plasma_physics ελληνικα

pi ,

, ,

pi pi, pi pi , pi

pi. pi. pi pi

...

-

pi-

.

pi pi pi -

pi pi

pipi.

pi pi

pi pi pi. pi , -

, , (1) -

pi (2) pi

pi , -

. pi

pi pi

pi. pipi -

pi

. pi pi

pi . pi pi -

pi .

pi

pi pi-

.

pi pi.

pi

. pipi

pi. pi-

pi pi pi

1-6.

pi 10 (3

). pipi

4pi pi pi

.

pi -

pi pi .

pi pi .

. . pi

pi pi pi pi pi-

, pi . ,

. . pi

pi pi.

pi pi pi pi

. Heinz Isliker pi

pi pi-

.

30 2000

1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 pi pi . . . . . . . . . . . . . . . . 1

1.1.1 pi pi . . . . . . . . . . . . . 2

1.1.2 pi . . . . . . . . . . . . . . . 4

1.2 pi pi . . . . . . . . . . . 7

1.2.1 Debye . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2 pi . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.3 . . . . . . . . . . . . . . . . . . . . . 10

1.2.4 . . . . . . . . . . . . . . . . . . . . . . 13

1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 -

pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 -

pi . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 pi . . . . . . . . . . 20

2.2.2 pi

. . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 pi . . . . . . . . . 23

2.3 pi . . . 24

2.3.1 pi . . . . . . . . . . 24

2.3.2 pi -

pi . . . . . . . . . . . . . . . . . . 32

2.4

pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4.1 pi . . . . . . . . 34

2.4.2 pi . . . . . . . . 35

62.4.3 pi pipi -

. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.1 pi . . . . . . . . . . . . . . . 40

2.5.2 . . . . . . . . . . . . . . . . . . . . . . . . 42

2.5.3 pi . . . . . . . 43

2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 pi . . . . . . . . . . . . . . . . . . . . . . 533.1 pi pi . . . . . . . . . . . . . . . . . 53

3.1.1 . . . . . . . . . . . . . . . . . . . . . . 58

3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2.2 pi . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 (MHD) . . . . . . . . . . . . 65

3.4 . . . . . . . . . . . . . . . . . . . . . 68

3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2 - pi . . . . . . . . . . . . . . . . . 78

4.2.1 . . . . . 78

4.2.2 . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.3 . . . . 83

4.2.4 . . . 86

4.3 pi . . . . . . . . . . . . . . . . . . . . 88

4.3.1 pi

pi ( -

) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.3.2 pi

pi ( -

) . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.3.3

pi pi -

pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.4

pi pi pi . . 96

4.4 Alfven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

74.5 . . . . . . . . . . . . . . . . . . . . 101

4.6 pi . . . . . . . . . . . . . . . . . . . . 103

4.7 . . . . . . . . . . . . . . . . . . . . . 105

4.7.1 pi pi . . . . . . . . . . . . . . 107

4.7.2 Bernstein . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

5 pi pi . . . . . . . . . . . . . . . . . . .1175.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 pi (MHD) . . . . . . . . . . . . . 117

5.2.1 . . . . . . . . . . . . . . 118

5.2.2 . . . . . . . . . . . . . . . . 120

5.2.3 pi pi . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2.4 -pinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2.5 -pinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.2.6 pi- . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.5 - . . . . . . . . . . . . 130

5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.7 ` . . . . . . . . . . . . . . . . . . . . . . . 139

5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6 pi . . . . . . . . . . . . . . . . . . . .1496.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.2 pi - . . . . . . . . . . . . . . . . . . . . 150

6.3 . . . . . . . . . . . . . . . . . . . . . . . 157

6.3.1 - . . . . . . . . . . . . . . . . . . . . . . . . 158

6.3.2 . . . . . . . . . . . . . . . . . . . . . 163

6.3.3 pi . . . . . . . . . . . . . . . . . . . . . 170

6.4 . . . . . . . . . . . . . . . . . . . . 174

6.4.1

KdV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.4.2 Zakharov . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

87 I : pi pi-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189

8 II : pi 229

9 III : . . . . . . . . . . . .237

10 IV : Bessel . . . . . . . . . . . . . . . . . . . .239

11 V : Maxwell . . . . . . . . . . . . . . . . . . . . . .241

12 VI : . . . . . . . . . . . . . . . . .245

13 VII : ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257

pi pi . . . . . . . . . . . . . . . . . .262

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .267

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .269

1.

1.1 pi pi

pi (plasma) pi pi 1929

pi Tonks Langmuir pi

. pi pi

pi pi

.

pi pi pi. pi,

-

106107 (). , pi pi-

pi pi , pi

, (). pi

pi pi pi pi -

, pi -

pi E 2mec2 (-), pi me c , pi pi .

pi pi pi (. 1.5),

(300 ), pi

, , pi pi pi

, pi

pi pi pi.

pi pi -

pipi pi

(108K) pi (1015cm3) - pi pi pi pi

pi . pi (2)

(3) pi

14.1 MeV (1eV = 1.6 1012 erg , 11.605 K). pi pi pi -

2

40%. , pi pi

pi pi pipi pi

pi .

pi -

. 99%

pi pi. , -

pi pi pi

, -

, pi

pi pi pi

pi, pi pi

pi.

pi pi

, .

1.1.1 pi pi

pi pi .

pi -

pi pi -

.

, .

pi pi -

.

, pi pi ..pi. pi pi-

pi pi , pi,

pi (glow discharge) pi 1 torr, pi-

109 1011cm3 104K. pi pi

.

eV,

104 105 . pi -

, pi, pi

1023cm3. pi pi. pi pi pi pi

(quantum plasma). pi -

, pi

.

1.1 pi pi 3

70 500 km pi pi pi , .

. pi pi

106cm3 2000 . pi ,

pi, .

pi pi

pi pi

pi. pi pi pi-

.

250-700 km/s, pi 1 10 cm3, 105 . pi -pi 10-13 RE, pi RE =6400 km pi , pi pi pi

.

pi pi , -

pi pi 1010 1011 cm3 - 104 . pi pi 109-108 cm3 106 . -,

pi. pi 102 gr/cm3

107 , pi - pi , pi , pi

pi 105 108 gr/cm3, pi - pi 1028 1031 cm3.

, pi pi pi pi

pi 102 1 cm3 104 . -pi , pi pi

pi, pi, pi pipi -,

, pi , pi pi-

.

1.1: pi pi (Te, Ti) pipi pi - ; (Te - Ti ).

: pi pi pi pi-

, -

pi pi, pi..

4

10

1

1

5

1010

1010 10 10

20 30

n (cm )-3

(

)

1.1. pi pi pi pi.

Ve, Vi

. Ve = 4.2 107T 1/2e < 3 1010 cm/sec Te < 6 109 ' 500 keV. pipi , - mec

2u kBTe.

Vi = 9.8 105T 1/2i < 3 1010 cm/sec Ti < 1013K u 900 MeV.pi pi Te < 500 keV Ti < 1 GeVpi

pi (pi 1.1).

1.1 pi pi -

pi pi pi pi .

- pi pi -

pi pi pi

pi pi.

1.1.2 pi

pi pi :

pi . -

1.1 pi pi 5

Coulomb, pi

pi pi

pi pi, pi

pipi . pi

pi pi -

pi pi pi pi

.

Coulomb. ,

pi pi

pi.

pi pi pi

pi pi

.

pi pi pi pi

pi pi , pi cgs,

pi pi , pi ,

mjdvjdt

= F L = qj [E(r, t) +vj B(r, t)

c] (1.1)

pi FL Lorentz, mj qj E(r, t),B(r, t) pi. -

pi (pi.. pi ,

.) pi pi

pi pi -

pi.

-

pi, pi.

pi pi pi -

Maxwell pi (1.1). Maxwell

E(r, t) = 4pi(r, t) (1.2) B(r, t) = 0 (1.3)

E(r, t) = 1c

[B(r, t)

t

](1.4)

6 1.

B(r, t) = 4pic

J(r, t) +1

c

E(r, t)

t(1.5)

pi pi -

pi () (J ) - r t. pi pi pi

J(r, t) =

N j=1 vjqj(r, t)

V(1.6)

(r, t) =

N j=1 qj(r, t)

V(1.7)

pi vj (1.1), N

pi V pi r. (1.1)-(1.7) pi +4 pi, pi pi pi.

pi, pi

...

pi pi -

, +4 pi .

Vlasov-Maxwell pi -

pi pi pi. pi-

pi pipi pi pi pi

pi pi , pi -

Vlasov-Maxwell pi

pi.

pi pi -

pi pi

pi pi pi-

. pi pi

pi.

pipi pi-

.

1.2 pi pi 7

1.2 pi pi

1.2.1 Debye

pi

( 1/r2), pi-pi pi .

, pi-

pi.

pi pi -

pi. qT > 0 pi. qT pi . pi pi pi qT, pi - pi Maxwell- Boltzmann

(n = A exp(U/kBT ) pi U )

ne = n0 exp

[e

kBTe

], ni = n0 exp

[ ekBTi

](1.8)

pi kB Boltzmann, Tj ( j - (j = e) (j = i)) ,n0i = n0e = n0 pi pi pi qT pi qT, pi - pi. pi Poisson

2 = 4pi = 4pie(ne ni) 4piqT(r) (1.9)pi (r) (x)(y)(z) Dirac. ne, ni pi (1.8) pi e

8 1.

pi (1.10)

2 = 2D

,

, pi,

pi ,

(r) =(qTr

)exp

( rD

). (1.12)

1.2: pi (1.12).

pi (1.12) pi pi

pi pi pi pi r pi pi ( 1/r). pi pi qT. Debye (Debye shielding)

pi pi .

Debye

Debye

1.2. Debye pi pi

Debye pi-

, pi pi -

.

pi pipi . pi

pi pi Te = Ti, - 43 pi

, pi pi -

pi

.

pi pi

pi. pi pi

1.2 pi pi 9

pi pi.

pi pi

pi, pi .

,

e > 1 (1.14)

pi 0 pi pi - Debye. pi pi pi

pi pi pi 0 >> 1.

1.2.2 pi (plasma frequency)

pi pi

j =

(4pin0e

2

mj

)1/2(1.15)

pi pi Langmuir.

pi pi pi-

pi.

, pi

pi . pi pi

pi L pi pi

10 1.

L

+ + - -

+ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ +

- -- -- -- -- -- -- -

- -

- -- -

- -- -

1.3. pi pi .

pi = n0e pipi

|E| = 4pin0e. (1.16)

me = e|E| = 4pie2n0 (1.17) pi pi -

pi pi e = (4pin0e2/me)

1/2.

, pi -

pi pi

p =

4pin0e2

me+

4pin0e2

mi. (1.18)

1.2.3

pi

pi

ei = n0scv,

pi sc pi pi-. , pi -

v, pi pi , pi pi

(pib2)v ( pipi pi pi pi pi b). -

, ei, n0(pib

2)v. pi pipi

1.2 pi pi 11

bV

1.4. e.

sc pib2. (1.19) pi pi

(cross section) pi-

pi pi.

pi v pi

e (. 1.4). pi Coulomb, pi

pi, pi , pi b pi . pipi (impact parameter).

