Reduced DIA equations for the weak warm beam instability in the strong mode-coupling limit

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Abstract
The direct interaction approximate (DIA) is considered for a one‐component, one‐dimensional turbulent plasma excited by wave–particle interactions in the limit of a weak warm beam instability. The situation where strong mode coupling effects take place is explicitly studied. In this limit the DIA equations for the nonlinear evolution of the instability are reduced to a dimensionless set of equations for the spectral harmonics of the correlation functions in which all time dependence has been scaled out. The distinctive polarization terms and mean field coupling renormalization effects of the DIA are seen to be of the same order as the usual quasi‐Gaussian or ‘‘test particle’’ renormalization effects. From these equations it is shown that the true growth rate γk is given by γk =A[k]γqlk, where γqlk is the quasilinear growth rate; the renormalizing function A[k] is shown to be larger than unity with A[k] =O(1), corresponding to a net increase of energy transfer between the waves and the particles as compared with the quasilinear predictions. The function A[k] may be a slowly varying function of k, but is independent of beam parameters. The basic physical effects are the promotion of mode‐coupling effects to the same order as the quasilinear terms by the intermediation of resonant particles in the mode‐coupling matrix elements, plus the consideration of the non‐Gaussian character of the self‐consistent electric field fluctuations. Arguments are presented that the reduced DIA equations are exact in this limit.