DRIFT MOTION OF CHARGED PARTICLE IN RESONANCE CONDITIONS

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DRIFT MOTION OF CHARGED PARTICLE IN RESONANCE CONDITIONS

V. Milantiev, A. Miroshnikov

To cite this version:

V. Milantiev, A. Miroshnikov. DRIFT MOTION OF CHARGED PARTICLE IN RESO- NANCE CONDITIONS. Journal de Physique Colloques, 1979, 40 (C7), pp.C7-611-C7-612.

�10.1051/jphyscol:19797296�. �jpa-00219285�

JOURNAL DE PHYSIQUE CoZZoque C7, supptdment au n07, Tome 40, J u i Z Z e t 1979, page C7- 6 1 1

DRIFT MOTION OF CHARGED PARTICLE IN RESONANCE CONDITIONS

V.P. Milantiev end A.G

.

Miroshnikov.

P a t r i c e L d a U n i v e r s i t y , Moscow W-302, U.S.S.R.

The relativistic dynamics of a charged

particle immersed in a strong magnetic (5)

4

field

B,(zft)

in the .presence of

H.F.

qunsi-

d4,

=

a.

+

{ a, eL8* + a, e

aie,

+

rnonochromatfc waves is considered under

different resonance conditions. A slow

+ ~ ( a , e

^{i Q,}

+a,eLL

+ U ~ Q " ' + c c ) f a , , changing "weak" /1/ electric field

g(?7t)

5'

is assumed. The H.F. fi elds m e expressed

d e -

Q L

(7)

ar-

as:

E, = &:(qt.ds,X:je'"

+,,c,

(,,

-3 ^\ ' I

B,

=

$

( ~ , , t , d ~ , H ; )

eLes +

^C.C. d65 ⁺

- L ~ A

-&i- = h , L Ti,(& ^.+

^&+

e

' ) z ~ , + A , (9) Here C.C. represent complex conjugate. 2m.f

Qnantities

, &

are assumed to be

where

Qz

⁼

ej ^&,

^{do =}

^{- - e} moct

^A,

'

slow changing functions of coordinates

and tine. Fast changing nave phases B S ( % ~ )

f=

Functions 0,

, A,

have the form of d,, are defined by equations:

d6i 'ds['ril,t) +

d5,

jg(?&) with coefficients CJ~ and

d;

respectively - = -

d t d t

⁽²⁾

( J

^{=I 9 . .}

.

,5).hcplicit forms of the coeffi-

where

cients are omitted here (see /3/).

at. ,

Considering resonances in functions

q ,

are the frequencies and wave vectors of

A

it is necessary to separate the the quasimonochromatic waves ( ? ) respec- "resonance" phase combination

tively. The particle momentum is presen- ted a8 follows:

where 6,,

-

cyclotron bhase rotation,

- ? + 4

a"

^{+ -9}

~,=e,+id,; 4,; 2 , 4, , 4, -

local unit vectors connected to the field lines of the field;

8, , P' -

longitudinal and the transversal momentum components of the

-3

particle relative to 60

.

The particle motion equation in the

4 4 +

fields

E o , so ,

E+

,

^0- is expressed

= nCBO + + MA 6'M s (hl, B p )

for which nod3 + + .

.

+ n,,, 13, = A

.

^Where

^Me

^are

fixed whole numbers defined by equations (5-9). Consequently is considered to be among the slow changing variables satisfying equation:

m e r e

p4='"ef(e _' !!=-]ai,

_x.

^GL,Q~, L

^are

Larmor radius, particle displacement in the R.F. field and the characteristic

as scale length respectively /3,4/. Then

averaging

/3/

can be oerformed over

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19797296

remaining fast phases in equations (5-9).

In /?i/ it was shown that for the case of one quasimonochromatic wave in the 1st approximation over small parameter

S/1

the resonance phase combinations are:

8,'Bi

, 2 R o + 0 ,

, 30,r@, , ^8, +dBi

(11) ,In the present paper, with a set of quasimonochromatic waves (1 ) apart from (11 ) the following phase combinations are also in resonance:

Bs'fle

, B P & @ S * Q ~

,

a 6 0 * 6 s ~ ' ~ (72) (s f 1).

In

higher order approximations more complicated resonances are possible.

In general, the averaged equations have the form:

--:

dw

(~l,l'~)

+ N > - + A $ ~ ~ J w

rJ.73inq ⁽¹⁴⁾

dt

where

K=[z, %c); ^{y -}

corresponding resonance phase combination.

For example,,in the case of resonance between two waves

((#=a-4,

Y6-J), .+A ) the following equations are obtained:

'' ^&-3, + f l o + & ~ ~ + & & ( J

(18)

-8-

=

3

Coefficients

8o , , M O

are defi- ned by the known expressions of the drift approximation / I / . Exact forms of the remaining coefficients are very compli- cated. The equations are much slm~lfied

in soule particular cases. Equations (13) give the possibility to write the corres- ponding drift kinetic equation in reso- nance conditions / L C / .

R E F E R E N C E S

/I/

A.I. M

o r o z o v, L.S. S o 1 o

-

v i e v, in Reviews of Plasma Physi- ,

cs, edited by M.A. Leontovich (Plenum, New York, 1966) Vol. 2.

/2/ N.N.

B

o ^g o 1 u b o v, Y.A. M i t- r o p o 1 s k i, Asymptotic method in the theory of nonlinear oscilla- tions, Moscow, 1973.

/3/ V.P. I i 1 a n t i e v, Isv. VUZov, Fisika,

5, 63,

1977.

/4/

V.P.

M i 1 a n t i e v, JEFF,

z,

159, 1977.