Coulomb pi

pi pi b. Coulomb pi

F = e2

r2er (1.20)

pi er -

pi . F pi

, ,

pi pipi t ' b/v. - - -

(mev) = |Ft| = e2/(bv). pi- > 90, mev, ,

12 1.

(mev) = mev = e2/(bv), b = e2/(mev

2) pi

sc pib2 = pie4

m2ev4

(1.21)

ei n0pie4

m2ev3. (1.22)

pi -

pi

Ve = (kBTe/me)1/2,

ei n0pie

4me(kBTe)3

. (1.23)

pi pi pi

pi 90, ei pipi pipi (bmax/bmin) (. 1.6). bmin = b - pi pi pi pi pi pi,

pi pi -

, pi pi,

. b Debye, Coulomb , pi -

pi Debye. pi

( pi) D/b 10-20 pi pi

ei = [2.9 105]n0T3/2e sec1 (1.24)(pi 1.1 ln =10) pi pi pi - cm3 eV. (1.15) (1.22) pi (e/ei) = 0 1 (pi .(1.14)), pi pi pipi pi pi. pi

pi

t 1e . .

1.2 pi pi 13

VI I

rL V

VII

B

1.5. pi.

1.2.4

-

pi pi pi (1.1)

mjdvjdt

= qj(vj B/c). (1.25) pi B = B0ez vj

pi pi pi,

vj = vje + vje, (1.25)

dvj

dt= 0 (1.26)

dvjdt

=

(qjB0mjc

)vj ez (1.27)

pi pi pi -

(cyclotron frequency)

j |qj |B0mjc

(1.28)

14 1.

rLj = (v/j ), pi Larmor.

1.3

pi pi-

, pi pi-

pi pi pi . 1.1

pi pi pi-

.

pi :

Debye (Dj ), pi pi pi

pi .

pi pi (0) pi pi pi 0 >> 1.

pi (j ), pi pi.

(ei). pipi pi

pi.

(j ) Larmor(rLj ), pi pi pi pi.

1.3 15

1.1.

e 1.8 107B rad/sec

i 9.58 103B rad/sec

pi

pi e 5.64 104n

1/2e rad/sec

pi i 1.32 103n

1/2i rad/sec

pi

ei 2.91 106ne (ln 0) T

3/2e sec

1

-pi

ii 4.8 108ni (ln 0) T

3/2i sec

1

pi-pi

Larmor rLe 2.38 T1/2e B

1cm

Larmor pi rLi 1.02 102T

1/2i B

1cm

Debye Dj 7.43 102T

1/2j n

1/2j cm

Ve 4.19 107T

1/2e cm/sec

pi Vi 9.8 105T

1/2i cm/sec

( pi c.g.s. pi pi eV, 0 pi (1.14))

pi pi :

pi -

pi 0 >> 1.

16 1.

1.1. pi , Debye, pi

pi pi (pi Te = Ti) pi 1.1.

1.2. pi -

. pi pi pi 1eV,

pi n0 = 105cm3. pi

;

1.3. pi mjv2/2.

pi pi, pi

pi n0j = exp(mjv2/(2kBTj)). pi qT, qT mjv

2/2 e, pi pi qT. pi (1.8) pi 1.2.1.

1.4. pi, pi pi x = d pi , pi ( ).

pi n0 q. () pi Poisson pi = 2pinq(d2 x2). () d >> D pi pi pi (x = 0) pi .

1.5. pi Saha (pi pi

pi

T ) pi pi pi pi (T = 300 pi ). ( pi

300 , 13.6 eV pi

n0 = 5 1025m3).1.6. pi pi pi 1.2.3

pi pi . , pi

.

1.7. pi pi pi -

pi pi n = 106m3 100 .pi Debye.

17

1.8. 100 eV -

pi 1000 Gauss pi

. pi pi pi

pi pi pi pi

pi.

1.9. (1) pi

pi .

(2) pi pi pi.

pi

pi ;

1.10. pi pi

pi , pi

V e V i , V e >> V i - Te. pi (kBTe/e) ln (Ve/Vi).

2.

2.1

pi pi pi pi

/ .

pi pi -

(1.1), pi

pi pi pi pi, , -

pi pi

pi.

pi pi pi

pi.

pi : (1)

(pi.. , Tokamak ..pi). (2)

pi , pi, pi-

..pi. (3) pi

(lasers, ..). (4) -

. (5) pi. (6)

pi.

pi

pi ( 1.1),

Lagrange,

Lj = mjc2

1 (v/c)2 + (qj/c)v A qj (2.1)pi A , j = e i. pi A

B = A (2.2)

E = (1/c)(A/t) (2.3)

20 2. /

pi pi

, (2.1) pi

Lj = mjv2/2 + (qj/c)v A qj. (2.4)

2.1: pi (2.4), pi -

(1.1).

pi Lagrange pi pi

Hamilton

Hj =

piqi Lj ,pi pi = L/qi, pi, qi pi. pi

(v c), Hamilton pi

Hj = [c2(p qjA/c)2 +m2jc4]1/2 + qj. (2.5)

(2.5) Hamilton (pi =(H/qi), qi = H/pi) pi pi pi v c v

2.2 21

F j = (qj/c)vj B0 . v = (v, v), pi v

v , d(mv2/2)/dt = 0 d(mv

2/2)/dt = 0. (2.6)

pi pi

xj(t) = xj0 (vj/j) sin(jt+ ) (2.7)yj(t) = yj0 (vj/j) cos(jt+ ) (2.8)zj(t) = zj0 + vjt (2.9)

2.2: pi (2.7)-(2.9).

(.1.5),

pi. -

pi v

z, rLj j .

v = 0, pi pi pi pi

pi m = IS/c, pi I pi pi (I = qj/2pi) S pi pipi (S = pir2Lj ). S I - pi pi pi rLj = v/jpi pi

mj =mjv

2/2

B0(2.10)

mj = (qj/2c)(rLj v) (2.11) pi (2.11) pi pi

pi pi-

pi

( ).

2.2.2 pi

(B = B0ez, F = .)

22 2. /

mj(dvj/dt) = (qj/c)(vj B) + F (2.12)

mj(dvj/dt) = F j (2.13)

mj(dvj/dt) = (qjB0/c)(vj ez) + Fj . (2.14) v v = u + wDj , pi

wFDj = (c/qj)(F B)/B20 (2.15) (2.14)

mj(du/dt) = (qj/c)(uB) (2.16) pi ( v = F = 0)

pi wFDj . -

wFDj (drift velocity) pi . pi

x B

+

2.1. pi pi pi. pi , pi pi wEDj .

(guiding center)

pi wFDj . -

F F,

pi

pi .

2.2 23

: (1) pi (F j = qjE) (2) pi (F j = mjg). pi

wEDj = c(E B)/B20 (2.17) pipi

. pi -

pi, pi

(pi wED), pi

pi (. 2.1).

pi pi

wgDj = (cmj/qj)(g B0)/B20 , (2.18)

pi g pi . pipi

-

pi J = n0e(wgDj wgDe).

pi (gravitational instability),

pi pipi .

2.2.3 pi (B0 = .,E = 0)

Fc

Rc

2.2. pi

24 2. /

, , pi pi (

pi Rc) - , pi pi-

pi . pi

pi

F c = (mv2/Rc)er, , . (2.15),

wcDj =v2(er ez)jRc

(2.19)

pi ez = B/B0. -pi (curvature drift)

.

2.3 pi

2.3.1 pi

rLo

O

2.3. pi. pi.

pi pi rL ( B/rL

2.3 25

pi 0 pi pi

. pi pi

B(r) pi pi (1.1), pi pipipi ,

mj(dv/dt) = (qj/c)[(v B0) + v (r 0)B]. (2.21)pi (2.21)

vB0, pi pi, -

,

pi v = v0 + v1, pi

mj(dv0/dt) = (qj/c)(v0 B0)(pi . 2.3). pi pi pi pi

pi v1. v (2.21)

pi

mj(dv1/dt) = (qj/c)(v1 B0) + F j . (2.22) F j

F j =< (qj/c)v0 (rL0j 0)B > (2.23)pi r = rL0j + r, rL0j Larmor pi pi |B0|, r pi pi pi pi

< f >= (fdt)/T , pi T pi

. F j pi

pi.

-

pi , (1) (pertur-

bation theory) (2) (averaging method).

pi

.

pi , (pi . 2.4) -

pi B = (Br, B, Bz) (2.23) -

F j = Fjez + Fj . (2.23)

pi pi, B pipi v0 F j , Br F Bz F,

26 2. /

rLo

Vo

eZ

2.4. pi pi pi Fj .

Fjez =< (qj/c)(v0 rL0j)(Br/r) > (2.23)

Fj =< (qj/c)(rL0j(Bz/r))(v0 ez) > (2.23)

2.3: pi (2.23) (2.23).

pi F j

pi

.

.

(2.11) (2.23) F j pi pi

(. . 2.4)

F j = 2m < Br/r > ez = 2m(Br/r)ez, (2.24)

Br/r pi 0 . pi- Br/r pi Maxwell B = 0 pi

Brr

+Brr

+Bzz

+1

r

B

= 0. (2.25)

(2.25)

r, pi (2.25)

1

2pir

1

r

B

rd = 0

2.3 27

pi .

r 0 pi Br = 0

limr0

Brr

=Brr

.

pipi pi (2.25) pi

2 limr0

1

2pir

rd

Brr

= limr0

1

2pir

rd

Bzz

2Brr

= Bzz

(2.26)

F j pi (r = 0)

Fj = m(Bzz

). (2.27)

-

pi (Bz/z 6= 0), (qjB0/mjc) Larmor rL0j pi v1

mj

(dv1

dt

)= m

(Bzz

). (2.28)

pi

pi pi .

(2.28) -

pi pi pi ,

pi pi pi -

(|B/B|

28 2. /

P

P

O

2.5. () pi () pi

mjv20/2

Bmin=mjv

2/2

B(2.29)

pi

mjv2 sin2 0/2

Bmin=mjv

2 sin2 /2

B. (2.30)

(pitch angle). pi (2.30)pi , pi pi-

, , pi ,

sin2 =

[B

Bmin

]sin2 0. (2.31)

P .2.5, pi = 90o, pi .

P , pi pi pipi .

pi pi pi

Van-Allen, pi pi pi -

60. pi

Bmax pi pi pi pi-

. pipi pi pi

pi

m = sin1

(BminBmax

)1/2. (2.32)

pi pi am - pi van Allen

2.3 29

. pi -

m

VII

Vx

Vy

V

VI I

2.6. pi pi pi pi . - pi m pi pi (loss cone ).

pi pi van Allen

.

pi pi .

2.6. m pi - , pi

(loss cone).

pi pi pi

pi pi -

z pi pi Bz/z > 0, , -

pi .

pi pi , a = 90o ( pi ).

pi . -

Fj (. 2.23).

rL0j , v0 ez pi (.

