Structure, Equilibrium and Pinching of Coronal Magnetic Fields

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

2. Acknowledgements Collaborators • on structure: • Pascal Démoulin (Paris-Meudon Observatory, France) • Gunnar Hornigand • 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 InstitutePotsdam,Germany) • Tibor Törok(Mullard Space Science Laboratory, UK)

3. Outline • 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). • 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). • Summary.

4. 2D case: field line connectivity and topology normal field line BP separtrix field line NP separtrix field line • Flux tubes enclosing separatrices split at null pointsor "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. Generic magnetic nulls in 3D 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 flowsonly Sustained by either field-aligned or spiral field-crossing flows • Magnetic nulls are local topological features: • field lines emanating from nulls formseparatrix surfaces.

6. Field line structure at Bald Patches (BPs) in 3D 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.

7. Extra opportunity in 3D: 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 notapplicable anymore; other approach is required. Nevertheless, thin QSLs are as importantas genuine separatrices for this process.

8. Outline • 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). • 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). • Summary. Titov et al., JGR (2002)

9. 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

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

11. 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.

12. 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.

13. Orthogonal parquet (complete description of magnetic connectivity) • Construction • The major and minor axes of infinitezimal ellipses define on the photospere twofields of directions orthogonalto 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 ~ Q1/2. • Such separatrices devide the photosphere on domains with a simple structure of parquet.

14. Critical points of orthogonal parquet • The orthogonality is violated if a mapped ellipse degenerates into a circle. • This occurs at two types of (critical) points: Proof Look at your fingerprints! • Such separatrices devide the photosphere on domains with a simple structure of parquet.

15. Outline • 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). • 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). • Summary. Titov & Hornig, COSPAR (2000); Titov et al., JGR (2002)

16. 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 usualNS-direction).

17. SquashingfactorQ Crescent strips of high Qconnect sunspots of the same polarity.

18. Expansion-contraction factorK Blue and red areas are connected byflux tubes tobridge the regions of weak and strong photospheric fields.

19. Hyperbolic Flux Tube (HFT) (its spread from N- to S-footprint) • 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.

20. 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.

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

22. Simple domains of orthogonal parquet • 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.

23. Outline • 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). • 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). • Summary. Titov & Démoulin, A&A (1999); Kliem et al., Török et al., A&A (2004)

24. Twisted force-free configuration Basic assumptions: • a/R << 1 and a/L << 1; • outside the tube the field isB=Bq+BI+BI0; • inside the tube it is approximately the field of a straight flux tube.

25. 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 thatthe total equilibrium current

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

27. Squashing factor Q „fishhooks“ withQmax~ 108

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

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

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. 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. Current sheets around a kinking tube

33. Outline • 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). • 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). • Summary. Titov et al., ApJ (2003); Galsgaard et al., ApJ (2003)

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

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

36. Deformations of the mid part of HFT • 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.

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

38. Pinching system of flowsin 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.

39. 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.

40. Force-free pinching of HFT • 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? • 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.

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