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.

The interplay between toroidal drift-wave turbulence and tokamak profiles is investigated using a wave-kinetic description. The coupled system is used to investigate the interplay between marginally stable toroidal drift-wave turbulence and geodesic acoustic modes (GAMs). The coupled system is found to be unstable. Notably, the most unstable mode corresponds to the resonance between the turbulent wave radial group velocity and the GAM phase velocity. For a low-field-side ballooned drift-wave growth, a background flow shear breaks the symmetry between inwards- and outwards-travelling instabilities. Although this turbulence–GAM coupling may not be the primary driver for avalanches in standard core ion temperature gradient simulations, this mechanism is generic and displays many of the expected features, and should be of interest in several other regimes, which include towards the edge or in the presence of energetic particles.

This paper presents a model for zonal flow generation based on a wave kinetic equation coupled to a poloidal momentum equation in a regime where wave trapping matters. Several models of the wave collision operator have been tested: Krook, diffusion and diffusion plus an instability growth rate. Conditions for zonal instability have been identified. It is found that a zonal instability is possible in all cases. However the force is a power law of the zonal velocity, so different from the quasi-linear case of random phases that produces a force that is linear in velocity. Also the zonal force may change sign, leading to flow radial profiles that are not sinusoidal.

We complete the 2D 2-fields turbulence model previously used with an interchange-like instability by slightly modifying the parallel loss terms to drive drift wave instabilities. We show that the instability driven by temperature fluctuations of the sheath losses is identical to that of the drift wave turbulence. The linear analysis is performed and used to select control parameters that yield identical maximum growth rates for the interchange alone and drift wave alone instability. Combining the two instabilities doubles the maximum growth rate. The non-linear simulations are used to analyse the SOL width. The simulations allow one to identify a low field side SOL region where interchange and drift wave are unstable and a high field side SOL region where only the drift wave is unstable. The SOL profiles appear exponential in the region close to the source but depart from a simple exponential fall-off in the far SOL. The low field side SOL width is found to be larger in the interchange alone case, slightly smaller when both instabilities are present and finally narrower when only the drift waves. For the high field side SOL, without interchange, the drift wave SOL width is observed to be identical to that on the low field side and larger than that when both instabilities at play. The Sherwood dimensionless parameter, ratio of convective particle flux divided by the diffusive particle flux, is used to compare the efficiency of turbulent transport. The profiles of the Sherwood parameter for time and flux surface averaged transport indicate that turbulent transport is dominant close to the separatrix but is less effective towards the far SOL. The Sherwood parameter evolution, determined with the flux-surface averaged transport, indicates that outward avalanche transport with corrugations governs the case with interchange only. When combining the two instabilities, outward avalanche transport is less pronounced and inward avalanche transport is observed, reducing the overall turbulent transport efficiency. The avalanche transport with drift waves only compared to interchange only is found to be inhibited.