2.4) qjv0 ez = (rL0j/rL0j)v0 (2.23)

30 2. /

Fj = (|qj |/c)v0

rL0jBzr

. (2.33)

rL0

x

y

e1

e2

2.7. pi F

pi pi < rL0jBzr > -

pi pi

(e1, e2) Larmor(pi . 2.7)

rL0j = rL0j cose1 + rL0j sine2,

x = r cos, y = r sin

pi

r=x

r

x+y

r

y= cos

x+ sin

y.

pi (Bz/y) = 0, B =B(x)e3. pi < sin cos >= 0 < cos

2 >= 1/2 (2.33) pi

Fj = (|qj |/c)v0rL0je1 < cos2 > Bzx

= |qj |v0rL0j2c

Bzx

e1 = mBzx

e1.

pi pi pi

B = (0, 0, Bz(x)), pi pi

2.3 31

Fj = m2B

B2, (2.34)

Bx(0) = By(0) = 0. pi pi

pi pi

(pi . 2.8) pi (. . 2.15)

+q

B

( )2

2

x x x x x x x x x x x x x

x x x x x x x

x x x x

x x x

x x x x

2.8. pi pi .

wBDj = (c

qj

)(mB

)((B2/2)BB2

). (2.35)

pi

v = v1 + wBDj ,

pi v1 pi (2.28) wBDj pi

(2.35).

2.4: pi (2.33)

(2.34).

2.5: -

pi pi

pi pi pi pi F = (F ,F) =(m[(B )B]/B,m(B2)/2B).

32 2. /

pi (2.2.3) pi,

pi -

pi .

pi pi ,

pi pi -

pi pi .

pi pi B = B(r)e. B = 0,

(B)z = 1r

r(rB) = 0 (2.36)

pi B (1/Rc), pi Rc pi pi, pi pi pipi

B2

B2= Rc

R2c

B2/B2 (2.35) pi

pi ( 2.19)

wB+cDj =mjc

qj

Rc BR2cB

2

[v2 +

1

2v2

]. (2.37)

2.3.2 pi

pi [B = B0ez, E = Ex(x)ex]

pi E =E0 cos(kx)ex, pi pi

vxj =qjmj

Ex(x) +qjB0mjc

vyj (2.38)

vyj = qjB0mjc

vxj. (2.38)

(2.38)

vxj = 2j vxj jEx(x)

Bc (2.39)

vyj = 2j vyj 2jEx(x)

Bc (2.39)

pi pi (Ex/B)c

2.3 33

pi ,

(2.39) x pi pi pi ,

< vyj >= 2j < vyj > 2j

E0c

B0< cos(k(x0 + rL0j cos(jt))) > . (2.40)

pi pi pi

. pi < vyj >' 0 (pi (2.39) . 2.9),

< vyj >= E0cB

< cos(k(x0 + rL0j cos(jt))) > (2.41)

pi cos(k(y0 + rL0j cos(t))) , krL0j 1, pi Taylor pi pi cos e = 1 e2/2 sin e = e, e = krL0j , , pi ,

wED < vyj > = cE BB2

[1 (1/4)k2r2L0j ]= wED[1 (1/4)k2r2L0j ]. (2.42)

2.6: pi (2.42).

wED pi pi pi

+

+

x

y

Ex

2.9. pi pi.

pi. pi, , -

pi pi .

34 2. /

pipi ,

k2r2L0j - pi Larmor.

2.4

pi

2.4.1 pi

pipi pi

pi pi (B/B

2.4 35

B = 0,

() = (mv22B

) = 0. (2.46)

pi pi , B/B

36 2. /

vyj = 2j(vyj +

ExBc

)(2.47),

Ex = iEx.

wPD i

j

ExBc (2.48)

pi pi, (2.47) (2.47)

vxj = 2j (vx wPD) (2.49)vyj = 2j (vy wED) (2.49)

pi wED = c(EB)/B2 = (Ex/B)c. (2.49)- (2.49) pipi pi

, pi

vx = veijt + wPD (2.50)

vy = iveijt + wED. (2.50) pi (2.50) -

(2.50)

vx = 2j vx + (2j 2)wPD (2.51)vy = 2j vy + (2j 2)wED (2.51)

pi pi (2.49) - (2.49) pi

2j >> 2. pi pi ( 0)

2

2.4 37

pi pi, pi

pi pi. -

, Lorentz v B pi. pi

..., wPD pi pi pi .

2.4.3 pi pipi

pi pipi

E = (0, E(t), 0) B = (0, 0, B(t)) x. pi pipi, E(t) = B(t),pi (v/c),

mdv

dt= qE,

pi pi

E. pi ( - Laser), pi

pipi pi

d(mv)

dt= q

[E +

(v B)c

], (2.53)

pi = (1 (v/c)2)1/2 = (12)1/2. x wmD .

, pi

A() = (0, a cos(), 0) pi = t x/c pi pipi.

(2.53), (2.2) (2.3) pi

( 2.2).

d

dt

[x

(1 2)1/2]

=qB

mcy = q

mc2ydA

d(2.54)

d

dt

[y

(1 2)1/2]

=q[EcBx]

mc= q

mc2(c x)dA

d(2.55)

38 2. /

d

dt

[z

(1 2)1/2]

= 0 (2.56)

d

dt

[c

(1 2)1/2]

=qE

mcy = q

mc2ydA

d(2.57)

pi E = B = (dA/d)/c. y pi . (2.54) (2.57) -

[1 x

c

]= (1 2)1/2, (2.58)

pi . pi-

d

dt= 1 x

c(2.59)

pi .(2.55) pi

y/c

(1 2)1/2 = qA

mc2. (2.60)

y pi (2.60) (2.58) (2.59) pi (2.54)

d

dt

(x/c

(1 2)1/2)

=( qmc2

)2AdA

d

d

dt,

pi

x/c

(1 2)1/2 =1

2

(qA

mc2

)2.

pi , pi (2.58)

(2.59),

x =c

2

(qA

mc2

)2 ddt

=c

2

(R

)2 ddt

cos2() (2.61)

y = c(qA

mc2

)d

dt= c

(R

)d

dtcos() (2.62)

pi (2.56)

z = 0, (2.63)

pi R = (qa/mc2). x y

pi T = (2pi/)

2.4 39

< x >=

T0 xdt T0 dt

=

x dtd d

d(1 + yc

dtd

) = c(R2

)21 +

(R2

)2 (2.64)

< y >= 0, (2.65)

< x > x. R < , ey, - pi, pi pi

wmD =q2

2m2c2< E B > (2.66)

x. pi -

wmD pipi pi Doppler

pi pi pi

, -

pi pi Doppler . 1

. pi

pipi pi

pi .

, pi pi -

mdv

dt= qE

pi pi pi

E = (0, E0(y) cos(t), 0). pi pi, -

y = (qE0/m2) cos(t). pi E0(y) pi , pi

E0(y) = E0(0) + yE0(0)

y+

= E0(0) qE0(0)m2

E0(0)

ycos(t) + (2.67)

1 pi pi pi Kibble, T.W.B., Phys. Rev., 138,B740, 1964.

40 2. /

pi pi

d2y

dt2=

q

mcos(t)

[E0(0) qE0(0)

m2E0(0)

ycos(t)

]. (2.68)

pi (2.68) d2y

dt2

=

( q2m

)2 E2y(0)y

= Fp. (2.69)

pi pi pi pi pi

(2.68) pi pi pi pi

pi

pi. ponderomotive force

pi - pi pi

pi pi (pi 6.4.2).

2.5 (Adiabatic Invariants)

pi pi pi-

(pi..

B/s

2.5 41

pi W . pi pi - pi .(2.71)

x

(x)

W

x1 x2

2.10. .

T =

dt =

dx

[(2/m)(W )]1/2 , (2.72)

pi . -

J (action integral)

J pdq =

m

[2

m(W )

]1/2dx, (2.73)

pi p = m(dx/dt) q = x - (2.71). J , (dJ/dt) = 0, pi pi. pi J

dJ

dt= 2

d

dt

x2(t)x1(t)

[2m(W )]1/2dx (2.74)

pi x1 x2 pi (. 2.10).

x2(t)x1(t)

[2m(W )]1/2dx pi

1

2

dJ

dt= [2m(W )]1/2x2

dx2dt

[2m(W )]1/2x1dx1dt

+

x2x1

t[2m(W )]1/2dx. (2.75)

[2m(W )]1/2 x1 x2,

t[2m(W )]1/2 = m

[2m(W )]1/2[W

t t

]. (2.76)

42 2. /

pi (W/t) (2.70) (2.71)

W

t= m

dx

dt

d2x

dt2+

t= x

x+

t.

(2.76) pi (2.71)

t[2m(W )]1/2 =

x.

pi

dJ

dt=

x2x1

(x

)dx = 0, (2.77)

(x1) = (x2) x1 x2 pi (x) pi .

2.11. , pi l ,pi pi pi pi .

2.5.2

pi pi pi

.

pi

2.5 43

W =p2

2m+m2q2

2

J pi pi (. (2.73))

J = 2

2W/m/

2W/m/(2mW m2q2)1/2dq = W

2pi. (2.78)

pi pi

(. 2.11) W pi . -

, =g/l,

. pi pi pi-

pi pi 2.9.

pi -

, pi pipi

(pi Wilson-Sommerfeld). -

pi pi.. pi pi-

pi pi . pi

pi pi ,

, .

pi -

pi pi.

2.5.3 pi

pi -

pi.

pi

, pi -

. pi

: (1) pi

j . (2) pi pi-

(pi pi 2.3.1) pi.

(3) pi

, pi pi pi pi-

pi pi (pi . 2.12).

pi pi , pi ,

.

44 2. /

N

S

2.12. , (1) pi pi, (2) , (3) pi .

pi : pi. -

pi . -

Lagrange, pi-

pi z = z0 (. (2.4)) B ez (A = A(r)) :Lj = mj r

2/2 +mjr22/2 + qrA/c. (2.79)

pi ,

A = B(t)r/2, pi

Lj = mj r2/2 +mjr

22/2 + qBr2/(2c), (2.80)

pi pr p

pr = mr p = mr2+ qBr2/(2c).

pi pi ,

md2r

dt2= mr2 +

qBr

c(2.81)

dpdt

= 0. (2.82)

pi pi pi

r pi pipi (2.81) pi

= .pi pi

pi (t) pi pi

(t) = (t)t. pi ,

J1 =

[prdr + pd] . (2.83)

2.5 45

, pr pipi ,

J1 = 2pip = pimr2. (2.84) W = mr

22/2,

J1 = 2piW

= 2pimc

|q| m (2.85)

pi pi m (2.11). pi pi pi

,

Lagrange (pi 2.16).

x=0, Bmin x

x=x , Bmax0

I0

I0

2.13. pi , a b.

: .

pi pi pi pi

pi (.2.13)

. pi-

, (longitudinal

adiabatic invariant)

J2 =

vds (2.86)

pi pi . pi pi 2.3,

pi -

pi ((Bz/z) 6= 0) pipi .(2.28) = mB. pi pi

, -

( . (2.73), pi W )

46 2. /

J2 =

[2m(mBturn mB)]1/2ds = .

[Bturn B]1/2ds = , (2.87)

pi Bturn pi .pi pi ,

(2.87) pi pi -

.

