Quasi-linear analysis of the gyro-water-bag model
Résumé
The energy confinement time in controlled-fusion devices is governed by the turbulent evolution of low-frequency electromagnetic fluctuations of nonuniform magnetized plasmas. The necessary kinetic calculation of turbulent transport consumes much more computer resources than fluid simulations. An alternative approach is based on water-bag-like weak solution of collisionless kinetic equations, allowing to reduce the Vlasov equation into a set of hyrodynamic equations while keeping its kinetic behaviour. In this paper we apply this concept to gyrokinetic modeling, and focus on the weak turbulence theory of the gyro-water-bag model. As a result we obtain a set of nonlinear diffusion equations where the source terms are the divergence of the parallel fluctuating Reynolds stress of each bag. These source terms describe the process of correlated radial scattering and parallel acceleration which is required to generate a sheared parallel flow and may have important consequences for the theory of both intrinsic rotation and momentum transport bifurcations which are closely related to confinement improvements and internal transport barrier dynamics in tokamaks. Using the kinetic resonance condition our quasilinear equations can be recast in a model whose the mathematical structure is the same as the famous Keller-Segel model, widely used in chemotaxis to describe the collective transport (diffusion and aggregation) of cells attracted by a self-emitted chemical substance. Therefore the second result of the paper is the derivation of a set of reaction-diffusion equations which describes the interplay between the turbulence process in the radial direction and the back reaction of the zonal flow in the poloidal direction.