L = 6 × 104 km

70 Gauss

50 Gauss

WB = 1032 ergs

:Magnetic Energy Conversion

ideal MHD reconnection

(< 10%)inefficient (100%)efficient

currents

Force-free fields: j || B

How is Energy Stored?

β = 10–3

:Current sheets

reconnection whenj > jcritical

emerging flux model sheared magnetic fields

j × B ≈ 0∇p ≈ 0

How Much Energy is Stored?

B = Bphotospheric + Bcoronal

invariantduring CME

source ofCME energy

jcorona

+

–

photospheric sources

Hα imageplage

prominence

& (1997)from Gaizauskas Mackay

model with magnetogram

B from corona ≈ B from photosphere

free magnetic energy≈ 50% of total magnetic energy

currents currents

.

(a) (b) (c)

impossibletransition

(a)(b)

(c)

time

ideal

resistive

forbidden

Aly - Sturrock Paradox

. .

time

7

6

5

4

3

2

1

00 2 4 6

Flux-Rope Height h

Source Separation 2λ

current sheetforms

critical point

8

jump

4–4 0

4–4 0

4–4 0

x

y

8

0

y

8

0

y

8

0

λ = 0.96

λ = 0.96

λ = 2.50time

A Storage Model

. .

t = 36 s

t = 87 s

x

y

y

x

t = 129 s

log P 0–2

0.0

8.0

4.0

150 300 4500

t (sec)

Numerical Simulation of Critical Point Configuration

t = 0 s

initial condition: V = 0

energy equation: Ohmic heating no coolingresistivity: uniform, S = 500

.

Three Dimensional Equilibria

dipole (strapping field)

flux rope 0

1

–1

distancefrom Sun1 2 3 4

flux rope hoop force

dipole attraction(strapping field)

Equilibrium is unstable to torus* instability

* see Bateman (1973): instability if r –α; α > 3/2 : for dipoleα = 3

unstable example

Van Tend & Kuperus (1978)

.

line-tiedsurface field

0

1

–1

distancefrom Sun1 2 3 4

hoop force

dipole attraction

effect of line-tying

stableequilibrium

How to Achieve a Stable Equilibrium

flux rope

Line-tying creates a second, stable equilibrium

Key factor: Line-tying

Three-Dimensional Configuration of Titov & Démoulin (1999)

3. line-current

1. flux rope

2. magnetic   charg

es

3 field sources

Initial Configuration(Titov & Démoulin)

Normal Field at Surface

flux-rope footprint

I0 = I

I : flux rope current

I0 : sub-surface line current

. ..

Török, Kliem, & Titov (2003)

Kink Stability Analysis of T&D Configuration

j contours

Φ loop = 2.1 π Φ loop = 4.9 π

twist (Φ loop/π)

Stable Case Unstable Casej contours

R = 2.2

R = 3.4

surface

image

flux rope current

stationarybackground

source

flux rope arc

perturbedposition

line current& q sources

3D Line-Tied Solution by Method of Images

Solution for B interms of incompleteelliptical integrals

Force Equation

€

F(h,R,ω ) =J 2

c2Rln

8R

a

⎛

⎝ ⎜

⎞

⎠ ⎟+

1

2ln

tan2(ϕ / 4)− tan2(ω / 4)

1− tan2(ϕ / 4)tan2(ω / 4)

⎡

⎣ ⎢

⎤

⎦ ⎥+ li / 2 – 3/ 2

⎡

⎣ ⎢

⎤

⎦ ⎥–

2qLJ / c

ρ+2 + L2

[ ]3 / 2

⎧

⎨ ⎪

⎩ ⎪

⎫

⎬ ⎪

⎭ ⎪ˆ R

€

+IJ

c2Ψ Ro ,ρ+ ,π − ϕ o ,θ+[ ] – Ψ Ro ,ρ − ,ϕ o ,θ −[ ]+

J

IΨ R,ρ i ,ϕ ,θ i[ ]

⎧ ⎨ ⎩

⎫ ⎬ ⎭ˆ R

€

+JIo

c2

h − R + d

ρ+2 sin(ω )ˆ x

arc field function:

force due to ±q sources

stationary background

line current

€

B(x = 0) =j

c

1

ρ + rF

α +θ

2,p

⎛

⎝ ⎜

⎞

⎠ ⎟+ F

α –θ

2,p

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

⎧ ⎨ ⎩

–1

ρ – rE

α +θ

2,p

⎛

⎝ ⎜

⎞

⎠ ⎟+ E

α –θ

2,p

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

€

–p2 sin

α +θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟cos

α +θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟

1 – p2 sin2 α +θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟

–p2 sin

α –θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟cos

α –θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟

1 – p2 sin2 α –θ

2

⎛

⎝ ⎜

⎞

⎠ ⎟

⎤

⎦

⎥ ⎥ ⎥ ⎥

⎫

⎬

⎪ ⎪

⎭

⎪ ⎪

ˆ x €

≡j

cΨ(r ,ρ ,α ,θ )ˆ x

generalized hoop force

: arc coordinates

Other Equations

Current, I, from flux conservation: Requires numerical integration

Radial variation: Assume Lundquist solution, a = a0 (I/I0), li = 1

Ignore tapering (end effect occuring over a distance of a – a0)

force

arc length

apex surface

side view

(x = 0)

end

view

(y =

0)

initial displaced

Line-Tied Displacements

line current field

Effect of Line Current on Twist

flux rope

line current

j

B j

B

j

B

–F F

line current

rotation out of planeupon eruption

Principal Results of 3D Analysis

4. Existence of lower equilibrium

3. Aneurism-like evolution

2. Out-of-plane twisting motion

1. Eruption without escape

QuickTime™ and aGIF decompressor

are needed to see this picture.

QuickTime™ and aPhoto decompressor

are needed to see this picture.

Kliem & Török (2004)

current density

Numerical Simulation of an Unstable Flux Rope

Titov & Démoulin (1999)Török & Kliem (2005)

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are needed to see this picture.

Simulation of “Torus*” Instability

*nonhelical kink(see Bateman 1973)

1. no subsurface line current2. subcritical twist for helical kink3. torus center near surface

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What is the efficiency of energy conversion as afunction of reconnection rate?

How does reconnection work in a current sheetwhose length grows at a rapid rate?

Can we determine the equilibria and stability propertiesof more realistic line-tied field configurations?