: . -

pi pi

pi (.2.12).

pi R pi. pi pi

J3 =

(p +

q

cA)ds. (2.88)

(2.88) pi pi

R =2pimwBDj Rc

qpiR2cB/c

pi Rc pi pi pi-. wBDj pi (2.37) ,

R =

(vjRc

)2,

v = jrL, pi,

R =

(rLRc

)2.

pi Larmor pi pi pi-

R, pi (2.88) pi . pi

J3 =q

c(piR2cB) , (2.89)

pi pi pi pi pi pi (pi .2.12).

2.6 47

2.6

pi.

pi .

pi j = |qj |B/mjc.

pi , ,

wFDj = (c/q)(F B)/B2. pi pi pi pipi 2.1.

2.1:

)

pi wEDj = cEBB2

0

) pi

wgDj = (cmj/qj)(g B0/B

20)

) pi

pi wcDj =v2(erez)

jRc

)

wBDj = cqj

mB

(B2/2)B

B2

)

pi pi wB+cDj = mjc

qj

RcBR2

cB2

v2 +12v2

)

pi wEDj = cEBB2

[1 (1/4)k2r2L0j ]

)

pi wPD = 1

jB0

dEdt

c

)

v ' c pipi

wmDe =e2

2m2c2< E B >

, pi pi pi pi ,

48 2. /

J =pdq pi pi .

pi pi pi -

pi, , pi..

) pi (qj/2c)(rLj v),)

(Bturn B)1/2ds,

) (q/c)piR2B. pi pi -

pi

pi .

49

2.1. pi pi-

pi .

mj(dv/dt) = m(Bz/z), pi pi pi .

2.2. pi

pipi ( (2.54)-(2.57)).

2.3. pi pi -

pi, pi pi

B(r) = (0/2pi)3/22ME/r

3er pi ME = 2.43 1036statampere cm2, pi (drift ve-locity) (pi 2 109cm pi - ) 104eV (pitch angle) = 45o.

2.4. pi Hamilton

pi, La-

grange pi L = (mv2/2) + (qA v/c) q.2.5.

pi

pi. pi pi

;

2.6. pi -

pi , pi

pi pi, pi-

Bmin = 200 Gauss, pi Bmax = 4000 Gauss.

2.7. q pi, pi

pi pi

v = (E B)/B2, pi v/c

50 2. /

( ), -

eimev, pi ei . (1) pi vD, , v > vD, pipi

pi . ( vD - (run-away velocity). ,

pi-

pi pi (test particle)).

(2) pi, ED, pi

. pi ED pi Dreicer. (3) , ED pi.

2.9. pi pi -

pi (pi.. d/dt

51

1 keV (v = v ). - vm = 10km/sec.() pi pi , -

B1

B2

Vm Vm

L=10 km10

2.14. pi Fermi .

, pi pi .

() pi ;

(pi : (1) -

. (2) pi -

pi . (3) pi T pi pi L pi .)

2.14. q m t = 0 U pi. pi pi-

pi B = B0(1 + ax)ez, pi B0 a . t = 0 - (x = x0, y = z = 0), pi x0 > 0.() Larmor .

() pi ( -

).

() pi pipi pi

;

() pipi pi z = z0, pi pi

pipi z = z0.() pi pi -

pi pipi z = z0 x = x0, y = 0 t = .

52 2. /

2.15. pi pi

Bz = B0(1 + a2z2) pi B0 a -

.

() pi z pi - z = 0 v2 = 3v20 = 1.5v

20;

() pi-

pi pi .

() pi -

.

() pi (longitudinal adi-

abatic invariant).

2.16. pi pi pi -

,

Langrange pi

2.5.3

1. pi

pi

B(r, t) = B0(1 ret)ez

pi (

3.

3.1 pi

pi 1.1 pi pi

pi pi ( )

pi pi

() pi. -

pi pi, pi pi

pi pi .

-

, -

6N0- pi N0 . pi pi, - pi pi

Hamilton, . 2

t=t

t=t

0

1

xi

vi

3.1. 6N0-

(xj(t),

2 pi pi pi pi . .

54 3.

vj(t)). - pi (pi . 3.1). Liouville

Hamilton ,

t = 0 N0 V , pi pi (pi . 3.2).

t1

t0

x

=

V =V

1

1

0

0

dx

dv , V0 0

v i

i

i i

3.2. pi pi pi .

pi pi , pi

pi .

pi, pi . , 6N0- , pi 6-

, pi

pi pi .

pi -

dxdv t pi pi f(x,v, t)dxdv. f(x,v, t) - (distribution function) pi pi

.

: f(x,v, t)dxdv/N0 pi - [x, x + dx] [v, v + dv] t. pipi pi-

3.1 pi 55

,

N0 pi,

f(x,v, t)dxdv = N0.

pi 3

pi pi

f(x,v, t) Boltzmann

f

t+ v xf + 1

mF vf =

(f

t

)c

, (3.1)

pi F pi pi ,

(f/t)c . x v pi

x x

ex +

yey +

zez

v vx

ex +

vyey +

vzez.

pi x. pi pi pi :

pi

pi ; pi Vlasov 1947

pi pi pi . -

pi pi pi Vlasov, -

Boltzmann (3.1), pi pi pi Klimon-

tovich pi .

pi pi pi 6-

. Nj pij (j = e, i) x,v t :

Nj(x,v, t) =

N0i=1

(x xi(t))(v vi(t)) =N0i=1

Si, (3.2)

pi xi(t) vi(t) i Si = (x xi)(v vi) pi pii .3 pi Mandl, . 7.

56 3.

,

pi , (dN/dt) = 0

Njt

= N0i=1

xi xSi N0i=1

vi vSi (3.3)

(3.3) -

xh(x b) =

bh(x b).

vi (3.3) pi

pi (. 1.1),

Njt

=

i

vi xSi

i

qjmj

[Em(xi, t) +

vi Bm(xi, t)c

]vSi, (3.4)

pi Em Bm pi pi (xi, vi)

t. , (a)(x a) = (x)(x a), pi

Njt

= v xNj

qjmj

[Em(x, t) +

v Bm(x, t)c

]vNj (3.5)

3.1: pi (3.5).

(3.5) Klimontovich

Maxwell pi, pi , -

pi . pi

,

pi pi pi pi-

. pi pi

pipi pi. Nj - pi pi

3.1 pi 57

(x). pi - pi vx t,

f(x,v, t) =< N(x,v, t) > .

< N > , pi pi pi

N pi .

Nj(x,v, t) = fj(x,v, t) + Nj (3.6)

Em(x, t) = E(x, t) + E (3.6)

Bm(x, t) = B(x, t) + B (3.6)

pi < N > f , < Bm > B < Em > E, < N >=< E >=< B >= 0, . (3.5), . (3.6) pi , pi

fjt

+ v fj + qjmj

[E(x, t) +

v B(x, t)c

]vfj

= qjmj

(E +

v Bc

)vN

(3.7)

pi Nj ,Bm

Em pi pi pi pi. - pi pi pi (-

) pi

, -

. Nj(x,v, t), Bm(x,v, t) Em(x,v, t)

pi .

(3.7) pi pi -

(x,v), pi pi pi . -

pi pi pi

pi, . -

(3.7)

pi , pi

pi . ,

58 3.

pi pi

Vlasov

fjt

+ v xfj + qjmj

[E(x, t) +

v B(x, t)c

]vfj = 0. (3.8)

pi pi, (3.8) pi

pi pi

(t

3.1 pi 59

, pi -

pi, pipi pi

, pi Gauss

g(u) = (2pi < u2 >)1/2exp

( u

2

2 < u2 >

).

u vx < u2 > < v2x >

f(vx) =n0

(2pi < v2x >)1/2

exp

( v

2x

2 < v2x >

),

pi n0 pi . pi , -

m < v2x >= kBT , pi kB Boltzmann T pi,

f(vx) = n0

(m

2pikBT

)1/2exp

( mv

2x

2kBT

). (3.11)

(3.11) Maxwell (-

pi . 3.3). pi f(vx) (v2 = v2x + v

2y + v

2z),

v

f(v)

Vi v

f(v)

uo

() ()

3.3. () pi, Maxwell, () pi Maxwell pi pipi V0

60 3.

fm(v) = n0

(mj

2pikBTj

)3/2exp

((1/2)mjv

2

kBTj

)=

= n0

(mj

2pikBTj

)3/2exp

( v

2

2V 2j

), (3.12)

pi Vj (kBTj/mj)1/2 , pi , - .

pi pi pi -

u0 (3.12)

f(v) = n0

(mj

2pikBTj

)3/2exp

((1/2)mj(u0 v)

2

kBTj

)(3.13)

pi Maxwellian (pi . 3.3). - 3.2 pi pi pi -

.

pi pi -

.

pi pi

(pi . 4). pipi

pi pi Dirac, pi .

3.2

pi pi Vlasov-Maxwell

pi pi pi pi pi pi -

pi . ,

Vlasov, pi

pi, pi pi pi ,

pi.. . pi

pi pi , pi -

pi . pi

pi -

pi pi

pi pi pi pi .

pi pi,

nj(x, t) =

fj(x,v, t)dv (3.14)

3.2 61

,

uj(x, t) =1

nj(x, t)

vfj(x,v, t)dv (3.15)

pi

Pj(x, t) = m

(v uj)(v uj)fj(x,v, t)dv. (3.16)

Maxwell pi pi vrms

v2rms =< v2 >=

fm(v)v2d3v

fm(v)d

3v

=

(3kBT

m

)(3.17)

Eav =

1

2mjv

2fm(v)d3v

fm(v)d

3v

=3

2kBTj =

(3.18)

=3

2mjV

2j = 3

(1

2mjV

2j

),

Vj pi 3.1.1.

3.2.1

pi ;

pi

pi.

pi pi pi-

pi pi

pi. pi -

pi pi pi pi, pi

pi P = PI, pi I pi. . (3.16) uj = 0 f = fm, pi pi pi pi

P = nkBT. pipi,

62 3.

P = C

pi C , = nom pi Cp/Cv.

PP

= n0n0

,

pipi pi pi P ' nkBT , = 1.

r pi

=(2 + r)

r.

pipi -

. pi -

pi pi pi pi

pi pi pi pi

.

pi pi pi pi

pi pi pi pi pi-

pi ,

.

3.2: pi pi

pi (P = nkBT ).

3.2.2 pi

pi pi,

pi pi pi pi pi.

Vlasov pi-

j,

njt

+ (njuj) = 0. (3.19)

3.3: pi pi Vlasov

pi. (

pi.)

3.2 63

pi Vlasov

v pi , (pi pi

)

t

dv(vfj) +

dv v(v x)fj

+qjmj

dv v

{[E +

v Bc

]vfj

}= 0. (3.20)

, pi pi

t

(vfj)dv =

t(njuj). (3.21)

dv v(v fj) =

dv vvfj = (nj < vv >), (3.22)

pi ( -

pi pi f 0 |v| ) (qj/mj)njE(x, t) (qj/cmj)njuj(x, t) B(x, t) . j pi

(njuj)

t+ (nj < vv >) = qj

mjnj

[E +

uj Bc

]. (3.23)

< v v > pipi pi. pi pi (=0)

uj pi

pi pi f = n0(v uj),

< vv >=1

nj

nj vv (v uj) dv ujuj . (3.24)

fj pi (pi.. Maxwell),

< vv >=

vxvx vxvy vxvzvyvx vyvy vyvzvzvx vzvy vzvz

=

V 2j 0 00 V 2j 0

0 0 V 2j

. (3.25)

pi , -

< v2x >=< v2y >=< v

2z >= V

2j , pi Vj

. pipi < vv >= V 2j I

64 3.

