AAU

Christian Pötzsche (Feb 2011)

Slow integral manifolds for Lagrangian fluid dynamics in unsteady geophysical flows

(with J. Duan and S. Siegmund) Physica D 233, 2007, 73-82

The authors consider Lagrangian motion of fluid particles in unsteady gravity currents in geophysical flows. The vertical motion of fluid particles, especially the induced vertical mixing in these currents, is partially responsible for the ocean thermohaline circulation, and thus plays a role in the global climate dynamics. First, a reduced dynamical system for slow variables is derived for a nonautonomous multiscale system. The reduced system, still nonautonomous, is the original system restricted to a center-like nonautonomous invariant manifold (so-called slow manifold) which holds slow motions of the system. An algorithm is also presented to obtain an approximation of the non-autonomous slow manifold. A novelty here is that the reduction principle applies to nonautonomous multiscale systems which satisfy conditions that are true only locally in space (as in many physical cases). This makes the reduction principle applicable to real physical systems.

Then, this invariant manifold reduction principle is applied to an approximate conceptual Lagrangian model of gravity currents and a reduced nonautonomous system for slow vertical motion is obtained. This reduced system may be useful as a conceptual tractable tool for understanding some features of vertical mixing in unsteady gravity currents.

Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients

Journal of Mathematical Analysis and Applications 289, 2004,
317-335

Unifying ordinary differential and difference equations, we consider linear dynamic equations on measure chains or time scales, which possess an exponential dichotomy uniformly in a parameter, and show that this dichotomy is robust, if the mentioned parameter changes slowly in time. Here, the equations can be infinite dimensional and are not assumed to be invertible.

Slow and fast variables in nonautonomous difference equations

Journal of Difference Equations and Applications 9(5), 2003,
473-487

In this paper we present an existence and smoothness result for center-like invariant manifolds of nonautonomous difference equations with slow and fast state-space variables. This result can be seen as a first step to obtain Fenichel‘s geometric theory for difference equations. Here, our basic tool is an abstract integral manifold theorem.

This paper is dedicated to Prof. George R. Sell on the occasion of his 65th birthday.