Zonal Flow Generation in Toroidally Confined Plasmasthrough Modulational Instability of Drift Waves
Creators: name: Abdullatif, Raden Farzand
Publisher: name: The Australian National University
Year: 2017
Subjects: Zonal Flow
Drift Wave
Hasegawa-Mima Equation
Modulational Instability
Ion Temperature Gradient
Nonlinear Schrödinger Equation
Hasegawa-Wakatani Equation
Ginzburg-Landau Equation
Multiple-scale Perturbation Analysis
Derivative Perturbation Analysis

description: The study of zonal flow constitutes an important part in the venture for achieving a controlled fusion reactor because of its role in mitigating turbulent transport. In this thesis, generation of zonal flow by modulational instability is discussed in a simple slab geometry, both in cold and hot ion models. In the case of cold ions with no resistivity, analysis on the Hasegawa-Mima equation results in a real nonlinear Schr¨odinger equation, derived both heuristically and formally, through multiple length and time scale asymptotic expansions. In the collisional case, a nonlinear Schr¨odinger equation with growth, otherwise known as the Ginzburg-Landau equation, is derived. In the simplest analysis it is shown that when a modulational instability criterion is fulfilled, zonal flow will be spontaneously generated from drift waves, with the growth rate of zonal flows increasing linearly with the zonal flow wave number. The growth rate cannot go up indefinitely but should peak at some zonal flow wave number, which is found using the nonlinear Schr¨odinger equation. Nonlinearity is the key ingredient in the modulational instability generation of zonal flows. It is confirmed in this thesis that the potential perturbation of drift-wave has to be split into a fast fluctuating part and a slowly varying surface-averaged part, and thus the original Hasegawa-Mima equation is modified. Unless such modification is made, the nonlinear interaction in the original Hasegawa-Mima equation vanishes. In the formal derivation, analyses on the equation are carried out order by order. It is found that the nonlinear interplay appears at higher orders between the fast and slow component of the fluctuation. In the hot ion case, the ion temperature gradient is taken into account. The equation describing the ion temperature gradient (ITG) is derived in this work by the multiple scale asymptotic expansion from the fundamental set of ion and electron equations of motion, ion continuity equation, and the heat-pressure balance equation. Assuming no collisions, in this model of ITG equation it is found that the ITG mode is unstable at a certain domain value of temperature gradient. The nonlinear Schr¨odinger equation derived by further analysis of the ITG equation gives the condition for the modulational instability of the ITG mode. This condition is reduced to that of drift-waves when the temperature gradient is set to zero. The Ginzburg-Landau equation is derived in this work from a set of potential and density fluctuation equations, known as the Hasegawa-Wakatani equations. These equations take account of resistivity and viscosity of the plasma, which are subsequently found to cause a phase shift between the potential and density amplitude. At a particular order in the asymptotic expansion analysis, it is found that resistivity and viscosity compete. When resistivity prevails, due to a phase shift between the potential and density amplitude, appearing from resistivity, the drift wave linearly grows. It is found that such drift-waves fall between a range of wavenumbers.