Current sheets formation along magnetic separators in 3d
description
Transcript of Current sheets formation along magnetic separators in 3d
Current sheets formation along magnetic separators in 3d
Dana Longcope
Montana State University
Thanks: I Klapper, A.A. Van BallegooijenNSF-ATM
The Coronal Field1 MK plasma(TRACE 171A)
Magnetic field @ photosphere (MDI)
8/10/01 12:51
8/11/01 9:25
Lower boundary: Bz confined to source regions
Corona: complexinter-connectionsbetween sources
Bz=0
The Coronal Field
8/10/01 12:51 8/11/01 17:39
8/11/01 9:25 1 MK plasma(TRACE 171A)
Magnetic field @ photosphere (MDI)
Evolution: lowerboundary changesslowly
(+30 hrs)
Outline
I. Lowest energy magnetic field contains current sheets localized to separators (Flux-Constrained Equilibrium)
II. Boundary motion drives field singular equilibrium via repeated Alfven wave reflection
I. EquilibriumForce-Free Equilibrium: Minimizes Mag. Energy*
xdWz
32
0
||2
1)}({
AxA
Constraints: (minimize subject to…)• None
• Ideal motion (line-tied to boundary)
AAA 0 B
BξA 0)( BB
potential
general FFF
*subject to BC: Bz(x,y,0) = f(x,y)
A new type of constraint
Boundary field Bz(x,y,0) = f(x,y): assume discrete sources
Bz=0
(Longcope 2001, Longcope & Klapper 2002)
A new type of constraint
The domain graph
Summarizes the Magnetic connectivity
N5
N4
N6 P3
P2
P1
Constrain coronal flux interconnecting sources
Structure of Constraint
N5
N4
N6 P3
P2
P1• Domain D16 connects P1 N6
• Flux in Domain D16:
(want to specify this)
• Flux in source 6:
(set by BC)
Structure of Constraint
N5
N4
N6 P3
P2
P1• Domain D16 connects P1 N6
• Flux in Domain D16:
(want to specify this)
• Flux in source 6:
(set by BC)
• Inter-related through
incidence matrix of graph:
i.e. �
A
A: Nc = Nd – Ns + 1
Structure of Constraint
N5
N4
N6 P3
P2
P1Q: How many domain fluxes
ab may be independantly specified?
Number of domains
Number of sources
here Nc = 7 – 6 + 1 = 2
Structure of Constraint
N5
N4
N6 P3
P2
P1
A: Nc = Nd – Ns + 1
Specifying fluxes of Nc chords reduces graph to a tree
Q: How many domain fluxes
ab may be independantly specified?
Structure of Constraint
N5
N4
N6 P3
P2
P1
A: Nc = Nd – Ns + 1
Specifying fluxes of Nc chords reduces graph to a tree … all remaining fluxes follow from flux balance: … etc.
Q: How many domain fluxes
ab may be independantly specified?
How to apply constraintsTopology of the potential field:*
• Extrapolate from bndry:
• Locate all magnetic null points B=0
0 B
• Trace spine field lines to spine sources
*same topology will apply to non-potential fields
The skeleton of the field
• Trace all fan field lines from null• Form sectors• Joined at separators• A separator connects + - null points
The skeleton of the field
• Trace all fan field lines from null• Form sectors• Joined at separators• A separator connects + - null points
The skeleton of the field
• Trace all fan field lines from null• Form sectors• Joined at separators• A separator connects + - null points
The skeleton of the field
• Trace all fan field lines from null• Form sectors• Joined at separators• A separator connects + - null points
Individual Domains Domain linking PaNb must be bounded by sectors +’ve sectors: Pa @ apex -’ve sectors: Nb @ apex
Sectors intersect @ closed separator circuitCircuit encircles domain
N5
N4
N6 P3
P2
P1
Formulating the constraint
Locate separator circuit Qi encircling domain Di:
0)}({ iQ
ii dF lAxAConstraint functional:
The Constrained Minimization
Minimize:Lagrange multiplier
Non-potential field: separator curve: Qi
annular ribbon xi(
xdScN
ii
iii
3
1
)()(
xA
ddFiQ
iii
ii
ii
xAxA )()}({
Singular density
-function
cN
iii FWC
1
)}({)}({)}({ xAxAxA All con-straints
• Vary
• Require stationarity: C = 0
• Get Euler-Lagrange equation
The Variation)()()( xAxAxA
CCC )}({)}({ xAxA
cc N
ii
N
ii
ii S
11
)()()( xJx
B
Singular current density, confined to separator ribbon i
Flux Constrained Equilibria
• Potential field (w/o constraints): i=i(v)
• Non-potential field: i=i(v) +i i=1,…,Nc
• Free Energy in flux-constrained field:
• General FFF:
0)()v( iWWW
)()v()(i
FFF WWW
N4
N2 P3
P1
23= 1
Flux Constrained Equilibria
• Min’m energy subject to Nc constraints
Nc fluxes are parameters: i
• Current-free within each domain
• Singular currents* on all separatorsN4
N2 P3
P1
1 1(v)
* Always ideally stable!
