A self-consistent model of the plasma staircase and nonlinear Schr\"odinger equation with subquadratic power nonlinearity

TL;DR

This paper introduces a self-consistent model of plasma staircase formation in tokamak plasma, linking nonlinear Schrödinger equations with Levy noise to explain avalanche dynamics and jet flow interactions, validated by gyrokinetic simulations.

Contribution

It develops a novel theoretical framework connecting nonlinear Schrödinger equations with Levy noise to describe plasma self-organization and avalanche behavior.

Findings

Avalanche size distribution follows a power-law with exponent ~2.56.

Plasma staircase saturates at marginal stability, exhibiting SOC-like behavior.

Model predictions are validated by gyrokinetic simulations.

Abstract

A new basis has been found for the theory of self-organization of transport avalanches and jet zonal flows in L-mode tokamak plasma, the so-called "plasma staircase." The jet zonal flows are considered as a wave packet of coupled nonlinear oscillators characterized by a complex time- and wave-number dependent wave function; in a mean-field approximation this function is argued to obey a discrete nonlinear Schr\"odinger equation with subquadratic power nonlinearity. It is shown that the subquadratic power leads directly to a white L\'evy noise, and to a L\'evy-fractional Fokker-Planck equation for radial transport of test particles (via wave-particle interactions). In a self-consistent description the avalanches, which are driven by the white L\'evy noise, interact with the jet zonal flows, which form a system of semi-permeable barriers to radial transport. We argue that the plasma staircase saturates at a state of marginal stability, in whose vicinity the avalanches undergo an ever-pursuing localization-delocalization transition. At the transition point, the event-size distribution of the avalanches is found to be a power-law $w_\tau (\Delta n) \sim \Delta n^{-\tau}$, with the drop-off exponent $\tau = ({\sqrt{17}} + 1)/{2} \simeq 2.56$. This value is an exact result of the self-consistent model. The edge behavior bears signatures enabling to associate it with the dynamics of a self-organized critical (SOC) state. At the same time the critical exponents, pertaining to this state, are found to be inconsistent with classic models of avalanche transport based on sand-piles and their generalizations, suggesting that the coupled avalanche-jet zonal flow system operates on different organizing principles. The results obtained have been validated in a numerical simulation of the plasma staircase using flux-driven gyrokinetic code for L-mode Tore-Supra plasma.