Structure, Equilibrium and Pinching of Coronal Magnetic Fields

Structure, Equilibrium and Pinching of Coronal Magnetic Fields

Slava TitovSAIC, San Diego, USA

Seminar at the workshop „Magnetic reconnection theory“Isaac Newton Institute, Cambridge, 18 August 2004

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AcknowledgementsCollaborators

• 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)

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Outline1. 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.

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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!

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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

6field 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 BPsTitov et al. (1993);Bungey et al. (1996);Titov & Démoulin (1999)

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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.

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connectivity? • Structural features of coronal magnetic fields:






6. Summary.

Titov et al., JGR (2002)

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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

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Again two possibilities: •Jacobi matrix:

•inverse Jacobi matrix:

Field line mapping: local description

Not tensor!

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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

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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

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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 ~ Q1/2.

Orthogonal parquet(complete description of magnetic connectivity)

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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:

Critical points of orthogonal parquet

Proof

Look at your fingerprints!

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6. Summary.

Titov & Hornig, COSPAR (2000); Titov et al., JGR (2002)

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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).

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

Squashing factor Q

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Blue and red areas are connected by flux tubes

Expansion-contraction factor K

to bridge the regions of weak and strong photospheric fields.

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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)

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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

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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

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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

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6. Summary.

Titov & Démoulin, A&A (1999); Kliem et al., Török et al., A&A (2004)

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Construction of the model Magnetogram

•a/R << 1 and a/L << 1;

•outside the tube the field is B=Bq+BI+BI0;

•inside the tube it is approximately the field of a straight flux tube.

Twisted force-free configuration

Basic assumptions:

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Matching condition is in the vicinity of the tubeor 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

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Stability criterion:

constant.

Equilibrium current

Minor radius changes with according to

to keep the number of field-line turns

unstable

Checked and improvednumerically byRoussev et al. (2003)

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„fishhooks“with Qmax~ 108

Squashing factor Q

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

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HFT in twisted configuration

(its spread from N- to S-footprint)

Variation of cross-sections along a twisted HFT:

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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?

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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!

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Current sheets around a kinking tube

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6. Summary.

Titov et al., ApJ (2003); Galsgaard et al., ApJ (2003)

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NB:sunspots crossingthe HFT footprintsin opposite directions,must generateshearing flows in between.

Simplified (straightened) HFT

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Two extremes: turn versus twist

Twisting shears muststrongly deform the HFTin the middle.

Turning shears mustrotate the HFT as a whole

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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

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Comparison with numerics

No currentin the middle!

Current sheetin the middle!

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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

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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

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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 flares1. 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?

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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!

Documents

Structure, Equilibrium and Pinching of Coronal Magnetic Fields