Approximate Inertial Manifolds for Ageostrophic Flows

We develop mathematical and numerical methods for nonlinear dynamical systems in nature and engineering. Our approach combines applied mathematics, dynamical systems theory and numerical methods to produce algorithms directly applicable to experimental and numerical data sets. Areas of current interest include nonlinear vibrations of multi-degree-of-freedom structures, rigorous model reduction in very high dimensional systems, complicated transport processes in the ocean and the atmosphere. The author elucidates in a concrete way dynamical challenges concerning approximate inertial manifolds (AIMS), i.e., globally invariant, exponentially attracting, finite-dimensional smooth manifolds, for nonlinear dynamical systems on Hilbert spaces. The goal of this theory is to prove the basic theorem of approximation dynamics, wherein it is shown that there is a fundamental connection between the order of the approximating manifold and the well-posedness and long-time dynamics of the rotating Boussinesq and quasigeostrophic equations.