Towards a Dynamical Systems Theory of Fluid Chaos

Mark Paul

Abstract:

In this talk, I discuss the exciting advances of a dynamical systems theory approach for building a physical understanding of chaotic fluid motion.  This talk focusses on the recent insights gained from the use of two powerful ideas: covariant Lyapunov vectors (CLVs) and exact coherent structures (ECS). At the heart of a dynamical systems approach to fluid dynamics is a state-space description of the fluid motion. The fluid motion in physical space is described as a trajectory through a very high, and possibly infinite, dimensional state-space. This state-space is often populated by an infinity of unstable exact solutions to the governing equations such as equilibria and periodic orbits. These unstable exact solutions are examples of ECS and serve to create a skeleton in state-space upon which a trajectory must navigate. The linear stability of the nonlinear trajectory through the state-space is described by the CLVs.  Taken together, the ECS and CLVs provide an unprecedented and insightful picture of chaotic fluid motion that is currently impossible to achieve using the physical space description alone. I will discuss how these ideas are being pushed forward by my research group to provide new insights into models of fluid motion and well as the chaotic dynamics of Rayleigh-Benard convection for experimental conditions.  These insights suggest the possibility of extending this approach to larger and  much more complicated systems that are of intense interest such as fluid turbulence and the dynamics of the atmosphere.

This work has been funded by NSF fund numbers CMMI-2138055 and CBET-2151389.