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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)
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Slava Titov
Institute for Theoretical Physics IV
Ruhr-Universität Bochum
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May 13, 2003
Page 1 of 22
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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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Page 3 of 22
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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
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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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Page 10 of 22
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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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Page 11 of 22
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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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Page 14 of 22
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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
Title Page
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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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Page 17 of 22
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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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Page 18 of 22
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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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Page 19 of 22
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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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Page 21 of 22
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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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