Central to modern geophysical fluid dynamics are computational solutions to Navier-Stokes equations. A continuing challenge is to solve them for ever larger range of spatial scales at which turbulence reigns. Based on both multigrid and renormalization group methods, a new computational paradigm that can meet this challenge has recently emerged, called systematic upscaling. This is a methodical approach to derive, from a given system of fine-scale physical equations, equivalent numerical equations at increasingly coarser scales. Switching back and forth between all scales, calculations can be restricted to only small representative regions at each scale. The derived equations can be either deterministic or stochastic, possibly changing with scale. The proposed algorithmic development will use this methodology to computationally derive, from the deterministic Navier-Stokes equations at the fine viscous scale, stochastic equations for simulating turbulent flows at the large geophysical scale. Unlike current turbulence modeling based on ad-hoc, limited-accuracy assumptions, this will be a systematic derivation that recognizes the stochastic nature of the problem and can in principle attain any desired accuracy at reasonable cost. The project will also involve new algebraic multigrid solvers for unstructured discretization of fluid problems, and will introduce new multigrid-based highly adaptive discretization schemes for time-dependent flows, potentially achieving better combinations of minimum-dissipation Lagrangian with conservative Eulerian formulations.
To understand the behavior of the atmosphere and oceans, and to predict weather and environmental effects of human activities, complicated fluid flows need to be simulated on supercomputers. The main difficulty is the turbulent or unstable nature of the flows, breaking down into myriads of small, medium and large eddies within eddies, making them impossible to simulate on any present or future computer. Various approximation methods have met with only partial success. The objective of the proposed research is to develop for such flows a new computational methodology, called systematic upscaling, designed to derive, from known underlying physical laws, the relation that govern the behavior of large-scale averages. The derived relations involve an element of chance, correctly assigning a measure of likelihood to each of the predicted events. The research project also makes basic contributions to the general development of multiscale computation, potentially impacting many other fields, including biochemistry to nanotechnology. The research team fits the aims of this project: Prof. Brandt led the development of multiscale computational methods that are already widely used, including recently the introduction of systematic upscaling in problems of statistical mechanics and molecular dynamics. Prof. McWilliams has pioneered computational studies of geophysical turbulence and has been a leader in incorporating the results of such studies into standard oceanic calculations. The research project will be combined with the classroom and research training at UCLA's Departments of Mathematics and Atmospheric and Oceanic Sciences, which have many students from under represented groups. Results will be disseminated by conference lectures and publications.
