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Hyperbolic flux tubes:

the criteria for their existence

and magnetic pinching

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

(5. MHD-Tage, Dresden, November 25-26, 2002)

Title Page

Slava Titov

Institute for Theoretical Physics IV

**Ruhr**-Universität **Bochum**

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May 13, 2003

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Collaborators:

Gunnar Hornig (**Ruhr**-Universität **Bochum**, Germany),

Pascal Démoulin (Paris-Meudon Observatory, France),

Klaus Galsgaard and Thomas Neukirch (St Andrews University, Scotland).

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Contents

1. Introduction.

Magnetic topology and field-line connectivity.

Description of the field-line connectivity.

2. Hyperbolic Flux Tubes (HFTs) in quadrupole configurations.

3. Magnetic pinching of HFTs (current layer formation in HFTs).

4. Summary.

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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Page 2 of 22

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1. Introduction

Q: What are the generic conditions for the formation of current sheets

in large-scale solar flares?

A: The configuration must contain

1. special geometrical features (hyperbolic magnetic flux tubes),

2. special type of shearing photospheric motions in their vicinity.

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

Topological paradigm:

current sheets are due to the presence of magnetic nulls and separatrix

lines/surfaces.

Observational evidences:

however, many flares occur in magnetic configurations without nulls.

The clue to the necessary generalization is in geometrical approach based

on the concept of magnetic field-line connectivity.

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Magnetic topology and field-line connectivity

in 2D and 2 1 2 D cases

field-line connectivity

topology

Introduction

normal

field line:

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

N

separatrix

field line:

corona

photosphere

null point

S

separatrix

field line:

N

NS-direction

S

N

bald

patch

point

SN-direction

S

Summary

Appendix

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Page 4 of 22

N

S

N

S

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• Flux tubes enclosing separatrices split at null and “bald patch” points.

• Current sheets are formed along the separatrices by suitable photospheric

motions.

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Extra opportunity in 3D: squashing instead of splitting

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

Title Page

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• Metric is needed ⇒ QSL is not topological

but geometrical object.

• For current layer formation thin QSLs must be

as important as genuine separatrices.

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Description of the field-line connectivity

Field-line mapping: global description

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

Title Page

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Cartesian coordinates ⇒ distance between footpoints.

Coronal magnetic field lines are closed ⇒ mapping Π:

from positive to negative polarity

+Π −

= (X − (x + , y + ), Y − (x + , y + )),

from negative to positive polarity

−Π +

= (X + (x − , y − ), Y + (x − , y − )).

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Field-line mapping: local description

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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( ∂X−

∂x +

∂Y −

∂x +

( ∂X+

)

∂X −

∂y +

∂Y −

∂y +

∂x −

∂X +

∂y −

∂Y +

∂x −

∂Y +

∂y −

)

≡

( )

a b

c d

= 1 (

d −b

∆ +

−c a

∆ + = ad − bc

)

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Invariants of the field-line mapping Π

The degree of squashing of elemental flux tubes in QSLs

(Titov, Démoulin & Hornig, 1999)

Q = (a2 + b 2 + c 2 + d 2 )

|ad − bc|

and the degree of expansion (K < 0) or contraction (K > 0)

(Titov & Hornig, 2002)

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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|K| ≡= | lg |ad − bc||.

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Here a, b, c and d are the elements of the Jacobi matrix

(irrespective of the direction of the mapping Π)

( ∂X ∂X

) ( )

∂x ∂y a b

D =

= .

c d

∂Y

∂x

∂Y

∂y

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2. Hyperbolic Flux Tubes (HFTs)

in quadrupole configurations

Magnetogram

Introduction

Source depth = 0.1

internal flux

total flux

= 0.4

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

Contours ∣ of B z :

B

∣ z ∣∣

B zmax

= 10

−0.5 n ,

n = 1, . . . , 5

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Page 9 of 22

Magnetic topology is trivial:

• no nulls of magnetic field,

• no field lines touching the photosphere

(the field at the inversion line has the NS-direction).

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Squashing degree Q

(Titov & Hornig, 2002)

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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Expansion-contraction degree K = lg |∆ + |

(Titov & Hornig, 2002)

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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Structure of Hyperbolic Flux Tubes (HFTs)

(Titov, Hornig & Démoulin, 2002)

Magnetic flux surface Q = const = 10 2

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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• The HFT is a union of two intersecting QSLs.

