Magnetic field lines and their reconnection

Dana Longcope

Montana State University

Reconnection: the magnetosphere

Courtesy R. Winglee

Dungey 1962

Reconnection: solar corona

Courtesy TRACE team

Kopp &

Pneman

1976

Reconnection: the heliosphere

Courtesy J. Gosling

Reconnection: Parallel wires in vacuum

The X-point (null point): B = 0

Reconnection: Parallel wires in vacuum

Separate wires slowly

B da Lz Bydx

Ez = d /dt

d

dtE dl LzEz(0)

• Good conductor: E = 0

• Good moving conductor: E + v B = 0

• Ez + (v B)z = 0

• = const.

B = 0

Ez = d /dt

• Mag. Energy >> plasma energy ( <<1)

• Move slowly

• Min. magnetic energy

+Bi

-Bi

0J B 2Bx(x,0 ) (y) ˆ z

Magnetic field discontinuity: tangential discontinuity

J B B B 12

ˆ y y

B20

Magnetic equilibrium

Minimum magnetic energy state

wire separation

net

cu

rren

t

Ics

1

0

B dl2

0

Bx(x,0 )dxL

L

Work done on wires

wire separation

FFx x

w/ cs

w/o cs

= const

Ez ≠ 0W

Good vs. Fair conductance

0J B 2Bx(x,0 ) (y) ˆ z

Can E = 0 when J = ∞?

Assume E = 0…

Ohm’s law• Q: What makes a conductor “good”? (E=0)

(e.g. copper or gold? a plasma?)

• A: electrons move to eliminate E• Q: What might limit “goodness”? (E≠0)

• A: electrons cannot respond effectively

momentum eq. of electron fluid

mene

dve

dt

t P e eneE eneve B mene ei(vi ve)

E vi Bme

e

dve

dt

1

ene

t P e

1

ene

J Bme ei

e2ne

J

drag

Hall term 1/ eelectron inertia

J ene(vi ve)

Generalized Ohm’s law

E vi Bme

e

dve

dt

1

ene

t P e

1

ene

J B eJ

(i) (iv) (v)

(i)

(iv)

vB

B2 / 0enel

vl

B / 0ene

v

vA

l

c / pi

(i)

(v)

vB

eB / 0l

l v 0

e

Rm

(ii) (iii)

(i)

(ii)

vB

mevB / 0e2nel

2

l 2

me / 0e2ne

l

c / pe

2

Importance of term dependson length scaleof solution

On large scalesplasma is ideal conductor*

What’s really important?

(i)

(iii)

vB

pe /enel

vl

kBT /eB

v

cs

l

i * except where B=0

How large is “large”?Earth’s

magnetotailSolar

corona

Externalscale

L 108 m 108 m

Collisionlessskin depth

c/ pe 104 m 0.1 m

ion skin depth

c/ pi 106 m 10 m

Ion gyro-radius i 105 m 0.1 m

LL

All small :E+v B=0

Good vs. Fair conductance

Scale of ``external’’ solution: large

Solution develops smallerscale spontaneously

Good vs. Fair conductance

E + v B ≠ 0

Ideal region

diffusion region

E = - v BExternal Eset by inner solution

≠ 0

E = - v B ≠ 0

Ideal region

Work done on wires

Ez ≠ 0

W

Ez = 0

separate wires slowly

Build current, store energy

release energy

IcsE dt Ics

d

dtdt IcsdW Spontaneous & irreversible

Ez = -d /dt = vy Bx

2L

Total reconnection

time

LBxrx

d /dt

L Bx

vyBx

rx

L

vi

vA

vi

A

A

MAi

MAi

v i

vA

A

L

vA

MAi << 1 : Slow reconnection

MAi ~ 1 : Fast reconnection

2

2

vi

vo

Mass conservation:vi = vo

MAi

vi

vA

vo

vA

Aspect ratio of diffusion region

~1

Classic Ohm’s law

E vi B me

dve

dt

1

ene

t P e

1

ene

J B eJ

E z eJz ~eBi

0

Ez viBi

2

v i ~e

0

MAi

vi

vA

e

vA 0

e

0LvA

LLu 1 L

MAi

MAi Lu 1/ 2 L

2

2L

Ideal region

diffusion region

Lu 1/2 L

advection balancesdiffusion

from mass conservation

&

Rmv i 0

e

~ 1

2 = 2L

2

vi

vo

Special case: Sweet-Parker

uniform: =L

MAi Lu 1/ 2 LLu 1/ 2 1

LLu1/ 2

LLu1/ 2 1

v. slow reconnection

v. great aspect ratio

Collisionless reconnection

E vi B me

dve

dt

1

ene

t P e

1

ene

J B eJ

2

MAi ~12

2L

from mass conservation

c/ pi from Hall term

Drake et al. 2008

c/ pi

2

How does small region affect external field?

2

It creates bent field lines…

… what next?

SMSSMSCS

FMRWFMRW

v vv v

Response to bendRiemann problem for 1D current sheet (CS)

Lin & Lee 1994

t=0

hot & dense

Petschek reconnection

vA vA

SMS

SMS

SMS

SMS

vi

External solution requires current -current appears in SMSs

Q: Will resistivity always result in slow (Sweet-Parker) reconnection?

Bis

kam

p&

Scw

artz

20

01

A: Yes, if is uniform in space…

But not, when (x) is locally enhanced*

JzSMS

enhanced

*as by micro-instability

Q: Is slowly reconnecting sheet stable?

Bhattacharjee et al. 2009 (vertical scale is expanded)

Lu = 6.3 X 105

A: No. Subject to resistive instability: tearing mode

• Solution becomes time-dependant• Smaller diffusion region(s) develop w/ larger ~ MAi

Q: what triggers reconnection in the CS?

Reeves & Forbes 2005?

SeparatricesThe X-point (null point): B = 0

B(x,y)0 B'

B' 0

x

yL

d

ds

x

y

x(s)

y(s)

1

1emB 's

approaches null as s ±∞

e-values = B’

Bold new world: 3d

B(x,y,z)

Bx

x

Bx

y

Bx

z

By

x

By

y

By

z

Bz

x

Bz

y

Bz

z

x

y

z

Ld

ds

x

y

z

B(0,0,0) 0

Null point (@ origin)

Mij

det(M) < 0: “positive” null point

2 pos. e-values (fan surface)

1 neg. e-value (spines)

fan = separatrix

det(M) > 0: “negative” null point

(reverse all arrows)

B 0 Tr(Mij ) i

July 31, 2007

B

July 31, 2007

Fans & Null-Null Lines

Feb. 19, 2010

separatrix between new & old positive flux

Feb. 19, 2010

1:592:05

1:59

21 MK

42 MK

RHESSI

2:05

July 31, 2007

July 31, 2007

July 31, 2007

July 31, 2007

July 31, 2007

A Real Example

PHOTOSPHERE

CORONA

TRACE 171A

movie

SOHO/MDI

July 31, 2007

July 31, 2007

July 31, 2007

The separator

Solar-B Science Meeting Kyoto, Nov. 9, 2005

Reconnection observed

24 hour delay

Reconn’n burst1017 Mx/sec

Heating ~ 5 x 1026 erg/sec

emergence

= 1021 Mx

0.1 GV1 GV

Summary

• Plasma behaves as “perfect conductor” on large* scales

• Perfect conduction leads to spontaneous development of small scales – sows seeds of its own violation

• Non-ideal response: reconnectionattempts to eliminate small scales

* In space, all scales are large