Structure of nonlocality of plasma turbulence

Abstract

Various indications on the weakly nonlocal character of turbulent plasma transport both from experimental fluctuation measurements from Tore Supra and observations from the full-f, flux-driven gyrokinetic code GYSELA are reported. A simple Fisher equation model of this weakly nonlocal dynamics can be formulated in terms of an evolution equation for the turbulent entropy density, which contains the basic phenomenon of radial turbulence spreading in addition to avalanche-like dynamics via coupling to profile modulations. A derivation of this model, which contains the so-called beach effect, a diffusive and convective flux components for the flux of turbulence intensity, in addition to linear group propagation is given, starting from the drift-kinetic equation. The proposed model has the form of a transport equation for turbulence intensity, and may be considered as an addition to transport modelling. The kinetic fluxes given, can be computed using model closures, or local gyrokinetics. The model is also used in a particular setup that represents the near edge region as a relatively stable zone between the core and edge region where the energy injection is locally more substantial. It is observed that with constant, physical coefficients, the model gives a convincing qualitative profile of fluctuation intensity when the turbulence is coming from the core region with either a group velocity or a convective flux.

Nonlinear dynamics of phase space zonal structures and energetic particle physics in fusion plasmas

“…The same results that hold for equation (41) with broad and narrow frequency spectrum fluctuations can be obtained for equation (36), given the mapping of Equations (38) and (39). Thus, one can demonstrate, respectively, quasilinear transport in the radial direction in the case of overlapping resonances, and nonlinear oscillations due to wave-particle trapping in the presence of a nearly periodic large amplitude wave 7 .…”

“…. Following [24,25], it is possible to show that, in the case of many waves characterized by a broad frequency spectrum with overlapping resonances, equation (41) reduces to the quasilinear theory of a weakly turbulent plasma [121,122]. In the opposite limit of a narrow frequency spectrum of nearly periodic nonlinear fluctuations (see section 5.3), equation (41) describes the oscillations of particles that are trapped in the wave, which, however, do not decay in time as expected as a consequence of phase mixing.…”

“…Following [24,25], it is possible to show that, in the case of many waves characterized by a broad frequency spectrum with overlapping resonances, equation (41) reduces to the quasilinear theory of a weakly turbulent plasma [121,122]. In the opposite limit of a narrow frequency spectrum of nearly periodic nonlinear fluctuations (see section 5.3), equation (41) describes the oscillations of particles that are trapped in the wave, which, however, do not decay in time as expected as a consequence of phase mixing. Therefore, in the scalar field δϕ k 0 oscillations are also predicted to continue indefinitely rather than to fade away, as in actual physical conditions [123], due to the fact that the processes described in figure 4 do not account for k 0 -harmonics generation produced by spatial bunching [124].…”

“…4 do not account for k 0 -harmonics generation produced by spatial bunching [124]. Equation (41) can also be used to analyze the adiabatic frequency chirping (| ω| ω 2 B ) of phase space hole-clump pairs due to the balance of EP power extraction by PSZS with a fixed background dissipation [81,82,83,84,125]. In particular, two limiting cases have been investigated in the literature: (i) the ω B t 1 limit [81,82,83], where a recursive solution can be adopted near marginal stability; and (ii) the ω B t 1 limit [84,125], where the lowest order solution corresponds to a constant distribution function inside the wave-particle trapping separatrix (hole/clump), which is slowly shifting in velocity space (chirping).…”

“…Meanwhile, they describe the corrugations of equilibrium radial profiles [35][36][37] as a consequence of fluctuation-induced transport processes, due to emission and reabsorption of (toroidal equilibrium) symmetry-breaking perturbations [15]. In general, the time scale of ZS and PSZS evolution is the same as that of the underlying fluctuations, with important consequences on the meso-scale dynamics of SAW/ DAW and DWT, which could become nonlocal and be characterized, e.g., by avalanches [38][39][40][41]. For understanding and explaining different behaviors of PSZS, it is important to compare the wave-particle trapping time, τ B , with the characteristic time of nonlinear evolution of PSZS, τ NL .…”