II. Approach to Equilibrium
• Separator defined
through footpoints
No locally
distinguishing
property*
• Singularity must build up through repeated reflection of information between footpoints
* In contrast to 2 dimensions: B=0 @ X-point
(Longcope & van Ballegooijen 2002)
Dynamics Illustrated
Equilib. fieldmaps sources tomerging layer
Long (almost straight)coronal field (RMHD)Maps between merging layers
Sourceson end planes
Dynamics Illustrated
N3
N4
Sep’x from nullon p-sphere
Dynamics Illustrated
Sources move (rotation)
N3
N4
Dynamics Illustrated
Sources move (rotation)
N3
N4
Dynamics Illustrated
Sources move (rotation)v
B
Initiates Alfven pulse (uniform rotation)
N3
N4
Dynamics Illustrated
Sources move (rotation)v
B
Initiates Alfven pulse (uniform rotation)
N3
N4
Dynamics Illustrated
Sources move (rotation)v
B
Initiates Alfven pulse (uniform rotation)
N3
N4
Dynamics Illustrated
Sources move (rotation)v
B
Initiates Alfven pulse (uniform rotation)
N3
N4
Dynamics Illustrated
v
B
Impact at Opposing End
v
B
P2
P1
Motion at mergingheight mapped down to photosphere
c.c rotation
Impact at Opposing End
P2
P1
Photosphere:fixed source positions,
moveable interiors
Merging height:No motion across sep’x
Free motion w/in source-regions
Impact at Opposing End
P2
P1
Photosphere:fixed source positions,
moveable interiors
Vorticitysheet @ sep’x
Merging height:No motion across sep’x
Free motion w/in source-regions
c.clockwise motion ineach region
Impact at Opposing End
P2
P1 N3
N4
Image of opposing separator is distortedby boundary motion
Impact at Opposing End
P2
P1 N3
N4
Image of opposing separator is distortedby boundary motion
Impact at Opposing End
P2
P1 N3
N4
Image of opposing separator is distortedby boundary motion
Impact at Opposing End
P2
P1 N3
N4
Intersection of separatrices: The Separator Ribbon
The Reflected Wave
Singular Alfven pulse:Voricity/Current sheetconfined to
P2
P1
The Reflected Wave
Separator ribbon left in wake of reflection
Repeated Reflection
z=0 z=L
• CS along reflects from z=0• CS along reflects from z=L• Repeated reflection retains only current on separator ribbon• Wave w/ current on ribbon - perfectly reflected
The Final Current Sheet
Interior CS(z=0)
B
Helical pitch, maps
z=0 z=L
The Final Current Sheet
Interior CS Helical pitch, maps
z=0 z=L
B
The Final Current Sheet
Interior CS Helical pitch, maps
z=0 z=L
B
The Final Current Sheet
Interior CS Helical pitch, maps
z=0 z=L
B
The Final Current Sheet
Interior CS Helical pitch, maps
z=0 z=L
Flux constrainedequilib.I set by e.g 23
B
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
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.
Individual Domains• Field lines from one source fan outward… topological ball of field lines• Opposing sources at surface of ball• Sectors partition ball
Individual Domains• Field lines from one source fan outward… topological ball of field lines• Opposing sources at surface of ball• Sectors partition ball
Individual Domains• Field lines from one source fan outward… topological ball of field lines• Opposing sources at surface of ball• Sectors partition ball
Individual Domains
Domain = Intersection of 2 balls… Intersection is a closed separator circuit
Circuit girdles domain
Negative sector in negative ball
Positive sector in positive ball
A 3d example•Ns=6 sources•Nd=7 domainsNc=2 circuits•4 nulls (A1 …B2)•2 null-null lines C2
C1
A current sheet
• Isolating loop Q1 links domain D34
• Current ribbon for (v) -- vertical