長江後浪推前浪 ( cháng jiāng hòu làng tuī qián làng )

38

⇒k?⇢i ⌧ 1

r2− k2Di

ni

n0(− ) = −4⇡e(n

i

+1

2⇢2i

r2?n

inho

i

− ne

)

−k2Di

(Φ− Φ) ⇠!2pi

⌦2i

r2?Φ

⇠ O(1)

n(x) = n +12⇢2

t

1Tr2

?nT

0 500 1000 1500 2000 2500 3000 3500 4000 45000

0.5

1

1.5

2

2.5

3

t(a/cs)

χ i (cs ρ

s2 /a)

0 500 1000 1500 2000 2500 3000 3500 4000 4500!18

!16

!14

!12

!10

!8

!6

t(a/cs)

ln(r

adia

l m

ode)

(vE× B

/cs)

0 500 1000 1500 2000 2500 3000 3500 4000 45000

0.5

1

1.5

2

2.5x 10

!3

t(a/cs)

radia

l m

ode (

vE× B

/cs)

ni

|particle

ni

= 1 +1

2⇢2i

1

pi

r2?pi

v

E⇥B

⇡ −1

2b⇥ r?pi

pi

cTi

eB

J

E⇥B

? (x) =X

↵

q↵

hZ

v

E⇥B

(R)F↵

(R)δ(R− x+ ⇢)dRdµdvki'

J? =c

Bb⇥rp+ en

i

⇢2i

2

r2

?vE⇥B

+v

E⇥B

pi

r2?pi

�

J? ⇡ c

Bb⇥ (rp)

1− 1

4⇢2i

r2?p

p

�

rB2

8⇡+ p

✓1− 1

4⇢2i

r2?p

p

◆�⇡ 0

ngc

i

+ npol

i

+ ninho

i

= ngc

e

n(r)

n0=

1

2 tanh[(r r0)/w]

2+ n

s

ni

⇡ ne

⇢2s

r? · nr?eφ

Te

= −δn

e(r)

Ti

=Z

r

0

dr01

2nr0

Zr

0

0

sds[n00(s) + n0(s)/s],

E(r) =Ti

2nre

Zr

0

sds[n00(s) + n0(s)/s]

enE? = (1/2)(Te

/Ti

)r?pi

npol

i

+ ninho

i

= 0 ngc

i

= ngc

e

n = ngc + δn = ngc +1

2Ti

⇢2i

r2?n

gcTi

= ngc +⇢2i

2Ti

✓@2ngcT

i

@2r+

1

r

@ngcTi

@r

◆

pi

⌘ ngcTi

⇢i

w r0

Shot 86470Three 200 ms sweepst=12.2-12.8 s

EF15.295-2c

JET Ohmic Discharge.

NSTX Discharge

Corresponding steep gradient region

C-MOD Discharge

0

200

400

600

0.9 0.92 0.94 0.96 0.98 1

−80

−60

−40

−20

0

20

40

ρ

Shot

: 112

0803

014

Tim

e:0.

922s

−0.

999s

T z [e

V]E r [k

V/m

]

(a) H-Mode, LFS (b) H-Mode, HFS

0

200

400

600

0.9 0.92 0.94 0.96 0.98 1

−80

−60

−40

−20

0

20

40

ρSh

ot: 1

1208

0301

4 T

ime:

0.92

2s −

0.99

9s

Tpol

Ttor

Tpol

Ttor

Er

Er Vpol

Er Vtor

Er dia

Er

Er VtorEr dia

Er Vpol

J?(x)

encs

⇡ b⇥ ⇢s

rp?nT

e

J?(x)

encs

=1

n

Zv?cs

F (x,v)dv

v✓

k?⇢i

r2A− 1

v2A

@A?@t2

= −4⇡

c

X

↵

q↵

ZvF

↵

dvkdµ

@F↵

@t+

vkb− c

B0r(φ− 1

cv? ·A?)⇥ b0

�· @F↵

@x− q

m

"r(φ− 1

cv? ·A?) · b+

1

c

@Ak

@t

#@F

↵

@vk= 0

µ = v2?/2v

T

p

= (mc/eB2)(@2A?/@

2t)

