Collection Nonlinear and stochastic methods in climate and geophysical fluid dynamics
Organisateur(s)
Date(s)
29/04/2024
A variational approach to closure of nonlinear dynamical systems
De Honghu Liu
In this talk, we discuss the closure problem of nonlinear evolution systems, with a focus on systems subject to autonomous forcings and placed in parameter regimes for which no slaving principle holds. Adopting a variational framework, we will show that efficient parameterizations can be explicitly determined as parametric deformations of invariant manifolds; such deformations themselves are optimized by minimization of cost functionals naturally associated with the dynamics. The minimizers are objects, called the optimal parameterizing manifolds, that are intimately tied to the conditional expectation of the original system, i.e. the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables. Applications to the closure of low-order models of Atmospheric Primitive Equations will then be discussed. The approach will be finally illustrated---in the context of the Kuramoto-Sivashinsky turbulence with many unstable modes---to provide efficient closures without slaving for a cutoff scale placed within the inertial range and the reduced state space just spanned by the unstable modes. This talk is based on joint work with Mickael D. Chekroun (UCLA) and James McWilliams (UCLA).