(nj < vv >) = (njV2j I) = (PjI)/mj =

Pjm

, (3.26)

pi Pj = njmjV2j = njkBTj pi.

3.4: pi (3.25)

Maxwellian.

pipi, pi pi

uj pi pi

uj ,

< vv >= ujuj +Pjmjnj

I. (3.27)

(3.27) pi pipi-

(nj < vv >) = ( uj)(njuj)+ (uj )(njuj) + 1

mjPj . (3.28)

(3.23)

pi (. 3.19),

mjnjujt

+mjnj(uj )uj = Pj+qjnj[E +

uj Bc

].(3.29)

(. 3.19), (. 3.29),

Maxwell ( (1.2) - (1.5))pi pi pi pi

pi . pi pi

pi (x, t) = e(ni ne), pi J(x, t) = e(niui neue). (3.29) pi E = B = 0, pi pipi pi Kj =m

(f/t)cvdv pi pi pi (j = e) (j = i) .

3.5:

3.3. MHD 65

qjmj

dv vE vfj = qj

mjnjE

qjmj

dv v

[v Bc

]vfj = qj

mjnj

uj Bc

.

3.3 (MHD)

pipi pi

pi pi (-

) pi pi

. -

(MHD). pi 3.1

Vlasov pi pi

>> ei (pi ei ). - ( ' 0) pipi pi

fjt

+ v fj + qjmj

(E +

v Bc

)vf =

(fjt

)c

. (3.30)

(. 3.19) pi

njt

+ (ujnj) =dv

(fjt

)c

.

pi pi-

dx. pi - ,

,

njt

+ (ujnj) = 0. (3.31) ( 3.29) pi

pi

mjnj

(ujt

)+ mjnj(uj )uj

= Pj + qjnj[E +

uj Bc

]+ Kj . (3.32)

66 3.

pi - , -

, Ke = K i. pi pi -

, pipi pi . , pi

m = neme + nimi ' nimi(pi me

3.3. MHD 67

3.6: pi -

(3.35).

, pi-

.

Ohm pi.

pi (3.32) qj/mj , j = i, e pi

J

t=

e

2meP +

e2mmemi

(E +

uBc

) emec

J B + eme

K i (3.36)

3.7: pipi (3.32)

qj/mj pi , (3.36).

pi pi .

(3.36) pi-

: (1) (nj/t) uj pi (2). (2) ni ' ne Pe ' Pi ' P/2. pi pi me

68 3.

(

mimec

)J B + mi

2meP J

(3.38)

(3.33), (3.34), (3.35) (3.38) pi

(1.4) (1.5) pi pi

(14 14 (m, q, u, J , E B) pipi pi , pi m). pi pi

(MHD) pi.

3.4

pipi pi -

pipi pi

.

E(r, t) = 1c

[B(r, t)

t

]

E

L(B

c

), (3.39)

pi L 1 , , pi E B . - u = dx/dt = L. pi pi 1c Et

/|B|

(L

c

)2(uc

)

3.4 69

(

e

)2 ( cu

)2: 1 : 1 :

(

e

)(ee

)( cu

)2:

(

i

)(csu

)2:

(

e

)(eie

)( cu

)2, (3.42)

pi cs = (P/)1/2 , Ki

nmi|uiue|ei = miJei/e ei . pi (3.42), 0 ,(

e

)

70 3.

(3.43)-(3.46) pi pi pi-

pi pi,

.

3.5

pi pi pi ( pi-

pi pi pipi pi .(1.1) -(1.7)) pi Vlasov-Maxwell. pi-

pi pi pi

. - Vlasov-Maxwell pi

pi pi pi pi pi

.

pipi pi pi

pi pi pi, pi

uj , pi Pj , pi nj ..pi. (

). pipi pi -

Maxwell

pi. pi pi

pi ,

pi pi pi

pi. pipi pi

, P = 0. -

pi pi -

pi pi

(J(x, t), (x, t)). pi pi.

, pi pi -

J , pi pi pi . pi pi-

pi (MHD). pi ,

pi pi 3.4, pi

.

pi pi pi

pi pi pi

pi :

3.5 71

1. (Vlasov-Maxwell)

fjt

+ vxfj +qjmj

E(x, t) +v B(x, t)

c vfj = 0

E(x, t) = 4pi(x, t)

B(x, t) = 0

E(x, t) = 1

c

B(x, t)

t

B(x, t) =4pi

cJ(x, t) +

1

c

E(x, t)

t

J(x, t) = j=e,i

qj vfj(x, v, t)dv

(x, t) = qj fj(x, v, t)dv

2.

njt

+ (njuj) = 0

mjnjujt

+ mjnj(uj )uj = Pj + qjnj

E(x, t) +uj B(x, t)

c

E(x, t) = 4pi(x, t)

B(x, t) = 0

E(x, t) = 1

c

B(x, t)

t

B(x, t) =4pi

cJ(x, t) +

1

c

E(x, t)

t

(x, t) = e(ni ne)

J(x, t) = e(niui neue)

3. (MHD)

mt

+ (mu) = 0

qt

+ J = 0

mu

t= P + qE +

1

cJ B

72 3.

memime2

J

t=

mi2me

P + E +1

cuB

mimec

J B J

E = 1

c

B

t

B =4pi

cJ +

1

c

E

t

4.

mt

+ (mu) = 0

mu

t= P +

1

cJ B

(uB) =B

t

B =4pi

cJ

73

3.1. pi 6- pi-

pi N(x,v, t), , x v , pi pi

.

3.2. pi -

, () pi

pi pi -

pi pi (pi -

), () pi

, () pi pi nb vb pi pi pi pi - pi pi n0 >> nb Ve , () x, < vx >, () pi pi pipi, pi

pi.

4.

4.1

pi pi

pi . pi

pi

pi, , pi

pi pi .

pi pipi , pi ,

pi ( Fourier). pi pi,

pi pi -

( pi

, = J = 0). Maxwell pi pi

2 1c22

t2= 0 (4.1)

pi = E,B.

(4.1)

(r, t) = Aei(krt) (4.2)

pi A . (4.2) (4.1) -pi pi k pi,

2 = c2k2.

pi pi pipi pi

(/k)

76 4.

(/k) . pi pi

pi.

4.1: pi -

. ; -

pi ;

pi .

3 pi pi pi pi

pi (Vlasov-Maxwell

pi, ,

MHD pi pi).

pi

pi. ,

pi (pi.. pi), pipi

. pi

pipi pi ( pi)

pi pi pi pi .

pi pi pi

:

1. pi pi, pi

pi pi pi

s = s0 + s1, pi s0 , pi pi, s1 .

2. pi s1

s1 = s10ei(krt), (4.3)

pipi pi

pi (s0),

3. pi pi

pi pi (k). pipi

4.1 77

pi, s0, pi . pi

6.

pipi pi pi

,

t+ (u) = 0, (4.4)

Euler

[u

t+ (u )u

]= P (4.5)

P = kBT/m, pi pi P pi.

(4.4) (4.5) -

= 0+10 exp(i(k r t)) ( pi pi0 ), u1 = u10 exp(i(k r t)) ( ,u0 = 0, ). pi

i0u1 = kBTm

ik1 ( Euler)

i1 + 0ik u1 = 0 ( ) pi k = kex u1 = u1ex,

i0u1 =kBT

mik0iku1i

2u1 = k2 kBT

mu1

pi

= kcs, (4.6)

pi

cs =

(P

)1/2=

(P00

)1/2=

(kBT

m

)1/2

78 4.

. (4.6) pi -

pi (sound waves)

.

pi pi -

pi pi

pi pi (pi..

, pi).

pi pi -

, -

, pi pi.

pi pi (E1 6= 0,B1 = 0) pi pi pi k E1, B1 6= 0, E1 6= 0. pi pi pi pi pi pi

pi ( pi T. Stix,

Theory of plasma waves), pi

pipi.

4.2 - pi (B0 = 0)

4.2.1

pi ( -

pi ), Vlasov pi

fjt

+ v fj + qjmj

E vfj = 0 (4.7)

pi (4.7) Poisson (.(1.2)) (pipi Maxwell pi

pi pi ) E =E1, pi pi ,

fj = fj0 + fj1, pi

fj1t

+ v fj1 + qjmj

E1 vfj0 = 0 (4.8)

E1 = 4pie[

fi1dv fe1dv

](4.9)

4.2 - pi (B0 = 0) 79

pi ,

pi pi, pi -

pi pi (fi1 = 0). pi fe1 E1

fe1 = f10ei(krt) (4.10)

E1 = E10ei(krt) (4.11)

(4.8)-(4.9) pipi pi

(/k ) xxVe

xfeo( )

O

4.1. pi pi pi

ife1 + ik vfe1 eme

E1 vfe0 = 0 (4.12)

ik E1 = 4piefe1dv. (4.13)

pi (4.12) pi -

fe1 = ieme

E1 vfe0k v .

80 4.

pi

pi, pi pi pi -

x, pi.

fe1 = ieme

E1xfe0vx

kxvx . (4.14) .

(4.13) fe1 pi (4.14) pi - pi

1 +2en0k2

dv

fe0vx(

kx

) vx

= 0. (4.15)

pi ,

/kx >> vx (. . 4.1), pi pipi vx

1 2e

n0k2

dvfe0((

kx

) vx

)2 = 0. (4.16) pi [/kx vx] pi pi . - [vx/(/kx)]

2

1 2e

n02

dvfe0

(1 +

2vxkx

+3v2xk

2x

2

)= 0. (4.17)

pi fe0 Maxwellian,, -, pi k2V 2e

4.2 - pi (B0 = 0) 81

kx

e

3Ve

4.2. pi pi pi.

Dirac, pi

(4.18). pi ;

( pi , ni =ni0)

net

+ (neue) = 0 (4.19)

meneuet

+mene(ue )ue = P neEe (4.20)

E = 4pie(ni ne) (4.21) pi pi pi (4.1)

n = n0 + n1, ue = ue1, E = E1 (4.18), P = 3n0kBTe ni0 = ne0 = n0.

pi E1x u1E1, pi pi, (4.19)-(4.21) pi

in1 + in0kxu1x = 0 (4.22)imen0u1x = en0E1x 3ikBTkxn1 (4.23)ikxE1x = 4pien1. (4.24)

82 4.

(4.22) pi u1x (4.24) pi n1 (4.23)

(2 2e 3k2xkBT/me)E1x = 0. (4.25)pi E1x , (4.25) pi

2 = 2e + 3k2xkBT/me. (4.26)

pi Vg = 3(kDe)Ve.

4.3:pi pi

pi pi,

pi, Te = Ti = 0.

4.2.2 pi

pi pi pi . pi-

pi Poisson

D = 0

D = E E,pi E pi pi pi ( pi pipi).

pi (E E) = 0, pi (4.24)

ikxE1x

(1 +

4pien1E1x

)= 0,

n1 (4.22), (4.23), pi

pi E , pi

E = 1 2e

2 3k2xVe. (4.27)

pi pi pi

pi (pi (4.27) (4.25))

E = 0( pi pi pi-

pi VI).

