| ~ 14 G d|B|/dz ~ 0.3 10-4 G/km 1 isolated region of normal polarity same as in Bommier et al. (1994)

– l + Magnetic dips measured in the photosphere i THEMIS Ha spine barb constant-a model – i + THEMIS Ha THEMIS B// barb spine THEMIS observations – l + parasitic polarities THEMIS vector B Lopez Ariste, Aulanier, Schmieder & Sainz-Dalda (2006)

Overview 1.) Physics of the solar corona 2.) 3D MHD simulations 1.1.) Force-free fields 1.2.) Free magnetic energy 1.3.) Extended currents & magnetic shear 1.4.) Current sheet formation & magnetic reconnection 2.) 3D MHD simulations 2.1.) The LESIA code 2.2.) Twisting flux tubes & X-ray sigmoïds 2.3.) Current sheets & reconnection in quasi-separatrices 3.) 3D MHS filament models 3.1.) Extrapolation applied to multi-l observations 3.2.) Comparison with magnetic field measurements 5.) Conclusion

MHD models of the solar corona are VERY MUCH Conclusions MHD models of the solar corona are VERY MUCH constrained by observations ! Many reasonable models of solar features (even those that use the right & well-known orders of magnitudes) FAIL the observational test…