# Excitation and control of chirped nonlinear ion acoustic waves

## Abstract

Large amplitude ion acoustic waves are excited and controlled by a chirped frequency driving perturbation. The process involves capturing into autoresonance (a continuous nonlinear synchronization) with the drive by passage through the linear resonance in the problem. The transition to autoresonance has a sharp threshold on the driving amplitude. The theory of this transition is developed beyond the Korteweg-de-Vries limit by using the Whitham’s averaged variational principle within the water bag model and compared with Vlasov-Poisson simulations.

###### pacs:

52.35.Mw, 52.35.Sb, 52.35.FpWaves in continuous media can be excited by a variety of processes involving resonant wave interactions. The approach requires phase matching and the examples range from optical fiber or superconducting parametric amplifiers (1); (2) to the formation of electrostatic waves in stimulated Raman (SRS) and Brilluin (SBS) scattering in laser plasma interactions (3) and more. The nonlinearity, as well as variation of the parameters of the background medium shift the frequencies and the wave vectors of the excited waves and, thus, tend to destroy the phase matching in resonant interactions leading to saturation of the excitation process. Nevertheless, under certain conditions, the nonlinearity and the variation of system parameters may work in tandem to dynamically preserve the phase matching. This phenomenon is called autoresonance (4). It was studied in many applications, such as particle accelerators (5), fluids (6), planetary dynamics (7), atomic systems (8), optics (9) and more. In plasmas, the autoresonance idea was used for generation of large amplitude plasma waves in beat-wave accelerators (10), excitation of the diocotron modes in pure electron plasmas (11), in recent experiments at CERN on formation of cold antihydrogen atoms (12), as well as in the SRS and SBS theory (13); (14).

The ion acoustic waves are low frequency longitudinal plasma density oscillations. They were predicted by Tonks and Langmuir (15) in 1929 on the bases of the fluid theory and observed experimentally by Revans in 1933 (16). Since then, this important branch of plasma waves was studied in a variety of different contexts, such as laser-plasma interactions (3) and dusty (17), ionospheric (18), and ultra cold (19) plasmas. The SBS is one of the most important resonant three-wave interaction processes in laser fusion plasmas involving ion acoustic waves (3); (20). It describes the decay of the incident high power laser radiation (the pump) in the plasma into the scattered electromagnetic wave and an ion acoustic wave. The process is one of the causes of depleting and redirecting the incident laser flux. Despite its importance, the theoretical understanding of this phenomenon is still incomplete for plasmas characteristic of many present experiments, the reason being the complexity involving such factors as the nonlinearity (21), plasma nonuniformity and time dependence (22), and the effects of resonant particles (23). All these factors affect the phase matching condition between the waves, while the kinetic effects lead to Landau damping, resonant trapping of plasma particles and consequent saturation. Can one preserve the phase matching and avoid saturation of the ion acoustic wave by autoresonance in the system? This work is devoted to studying this question in the small ion temperature limit limit. Previous related work used the long wave length fluid-type, driven KdV model (24); (25). We will advance the analysis by allowing for arbitrary nonlinearity, longer wave length () and finite ion temperature.

This work is motivated by numerical simulations of the following one-dimensional Vlasov-Poisson system describing a driven ion acoustic wave

(1) |

Here and are the ion distribution and the electric potential and , where is a small amplitude wave-like driving potential, having a slowly varying frequency . All dependent and independent variables in (1) are dimensionless, such that the position, time, and velocity are rescaled with respect to the Debye length , the modified inverse plasma frequency , and the modified electron thermal velocity . The distribution function and the potentials are rescaled with respect to , and , respectively (here ). We assume that the electrons are Boltzmanian in the combined driven and driving potentials. We also assume spatial periodicity of period associated with the driving wave and solve the time evolution problem, subject to the simplest initial equilibrium: and , where . Note that and the driving parameters fully define our rescaled, dimensionless problem. We applied our Vlasov code (26) for solving this problem numerically and show the results of the simulations in the driving wave frame in Fig. 1 for , i.e. . We increase the driving frequency, , and use parameters , , , and . Our simulations show that after the driving frequency passes the linear ion acoustic frequency, , the system phase-locks to the drive and one observes the formation of a growing amplitude autoresonant deformation of the ion distribution in the figure. The associated density perturbation comprises a continuously phase-locked, growing amplitude ion acoustic wave. Note that at all stages in this example, the driving phase velocity (its location is indicated by the straight red line in the figure) is well outside the ion distribution and, thus, the effect of resonant particles is negligible, until the final stage at , when some resonant particles can be seen in the simulations. We found that the phase-locking was lost beyond this stage. Figures 2a,b show the time evolution of the amplitude of the first harmonic of the electric potential of the ion acoustic wave and the phase mismatch between the driven and driving waves as obtained in the simulations (full lines) and theory (dotted lines) presented below. Importantly, we also found that the autoresonant excitation as seen in Fig. 1 took place only if the driving amplitude exceeded a threshold ( in our example). Below the threshold, the excitation saturates (see Figs. 2c,d for ) and the phase-locking discontinues.

