Physics – Plasma Physics
Scientific paper
Feb 1991
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1991phdt........78m&link_type=abstract
Thesis (PH.D.)--THE UNIVERSITY OF TEXAS AT AUSTIN, 1991.Source: Dissertation Abstracts International, Volume: 52-04, Section: B,
Physics
Plasma Physics
Scientific paper
A stressed hydrodynamical system undergoes a series of changes in its behavior as the control parameter (proportional to the stress) is increased. We take the specific case of a slab of fluid bounded by rigid vertical walls horizontally, but of infinite extent in height and depth, that is subject to a vertical gravitational field and a horizontal temperature gradient imposed by maintaining different uniform temperatures on the side walls. This is called the vertical slot convection (VSC) problem. This system exhibits a one dimensional buoyancy-driven convective flow at any finite temperature difference. This flow, in turn, becomes unstable as the dimensionless control parameter, the Grashof number, is increased. We derive 2-D vorticity/stream function equations in the Boussinesq approximation (eliminating pressure waves, but allowing buoyancy forces). We linearize these and perform a thorough linear stability analysis, obtaining accurate eigenvalues, neutral stability curves, and eigenfunctions for the VSC problem and several related and limiting cases, including the Benard problem. This demonstrates the role of each term in the equations. We constructed a fully nonlinear pseudo-spectral semi-implicit fluid simulation code using a Fourier-Chebychev decomposition and working at a Prandtl number of 7.5 (water), we use the linear eigenfunctions at the linearly critical wave number, alpha = 1.383, as perturbations to the 1-D basic shear flow. Above the critical Grashof number, G_ c, the system becomes unstable to the onset of secondary vortices. The Landau expansion theory is applied and checked in the near supercritical regime. At higher Grashof numbers a second bifurcation (Hopf) leading to oscillatory behaviors is found. Above G_ h the oscillations become anharmonic and evolve into nonlinear thermal relaxation oscillations which are analyzed in detail. These show similarities to the well-known 'sawtooth' oscillations found in tokamak discharge. After a probable third bifurcation we find an apparently chaotic attractor that resembles the well-known Lorenz attractor. Preliminary results are presented to support this interpretation. Finally we examine extensions to other parameter regimes and consider future extensions and applications to plasma physics in solar and astrophysical problems.
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