Particle diffusion in random fields: Time-nonlocal description and numerical simulations

Abstract

The theory of time-nonlocal random processes formulated in terms of the non-Markovian Fokker–Planck equation is used to describe the results of numerical simulations of particle diffusion in the random longitudinal field with given statistical properties. The simulations of particle motion were performed for the wide range of particle velocity and random field parameters. It is confirmed that conventional quasilinear theory in the approximation disregarding the time and velocity dependence of the diffusion coefficient in the velocity space can be used only in the case of small intensity and large width of turbulent field spectrum. The increase of the intensity as well as the decrease of the spectral width lead to considerable deviation of the results of simulations (such as saturation and frequent oscillation of the mean-square velocity displacement) from the predictions of the quasilinear theory. It is shown that in the case of small intensities these deviations can be successfully described in terms of non-Markovian generalization of the quasilinear approximation. In the case of high field intensity the description of these features would require more consistent account for the diffusion coefficient velocity dependence and time-nonlocal effects.

Progress on anomalous transport in tokamaks, drift waves and nonlinear structures

“…The fact that the diffusion in resonance broadening is in velocity space means no significant difference here since it actually enters only as a dissipation. In general we have diffusion both in real space and in velocity space [29]. This type of renormalization also predicts that the mean square velocity perturbation due to turbulence reaches a stationary asymptotic value where no average energy transfer between turbulent fields and particles occurs [29].…”

“…Thus it is natural that the flow gets a slow poloidal variation due to this, Such modes are usually called geodesic acoustic modes (GAMs). The relation between aspects of fluid closure in coherent and turbulent systems has been considered in the framework of a generalized Mattor-Parker system [22,23] and renormalized turbulence theory [28,29]. Recent results on particle pinches will also be discussed and related to fluid closure or self-consistent relaxation of velocity space.…”

Turbulence and Microprocesses in Inhomogeneous Solar Wind Plasmas

“…Some authors studied particle diffusion in given packets of waves with random phases and found discrepancies between the velocity diffusion coefficients calculated numerically and those predicted by the quasilinear theory [132][133][134][135][136][137]. Besides, the non classical nature of the diffusion at work was revealed [138,139] as, for example, the nonlinear time dependence of the particles' mean-square velocity variations (∆v) 2 . Another approach was proposed by the authors [77] as an alternative to the quasilinear theory.…”

Velocity diffusion in plasma waves excited by electron beams

“…Most of the related works consider the motion of particles in given wave fields, assuming or not that the waves are randomly phased. Some authors [26][27][28][29][30][31] observed a series of discrepancies between the quasilinear and the numerically calculated velocity diffusion coefficients and even noted the non-classical character of the diffusion at work [31][32][33], when no linear growth with time of the mean-square velocity displacement v 2 was observed. In some cases, the saturation of v 2 with time was explained by the fact that particles cannot diffuse outside the region limited by the waves' phase velocities.…”