Quantifying long time statistical properties of a few fluid models

Florida State University, Tallahassee FL

Investigators

Abstract

Wang DMS-1008852 The principal investigator and colleagues study the issue of quantifying the long-time statistical properties of a few prototype fluid systems via long-time statistical properties of suitable discrete dynamical systems related to temporal and/or spatial approximations. The physical problems considered are the Rayleigh-Benard convection at large Prandtl number and/or small Ekman number regime, and a few related simplified models. In particular, the methodology developed is applied to numerically quantify an important physical long-time statistical quantity, the averaged heat transport, in a few convection models. The key issue here is the design, analysis and implementation of schemes that are efficient and convergent (in the sense that the stationary statistical properties of the discrete system converge to those of the underlying system). Approximating long-time behavior of large complex systems is a well-known challenge because small errors could accumulate and amplify. Additional difficulties related to multiple scales (induced by large Prandtl number, small Ekman number, large Rayleigh number), and generalised dynamical system (such as the 3D Boussinesq system) are also addressed. Suitable random perturbations of the fluid systems are considered in order to ensure convergence to the physically relevant long-time behaviour. Quantifying long-time statistical properties is of great importance in applications. Besides well-known applications in classical turbulence theory, it is also extremely important in climate studies because the predicted climate is the long-time statistical behaviour of the underlying climate model. The models to be investigated, although far from practical climate models, share several important mechanisms that are crucial to realistic climate models, such as energy-preserving nonlinear advection, rotation, convection, dissipation/damping and forcing. A clearer understanding of long-time statistical behavior in this setting helps us better understand many geophysical fluid phenomena, and provides guidelines for accurate numerical study of climate changes. The project also provides abundant opportunities for graduate students, including student from underrepresented group, to participate in the modeling, analysis, and computation of many physically motivated problems.