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MHD Simulations of 3D ReconnectionTriggered by Finite Random Resistivity

Perturbations

T. YokoyamaUniv. Tokyo

in collaboration with H. Isobe (Kyoto U.)

Lab.-Hinode Reconnection Workshop

2008.11.29. Hayama

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“Size-gap” problem in solar flares

Spatial size necessary for the anomalous resistivity: = i1 m

; thickness of the current sheet

i; ion-gyro radius

Spatial size of a flare– 104 –105 km

Enormous gap with a ratio of 107 !

(c.f. Tajima Shibata 1997)

For a rapid energy release by the fast reconnection, an anomalous resistivity is expected near the neutral point.

It is hard to believe that a laminar flow structure is sustained in a steady state manner.

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Fractal Current Sheet model

~1-10 m

~104 km

>1 km

Global current sheet

Tajima & Shibata (1997)

“Turbulent reconnection”:

Matthaeus & Lamkin (1986), Strauss (1988), Lazarian & Vishniac (1999), Kim & Diamond (2001) ...

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Our final goal (but far away …)

To investigate the physical nature (evolution, saturation, steadiness etc. ) of the 3-dimensional turbulent magnetic reconnection

This study

By using the resistive MHD simulations, the evolution is studied about a current sheet with initially imposed finite random perturbations in the resistivity.

We focus on:– (1) Evolution of 3D structures– (2) Energy release rate

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Model

4300/Am CR

4.3/ SA CC

Periodic condition on all the boundaries

• 3D MHD eqs

with the uniform resistivity

plus finite resistivity perturbations

spatially random; 50% max. amp.

during t/(/Cs)<4

128

10

1.0

32

x

y

zgrid: 256 x 256 x 256

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xy-plane (z=0)

Jz

Vx Vy Vz

Bx By Bzx

y

Tiny structures evolve first. They are overcome by the larger wavelength mode, whose size are roughly consistent with the most unstable mode of the tearing instability. In later phase, the magnetic islands in the sheet coalesce with each other to form a few dominant X-points.

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zy-plane, x=0

z

y

Jz

Vx Vy Vz

Bx By Bz

• "Turbulent" structures

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Evolution in the current sheet (zy-plane, x=0)

Jz

Vx Vy Vz

Bx By Bz

z

y

• Many tiny concentrations in the current are all X-points. They are associated with bipolar structure of the reconnected magnetic fields (Bx) and outflows (Vy).

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z

y

Bx Vz

Reconnection X-point

x

y

Jz Pcontour:Jz

•A pair of z-directional flows into the X-point evolve for each of the current concentrations. It controls the evolution of 3D structure in the z-direction.

yy

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Power spectrumBx(x=0) t/s=1, 10, 50, 100, 200

•Inverse cascade•Evolution of “power-law”-like spectrum

ky(Ly/2)

K

IK

time time

kz(Lz/2)

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MA

t / (/Cs)

Vx Vy Vx Vy

random perturbation

Comparison with the Sweet-Parker reconnection

t=90 t=90

random perturbation

SP

Sweet-Parker

A

2m

A 42 C

BS

dtdE

M

•The energy release rate is slightly larger than the Sweet-Parker reconnection.

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Discussion: divided-sheet vs. Sweet-Parker

L

• single current sheet

2/1

2/1

AmA LC

RM

• N-divided multi current sheet

NLL /2/12/1

2/1

2/1

mA

mA RNCL

RM

(e.g. Lazarian & Vishniac 1999)

When the current sheet is divided into N parts, the length of each part becomes shorter. Then, keeping the aspect ratio of the diffusion region, the total amount of the released energy increases like this proportional to the square-root of the dividing number N.

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MA

t / (/Cs)

Vx Vy Vx Vy

3D perturbation

Comparison with the 2D turbulent case

t=110 t=110

2D

2D perturbation

３ D

• The energy release is weaker in the 3D case than 2D.

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MA

t / (/Cs)3D perturbation

Comparison with the 2D turbulent case

2D

2D perturbation

３ D

• The energy release is weaker in the 3D case than 2D.

Jz Bx Jz Bx

t=110 t=110

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Discussion: 3D vs. 2DWhy the energy release became less effective in the 3D case than in

the 2D case ?

In case of 3D, the distribution of the current density is not uniform with scattering concentrations here and there in the sheet. The filling factor is smaller than the 2D case. As a result, the efficiency of the energy release might be smaller.

Current sheets

Jz

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Dependence on diffusivity

1.d-3

1.d-2

1.d-1

t / (/Cs)

A

2ki

kiA; 42 C

BS

dtdE

M

/AM

• The reconnection rate is: 　　　　　i.e., has a Sweet-Parker like dependence on Rm.

2/1m

RM A

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Summary

• 3D structures in the z-direction evolve in time. A pair of Inflows toward the z-direction play role for this evolution.

• Slight increase (3%) of energy-release rate above the Sweet-Parker rate is observed. We interpreted it is due to the spatial division of the current sheets by the magnetic islands.

• Energy-release rate is reduced in 3D random case compared with 2D random case, presumably because of the reduction of the filling factor of the diffusion regions in the additional dimension.

• The dependence on the magnetic Reynolds number is consistent with that of the Sweet-Parker model.

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D

D

D

anom0

2Danom0

0

2for

21for

1for

)1(

D0D

/

V

J

uniform

anomalous

MA

t / (/Cs)

Vx Vy Vx Vy

• anomalous resistivity

)/(3 00D0 JV •uniform resistivity

Case with anomalous resistivity

t=140 t=140

• In case with the anomalous resistivity, Petschek like structures (i.e. pairs of slow-mode shocks) appear, leading to the enhanced reconnection rate.

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MA

t / (/Cs)

Vx Vy Vx Vy

Bz0/By0=0

Guide field (preliminary)

t=150 t=228

Bz0/By0=0.1

Bz0/By0=0.1

Bz0/By0=0

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Bz0/By0=0.1xy-plane (z=0)

x

y

Jz

Vx Vy Vz

Bx By Bz

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Bz0/By0=0.1zy-plane (x=0)

z

y

Jz

Vx Vy Vz

Bx By Bz

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MA

t / (/Cs)

Vx Vy Vx Vy

Bz0/By0=1

Guide field

t=110 t=180

Bz0/By0=2

Bz0/By0=0.1

Bz0/By0=0

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MA

t / (/Cs)

Vx Vy Vx Vy

random perturbation

Comparison with the simple diffusion

t=130 t=130

Random

simple diffusion

simple diffusion

A

2m

A 42 C

BS

dtdE

M

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Tanuma et al. (2001)•2D MHD simulations: Resistivity: uniform (Rm=150) + current-dependent

• Current-sheet thinning by the secondary tearing

• Impulsive nature of the inflow induced by the ejections of mag. islands

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In the context of emerging flux study

(Shimizu & Shibata 2006 private commu.)