Sept. 13, 2007

A Global Effect of Local Time-Dependent

Reconnection

In Collaboration with

Dana Longcope

Eric Priest

Sept. 13, 2007

1. Introduction - Reconnection on Sun(a) Solar Flare:

Sept. 13, 2007

An Effect:seismicflare waveIn

photosphere (MDI)

(8-100 km/s

70 mins)

Sept. 13, 2007

Chromospheric flare wave:

Sept. 13, 2007

Yohkoh- glowing in x-rays

Coronal Heating:

Sept. 13, 2007

Hinode X-ray Telescope (1 arcsec):

Sept. 13, 2007

Close-up How heated?

Sept. 13, 2007

Coronal Tectonics Model

--> web separatrix surfaces

(updated version of Parker nanoflare/topological dissipation)

Corona filled w. myriads of J sheets, heating impulsively (see talk by Haynes and by Wilmot-Smith)

Each ‘Coronal Loop’ --> surface in many B sources

As sources move --> J sheets on surfaces --> Reconnect --> Heat

Sept. 13, 2007

- there are many nulls in corona ??

Some j sheets at nulls/separators (see Pontin

talk)

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

“X marks the spot”

Many sheets also at QSLs-non-nulls

Sept. 13, 2007

If reconnection heating coronaat many sheets,

1. How does energy spread out ?-- conduction along B-- waves across B

2. If reconnection time-dependent, how much energy liberated locally/globally?

Simple model problem

Sept. 13, 2007

2. Magnetic fields in 2D

∇⋅B = 0contours are field lines

A(x, y) = 12 ′B [x2 −y2 ] = 1

2 ′B r2 cos(2φ)

X-point:

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

B(x, y)=∇A×z=∂A∂y

x−∂A∂x

y

A = const

current:

A - flux function

J =∇×B =−z∇2A=0j

Bx=B’y, By=B’x

By+iBx=B’(x+iy)=B’w

€

4π

Sept. 13, 2007

branch cut

€

By + iBx = B' w2 − Δ2 = dΨ /dw

branch pointbranch point

w =−Δ w =+Δ

A Current Sheet(Green 1965)

Current I0

€

=Δ2B' /4A(x, y)=Re Ψ(x+iy)[ ] J =−z∇2A=0j

€

4π

? large r

Sept. 13, 2007

At large r

X-point fieldB0 =−B'[yx+ xy]

Sept. 13, 2007

At large r

perturbation: line current

WM = B 2∫ d2x :

I 02

r

L

∫ dr : I 02 ln(L)Most of perturbation

energy to distance L

Lots of energy far from CS

~ ~

€

Bφ = −∂A

∂r=

I0

2π r

€

8π

Sept. 13, 2007

Suppose sheet reconnects

Local process but has global consequences:

Decrease I --> B must change at great distance

How ??

Sept. 13, 2007

3. Model for effect of reconnection

Linearize about X-point B0 :

is “turned on” current diffuses i.e. reconnection

Assume a small current sheet,i.e. (natural diffusion length)

so that sheet rapidly diffuses to circle anddynamics can be approximated by linearising about X

€

Δ << lη

Assume B1 @ t=0 is due to current sheet

B0 =−B'[yx+ xy]

€

∂B1

∂t=∇ × (v1 × B0) + η∇ 2B1,

ρ 0

∂v1

∂t= j1 × B0.

Sept. 13, 2007

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Equations

vA,02 =

B02

4πρ0

=( ′B )2

4πρ0

r2 =ωA2r2

B0 =−B'[yx+ xy]

€

∂B1

∂t=∇ × (v1 × B0) + η∇ 2B1,

ρ 0

∂v1

∂t= j1 × B0.

€

×B0

€

ρ0

∂(v1 × B0)

∂t= − j1 B0

2

€

EV ˆ z = −v1 × B0

Use only axisymmetric (m=0) part

€

B1φ

natural variable

Sept. 13, 2007

Natural New Variables

r

I€

EV (r, t) ˆ z = −v1 × B0

I(r, t) = rB1φ = −r∂A1

∂r

€

Ienclosed (r, t) =∇ × B1

4π⋅dS =

B1

4π⋅ds = 1

2∫∫ rB1φ = 1

2I(r, t)

r

(r)

Δ

I0Expect:

Sept. 13, 2007

• Combine:

Governing equations

EV

EV

I I

I

I I I

wave diffusive

• Natural length-scale:

€

lη2 = η t =

η

ωA

Induction

Motion

Sept. 13, 2007

Nondimensionalise

∂C

∂t=

∂U

∂R+

∂

∂Re−2 R ∂C

∂R⎡⎣⎢

⎤⎦⎥

∂U

∂t=

∂C

∂R

l =

ωA

t → ωAt

U → U /ωA

R =ln(r / l )

----->

EV

EV

EV

EV

I

I

I

I

I

EV EV

I

? large r and small r

Sept. 13, 2007

4. Limits(i) Large r (wave) limit: when

∂C

∂t=

∂U

∂R+

∂

∂Re−2 R ∂C

∂R⎡⎣⎢

⎤⎦⎥

∂U

∂t=

∂C

∂R

C(R, t) =C0 −F(t mR)U(R,t) =±F(t mR)

telegraphersequations

R

EV

EV

EV

I

I

I

I I0

I0

R =ln(r / l ) ? 1>>

Sept. 13, 2007

4. Limits (i) Large r (wave) limit: when

∂C

∂t=

∂U

∂R+

∂

∂Re−2 R ∂C

∂R⎡⎣⎢

⎤⎦⎥

∂U

∂t=

∂C

∂R

C(R, t) =C0 −F(t mR)U(R,t) =±F(t mR)

telegraphersequations

rightward

leftward

(FastMagneto-Sonicwaves)

