Zonal Flow Generation in Toroidally Confined Plasmasthrough Modulational Instability of Drift Waves
Date
2017
Authors
Abdullatif, Raden Farzand
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Abstract
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.
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Keywords
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
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