Wave trapping and E × B staircases

Abstract

A model of E×B staircases is proposed, based on a wave kinetic equation coupled to a poloidal momentum equation. A staircase pattern is idealised as a periodic radial structure of zonal shear layers that bound regions of propagating wave packets, viewed as avalanches. Wave packets are trapped in shear flow layers due to refraction. In this model an E × B staircase motif emerges due to the interaction between propagating wave packets (avalanches) and trapped waves in presence of an instability drive. Amplitude, shape, and spatial period of the staircase E × B flow are predicted as functions of the background fluctuation spectrum and the growth rate of drift waves. The zonal flow velocity radial profile is found to peak near its maxima and to flatten near its minima. The optimum configuration for staircase formation is a growth rate that is maximum at zero radial wave number. A mean shear flow is responsible for a preferential propagation speed of avalanches. It is not a mandatory condition for the existence of staircase solutions, but has an impact on their spatial period.

Self-consistent model of the plasma staircase and nonlinear Schrödinger equation with subquadratic power nonlinearity

“…Concerning the units, "a" is the tokamak minor radius, "c s " is the ion acoustic speed, and ρ is the dimensionless radial coordinate. plasma to generate staircase structures proves to be an established fact [3,4], it has been confirmed computationally using different numerical codes [11][12][13][14][15][16]; discussed theoretically, especially invoking flux-gradient time delay, flux landscape bistability or wave trapping [17][18][19][20][21][22]; and observed experimentally other than on Tore Supra also on DIII-D [16]; KSTAR [23]; and lately in HL-2A L-mode discharges [24].…”

“…In this regard, the interpretation of the parameter γ as "adiabatic" index finds its justification in the two conserved quantities of the NLSE model: the Hamiltonian [Eqs. (22) and ( 23)] and the total probability [Eq. (27)].…”

Predicting the Z-pinch Dimits shift through gyrokinetic tertiary instability analysis of the entropy mode

“…2016; Garbet et al. 2021) that may extend well above the Dimits regime. This raises doubts that the breakdown of these barriers are sufficient to capture the Dimits shift.…”

“…In effect, this creates transport barriers with limited drift waves between these points, limiting overall transport. However, an associated nonlinear effect also seems to be that zonal flows eventually saturate into so called E × B-staircases (Dif-Pradalier et al 2010;Peeters et al 2016;Garbet et al 2021) that may extend well above the Dimits regime. This raises doubts that the breakdown of these barriers are sufficient to capture the Dimits shift.…”

Global E × B flow pattern formation and saturation

“…Besides, inhomogeneous mixing of generalized PV and a feedback process via a Rhines scale dependent mixing length were suggested as the staircases' formation mechanism in a reduced model [13,14] based on the 2D Hasegawa-Wakatani (HW) system of equations for drift-wave turbulence. Various theories and models proposed different zonal flow staircase generation mechanisms [15][16][17][18]. According to our knowledge, there is so far no clear answer as to how the staircase-like corrugations are generated in tokamak plasmas.…”