Reconnection Rate in Collisionless Magnetic Reconnection under Open

Boundary Conditions

Zhiwei Ma and Jun Huang

Zhejiang University, Hangzhou, China

US-China Magnetic Fusion Collaboration Workshop,

Austin, Texas, May 5, 2008

Outline1. Introduction2. A model case for the different boundary conditions3. Results for the different system sizes4. Results for the different initial current sheet

thicknesses 5. Summary

What is the magnetic reconnection?

The basic process of magnetic reconnection

0tt 1tt

1. Introduction

Introduction of Magnetic Reconnection

In the steady-state models, the spatial scale and reconnection rate is determined by the resistivity

Reconnection rate

A. Sweet-Parker model (Y-type geometry)for the uniform distribution of the resistivity

2/1~

B. Petschek model (X-type geometry) (1964)

with a localized resistivity

Reconnection rate ln~

In the collisionless plasma, the are a complicated or multiple-layer structure in the diffusion region

In the GEM challenge, it is found that the reconnection rates from the Hall MHD model and the full particle are nearly the same. We should ask what controls the reconnection rate:

Local plasma parameters? such as mass ratio or Global parameters? such as system sizesor Initial or boundary conditions?

One argument: independent system size– M. Shay et al. Geophys.Res.Lett,26, 2163(1999)

Vrec ~ 0.1VA

– Huba etal, Phys.Rev.Lett 2004

The other arguments: system dependence

– Wang, Ma & Bhattacharjee, Phys. Rev. Lett, 87, 265003(2001)

Vrec ~ (d/L)αVA, a =1/2

– Fitzpatrick, Phys. Plasmas, 11, 937(2004) a =3/2

– Recent experiment (MRX, PPPL)

System size dependences

2. A model case for different boundary conditions

a. Zero-gradient (ZG) boundary condition

2Ay/z2=0 used in MHD simulations

b. Electromagnetic (EM) wave boundary condition

c. Magnetosonic (MS) wave boundary condition

2Ay/ztc2Ay/z2 =0 W. Daughton et al

2Ay/zt(VA2+Cs2)1/22Ay/z2 =0

Fig4 ： Plasma flow distributions in the x-z planeunder (a) EM and (b) MS conditions

0x xB BV

t z

for the upper inflow boundary 0xB

z

so 0x xB BV

t z

3. The reconnection rates with the different system sizes

25 25 , 50 25 , 100 25x z i i i i i iL L d d d d d d Fig7 ： The time evolutions of the reconnection rate for the cases

Fig 8 ： The current density distributions when the reconnection rate reach their maximum values for all three cases.

yJ

yJFig 8 ： The current density distributions when the reconnection rate are nearly quasi-steady states for all three cases.

Fig10 ： The thicknesses and lengths of the current sheets at the quasi-steady states.

Fig 11 ： The dependence of (a) the lengths and (b) the reconnection rates on the system sizes.

Fig 12 ： The time evolutions of the reconnection ratesunder the different initial current sheet thicknesses a=0.5, 1.0 , 2.0 and 3.0.

4. Rates for different initial current sheets

Fig13 ： The different snapshots of the current distributions for the case a=3.0di.

J y

Fig 14 ： The dependences of the thicknesses and lengths of the diffusion regions on the system sizes.

Fig 15 ： The dependences of the reconnection rates on the initial current sheet thickness. Er ~ (di/a)1/2 is in good agreement with the result obtained by Wang, Ma, and Bhattacharjee (2001).

Er ~ (di/a)1/2

5. Summarya. Under the MS boundary condition, the quasi-steady state is

achieved with a higher reconnection rate than that under the EM boundary condition.

b. With the given initial current sheet thickness, it is found that (1) the maximum reconnection rates are nearly independent in the system sizes of the simulations, but (2) the reconnection rate in the quasi-steady state are nearly inverse relationship to the lengths of the system sizes and the lengths of the diffusion region linearly increase with the increase of the system sizes.

c. With the given system size, the maximum reconnection rates have the square-root dependence on the initial current sheet thicknesses or Er ~ (di/a)1/2 . There is no steady state observed when the initial current sheet thickness is larger than 2di , instead of the formation of the second island after the reconnection rates reach their maximum values.