Structure Equilibrium and Pinching of Coronal Magnetic Fields

Structure, Equilibrium and Pinching of Coronal Magnetic Fields Slava Titov SAIC, San Diego, USA Seminar at the workshop „Magnetic reconnection theory“ Isaac Newton Institute, Cambridge, 18 August 2004

Acknowledgements Collaborators • on structure: Ø Pascal Démoulin (Paris-Meudon Observatory, France) Ø Gunnar Hornig and Ø Eric Priest (University of St Andrews, Scotland) • on pinching: Ø Klaus Galsgaard and Ø Thomas Neukirch (University of St Andrews, Scotland) • on kink instability and pinching: Ø Bernhard Kliem (Astrophysical Institute Potsdam, Germany) Ø Tibor Törok (Mullard Space Science Laboratory, UK) 2

Outline 1. Introduction: magnetic topology and field-line connectivity? Structural features of coronal magnetic fields: Ø topological features - separatrices in coronal fields; Ø geometrical features - quasi-separatrix layers (QSLs). 2. 3. 4. 5. Theory of magnetic connectivity in the solar corona. Quadrupole potential magnetic configuration. Twisted force-free configuration and kink instability. Magnetic pinching of Hyperbolic Flux Tubes (HFT=QSL+QSL, intersected). 6. Summary. 3

2 D case: field line connectivity and topology normal field line NP separtrix field line BP separtrix field line ØFlux tubes enclosing separatrices split at null points or "bald-patch" points. ØThey are topological features, because splitting cannot be removed by a continous deformation of the configuration. ØCurrent sheets are formed at the separatrices due to photospheric motions or instabilities. All these 2 D issues can be generalized to 3 D! 4

Generic magnetic nulls in 3 D Skewed improper radial null Skewed improper spiral null Stationary structure of both types of nulls can be sustained by incompressible MHD flows. Titov & Hornig 2000 Sustained by field-aligned flows only Sustained by either field-aligned or spiral field-crossing flows Magnetic nulls are local topological features: field lines emanating from nulls form separatrix surfaces. 5

Field line structure at Bald Patches (BPs) in 3 D Global effects of BPs Titov et al. (1993); Bungey et al. (1996); Titov & Démoulin (1999) BP criterion: magnetic field at BPs is directed from S to N polarity. BPs are local topological features: field lines emanating from BPs form separatrix surfaces. 6

Extra opportunity in 3 D: squashing instead of splitting Essential differences compared to nulls and BPs: • squashing may be removed by a suitable continuous deformation, • => QSL is not topological but geometrical object, • metric is needed to describe QSL quantitatively, • => topological arguments for the current sheet formation at QSLs are not applicable anymore; other approach is required. Nevertheless, thin QSLs are as important as genuine separatrices for this process. 7

Outline 1. Introduction: magnetic topology and field-line connectivity? • Structural features of coronal magnetic fields: Titov et al. , JGR (2002) Ø topological features - separatrices in coronal fields; Ø geometrical features - quasi-separatrix layers (QSLs). 2. 3. 4. 5. Theory of magnetic connectivity in the solar corona. Quadrupole potential magnetic configuration. Twisted force-free configuration and kink instability. Magnetic pinching of Hyperbolic Flux Tubes (HFT=QSL+QSL, intersected). 6. Summary. 8

Field line mapping: global description Construction • Cartesian coordinates ==> distance between footpoints. • Coronal magnetic field lines are closed ==> field-line mapping: from positive to negative polarity from negative to positive polarity 9

Field line mapping: local description Again two possibilities: • Jacobi matrix: Not tensor! • inverse Jacobi matrix: 10

Squashing factor Q Geometrical definition: Elemental flux tube such that an infinitezimally small cross-section at one foot is curcular, then circle ==> ellipse: Q = aspect ratio of the ellipse; Q is invariant to direction of mapping. Norm squared, Priest & Démoulin, 1995 Definition of Q in coordinates: where a, b, c and d are the elements of the Jacobi matrix D and then Q can be determined by integrating field line equations. 11

Expansion-contraction factor K Geometrical definition: Elemental flux tube such that an infinitezimally small cross-section at one foot is curcular, then circle ==> ellipse: K = lg(ellipse area / circle area); K is invariant (up to the sign) to the direction of mapping. Definition of K in coordinates: where a, b, c and d are the elements of the Jacobi matrix D and then Q can be determined by integrating field line equations. 12

Orthogonal parquet (complete description of magnetic connectivity) Construction ØThe major and minor axes of infinitezimal ellipses define on the photospere two fields of directions orthogonal to each other. ØA family of their integral lines forms an orthogonal network called parquet. ØParameterization of the lines such that the aspect ratio of tiles ~ Q 1/2. 13

Critical points of orthogonal parquet ØThe orthogonality is violated if a mapped ellipse degenerates into a circle. ØThis occurs at two types of (critical) points: Proof I-point Y-point Look at your fingerprints! • One separatrix emanates. • Three separatrices emanate. • I-point is at the common side • Y-point is a vertex of six of two adjoint triangles. adjoint tetragons. 14

Outline 1. Introduction: magnetic topology and field-line connectivity? • Structural features of coronal magnetic fields: Ø topological features - separatrices in coronal fields; Titov & Hornig, COSPAR (2000); Ø geometrical features - quasi-separatrix layers (QSLs). Titov et al. , JGR (2002) 2. 3. 4. 5. Theory of magnetic connectivity in the solar corona. Quadrupole potential magnetic configuration. Twisted force-free configuration and kink instability. Magnetic pinching of Hyperbolic Flux Tubes (HFT=QSL+QSL, intersected). 6. Summary. 15

Magnetogram Model: four fictituous magnetic charges placed below the photosphere to give Magnetic topology is trivial: • no magnetic nulls in the corona; • no BPs (the field at the inversion line has usual NS-direction). 16

