1. 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

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

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

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

5. 5 Generic magnetic nulls in 3D Magnetic nulls are local topological features: Skewed improper radial null Skewed improper spiral null field lines emanating from nulls form separatrix surfaces. 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

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

7. 7 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. Extra opportunity in 3D: squashing instead of splitting Nevertheless, thin QSLs are as important as genuine separatrices for this process.

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

9. 9 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 Field line mapping: global description

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

11. 11 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. 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. Squashing factor Q Norm squared, Priest & Démoulin, 1995

12. 12 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. 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. Expansion-contraction factor K

13. 13 Construction  The major and minor axes of infinitezimal ellipses define on the photospere two fields of directions orthogonal to each other.  A family of their integral lines forms an orthogonal network called parquet.  Parameterization of the lines such that the aspect ratio of tiles ~ Q 1/2. Such separatrices devide the photosphere on domains with a simple structure of parquet.domains with a simple structure of parquet Orthogonal parquet (complete description of magnetic connectivity)

14. 14 I-point Y-point One separatrix emanates. I-point is at the common side of two adjoint triangles. Three separatrices emanate. Y-point is a vertex of six adjoint tetragons.  The orthogonality is violated if a mapped ellipse degenerates into a circle.  This occurs at two types of (critical) points: Such separatrices devide the photosphere on domains with a simple structure of parquet.domains with a simple structure of parquet Critical points of orthogonal parquet Proof Look at your fingerprints!

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

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

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

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

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

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

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

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

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

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

25. 25 (5)(5) 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 Equilibrium condition

26. 26 Stability criterion: (5)(5) constant. Equilibrium current Minor radius changes with according to to keep the number of field-line turns unstable Checked and improved numerically by Roussev et al. (2003)

27. 27 „fishhooks“ with Q max ~ 10 8 Squashing factor Q

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

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

30. 30 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?

31. 31 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!

32. 32 Current sheets around a kinking tube

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

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

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

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

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

38. 38 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. Pinching system of flows in quadrupole configuration

39. 39 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. Basic kinematic estimates

40. 40 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. Force-free pinching of HFT Implications for solar flares 1.The free magnetic energy is sufficient for large-scale flares. 2.The effect of Spitzer resistivity is negligibly small. 3.The current density is still not high enough to sustain an anomalous resistivity by current micro-instabilities. 4.Tearing instability? 5. underestimated?

41. 41 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 the theory reveals HFT that is a union of two QSLs. 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. Summary Thank you!