Turbulence: Dynamics, Geometry, and Topology
ABSTRACT :
Turbulence is widely regarded as the last unsolved problem in classical physics. Despite the availability of accurate governing equations, a tractable framework that describes both dynamical and statistical properties of turbulence has eluded scientists. In this talk, we discuss how chaos theory (developed for low-dimensional dynamical systems) may find application in solving this outstanding problem. Specifically, we present a combined experimental and numerical study of a moderately turbulent flow in an electromagnetically driven shallow fluid layer. Turbulent evolution in this system is punctuated by intervals of dramatic coherence, which is a signature of unstable nonchaotic solutions (e.g., steady or time-periodic flows) of the Navier-Stokes equation. Often termed Exact Coherent States (ECS), such simple solutions form the chaotic saddle whose geometry and topology underpin turbulent dynamics. For specific examples, we demonstrate that turbulent evolution can be quantitatively and robustly forecast using ECS. We also compare statistical properties of turbulence with those of ECS.