Squashing factor Q Crescent strips of high Q connect sunspots of the same polarity. 17

Expansion-contraction factor K Blue and red areas are connected by flux tubes to bridge the regions of weak and strong photospheric fields. 18

Hyperbolic Flux Tube (HFT) (its spread from N- to S-footprint) Geometrical properties of HFTs: Øthey consist of two intersecting layers (QSLs) ; Øeach of the layers stems from a crescent strip at one polarity and shrinks toward the other; Øthe crescent strips connect two sunspots of the same polarity. 19

Mid cross-section of HFTs Variation of cross-sections along an HFT This is a general property that is valid, e. g. , for twisted configurations as well. 20

Field lines in HFTs Physical properties of HFTs: Øany field line in HFT connects the areas of strong and weak magnetic field on the photosphere (see the varying thickness of field lines); ==> Øany field line in HFT is stiff at one footpoint and flexible at the other; ==> ØHFT can easily "conduct" shearing motions from the photosphere into the corona! 21

Simple domains of orthogonal parquet General properties: ØTwo pairs of Y-points and three pairs of I-points. ØThe mostly distorted areas of the field line mapping are indeed smoothly embedded into the whole configuration. 22

Outline 1. Introduction: magnetic topology and field-line connectivity? • Structural features of coronal magnetic fields: Ø topological features - separatrices in coronal fields; Ø geometrical features - quasi-separatrix layers (QSLs). Titov & Démoulin, A&A (1999); 2. 3. 4. 5. Kliem et al. , Török et al. , A&A (2004) Theory of magnetic connectivity in the solar corona. Quadrupole potential magnetic configuration. Twisted force-free configuration and kink instability. Magnetic pinching of Hyperbolic Flux Tubes (HFT=QSL+QSL, intersected). 6. Summary. 23

Twisted force-free configuration Construction of the model Magnetogram Basic assumptions: • a/R << 1 and a/L << 1; • outside the tube the field is B=Bq+BI+BI 0; • inside the tube it is approximately the field of a straight flux tube. 24

Equilibrium condition Matching condition is in the vicinity of the tube or the force balance: where is due to and is due to curvature of the tube. is the internal self-inductance per unit length of the tube. From here it follows that the total equilibrium current 25

Equilibrium current Stability criterion: unstable Checked and improved numerically by Roussev et al. (2003) Minor radius changes with according to keep the number of field-line turns constant. 26

Squashing factor Q „fishhooks“ with Qmax~ 108 27

HFT in twisted configuration „Fishhooks“ are outside of the flux rope: 28

HFT in twisted configuration (its spread from N- to S-footprint) Variation of cross-sections along a twisted HFT: 29

Implications for sigmoidal flares Soft X-ray images of sigmoids S-shaped (positive current helicity ) Z-shaped (negative current helicity ) Short bright and long faint systems of loops? 30

Implications for sigmoidal flares Perturbed states due to kink instability S-shaped (positive current helicity ) Z-shaped (negative current helicity ) Sigmoidalities of the kink and HFT are opposite! 31

Current sheets around a kinking tube 32

Outline 1. Introduction: magnetic topology and field-line connectivity? • Structural features of coronal magnetic fields: Ø topological features - separatrices in coronal fields; Ø geometrical features - quasi-separatrix layers (QSLs). 2. 3. 4. 5. Theory of magnetic connectivity in the solar corona. Titov et al. , Ap. J (2003); Galsgaard et al. , Ap. J (2003) Quadrupole potential magnetic configuration. Twisted force-free configuration and kink instability. Magnetic pinching of Hyperbolic Flux Tubes (HFT=QSL+QSL, intersected). 6. Summary. 33

Simplified (straightened) HFT NB: sunspots crossing the HFT footprints in opposite directions, must generate shearing flows in between. 34

Two extremes: turn versus twist Turning shears must rotate the HFT as a whole Twisting shears must strongly deform the HFT in the middle. 35

Deformations of the mid part of HFT Assumed photospheric velocities: is a velocity of sunspots, is a length scale of shears, is a half-length of the HFT. Velocity field extrapolated into the coronal volume: 36

Comparison with numerics No current in the middle! Current sheet in the middle! 37

Pinching system of flows in quadrupole configuration Mechanism of HFT pinching: photospheric vortex-like motion induces and sustains in the middle of HFT a long-term stagnation-type flow which forms a layer-like current concentration in the middle of HFT. 38

Basic kinematic estimates Current layer parameters for the kinematically pinching HFT: the width is the thickness is where the dimensionless time or displacement of sunspots is The longitudinal current density in the middle of the pinching HFT is where and are initial longitudinal magnetic field and gradient of transverse magnetic field, respectively. 39

Force-free pinching of HFT Current density in the middle of HFT is Here and depend on the half-distance between spots, half-distance between polarities, source depth and magnetic field in spots. 1. 2. 3. 4. 5. Implications for solar flares The free magnetic energy is sufficient for large-scale flares. The effect of Spitzer resistivity is negligibly small. The current density is still not high enough to sustain an anomalous resistivity by current micro-instabilities. Tearing instability? underestimated? 40

Summary 1. The squashing and expansion-contraction factors Q and K are most important for analyzing field line connectivity in coronal magnetic configurations. 2. The application of theory reveals HFT that is a union of two QSLs. Thank you! 3. HFT appears in quadrupole configurations with sunspot magnetic fluxes of comparable value and a pronounced S-shaped polarity inversion line. 4. A twisting pair of shearing motions across HFT feet is an effective mechanism of magnetic pinching and reconnection in HFTs. 5. In twisted configurations the HFT pinching can also be caused by kink or other instability of the flux rope. 41