Current Sheet Formation Along Magnetic Separators in 3d

1
Current Sheet Formation Along Magnetic Separators
in 3d

• Dana Longcope
• Montana State University

Thanks I. Klapper, A.A. van Ballegooijen NSF-ATM
2
The Coronal Field
1 MK plasma (TRACE 171A)
8/11/01 925
Corona complex inter-connections between sources
Bz0
Lower boundary Bz confined to source regions
8/10/01 1251
Magnetic field _at_ photosphere (MDI)
3
Outline

• I. GEOMETRY
• Lowest energy equilibrium contains
• current sheets localized to separators
• (Flux-Constrained Equilibrium)
• II. DYNAMICS
• Boundary motion drives field ? singular
• equilibrium via repeated Alfven wave
• reflection from boundaries

4
I. Equilibrium
Force-Free Equilibrium Minimizes Mag. Energy
subject to BC Bz(x,y,0) f(x,y)

• Constraints (minimize subject to)
• None (? absolute minimum)
• Ideal motion (line-tied to boundary)

potential
?
general FFF
?
5
A new type of constraint
(Longcope 2001, Longcope Klapper 2002)
Bz0
Boundary field Bz(x,y,0) f(x,y) assume
discrete sources
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A new type of constraint
Constrain coronal flux interconnecting sources
The domain graph
N4
P1
P2
N5
N6
P3
Summarizes the Magnetic connectivity
7
Structure of Constraint

• Domain D16 connects P1 ? N6
• Flux in Domain D16 y16
• (want to specify this)
• Flux in source 6 F6
• (set by BC)

N4
P1
P2
N5
N6
P3
8
Structure of Constraint

• Domain D16 connects P1 ? N6
• Flux in Domain D16 y16
• (want to specify this)
• Flux in source 6 F6
• (set by BC)
• Inter-related through
• incidence matrix of graph
• i.e. - y16 - y26 - y36 F6

N4
P1
P2
N5
N6
P3
9
Structure of Constraint

• Q How many domain fluxes
• yab may be independantly specified?

N4
P1
P2
N5
N6
P3
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Structure of Constraint

• Q How many domain fluxes
• yab may be independantly specified?

N4
P1
A Nc Nd Ns 1
P2
N5
N6
P3
Number of domains
Number of sources
here Nc 7 6 1 2
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Structure of Constraint

• Q How many domain fluxes
• yab may be independantly specified?

y14Y2
N4
P1
y34Y1
A Nc Nd Ns 1
P2
N5
Specifying fluxes of Nc chords reduces graph to a
tree
N6
P3
12
Structure of Constraint

• Q How many domain fluxes
• yab may be independantly specified?

y14Y2
N4
P1
y34Y1
A Nc Nd Ns 1
P2
N5
Specifying fluxes of Nc chords reduces graph to a
tree all remaining fluxes follow from flux
balance y36F3-Y1 etc.
N6
P3
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How to apply constraints

• Topology of the
• potential field
• Extrapolate from
• bndry
• Locate all magnetic
• null points B0

• Trace spine field lines
• to spine sources

same topology will apply to non-potential fields
14
The skeleton of the field

• Trace all fan field
• lines from null
• Form sectors
• Joined at
• separators
• A separator
• connects ? -
• null points

15
The skeleton of the field

• Trace all fan field
• lines from null
• Form sectors
• Joined at
• separators
• A separator
• connects ? -
• null points

16
The skeleton of the field

• Trace all fan field
• lines from null
• Form sectors
• Joined at
• separators
• A separator
• connects ? -
• null points

17
The skeleton of the field

• Trace all fan field
• lines from null
• Form sectors
• Joined at
• separators
• A separator
• connects ? -
• null points

18
Individual Domains
Domain linking Pa?Nb must be bounded by
sectors -ve sectors Pa _at_ apex ve
sectors Nb _at_ apex
Sectors intersect _at_ closed separator
circuit Circuit encircles domain Every domain is
encircled
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Formulating the constraint
N4
P1
y34Y1
P2
N5
N6
P3
To constrain flux Yi locate circuit around
domain Di
Constraint functional
20
The Constrained Minimization
Lagrange multiplier
Minimize
All con- straints
Non-potential field separator curve Qi ?
annular ribbon xi(x,h)
d-function
Singular density
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The Variation

• Vary
• Require stationarity dC 0
• Get Euler-Lagrange equation

Singular current density, confined to separator
ribbon i
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Flux Constrained Equilibria
P1
N4
y23 Y1
P3
N2

• Potential field (w/o constraints) YiYi(v)
• Non-potential field YiYi(v) DYi
  i1,,Nc
• Free Energy in flux-constrained field
• General FFF

23
Flux Constrained Equilibria

• Minm energy subject to Nc constraints
• ? Nc fluxes are parameters Yi
• Current-free within each domain
• Singular currents on all separators

P1
N4
P3
N2
Y1 ltY1(v)
Always ideally stable!
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II. Approach to Equilibrium
(Longcope van Ballegooijen 2002)