4.2 - pi (B0 = 0) 83

4.2.3

-

pi pi,

pi pi

, pi

pi.

, (3.19) (3.29),

net

+ (neue) = 0 (4.28)

meneuet

+mene(ue )ue = Pe eneE (4.29)

nit

+ (niui) = 0 (4.30)

mini

(uit

)+mini(ui )ui = Pi + eniE (4.31)

E = 4pie(ni ne). (4.32)pi pi Pj =

jkBTjnj , pi j = e, i. e i pi -, pi pi pi,

pi -

i = 3, ,

pi , i = 5/3. , pi- , e = 1. pi (4.28) - (4.32)

ne1t

+ ne0 ue1 = 0 (4.33)

ni1t

+ ni0 ui1 = 0 (4.34)

mene0ue1t

= ekBTene1 ene0E1 (4.35)

mini0ui1t

= ikBTini1 + eni0E1 (4.36)

84 4.

E1 = 4pie(ni1 ne1). (4.37) pi pi

ne0 = ni0 = n0. pi , , pi pi ,

( 0), ni1 = ne1 pi pipi ui1 = ue1.

pi Poisson

pi pi

pi (E1 = E1ex), pi . pipi pipi pi-

(4.35) - (4.36) pi pi x, - pi pi t (4.33)

(me +mi)2ne1t2

= (ekBTe + ikBTi)2ne1x2

2ne1t2

= c2s2ne1x2

,

pi c2s (P/) = [ekBTe + ikBTi]/[me +mi]. pi pi pipi pi

( )

2 = k2c2s , (4.38)

pi

( 4.6).

4.4: pi pi-

pipi pi

;

pi ,

pi pi pi

, ne1 6= ni1. pi mene0(ue/t) = 0,

me me

4.2 - pi (B0 = 0) 85

i

cs

kx

4.3. pi pi ( Ti 0).

ne1x

= en0ekBTe

E1 (4.39)

, (pi (4.34)

(4.36))

mi2ni1t2

= ikBTi2ni1x2

+ en0E1x

.

pi pipi -

(4.39), (4.40) ne1, ni1, pi Poisson (4.37), pi pi

2 = k2xikBTimi

+k2xekBTe/mi1 + ek2x

2De

(4.40)

pi kDe = 0, (4.37), , Ti = 0 (kDe)

2 >> 1, = i, pii pi .

pi pi pi,

pi pi

(Vj = 0).

86 4.

4.2.4

pi -

pi

. pi

B1 6= 0. pi - pi pi ,

pi pi . -

pi pi (Te = 0 Pe = 0), ,

Ve

4.2 - pi (B0 = 0) 87

c( (E1)) = 4pin0e2

mecE1 +

1

c

2E1t2

. (4.49)

pi ei(krt) E1 k, pi

pi

2 = 2e + c2k2. (4.50)

k

i

e

c

c

s

3Ve

4.4. pi pi () , () pi () ( Ti = 0).

. 4.4 pi

- pi.

pi -

pi .

3Ve .

pi pi pi.

pi pi

n ck

1 2e

2.

88 4.

k < e, pi- (exp(i(kr t)) pi pi pi. pi

, pi ( = e) - pi. pi

,

pi.

4.3 pi (B0 6= 0)

pi pi pi

pi (B1 = 0), - (B1 6= 0), . pi

pi (k B0 = 0) pi (k B0 = 0). pi - pi pi (k E1 = 0) , pi (k E1 = 0) .

4.5:

.

pi pi

pi , pi pi

.

4.3.1 pi

pi (

)

pi pi pi-

pi pi (Te = Ti = 0 Pe = Pi = 0) pi .

pi

k E1 = 0, pi k B0 = 0. pi pi pi pi

4.3 pi (B0 6= 0) 89

net

+ (neue) = 0 (4.51)

meneuet

+mene(ue )ue = eneE1 enec

ue B (4.52)

E1 = 4pie(n0 ne), (4.53)pi ni0 = ne0 = n0.

4.6: (1.3)-(1.5) -

pi Poisson;

E1 = E1ex, k = kex B = B0ez, ue pipi (ex, ey) pi (4.52).pi (4.51) - (4.53)

ine1 + ikn0uex = 0 (4.54)

imeuex = eE1 ecueyB0 (4.55)

imeuey = ecuexB0 (4.56)

ikE1 = 4piene1 (4.57) , (4.55) - (4.56),

uex =eE1/ime(

1 2e2

) (4.58) (4.54) uex, pi (4.57)(

2

2e+ 1 +

2e2e

)E1 = 0, (4.59)

2 = 2e +2e 2UH. (4.60)

pi-

pi pi

pi. pi pi

pi pi.

90 4.

4.3.2 pi

pi pi (

)

, -

pi Poisson pi Maxwell,

pi pipi pi -

pi ,

, ne1 = ni1. - pi pi

( pi ne1,ue,ui,E1 pi pi )

men0uet

= ekBTene1 en0E1 en0c

ue B0 (4.61)

ne1t

+ n0 ue = 0 (4.62)

min0uit

= ikBTine1 + en0c

ui B0 + en0E1 (4.63)

ne1t

+ n0 ui = 0 (4.64)

pi ez

pipi (ex, ez) pi pi - pi (k E1), pi (4.61) (4.63), pi pi

( exp[i(k r t)])

in0(meue +miui) = ik(ekBTe + ikBTi)ne1+en0c

(ui ue)B0. (4.65)

pi (4.65) -

, k ue1 = k ui1 = ne1/n0, (4.65), pi

in0(me +mi)ne1n0

= ik2(ekBTe + ikBTi)ne1

+en0c

k [(ui ue)B0]. (4.66)

4.3 pi (B0 6= 0) 91

k [(uiue)B0] = kx[(uiy uey)B0], pi pipi uiy uey ne1. pi pipi (4.61) k

imen0(k ue) = en0c

k (ue B0). (4.67)

(4.67) pi ,

pi uex uez uey,

uex = ieuey (4.68)

uez = +ueykzkxie

(1

2

2e

). (4.69)

(4.62) pi

kxuex + kzuez = ne1n0

,

ne1n0

=

[kx

( ie

)+ kz

(+ie

)kzkx

(1

2

2e

)]uey. (4.70)

(4.70) pi uey - (4.68) (4.69) pi uex uez ne1. uex uez (4.66) me

92 4.

1 kxc2s

2 ie

2

2i

2= 0, (4.72)

pi 2i

4.3 pi (B0 6= 0) 93

E1, uB0 = 0. . (4.75)-(4.78) pi . (4.43)-(4.45), (4.48) pi 4.24, pi pi

pipi pi (4.75)-(4.78) (4.50)

2 = 2e + k2c2. (4.79)

E1 B0, pi

pi.

() E1 = Exex + Eyey (extraordinary wave). pi

pi pipi (xy). (4.75)-(4.77) pi

imeux = eEx ecuyB0 (4.80)

imeuy = eEy + ecuxB0 (4.81)

ikEy =i

cB1 (4.82)

ikB1 = 4pien0c

uy icEy (4.83)

0 = 4pien0c

ux icEx (4.84)

(4.80)-(4.84) pi pi

pi . (4.82) pi B1, (4.83) (4.84)( B1) ux, uy Ex, Ey. pi , (4.80) (4.81) pi-

ime

(i

4pin0e

)+ e eB0c

(ik2c2

4pin0e+ i4pin0e

)

eB0c(

i4pin0e

)ime

(ik2c2

4pin0e+ i4pin0e

)+ e

ExEy

=

0

0

. (4.85)

94 4.

(pi pi

Ex Ey ) pi :

1

x

y

10B , B

k

Z

4.5. pi pi .

(1

2

2e

)(1 +

k2c2

2e

2

2e

)+2ek

2c2

4e

22e4e

= 0 (4.86)

(4.86) pi n2 = k2c2/2

n2 =k2c2

2= 1

2e

2

2 2e2 2UH

(4.87)

.

(4.87) pi (k) - (4.85),

pi Ex Ey -pi pi (. 4.5).

4.8: pi (4.86) (4.87).

4.3 pi (B0 6= 0) 95

pi pi

O-mode ( ) X-mode ( ).

X-mode pi (resonance) pipi (cut-off). k 0, pi pipi pipi,

k , . pi (4.87) = UH k , -. pipi pipi pi ,

k 0,

k

X-mode

X-mode

O-mode

R

UH

e

L

/k = c

4.6. pi = (k) (Xmode) (O mode).

2 = 2e2 2e2 2UH

4 22UH 2e2 + 4e = 0. (4.88) (4.88) = L = R

L = e2

+2e +

2e/4 (4.89)

96 4.

R = +e2

+2e +

4e/4 (4.90)

pi (.4.6)

pi pi

, pi pi pi-

pi (4.79). pi .4.6

pi 0 < < e pi0 < < L UH < < R . pi pi pi (stop bands).

4.3.4 pi

pi pi

pi pi

pi pi pi

ik E1 = ic

B1 (4.91)

ik B1 = 4pin0ec

u ic

E1 (4.92)

imeu = eE1 ecuB0, (4.93)

pi ue = u, Te = Ti = 0, J = en0u, pi pi (4.3.3).

pi (4.91)-(4.93), pipi

, pi pi

pipi (x, y), k B0ez. pi pi- u, E1 B1 (ux, uy, 0), (Ex, Ey, 0) (Bx, By, 0), (4.91)-(4.93) pi

ikEy = icBx (4.94)

+ikEx =i

cBy (4.95)

4.3 pi (B0 6= 0) 97

ikBy = 4pin0ec

ux icEx (4.96)

ikBx =4pin0e

cuy i

cEy (4.97)

imeux = eEx ecB0uy (4.98)

imeuy = eEy + ecB0ux. (4.99)

(4.94)-(4.99) pi

pi pi pi pi

pi(1 +

k2c2

2e

2

2e

)=2e4e

( k

2c2

)2, (4.100)

pi pi

pi pi pi

n2 =k2c2

2= 1

2e/

2

1e/ . (4.101)

(4.101) , R ( R pi(right circularly polarised)) + L ( pi (left circularly polarised)). pipi -

pi pi E1 ez. pi R pipi pi . pi pi ,

1e/ = 0, k . pipi pi E1 pi . L , pi -

(k ). pi (4.101) pi pi pipi, k = 0,

R

L

= e2

+2e +

2e/4 (4.102)

pipi pi -

(. (4.89), (4.90)).

