Mesoscopic Transport Events and the Breakdown of Fick’s Law for Turbulent Fluxes

Abstract

This paper presents a pedagogical review of the physics of mesoscopic transport events and their role in the breakdown of Fick's Law for turbulent transport in magnetically confined plasma. It is now clear that the conventional picture of localized turbulence and quasi-linear calculation of fluxes fails to address and account for the phenomenology of tokamak transport. One key issue is the observed departure from the expected gyro-Bohm transport scaling. The causes of this breakdown of Fickian thinking include turbulent avalanching and pulse propagation (turbulence spreading). Both are mesoscopic transport events, and both tend to de-localize the flux-gradient relation. Turbulence spreading is the process of self-scattering and expansion of a slug or other local exciton of turbulence. Spreading is described by theoretically-motivated, phenomenological reaction-diffusion models for the turbulence activity (intensity) field, much in the spirit of Ginzburg-Landau theory. Such models imply that spreading will occur by propagation of intensity fronts. After discussing the basic theory, this paper presents several critical tests of turbulence spreading models using gyrokinetic simulation. Applications include rho-star scaling, penetration of transport barriers and core-edge coupling. Relevant experiment-theory comparisons are addressed, as well. Avalanching refers to a process whereby correlated topplings of nearby localized cells overturn sequentially and drive a burst of transport. Avalanching is a process intrinsic to systems that support a broad range of scales l between a cell size ∆ and system size L, i.e. ∆ < l < L. Avalanching is also a natural way to produce transport events on scales that exceed the cell size or correlation length. Therefore, the PDF (probability distribution function) of avalanches as a function of l is a crucial quantity, necessary for predicting confinement in a system like ITER, with a very large-scale separation between L and ∆. Avalanching emerged from the theory of selforganized criticality but is a more general phenomenon. The paper traces the intellectual prehistory of avalanching through the advent of self-organized criticality. Special focus is devoted to reduced continuum models of avalanching. The physics of avalanching in confined plasma is discussed in detail, via several multi-faceted comparisons to flux-driven fluid and gyrokinetic simulations. The dominance of bursty, large transport events in the flux is identified. Evidence for avalanching in basic and confinement experiments is summarized. The paper concludes with sections on selected special topics, a discussion of the relation between turbulence spreading and avalanching, and a list of possible future directions. Throughout the paper, an effort is made to set fusion theory and phenomenology in the context of ideas discussed in the broader scientific community.

Effects of plasma turbulence on the nonlinear evolution of magnetic island in tokamak

“…Turbulence spreading 36 – 38 is expected to play an important role for the evolution of an MI 39 , since the accompanying heat or particle flux can change the pressure and current profile inside the MI. Detailed observation of the dynamics of turbulence spreading would be helpful to understand its effect on the MI evolution.…”

Dynamics of potential vorticity staircase evolution and step mergers in a reduced model of beta-plane turbulence

“…Staircase formation in the Hasegawa-Wakatani system are discussed in Refs. [9,10]. These studies are primarily computational.…”

“…Here we added to the eddy diffusivity (the first term on the rhs), a conventional collisional diffusivity D that may be associated with the molecular viscosity ν in eq.(1). We prefer to consider it as a modest additive regularization of the turbulent diffusivity, D. Applying similar argument to the turbulent part of PV and adding the terms responsible for its production, damping and unstable growth (see [9,10] for further details), we write an evolution equation for the potential enstrophy ε as follows:…”

“…In a number of respects, our approach here is along the lines of [8] but as the model is different, so are the results. More about the relation of the present model to that developed by [8] can be found in companion papers [9,10].Apart from fluid mechanics, a promising area for applications of the staircase concept is plasma transport in magnetic fusion devices, such as tokamaks. The idea of spontaneously formed transport barriers (N.B.…”

Bifurcation phenomena in magnetically confined toroidal plasmas

“…Interplay between turbulent transport and MHD instability has been recognized to be a crucial issue in toroidal plasma and various theoretical works on this topic have been done [237][238][239][240][241]. Turbulence spreading corresponds to the spatio-temporal propagation of turbulence from a region where it is locally excited to a region of weaker excitation such as transport barrier [90], magnetic island [242], and scrape-off-layer [243], and it plays an important role in determining the turbulence penetration into these regions [244,245] (see review [246]). When the turbulence spreading into these regions is large enough, the discontinuity of temperature gradient at the boundary becomes small or disappears, while the discontinuity of T e or T i gradient (second derivative of temperature) could be large when the turbulence spreading is shielded.…”