Abstract:
Various contemporary research topics involve the understanding, modeling and prediction of complex turbulent systems with an important number of instabilities.  Topics of this kind include the probabilistic quantification of low-Re laminar but unstable fluid flows, the reduced-order modeling of low-dimensional quantities of interest for high-Re turbulent flows with a very large number of persistent instabilities, as well as the quantification of low-dimensional, intermittent structures, such as extreme events, that occur in wave systems characterized by broad spectra.  In this talk, three recent approaches involving the above topics will be discussed: 1.) The dynamically orthogonal (DO) field equations for the detailed stochastic modeling of low-dimensional, attractors of dynamical systems that have a small number of instabilities, 2.) The reduced-order modified quasilinear gaussian (ROMQG) closure for the modeling of low-dimensional quantities of interest for turbulent systems characterized by a very large number of instabilities (joint work with Andy Majda - NYU), and 3.) A blended approach for the quantification and prediction of low-dimensional, intermittent events in a dispersive, nonlinear waves with broad spectra (joint work with William Cousins - MIT).