• Separator defined
• through footpoints
• ? No locally
• distinguishing
• property
• Singularity must build up through repeated
  reflection of information between footpoints

In contrast to 2 dimensions B0 _at_ X-point
25
Dynamics Illustrated
Sources on end planes
Long (almost straight) coronal field (RMHD) Maps
between merging layers
Equilib. field maps sources to merging layer
26
Dynamics Illustrated
N3
S-
N4
Sepx from null on p-sphere
27
Dynamics Illustrated
N3
N4
Sources move (rotation)
28
Dynamics Illustrated
N3
N4
Sources move (rotation)
29
Dynamics Illustrated
Dq
N3
N4
Bf
Sources move (rotation)
vf
Initiates Alfven pulse (uniform rotation)
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Dynamics Illustrated
Dq
N3
N4
Bf
Sources move (rotation)
vf
Initiates Alfven pulse (uniform rotation)
31
Dynamics Illustrated
Dq
N3
N4
Bf
Sources move (rotation)
vf
Initiates Alfven pulse (uniform rotation)
32
Dynamics Illustrated
Dq
N3
N4
Bf
Sources move (rotation)
vf
Initiates Alfven pulse (uniform rotation)
33
Dynamics Illustrated
Bf
vf
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Impact at Opposing End
P1
S
P2
c.c rotation
Bf
Motion at merging height mapped down to
photosphere
vf
35
Impact at Opposing End
P1
S
P2
Merging height No motion across sepx S Free
motion w/in source-regions
Photosphere fixed source positions, moveable
interiors
36
Impact at Opposing End
c.clockwise motion in each region
P1
S
Vorticity sheet _at_ sepx
P2
Merging height No motion across sepx S Free
motion w/in source-regions
Photosphere fixed source positions, moveable
interiors
37
Impact at Opposing End
P1
N3
S
S-
N4
P2
Image of opposing separator is distorted by
boundary motion
38
Impact at Opposing End
P1
N3
S
S-
N4
P2
Image of opposing separator is distorted by
boundary motion
39
Impact at Opposing End
P1
N3
S
S-
N4
P2
Image of opposing separator is distorted by
boundary motion
40
Impact at Opposing End
P1
N3
S
S-
N4
P2
Intersection of separatrices The
Separator Ribbon
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The Reflected Wave
P1
S
P2
Singular Alfven pulse Voricity/Current
sheet confined to S
42
The Reflected Wave
S
S-
Separator line in initial field Separator ribbon
left in wake of reflection
43
Repeated Reflection
S
S-
z0
zL

• CS along S reflects from z0
• CS along S- reflects from zL
• Repeated reflection retains only current on
• ?? separator ribbon
• Wave w/ current on ribbon - perfectly reflected

44
The Final Current Sheet
S
S-
B
Helical pitch, maps S ? S- z0 ? zL
Interior CS (z0)
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The Final Current Sheet
S
S-
B
Helical pitch, maps S ? S- z0 ? zL
Interior CS
46
The Final Current Sheet
S
S-
B
Helical pitch, maps S ? S- z0 ? zL
Interior CS
47
The Final Current Sheet
S
S-
B
Helical pitch, maps S ? S- z0 ? zL
Interior CS
48
The Final Current Sheet
S
S-
B
Helical pitch, maps S ? S- z0 ? zL
Flux constrained equilib. I set by e.g y23
Interior CS
49
Conclusions

• New class of constraints domain fluxes
• Flux constrained equilibria have CS on
• all separators
• Equilibrium is approached by repeated
• Alfven wave reflections from boundary

50
Implications

• Coronal field tends toward singular state
• Current sheets are ideally stable
• Magnetic reconnection can
• Eliminate constraint
• Decrease magnetic energy
• Free energy depends on flux in NC different
  fluxes within corona.

51
Individual Domains

• Field lines from one source
• fan outward topological
• ball of field lines
• Opposing sources at
• surface of ball
• Sectors partition ball

52
Individual Domains

• Field lines from one source
• fan outward topological
• ball of field lines
• Opposing sources at
• surface of ball
• Sectors partition ball

53
Individual Domains

• Field lines from one source
• fan outward topological
• ball of field lines
• Opposing sources at
• surface of ball
• Sectors partition ball

54
Individual Domains
Negative sector in negative ball
Positive sector in positive ball
Domain Intersection of 2 balls Intersection
is a closed separator circuit Circuit girdles
domain
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A 3d example

• Ns6 sources
• Nd7 domains
• ?Nc2 circuits
• 4 nulls (A1 B2)
• 2 null-null lines

C1
C2
56
A current sheet

• Isolating loop Q1 links domain D34
• Current ribbon for y34 lt y(v) -- vertical

57
The Coronal Field
8/11/01 925
1 MK plasma (TRACE 171A)
Evolution lower boundary changes slowly
8/10/01 1251
8/11/01 1739
Magnetic field _at_ photosphere (MDI)
(30 hrs)