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