Thought in 2000: Magnetic helicity is an important theoretical concept Pascal Démoulin but there is no way to estimate it from observations

Few typical observations Corona ( T ~ K ) Photospheric magnetic field ( [10 -3, ] T ) EIT / SOHO MDI / SOHO TRACE Coronal loops (N ~ p cm -3 ) B // < 0 B // > 0

What represent magnetic helicity ? Few simple examples: Twisted flux tube Sheared arcadeBraided flux tubes X rays & UV emissions : trace of field lines In the corona: e.g. sigmoids Indeed present in any non potential magnetic configuration X rays UV B n > 0 B n < 0

Basic definition of H Magnetic vector potential Bad for application to observations ! ( Elsasser 1956 ) Magnetic field confined in a volume Physically meaningful ONLY if invariant by gauge transformation NOT for the corona ! B B n = 0 n on S S

Equivalent definition of H Double summation over the volume without the vector potential Involves only magnetic field & spatial position Summation over all the magnetic flux tubes pairs Simple example: Two inter-linked flux tubes ( Moffatt 1969 ) Gauss linking number

More general definition of H ( Berger & Fields 1984 ) ( Finn & Antonsen 1985 ) gauge transformation Same tangential vector potential Same magnetogram (normal component) Invariance of gauge implies: (usually potential field) Close outside by the same field Coronal field Reference field

Practical computation of H coronal with a linear force free field determined to best fit the coronal loops Summation over the spatial Fourier modes ( Berger 1985, Démoulin et al. 2002, Green et al. 2002, Nindos & Andrews 2004, Mandrini et al ) H max (AR) ~ 0.2 (magnetic flux) 2 Comparable with a twisted flux tube having 0.2 turn Moderate global magnetic helicity content ! (H more concentrated in the AR core) Computed field lines Coronal loops Coronal loops

Practical computation of H coronal ( Démoulin et al 2006 ) without a coronal magnetic model: needs the connectivities += Function only of : - foot point position - above / below (use Gauss linking number ) Summation over all the coronal pairs of flux tubes Arcade-like closing field General case Open configurationClosed configuration ( Show only one pair of flux tubes )

Practical computation of H coronal ( Démoulin et al 2006 ) without a coronal magnetic model: needs the connectivities Summation over all the coronal pairs of flux tubes Applications: Needs many observed loops => Observations in a large range of coronal temperatures TRACE, Hinode, SDO spacecraft Simple case : with only arcade shaped flux tubes ( One pair of flux tubes ) Coronal loops ( UV, TRACE )

Magnetism is to astrophysicists what sex is to psychoanalysts ( van den Hulst, 1965 ) Magnetic helicity is for magnetic field what women are for life Without women, life would be boring ! Without magnetic helicity : - No build up of large scale magnetic field ( dynamo ) - No magnetic flux tube can cross the convective zone (need twist) - No sigmoids - No CMEs - No solar cycle …. Without magnetic helicity, magnetic field would be boring !

Cascade to large scales  very low dissipation dissipate on the global resistive time scale (> 100 years ) ( Frisch et al. 1975, Alexakis et al ) Inverse cascade of H k Cascade large scales small scales injection => Link the physics of : * the convective zone ( dynamo ) * the corona ( sigmoids, CMEs ) * the interplanetary space ( magnetic clouds, ICMEs ) even with important magnetic energy release ( e.g. in a flare, Berger 1984 ) H is a conserved quantity

3D MHD turbulence 3D Hydrodynamic turbulence small scale B small scale vortexes + large scale organized B ( as in the corona ! ) (only direct cascade to small scales) ( Biskamp 1993, Seehafer 1994, Brandenburg 2001 ) Inverse cascade of H Power spectrum of energy during before & after small scales large scales k -3/2 ( Kraichnan ) Reconnection of two twisted flux tubes ( Milano et al )  

Magnetic / current helicity Curl operation current magnetic measured : horizontalvertical ( Abramenko et al Bao & Zhang 1998, Bao et al ) Hemispherical dominance (70-80 %) :H c < 0 north hemisphere H c > 0 south hemisphere ( Pevtsov et al Hagino & Sakurai 2004 ) Also for  best so for magnetic helicity Properties : H and H c have the same sign : always true ? H c is not conserved

Coronal Mass Ejection (CME ) Destabilization & launch of a coronal magnetic structure in the interplanetary space EIT, LASCO/ SOHO 5 dec CME Coronagraph occulting disk

H conservation : linking coronal & interplanetary physics Measurements of the 3 components of B + flux rope model AR 7912, 14 Oct days later time (h) Magnetograms + coronal loops + extrapolation before CME after CME MC Remote sensing but global In situ but local CME -> H Magnetic Cloud ->  H corona Data : X rays Computed field lines

H conservation :  H corona ~ H Magnetic Cloud ? ( Mandrini et al. 2005, Luoni et al Dasso et al ) large event 14 Oct tiny event 11 May   H corona   |  H corona |   H cloud   |H cloud |  3.0 ~ factor ~ 2 L cloud = 0.5 AU L cloud = 2 AU Units : Mx 2 Units : Mx 2 Large range of H cloud : 5 orders of magnitude ! Number of events Log 10 H cloud (Mx 2 ) ( Lynch et al ) Quantitative link between CME & MC -> relate the physics involved in in both domains

Are CMEs a consequence of magnetic helicity accumulation ? ( Rust 1994, Low 1997 ) * injection with dominant hemispheric sign ( 0 south) * sign independent of the solar cycle * negligible dissipation ( from theory ) * few reconnections between north / south hemispheres * few reconnections with coronal holes To limit the buildup, magnetic helicity has to be ejected via CMEs Conjecture:

Upper bound on magnetic helicity ( Flyer et al. 2004, Zhang et al ) Family of axisymmetric force-free field outside a sphere Upper bound of total magnetic helicity for force free fields (for a fixed Bn at the boundary) Conjecture: But a CME can still occur below this upper bound … Azimuthal flux ( As for magnetic energy but H is a conserved quantity so its accumulation is easier with a given sign of H flux ) Helicity

Does a CME occur when H > H threshold ? ( Phillips et al ) * Not important in the breakout model ( inject H with opposite sign on the sides of the central arcade globally H = 0, still a CME ) ( Kusano et al ) * Major effect if injection of opposite H around PIL ( cancellation of opposite H = > CME ) ( Amari et al. 2003a,b ) * Necessary but not sufficient condition ( can get a CME with H = constant ) * Yes H threshold ~ flux 2 ( Jacobs et al ) ( large scale dipole, slight influence of the solar wind model ) Controversial results from MHD simulations !

What observations tell us ? ( Nindos & Andrews 2004 ) Synthesis of present works: * accumulation of H coronal is one condition to launch a CME * other ingredients are also present: - photospheric motions: where H is injected ? - magnetic topology: where is reconnection “allowed” ? More helicity needed to get a CME ( ~ factor 4) Flares with CMEs Flares without CMEs Linear fff which best fit loops, get H coronal for flaring ARs (M & X flares) H computed field lines coronal loops

Photospheric flux of magnetic helicity (Berger & Fields 1984 ) emergencehorizontal motionsHelicity flux horizontal ( transverse ) field component vertical ( normal ) field component => Needs magnetic + velocity fields Do we need the 3 components of B ? Do we measure only the last term with longitudinal magnetograms ? How do we measure them ?

Photospheric velocity ( Schuck 2005 ) * Feature tracking ( Strouss 1996 ) ( Schuck 2006 ) ( Welsch et al ) ( November and Simon 1988 ) * Local Correlation Tracking ( LCT ) * Solve the Induction Equation + LCT - minimize the input of LCT - minimize computation time ( FFT ) - cross correlation, rigid translation - differential LCT, include linear deformation within the apodising window * Doppler velocities ( Kusano et al ) - Differential Affine Velocity Estimator ( DAVE ) Only the longitudinal component ! Need a very precise measured B to remove the flow // to B ( no contribution to E & H flux ) => not used Needs high spatial resolution to follow individual features Efficient even with noisy data !

Footpoint motions Simple example: emerging flux tube All the previous methods derive : - the photospheric footpoint motions of the magnetic flux tubes ( u ) - NOT the plasma motions ( v ) Corona Photosphere emergence Corona Photosphere emergence horizontal motions footpoint motion

Photospheric flux of magnetic helicity * With plasma motions ( v ) vertical motions horizontal motions two contributions * With the footpoint motions of flux tubes ( u ) ( Démoulin &Berger 2003 ) = > Full helicity flux from longitudinal magnetogram time series Derived from longitudinal magnetograms ( close to centre disk )

Photospheric flux of magnetic helicity ( Cheung et al ) Flux increase: emergence AR D MHD simulation (Chae 2004 ) Emergence of a twisted flux tube Similar peak of helicity flux Helicity flux

Flux density of magnetic helicity Flux density : All previous studies with G A maps : simultaneous injections of both sign of magnetic helicity. True ? ( Chae 2004 ) ( Nindos et al ) G A & B n ( Kusano et al ) G A & velocity Does it had a physical meaning ? Total H flux : well established physical meaning

Simplest example: a translated magnetic flux tube => G A is NOT a good proxy of the flux density ! ( Pariat et al ) G A introduce fake signal of both signs in equal amount Only the total flux of helicity is reliable v ( Kusano et al ) Example of an observed AR --> v Flux tube v Photosphere While no helicity is injected ! GAGA

Flux density of magnetic helicity + => Double integration on the magnetogram ( Pariat et al ) A better proxy of the helicity flux density is : Helicity flux density  summation of the relative rotation of all the elementary flux tubes, weighted by their magnetic fluxes Magnetogram + velocity ( arrows ) Rotation rate  x x’  B // > 0 B // < 0

Example: emerging flux tube Positive helicity flux covered by fake signal in G A maps Very weak fake signal with G  Factor 5 to 10 difference Weakly twisted flux tube : 0.1 turn ( small amount of helicity ) emergence GG GG

Flux density of magnetic helicity ( Pariat et al ) strong fake signalMore homogeneous GG GG B n magnetogram + velocity (arrows) B // > 0 B // < 0 GG AR 8210 AR 8375 GG GG Magnetic helicity injection in ARs : much more coherent than previously thought => Constraint on the dynamo models

Evolution of helicity flux density ( Pariat et al ) dominantly fake signal => Can follow the evolution of magnetic helicity injection in ARs Example : evolution of AR 9114 during 6 days Coherent evolution GG GG

Few next steps Themis, Hinode (solar B), SDO, STEREO * Detailed photospheric flux map : constraints on : - emerging flux tubes => dynamo - physics of flares and CMEs * Broad temperature coverage of the corona => field line linkage => H corona * Stereoscopy : avoid ambiguity in the loop crossing (front / back) * Multi-spacecraft observations of a magnetic clouds Now magnetic helicity is a measurable quantity ! * in the photosphere (maps of helicity flux ) * in the corona (extrapolation or summation of loop helicities ) * in magnetic clouds ( => CMEs) Much more is still expected and needed due to the complexity of magnetic helicity and its multi-faced nature Conclusion