Recurrence of Nonlinear Ion Acoustic Waves

Abstract

TIHE ~FIG. 4 # Magnetic moment, and particle motion along mirror field when electric field is resonant, flute, and moderate amplitude, e<p/(mv 0 2 /2) = OM.mirror machine. 4 For low and intermediate values of <p 9 resonant modes act on the particles quite differently than in the nonresonant case* Where oo -IQ = 0, the system is strongly nonlinear, and near the ends of the system where the nonlinear terms in (1) are small compared to u> -IQ 9 the system is essentially linear. For such an implied change in phase-space topology the particle is almost always forced to cross singular surfaces during one mirror transit and thereby scatter appreciably in a number of transit times. Figure 4 illustrates this well as the field configuration and initial particle phases are identical to those of Fig. 3, except that in Fig. 4 the potential <p is 50 times smaller. Typically, the cumulative changes in p. are sufficient to cause particle loss at time Qt < 10 4 .In summary, what these many computer runs have shown is that the containment properties of weak mirrors seem to be completely consistent with those calculated with the superadiabatic theory at least for times of order 10 5 /27r cyclotron periods. Furthermore, the overall results are that containment is a strong function of orbit phase, and if almost everywhere along the trajectory the conditions O 0 »aJ»cu bounce are satisfied, then moderate values or a strong Z variation of cp lead to the most rapid particle losses.The recurrence time of ion acoustic waves in a plasma has been obtained, as a function of the parameters describing the plasma, on the basis of the Korteweg-de Vries (KdV) equation and Whitham's extended KdV equation. Direct numerical solutions of the ionic Vlasov equation have also been obtained which quantitatively confirm the predictions of the KdV equation.Recurrence of the initial state in nonlinear wave propagation (implying an absence of relaxation to a state of energy equipartition) was first observed by Fermi, Pasta, and Ulam 1 in numerical experiments on nonlinear lattices, and was subsequently explained in terms of solitons and the Korteweg-de Vries (KdV) equation by Zabusky and KruskaL 2 Recent experimental observations 3,4 of recurrence and KdV-type behavior of signals propagating on nonlinear transmission lines have led to renewed interest in this phenomenon. We shall report on a theoretical-numerical analysis of recurrence of ion acoustic waves in plasmas in a regime where solitons have been observed 5 1308 VOLUME 29, NUMBER 19 PHYSICAL REVIEW LETTERS 6 NOVEMBER 1972 in the laboratory. It has recently been shown 6 that weakly nonlinear and dispersive ion acoustic waves in a plasma with warm ions and isothermal electrons are described by the KdV equation, v t + cu x + $(y + l)vv x + hcX D 2 (l+3/e) ml v xxx = 0, (1)where v is the ion fluid velocity, c = c 0 (l +3/0) 1/2 is the linear nondispersive ion acoustic velocity with c 0 =(k B T e /mi) 1/2 , 6 = T e /T { is the ratio of electron to ion temperature, y = (1 +9/0)(l + 3/0)" 1 is the effec...

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