Interaction of Ions with a Spectrum of Electrostatic Waves

A single electrostatic, lower-hybrid wave, propagating across a magnetic eld,can energize ions only if the ion dynamics becomes chaotic [1]. We show that low energy ions, which would not gain energy from a single wave, can gain energy from two or more waves provided the wavelengths and frequencies are chosen appropriately. An appropriate choice can also lead to an extraction of ion energy by the waves. In this paper we give results of our analytical and numerical studies on the dynamics of ions when interacting with two electrostatic waves in the lower-hybrid frequency range. We nd that, for appropriate choices of the wave parameters (amplitudes, frequencies, and wavelengths), the dynamical phase space cannot be distinctly divided as in the case of one electrostatic wave. We analytically deduce, and numerically observe, a new phenomenon of nonlinear, coherent energization by which ions, with initially low energies, started o in the coherent part of phase space can be energized into the chaotic phase space. Thus, low-energy ions, which would be una ected when interacting with a single wave, can be energized by two (or more) waves. In particular, for two waves, the wave frequencies have to be separated by an integer ( 3) multiple of the ion cyclotron frequency, and the higher frequency wave should have the shorter wavelength. For an appropriate choice of the wave parameters, the waves can also be used to remove energy from the ions by this nonlinear process. Since the frequency and wavelength spectra of lower-hybrid waves can be externally controlled in tokamak experiments, it should be possible to either accelerate or decelerate ions of a particular species, and from a predetermined region of phase space. Experiments can be set up to test these nonlinear, coherent energization or de-energization of ions. The motion of an ion, of charge Q and mass M, interacting with two plane electrostatic waves, propagating perpendicularly (along x̂) to an ambient, uniform, magnetic eld ~ B = B0ẑ, is given by: dx d 2 + x = 1 sin (x 1 ) + 2 sin ( x 2 ) (1) where x is the normalized (to the wavelength of the rst wave 2 =k1) position, is the time normalized to the inverse of the ion-cyclotron frequency = QB0=M , 's are the wave frequencies normalized to , = k2=k1, i = QEik1=(M ), and Ei is the electric eld. A parameter which is useful for describing the ion dynamics is the normalized ion Larmor radius = p x + _ x where _ x = dx=d . In general, it has been found that an ion gains energy from a single wave only if its motion becomes chaotic [1]. The ion motion becomes chaotic for > th =4 provided the initial = 0 satis es: p < 0 < (2= ) 1=3 (4 ) The left-hand side of the above inequality gives the lower bound, in , of the chaotic phase space. For ions whose energies are below the lower energy bound of the chaotic region, we expect that their dynamics can be determined analytically. Towards that end, we carry out a perturbation analysis of Eq. (1) using the method of multiple time scales [2]. A more comprehensive and general analytical treatment using the Lie transform perturbation technique has also been developed and is presented elsewheres [3]. The perturbation parameter, in the method of multiple time scales, is the normalized amplitude of the waves. In our analysis we assume that neither 1 nor 2 is an integer, i.e. the wave frequencies are not an integer multiple of the ion-cyclotron frequency. However, we will assume that the di erence in the frequencies of the two waves is an integer multiple of the ion-cyclotron frequency, i.e. 1 2 = N , an integer. (The analysis can be generalized to the case when 1 + 2 = N , and also for 1 and 2 are integers [3].) Our analysis breaks down in the vicinity of the chaotic regime. Upon carrying the multiple time scale analysis to second order in the amplitudes, we nd that an approximate solution of (2) is given by: x( ) ( ) sin f + ( )g : (2) The evolution equations for ( ) and ( ) are: