Nonlinear particle dynamics in a broadband turbulence wave spectrum

Abstract

In the statistical quasilinear theory of weak plasma turbulence, charged particles moving in electrostatic fluctuations diffuse in velocity, i.e. the velocity variance 〈Δv²(t)〉 increases linearly with time t, for times long compared with the auto-correlation time τ_ac of the field, which may be estimated as the reciprocal of the spectral width of the fluctuations. Recent test-particle simulations have revealed a new regime at very long timescales t[Gt ]τ_ac where quasilinear theory breaks down, for intermediate field amplitudes. As this behaviour is not consistent with a diffusion on quasilinear timescales, the problem of the motion of particles in a broadband wave field, for the case of a slowly growing field, is considered here from a purely dynamical point of view, introducing no statistics on the field and no restriction on the amplitude of this field. By determining, on a given timescale, and in the frame of wave–particle interaction, the spectral width over which waves interact efficiently with a particle, a new timescale is found: the nonlinear time of wave–particle interaction τ_NL∝ (spectral density of energy)^−1/3[Gt ]τ_ac. This is the correlation time of the dynamics. For times shorter than τ_NL, the particles trajectories remain globally regular, and do not separate: they follow a quasifractal set of dimension 2. For times long compared with τ_NL, there appears a ‘true’ diffusive regime with mixing and decorrelation, due to nonlinear mixing in phase space and the localization of the wave–particle interaction. These theoretical results are confirmed by a numerical study of the velocity variance as a function of time. In particular, the particle dynamics really do become diffusive on timescales several orders of magnitude longer than that predicted by quasilinear theory (namely [Gt ]τ_NL[Gt ]τ_ac). Finally, deviations from the quasilinear value of the diffusion coefficient and wave growth rate, discussed in the literature, are explained.