Chaos, Ergodic Gheory and Multifractal Singularities of Stochastic Differential Equations

Friday, November 2, 2001 - 11:00am - 12:00pm
Keller 3-180
Daniel Schertzer (Laboratoire de Modelisation en Mecanique (LMM))
In collaboration with S. Lovejoy.

Nonlinear differential equations in low space dimensions (e.g. Energy Balance Models) have been popular and helpful to show the strong limitations of classical methods in climate dynamics. However, the understanding of dynamics over a wide range of time and space scale is indispensable. It has been therefore argued that high dimensional dynamical systems, e.g. partial differential systems, are required and could be analyzed in the framework of the ergodic theory of chaos.

Existence and uniqueness of physical invariant measures have been obtained for large-scale approximations (e.g. quasi-geostrophic approximation). Nevertheless, these derivations depend on the regularity of the deterministic system, its physical relevance, as well as that of the pertubative noise, which is usually considered as gaussian and white in time.

Nevertheless, we show that: large-scale approximations are rather incompatible with:

- a scaling anisotropic regime from planetary scale down to km scale,

- colored and strongly non-gaussian noises.

We show that these properties, which rather correspond to the appearance of multifractal singularities, are rather well empirically supported. We discuss the corresponding (stochastic) models.