Volume 53A, number 1 PHYSICS LETTERS 19 May 1975

P A R T I C L E ACCEI F R A T I O N BY A N O N - L I N E A R L A N G M U I R W A V E IN

AN I N H O M O G E N E O U S PLASMA

V.I. KARPMAN, J.N. ISTOMIN and D.R. SHKLYAR lzmtran, Moscow Region, 142092, USSR

Received 17 April 1975

An expression for the average particle distribution function disturbed by a non-linear Langmuk wave in an inho. mogeneous plasma is derived. Particle acceleration due to the resonant wave-particle interaction is investigated.

Let us consider a wave excited by an external sta- tionary source at the point x = 0 and propagating in the direction x > 0. The equations for the wave elec- tric field are

/I! ,l} E~(x , t )=E(x)exp i k ( x ' ) d x ' - 6 o t + C ( x (I)

co2 = ~2 + 3k2(x)v 2, co 2 = 4ne2n(x)[m (2)

where E(x ), lc(x ) and ~ x ) are slowly varying quanti- ties. The width of the resonant region is determined by the condition

Iv - v ~ l ~ l / k r ' ¢ v e ( 3 )

where r(x) is a characteristic non-linear time which is of the order of the bounce period of trapped particles,

T = (m/eEk)l /2 (4)

and v~(x) = co[k(x) is the local phase velocity. The trapped particles are dragged by the wave with the phase velocity, thus being accelerated with phase ac- celeration v~o do~o/dx.

The untrapped particles interact resonantly with the wave only during a restricted time T ~ 1/31z, where

k do~ = _ co2 dk 0 = (2otr2)-l; a =2 v~ dx 2k 2 dx" (5)

(For A t > Tthe change of phase velocity Av~o is greater than 1/kT).

An untrapped particle which has a velocity v at x = 0 interacts resonantly with the wave in the vicinity of the point x r for which k(xz) = co]v (x r > 0). The width of the region of resonant interaction (Ax) z is of the order of (Ax~ t = T % ( x r) = 0~[(2 lal r)*. We shall suppose that variations of a and r are small enough

(AX)r d~" (Ax)r da z dx < 1 ' 0t dx < 1 (6)

and, thus, r, a and 0 can be considered, in the first ap- proximation, as constants in the interval of resonant interaction (Ax) r.

The equations of motion for the resonant particles in the vicinity (Ax)r of the resonant point x z can be approximately transformed to the form [1,2]

dz du 101(cosz- 1/0) (7) d-O = ZU; d-'O =

where

z = / R(X') dx' - w t + ~o(x) + rr; u =

0 (8)

dO _ ~/[~/¢(x) dx 6o

The dimensionless quantities 2u, z and 0 play the roles of velocity, coordinate and time, respectively, in the reference frame moving with the phase velocity. As far as this frame moves with the phase acceleration, the force in the second eq. (7) contains two terms: the force from the wave and inertial force which is equal to -0/101 in our units. By virtue of(6) the parameter 0 may be considered as constant. Thus the energy con- servation equation follows from (7):

e = u 2 + y ( z ) = c o n s t , y ( z ) = IOl(z/O- sin z). (9)

* For simplicity we neglect here the external field which pro- duces inhomogeneities. A mote general treatment taking this field into account leads to the rome result.

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Volume 53A, number 1 PHYSICS LETTERS 19 May 1975

# ,

Fig. 1. The potentialy(z) for/3 > 0.

The potential energy y (z) is shown in fig. 1. It is seen that the potential troughs, in which the particles can be trapped, exist only for 1/31 > I. The averaged trapped particle density is defined by [2]

_ I b l n T ~T----T~ fT (10)

where fT is the averaged distribution function of trapped particles, and

1 Za b=~-~ /" (11)

g a t

The meaning of the quantities z a, z a and a is clear from fig. 1. In a weakly inhomogeneous plasma

(i~1 >~1)

4 1 ~ 2 b = ~- Vt-~ sign/3.

If the depths of the potential troughs decrease with in- creasing x, the particles are trapped only in the vicinity of the wave source. Thus, in this case fT =fo (o~(0), 0), where fo (o, x) is the undisturbed distribution function.

By solving eqs. (7) and calculating the average varia- tion of the distribution function 6 f = f(o,x) -.to(O,x), one gets

6f(v, x) 0( 0 a [

(12)

+ 6 ( o - %(x) b(x) o VT(X) - [ o ( O , x ) ] •

Here Xr(X, u) is the point of resonant interaction. For a Maxwellian distribution fo =fM it is def'med by o2(xr) = o 2 + 202 log[n(Xr)/n(x)]. 0(Xr) = 1, if 0 < x r < x; otherwise 0(xr) = 0. The term with the 6- function describes the contribution from the trapped

particles, and the first term comes from the untrapped particles.

A detailed derivation of eq. (12) will be published separately. The variation in the distribution function does not lead to an additional current, i.e. 6](x) = foSf do = 0 (in accordance with the general kinetic equation for stationary case). However, the average variations of the concentration 6n(x) and of the energy flux S(x) = f(mv2/2)b6f do are different from zero. The electrostatic field which could arise due to an(x) =~ 0 depends on the boundary conditions and will be considered elsewhere. The energy flux leads to a wave damping associated with the particle accelera- tion. By using (12) one has

aS= m ~ ~ [fT(X) _fo(V~o,x) ] ~x k3r3[3

where all variables are taken at the point x. The terms wi th f T a n d f o are associated here with trapped and untrapped particles, respectively. For/3 > 0 the trapped particles are accelerated and untrapped ones decel- erated by the wave. For/3 < 0 the effects will be op- posite. It should be noted also that aS/ax = 0 for 1/31 < 1 because in this case b = 0 (trapped particles are absent and the average acceleration of untrapped particles vanishes).

From (12) it follows, also, that, under certain con- ditions the beams can be formed by the particles which are detrapped due to the wave amplitude de- crease. For instance, if the wave propagates in the di- rection of a density gradient (dn/dx > 0,/3 > 0), then the velocity interval where af(o,x)/Oo > 0 will be ap- proximately determined by the condition

2- [ 1, for (TLro)(COro) 2 >> 1 02 - op(O) >

log wr o ~ , for I L J where r o = ¢(0), T L is the Landau growth rate, and o~(0) < o < u~o(x ). Possible instabilities connected with such beams will be considered separately. Nu- merical investigations of some effects similar to those considered above have been published recently in [3].

[1] V.I. Kazpman and D.R. Shklyar, Zh. Eksp. Teor. Fiz. 67 (1974) 102.

[2] V.I. Karpman, J.N. Istomin and D.R. Shklyar, Physica Seripta (to be published, April 1975).

[3] H.H. Klein and W.M. Manheimer, Phys. Rev. Lett. 33 (1974) 953.

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