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

TL;DR

This paper rigorously analyzes a two-layer quasi-geostrophic turbulence model, proving stability bounds, existence of attractors, and inertial manifolds, thus providing a theoretical foundation for observed long-term behaviors.

Contribution

It establishes the existence of bounded solutions, attractors, and inertial manifolds for the model, advancing understanding of its long-term dynamics.

Findings

Solutions are bounded independently of initial data.

Existence of a finite-dimensional attractor is proven.

An inertial manifold for the system is constructed.

Abstract

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.