Nonlinear theory of resonant beam-plasma interaction: Nonrelativistic case

10.1007/s11452-008-2003-7

“…(18) can be replaced with constants, (20) where q αn are certain dimensionless parameters describing the coupling between the plasma and beam waves. In the limit given by inequalities (19), the parameters q αn coincide with the corresponding parameters in the potential approximation used in [1].…”

“…A nonlinear theory of the instabilities of nonrelativistic electron beams in finite-size plasma systems is presented in our earlier paper [1], where we have developed fairly efficient analytic methods and applied them to analyze resonant instabilities that occur in the regime of the collective Cherenkov effect. The nonlinear analytic theory of collective Cherenkov instabilities is based on the method of expansion of the trajectories of the beam and plasma electrons in powers of the small parameter describing the interaction between and beam and plasma subsystems.…”

“…(1), we took into account the fact that, in a strong external longitudinal magnetic field, the transverse electron motion is forbidden. We also assumed that the beam and plasma are infinitely thin in the waveguide cross section (for a more detailed formulation of the problem, see [1]). At the initial instant, the distribution functions satisfy the conditions (2) where n 0 α are the unperturbed densities of the particles of species α , p 0 α are their unperturbed momenta such that p 0 p = 0 and p 0 b = m (1u 2 / c 2 ) -1/2 , and u is the unperturbed velocity of the beam electrons.…”

“…In order for the series to be convergent, it is necessary that the inequality | | < 1 be satisfied (this inequality, as well as geometric series (15), is understood in an operator sense, i.e., as a result of action on a time-dependent function). In Cherenkov beam instabilities, the fastest growing perturbations are those having frequencies close to nk z u [1]; consequently, from the expression for , we can see that the applicability condition for transformation (13) reduces to the obvious inequality (16) The quantities q nd 2k /dt 2k are transformed in an analogous manner. Now, we introduce the operators (17) in terms of which to write Eqs.…”

“…We further note that, if the electron plasma density exceeds the beam density, then the beam instability saturates in a nonlinear fashion predominantly due to the nonlinear processes occurring in the beam, in which case the plasma can be described in the linear approximation [1,4,5]. In this approximation, the intensity of the generation of the highest longitudinal modes is low [6]; consequently, in Eqs.…”