混合模拟基本方程与无量纲化 基本方程

0

( )

1

1

ii

e p

i e

e e

dm q

dt

n n n

en en

t

pne

vE v B

J B

J u u

BE

E u B

电子状态方程

1

0 0 0 0

0

0 020 0

,

: , : , : ,

:

,2

i

A

e

i ei

t V V

n n J en V B V B

p p

eB p

m B

-1A A i

A

： 速度: ,长度: ,

磁场: ,电场

而

无量纲化

1

1 1

2

ii

e i

e e e

d

dt

n

t

pn

vE v B

u u B

BE

E u B

+电子状态方程

方程化为

一些基本关系式

20 0

0 0 0

00

0

0 0

2/( 2 )]

/

2

2

2 1

ithii i

A i

i

i

ithii

A i

i p

RTvp B

V B m n

pB

RTv

V p

m n

[利用

可得

因此

02

0 0 00 0

02

0 0 0

i iA

i i

i

pi

B m mV

qB n qm n

m c

n e

Bow shock and magnetosheath

n

n

Q-// shock

Q- shock

Bn

Bn

ElectronsElectrons

IonsIons

Ion foreshock boundaryIon foreshock boundary

BB

[after [after Tsurutani and Rodriguez, 1981Tsurutani and Rodriguez, 1981]]

Ion cyclotron waves

Ion cyclotron waves downstream of quasi-perpendicular shock

[From Fuselier et al. ,JGR, 1997]

The injection velocity can be described as

))1()/(

)/(1( 2 rr

mq

mqVVVV

H

MIswHMIinject

Here is the compression ratio at the shock. For a supercritical, high Mach number, nearly perpendicular shock, is about 0.3 and the injection speeds for He2+ and O6+ are about and .We choose the injection speeds of He2+ and O6+ as 2.5 ( is the local Alfvén speed) and 2.9 , respectively. The injection angle is .

swH VVr

r

swV44.0 swV51.0

AV

085AV

AV

Observations of He2+ ring-beam distribution downstream of quasi-perpendicular shocks [Fuselier et al., JGR, 1997]

Proton and He2+ distributions from self-consistent hybrid simulations[McKean et al., 1995]

1. The evolutions of velocity distributions for He2+ and other heavy ions from ring-beam distributions downstream of quasi-perpendicular shocks at different conditions(proton temperature anistropy)

Hybrid simulations

In hybrid simulations the ions are treated kinetically while the electrons are considered as massless fluid. In our code, the particles are advanced according to the well-known Boris algorithm while the electromagnetic fields are calculated with an implicit algorithm.

··

· ·

· · ·

· ·

·

Particles in anywhere<===>Fields in GridsParticles in anywhere<===>Fields in Grids interpolationinterpolationSolve particles and fields self-consistentlySolve particles and fields self-consistently

0 50 100 150 2000.8

1.2

1.6

2.0

T | p

/T||p

T||p

/T||p0

pt

(c)

0 50 100 150 2000

2

4

6

8

pt

T||/T||0

T | /T||

(b)

0 50 100 150 2000.00

0.01

0.02

0.03

0.04

0.05

pt

(a)

5.0|| p||/ 1.7p pT T

Case 1:

Time evolution of He2+ and O6+ temperature anisotropies

k diagram obtained by FFT of magnetic field

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

5.833E-4

0.001167

0.001750

0.002333

0.002917

0.003500

pt=0

(a)

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

3.333E-4

6.667E-4

1.000E-3

0.001333

0.001667

0.002000

pt=160

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

3.333E-4

6.667E-4

1.000E-3

0.001333

0.001667

0.002000

pt=80

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

5.833E-4

0.001167

0.001750

0.002333

0.002917

0.003500

pt=0

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

3.333E-4

6.667E-4

1.000E-3

0.001333

0.001667

0.002000

pt=160

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

3.333E-4

6.667E-4

1.000E-3

0.001333

0.001667

0.002000

pt=80

The He2+ and O6+ velocity distributions at different times.