• Source depth → 0 ⇒ the surface Q = Q max /2 collapses into

two genuine separatrix surfaces intersecting along the separator.

• Any field line in HFTs connects the areas of strong and weak magnetic

field on the photosphere.

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A half of the surface Q = const = 10 2

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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Page 13 of 22

Cross-sections of an HFT:

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3. Magnetic pinching of HFTs

(Titov, Galsgaard & Neukirch, 2003)

Q: What is the reason of current layer formation in HFTs

(magnetic pinching of HFTs)?

A: It is a twisting pair of shearing motions across HFT feet.

A half of the surface Q = const = 10 2

and the pinching system of plasma flows

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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“Straightened” version of HFT

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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Page 15 of 22

Q: What happens if sunspots start to cross HFT feet in opposite directions?

A: It will generate shearing flows in between.

The resulting HFT deformation is shown in the next slide.

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Non-pinching and pinching shears across HFT feet

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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Extrapolated velocity field:

[ (

1 −

L)

z tanh

v = V s

2

( y

l sh

)

ˆx ∓

where V s is a characteristic velocity of sunspots,

l sh is a characteristic scale of shear,

L is a half-distance between the polarities.

(

1 + z ) ( ) ] x

tanh ŷ

L l sh

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Basic estimations

Current layer parameters for the kinematically pinching HFT

The width is

The thickness is

∆ ≃ 2.5 l sh .

δ ≃ 2 √ 2 e −¯t l sh ,

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

where

¯t = V st

2 l sh

.

The longitudinal current density in the middle of the pinching HFT is

jz ∗ ≡ j z | r=0

≃ e 2¯t h (

1 + B )

‖

,

µ 0 2hL

where B ‖ is a longitudinal magnetic field and h is a gradient of transverse

magnetic field.

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Current density in the middle of the force-free pinching HFT

jz ∗ eq ≃ e 2¯t h (

1 + B ) [ (

‖

1 + e 2¯t 0.91 hl sh

+ 0.57 l ) ] 2

sh

.

µ 0 2hL

B ‖ L

Here B ‖ and h depend on the half-distance l between the spots, source

depth d and and magnetic field in the spots B s .

At ¯t = 2.5

• The effect of Spitzer resistivity

is negligibly small.

• The current density j ∗ z eq is still

not high enough to sustain an

anomalous resistivity by current

micro-instabilities.

• Tearing instability?

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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4. Summary

1. The degrees of squashing Q and expansion-contraction K of elemental

flux tubes provide the most important information on field-line

connectivity.

2. Application of the theory to quadrupole potential configuration shows

its high efficiency in analyzing of magnetic field structure in the solar

corona.

3. The application reveals Hyperbolic Flux Tube (HFT) that is a union

of two intersecting Quasi-Separatrix Layers (QSLs).

4. HFT appears in quadrupole configurations with sunspot magnetic

fluxes of comparable value and a pronounced S-shaped polarity inversion

line.

5. A twisting pair of shearing motions across HFT feet is a reason of

magnetic pinching and reconnection in HFTs.

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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5. Appendix: the related works

1. Priest & Démoulin (1995), QSL theory.

2. Sweet (1969), the source model of quadrupole configuration.

3. Gorbachev & Somov (1988, 1989), an attempt to generalize the

source model for the case of distributed flux, H α -ribbons and knots,

photospheric vortex flows.

4. Galsgaard & Nordlund (1996), current layer formation by stagnation

flow due to “interlocking” double-shear flows in braiding flux model.

5. Cowley, Longcope & Sudan (1997) current layer formation by photospheric

stagnation flows.

6. Syrovatskii (1981), 2D current sheet formation as a hyperbolic pinch

process.

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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Comparison with Gorbachev-Somov model

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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Field-line structure in HFT and K-distribution

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Separatrix surfaces in the 4-source model

Introduction

Hyperbolic Flux Tubes . . .

Magnetic pinching of . . .

Summary

Appendix

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e 1 , e 2 , e 3 , e 4 – magnetic “charges” (≡ sunspots)

(Sweet, 1958), (Baum & Bratenahl, 1980)

Generalization for distributed sources?

(Gorbachev & Somov, 1988, 1989).

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