EV

EV

EV

EV

I I

I

I I0

R

tI0

I0I

R =ln(r / l ) ? 1>>

Sept. 13, 2007

(ii) Small r (diffusive) limit: when

U = ωAC

J(r, t) =1r∂C∂r

=C0

2texp −

r2

4t⎛

⎝⎜⎞

⎠⎟

Current density:

classic diffusion:(from wire @ t=0)

EV

EV

EV

<<

I0

2 t

r

I0

I0I

I

I

I

Id I0

I(i.e. r small)

jd

r

jd

Sept. 13, 2007

(iii) Numerical Solution

∂C

∂t=

∂U

∂R+

∂

∂Re−2 R ∂C

∂R⎡⎣⎢

⎤⎦⎥

∂U

∂t=

∂C

∂R

IE IE IE IE IE IE R

R=20R=-7

E=0I=0

E=0I’=0

Use a uniform staggered grid in R,advancing I and E alternately

EVEVI I I

(11 orders of magnitude in r)

,

Sept. 13, 2007

5. Numerical Solution

∂C

∂t=

∂U

∂R+

∂

∂Re−2 R ∂C

∂R⎡⎣⎢

⎤⎦⎥

∂U

∂t=

∂C

∂R

IE E IE IE IE IE R

R=20R=-7

E=0I=0

E=0I’=0

initial condition@ t=0.001/ωA

Diffn term advanced implicitly in an operator splitting method

EVEV

EV

I

I I I

I I0 I0I0

Sept. 13, 2007

Numerical Solution for I(------- diffusive solution)

(Departs slightly from diffusive solution at small r)

I

log r

Sept. 13, 2007

Numerical Solution

Diffusive solution

R = 12 lnt

I(r)

€

Location where I = 2

3I0

R

t

€

R = ωA t

Transition: diffusive towave solution

Wave solution

Sept. 13, 2007

C(R,t) =C0 −F(t−R)U(R,t) =F(t−R)

C(R, t)+U(R,t) =C0

EV

EV

EV

I

I I

I I

Wave solution:

Numerical solutions

forI

and Eu

Sept. 13, 2007

I

Magnetic Field:

Sheath ofCurrentpropagates outleaving I = 0 behind

Sept. 13, 2007

Ev --> v

U =v1⋅∇A0 =rB'[vr cos2φ+ vφ sin2φ]EV

EV

I

But flow near X does not disappear-- it slowly increases !

In wake of sheath -a flow

Sept. 13, 2007

(diffusive limit)

Flux function

A?A

-->X-point Electric field

I

Numerical solution has E const (driven by outer wave soln)

Diffusive soln has E decreasing in time

€

E1(0, t) = −∂A1

∂t− v1 ⋅∇A1

log rt

A(0,t) A(r,t)

Sept. 13, 2007

J(r, t) =1r∂C∂r

=C0

2texp −

r2

4t⎛

⎝⎜⎞

⎠⎟II

5. Resolving the Paradox-- a 3rd regime -- at later times - eg in j(r,t):

J(0, t) ;

C0

2t

newregime

~I

Diffusive solutionj

t log r

jd

jd(0,t)

Sept. 13, 2007

Resolving the Paradox

j

EV

EV

I I

I

But what is j(r,t)?

At large t

€

∂j

∂t+

ωA2r2

ηj =

∂j

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟0

€

(i.e., t >1/ωA )

€

[using

jd (0,1/ωA ) = I0ωA /η ]

€

0 = r∂

∂r(Ev + η j)

€

Ev (r, t) = η ( j(0, t) − j(r, t))

advection - diffusion

I0ja/d (r,t)

Sept. 13, 2007

New Regime

€

ωA

I0EV

I0I

Ija/d (r,t)

Peak in j at X-point remains

-- produces a steady E (independent of )

€

Sept. 13, 2007

Energetics

Wm =18π

B∫2d3x= 1

4 C2 drra

b

∫

Wk =ρ0

2v∫

2d3x= 1

4ωA−2 U 2 dr

ra

b

∫EV

IMagneticenergy

Kineticenergy

Poyntingflux

Ohmicheating

EV

For an annular region

(in or out) (negative)

Sept. 13, 2007

Energetics

Heat (integrated from t=1/ )

(i)K.E

€

ωA t

Heat

(ii) Large t K.E

€

ωA t

€

ωADiffusion -->Magnetic energy into Ohmic heat

€

~ log lη /Δ (ie stored near sheet)€

Small t (<1/ωA )

Magnetic energy into K.E

Sept. 13, 2007

6. SummaryResponse to enhanced in current sheet (CS):

(i) Diffusion spreads CS out(ii) Wave (Lorentz force) carries current out at vA - as sheath

• Most magnetic energy is converted into kinetic energy in wave --

• Coronal heating -- reconnection + wave

may later dissipate.

(iii) Peak in j at X remains --> steady E independent of i.e. fast

€

Sept. 13, 2007