-4 -2 0 2 40

1

2

3

4

5

v | /v

A

v||/v

A

(a)

-4 -2 0 2 40

1

2

3

4

5

v | /v

A

v||/v

A

(b)

Typical trajectories of He2+ and O6+ particles. When the dispersion relation is dispersiveless, the trajectories of the particles are

k

2 2||( )Av v v const

constvvv A 22

|| )(

5.0|| p

||/ 2.5p pT T Case 2

0 50 100 150 200

1.0

1.5

2.0

2.5

pt

T||p

/T||p0

T | p

/T||p

(c)

0 50 100 150 2000

2

4

6

8

T||/T||0

T | /T||

pt

(b)

0 50 100 150 2000.000

0.015

0.030

0.045

0.060

/B

2 0

pt

(a)

Time evolution of He2+ and O6+ temperature anisotropies

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

3.000E-4

6.000E-4

9.000E-4

0.001200

0.001500

0.001800

pt=180

(b)

-4 -2 0 2 40

1

2

3

4

5

v

||/v

A

v | /v

A

0

4.333E-4

8.667E-4

0.001300

0.001733

0.002167

0.002600

pt=180

(a)

The He2+ and O6+ velocity distributions

5.0|| p||/ 3.0p pT T

Case 3

0 50 100 150 200

1.0

1.5

2.0

2.5

3.0

T||p

/T||p0

T | p

/T||p

pt

(c)

0 50 100 150 2000.00

0.02

0.04

0.06

0.08

0.10

2 /B

2 0

pt

(a)

0 50 100 150 2000

2

4

6

8

T||/T||0

T | /T||

pt

(b)

Time evolution of He2+ and O6+ temperature anisotropies

k diagram obtained by FFT of magnetic field

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

5.833E-4

0.001167

0.001750

0.002333

0.002917

0.003500

pt=0

(a)

-4 -2 0 2 40

1

2

3

4

5

pt=80

v||/v

A

v | /v

A

0

4.000E-4

8.000E-4

0.001200

0.001600

0.002000

0.002400

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

2.333E-4

4.667E-4

7.000E-4

9.333E-4

0.001167

0.001400

pt=160

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

4.000E-4

8.000E-4

0.001200

0.001600

0.002000

0.002400

pt=160

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

3.333E-4

6.667E-4

1.000E-3

0.001333

0.001667

0.002000

pt=80

-4 -2 0 2 40

1

2

3

4

5

v||/v

A

v | /v

A

0

5.000E-4

1.000E-3

0.001500

0.002000

0.002500

0.003000

pt=0

(b)

The He2+ and O6+ velocity distributions at different times

2 2||( )pv v v const

2 2||( )pv v v const

-4 -2 0 2 40

1

2

3

4

5

v | /v

A

v||/v

A

(b)

-4 -2 0 2 40

1

2

3

4

5

v | /v

A

v||/v

A

(a)

Typical trajectories of He2+ and O6+ particles2 2

||( )pv v v const

Conclusions

1. The temperature anisotropy of protons and He2+ ring-beam distribution can excite proton cyclotron waves and helium cyclotron waves.

2. Helium cyclotron waves can scatter He2+

and O6+ from ring-beam distributions to shell-like distributions.

Conclusions3. Only the waves are dominated by the heli

um cyclotron waves, there exist He2+ and O6+ shell-like distributions. While the waves are dominated by the proton cyclotron waves, their effects is to heat He2+ and O6+ .

4. There are observations [Fuselier, GRL, 1989], which shows the shell-like distributions for He2+ and O6+ particles. Therefore, helium cyclotron waves should exist in the downstream of quasi-perpendicular shock.

2. Ion cyclotron waves in magnetosheath/plasma depletion layer

Formation of terrestrial plasma depletion layer(PDL)

Adjacent to the sunward side of the low-shear dayside magnetopause, there exists a region characterized by reduced plasma density and increased magnetic field strength. This region is called plasma depletion layer (PDL), which is formed by the stretching of magnetic field lines and consequent pileup of the magnetic field as the magnetosheath plasma approaches the magnetopause.

Characteristics of magnetic fluctuations in PDL

The magnetic fluctuations are predominantly transverse to the background field and their frequencies can up to the proton gyrofrequency, which are mainly electromagnetic ion cyclotron waves with left-handed polarization. The ion cyclotron waves are considered to be excited by the H+ and He2+ temperature anisotropies.

Ion cyclotron waves inside the PDL have three different spectral categories, called LOW, CON and BIF respectively. Anderson et al. [1994] defined these categories as follows: (1) LOW, a continuous spectrum of the ion cyclotron waves with main power below , (2) CON, a continuous spectrum with mainly power extending from up to ; (3) BIF, a spectrum characterized by two activity peaks, one above and one

below .

0.5 p

p

0.5 p

0.2 p

Theoretical model

The linear Vlasov theory can only calculate the growth rate of ion cyclotron waves excited by proton and helium temperature anisotropies. It find that the growth rate has two peaks in low beta plasma, and has only one peak in high beta plasma.

