Modeling Magnetic Reconnection in a Complex

Solar CoronaDana Longcope

Montana State University

&

Institute for Theoretical Physics

The Changing Magnetic Field

TRACE 171: 1,000,000 K

8/10/01 12:51 UT

8/11/01 17:39 UT

8/11/01 9:25 UT (movie)

THE CORONAPHOTOSPHERE

Is this Reconnection?

TRACE 171: 1,000,000 K

8/10/01 12:51 UT

8/11/01 17:39 UT

8/11/01 9:25 UT (movie)

THE CORONAPHOTOSPHERE

Outline

1. Developing a model magnetic field2. A simple example of 3d reconnection3. The general (complex) case ---

approached via variational calculus.4. A complex example

The Sun and its field

Focus on the p-phere

And the corona just above

Modeling the coronal field

Example: X-ray bright points

EIT 195A image of “quiet” solar corona (1,500,000 K)

Example: X-ray bright points

Small specks occur above pair of magnetic poles (Golub et al. 1977)

Example: X-ray bright points

When 2 Poles Collide

All field lines from positive source P1

All field lines to negative source N1

Regions overlap when poles approach

When 2 Poles Collide

Stress applied at boundary

Concentrated at X-point to form current sheet

Reconnection releases energy

How it’s done in 2 dimensions

A Case Study

TRACE & SOI/MDI observations 6/17/98(Kankelborg & Longcope 1999)

The Magnetic ModelPoles Converging: v = 218 m/secPotential field: - bipole - changing

1.6 MegaVolts (on separator)

Mx 101.1 19

Mx/sec 106.1 14Mx 106.0 19

Reconnection Energetics

Flux transferred intermittently:

Current builds between transfers

Minimum energy drops @ transfer:

LI /LE /2

21

Post-reconnection Flux Tube

Flux Accumulated over

Releases stored Energy

Into flux tube just inside bipole (under separator)

Mx 108.1 17

min. 20t

ergs 106.1 26 IE Projected to bipole location

Post-reconnection Flux Tube

Flux Accumulated over

Releases stored Energy

Into flux tube just inside bipole (under separator)

Mx 108.1 17

min. 20t

ergs 106.1 26 IE

A view of the model

More complexity

From p-spheric field (obs).

Find coronal coronal field

Defines connectivity

Minimum Energy: Equilibrium

• Magnetic energy

• Variation:

• Fixed at photosphere:

Potential field

V

xdW 32||2

1A

),()0,,( yxfyxBz

0)( A

)()()( xAxAxA 0W

Minimization with constraints

• Ideal variations only

force-free field

• Constrain helicity

( w/ undet’d multiplier constant- fff

BA

0)( BB

V

xdH 3)()( xBxA

BB

A new type of constraint…

Photospheric field: f(x,y) -- the sources

…flux in each domain

Domain fluxes

• Domain Dij connects sources Pi & Nj

• Flux in source i:• Flux in Domain Dij

• Q: how are fluxes related:

A: through the graph’s incidence matrix

�

A

iji

The incidence matrix

•Ns Rows: sources•Nd Columns: domains Nc = Nd – Ns + 1 circuits

The incidence matrix

36343

�

A

Reconnectionpossible allocation of flux…

Reconnection… another possibility

ReconnectionRelated to circuit in the domain graph

Must apply 1 constraint to every circuit in graph

Separators: where domains meet

4 distinct flux domains

Separators: where domains meet

4 distinct flux domains

Separator at interface

Separators: where domains meet

4 distinct flux domains

Separator at interface

Closed loop encloses all flux linking P2N1

Minimum W subj. to constraint

Constraint on P2N1 flux

Current-free within each domain

current sheet at separator

Minimum W subj. to constraint

2d version: X-point @ boundary of 4 domains becomes current sheet

A complex example

Ns = 20

A complex example

Ns = 20 Nc = 33

The original case study

Approximate p-spheric field using discrete sources

The domain of new flux

Emerging bipole P01-N03

New flux connects P01-N07

Summary

• 3d reconnection occurs at separators

• Currents accumulate at separators

store magnetic energy

• Reconnection there releases energy

• Complex field has numerous separators