d

dt

⌧Z(1

2v2k + µ)(m

e

Fe

+mi

Fi

)dvkdµ+!2ci

⌦2i

|r?Φ|2

8⇡+

|rAk|2

8⇡

�

x

= 0

⌘ − v? ·A?/c

v

L

p

= −(mc2/eB2)(@r?φ/@t)

k2?⇢2i

⌧ 1r2φ+!2pi

⌦2i

r2?φ = −4⇡

X

↵

q↵

ZF

↵

dvkdµ

!2 ⌧ k2?v2A

µB

⌘ µ/B ⇡ const.

b

⇤ ⌘ b+vk

⌦↵0

b0 ⇥ (b0 ·r)b0 b = b0 +r⇥ A

B0

F↵

=

N↵X

j=1

(RR

↵j

)(µ µ↵j

)(vk vk↵j)

@F↵

@t+

dR

dt· @F↵

@R+

dvk

dt

@F↵

@vk= 0

⌦↵0 ⌘ q

↵

B0/m↵

c

⌘ − v? ·A?/c

dR

dt= vkb

⇤ +v2?

2⌦↵0

b0 ⇥rlnB0 −c

B0rΦ⇥ b0

dvk

dt= −v2?

2b

⇤ ·rlnB0 −q↵

m↵

✓b

⇤ ·r+1

c

@Ak

@t

◆

v? ·A? = 1

2⇡

eB0

mc

Z 2⇡

0

Z⇢

0

δBkrdrd✓

Gyrokinetic Current Densities

J

gc

(x) = Jkgc(x) + J

M

?gc

(x) + J

d

?gc

(x)

=X

↵

q↵

hZ

F↵gc

(R)(vk + v? + v

d

)δ(R− x+ ⇢)dRdvkdµi'

7 10

p↵? = m

↵

Z(v2?/2)F↵gc

(x)dvkdµ

p↵k = m

↵

Zv2kF↵gc

(x)dvkdµ

J?gc

= J

M

?gc

+ J

d

?gc

J?gc

=c

B

X

↵

b⇥rp↵

=c

B

X

↵

hb⇥rp

↵? + (p↵k − p

↵?)(r⇥ b)?

i

J

d

?gc

=c

B

X

↵

hp↵k(r⇥ b)? + p

↵?b⇥ (rlnB)i

J

M

?gc

(x) = −X

↵

r? ⇥ cb

Bp↵?

v

d

=v2k

⌦↵

b⇥ (b · @

@R)b+

v2?2⌦

↵

b⇥ @

@RlnB

p↵

= p↵k = p

↵?

ρv - ion

Rx

b - out of the board

k2?⇢2i

⌧ 1 F ! F ! Ak ! Ak v? ·A? ! 0

r2?Ak = −4⇡

cJk

J? =c

B

X

↵

b⇥rp↵

b ⌘ B

BB = r⇥A

B = B0 + B

d

dtr2

?φ− 4⇡v2A

c2r · (Jk + J?) = 0

d

dt⌘ @

@t− c

Brφ⇥ b ·r

dp↵

dt= 0

Ek ⌘ −1

c

@Ak

@t− b ·rφ = ⌘Jk ! 0

! = ±kkvA

r2?A? − 1

v2A

@2A?@t2

= −4⇡

cJ? !2 ⌧ k2?v

2A

J? =c

B

X

↵

b⇥rp↵

r · (Jk + J?) = 0

d

dtr2

?φ+v2A

c(b ·r)r2

?Ak − 4⇡v2A

c2r · J? = 0

Ek ⌘ −1

c

@Ak

@t− b ·rφ = 0

d

dt⌘ @

@t− c

Brφ⇥ b ·r

@Ak

@t! 0 ! 0