Magnetic Reconnection and its Applications

Zhi-Wei Ma Zhi-Wei Ma

Zhejiang UniversityZhejiang University

Institute of Plasma PhysicsInstitute of Plasma Physics

Chengdu, 2007.8.8

Outline1. Numerical Scheme2. Steady-state reconnection

A. Sweet-Parker modelB. Petschek model

3. Time-dependent force reconnectionA. Harris sheetB. MagnetotailC. Solar corona

4. Magnetic reconnection with Hall effects A. Harris sheetB. MagnetotailC. Solar corona current dynamicsD. Coronal mass ejection

5. Summary

Numerical Scheme

Euler's method

( , )y

f y tt

Consider the general first-order ordinary differential equation,

t

y

1nx nx 1nx

The standard fourth-order Runge-Kutta method takes the form:

round-off error =O( /h)7~ 10 for single precision

16~ 10 for double precision

dyy

dt

Global integration errors associated with Euler's method (solid curve) and a fourth-order Runge-Kutta method (dotted curve) plotted against the step-length. Double precision calculation.

                                                

        

The equation for the shock propagation:

n+1/2**

Hall MHD Equations

0/ Vdtd

BJPdtVd�

/

)()1(

/

BVEJ

VppVtp

/)(i PBJdJBVE

/B t E

Rouge-Kutta Scheme (4,4)

Dispersion Properties for Different Schemes

Fast rarefaction wave (FR),Slow compressional wave (SM),Contact discontinuity (CD)Slow shock (SS)

What is magnetic reconnection?

Magnetic energy converts into kinetic or thermal energy and mass, momentum, and energy transfer between two sides of the central current sheet.

0tt 1tt Another key requirement:Time scale must be much faster than diffusion time scale.

1. Steady-state Reconnection

A. Sweet-Parker model (Y-type geometry)

Reconnection rate Time scale

2/1~2/1~

B. Petschek model (X-type geometry)

Reconnection rate and time scale are

weakly dependent on resistivity.

ln~

Difficulties of the two models

For Sweet-Parker model– The time scale is too slow to explain the

observations.– Solar flare

14 12~ 10 10 yearssp ~

hour~

Substorm in the magnetotail

108 10~10~

dayssp ~

hour~

For Petschek model

The time scale for this model is fast enough to explain the observation if it is valid. But the numerical simulations show that this model only works in the high resistive regime. For the low resistivity , the X-type configuration of magnetic reconnection is never obtained from simulations even if a simulation starts from the X-type geometry with a favorable boundary condition.

Basic problem in both models is due to the steady-state assumption. In reality, magnetic reconnection are time-dependent and externally forced.

410

2. Time-dependent force reconnection

A. Harris Sheet

)cos1()( 0 kxvxv

Resistive MHD Equations

0/ Vdtd

BJpdtVd

/

)()1(

/

BVEJ

VppVtp

JBVE

/B t E

5/130 )( RAN 5/1 Nor

New fast time scale in the nonlinear phase (Wang, Ma, and Bhattacharjee, 1996)

B. Substorms in the magnetotail

Observations (Ohtani et al. 1992)

Time evolution of the cross tail current density at the near-Earth region (Ma, Wang, & Bhattacharjee, 1995)

C. Flare dynamics in the solar corona

Time evolution of maximum current density(Ma and Bhattacharjee, 1996)

(Ma and Bhattacharjee, 1996)

Brief summary for time-dependent force reconnection

1. New fast time scale is obtained for time-dependent force reconnection.

2. The new time scale is fast enough to explain the observed time scale in the space plasma.

3. The weakness of this model is sensitive to the external driving force which is imposed at the boundary.

4. The kinetic effects such as Hall effect are not included, which may become very important when the thickness of current sheet is thinner than the ion inertia length.

3. Magnetic reconnection with Hall effects

2/1~Resistive term

)(

2

BJ

J

JBvE

pd

dt

dd

i

e

Inertia term ~ ed

Hall term ~ id

Spatial scales

If , the resistivity term is retained (resistive MHD).

If , both the resistivity and Hall terms have to be included (Hall MHD).

If , we need to keep the Hall and inertia terms and drop the resistive term (Collisionless MHD).

For solar flare,

For magnetotail,

1/ 2

14 12 4

5 10 1 10

where 10 10 , 10

id m a m

a km

id

ei dd ~

ei dd

1/ 2

10 8 4

50 500 1

where 10 10 , 10

id km a km

a km

A. Harris Sheet

(Ma and Bhattacharjee, 1996 and 2001, Birn et al. 2001)

1. X-type vs. Y-type2. Decoupling3. Separation4. Quadruple B_y5. Time scale 6. Reconnection rate7. No slow shock

Time evolution of the current density in the hall (dash line) and resistive MHD (solid line)

The GEM challengeresults indicate that the saturated level from HallMHD agrees with one obtained from hybrid andPIC simulation.

B. Hall MHD in the magnetotail (Ma and Bhattacharjee, 1998)

1. Impulsive growth2. Quite fast disruption3. Thin current sheet4. Strong current density5. Fast time scale6. Fast reconnection rate

Explosive trigger of substorm onset

With increasing computer capability, we are able to further enhance our resolution of the simulation to reduce numerical diffusion. In the new simulation, explosive trigger of substorm onset is observed due to breaking up extreme thin current sheet.

The tail-ward propagation speed of the x-point orDisruption region ~ 50km/s

Zhang H., et al., GRL, 2007

Reconnection rate ~ 0.1

Density depletion and heat plasma around the separatrices

C. Flare dynamics

1. Geometry2. Electric field(Bhattacharjee, Ma &Wang, 1999)

Time evolution of current density and parallel electric field

D. Coronal mass ejection or flux rope eruption

Initial Geometry

CatastropheOr loss equilibrium

Hall MHD RunMHD run

Flux rope region

Total energyThermal energyMagnetic energyKinetic energy

Comparison between Hall and Full PIC simulation

Spontaneous Reconnection

– Periodic boundary condition

– Open boundary condition

Periodic boundary condition(Hall MHD)

Open boundary condition (Hall MHD)

Periodic boundary condition (PIC)

Open boundary condition (PIC)[Daughton and Scudder; Fujimoto; PoP, 2006)]

Summary

Hall MHD vs. Resistive MHD

1. Time scale and reconnection rate: Fast with very weak dependence of the resistivity

vs. Fast with a suitable boundary conditions1. Geometry: X-type vs. Y-type 2. Decoupling Motion of ions and electrons: yes vs. no3. Spatial scale separation of electric field and current

density: Yes vs. No4. Magnitude and distribution of parallel electric field:

strong and broad vs. weak and narrow5. Quadruple distribution of B_y: yes vs. no6. No slow shock for both cases, which is different from

Petschek’s model

Hall MHD vs. Full Particle

1. Periodic boundary condition: Nearly identical

Fast, time-dependent, x-type.

2. Open boundary condition:

Slow and steady vs. Fast and unsteady in the transition period & Slow and steady in the late phase

ThankThanks!!!s!!!