Long-time Behavior of a Two-layer Model of Baroclinic Quasi-geostrophic Turbulence

We study a viscous two-layer quasi-geostrophic beta-plane model that is forced by imposition of a spatially uniform vertical shear in the eastward (zonal) component of the layer flows, or equivalently a spatially uniform north-south temperature gradient. We prove that the model is linearly unstable, but that non-linear solutions are bounded in time by a bound which is independent of the initial data and is determined only by the physical parameters of the model. We further prove, using arguments first presented in the study of the Kuramoto-Sivashinsky equation, the existence of an absorbing ball in appropriate function spaces, and in fact the existence of a compact finite-dimensional attractor, and provide upper bounds for the fractal and Hausdorff dimensions of the attractor. Finally, we show the existence of an inertial manifold for the dynamical system generated by the model's solution operator. Our results provide rigorous justification for observations made by Panetta based on long-time numerical integrations of the model equations

Nonlinear dynamics of a dispersive anisotropic Kuramoto–Sivashinsky equation in two space dimensions

A Kuramoto–Sivashinsky equation in two space dimensions arising in thin film flows is considered on doubly periodic domains. In the absence of dispersive effects, this anisotropic equation admits chaotic solutions for sufficiently large length scales with fully two-dimensional profiles; the one-dimensional dynamics observed for thin domains are structurally unstable as the transverse length increases. We find that, independent of the domain size, the characteristic length scale of the profiles in the streamwise direction is about 10 space units, with that in the transverse direction being approximately three times larger. Numerical computations in the chaotic regime provide an estimate for the radius of the absorbing ball in ℒ2 in terms of the length scales, from which we conclude that the system possesses a finite energy density. We show the property of equipartition of energy among the low Fourier modes, and report the disappearance of the inertial range when solution profiles are two-dimensional. Consideration of the high-frequency modes allows us to compute an estimate for the analytic extensibility of solutions in ℂ2. We also examine the addition of a physically derived third-order dispersion to the problem; this has a destabilizing effect, in the sense of reducing analyticity and increasing amplitude of solutions. However, sufficiently large dispersion may regularize the spatio-temporal chaos to travelling waves. We focus on dispersion where chaotic dynamics persist, and study its effect on the interfacial structures, absorbing ball and properties of the power spectrum