4.4 99

pi pi

( Alfven) , , pi

pi pi (fast magnetosonic waves).

pipi pi Te =Ti = 0 . E1 =Exex B1 = Byey, pi, k = kez B0 = B0Ez. ui ue pipi (x, y) (.4.8).

x

y

z

1

0B , k

e1B , V

iV

4.8. Alfven.

pi pi Alfven ( -

pi

pi) pi ,

ik E1 = ic

B1 (4.103)

ik B1 = 4picn0e(ui ue) i

cE1 (4.104)

0 = en0E1 en0c

ue B0 (4.105)

100 4.

imin0ui = en0E1 + en0c

ui B0. (4.106)

, uey - E1 B0 ey, - pi uex (. 2.52) ex, - (. . (4.105)). pi ey E1 B0 , pi ey, uiy = uey. pi ex pi

n0e(ui ue) ' n0eui (pi . 4.104) pi - B1 pi. ( pi pi

ue, pi pi - uex 0, uey = uiy.) (4.103)-(4.106) pi

ik2c

Ex =4pin0e

cuix i

cEx (4.107)

imiuix = eEx + ecuiyB0 (4.108)

imiuiy = ecuixB0. (4.109)

pi (4.109)

uiy = iiuix

uiy (4.108)

uix =

(e/m

i + i2i /)E1,

uix (4.107) - pi

1 k2c2

2+

2i2i 2

= 0. (4.110)

4.5 101

pi m n0mi pi . - Alfven vA = (B

20/4pim)

1/2, pi Alfven pi

2 =k2v2A

1 + (v2A/c2). (4.112)

Alfven pi pi

, pi = kvA. pi -

Alfven (. 4.6).

cs = (P/)1/2 -

pi vA = ((B2/4pi)/)1/2, pi

Alfven pi pi B2/8pi. pi pi Alfven

( = kcT, cT =T/m, pi T

m ). Alfven pi

pi pi

pi (.4.9).

y

z

Btot

Btot

B1

B0

B1

B0

B0

4.9. pi pi Alfven (Btot pi).

102 4.

4.5 (Fast Magnetosonic)

pi pi

pi pi pi pi-

(Te = Ti = 0,me = 0 E1 B0 = k E1 = 0). B0 = B0ez,k = key, E1 = E1ex B1 = B1ez. pi

ik E1 = ic

B1 (4.113)

ik B1 = 4picn0e(ui ue) i

cE1 (4.114)

0 = en0E1 en0c

ue B0 (4.115)

imin0ui = en0E1 + en0c

ui B0. (4.116)

pi (4.115) pi

E1 B0 k, ( pi ) . pipi pi ex

pi pi E1. pi pi ex pi B1pi z. pi pipi pipi

ikE1 = icB1 (4.117)

ikB1 =4pin0e

cuix i

cE1 (4.118)

imiuix = eE1 + ecuiyB0 (4.119)

imiuiy = ecuixB0. (4.120)

4.9: pi pi

Alfven , pi

.

4.6 pi 103

pi pipi pi pipi -

(4.107)-(4.109) pi pi pi

Alfven, pi ( -

,

104 4.

, pi pi -

. pi pi . 4.10.

pi pi -

, , pi-

pi , ,

pi wD.pi k = (0, ky, kz), ky >>

kz 0. pi pi pi k (Ey >> Ez) Eyey B0ez pi ex pi .

(u1 ue1)

in1 + (n0u1x)x

= 0. (4.122)

pi ( pipi pi)

u1x x kzu1z,pi kz pipi . pi k1yu1y u1y pi , pi pi pi u1y (. 2.52). pipi u1x pi EB, , pi (2.17) pi E1 ' E1y,

u1x =E1c

B0. (4.123)

pi

n1 =1

i

n0x

E1B0c. (4.124)

(4.20) ez

imen0u1, pi , pi P = kBTn,

0 ' kBTeikzn1 en0Ez' kBTeikzn1 en0E1(kz/ky) (4.125)

n1 =ien0E1kBTeky

. (4.126)

(4.126) (4.124) pi

E1 pi

4.7 105

=kyV

2e

eLn, (4.127)

pi

Ln (

1

n0

dn0dx

)1> 0

( dn0dx < 0, . . 4.10). (. .(2.15), F = P/n0)

wnDe =c

qe (P/n0 B0)

B20

wnDe =V 2eeLn

pi (4.127) pi

kywnDe . (4.128) , pi -

. pi

, pi -

pi . pi -

pi ,

pi pi

pipi .

4.7

pi pi pi -

pi pi.

3 pi pi , -

. pi pi

pi

pi ( pi).

pi Larmor . pi

pi pi pi pi

pi pipi pi-

pi, pi Vlasov.

106 4.

pi pi -

pi -

e e. pi pi pi

pi, pi-

pipi E = E0ei(krt)ex, pi

pipi UH- pi pi

4.3.1.

pi

pi pi pi x0 = 0 pi pi (. pi 2.2.1)

x+2ex = 0. (4.129)

, pi pi, pi

x+2ex =eme

E0 exp(i(kx t)). (4.130)

pi pi (4.130)

kx krL sin(et), (4.129),

x+2ex =eme

E0 exp(i(krL sin(et) t)). (4.131)

, Bessel

(pi pi IV) pi ,

exp(z(x 1/x)/2) =

n=

xnJn(z). (4.132)

z = krL x = exp(iet)

exp(ikrL sin(et)) =

n=

Jn(krL) exp(inet), (4.133)

(4.131)

x+2ex =emE0

n=

Jn(krL) exp[i( ne)t]. (4.134)

(4.134)

4.7 107

x = eme

E0

n=

Jn(krL) exp(i( ne)t)2e ( ne)2

(4.135)

pi pi .

pi

e pi Jn(krL)[

2e ( ne)2]1. pi

pi pi .

= (n1)e, pi n = 1,2,3, ... krL 0, pipi pi, Bessel

pi pi (krL/2)n/n!, pi pi

n n = 0. n = 0 uxpi pi pi pi (4.135) (4.58).

pi, pipi pi krL , .(4.135) (4.58) pi 4.3.1 -

pi (4.60) , pi pi

e.

4.7.1 pi pi

pi pi -

pi . (4.15), fe0 Maxwellian, pi W :

W (z) =12pi

x

x z exp(x2

2)dx (4.136)

(4.15) pi pi pi ,

pi Maxwellian,

1 +k2Dk2W

(

kVe

)= 0 (4.137)

pi kDj = (4pine2/kBTj)

1/2 Debye.

W . zpi , pipi

(pi pi ) pi

pi . |z| < 1, pi pi pi

108 4.

W (z) = i(pi

2

)1/2z exp

(z

2

2

)+ 1 z2 + z

4

3

+ (1)n+1z2n+2

(2n+ 1)!! , (4.138)

pi (2n+ 1)!! = (2n+ 1) (2n 1) ... 3 1. |z| > 1

W (z) = i(pi

2

)1/2z exp

(z

2

2

) 1z2 (2n 1)!!

z2n (4.139)

pi pi pi-

, pi pi pi

pi .

pi pi (4.139) pi-

pi pi pi pi-

pi 4.2.1: (4.139) (4.136)

( k2V 2e

4.7 109

1 +j

k2Djk2{

1 +n

nj

[W

( nj

|k|(kBTj/m)1/2

) 1]n(b0j)

}= 0

(4.141)

pi

b0j k2kBTj/(mj2j ),n(b0j) In(b0j) exp(b0j),

In pi (modified) Bessel. - n, pi pi. -

:

() 0 n 1() 0(b0j) pi pi

0(0) = 1() n 6= 0 n(b0j) n(0) = 0,

b0j , -

n(b0j) (2pib0j)1/2 exp(n2/2b0j), (4.142) b0j .

(4.142) pi pi pi

n(b0j) n. n(b0j) b0j = n2 |nj | = k(kBT/m)1/2 n(b0j) 1/|n|.

. (4.142) B0 0. pipi j 0, pi

n=

dx

|j | ,

pi nj x. pi pi pi pipi - B0 6= 0 pi pi pi pipi pipi B0 = 0.

110 4.

( -

pi pi). pi

pi (pi (4.141))

1

1

2

2

3

3

4

4

5

5

6

k rL

UH

/

e

ee2

( / ) = 10

10

10

10

10

5

5

5

5

5

1

1

1

L

4.11. pi Bernstein.

1 +k2Dek2

[1 0(b0e)] (4.143)

kDek2

ne

[1W

( ne|k|Ve

)]n(b0e) = 0,

pi ' ne, n = 1,2, ... . = n(k) = ne(1 +n) pi

|k|Ve|ne|

4.8 111

n =k2Den(b0e)

k2 + k2De[1 0(b0e)].

.4.11 pi pi /e krL ().

4.8

pi Vlasov-

Maxwell pi pi

pi ei(krt) pi pi. 4.1 pi

pi.

pi pi pi pi

pi (-

pi, pi, pi ..pi.).

pi , pi

pi pi . -

pi pi

pi (Omode) (Xmode), pi pi-

pi pi -

.

pi pi

Alfven pi pi-

, pi ( Alfven

(convection zone) -

pi

pi ).

pi pi : (1)

pi pi, pi

; (2)

pi , pi pi pi ,

pi ; (3) pi

pi pi ;

pi pi 6.

112 4.

4.1: pi

= kc

= kcs

B = 0

( pi) 2 = 2e + 3k2V 2e

( ) 2 = k2 ikBTimi +k2ekBTe/mi1+ek22De

2 = 2e + c2k2

4.8 113

4.1: pi

B 6= 0

pi

pi 2 = 2UH = 2e +

2e

(k = 0, k 6= 0) 2 = 2LH = ie

(k = 0, k 6= 0)(1 2

2e

)(1 + k

2c2

2e 2

2e

)+

2ek

2c2

4e 22e

4e= 0

(k = 0, k 6= 0)(1 + k

2c2

2e 2

2e

)=

2e

4e

( k2c2

)2 Alfven (k 6= 0, k = 0) = kvA(1 + (v2A/c2))1/2

(k = 0, k 6= 0) = kvA(1 + (v2A/c2))1/2

pi

(k 6= 0, k 0) = kV 2e /eLn

Bernstein (k = 0, k 6= 0) = ne(1 +n)

114 4.

4.1. -

:

uet

+ ue

(uex

)=

(e

me

)E eiue

pi ei . () pi pi. ()

.

4.2. H.Alfven pi pi pi Nobel -

pi. 1940 pi

pi pi -

, pi, pi pi pi n0 . pi pi -

pi -

pi ( ,

).

4.3. pi -

, pi pi, pi

: B = B0ez + B1ey, uj = u1yey, E = E1xex, k = kez.pi pi Alfven pi-

pi pi. pi

. (4.112), pi pi pi

, .

4.4. pi pi pi pi -

f0,e =n0

(2piV 2e )1/2

exp(v2x/2V 2e ) + nb(vx Vb),

pi n0, nb, Ve Vb Vb >> Ve. pi pi pi

pi.

4.5. pi pi

pi = 0 =pi/2, pi pi . 4.12 (k,E, pipi x, z). ne1 = ni1, Te = Ti c

2s

.

115

x

y

z, 0B

E, k

4.12. pi pi pi pi 4.5

4.6. Faraday, ,

pi pi

pi B0 pi pi

= 1.5 101120

0B(z)ne(z)dz,

pi 0 pi.

4.7. pi pi -

pi pi -

pi .

pi pi pi (fj0 = fj0(|v|2)). pi fe0 = fi0 = (v)( pi);

4.8. pi .

4.13 pi pi

pi pi pi

.

() pi pi pi n xn, d

2xn/dt2 -

xn1, xn xn+1, pi l.() pi: pi -

pi pi . pi

116 4.

pi pi pi

. pi pi-

n z, pi (n 1) z a ..., pi x pi pi:

xn(t) = x(z, t)

xn1(t) = x(z a, t)xn+1(t) = x(z + a, t).

