Hopf fibrations for oceanic waves and turbulent pipe flows

Math Physics Seminar
Thursday, September 18, 2014 - 2:00pm
1 hour (actually 50 minutes)
Skiles 006
School of Civil and Environmental Engineering, Georgia Tech
I propose a generalization of Hopf fibrations to quotient the streamwise translation symmetry of water waves and turbulent pipe flows viewed as dynamical systems. In particular, I exploit the geometric structure of the associated high dimensional state space, which is that of a principal fiber bundle. Symmetry reduction analysis of experimental data reveals that the speeds of large oceanic crests and turbulent bursts are associated with the dynamical and geometric phases of the corresponding orbits in the fiber bundle. In particular, in the symmetry-reduced frame I unveil a pattern-changing dynamics of the fluid structures, which explains the observed speed u ≈ Ud+Ug of intense extreme events in terms of the geometric phase velocity Ug and the dynamical phase velocity Ud associated with the orbits in the bundle. In particular, for oceanic waves Ug/Ud~-0.2 and for turbulent bursts Ug/Ud~0.43 at Reynolds number Re=3200.