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

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.