-

n pi . -

2 ' 2pe(1 + 32D/(2pi/)2), >> D.() pi pi pi pi

pi (g/l)1/2;

4.13. pi.

4.9. pi pi (pulsar) pipi

pi , pi

pi. pi

1, 2 (1,2 >> e, 1 2) pi pi pi .

5.

5.1

pi pi , -

pi . pi, pi

(pi.. -

pi pi, , -

), , pipi

pi pi (

pipi pi pi

pi) . pi -

pi

pi (pi.. ,

pi).

-

pi pi . pi

, pi pi pi-

( pi) ( pi-

, pi. pi pi pi

, -

(. 5.1)). -

pi pi pi

pi ,

pi .

5.2 (MHD) pi

(MHD) (.

(3.33), (3.34), (3.35) (3.38)) pi -pi pi

(/t = 0).

118 5.

()

()

()

()

5.1. pi () pi, () pi,() , () .

pi -

pi

Vlasov-Maxwell ( ).

pi , pi pi

pi . pipi

pi pi

pi .

pi pi,

, pi pi

pipi , -

pi pi.

-

pipi pi -

pi.

5.2.1

pi-

, pi pi pi pi.

: -

pi ;

pi

Ohm ( pi pi pi 3.4)

J = [E + uB/c] Maxwell ( pi,

, pi pi, pi )

5.2 () pi 119

B = 4pic

J (5.1)

E = 1c

B

t. (5.2)

(5.2) pi pi

Ohm pi pi (5.1)

B

t= (uB) + c

2

4pi2B. (5.3)

(5.3) pi

pi pi.

(uB) pi , pi, (c2/4pi)2B pi pi .

pi x0

t0 , (pi (5.3)), ( (5.3)). u = 0 ( , ), B ' 0 ( pi - pi ), (B/B) pi , (5.3) pi

B

t=

c2

4pi2B. (5.4)

pi pi

L pi pi

B

t c

2

4pi

B

L2(5.5)

pi B = B0 exp(t/), pi = 4piL2/c2

(diffusion time).

- (ei) (pi 3.3), , -

pi .

pi , , pi

pi.

120 5.

5.2.2

u

u

u

S

dll

B

5.2. pi 4S pi u.

pi pi pi pi-

4S (. 5.2), pi pi pi pi pi m =

B dA. pi

pi u m

m =

SdA B +

SdA B. (5.6)

pi pi pi ,

ei 0, (c2/4pi)2B ' 0, B

t= (uB). (5.7)

(5.6) (5.7) pi dA =u dl (pi . 5.2)

m =

SdA( (uB)) +

cB(u dl). (5.8)

Stokes

A (B C) = (AB) C pi m = 0, - pi pi (frozen

in), pi (collisionless plasma).

5.2 () pi 121

5.2.3 pi pi

pi pi

MHD, pi -

pi (/t) u = 0,

0 = P + 1cJ B (5.9)

B = 4pic

J . (5.10)

(5.9) (5.10)

(P +

B2

8pi

)=

1

4pi(B )B. (5.11)

5.1: (5.9) (5.10)

(5.11).

pi ,

(B )B = 0, pi pi pi (P +B2/8pi) = 0, , pi( ) .

J pi J B, pi (5.10)

B = f(x)B. (5.12) pi pi pi

(force free), pi

(J B = 0, P = 0, . . (5.9)). pipi - pi pi

pi.

5.2.4 - pinch

- pinch (. 5.3) pi pi pi . pi

(Jext) pi pi , pi

122 5.

J

5.3. - pinch.

pi pi , pi -

pi

Jext. pi- 5.4. pi pi, pi

pi pi pi pi

J B. pi , 0 < r < a, pi pi. r = a pi J = J0(r a) ( ;) (J B)/c pi pi, r = a P = (J B)/c.

5.2.5 Z - pinch

pi pi-

pi pi pi J B(. 5.5). pi

P =1

cJ B = 1

4pi(B)B (5.13)

P

r= B

4pir

(rB)

r. (5.14)

(5.14) , pi pi Z - pinch pi ( 5.2).

5.2 () pi 123

B = 0

B = 0

P

r r0

5.4. pi (P 6=0, r < a), pi pi (B2/8pi 6= 0, r > a).

5.2.6 pi

pi pi pi

pi pi.

pi pi

pi P + B2/8pi = . pi pi (n 6= 0, P 6= 0), - pi pipi pi pi

(n P ). pi pi . -

pi

=nkBT

(B2/8pi)=

pi

pi(5.15)

124 5.

J=J(r)z

B

5.5. pi pi Z - pinch.

pi (103 106, low-) pi, pi -

1 (high-). pi ,(nkBT 1, pi pi .

pi pi

, pi, ,

.

pi pi

.

pi , pi , ..pi.,

pi pi .

5.3

pi pi pi,

pi.. pi pi,

, pi Maxwellian,

pi pi .

, pi

pi pi pi -

4. pi pi pi pi

pi , pi pi pi

5.3 125

pi pi .

, pi 6,

pi pi pi pi

pi . pi -

pi

pi , ,

pi pi

.

pi pi

pi pi

pi. pi:

1. (Streaming Instabilities).

pi pi , -

, pi pi

nb ub >> Vj , pi pi- (mnbu

2b)/2.

pi. pi

pi , pi -

pi pi, pi

pi pi ub 0. - , ,

,

pi pi

pi pi pi -

(wave relaxation time) pi pi

(collision time).

2. Rayleigh - Taylor.

pi (pi.. )

pi (n 6= 0) pi (B 6= 0) pi pi pi

pi.

3. (Kinetic Instabilities).

Maxwellian,

pi pi

pi,

Maxwellian (pi.. pi pi

pi pi (T >> T), pi pi pi

(T = T)).

126 5.

4. (Universal Instabilities).

pi pi pi pi

pi ( pi

pi). pi pi pi-

pi pi

pi pi pi .

pi pi -

pi 1, 2 3. pi 4

, pi

pi .

5.4 (Two Stream Instabilities)

u

f(u)

uo

(/k)

5.6. u0 .

pi pi pi pi

pi (. 5.6). pi-

pi pi.

( ), -

u0 pi (B0 = E0 = 0). - 0

5.4 127

. 5.6). pi pi

, pi -

pi (Vj

128 5.

, pi i

5.4 129

()

y x0

1

F(x, y)

Fmin

()

y x0

1

F(x, y)

Fmin

5.7. () pi , pi pi ,() pi pi pi .

pi (5.23) (5.7)

(5.7) pi (y). pi .(5.7) pipi , .(5.7)

pi , pi -

. pi

Fmin(x, y) > 1 (5.24)

pi Fmin(x, y) pi pi (F/x) = 0. pi - (5.24) pi ( x

130 5.

Fmin = 2i

(me/mi)2/3k2u20+

2ek2u20

.

Fmin

|ku0| e.pi pi

e/u0 < k < e/u0, pi .

pi .

pi pi -

60, pi , pi pipi-

pi ,

pi.

6 -

pi.

5.5 -

pi-

. pi pi pi-

pi pi .

pi pi pi

(pi pi).

pi pi

pi pi -

pi pi.

. 5.8 pi . -

pi ,

Te 6= Ti 6= 0. - vx = vb ( vy = vz = 0) pi. pi

. Vlasov - Maxwell

5.5 - 131

b

f(v )x

vx

()

2 1 3 4

/kb

5.8. pi-. vb. pi pi pi pi- .

. pi

(fj(vx) =fj dvydvz)

fe(vx) = f0e(vx) + fbe(vx) (5.26)

fi(vx) = f0i(vx) (5.27)

pi

f0j =n02piVj

exp

[ v

2x

2V 2j

](5.28)

Maxwell (j = e) (j = i), n0 pi pi Vj . pi

fbe(vx) =nb2piVeb

exp

[(vx vb)

2

2V 2eb

], (5.29)

132 5.

pi nb pi (nb

5.5 - 133

pi,

pi 1 pi

fe = n0(vx) + nb(vx vb) (5.32) pi pi .

pipi pi pipi -

pi 5.4. pi pi pi

pi 4.2.1 pi (4.16)

1 2e

n0k2

dvx

fe(k vx)2

= 0 (5.33)

1 2e

2

2b

( kvb)2 = 0, (5.34)

. (5.21), pi

( pi

5.4, e b i e, . . (5.22))

max ' 31/2

2

(nbn0

)1/3e (5.35)

r ' e kVb. pipi pi-

(3) (1), (nbn0

)1/3>>

Vebvb, (5.36)

pi pi pi

. pi pi pi

(5.32).

pi Landau . -

pi ( /k ' Ve), pi 2 . 5.8, pi Maxwell pi

(f0e/vx)(k vx

) dvx (5.37)

134 5.

pipi pi pi /k =vx. Landau pi pi (5.37) pi , -

pi Landau (Landau damping): pi = r + i,pi ||

5.5 - 135

Le

= pi/8 exp(3/2)

(ekVe

)3exp

[

2e

2k2V 2e

]. (5.39)

pi pi , -

pi pi

/k Maxwell (.5.10).

(/k)

x

xf( )

O

A

B

5.10. pi pi pi , pi pi .

pi f0/v , /k, pi pi pi vx /k, , pi -

vx /k, pi pi . pi pi

pi pi.

-. pi 3 .

5.8 nb 6= 0. pi , pi /k ' vb. pi- (5.30) pi (

(/k) Ve) pi pi (. 5.31). pi -

, nb

136 5.

. (5.30):

. (5.38), Maxwell

(. (5.29)), pi

b = pi

(nbn0

)2 (ek

)2e

[fbevx

]vx=

k

. (5.40)

/k < vb . pi pi

E(x) = (E0ebt)ei(kxrt), (5.41)

pi r = e + 3/2kVe,

b =(pi

2

)1/2(nbn0

)e

(vbVeb

)2(5.42)

pi (/k) ' vb vb >> Veb.

5.3: pi pi (5.42) -

(5.29) (5.40).

pi

pi 3 pi . -

pi pi 4.

pi pi pi

.

pi pi -

. pi pi pi

E(, k) = Er + iEi = 0 = r + i pi pi ||

5.6 137

= Ei(r, k)Er/|=r

. (5.43)

pipi pi (5.30) pi

pi pi pi pi-

.

5.6

pi pi -

( pi-

), pi pi, pi pi

pi pi .

vvd

f(v)

5.11. pi vd > cs .

, k 0 pi ,

pi = kcs ( 4.37). , Maxwell (

Te >> Ti) vd 6= 0 (.5.11), pi -

vd. pi ( )

138 5.

fi(v) =ni2piVi

exp

[ v

2

2V 2i

](5.44)

fe(v) =ne2piVe

exp

[(v vd)

2

2V 2e

](5.45)

k vd, pi pi

1 +2en0k2

dv(fe/v)

(k/v) +2in0k2

dv(fi/v)

(k/v) = 0. (5.46)

Vi

5.7 139

= (pi/8)1/2(memi

)1/2

k{

(cs vd) + cs(miT

3e

meT 3i

)1/2exp

( Te

2Ti

)}. (5.50)

Te >> Ti ( pi pi Landau pi ), k , vd > cs,

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