Hybrid simulations To explain the generation mechanisms of the three

spectral categories in the PDL, We perform one-dimensional (1-D) hybrid simulations to investigate the spectrum evolution of the ion cyclotron waves

excited by the H+ and He2+ temperature anisotropies. Parameters:

p|| ||/p pT T ||/T T case

1 0.1 3.8 4.56 0.04

2 0.1 3.8 1.0 0.04

3 0.1 1.0 4.56 0.04

4 0.1 3.8 -- --

5 0.3 3.8 4.56 0.04

6 0.3 3.8 1.0 0.04

7 0.3 1.0 4.56 0.04

Time evolution of the H+ and He2+ temperature anisotropies for Run 1. The solid and dash lines represent the H+ and He2+ temperature anisotropies, respectively.

|| 0.1p

||/ 3.8p pT T

||/ 4.56T T

0.04

Case 1

The frequency spectrum of the excited ion cyclotron waves during four different time periods for case 1. The solid, dash, dot and dash dot lines represent the periods, (A) from to , (B) from to , (C) from to , and (D)from to , respectively.

|| 0.1p

||/ 3.8p pT T

||/ 4.56T T

0.04

0pt

51.2pt 51.2pt 102.4pt

102.4pt 153.6pt

204.8pt 256.0pt

The frequency spectrum of the excited ion cyclotron waves during four different time periods for (a) case 2 and (b) case 3. The solid, dash, dot and dash dot lines represent the periods, (A) from to (B) from to , (C) from to , and (D)from to , respectively.

|| 0.1p

||/ 3.8p pT T

||/ 1.0T T

0.04

0pt 51.2pt

51.2pt

102.4pt 102.4pt 153.6pt

204.8pt 256.0pt

|| 0.1p

||/ 1.0p pT T

||/ 4.56T T

0.04

0.04

Case 2

Case 3

The frequency spectrum of the excited waves during four different time periods for case 4, The solid, dash, dot and dash dot lines represent the periods (A) from

to , (B) from to , (C) from to , and (D) from to , respectively.

|| 0.3p

||/ 3.8p pT T

||/T T

0pt 51.2pt 51.2pt

102.4pt 102.4pt

153.6pt

204.8pt 256.0pt

Case 4

(a)The time evolution of the H+ and He2+ temperature anisotropies for case 5, the solid and dash lines represent the H+ and He2+ temperature anisotropies, respectively. (b) The frequency spectrum of the excited waves during four different time periods for case 5, The solid, dash, dot and dash dot lines represent the periods (A) from

to , (B) from to , (C) from to , and (D) from to , respectively.

|| 0.3p

||/ 3.8p pT T

||/ 4.56T T

0.04

0pt 51.2pt 51.2pt

102.4pt 102.4pt

153.6pt 204.8pt 256.0pt

Case 5

The frequency spectrum of the excited ion cyclotron waves during four different time periods for (a) case 6 and (b) case 7. The solid, dash, dot and dash dot lines represent the periods, (A) from to (B) from to , (C) from to , and (D)from to , respectively.

|| 0.3p ||/ 3.8p pT T

||/ 1.0T T

0pt 51.2pt

51.2pt

102.4pt 102.4pt 153.6pt

204.8pt 256.0pt

|| 0.3p

||/ 1.0p pT T

||/ 4.56T T

0.04

0.04

Case 6

Case 7

Conclusions

1. The H+ and He2+ temperature anisotropies excite the proton cyclotron waves and helium cyclotron waves, respectively. In their linear growth stage, the dominant frequency of the proton cyclotron waves is larger than the helium gyrofrequency, while the dominant frequency of the helium cyclotron waves is smaller than the helium gyrofrequency.

2. When beta is small, the BIF category is formed in the spectrum after the helium cyclotron waves are excited, which correspond to the dominant frequencies of the helium cyclotron waves and proton cyclotron waves.

Conclusions

3. When beta is large, the dominant frequency of the proton cyclotron waves decreases in the nonlinear stage, and their frequency band merge with that of the helium cyclotron waves after the helium cyclotron waves are excited. This can explain the CON category observed in the PDL.

4. The LOW category is formed at the quasi-equilibrium stage after their frequencies are drifted to smaller than the helium gyrofrequency.

1.Ion cyclotron waves downstream of quasi-perpendicular shock(Lu et al., GRL, 32, 2005;Lu et al., JGR, 2006)

2. Ion cyclotron waves in magnetosheath/plasma depletion layer(Lu, et al., JGR, 111, 2006)

Thank you ！