Volume 53A, number 1 PHYSICS LETTERS 19 May 1975 PARTICLE ACCEI FRATION BY A NON-LINEAR LANGMUIR WAVE IN AN INHOMOGENEOUS PLASMA V.I. KARPMAN, J.N. ISTOMIN and D.R. SHKLYAR lzmtran, Moscow Region, 142092, USSR Received 17 April 1975 An expression for the averageparticle distribution function disturbed by a non-linear Langmuk wave in an inho. mogeneous plasma is derived. Particle acceleration due to the resonant wave-particleinteraction 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')dx'-6ot+C(x (I) co2 = ~2 + 3k2(x)v2, 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/kr'¢ 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 At> Tthe change of phase velocity Av~ois 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=u2+y(z)=const, 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. 101