二维电磁模型

二维电磁模型基本方程与无量纲化基本方程

0 0 0

0

( )

0

idm qdt

t

t

vE v B

BE

EB J

B

E

1pe

2

0 0 0

22 0 0

0

,

: , : , :

,

pe

e pe e pepe

pe ee e

t

m mJ en E B

e e

n e eB

m m

-1：长度:网格距 ,速度:

而

无量纲化

2

( )

0

id

dt

t

ct

vE v B

BE

EB J

B

E

方程化为

二维时的方程

2

2

2 2

-( )

x z

y z

yxz

x zx

y zy

y xzz

d

dtB E

t y

B E

t xEEB

t y x

E Bc J

t y

E Bc J

t xB BE

c c Jt x y

vE v B

时间上利用蛙跳格式

网格划分

计算步骤

稳定性条件

22 2

1 11 ( ) ( )c t

x y

例子：磁场重联的二维粒子模拟

Magnetic reconnection rapidly converts magnetic energy into plasma energy, which leads to heating and acceleration of ions and electrons.

Topology changes of magnetic field lines during magnetic reconnection

(1) (2) (3)

Particle simulations[Fu and Lu, 2006]

With 2D particle-in-cell simulations we investigate the influence of the guide field on the electron acceleration near X-point and O-point.

Initial conditions and boundary conditions:

0 0 0

20

( ) tanh[( ) / ]2

( ) sec [( ) / ]2

yz

yb

LB y B y B

Ln y n n h y

x ze e

Initial conditions1D Harris current sheet in the (x,y) plane

Initial flux perturbation is introduced

0( , ) cos[2 ( ) / ]cos[ ( ) / ]2 2

yxx y

LLx y x L y L

Boundary conditionsX direction: periodic Y direction: ideal conducting boundary condition for EM fields, Reflection condition for particles

current sheet

Parameters

0 0 0

0

(12.8 / ) (6.4 / )

256 128

0.05 / , 0.0015

/ 100

20 , /

0.5 /

5

x y pi pi

x y

pi i

i e

A A i

pi

b

L L c c

N N

x y c t

m m

c V V B n m

c

n n

0 2 4 6 8 10 120

1

2

3

4

5

6

Y

X

it=0

0 2 4 6 8 10 120

1

2

3

4

5

6

Y

X

it=15

0 2 4 6 8 10 120

1

2

3

4

5

6

Y

X

it=20

0 2 4 6 8 10 120

1

2

3

4

5

6

Y

X

it=25

Time evolution of the reconnection flux for

0 00,0.5,1.0zB B

Contours of (a) ,(b) , (c) ,(d) at for

0 0zB xE yE zE zB

19it


0 00.5zB BxE yE zE 0z zB B

20it


0 0zB BxE yE zE 0z zB B

23it

Typical trajectories in (x,y) plane, one passes through X-point (from to , the other is trapped near O-point (from to

for 0 0zB

17.5it

20it 18.5it

19.5it

The time evolution of (a) the kinetic energy, (b) , (c) ,(d) for The electron passes through X-point .

0 0zB xv yv zv

Comparison with others[Hoshino,2005; Pritchett, 2006]

The time evolution of (a) the kinetic energy, (b) , (c) ,(d) for The electron is trapped near O-point .

0 0zB xv yv zv


for 0 00.5zB B

15it

23it 18it

21it


0 00.5zB Bxv yv zv


for 0 01.0zB B

17.5it

21.5it 19it

19it


0 01.0zB Bxv yv zv

The positions of the energetic electrons

Comparison with others [Pritchett, 2006]

no guide field

With guide field 1.0B0

Discussion

1.Observations of energetic electron in ion diffusion region in magnetotail [Oieroset, 2002]

Figure 5 shows the plasma temperature, magnetic field vectors, high-speed flows and energetic electron differential fluxes. The bottom four panels denote electron differential fluxes obtained from the RAPID on the four satellites from 35.1 to 244.1keV.

A depletion in the energetic electron fluxes in the diffusion region was detected by all the four satellites. The duration is about 162s. Please note that the first two low energy channels of c3 is not well calibrated (Private communication from Q. G. Zong), while the other energy channels have the same depletion as other satellites.

Similarly, a local minimum of the plasma temperature can also be found near the center of diffusion region

By WIND satellite, the fluxes of energetic electrons up to ~300keV peak near the center of the diffusion region and decrease monotonically away from this region. No secondary acceleration was found in the reconnection.Note:the initial guild field is about 50% of the total magnetic field magnitude during the magnetic reconnection.

From Øieroset et al PRL 2002 Fig 1.

From Øieroset et al NATURE 2001 Fig 2.

Potential applications in Solar atmosphere [Drake et al., 2006]?

二维电磁模型

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