This project is focused on the discovery of driving forces that support the Kraichnan theory of 2-D fully developed turbulence. In recent work the investigators and their collaborators have identified critical wave numbers expressed as averages of different norms of the solution to the Navier-Stokes equations (NSE). These wave numbers provide necessary, and nearly sufficient conditions for the Kraichnan theory to hold. This team has also localized to some extent the global attractor of the NSE in a plane spanned by two of these norms (one being the energy), to help understand which driving forces produce these conditions. The proposed work will combine this analysis with computational optimization to zero in on such forces, and then study in detail the turbulent features they produce. The mathematical treatment of turbulence is largely driven by the heuristic theories of Kolmogorov, Batchelor and Kraichnan. The approach taken in projecting the attractor however, seems to be entirely new. The information provided by this analysis will guide the computational component which otherwise would be confronted with a vast landscape of possible driving forces to consider.
Turbulence is readily observed in three-dimensional physical space domains. Most people think of a bumpy plane rides (in this case the domain is the volume around the airplane). Like the swirls generated by rocks in a stream, rapidly changing patterns form in the air around the plane. Turbulence theories do not attempt to predict the precise development of these patterns, but rather find (a) consistent laws which describe how, on average, energy is transferred to smaller length scales, and (b) critical length scales at which this this phenomenon changes. True 2-D flows in nature are less prevalent. The most prominent example, the earth's atmosphere, is actually a thin 3-D domain, whose behavior approaches that of a 2-D flow. The fate of energy over different length scales is more complicated for 2-D flow, though that of 3-D flow is in some sense embedded into it. Though 2-D experiments are difficult to carry out in the laboratory, they allow for much finer study on a computer. Of all 2-D flows, the one studied in this project is arguably the most amenable to analysis and efficient simulation. Yet it is fundamental, not only to 2-D and nearly 2-D flows such as the atmosphere, but also to 3-D turbulence due to universality.
