Extensive chaos and complexity of two-dimensional turbulence

grantor: University of TorontoThis thesis concerns how the dynamics of 2D fluid systems may become complex due to various physical parameters such as the forcing intensity and scale, the dissipation mechanism, and the system size. These complexity-determining factors are found to act collectively to give rise to the dynamical complexities. The fluid models in this study are shallow layers of viscous incompressible fluids on rectangular domains governed by the familiar 2D Navier-Stokes equations with and without a hypoviscosity term (linear Ekman drag is added to model large-scale dissipation). Three main analyses are carried out. One is the asymptotic analysis of the 2D Navier-Stokes equations driven by a monoscale forcing. The results obtained include a dynamical constraint and possible scaling laws for the energy spectrum. These are then generalized to systems with a general viscosity and extended to systems with Ekman drag. The second analysis concerns the estimation of the attractor dimension of the systems under consideration. We employ the technique developed by Constantin-Foias-Temam for the calculation of attractor dimension of dissipative dynamical systems. The present investigation focuses on the optimal estimation of the dimensionality and its extensivity. It is found, in general, that the estimates do not depend on the physical parameters in a fixed functional form, but rather take different expressions in different regions of parameter space. In particular, the attractor dimension of the Navier-Stokes equations (with or without Ekman drag) is shown to grow linearly with the domain area (for a sufficiently large domain) if the kinematic viscosity and the forcing density and its scale are held fixed. We also show that slightly super-extensive behaviour prevails for a wide range of the parameters. The third investigation concerns the stability problem (both linear and nonlinear) of simple laminar stationary flows. The analysis examines how the flows may become unstable and explores the properties of some unstable eigenmodes. The familiar Fourier expansion method is used for this study. It is found that (in)stabilities depend on the forcing scale in a peculiar and nontrivial way.Ph.D