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.
The PI and his collaborators investigated the issue of how to quantify the long time statistical properties, i.e., the "climate", of a few prototype fluid problems that are central to our understanding of more complete (and complex) models for the atmosphere, ocean and the earth's mantle. Most models under investigation possess chaotic or turbulent behavior. On the one hand, it is of great interest to study the long time behavior of chaotic or turbulent systems due to our desire to make long range forecast. On the other hand, such kind of research is notoriously difficult due to the intrinsic instability, uncertainty in the initial data, and model errors. The PI has proposed and analyzed a general methodology in constructing numerical algorithms that are able to capture the long time statistical properties of a family of physical models. He and his team have developed several highly efficient numerical schemes that are able to capture the "climate" of a few prototype fluid models. The results have been published in more than 20 papers, most of them appeared in top journals in the area. A survey paper is also underway. The work by the PI and his team has attracted a lot of attention from the scientific community. As a result, the PI has been invited to deliver more than 35 talks and lectures on topics related to the results derived under the support of this grant. A mini-conference devoted to the long time stochastic and statistical approximations for turbulent dynamical systems was held at Fudan University in China. Three graduate students and one undergraduates student are trained under the support of this grant. This includes an African American female doctoral student who will graduate this academic year. The project also involved several junior (female) faculty members from several schools in the United States, China and the United Kingdom. Most of them are now tenured associate professors. Their collaborative work with the PI on this project contributed to the scientific development of these junior faculty members and the development of scientific networks in the country and around the world.