Our theory of the autoresonant evolution of the ion acoustic wave illustrated in Figs. 1 and 2 is based on the water bag model (27) of the ion distribution. We assume that the distribution is constant, , between two trajectories in phase space and vanishes outside these trajectories (see Fig. 3). In this case, the problem can be described by the following set of the momentum and Poisson equations

(2) | |||

If one defines and , Eqs. (2) yield

(3) | |||

Thus, our water bag model is isomorphic to the fluid limit of the driven ion acoustic waves with Boltzmanian electrons, the adiabatic ion pressure scaling , and .

Next, we observe that by defining the auxiliary potentials via , Eqs. (2) can be derived from the variational principle with the following 3-field Lagrangian

(4) | |||

This Lagrangian can be used in the Whitham’s averaged variational principle (28) for studying the fluid limit of the driven-chirped ion acoustic waves. The idea is to average (4) over the fast oscillations in the problem to get a new Lagrangian characterizing adiabatic modulations of the autoresonant wave parameters and, thus, describe the slow evolution of the system trapped in resonance with the driving wave. In studying the aforementioned autoresonance threshold phenomenon, we limit our theory to a weakly nonlinear evolution stage and, consequently, write the truncated harmonic decomposition of the three potentials (28): , . Here, the amplitudes , , are assumed to be slow functions of time, the wave phase and auxiliary phases (necessary because enter the Lagrangian via space/time derivatives only) are assumed to be fast, but , are constants (given by initial conditions), and , are slow. Furthermore, we assume that the amplitudes of the zero and second harmonics scale quadratically with the amplitudes of the first harmonics. Then, the substitution into (4) (with approximated as , where the phase mismatch is assumed to be slow), averaging over between and , and truncating at the fourth order in terms of the fundamental harmonic amplitudes yields the averaged slow Lagrangian

(5) |

in the problem. Note that in our problem are given. Taking variations with respect to all slow amplitudes, but yields six algebraic equations , where represents the set . These equations allow to eliminate all from the problem and, after the substitution back into (5), obtain a new slow Lagrangian

(6) |

involving the remaining amplitude and phases (via its derivatives and ) and (via their derivatives only). Taking the variations with respect to yields two algebraic equations , which allow to eliminate and obtain the final Lagrangian of form

(7) |

The evaluation of (to ) in our problem involves a lengthy algebraic manipulation, which we performed by using Mathematica (29). Here, we present the final result in the limit :

(8) |

where , , and . Next, we use (8) and take variations with respect to and to get

(9) |

(10) |

At this stage, we write , assume proximity to the linear resonance , and expand and in Eqs. (9) and (10) around to lowest significant order in to get

(11) |

(12) |

Then, after evaluating and , we have

(13) |

and (assuming passage through the linear resonance )

(14) |

where . Finally, we define , rescaled time , and rescaled driving amplitude . Note that Eqs. (13) and (14) can be combined into a single equation for :

(15) |

This one-parameter, nonlinear Schrodinger-type equation is characteristic to passage through linear resonance in many dynamical systems and predicts transition to autoresonance for (4). We illustrate this transition phenomenon in Fig. 4, showing the evolution of for just below and above . Returning to the original parameters in our driven ion acoustic wave problem (), we obtain

(16) |

The numerical solutions of Eqs. (13) and (14) shown in Fig. 2 ( in this example) are in an excellent agreement with Vlasov-Poisson simulations until the amplitude of the wave becomes large in violation of our assumption of weak nonlinearity. Finally, we found that if one uses a linear expansion (the usual assumption in deriving the KdV limit in the problem) instead of the fourth order expansion used above, then in the denominator in the right hand side in Eq. (16) should be replaced by . This yields a significant difference in the threshold at small .

In summary, we have studied autoresonant excitation of nonlinear ion acoustic waves in the fluid approximation by passage through the linear resonance in the problem. Weakly nonlinear Whitham’s averaged variational principle was used in the theory of the autoresonant transition, yielding a good agreement with Vlasov-Poisson simulations. Extension of the variational approach to include larger ion acoustic wave amplitudes, resonant kinetics, and spatial nonuniformity effects seem to be important directions for future research. This work was supported by the Israel Science Foundation.

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